Complex-Valued Modeling in Economics
and Finance
Sergey Svetunkov
Complex-Valued Modeling
in Economics and Finance
Sergey Svetunkov
National Mineral Resources
University - Mining University
St. Petersburg, Russia
ISBN 978-1-4614-5875-3
ISBN 978-1-4614-5876-0 (eBook)
DOI 10.1007/978-1-4614-5876-0
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2012954285
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Preface
As early as the mid-eighteenth century, mathematicians discovered a way to use
models involving complex variables. Since then, work with complex variables has
been progressing, with the theory of complex variables emerging as a branch of
mathematics. Nowadays this theory is widely implemented in all of the natural
sciences as work with complex variables makes it possible to describe adequately
more complicated processes than do real variables. Economics as an object of
scientific research and cognition is no less complicated than the natural sciences,
which is why complex variables may be applied in economics to give a more
precise description of the processes involved and to build even more complicated
models than those that can be built using real variables. Complex variables are
sometimes used in certain branches of economic and mathematical simulation, but
this study considers application of models of economic and mathematical simulation exclusively in the form of models of complex variables.
Models and mathematical methods of working with complex variables are
considered in the study not as some alternative to real-variable models and methods
but as an instrument complementing and expanding the existing arsenal of economic and mathematical modeling. It is of principal importance that complex
economics provides the economist with a new instrument of research, and the
more flexible the instrument, the more diverse the tasks that can be solved by
the researcher
The first chapter of this work presents basic principles of complex economics
and certain data from the theory of functions of complex variables that are
necessary for an understanding of further actions in the formation of complex
economic theory.
Very often, understanding a certain meaning of mathematical operations
requires graphic methods of describing these operations. Since in the study in
question complex economics involves functions of complex variables, it is necessary to know the characteristics of these functions including their graphical
v
vi
Preface
representation. This task is solved in the second and third chapters of the study,
which consider conformal mappings of basic functions of complex variables. These
chapters do not simply state the respective branches of the theory of functions of a
complex variable encountered in various textbooks. Textbooks on the theory of
functions of a complex variable, in the section devoted to conformal mappings, do
not consider, for instance, exponential complex functions with a complex coefficient; the need for such treatment has not existed. Textbooks consider exceptional
conformal mappings of this exponential complex function with real exponent. For
the purposes of complex economics, there is a need to use complex exponents not
only for exponential functions.
The fourth chapter presents an instrument of practical application in complex
economics – complex econometrics. The chapter provides only the basic principles
of complex econometrics because it is practically impossible for a group of
scientists working in this field to develop or adapt to complex econometrics all
the branches of real-variable econometrics. Moreover, it is simply impossible to
present the entire scientific discipline of econometrics in one chapter. Here we will
substantiate and adapt to complex economics the basic sections of correlation and
regression analysis of mathematical statistics – calculation of complex coefficients
of pair correlation, least-squares method for evaluating coefficients of complex
models, method of construction of confidence limits for obtained statistical
estimates, and new coefficients showing the adequacy of econometric
constructions. The results obtained are sufficient for solving subsequent tasks of
complex economics and developing complex econometrics. The ideas of statistical
characteristics of complex random variables currently in use in mathematical
statistics has led to a deadlock. This can be seen in calculations of complex
coefficients of pair correlation – the obtained contradictory results that follow
from standard situations testify to their erroneous character. This made it necessary
to devise other principles of statistics of complex random variables that underlie
new and consistent conclusions and recommendations.
The fifth chapter contains the results of an investigation of one of the simplest
types of economic models of complex variables – production functions of complex
arguments, where actual production results depend on a complex argument, that is,
production resources represented in the form of a complex variable. These
functions possess some very important properties applicable in the successful
solution of certain economic tasks. Here, we also demonstrate one remarkable
feature of complex argument models – the sustainability of their assessments
under multicollinearity.
Chapter 6 discusses production complex variable functions, more complicated
models than complex argument ones. Here, the complex production result is
represented in the form of dependence on a complex resource. Since functional
relationships between two complex variables may have various forms, this chapter
considers the basic ones.
Preface
vii
The seventh chapter uses a case that has been insufficiently explored in the
mathematics of complex variables; the existing theory operates with only one
complex variable and is therefore called the theory of functions of a complex
variable. Chapter 7 involves multifactor complex models, i.e., models of several
complex variables. The development of the theory of functions of several complex
variables in mathematics finds very little application in economics. This is why in
the sixth chapter we state for the first time the principles and approaches of the
theory of multifactor functions of complex variables. This was necessitated by our
wish to build complex production functions that are more applicable to real
economic processes, and that could be done only by increasing the explanation
factors used in complex economic models. This chapter presents the properties and
characteristics of simple multifactor complex models. The variety of possible
applications of the models and methods of complex economics is not limited only
to production function models. However, production functions provide a good
example for seeing the advantages and disadvantages of complex models. In
Chaps. 5 and 6 this is done by comparing such models with basic models of
production functions with real variables.
Another good example that demonstrates the advantages of complex variable
models is their application in the analysis of stock markets. Chapter 8 shows how to
use complex indices of economic conditions and how to use the properties of
complex numbers to obtain phase portraits of stock markets that could allow us to
reveal laws that remain hidden with the use of real variables. The materials of Chap.
9 aim at showing other ways of developing complex economics than those specified
in previous chapters.
In this study we make reference to the literature used in the form of footnotes. At
the end of the study we give a complete list of all the publications by those scientists
working on the formation of complex economics. If necessary, the reader may refer
to these sources. The large volume of scientific results provided by this study could
not have been obtained without the support of the Russian Foundation of Fundamental Research. The grants allocated by the foundation on a competitive basis
from 2006 till 2010 rendered invaluable financial and moral assistance.
The main ideas, hypotheses, and materials stated in this study belong to the
author; however, these hypotheses would never have become a well-balanced
theory without the active involvement of a group of scientists, chief among
whom was Dr. Ivan S. Svetunkov. This collaboration laid the foundations of
complex economics on which basis many other scientific results were obtained.
We are extremely grateful to Prof. G.V. Savinov who reviewed my first works,
written in cooperation with I.S. Svetunkov, and later formulated a number of
interesting proposals published in various articles. A very important contribution
in the creation of various sections of complex economics was made by Dr. T.V.
Koretskaya, E.V. Sirotina, and A.F. Chanysheva. Their work and the results
obtained by them are discussed in the corresponding sections of the book. Some
partial conclusions, recommendations, and new scientific results obtained by other
young scientists are stated in various parts of the book.
viii
Preface
Insofar as the materials presented in this study are new and this is the first time
they have been systematized in this way and used for the proposed scientific
purposes, the author understands that some points may be subject to debate or
contain inaccuracies. It is every scientist’s mission, having obtained a new scientific
result, to encourage comprehensive scientific discussion and get to the truth. This is
why any constructive criticism of the study is welcome. Comments and remarks
may be mailed directly via www.sergey.svetunkov.ru.
St. Petersburg, Russia
Sergey Svetunkov
Contents
1
2
3
Theoretical Basis of Complex Economy . . . . . . . . . . . . . . . . . . . . .
1.1 Complex Economies as a New Branch of Economics
and Mathematical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Basic Concepts of the TFCV . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Axiomatic Core of the Theory of the Complex Economy . . . . . .
1.4 Basic Model of a Complex Economy . . . . . . . . . . . . . . . . . . . .
1.5 Some Data on Minkowsky’s Geometry . . . . . . . . . . . . . . . . . . .
1.6 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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59
62
Conformal Mappings of Functions of a Complex Variable . . . . . . . .
3.1 Power Functions of a Complex Variable . . . . . . . . . . . . . . . . . . .
3.2 Exponential Functions of Complex Variables . . . . . . . . . . . . . . . .
63
63
75
Properties of Complex Numbers of a Real Argument
and Real Functions of a Complex Argument . . . . . . . . . . . . . . . . .
2.1 General Problem of Conformal Mapping
in Complex-Valued Economics . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Complex Functions of a Real Argument . . . . . . . . . . . . . . . . . .
2.3 Functions of a Complex Argument: Linear Function . . . . . . . . .
2.4 Power Function of a Complex Argument
with a Real Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Exponential Function of Complex Argument
with Imaginary Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Power Function of Complex Argument
with Complex Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Exponential Function of a Complex Argument . . . . . . . . . . . . . .
2.8 Logarithmic Function of a Complex Argument . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
x
Contents
3.3
3.4
4
5
6
Logarithmic Functions of Complex Variables . . . . . . . . . . . . . . .
Zhukovsky’s Function and Trigonometric
Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
Principles of Complex-Valued Econometrics . . . . . . . . . . . . . . . . .
4.1
Statistics of Random Complex Value:
Standard Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Method of Least Squares of Complex Variables:
Standard Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Correlation Analysis of Complex Variables:
Contradictions of the Standard Approach . . . . . . . . . . . . . . . . .
4.4
Consistent Axioms of the Theory of Mathematical
Statistics of Random Complex Variables . . . . . . . . . . . . . . . . .
4.5
Least-Squares Method from the Point of View
of the New Axiomatic Theory . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Complex Pair Correlation Coefficient . . . . . . . . . . . . . . . . . . .
4.7
Interpretation of Values of Complex Pair
Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8
Assessments of Parameters of Nonlinear Econometric
Models of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . .
4.9
Assessment of Confidence Limits of Selected Values
of Complex-Valued Models . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Balancing Factor in Evaluating the Adequacy
of Econometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. 103
. 112
. 114
. 119
. 128
. 135
. 142
Production Functions of Complex Argument . . . . . . . . . . . . . . . . . .
5.1
Fundamentals of Production Functions
of a Complex Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Linear Complex-Valued Model of a Complex
Argument and Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
Linear Production Function of a Complex Argument . . . . . . . . .
5.4
Power Production Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
Exponential Production Function of Complex Argument . . . . . . .
5.6
Logarithmic Production Function of Complex Argument . . . . . .
5.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Production Functions of Complex Variables . . . . . . . . . . . . . . . . . .
6.1 General Provisions of the Theory of Production
Functions with Complex Variables . . . . . . . . . . . . . . . . . . . . . .
6.2 Linear Production Function of Complex Variables . . . . . . . . . . .
6.3 Model of Power Production Function of Complex
Variables with Real Coefficients . . . . . . . . . . . . . . . . . . . . . . . .
83
86
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143
146
155
164
172
175
179
180
. 181
. 181
. 185
. 194
Contents
Power Production Complex-Valued Functions
with Real Coefficients of the Diatom Plant
and Russian Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Coefficients of Elasticity of the Complex Exponential
Production Function with Real Coefficients . . . . . . . . . . . . . . . .
6.6 Power Production Function of Complex Variables
with Complex Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Logarithmic Production Function of Complex Variables . . . . . . .
6.8 Exponential Production Function of Complex Variables . . . . . . .
6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
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7
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Multifactor Complex-Valued Models of Economy . . . . . . . . . . . . .
7.1 General Provisions of Complex-Valued Model Classification . . .
7.2 Linear Classification Production Function . . . . . . . . . . . . . . . . .
7.3 Classification Production Function of Cobb-Douglas Type . . . . .
7.4 Elasticity and Other Characteristics of a Classification
Production Complex-Valued Function . . . . . . . . . . . . . . . . . . . .
7.5 Classification Power Production Function . . . . . . . . . . . . . . . . .
7.6 The Shadow Economy and Its Modeling by Means
of Complex-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Formation of Complex, Multifactor Models
of Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 246
. 256
8
Modeling Economic Conditions of the Stock Market . . . . . . . . . . .
8.1 Stock Market Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Phase Plane and K-patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Mathematical Models of K-Patterns . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9
Modeling and Forecasting of Economic Dynamics
by Complex-Valued Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1 Ivan Svetunkov’s Model for Short-Term Forecasting . . . . . . . . .
9.2 Complex-Valued Autoregression Models . . . . . . . . . . . . . . . . . .
9.3 Solow’s Model of Economic Dynamics and Its
Complex-Valued Analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Modeling Regional Socioeconomic Development . . . . . . . . . . . .
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 259
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269
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Chapter 1
Theoretical Basis of Complex Economy
In the first chapter of the study we state axiomatic provisions of the theory of
economic modeling using methods of the theory of functions of a complex variable
(TFCV). Here we state the principles of economic modeling using complex variable
functions, introduce the main concepts of the theory of complex economies, and
explain the essence of the basic model of the new theory of economic modeling. We
show that the methods of the new modeling theory are not alternative but complementary to the existing methods of economic and mathematical modeling due to
new models and modeling methods. For the reader to understand the ideas of the
study the first chapter provides the data of the complex variable function theory
underlying the proposed economic modeling theory. The concept of complex
variable is briefly explained, and its basic properties and the essence of mathematical operations are specified, as are certain data on the Minkowsky geometry and the
Laplace transform.
1.1
Complex Economies as a New Branch of Economics
and Mathematical Simulation
Problems of decision making that permanently arise in the course of economic
management at any level of a hierarchy – from the workplace and production area
to the global economy –require the processing of significant arrays of information
to provide information support of the decision-making process. If at the lowest level
of the decision-making hierarchy – i.e., in the workplace – it is enough to have
intuitive expert assessments, since the decision-making problem is trivial and the
quantity and composition of the information being processed with respect to the
situation seem to be elementary, every higher level of the hierarchy involves ever
more complicated tasks. Here it is not possible to do without mathematical methods
that are becoming increasingly more complicated with the increase in the
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
DOI 10.1007/978-1-4614-5876-0_1, # Springer Science+Business Media New York 2012
1
2
1 Theoretical Basis of Complex Economy
complexity of problems, which changes for every next higher level of economic
management.
Nowadays the arsenal of mathematical methods and models used in economics
is quite vast. However, this variety by no means guarantees a successful solution of
management problems. On the contrary, there are at least tens of directions of
economics and mathematical modeling where models of real variables cannot
exceed their capacities and describe economics somewhat incompletely. Under
these conditions, economists either fully refuse to use mathematical methods or
transfer economics and mathematical models from the sphere of practical solutions
to the sphere of the theory of conditional objects having little to do with a real
economy. In this case scientists must impose restrictions and assumptions that
could transform these models from a set of abstract images into a set of idealized
images possessing properties that no real economic object possesses. Let us take a
model of “an eternal individual” as being similar to those in the annals of various
theoretical branches of economics. The property of “eternal” life cannot be
attributed to any real individual; moreover, it completely contradicts reality. The
construction of such models and serious discussion thereof in scientific circles
demonstrate an inability of modern science to make any progress in the solution
of practical tasks of economic modeling.
The limitation of economics and mathematical models of real variables are clear.
Attempts to develop them by including in the models new variables or complicating
the computational tools using more powerful computational equipment is an
important trend in the improvement of economics and mathematical modeling
that must not be denied. However, nowadays there is a palpable need for other
principles of economics and mathematical modeling, these principles being
represented by TFCV.
It should be noted that economists have long faced situations where, during the
development and implementation of certain models, they had to calculate imaginary roots. The bravest ones studied the behavior of these complex variable models
as one of the most interesting phenomena in economic modeling but never went
further. Nobody provided any practical recommendations and suggestions for a
wider application of complex variables.
From the economic literature we know about attempts to apply the Laplace
transform to economic modeling, where the simulation of complex processes
described by real variable models involves their transformation into complex
variable models that provide easier operation. A solution of the problem in the
area of complex variables is followed by the inverse transformation into the area of
real variables.
Laurent Z-transforms applicable to the problem of predicting socioeconomic
dynamics were proposed by V.R. Semyonychev [1] as a modification of the
discrete Laplace transform. In this case a nonlinear model is transformed by
means of a Z-transform into a complex variable model, which provides reparameterization of the original model. This simplifies the evaluation of the coefficients of
the original, nonlinear trend. Here a TFCV tool is used as an instrument, to support
the application of real variable models in economics.
1.1 Complex Economies as a New Branch of Economics and Mathematical Simulation
3
There are other particular examples of using complex variables for modeling
special economic problems. However, they do not refer to the presentation of the
economy as an object for modeling by TFCV methods.
It should be noted here that nowadays it is impossible to calculate anything in
natural, engineering, and technical sciences without complex variables. Problems
of hydro- and gas dynamics, the theory of elasticity, calculation of electric contours
and electric transition processes, the physics of the micro and macro world, aircraft
building, and many other branches of modern science use complex variables as the
basic mathematical instrument of modeling. However, there is no such instrument
in economics.
About 100 years ago scientists started using complex variable functions theory
to describe nonuniform fields, to model complex flows, and to describe rotating
fields, and they began obtaining complex-variable models that much more easily
describe complex objects and phenomena than real-variable models. The TFCV
providedscientists with a convenient instrument of complex-object modeling. However, economists still ignore the power and variety of the instruments of this theory.
I think that all of this is due primarily to the habit of using a number of general
scientific methods and principles like the analogy method and the principle of
simplicity.
The principle of simplicity teaches us to use simple models if there is no need to
use more complex ones. The analogy method stubbornly gets scientists who think
about the very possibility of using the TFCV in economics to search for rotating
fields and the economic meaning of the real and imaginary constituents of a
complex variable. Since it is convenient to investigate a model of an immortal
individual, no results will carry any consequences for economic practice, but these
models make it possible to conduct numerous modeling experiments, obtain various
trajectories of the calculated variables giving them various names from among
concepts in economics, certain scientists believe that the situation with the
instruments of economic modeling is satisfactory and it is unreasonable to “multiply entities in excess of those needed” – it is sufficient to apply the available
mathematical tools.
And if we try to use the analogy method to find situations in economics where
complex variable models describe economic processes, such as, for example,
transition processes that take place in AC circuits with rotating electromagnetic
fields, then we will not find such situations.
Scientists who study economics and mathematical modeling miss an obvious
fact – the complex variable itself may be considered as a model that characterizes
the properties of an object in a more complex way as it consists of two real
variables, not one as in real-variable models.
When we study an economic index like gross profit, G, we understand that it
allows us to evaluate only one aspect of a complex economic phenomenon that
represents the results of an economic process. It is no accident that when it comes to
decision making, nobody will be satisfied with just maximum gross margin as a way
of understanding a situation and making the right decision. It is only in modern
economic theory that company behavior is explained on the basis of maximum
4
1 Theoretical Basis of Complex Economy
gross profit. The real economy involves another, no less important, index – production costs C. Then, correlating the gross margin with the production costs we obtain
profitability. Since it is profitability that shows both costs and results, i.e., is an
index of the economic efficiency of the production process, it is used as another
parameter for economic decision making.
In real economic practice, to describe some production process using models of
real variables, scientists must simulate both the gross margin and the production
costs. Since it is not convenient and rather costly to build two models, scientists
build one model adding the gross income and the production costs, which give gross
output, the very parameter considered in economic and mathematical modeling as
the main production result.
The desire to simultaneously model two economic variables – gross income and
production costs – is easily satisfied if we consider the production result as a
complex number. In this case this complex number behaves as a model showing
the production results. For the case under consideration it may be represented as
follows:
z ¼ G þ iC
(1.1)
where i is an imaginary unit possessing the following property: i2 ¼ 1.
Considering and modeling the new number z we take into account both the gross
income G and the production costs C as they are integral characteristics of the
complex number. This means that in dealing with one complex variable, one will be
dealing with two real variables. Therefore, the use of a complex variable like (1.1)
as a model connecting two economic variables in one unit makes it possible, on the
one hand, to obtain a more compact notation and, on the other hand, include in an
economic and mathematical model more detailed information on the modeled
object, and, in addition, consider them in interrelation.
But if these had been the only innovations introduced in economics and mathematical modeling by application of complex variables, then perhaps it would not
have been worth the trouble. Economic parameters and processes simulated by
means of complex variables are much more extensive than may appear at first
glance. In fact, if we just add up the real and imaginary parts of variable (1.1) we
obtain the known value – the gross margin:
Q¼GþC
(1.2)
And if we find the ratio of the real to the imaginary part, we get the arctangent of
the polar angle of the complex number (1.1) and . . . profitability at cost:
r¼
G
C
(1.3)
Thus, modeling the behavior of only one complex variable makes it possible to
study the character of change not only of the two original variables but also of a
1.1 Complex Economies as a New Branch of Economics and Mathematical Simulation
5
number of their additional derivative parameters. In the case under consideration
we model four important economic parameters simultaneously.
But that’s not all! Complex numbers may be presented not only in arithmetic but
also in exponential and trigonometric forms. For that, a complex number considered
on a complex plane is presented in polar coordinates. It is characterized by the absolute
value and a polar angle. The absolute value of a complex number (1.1) determined as
R¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G2 þ C2
(1.4)
is unique in the system of technical economic analysis and represents a new
economic parameter showing the production scale. Its usage in practice may
enlarge the diagnostic apparatus of, for example, a branch of economics such as
the analysis of economic activity. The ratio of gross margin Q to scale R may also
provide an additional production characteristic whose properties may be useful for
economic analysis. There can be even more examples. Each application of
complex-variable models provides ever new possibilities for more detailed economic modeling.
Thus, even a simple presentation of economic parameters and factors in the form
of a complex number (1.1) provides many new possibilities for economic and
mathematical modeling. However, mathematical operations with complex numbers
provide nontrivial results compared to operations with real numbers. For that very
reason there is a branch of mathematics called the theory of functions of a complex
variable (TFCV). Use of this new mathematical tool in economics extends the
instrumental basis of economic modeling since complex-variable models describe
the interrelation between variables in a different way than real-variable models.
Often it is easier to describe very complicated relations between real variables using
TFCV models and methods than using models of real variables.
As follows from the TFCV, any complex function may ultimately be represented
as a system of two functions with real variables, but these functions with real
variables are often so complicated that they cannot be applied in practice – simple
models of complex variables have very complicated analogs in the world of real
variables. Appropriate examples will be given later in this study.
Why do complex-variable models appear to be more preferable to real-variable
ones in simulating complex processes? What is the “sacred” meaning of this
property?
To answer this question, let us look at a geometric interpretation of each number.
A real number represents a point on a numerical axis having a zero point and going
to + or – infinity (Fig. 1.1, point Y).
This real number is characterized by the distance from the zero point to this
number. If the number is to the left of the zero point, it will be negative; if it is to the
right of the zero point, it will be positive.
As follows from mathematical notation (1.1), a complex number represents a
point not on an axis but on a complex plane. Therefore, to define it a given point
on a complex plane unambiguously, it is not sufficient to have one characteristic.
6
1 Theoretical Basis of Complex Economy
Imaginary part
z
Y
−∞
0
+∞
0
Real part
Fig. 1.1 Geometric interpretation of real (y) and complex (z) numbers
We need two coordinates – a segment on the axis of the real part and a segment on
the axis of the imaginary part (Fig. 1.1, point z).
When we perform any mathematical operations with two real variables, we do it
only with these variables, and when we perform similar operations with two
complex numbers, for example, multiply one complex number by another, we
perform a mathematical operation with four real numbers simultaneously.
It should be noted that the foregoing discussion does not mean that mathematical
operations with complex numbers are better than the same operations with real
numbers or that complex-variable models are better than real-variable models. It
only means that mathematical operations with complex numbers give different
results, whereas mathematical models of complex economic variables simulate
different economic processes. In some cases models of complex variables will
better describe economic processes than models of real variables; in other cases
they will do it worse.
But it is evident that the presentation of a pair of economic parameters in the
form of a complex number, as was done in the case of production result (1.1),
presents economists with a new opportunity to use the theory of complex-variable
functions for the purpose of economic simulation. In this theory, functions where
variables are complex numbers are called complex functions. Since the material
developed and proposed in this study demonstrates the application of the TFCV in
economics and mathematical modeling, we propose a short but precise name –
complex economics – as a general definition of this new branch of economics and
mathematical modeling.
Thus, complex economics is a branch of economics and mathematical modeling
where variables are complex values of economic parameters.
1.2
Basic Concepts of the TFCV
Since in economic and mathematical modeling complex variables have not been
used as independent-variable operations that would allow for the formation of
original economic and mathematical models, few economists are familiar with
1.2 Basic Concepts of the TFCV
7
their properties and basic rules for dealing with them. This is why the object of this
section is to state the basic concepts of the TFCV without which an untrained reader
would not understand complex economics. Those who are familiar with the TFCV
may skip this section.
In mathematics we often need to solve problems that do not have roots in the
area of real numbers. For example, we need to find the root of the equation
x2 þ 4 ¼ 0:
The solution gives the following roots:
x1;2 ¼
pffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
16
¼ 2
1:
2
It is evident that in the area of real numbers this equation does not have solutions
because there is no square root of minus one in this field. However, since impossibility of solving such problems leads to considerable limitations on computational
capabilities, the so-called “imaginary unit” was introduced in mathematics, i.e., the
pffiffiffiffiffiffiffi
number i ¼
1. The square root of this number will definitely be equal to minus
one: i2 ¼ 1.
Then the preceding equation has the following solution, which is imaginary:
x1;2 ¼ 2i
We can work further with the obtained imaginary number just as with the
solution of the foregoing quadratic equation. But roots of a quadratic equation
may not be solely real or imaginary. They may contain both real and imaginary
parts. For example, if we solve the equation
x2 þ x þ 2; 5 ¼ 0
then its roots, taking into account the introduced concept of “imaginary number,”
will be represented by two numbers:
x1;2 ¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 4 1 2; 5
2
The first root of the equation under consideration is
x1 ¼ 0; 5 þ 1; 5i;
while the second one is
x2 ¼ 0; 5
1; 5i:
8
1 Theoretical Basis of Complex Economy
Because the roots of the equation represent a number consisting of two parts in
which there are both a real and an imaginary part, we call it a complex number.
Thus, a complex number may be said to be a numerical pair consisting of two
parts – a real and an imaginary one:
z ¼ x þ iy;
(1.5)
where x is a real part of the complex number, iy is its imaginary part, x and y are real
numbers, and i is the imaginary unit that, as shown above, satisfies the equality
i¼
pffiffiffiffiffiffiffi
1 or i2 ¼
1:
(1.6)
Complex numbers allow for the same operations as real ones. However, taking
into account specific properties of the imaginary unit, these operations have an
original character not inherent in operations with real numbers.
The main problem that an economist faces when representing an economic
parameter in the form of a complex number is the complexity of the economic
interpretation of the imaginary part. The main question may be stated as follows:
where in a real economy would one meet imaginary numbers in general and
imaginary units in particular? What is the meaning of an imaginary unit? The
answer is that there is no meaning, neither economic nor physical. An imaginary
unit is a mathematical rule, nothing more.
Where in real life do economists come across imaginary numbers and imaginary
units? Nowhere! Where in real life could an economist encounter logarithms?
Again, nowhere! There are no logarithms in the world around us. A logarithm is
a mathematical rule that makes it easy to solve practical problems, including in
economics. In the same way, an imaginary unit, which, as we have already
mentioned, can be considered a mathematical rule, may be used to solve in a very
convenient way a whole range of applied tasks in various spheres of human activity.
Using rules specified by conditions (1.5) and (1.6) we can use new mathematical
operations, obtain new mathematical results, and form new mathematical models. It
should be noted right away that in no sphere of the natural sciences are there
processes where one can observe imaginary numbers or units. Complex numbers
represent a mathematical model that may or may not describe some actually
existing phenomena. If scientists decide to use complex variables for modeling
real processes, they set the rules in advance according to which they always
associate one component of a complex process to the real part and the other to
the imaginary part of the complex variable.
In the same way, there are no phenomena in an economy that could be associated
to a real or imaginary part because there are no phenomena where these parts are
explicitly delineated. Like scientists in other fields, we will set rules according to
which we will be able to represent economic phenomena in the form of complex
numbers and complex variables. And wherever this representation of a complex
socioeconomic object allows us to obtain a more precise description of its
1.2 Basic Concepts of the TFCV
9
Fig. 1.2 A complex number
(1.5) considered as a vector
Imaginary part
Z
y
r
φ
0
x
Real part
properties, we will use models of a complex variable or several complex variables
instead of models of real variables. Complex numbers may be represented
graphically as in Fig. 1.1. The representation of a number on a plane but not on a
numerical axis provides a number of new properties of complex numbers that are
rather important for their further application in theory and practice, which is why
we now turn to its graphical interpretation.
Since, unlike a real variable, a complex variable consists of two parts, it is these two
parts that determine a complex plane. In the graph in Fig. 1.2 two axes are plotted that
by definition are orthogonal – the axis of the real part and that of the imaginary part of a
complex number. It should be pointed out that we have a plane in a Cartesian
coordinate system and on its axes we place real numbers x and y. On the horizontal
axis we place the real part of the complex variable and on the vertical axis its
imaginary part.
Any point lying on a complex plane determined by the aforementioned axes
represents a complex number, even if this point is on the axis of real numbers. In
this case it represents a complex number with the imaginary part equal to zero.
In a Cartesian plane a complex number (1.5) may be considered a vector
(Fig. 1.2) that comes from the point of origin and ends at the point (x, y). Then
any complex number may be represented in polar coordinates by means of the
vector length and the polar angle:
z ¼ x þ iy ¼ r ðcos ’ þ i sin ’Þ:
(1.7)
Here, z is a polar angle (Fig. 1.2) and r a polar radius that in the present case is
called the absolute value of a complex number (vector length). Clearly the absolute
value of a complex number is equal to
r¼
The polar angle may be found as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2 þ y2 :
y
’ ¼ arctg þ 2pk:
x
(1.8)
(1.9)
10
1 Theoretical Basis of Complex Economy
where k is a whole number. Sometimes a polar angle is called an argument of a
complex number.
Two complex numbers are equal to each other only when their real and imaginary parts are equal. This means that, for example, the equation
yr þ iyi ¼ fr ðxr Þ þ ifi ðxi Þ
may be considered a more compact form of the following set of equations:
(
yr ¼ fr ðxr Þ;
yi ¼ fi ðxi Þ:
The trigonometric form of a complex variable is especially convenient for
multiplying complex numbers by each other. In addition to complex number
(1.7), let there, be another complex number:
w ¼ rðcos c þ i sin cÞ:
By multiplying z and w we will calculate the product of these variables. Skipping
the elementary derivation we obtain the following result:
zw ¼ rrðcos ’ þ i sin ’Þðcos c þ i sin cÞ ¼ rrðcosð’ þ cÞ þ i sinð’ þ cÞÞ:
This formula is known as Moivre’s formula; it says that the absolute value of the
product of complex numbers is equal to the product of modules, and its argument is
equal to the sum of the arguments of the factors. Moivre’s formula makes it much
easier to perform such operations with complex numbers as involution and extraction of the square root of a complex number. In fact, to find the square of a complex
number, we need to square its module and multiply the polar angle by 2.
In 1748, in his book “Introduction to the Analysis of Infinitesimals,” L. Euler
proved the formula bearing his name, that is [2]
ei’ ¼ cos ’ þ i sin ’:
(1.10)
Using Euler’s formula any complex number z with module r and argument ’
may be written in the following (exponential) form:
z ¼ rei’ :
(1.11)
This form is also very convenient for multiplying two complex numbers z1 and z2
and performing other operations. Indeed, using (1.11) again multiply complex
number z by another complex number w:
z1 z2 ¼ r1 ei’1 r2 ei’2 ¼ r1 r2 eið’1 þ’2 Þ :
(1.12)
1.2 Basic Concepts of the TFCV
11
Since the absolute value of a complex number may be presented in exponential form
r ¼ eln r ;
complex number (1.11) may be presented in another form, namely,
z ¼ eln rþi’ ;
which makes it possible to calculate the logarithms of a complex number.
Taking into account the fact that an argument of complex number (1.9) is
determined up to a term that is a multiple of 2p, the logarithm of complex number
z may easily be calculated as follows:
ln z ¼ lnðeln rþi’ Þ ¼ ln r þ i’ þ 2pk
(1.13)
That is, the logarithm of a complex number is a periodic function. As a rule, in
practice we use the main logarithm value taking k ¼ 0. The polar angle of complex
number ’ is called the argument of a complex number for short and is denoted by
’ ¼ Argz :
(1.14)
The argument of a complex number is not determined uniquely but up to a term
multiple of 2p:
Argz ¼ arg z þ 2pk;
(1.15)
where k is a whole number and argz is the main value of the argument determined
by the condition
p < arg z p:
(1.16)
For brevity’s sake and to formalize the language of mathematical problems, the
name of the real part of complex number z is abbreviated and referred to as Re(z).
The imaginary part is referred to as Im(z). Taking into account these designations it
may be easy to specify some range in a complex plane z on which the set of points
satisfies the condition
Im z3 > 4:5
To find this range, we should find the imaginary part of the complex variable (z3)
and then substitute the value found into the specified inequality. Incidentally,
describing a set of points in a plane using, for example, a Cartesian coordinate
system and applying real numbers is much more difficult and would involve a
system of several nonlinear inequalities. It is evident that complex variables
provide a compact representation of this problem.
12
1 Theoretical Basis of Complex Economy
A complex plane consists of various areas in which a function of a complex
variable may be determined. The complex variable function is said to be set to
w ¼ f ðzÞ
(1.17)
if a law is specified according to which each point z from the range of permissible
values is associated with a certain point or an aggregate of points w. In the first case,
where there is a correlation with one point, the function (1.17) is called singlevalued; in the second case, where there is a correlation of each point from z with a
set of points from w, the function is called multivalued.
If z ¼ x þ iy and w ¼ u þ iv, then setting the complex variable function w ¼ f ðzÞ
is equal to setting two functions of two real variables:
(
u ¼ uðx; yÞ;
v ¼ vðx; yÞ:
(1.18)
Using complex-number theory one can provide the functional dependence for
any pair of real numbers. Situations where one can establish a functional association for a pair of values are quite frequent in economics.
Thus, for example, such parameters as total costs involved in production C and
gross income G may serve as the results of any production activity. Then the
complex variable of production results may be represented in the following form:
G þ iC:
Production resources used as the real subjects of economic activity are quite
diverse. However, all this diversity in the theory of production functions is limited
to two resources: capital K and labor L, which also may be represented as one
complex variable:
K þ iL:
So far we have not discussed what variables refer to real or imaginary parts of
complex numbers. We will do this later on. The important thing is that using an
imaginary unit we can to combine two economics parameters into one complex
number. Just this presentation of economic parameters of production shows that the
model connecting production resources with production results may have the form
of a complex-variable function:
G þ iC ¼ f ðK þ iLÞ:
(1.19)
Any model generated by this relationship (1.19) will describe a real production
process with a certain degree of accuracy. And it will do so in a different way than
models of real variables.
1.2 Basic Concepts of the TFCV
13
Besides these pairs of values, other pairs of economic parameters may also be
identified. Linking them together by an imaginary unit we obtain complex economic variables. Using them to perform mathematical operations will give us other
results than those that those obtained by economists today using models of real
variables.
More often, in economics several parameters (more than two) depend on several
factors (more than two). This is why we would like to simulate this dependence, i.e.,
to use a mathematical equation to correlate an aggregate of economic parameters
with the aggregate of factors influencing them – to use hypercomplex numbers.
However, the attempt to introduce a system of numbers containing three units did
not yield positive results. We succeeded in building a system of numbers with four
imaginary units. In this case we get a so-called system of quaternions, i.e., numbers
such as
A ¼ a þ ib þ jc þ kd;
(1.20)
where a, b, c, d are real numbers and i, j, k are imaginary units.
Operations with quaternions are complicated, which does not allow for their
application to any practical purposes and is still an area of idealized research. In the
field of quaternions, the commutative property of multiplication does not work,
which leads to various confusions. Thus, for the equation
x2 þ 1 ¼ 0
there is an endless number of roots:
X ¼ ip þ jq þ kr; where p2 þ q2 þ r 2 ¼ 1:
That is why the quite evident desire to describe the dependence of a certain
complex indicator represented by quaternion (1.20) from another economic indicator represented in the form of another quaternion is not yet feasible.
In the present work we will use only complex numbers like (1.5) or consider
functions of complex variables (1.17). If necessary, we will use the conclusions and
proposals of the TFCV.
The last important property of complex variables that should be mentioned here
is the concept of infinity. For real variables it is evident (Fig. 1.1). When the
numerical axis tends to the right, to the area of positive numbers, this means plus
infinity. If it tends to the left of the zero point to the area of negative numbers, it
indicates a tendency to minus infinity. If we try to determine in the same way
infinity for a complex variable, then we will fail as complex numbers are presented
not on a numerical axis but in a complex plane, where each of the axes determining
the complex plane tends to infinity. The axis of real numbers has both plus infinity
and minus infinity, as does the axis of imaginary numbers. Then the question is how
to define an infinite complex number.
14
1 Theoretical Basis of Complex Economy
This question was answered by B. Riemann. Let us consider a sphere S touching
a complex plane at zero point [3]. Let P be a point of sphere S opposite the zero
point. Each point z of complex sphere S is associated with point M, which is the
point of intersection of sphere S with the segment connecting points z and P. The
sequence {zn} converging to infinity corresponds to a sequence of points of sphere S
converging to point P. This is why point z ¼ 1 is associated with point P on a
Riemann sphere.
1.3
Axiomatic Core of the Theory of the Complex Economy
All theories are based on certain assumptions taken without proof. The first group of
such assumptions refers to axioms, which are known to be taken without proof
because they are obvious. The second group represents postulates, which are
conclusions made in other branches of science and therefore taken without proof
as the proof has already been done by other researchers.
In the theory of a complex economy let us postulate provisions of the theory of
functions of complex variables taking its basic recommendations without proof;
anyone who has doubts may always consult the numerous textbooks and scientific
works on this branch of mathematics.
We will also postulate the main conclusions of theoretical economics or applied
sections of economics. For example, we need not prove the interrelation between
the wages of a worker and his productivity; this has already been done in the branch
of economics that deals with labor management.
Let us present those provisions that seem evident and therefore do not require
proof, i.e., are axiomatic, but without which complex economics as a theory cannot
exist.
The first axiomatic provision says that practically all economic indicators used
by economists to make judgments about the economy represents some generalized
or aggregated values that may easily be presented as the sum of two terms that, with
a certain degree of confidence, may be called the “active part” and the “passive
part.” This idea is based on numerous classifications adopted in economics.
For example, according to a given classification, the labor resources of any
enterprise may be divided into an active part (industrial and production personnel)
and a passive part (nonproduction personnel). Or a family’s expenses may be
divided into an active part, connected with the immediate satisfaction of current
needs, and a passive part, connected with the satisfaction of future needs. The same
may be said of the division of the final product of any country, which may be
represented in the form of two constituents – consumption (active part) and
accumulation (passive part).
Since the active and passive parts of some indicator or factor have different
influences on other economic indicators, their total influence can very logically
represented in the form of a complex variable, where the active component will be
said to be its real part and the passive one will refer to its imaginary part.
1.3 Axiomatic Core of the Theory of the Complex Economy
15
In the aforementioned example of electrical engineering, for instance, in
alternating current modeling, the active part will be real and the reactive imaginary.
Alternating currents occur in a rotating electromagnetic field that creates electric
current, voltage, capacity, and power variables in the conductor. Transmission of
the electric power along some circuits encounters active and reactive resistance.
This could seem like the semantic content of the real and imaginary parts of a
complex variable. However, in practice, classification of the active part of electricity parameters as the real part of a complex number, as well as the classification of
the reactive components as the imaginary part is conditional; one could just as
easily swap them, that is, classify the active power as the imaginary part and the
reactive as the real one. Actually, the very concepts of “active part” and “reactive
part” represent a rule, a matter of preliminary agreement on the part of scientists.
For example, if we classify an active current as the imaginary part of a complex
variable and its reactive component as the real part, i.e., do the opposite of what is
currently accepted in electric power engineering, the form of the applied mathematical models will change slightly but the computational process and, more
importantly, the results will not change at all. It’s just that when the theory of
functions of complex variables was used for the first time in electrical engineering
scientists agreed on how to classify each part. This has become a RULE that is now
beyond question. Some scientists even seriously believe that according to its
physical properties, the reactive part of electromagnetic power exactly matches
the imaginary part of a complex variable. This is certainly not the case.
Similarly, let us, in the theory of the complex economy, agree that in a complex
economy there is a RULE according to which the active part of an economic
indicator will refer to the real part of a complex number and the passive one to its
imaginary part.
Now we should mention several important conditions restricting the scope of
complex numbers in economics. To use the tools of the theory of functions
of complex variables in economics, a combination of two economic indicators
in one complex variable should satisfy the following obvious conditions defined by
the features of complex numbers:
1. These indicators should be two characteristics of the same process or phenomenon, i.e., they should show various sides of this phenomenon.
2. In addition, they should have the same dimension or be dimensionless. Also,
they should be of the same scale.
Why is it necessary to take into account the first condition if, according to the
rules, the real and imaginary parts are independent (orthogonal) of each other?
Should they be considered two sides of the same coin? Yes, they should, because
any complex variable that is formed from two real ones will further be considered
an independent variable unit. Figuratively speaking, it carries within itself information about its two components and demonstrates the influence of each on some
result. These values should reflect various aspects of the same phenomenon;
otherwise their combination in one variable becomes senseless. These variables
may be in a close functional dependence or be completely independent, but the
16
1 Theoretical Basis of Complex Economy
main condition is that they should carry information about a certain common
process. Such characteristics of a complex number as its module and argument
make sense only when the components of the complex number reflect a common
content.
The second condition requiring the same dimension of the components of a
complex number stems from the specific properties of complex numbers. In fact,
how can the absolute value of complex number (1.8) be calculated if the real and
imaginary parts have different dimensions, for example, dollars and pieces? It is
impossible to square them as (dollar)2 cannot be added to (piece)2. In the same way,
when calculating the polar angle, we should find the ratio of the imaginary to the
real part and then the arctangent of the result. If the real and imaginary parts have
different dimensions, then nothing can be done as the tangent of an angle is a
dimensionless value; it cannot be measured in dollars/pieces.
In economics, a considerable part of parameters may be transformed with
monetary units of measurement, for example, labor costs can be determined not
in “man-hours” but in the cost of wages – the size of the wage fund at the enterprise
or subdivision. This is why for most real economic problems this condition seems
quite feasible. However, when each of the variables has its own unit, they should be
brought to some relative dimensionless values using the best method for the
selected model.
1.4
Basic Model of a Complex Economy
Economic and mathematical models operating with real variables are based on the idea
that some economic parameter y is represented as being dependent on another parameter x. This dependence may be described by means of a function when parameter x is
correlated with only one parameter y:
y ¼ f ðxÞ :
(1.21)
Since each of the variables of the function (1.21) is an aggregate of real numbers
that can be shown as an array of points on the numerical axis going through the zero
point from minus infinity to plus infinity, and the distance from zero to this point is a
number in the selected scale and shows the number itself, model (1.21)
demonstrates that for each point on the real variable axis x there is one and only
one point on the real variable axis y. These axes may be plotted on one plane in any
order, for example, parallel to each other. However, the most informative is the
location of these numerical axes when they cross each other at a right angle
(perpendicular to each other) and their crossing point is the zero point on each of
the axes. In this case the model may be examined in a Cartesian coordinate system.
Model (1.21) may be made more complicated in any number of ways, by adding
new variables or by making it multifactorial. Then parameter y will depend on
1.4 Basic Model of a Complex Economy
xi
17
B’
B
yi
A’
A
0
xr
0
yr
Fig. 1.3 Conformal mapping of points of complex plane x on complex plane y
several variables, and the graph of this dependence will be three-dimensional, fourdimensional, or generally multidimensional subject to the number of variables.
Function (1.21) is basic for constructing models in the area of real variables, its
graphical interpretation on the Cartesian plane being its additional characteristic.
In the same way, in a complex economy we consider the basic model of
dependence of one complex variable on another. If we know the properties of this
dependence, then we can determine which processes it can describe, as well as
switch to multifactor complex models and to systems of complex models. However,
multifactor complex models and systems of complex equations are not as common
as their analogs in the area of real variables.
A very basic model represents a functional dependence of one complex variable
yr + iyi on another xr + ixi:
yr þ iyi ¼ f ðxr þ ixi Þ :
(1.22)
In the TFCV, this function is called complex.
As any complex variable represents a point in a Cartesian coordinate system,
equality (1.22) means that one point in a complex plane of variables x is correlated
with a point (and in some cases several points) in the complex plane of variables y.
This correlation is shown in Fig. 1.3.
In this figure, to the left we see a complex plane of variables x with two points A
and B. According to rule (1.22) each of these points is correlated with some point in
the plane of complex variables y. Point A is correlated with point A’, and point B is
correlated with point B’. Thus, using complex function (1.22) any set of points in
complex plane x is mapped onto complex plane y. This is why a graphic representation of a complex function is called conformal mapping. This concept is quite
suitable for purposes of the present study, though we could use a clearer mathematical definition of conformal mapping, for example, a mapping of a neighborhood of
point x0 onto a neighborhood of point y0 provided by function (1.22) is called
conformal if in point x0 it (the mapping) possesses a property to preserve the angles
between lines and the property of permanent stretching [4]. This means that not all
the mappings of the points of one complex plane onto another plane by means of
function (1.22) will be conformal, only those where the curves in the first plane
transfer to curves in the other plane, so that the angle between the tangents to these
curves in the first plane would correspond to some angle to tangents in the second
18
1 Theoretical Basis of Complex Economy
The complex variable Х
Xi
10
8
6
4
2
1
0
-2
2
4
6
8
Xr
10
-4
-6
Fig. 1.4 A change in complex variable x
plane, and an infinitesimal circle with the center at point x0 would correlate with an
infinitesimal circle of the second complex plane with the center at point y0.
Complex functions to be considered in the present work possess this property,
which is why we will consider conformal mapping as a graphical interpretation of
the functional dependence between two complex parameters.
If by means of complex function (1.22) each point in complex plane x is
correlated with one and only one point in complex plane y, this conformal mapping
is called univalent. But in the TFCV we frequently face situations where, using
complex function (1.22), each point in complex plane x is correlated with several
points in complex plane y. As we have already mentioned, this function is called
multivalued and a graphical representation of the conformal mapping is called
multivalent. The phenomenon of multivalence is to be considered more in detail
below, in the section where multivalence is revealed as a property of a complex
function.
From Fig. 1.3 it is easy to see that a graphical visualization of functions of
complex variables is worse than that of functions of real variables. An economist
studying empirical data who uses graphical analysis of the conformity of real
variables to each other gets an idea of the type and direction of this correlation and
knows exactly what form – quadratic, exponential, etc. – the linear relationship has.
If he or she tries to study conformal mapping of one real changing complex
variable onto the plane of another complex variable using some function and does
this graphically, then most often he or she will not obtain any idea about a
correlation between the complex variables.
Thus, for example, the graph in Fig. 1.4 shows a change in complex variable
x ¼ xr + ixi. Figure 1.5 shows a change in another complex variable y ¼ yr + iyi.
The researcher knows that between these two complex variables there is some
correlation whose character she does not know. Her task is to determine what form
of complex function should be applied to model this correlation.
1.4 Basic Model of a Complex Economy
19
The complex variable Y
20
Yi
15
10
5
Yr
-6
-4
-2
0
2
4
6
8
-5
Fig. 1.5 Change in complex parameter y corresponding to change in complex parameter x in
Fig. 1.4
A visual comparison of the graphs in Figs. 1.4 and 1.5 creates a stable impression
that if there is a relationship between these two complex variables, it has a complex
nonlinear character.
However, in fact we have a simple linear complex function like
yr þ iyi ¼ ð1 þ iÞðxr þ ixi Þ:
This example clearly shows how difficult it is to use complex variable functions
in practice, at least where a graphical interpretation of ongoing processes is
required.
On the other hand, in modeling the economy, we more often deal with smooth
trends of indicator changes, for example, a smooth trend of change in one complex
indicator x will be transformed into a smooth trend of another complex indicator y if
the same linear relationship exists between them that is not revealed visually in the
graphs of Figs. 1.4 and 1.5.
According to the properties of complex numbers, basic model (1.21) can be
represented as a system of two equalities with real variables:
(
yr ¼ fr ðxr ; xi Þ
:
(1.23)
yi ¼ fi ðxr ; xi Þ
For example, the simplest linear complex-value function
yr þ iyi ¼ ða0 þ ia1 Þðxr þ ixi Þ
can be represented as a system of two equalities:
(
y r ¼ a0 x r a1 x i
yi ¼ a0 xi þ a1 xr
:
(1.24)
(1.25)
20
1 Theoretical Basis of Complex Economy
It follows from the foregoing discussion that change in one of the indicators of
complex variable x leads to a change in both the real and imaginary parts of
complex variable y, that is, complex-value functions are multifactor by definition.
In the domain of real numbers we also encounter multifactor models, for
example,
y ¼ 5 þ 3x1 þ 2x2 :
(1.26)
However, in the domain of real variables, let us set the following problem for
model (1.26): find a pair of values of influencing variables (x1, x2) for a given value
y*. This problem does not have just one solution, since from equality (1.26), for the
given y*, one can obtain an equation with two unknowns:
3x1 þ 2x2 ¼ y
5:
(1.27)
For complex-value function (1.22) this problem (in the absence of multivalence)
can easily be solved, for example, for linear function (1.24), given known values of
yr* and yi*, one can easily find only one pair of values of complex variable x:
xr þ ixi ¼
yr þ iyi
:
a0 þ ia1
(1.28)
For example, if an enterprise wishes to find the best combination of production
resources to obtain a certain profit, in the domain of real variables we should solve
an optimization problem, and in a complex-value economy we should build an
inverse function, as is done in (1.28).
The model of a complex argument is a special case of the basic model (1.22):
yr ¼ f ðxr þ ixi Þ :
(1.29)
This model can be represented in general form as a complex-value function
where the imaginary part is equal to zero:
yr þ i0 ¼ f ðxr þ ixi Þ :
(1.30)
This means that it is a function of real variables but presented in a complex form.
For a whole series of economic tasks the inverse function is of interest:
xr þ ixi ¼ f ðyÞ :
(1.31)
It is a function of complex variables with a real argument. A simple example of
this function may be as follows:
xr þ ixi ¼ ya0 þia1 :
(1.32)
1.5 Some Data on Minkowsky’s Geometry
21
Here, a change in one variable entails a simultaneous change in two variables. If
we model such a situation using real variables, then we should use a system of two
equations. A complex economy provides compact modeling of complicated processes, and this is not its only advantage.
1.5
Some Data on Minkowsky’s Geometry
Complex variables “open the door” to a fascinating world of various ideas about the
world around us and its models. All the strength of this mathematical tool is
demonstrated in theoretical physics, especially in the theory of relativity. Application of the TFCV gives the physicist a graphical interpretation of space curvature
and time deceleration or acceleration. This interpretation is connected with the
name of Minkowsky – an outstanding mathematician from Konigsberg.
Since this tool may also be used in economics, Minkowsky’s geometry should be
considered in the first chapter of a study containing the methodological foundations
of a complex economy. The first successful attempt to apply the properties of
Minkowsky’s geometry to a solution of economic problems was taken up by Ivan
Svetunkov to verify the adequacy of models in the real world. The main results of
this approach are specified in Sect. 4.10.
First, it should be noted that so far the complex number yr þ iyi has been
considered in the Euclidean plane, i.e., in a plane where we place real numbers
on x and y axes. On the vertical axis of the Euclidean plane we deliberately wrote
“imaginary part” or “iyi” as in Fig. 1.6. The imaginary number is shown by means
of a real number.
That is, a complex number consisting of a real and an imaginary part is
represented as a real number in vector form. Then what is the difference between
a complex number and a two-dimensional vector? Is there any point in using the
theory of complex variable functions where one could use, for example, vector
algebra? In this geometrical interpretation these questions are obvious, but there are
no simple unambiguous answers.
To understand the power of the TFCV, we should switch from the Euclidean
plane to a pseudo-Euclidean plane. To do that, on the horizontal axis we place real
numbers (real part of a complex variable) and on the vertical axis an imaginary
number (imaginary part of a complex variable). Then we obtain a complex plane
(1.7), and points therein will have an absolutely different interpretation than points
in a Euclidian plane. This difference means the following. For the Euclidean plane
the scale of each of its axes (x- and y-axes) is determined by a one-unit-interval
segment. On the horizontal axis, to the right of the zero point we place a “+1”
segment, to the left a “–1” segment. In the same way, on the vertical axis, above the
zero point we place a “+1”, below a “–1.” Then the coordinates of the one unit point
are (1;1).
However, a complex number consists of a real and an imaginary part and a
complex unit should have the coordinates (1;i). This is why along the horizontal
22
1 Theoretical Basis of Complex Economy
Fig. 1.6 Complex number
in Euclidean plane
iyi
yr + iyi
1
-1
0
-1
1
yк
axis we should place real numbers and on the vertical axis not a “+1” or “–1” but
imaginary numbers. Then in the complex plane, on the horizontal axis, we place a
“+1” to the right and “–1” to the left of the zero point, but on the vertical axis we
place “+i” above and “–i” below the zero point. Then any imaginary component of
a complex number will be placed on this axis in the following way: we should
multiply i by y y times. The principal difference in the geometric interpretation is
that distances in this pseudo-Euclidean complex plane will be mapped in a
completely different way than in the Euclidean plane; therefore, curves will also
be mapped differently.
Let us ask one question: what distance does a vector drawn from the zero point to
the point corresponding to the complex number yr þ iyi have?
Or, in other words, what length does it have? The obvious answer turns out to be
incorrect. The length of the vector drawn from the zero point to the given one does
not characterize the distance, as it does in the Euclidean plane. Since by definition
this distance is equal to the square root of the sum of squares of the point
coordinates, we obtain the following expression for the complex pseudo-Euclidean
plane:
j zt j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2r þ ðiyi Þ2 ¼ y2r y2i :
(1.33)
Thus distances and lengths in the complex pseudo-Euclidean plane have a
different interpretation than the same characteristics of Euclidean planes.
Fascinating properties follow from formula (1.33). For example, when the absolute value of the real part of a complex variable is more than the absolute value of the
imaginary part, the vector length is a real number. When the absolute value of the
real part of a complex variable is less than the absolute value of the imaginary part,
the radicand (1.33) becomes negative and the distance then becomes an imaginary
number! And when the absolute value of the real part of a complex variable is equal
to the absolute value of the imaginary part, if follows from (1.33) that the distance
from the point to the origin of coordinates is equal to zero!
It is known that in the Euclidean plane only a zero vector [with coordinates
(0;0)] can have zero length, and in the pseudo-Euclidean plane (Fig. 1.7) nonzero
1.6 Laplace Transform
23
Fig. 1.7 Complex number
in pseudo-Euclidean plane
yi
iyi = yr
i
yr + iyi
yr
-1
0
1
-i
iyi = -yr
vectors can have zero length. There are many vectors in a plane whose lengths are
equal to zero, and all of them, as follows from (1.33), will satisfy the condition
jyr j ¼ jyi j
(1.34)
or, as follows from the preceding expression, one of the following conditions:
yr ¼ yi ;
yr ¼
yi :
(1.35)
(1.36)
Vectors whose coordinates satisfy condition (1.35) or (1.36) lie on the
corresponding axes in the pseudo-Euclidean plane and have zero lengths. These
lines are isotropic. In Fig. 1.7 isotropic lines are shown as dotted lines. They
divide the plane into four sectors. In this pseudo-Euclidean plane all the vectors
with real lengths will lie either in the right or in the left sector, though vectors with
imaginary lengths will lie either in the upper or in the lower sectors. In some
economic problems this property may appear rather useful in the identification of
various objects.
Now economists can understand how physicists imagine the curvature of space
and time as well as the basis for the astonishing hypotheses regarding the instantaneous traversal of a point over huge distances in space because there are numerous
points in the pseudo-Euclidean space where the distance between them is equal to
zero! We do not insist on extending Minkowsky’s geometry to the economic space,
however, the mere possibility of using this new mathematical tool for economic
modeling should be noted.
1.6
Laplace Transform
As was said at the very beginning of this study, the use of complex variables in
economic modeling is negligible. In this smallest branch of economic and mathematical methods and models the section on the Laplace transform is the most
24
1 Theoretical Basis of Complex Economy
popular. Laplace’s transforms are actively used to solve systems of differential and
integral equations, calculate transfer functions of dynamics systems, calculate
transitional electromagnetic processes in electric contours, etc. This transform
underlies so-called operational calculus whose tools may also be applicable in
several economic problems. Let us show the essence of this transform to determine
its applicability in economics.
It is obvious that multiplying a complex number by a real number we can obtain
a new complex number. In the same way, multiplying any real variable function f(x)
by some complex number we can obtain a complex function of a real variable:
ðpr þ ipi Þf ðxÞ ¼ pr f ðxÞ þ ipi f ðxÞ :
(1.37)
In general, there are many methods of transferring a real number or its function
to the domain of complex numbers. For example, we can transform a real variable
function f(x) into a complex function of a real variable by a more complicated
method than multiplication, for example:
eðpr þipi Þf ðxÞ ¼ Rei’ ; R ¼ epr f ðxÞ ; ’ ¼ pi f ðxÞ:
(1.38)
However, any such transformation, a transfer of operations from the domain of
real numbers to the domain of complex ones, should make sense – it should lead to
new results or help solve some problems. That is why from the great diversity of
such possible transformations the most popular is the so-called Laplace transform.
Its essence is as follows.
Let there be a function f(t) in the domain of real variables about which we know
that it satisfies the following conditions:
1. It is integrable in any finite interval of the time axis t.
2. It is equal to 0 for negative t, f(t) ¼ 0 for t <0.
3. If t increases, the absolute value of the function can also increase but not faster
than some exponential function jf ðtÞjeat :
This function is called the original function.
Let there also be some complex number p. We call Laplace’s transform of real
variable function f(t) a complex variable function F(p) determined by the formula
FðpÞ ¼
1
ð
f ðtÞe
pt
dt :
(1.39)
0
This function, resulting from transform (1.39), is called the image function. The
correlation between the original function and the image function is written using
various mathematical symbols. We will use the following notation:
f ðtÞ ¼ FðtÞ :
(1.40)
1.6 Laplace Transform
25
It should be noted that for any original function image F(p) is determined in the
semiplane
Re p > M :
(1.41)
Laplace’s transform turned out to be a very efficient tool for solving numerous
complex mathematical problems using differential and integral equations since the
main operations with the original functions correspond to mathematical operations
with image functions that are easier than those with the originals.
It would not make sense to go into more detail regarding Laplace’s transform
and the rules and properties of the interrelations of the original functions and the
images as there are many specialized publications on operational computation, let
us concentrate here on demonstrating the use of Laplace’s transform in solving
problems.
For example, say we need to solve the differential equation
x0
x¼1
(1.42)
for initial conditions x(0) ¼ 1 (Kochi’s problem).
To solve this problem, let us use Laplace’s transform. We know that the original
unit function
f ðtÞ ¼ f0; if t < 01; if t 0ð1:43Þ
corresponds to the image function
FðtÞ ¼
1
:
p
(1.44)
By means of Laplace’s transform, for any constant o > 0
f ðotÞ ¼
1 p
Fð Þ
o o
(1.45)
and the derivative of any image function has the following mapping in a complex
plane:
f 0 ðtÞ ¼ pFðpÞ
f ð0Þ
(1.46)
Taking into account all this, a mapping of the problem of the solution of a
differential equation of real variables (1.42) to the domain of complex variables
using transforms (1.44)–(1.46) will have the following form:
pXðpÞ
f ð0Þ
XðpÞ ¼
1
:
p
(1.47)
26
1 Theoretical Basis of Complex Economy
This means that we have an elementary algebraic equation for x(p) instead of a
differential equation (1.42). Substituting the initial conditions x(0) ¼ 1 into this
algebraic equation we get a simple solution:
ðp
1ÞXðpÞ
1¼
1
pþ1
2p ðp 1Þ
2
! XðpÞ ¼
¼
¼
p
pðp 1Þ
pðp 1Þ
p 1
1
: (1.48)
p
From the image obtained it is easy to move on to the originals since it is known
that an exponential function has the following mapping:
elt ¼
1
p
l
(1.49)
and it is the first term on the right-hand side of (1.47) that has this form.
Then the original of the first term will be written as follows: 2et .
The second term on the right-hand side of solution (1.48) represents just the
image of the unit function (1.43).
Then the following will be the solution to the original differential equation
(1.42):
xðtÞ ¼ 2et
1:
(1.50)
This example can easily persuade us that mathematical operations with images
of various functions of real variables meeting the aforementioned requirements are
much easier than operations with the originals. And since Laplace’s transform for a
wide variety of functions of real variables that can serve as original functions were
calculated long ago and tabulated, the solution to the most diverse and complex
problems of operational calculus is considerably simplified.
In economics, use of differential equations, much less systems of differential and
integral equations, is quite rare and is found mainly in the construction of idealized
mathematical models, but when it does happen, scientists achieve success when
they use Laplace’s transforms.
References
1. Semyonychev VK, Semyonychev EV (2006) Information systems in economics. Econometric
modeling of innovations. Part I. Publishing house of Samara State Aerospace University,
Samara
2. Euler L (1961) Introduction in analysis of infinitesimals, vol 1. State Publishing House of
Literature in Physics and Mathematics, Moscow
3. Shabunin MI (2002) Theory of functions of complex variable. Unimediastyle, Moscow
4. Krasnov ML, Kiselev AI, Makarenko GI (2003) Complex variable function: problems and
examples with detailed solutions. Editorial URSS, Moscow
Chapter 2
Properties of Complex Numbers of a Real
Argument and Real Functions of a
Complex Argument
Important results can be obtained if we apply simple complex-value models in
economic modeling – complex functions of a real argument and real functions of a
complex argument. This chapter focuses on the properties of these models and the
possibility of using them in economic practice.
Complex models of a real argument represent the dependence of a complex
variable on a real argument. This dependence can be obtained only if one uses a
function that transforms real variables into complex ones. The Laplace transform is
a well-known transformation method; however, this chapter focuses on other
methods widely applied in economics. Real models of complex argument solve
another problem – the transformation of a complex variable into a real one. The
properties of the simplest models of this type are considered in this chapter with
respect to economic modeling.
2.1
General Problem of Conformal Mapping
in Complex-Valued Economics
Before using a tool of the theory of functions of a complex variable (TFCV) in
economics it is necessary to study the properties of this tool [1]. One of the methods
for understanding these properties is provided by conformal mapping of points from
one complex plane to another. With reference to various cases of the TFCV,
conformal mapping provides problems of varying degrees of complexity. We will
consider the simplest cases since an understanding of conformal mappings of
elementary complex-valued functions will allow researchers to choose the proper
complex-valued function for modeling.
Thus, we can say that conformal mapping is a convenient graphical method for
understanding how, by means of a given function, one complex variable in a
complex plane of an argument is mapped to another complex variable modeling
the value of the variable of the complex result.
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
DOI 10.1007/978-1-4614-5876-0_2, # Springer Science+Business Media New York 2012
27
28
2
Properties of Complex Numbers of a Real Argument. . .
Since we work in the sphere of complex numbers, any real number may be
represented as a complex number with zero imaginary part. Then we obtain three
types of functions to be used in economic modeling.
The first type represents the relationship between a complex variable and a real
argument:
yr þ iyi ¼ Fðxr þ i0Þ ¼ f ðxr Þ þ if ðxr Þ:
(2.1)
It is a complex function of a real variable.
The second type comes up when a complex argument is associated with a real
result:
yr þ i0 ¼ Fðxr þ ixi Þ ¼ fr ðxr ; xi Þ þ ifi ðxr ; xi Þ ,
(
yr ¼ fr ðxr ; xi Þ;
fi ðxr ; xi Þ ¼ 0:
(2.2)
It is a function of a complex argument.
The third type is the relationship between a complex variable and a complex
result:
yr þ iyi ¼ Fðxr þ ixi Þ ¼ fr ðxr ; xi Þ þ ifi ðxr ; xi Þ:
(2.3)
It is a complex function (a function with complex values).
The TFCV considers mainly conformal mappings of the third type. However, in
economics we can use all three as models of a complex-valued economy. This is
why it is essential to examine in depth the properties of all three types of functions.
This chapter will focus on the properties of the first two types.
2.2
Complex Functions of a Real Argument
The complex functions of a real argument represent a certain “mapping” of a set of
real numbers on a numerical axis to the plane of complex variables:
yr þ iyi ¼ Fðx þ i0Þ ¼ f ðxÞ þ if ðxÞ:
(2.4)
This function transforms real variables and the respective functions to complex
variables and the respective functions.
Situations where one variable influences two others are quite frequent in economics. For example, in marketing, consumers are grouped in particular categories
– segments where the basic indicator is a similar reaction of all consumers of this
segment to a product and its marketing support. This means that consumers with
similar levels of income (if we categorize by income) will react similarly to a given
price and buy the same quantity of the product at that price. This in turn means that
2.2 Complex Functions of a Real Argument
29
the price yr and the consumption volume yi depend on the level of income x. With
this knowledge, one can look at the reaction to goods by consumers from various
segments as being subject to an increase in income of each segment and model this
reaction by a function of a real argument (2.4).
The variety of possible functions of a real argument that may be put forth to
model the aforementioned economic processes is limited only by the imagination of
the researcher creating the model. This is why in this section we deal only with the
simplest functions and their properties.
A linear model of a real argument,
yr þ iyi ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þðx þ i0Þ ¼ ða0 þ b0 xÞ þ iða1 þ b1 xÞ;
(2.5)
is of little interest because any change in the argument entails a directly proportional change in the real and imaginary parts of the complex result. This means that
for any change in the real argument – linear or nonlinear – we have a line in a
complex plane whose slope and position thereon is completely determined by the
values of a complex proportionality coefficient.
Nonlinear transformations of a real variable to a complex plane are of practical
interest. The first of these methods is the complex involution of a real argument:
yr þ iyi ¼ ða0 þ ia1 Þxðb0 þib1 Þ :
(2.6)
The proportionality coefficient that can be placed before the argument that is
subject to involution can be not only complex, but also real or imaginary. Let us
represent the complex function of a real argument in exponential form, then (2.6)
will be written as
yr þ iyi ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
iðarctga1 þb1 ln xÞ
0
a20 þ a21 xb0 e
:
(2.7)
To simplify the notation of the complex proportionality coefficient, let us write it
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
as A ¼ a20 þ a21 and the polar angle as a ¼ arctg aa10 :
The equality of the real and imaginary parts of this equation may be represented
as the following system:
(
yr ¼ Axb0 cosða þ b1 ln xÞ;
yi ¼ Axb0 sinða þ b1 ln xÞ:
(2.8)
It is clear that both the real and imaginary parts of this complex real variable
function change with an increase in the argument according to the cosine (real part)
and sine (imaginary part) law. Taking into account the fact that the real argument in
these trigonometric functions is not direct but a logarithm, with a uniform increase in
the real argument, periods of oscillation of both the real and imaginary parts of the
function under consideration will increase. A logarithm limits the function domain;
30
2
Properties of Complex Numbers of a Real Argument. . .
since a logarithm of zero does not exist, the zero point is not included in the function
domain.
If we consider the result (2.7) in the complex plane, the points of this function
will be located as follows. The module of this function
r ¼ Axb0
(2.9)
will increase with an increase in the argument x > 0 for b0 > 0 and decrease for
b0 < 0, and the polar angle will increase,
’ ¼ a þ b1 ln x;
(2.10)
if b1 > 0 and decrease (move in a clockwise direction) if b1 < 0.
Hence, it is easy to see that in the complex plane function (2.6) is mapped subject
to the values of the complex exponent in the form of a convergent or divergent
spiral.
Let us consider a special case of function (2.6), where time t acts as the
argument:
yr þ iyi ¼ ða0 þ ia1 Þtðb0 þib1 Þ :
(2.11)
This function represents a complex trend and may be used in practice in certain
economic situations.
As follows from the aforementioned properties of the function under consideration, the character of a complex trend will be fully determined by its coefficients.
Here are some interesting types of such trends.
Thus, if we use the trend
yrt þ iyit ¼ tð
0;5þi10Þ
;
(2.12)
then each of the components of the complex-valued trend will look like Figs. 2.1
and 2.2.
The same form of the trend but with other coefficients
yrt þ iyit ¼ tð0;25þi0;35Þ
(2.13)
models completely different dynamics (Figs. 2.3 and 2.4).
Trends like those shown in Figs. 2.3. and 2.4 are quite frequent in the domain of
real variables; however, models describing the dynamics of trends like 2.1 and 2.2
are quite rare in studies on socioeconomic processes, except for stock markets.
The next model of a real argument may be a complex exponential function of the
real argument. It may be presented as follows:
yr þ iyi ¼ ða0 þ ia1 Þeðb0 þib1 Þx :
(2.14)
2.2 Complex Functions of a Real Argument
Fig. 2.1 Dynamics of real
part of complex trend (2.12)
31
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0
5
10
15
20
25
0
5
10
15
20
25
20
25
-0.4
-0.6
Fig. 2.2 Dynamics of
imaginary part of complex
trend (2.12)
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
Fig. 2.3 Dynamics of real
part of complex trend (2.13)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
32
2
Fig. 2.4 Dynamics of
imaginary part of complex
trend (2.13)
Properties of Complex Numbers of a Real Argument. . .
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
The base of an exponential function may also be different, for example, a
complex number, but we will not consider these variants.
In the exponential form function (2.14) may look like this:
yr þ iyi ¼ Aeb0 x eiðaþb1 xÞ :
(2.15)
As we see, the module of this function varies according to the exponential law
and variations in the polar angle are directly proportional to variations in the
argument. Since the complex coefficient of an exponent can take various values,
a modeled function can describe different variants of the dynamic whose details
differ from function (2.7), but, similarly to that function, its mapping to the complex
plane is spiral.
If we consider the real and imaginary parts of this complex function separately,
we will have a system of equations:
(
yr ¼ Aeb0 x cosða þ b1 xÞ;
yi ¼ Aeb0 x sinða þ b1 xÞ:
(2.16)
Now the differences between a complex exponential function of a real argument and
a complex power function of a real argument are evident. The real and imaginary parts
of an exponential function vary according to the cosine and sine laws with a constant
period of oscillations, the oscillation range varying with the change in the argument. If
b0 > 0, then the oscillation range increases with the growth of the argument; if b0 < 0,
then the oscillation range decreases.
A complex trend model is a simple variant of this model.
For example, for low positive values of coefficients of a complex exponent like
yrt þ iyit ¼ eð0;15þi0;05Þt
(2.17)
2.2 Complex Functions of a Real Argument
Fig. 2.5 Dynamics of real
part of trend (2.17)
33
14
12
10
8
6
4
2
0
Fig. 2.6 Dynamics of
imaginary part of trend (2.17)
0
5
10
15
20
25
0
5
10
15
20
25
30
25
20
15
10
5
0
each of the components is described for t ¼ 1,2,. . .22 with an increasing area
(Figs. 2.5 and 2.6).
For other coefficients a considerably more complex cyclical dynamics can be
modeled, for example, if the model has the form
yrt þ iyit ¼ eð0;05þi60Þt
(2.18)
then the dynamics of the real and imaginary constituents of the complex-valued
trend takes the form shown in Figs. 2.7 and 2.8.
It is seen from these figures that the function models the oscillation process with
increasing amplitude at a constant oscillation frequency.
We could continue looking at similar elementary complex functions of real
arguments, but this goes beyond the problems covered by our study. Thus, we
will consider several special nonstandard functions that are subspecies of those
mentioned previously.
First is an exponential-power function with an imaginary exponent:
yr þ iyi ¼ xix :
(2.19)
34
Properties of Complex Numbers of a Real Argument. . .
2
Fig. 2.7 Dynamics of real
part of trend (2.18)
3
2
1
0
-1
0
5
10
15
20
25
-2
-3
-4
Fig. 2.8 Dynamics of
imaginary part of trend (2.19)
3
2
1
0
0
10
20
30
-1
-2
-3
Let us present this function in exponential form:
yr þ iyi ¼ eix ln x :
(2.20)
This helps us to determine a change in the real and imaginary parts with the
growth of the argument:
(
yr ¼ cosðx ln xÞ;
yi ¼ sinððx ln xÞ:
(2.21)
They change according to the cosine and sine laws with an increasing oscillation
period. A logarithm limits the function domain; since a logarithm of zero does not
exist, the zero point is not included in the function domain.
Since the module of this function is equal to one, in a complex plane the function
represents a unit circumference.
An exponential-power function with a complex exponent is an expected development of this function:
2.2 Complex Functions of a Real Argument
yr þ iyi ¼ xðxþixÞ ¼ xð1þiÞx :
35
(2.22)
In this case, the right-hand side of the equality is easily represented in exponential form:
yr þ iyi ¼ xx eix ln x :
(2.23)
This model has a domain in the positive part of real numbers since a logarithm of
a negative number, as well as a logarithm of zero, does not exist.
Let us present the real and imaginary parts of this complex function separately:
(
yr ¼ xx cosðx ln xÞ;
yi ¼ xx sinðx ln xÞ:
(2.24)
The module of this complex function increases sharply with an increase in the
argument; this is why the real and imaginary parts of the function represent an
oscillatory function with increasing oscillation period and sharply growing oscillation range. In a complex plane this function is shown as a sharply diverging spiral.
This feature gives the function little applicability in economic modeling, though the
initial part of the function could be of interest. The module of a function in the
positive neighborhood of the zero point is close to one (any number to the zero
power is equal to one); however, with an increase in the argument it will first
decrease and then increase. The module of the complex function reaches its
minimum value at the point where the first derivative is equal to zero:
dr
¼ ðxx Þ0 ¼ 0
dx
After solving this equation and using the Leibniz-Bernoulli formula we have
r 0 ¼ xx ð1 þ ln xÞ ¼ 0:
Since |x| > 0, the module of the complex function reaches its minimum value at
the point x ¼ e 1.
The dynamics of the polar angle u with changes in the argument within the
interval [0;1) is complicated since it is determined by the following equality:
y ¼ x ln x:
The first derivative of this relationship with respect to the argument will have the
following form:
dy
¼ ðxÞ0 ln x þ xðln xÞ0 ¼ ln x þ 1;
dx
36
Fig. 2.9 Real parts of
complex function (2.22)
2
Properties of Complex Numbers of a Real Argument. . .
yк
2
1.5
1
0.5
0
-0.5
0
0.5
1
1.5
2
2.5
x
-1
-1.5
-2
which means that the polar angle reaches its minimum value at the same point as the
module of the complex point x ¼ e 1.
Thus, for the argument x¼e 1 the complex function under consideration reaches
its minimum values of both the module and polar angle. In the complex plane this
will be shown with an increase in the argument as follows. The curve starts its
movement in the clockwise direction from the neighborhood of a point with the
coordinates xr ¼ 1, xi ¼ 0 until it reaches the point where both the module and the
argument take their minimum values. The module then is equal to
e
rmin ¼ xx ¼ ðe 1 Þ
1
¼
1e
1
e
and the polar angle to
ymin ¼ x ln x ¼
1
:
e
Then, with growth of the x module of the complex function, its polar angle starts
growing, too. In the complex plane this growth is revealed in movement along the
same line but in a counterclockwise direction.
Then the function module starts growing sharply, which leads to an increase in
the values of the real and imaginary parts of the function under consideration. The
separate dynamics of the real and imaginary parts of this complex function at low
values of the real argument x ¼ [0;2) are of more interest. This dynamics is given in
Fig. 2.9.
It is possible to narrow the spiral span and increase or reduce the rotation
frequency by involution of the complex proportionality coefficient, which is different from the complex unit:
2.2 Complex Functions of a Real Argument
Fig. 2.10 Imaginary parts
of complex function (2.22)
37
yi
1.2
1
0.8
0.6
0.4
0.2
x
0
-0.2
0
1
2
3
-0.4
yr þ iyi ¼ xðb0 þib1 Þx :
(2.25)
For different values of the real and imaginary parts of this coefficient, for the
function under consideration there one obtains a great variety of spirals in the
complex plane, as well as various types of dynamics of the real and imaginary parts
of the complex function (Fig. 2.10).
We can continue the logic of the real argument transformation to the complex
plane by suggesting a complex exponential-power function with a complex base:
yr þ iyi ¼ ðx þ ixÞx ¼ ðxð1 þ iÞÞx :
(2.26)
Its exponential form will look like this:
pffiffiffi x p
yr þ iyi ¼ ð 2xÞ ei4x :
(2.27)
Then for the real and imaginary parts of this complex function we have
8
pffiffiffi x
p
>
>
>
< yr ¼ ð 2xÞ cos 4 x ;
pffiffiffi x
>
p
>
>
x :
: yi ¼ ð 2xÞ sin
4
The zero point of the real argument is included in the function domain. In
general this function looks like a spiral; however, at low values its behavior is
complicated as its polar angle increases with the growth of the real argument and its
module first decreases, reaches its minimum, and starts increasing again.
The first derivative of the module is
38
2
Properties of Complex Numbers of a Real Argument. . .
yi
7
6
5
4
3
2
1
yr
0
-0.5
0
0.5
1
1.5
Fig. 2.11 Function (2.26) in complex plane at low values of real argument
pffiffiffi
pffiffiffi x
x
r 0 ¼ ð 2xÞ pffiffiffi þ lnð 2xÞ :
2x
If we set it equal to zero we obtain a point where the module takes minimum
values:
p1ffi
e 2
x ¼ pffiffiffi ¼ 0:348652215
2
Taking into account these specifics, for initial values of the real argument the
complex function will have nonlinear dynamics (Fig. 2.11).
The last elementary function of a real argument is a complex exponential-power
function with a complex base and complex exponent:
yr þ iyi ¼ ðx þ ixÞðxþixÞ ¼ ðxð1 þ iÞÞð1þiÞx :
(2.28)
In exponential form it looks like this:
pffiffiffi ð1þiÞx ipð1þiÞx
pffiffiffi x
yr þ iyi ¼ ð 2xÞ
e4
¼ ½ð 2xÞ e
p
4x
pffiffiffi ix p
½ð 2xÞ ei4x :
(2.29)
The module of this complex function of a real argument will be
pffiffiffi x
R ¼ ð 2xÞ e
p
4x
:
(2.30)
And its polar angle will have the following form:
pffiffiffi
p
y ¼ x lnð 2xÞ þ :
4
(2.31)
2.2 Complex Functions of a Real Argument
39
It is easy to see that the zero value of the real argument is not included in the
function domain.
The behavior of the module of this function is more complicated than that of the
previous ones. For an argument close to zero the module will be close to one, then it
gets lower up to a certain value, after which it starts increasing again, but not
sharply as in the case of the previous function.
To determine the point where the module of complex function (2.30) takes its
minimum value, we should find its first derivative:
pffiffiffi x
R0 ¼ ½ð 2xÞ 0 e
p
4x
pffiffiffi x
þ ð 2xÞ ðe
p
4x
pffiffiffi x 1
pffiffiffi
Þ0 ¼ ð 2xÞ pffiffiffi þ lnð 2xÞ e
2
p
4x
p pffiffiffi x
ð 2xÞ e
4
p
4x
;
which should be equal to zero. Then, solving the equation we find the point at which
the module is at its minimum:
3
p 22
e 4
x ¼ pffiffiffi :
2
The polar angle also varies nonlinearly – it decreases from values close but not
equal to zero (the points are in the fourth quadrant of the complex value) and then
grows. To determine the minimum value of the polar angle, let us find its first
derivative of the real argument:
0
3
pffiffiffi
pffiffiffi
pffiffiffi
dy
p
p
p þ 22
¼ ðxÞ0 lnð 2xÞ þ
:
¼ lnð 2xÞ þ
þ x lnð 2xÞ þ
dx
4
4
4
After setting it equal to zero and solving the equation we find the value of the real
argument for which the polar angle reaches its minimum value:
e
3
pþ22
4
x ¼ pffiffiffi
2
With an increase in the argument, the conformal mapping of the function under
consideration in a complex plane takes place in a spiral moving in a clockwise
direction, as shown in Fig. 2.12.
With the further growth of the argument, the function module increases sharply,
as does the polar angle, the function itself continuing its spiral movement in the
clockwise direction.
The elementary complex exponential-power function with complex base and
complex exponent (2.28) can be represented in a form more applicable to practical
purposes, namely:
yr þ iyi ¼ ðxða0 þ ia1 ÞÞðb0 þib1 Þx :
(2.32)
40
2
Properties of Complex Numbers of a Real Argument. . .
Fig. 2.12 Part of conformal
mapping of function (2.28)
for low value of argument
yi
1
0.8
0.6
0.4
0.2
xr
-2
-1.5
-1
0
-0.5
0
-0.2
0.5
1
1.5
-0.4
-0.6
-0.8
Its exponential form will have the form
yr þ iyi ¼ ½ðAxÞb0 x e
b1 xa
½ðAxÞib1 x eib0 xa :
(2.33)
Hence, for the module of this complex function of a real argument
R ¼ ðAxÞb0 x e
b1 xa
(2.34)
and for the polar angle
y ¼ x½b1 lnðAxÞ þ b0 a:
(2.35)
The function domain lies in the area of positive arguments, which clearly follows
from (2.35).
Changing the values of function coefficients (2.32) we can obtain a great
diversity of conformal mappings and variations of the real and imaginary parts of
this function that have an oscillatory character.
As we see from (2.35), the polar angle of this complex function of a real
argument depends largely on the constant b1. The higher the values of this constant,
the more rapid is the increase in the polar angle with the increase of the argument,
and the faster is the turnover of the function values in the complex plane. This
coefficient also influences the change in the module of the function under consideration, but for a low value of a1 and high value of a0 this influence decreases.
The coefficient b0 is responsible for the growth in the function module. At its
positive values the module increases sharply.
For various coefficient values the function behaves in a different way – it
converges to zero and diverges, changes values around some circumference,
changes chaotically, etc.
2.2 Complex Functions of a Real Argument
Fig. 2.13 Dynamics of real
part of function (2.36)
41
yr
10000
8000
6000
4000
2000
0
x
-2000 0
5
10
15
5
10
15
-4000
-6000
-8000
-10000
Fig. 2.14 Dynamics of
imaginary part of function
(2.36)
yi
10000
8000
6000
4000
2000
0
-2000 0
x
-4000
-6000
-8000
-10000
It is interesting that the function can also model the process of reaction of some
system to an external influence with further stabilization at the previous level. This
function behaves in this way, for example, for the following coefficients:
yr þ iyi ¼ ðxð1
iÞÞð
1;5þi6Þx
:
(2.36)
Subsequent change in the real and imaginary parts of this function with the
growth of the argument within 0 < x 10 subject to the argument is shown in
Figs. 2.13 and 2.14.
According to the results of this section we can draw the conclusion that complex
functions of a real argument model a great diversity of cyclical dynamics.
Numerous functions of complex arguments are not limited at all to only the
aforementioned types. However, it is not possible to consider all functions within
42
2
Properties of Complex Numbers of a Real Argument. . .
the framework of this study; this would be at odds with the purpose of the present
study, where we state only the basics of the application of the TFCV to solutions of
economic problems.
The superposition of elementary complex functions provides vast possibilities
for the generation of new functions. A simple example is the case where the
complex power function of a real argument
zr þ izi ¼ ðc0 þ ic1 Þxd0 þid1
is added to by a complex function of a real argument (2.32)
zr þ izi ¼ ðc0 þ ic1 Þxd0 þid1 þ ðxða0 þ ia1 ÞÞðb0 þib1 Þx :
(2.37)
If, for example, for the second term of this function we use coefficients like those
proposed in (2.36), the resulting model will describe the dynamics of some nonlinear process, which may chaotically deviate from its previous trajectory on a certain
segment under a certain external influence, but due to the stability of the object it
returns to its former trajectory. It is evident that instead of the power function, the
first term may be represented by other forms, for example, by a step function. With
the proper selection of parameters, with the assistance of such a superposition, we
model the transition from one stationary state to another.
The real argument itself can be presented in complex functions as a real function
of a real argument, for example, sinx or cosx.
It is evident that the variety of complex functions of a real argument is enormous, and it is impossible to cover them in one section or chapter.
2.3
Functions of a Complex Argument: Linear Function
Since it is possible to transform a real argument to a complex plane using particular
functions, a reverse transformation procedure is also possible – from the field of
complex variables to the numerical axis of real variables. The relationship between
a complex argument and a real result will represent a function of a complex
argument:
y ¼ f ðxr þ ixi Þ ¼ fr ðxr ; xi Þ þ ifi ðxr ; xi Þ:
(2.38)
Since there is a complex number in the right-hand side of this equality and a real
one in the left-hand side, the function of the complex argument may be written as
follows:
y þ i0 ¼ fr ðxr ; xi Þ þ ifi ðxr ; xi Þ:
(2.39)
2.3 Functions of a Complex Argument: Linear Function
43
Hence we have a system of two real equations:
(
y ¼ fr ðxr ; xi Þ;
0 ¼ fi ðxr ; xi Þ:
(2.40)
The first equality of the system represents an equation of some surface in a threedimensional space, the second one a line in the argument’s plane. Since the problem
is considered in a three-dimensional coordinate system, for the second equation of
system (2.40) the equality is valid for any y value. This means a surface in threedimensional space that is not crossed by the y-axis, i.e., the y-axis is parallel to this
surface, the surface itself being perpendicular to the complex plane of the argument.
Since these two equations are combined in a system, they are simultaneously
satisfied. Geometrically this means that system (2.40) and the initial function (2.38)
represent an intersection of two planes in three-dimensional space – the first and
second equations of system (2.40). The perpendicular nature of the second equation
of system (2.40) means that the aggregate of the points lying on the surface of the
first equation of system (2.40.) must be projected onto the plane of the complex
argument as a line described by the second equation of system (2.41).
Let us sequentially consider the main functions of a complex argument and their
graphical interpretation in order of increasing complexity, bringing each of them to
the form (2.40).
The first such model to be used in economics is a linear function of a complex
argument with a zero free term:
y ¼ ðb0 þ ib1 Þðxr þ ixi Þ:
(2.41)
If we single out the real and imaginary parts of this function and group them, we
have
(
y ¼ b0 x r b 1 x i ;
0 ¼ b1 x r þ b0 x i :
(2.42)
The first Eq. (2.42) is that of a plane in space passing through a zero point. The
slope angle and position of the plane in space is fully determined by the signs and
values of the coefficients of the complex proportionality coefficient.
The second equation of the system under consideration represents an equation of
a line in the plane of the argument:
xr ¼
b0
xi :
b1
(2.43)
This straight line originates from the zero point, and its location in the particular
quadrant of the complex plane is determined by the values of the real and imaginary
44
2
Properties of Complex Numbers of a Real Argument. . .
parts of the complex proportionality coefficient. Since the second equation of
system (2.42) should be considered in space, it represents a plane parallel to the
y-axis and perpendicular to the complex plane.
The two crossing planes form a line, meaning that the linear function of complex
argument (2.41) represents a line in three-dimensional space (0y;0xr;0xi) passing
through the zero point.
If we now consider a linear function of a complex argument with a free complex
coefficient:
y ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þðxr þ ixi Þ:
(2.44)
then singling out the real and imaginary parts of this function and grouping them as
in the previous case, we have
(
y ¼ a0 þ b 0 x r
b1 x i ;
0 ¼ a1 þ b1 x r þ b0 x i :
(2.45)
It is clear that the nature of the function has not changed – both the first and the
second equations are planar equations – only the location of the planes in space has
changed, as has the location of the line resulting from the planes’ intersecting. It
follows from the first equation that the plane does not pass in space through the zero
point and crosses the y-axis at the point y ¼ a0. The second equation shows that the
line in space of the complex argument does not pass through the zero point either
since
xr ¼
a1
b1
b0
xi :
b1
(2.46)
And on the axis of real values of the complex argument this line passes through
the point xr ¼ ab11 :
Thus, the linear function of a complex argument with free complex coefficient
(2.44) represents the equation of a line in three-dimensional space “a complex plane
of an argument – an axis of a real number.” Or put another way, any line in threedimensional space may be described by a linear function of complex argument
(2.44).
It is appropriate to recall that in the Cartesian coordinate system the equation of a
line is also determined by the intersection of two planes and may be written as
follows:
(
a1 y þ b1 xr þ c1 xi þ d1 ¼ 0;
a2 y þ b2 xr þ c2 xi þ d2 ¼ 0:
(2.47)
2.4 Power Function of a Complex Argument with a Real Exponent
45
As follows from (2.47), a line in a Cartesian coordinate system is defined by
eight coefficients; the same line in the form of a linear function of a complex
argument, as follows from (2.44), is defined by only four coefficients and is
represented in the form of a linear equation. We can again see that to actions with
complex numbers correspond actions with real numbers, and functions of complex
variables often represent a more convenient form of notation than those of real
numbers.
It should be noted that in the Cartesian coordinate system the equation of a line
passing through two different points P1(y1,xr1,xi1) and P2(y2,xr2,xi2) will be written
as follows:
y
y2
y1
xr
¼
y1 xr2
xr1
xi
¼
xr1 xi2
xi1
:
xi1
(2.48)
With reference to the line described by the function of the complex argument
(2.44), the equation of the line for these two points will be written as follows:
y
y2
y1
ðxr þ ixi Þ ðxr1 þ ixi1 Þ
:
¼
y1 ðxr2 þ ixi2 Þ ðxr1 þ ixi1 Þ
(2.49)
The specifics of a linear function of a complex argument with reference to some
economic problems will be considered in other chapters of this book.
2.4
Power Function of a Complex Argument
with a Real Exponent
The linear function of a complex argument can be applied in many cases of
economic modeling since in accordance with the general scientific principle
“from the simple to the complex,” to study some complex object, first simple
models including linear ones are used, after which models become increasingly
complex as the object’s properties become clearer for a more adequate description
of complex processes.
The power function of a complex argument is more complex than a linear one,
its general form being
y ¼ ða0 þ ia1 Þðxr þ ixi Þðb0 þib1 Þ :
(2.50)
It is easy to see that if the exponent of the model is equal to one, it is turned into
an elementary linear model of the complex argument (2.41).
Let us consider function (2.50) sequentially in order of increasing complexity
depending on the exponent – real, imaginary, or complex.
46
2
Properties of Complex Numbers of a Real Argument. . .
The first of the possible models determined by the equality (2.50) is one with a
real exponent:
y ¼ ða0 þ ia1 Þðxr þ ixi Þb0 :
(2.51)
To understand the properties of this function, let us represent the complex
proportionality coefficient and complex resource variable in exponential form.
Then we have
b0
y ¼ aeia ðrei’ Þ ¼ ar b0 eiðaþb0 ’Þ ;
a¼
(2.52)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1
xi
a20 þ a21 ; a ¼ arctg ; r ¼ x2x þ x2i ; ’ ¼ arctg :
a0
xr
Hence we have a system of equations for the real and imaginary parts of the
function under consideration:
(
y ¼ ar b0 cosða þ b0 ’Þ;
0 ¼ ar b0 sinða þ b0 ’Þ:
(2.53)
It follows from the last equality that it holds for the following conditions:
sinða þ b0 ’Þ ¼ 0 ! a þ b0 ’ ¼ pk;
(2.54)
where k is a whole number.
It should be noted that for values of the polar angle of the function determined by
these conditions, its cosine takes the following values:
(
cosða þ b0 ’Þ ¼ 1; 8 k ¼ 0; 2; 4; . . .
cosða þ b0 ’Þ ¼
1; 8 k ¼ 1; 3; 5; . . .
(2.55)
If, for example, we consider the polar angle in the complex plane of the
argument from 0 to 2p, at б ¼ 0 and b0 ¼ 1, we have that y is positive for ’ ¼ 0
and ’ ¼ 2p and negative for ’ ¼ p. For any a and for b0 6¼ 0 (with exponent
b0 ¼ 0, the function is turned into the point y ¼ acosa ¼ a0) function (2.51),
subject to the values of coefficients a and b0 and polar angle ’, takes both positive
and negative values.
Since it follows from the second equation of system (2.53) that the relationship
between the real and imaginary parts of the complex argument in this function is a
constant value regardless of the values of y, in three-dimensional space this
equation is represented by a plane parallel to the axis of the real variable 0y and
perpendicular to the plane of the complex argument.
2.4 Power Function of a Complex Argument with a Real Exponent
47
The first Eq. (2.53) determines the change in the y depending on the change in
the two factors xr and xi, which may be represented in three-dimensional space in
the form of some surface.
The complex proportionality coefficient changes the surface scale and slopes,
which is why its influence on the result is negligible and we may consider this
coefficient to be equal to one. For this reason let us consider a simplified analog of
function (2.51):
y ¼ ðxr þ ixi Þb0 :
(2.56)
Then the real and imaginary parts of this function will have the form
(
y ¼ r b0 cos b0 ’;
(2.57)
0 ¼ r b0 sin b0 ’:
Let us consider the influence on the real part of model (2.56) of exponent b0, i.e.,
equation of the first function of system (2.57), since the second equation describes
the linear relationship between the real and imaginary parts of the complex argument, as
0 ¼ r b0 sin b0 ’ ! sin b0 ’ ¼ pk ! ’ ¼ arctg
xi
¼ const:
xr
Here three variants of the function’s behavior are possible:
1. When the exponent is negative, b0 < 0;
2. When the exponent lies within the range: 0 < b0 < 1.;
3. When the exponent is higher than 1, b0 > 1.
Let us consider the first case where the exponent of the power function of
complex argument b0 is less than zero. For convenience, let us give the full form
of the first equation of the system:
y¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffib0
xi
x2r þ x2i
cos b0 arctg
:
xr
(2.58)
Since the second equation of system (2.57) states that the function under
consideration is projected in the complex plane of the argument to a line passing
through the zero point and having a constant value of the polar angle, let us consider
first what values the factor containing this constant polar angle can take. Cosine is a
periodic function, but the argument of this function does not change; it is always
constant due to the constant nature of the polar angle. The surface described in
three-dimensional space by the first equation of system (2.57) is nonlinear and its
character is determined by coefficient b0, since it characterizes the frequency of
48
2
Properties of Complex Numbers of a Real Argument. . .
oscillations – the higher it is by module, the more uneven (“corrugated”) is the
surface.
Since subsequently we will not need the type of this surface, we will consider
only the nature of the lines in the space made by system (2.57).
Let us first consider the situation where the polar angle of the argument in the
complex plane is equal to zero. This is possible when xr > 0, xi ¼ 0. Since the
cosine of zero is one, (2.58) will look as follows:
y ¼ xbr 0 :
(2.59)
In the (y,xr) plane, this function will represent a hyperbola that decreases from
plus infinity to zero with the growth of the module of the argument.
Now let us suppose that the polar angle of the complex argument is p/4, i.e., have
the following form:
pffiffiffi b0
p
y ¼ ðxr 2Þ cos b0
:
(2.60)
4
The function is positive for 2 < b0 < 0, equal to zero for b0 ¼ 2, negative
for 6 < b0 < 2, etc. Absolute values of the function for this case also decrease
with the growth of the module of the complex argument, as previously, but in the
case of negative values of the function, it decreases from minus infinity to zero.
Let us consider another case where the real part of the complex argument is
equal to zero: xr ¼ 0, and its imaginary part is positive: xi > 0. The polar angle of
the complex argument is equal to p/2 and the function looks like this:
p
y ¼ xbi 0 cos b0 :
(2.61)
2
If the coefficient b0 lies within the range 1 < b0 < 0, then the cosine of
function (2.61) will be positive, which means the function has a positive character.
If the exponent is equal to b0 ¼ 1, then cosine becomes zero and the function
also becomes equal to zero. If the values of this coefficient are within the range
3 < b0 < 1, then the cosine of the function becomes negative like the function
itself. Since the cosine is a periodical function, then with the subsequent increase of
the module of the values of the exponent b0, the function becomes both positive and
negative. With the growth in the values of the function argument (2.61), the
absolute values of the function also decrease with the hyperbola.
Continuing on, it is clear that the power function of a complex argument with
negative real exponent decreases in its absolute values according to the hyperbolic
law with an increase in the argument’s module. With this, the sign of the function is
determined by the result of multiplying the exponent by the polar angle. In some
cases the function is equal to zero.
In the second case, where the exponent of the function of the complex argument
lies within the range 0 < b0 < 1, the function behaves slightly differently:
y ¼ xbr 0 :
(2.62)
2.4 Power Function of a Complex Argument with a Real Exponent
49
However, since the exponent is positive and not greater than one, with the
growth of the module, the function increases nonlinearly from zero to plus infinity
according to the exponential law with a negative second derivative.
If the polar angle of the complex argument is p/4, then when the variables are in
the first quadrant of the complex plane and xr ¼ xi, then function (2.58) takes the
following value:
pffiffiffi b0
p
:
y ¼ ðxr 2Þ cos b0
4
(2.63)
The function will be positive for an exponent lying within 0 < b0 < 2 and equal
to zero at b0 ¼ 2. It will be negative for 2 < b0 < 6, etc. With the growth of the
module of the complex argument, absolute values will behave similarly to (2.62).
It makes no sense to examine this case further since it is clear that the function
will behave just like this – its absolute values will increase nonlinearly from zero,
and the function sign will be determined by the exponent of the function.
In the third case of the power function with a real exponent, the exponent b0 > 1,
function (2.58) will take negative or positive values, as well as values equal to zero,
depending on the result of multiplying the exponent by the polar angle since the
cosine of an angle may be both positive and negative and be equal to zero.
However, by its absolute value, with the growth of the complex argument, the
function will tend from zero to infinity according to the exponential law with
positive second derivative.
We can now go back to the model with a complex proportionality coefficient
under consideration (2.51):
y ¼ ða0 þ ia1 Þðxr þ ixi Þb0
and pay more attention to the influence of this coefficient on the function behavior.
Values of this complex proportionality coefficient influence both the module of
the function and the polar angle.
For various values of the proportionality coefficient, the module of the function
of the complex argument equal to R ¼ ar b0 is presented on various scales.
When the values of this proportionality coefficient vary, the polar angle also
turns in the plane of the complex argument:
’¼
pk
b0
a:
This is why variations in the proportionality coefficient move the power function
curve in various parts of space symmetrically to the y-axis and change the row curve
scale.
50
2
2.5
Properties of Complex Numbers of a Real Argument. . .
Exponential Function of Complex Argument
with Imaginary Exponent
Having considered the case where the power function of complex argument (2.50)
is represented by a real exponent and the function represents a line of the power
function in three-dimensional space, let us move on to a more complicated case
where the real part of this function is equal to zero and the exponent is imaginary:
y ¼ ða0 þ ia1 Þðxr þ ixi Þib1 :
(2.64)
In this case another relationship besides the power one will be modeled, though
the complex argument is subject to involution.
Since the influence of the proportionality coefficient on the result in this case
will remain the same, let us consider it to be equal to one:
a0 þ ia1 ¼ 1:
Then the function in exponential form will look like this:
y ¼ ðrei’ Þ
ib1
¼e
’b1 ib1 ln r
e
;
(2.65)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where r ¼ x2x þ x2i ; ’ ¼ arctg xxri :
From this follows a system of equations for the real and imaginary parts of the
function under consideration:
8
>
>
>
<y ¼ e
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos b1 ln x2r þ x2i ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
>
>
b1 arctgxri
>
sin b1 ln x2r þ x2i :
:0¼e
x
b1 arctgxri
(2.66)
The first equation of this system represents a description of some nonlinear
function in three-dimensional space, to be discussed later.
The second equation of this system represents a nonlinear surface perpendicular
to the plane of a complex argument, where all the lines lying in that plane are
parallel to the 0y-axis.
The intersection of these two surfaces is simply a graphical interpretation of
function (2.64) in the space.
Let us consider what the second equation looks like in the plane of a complex
argument:
0¼e
x
b1 arctgxri
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin b1 ln x2r þ x2i :
(2.67)
2.5 Exponential Function of Complex Argument with Imaginary Exponent
51
The first factor can be equal to zero only if its exponent is equal to infinity.
Variants when b1 is equal to infinity are not considered here because they are
meaningless. The arctangent is known to lie within a range of –p/2 to + p/2.
Therefore, the first factor (2.67) will never be equal to zero and the equality holds
when the second factor is equal to zero:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin b1 ln x2r þ x2i ¼ 0:
(2.68)
This equality holds when
b1 ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2r þ x2i ¼ pk:
(2.69)
Hence,
2pk
x2r þ x2i ¼ e b1 :
(2.70)
This means that the imaginary part (2.64) is equal to zero when the values of the
2pk
complex argument in the complex plane lie on a circumference with radius e b1 . In
particular, if k ¼ 0, then the equality holds when
x2r þ x2i ¼ 1;
(2.71)
i.e., when the points in the complex plane lie on a one-unit circumference.
k may take any whole values, which means a family of circumferences in a
complex plane of arguments. In the three-dimensional space it is a cylindrical
surface perpendicular to the plane of the complex argument.
Now, let us consider the first equation of system (2.66) referring to the real part
of the function:
y¼e
x
b1 arctgxri
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos b1 ln x2r þ x2i :
(2.72)
This equation describes a nonlinear surface in space, but since the type of this
surface will not be used subsequently, we should consider what line on this surface
is cut off by the cylinder, since the function of complex argument (2.64) is an
intersection of two nonlinear surfaces one of which is a cylinder (2.70). Thus, let us
consider the behavior of (2.72) in the case where the variables xr and xi lie on some
circumference.
In this case the logarithm argument is a constant, which is why the nature of this
curve is completely determined by the first factor (2.72), which represents the
exponent.
Since a complex argument changes its values in a circumference, in the initial
point where the minimal component is equal to zero, the first factor is equal to one
52
2
Properties of Complex Numbers of a Real Argument. . .
since any number to the zero power equals one. Then, for a polar angle equal to
zero, the function will take the following values:
yð0Þ ¼ cosðb1 ln rÞ:
(2.73)
With the growth of the values of the imaginary component xi on the circumference and the respective decrease in the real component xr (increasing polar angle),
the polar angle of the complex argument tends from zero to p/2. In the extreme
point, when the real component of the complex argument is equal to zero, the
function will have the following form:
y
p
¼e
2
b1 p2
cosðb1 ln rÞ:
(2.74)
In the interval between these two points the function will vary exponentially
from points with coordinates determined by (2.73) to points determined by (2.74).
Further movement of the complex argument on the circumference corresponds
to a variation in the polar angle from p/2 to p. In the extreme point, when the polar
angle is equal to p, which means that the imaginary component is equal to zero and
the real constituent xr ¼ r, the function will take the following values:
yðpÞ ¼ e
b1 p
cosðb1 ln rÞ:
(2.75)
Continuing along the circumference and arriving at the point where the real part
is equal to zero and the imaginary xi ¼ r, we obtain a function value equal to
3
3p
y
p ¼ e b1 2 cosðb1 ln rÞ:
(2.76)
2
Completing the movement along the circumference in the point where the
imaginary part is equal to zero and the real one is equal to the radius, the function
takes the following value:
yð2pÞ ¼ e
b1 2p
cosðb1 ln rÞ:
(2.77)
Now it is clear what the power function of a complex argument represents if the
exponent is an imaginary number – this exponent is located on the cylinder surface.
Completing the full circle equal to 2p we see that the function differs from its initial
point by e b1 2p times.
If the coefficient b1 is positive, then the function tends to zero; if the exponent is
negative, then the function tends to infinity, making turn after turn on the cylinder
surface, if the complex argument makes rotational movements in the complex
plane. However, since in economics we do not observe such rotational movements,
meaning variations in the polar angle of the complex argument within a range
0 ’ 2p, function (2.64) should be considered an exponent on the cylinder
surface making one complete turn thereon.
2.6 Power Function of Complex Argument with Complex Exponent
2.6
53
Power Function of Complex Argument
with Complex Exponent
When a complex argument of a power function is raised to a real power, in threedimensional space this represents a curve described by an exponential function and
lying in space in a plane perpendicular to the plane of the complex argument.
If the exponent of this function is an imaginary number, then it represents an
exponent varying with the increase in the polar angle of the complex variable and
lying in the space on the cylinder surface perpendicular to the complex argument
plane.
Now let us consider the nature of a power function of a complex argument where
the exponent is complex:
y ¼ ða0 þ ia1 Þðxr þ ixi Þðb0 þib1 Þ :
(2.78)
This function, taking into account previously introduced designations and the
assumption that the complex proportionality coefficient is equal to one, may be
written in exponential form as follows:
y ¼ r b0 eib0 ’ e
’b1 ib1 ln r
e
¼ r b0 e
’b1 iðb0 ’þb1 ln rÞ
e
:
(2.79)
The real and imaginary parts of this function may be written as a system of
equations:
(
y ¼ r b0 e
b0
0¼r e
b1 ’
b1 ’
cosðb0 ’ þ b1 ln rÞ;
sinðb0 ’ þ b1 ln rÞ:
(2.80)
Again, we have equations of complex nonlinear surfaces in the space, the second
equation describing the surface perpendicular to the complex argument plane. As in
the previous cases, let us examine the properties of function (2.78) with the
condition that the imaginary part of the complex argument function is equal to zero.
The second equation will be equal to zero when the argument is equal to zero and
when the sine is equal to zero:
sinðb0 ’ þ b1 ln rÞ ¼ 0:
(2.81)
In the zero point the function itself is equal to zero, which is why (2.81) is of
interest and may be written as follows:
b0 ’ þ b1 r ¼ pk:
(2.82)
54
2
Properties of Complex Numbers of a Real Argument. . .
Here, as in the previous case, k is a whole number.
Let us take k ¼ 0. Then (2.82) can be written as follows:
r¼
b0
’:
b1
(2.83)
It is evident that we have obtained Archimedes’ spiral, where with the change in
the polar angle within the range 0 ’ 2p the coefficient before the polar angle
should always be positive since the module of the complex argument cannot be
negative. This means that the signs of the real and imaginary parts of the complex
exponent should be different.
We do not consider rotational processes that practically do not exist in economics, which is why in the space under consideration the second equation of system
(2.80) indicates one turn of Archimedes’ spiral, that is, a nonlinear surface in
Archimedes’ spiral perpendicular to the complex argument plane. This surface
“cuts off” a nonlinear curve on the other surface represented by the first equation
of system (2.80).
We are not interested in the type of the surface described in the space by the first
equation of system (2.80), but in the line on this surface that is cut off by
Archimedes’ spiral.
To understand this, we substitute (2.83) into the first equation of system (2.80)
and get
y¼
b0
b0
’ e
b1
b1 ’
b0
’ :
cos b0 ’ þ b1 ln
b1
(2.84)
Since it was shown previously that b0 and b1 have different signs, let us first take
b0 > 0 and b1 > 0. For this case with a growing polar angle:
b0
– The first factor ð bb10 ’Þ increases according to the power law;
– The second factor e b1 ’ increases according to the exponential law;
– The third factor cosðb0 ’ þ b1 lnð bb10 ’ÞÞ varies nonlinearly depending on the
modules of the values of coefficients b0 and b1. If the module of the complex
cosine argument increases with the growth of the polar angle, this factor
decreases up to zero, after which it becomes negative.
Thus, on the whole, (2.84) describes a function increasing up to a certain limit
with its subsequent decrease to zero and further to the negative range. This line is
located in the space on the nonlinear surface of Archimedes’ spiral.
If we now change signs of the coefficients to the opposite ones and set b0 < 0,
b1 > 0, then the picture will look as follows:
b0
– The first factor ð bb10 ’Þ decreases according to the power law;
– The second factor e b1 ’ decreases according to the exponential law;
2.7 Exponential Function of a Complex Argument
55
– The third factor cosðb0 ’ þ b1 lnð bb10 ’ÞÞ behaves in the same way as in the first
case as the cosine is a symmetrical function.
On the whole, with such signs of the coefficients, the function decreases with the
growth of the argument and becomes negative as it travels along Archimedes’
spiral.
Various combinations of coefficients give various forms of a curve in space. If
the imaginary part of a complex exponent is equal to zero, then the function
represents points lying on a line of the exponential function in space in a plane
perpendicular to the plane of the complex argument. When the real part is close to
zero, then the curve represents an exponent lying on the cylinder surface. If the
exponent is equal to one, then we have a linear function of a complex argument.
To conclude our study of the properties of this function, it should be noted that
the coefficients of a function may be easily estimated by two points.
Let there be two points (xr1,xi1,y1) and (xr2,xi2,y2) available to an economist
disposaltwo , and she thinks that there is a relationship between these variables that
may be described by a model in the form of a power function of complex argument
(2.78). Substituting these values into the function and dividing the left- and righthand sides by each other we obtain the following equation:
y1 xr1 þ ixi1 ðb0 þib1 Þ
:
¼
y2 xr2 þ ixi2
(2.85)
Here we can derive the exponent
b0 þ ib1 ¼
ln yy12
i1
ln xxr1r2 þix
þixi2
:
(2.86)
Knowing this value we can easily find the value of the proportionality coefficient
(a0 + ia1).
Thus, for example, if an economist wants to build a model in the form of a power
function of a complex argument and she has two points at her disposal in threedimensional space – (2; 3; 5) and (2.5; 4.7; 15), then she can easily do this using
(2.85) and (2.86), and model (2.78) passing through these two points in threedimensional space will have the following form:
y ¼ ð 0; 014
2.7
i0; 082Þðxr þ ixi Þð2;648
i0;674Þ
:
Exponential Function of a Complex Argument
It is clear from the aforementioned properties of the power function of a complex
argument that it can be used for modeling various complex nonlinear relationships
in three-dimensional space. But this model hardly covers the entire possible variety
56
2
Properties of Complex Numbers of a Real Argument. . .
of functions of a complex argument. One of the simple nonlinear functions of a
complex argument that differ in their properties from the power function is the
exponential function
y ¼ ða0 þ ia1 Þeðb0 þib1 Þðxr þixi Þ :
(2.87)
To study its properties let us first consider a situation where the proportionality
coefficient is represented as a real coefficient, then when it is the imaginary part, we
can consider the whole function (2.87).
The exponential function of a real argument with a real exponent coefficient will
look as follows:
y ¼ ða0 þ ia1 Þeb0 ðxr þixi Þ :
(2.88)
In exponential form it will be written as follows:
y ¼ aeb0 xr eiðaþb0 xi Þ ;
(2.89)
where the following equalities hold for the real and imaginary parts:
(
y ¼ aeb0 xr cosða þ b0 xi Þ;
0 ¼ aeb0 xr sinða þ b0 xi Þ:
(2.90)
The imaginary part can be equal to zero in two cases –with a positive exponent
coefficient xr ! -1 and when
a þ b0 xi ¼ pk:
(2.91)
Situations where one or all of the factors tend to infinity do not exist in
economics, so we will concentrate on equality (2.91).
The last condition represents a combination of equations of a line in a complex
plane of the argument parallel to the axis of real values of the complex argument
since it follows from (2.91):
xi ¼
pk a
:
b0
(2.92)
In a simple case where k ¼ 0 there is a line
xi ¼
in the complex plane.
a
b0
(2.93)
2.7 Exponential Function of a Complex Argument
57
Since we are looking at the problem of presenting a function in threedimensional space, (2.93) represents a plane perpendicular to the plane of a
complex argument and parallel to the axes of the real part of the complex argument
xr and function y.
The first Eq. (2.90) describes a nonlinear surface in three-dimensional space. Let
us consider it.
If the real part of a complex argument is a constant value xr ¼ d ¼ const, then
the function varies by the cosine law:
y ¼ aeb0 d cosða þ b0 xi Þ:
(2.94)
If the imaginary part of the complex argument is constant xi ¼ g ¼ const, then
the function varies according to the exponential law:
y ¼ aeb0 xr cosða þ b0 gÞ:
(2.95)
Since the last condition is a restriction (2.93) that follows from the fact that the
imaginary part of the function under consideration is equal to zero, then in threedimensional space, the exponential function of a complex argument with a real
exponent coefficient represents an exponent:
y ¼ ae
b0 x r
pk a
¼ aeb0 xr :
cos a þ b0
b0
(2.96)
Let the exponential function of a real argument have an imaginary coefficient of
the exponent:
y ¼ ða0 þ ia1 Þeib1 ðxr þixi Þ :
(2.97)
In exponential form this will look like
y ¼ ae
b1 xi iðaþb1 xr Þ
e
;
(2.98)
where for the real and imaginary parts of the function
(
y ¼ ae
0 ¼ ae
b 1 xi
b1 xi
cosða þ b1 xr Þ;
sinða þ b1 xr Þ:
(2.99)
If the imaginary part of the function under consideration equals zero, then
a þ b1 xr ¼ pk
(2.100)
58
2
Properties of Complex Numbers of a Real Argument. . .
or
xr ¼
pk a
:
b1
(2.101)
This means we have again obtained an equation of lines that, in a plane of a
complex argument, are parallel to the imaginary axis. We can limit ourselves to the
case where k ¼ 0. In the space under consideration this equation means a plane
perpendicular to the plane of a complex argument and passing through line (2.101).
This means that on a complex nonlinear surface described by the first equality
(2.99) there is a curve with a constant value of xr. It is clear from the first equation of
system (2.99) that this curve is described by the exponent
y ¼ Ce
b 1 xi
;
C ¼ a cosða þ b1 xr Þ;
xr ¼ const:
Now it is clear what will represent the full exponential function of a complex
argument. Let us present again the complex values of the model – the
proportionality coefficient and the complex argument – in exponential form.
Grouping the constituents of the module and the polar angle we get
b1 xi iðaþb1 xr þb0 xi Þ
y ¼ aeb0 xr
e
:
(2.102)
Let us represent this model as an equality system of real and imaginary parts:
(
y ¼ aeb0 xr
0 ¼ ae
b1 xi
b0 x r b1 x i
cosða þ b1 xr þ b0 xi Þ;
(2.103)
sinða þ b1 xr þ b0 xi Þ:
From the last equality we can easily get
xi ¼
1
ðpk
b0
a
b1 xr Þ:
(2.104)
This equation describes a family of parallel lines in a complex plane of the
argument. In the simple case, where k ¼ 0, it is a line in the plane and surface
perpendicular to the complex plane of the argument in the space. Both the line and
the plane are defined in the whole range of the problem.
This plane cuts off some line on the surface defined by the first equation of
system (2.103):
y ¼ aeb0 xr
b1 x i
cosða þ b1 xr þ b0 xi Þ:
(2.105)
Substituting (2.104) into this equation we get
y ¼ ae
b0 xr
b1
b0 ðpk
a b 1 xr Þ
cosða þ b1 xr þ ðpk
a
b1
b1 xr ÞÞ ¼ aeb0
ða pkÞþ
b2 þb2
0 1x
r
b0
:
(2.106)
2.8 Logarithmic Function of a Complex Argument
59
This means that we have an exponent in the space located in a plane perpendicular to the complex plane of the argument.
2.8
Logarithmic Function of a Complex Argument
Let us now examine the properties of the logarithmic function of a complex
argument. The logarithm of a complex variable is known as a periodical function,
which is why when we study it we should specify what part of the function is being
studied. It was determined in the first chapter of this book that from the entire
combination of logarithmic values we will consider only the main values.
The logarithmic function of a complex argument may be presented in its general
form as follows:
y ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðxr þ ixi Þ:
(2.107)
If we apply the formula of a logarithm of a complex argument to the model under
consideration, we get
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
y ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þ ln x2r þ x2i þ iarctg
:
xr
(2.108)
Let us consider the variant where the imaginary part of the complex
proportionality coefficient is equal to zero:
y ¼ ða0 þ ia1 Þ þ b0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
2
2
ln xr þ xi þ iarctg
:
xr
(2.109)
Opening the brackets and grouping the real and imaginary parts of this equation
we get
y ¼ a0 þ b0 ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
x2r þ x2i þ i a1 þ b0 arctg
:
xr
(2.110)
Two equalities for the real and imaginary parts follow from the preceding
equation:
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
< y ¼ a0 þ b0 ln x2r þ x2i ;
x
>
: 0 ¼ a1 þ b0 arctg i :
xr
(2.111)
60
2
Properties of Complex Numbers of a Real Argument. . .
The second equation requires a constant polar angle in the plane of the complex
argument:
arctg
xi
¼
xr
a1
:
b0
(2.112)
This indicates an equation of the line passing through the neighborhood of the
zero point but not including it. The zero point does not exist for the first equation as
well since a logarithm of zero does not exist.
The first equation of system (2.111) describes a nonlinear surface in threedimensional space. We are interested in the location of the line on this surface
that satisfies condition (2.112). Thus, let us consider the equation
y ¼ a0 þ b0 ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2r þ x2i
for xi ¼ dxr :
If we substitute this into the equation, we get
y ¼ a0 þ b0 ln xr
pffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ d2 :
(2.113)
Therefore, the logarithmic function of a complex argument with a real
proportionality coefficient represents in three-dimensional space a logarithmic
function passing through the zero point and lying in a plane perpendicular to the
complex plane of the argument.
Now let us consider the second extreme version, when the real part of the
complex proportionality coefficient is equal to zero:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
y ¼ ða0 þ ia1 Þ þ ib1 ln x2r þ x2i þ iarctg
:
xr
(2.114)
Grouping the real and imaginary parts of this function we get the following
system:
(
b1 arctg xxri ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 ¼ a1 þ b1 ln x2r þ x2i :
y ¼ a0
(2.115)
The fact that the imaginary part equals zero means that in the complex plane of
the argument the function is determined on a circumference since this equality can
easily be made as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2r þ x2i ¼ e
a1
b1
¼ const ¼ d:
(2.116)
2.8 Logarithmic Function of a Complex Argument
61
The zero point does not serve to define the function because a logarithm of zero
does not exist. In the space under consideration the second equation of system
(2.115) represents a cylinder surface. This cylinder surface cuts off a curve in the
plane of the first equation, which we are interested in.
Since the polar angle in the plane of a complex argument varies on a circle, on
the surface described by the first equation of system (2.115) a curve is defined that
represents an arctangent function lying on the surface of the cylinder perpendicular
to the complex plane of the argument.
The general logarithmic function of complex argument (2.107) represents a
complex superposition of these two functions. After opening the brackets in the
right-hand side of equality (2.107) and grouping the real and imaginary parts we get
y ¼ a0 þ b0 ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2r þ x2i
b1 arctg
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
xi
þ i a1 þ b1 ln x2r þ x2i þ b0 arctg
:
xr
xr
(2.117)
This equality holds only when the real and imaginary parts are equal to each
other:
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
>
>
< y ¼ a0 þ b0 ln x2r þ x2i b1 arctg
xr
:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
>
2
>
2
: 0 ¼ a1 þ b1 ln xr þ xi þ b0 arctg
xr
(2.118)
The second equation of the system describes a curve in the plane of the complex
argument that does not include the zero point:
0 ¼ a1 þ b1 ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
x2r þ x2i þ b0 arctg :
xr
(2.119)
The approximate form of the function can be imagined from the location of a
function with these coefficients in the plane:
0¼
3 ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
x2r þ x2i þ 0; 2arctg :
xr
(2.120)
This function is given in Fig. 2.15.
The first equation of system (2.118) represents a complex surface. Its general
form may look like this: in space there are a great number of lines like those shown
in Fig. 2.15 that are parallel to each other and increase on the S-axis or decrease
with the growth of the argument, depending on the function coefficient. This
surface intersects with another one perpendicular to that of the complex argument
and passing in the plane of the complex argument through the points determined by
62
Properties of Complex Numbers of a Real Argument. . .
2
xi
6
5
4
3
2
1
xr
0
-20
-10
-1
0
10
20
30
-2
-3
Fig. 2.15 Line (2.120) in plane of complex argument
line (2.119). Intersection of these two planes gives a line in space that has the form
of the line in Fig. 2.15.
The functions of a complex argument studied here do not exhaust their full
range, but of those that can be used in economic practice, the previously mentioned
function are fundamental.
Reference
1. Kasana HS (2005) Complex variables: theory and applications. PHI Learning, New Delhi
Chapter 3
Conformal Mappings of Functions
of a Complex Variable
To understand the meaning of a model applied in economics, one should know its
basic properties including graphical characteristics. Conformal mapping shows
graphically how one complex variable is mapped to another complex variable by
means of a particular complex function. This chapter not only considers the
corresponding section of the complex variable function theory from the point of
view of the economy but presents conformal mappings of functions that are not
considered or practically applied in the TFCV but that can, however, be widely
applied in economic modeling, for example, the power complex-value function
with complex exponent.
3.1
Power Functions of a Complex Variable
Since functions of a complex variable have analogs in the domain of real variables,
models of complex variables are often considered a convenient notation of complex
relationships y of the domain of real numbers. This property of describing in a
simple way complex relationships in the domain of real numbers is clearly
demonstrated by functions of a complex variable that are to be considered in this
chapter [1]. We are interested in those properties of functions of a complex variable
and those parts of the TFCV that may be applied in economic modeling. From this
point of view we will consider the basic functions of a complex variable.
The power function of a complex variable is the most popular in the practice of
economic modeling. As we did previously, let us denote the explanatory complex
variable as follows:
z ¼ xr þ ixi ¼ rei’ ;
(3.1)
where
xr – real part of the complex variable;
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
DOI 10.1007/978-1-4614-5876-0_3, # Springer Science+Business Media New York 2012
63
64
3 Conformal Mappings of Functions of a Complex Variable
xi – imaginary part of the complex variable;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r – module of the complex variable, r ¼ x2r þ x2i
’ – polar angle of this variable (argument of a complex variable Argz):
’ ¼ Argz ¼ arctg
xi
;
xr
i – imaginary unit.
Let us represent the resulting variable in the form of another complex variable:
w ¼ yr þ iyi ¼ reiy ;
(3.2)
where
yr – real part of resulting complex variable;
yi – imaginary part of resulting complex variable;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r – module of complex variable, r ¼ y2r þ y2i ;
u – polar angle of this variable;
y ¼ Argw ¼ arctg
yi
:
yr
We will consider that, according to general axiomatic assumptions, all the
variables considered in this study are nondimensional or have one dimension and
one scale. In addition, they are defined in the entire domain of the complex plane.
Taking into account the introduced symbols, the power function of complex
variables will have the following form:
yr þ iyi ¼ ða0 þ ia1 Þðxr þ ixi Þb0 þib1 :
(3.3)
In exponential form, using (3.1) and (3.2), a power model of complex variables
(3.3) may be represented as follows:
b0 þib1
reiy ¼ ða0 þ ia1 Þðrei’ Þ
:
(3.4)
Let us first consider a simple form of the power function of complex variables,
when all the coefficients are real numbers, subsequently making the model increasingly complex. Let the coefficient of proportionality be equal to one, a ¼ 1, and the
exponent is a positive integer b ¼ n. The model will have the form
yr þ iyi ¼ ðxr þ ixi Þn :
(3.5)
3.1 Power Functions of a Complex Variable
65
It is more convenient to consider the properties of this function in exponential
form:
reiy ¼ r n ein’ ;
where due to the properties of the equality of complex numbers
(
r ¼ rn ;
y ¼ n’:
Then it is evident that the conformal mapping provided by function (3.5) results
in stretching of the module of the complex variable z to the nth power and an
increase in the polar angle of the complex variable z n times. If we consider the
complex variable z as a vector in polar coordinates, the vector will rotate by an
angle
ðn
1Þ’:
Since the polar angle is defined up to a period, all of its functions are periodical.
From this it obviously follows that complex numbers z1 and z2 with equal modules
and arguments that differ from one another by a multiple of 2p/n move at mapping
(3.5) to a single point.
In the TFCV, there is a concept of mutually monosemantic or univalent
mapping. It has the following definition: if the function w ¼ f(z) is monosemantic
or single-valued on the set M and two different points on this set M of the complex
plane w are always correlated with different points N in the complex plane z, then
this mapping is called univalent. Otherwise, the mapping is multivalent. For the
exponential function under consideration, multivalency is valid only in particular
sectors of the complex plane z, namely, in those sectors where a condition is true
restricting the value of the initial variable’s polar angle:
k2p=n < ’ < ðk þ 1Þ2p=n;
where k is a positive integer.
If the exponent of function (3.5) is not a whole number b 6¼ n, which is expected
for most economic problems, then the nature of the arguments will not change. The
module of the mapped variable z grows to a power b and the angle turns counterclockwise b times, except that under conditions of univalency one should insert
nonwhole b instead of positive integer n:
k2p=b < ’ < ðk þ 1Þ2p=b:
With the values of the complex factor z, when the polar angle u goes beyond the
restrictions (3.6), the function becomes multivalent, i.e., there is a set of points in a
66
3 Conformal Mappings of Functions of a Complex Variable
The complex plane W
xi
yi
2
5
4
3
2
1
a
w1 = w2
y1i=y2i
1
θ
ϕ
0
0
y1r=y2r
1 2 3 4 5 xr
yr
The complex plane Z
w = zna
Fig. 3.1 Model (3.5) for b ¼ 3,128 gives the same result y1i ¼ y2i and y1r ¼ y2r for various
x1r ¼ 5; x1i ¼ 2 and x2r ¼ 2; x2i ¼ 5
complex plane of explanatory variables z that, by means of (3.5), will be mapped in
the complex plane of results w to the same point. For example, with the exponent
b ¼ 3,128, the same result w(yr,yi) may be obtained if we take five xr and two xi
units or take a smaller number of xr units, for example, two, but increase the number
of xi units to five (Fig. 3.1).
In Fig. 3.1 the first point of the complex plane z(5,2) is mapped in the complex
plane of results to point w1. Another point in the complex plane z(2,5) is mapped in
the complex plane of results to point w2. The coordinates of points w1 and w2
coincide with each other. This is why the complex variable z coming from the first
point z(5,2) around the circumference to point z(2,5) is mapped, by means of an
exponential function with the exponent b ¼ 3,128, to a circumference of a complex
plane of the results w with another radius, forming a complete cycle, beginning
from point w1 and ending at point w2.
If we continue the counterclockwise movement in the complex plane z around
the circumference shown in the figure, the conformal mapping of this movement to
the plane of the results w will again correspond to a counterclockwise movement
around the circumference from the point w1 ¼ w2. And again, the points in the
resulting plane will pass through the complete cycle while in plane z the points pass
only the sector of the circumference.
This is what the multivalence of conformal mapping means – in a complex plane
z there is a set of points that are mapped to another complex plane w at the same
point. That is, the same result may be obtained for various combinations of the real
and imaginary components of the complex factor.
Since model (3.5) represents the equality of the real and imaginary parts, they
comprise a system of two real equations:
3.1 Power Functions of a Complex Variable
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffib
xi
>
>
2 þ x2
>
x
y
¼
cos
barctg
;
< r
r
i
xr
qffiffiffiffiffiffiffiffiffiffiffiffiffiffib
>
xi
>
>
2
2
: yi ¼
sin barctg
xr þ xi
:
xr
67
(3.6)
It is virtually impossible to imagine a situation in economics where a complex
argument would change cyclically around a circumference that has been set for
good. More often are situations either involving the growth of a complex argument
that is becoming linear or nonlinear dynamics.
In the first case, where the complex argument shows linear changes, we expect
two variants:
– When a line passes through the zero point in the argument’s complex plane, and
the polar angle of the complex argument is a constant value;
– When the line does not pass through the zero point and the polar angle of the
complex argument changes.
In the first case with the growth of the argument’s module the real and imaginary
parts of the complex function will change nonlinearly according to the exponential
law, depending on exponent b, but in the complex plane the result will be seen on
the line.
In the second case the real and imaginary components change nonlinearly
according to a more complicated law, as they represent the result of multiplication
of two components – the power function and the periodical component. In the
complex plane they will represent smooth nonlinear functions monotonically
increasing or decreasing depending on the argument’s values and on how the
polar angle changes: it will decrease (if the line passes above the zero point) or
increase (if the line passes below the zero point).
If the argument changes nonlinearly, then the function under consideration will
model various nonlinear relationships in the complex plane of the result. Figure 3.2
shows how the transition of the argument from point 1 to point 2 will be displayed
in the complex plane of the result.
Let us consider particular features of the behavior of power function (3.5) with
exponent b lying within the limits
0 < b < 1:
(3.7)
Let b ¼ 1/n, where n is any positive whole number. This function is very well
studied in the TFCV; its behavior has been studied for closed curves lying in a
complex plane having (Fig. 3.3) or not having (Fig. 3.4) the point z ¼ 0 inside [2].
In the first case, mapping of the complex variable by means of function (3.5)
gives n continuous and single-valued functions called branches of a multivalued
function (3.5) depending on condition (3.7), each of which takes one of the values
ffiffi
p
n
z since, in circumvention of a closed curve having point z ¼ 0 inside, the
argument of the complex variables of the factors receives an increment 2p, and
68
3 Conformal Mappings of Functions of a Complex Variable
xi
yi
w2
2
θ2
1
ϕ1
w1
θ1
ϕ2
xr
0
yr
0
Fig. 3.2 Conformal mapping of w ¼ zb at 1 < b < 1.5 when argument moves from point 1 to
point 2
z
The branch for
ϕ0+2π ≤ϕ ≤ϕ0 +4π
xi
yi
The branch for
ϕ0≤ϕ ≤ϕ0 +2π
1
0
ϕ0
xr
The branch for
ϕ0+4π ≤ϕ ≤ϕ0 +6π
θ0
w
0
yr
Fig. 3.3 Conformal mapping of closed curve in complex plane (xr,xi) having zero point inside by
means of function w ¼ zb to complex plane (yr,yi) at 0 < b < 1
yi
xi
z
w
1
1
ϕ
θ
xr
0
0
yr
Fig. 3.4 Conformal mapping of closed curve in complex plane (xr,xi) without zero point inside by
function w ¼ zb to complex plane (yr,yi) at 0 < b < 1
pffiffi
the point n z with this argument in the plane of a complex result does not return to its
initial position until the vector on the complex plane of the factor makes an n-times
circumvention of the closed curve.
In other words, a vector in the factor z plane, having made a complete circumvention, is mapped to the complex plane of the result w only in the form of one
nonclosed curve. If a vector of factor z makes another circumvention, then in the
3.1 Power Functions of a Complex Variable
69
complex plane of the result w this will be shown by points lying on the next curve
similar in form to the first one but following it in the complex plane in a counterclockwise direction. In this way branches in the complex plane of the result will
grow until they finally close. But this can occur only when the vector of factor z
makes a circumvention in its plane exactly n times.
In the second case, where the closed curve does not have the zero point inside
(Fig. 3.4), the argument of the complex variable of resources does not receive an
increment 2p because it does not make a full circumvention around the origin of
coordinates, which is why function (3.5) under condition (3.7) correlates each point
in the factor plane with only one point in the plane of complex results.
This means that in the second case the closed curve in the resulting complex
plane w will correspond to the closed curve in the plane z.
Without loss of generality, these ideas may be extended to other cases of
condition (3.7), i.e., when the exponent b ¼ 1/n takes any values on the specified
interval including the cases when n is not a whole number.
Which of these two possible variants should be attributed to economic practice
when complex power functions can be used? Certainly, a considerable, if not the
largest, part of economic indicators is considered only in the first quadrant of a
Cartesian coordinate system since they are essentially nonnegative (prime cost,
volume of output, quantity of energy consumed, labor productivity, number of
employees, etc.).
Gross profit as an indicator of production efficiency may also be negative if the
company operates at a loss, but cases of negative economic variables are quite rare.
Thus, economists are most interested in the nature of conformal mapping using a
model of a nonclosed curve lying in the first quadrant of a complex plane of factors,
i.e., when the initial variables of the model are positive.
The real and imaginary parts of the function under consideration will represent a
system of two equations:
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n
>
1
xi
>
>
2 þ x2
>
y
arctg
¼
x
cos
;
r
<
r
i
n
xr
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n
>
>
1
xi
>
2
2
>
arctg
xr þ xi sin
:
: yi ¼
n
xr
If n tends to infinity, then the real part of the function tends to one and the
imaginary part to zero. In all other cases the function models nonlinear dynamics of
the real and imaginary parts with an increase in the argument. For example, if in a
complex plane of an argument its values are on a line passing below the zero point
described, for example, by the equation
xi ¼ xr
3:55;
(3.8)
70
3 Conformal Mappings of Functions of a Complex Variable
yi
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
yr
-0.4
-0.6
-0.8
Fig. 3.5 Function (3.5) for argument (3.8) and exponent b ¼ 1/3
1.2
yi
1
0.8
0.6
0.4
0.2
yr
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Fig. 3.6 Function (3.5) for argument (3.9) and exponent b ¼ 1/3
then for n ¼ 3 and xr ¼ 1,2,3,. . .40 on the complex plane the points of the function
will lie on a curve shown in Fig. 3.5.
For the same exponent but in the case where the argument is described by a line
passing above the zero point, for example,
xi ¼ xr þ 3:55;
(3.9)
a line will describe an increasing curve of another form, shown in Fig. 3.6.
If the exponent is negative, then any increase in argument z will definitely lead to
a decrease in function w values – an increase in the argument r module leads to a
decrease in the function c module, and an increase in the polar angle j leads to a
rotation of the conformal mapping in the opposite direction, as for the case
3.1 Power Functions of a Complex Variable
71
yi
0.3
0.25
0.2
0.15
0.1
0.05
yr
0
-0.05 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-0.1
-0.15
Fig. 3.7 Function (5.5) for argument (3.8) and exponent b ¼
r ¼ r b; y ¼
1/3
b’:
(3.10)
Are there cases in economics when a complex power function has a negative
exponent? Such cases are quite possible in economic practice. For example, in
production, when additional involvement of labor, resources, and capital only
diminishes production results (the module of the complex variable of production
results decreases) or, in other words, the redundancy of employees and the closure
of some production sites (for example, noncore divisions) has a positive effect on
production results, the volume of gross output increases and the gross margin
grows. These cases are not rare in economics, especially in times of crisis.
This means that a complex power function with negative exponent has the right
to exist in a complex economy.
The conformal mapping of a power function with negative exponent will again
be nonlinear. If an argument increases according to a linear law (3.8) and within the
same limits, and the exponent b ¼ 1/3, then the function will change its values in
the clockwise direction from yr ¼ 0.658, yi ¼ 0.278 to a point with the coordinates
yr ¼ 0.254, yi ¼ 0.064 (Fig. 3.7).
When the argument line passes above the zero point, as described by relationship
(3.9), a decrease in the module will be shown by another curve (Fig. 3.8) in a
counterclockwise direction from a point with the coordinates yr ¼ 0.539, yi ¼
0.261 up to a point with the coordinates yr ¼ 0.245, yi ¼ 0.069.
Let us consider a power function of complex variables with imaginary exponent
and coefficient of proportionality equal to one:
yr þ iyi ¼ ðxr þ ixi Þib :
(3.11)
Its exponential form is as follows:
ib
reiy ¼ ðrei’ Þ ¼ e
b’ ib
r ¼e
b’ ib ln r
e
:
(3.12)
72
3 Conformal Mappings of Functions of a Complex Variable
0
0
0.1
0.2
0.3
0.4
0.5
0.6
yr
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-yi
Fig. 3.8 Function (5.5) for argument (3.9) and exponent b ¼
1/3
Here exponent b is any real number.
Then the module of the function under consideration will represent the following
relationship:
r¼e
b’
; the argument being y ¼ b ln r:
It is easy to see that in this model, the module of function c depends on the
correlation between factors xr and xi (of argument ’), and the argument of the
function changes depending only on changes in the module of the complex
variables of the factors.
Therefore, when the module of the complex variable of argument r is a constant
value with only the polar angle changing, i.e., the argument changes its values along
a circle, by means of function (3.11) it is mapped to the complex plane w in the form
of a line at an angle equal to blnr. The points on the line increase with the increase
in the polar angle if the exponent b is negative and decrease if it is positive.
If the complex argument changes linearly and this line passes through the zero
point but does not include it within, then that indicates stability of the polar angle
and the growth of the argument module. Conformal mapping of these points to the
complex plane w takes place in the form of a circumference since the module is not
changed and the polar angle changes with changes of the argument’s module.
Of greater interest is the situation where an argument represents a line passing
either above or below the zero point. In this case, both the polar angle of the
argument and its module will change. Then function (3.11) will model the nonlinear
dynamics of both the real and imaginary parts of the complex function, since they
may be presented as a system:
8
>
>
>
< yr ¼ e
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos b ln x2r þ x2i ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
>
>
bartgxri
>
sin b ln x2r þ x2i :
: yi ¼ e
x
bartgxri
(3.13)
3.1 Power Functions of a Complex Variable
73
yi
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
yr
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Fig. 3.9 Function (3.11) for argument (3.8) and exponent b ¼ 1/3
yi
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
yr
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Fig. 3.10 Function (3.11) for argument (3.9) and exponent b ¼ 1/3
If the argument is described by a line passing below the zero point as defined by
(3.8), then with the exponent b ¼ 1/3 the conformal mapping of function (3.11) will
have the form of a nonlinear function, shown in Fig. 3.9. The function changes its
values from right to left in a counterclockwise direction from a point with the
coordinates yr ¼ 1.407, yi ¼ 0.491 to a point with the coordinates yr ¼ 0.186,
yi ¼ 0.759.
If a line in the plane of the complex argument passes above the zero point as
defined by (3.9), then, within the same range of values of the argument and with the
exponent b ¼ 1/3, the function will change again, decreasing the module with an
increase of the polar angle. However, the form of this nonlinear change will differ
from Fig. 3.9: it is smoother, which is clearly seen from Fig. 3.10.
For negative values of exponent (3.11) other forms of the conformal mapping
will be modeled.
Let us now consider a general model of a complex power function with complex
coefficients:
74
3 Conformal Mappings of Functions of a Complex Variable
yr þ iyi ¼ ða0 þ ia1 Þðxr þ ixi Þb0 þib1 ;
(3.14)
which combines all the aforementioned variants of a conformal mapping and
includes a complex coefficient of proportionality.
The effect of the complex coefficient of proportionality on the behavior of this
function is easy to understand if it is represented in exponential form:
a0 þ ia1 ¼ aeia :
This coefficient changes the function module a times and shifts the polar angle
by a.
Let us present model (3.14) in exponential form:
w ¼ aeb0 ln r
b1 ’ iðaþb0 ’þb1 ln rÞ
e
:
(3.15)
Then, for the real and imaginary parts of the complex function under
consideration
(
yr ¼ aeb0 ln r
yi ¼ ae
b1 ’
b0 ln r b1 ’
cosða þ b0 ’ þ b1 ln rÞ;
sinða þ b0 ’ þ b1 ln rÞ:
(3.16)
It is evident from the equalities obtained that for various combinations of the
complex coefficients (a0 + ia1) and (b0 + ib1) function (3.14) will describe the
most varied forms of relationship including cyclical ones. From this point of view
complex model (3.14) is universal and can be used for numerous economic
applications.
To get an idea of the possible forms of a conformal mapping of this function, let
us consider the following model:
yr þ iyi ¼ ðxr þ ixi Þ
0:5þi
:
(3.17)
In the first case line (3.8) passing in the plane of the complex argument below the
zero point (Fig. 3.11) is mapped to the complex plane w, and in the second case line
(3.9) passes in the plane of the complex argument above the zero point. This second
case is shown in Fig. 3.12.
Comparing these two figures we see that the position of the line in the plane of
the complex argument considerably affects the form of the conformal mapping.
Thus, it should be stated that a complex power function with complex exponent
may be used to model the most varied nonlinear processes in an economy.
3.2 Exponential Functions of Complex Variables
75
yi
2.5
2
1.5
1
0.5
yr
0
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.5
Fig. 3.11 General form of conformal mapping of exponential complex function for argument
(3.8) and complex exponent ( 0.5 + i)
yi
0.14
0.12
0.1
0.08
0.06
0.04
0.02
yr
0
-0.1
-0.05
-0.02 0
0.05
0.1
-0.04
Fig. 3.12 General form of conformal mapping of exponential complex function for argument
(3.9) and complex exponent ( 0.5 + i)
3.2
Exponential Functions of Complex Variables
Exponential functions of real variables are not the only class of functions used in
modern economics, though they are dominant among economic models of real
numbers. In particular, it is exponential functions that are widely applied in the
theory of production functions since exponents have a clear economic interpretation
and the differentiability of functions makes it possible to judge the modeled
production processes by the values of the first and second derivatives. These models
are fundamental for modeling economic dynamics, for parametric models in price
formation, etc. However, since other functions are also used in the economics
of real variables, in a complex economy it is necessary to study the properties of
complex functions that are similar to those of real variables and the possibility
of using them in the practice of mathematical economic modeling.
76
3 Conformal Mappings of Functions of a Complex Variable
Let us consider exponential functions taking the following exponential complex
function as an example:
yr þ iyi ¼ ða0 þ ia1 Þeðb0 þib1 Þðxr þixi Þ :
(3.18)
The properties of other exponential functions will be similar to this one, which is
why we can extrapolate them to the entire class of exponential functions of complex
variables.
Let us agree that initially the exponent is a real number, i.e., b1 ¼ 0. Then the
function in exponential form will look like this:
yr þ iyi ¼ aeia eb0 ðxr þixi Þ ¼ aeb0 xr eiðaþb0 xi Þ :
(3.19)
That is, the function module depends only on the changes of the real part of the
complex argument:
r ¼ aeb0 xr :
(3.20)
The polar angle is wholly defined by changes in the imaginary part of the
complex argument:
y ¼ a þ b0 x i :
(3.21)
Let us first consider the simplest situation, where one of the factors xr or xi is a
constant value, with only the other factor changing.
If the real part of the complex argument remains constant, i.e., xr ¼ Х ¼ const,
but the imaginary part xi increases, then in the plane of the complex argument this
indicates a line parallel to the axis of imaginary values and crossing the line of real
numbers in point X. A conformal mapping of this line will represent a circumference with constant radius:
r ¼ aeb0 X
(3.22)
and a polar angle (3.21) that increases with an increase in the imaginary part of the
positive coefficient b0 and decreases if it is negative.
In the second case, where the line in the complex plane of the argument is
perpendicular to the axis of the imaginary numbers at xi ¼ X ¼ const and changing
xr, function (3.19) is mapped in the plane w as a line since it is characterized by a
constant value of the polar angle
y ¼ a þ b0 xi ¼ a þ b0 X;
with the radius changing according to exponent (3.20).
3.2 Exponential Functions of Complex Variables
77
When the complex argument changes its values linearly and is not parallel to any
axis of the plane of the complex argument, function (3.19) models a more complex
line. The real and imaginary parts of this function will look as follows:
(
yr ¼ aeb0 xr cosða þ b0 xi Þ;
(3.23)
yi ¼ aeb0 xr sinða þ b0 xi Þ:
The nature of the conformal mapping depends on the sign of exponent b0. If
b > 0, a linear increase in the values of the complex argument will be followed by a
considerable increase in the module of the function. Since its polar angle also
increases, in the complex plane w, this will mean movement along a divergent
spiral, the real and imaginary parts of the complex function changing by sine and
cosine respectively with a nonlinear increase of the oscillations.
When the exponent b0 < 0, the module of the function decreases exponentially,
and in the complex plane it will look like a spiral that tends to zero, and the real and
imaginary parts considered separately will represent a damped oscillatory process.
Now let us consider the case where the exponent is multiplied by the imaginary
coefficient b1 and the real one is equal to zero, b0 ¼ 0:
yr þ iyi ¼ ða0 þ ia1 Þeib1 ðxr þixi Þ ¼ ða0 þ ia1 Þe
b1 xi ib1 xr
e
:
(3.24)
The module of this function represents an exponential dependence on the
imaginary part of the complex argument
r ¼ ae
b1 xi
;
(3.25)
and the polar angle represents a linear dependence on the real part of the argument:
y ¼ a þ b1 xr :
(3.26)
If we compare (3.25) with (3.20), and (3.26) with (3.21), we will see that the
function has varied “symmetrically” – now the imaginary part of the argument
influences the module of the function and the real part influences the polar angle.
This means that exponential function (3.24) will map a linear increase of the
complex argument to the complex plane w in the form of a spiral, which is divergent
if coefficient b1 ¼ 0 and convergent to zero if b1 > 0.
Now we can consider function (3.18) with a complex exponent:
yr þ iyi ¼ ða0 þ ia1 Þeðb0 þib1 Þðxr þixi Þ ¼ ða0 þ ia1 Þeðb0 xr
b1 xi Þ iðb1 xr þb0 xi Þ
e
:
(3.27)
Presenting all the complex variables and coefficients of function (3.25) in
exponential form and grouping the components of the polar angle and those of
the module we have
78
3 Conformal Mappings of Functions of a Complex Variable
reiy ¼ aeb0 xr
b1 xi ia iðb1 xr þb0 xi Þ
e e
:
(3.28)
Whence for the module of the complex function,
r ¼ aeb0 xr
b1 xi
(3.29)
and for the polar angle of this function,
y ¼ a þ b1 x r þ b0 x i :
(3.30)
The case is of interest when an argument represents a line lying in a complex
plane, i.e.,
cr xr þ ci xi ¼ d ! xi ¼
d
cr xr
:
ci
(3.31)
Let us substitute (3.31) into the module of function (3.29):
r ¼ ae
b0 xr b1
cr xr
ðci b0 þ b1 cr Þxr
ci
ci
¼ ae
d
b1 d
:
(3.32)
The module of an exponential function with a complex proportionality coefficient varies by exponent again and may increase depending on the coefficient or
decrease depending on its values.
If now we substitute (3.31) into (3.30), then it is easy to see how the polar angle
of this function varies with linear variations of the complex argument:
y ¼ a þ b1 x r þ b0
d
cr xr
ðb1 ci
¼aþ
ci
cr b0 Þxr þ b0 d
:
ci
(3.33)
It varies linearly.
Therefore, a general exponential function with complex exponent models a
divergent spiral process or one that converges to zero (depending on the coefficient
values) if the complex argument varies linearly.
For nonlinear variations of the argument the form of the line of the conformal
mapping will change.
3.3
Logarithmic Functions of Complex Variables
Let is consider the possibility of applying logarithmic functions of complex
variables in a complex economy. In general, the function of a complex variable
will have the following form:
yr þ iyi ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðxr þ ixi Þ:
(3.34)
3.3 Logarithmic Functions of Complex Variables
79
Obviously, other logarithm bases (not just natural but decimal, binary, etc.) are
possible, but this does not influence our subsequent arguments or the basic
properties of the function itself, which is why we will consider the natural basis
of logarithms as the most convenient basis for our analysis.
It should be noted immediately that the logarithmic function of complex variables,
as well as the function of real variables, does not exist in the point xr ¼ 0, xi ¼ 0, but,
unlike the logarithmic function of real variables, it exists for negative values of the
argument. This is why in our subsequent reasoning in this section we exclude this zero
point from our consideration by default. If we need to consider the situation where
some line displayed in a complex plane issues from the zero point, then we will think
that it comes from the neighborhood of this point.
Since the free term on the right-hand side of the equality characterizes the initial
conditions and its effect on the complex result does not go further, let us consider
the function without it:
yr þ iyi ¼ ðb0 þ ib1 Þ lnðxr þ ixi Þ:
(3.35)
From the TFCV it is known that a logarithm of the complex number z may be
represented as the sum
ln z ¼ lnjzj þ i arg z ¼ ln r þ ið’ þ 2kpÞ;
(3.36)
where k is a whole number.
This means that the complex number z 6¼ 0 has an infinite number of logarithms
(functions with an infinite number of values), as the real part of a logarithmic
function is defined by a single value and its imaginary part is defined up to a
multiple of 2p.
It is clear from (3.36) that if the real or imaginary part of a complex argument of
a logarithmic function is negative, then a logarithm of this function exists. This is an
additional argument in favor of using functions of complex variables in economics
since a logarithmic relationship may be extrapolated to a range of negative values of
an argument, unlike functions of real variables.
In the theory of complex variables there is a concept known as the “main
logarithm value,” when k ¼ 0. We will just use this main value in models of a
complex economy.
Applying the property of a logarithm of a complex number to the function under
consideration (3.35), we get
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xi
2
2
yr þ iyi ¼ ðb0 þ ib1 Þ ln xr þ xi þ iarctg
¼ ðb0 þ ib1 Þðln r þ i’Þ: (3.37)
xr
Let us consider the case where the imaginary part of a complex coefficient is
equal to zero. Then the function will look as follows:
80
3 Conformal Mappings of Functions of a Complex Variable
yr þ iyi ¼ b0 ln r þ b0 i’
(3.38)
yr ¼ b0 ln r
(3.39)
yi ¼ b0 ’:
(3.40)
or
If a complex argument represents points lying on a circumference, then the
module of the argument will also be a constant value. This indicates the constant
nature of the real part of the function. The imaginary part will change linearly since
it represents a linear function of the polar angle of the argument. That is, the
circumference in a complex plane of an argument is mapped to the complex
plane w in the form of a line perpendicular to the axis of real variables.
If a complex argument represents a line passing in the neighborhood of the zero
point, then the polar angle of the argument will be a constant, ’ ¼ const, which will
lead to the constant nature of the function yi ¼ const. Therefore, in this case the
function is also mapped in the complex plane w as a line, though it is perpendicular
to the axis of the imaginary part of the function.
But if a complex argument is represented as a line passing beyond the zero point,
then the function in question will provide another mapping of this line. Let it be
cr xr þ ci xi ¼ d:
(3.41)
Then, according to (3.39), the real part of the function will represent
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi
d
c
x
r
r
:
yr ¼ b0 ln x2r þ
ci
(3.42)
This means that with a linear increase in the argument, the real part of the
function increases logarithmically.
The imaginary part of the function, taking into account the linear nature of the
change in the argument, will change as follows:
yi ¼ b0 arctg
xi
d cr xr
d
¼ b0 arctg
¼ b0 arctg
ci xr
xr
ci xr
cr
:
ci
(3.43)
This change follows the arctangent of the inverse value.
A simultaneous change in the real and imaginary parts of a function in a complex
plane will provide a picture of smooth curves.
Figure 3.13 shows this mapping for the case where the complex argument
changes according to the linear law
3.3 Logarithmic Functions of Complex Variables
81
yi
1.5
1
0.5
yr
0
-0.5
-0.5
0
0.5
1
1.5
2
2.5
3
-1
-1.5
-2
-2.5
Fig. 3.13 Function (3.38) for linear change in argument
1; 5x þ 2 ¼ 0;
2xr
where proportionality coefficient b0 ¼ 1.5.
Here the real part of the argument changes from xr ¼ 0.1 to xr ¼ 4.0, with an
interval of 0.1.
Let us consider a situation where the real part of the complex proportionality
coefficient of model (3.35) is equal to zero, i.e., the following function is used:
yr þ iyi ¼ ib1 lnðxr þ ixi Þ ¼
b1 ’ þ ib1 ln r;
(3.44)
where
yr ¼
b1 ’;
yi ¼ b1 ln r:
Here we get properties symmetrical to those of model (3.38) – the same
circumference in the complex plane will be mapped to the complex plane w in
the form of a line parallel to yr, and the line issuing from the neighborhood of the
zero point on the argument axis will be mapped as a line parallel to the yi-axis.
Now we can consider the properties of model (3.35) where both the real and
imaginary parts of the proportionality coefficient are not equal to zero. After
opening the parentheses (3.37) we have
xr þ ixi ¼ b0 ln r
b1 ’ þ iðb0 ’ þ b1 ln rÞ;
where for the real part of the equality
yr ¼ b0 ln r
b1 ’
(3.45)
82
3 Conformal Mappings of Functions of a Complex Variable
yi
2.95
2.9
2.85
2.8
2.75
2.7
-yr
-7
2.65
-6
-5
-4
-3
-2
-1
0
Fig. 3.14 Function (3.48) for linear changes in argument
and for the imaginary part
yi ¼ b1 ln r þ b0 ’:
To understand the behavior of this model, let us represent it in exponential form.
The tangent of the polar angle for the function under consideration will be
tgy ¼
b1 ln r þ b0 ’
:
b0 ln r b1 ’
(3.46)
The module of the complex result is
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðb20 þ b21 Þðln2 r þ ’2 Þ:
(3.47)
The function in exponential form will look like this:
w¼
0’
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiiarctgbb1 lnln rþb
0 r b1 ’
:
ðb20 þ b21 Þðln2 r þ ’2 Þe
As is clear, complex-form smooth curve is being modeled that depends on the
proportionality coefficient and on the position of the argument points in the
complex plane. An example of how the function
w¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b ln rþb ’
iarctgb1 ln r b0 ’
0
1
ðb20 þ b21 Þðln2 r þ ’2 Þe
(3.48)
maps linear changes in the complex argument xr 2xi 2 ¼ 0 to the complex plane
is given in Fig. 3.14.
3.4 Zhukovsky’s Function and Trigonometric Complex Functions
3.4
83
Zhukovsky’s Function and Trigonometric
Complex Functions
The aforementioned simple complex functions in the TFCV are also supplemented
by Zhukovsky’s function and trigonometric functions. Since an economy is very
diverse and the presence in that economy of relationships that can be described by
these functions cannot be excluded, let us study their properties.
The fractional-rational function
w¼
1
1
zþ
2
z
(3.49)
was thoroughly studied by N.E. Zhukovsky, which is why in the TFCV it is called
Zhukovsky’s function.
To determine the range of univalency of a conformal mapping of this function,
we should first determine if there is a range of multivalency, i.e., if there are such z1
and z2 that, by means of Zhukovsky’s function, transfer to one point w. To answer
this question, we equate Zhukovsky’s functions for these two complex variables:
1
1
z1 þ ¼ z2 þ ! ðz1
z1
z2
z2 Þ 1
1
z1 z2
¼ 0:
(3.50)
If this equality is valid, then the function is multivalent; if not, then each of the
values of the complex variable of factors is associated with only one complex value
of the result.
Since by the problem condition z1 6¼ z2, the function is univalent except for the
points for which z1z2 ¼ 1. Let us see which points do not satisfy the univalency
condition, i.e., where z1z2 ¼ 1. For that let us represent complex variables in
exponential form:
z1 z2 ¼ r1 eiy1 r2 eiy2 ¼ 1 $ r1 r2 eiðy1 þy2 Þ ¼ 1:
This equality will be equal to 1 if
– The sum of polar angles is equal to zero – y1 ¼ y2, which means symmetry of
points with respect to the real number axis,
– Modules of complex variables z1 and z2 are equal to 1.
Thus, Zhukovsky’s function is univalent in the entire complex plane, except for
points of the unit circumference.
Since any complex number may be represented both in arithmetic and trigonometric forms, let us represent z as follows:
z ¼ r cos ’ þ ir sin ’:
(3.51)
84
3 Conformal Mappings of Functions of a Complex Variable
If we substitute this expression into Zhukovsky’s function we have
1
1
1
1
zþ
¼
r cos ’ þ ir sin ’ þ
2
z
2
r cos ’ þ ir sin ’
1
r cos ’ ir sin ’
¼
r cos ’ þ ir sin ’ þ 2
2
r ðcos2 ’ þ sin2 ’Þ
w¼
or
1
1
1
w¼
rþ
cos ’ þ i
r
2
r
2
1
r
sin ’:
(3.52)
If we consider a circumference, r ¼ r0 6¼ 1, in a plane, Zhukovsky’s function is
mapped in the resulting plane in the form of ellipses with semiaxes:
1
1
1
a¼
; b¼
r0 þ
r0
2
r0
2
1
:
r0
(3.53)
For r0 ! 1 the ellipse shrinks to the interval [ 1,1] of the real axis of the
complex plane, for r0 ! 0; as well as for r0 ! 1, it tends to infinity. Only when
jzj < 1, i.e., the points lie inside a unit circumference, with an increase in the polar
angle of the argument, does the ellipse go g around in the negative direction, and
when jzj > 1 – positive.
It is easy to see from (3.52) that for r > > 1 Zhukovsky’s function almost
coincides with the complex function, and on the interval r < 1 it becomes nonlinear. That is, if a complex argument varies linearly, then Zhukovsky’s function will
vary practically linearly, with the exception of an interval of values of the argument
inside a unit circumference. Here the function becomes nonlinear.
With reference to economic problems, this function may be used in econometrics. Then it is convenient to represent it in the following way:
1
w ¼ ða0 þ ia1 Þ z þ
z
(3.54)
since the complex coefficient of proportionality promotes the generation of conformal mapping in the form of ellipses of various scales and forms.
As an example, Fig. 3.15 shows a conformal mapping of Zhukovsky’s function
to the complex plane w for the linearly varying argument z when the imaginary
component represents a dependence on the real component of the argument calculated by the formula
xi ¼ 0; 4 þ xr ; 7 < xr < 7:
For other methods of linear variations in the argument inside a unit circumference, the function takes other nonlinear forms.
3.4 Zhukovsky’s Function and Trigonometric Complex Functions
85
5
4
3
2
1
-4
-3
-2
0
-1 0
-1
1
2
3
4
-2
-3
-4
Fig. 3.15 Zhukovsky’s function for linearly varying argument values ( 7 < xr < 7)
This means that Zhukovsky’s function may be used in modeling particular
economic processes when in the area of change in the real and imaginary parts of
the argument from 1 to +1 the instability of linear dynamics is modeled.
Zhukovsky’s function is convenient for studying trigonometric functions of
complex variables.
Since the following two equalities follow from Euler’s formula:
eix ¼ cos x þ i sin x; e
ix
¼ cos x
i sin x;
it is easy to derive formulae for the calculation of sines and cosines:
sin x ¼
eix
e
2i
ix
; cos x ¼
eix þ e
2
ix
:
These formulae are used in the TFCV to determine the trigonometric functions
of the complex variable z:
sin z ¼
eiz
e
2i
iz
; cos z ¼
eiz þ e
2
iz
:
(3.55)
For these functions all trigonometric correlations are valid, they are periodical
with period 2p, etc. To provide the essence of a conformal mapping of the first
function, it can be represented as follows:
sin z ¼
eiz
e
2i
iz
iz
¼
1
2
eiz
þ ie
i
iz
¼
1
2
eiz
i
þ iz
e
i
¼
1
1
z1 þ
:
2
z1
(3.56)
Here z1 ¼ ei :
Thus, a mapping of the function sinz can be considered a superposition of other
mappings – an exponential function of a complex variable and Zhukovsky’s
86
3 Conformal Mappings of Functions of a Complex Variable
function. Since currently it is not clear what economic practices trigonometric
functions of complex variables are applicable in, we can only state this fact.
In conclusion, it should be noted that the cosine of a complex variable differs
from the sine only in the shift due to the obvious equality
p
cos z ¼ sin z þ
:
2
The sine and cosine formulae serve as the basis for defining other trigonometric
functions.
References
1. Kasana H.S. (2005), Complex Variables: Theory And Applications. New Delhi: PHI Learning
Pvt. Ltd
2. Cohen H. (2007), Complex Analysis with Applications in Science and Engineering. Springer
Chapter 4
Principles of Complex-Valued Econometrics
Mathematical statistics has paid little attention to the processing of random complex
values; however, such statistical processing is crucial for models and methods of
complex variables to be used in practical economics. That is why in this chapter we
propose principles of a new mathematical apparatus for statistical processing of
economic data, namely, principles of complex-valued econometrics – regression
and correlation analysis – and adapt the least-squares method to complex random
variables. We also derive the formula for a pair correlation coefficient for two
random complex variables and provide an interpretation of its values. The method
of estimation of confidence limits of complex-valued econometric models is also
provided.
4.1
Statistics of Random Complex Value: Standard Approach
Econometrics is one of the developed and most in-demand branches of economic
and mathematical modeling. Economists at all levels of activity – from the macro to
the micro level – continuously face problems associated with the description,
explanation, and forecasting of various trends in socioeconomic dynamics or
relationships between the factors. Econometrics is a branch of economics that
involves methods of processing f statistical data to build economic models giving
a quantitative description of the laws of economic relationships. Modern econometrics is based on methods of mathematical statistics, above all methods of
regression and correlation analysis.
The main goal of these branches of mathematical statistics is to discern the
interrelations between random factors, the estimation of these interrelations, the
selection of a proper form of regression model, the estimation of model coefficient
values, and the assessment of the reliability of the obtained results.
The set of regression models is determined by well-known mathematical functions
of a variable whose number barely exceeds two tens. In terms of the types of
socioeconomic dynamics this number is infinitely small since economic dynamics is
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
DOI 10.1007/978-1-4614-5876-0_4, # Springer Science+Business Media New York 2012
87
88
4 Principles of Complex-Valued Econometrics
varied and one can count several tens of thousands of market factors. This is why the
problem extending the range of models to be used in economics remains very relevant.
It is this very problem that the complex-valued functions considered in previous solve
providing an tool that describes the dependencies between variables in a different way
compared to real-variable models. Various elementary functions of complex
variables, with rare exceptions, allow economists to model such nonlinear
interrelations that have no analogs in the econometrics of real variables, or these
analogs are so complex that their practical application is pointless. Thus, using
elementary models of complex variables we considerably extend the instrumental
base of econometric research. Properties of elementary models of functions of complex variables and complex arguments are such that they can be used to describe the
most varied economic processes – intensive and extensive, efficient and stagnating.
To make models of a complex-valued economy work in reality we appeal to
econometrics. However, since models of complex variables are not practically used
in the economy, modern econometrics will not help us solve this problem, so we
should turn to mathematical statistics.
In practice, problems of processing statistical data of a random complex value
are quite rare – they are more often in recognition of certain signals. This is why it is
not easy to solve the problem of building econometric models of complex variables
on the basis of the results of mathematical statistics.
Nevertheless, in mathematical statistics there are methods concerning just the
statistics of a random complex value that can be applied to the solution of a given
problem. According to the available literature, interest in the statistical processing
of observations about variations of complex variables appeared in the 1950s and
1960s [1–4]. Later other properties of random value statistics were discovered, but
these studies are insufficient and do not provide the necessary knowledge for
purposes of complex-variable econometrics.
First of all let us mention the fundamental concept of the statistics of a complex
variable – its mathematical expectation. The expectation of a complex random
value z ¼ x + iy is a complex number:
MðzÞ ¼ MðxÞ þ iMðyÞ:
(4.1)
Since any complex number may be represented as a pair of real numbers,
mathematical statistics assumes that the complex representation of random
functions is simply convenient for analyzing the mathematical form of a mapping,
which may always be transferred to the real form. If is for that reason that functions
of variance, correlation, and covariation are presented in mathematical statistics as
simple and nonrandom real characteristics of random processes and functions
regardless of the form of their mathematical representation.
Then, following the approach adopted in mathematical statistics, let us determine the variance of a complex random value as a real number. “The variance of a
complex random value is the expectation of the square of the Absolute value of the
corresponding centered value” [5]:
4.1 Statistics of Random Complex Value: Standard Approach
89
DðzÞ ¼ M½jzj
_ ¼ M½x_ 2 þ y_2 ¼ M½x_2 þ M½y_2
(4.2)
where
DðxÞ ¼ M½x_ 2 ¼ M½ðx
xÞ2 ;
(4.3)
DðyÞ ¼ M½y_2 ¼ M½ðy
yÞ2 :
(4.4)
This means that the variance of a complex random value equals the sum of
fvariances of its real and imaginary parts.
A correlation moment is an important characteristic used in mathematical
statistics. Since according to the above-mentioned definition, a correlation moment
is a real number, the expectation of a product of two random complex values
M½ðx1 þ iy1 Þðx2 þ iy2 Þ
may not be considered a correlation moment since the product of two complex
variables will be a complex number.
Incidentally, in mathematical statistics of real variables the correlation moment
of two equal random values is equal to the variance; however, this does not follow
from the preceding formula.
That is why it was decided to consider the correlation moment of a complex
random value as the mathematical expectation of the multiplication of one variable
by the conjugate of another variable:
mzz ¼ M½ðx1 þ iy1 Þðx2
iy2 Þ:
(4.5)
If we perform the multiplication and group the terms, we get
mzz ¼ M½x1 x2 þ M½y1 y2 þ iðM½y1 x2
M½x1 y2 Þ ¼
mx1 x2 þ my1 y2 þ iðmy1 x2
mx1 y2 Þ:
(4.6)
If z1 ¼ z2 ¼ z, then the last term of (4.6) with the imaginary unit becomes equal
to zero and the correlation moment is equal to variance (4.2), which was necessary
to obtain for a complete analogy with the statistics of real random variables.
Further, in mathematical statistics there is a concept of a complex random
function:
z ¼ xðtÞ þ iyðtÞ:
(4.7)
For this we calculate the variance in a similar way:
Dz ðtÞ ¼ Dx ðtÞ þ Dy ðtÞ
(4.8)
90
4 Principles of Complex-Valued Econometrics
and the correlation function
mz ðt1 ; t2 Þ ¼ mx ðt1 ; t2 Þ þ my ðt1 ; t2 Þ þ iðmxy ðt1 ; t2 Þ
mxy ðt2 ; t1 ÞÞ:
(4.9)
This is everything that mathematical statistics gives economists if they want to
use its tools for complex-valued econometrics. As follows from the given materials,
there is a rule according to which all the basic parameters characterizing random
processes, such as variance, correlation moment, covariation, etc., are real values.
This rule does not result from a strict mathematical proof but from the logical
conclusion that all the characteristics of variability of a random value are real ones
regardless of the form of presentation of random variables – real or complex.
4.2
Method of Least Squares of Complex Variables:
Standard Approach
As we can see from the previous section, mathematical statistics does not answer
those questions that can be answered using econometric tools, these questions are as
follows:
– How can the relationship between two random complex variables be
determined?
– If we reveal this relationship, how do we select the required model of complex
variables and estimate its parameters on statistical data?
The first problem in the range of real variables is solved by correlation analysis,
the second one by regression. Mathematical tools of regression-correlation analysis
are unique and based on a uniform methodology.
We have not found any available works in mathematical statistics containing the
results of complex-value-regression-correlation analysis. That is why we had to
solve this problem independently [6]. The only scientific publication referring to the
problem in question is a work published in 2007 [7], but, first of all, when there was
a vital necessity to develop the tools of regression analysis for the evaluation of
coefficients of complex-valued models this article was unavailable, and, second, it
is based on the conditions stated in the previous sections, which, as we will see later,
may not be axiomatic for the econometrics of complex variables.
Since the main computational coefficients of correlation analysis are based on
regression analysis methods, let us start our examination of the conditions of
econometrics of complex variables with the main task of regression analysis – the
evaluation of regression model coefficients.
The least-squares method (LSM) is the most popular method in modern econometrics, as it possesses a number of remarkable properties and is highly standard.
We are not going to speak here about the specifics of the distribution of a random
complex variable, taking only its normal distributions. Let us assume a priori that in
4.2 Method of Least Squares of Complex Variables: Standard Approach
91
this chapter we will consider only stationary processes having a normal distribution.
These and other assumptions usually made to substantiate regression-correlation
analysis methods will be considered as given by default so as not to repeat them
henceforth.
Let us designate the values of some complex variable varying in time:
yrt þ iyit :
(4.10)
The reason for this dynamic set of complex variables is another random variable
that is a complex argument:
xrt þ ixit :
(4.11)
For example, the amount and price of an acquired product combined into a
complex variable (4.10) are explained by a consumer’s cash income and
accumulated income (property), which may be presented in the form of complex
variable (4.11). We can provide quite a few pairs of related socioeconomic
indicators; some of them will be discussed in subsequent chapters of this study.
Values of the dynamic range (4.10) should be approximated by some regression
model to calculate the computational values of this complex variable:
^
^
Fðxrt þ ixit Þ ¼ yrt þ iyit :
(4.12)
The method of approximation of the calculated values to actual ones is
represented by the difference between them:
ðyrt þ iyit Þ
^
^
ðyrt þ iyit Þ ¼ ert þ ieit :
(4.13)
This may be defined as a complex approximation error. Depending on the values
taken by the model coefficients, error (4.13) can also be different. This is why the
econometric model coefficient should provide the average minimum error (4.13).
This requirement should be met on the entire range of values of t. However, this
general wish should be made a strict form of mathematical criterion. The problem is
that if the range of real numbers provides an easy comparison between two
numbers, then the range of complex numbers makes it impossible.
In fact, it is no use looking for an answer to the question of which number is
more, z1 or z2, if they take the following values:
z1 ¼ 2 þ i3; z2 ¼ 3 þ i2:
It is possible to compare only real numbers, i.e., we can say which number has a
higher real or imaginary part, or we can compare the absolute value of complex
variables or their polar angles (which are real numbers), but it is not possible to
compare two complex numbers.
92
4 Principles of Complex-Valued Econometrics
Here we can use assumptions that all the measures of the variability of complex
random variables are real numbers. In particular, the variance of the complex approximation error (4.13) will also be a real number calculated by variance formula (4.2).
The least-squares method for real variables minimizes the variance of the approximation error of the actual values by the calculated ones. With reference to the complex
approximation error, the minimum of its variance will be determined by the following
criterion:
X
F¼
ðe2rt þ e2it Þ ! min:
(4.14)
t
If we write down the approximation error in exponential form, we get
ert þ ieit ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
eit
e2rt þ e2it eiarctgert :
Hence it is easy to see that the LSM criterion for the evaluation of coefficients of
econometric complex-valued model (4.14) means a minimization of the absolute
value of the complex approximation error. It is seen from (4.14) that the polar angle
of the complex approximation error has no meaning, i.e., when we use (4.14)
corresponding to the standard formulation of the problem, some information on
the nature of the complex variable, its complex regression model, and complex
approximation error gets lost.
It is evident that the sum (4.14) is determined by the values that the complex
coefficients of the econometric model take. Let an econometric model have a pair of
complex coefficients, as is characteristic for most analytical functions considered in
Chap. 2, i.e.
a0 + ia1
b0 + ib1
the sum (4.14) represents a function of these parameters.
In the general case the LSM criterion for the evaluation of these coefficients of a
complex-valued model will be written as follows:
Fða0 ; a1 ; b0 ; b1 Þ ¼
X
ðyrt
Re½Fðxrt þ ixit ÞÞ2 þ
X
ðyit
ImFðxrt þ ixit ÞÞ2 ! min :
(4.15)
With reference to a linear complex-valued model:
yrt þ iyit ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þðxrt þ ixit Þ
(4.16)
this criterion will look like this:
Fða0 ; a1 ; b0 ; b1 Þ ¼
X
ðyrt ½a0 þ b0 xrt b1 xit Þ2
X
þ
ðyit ½a1 þ b1 xrt þ b0 xit Þ2 ! min :
(4.17)
4.2 Method of Least Squares of Complex Variables: Standard Approach
93
To find the minimum of such a function of real variables it is necessary to
calculate the first partial derivatives of the function for the variables, set them equal
to zero, and solve the resulting system of equations. If there are doubts that the
extremum is really the function minimum, it is necessary to build a Hesse matrix
and make sure that it is positively determined. The system of normal equations for
the evaluation of coefficients of a linear complex-valued model for a number of
observations T looks like this:
X
X
8X
yrt ¼ a0 T þ b0
xrt b1
xit ;
>
>
>
X
X
X
>
>
<
xit ;
xrt þ b0
yit ¼ a1 T þ b1
X
X
X
X
X
X
(4.18)
>
xrt þ b0 ð
xit ;
yrt xrt þ
yit xit ¼ a0
x2rt þ
x2it Þ þ a1
>
>
>
X
X
X
X
X
>
:X
yit xrt
xit þ b1 ð
xrt :
yrt xit ¼ a0
x2rt þ
x2it Þ þ a1
This is quite a bulky system of four equations with four unknowns. Since modern
computation equipment makes it possible to work with complex variables – for
example, Microsoft Excel has an “engineering calculations” section that allows you
to perform basic operations with complex variables – we would like to convert the
system of normal (4.18) to a system of complex equations. Then the calculations for
practical purposes would be much easier. And this can be done.
It is easy to see that system (4.18) is equivalent to a system of two complex
equations with two complex coefficients:
X
8X
yrt þ iyit ¼ ða0 þ ia1 ÞT þ ðb0 þ ib1 Þ
ðxrt þ ixit Þ;
>
>
<X
X
ðyrt þ iyit Þðxrt ixit Þ ¼ ða0 þ ia1 Þ
ðxrt ixit Þ
>
X
>
:
þ ðb0 þ ib1 Þ
ðxrt þ ixit Þðxrt ixit Þ:
(4.19)
It is a system of two linear complex-valued equations with two complex
coefficients that has a simple solution. But it can also be simplified, as in the case
where all the initial variables are centered with respect to their averages, the free
complex coefficient is equal to zero. In this case from (4.19) we can derive a
formula for the complex proportionality coefficient:
P
ðyrt þ iyit Þðxrt
b0 þ ib1 ¼ P
ðxrt þ ixit Þðxrt
ixit Þ
:
ixit Þ
(4.20)
Let us see how the LSM is applied to the data given in Table 4.1.
After centering these variables with respect to their averages we obtain the
numerator (4.20)
171,373-i41,862 and its denominator: 130,819.
94
4 Principles of Complex-Valued Econometrics
Table 4.1 Data of
conditional example
t
1
2
3
4
5
6
7
8
9
10
11
12
Average
xrt
1.0
1.2
1.3
1.3
1.2
1.5
1.4
1.6
1.4
1.7
1.5
1.8
1.40
xit
2
3
7
1
2
2
3
1
8
3
3
2
1.75
yrt
4.250
4.832
6.243
4.323
4.512
3.625
5.094
4.716
6.694
3.567
3.305
5.298
4.705
yit
5.100
3.854
1.354
6.506
5.164
10.500
3.918
6.602
2.632
11.874
11.810
5.356
5.558
Substituting these values into the formula for finding complex coefficients, we get
b0 þ ib1 ¼
171:373 i41:862
¼ 1:31
130:819
0:32i
Now it is easy to calculate the values of the free complex term of the linear
regression relationship because for evaluations of the LSM of linear relationships
an equality with respect to the averages always takes place:
yr þ i
yi ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þð
xr þ i
xi Þ:
The complex free coefficient for this model built on the data of Table 4.1 looks
like this:
a0 þ ia1 ¼ yr þ i
yi ðb0 þ ib1 Þð
xr þ i
xi Þ ¼
4; 705 i5; 558 ð1; 31 i0; 32Þð1; 4 þ i1; 75Þ ¼ 2; 3
i7; 4:
The linear complex-valued model looks like this:
yrt þ iyit ¼ ð2; 3
i7; 4Þ þ ð1; 31
i0; 32Þðxrt þ ixit Þ:
The obtained evaluations of the standard LSM will provide, on the considered
set of initial data, values of approximation errors with minimum variance, the latter
being a real number in the case under consideration.
4.3 Correlation Analysis of Complex Variables: Contradictions of the Standard. . .
4.3
95
Correlation Analysis of Complex Variables:
Contradictions of the Standard Approach
With respect to the expansion of econometrics by including functions of complex
variables, we should add to that all the other econometric attributes: not only
regression but also correlation analysis.
It should be noted that in mathematical statistics correlation analysis means a
combination of approaches, methods, and techniques aimed at determining the
degree and nature of the relation between two random factors. If this combination
is intended for studying multiple relationships, then this involves multiple
correlations.
Let us consider the possibility of transforming and adapting the basic conditions
of correlation analysis to the econometrics of complex variables. As follows from
the literature on the application of the TFCV in various branches of science, this
problem has not been posed or solved in its full scope.
The point is that to solve problems of hydrodynamics or gas dynamics, the
theory of cumulative charge, the theory of elasticity, to calculate electric contours
and other practical calculations, where the TFCV has been successfully applied, the
real and imaginary parts of complex variables have clear semantic interpretations,
and the models used are deterministic. For example, if electrical engineering studies
alternating currents, then active resistance R refers to the real part and reactive
resistance of, for example, coil L refers to the imaginary part of full resistance. Then
the model of full resistance Z for these two elements joined successively will look
like this:
Z ¼ R þ iL
The values of active and reactive resistance are easily measured using the
appropriate tools, taking into account the measurement error. And there is no
need to reveal the degree of correlation between the complex resistance and
consumed complex power, as this relationship is uniquely determined by the
objectively existing Kirchhoff’s law. If complex resistances are connected to a
complex network, the processing of the network characteristics will depend on the
connection of the elements – successive or parallel. A model of the most complicated electrical contour is deterministic since all the elements and characteristics –
resistances, currents, voltages, power, etc. – are defined therein. It is only when we
move to huge systems, influenced by many random factors, are the calculated
characteristics resulting from the properties of the mathematical model of an
electrical contour observed with some error.
However, the task of correlation analysis in these cases is to determine the
relationship between the errors of the calculated characteristics of electrical circuits
and random external factors. In this case, there can be no talk of a correlation
between complex variables, for example, active and reactive current.
96
4 Principles of Complex-Valued Econometrics
Economics is another matter; here, deterministic functional relationships practically do not exist, and even revealed relationships change their force,
characteristics, and even direction with time; in economics, laws do not have a
quantitative interpretation, and models describing them are not universal and do not
take into account the actual variety of factors existing in the economy. The
discovery of these relationships at any particular period of time does not mean
the discovery of any law since at some point this law may change its force and
instead of a linear dependence we might be faced with a nonlinear one. This is why
there is an objective necessity to regularly revise previous research on the relationship among the same economic indicators. In this situation regression and correlation analysis is practically the only tool for studying real economic phenomena and
the relationships among them, since in a changing economy there are no and there
may be no frozen relationships. The results of regression and correlation analysis
are regularly reviewed because, once revealed, they can and do change.
Therefore, expanding the instrumental base of econometrics by including in it
models of complex variables and considering the proper mathematical tools to
estimate the indicators of econometric models of complex variables (LSM), we
must give thought to the problem of discovering the interrelations among random
complex variables. Based on the principles and approaches of correlation analysis,
it is necessary to derive the coefficients applicable to complex variables that could
characterize the interrelations among complex factors.
It should be noted that modern correlation analysis does not provide an exhaustive solution to this problem, even with respect to random real variables. The most
popular tool in this area of mathematical statistics is the pair correlation coefficient,
which shows how closely the relationship between two random values approaches a
linear one, if, of course, this relationship exists. The researcher should put forward
and substantiate a hypothesis on the existence of a linear relationship between the
factors and then, using the pair correlation coefficient, confirm or refute it. This is
why the proposed correlation coefficients between complex variables may only
serve as an additional argument confirming (or refuting) the linear relationship
between complex variables.
We were unable to find methods and techniques of application of correlation
analysis to complex variables, though there are general approaches, some of which
were stated in the first section of this chapter. It was shown that in modern
mathematical statistics, the variance of a complex random value equals the sum
of variances of its real and imaginary parts, i.e., is a real value like the variability
measure. The expectation of the product of deviation of one of the values from its
expectation by the conjugate deviation of the other is called the correlation momentum of two complex random values (covariation).
The correlation momentum on whose calculation we must rely was presented in
the first section in the following form (4.5):
mzz ¼ M½ðx1 þ iy1 Þðx2
or
iy2 Þ
4.3 Correlation Analysis of Complex Variables: Contradictions of the Standard. . .
mzz ¼ mx1 x2 þ my1 y2 þ iðmy1 x2
mx1 y2 Þ:
97
(4.21)
For the discrete case, which we must deal with in econometrics,
"
1 X
ðyrt
mzz ¼
n t
þið
X
t
ðxrt
yr Þðyit
xr Þðyit
yi Þ þ
X
ðxrt
yi Þ
X
ðyrt
t
t
xr Þðxit
yr Þðxit
xi Þ
#
(4.22)
xi Þ :
It is known that the pair correlation coefficient for real variables can be found
using the correlation momentum in the following way:
rXY ¼
mXY
:
sX sY
(4.23)
One can use this formula for complex random values. To do this, we should
substitute the correlation momentum of two random values into the numerator of
formula (4.22).
To calculate the pair coefficient correlation (4.23), it is necessary to determine its
denominator. Since the standard approach in mathematical statistics assumes that
the variance of a complex random value equals the sum of the variances of its real
and imaginary parts, we get
1 X
s2X ¼ ð
ðxrt
n t
xr Þ þ
X
ðxit
xi Þ Þ;
1 X
ðyrt
s2Y ¼ ð
n t
yr Þ þ
2
X
ðyit
yi Þ Þ:
2
t
t
2
2
(4.24)
(4.25)
Substituting these values into the denominator (4.23) and taking into account the
fact that the standard deviation represents the square root of the variance, we obtain
a formula for calculating a sample value of the coefficient of correlation of two
complex random values:
rXY ¼
P
t
ðyrt
yr Þðyit
P
P
P
ðyrt
yi Þ þ ðxrt xr Þðxit xi Þ þ ið ðxrt xr Þðyit yi Þ
t
t
t
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
P
P
P
½ ðxrt xr Þ2 þ ðxit xi Þ2 ½ ðyrt yr Þ2 þ ðyit yi Þ2
t
t
t
yr Þðxit
xi Þ
:
t
(4.26)
It is clear that the obtained coefficient is a complex number since its imaginary
component will be equal to zero only in particular cases for a small number of
phenomena.
98
4 Principles of Complex-Valued Econometrics
Since all the variables used in the calculation of the pair correlation coefficient
(4.26) represent values centered with respect to their averages, we will use a
simplified form, taking into consideration that all the variables are preliminarily
centered. Then formula (4.26) may be represented like this:
rXY ¼
P
t
P
ðyrt yit þ xrt xit Þ þ ið ðxrt yit yrt xit ÞÞ
t
ffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
P
P
2
2
ðxrt þ xit Þ ðyrt 2 þ yit 2 Þ
t
(4.27)
t
Here we will not examine the properties of the derived pair correlation coefficient;
we would like to draw attention to its substance. It is complex! The degree of
approximation of the relationship between two random complex variables to a
linear one turns out to be a complex number, though the standard approach of
mathematical statistics to random complex variables assumes axiomatically that all
the measures for complex random variables are real ones. This is the first paradox of
the standard approach to the econometrics of complex random variables.
It is known that the pair correlation coefficient between two random real
variables was derived by another method [8], and later scientists discovered that
it may be represented via the correlation momentum and variances. Originally, the
pair correlation coefficient was determined via regression coefficients. Let us see
how this coefficient can be derived via regression coefficients with respect to
random real variables. Regression coefficients of real variable Yt to Хt and Хt to
Yt can be found using LSM, which for the model
Yt ¼ a 0 þ a 1 X t
(4.28)
suggests solving a system of normal equations:
8X
X
>
Y
¼
na
þ
a
Xt
t
0
1
>
<
t
t
X
X
X
>
Xt2
Xt þ a 1
Y t Xt ¼ a0
>
:
t
and for the model
t
(4.29)
t
Xt ¼ b0 þ b1 Yt
(4.30)
it suggests solving another system of normal equations:
8X
X
>
Yt ;
Xt ¼ nb0 þ b1
>
<
t
t
X
X
X
>
Yt2 :
Yt þ a1
Yt Xt ¼ a0
>
:
t
t
t
(4.31)
4.3 Correlation Analysis of Complex Variables: Contradictions of the Standard. . .
99
To find regression coefficients a1 and b1 initial variables are centered with
reference to their averages:
Xt
Yt
X;
Y:
Taking account the fact that the sum of deviations of any variable from its
average is equal to zero, the equalities for the first equations of systems of normal
equations will hold only if their free terms (a0 and b0, respectively) are equal to
zero. Then the first system of normal equations for centered variables will become
one equation:
X
t
ðYt
t
YÞðX
¼ a1
XÞ
X
ðXt
2
XÞ
¼ b1
XÞ
X
ðYt
2
YÞ
t
as well as the second one:
X
t
ðYt
t
YÞðX
t
Hence it is easy to calculate the values of regression coefficients a1 and b1 via
centered variables. The geometric mean of regression coefficients a1 and b1
r¼
pffiffiffiffiffiffiffiffiffi
a1 b1
(4.32)
will represent the pair correlation coefficient:
t XÞ
ðYt YÞðX
t
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi:
r
r¼ P
2P
2
ðXt XÞ
ðYt YÞ
P
t
(4.33)
t
It is clear that a coefficient value whose absolute value equals zero is obtained
only when a1 ¼ 1/b1, i.e., when there is functional linear relationship between the
variables.
Let us now derive a formula for the pair correlation coefficient between two
random complex variables as the geometric mean of two complex regression
coefficients. For that let us substitute into the appropriate formulae the values of
the complex coefficients of regression obtained previously with the variance minimization if we consider to be a real characteristic of the process (4.2). This
approach allowed us in Sect. 4.2. to obtain a system of equations that can help us
calculate complex coefficients that minimize the values of this variance. Since we
use centered variables and therefore get rid of the free term, we can directly use
formula (4.20) obtained previously.
100
4 Principles of Complex-Valued Econometrics
The complex regression coefficient of linear dependence of complex random
variable Y on another complex random variable X will be calculated as follows, by
means of LSM:
P
ðyrt þ iyit Þðxrt
a¼P
ðxrt þ ixit Þðxrt
ixit Þ
¼
ixit Þ
P
ðxrt þ ixit Þðyrt
b¼P
ðyrt þ iyit Þðyrt
iyit Þ
¼
iyit Þ
P
ðyrt þ iyit Þðxrt ixit Þ
P 2
:
ðxrt þ x2it Þ
(4.34)
If we consider the inverse dependence, i.e., that of complex random variable X
on another complex random variable Y, then the complex coefficient of this
regression will be found similarly by the standard LSM:
P
ðxrt þ ixit Þðyrt iyit Þ
P 2
:
ðyrt þ y2it Þ
(4.35)
Let us now substitute formulae (4.34) and (4.35) into formula (4.32) to calculate
the complex pair correlation coefficient:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
uP ðx þ ix Þðy
iyit Þ ðyrt þ iyit Þðxrt ixit Þ
rt
it
rt
u
:
r ¼ t
P
P
ðxrt 2 þ xit 2 Þ ðyrt 2 þ yit 2 Þ
t
(4.36)
t
The first factor of the numerator may be put in the following form:
X
ðxrt þ ixit Þðyrt
X
ðxrt
iyit Þ ¼ i
X
ðxit
X
ðxrt
ixrt Þðyrt
iyit Þ:
(4.37)
The second factor of the coefficient numerator (4.36) may also be transformed in
the following way:
ixit Þðyrt þ iyit Þ ¼ i
ixit Þðyit
iyrt Þ:
(4.38)
Now we can be sure that these two factors of the numerator radicand (4.36) are
equal to each other, which is why this numerator may be written as follows:
i
X
ðxrt
ixit Þðyit
iyrt Þ:
(4.39)
Taking this into account, the formula for calculating the pair correlation coefficient of two random complex variables (4.36) may be written as follows:
P
i ðxrt ixit Þðyit iyrt Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi:
p
r¼ P
P
ðy2rt þ y2it Þ ðx2rt þ x2it Þ
(4.40)
The imaginary unit in the numerator is not very convenient for understanding the
properties of this coefficient. However, if we expand the expression in the
4.4 Consistent Axioms of the Theory of Mathematical Statistics of Random. . .
101
numerator under the summation sign and multiply the resulting expression by an
imaginary unit, grouping the real and the imaginary parts, we obtain the following
formula for the numerator:
i
X
ðxrt
ixit Þðyit
iyrt Þ ¼
X
ðxrt yrt þ xit yit Þ þ i
X
ðxrt yit
xit yrt Þ:
(4.41)
Substituting it into the formula, we obtain for the pair correlation coefficient
r¼
P
P
ðxrt yrt þ xit yit Þ þ i ðxit yrt xrt yit Þ
ffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
P 2
P
ðyrt þ y2it Þ ðx2rt þ x2it Þ
(4.42)
Since this coefficient was derived from the premises of the standard approach,
we should expect its complete identity with that derived previously via the correlation momentum of two complex random variables (4.28). This formula is given
here to simplify comparison of the two variants of the same formula:
r¼
P
t
P
ðyit yrt þ xit xrt Þ þ ið ðxrt yit yrt xit ÞÞ
t
ffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
P
P
2
2
ðyrt þ yit Þ ðxrt 2 þ xit 2 Þ
t
(4.43)
t
The denominators of the two coefficients (4.42) and (4.43) coincide. The imaginary parts of the numerators differ from each other by minus one. But their real
parts differ from each other in principle – in formula (4.42) we sum up the product
of the real parts of the two variables and that of the imaginary parts of the variables.
In formula (4.43) we sum up the product of the real and imaginary parts of one
complex variable with that of another complex variable.
This means that the standard approach, which assumes that all the statistical
measures of variability of complex variables should be real, lead us to the second
paradox – the formulae of the pair correlation coefficient of two random variables
do not coincide, though this is the case for the domain of real variables.
There is a well-known scientific principle that says that if there are paradoxes in
a theory, one should pay special attention to the consistency of the theory’s axioms,
which should be done in our case.
4.4
Consistent Axioms of the Theory of Mathematical
Statistics of Random Complex Variables
It is evident from the discussion of the previous section that the rule introduced in
mathematical statistics with respect to complex variables does not hold in full. As
shown in the first section of this chapter, it is accepted in mathematical statistics that
a complex representation of random functions is simply a mathematical form
102
4 Principles of Complex-Valued Econometrics
convenient for analysis and may always be transformed into the form of real
functions. This is why it is recommended that when we examine random complex
variables, we should consider the parameters of variation of random complex
variables to be real. And it for this very reason that functions of variance, correlation, and covariation represent simple and nonrandom real characteristics of complex random processes and functions.
In the domain of real numbers, there is no operation for extracting the square root
of a negative number – it was this operation that gave birth to the TFCV as a branch
of mathematics. Then why should we look back to the rules adopted in the area of
real variables to deal with operations with complex ones? Why can we not consider
complex variance, complex correlation momentum, etc.? Simply because this very
variance means a particular measure of variability, and the meaning of the measure
itself is demonstrated fully when it is a real number? But for a complex series
represented by a complex variable series it is not a simple but a complex measure
that is an adequate measure of variability, which may always be considered a
convenient form of representing two real measures of variability. The real part of
a complex variance characterizes one side of the variance of a complex random
variable and the imaginary part characterizes the other one.
This view changes the axiomatic core of the theory of mathematical statistics of
complex random variables – all the measures of variability of these variables can be
complex, showing the complex nature of the modeled processes.
With this axiomatic basis, the correlation momentum is considered in a different
way than is accepted in standard mathematical statistics, which leads to conflicting
results, i.e., as follows from the substance of the correlation momentum:
mzz ¼ M½ðx1 þ iy1 Þðx2 þ iy2 Þ:
(4.44)
It is evident that new axioms of the theory will entail results that are different
from the standard formulation of the problem. However, the consistency of the
results and recommendations of the new theory should be verified.
First of all, let us redefine a basic concept that will be used subsequently, namely,
the concept of variance. Let us define the variance of a complex random value as a
complex number. To differentiate the newly introduced concept from the generally
accepted one we will add to it the word “complex” in this and all other cases. Thus,
the complex variance of a complex random value will be the expectation of the
square of the corresponding centered value:
DðzÞ ¼ M½z_2 ¼ M½x_ 2
_ ¼ M½x_ 2
y_2 þ i2x_y
_
M½y_2 þ 2iM½x_y;
(4.45)
where
DðxÞ ¼ M½x_ 2 ¼ M½ðx
xÞ2 ;
(4.46)
DðyÞ ¼ M½y_2 ¼ M½ðy
yÞ2 :
(4.47)
4.5 Least-Squares Method from the Point of View of the New Axiomatic Theory
103
This means that complex variance may also be a real negative value or it may be
an imaginary one. Since in Minkowsky’s geometry distances may be negative or
even imaginary, there is nothing extraordinary in the result obtained; for complex
variables that is how it should be if we consider them fully.
Obviously, complex correlation momentum (4.44) will take a different form:
mzz ¼ M½x1 x2 M½y1 y2 þ iðM½y1 x2 þ M½x1 y2 Þ
¼ mx1 x2 my1 y2 þ iðmy1 x2 þ mx1 y2 Þ
It is clear again that the complex correlation momentum may also be both
negative and imaginary, and complex.
We will not give an interpretation of the values of variance and correlation
momentum here.
We are interested in applied matters: how does one use these basic concepts to
solve a problem of evaluating coefficients of complex-valued economic models and
what form should a complex pair correlation coefficient have? Could there be any
contradictions that might refute the new axioms of the theory?
Let us find answers to these questions.
4.5
Least-Squares Method from the Point of View of
the New Axiomatic Theory
In regression analysis the problem of evaluating the coefficients of regression
models is considered with respect to simple linear monofactor models, after
which the problem gets complicated as one moves toward nonlinear functions.
That is why, in the case of the econometrics of complex variables, we start with the
solution of a problem for a simple linear model of complex variables. Its parameters
are to be assessed using LSM. This model has the following form:
y^rt þ i^
yit ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þðxrt þ ixit Þ:
(4.49)
The complex variance of the errors of approximation of the actual values of a
random complex variable under this model will look like this:
f ðzÞ ¼
¼
X
t
X
t
½yrt þ iyit
½Yt
A
ða0 þ ia1 Þ
2
BXt ¼
X
t
2
ðb0 þ ib1 Þðxrt þ ixit Þ :
ðYt2 þA2 þ B2 Xt2
2AYt
2BYt Xt þ 2ABXt Þ:
(4.50)
Let us consider each of the terms on the right-hand side of the last equality
separately taking into account the properties of complex numbers:
104
4 Principles of Complex-Valued Econometrics
X
Yt2 ¼
X
ðyrt þ iyit Þ2 ¼
X
y2rt
X
A2 ¼
X
ða0 þ ia1 Þ2 ¼
X
a20
t
t
X
t
X
B2 Xt2 ¼
¼ b20
X
t
t
t
b21
t
2
X
X
2a0
2b0
þi
2
X
t
t
b0 b1
X
2
X
yrt þ 2a1
2
X
t
xrt yrt þ 2b1
2b0
X
xrt yit
xrt
X
t
t
þ i 2a1 b0
X
t
þ
b20
t
(4.51)
X
a21 þ i2
X
a0 a 1 ;
(4.52)
t
t
b21 þ i2b0 b1 Þðx2rt
ðb2o
x2it
4b0 b1
t
X
x2it þ i2xrt xit Þ
xrt xit
t
b21
xrt xit
t
X
#
xrt xit ;
t
(4.53)
ða0 þ ia1 Þðyrt þ iyit Þ ¼
X
t
"
yit þ i
2a1
X
yrt
X
2a0
yit ;
t
t
(4.54)
#
ðb0 þ ib1 Þðyrt þ iyit Þðxrt þ ixit Þ ¼
xrt yit þ 2b0
2b1
X
t
xrt yrt
X
xit yit þ 2b1
2b0
X
t
t
X
xit yrt
xit yrt þ 2b1
X
(4.55)
t
#
xit yit ;
t
ða0 þ ia1 Þðb0 þ ib1 Þðxrt þ ixit Þ
2a1 b1
X
xrt
t
X
yrt yit ;
X
X
t
ABXt ¼ 2
"
x2it
X
t
x2it þ b21
X
t
X
X
t
X
¼ 2a0 b0
t
t
BYt Xt ¼
t
"
x2rt
AYt ¼
t
t
X
b20
t
t
X
x2rt
t
þ i2 b0 b1
2
X
y2it þ i2
t
ðb0 þ ib1 Þ2 ðxrt þ ixit Þ2 ¼
x2rt
"
t
X
xrt þ 2a0 b1
2a1 b0
X
xit
2a0 b1
t
X
t
xrt þ 2a0 b0
X
xit
(4.56)
t
X
t
xit
2a1 b1
X
t
#
xit :
Substituting various values of coefficients of the complex-valued model into the
complex variance (4.50), we obtain different values of complex variances. Since it
is impossible to compare complex numbers with each other, it is impossible to
suggest a combination of coefficients for which the complex variance is at a
minimum. There is no concept of a minimum of a complex-valued function;
therefore, there is no concept of a complex variance minimum.
One can find the minimum of the real part of a complex variance. One can find
the minimum of its imaginary part. However, one cannot find the minimum of a
complex variance. As a matter of fact, the situation is simpler since we are dealing
4.5 Least-Squares Method from the Point of View of the New Axiomatic Theory
105
with a complex-valued function of the dependence of the complex variance on the
values of complex coefficients.
Let us first see what the application of the first criterion – the minimum of the
real part of a complex variance – will lead to. For that we should calculate the first
derivative of the real part of complex variance (4.50) and set it equal to zero.
Let us calculate the first derivatives of this part for each of the coefficients a0, a1,
b0, b1. Using Riemann–Cauchy formulae it is easy to calculate partial derivatives
for a0, then for a1, b0, and b1:
@Reðf ðzÞÞ
¼ 2 a0 n
@a0
@Reðf ðzÞÞ
¼2
@a1
X
t
yrt þ b0
X
a1 n þ
yit
X
@Reðf ðzÞÞ
¼2
@b1
¼
a0
X
xit
t
b1
X
xrt
b0
t
xit ;
X
t
X
t
xit ;
xrt yrt þ
X
xit yit
t
t
X
t
a1
X
t
X
X
X
@Reðf ðzÞÞ
xrt xit
x2it 2b1
x2rt b0
¼ 2 b0
@b0
t
t
t
X
X
þ a0
xrt a1
xit ;
t
b1
t
b1
t
xrt
X
t
x2rt þ b1
xrt ;
X
x2it
2b0
X
t
t
xrt xit þ
X
t
xrt yit þ
X
xit yrt
t
where n is the number of observations, t ¼ 1,2,3, . . ., n. If we set each of the partial
derivatives equal to zero and group them, we get the following system of equations:
X
X
8X
yrt ¼ na0 þ b0
xrt b1
xit ;
>
>
>
>
t
t
t
>
>
X
X
X
>
>
>
yit ¼ na1 þ b1
xrt þ b0
xit ;
>
<
t
t
t
X
X
X
X
X
>
ðx2rt
xit þ b0
xrt a1
xit yit ¼ a0
xrt yrt
>
>
>
>
t
t
t
t
t
>
>
X
X
X
X
X
>
>
>
ðx2rt
x
þ
b
x
þ
a
x
y
¼
a
x
y
þ
rt
1
it
1
it rt
0
rt it
:
t
t
t
t
t
x2it Þ
2b1
x2it Þ þ 2b0
X
xrt xit ;
t
X
xrt xit :
t
(4.57)
This system can easily be solved since we have four equations with four
unknowns.
106
4 Principles of Complex-Valued Econometrics
Now let us study another criterion and minimize the imaginary part of the
complex variance as a complex-valued function of complex coefficients. We will
get four levels corresponding to four partial derivatives:
@Imðf ðzÞÞ
¼ 2 a1 n
@a0
X
yit þ b1
X
xrt þ b0
X
xit ;
@Imðf ðzÞÞ
¼ 2 a0 n
@a1
X
yrt þ b0
X
xrt
X
xit ;
t
t
X
@Imðf ðzÞÞ
x2rt
¼ 2 b1
@b0
t
X
t
xit yrt þ a1
X
t
X
@Imðf ðzÞÞ
¼ 2 b0
x2rt
@b1
t
þ a0
X
xrt
a1
X
t
t
b1
t
xrt þ a0
b0
X
X
X
t
x2it
xit :
b1
t
x2it þ 2b0
xit ;
2b1
t
t
X
t
t
X
xrt xit
t
xrt yit
t
t
xrt xit
X
X
t
xrt yrt þ
X
xit yit
t
When we set these partial derivatives equal to zero and group them, we obtain
X
X
8X
yit ¼ na1 þ b1
xrt þ b0
xit ;
>
>
>
>
t
t
t
>
>
X
X
X
>
>
>
yrt ¼ na0 þ b0
xrt b1
xit ;
>
<
t
t
t
X
X
X
X
X
>
xrt yit þ
xit yrt ¼ a0
xit þ a1
xrt þ b1
ðx2rt
>
>
>
>
t
t
t
t
t
>
>
X
X
X
X
X
>
>
>
xrt yrt
xit yit ¼ a0
xrt a1
xit þ b0
ðx2rt
:
t
t
t
t
t
x2it Þ þ 2b0
x2it Þ
2b1
X
xrt xit ;
t
X
xrt xit :
t
(4.58)
It is clear that we have the same system as (4.57), only the sequence of the
equations has changed in accordance with the order of calculation of the first
derivatives with respect to the imaginary part of the complex-valued function of
the variance (4.50).
It turns out that any of the criteria (of the minimum of the complex-valued
function with respect to its real and imaginary parts) gives us the same result. It is
this conclusion that follows from the Riemann–Cauchy rule (in some works on the
TFCV it is called the d’Alembert-Euler rule).
4.5 Least-Squares Method from the Point of View of the New Axiomatic Theory
107
Since in the first and second cases we used the criterion of minimization of the
sum of the squares of the deviations, the result obtained can be called a complex
least-squares method. The minimum of complex variance (4.50) for the complex
coefficients of complex-valued model (4.49) agrees with the LSM criterion
@f ðzÞ
¼ 0;
@ða0 þ ia1 Þ
f ðzÞ ! min $
@f ðzÞ
>
>
:
¼ 0:
@ðb0 þ ib1 Þ
8
>
>
<
Similarly to (4.19), let us make the obtained system of equations convenient for
practical application in software products that allow for direct operations with
complex variables. It is easy to see that system (4.18) is equal to the system of
two complex equations with two complex coefficients:
X
8X
ðy
þ
iy
Þ
¼
ða
þ
ia
Þn
þ
ðb
þ
ib
Þ
ðxrt þ ixit Þ;
>
rt
it
0
1
0
1
>
<X
X
ðyrt þ iyit Þðxrt þ ixit Þ ¼ ða0 þ ia1 Þ
ðxrt þ ixit Þ
>
X
>
:
þ ðb0 þ ib1 Þ
ðxrt þ ixit Þðxrt þ ixit Þ:
If we now compare the obtained system with the similar system obtained in
Sect. 4.2, we will easily see the difference between them – in the second equation of
the obtained system there is no multiplication by a conjugate variable, as there is in
the standard approach:
X
8X
ðyrt þ iyit Þ ¼ ða0 þ ia1 Þn þ ðb0 þ ib1 Þ
ðxrt þ ixit Þ;
>
>
<X
X
ðyrt þ iyit Þðxrt ixit Þ ¼ ða0 þ ia1 Þ
ðxrt ixit Þ
>
X
>
:
þ ðb0 þ ib1 Þ
ðxrt þ ixit Þðxrt ixit Þ:
Since the use of complex variables gives the researcher more diverse modeling
options than real-variable models, the family of linear complex-valued models is
not limited only to model (4.49). Options are possible when only a real coefficient
or only an imaginary one is used instead of a complex coefficient, or perhaps a real
argument instead of a complex argument, or vice versa – the model of a complex
argument describes the behavior of a real variable. This diversity was used in the
second and third chapters of the present study.
Let us show how this approach can be used to implement a complex LSM for a
model of a complex argument:
yt ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þðxrt þ ixit Þ:
(4.59)
A complex-valued function whose minimum agrees with LSM evaluations of
coefficients of a linear model of a complex argument will be written as follows:
108
4 Principles of Complex-Valued Econometrics
f ðzÞ ¼
¼
X
t
X
t
½yt
ða0 þ ia1 Þ
½yt
A
ðb0 þ ib1 Þðxrt þ ixit Þ
2
BXt ¼
X
t
2
ðy2t þA2 þ B2 Xt2
2Byt Xt þ 2ABXt ÞÞ:
2Ayt
Let us consider each of the terms of the right-hand side of the last equality
separately, except for the square of the real variable y2t since this term has already
been presented in a form convenient for the calculation of derivatives:
X
t
X
t
¼
"
X
B2 Xt2 ¼
b20
X
A2 ¼
t
x2rt
t
ða0 þ ia1 Þ2 ¼
b21
X
x2rt
X
b20
x2rt
b0 b1
X
x2it
þ b20
t
t
2
X
t
2
X
Byt Xt ¼
t
2b0
X
t
2
X
t
2a0 b0
X
xrt
t
"
i 2a1 b0
2
t
X
t
t
X
ðb2o
t
X
x2it
(4.60)
b21 þ i2b0 b1 Þðx2rt
x2it þ i2xrt xit Þ
t
b21
X
X
a0 y t
xrt xit þ
#
xrt xit ;
t
2
t
X
t
xrt xit
X
a0 a 1 ;
4b0 b1
t
2ia1
X
t
t
"
xit yt þi
yt Þ;
(4.62)
X
2b1
xrt yt
2b0
X
#
t
t
(4.63)
xit yt ;
ða0 þ ia1 Þðb0 þ ib1 Þðxrt þ ixit Þ ¼
2a1 b1
X
xrt
2a1 b0
t
X
t
a21 þ i2
ðb0 þ ib1 Þyt ðxrt þ ixit Þ ¼
X
t
X
a20
t
þ b21
X
Ayt ¼
X
xrt yt þ 2b1
ABXt ¼ 2
x2it
t
t
X
X
ðb0 þ ib1 Þ2 ðxrt þ ixit Þ2 ¼
t
i2 b0 b1
X
xrt þ 2a0 b1
X
xit
2a0 b1
t
X
t
xrt þ 2a0 b0
X
t
X
xit
2a1 b1
t
xit þ
X
t
(4.64)
#
xit :
Let us use the obtained components to calculate the first partial derivatives of the
real part of complex-valued function (4.59) for each of the coefficients a0, a1, b0, b1.
Then we obtain four levels corresponding to four partial derivatives and four
variables of the problem under consideration:
@Reðf ðzÞÞ
¼ 2 a0 n
@a0
X
t
yt þ b0
X
t
xrt
b1
X
t
xit ;
4.5 Least-Squares Method from the Point of View of the New Axiomatic Theory
@Reðf ðzÞÞ
¼2
@a1
X
@Reðf ðzÞÞ
¼ 2 b0
x2rt
@b0
t
b0
X
a1 n
b1
X
xrt
X
b0
x2it
2b1
t
X
X
xrt xit
t
t
xit ;
t
t
109
xrt yt þ a0
X
X
a1
xit ;
xrt
t
t
X
X
X
@Reðf ðzÞÞ
¼ 2ð b1
x2rt þ b1
x2it 2b0
xrt xit
@b1
t
t
t
X
X
X
xrt Þ:
xit a1
xit yt a0
þ
t
t
t
If we set each of the derivatives equal to zero and group them, we get
X
X
8X
y
¼
na
þ
b
x
b
xit ;
>
t
0
0
rt
1
>
>
>
t
t
t
>
>
X
X
>
>
> 0 ¼ na1 þ b1
xrt þ b0
xit ;
>
<
t
t
X
X
X
X
>
xit yt ¼ a0
xit þ a1
xrt þ b1
ðx2rt
>
>
>
>
t
t
t
t
>
>
X
X
X
>X
>
>
x
y
¼
a
x
a
x
þ
b
ðx2rt
rt t
0
rt
1
it
0
:
t
t
t
t
x2it Þ þ 2b0
x2it Þ
2b1
X
xrt xit ;
(4.65)
t
X
xrt xit :
t
Similar equalities can be obtained by finding partial derivatives of the imaginary
part of complex-valued function (4.64), setting them equal to zero, and using other
variants resulting from the d’Alembert–Euler condition.
If now we compare system (4.65) with system (5.4.19) we may be certain that
system (4.65) is easily obtained from (4.57) if we substitute yit ¼ 0 into the latter.
In a whole series of cases, instead of linear model (4.58) we can use its simpler
analog – a linear model of a complex argument without a free term:
yt ¼ ðb0 þ ib1 Þðxrt þ ixit Þ:
(4.66)
With reference to this case, a complex-valued function LSM will look as
follows:
X
f ðzÞ ¼
yt
t
b0 þ ib1 Þðxrt þ ixit Þ
2
¼
X
t
ðy2t
2Byt Xt þ
B2 Xt2
:
(4.67)
Let us use (4.61) and (4.63) to find partial derivatives of the real part of this
function for each of the coefficients:
110
4 Principles of Complex-Valued Econometrics
X
@Reðf ðzÞÞ
¼ 2 b0
x2rt
@b0
t
@Reðf ðzÞÞ
¼2
@b1
b1
X
b0
x2it
2b1
t
x2rt
t
X
þ b1
X
xrt xit
t
X
x2it
2b0
t
X
xrt yt ;
t
X
t
xrt xit þ
X
t
xit yt ;
where the LSM system is to be written in the following form:
8X
X
>
ðx2rt
x
y
¼b
rt
t
0
>
<
t
t
X
X
>
ðx2rt
x
y
¼
b
>
it
t
1
:
x2it Þ
X
xrt xit ;
t
x2it Þ þ 2b0
t
t
2b1
X
(4.68)
xrt xit :
t
If we compare this system of normal equations with system (4.57) we see that
(4.68) can be obtained without calculating derivatives but setting the components
missing in (4.66) equal to zero.
Let us now show the interrelations between the LSM problems for real and
complex variables.
As is known, for a simple linear monofactor model of real variables
y ¼ a þ bx;
the system of normal LSM equations has the following form:
8X
X
>
yt ¼ an þ b
xt ;
>
<
t
t
X
X
X
>
yt xt ¼ a
xt þ b
x2t :
>
:
t
t
(4.69)
t
If now we substitute complex variables and complex coefficients into this system
of normal equations instead of real variables and coefficients, then for a linear
complex-valued monofactor function we get the same system of normal equations
(4.57) as previously. In fact, the first equation of system (4.69), if we substitute
complex variables and complex coefficients into it, will have the form
X
t
yrt þ i
X
t
yit ¼ na0 þ b0
X
t
xrt
b1
X
t
X
X
xit :
xrt þ b0
xit þ i na1 þ b1
t
t
Dividing the real and imaginary parts of the equality we get the first part of LSM
system (4.57):
8X
X
X
>
y
¼
na
þ
b
x
b
xit ;
rt
0
0
rt
1
>
<
t
t
t
X
X
X
>
xit :
x
þ
b
y
¼
na
þ
b
>
rt
0
it
1
1
:
t
t
t
(4.70)
4.5 Least-Squares Method from the Point of View of the New Axiomatic Theory
111
If we substitute complex variables and complex coefficients into the second
equation of system (4.69), it will be a more complicated equation, which is why we
divide it into separate components. Following substitution the left-hand side of the
equality will have the form
X
t
X
ðyrt þ iyit Þðxrt þ ixit Þ ¼
yrt xrt
t
X
t
yit xit þ i
X
t
yit xrt þ
yrt xir :
X
t
The first term of the right-hand
side of the second
(4.69) will have the form
P
P
P
P
a0 xrt a1 xir þ i a1 xrt þ a0 xit , and the second will look like this:
t
b0
X
t
t
x2rt
X
t
x2it
t
t
X
X
X
X
2
2
xrt xit þ i b1
xrt
xit þ 2b0
xrt xit :
2b1
t
t
t
t
If we group the real components into one equality and the imaginary ones into
another, we get two more equations:
8X
X
X
X
X
>
ðx2rt
x
þ
b
x
a
x
y
¼
a
x
y
it
0
rt
1
it
it
0
rt
rt
>
<
t
t
t
t
t
X
X
X
X
X
>
ðx2rt
xrt þ b1
xit þ a1
xit yrt ¼ a0
xrt yit þ
>
:
t
t
t
t
t
x2it Þ
2b1
x2it Þ þ 2b0
X
xrt xit ;
t
X
xrt xit :
t
(4.71)
Combining the systems of equations (4.70) and (4.71) into one system it is easy
to see that we have a system of normal LSM equations for linear functions of
complex variables (4.57).
The direct parallels between the method suggested in this section and LSM
applied to models of real variables constitute one of the arguments in favor of the
method under consideration, but not in favor of the one that follows from the
standard formulation of the problem adopted in mathematical statistics.
Since the solution of a system of four equations with four variables is not the
most enjoyable problem, the procedure for assessing the coefficients of linear
complex-valued function (4.49) by means of LSM can and should be simplified.
To do this, after preliminary centering of the initial variables of the problem with
respect to their averages:
0
yr ¼ yr
0
yr ; yi ¼ yi
0
yi ; xr ¼ xr
0
xi ; xi ¼ xi
xi ;
let us simplify model (4.49):
y0rt þ iy0it ¼ ðb0 þ ib1 Þðx0rt þ ix0it Þ:
(4.72)
112
4 Principles of Complex-Valued Econometrics
To find the coefficients, it is sufficient to solve the following system of two
equations with two variables:
X
X
y0 rt x0 rt
y0it x0 it ¼ b0
ðx0rt 2
X
X
X
y0it x0 rt þ
ðx0rt 2
y0rt x0 it ¼ b1
(X
x0it 2 Þ
2b1
x0it 2 Þ þ 2b0
or, for work with complex variables,
X
X
P
ðyrt þ iyit Þðxrt þ ixit Þ
:
b0 þ ib1 ¼ P
ðxrt þ ixit Þðxrt þ ixit Þ
x0 rt x0 it ;
x0 rt x0 it ;
(4.73)
(4.74)
Let us show the applicability of LSM to an example whose initial data are given
in Table 4.1. The numerator of the preceding fraction for finding the complex
proportionality coefficient is equal to
163; 905 þ i31; 048 and the denominator is equal to
129; 681
i6; 350:
The complex proportionality coefficient will be equal to
b0 þ ib1 ¼ 1,249
i0,301:
It should be noted that the standard approach used in Sect. 4.2 provided other
values of the complex proportionality coefficient:
b0 þ ib1 ¼ 1,257
i0,295
Clearly, the differences are not as great as might be expected, but they do exist,
since the minimization criteria are different.
4.6
Complex Pair Correlation Coefficient
An attempt to calculate the coefficient of a pair correlation between two random
complex variables based on the standard axiomatic core of the theory of mathematical statistics of complex random variables showed contradictory results and
became the basis for the creation of a new axiomatic core of the theory of
complex-valued econometrics.
It was shown in Sect. 4.3 how to derive a formula for calculating a pair
correlation coefficient using two methods. First we used a well-known formula in
mathematical statistics:
4.6 Complex Pair Correlation Coefficient
rXY ¼
113
mXY
:
sX sY
(4.75)
Let us substitute into this formula values of the complex correlation momentum
and complex variances from Sect. 4.4 that result from the new axiomatic theory. To
simplify the form, let us assume that all the initial variables are centered with
respect to their variables:
P
ðyrt þ iyit Þðxrt þ ixit Þ
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
P
P
ðxrt þ ixit Þ2 ðyrt þ iyit Þ2
rXY
(4.76)
Expanding the numerator and grouping the real and imaginary parts we obtain
X
ðyrt xrt
yit xit Þ þ i
X
ðxrt yit þ yrt xit Þ:
(4.77)
Now in the denominator, let us write the expression under the summation
symbol to the second power and group the real and imaginary parts:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
X
X
2X
ðxrt þ ixit Þ
ðx2rt x2it þ i2xrt xit Þ
ðyrt þ iyit Þ2 ¼
ðy2rt y2it þ i2yrt yit Þ:
(4.78)
Substituting the numerator and denominator into the initial formula (4.76) we
obtain a formula for a complex pair correlation coefficient that may be used in the
absence of a way to work with complex numbers:
rXY
P
P
ðyrt xrt yit xit Þ þ i ðxrt yit þ yrt xit Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi:
¼ pP
P
ðx2rt x2it þ i2xrt xit Þ ðy2rt y2it þ i2yrt yit Þ
(4.79)
It is easy to determine the differences of the obtained formula from the one
following from the standard formulation of problem (4.28) if we write this formula
here:
rXY ¼
P
t
P
ðyrt yit þ xrt xit Þ þ ið ðxrt yit yrt xit ÞÞ
t
ffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
P
P
2
2
ðxrt þ xit Þ ðyrt 2 þ yit 2 Þ
t
(4.80)
t
Both numerators and denominators differ from each other.
Let us now derive a formula for calculating the pair correlation coefficient of
complex variables via the geometric mean of the product of complex coefficients of
regressions (4.32).
With respect to the complex regression coefficient of X to Y designated as a, we
obtain
114
4 Principles of Complex-Valued Econometrics
P
ðyrt þ iyit Þðxrt þ ixit Þ
a¼P
:
ðxrt þ ixit Þðxrt þ ixit Þ
(4.81)
Now let us consider a regression inverse to the given one, i.e., a complex
regression of Y to X:
X ¼ X0 þ bY;
(4.82)
where X0 and b are complex coefficients of the equation of regression.
The complex proportionality coefficient b may also be found by means of
complex LSM:
P
ðyrt þ iyit Þðxrt þ ixit Þ
:
b¼P
ðyrt þ iyit Þðyrt þ iyit Þ
(4.83)
Since the pair correlation coefficient represents the geometric mean of regression coefficients (4.32), let us find the geometric mean of regression coefficients
(4.81) and (4.83):
rXY ¼
P
pffiffiffiffiffiffiffiffiffi
ðyrt þ iyit Þðxrt þ ixit Þ
:
a1 b1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
P
ðxrt þ ixit Þ2 ðyrt þ iyit Þ2
(4.84)
It is evident that we get the same formula as (4.76) derived via the complex
correlation momentum and complex variance. It may be transformed into (4.79) if
the economist has no way of working with complex numbers and must work with
real numbers only.
It is crucial that the same result was obtained using two different approaches to
the derivation of a formula for calculating a complex pair correlation coefficient
between two random complex variables, in contrast to attempts made for the
standard formulation of the problem.
Neither of the paradoxes found in Sect. 4.3 has been revealed for the new
axiomatic core of the theory of complex-valued econometrics. This is why subsequently we will just on this axiomatic core – all the measures of variability of
random complex variables are complex, showing the complex nature of the process.
4.7
Interpretation of Values of Complex Pair
Correlation Coefficient
Having determined a method for correctly calculating a complex pair correlation
coefficient, we should provide an interpretation of the values it can take. A linear
relationship between two complex variables in the domain of real variables means
4.7 Interpretation of Values of Complex Pair Correlation Coefficient
115
that both the real and imaginary parts of one complex variable function as twofactor linear dependencies on the real and imaginary parts of another complex
variable. This is why if one variable varies nonlinearly, then the other will do the
same, and it will be difficult to determine this dependence visually. If the dependence under consideration is not functional but regression, the scattering of the
points in the complex planes is to an even lesser extent associated with a linear
dependence. This is why a visual analysis of the dependence between variables is
hard, and one can judge the linear relationship of two complex variables only by the
calculated characteristics, primarily the complex pair correlation coefficient.
As follows from the discussion of the previous section, a complex pair correlation coefficient represents the geometric mean of two complex coefficients of
regression:
pffiffiffiffiffi
rXY ¼ ab:
(4.85)
This is why, for a strictly functional linear complex-valued relationship
Y ¼ aX; X ¼ bY;
the following equality will be evident:
1
a¼ ;
b
(4.86)
where
1
ab ¼ b ¼ 1:
b
That is, a complex pair correlation coefficient for a linear functional relationship
is
rXY ¼ ð1 þ i0Þ:
(4.87)
This means that for a linear functional relationship between two complex
variables the absolute value of the real part of the complex pair correlation coefficient is equal to one, and its imaginary component is equal to zero.
Therefore, the square of a complex pair correlation coefficient (complex coefficient of determination) for a linear relationship will always be equal to a real unit.
But in what cases of a linear functional relationship between two complex variables
will the square root of the determination coefficient take values of “plus one” and in
what cases “minus one”?
To answer this question, let us present the complex proportionality coefficients
in arithmetic and exponential forms:
116
4 Principles of Complex-Valued Econometrics
a ¼ a0 þ ia1 ¼ aeia ;
(4.88)
b ¼ b0 þ ib1 ¼ beib :
(4.89)
ab ¼ ra rb eiðaþbÞ ¼ ra rb cosða þ bÞ þ ira rb sinða þ bÞ:
(4.90)
Their product will be
Since for a linear functional relationship we have (4.87), i.e., the imaginary part
of the complex pair correlation coefficient is equal to zero,
a þ b ¼ 2pk; ab ¼ 1:
(4.91)
Let us consider a simple case where k ¼ 0. Then the complex pair correlation
coefficient is determined as the square root of
rXY ¼
pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ab ¼ ra rb cosða þ bÞ:
(4.92)
Since the absolute value of each proportionality coefficient is positive by default,
whether the pair correlation coefficient is equal to “plus one” or “minus one” will be
determined by the cosine of a.
We are interested in the case where the radicand may be
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð 1Þð 1Þ:
(4.93)
Then it is possible to determine the characteristics of the complex
proportionality coefficient. Based on equality (4.91), the cosine of radicand (4.92)
may be written as follows:
cosða þ ð aÞÞ ¼ cos a cosð aÞ
sin a sinð aÞ:
Then it is easy to determine that the case we are interested in, (4.93), is
determined by a polar angle of the complex proportionality coefficient a0 þ ia1
lying within the range
3
5
p a p:
4
4
(4.94)
For this case the real component of the complex proportionality coefficient is
always nonpositive:
a0 0;
and its imaginary part is always not less than the real one:
(4.95)
4.7 Interpretation of Values of Complex Pair Correlation Coefficient
117
a0 a1 :
(4.96)
These conditions are valid for the following coefficients:
1 þ i; 1 þ 0i; 10
i9; 999 . . .
What does a linear complex-valued relationship with such a complex pair
correlation coefficient mean? To answer this question, let us present a linear
functional complex-valued relationship as a system of equalities of real and imaginary parts:
y r ¼ a0 x r
a1 x i ;
(4.97)
yi ¼ a1 xr þ a0 xi :
(4.98)
According to conditions (4.95) and (4.96), the coefficient a0 is always
nonpositive, and the imaginary part a1 can take both positive (1) and negative (2)
values. Considering variations of the complex argument in the first quadrant of the
complex plane, we obtain that with the simultaneous growth of the real and
imaginary parts of the argument, the real part of the complex result Yr decreases,
and the imaginary part Yi can both increase and decrease due to (4.96).
Thus, the real part of the complex pair correlation coefficient rr testifies to the
degree of approximation of the dependence of two random complex variables to a
linear dependence, and the interpretation of its values is similar to that of the values
of the pair correlation coefficient in the domain of real numbers.
Now it is necessary to determine the meaning of the imaginary component of the
complex pair correlation coefficient ri. It definitely follows from (4.87) that the
imaginary component will be equal to zero only if there is a linear functional
relationship between two complex variables. In all other cases it will not be equal
to zero.
The case where
rXY ¼ ð0 þ iÞ
(4.99)
is the extreme manifestation of this component of the complex pair correlation
coefficient.
It follows then that
ab ¼ ra rb eiðaþbÞ ¼ ra rb cosða þ bÞ þ ira rb sinða þ bÞ ¼
1 þ i0:
(4.100)
This means that
a þ b ¼ ð2k
1Þp; ra rb ¼ 1:
(4.101)
118
4 Principles of Complex-Valued Econometrics
What does this equality mean? If we consider the situation where k ¼ 1, then the
complex proportionality coefficient a0 þ ia1 ¼ 1 þ i0 should be associated with
coefficient assessments such as b0 þ ib1 ¼ 1 þ i0 , so that the complex pair
correlation coefficient could take values (4.99) by the LSM as applied to the inverse
dependence of the argument on the result. Or the coefficient b0 þ ib1 ¼ 1 i
should correspond to the coefficient a0 þ ia1 ¼ 1 þ i, a vector whose direction is
opposite to that of the first one in the complex plane.
It is clear now in what case the real part of the complex pair correlation
coefficient will be equal to zero and the absolute value of its imaginary part will
be equal to one. If we find the regression of the complex argument to the complex
result
yr þ iyi ¼ ða0 þ ia1 Þðxr þ ixi Þ;
then the proportionality coefficient a0 þ ia1, found by LSM, will model some linear
sequence y^r þ i^
yi . If we find the inverse regression of the complex result to the
complex argument
xr þ ixi ¼ ðb0 þ ib1 Þðyr þ iyi Þ;
then LSM will give a complex coefficient b0 þ ib1 such that its application to the
regression
yr þ iyi ¼
xr þ ixi
b0 þ ib1
will model a series of points y^0 r þ i^
y0 i , turned with respect to the initial series y^r
þi^
yi at an angle p, i.e., backward.
This is possible if there is no linear relationship whatsoever.
To understand the essence of the intermediate values (from zero to one by a
absolute value) of the imaginary component of the complex pair correlation coefficient, one can use the results of empirical studies. The main idea was as follows.
A linear functional relationship between two complex variables was plotted. The
real part of the complex pair correlation coefficient was one, and the imaginary one
– zero.
The error variance of this linear relationship, determined in fractions of the
modeled value Y, was subsequently increased. In other words, the functional
relationship was replaced by a regression complex-valued one with an increasing
variance. The higher the variance of the approximation error between the calculated
and actual values of complex Y, the higher the imaginary component of the complex
pair correlation coefficient and the lower its real part. This means that the growth of
the imaginary component indicates a growth in the variance of the approximation of
a random variable by a linear complex-valued model.
Since the coefficient itself characterizes the degree of approximation of the
relationship under study to the linear complex-valued function, in practice we
4.8 Assessments of Parameters of Nonlinear Econometric Models of Complex Variables
119
should stick to the following rule. If the absolute value of the real part of a complex
pair correlation coefficient is close to one and the imaginary part is close to zero,
then one can safely use a linear complex-valued regression relationship. If the
absolute value of the imaginary part is far from zero, a linear relationship will
give a high variance.
If the absolute value of a complex pair correlation coefficient is lower than 0.8 and
the imaginary part is too low, then one should choose a nonlinear complex-valued
regression model since the linear model will describe the relationship badly. A low
value of the absolute value of the imaginary part shows that there is some nonlinear
regression relationship between the variables. However, if the imaginary part of the
complex pair correlation coefficient is rather high, for example, greater than 0.5, then
the existence of any relationship between random complex variables is quite doubtful.
4.8
Assessments of Parameters of Nonlinear Econometric
Models of Complex Variables
The general principles of LSM, determined in Sect. 4.5 of this chapter on econometric complex-valued models, when applied to problems related to the assessment
of parameters of econometric models, require individual consideration with respect
to each complex-variable model. LSM adapted for a simple linear model of
complex variables gives rise to general principles applicable to linear complexvalued econometric models. This could underlie an approach that would make it
possible to use LSM for the assessment of selected coefficients of nonlinear
econometric models of complex variables if they are represented in a linear form.
In the general case, in the domain of real variables, the procedure for assessing
the coefficients of nonlinear models is not easy. For additive nonlinear models the
creation of a system of normal equations is not difficult, for example, for a nonlinear
additive model like
y ¼ a1 x2 þ a2 ln x þ a3 cos x:
The system of normal equations by LSM will have the form
X
X
X
8X 2
x 4 þ a2
x2 ln x þ a3
x2 cos x;
yx ¼ a1
>
>
<X
X
X
X
ln x cos x;
ln2 x þ a3
x2 ln x þ a2
y ln x ¼ a1
>
X
X
X
>
:X
y cos x ¼ a1
x2 cos x þ a2
ln x cos x þ a3
cos2 x:
Assessments of coefficients of this model will possess all the remarkable
properties of LSM assessments. However, it is very hard to use LSM for assessing
the parameters of a simple power model.
Thus, to assess the parameters of a simple power model
120
4 Principles of Complex-Valued Econometrics
yt ¼ a0 xat 1
using LSM, we should solve the system of nonlinear equations
a0 xat 1 Þ2
¼ 0;
@a0
a1 2
>
>
: @ðYt a0 xt Þ ¼ 0:
@a1
8
@ðYt
>
>
<
Calculating the derivatives we get
8X
X
a 1 xt
>
y
e
a
e2a1 xt ¼ 0;
t
0
>
<
t
t
X
X
>
yt xt ea1 xt a20
xt e2a1 xt ¼ 0:
>
: a0
t
t
Although nowadays it is not hard at all to solve such a system of two nonlinear
equations – it is rather easy using numerical methods – when econometrics was first
created, it was practically unsolvable. Nowadays many practicing economists who do
not have proper expertise in the use of mathematical methods find it difficult to solve
similar problems, unless, of course, they have the software applications to do it.
It is for this precise reason that another method was proposed to solve the
problem set – the linearization of the initial nonlinear model. It was less precise
but much simpler. For the power function under consideration, we take a logarithm
of the left- and right-hand sides of the initial equality to any base and obtain a linear
model (here we do it to the natural base):
0
0
0
ln yt ¼ ln a0 þ a1 ln xt , yt ¼ a0 þ a1 xt :
The coefficients of this linear model can be found easily using LSM. After
assessing the parameters of this model it is easy to return to the power model –
0
find the coefficient a0 using the known value of a0 :
0
a0 ¼ ea0 :
It has long been known that assessments of the parameters of an original power
model found in this way will be shifted, as the minimized squared deviations are not
of the power function but of its linear analog. However, in the vast majority of cases
this is not a big problem since models in which the parameters were found by a
similar method describe the original series quite well.
It is evident that if we use LSM to assess parameters of econometric models of
real variables, direct application of this method will create certain computational
burdens connected with the need to solve systems of nonlinear equations, whereas
4.8 Assessments of Parameters of Nonlinear Econometric Models of Complex Variables
121
for the case of nonlinear econometric models of complex variables the situation is
exacerbated due to the properties of nonlinear complex-valued functions considered in Chaps. 2 and 3.
This is why, if we want to apply LSM in the econometrics of complex variables,
we should develop practical techniques for assessing the parameters of each of the
previously considered econometric models of complex variables using an approach
similar to that used in the econometrics of real variables.
Let us consider this possibility and sequentially examine the functions of
complex variables according to the same procedure that was used to study the
conformal mapping of these functions in Chap. 3, bearing in mind that the complex
models of a real argument and models of a complex argument are a special case.
A complex-valued power function with real coefficients will be the first to
consider. It has the following form:
yrt þ iyit ¼ a0 ðxrt þ ixit Þb0 :
(4.102)
Before we show one way of applying LSM to this function, one interesting
feature of this and similar models should be noted. If we present each of the
complex variables in exponential form and substitute them into model (4.102),
we obtain
Ryt eiyyt ¼ a0 Rbxt0 eib0 yxt ;
(4.103)
where
Ryt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
yit
xit
y2rt þ y2it ; Rxt ¼ x2rt þ x2it ; yyt ¼ arctg ; yxt ¼ arctg :
yrt
xrt
Two complex variables presented in exponential form are equal to each other
only if their moduli and arguments are. This is why
Ryt ¼ a0 Rbxt0 ; and yyt ¼ b0 yxt :
Therefore, for each observation t one can find the value of the exponent b0:
b0t ¼
yit
yyt arctg yrt
¼
yxt arctg xxrtit
(4.104)
and of the proportionality coefficient a0:
a0t ¼
Ryt
b0t
Rxt
¼
Ryt
yyt
Ryxtxt
:
(4.105)
122
4 Principles of Complex-Valued Econometrics
Thus, to assess the values of the coefficients of a complex-valued power function, it is not necessary to use some set of observations – the coefficients are
assessed at each observation!
This possibility again demonstrates the difference in the properties of models of
complex variables from those of real ones – there is a unique possibility to assess
the coefficients of nonlinear models at each observation and give an economic
interpretation of the modeled process, not generally for a period but for each
observation, if only the coefficients carry economic meaning and the modeled
relationship is described by this very function.
This property makes it possible to analyze two series of values {a0t} and {b0t},
which in the case of reversible random process represent values of coefficients
selected from some universe of coefficients of model (4.102) describing the expectation of the relationship under study. Since econometrics a priori assumes that we
are dealing with a normal distribution of probabilities and the selected values of
coefficients a0 and b0 found according to (4.104) and (4.105) vary around some
expectations a0 and b0, the best assessment of the searched parameters of model
(4.102) will be their averages:
b0 ¼
T arctg yit
1X
yrt
T t¼1 arctg xxrtit
(4.106)
and
a0 ¼
T
1X
T t¼1
1
ðy2rt þ y2it Þ2
ðx2rt þ x2it Þ
y
arctgy it
1
rt
2arctg xit
xrt
:
(4.107)
For these assessments it is easy to calculate their variance and confidence limits,
which is why nonlinear model (4.102) seems to be quite convenient and easy for
modeling various economic processes.
Since this section aims at looking for approaches to using LSM in application to
this model, let us show its solution. This problem may occur if researchers must not
only calculate the parameters of an econometric model but build this model to
describe some average trend. Then LSM is essential. First of all, in this case the
original model should be linearized. Then we get
lnðyrt þ iyit Þ ¼ ln a0 þ b0 lnðxrt þ ixit Þ:
Since we are working with the principal values of logarithms, the expression
obtained may be represented as follows:
ln Ryt þ i’yt ¼ ln a0 þ b0 ðln Rxt þ i’xt Þ:
In the linear form the model has the form
4.8 Assessments of Parameters of Nonlinear Econometric Models of Complex Variables
123
ln Ryt þ i’yt ¼ ðln a0 þ i0Þ þ ðb0 þ i0Þðln Rxt þ i’xt Þ:
Taking into account this form and the previously obtained LSM system for linear
model (4.58) we can substitute these values into the LSM system (including zero
ones) and obtain
8X
X
>
ln Ryt ¼ n ln a0 þ b0
ln Rxt
>
<
t
t
X
X
:
(4.108)
>
’yt ¼ b0
’it
>
:
t
t
Hence
b0 ¼
1
ln a0 ¼
n
X
X
t
’yt
t
ln Ryt
X
’xt ;
(4.109)
t
P
t
P
t
’yt X
’xt
t
ln Rxt :
(4.110)
In the general case assessments (4.108) will differ from (4.106) and (4.107). In
the present work we are not going to compare them to each other or give
recommendations. We need only specify that use of LSM for a linearized model
will result in a shift of assessments (4.109) and (4.110) compared to (4.106) and
(4.107).
Now let us show how to find the parameters of a power model of complex
variables with complex coefficients that represents the most general form of power
functions of complex variables:
yrt þ iyit ¼ ða0 þ ia1 Þðxrt þ ixit Þðb0 þib1 Þ :
(4.111)
It will not be possible to find coefficients at each observation since the model has
four coefficients and not two as in model (4.102). It is appropriate to state that any
complex-valued function with two coefficients (either real or imaginary) may be
built only for one observation. The explanation for this is quite simple. Complexvalued functions are simply systems of two real functions, and if at some observation t economists have data for xrt, xit, yrt и yit, then they can substitute them into the
function, set the left- and right-hand sides of the equality equal to each other, and
obtain two equalities with two unknown coefficients, which can be calculated
immediately.
As for the power function, its coefficients are found at one observation for such
of its variants as a model with real coefficients (4.102) and such models as
yrt þ iyit ¼ a0 ðxrt þ ixit Þib1 ;
124
4 Principles of Complex-Valued Econometrics
yrt þ iyit ¼ ia1 ðxrt þ ixit Þb0 ;
yrt þ iyit ¼ ia1 ðxrt þ ixit Þib1 :
After obtaining the coefficients of these models for each observation t it is easy
to calculate their averages, variances, etc.
Let us return to the complex-valued power model with complex coefficients
(4.108) and present it in linear form taking natural logarithms of the left- and righthand sides of the equality:
lnðyrt þ iyit Þ ¼ lnða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðxrt þ ixit Þ:
(4.112)
As previously, we use the principal values of the logarithms. For the complex
argument the logarithm will have the following form:
lnðxrt þ ixit Þ ¼ ln Rx þ i’x ;
(4.113)
where Rx is the absolute value of the complex variable of the determinant and ’x is
its polar angle. For a complex variable of the modeled result the logarithm will be
written as follow:
lnðyrt þ iyit Þ ¼ ln Ry þ i’y ;
(4.114)
where Ry is the absolute value of the complex variable of the result and ’y is its
polar angle. For the complex proportionality coefficient of model (4.108) the
logarithm will be written as follows:
lnða0 þ ia1 Þ ¼ ln Ra þ i’a ;
(4.115)
where Ra is the absolute value of the complex proportionality coefficient and ’a is
its polar angle.
It is not necessary to write the complex coefficient (b0 + ib1) in exponential
form. With these designations we obtain the following linear model of the power
function of complex variables with complex coefficients:
ln Ry þ i’y ¼ ðln Ra þ i’a Þ þ ðb0 þ ib1 Þðln Rx þ i’x Þ:
(4.116)
To make subsequent operations simple, let us introduce the following
designations:
lnða0 þ ia1 Þ ¼ ln Ra þ i’a ¼ A0 þ iA1 :
(4.117)
It is clear that the model obtained is a linear model of complex variables with
complex coefficients:
ln Ry þ i’y ¼ ðA0 þ iA1 Þ þ ðb0 þ ib1 Þ ðln Rx þ i’x Þ:
4.8 Assessments of Parameters of Nonlinear Econometric Models of Complex Variables
125
The coefficients of this model can be found using LSM adapted to
linear complex-valued models. Substituting the values of model (4.116) into the
system of normal equations (4.58), taking into account the designations (4.117), we
have
8X
X
X
>
’xt ;
ln Rxt b1
ln Ryt ¼ TA0 þ b0
>
>
>
t
t
t
>
>
X
X
X
>
>
>
’yt ¼ TA1 þ b1
ln Rxt þb0
’xt ;
>
>
>
>
t
t
t
>
>
X
X
X
X
>
>
>
ln
R
ln
R
’
’
¼
A
ln
R
A
’xt
yt
xt
0
xt
1
>
yt
xt
<
t
t
t
t
X
X
2
2
>
’t ln Rt
Þ
2b
ðln
R
’
þb
>
xt
1
0
xt
>
>
>
t
t
>
>
X
X
X
X
>
>
>
ln Rxt
’xt þ A1
’xt ln Ryt ¼ A0
’yt ln Rxt þ
>
>
>
>
t
t
t
t
>
X
X
>
>
>
>
’t ln Rt :
ðln2 Rxt ’2xt Þ þ 2b0
: þ b1
(4.118)
t
t
By solving this system one can find the values of the unknown coefficients A0,A1,
b0, b1. The result will be as follows:
yrt þ iyit ¼ eA0 þiA1 ðxrt þ ixit Þðb0 þib1 Þ :
To obtain a model like (4.108) the proportionality coefficient should be taken out
of exponential form and rewritten in arithmetic form. To do this we should find a0
and a1 according the available values of A0 and A1:
a0 ¼ e cos A1 ;
a1 ¼ eA0 sin A1 :
One of the subforms of power functions of complex variables is a power function
of a complex argument. In the general case it may be represented as follows:
yrt ¼ ða0 þ ia1 Þðxrt þ ixit Þðb0 þib1 Þ :
(4.119)
Taking the natural logarithms of the left- and right-hand sides of the equality we
obtain the following linear model:
ln yrt ¼ lnða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðxrt þ ixit Þ:
(4.120)
Taking into account previously introduced designations we have
ln yrt ¼ ðln Ra þ i’a Þ þ ðb0 þ ib1 Þðln Rx þ i’x Þ
¼ A0 þ iA1 þ ðb0 þ ib1 Þðln Rx þ i’x Þ:
(4.121)
126
4 Principles of Complex-Valued Econometrics
Compared to the model of complex variables, this one does not have an
imaginary part in the left-hand side of the obtained equality. This is why the system
of normal equations for a power function of complex argument may be represented
as follows:
8X
X
X
>
ln yrt ¼ TA0 þ b0
ln Rxt b1
’xt ;
>
>
>
t
t
t
>
>
X
X
>
>
>
0 ¼ TA1 þ b1
ln Rxt þb0
’xt ;
>
>
>
>
t
t
>
>
X
X
X
X
>
>
>
ln
y
ln
R
¼
A
ln
R
A
’
þ
b
ðln2 Rxt
rt
xt
0
xt
1
0
>
xt
<
t
t
t
t
X
>
2b1
’xt ln Rxt
>
>
>
>
t
>
>
X
X
X
>
>
>
ln yrt ’xt ¼ A0
’xt þ A1
ln Rxt
>
>
>
>
t
t
t
>
X
X
>
>
>
>
’xt ln Rxt :
ðln2 Rxt ’2xt Þ2b1
: þb1
t
’2xt Þ
(4.122)
t
Solving the system we find LSM assessments for the power function of a
complex argument with complex coefficients.
Let us now consider the technique of assessing the parameters of a model of an
exponential function of complex variables, which can be written in the following
general form:
yrt þ iyit ¼ ða0 þ ia1 Þeðb0 þib1 Þðxrt þixit Þ :
(4.123)
Again, let us make this linear by taking logarithms of the left- and right-hand
sides of the equality:
lnðyrt þ iyit Þ ¼ lnða0 þ ia1 Þ þ ðb0 þ ib1 Þðxrt þ ixit Þ:
(4.124)
Taking into account the introduced designations (4.114) and (4.115):
ln Ryt þ i’yt ¼ A0 þ iA1 þ ðb0 þ ib1 Þðxrt þ ixit Þ:
(4.125)
Again, assessing the model parameters is not hard since the system of normal
equations has the following form:
4.8 Assessments of Parameters of Nonlinear Econometric Models of Complex Variables
X
X
8X
ln Ryt ¼ TA0 þ b0
xrt b1
xit ;
>
>
>
> t
t
t
>
>
X
X
X
>
>
>
xit ;
x
þb
’
¼
TA
þ
b
rt
0
1
1
>
yt
<
t
t
t
X
X
X
X
X
>
ðx2rt
xit þ b0
xrt A1
’yt xit ¼ A0
ln Ryt xrt
>
>
>
>
t
t
t
t
t
>
>
X
X
X
X
X
>
>
>
ln
R
x
þ
’
x
¼
A
x
þ
A
x
þ
b
ðx2rt
yt it
0
it
1
rt
1
:
yt rt
t
t
t
t
t
x2it Þ
2b1
X
127
xrt xit ;
t
x2it Þ þ 2b0
X
xrt xit :
t
(4.126)
Solving this system we find the parameters of the original model.
A similar method may be applied to make LSM assessments of the parameters of
a simpler model – an exponential function of a complex argument:
yrt ¼ ða0 þ ia1 Þeðb0 þib1 Þðxrt þixit Þ :
(4.127)
Using logarithms and particular designations we obtain the following linear
function:
ln yrt ¼ A0 þ iA1 þ ðb0 þ ib1 Þðxrt þ ixit Þ:
(4.128)
Then the system of normal equations for a linear exponential function of a
complex argument will look like this:
X
X
8X
ln yrt ¼ TA0 þ b0
xrt b1
xit ;
>
>
>
>
t
t
t
>
>
X
X
>
>
>
xit ;
xrt þb0
>
< 0 ¼ TA1 þ b1
t
t
X
X
X
X
>
ðx2rt
xit þ b0
xrt A1
ln yrt xrt ¼ A0
>
>
>
>
t
t
t
t
>
>
X
X
X
X
>
>
>
ln
y
x
¼
A
x
þ
A
x
þ
b
ðx2rt
rt it
0
it
1
rt
1
:
t
t
t
t
x2it Þ
2b1
x2it Þ þ 2b0
X
xrt xit ;
t
X
xrt xit :
t
(4.129)
A solution of this system would provide an economist with the necessary values
of model coefficients.
It is just as easy to find the parameters of a model of a logarithmic function of
complex variables with the general form
yrt þ iyit ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðxrt þ ixit Þ:
(4.130)
It should be noted with respect to this model that it is presented in additive form
and need not be transformed into a linear model, in contrast to all the aforementioned nonlinear models. One should only find the principal value of a logarithm of
128
4 Principles of Complex-Valued Econometrics
a complex argument. This circumstance shows that the assessments of the complex
coefficients of this model will not be shifted since the value to be minimized refers
to the sum of squares of complex-valued deviations of model (4.130) from actual
values but not its linear analog.
Taking into account designations (4.110), the system of normal equations for
this model, which is relevant for logarithmic operations, will have the form
8X
X
X
>
yrt ¼ Ta0 þ b0
ln Rxt b1
’xt ;
>
>
>
t
t
t
>
>
X
X
X
>
>
>
yit ¼ Ta1 þ b1
ln Rxt þb0
’xt ;
>
>
>
>
t
t
t
>
>
X
X
X
X
X
>
>
>
ðln2 Rxt
’
þ
b
ln
R
a
y
’
¼
a
y
ln
R
0
xt
1
it
0
rt
xt
>
xt
xt
<
t
t
t
t
t
X
>
2b1
’xt ln Rxt ;
>
>
>
>
t
>
>
X
X
X
X
X
>
>
>
yrt ’xt þ
yit ln Rxt ¼ a0
’xt þ a1
ln Rxt þ b1
ðln2 Rxt
>
>
>
>
t
t
t
t
t
>
X
>
>
>
>
þ2b
’
ln
R
0
xt
xt
:
’2xt Þ
:
’2xt Þ
t
(4.131)
The solution to this system will provide an economist with the needed values of
assessments of model coefficients.
Thus, if we know the properties of the conformal mappings of simple functions
of complex variables, then we can use them to solve various problems of modern
econometrics by finding the values of the coefficients of econometric models using
LSM, as proposed in the present section. Examples of building nonlinear econometric models will be given in subsequent chapters, along with solutions of applied
economic problems.
4.9
Assessment of Confidence Limits of Selected
Values of Complex-Valued Models
Since in this chapter we are looking at problems associated with building econometric models applicable exclusively to reversible processes, it is assumed a priori
that researchers deal with selected values of random values. And since selected
values are being assessed, it is necessary to determine the degree to which these
selected values can be trusted, how close they are to their true values, the mathematical expectation. In mathematical statistics this problem has been successfully
solved. Hence the natural wish to extrapolate the methods and approaches of
mathematical statistics to complex-valued models.
4.9 Assessment of Confidence Limits of Selected Values of Complex-Valued Models
Fig. 4.1 Example of
confidence domain of
complex variables
129
yi
yi + σitα
yi
yi − σitα
yr
0
yr − σrtα
yr
yr + σrtα
It is clear that if a researcher aims at studying a simple stationary process that
reveals itself in a sample of a random normally distributed value Y, then calculating
the average
1
Y ¼
n
n
X
Yi
(4.132)
i¼1
and the variance of deviations of actual observations from this average s2, one can
find the interval where the true value of Y lies:
Y
sta Y Y
sta :
(4.133)
If instead of the real case we consider a complex variable, the argument should
not be violated at first glance –the averages are calculated separately for the real and
imaginary parts, the confidence limits are defined for them, and then the confidence
limits of the values of the two components of the complex value are found:
yr
yi
sta yr yr
sta yi yi
sta ;
sta :
(4.134)
Figure 4.1 graphically demonstrates this procedure. In a complex plane the
confidence domain represents a rectangle with sides defined by the confident limits
(4.134). The center of this confidence domain is a point in the complex plane with
the coordinates ðyr ; yi Þ. However, we obtain a rectangle if the random variables are
independent of each other, and in a complex-valued economy one of the principles
is that of the dependence of the real and imaginary parts of a complex variable.
They should show various sides of the same phenomenon or object. This is why the
premise of the independence of yr and yi from each other that underlies the method
for determining confidence limits is not true in our case.
130
4 Principles of Complex-Valued Econometrics
By this logic, the confidence domain should represent a cloud of scattered
possible and valid values that should not have sharp corners. In the general case
this cloud should take the form of an ellipse.
The aforementioned standard procedure for finding confidence limits (4.134) is
not the only one in mathematical statistics; it is just the most popular one. In
multidimensional statistics the option of combining the distribution of random
values has been studied, and this very method may be applied in a complexvalued economy. Since a complex variable is a two-dimensional value, it will be
most suited for T2 Hotelling statistics representing a locus of the points of an
ellipsoid of a confidence domain for two random normally distributed values.
The confidence interval for expectation m of a real random value is constructed
by means of a t statistic that has a t-distribution with n 1 degrees of freedom:
t¼
x
_
s
m pffiffiffi
n:
(4.135)
This equality may be written in the following equivalent form:
t2 ¼ nð
x
1
mÞC ð
x
mÞ;
(4.136)
where C is a matrix inverse to that of covariance evaluations.
To find a combined confidence domain for a two-dimensional random value
using multidimensional statistical methods, namely Hotelling’s distribution, we
introduce an analog form of (4.136):
ns2 s2
T ¼ 2 2 rt it 2
srt sit srit
2
ðyrt
s2rt
yr Þ2
þ
ðyit
s2it
yi Þ2
2
srit ðyrt
yr Þðyit
s2rt s2it
yi Þ
:
(4.137)
To calculate T2, we first determine selected evaluations of the expectations of
random values yr ; yi as the averages of the series yrt and yit, and then selected
variances s2rt ; s2it ; s2rit . To simplify formula (4.137), it is presented in matrix form:
T 2 ¼ nðY
T
mÞ C 1 ðY
mÞ;
(4.138)
where m is a vector of expectations of two-dimensional random vector Y and Y is a
vector of the average values (selected evaluations) of the expectations of twodimensional random vector Y.
Hotelling connected T2 with distribution F, which for the two-dimensional case
has the following form:
Ta2 ¼
The Fa statistic has 2 and n
2ðn 1Þ
Fa :
n 2
2 degrees of freedom.
(4.139)
4.9 Assessment of Confidence Limits of Selected Values of Complex-Valued Models
131
The confidence domain of a random complex-valued variable and its
characteristics obtained by means of Hotelling’s statistics can also be applied to
the evaluation of confidence limits of selected evaluations of regression
relationships between complex-valued variables. This approach was adapted for a
complex-valued economy by A.F. Chanysheva.
Since any nonlinear elementary single-factor complex-values function can be
transformed into a linear one by various methods (}4.8):
yr þ iyi ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þðxr þ ixi Þ;
(4.140)
it may be represented as an equation with one coefficient by centering the variables
of the linear model with respect to their averages:
y0r þ iy0i ¼ ðb0 þ ib1 Þðx0r þ ix0i Þ:
(4.141)
As a result, the problem of determining the confidence domain of a regression
model boils down to constructing a confidence domain for the two-dimensional
random value B ¼ (b0, b1). LSM evaluations are the best way to evaluate complex
proportionality coefficients for stochastic normally distributed values.
The possibility of finding a pair of coefficients for each observation is a specific
feature of a simple linear complex-valued model without a free term:
b0t ¼
yrt xrt þ yit xit
;
xrt 2 þ xit 2
(4.142)
b1t ¼
yit xrt yrt xit
:
xrt 2 þ xit 2
(4.143)
Then, the method of finding a confidence domain of a complex proportionality
coefficient proposed by Chanysheva is reduced to the following procedure:
1. Point estimates of the expectations of random values b^0 and b^1 are found using
LSM: B^ ¼ b^0 þ ib^1 :
2. A random series of the complex variable b0t + ib1t is formed using (4.142) and
(4.143).
3. The two-dimensional covariance matrix C is calculated for the random vector
B ¼ (b0, b1) according to
C ¼
1
n
1
n
X
t¼1
1
ðbkt
b^k Þðblt
b^l Þ; k ¼ 0; 1; l ¼ 0; 1
(4.144)
and the inverse matrix C .
4. Using (4.135), the interval evaluation for the expectation of the random vector
B ¼ (b0, b1) is calculated.
132
4 Principles of Complex-Valued Econometrics
The confidence limits of the expectation of the two-dimensional random vector
B with confidence probability P are described by the following equation:
ðB
1
T
mÞ C ðB
mÞ ¼
2ðn
nðn
1Þ
F1
2Þ
P;2;n 2 :
(4.145)
The confidence domain for m is defined by an inequality:
ðB
1
T
mÞ C ðB
mÞ<
2ðn
nðn
1Þ
F1
2Þ
P;2;n 2 :
(4.146)
Its random
Equation (4.145) defines an ellipse with its center in random point B.
1
sizes and the directions of its principal axes are defined by C and the number
F1–P,2,n–2. The confidence domain represents a set of internal points of this random
ellipse. According to the theory, this ellipse covers the unknown point m with
probability P. However, Chanysheva shows that in practice this is not the case. A
considerable part of the points go beyond the ellipse and this domain does not
appear to be a confidence domain. This is why she proposes a correction factor [9]:
H¼
ðn 2Þ2
2ðn 1Þ;
(4.147)
which should be taken into account to connect Hotelling’s T2 statistics with F
statistics:
T2 ¼ H
2ðn
nðn
1Þ
F1
2Þ
P;k;n k
¼
n
2
n
F1
P;2;n 2 :
(4.148)
Then the equation of an ellipse (4.145) will have the form
ðB
1
T
mÞ C ðB
mÞ ¼
n
2
n
F1
P;2;n 2 :
(4.149)
Let us demonstrate the essence of the procedure for calculating the confidence
limits of a complex proportionality coefficient of a complex-valued regression
model on some conditional data. The original data of the conditional example for
random values are given in Table 4.2 and were selected to have similar dimensions
and normal distributions. Therefore, they satisfy the initial assumptions for solving
the problem of determining the confidence limits of LSM evaluations.
Before calculating the coefficients of this linear regression model, let us first
center the original data with respect to their averages to represent the model as
(4.141).
LSM applied to the centered data of this table made it possible to construct a
linear regression model:
4.9 Assessment of Confidence Limits of Selected Values of Complex-Valued Models
133
Table 4.2 Observations
of random values
yi
78.4
81.4
75.8
79.2
65.4
71.4
67.8
64.6
74.2
71.2
63.4
74.0
80.2
72.9
81.4
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
^
^
r
i
xr
50
49
42
56
47
51
50
56
48
50
46
57
54
51
53
xi
20
20
21
23
21
22
24
27
23
25
26
25
23
28
27
yr
25.6
25.4
22.6
27.3
29.5
29.0
22.7
31.0
26.6
30.1
30.0
23.7
30.4
22.6
25.0
y þi y ¼ ð0; 961457 þ i0; 981518Þ ðxr þ ixi Þ
Now, using the fact that the real and imaginary parts of a complex
proportionality coefficient are easy to calculate at any observation t using (4.142)
and (4.143), we can obtain these values.
Comparing the dynamic series of the complex regression coefficient with an
evaluation of it obtained on the whole set by means of LSM, we can see that they in
fact represent evaluations of the expectations of this coefficient’s values.
Now we have all the values to find a covariance matrix for this series, as well as
an inverse one, using (4.144). Since later on we will use the inverse covariance
matrix, its values are of utmost interest (Table 4.4)
Let us define the confidence probability at a level of 99 %. For k ¼ 2 and n ¼ 15
the tables give Fa ¼ 5.8. Substituting all the obtained characteristics into the
modified Hotelling statistic (4.148), we obtain the equation of an ellipse (4.149)
describing the confidence limit for the regression coefficient b0 þ ib1 :
157; 55b20 þ 49; 51b21
255; 614b0
45; 615b1
56; 06b0 b1 þ 144; 4 0:
Knowing the confidence domain for the regression model coefficients we can
easily construct the confidence domain for the calculated values of complex Y. For
this the confidence limit equation for coefficients B should be multiplied by the
complex variable xr þ ixi :
ð157; 55b20 þ 49; 51b21
255; 614b0
45; 615b1
56; 0:6b0 b1 þ 144; 4Þðxr þ ixi Þ 0:
It is true that this method of building confidence limits (which we called
decomposition limits) is not quite correct since the complex argument is also a
134
4 Principles of Complex-Valued Econometrics
Table 4.3 Evaluations of
regression coefficient for
any observation
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Table 4.4 Matrix inverse
to covariance one
157.55109
28.03279
28.0328
49.50967
Table 4.5 Matrix inverse
to covariance one
0.0844132
0.024437
0.02444
0.039274
b0
0.982069
1.0253998
1.152162
0.9140683
1.0415244
0.9895694
0.8989625
0.9002417
1.0525014
1.0513292
1.0847148
0.8252822
1.0113657
0.9431685
0.9963899
b1
1.175172
1.242229
1.228133
1.038601
0.926152
0.9737
0.92503
0.719339
1.041949
0.898959
0.76597
0.935525
1.053599
0.912438
1.028254
random value and it makes its own variance to the modeled result. We think it
would be more correct to use the aforementioned procedure directly for the
calculated values of the complex result y^r þ i^
yi . The logic of this approach is as
follows.
1. LSM is used to evaluate the values b^0 and b^1 : B ¼ b^0 þ ib^1 and y^r þ i^
yi ,
respectively.
2. The random series of the complex variable yrt þ iyit is the original one and is at
the researcher’s disposal.
3. Thetwo-dimensional covariance matrix C is calculated using the formula
C ¼
1
n
1
n
X
t¼1
ðykt
y^k Þðylt
y^l Þ; k ¼ r; i; l ¼ r; i
1
as well as the inverse matrix C :
For the conditional example under consideration the inverse covariance
matrix is shown in Table 4.5.
4. Formula (4.135) is used to calculate the interval evaluation for the expectation of
the random vector Yr þ iYi :
4.10
Balancing Factor in Evaluating the Adequacy of Econometric Models
135
Using it, one can also determine the confidence limits for the modeled random
complex variable:
186; 9
0; 475Yr þ 0; 084Yr 2
4; 92Yi
0; 048Yr Yi þ 0; 039Yi 2 ¼ 0:
The confidence limits will look like an ellipse and, as studies have shown, the
actual values fall within this confidence domain.
The approach proposed by Chanysheva can be extended to the problem of
determining the confidence limits of other evaluations of sample value parameters.
4.10
Balancing Factor in Evaluating the Adequacy
of Econometric Models
The TFCV provides economists with the possibility not only of describing more
precisely particular complex socioeconomic processes but also of solving problems
that in the domain of real variables are too huge or unsolvable.
In modeling various processes, modern researchers face the problem of
evaluating the adequacy of the obtained model. Various coefficients or factors are
usually calculated to determine how well the obtained model describes the original
series of data. Based on these factors conclusions are drawn on the degree of the
model’s adequacy. The average approximation error is a coefficient characterizing
the compliance of the modeled processes with the actual ones. It is recommended
that it be calculated using one of the following formulae:
Afirst
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uP
un
u ðyt y0 t Þ2
100% tt¼1
;
¼
n
y
Asecond ¼
n
0
100% X
yt y t :
y
n
t
t¼1
(4.150)
(4.151)
Here, Afirst and Asecond are values of the average approximation error, y is the
average of the original series of data, yt is theactual value at observation t, y0 t is the
calculated value at observation t, and n is the number of observations.
In most cases these coefficients provide good results, and they are sufficient for
evaluating model adequacy, though their values differ from each other. However,
according to Ivan Svetunkov, there are a number of situations in practice in which
none of the aforementioned formulae gives correct information on the properties of
the constructed models and therefore misleads the researcher regarding model
accuracy. This can happen in two situations:
136
4 Principles of Complex-Valued Econometrics
1. Calculation of the average approximation error for a series whose average is
close to zero;
2. Calculation of the average approximation error for a series of data where there
are values close to zero.
Actually, it follows from formula (4.150) that if the average y of a series of data is
close to zero, then the approximation error becomes very high and stops showing
the real properties of the model however adequate it is to the real process.
Researchers will face these situations when they, for example, center the original
series with respect to its average.
On the other hand, it is clear from (4.151) that if in a series there are some values
yt close to zero, then the approximation error also becomes overinflated regardless
of the adequacy of the constructed model as some errors represent a quotient where
the denominator is close to zero. At the same time, if y0 t equals zero (or is close to it),
then, as is clear from (4.151), the coefficient stops taking into account the difference
between the actual and the calculated values because we have unity under the
summation sign.
In these situations it is not possible to estimate the degree of adequacy of the
modeled series by (4.150) and (4.151). This problem is especially serious when the
original series contains both negative and positive values, and it should be reduced
to dimensionless quantities. The results will be both positive and negative, as well
as close to zero. It is difficult to find solutions to this problem in the domain of real
variables. Ivan Svetunkov proposed considering actual yt and calculated y0 t values
of variables not in the form of independent series but as complex numbers [10]:
zt ¼ yt þ iy0 t .
If we plot points of this series in a complex plane we will obtain a particular
aggregate of it lying around the line issuing from the point of origin at a 45 angle,
and the closer the calculated values are to the actual ones, the closer the series
points are to this line.
Another picture is when these points are in the pseudo-Euclidean plane where
we plot real numbers on the horizontal axis and imaginary ones on the vertical axis.
The length of the vector zt ¼ yt þ iy0 t in this plane can be found by the formula
j zt j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2t þ ðiy0 t Þ2 ¼ y2t y0t 2:
(4.152)
The specifics of presenting complex variables in the pseudo-Euclidean plane
were considered in the first chapter; here we will merely recall that in the Euclidean
plane only a zero vector [with coordinates (0;0)] can have zero length, and in the
pseudo-Euclidean plane, as we see from formula (4.152), nonzero vectors can also
have zero length. For example, in the pseudo-Euclidean plane, the absolute value of
the complex number 2 + 2i will be
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
R ¼ 22 þ ð2iÞ2 ¼ 22 ð2Þ2 ¼ 4 4 ¼ 0:
4.10
Balancing Factor in Evaluating the Adequacy of Econometric Models
Fig. 4.2 Representation of
complex number in pseudoEuclidean plane
137
y′
iy = y′
yt + iyt′
i
0
1
y
iy = − y′
There many vectors with zero length in a plane, all of them satisfying the
condition jyt j ¼ jy0 t j more precisely, one of the following two conditions:
yt ¼ y0 t ;
yt ¼
y0 t :
(4.153)
(4.154)
Thus, if the coordinates of the vectors satisfy (4.153) or (4.154), then they lie on
the respective lines of the pseudo-Euclidean plane and have zero lengths. These
lines are called isotropic. In Fig. 4.2 isotropic lines are shown as dotted lines.
They divide the plane into four sectors:
0
jyt j>jy t j
0
jyt j<jy t j
(
(
yt >0
yt <0
yt >0
yt <0
right sector;
left sector;
top sector;
bottom sector:
It is clear from (4.152) that in the pseudo-Euclidean plane the vector length can
also be:
– A real number if jyt j>jy0 t j;
– An imaginary number if jyt j<jy0 t j:
In a plane all vectors with real lengths will lie either in the right or left sector,
while vectors with imaginary lengths will be either in the top or in the bottom
sector.
For us these properties are interesting due to the opportunity it gives us to
compare the actual values with the calculated ones and obtain information on the
conformance of our model to reality:
138
4 Principles of Complex-Valued Econometrics
1. If the absolute value of the actual data is greater than that of the calculated data,
i.e., jyt j>jy0 t j, then the absolute value of the complex number zt ¼ yt þ iy0 t in the
pseudo-Euclidean plane will be a real number.
2. If the absolute value of the actual data is lower than of the calculated data, i.e.,
jyt j<jy0 t j, then the absolute value of the complex number zt ¼ yt þ iy0 t in the
pseudo-Euclidean plane will be an imaginary number.
3. If the actual and calculated data are equal, then the absolute value of the complex
number zt ¼ yt þ iy0 t will be equal to zero.
Taking into account these specific features of complex numbers in the pseudoEuclidean plane, one can use them to solve the problem of the adequacy of the
evaluation of an economic model. For that, let us use a correlation between two
modules: in the pseudo-Euclidean plane R2 and in the Euclidean plane R1:
R1t ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2t þ y0t 2;
R2t ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2t y0t 2:
It should be noted that:
1. R2t can be both an imaginary and a real number, while R1t can only be a real
number.
2. R2t ¼ 0 if the absolute value of the actual values is equal to that of the calculated
ones.
3. Always, R1t 0, where R1t ¼ 0 only when yt ¼ y0 t ¼ 0.
pffiffiffiffiffiffi
4. If the actual values are equal to zero, then
R1t ¼ y0t 2 ¼
p
ffiffiffiffiffiffiffiffiffi
ffi
y0t ; R2t ¼
y0t 2 ¼ iy0t :
pffiffiffiffiffi
pffiffiffiffiffi
5. If the calculated values are equal to zero, then R1t ¼ y2t ¼ yt ; R2t ¼ y2t ¼ yt :
Using these properties, one can obtain the balancing factor of a series of actual
data and a series of calculated ones proposed by Ivan Svetunkov:
n pffiffiffiffiffiffiffiffiffiffi
P
2
0
yt
t¼1
yt 2
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
y2t y0t 2
t¼1
n
B¼P
n pffiffiffiffiffiffiffiffiffiffi ¼ n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
P
y2t þy0t 2
y2t þ y0t 2
t¼1
n
(4.155)
t¼1
Since the numerator of formula (4.155) contain a radicand that may be negative
for certain t, factor B will be complex in the general case.
Here, two extreme cases are possible.
The first occurs when the balancing factor is a real number. As we see from
(4.155), this is possible if a model has systematic deviations so that jyt j>jy0 t j. Then
the radicand of the numerator will always be positive.
4.10
Balancing Factor in Evaluating the Adequacy of Econometric Models
139
The second one is when the balancing factor is imaginary. It follows from
(4.155) that this can occur when a model has systematic deviations from the real
data jyt j<jy0 t j.
If a model describes an actual series of data so that the approximation error is
both positive and negative and the swing is almost the same, then the imaginary part
of the complex balancing factor will be equal to its real part.
The proposed balancing factor possesses quite logical properties.
First of all, it follows from (4.155) that both the numerator and the denominator
of the factor are always positive. Therefore, both the real and the imaginary parts of
the balancing factor are nonnegative quantities.
Besides this evident property, the balancing factor has other characteristics. If a
model ideally describes actual values, and the approximation error is zero, the
numerator of the balancing factor will be equal to zero and the factor itself will be
equal to zero, too. If a model describes a real series of values with some slight
approximation error, then the numerator of formula (4.155) of the balancing factor
will not be high, and after being divided by the denominator will be close to zero.
Therefore, the closeness to zero of the real and imaginary parts of this factor
testifies to the high degree of adequacy of the model.
If a model poorly describes the original series of data, the factor numerator will
be far from zero, its value tending to the denominator, but it will never be greater
than the latter. Thus, both the real and imaginary parts of the balancing factor will
not exceed one.
Thus, if a balancing factor has the real and imaginary parts close to each other,
this means that the model does not have a shift. If the values of the real and
imaginary parts of the factor are close to zero, this indicates a good approximation
of the model.
For example, if as a result of calculation B ¼ 0.5 + i0.1, then this means that the
actual values according to the absolute value are greater than the calculated values,
and the model is shifted, and as the real part is much greater than zero, then this
means that the model describes the original series of data poorly.
Let us consider now the practical applicability of the coefficient proposed by
Ivan Svetunkov, namely, the production function of complex variables for the
Diatom plant in the town of Inza, Russia [11], which described the dependence of
the gross margin and production costs (complex production result) on capital and
labor (complex production resources). Then three parameters are to be modeled:
– Gross profit,
– Production costs,
– Gross output.
Since the data on the production process at this plant constitute a commercial
secret, all the original data were presented in nondimensional quantities, which are
occasionally given in subsequent chapters of this study. Here we are not interested
in these data. The plant operates in a tough competitive environment, which is why
there are months without income and even losses. This means that the original data,
when presented in nondimensional quantities, are not far from zero, taking both
140
4 Principles of Complex-Valued Econometrics
Table 4.6 Actual and
calculated data of gross
margin of Diatom plant
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Actual profit
0.012308
0.058620
0.165870
0.007310
0.007612
0.000110
0.063887
0.023063
0.050557
0.026168
0.062789
0.140044
0.085624
0.123722
Modeled profit
0.049609
0.032465
0.038947
0.012894
0.008581
0.025115
0.010122
0.038876
0.037702
0.054776
0.074427
0.097013
0.114934
0.120919
positive and negative values, sometimes close to zero. Nondimensional original
data on the gross profit and its calculated values obtained by our model are given in
Table 4.6.
Let us first calculate approximation errors (4.150) and (4.151) as is done in
econometrics today (14). We obtain
Afirst ¼ 173; 7122%; A second ¼ 1678; 0838%;
respectively.
As is clear, the approximation errors obtained are simply enormous, and if Afirst is
more than 100%, then Asec ond shows an error of more than 1,000 %. The formal
conclusion is clear: the model should not be used.
However, it is clear that the original series corresponds to the second case where
these coefficients give a distorted view – some values of the series are close to zero,
and it is division by a value close to zero that distorts the picture.
If we calculate a coefficient of determination between the calculated and actual
values, the picture will be somewhat different: R2 ¼ 0; 6309. This means that the
model is quite good at describing real values, not as bad as follows from the values
of the average approximation errors in percentage terms.
Now let us calculate the balancing factor proposed by Ivan Svetunkov (4.155). It
turned out to be equal to B ¼ 0.3749 + .02452i.
This means that the model does not describe the original series of data in the best
possible way because the real part of the balancing factor is not close to zero, and
the imaginary part, though less than the real one, also cannot be considered as close
to zero. However, the real and imaginary parts of the complex coefficient do not
differ greatly from each other – only by 0.13. That is, there is a shift of the model
with respect to the actual values, but it is not significant.
Values of the real and imaginary parts of the balancing factor also characterize to
a certain extent the accuracy of approximation. It is interesting that the coefficient
4.10
Balancing Factor in Evaluating the Adequacy of Econometric Models
141
of determination indicates that 63 % of the changes in the real values are modeled
by the model, but 37% cannot be explained. incidentally, the real part of the
balancing factor estimated in percentage terms is 37.5 %. Clearly, this is not just
a coincidence. The values of the real and imaginary parts of the balancing factor
may really show how accurately the model describes real values.
Thus, the balancing factor is free from the aforementioned drawbacks of the
average absolute approximation errors and provides the researcher with additional
information on the adequacy of econometric models of real variables.
Ivan Svetunkov’s balancing factor, which can be used to estimate the adequacy
of econometric models of real variables, may be applied to complex-valued econometric models.
In fact, a complex-valued econometric model can be regarded as a system of two
models of real variables – a separate equation for the real part and a separate one for
the imaginary part. Then we can build a pseudo-Euclidean plane separately for the
real part of the model and compare the real values with the modeled ones and do the
same for the imaginary part of the complex variable and imaginary part of the
complex-valued model.
To estimate the adequacy of a model’s ability to describe real parts, we should
calculate the their balancing factor:
Br
¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
y2rt y0rt2
t¼1
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
P
y2rt y0rt2
t¼1
(4.156)
And to estimate the adequacy of a model’s ability to describe the imaginary parts
of complex variables, we should calculate the balancing factor of the imaginary
part:
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
y2it y0it2
Bi ¼ t¼1
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
y2it þ y0it2
(4.157)
t¼1
Both coefficients are independent and should therefore be considered in isolation, outside their relation. In other words, they should not be added to each other
and on that basis make any general conclusions about the complex-valued model.
Therefore, the general idea of the adequacy of complex-valued economic models
consists of a notion of how well the real part is described and how well the
imaginary part is described.
142
4 Principles of Complex-Valued Econometrics
References
1. Arens R (1957) Complex processes for envelopes of normal noise. IRE Trans Inform Theory
IT-3:204–207
2. Goodman NR (1963) Statistical analysis based on a certain multivariate complex Gaussian
distribution. Ann Math Statist 34:152–176
3. Reed IS (1962) On a moment theorem for complex Gaussian processes. IRE Trans Inform
Theory IT-8:194–195
4. Wooding RA (1956) The multivariate distribution of complex normal variables. Biometrika
43:212–215
5. Wentzel ES (2010) The probability theory. KnoRus, Moscow
6. Svetunkov SG (2007) Method least squares for econometric models of complex variables.
Economic cybernetics: systems analysis in economics and management. SUEF, St. Petersburg,
pp 86–91
7. Tavares GN, Tavares LM (2007) On the statistics of the sum of squared complex Gaussian
randm variables. IEEE Trans Commun 55(10):1857–1862
8. Slutsky EE (2010) Economic and statistical works: selections. AST, Moscow
9. Chanysheva AF (2009) Finding the interval estimation of complex regression equations. In:
Economic forecasting: models and methods: proceedings of the international scientific and
practical conference, 5–6 April 2009, VSU, Voronezh
10. Svetunkov IS (2011) New coefficients of econometrics models quality estimation. Applied
Econometrics 24(4):85–99, Publishing House “Market DS”
11. Svetunkov SG, Svetunkov IS (2008) Production functions of complex variables. Publishing
House LKI, Moscow
Chapter 5
Production Functions of Complex Argument
The theory of production functions most often considers the dependence of the
production result (output) on two production resources – capital and labor. This
chapter considers the main models of production functions with a complex argument, when the real production result depends on the complex argument. Capital
refers to the real part of the complex argument and labor to the imaginary one.
Models of this type considerably expand the instrumental base of the theory of
production functions. In addition, this chapter demonstrates an important property
of linear complex models – their stability under conditions of multicollinearity.
We subsequently consider production functions of a complex argument in the
form of linear, power, exponential, and logarithmic models, study their properties,
and derive elasticity coefficients. Models of each type are built on the basis of
methods of complex-valued econometrics with real data of production systems of
various levels.
5.1
Fundamentals of Production Functions
of a Complex Argument
In order to model the operation of an enterprise, region, branch of industry, or
country in general, it is necessary to use the dependence of production results on the
resources used without studying the very process of transformation of the resources
into the production result. At each of the production hierarchy levels for modeling
this relationship we proceed from a number of various assumptions and admissions
that are of no any interest for the purposes of our research. We assume that we have
some production structure where modeling is considered only as a process of
transformation of production resources into a production result.
It is proposed that production be considered a “black box” where you have
resources x1, x2, x3, . . ., xn coming in and some production result y1, y2, y3, . . . ym
going out:
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
143
DOI 10.1007/978-1-4614-5876-0_5, # Springer Science+Business Media New York 2012
144
5 Production Functions of Complex Argument
Y ¼ f ðXÞ :
(5.1)
In economics such a relationship is called the “production function.” In the
theory of production functions, which uses real variables, the production result is
considered one real variable varying with respect to several production resources,
i.e., we have a multifactor relationship:
y ¼ f ðxi Þ; i ¼ 1; 2; 3; . . . n:
(5.2)
Various resources are used in manufacturing. They include labor resources,
financial resources, capital resources (machines, equipment, etc.), intellectual
resources, and others. In addition, each of the resources represents some aggregate
value, for example, labor resources can be divided into production and
nonproduction personnel. However, production personnel are also divided into
workers and students, engineering and technical personnel, nonmanual workers, etc.
If you try to make a model closer to economic realities, it is necessary to include
therein all these resources in as much detail as possible, which is why from the wide
array of production resources we focus on two basic and highly aggregated ones –
labor L and capital C. The reason for this is the following.
To produce some quantity of goods Q, besides capital and labor, one should
expend a certain quantity of material and financial resources, heating, and electric
power. The quantity of these resources used in production can be considered to a
certain extent directly proportional to the output. Nevertheless, capital and labor,
though varying with changes in output, do not vary according to a linear law; in
addition, these two resources are interchangeable – increasing capital resources and
implementing new technologies can result in increasing labor productivity and
reduction in labor costs. This means that with higher capital resource and lower
labor costs one can obtain the same output.
This interchangeability is inherent in any level of production hierarchy, for
example, at the enterprise level one can achieve an increase in production with
unchanged capital resources but increasing labor costs and organizing work in two
or three shifts. The same production volume can be achieved if we increase the
capital by implementing more productive equipment while making the number of
employees unchanged. However, in the first case the prime cost, gross margin, and
profitability will considerably differ from the same parameters in the second option,
though these specifics are not discussed in the theory of production functions.
For the purposes of our scientific research it is important that from among the
whole variety of resources production functions include two resources – labor and
capital:
Q ¼ f ðK; LÞ:
(5.3)
We will not review here parts of the theory of production functions since the
reader must know the theory or, if necessary, can familiarize himself with it from
5.1 Fundamentals of Production Functions of a Complex Argument
145
various studies and textbooks. Here, we consider the possibility and specifics of
applying complex-valued models to problems of the theory of production functions.
Since (5.3) represents a mapping of points of the resource plane to the real axis
of production results, we could replace the plane of real variables by that of
complex ones to preserve the meaning of the problem but change its form. The
production function of a complex argument will look as follows:
Q ¼ f ðK þ iLÞ:
(5.4)
Association of labor resources with the imaginary part is not due to any economic reasons – only those of convenience. However, in Chap. 1, we introduced a
rule according to which the real part of complex variables is associated with the
active part of an economic indicator, and the imaginary part with a passive one. To
some extent this rule is followed in the case under consideration; since labor
resources in the production process are more subject to various changes compared
to capital resources, they could be associated with the imaginary part.
In accordance with TFCV rules, the equality sign in (5.4) means that the
production result represented in the form of a real variable can in reality be
represented as a complex variable with the imaginary part equal to zero:
(
Q þ iQi ¼ f ðK þ iLÞ;
(5.5)
Qi ¼ 0:
Because in economics smooth functions exist only in the form of ideas of the
most reckless idealists and are never encountered in practice, there is no precise
functional compliance between resources and production result (5.4), only a regression relationship. Taking into account this fact, while constructing a production
function of a complex argument, we could obtain the following relationship:
Q ¼ f ðK þ iLÞ þ er þ iei :
(5.6)
Hence, it follows that the production functions of a complex argument (PFCA)
represent a conformal mapping of a set of points in a complex plane of production
resources to the complex plane of the production result, where along the real axis
we plot Q, and along the imaginary one we plot the deviation еi. In the general case,
this mapping has the form of a linear regression coinciding with the axis of real
numbers Q, with the deviation being random. This conformal mapping is shown in
Fig. 5.1.
If we use the least-squares method (LSM) to find PFCA coefficients, then
X
ðer þ iei Þ ¼ 0:
(5.7)
The specifics of the complex approximation error and methods of measuring the
degree of approximation were stated in the previous chapter and will not be
discussed in this and subsequent chapters.
146
5 Production Functions of Complex Argument
L
ei
K
0
Q
0
Fig. 5.1 Conformal mapping of PFCA of points in complex plane of production resources to
complex plane of results
The formal mathematical basis for PFCA will be any elementary complexvalued function from among a set of known ones. The properties of the principal
functions were studied in the Chap. 2.
In this chapter we will gradually consider PFCA on the basis of the general
scientific principle of from the simple to the complex and therefore start with the
linear production function of a complex argument. After obtaining particular results
we will move on to more complex models. However, before that, we should pay
attention to the problem domain – all the variables, both the production resources
and the production result, and positive values.
5.2
Linear Complex-Valued Model of a Complex Argument
and Multicollinearity
Before we begin examining the specific features of functions of a complex argument as production process models, let us devote some attention to a peculiarity of
these functions. In Chap. 2 we studied conformal mappings of these models, which
showed that they represent an equation of a line in three-dimensional space. This
peculiarity of complex argument models may in certain cases be considered a
considerable advantage compared to models of real variables, especially for a linear
model of a complex argument. Let us show this.
Any complex-valued model is multifactor since a complex argument consists of
two real variables consisting of the real and imaginary parts. This is why, before
considering models of the production function of complex argument, it seems
logical to carry out a comparative analysis of complex argument functions with
two-factor models of real variables, which are widely used by economists in
economic modeling.
While building multifactor models of socioeconomic dynamics (including twofactor ones), scientists practically always come up against the fact that many, and
very often all, pair correlation coefficients between variables included in the models
are close to one in their modulus. This phenomenon has been called multicollinearity. It takes place when model factors have similar monotonic, almost linear,
trends in their dynamics.
5.2 Linear Complex-Valued Model of a Complex Argument. . .
147
There is one big problem, however. Attempts to use LSM for estimating the
coefficients of multifactor models under conditions of multicollinearity results in a
degeneration of the system of normal equations using LSM – coefficients of the
equations of these systems appear to be close to each other.
This problem of building multifactor models under conditions of multicollinearity
has occupied scientists’ minds since the middle of the twentieth century, with scientists proposing various modeling methods under conditions of multicollinearity.
There is even a special branch of mathematical statistics – robust statistics – aimed
at building multifactor models under conditions of multicollinearity.
Let us show how to obtain satisfactory LSM estimations of a multifactor model
of real variables under conditions of multicollinearity without complex mathematical procedures [1].
First of all let us write down an LSM system of a normal equation for a
multifactor linear model:
Yt ¼ a0 þ a1 x1t þ a2 x2t þ . . . þ an xnt ;
(5.8)
omitting the summation limits to make the notation simpler, and taking into account
the fact that it is valid for all t beginning from t ¼ T:
X
X
8X
>
xnt
x1t þ . . . þ an
Yt ¼ a0 T þ a1
>
>
X
X
X
X
>
>
2
<
Yt x1t ¼ a0
x1t þ a1
x1t þ . . . þ an
xnt x1t
>
...
>
>
>
>
: X Y x ¼ a X x þ a X x x þ . . . þ a X x2 :
t nt
nt
0
1
1t nt
n
(5.9)
nt
If we present this system in the form of equations in segments, we obtain
8
a0
a1
an
>
1 ¼ P þ P þ ... þ P
>
>
Yt
Yt
>
>
P
P Yt
>
T
>
x1t
xnt
>
>
>
>
a
a
>
> 1 ¼ P 0 þ P 1 þ . . . þ Pan
>
<
PYt x1t
PYt x2 1t
P Yt x1t
x1t
x1t
xnt x1t
>
>
>
>
...
>
>
>
>
a0
a1
an
>
>
>
þP
þ ... þ P
1¼P
>
>
Y
x
Y
x
>
P t nt P t nt
PYt x2 nt :
:
xnt
x1t xnt
(5.10)
xnt
Since every equation of this LSM system represents an equation of a hyperplane
in a hyperspace of coefficients ai, the solution will be a intersection of the
hyperplanes in the hyperspace. However, if the coefficients of the equations of
these hyperplanes are close to each other, the latter will be almost parallel, which
takes place under conditions of multicollinearity . This means that at the slightest
148
5 Production Functions of Complex Argument
shift of at least one of the hyperplanes in the hyperspace the intersection of all the
hyperplanes will significantly change its coordinates, which are the required LSM
estimations. Hence, the solution of a system of normal equations rounded to the
tenth decimal place will give certain values of the coefficients of the multifactor
model, and rounded to the eighth place will give other values of the same multifactor linear model, sometimes opposite in sign.
Equations of the LSM system in (5.10) are represented as equations of
hyperplanes in segments in the hyperspace of model coefficients. If in the singlefactor case LSM estimations represent an intersection of two equations of LSM
systems, since there are only two unknown parameters, a0 and a1, and the problem
can be presented in a plane, then with two factors the number of model coefficients
will be three – a0, a1, and a2. This problem of estimating the parameters of a
multifactor model should be considered not in a plane but in three-dimensional
space. Actually, the number of unknown parameters becomes three and they can be
shown as axes of a three-dimensional space 0a0, 0a1, and 0a2. In this case LSM
conditions represent a system of three equations with three unknowns, each of them
being an equation of a plane in space. The solution of an LSM system in this case
will represent the intersection of three planes in this space of coefficients. The
coordinates of this point will give the values of model coefficients.
If we present the problem of finding the coefficients of single-factor models
under conditions of multicollinearity by LSM in the form of system (5.10), we see
that the values of the segments obtained, cut off by LSM hyperplanes on each of the
axes of the coefficients’ hyperspaces, will almost always practically coincide,
which means that the hyperplanes are almost parallel to each other. Apparently,
this is why the solution of an LSM system, which represents an intersection of these
practically parallel hyperplanes in a hyperspace, is quite unstable – the slightest
rounding error may result in a new intersection due to a slight shift of the
hyperplanes in the hyperspace, and this point will be considerably remote from
the original one. Here, the solution of an LSM system as an intersection of
hyperplanes changes in such a way that it is not only the absolute value of a
multifactor model’s coefficients that is distorted, but the sign itself of these
coefficients, which is a universal consequence.
Therefore, the consequences of multicollinearity are due to the unacceptability
of the existing algorithm for the estimation of multifactor models’ parameters in
this case. Then research should be aimed not at the struggle with objective reality –
strong collinearity of virtually all indicators and factors of socioeconomic dynamics
(which is confirmed by the false-correlation phenomenon), not at the improvement
of mathematical algorithms of work with weakly justified matrices, but at the
improvement of the involved coefficient estimation tool.
In order to increase the stability of estimations of parameters of multifactor
models it is necessary to separate the hyperplanes of an LSM system of normal
equations eliminating their practical parallelism. Therefore, we should create a
situation where segments on the axes of the hyperspaces of model-parameter
estimations differ from each other as much as possible but do not coincide, as
always happens under conditions of multicollinearity. The best thing would be to
5.2 Linear Complex-Valued Model of a Complex Argument. . .
149
propose a method of obtaining LSM estimations providing perpendicularity of the
hyperplanes – this would make it possible to obtain stable estimations of model
coefficients.
To solve this problem, recall the formula for finding the angle between two
planes in a three-dimensional space. If we represent one plane by
Ax þ By þ Cz þ D ¼ 0;
(5.11)
A0 x þ B0 y þ C0 z þ D0 ¼ 0;
(5.12)
and the second plane by
the cosine of the angle between them can be found by the formula [2]
AA0 þ BB0 þ CC00
cos g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
A2 þ B2 þC2 A0 2 þ B0 2 þC0 2
(5.13)
If the planes are almost parallel, the angle between them is close to zero and the
cosine of the angle is close to one. If the planes are perpendicular, the angle between
them is 90 and the cosine is zero. This means that to obtain stable LSM estimations
under conditions of multicollinearity, transformations are necessary such that the
numerator is equal (or close) to zero or the denominator tends to infinity.
Using the formula for the cosine of the angle between planes (5.13), one can find
the cosine of the angle between the planes of system (5.9) in the space of
coefficients of the econometric model.
The cosine of the angle between the first and second planes described by the first
and second equations of the system of normal equations, respectively, will be
determined as follows:
P
P P
P P
T x1t þ x1t x21t þ .. . þ xnt x1t xnt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
cos g12 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
P
P
P
P
2
T 2 þ ð x1t Þ2 þ . .. þ ð xnt Þ2 ð x1t Þ2 þ ð x21t Þ þ ... þ ð x1t xnt Þ2
(5.14)
It has been stated that the cosine can be equal to zero in only two cases – where
the numerator is equal to zero and where the denominator tends to infinity.
If the second condition – setting the denominator to infinity without increasing
the numerator in the same direction – is impossible, the first one is quite feasible. To
do that, the original variables should be centered around their average. This operation represents a transformation of the original series {xt} where the average of this
series is subtracted from each value thereof:
0
xt ¼ xt
x:
(5.15)
150
5 Production Functions of Complex Argument
As follows from the elementary principles of mathematical statistics, the sum of
this centered series will be equal to zero. This is why in formula (5.14) the
following sums will be equal to zero:
X
X
X
x1t ¼ 0;
x2t ¼ 0; . . . ;
xnt ¼ 0:
(5.16)
Since all of them are factors of each term (5.14), the numerator will be equal to
zero. The denominator will always be greater than zero. This means that the cosine
of the angle between the first and second hyperplanes is equal to zero, i.e., the angle
between them is 90 and these two hyperplanes are perpendicular to each other. In
turn, this testifies to the fact that the intersection of these hyperplanes will be
defined quite stably regardless of rounding errors.
It is clear that with (5.15) for all the model coefficients, the angles between the
first and the other hyperplanes of the hyperspace of the unknown coefficients that
can be found by an LSM system of normal equations will be correct, and the first
hyperplane will be perpendicular to the other hyperplanes. In centering with respect
to the averages, the cosines of angles between other hyperplanes of system (5.9), for
example the second and the third hyperplanes, will not be equal to zero, but they
will be less than in the case where uncentered variables are used, which testifies to
the fact that the hyperplanes intersect at a wider angle, and the intersection of all the
hyperplanes will be more stable to the rounding errors and new information.
In many cases this simple procedure gives satisfactory results, which is why in
conditions of multicollinearity all the original variables should always be centered
in advance. However, in practice, there are situations where this approach does not
give the needed results. An example is provided by M. Glushenkova and A.
Zemlyanaya. Let us observe an socioeconomic process described by a functional
two-factor linear model with a zero free term:
yt ¼ 7:3x1t þ 2x2t :
(5.17)
Factors x1t and x2t in the period under consideration vary linearly, and this
variation is described by a linear functional dependence of one factor on the other:
x2t ¼ 0:273972603x1t :
(5.18)
It is evident that in this case, the pair correlation coefficient between them will be
equal to one, which testifies to an extreme case of multicollinearity – a linear
functional relationship between them. Table 5.1 shows the values of the resulting
parameter yt calculated according to formula (5.17) for linearly varying variables. It
should again be noted that all the data in Table 5.1 result from the functional
relationship (5.17) between all the variables.
Now let us pose another problem, namely, to build a multifactor linear model on
the data from the table, without knowing the coefficients (5.17), which means
calculating the model coefficients according to the table data:
yt ¼ a1 x1t þ a2 x2t :
5.2 Linear Complex-Valued Model of a Complex Argument. . .
151
Table 5.1 Data of conditional example
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Calculated value Yt,
obtained by (5.17)
15.69589
31.39178
47.08767
62.78356
78.47945
94.17534
109.8712
125.5671
141.263
156.9589
172.6548
188.3507
204.0466
219.7425
235.4384
251.1342
266.8301
282.526
298.2219
313.9178
x2t
0.547945205
1.095890411
1.643835616
2.191780822
2.739726027
3.287671233
3.835616438
4.383561644
4.931506849
5.479452055
6.02739726
6.575342466
7.123287671
7.671232877
8.219178082
8.767123288
9.315068493
9.863013699
10.4109589
10.95890411
x1t
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
Since there is a strict functional relationship between all the variables of the
problem, one might think that it would be easy to solve it since there is no random
error or uncertainty; the problem is completely determined. Moreover, one should
expect accuracy as a result of applying LSM. Let us use it to find estimations of the
original model from the table data, taking into account that the free term is equal to
zero. The system of normal equations for this model will have the form
(X
X
Yt x1t ¼ a1
Yt x2t ¼ a1
X
X
x21t þ a2
X
x2t x1t
X
x1t x2t þ a2
x22t
The equations that may help to calculate the coefficients calculated for Table 5.1
data will look like the following system:
(
90094:411 ¼ a1 11480 þ a2 3145:206
:
24683:403 ¼ a1 3145; 206 þ a2 861:700
(5.19)
The next step is to solve the system and obtain the unknown coefficient values.
But before that, let us present this system of equations as a system of equations in
segments, as is done in (5.10):
152
5 Production Functions of Complex Argument
a1
a2
þ
7:8479452 28:645 :
a1
a2
>
:1 ¼
þ
7:8479452 28:645
8
>
<1 ¼
(5.20)
We see that this system of equations is degenerated and does not have solutions.
The planes of LSM equations are not only parallel to each other, they fully coincide,
i.e., they do not intersect. Therefore, multifactor LSM does not allow for solving the
set problem.
Now let us center the original data given in Table 5.1 around their averages
hoping for an improvement in the situation. Then the system of normal equations
will look like this:
(
20875:53 ¼ a1 2660 þ a2 728:77
:
(5.21)
5719:32 ¼ a1 728:77 þ a2 199:66
Clearly, the values of this system differ from the uncentered ones (5.19).
However, will it be possible to solve the problem? Let us again present the system
obtained in the form of a segment system:
a1
a2
þ
7:8479452 28:645
a1
a2
>
:1 ¼
þ
7:8479452 28:645
8
>
<1 ¼
It is easy to see that this system coincides with (5.20), which tells us that the
system cannot be solved.
Between all the variables there is a strict linear relationship that, in the threedimensional space of the original variables, represents points lying on one line, and
since these points lie on one line, there is an infinite number of planes that this line
will belong to. System (5.20) demonstrates this fact – it makes it possible to
infinitely combine pairs of values of planar model coefficients in three-dimensional
space, i.e., it allows one to obtain an infinite set of equations of planes intersecting
and containing a line. One such plane is that described by (5.17). However, it is not
possible to single this plane out of the infinite number of others using LSM.
We have considered an extreme case that does not occur in actual economic
practice, but situations close to the it exist everywhere – where pair correlation
coefficients between factors and modeled results are close to unity in the modulus,
for example they are equal to 0.97. In these cases our task is to estimate the
coefficients of a multifactor linear model that does not have a satisfactory solution
in the domain of real variables.
Linear models of a complex argument make it possible to solve this problem. In
the general case, linear models of a complex argument have the form
yr ¼ ða0 þ ia1 Þðxr þ ixi Þ þ ðb0 þ ib1 Þ:
(5.22)
5.2 Linear Complex-Valued Model of a Complex Argument. . .
153
After centering the original variables with respect to their averages, this model
will look much easier:
yr ¼ ða0 þ ia1 Þðxr þ ixi Þ:
(5.23)
Taking out the real and imaginary parts of this equality we obtain a system of
two equations:
(
yr ¼ a0 xr a1 xi ;
0 ¼ a1 xr þ a0 xi
(5.24)
where
8
< y r ¼ a0 x r a1 x i ;
a0
: xr ¼
xi
a1
(5.25)
The first equation of the obtained system tells us that the result linearly depends
on the factor variables, and the second one testifies to the fact that the linear
functions of a complex argument are characterized by a linear functional relationship between the factor variables. In other words, the linear functions of a complex
argument may exist only when a linear relationship exists between the model
factors, i.e., under the extreme condition of multicollinearity – a linear functional
dependence between factors! Scientists struggle to eliminate the consequences of
multicollinearity, and a function of a complex argument introduced in scientific use
should not be used in other cases!
Let us apply LSM to the estimation of coefficients of complex-valued model
(5.23), as was shown in Chap. 4. There, we derived a formula for assessing linear
models of a complex argument:
8X
X
>
x
y
¼a
ðx2rt
rt
t
0
>
<
t
t
X
X
>
xit yt ¼ a1
ðx2rt
>
:
t
t
x2it Þ
2a1
X
xrt xit ;
t
x2it Þ þ 2a0
X
xrt xit :
t
With reference to the data of Table 5.1 this system has the form
(
or, in segments,
90094:41 ¼ a0 10618:29
a1 6290:41;
24683:4 ¼ a1 10618; 29 þ a0 6290:41;
(5.26)
154
5 Production Functions of Complex Argument
a0
a1
;
8:4848 14:3225
a1
a0
>
:1 ¼
þ
:
2:3246 3:9239
8
>
<1 ¼
(5.27)
It is clear that we have obtained a stable system of normal equations, which, if
solved, gives us the following complex-argument model:
yr ¼ ð7:3
i2Þðx1 þ ix2 Þ
(5.28)
If we open the brackets and group the real and imaginary parts of the obtained
equality, it will be transformed into the following one:
yr ¼ 7:3x1 þ 2x2 þ ið 2x1 þ 7:3x2 Þ:
(5.29)
The real part of the equality fully corresponds to the original function (5.17), and
the imaginary part can easily be transformed into the following equality:
0 ¼ ið 2x1t þ 7:3x2t Þ ! x2t ¼
2
x1t ¼ 0:273972603x1t :
7:3
(5.30)
This means that the original dependence between the factors of (5.18) is
identified with absolute precision. M. Glushenkova and N. Zemlyanaya made
numerous calculations for this and other conditional examples setting the variance
in factors, thereby decreasing the correlation between them, as well as introducing
variance to the equation of the dependence, transferring it from a functional to a
regression dependence. Each time, estimations of the linear function of a complex
variable appeared stable to these variances.
To conclude this section, it should be noted that there is a fundamental difference
between a linear two-factor model of real variables such as
yt ¼ a0 x1t þ a1 x2t
(5.31)
and a linear model of a complex argument:
yt ¼ ða0 þ ia1 Þðx1t þ ix2t Þ
(5.32)
Although both models (5.31) and (5.32) have two variables and two coefficients,
they describe different figures. The model of real variables (5.31) represents an
equation of a plane in the space of original variables. The model of complex
argument (5.32) is an equation of a line in this space.
This means that in a situation where there is no linear relationship between
variables x1t and x2t the complex-argument model will provide poor results, which
is why a model of real variables should be used. In this case the observed points are
located around some plane in space with some deviation from the line; the smaller
5.3 Linear Production Function of a Complex Argument
155
the pair correlation coefficient, the larger the deviation of the points in the plane
described by (5.31) from the line. However, if the points in three-dimensional space
are on the same straight line, they should be described by a linear complexargument model.
Once again, it follows that models of complex variables are different from those
of real variables and usage thereof in economic modeling extends the instrumental
base of the economy, facilitating the solution of the complicated problems
associated with modeling socioeconomic processes.
In the cases, when the system of equations (5.25) is violated than the suggested
approach becomes non effective.
5.3
Linear Production Function of a Complex Argument
Economic dynamics are complex and diverse. The version of its development
described by the linear character of variations in its indicators cannot be excluded.
In this case a linear PFCA can be used for various purposes of economic analysis. It
is the properties of linear PFCA specified in early 2005 that served as the basis for
the formation of a complex-valued economy [3]. Its main features are well studied
and described in the study published in 2008 [4], which is why here we will focus
only on its most important properties. To build this simple function, let us use three
variables describing the production process: production volume Qt, labor costs Lt,
and capital expenditures Kt. Let us present the production resources in the form of a
complex variable.
Let us examine the linear function of a complex argument without a free term,
bearing in mind that, if necessary, it is easy to get rid of it by centering the original
variables around their averages. The model of the linear PFCA will have the form
Qt ¼ ða0 þ ia1 ÞðKt þ iLt Þ:
(5.33)
Multiplying the two factors on the right-hand side of equality (5.33) and
grouping the real and imaginary parts separately, we obtain
Qt ¼ ðKt a0
Lt a1 Þ þ iðLt a0 þ Kt a1 Þ:
(5.34)
Thus, the production function (5.34) can be represented in the form of a system
of two equations:
Qt ¼ Kt a0
Lt a 1 :
(5.35)
0 ¼ Lt a0 þ Kt a1 :
(5.36)
and
156
5 Production Functions of Complex Argument
Since it is clear from the discussion in the previous section that (5.33) represents
an equation of a line in three-dimensional space, in the resource plane it will
describe the following kind of linear projection:
a0
Lt :
a1
Kt ¼
(5.37)
Since labor resources and capital resources are positive by definition, (5.37)
indicates that one of the proportionality coefficients should be negative. Since most
often, growth of resources leads to growth of production results, to meet this
condition, coefficient a1 in (5.35) should be negative.
This means that if the coefficients and resources are positive, we should use the
following linear PFCA:
Qt ¼ ða0
ia1 ÞðKt þ iLt Þ:
(5.38)
Now it is quite simple to find the coefficients of this model and their economic
meaning. To do that, let us find the complex proportionality coefficient from the
production volume and production resources:
a0
ia1 ¼
Qt
Qt ðKt iLt Þ
¼
:
Kt2 þ L2t
Kt þ iLt
(5.39)
This equality, as follows from the properties of complex numbers, holds only if
the real and imaginary parts of the complex numbers of the left- and right-hand
sides of the equality are equal to each other (5.39). This property makes it possible
to obtain the formulae for the calculation of each coefficient – by opening the
brackets and grouping the real and imaginary parts separately. Then we obtain a
formula for the calculation of each of the coefficients. For the real part of the
proportionality coefficient:
a0 ¼
Qt Kt
Kt2 þ L2t
a1 ¼
Kt2
(5.40)
and for its imaginary part:
Qt Lt
:
þ L2t
(5.41)
It is evident from the given formulae that a pair of coefficient values is calculated
when there is at least one observation of both resource and production result values.
This property distinguishes the proposed function from its analogs in the domain of
real numbers, where in order to find two unknown coefficients you should have at
least two observations. This property of production function (5.38) is easy to
5.3 Linear Production Function of a Complex Argument
157
explain: the function has only one unknown complex coefficient, which is why its
values can be easily determined by one observation. If we wanted to find the
coefficients of a more complicated linear model of a complex argument, for
example,
Qt ¼ ðb0 þ ib1 Þ þ ða0 þ ia1 ÞðKt þ iLt Þ;
then we would have a model with two complex coefficients, and we would need two
observations to find the their values. However, since we are studying a simple
function without a free term, this means that for its practical use one should first
center the original variables with respect to their averages.
The obtained formulae (5.40) and (5.41) not only make it possible to find the
numerical values of the coefficients on the basis of the known values of costs and
expenditures, but they also give an economic interpretation to coefficients a0 and a1
if the production result is well modeled by this linear function.
The denominators of the formulae are the same and characterize the magnitude
of the resources involved. These formulae differ only in their numerators, which
makes it possible to understand the meaning of each coefficient.
The numerator of the real coefficient a0 characterizes the use of capital
resources, the numerator of the imaginary part characterizes that of labor resources.
This is why it makes sense to name them as follows: coefficient a0 is the
coefficient of use of capital resources and coefficient a1 that of the use of labor
resources.
Let us build a linear PFCA according to the statistical values of the produced
national income, the value of the fixed production assets, and the mean average
number of industrial and production workers in the Soviet Union from 1972 till
1989. These data, expressed in relative values, as well as those of resource usage
coefficients, and calculated according to (5.40) and (5.41), are given in Table 5.2.
To calculate the coefficients of model (5.33) all the original table data are
oriented relative to their averages.
According to variations in the complex proportionality coefficient in time, one
can see that for observations from 1981, 1982, and 1983 the complex
proportionality coefficient differs significantly from the values of the whole series.
This is easy to explain. The original data were centered around their arithmetic
averages, and therefore, in the points where the values assigned to observations of
resources approach the arithmetic averages, their difference is close to zero, and
division by a small value provided by formulae (5.40) and (5.41), yields considerable coefficient values.
Actually, for example, for 1981, the centered relative value of capital resources
of the Soviet Union was 0.015389, though for 1984 it was 0.432389. If we ignore
the observations of the complex coefficient of these years we notice that the real
part of the complex proportionality coefficient varies around 0.53422, and its
imaginary part around 0.04278, with a slightly decreasing trend for the whole
observation period.
158
5 Production Functions of Complex Argument
Table 5.2 Calculation of coefficients of resource use for Soviet economy from 1972 to 1989 [5]
Resource use
coefficient
Year
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
National
income, Qt
1
1.079
1.130
1.159
1.232
1.295
1.361
1.399
1.476
1.554
1.671
1.750
1.819
1.847
1.875
1.914
2.014
2.097
Fixed production
assets, Kt
1
1.091
1.193
1.292
1.393
1.496
1.612
1.724
1.846
1.973
2.107
2.247
2.390
2.518
2.649
2.778
2.904
3.024
Annual average number
of industrial and
production workers Lt
1
1.013
1.029
1.049
1.073
1.091
1.109
1.124
1.136
1.147
1.159
1.165
1.169
1.174
1.178
1.175
1.151
1.122
a0
0.553187
0.521700
0.526128
0.562921
0.537856
0.523597
0.510070
0.591209
0.530158
0.200023
0.822334
0.713303
0.641294
0.546466
0.484325
0.456662
0.502928
0.524799
a1
0.066240
0.061203
0.058947
0.055536
0.039692
0.026845
0.008363
0.02362
0.10133
0.420264
0.244040
0.124065
0.080584
0.057859
0.044366
0.033584
0.019308
0.003609
Table 5.3 Original data for building production functions and calculated data of resource use
coefficients [5]
Gross domestic
product, Qt
Investments in fixed
capital, Kt
Number of people
participating in
economy, L t
Resource use
coefficient
Absolute
values
(billions
Year of rubles)
Absolute
values
Relative (billions
values
of rubles)
Absolute
values
Relative (millions
values
of rubles)
Relative
values
a0
a1
1998 2630
1999 4823
2000 7306
2001 8944
2002 10834
2003 13285
2004 16779
1
1.834
2.778
3.401
4.119
5.051
6.380
1
1.651
2.860
3.702
4.331
5.371
7.627
1
0.986
1.009
1.014
1.041
1.035
1.053
0.00634
0.01232
0.00901
0.07636
0.04457
0.00944
0.00649
407.1
670.4
1165.2
1504.7
1762.4
2186.2
3105.1
63.6
62.7
64.2
64.5
66.2
65.8
66.9
0.89868
0.78225
0.78447
1.19895
1.12936
0.97630
0.74852
The latter means that in the Soviet Union the national income was provided by a
small number of workers, meaning a slight growth of labor productivity.
Let us see now how the linear PFCA (5.38) behaves if we consider Russia as an
example, and if we can use it for these purposes. Table 5.3 shows the corresponding
5.3 Linear Production Function of a Complex Argument
159
statistical data for 1998–2004. Gross domestic product is taken as the result of
production function Qt, the number of people involved in the economy as labor
costs Lt, and investments in the fixed capital as capital Kt. The last two columns of
this table give the results of calculation of the resource usage coefficient.
Again, we are looking at the effect of the scale of centered values on model
coefficients in the middle of the segment under consideration. For 2001–2002 the
coefficient is greatly affected by centering results. It is evident that it is difficult to
draw a conclusion on the nature of the production process on the basis of the data
obtained – either we need longer series or we should get rid of the free complex
coefficient in a different way, avoiding centering with respect to the averages.
Thus, the simplest model of complex argument (5.38), despite its coefficients’
having a simple economic meaning, is not universal and, therefore, of little interest
and use for describing real economic situations.
Linear PFCAs with a free term are more complicated:
Qt ¼ ðb0 þ ib1 Þ þ ða0 þ ia1 ÞðKt þ iLt Þ:
(5.42)
This free term removes stringent requirements from the original data and initial
points. This is why it shows real processes more clearly. However, economic
processes are never linear or nonlinear according to some set form. All the processes in an economy change, as do the resulting proportions and quantitative laws.
Since there could be situations that describe models like (5.42), we should
consider its properties and methods of finding coefficients.
For that let us consider model (5.42), not only at moment t but also in the
following observation:
Qtþ1 ¼ ðb0 þ ib1 Þ þ ða0 þ ia1 ÞðKtþ1 þ iLtþ1 Þ:
(5.43)
If we delete the left-hand side of equality (5.42) from the left-hand side of
equality (5.43) and the right-hand side of equality (5.42) from the right-hand side
of equality (5.43), we obtain
DQt ¼ ða0 þ ia1 ÞðDKt þ iDLt Þ:
(5.44)
Hence, we can derive a formula for calculating the complex proportionality
coefficient.
For the real part of the proportionality coefficient this will have the form
a0 ¼
DQt DKt
;
DKt2 þ DL2t
(5.45)
a1 ¼
DQt DLt
:
DKt2 þ DL2t
(5.46)
and for its imaginary part:
160
5 Production Functions of Complex Argument
Table 5.4 Calculation of resource extensiveness coefficients for Soviet Union from 1972 to 1989
Resource
use coefficients
Year
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
Increment of
national
income DQt
Increment of
fixed production
assets, DKt
0.079
0.051
0.029
0.073
0.063
0.066
0.038
0.077
0.078
0.117
0.079
0.069
0.028
0.028
0.039
0.100
0.083
0.091
0.102
0.099
0.101
0.103
0.116
0.112
0.122
0.127
0.134
0.140
0.143
0.128
0.131
0.129
0.126
0.120
Increment of annual average
number of industrial and
production workers DLt
0.013
0.016
0.020
0.024
0.018
0.018
0.015
0.012
0.011
0.012
0.006
0.004
0.005
0.004
0.003
0.024
0.029
a0
a1
0.850769 0.121538
0.487992 0.076548
0.281443 0.056857
0.684142 0.162568
0.593524 0.103723
0.555588 0.086212
0.333307 0.044639
0.625100 0.061485
0.609600 0.052800
0.866188 0.077569
0.563251 0.024139
0.48214 0.013486
0.218417 0.008532
0.213541 0.006520
0.302162 0.00703
0.765864 0.14588
0.65350 0.15793
Since in the denominator of each formula we have the same value characterizing
the change in the return of the resource scale, the meaning of the coefficients is
determined by their numerators.
The first coefficient comprising the real part of the complex proportionality
coefficient shows the increment of capital resources, and the second one that of
labor resources.
Let us estimate the values of the formulae for calculating the coefficients for the
above-mentioned examples.
Table 5.4 give the results of calculations for the economy of the Soviet Union.
The real part of the complex proportionality coefficient varies around some
average equal to 0.535. These variations have a wide range, however, the coefficient does not show explicit trends in the changes in values.
The imaginary part of the proportionality coefficient showing to some extent the
intensity of use of the labor force demonstrates a decrease in its values with time. If we
try to interpret this trend, we could say that year over year, in the Soviet Union the labor
forces were used less and less extensively, i.e., more and more intensively compared to
the previous year. This means a growth in labor productivity. This conclusion was
drawn using model (5.33). In the first approximation, this change of trend in the labor
force extensiveness coefficient can be described by a linear trend model:
a1t ¼ 0:1575
0:0137t
5.3 Linear Production Function of a Complex Argument
161
Table 5.5 Original data for building of a production function and calculated values of resource
use coefficients
Resource extensiveness
coefficients
Year
1998
1999
2000
2001
2002
2003
2004
GDP
increment,
DQt
Increase in
investments
in fixed capital, DKt
0.834
0.944
0.623
0.718
0.932
1.329
0.651
1.209
0.842
0.629
1.04
2.256
Increase in number of
people participating
in the economy, DL t
0.014
0.023
0.005
0.027
0.006
0.018
a0
38.7810
49.62157
104.9132
16.72674
161.5470
166.5680
a1
0.018330
0.017623
0.003678
0.029552
0.005410
0.010520
Then, with the help of linear PFCA, the production process in the former USSR
can be more or less successfully described by the model
DQt ¼ ð0:535 þ ið0:1575
0:0137tÞðDKt þ iDLt Þ
Where t ¼ T-1972, Т – current year.
It should be noted here that formulae (5.45) and (5.46) will tend to infinity if
capital and labor resource increments are minor and close to zero.
Let us see what this model of linear PFCA could provide for the example of
Russian economy. Since we are interested in the possibilities of this model we could
consider the data given in Table 5.3.
Table 5.5 shows the results of calculation of capital and labor resource extensity
coefficients. From the analysis of variations of these coefficients in time we can see
that neither the first, nor the second coefficient demonstrate any variation trend.
Therefore, linear PFCA cannot be applied for modeling of the economy of Russia
with the data used.
Naturally, it is not necessary to calculate coefficient values for each observation
but estimate their values on the whole set of observations using LSM.
Thus, for centered original data (i.e. for model (5.33)), to calculate LSM
estimations of model of complex argument one should solve the following complex
equation:
P
Qt ðKt þ iLt Þ
a0 þ ia1 ¼ P
ðKt þ iLt Þ2
(5.47)
With reference to the former USSR, LSM estimations of the complex
proportionality coefficient appeared to be as follows:
a0 þ ia1 ¼ 0:52936
i0:03976
162
5 Production Functions of Complex Argument
Calculation of coefficient values for each observation and calculation of their
average values given earlier provided the following values of the complex
proportionality coefficient:
a0 þ ia1 ¼ 0:53422 þ i0:04278
The values of the real part of the coefficient are close to each other, and those of
the imaginary part are different. This is why, taking into account the fact that LSM
estimations are free of the shortcomings of the calculation procedure at each
observation, economists who decide to use a model of a production function in
the form of a linear complex argument function should use LSM estimations.
Since we are considering a new production function, we should determine a very
important characteristic of the production function in general – coefficients of
elasticity of production results for resources. The linear function without a free term
Qt ¼ ða0 þ ia1 ÞðKt þ iLt Þ
is of interest.
An elasticity coefficient represents the following value for discrete values:
Dy
y
eyx ¼ Dx :
x
For a continuous function, the elasticity coefficient may be written as follows:
dy
y
eyx ¼ @x ¼
x
dy x
:
dx y
(5.48)
The elasticity coefficient for capital for the function under consideration will
look like this:
eK ¼
@Q K @Q
K
¼
:
@K Q @K ða0 þ ia1 ÞðK þ iLÞ
(5.49)
A partial derivative of the linear PFCA for capital will be equal to
@Q
¼ ða0 þ ia1 Þ:
@K
(5.50)
Substituting this value of the partial derivative into (5.49) we obtain the formula
for the coefficient of the production elasticity of capital:
eK ¼
K
:
K þ iL
(5.51)
5.3 Linear Production Function of a Complex Argument
163
This means that the coefficient of production elasticity of capital is a complex
value and varies with variations in the number of observations of a complex resource.
Similarly, one can determine the coefficient of production elasticity of labor:
eL ¼
@Q L @Q
L
¼
:
@L Q @L ða0 þ ia1 ÞðK þ iLÞ
(5.52)
Since the partial derivative of the output volume for labor is
@Q
¼ iða0 þ ia1 Þ;
@L
(5.53)
the necessary coefficient of elasticity of the output volume of labor will also be
complex:
iL
:
K þ iL
(5.54)
eK þ eL ¼ 1:
(5.55)
eL ¼
It is easy to see that
Since the elasticity coefficient of a production function shows the change in the
production result in percentage terms with a 1% change in the resource, let us
determine the essence of the coefficients obtained. Each of them, (5.51) and (5.52),
shows the change in the production result with changes in the resource under the
influence of the other resource.
If the complex argument changes by 1%, the production result, as follows from
(5.55), will also change by one.
Isoquantum is an important characteristic of production functions. It represents
an aggregate of points in the resource plane, each of which is associated with the
same value of the production result, i.e., Q ¼ Qc ¼ const.
The isoquantum equation for the model of production function without a free
term will have the form
K¼
Qc
a1 L
a0
:
(5.56)
The obtained equation shows that isoquantum represents a set of parallel lines
shifting upward to the right with an increase in the production volume.
164
5.4
5 Production Functions of Complex Argument
Power Production Function
A two-factor linear relationship in the domain of real variables represents an
equation of a plane in three-dimensional space. The linear complex argument
model, as was discovered previously, represents an equation of a straight line.
Nonlinear two-factor models of real variables represent an equation of nonlinear
surfaces in three-dimensional space, and nonlinear complex argument models,
accordingly, represent an equation of some curved line in three-dimensional
space. The specifics of these lines were considered in Chap. 2. Let us now consider
nonlinear models of a PFCA.
We start with the power model, which is traditional for the theory of production
functions. From the whole variety of complex-valued power functions we first take
the functions with real coefficients. It has the form
Qt ¼ aðKt þ iLt Þb :
(5.57)
Its linear form looks as follows:
ln Qt ¼ ln a þ b lnðKt þ iLt Þ:
(5.58)
For the real and imaginary parts of the linear function, using the principal
logarithm value, we obtain the following equalities:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8
>
< ln Qt ¼ ln a þ b ln L2t þ Kt2 :
L
>
: 0 ¼ barctg t
Kt
(5.59)
From the second equality we have an obligatory condition: Lt ¼ 0, which means
that it is not possible to use this model in modeling production processes.
After the introduction of a complex proportionality coefficient into the model,
Qt ¼ ða0 þ ia1 ÞðKt þ iLt Þb :
(5.60)
Taking the logarithms of the left- and right hand sides we get
ln Qt ¼ ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1
a20 þ a21 þ iarctg þ b lnðKt þ iLt Þ:
a0
(5.61)
Then, taking separately the real and imaginary parts of the obtained equalities,
we can represent the model as a system of two equalities:
5.4 Power Production Function
165
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8
>
< ln Qt ¼ ln a20 þ a21 þ b ln L2t þ Kt2 ;
>
: arctg a1 ¼
a0
barctg
Lt
:
Kt
(5.62)
It follows from the system above that the power PFCA (5.60) can be used when
there is a linear relationship between the resources with a constant angle between
them, i.e., the line goes through the zero point.
Incidentally, the relationship between the production resources and production
result is complicated and can be determined from the exponential form of model
(5.60):
Qt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b
a20 þ a21 ðL2t þ Kt2 Þ :
(5.63)
Let us now find the values of the elasticity coefficient of this function (5.60) for
capital and labor.
The coefficient of production elasticity for capital,
eK ¼
@Q K @Q
K
¼
;
@K Q @K ða0 þ ia1 ÞðK þ iLÞb
(5.64)
can be determined if we know the partial derivative of the function of a complex
argument of capital:
@Q
¼ bða0 þ ia1 ÞðK þ iLÞb 1 :
@K
(5.65)
Then the coefficient of production elasticity of capital will have the form
eK ¼
bK
K þ iL
(5.66)
Similarly, we can determine the elasticity coefficient of function (5.60) for labor:
eL ¼
ibL
:
K þ iL
(5.67)
If not one but the entire complex resource varies by 1%, the production result
will change by
e ¼ eK þ eL ¼ b:
(5.68)
Thus, the exponent of the function under consideration is the coefficient of
general elasticity of production for a complex argument.
166
5 Production Functions of Complex Argument
Now let us define the isoquantum equation. If Q ¼ Qc ¼ const, the isoquantum
equation for model (5.60) will have the form
L2t
þ
Kt2
¼
!1b
Qc
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
a20 þ a21
(5.69)
which means that isoquanta of model (5.60) represent circumferences in the
production resource plane with various diameters determined by production result
Qc: if it grows, the circumference radius grows too.
Let us now consider a model of PFCA with a complex proportionality coefficient
and imaginary exponent:
Qt ¼ ða0 þ ia1 ÞðLt þ iKt Þib :
(5.70)
Taking the logarithm and singling out the real and imaginary parts we obtain
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kt
>
>
ln
Q
¼
ln
a20 þ a21 barctg ;
<
t
Lt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
>
1
>
: arctg ¼ b ln L2t þ Kt2 :
a0
(5.71)
From the second equality we see that the relationship between the resources
represents a circumference. In three-dimensional space this means a model of onefourth of a cylinder perpendicular to the resource plane.
There is a complex nonlinear relationship between the production resources and
the production result:
Qt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a20 þ a21 e
barctgKLtt
:
(5.72)
Additional information on the function and its properties is obtained from the
elasticity coefficients. The elasticity coefficients of the production result for a
complex argument applicable to function (5.70) will be calculated as follows:
eK ¼
dQ
ðK þ iLÞ
:
dðK þ iLÞ
Q
(5.73)
To find it, we should calculate a derivative of this function for the complex
argument:
dQ
ða0 þ ia1 ÞðK þ iLÞib
¼
:
dðK þ iLÞ
dðK þ iLÞ
5.4 Power Production Function
167
To do that, let us represent function (5.70) in exponential form:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i
a hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L ib
iarctga1
0
L2 þ K 2 eiarctgK :
a20 þ a21 e
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiib
L
iarctga1
0
L2 þ K 2 e barctgK
¼ a20 þ a21 e
Qt ¼
According to the d’Alembert-Euler (Riemann-Cauchy) condition, we can find
the derivative of a complex-valued function by calculating the derivative of the real
part for each of the constituents of the complex argument. The real part of the
function will then have the form
Qt ¼ R cos y ¼ ae
barctgKL
cosða þ b ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1
L2 þ K 2 Þ; a ¼ a20 þ a21 ; a ¼ arctg :
a0
(5.75)
The first derivative will be
dQ
@Q
¼
dðK þ iLÞ @K
i
@Q
:
@L
(5.76)
The first partial derivative of the volume for capital,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
barctgKL
@
ae
cos
a
þ
b
ln
L2 þ K 2
@Q
;
¼
@K
@K
(5.77)
can be defined as the derivative of a complex function:
L
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@Q @ðae barctgK Þ
cos a þ b ln L2 þ K 2
¼
@K
@K
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@ðcosða þ b ln L2 þ K 2 ÞÞ
L
þ
ae barctgK :
@K
(5.78)
Its first part will be equal to
L
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@ðae barctgK Þ
cos a þ b ln L2 þ K 2
@K
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@ð barctg KL Þ
L
ae barctgK cos a þ b ln L2 þ K 2 :
¼
@K
(5.79)
Since
@ð barctg KL Þ
¼
@K
b
@ KL
@K
1 þ ðKL Þ
2
¼
b KL2
2
1 þ ðKL Þ
;
(5.80)
168
5 Production Functions of Complex Argument
the first part of derivative (5.78) will have the form
b KL2
1 þ ðKL Þ
ae
2
barctgKL
cosða þ b ln
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2 þ K 2 Þ ¼
b KL2
1 þ ðKL Þ
2
R cos y ¼ b
bL
R cos y:
K 2 þ L2
(5.81)
Now let us consider the second part of the derivative of complex function (5.78):
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@ cos a þ b ln L2 þ K 2
bK
L
ae barctgK ¼ R sin y 2
:
(5.82)
@K
L þ K2
Taking into account (5.81) and (5.82) the first partial derivative of the volume for
capital will be
@Q
bL
¼
R cos y
@K K 2 þ L2
bK
bR
R sin y ¼ 2
ðL cos y
K 2 þ L2
K þ L2
K sin yÞ:
(5.83)
Let us now find the first partial derivative for labor since it constitutes the second
term of the unknown derivative:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
barctgKL
@
ae
cos
a
þ
b
ln
L2 þ K 2
@Q
¼
:
(5.84)
@L
@L
Let us consider it as a derivative of a complex function:
L
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@Q @ðae barctgK Þ
cosða þ b ln L2 þ K 2 Þ
¼
@L
@L
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@ cos a þ b ln L2 þ K 2
L
þ
ae barctgK :
@L
(5.85)
The first part of this sum will be equal to
@ðae
barctgKL
@L
Þ
@ð barctg KL Þ
ae
@L
cos y ¼
barctgKL
cos y:
(5.86)
bK
;
K 2 þ L2
(5.87)
Since
@ð barctg KL Þ
¼
@L
b
@ KL
@L
1þ
2
ðKL Þ
¼
b
K
1þ
2
ðKL Þ
¼
for the first term (5.85) we have
bK
R cos y:
þ L2
K2
(5.88)
5.4 Power Production Function
169
The second part of the derivative of a complex function is
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
@ cos a þ b ln L2 þ K 2
bL
R ¼ R sin y 2
:
@L
L þ K2
(5.89)
Then the first derivative of the function of volume for labor under consideration
can be written taking into account (5.88) and (5.89) as follows:
@Q
¼
@L
bK
R cos y
K 2 þ L2
R sin y
bL
¼
L2 þ K 2
bR
ðK cos y þ L sin yÞ:
K 2 þ L2
(5.90)
Now we can obtain a general formula for the first derivative of a complex
function for a complex argument by substituting the values of the first, (5.83),
and second, (5.90), terms into (5.76):
@Q
bR
¼
ðL cos y
@ðK þ iLÞ K 2 þ L2
K sin yÞ
i
bR
ðK
cos
y
þ
L
sin
yÞ
:
K 2 þ L2
(5.91)
Grouping will give us
@Q
bR
¼
½Lðcos y þ i sin yÞ
@ðK þ iLÞ K 2 þ L2
iKðcos y þ i sin yÞ;
(5.92)
which is easily transformed into
@Q
ðL iKÞ
bReiy
¼ 2
:
¼
@ðK þ iLÞ K þ L2
L þ iK
(5.93)
Now it is possible to find the coefficient of elasticity of the required function for
a complex argument:
eKþiL ¼
dQ
Q
bReiy K þ iL
=
¼
¼ ib:
dðK þ iLÞ K þ iL L þ iK Reiy
(5.94)
The elasticity coefficient of the function under consideration is an imaginary
value!
The elasticity coefficient of the function under consideration for capital can be
calculated using the formula obtained for the first partial derivative of the function
for capital (5.83):
eK ¼
@Q Q
bR
= ¼
ðL cos y
@K K K 2 þ L2
K sin yÞ=
Reiy
bKðL cos y K sin yÞ
¼ 2
:
ðK þ L2 Þðcos y þ i sin yÞ
K
(5.95)
170
5 Production Functions of Complex Argument
Using the previously derived values of the partial derivative of the function for
labor (5.90) it is also easy to find the elasticity of the volume for this resource:
eL ¼
@Q Q
= ¼
@L L
bR
Reiy
¼
ðK
cos
y
þ
L
sin
yÞ=
K 2 þ L2
L
bLðK cos y þ L sin yÞ
:
ðK 2 þ L2 Þðcos y þ i sin yÞ
(5.96)
If we assume that the production result is a constant value, we can find an
equation of the isoquantum of this production function.
From (5.72), on the assumption of a constant result, we will have an obvious
equality:
barctg
Kt
¼ const:
Lt
(5.97)
This means that the isoquantum represents a straight line in the resource plane
issuing from the point of origin.
It is easy to notice that a change in the exponent from the real to the imaginary
symmetrically changed the properties of the respective parts.
Therefore, with a restriction on the form of resource change (cylinder), the
dependence of the production result on the nth resources shown in the space
represents a nonlinear curve located on the cylinder surface. The form of this
model is much more complicated than that of the model with a real exponent.
The power form of the PFCA can have a more complicated form if a complex
exponent is used:
Qt ¼ a0 ðKt þ iLt Þðb0 þib1 Þ :
(5.98)
Taking the logarithms of the left- and right-hand side of the function we obtain
ln Qt ¼ ln a0 þ ðb0 þ ib1 Þ lnðKt þ iLt Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
¼ ln a0 þ ðb0 þ ib1 Þðln L2t þ Kt2 þ iarctg Þ:
Kt
(5.99)
Then, singling out the real and imaginary parts yields
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
>
>
< ln Qt ¼ ln a0 þ b0 ln L2t þ Kt2 b1 arctg ;
Kt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
>
>
: 0 ¼ b0 arctg þ b1 ln L2t þ Kt2 :
Kt
(5.100)
It follows from the second equality of this system that model (5.98) is suitable
for modeling production processes characterized by a complex nonlinear
5.4 Power Production Function
171
relationship between the resources, which are not easy to represent in an explicit
form. Since for various values of coefficients b0 and b1 this relationship takes
various forms, it is more applicable in practice than previous models of power
PFCAs.
Power PFCAs with complex coefficients should be considered universal:
Qt ¼ ða0 þ ia1 ÞðKt þ iLt Þðb0 þib1 Þ
(5.101)
since when we take logarithms of the left- and right-hand sides of this equation and
single out the real and imaginary parts, we obtain
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
>
2 þ a2 þ b ln
>
¼
ln
a
L2t þ Kt2 b1 arctg ;
ln
Q
<
t
0
0
1
Kt
(5.102)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1
Lt
>
>
2
2
: 0 ¼ arctg þ b0 arctg þ b1 ln Lt þ Kt :
a0
Kt
Various combinations of this model make it possible to model most varied
nonlinear dependences of the production result from resources, which vary from
linear (for b1 ¼ 0) to complex nonlinear ones. Coefficients of these models should
be found using LSM, as was pointed out in Chap. 4.
However, use of a complex exponent makes it possible to find intermediate
values of this indicator. To do that, let us take the ratios of the left- and right-hand
sides of the model at close times. We obtain
Qt
¼
Qt 1
Kt þ iLt
Kt 1 þ iLt
1
b0 þib1
:
(5.103)
From here it is easy to find a complex exponent:
bt0 þ ibt1 ¼ ln
Qt
Kt þ iLt
= ln
:
Qt 1
Kt 1 þ iLt 1
Let us demonstrate the change in the exponent for this model for the Russian
economy, taking a longer series of data from 1995 to 2009. The results of calculation of the complex exponent are given in Table 5.6.
During the entire calculation period the complex exponent varies in value. This
is especially evident for the default situation of 1998–1999.
The calculated value of the coefficient has changed significantly. If we consider
the complex exponent for the following period, we see that its real and imaginary
parts did not vary so dramatically. This means that for this period model (5.98)
describes production more or less satisfactorily.
There is no need to carry out complex calculations and prove that the elasticity of
the function of the complex argument under consideration (5.98) will be equal to
eKþiL ¼ b0 þ ib1
(5.104)
172
5 Production Functions of Complex Argument
Table 5.6 Original data (in dimensionless quantities) for building a power PFCA and calculated
values of complex exponent
Coefficients
Year
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
GDP, Qt
1
1.406
1.640
1.841
3.376
5.114
6.261
7.574
9.246
11.919
15.127
18.843
23.274
29.001
27.371
Fixed assets, Kt
1
2.523
2.564
2.726
2.749
3.204
3.906
4.714
5.853
6.280
7.404
8.457
10.469
12.457
14.371
Economically active
population, L t
1
0.983
0.961
0.950
1.019
1.021
1.008
1.022
1.028
1.029
1.042
1.046
1.059
1.071
1.056
b0
0.373
5.833
1.836
15.272
2.715
1.016
1.016
0.924
3.609
1.450
1.653
0.990
1.266
0.404
b1
0.238
6.783
0.801
19.114
0.919
0.312
0.222
0.176
0.603
0.204
0.210
0.105
0.111
0.035
since this function can also be represented as follows:
Qt ¼ ða0 þ ia1 ÞðKt þ iLt Þb0 ðKt þ iLt Þib1 ;
(5.105)
which will definitely result in (5.104).
An indisputable advantage of PFCA is that its application is much broader than
that of linear models since for b0 ¼ 1 and b1 ¼ 0 model (5.98) is transformed into a
model of a linear PFCA, which testifies to the fact that the linear model of PFCA is
a special case of a power model. Incidentally, for b0 ¼ 1 and b1 ¼ 0 an inversely
proportional complex-valued relationship is modeled, which simulates stagnation –
an increase in production resources in this model results in a reduction of the
production result.
5.5
Exponential Production Function of Complex Argument
From among the family of possible models of exponential PFCAs let us consider an
exponential function with a natural base, taking into account that other bases are
decimal. Binary and other bases can be applied similarly.
Studies of this function start according to the principle of from the simple to the
complex. The model of an exponential function with real coefficients is the simplest
in this family:
Qt ¼ aebðKt þiLt Þ :
(5.106)
5.5 Exponential Production Function of Complex Argument
173
This function may easily be transformed as follows:
Qt ¼ aebKt eibLt :
(5.107)
Due to the fact that one can speak about an equality of complex variables only if
their real and imaginary parts are equal to each other, one can see that this model
means a system of two equalities:
(
Qt ¼ aebKt ;
2pk ¼ bLt ;
(5.108)
where k ¼ 0,1,2,3,. . .
Of course, it is more convenient to consider k ¼ 0. In any case, the second
equation of the system (5.108) testifies to one thing – labor resources are considered
here as a constant value. From the first equality it is clear that this model represents
a single-factor power dependence of the production volume on capital resources.
Both the first and second equations of the system show that this model describes the
effect of capital on production volume with a constant labor force, which influences
neither the result nor the capital.
An exponential function with an imaginary exponent has a similar meaning,
though it is symmetrical with reference to production resources:
Qt ¼ aeibðKt þiLt Þ :
(5.109)
This function can easily be transformed into
Qt ¼ aeibKt e
bLt
;
(5.110)
where
(
Qt ¼ ae bLt ;
2pk ¼ bKt ;
(5.111)
which again shows that it is the single-factor dependence of the production result on
labor, with constant capital resources, that is being modeled.
If now we use not the real but the imaginary proportionality coefficient, the
practical value of the model will not change:
Qt ¼ iaebðKt þiLt Þ :
(5.112)
This function can also be transformed into a convenient form:
Qt ¼ aeiðp=2Þ ebKt eibLt :
(5.113)
174
5 Production Functions of Complex Argument
The argument of the imaginary proportionality coefficient is determined up to
one period. Now it is easy to obtain a system of two equations characterizing the
real and imaginary parts of model (5.112):
8
< Qt ¼ aebKt ;
(5.114)
p
: ¼ bLt :
2
This means that model (5.112) implies that the capital resource is a priori
constant.
Let us make the model more complicated due to a complex proportionality
coefficient with a real proportionality coefficient of the exponent:
Qt ¼ ða0 þ ia1 ÞebðKt þiLt Þ
(5.115)
This complex-argument model can be represented as equalities of the modulus
and the argument, which comprises the following system of equations:
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
< Qt ¼ a20 þ a21 ebKt ;
a
>
: 2pk ¼ arctg 1 þ bLt :
a0
(5.116)
We see again that this model implies the a priori constant nature of the labor
resource [the second equation of system (5.116)] and the single-factor dependence
of the production result on capital resources.
Thus, in contrast to the model of a power PFCA, the model of an exponential
PFCA is not diversified and for practical purposes can be presented only with a
complex exponent:
Qt ¼ ða0 þ ia1 Þeðb0 þib1 ÞðKt þiLt Þ :
(5.117)
Singling out its modulus and argument, we obtain the following system of
equations:
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0 Kt
>
e
>
< Qt ¼ a20 þ a21
;
e b 1 Lt
(5.118)
a1
>
>
: 2pk ¼ arctg þ b0 Lt þ b1 Kt :
a0
It follows from the second equation that the model of an exponential power
function of a complex argument implies an a priori linear relationship between the
production resources.
The first equation shows that the exponential model is an analog of complexvalue function (5.117) in the domain of real variables (for a linear variation of
resources):
5.6 Logarithmic Production Function of Complex Argument
Qt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0 Lt
e
a20 þ a21 b K ¼ aeb0 Lt
e1 t
175
b 1 Kt
:
Its logarithm will look as follows:
ln Qt ¼ A þ b0 Lt
b1 K t :
(5.119)
Graphically, the model will represent an exponent located in a plane perpendicular to the resource axis, where all the points satisfy the second equality of system
(5.118).
It is not difficult to find coefficients of this model using LSM.
However, it is possible to discover if the model is suitable for describing real
economic production situations without appealing to LSM.
For that, let us divide the left- and right-hand sides of equality (5.117) at time
t by the the left- and right-hand sides of the same equality at the previous moment:
Qt
¼ eðb0 þib1 ÞðDKt þiDLt Þ :
Qt 1
(5.120)
Then we can determine the complex exponent basing on two values of the
original variables:
b0 þ ib1 ¼
ln QQt t 1
DKt þ iDLt
:
(5.121)
Let us demonstrate this possibility for the Russian economy. The original data
are given in Table 5.7.
It is seen from the table that the complex exponent of an exponential PFCA
varies, especially its real part. This testifies to the fact that this model is not suitable
for modeling the current production process.
We see that the power model of a complex argument considered in Sect. 5.4
possesses much more interesting properties than the exponential model, which is
applicable for practical purposes. There might also be cases where the exponential
functions of a complex argument will better describe some production process than
the power one, but they will be few in number. This is why in this section we do not
study elasticity coefficients of this production function.
5.6
Logarithmic Production Function of Complex Argument
A logarithmic function of a complex variable is a periodical function and, as we
agreed at the beginning of the book, we will use only the principal value of the
logarithm. In this case the logarithmic PFCA will have the form
Qt ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðKt þ iLt Þ:
(5.122)
176
5 Production Functions of Complex Argument
Table 5.7 Exponential PFCA and calculated values of complex exponent
Year
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
GDP, Qt
1
1.406
1.640
1.841
3.376
5.114
6.261
7.574
9.246
11.919
15.127
18.843
23.274
29.001
27.371
Fixed assets, Kt
1
2.523
2.564
2.726
2.749
3.204
3.906
4.714
5.853
6.280
7.404
8.457
10.469
12.457
14.371
Economically active
population, L t
1
0.983
0.961
0.950
1.019
1.021
1.008
1.022
1.028
1.029
1.042
1.046
1.059
1.071
1.056
Coefficients
b0
2.891
0.711
2.717
0.912
0.288
0.235
0.175
0.595
0.212
0.209
0.105
0.111
0.030
0.000
b1
0.002
1.562
0.046
7.959
0.004
0.005
0.004
0.001
0.001
0.002
0.001
0.001
0.001
0.000
In this function, the effect of the free term is evident – the real part of this
coefficient a0 characterizes a shift in the production result for initial values of the
variables, and the imaginary part of coefficient a1 characterizes the correction of the
imaginary part of the equality. Since there are no other interpretations or influences
on the results of modeling production, we can neglect this coefficient to consider the
model properties. Then the model of the logarithmic PFCA can be presented in the
following general form:
Qt ¼ ðb0 þ ib1 Þ lnðKt þ iLt Þ:
(5.123)
This function can easily be transformed into a form convenient for investigation,
namely, a system of two equalities, real and imaginary parts:
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
>
>
< Qt ¼ b0 ln L2t þ Kt2 b1 arctg ;
Kt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L
>
>
: 2pk ¼ b1 ln L2t þ Kt2 þ b0 arctg t :
Kt
(5.124)
The second equality of this system shows the relationship between production
resources, and the first one represents an analog of a complex-valued model on a set
of real numbers.
As we can see from the second equality, this model implies the most varied
forms of relationship between the production resources. These forms are determined first of all by the values of coefficients b0 and b1. Thus, for example, if
b0 ¼ 0, the relationship between the resources should be described by the equation
5.6 Logarithmic Production Function of Complex Argument
Table 5.8 Complex
proportionality coefficient
of logarithmic PFCA
177
Coefficients
Year
b0
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
0.470
1.239
1.445
1.578
2.831
3.961
4.346
4.726
5.139
6.392
7.483
8.766
9.871
11.468
10.252
b1
1.066
0.462
0.514
0.499
0.934
1.007
0.787
0.641
0.501
0.561
0.520
0.504
0.423
0.389
0.282
of a circle. If b1 ¼ 0, the relationship between the resources is a straight line. If
these coefficients are not equal to zero, the relationship between the production
resources can take the most diverse forms, for example isoquanta of “neoclassical
production functions.” From this point of view, the model of the logarithmic PFCA
is universal. The analytical properties of the logarithmic function of a complex
argument were studied in Chap. 2; therefore, it would be unnecessary to focus on
those results here.
Since the first equality of the system modeling the relationship between the
production resources and the production result also represents a combination of
the equation of a circle and the equation of an arctangent, it can, depending on the
values and signs of the coefficients, describe rather varied combinations of a
straight line and circle. A curve in space that results from the intersection of these
two nonlinear surfaces is complicated itself, and the model is able to describe
complicated trajectories of the development of production systems.
The obvious fact that (5.126) represents a system of two equations with two
unknowns is an important advantage, which is why the model coefficients can be
found by means of only one observation of the production process. That is, the
practical application of this model appears simple.
In fact, from (5.123) we derive a formula for the complex proportionality
coefficient:
b0 þ ib1 ¼
Qt
:
lnðKt þ iLt Þ
(5.125)
Again, let us apply this formula to the data on the Russian economy over the last
years. Table 5.8 gives only the results of calculation of the complex proportionality
coefficient.
178
5 Production Functions of Complex Argument
Table 5.9 Complex
proportionality coefficient
of logarithmic PFCA with
the free term
Coefficients
b1
b0
Year
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
0.444122
8.863934
3.191134
38.66129
11.36293
5.759372
7.008043
7.741272
37.98628
19.5134
27.95692
20.77293
32.95931
11.3934
0.444122
0.282994
10.30833
1.391758
48.387
3.844329
1.769688
1.528732
1.471488
6.349217
2.739132
3.559588
2.194427
2.880673
0.99884
0.282994
The results of the calculations of the coefficients of the logarithmic model show
that both parts of the complex coefficient increase in time. Beginning in 2001 they do
it almost linearly. This is why the logarithmic function can be applied in practice using
trends to describe variation in the coefficients over time. Then the model describing
properly the original production series in Russia will have the following form:
Qt ¼ ½ð0:9281t þ 2:9644Þ þ ið0:0498t
0:7611Þ lnðKt þ iLt Þ:
Obviously, the logarithmic model with a free term (5.122) will be more adequate
for real production processes:
Qt ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðKt þ iLt Þ:
To verify the possibility of modeling macro-level production in Russia, model
coefficients should be calculated. Let us use the differences between the left- and
right-hand sides of the exponents that differ from each other by a unit of time:
Qt
Qt
1
¼ ðb0 þ ib1 Þ ln
Kt
Kt þ iLt
1 þ iLt
:
(5.126)
1
Hence it is easy to calculate the complex proportionality coefficient for the
logarithmic PFCA with a complex free term (5.122):
b0 þ ib1 ¼
Qt Qt 1
:
þiLt
ln KtK1t þiL
t 1
(5.127)
Substituting these coefficients into (5.122), it is easy to calculate the value of the
free complex term. Let us show that this procedure is even possible for the same
example with respect to Russia. Calculations of the complex proportionality coefficient are given in Table 5.9.
5.7 Summary
179
Over time, both coefficients in the table vary in their values and signs. This
variation is virtually unpredictable and is characterized by a high variance; therefore, the conclusion will be as follows. The logarithmic PFCA with a free term
cannot be used for modeling this process.
It is again evident that the power PFCA is more interesting for researchers and
practicing economists than the logarithmic function of a complex argument.
In conclusion it should be noted that the logarithmic function is a periodic
function and one should always take that into account when using it in practice.
5.7
Summary
The theory of functions of a complex variable does not consider models of a
complex argument when the real variable depends on the complex one. This may
have been done in specific chapters of TFCV but they are not accessible for
analysis. Since a complex argument represents a two-factor relationship, one
could speak of an analogy between two-factor production functions of real
variables and models of a complex argument.
This chapter showed that models of a complex argument can operate as models
of production functions, that they can model various production processes. The
examples given in this chapter demonstrated that in certain cases particular models
are quite good at describing real production processes at the macro level, though
other models are not applicable. But the scope of production processes is not limited
to the examples with real data given in this chapter. There are many more. Real
production processes are so diverse that they cannot be fit into the Procrustean bed
of neoclassical functions as most economists do. These models often distort reality.
This is why an extension of the tools of the theory of production functions is
quite timely. PFCA turns out to be rather helpful here. The research whose results
are provided in Chap. 5 shows that each of the considered models of a complex
argument will be the best at describing some particular original production process.
Calculation of the trajectories of a production process for each of the considered
production functions of a complex argument on conditional examples shows that
they generate the most diverse forms – convex, concave, increasing, decreasing,
with asymptote, and without it. . . Readers can obtain these trajectories independently, substituting their own series into the model formulae and setting various
coefficient values. This is why it was senseless to provide the entire range in this
chapter – it was sufficient to show the main way to use complex-argument models
in economics. Its use in modeling production processes enriches the economist’s
arsenal of tools.
The possibility of building a linear two-factor model under conditions of
multicollinearity turned out to be an extremely important result that revealed in
this chapter new properties of the mathematical apparatus of the TFCV referring to
solution of economic problems. It was shown in this chapter that the method of real
variables does not provide a satisfactory solution of this problem, while TFCV
180
5 Production Functions of Complex Argument
demonstrated quite promising results. The two-factor linear model of real variables
represents an equation of a plane in three-dimensional space. This is why in a
situation where points lying in this space are on a straight line (multicollinearity)
LSM demonstrates an inability to find an equation of this plane. It is this straight
line in three-dimensional space that is described by a linear function of a complex
argument. This is why LSM applied to this function provides stable estimations of a
two-factor model under conditions of multicollinearity.
Since nonlinear functions of a complex argument often mean a linear dependence between the resources, the problem of the influence of multicollinearity on
the results of building two-factor nonlinear models can also arise in this case.
Nonlinear functions of a complex argument will provide stable values of model
coefficients since this is the only case where they may exist.
We should also point out an important feature that distinguishes models of a
complex argument from two-factor models of real variables. Models of real
variables describe a plane (in the linear case) or nonlinear surfaces in threedimensional space, and complex-argument models describe a straight line lying
in a plane (in the linear case) or a curve lying on the surfaces perpendicular to the
plane of production resources.
This fact shows that PFCAs has quite a narrow sphere of application compared
to real-variable models, but they describe those very processes that are poorly
described by models of real variables.
All the power of the TFCV, the significant advantages of models involving
TFCV tools are demonstrated in the building of model dependences of one complex
variable on another. Here, complex-argument models appear to be an “introduction” to the new apparatus of the theory of production functions, namely, the
apparatus of production functions of complex variables.
References
1. Svetunkov SG (1994) Modeling under conditions of multicolinearity. Indus Power Eng N
6:28–32
2. Corn G, Corn T (1984) Reference book on mathematics for scientists and engineers. Nauka,
Moscow
3. Svetunkov SG, Svetunkov IS (2005) Study of properties of production function of complex
argument (preprint). Publishing House SPbGUEF, St. Petersburg
4. Svetunkov SG, Svetunkov IS (2008) Production functions of complex variables. LKI, Moscow
5. Short-term economic parameters of the Russian federation (Goscomstat of Russia) http://www.
cir.ru. Accessed Oct 2004
Chapter 6
Production Functions of Complex Variables
The whole power of the apparatus of TFCV is demonstrated with reference to
economic modeling on production functions of complex variables (PFCVs) – when
the complex production result depends on the complex production resource. In this
chapter the complex production result is represented as a variable with the gross
margin as its real part and production costs as its imaginary part. Capital refers to
the real part of the complex resource and labor to the imaginarypart. The main
properties of linear, power, logarithmic, and exponential models of complex
variables are considered consistently. Numerous complex coefficients of effectiveness are derived and their economic interpretation is given. Here we show how
complex-valued econometrics can be used to model real production systems.
6.1
General Provisions of the Theory of Production
Functions with Complex Variables
Complex-argument functions represent a certain “truncation” of properties of
functions with complex variables – they described the dependence of a real variable
on a complex one that served as as the complex argument of a function. Mere
formulation of the problem with reference to one of the branches of economics – the
theory of production functions – has already yielded new scientific results. Therefore, even more diverse and exciting results are naturally expected from the use of
functions with complex variables, i.e., the dependence of one complex variable on
another. Since a complex variable inherently represents some two-factor model, the
focus will be on the dependence of one pair of economic indicators on another.
With reference to economic problems it is natural to assume that one pair – the
complex argument – can represent production indicators. Such a dependence that
connects production resources with the production result will be a production
function.
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
181
DOI 10.1007/978-1-4614-5876-0_6, # Springer Science+Business Media New York 2012
182
6 Production Functions of Complex Variables
Formally, production functions represent some sort of mathematical dependence
of the production result on the production resources given a number of assumptions.
Of the majority of production resources only two are used to build production
functions – production capital K in its various forms and labor L. These two
resources are interchangeable to a certain extent, which justifies their use.
In the previous chapter we used these very resources as one complex variable.
Material will be presented in this chapter that says there is not much difference in
terms of which variable we assign to the real and which one to the imaginary part of
the complex variable of a complex production resource (argument). Properties of
models of complex-argument production functions most often varied symmetrically without changing their essence.
In models of PFCVs, the procedure for assigning both the complex argument and
the complex result to the real or imaginary parts has a pronounced economic sense.
This is why we will stick to a particular rule of creation of a complex production
argument – capital will constitute the real part and labor the imaginary part. Then
for these functions, the complex production result will have the form
Kt þ iLt
(6.1)
Since all the examples to be considered in this chapter refer only to socioeconomic dynamics, all the variables have a regulation index t. If there is a need to
build certain production functions on other than a time set, the index will be easily
replaced by another.
Production performance can be demonstrated by the most diverse technical and
economic indicators. In the theory of production functions we use mainly one of
them – the volume of produced and sold products Qt. Clearly, in this case
assumptions arise regarding production functions: that the demand is unsaturated,
that the price is unchanged, etc.
However, in real economic practice nobody judges production performance only
by the production volume (output); one should have an idea of successful economic
activity, which is confirmed by various indicators of economic efficiency, primarily
gross margin Gt, production costs Ct, and, based on those, profitability Rt.
Since the gross margin, production costs, and gross output are connected with
each other by a simple relationship
Qt ¼ Gt þ Ct
then, copying any pair of these three, one can calculate the third parameter.
These indicators are also helpful to calculate another indicator of economic
activity – profitability:
Rt ¼
Gt Qt Ct
¼
:
Ct
Ct
6.1 General Provisions of the Theory of Production Functions. . .
a real variables
b complex variables
Kt
Lt
Kt
production
function
183
production
function
Qt
iLt
Gt
iCt
Qt=Ct+Gt,
Rt=Gt /Ct
Fig. 6.1 Structural scheme of production functions of real variables (a) and complex variables (b)
To create a complex variable of the production result we need a pair of variables
showing various sides of one process and having the same dimension and scale.
Because various combinations of production resources lead to various
combinations of production costs and gross margin, and therefore various gross
output and profitability, it is gross margin Gt and production costs Сt variables that
should be parts of the complex variable of the production result.
It is proposed to represent the complex variable of the production result including gross margin Gt and production costs Сt in the following form:
Gt þ iCt :
(6.2)
We deliberately assign the gross margin to the real part and the costs to the
imaginary part of the complex variable of the production resources. This procedure
is defined by the formation of the complex variable of the production resources
(6.1) whose meaning will become clear in the course of study of the corresponding
production functions. In addition, Chap. 1 introduced a rule – we will assign the
indicator reflecting the active part of the socioeconomic process to the real part and
the passive one to the imaginary part. Comparing the gross margin and production
costs with each other one could probably say that economists pay more attention to
the former; therefore, from this point of view it seems to be quite in line with the
rule introduced to assign the gross margin to the real part of the complex variable.
Figure 6.1 gives two structural schemes that can help us obtain a visual idea of
the difference between production functions of real variables and those of complex
variables.
Production functions of real variables model the influence of production
resources on the gross output and PFCVs first model the influence of production resources on the gross margin and production costs and then, on the basis of
this information, their influence on the gross output. The correlation between the
gross margin and costs characterize profitability.
It is evident now that PFCVs provide a more detailed description of the production process than those of real variables, which is why one expects more reliability
and accuracy from a description of production processes from complex-valued
models.
Figure 6.1b helps us to understand that in the general form PFCVs can be
represented as follows:
Gt þ iCt ¼ FðKt þ iLt Þ:
(6.3)
184
6 Production Functions of Complex Variables
Many functions can us help associate, by dependence (6.3), two complex
variables Gt + iCt and Kt + iLt. We will use only those that were considered in
Chap. 3, except for Zhukovsky’s function and trigonometric functions because it is
very difficult to imagine a production process described by these models.
Since production processes differ from each other:
– By hierarchy (enterprise, group of enterprises, regional manufacturing, world
production, etc.),
– By the specificity of the production (agriculture, machine-building, light industry, oil refinery, electric power engineering, etc.), and
– By national geographic features (surplus laboring or scarce labor population;
availability of sources of raw materials and infrastructure; warm, hot, or cold
climate, etc.),
there can be no uniform standard production function of complex variables that
would best describe all these diverse production processes changing only the
coefficient values depending on the situation. In each case economists should
choose the best indicators from the many available ones. That is why this chapter
will focus on PFCVs of various forms from among those elementary functions
whose conformal mapping was studied in Chap. 3.
It follows from (6.3) that using complex-valued functions one can immediately
model two economic indicators – gross margin and production costs; however, this
was said to be a model of three production performance. After all, the sum of the
gross margin and production costs simply represents gross output:
Gt þ Ct ¼ Qt
(6.4)
Function (6.4) can also be presented in a different form, representing a complex
result in exponential form:
Gt þ iCt ¼ Rt eiyt ;
(6.5)
where we obtain
8
>
< Gt ¼ Rt cos yt ;
Ct ¼ Rt sin yt ;
>
:
Qt ¼ Rt ðcos yt þ sin yt Þ:
(6.6)
Here the polar angle can be found in the following way:
yt ¼ arctg
Ct
1
¼ arctg :
Rt
Gt
That is, it shows the profitability of production – the higher the polar angle, the
less efficient is the production. If a company is profitless but not running at loss, i.e.,
with zero profitability, the polar angle tends to plus infinity.
6.2 Linear Production Function of Complex Variables
185
More often, none of model forms (6.6) is used in modern economic and
mathematical modeling. Moreover, nor is the entire whole system of dependence
of a pair of production results on a pair of production resources used. One should
note the fact that finding estimations of model coefficients (6.6), for example using
the least-squares method (LSM), is most often an extremely complicated problem.
For this, one should resort to numerical methods of solving systems of nonlinear
equations. For modern scientists armed with computer equipment and software this
is not an obstacle, but for an economist whose knowledge of mathematics is at the
level of a university course this task is insurmountable. Coefficients of complexvalued functions can be found quite easily. In Chap. 4 we showed how to use LSM
for these purposes.
New economic indicators appear in PFCVs that have not been encountered in the
theory of production functions, which is based on real variables. These are moduli
of complex variables and their polar angles. If polar angles are more or less
understood – they characterize resources from the point of view of the capital-tolabor ratio (the tangent of a polar angle represents the labor-to-capital ratio) and the
production result from the point of view of profitability at cost – the interpretation
of the moduli of these complex variables is quite hard. In fact, the moduli of these
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
variables are RGC ¼ G2 þ C2 ; RKL ¼ K 2 þ L2. What economic meaning do they
have? They show the production scale and the resource scale, respectively.
PFCVs possess another unique property. An inverse relationship follows
from (6.3):
Kt þ iLt ¼ f ðGt þ iCt Þ:
(6.7)
This means that if we describe some production process using a PFCV, we can
build an inverse function (6.7), and using this we could solve a problem that has not
arisen in the modern theory of production functions –achieving a desired gross
margin, production costs, or production volume. What labor and capital resources
would have to be used to obtain a given profitability level? Function (6.7) makes it
rather easy to answer these questions, though functions of real variables will be
more complicated and will be represented by a system of equations.
6.2
Linear Production Function of Complex Variables
The simplest – linear – function of complex variables will have the form
Gt þ iCt ¼ ðb0 þ ib1 ÞðKt þ iLt Þ:
(6.8)
Since this function has only one complex coefficient, namely, the proportionality
coefficient (b0 + ib1), it will be the subject of our study of this function. From
equality (6.8) we easily obtain
186
6 Production Functions of Complex Variables
b0 þ ib1 ¼
Gt þ iCt
:
Kt þ iLt
Multiplying the numerator and denominator of the right-hand side of the equality
by the conjugate term we obtain
b0 þ ib1 ¼
Gt þ iCt Kt
Kt þ iLt Kt
iLt Gt Kt þ Ct Lt þ iðCt Kt
¼
iLt
L2t þ Kt2
Gt Lt Þ
:
After grouping the real and imaginary parts it is easy to derive each of the
coefficients b0 and b1:
b0 ¼
Gt Kt þ Ct Lt
;
Lt 2 þ Kt 2
(6.9)
b1 ¼
Ct Kt Gt Lt
:
Lt 2 þ Kt 2
(6.10)
In contrast to the coefficients of the production functions of a complex argument,
it is not easy to give an economic interpretation of each of the coefficients of the
function of complex variables (6.8). These coefficients have the same denominator,
which shows the resource scale; however, the numerators (6.9) and (6.10) greatly
differ and do not have clear economic parallels.
The coefficient b0 will grow linearly with both the growth of the production
volume (G + C) the growth of the gross margin and production costs, given
constant overhead costs. If both resources and performance grow according to the
direct proportionality law, this coefficient will stay constant. In all other cases, its
dynamics are more complex.
As for b1, it will grow with the growth of the prime cost and, to a certain extent,
with the growth of the number of people employed in production. The latter
relationship is nonlinear. Growth of the gross margin will be demonstrated by a
reduction in the values of this coefficient.
If we take the first observation as a starting point, making all the other values
relative, b0 in the first observation will be equal to one and b1 to zero. However, not
only the initial value but any other, for example, the last one, can be taken as the
starting point. Then, for this year’s observation of the production process, the
coefficient b0 will be equal to one and b1 to zero. Since in the real economy,
situations where the linear dependence of some complex indicator on a complex
factor is modeled with a zero value of the free term are quite rare, the model can be
transformed into this form by preliminary centering of the original variables around
their averages.
Opening the brackets of equality (6.8) and grouping the real and imaginary parts
of the equality we have
Gt þ iCt ¼ ðb0 þ ib1 ÞðKt þ iLt Þ $ Gt þ iCt ¼ b0 Kt
b1 Lt þ iðb0 Lt þ b1 Kt Þ;
6.2 Linear Production Function of Complex Variables
187
where for the real part of the equality
Gt ¼ b0 Kt
b1 Lt
(6.11)
Ct ¼ b0 Lt þ b1 Kt :
(6.12)
and for the imaginary part
The expressions obtained have a simple economic meaning and determine in
what cases of modeling production can a linear PFCV be used (if we consider the
coefficients to be positive). Equation (6.11) uniquely specifies that with the growth
of the capital resource, the gross margin increases, and with the growth of labor
resources it decreases. It follows from (6.12) that production costs increase. This
testifies to the fact that the modeled process is characterized by a permanent return
on the capital resource and a decrease in the return on labor resources.
However, if coefficient b1 is negative, the increase in labor resources leads to an
increase in profits and costs, and growth of the capital resources leads in this case to
the modeling of a situation where the gross margin increases and the costs decrease.
This means that there is a permanent return on the labor resource and – regardless of
the correlation between b0 and b1 – increasing, permanent, or decreasing return on
the capital resource.
This means that linear a PFCV possesses certain analytical properties that can
facilitate study of the core significance of the production processes of various
hierarchy levels. Apparently, this can be done only if the linear PFCV provides a
good description of the modeled production.
Since all the variables are made dimensionless by division by their first values
(Gt/G1, Ct/C1, Kt/K1, Lt/L1), there are several important points to pay attention to.
The values obtained make it possible to determine some analog of the gross margin:
Gt þ Ct ¼ ðb0 Kt
b1 Lt Þ þ ðb0 Lt þ b1 Kt Þ ¼ ðb0 þ b1 ÞKt þ ðb0
b1 ÞLt :
(6.13)
We call this sum not the gross revenue but its analog for the following reasons.
All the input variables are scaled. This means that the gross profit Gt is divided by
the initial value of profit in the first year - G1 - and production costs Ct are divided
by the costs of the first year - C1. Their sum is not equal to the quotient of gross
revenue Qt to the value of gross revenue in the first year - Q1,
Gt Ct Gt C1 þ Ct G1
Gt þ Ct
Qt
þ
¼
6¼
¼
:
G1 C1
G1 C1
G1 þ C1 Q1
This is why (6.13) is just an analog of the revenue. For (6.13) to have the
necessary meaning, the gross margin and the prime cost should be dimensionless
relative to the gross revenue:
Gt Ct
; :
Q1 Q1
(6.14)
188
6 Production Functions of Complex Variables
In this case (6.13) means the gross revenue in relative values.
In the same way, the ratio
Gt b0 Kt b1 Lt
¼
Ct b0 Lt þ b1 Kt
(6.15)
is an analog of profitability, as here we calculate not the profitability itself. In
dimensionless values,
Gt =G0 Rt
¼ :
Ct =C0 R0
However, if the variables of the production result are transformed into dimensionless values by presenting them as the original value of the gross revenue, then
(6.15) will characterize profitability by the cost.
This is why for practical purposes all the original values of this and other models
of PFCVs should be transformed into dimensionless values by dividing their values
by the gross output at the initial time Q0:
Gt Ct Kt Lt
; ; ; :
Q1 Q1 Q1 Q1
Formulae (6.9) and (6.10) make it possible to find the corresponding coefficients
of the linear PFCV (6.8) for each observation. To do that one should only substitute
the available statistical data therein. Let us take an actually operating enterprise as
an example to calculate these coefficients.
The board of Inza Diatom plant (Ulyanovsk oblast of Russia) kindly supplied us
with the necessary statistical data concerning their enterprise. They were
preprocessed and systematized by I.E. Nikiforova. The absolute production values
concerning operation of the plant are given in Table 6.1.
A cursory analysis of the data supplied shows a high variance with respect to the
trends, and this is not surprising for a real production process – the results of
production activity are affected by numerous factors most of which remain
unidentifiable.
There is an interesting fact that a change in the capital resource takes place by
stages – July is characterized by the development of new production capacities, and
according to the table the return comes gradually reaching its maximum by
December.
All these circumstances convince us that no model can describe a production
process with an error lower than 10–20%. However, first of all, we are interested in
the mere opportunity to build complex-valued production functions and, secondly,
model general relationships between resources and production result, albeit with
some error.
Using the values of revenue, production costs, payroll, and fixed assets one can
build a production function like (6.8). For that we should first transform all the
6.2 Linear Production Function of Complex Variables
189
Table 6.1 Production activity of Diatom production enterprise by months
Profit
(thousands
Month
of rubles)
February
59
March
72
April
26
May
47
June
21
July
73
August
47
September 49
October
41
November 60
December 107
Costs
(thousands
of rubles)
2,604
3,178
1,146
2,059
897
3,202
2,045
2,152
1,804
2,615
4,736
Payroll
(thousands
of rubles)
213.5
231.3
289.1
246.1
266.6
294.1
396.4
310.2
402.4
511.9
439.4
Work
(man-hours)
52,100
51,347
57,095
62,898
62,742
57,005
61,662
64,484
63,071
64,599
63,905
Number
of people
354
357
364
401
400
404
437
457
454
465
460
Fixed assets
(thousands
of rubles)
4,263
4,263
4,263
4,263
4,263
5,684
5,684
5,684
5,684
5,684
5,684
values into dimensionless relative ones. The data for February for each of the
economic parameters will serve as the base. At this stage we will not transform
the variables into a uniform scale by dividing them by the output for the first month.
We will simply study dimensionless variables whose values for February are
considered to be one.
Since labor is characterized by three parameters – payroll, work, and number of
people involved – we should take the one that shows resource costs more accurately. To transform labor resource values into one dimension and scale, it is payroll
that will go with capital, because it is what characterizes labor costs in monetary
units.
Since it is necessary to center the original variables in order not to use the free
factor by using model (6.8) directly, Table 6.2 gives dimensionless variables
centered with respect to the averages. Since formulae (6.9) and (6.10) make it
possible to find the coefficients for each observation, let us use them to obtain two
series of coefficients with the dynamics shown in Table 6.3.
It is evident that the function coefficients do not remain constant, which is
expected with these initial data. But to answer the question as to whether this
model is suitable for studying production processes at a production plant, one
should determine if there is a trend in the changes of the coefficients. If there is a
trend in the change of at least one coefficient, it will demonstrate that the model is
shifted and is a bad one for describing production relationships. In this case one
should either change the model or modify it significantly.
The model coefficients vary chaotically, the spread of values is quite high, and
no trend of coefficient change is identified. This means that the model of a linear
production function of a complex argument can be used to describe the production
process under consideration but the approximation accuracy will be very slight.
To provide a more correct economic interpretation of the calculated economic
variables, the gross margin and production costs should be assigned dimensionless
190
6 Production Functions of Complex Variables
Table 6.2 Dimensionless
centered data for Diatom
production enterprise
Month
Profit
Costs
Payroll
Fixed assets
February
0.072419
0.077014
0.53332
0.18182
March
0.292758
0.297444
0.44995
0.18182
April
0.4869
0.48289
0.17922
0.18182
May
0.13097
0.13228
0.38063
0.18182
June
0.57165
0.57852
0.28461
0.18182
July
0.309707
0.306661
0.1558
0.151515
August
0.13097
0.13766
0.323355
0.151515
September
0.09707
0.09656
0.08039
0.151515
October
0.23267
0.23021
0.351458
0.151515
November
0.089368
0.081239
0.864339
0.151515
December
0.885978
0.895755
0.52476
0.151515
Table 6.3 Coefficients of
production functions (6.8)
according to data of Diatom
production enterprise
Month
February
March
April
May
June
July
August
September
October
November
December
Coefficient b0
0.17084
0.79429
2.686068
0.416793
2.354795
0.01806
0.50469
0.23606
0.79301
0.108772
2.025593
Coefficient b1
0.077546
0.329688
0.008219
0.145
0.50422
2.005393
0.168554
0.76258
0.320131
0.08433
1.10349
values. This can be done by dividing their values by the gross revenue, as was stated
previously (6.14). Let us compare the result obtained with the previous one.
The dimensionless values of the variables expressed as a function of gross
revenue values are again centered around their averages. These data are given in
Table 6.4.
With the changes in the original variables, the model coefficient values also
changed (6.8). To show this more clearly Table 6.5 provides two series of values of
the complex coefficient – those from Table 6.3 and those calculated according to
the data of Table 6.4.
The change in the scale of the variables caused a change in the values of the
model coefficients (6.8) but not a change in trends since both the first and second
cases involve a linear model but with different scaling rules. New coefficients vary
as chaotically as previously, almost repeating the dynamics of change in the
coefficients calculated according to Table 6.2.
The fact that a linear PFCV should not be used for this production is also
confirmed by the calculation of the complex pair correlation coefficient (4.84),
which for the applied data of the Diatom plant is
rxy ¼ 0:301931 þ i0:055403:
6.2 Linear Production Function of Complex Variables
191
Table 6.4 Dimensionless data for Diatom production enterprise. The profit and costs are divided
by gross revenue
Month
February
March
April
May
June
July
August
September
October
November
December
Profit
0.001604
0.006486
0.01079
0.0029
0.01267
0.006862
0.0029
0.00215
0.00515
0.00198
0.019629
Costs
0.075308
0.290854
0.47219
0.12935
0.5657
0.299867
0.13461
0.09443
0.2251
0.079439
0.875909
Payroll
0.53332
0.44995
0.17922
0.38063
0.28461
0.1558
0.323355
0.08039
0.351458
0.864339
0.52476
Fixed assets
0.18182
0.18182
0.18182
0.18182
0.18182
0.151515
0.151515
0.151515
0.151515
0.151515
0.151515
Table 6.5 Coefficients of production function (6.8) according to the production data of Diatom
production enterprise
According to Table 6.2
Month
February
March
April
May
June
July
August
September
October
November
December
Coefficient b0
0.17084
0.79429
2.686068
0.416793
2.354795
0.01806
0.50469
0.23606
0.79301
0.108772
2.025593
According to Table 6.4
Coefficient b1
0.077546
0.329688
0.008219
0.145
0.50422
2.005393
0.168554
0.76258
0.320131
0.08433
1.10349
Coefficient b0
0.12742
0.56069
1.328494
0.279659
1.431754
0.96716
0.34478
0.246949
0.54544
0.089557
1.550685
Coefficient b1
0.04043
0.21215
1.287556
0.125965
0.87016
0.984596
0.15258
0.49218
0.22047
0.013408
0.410326
Since the real part of the complex pair correlation coefficient is small, it is
evident that a linear complex-valued relationship should not be used to model
production at the Diatom plant. The proximity of the imaginary part to zero testifies
to the existence of a certain nonlinear dependence between the two complex
variables, but one can judge its character only after additional studies.
Let us see now whether this function can be used with the Russian economy.
Table 6.6 shows data on the cost of fixed assets in Russia, the level of active participation in the economy of the people, and the final product, which is divided into final
consumption and gross. Final consumption represents gross margin at the macro level,
which is why we will use this indicator as a real part of the complex production result.
Before building a model, let us calculate complex pair correlation coefficient
(4.84) between two complex variables – the complex resource (fixed assets and
active population) and the complex result (final consumption and gross margin):
rxy ¼ 0:991963
i; 0:005319
192
Table 6.6 Overall development of Russian economy
Fixed assets (Kt)
Economically active population (Lt)
Relative
values
1,000000
2,522632
2,563908
2,72589
2,749193
3,204385
3,906073
4,714465
5,852735
6,279659
7,403701
8,456677
10,46915
12,45701
Absolute values
(thousands of people)
70740
69660
68079
67339
72176
72332
71411
72421
72835
72909
73811
74156
75060
75892
Relative
values
1,000000
0,984733
0,962383
0,951923
1,020300
1,022505
1,009485
1,023763
1,029615
1,030662
1,043412
1,048290
1,061069
1,072830
Final consumption (Gt)
Absolute values
(millions of rubles)
1095820,9
1544658,8
1891846,7
2100663,3
3303947,9
4476851,0
5886861,0
7443199,0
9024756,0
11401444,0
14318964,0
17629743,0
21785787,0
27237356,0
Gross (Ct)
Absolute values
Relative values (millions of rubles)
1,000000
391588,4
1,409591
528694,9
1,72642
564244,2
1,916977
443978,2
3,015044
729214,5
4,085386
1365734,0
5,372101
1963110,0
6,79235
2169314,0
8,235612
2755048,0
10,40448
3558952,0
13,06688
4338731,0
16,08816
5748727,0
19,88079
8031682,0
24,85566
10642560,0
Relative values
1,000000
1,350129
1,440911
1,133788
1,862196
3,487677
5,013198
5,539781
7,035571
9,088502
11,07983
14,68053
20,51052
27,17792
6 Production Functions of Complex Variables
Year
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Absolute values
(millions of rubles)
5182040
13072378
13286272
14125670
14246427
16605251
20241428
24430544
30329106
32541444
38366273
43822840
54251541
64552706
6.2 Linear Production Function of Complex Variables
Table 6.7 Coefficients of
production function (6.8)
according to data of Table 6.6
193
Complex proportionality coefficient
Year
Coefficient (b0)
Coefficient (b1)
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
1.163592
2.230591
2.270221
2.87264
2.15053
1.224434
140.9579
1.484029
1.386099
2.007539
1.940288
2.285952
2.476524
2.683951
1.163592
1.43889
2.70000
2.50794
2.64283
1.90593
1.60172
34.74048
1.94932
1.54839
2.17582
2.23109
2.37343
2.21761
2.27884
1.43889
Since the real part of the complex pair correlation coefficient is close to one and
the imaginary one is close to zero, the linear complex-valued function will be quite
acceptable for modeling the dependence under consideration.
Let us find the coefficients of linear model (6.8) according to the data of this
table as was done previously, i.e., complex coefficients of model (6.11) and (6.12)
that change over time. The calculation results are given in Table 6.7.
It is seen from the table that the coefficients vary with respect to some average
values, except for those calculated for 2001. They differ considerably from the
whole series.
This is due not to some particular features from this year for Russia but to the fact
that to calculate the complex proportionality coefficients all the original data were
centered with respect to the averages. The averages of the original data of Table 6.6
fall exactly in the year 2001 and deviations from the average for this year are close to
zero, which means that the centered values of the variables for the year 2001 are close
to zero. And since in (6.11) and in (6.12) division is used, then with the denominator
close to zero, the quotient will be high, which we observe from the table.
If a researcher wishes to build a complex-valued model not for each observation
but for the whole series, it is necessary to find model coefficients using LSM. In
Chap. 4, certain operations were performed that could provide such a complexvalued model of the dependence of the final product of Russia on the labor
resources and fixed assets:
Gt þ iCt ¼ ð 1:4095
i5:3207Þ þ ð2:2105 þ i1:9415ÞðKt þ iLt Þ:
(6.16)
This model describes very well the process under consideration and can be used, for
example, for purposes of multivariant forecasting of economic development in Russia.
For that, it is necessary to find, in addition to the obtained coefficient values, the
194
6 Production Functions of Complex Variables
confidence interval of changes in the obtained point values. However, the process
under consideration can by no means be associated with a stationary one and methods
of mathematical statistics should not be used in this case formally in their full scope.
Moreover, this study does not consider the problem of applying complex-valued
models to irreversible processes of socioeconomic dynamics. The aim of this study
is to elaborate the principles of the application of methods and models of the TFCV to
economic problems.
6.3
Model of Power Production Function of Complex
Variables with Real Coefficients
In economics linear relationships are known to be quite rare, occurring mainly in
certain, rather short periods of time. This refers mainly to dynamic processes. In
most cases nonlinear dependences prevail, which, as a matter of fact, also operate
in a relatively small period of time, since one nonlinear trend replaces another.
A change in trends of the development of economic projects that we observe can be
explained by the evolution of those economic projects when they change their
structure, composition of the elements, relationships between the objects, and
others. In the same way, the production process develops according to a complex
cyclical trajectory and in particular periods of time can be described by various
nonlinear models. This is why economists should most often work with nonlinear
complex-valued models.
From among a great variety of possible complex-valued models, power PFCVs
occupy a special place. As in the theory of production functions of real variables
where power functions prevail due to their remarkable properties, in the theory of
complex-valued economies power functions play an important role. They are
universal, easy to use, and are very good at describing particular real production
situations. The previous chapter, which considered production functions of a
complex argument, showed that the power function possesses the most interesting
properties of all the functions already studied.
The general form of power PFCVs can have the form
Gt þ iCt ¼ ða0 þ ia1 ÞðKt þ iLt Þb0 þib1 :
(6.17)
Let us first study the simplest case – where the imaginary parts of complex
coefficients of this function are equal to zero and function (6.17) is a power
production function with real coefficients:
G þ iC ¼ aðK þ iLÞb :
(6.18)
6.3 Model of Power Production Function of Complex Variables. . .
195
The properties of the model of power PFCVs with real coefficients has been
studied in detail by Ivan Svetunkov. Here it is appropriate to specify only the most
important properties of this model.
With reference to our problem of mapping a complex variable of production
resources onto a complex plane of production results, there are restrictions due to
the core economic meaning of the variables. Naturally, these restrictions influence
linear complex-valued production functions, but in the above-mentioned example
going beyond these limits is hardly possible, though – with respect to nonlinear
models – quite probable.
Thus, the first group of restrictions is due to the fact that complex variables of
production resources lie in the first quadrant since K > 0 and L > 0, i.e., argument
ц of the complex variable of resources varies within a range of 0 to p2 . If it is equal to
zero, this means that no unit of labor resources is involved in production. If it
becomes equal to p2 , this means that only labor resources are involved in production
with capital resources being equal to zero. It is evident that in practice these cases
do not exist and we should exclude coordinate axes from the problem domain.
According to their economic meaning, complex variables of production performance cannot be defined on the entire complex plane. Though they lie in that plane
within wider ranges defined by the polar angle falling within the range from 0 to 34 p,
they cannot go beyond the limits. Thus production performance is defined in the
first and, partially, the second quadrants of the complex plane.
If the polar angle y of a complex variable of production performance is equal to
zero, it means that the production costs are equal to zero and the gross margin is at its
maximum. Such situations in actual economic practice are exceedingly rare, which
is why the limit in this part should be shown as a strict inequality. Since in the second
quadrant of the complex plane of production performance the gross margin plotted
on the axis of real numbers becomes negative, the enterprise is working at a loss –
the negative gross margin is numerically equal to gross losses at the company. In its
economic meaning, the negative gross margin (loss) cannot be higher than the
production costs: –G C. If no units of the goods produced are sold, then the
gross margin G is numerically equal to the sum of incurred production costs C, its
sign being negative. It is in this case where the polar angle of production performance becomes 34 p. The case where –G ¼ C is a rare but possible occurrence in
actual practice.
This is why any model of PFCVs, including the power function, should be
supplemented by the conditions imposed on the polar angles by complex variables:
0<’<
p
3p
; 0<y
:
2
4
(6.19)
However, due to the periodic character of polar angles and taking into account
the TFCV, this condition should have the form
p
3p
2pk < ’ < 2pk þ ; 2pk < y 2pk þ :
2
4
196
6 Production Functions of Complex Variables
From among the whole set of k due to the economic meaning of the variables, we
will use only k ¼ 0. Incidentally, the power production function of complex
variables should be monosemantic; otherwise, the model stops reflecting the real
economic situation. This means that exponent b should be limited so that the
extreme allowable value of the polar angle of production resources ’ is associated
with the extreme allowable value of the polar angle of production performance y.
Since for the power function under consideration y ¼ b’, the exponent should
satisfy the condition
0 < b’
3p
:
4
(6.20)
If the exponent is negative, b < 0, then any increase in production resources will
definitely lead to a reduction in production performance and vice versa – a reduction in labor and capital resources will cause an increase in production performance.
Here the polar angle of production performance becomes negative, which means
negative production costs – an impossible situation in economics. This is why we
consider only functions with positive exponents.
For a power function with real coefficients used as a model of production
processes, the increase in the radius and polar angle of the complex variable of
production resources (which means a greater growth in labor than in capital
resources) will mean an increase in the production performance with growth in
costs outpacing the gross margin. If we consider an inverse economic process –
capital rather than labor resource growth (which in the complex plane of production
performance means a decrease in the polar angle with simultaneous growth in the
variable radius), – we will have an increase in the production performance with
growth in the gross margin outpacing production costs.
Since in most real production processes investments in fixed capital lead to
improvements in production technology and productivity growth and a reduction in
scrap and production wastes, this means a reduction in costs with a simultaneous
increase in the gross margin. It is this process that is modeled by the production
complex-valued power function with real coefficients. If in this production it is
necessary to achieve a fast increase in the production volume, additional workers
will be hired with the same capital resources, and the enterprise will start to work in
two or three shifts. It is clear that allowances should be made for work in the second
and third shifts, which increasing costs per labor unit leading to growth in total
production costs. The gross margin starts decreasing. This process is modeled by
means of the production function under consideration.
Therefore, according to its properties, the production complex-valued power
function with real coefficients corresponds to real production processes. In periods
of stagnation, this model can also be used, but this will be shown subsequently.
Now let us consider the model properties.
Production function (6.18) can be presented in trigonometric form:
G þ iC ¼ a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib
L
L
K 2 þ L2
cos barctg
þ i sin barctg
:
K
K
(6.21)
6.3 Model of Power Production Function of Complex Variables. . .
197
This allows us to derive two simple formulae for calculating gross margin G and
production costs C:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib
L
K 2 þ L2 cos barctg
;
K
(6.22)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib
L
2
2
C¼a
K þL
sin barctg
:
K
(6.23)
G¼a
These formulae show how the profit and costs will be modeled with various
combinations of production resources. If the production technologies do not change
and the resources used increase, the proportions between the resources will not
change, nor will the polar angle in the complex plane of production resources [arg
(K + iL) ¼ const]. The modulus of the complex variable will increase. For this
case, as follows from (6.3.6) and (6.3.7), the gross margin and gross costs grow at
the same rate at increases in resources. The degree of this growth is determined by
the real coefficients of the model.
The power production function of complex variables with real coefficients has a
remarkable property. To find the coefficient values of function (6.18) it is sufficient
to have only one observation of the production process, since (6.13) and (6.23)
represent a system of two equations with two unknowns a and b, which can be
found using the formulae
b¼
arctg GC
;
arctg KL
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a ¼ exp ln
G2 þ C2
!
arctg GC pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
ln
K þL
:
arctg KL
(6.24)
(6.25)
One can see from (6.24) that coefficient b characterizes a relationship of two
well-known economic indicators – profitability at cost GC and capital-labor ratio KL .
This circumstance makes it possible to consider the model exponent as one of the
analytical features of the proposed model.
Let us show the extremely positive value of coefficient b to satisfy condition
(6.20) as bNQ:
bNQ ¼
3p
:
4arctg KL
(6.26)
If b ¼ bNQ, the company revenue becomes 0 (Q ¼ G + C ¼ 0) due to the fact
that in this case the gross margin is negative, G < 0, i.e., a loss that equals the
production costs |G| ¼ C. This is possible in a situation where no manufactured unit
is sold.
198
6 Production Functions of Complex Variables
Looking at exponent b as a variable lying within the range 0 < b < bNG, one
can examine the influence of this variable on the production resources with fixed costs
of production resources, for example, determine the conditions for reaching the
maximum of real parameters with changing real variables. Calculating the first
derivative of function (6.18) for variable b one can also find some other conditions [1].
Gross margin G reaches its maximum if the following condition is valid:
pffiffiffiffiffiffiffiffiffiffi
ln K 2 þL2
arctgKL
:
arctg KL
arctg
b ¼ bG ¼
(6.27)
Gross margin G is equal to costs C (i.e., profitability at cost is 100%) when the
exponent becomes
p
:
4arctg KL
b ¼ bprof ¼
(6.28)
Gross margin G is equal to zero, i.e., production is neither unprofitable nor
profitable, when
b ¼ bNG ¼
p
:
2arctg KL
(6.29)
Revenue from production Q becomes maximal if the exponent is
b ¼ bQ ¼
3p
4
arctgKL
pffiffiffiffiffiffiffiffiffiffi
ln K 2 þL2
arctg KL
arctg
pl
;
(6.30)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where l ¼ 1 if K 2 þ L2 < 1 and l ¼ 0 in all other cases.
Production costs C become maximal when
pm
b ¼ bC ¼
arctgKL
pffiffiffiffiffiffiffiffiffiffi
ln K 2 þL2
arctg KL
arctg
;
(6.31)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where m ¼ 0 if K 2 þ L2 < 1 and m ¼ 1 in all other cases.
Now, if we know these characteristic points, we can distinguish 13 areas and
points of change in exponent b that characterize the various variations in production
efficiency (Fig. 6.2). Along coefficient b’s axis we plot the numerical values
calculated for the conditional example for resource values equal to one. Let us
consider the characteristic points and the model with an increasing exponent
beginning from zero and in compliance with Fig. 6.2.
6.3 Model of Power Production Function of Complex Variables. . .
199
2.5
2
C
1.5
1
0.5
bG
0
-0.5
0
0.5
bprof
1
bQ
1.5
bNG
2
bC
2.5
bNQ
3
Q
-1
-1.5
-2
G
-2.5
Fig. 6.2 Coefficient b values
b 2 ½0; bG Þ is an area of highly profitable production. With growth in coefficient b
from zero to bG, the growth in production costs C is accompanied by even higher
growth in profit G. Profitability increases, and revenue Q increasing accordingly.
b ¼ bG is the maximal gross margin. Here gross margin G reaches its highest value.
b 2 ðbG ; bprof Þ means production is efficient but profit G decreases and costs C
grow. Revenue Q continues increasing. Profitability at cost G/C decreases.
The point b ¼ 1 is interesting because in this case K does not influence C and L
does not influence G, as we see from formula (6.18), G ¼ aK, C ¼ aL. And it is
clear that one can derive the revenue using the formula Q ¼ a(K + L).
b ¼ bprof is the point where profitability at cost is equal to 100%.
b ¼ bprof is where the profit G decreases but the company revenue Q is still growing
due to faster growth in production costs. Profitability is lower than 100% and
continues to fall.
b ¼ bQ is the point of maximal revenue of the company.
b 2 ðbQ ; bNG Þ indicates that production is still efficient despite the continued fall in
profit G with increasing costs. Revenue Q in this interval starts decreasing.
b ¼ bNG is the point of profitless production (known in the economic theory as
“critical point”). Here, G ¼ 0 and revenue Q is equal to costs C.
b 2 ðbNG ; bC Þ denotes inefficient unprofitable production. Profit is negative (loss)
but lower than costs in absolute value. In real production this may reflect the
situation where one has to sell goods at a lower price than the prime cost. Further
increase in the exponent means that costs C grow and revenue Q decreases.
b ¼ bC is the point of maximum costs. This is an extreme point where costs are at
their highest, gross margin is negative and lower than the costs in absolute value,
200
6 Production Functions of Complex Variables
i.e., fewer and fewer products are sold; therefore, loss is still lower than the
production costs but is increasing rapidly.
b 2 ðbC ; bNG Þ denotes extremely inefficient production. Costs, gross margin, and
revenue decrease, while loss grows.
b ¼ bNG indicates an absence of revenue and is the point of production termination
since the gross margin G is equal to costs C, the revenue Q being equal to zero.
This means that the whole production volume is unprofitable, and no product
unit is sold.
Since the economy is diverse, real practice admits the most diverse alternatives
besides those mentioned previously, namely:
– An alternative is possible when production resources take the following value:
–
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K 2 þ L2 ¼ 1:
– In this case bG ¼ 0;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
– if the production resources yield K 2 þ L2 < 1;
– bG becomes negative; under the given conditions it is impossible to attain
maximum profit;
– when the equality
arctg
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L
¼ ln K 2 þ L2
K
holds for the production resources, the values of the points
bNG ¼ bQ, bC ¼ bNQ become equal to each other;
arctg
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L
< ln K 2 þ L2 ;
K
when
the points take each other’s places:
bNG of bQ and bC of bNQ.
This means that additional revenue is gained at a loss.
This additional information allows the researcher to understand better the character of production if the resources take one of these values.
The point of maximum revenue bG makes it possible to obtain the analytical
character of the proposed power function of complex variables, namely, to determine the production efficiency level using the distance of the actual value b to point
bG. Ivan Svetunkov suggests using a parameter that shows this level and that can be
found by means of the formula
6.4 Power Production Complex-Valued Functions with Real Coefficients. . .
201
Table 6.8 Results of calculation of the Russian economy efficiency level
Year
S
G/C
1998 (%)
8.3
12.7
1999 (%)
20.7
25.5
2000 (%)
23.4
24.7
S¼1
b
bNG
2001 (%)
20.4
18.5
bG
:
bG
2002 (%)
17.8
14.4
2003 (%)
18.8
13.5
(6.32)
It is evident that S is positive when b < bNG, i.e., when company profits are
positive. The coefficient is close to zero if b is close to bNG, i.e., when the profit is
close to zero. S ¼ 0 only in the case of profitless production when b ¼ bNG. S is
equal to one when exponent b coincides with bG. In the zone of “highly profitable
production” [when b falls within ð0; bG Þ] S becomes higher than one. If b > bNG,
the proposed coefficient becomes negative, which shows the unprofitability of
production. For convenience S can also be represented in percentage terms just
by multiplying its value by 100%.
Studies of numerous conditional and real examples have shown that this coefficient correlates with the dynamics of profitability at cost, and the higher the
profitability, the closer the parameter values to unity and vice versa. Table 6.8
gives an example of the calculation of coefficient S for the Russian economy
according to data from the Russian Federal State Statistics Service from 1998 to
2003 and G/C ratio as a reflection of the average profitability.
As we see from the table, coefficient S can really be used as one of the
characteristics for estimating the production efficiency level since to a certain
extent it shows the average profitability of the production (the pair correlation
coefficient between S and G/C on this set of values is 0.71).
Since the proposed model has a clear economic meaning and shows real production processes in progress, its analytical properties should be more carefully studied
for other production processes as well.
6.4
Power Production Complex-Valued Functions with Real
Coefficients of the Diatom Plant and Russian Industry
The remarkable properties of the power PFCV mentioned in the previous chapter
require confirmation by real economic examples. Since the data of the Diatom plant
seem to be quite representative and in compliance with the above-mentioned
procedure, Ivan Svetunkov calculated a, b, bG, bNG, bQ, and S values for each
observation. These values are given in Table 6.9. Unfortunately, the requirements
of commercial secrecy do not allow us to furnish here the original data for this plant
for the period under consideration from 2004 to 2007, which is why we provide
only the results of the calculations.
202
6 Production Functions of Complex Variables
Table 6.9 Characteristics of power production function of complex variables with real
coefficients for Diatom production works
Quarter
Q1 2004
Q2 2004
Q3 2004
Q4 2004
Q1 2005
Q2 2005
Q3 2005
Q4 2005
Q1 2006
Q2 2006
Q3 2006
Q4 2006
Q1 2007
Q2 2007
a
0.939
1.928
1.643
1.558
1.127
1.344
1.335
1.728
1.272
1.476
1.403
1.218
1.383
1.502
b
1.241
1.294
1.332
1.288
1.283
1.332
1.273
1.329
1.308
1.348
1.349
1.326
1.397
1.385
bG
0.032
0.009
0.053
0.066
0.069
0.016
0.060
0.086
0.092
0.061
0.091
0.092
0.062
0.105
bQ
0.658
0.644
0.683
0.708
0.713
0.682
0.715
0.756
0.760
0.742
0.782
0.799
0.787
0.829
bNG
1.251
1.270
1.260
1.284
1.288
1.332
1.309
1.339
1.336
1.362
1.383
1.415
1.449
1.449
S (%)
0.80
1.92
6.02
0.28
0.40
0.01
2.86
0.78
2.30
1.06
2.60
6.75
3.72
4.77
G/C (%)
1.23
2.99
9.08
0.41
0.59
0.01
4.29
1.14
3.36
1.59
3.82
9.94
5.60
6.96
Based on the interpretation of the values of the given model characteristics and
proposed coefficient S, one can draw a conclusion on the low efficiency of the plant.
Exponent b is in zone 8 of the 13 previously considered and is close to the outer
value of bNG, which characterizes profitless activity. The low S value also testifies
to the fact. Since our task was to verify the analytical properties of the production
power function with real coefficients, the table gives the profitability indicator G/C.
It is easy to see that parameter S correlates with profitability, confirming the
suitability of the complex-valued model for economic analysis.
It is evident from the table that the proportionality coefficient and exponent vary
in time, but this variation is not crucial. Obviously, any model of socioeconomic
dynamics, however well it described a series of observations, at some point will
stop doing it satisfactorily since socioeconomic dynamic processes are subject to
evolution. This is why in forecasting it is necessary to adapt the model, i.e., to
correct the model coefficients in the event of changes in trends. The complexvalued production function with real coefficients for which coefficients are calculated for each observation shows this in its full scope.
Besides the fact that the model under study makes it possible to estimate
production efficiency, it can help to obtain recommendations on how to
increase production efficiency. On the basis of the last observation for the
Diatom plant and especially the calculated values of the coefficients of
the power PFCV, which looks has the form
G þ iC ¼ 1:502ðK þ iLÞ1:385 ;
(6.33)
we can provide for the company’s leadership recommendations that follow from the
model.
First of all let us determine what level of profits and costs the plant can attain if
the production is improved to the maximum degree. Certainly, this proposal is
6.4 Power Production Complex-Valued Functions with Real Coefficients. . .
203
idealized since we do not take into consideration all the possible factors and
conditions. This is why we consider the extreme case. Thus, if we improve
production to this unattainable extreme case, the model shows this when
S ¼ 100% or
b ¼ bG ¼ 0:105:
Let us take the same K and L resource values for Q2 of 2007 and calculate G and
C values for the exponent b ¼ bG ¼ 0.105. We will have the following in relative
values:
G ¼ 1:512;
C ¼ 0:172:
(6.34)
We see that production costs for the case under consideration drop considerably
since they relate to their original value for Q1 of 2004 and have been increasing for
the whole considered period from 1 onward. In (6.34) we see a more than fivefold
cost reduction! The type of production that provides such profits and costs is ideal,
the obtained calculated values being marginal. These unattainable values show the
direction for possible company development, in particular, they show that with
existing production technology and more rational usage of the available resources,
including labor resources, the Diatom plant could earn much bigger profits and
incur lower costs. But what should be done to attain this?
To answer this question, let us determine how the enterprise can obtain the
ultimate combination of profits and production costs, equal to G* ¼ 1.512;
C* ¼ 0.172; if it does not improve organizational and economic mechanisms,
what resources should be used? For that let us substitute these extreme values of
the gross margin and costs into formula (6.33), which shows the state of affairs at
the company. On this basis we can calculate the necessary sizes of capital and labor
to attain the necessary values of profits and costs without changing anything else at
the company. In absolute values they will be equal to
K ¼5:55 million rubles; L¼0:46 million rubles
(6.35)
For comparison, it should be noted that in Q2 of 2007, the cost of fixed
production assets at the Diatom plant was 2.92 million rubles, and payroll was
5.52 million rubles.
What does this comparison mean? To make the production process at the plant
more efficient, one should increase the fixed production assets there by 5.55/
2.92 ¼ 1.9 times, and the payroll should be reduced by 5.52/0.46 ¼ 12 times.
This can be done by reducing excessive labor resources and increasing productivity
at the expense of investments in fixed capital.
Now we can give an economic interpretation to the obtained results.
The calculated values of labor and capital make it possible to draw the conclusion that the company’s resources are used inefficiently: to increase production
efficiency, we should increase investments in fixed production assets and reduce
204
6 Production Functions of Complex Variables
labor costs. The given proportions do not specify that the resource changes should
be made on such a scale. Any model represents the result of abstraction, and many
real factors are not taken into account, which is why we called the values of the
gross margin and production costs calculated according to the ideal model
“extreme” values. The proportions obtained make it possible to determine the
main direction of actions to take. Since to attain efficient production it is necessary
to increase capital by 1.9 times and reduce labor resources by 12 times, it is evident
that labor resources are the main direction of improvement in company operations.
It is this indicator that is used at the plant inefficiently, which is why rationalization
of labor is an important factor in the improvement of production efficiency. It is
seen from the calculations that investments should not be ignored by economic
analysis, but the main focus of the company management, in terms of increasing
efficiency, is on labor resources. Company management should examine the ratio
between industrial, production, and other company personnel and the organization
of labor and wages since it is here where they would be able to find redundancies for
increasing production efficiency.
It should be noted that the criterion of maximum gross margin is the main one for
company operations; however, sometimes competitive market conditions are such
that companies are forced at all costs to be the leader in sales. This means that the
main criterion of company operations is maximum production volume. Understanding that production and sales volumes are related but nonetheless different
concepts, all the same we emphasize their closeness, not their differences, to
simplify the problem. Therefore, let us assume that the maximum production
volume criterion is associated with maximum revenue. Then we can make similar
calculations for the Diatom plant, where we use bQ instead of bG, i.e., when
b ¼ bQ ¼ 0:829
we get the following results.
Maximum company revenue will be Q ¼ 61.75 million rubles, but profit will be
G ¼ 27.36 million rubles, costs C ¼ 34.39 million rubles. This level can be
reached with an increase in fixed production assets K of up to 4.73 million rubles
and a reduction in labor costs L down to 3.6 million rubles. This means that to reach
maximum revenue we should increase the cost of the fixed production assets by
only 4.73/2.92 ¼ 1.6 times and reduce labor costs by 5.52/3.6 ¼ 1.53 times. This
means that the Diatom company works at almost maximum production volumes
with the given technology and production organization. If we recall that this
company is the only supplier in Russia of diatom insulating bricks for nonferrous
metallurgy, but that it has competitors from China, then it is understandable that in
striving not to allow its competitors enter the market the company ensures, to the
maximum degree, that it will have customers for its products.
Now it is interesting to compare the results and recommendations obtained by
means of the power PFCV having real coefficients with those provided by the
production function of Cobb-Douglas. Estimating the parameters of the power
production function of complex variables with real coefficients and the production
6.4 Power Production Complex-Valued Functions with Real Coefficients. . .
205
function of Cobb-Douglas with the data of the Diatom company using the LSM we
obtain the following models:
G þ iC ¼ 1:398ðK þ iLÞ1;319 ;
(6.36)
Q ¼ 2:348K 0:457 L0:543 :
(6.37)
The revenue approximation error for the first model (6.36) is 15.3% and for the
second one (6.37) it is 14.8%. This means that these two models approximate the
original data of the production under consideration with a practically similar
approximation accuracy.
Recommendations that can be obtained using the production function of CobbDouglas mean that to increase the revenue, the Diatom plant should increase
investments in fixed production assets and increase labor resources (number of
workers), with an increase in the number of workers being more desirable since the
elasticity coefficient (6.37) for labor is higher than for capital. The exponent value
of the Cobb-Douglas function, calculated using LSM, with labor resources L equal
to 0.543 shows that increasing the number of employees by 1%, one obtains a
revenue increase of 0.543%.
Thus, if we leave the value of the fixed production assets for the last year of
observation unchanged at an amount of 2.92 million rubles and increase the number
of employees, it is possible to double the gross output, as follows from CobbDouglas function (6.37) by increasing the labor resources to 2,187 people (leaving
current wages unchanged). If we substitute these values of capital and labor
resources into to our function (6.36), then we will model another result, that is,
gross margin will be negative and equal to 60.5 million rubles, and production
costs will be 191million rubles.
Gross output will be 130.55 million rubles, which means that the number of
employees should not be increased and, conversely, this number should be reduced
by optimizing the organization of labor.
Since the two production functions (6.42) and (6.43) possess an accuracy of
approximation that is almost equal to that of previous values of modeled production
(15.3% and 14.8%, respectively) but propose diametrically opposite
recommendations with respect to production development, there is a natural alternative: choose the model that actually describes the situation without distorting it.
To determine which recommendations are closer to the real situation at the plant
and which model adequately describes the production situation, we appealed to the
management of the Diatom plant. The general director, E.A. Nikiforov, who holds a
Ph.D. in economics, explained that the number of employees was excessive. This
was due to the fact that the company was a city-forming enterprise and to decrease
the level of social tension and reduce unemployment in the city the board decided to
provide employment to the maximum possible number of Inza residents using labor
resources in a not very efficient manner. In the opinion of the company director, the
labor resources at the enterprise are excessive. The situation with the company
206
6 Production Functions of Complex Variables
Table 6.10 Original statistical data for Russian industry
Year
1998
1999
2000
2001
2002
2003
2004
Production volume
(billions of rubles)
1,707
3,150
4,763
5,881
6,868
8,498
11,209
Annual average number
of production
workers (thousands
of workers)
13,173
13,077
13,294
13,282
12,886
12,384
11,977
Investments in fixed
capital in industry
producing goods,
million rubles
165,092
297,278
527,544
699,366
817,504
980,188
1,179,744
Profitabiliy level
of goods (works,
services) sold
0.127
0.255
0.247
0.185
0.144
0.135
0.179
capital resources is as follows. After the crisis of the 1990s, when the factory almost
shut down, its material and technical base was in decline and was partly ruined (the
unemployed Inza residents started removing the company’s fixed assets for sale in
the form of metal scrap). Only since the late 1990s did the plant start to recover,
primarily by investment in fixed capital. That is why speedy growth of investments
in fixed capital with an unchanged number of employees (due to socioeconomic
reasons) is a strategic direction of the development of the Diatom plant, which, as
we see from the above-mentioned example, is fully in line with the conclusions and
recommendations of the power PFCV with real coefficients (6.42) and does not
agree with the recommendations of the Cobb-Douglas production function (6.43).
From the foregoing example it does not mean that our function is always better
than the Cobb-Douglas function. But since in the example in question it adequately
described the production process but the Cobb-Douglas function did not, we can
conclude that there could be other production situations where the complex-value
function will be more efficient than the production function of real variables.
Let us verify if it is possible to apply power PFCVs with real variables at the
macro level. Let us take statistical data for Russian industry for the period
1998–2004 given in Table 6.10.
To build a power PFCV, we need data for gross margin and production costs,
which cannot be found for Russia in general in the statistical reference books of the
TF State Committee for Statistics. However, the necessary calculated values of
profit and costs can be found on the basis of production profitability values given in
the table. Instead of the fixed production assets, we will use investments in fixed
capital.
Transforming these data into relative values (1998) we can, using LSM, calculate the model coefficient values:
G þ iC ¼ 0:001ðK þ iLÞ3:457
(6.38)
A high exponent value should be frightening, as for the period under consideration the polar angle of the complex resource decreased almost eightfold! The polar
angle of the complex production result did not change. This is why the model
6.5 Coefficients of Elasticity of the Complex Exponential Production Function. . .
207
exponent (6.38) appeared to be so high. According to model recommendations, an
increase in investments in fixed capital would boost production efficiency.
The calculation of control points for this model is senseless since instead of fixed
assets the model uses investments.
6.5
Coefficients of Elasticity of the Complex Exponential
Production Function with Real Coefficients
Elasticity is one of the most important indices used in simulating socioeconomic
processes. By definition, elasticity is calculated as follows:
e1
0
Dy
y
:
¼
Dx
x
(6.39)
Elasticity (6.39) has a clear interpretation showing in terms of percentage the
change in the result with a 1% factor change. For example, in economic theory,
characterizing demand behavior often involves the price elasticity of the demand
volume, with elasticity showing consumer reaction (change of the consumption
volume) at slight changes of the price per product unit.
Moving from discrete values to constants, we can represent the elasticity as
follows:
e1
0
dy
dy x
y
¼
:
¼
dx dx y
x
(6.40)
This form of notation allows for calculating the elasticity for various differentiable functions including those of production. Since we are discussing the
properties of PFCVs, then, after becoming acquainted with the peculiarities of the
behavior of an exponential complex function with real coefficients, we would like
to focus on the resource elasticity for this function. For that, as follows from (6.40),
it is necessary to calculate the first derivatives of this function with respect to its
variables – labor and capital.
First, to calculate the derivatives, let us represent the model of the production
function (6.18) in exponential form:
b
G þ iC ¼ aðK þ iLÞb ¼ aðReiy Þ ;
where
(6.41)
208
6 Production Functions of Complex Variables
R¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K 2 þ L2 ;
y ¼ arctg
L
:
K
Now it is easy to group the model modulus and its polar angle:
by
G þ iC ¼ aRb ei :
(6.42)
With this in mind, the model of the exponential complex production function
may be represented in trigonometric form:
G þ iC ¼ aRb ½cosðbyÞ þ i sinðbyÞ:
(6.43)
This makes it possible to calculate the first and second (if necessary) partial
derivatives of the complex function with respect to resources – labor and capital.
According to the d’Alembert-Euler (Riemann-Cauchy) condition for finding complex function derivatives, it will be enough to take derivatives of its real and
imaginary parts. The real part of the model (6.43) may be represented in the
following form:
ReðG þ iCÞ ¼ aRb cosðbyÞ ¼ U:
(6.44)
This is why the derivative of the function (6.18) with respect to resources may be
calculated as follows:
@ðG þ iCÞ @U
¼
@ðK þ iLÞ @K
i
@U @ðaRb cosðbyÞÞ
¼
@L
@K
i
@ðaRb cosðbyÞÞ
:
@L
Let us calculate the first component of derivative (6.45), namely,
(6.45)
@ðaRb cosðbyÞÞ
@K
, as a derivative of a complex function:
@ðaRb cosðbyÞÞ @ðaRb Þ
@ cosðbyÞ
¼
cos by þ
ðaRb Þ:
@K
@K
@K
(6.46)
The first part of (6.46) takes the form
@ðaRb Þ
cosðbyÞ ¼ abRb
@K
or
1
K
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðbyÞ
K 2 þ L2
@ðaRb Þ
cosðbyÞ ¼ abRb 2 K cosðbyÞ:
@K
(6.47)
(6.48)
6.5 Coefficients of Elasticity of the Complex Exponential Production Function. . .
209
The second form of (6.46) is a derivative of the cosine of the argument in K. It
may also be calculated as follows:
@ cosðbyÞ
ðaRb Þ ¼
@K
aRb sinðbyÞ
@ðbyÞ
¼
@K
aRb sinðbyÞ
@ðarctgb KL Þ
:
@K
After applying the formula for calculating the derivative of the arctangent we
have
@ cosðbyÞ
ðaRb Þ ¼
@K
abRb sinðbyÞ
L
L
¼ abRb sinðbyÞ 2
2
R
K 2 ð1 þ KL 2 Þ
¼ abRb 2 L sinðbyÞ:
(6.49)
Then the partial derivative of the exponential complex production function with
real coefficients (6.46) for capital will be written as follows:
@ðaRb cosðbyÞÞ
¼ abRb 2 K cosðbyÞ þ abRb 2 L sinðbyÞ
@K
¼ abRb 2 ðK cosðbyÞ þ L sinðbyÞÞ:
(6.50)
The same procedure can be applied to the partial derivative of the exponential
@ðaRb cosðbyÞÞ
production function (6.18) for labor:
:
@L
Since this is a complex function derivative, it should be calculated as follows:
@ðaRb cosðbyÞÞ @ðaRb Þ
@ cosðbyÞ
¼
cosðbyÞ þ
ðaRb Þ:
@L
@L
@L
(6.51)
The summand (6.51) taking into account the derivative of the modulus may be
represented as follows:
@ðaRb Þ
cosðbyÞ ¼ abRb
@L
1
L
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðbyÞ ¼ abRb 2 L cosðbyÞ
2
K þ L2
(6.52)
The addend (6.51) includes the derivative of the cosine of the argument for labor.
This derivative may also be determined as follows:
@ cosðbyÞ
ðaRb Þ ¼
@L
abRb sin y
@ðarctg KLÞ
:
@L
After calculating the derivative of the arctangent we finally obtain for this
component
210
6 Production Functions of Complex Variables
@ cosðbyÞ
ðaRb Þ ¼
@L
abRb sinðbyÞ
K
¼
2
þ KL 2 Þ
abRb 2 K sinðbyÞ:
K 2 ð1
(6.53)
Then the derivative with respect to labor will have the form
@ðaRb cosðbyÞÞ
¼ abRb 2 L cosðbyÞ
@L
abRb 2 K sinðbyÞ
¼ abRb 2 ðL cosðbyÞ
K sinðbyÞÞ:
(6.54)
Now we can obtain the required formula of the first derivative of the production
function in question for the complex resource (6.45):
@ðG þ iCÞ
¼ abRb 2 ðK cosðbyÞ þ L sinðbyÞÞ
@ðK þ iLÞ
iabRb 2 ðL cosðbyÞ
K sinðbyÞÞ:
This huge expression can easily be simplified:
@ðG þ iCÞ
¼ abRb 2 ½ðK cosðbyÞ þ L sinðbyÞ
@ðK þ iLÞ
¼ abR
b 2
½ðK
iL cosðbyÞ þ iK sinðbyÞ
(6.55)
iLÞðcosðbyÞ þ i sinðbyÞ:
Since cosðbyÞ þ i sinðbyÞ ¼ eiby and R 2 ðK
much simplified:
@ðG þ iCÞ
¼ abRb eiby ðK þ iLÞ
@ðK þ iLÞ
1
1
, (6.55) can be
iLÞ ¼ KK2 þLiL2 ¼ KþiL
¼ bðK þ iLÞ 1 ½aRb eiby :
(6.56)
If we know the value of the first derivative of the exponential complex function
with real coefficients for a complex argument, we can determine the elasticity of
this function. It will have the following form:
e¼
@ðG þ iCÞ K þ iL
K þ iL
aRb eiby K þ iL
¼ bðK þ iLÞ 1 ½aRb eiby
¼b
¼ b:
@ðK þ iLÞ G þ iC
G þ iC
G þ iC K þ iL
(6.57)
Thus, the elasticity of the exponential complex function with real variables for a
complex resource is equal to the function exponent. This means that the model
under study possesses the following property: with a simultaneous 1% increase in
production resources, the production result will increase by b percent.
Remarkable properties of models of complex variables revealing their additional
advantages compared to models of real variables are demonstrated in the fact that
using an exponential complex production function we can determine the
6.5 Coefficients of Elasticity of the Complex Exponential Production Function. . .
211
contribution of each of the components of a complex resource per each component
of the production result – the gross margin G and the production costs C. Since a
real economy allows for an increase in, say, only labor resources with capital
resources unchanged, it is important to determine how the gross margin and the
production costs will change. To answer this question let us calculate the elasticity
coefficients of the complex result for each of the resources –labor and capital.
First let us calculate the first partial derivative of the complex function for
capital, using the same symbols:
@ðG þ iCÞ @ðaRb cosðbyÞÞ
@ðaRb sinðbyÞÞ
¼
þi
:
@K
@K
@K
(6.58)
For the first summand the following formula has been obtained in (6.50):
@ðaRb cosðbyÞÞ
¼ abRb 2 ðK cosðbyÞ þ L sinðbyÞÞ:
@K
Omitting huge calculations like those given previously by (6.58), we have the
following for the addend:
@ðaRb cosðbyÞÞ
¼ abRb 2 ðK sinðbyÞ
@K
L cosðbyÞÞ:
(6.59)
Substituting the obtained results into (6.58) we find the form of the first partial
derivative of the required function for capital:
@ðG þ iCÞ
¼ abRb 2 ½ðK cosðbyÞ þ L sinðbyÞÞ þ iðK sinðbyÞ
@K
L cosðbyÞÞ:
(6.60)
The expression in square brackets may be easily transformed into
K cosðbyÞ þ iK sinðbyÞ
¼ ðK
iðL sinðbyÞ þ L cosðbyÞÞ
iLÞðcosðbyÞ þ i sinðbyÞÞ ¼ ðK
iLÞeiby :
The factor before the square brackets in (6.60) may be presented as follows:
abRb
2
¼ ab
Rb
:
K 2 þ L2
Taking the previous expressions into account, it is easy to obtain a convenient
formula of the first partial derivative of the complex production result for capital:
@ðG þ iCÞ
Rb
¼ ab 2
ðK
@K
K þ L2
iLÞeiby ¼ bðaRb eiby Þ
K iL
:
K 2 þ L2
(6.61)
212
6 Production Functions of Complex Variables
Hence we have the elasticity coefficient of the exponential PFCV for capital:
eK ¼
@ðG þ iCÞ K
K iL
K
¼ bðaRb eiby Þ 2
@K
G þ iC
K þ L2 aRb eiby
¼ bð
K2
K 2 þ L2
i
LK
Þ:
K 2 þ L2
(6.62)
Since this elasticity coefficient is complex, its real part characterizing the effect
of capital on the gross margin and its imaginary part the effect of capital on
production costs, the former and the latter should be considered separately:
eK ¼ eGK þ ieCK :
(6.63)
The real part of the complex elasticity coefficient eGK characterizes the value by
which the gross margin will change with a 1% increase in capital:
eGK ¼ b
K2
:
K 2 þ L2
(6.64)
In the denominator of the real part of the complex elasticity coefficient (6.63)
there is a value that was previously called the scale of production resources. It is
evident that a fraction of an expression (6.64) is always lower than one, which is
why the real part of the elasticity for capital showing growth in the gross margin is
determined by the value of exponent b. Since exponent b for this function
characterizes the overall elasticity of the complex production result for a complex
production resource (6.57), (6.64) may be written as follows:
eGK ¼ e
K2
L2
!
e
¼
e
ð1
þ
Þ:
GK
K 2 þ L2
K2
(6.65)
This means that the gross margin elasticity coefficient for capital of the production function in question is always lower than the elasticity.
It follows from (6.64) that an increase in capital resources will always, in this
model, result in an increase in the gross margin – more or less depending on the
value of exponent b.
Let us consider now the imaginary component of the complex elasticity of the
production function for capital. It shows the effect of capital on production costs
and is equal to
eCK ¼
b
LK
¼
K 2 þ L2
e
LK
:
K 2 þ L2
(6.66)
The fraction of this expression is always positive, like coefficient b, which is
why the imaginary component of the complex elasticity for capital is always
6.5 Coefficients of Elasticity of the Complex Exponential Production Function. . .
213
negative. This means that the model under consideration describes production
processes under which any increase in capital results in a reduction in production
costs, i.e., to a reduction in the unit cost. Most real production processes do behave
in this way as capital increases lead mainly to an increase in labor productivity,
which results in cost reductions. It is evident from (6.66) that the degree of
reduction in production costs is determined by exponent b of the production
function and labor and capital values.
Since the elasticity of the complex production result for capital is a complex
number, it may be presented in exponential form. For that, let us calculate the
modulus of the complex number and its polar angle.
The modulus of the complex elasticity coefficient (6.62) is equal to
ReK ¼ bK;
(6.67)
and its polar angle is
’eK ¼ arctg
L
:
K
(6.68)
Since the sum of the gross margin and costs represents the sales volume, when
both the gross margin and production costs are adjusted to the gross output as a
result of preliminary scaling, the sum of the real and imaginary parts of the
elasticity coefficient for capital will characterize the elasticity of the output for
capital in the following way:
eQK ¼ erk þ eik ¼ b
K 2 LK
:
K 2 þ L2
(6.69)
It then follows that the elasticity of the output for capital may also be negative if
the numerator (6.69) is negative – for labor-intensive processes. However, with an
increasing capital resource and stable labor resource, the elasticity of the output for
capital becomes positive.
Similarly, we can calculate the first derivative of the exponential production
function for labor and determine on its basis the formula for calculating the
elasticity of the complex production result for this resource. If we omit the
calculations similar to those above, the following formula will result:
eL ¼
@ðG þ iCÞ
L
L2
LK
¼ bð 2
þi 2
Þ:
2
@L
G þ iC
K þ L2
K þL
(6.70)
The complex elasticity coefficient may be given an economic interpretation.
Since it is complex, it is more convenient to consider its real and imaginary parts:
eL ¼ eGL þ ieCL :
(6.71)
214
6 Production Functions of Complex Variables
The real part of the complex elasticity of the complex production result for labor,
which shows the contribution of labor to the change in the gross margin, is
eGL ¼ b
L2
L2
¼
e
:
K 2 þ L2
K 2 þ L2
(6.72)
Elasticity of the gross margin for labor is always positive and always lower than
the total elasticity e since a fraction of the expression (6.72) is always less than one.
The positive nature of the gross margin elasticity coefficient for labor means that
this function may be used when growth in labor resources leads to growth in the
gross margin.
The imaginary component of the complex elasticity of the production result for
labor (6.70), which shows the effect of labor on production costs, has the following
form:
eCL ¼ b
LK
:
K 2 þ L2
(6.73)
Since all the components of this part of the elasticity coefficient are positive, it is
evident that growth in labor resources will inevitably lead to growth in production
costs. This also corresponds exactly to actual production situations – the use of
additional human resources or growth in payroll costs to stimulate the employed
workers will increase total production costs.
Now, it is necessary to consider the contribution of the labor resource to changes
in production on the whole, i.e., to changes in output volume. For that let us sum up
the two parts of coefficient (6.70) and evaluate the contribution of the labor resource
to the combined growth in both gross margin and production costs, with the
assumption that the variables are properly scaled:
eQL ¼ eGL þ eCL ¼ b
LðL þ KÞ
:
K 2 þ L2
(6.74)
This means that a 1% increase in labor resources increases the production
volume by the above-mentioned value.
If now we consider the complex elasticity of the production result for labor
(6.70) in exponential form, the modulus of this complex value will be
ReL ¼ bL;
(6.75)
and the polar angle will be
’eK ¼ arctg
K
:
L
(6.76)
6.5 Coefficients of Elasticity of the Complex Exponential Production Function. . .
215
Table 6.11 Elasticity coefficients of complex production function for Russian economy
Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
b¼e
12.674
8.114
6.264
6.088
5.758
5.789
4.922
4.842
4.472
4.397
4.255
eGK
12.574
7.968
6.074
5.875
5.520
5.555
4.652
4.565
4.167
4.080
3.927
eCK
1.124
1.081
1.074
1.118
1.146
1.141
1.120
1.123
1.129
1.137
1.135
eGL
0.018
0.068
0.205
0.365
0.674
1.009
1.910
2.822
4.893
8.101
12.821
eCL
1.124
1.081
1.074
1.118
1.146
1.141
1.120
1.123
1.129
1.137
1.135
eK
11.449
6.886
4.999
4.757
4.375
4.414
3.532
3.442
3.038
2.943
2.792
eL
1.225
1.228
1.264
1.331
1.384
1.375
1.390
1.400
1.435
1.454
1.463
To conclude our study of the elasticity of the exponential PFCV with real
coefficients, we should mention one more important property of complex
coefficients of elasticity of the production result for resources. If we sum up
elasticity for capital (6.62) and elasticity for labor (6.70), we will obtain the total
elasticity (6.57):
K2
e ¼ eK þ eL ¼ b
K 2 þ L2
LK
i 2
K þ L2
L2
LK
þi 2
þb
K þ L2
K 2 þ L2
¼ b:
The result that we obtain on the basis of this chapter means that an exponential
complex production function with real coefficients possesses much more extensive
analytical capabilities for studying production processes than production functions
with real variables. If a similar model of an exponential production function with
real coefficients has only two elasticity coefficients, which are exponents of
resource indices, the complex model allows for the following calculations:
Total elasticity (6.57);
Complex elasticity of the production result for capital (6.63), which consists of two
real elasticity coefficients (6.64) and (6.66);
Elasticity of output Q for capital (6.69);
Complex elasticity of production result for labor (6.70) with two components (6.72)
and (6.73);
Output elasticity Q for labor ( 6.74).
In total there are seven different elasticity coefficients showing the most diverse
effects of the two resources on the complex production result and the gross output.
Let us show how to calculate the above-mentioned elasticity coefficients of a
complex production function with real coefficients for real data. Here we will use
previously mentioned data on the Russian economy from 1998 to 2008. The results
of calculation of the coefficients are given in Table 6.11.
The obtained elasticity values show the following specific features of the
development of the Russian economy for the period under consideration.
216
6 Production Functions of Complex Variables
First of all, it should be noted that in 1998 Russia experienced a default that
resulted in a devaluation of the ruble, a crucial reduction in the price of domestic
goods, considerable growth in the demand for them, and an essential recovery in
production and the economy in general. This is why the data for 1998 and 1999 are
so different from those for subsequent years.
The 1990s were characterized by a sharp drop in production volume. Labor and
capital resources were underutilized, basic production assets were mostly “frozen,”
and earnings of the citizens were extremely low. This is why the use of additional
capital resources in the economy after the default of 1998 and the growth in
material incentives resulted in a considerable increase in production volume,
gross margin, and production costs. This is reflected in all the elasticity values.
It should be noted that investments in basic capital aimed at restoring production
but not at modernization. For Russian entrepreneurs investments in innovations
were pointless given the absence of competition among the national producers and a
high price competitiveness of Russian products compared to exported products.
These specific features of the economic growth following the default of 1998 are
reflected in the dynamics of the elasticity coefficients. It is evident from Table 6.11
that the effect of capital on the reduction in production costs is quite low: the
elasticity coefficient eCK, reflecting this, demonstrates no pronounced dynamics
since the efficiency of capital in GDP growth reflected by output elasticity for
capital eK decreased. Investments in basic capital of the Russian economy, which in
1998 and 1999 led to excellent results in achieving high gross margins, as shown by
the elasticity values eGK for these years, start playing less important role in the
Russian economy.
Since at that time the elasticity coefficient of the gross margin for labor eGL
increased considerably, the basic contribution of labor resources to the increased
efficiency of the state economy becomes clear. It should also be noted that GDP
elasticity for labor eL grew but not as fast as that of the gross margin for labor eGL.
This means that labor productivity for the Russian economy in general during this
period increased, but the rates of this increase were not significant.
We do not intend here to conduct a thorough analysis of the economic situation
in the Russian economy. The calculation of elasticity coefficients using the Russian
economy as an example aimed at demonstrating the possibility of such calculations
and interpretation of the values of elasticity coefficients.
6.6
Power Production Function of Complex Variables
with Complex Coefficients
It was shown in previous sections that the power PFCV with real coefficients
possesses remarkable properties of compliance with actual production processes.
Its seven elasticity coefficients considerably expand economists’ analytical tools.
But the diversity of production observed in the economy cannot be described in its
6.6 Power Production Function of Complex Variables with Complex Coefficients
217
entirety by only this power function. The reality is much richer and more diverse,
and this diversity is partly described by the power model of a complex variable with
real coefficients. It follows from the conclusions of the previous section that this
model describes processes characterized by an increasing return on resources, a
decreasing return on resources, and a constant return on resources. However, there
are situations of crisis production or cyclical production, etc. This is why model
(6.18), being in many ways universal, is not always able to provide the best results
of production modeling.
Model (6.18) with real coefficients is one of the simplest among power PFCVs.
Function (6.17) with complex variables is the most general among the possible
power PFCVs:
G þ iC ¼ ða0 þ ia1 ÞðK þ iLÞðb0 þib1 Þ :
(6.77)
This general function can serve as the basis for the most varied types of function
that can be obtained by varying the four coefficients. One of these types is the power
production function with real coefficients (if imaginary parts of complex
coefficients are equal to zero); another one is a linear function of complex variables
(when the exponent is equal to a real number – unity).
The modeled production processes are diverse, which is why one can suppose that
in various cases one of the types of power functions can be the best. Hence, the
question arises as to how one selects the best option from among the above-mentioned
variety of production functions. For that, we can recommend the following procedure.
We use LSM to find parameters of a general power function with a complex
proportionality coefficient and complex exponent (6.77). Based on the value of the
coefficients a0, a1, b0, and b1 found by means of LSM, the researcher can choose the
production function for modeling. For example, if b1 ! 0 and a1 ! 0, then, based on
the principle of simplicity, one should use the power PFCV with real coefficients like
G þ iC ¼ a0 ðK þ iLÞb0 :
If the power coefficient values are close to zero, then one should use a linear
function with complex variables:
G þ iC ¼ ða0 þ ia1 ÞðK þ iLÞ:
According to the above-mentioned statistical data, one can build, for example, a
power production function with complex coefficients for the Diatom plant.
Calculations give it the following form:
G þ iC ¼ ð0:534 þ i0:363ÞðK þ iLÞ0:459
i0:355
:
We see that all the components of complex coefficients (both real and imaginary
parts) have huge values, which is why it would not be proper to use more simplified
models for the purposes of approximation and economic analysis. It should be noted
218
6 Production Functions of Complex Variables
that this model is quite good at approximating real data (the average approximation
error is 20%) taking into account the fact that investments K in fixed capital of the
Diatom factory for the period in question vary gradually, their consequences being
prolonged, which cannot help but reduce the accuracy of a description of the
production dynamics by any model.
Everything is somewhat different with the power production function with
complex coefficients for industrial production in Russia. Using LSM the following
model for the available statistical data was obtained:
G þ iC ¼ ð1:732
i0:213ÞðK þ iLÞ0:896þi0:351 :
Here, the imaginary component of the proportionality coefficient is small and
closer to zero compared with its real part.
It is true that the imaginary part of the complex proportionality coefficient equal
to ( 0.213) in its absolute value is eight times smaller than its real part (1.732). This
model is good at approximating real data – the average production result approximation error was (4.6%). However, for simplified calculations, due to small values
of a1, to describe industrial production in Russia, we can use the following model:
G þ iC ¼ aðK þ iLÞb0 þib1 :
Thus, calculation with the statistical data made in accordance with the proposed
approaches allows for selecting power production functions with various
coefficients – complex or real – in order to use a model of adequate complexity
in each case.
Since complex-valued power functions with complex coefficients can turn to be
the best for modeling some production processes, we should study their properties
more carefully. For that, let us single out the real and the imaginary parts from
equality (6.77):
G ¼ f ðK; LÞ;
C ¼ gðK; LÞ:
In formula (6.77) the complex variable ðK þ iLÞðb0 þib1 Þ can be represented in the
following way:
eðb0 þib1 Þ lnðKþiLÞ :
Now formula (6.77) will have the form
G þ iC ¼ ða0 þ ia1 Þ eb0 lnðKþiLÞþib1 lnðKþiLÞ :
(6.78)
The sum of the products in the exponent can be transformed taking account the
properties of logarithms of complex numbers using their principal values:
6.6 Power Production Function of Complex Variables with Complex Coefficients
G þ iC ¼ ða0 þ ia1 Þ eb0 lnð
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
K 2 þL2 Þþib0 arctgðKL Þþib1 lnð K 2 þL2 Þ b1 arctgðKL Þ
219
:
(6.79)
Now let us group the real and imaginary parts of the complex variable to obtain a
formula that would be convenient for further study:
G þ iC ¼ ða0 þ ia1 Þ eb0 lnð
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
K 2 þL2 Þ b1 arctgðKL Þ iðb0 arctgðKL Þþb1 lnð K 2 þL2 ÞÞ
e
:
(6.80)
On the right-hand side the complex coefficient ða0 þ ia1 Þ is multiplied by a
variable that can be designated as R. That is,
G þ iC ¼ ða0 þ ia1 ÞRei’ ;
(6.81)
where
pffiffiffiffiffiffiffiffiffiffiffi
K2þL2Þ b1 arctgðKL Þ
R ¼ eb0 lnð
;
(6.82)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L
’ ¼ b0 arctg
K 2 þ L2 :
þ b1 ln
K
(6.83)
Now if we present model (6.81) in trigonometric form, open the brackets, and
group the real and imaginary parts, we obtain
G þ iC ¼ Rðða0 cos ’
a1 sin ’Þ þ iða0 sin ’ þ a1 cos ’ÞÞ;
(6.84)
which means that two equalities are satisfied:
G ¼ Rða0 cos ’
a1 sin ’Þ;
(6.85)
C ¼ Rða0 sin ’ þ a1 cos ’Þ:
(6.86)
Thus, to calculate profit G and production costs C, the researcher can use
formulae (6.85) and (6.86).
According to the economic meaning of the problem under consideration, production costs cannot be negative but profit can (operating at a loss).
This means that (6.86) imposes restrictions on the limits of variation of the
complex proportionality coefficient factors:
a0 sin ’ þ a1 cos ’ > 0 ! tg’ >
a1
;
a0
(6.87)
where ’ is calculated from (6.83).
Finding function coefficients (6.77) is a simple task. In Chap. 4, we mentioned
ways of solving them using LSM.
220
6 Production Functions of Complex Variables
Let us give the following statement without proof (due to the considerable
bulkiness of the conclusion): the coefficient of elasticity of a power complexvalued production function with complex coefficients is equal to the exponent of
this function:
e ¼ b0 þ ib1 :
(6.88)
It follows from (6.39) that
Dy
¼
eyx y
Dx
x
(6.89)
or
DðG þ iCÞ ¼ ðb0 þ ib1 ÞðG þ iCÞ
DðK þ iLÞ
;
K þ iL
(6.90)
where we can find an answer to the question of what process of change in the gross
margin and production costs does a power production function with complex
coefficients model when production resources vary by 1%:
8
>
>
< DG ¼ ðb0 G
DKK þ DLL
DKL DLK
þ ðb0 C þ b1 GÞ
;
K 2 þ L2
K 2 þ L2
>
DKK þ DLL
ðDLK DKLÞ
>
: DC ¼ ðb0 C þ b1 GÞ
þ ðb0 G b1 CÞ
:
K 2 þ L2
K 2 þ L2
b1 CÞ
(6.91)
It is seen from the obtained equalities that both the gross margin and production
costs may decrease and increase with an increase in the resources given various
combinations of the exponent coefficient values. In contrast to the power function
with real coefficients, the elasticity of a function with a complex exponent gives
little information on the direction of the complex result variation with a 1% increase
in the production resources. Everything is determined by both the combination of
values of the real and imaginary parts of the complex exponent and the values of the
resources and the results.
However, since this variied combination makes it possible to model diverse
production processes, the complex-valued power function with complex
coefficients possesses rather high identification properties. It describes various
production processes – from efficient to unprofitable, from processes with an
increasing return on resources to those with a decreasing return.
If in practice calculations of production volumes are required but not a
company’s profits and costs, then one can derive a formula for the company’s
output volume for this function, taking into account that with correct scaling Q ¼ C
þG. Adding (6.85) and (6.86) we get
Q ¼ G þ C ¼ Rða0 cos ’
a1 sin ’Þ þ Rða0 sin ’ þ a1 cos ’Þ
6.7 Logarithmic Production Function of Complex Variables
221
or
Q ¼ Rðða0
a1 Þ sin ’ þ ða0 þ a1 Þ cos ’Þ;
(6.92)
which is the same.
This function will have other elasticity coefficients than those calculated in the
previous section for the power complex-valued model with real coefficients. Since
they do not have as clear a meaning as those considered in the previous section,
their calculation here is irrelevant.
The foregoing discussion shows that the complex-valued power production
function with complex coefficients can be a powerful instrument for production
process analysis and modeling.
6.7
Logarithmic Production Function of Complex Variables
In the theory of production functions, from among the whole set of nonlinear
models, economists prefer power models, as they are convenient to use and have
a simple economic interpretation. The same may be said of PFCVs: power
functions are convenient to use and have a simple interpretation of their parameters.
Nevertheless, in the real economy the most varied situations are possible when
these models fail to approximate real production. These cases should use production functions of another form. One such alternative model is that of a logarithmic
production function of complex variables. In this section we will focus on its
properties and features for practical use.
The general form of a production logarithmic function has the following form:
Gt þ iCt ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðKt þ iLt Þ:
(6.93)
The free term of this equality has a simple meaning – it corrects the initial model
conditions with respect to the real values of the variables. This is its only contribution to modeling the production situation. This is why, without loss of generality,
we consider the model without a free term:
Gt þ iCt ¼ ðb0 þ ib1 Þ lnðKt þ iLt Þ:
(6.94)
The properties of this model are fully revealed if we present the logarithm of the
complex variable in arithmetic form and multiply the value obtained by the
complex proportionality coefficient:
Gt þ iCt ¼ ðb0 þ ib1 Þðln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
Kt2 þ L2t þ iarctg Þ:
Kt
As a result, we obtain the following statement:
222
6 Production Functions of Complex Variables
Gt þ iCt ¼ ðb0 ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kt2 þ L2t
b1 arctg
Lt
Þ þ iðb1 ln
Kt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
Kt2 þ L2t þ b0 arctg Þ;
Kt
(6.95)
from which we derive two equalities characterizing the real and imaginary parts of
the model:
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
>
>
< Gt ¼ b0 ln Kt2 þ L2t b1 arctg ;
Kt
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L
>
>
: Ct ¼ b1 ln Kt2 þ L2t þ b0 arctg t :
Kt
(6.96)
With positive coefficients and growing labor costs, production costs also grow
and the gross margin may also grow depending on the coefficient values and
variable scale, but much slower than the costs. The function may demonstrate
another behavior under conditions where b0 < <b1 – then the gross margin
decreases with an increase in labor resources.
If capital resources grow and the model coefficients are positive, then the gross
margin and production costs will also increase.
However, a situation may arise where the model coefficients become negative.
For example, with negative b1 the gross margin will grow if there is a growth in
labor resources and the effect of capital is uncertain – it may lead either to an
increase or a reduction in the gross margin. Production costs may also behave
ambiguously – with growing labor resources they may both increase and decrease.
However, an increase in the capital resources will definitely result in a decrease in
production costs.
This means that a logarithmic function of complex variables can be suitable for
describing several different production situations.
Since model (6.94) has only two coefficients, it may mean, in reference to
complex-valued functions, that there is a possibility of estimating their values
with one statistical observation. Indeed, it definitely follows from (6.94) that
b0 þ ib1 ¼
Gt þ iCt
Gt þ iCt
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
2
lnðKt þ iLt Þ ln Kt þ L2t þ iarctg KLt
(6.97)
t
Multiplying the numerator and denominator of the right-hand side of the equality
by the value conjugate to the denominator we obtain
b0 þ ib1 ¼
Gt ln
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kt2 þ L2t þ Ct arctg KLtt þ iðCt ln Kt2 þ L2t
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ln2 Kt2 þ L2t þ arctg2 KLtt
Gt arctg KLtt Þ
:
(6.98)
6.7 Logarithmic Production Function of Complex Variables
223
Since the left- and right-hand sides of the complex equality mean the simultaneous equality of the real and imaginary components, let us obtain the formulae for
calculating each of the coefficients at observation t:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kt2 þ L2t þ Ct arctg KLtt
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
b0 ¼
;
ln2 Kt2 þ L2t þ arctg2 KLtt
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ct ln Kt2 þ L2t Gt arctg KLtt
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
b1 ¼
ln2 Kt2 þ L2t þ arctg2 KLtt
Gt ln
(6.99)
Using the previous data for the Inza Diatom factory, let us calculate these timevarying coefficients for the logarithmic production function of complex variables of
this factory. First, to nullify the effect of scale, let us center the production
performance and logarithms of the complex resource with reference to their
averages, which follows from (6.94). The results are given in Table 6.12.
Analysis of the coefficient dynamics shows that both these coefficients tend to
grow slightly with time. The dynamics of coefficient b0 can be described by a linear
trend, and its equation can be easily found by LSM:
b0t ¼ 0:1051t
0:8496:
The other coefficient, b1, has a less pronounced growth trend. Nevertheless, it
can also be represented in the form of a linear trend. LSM allows us to find
estimations of this trend:
b1t ¼ 0:0044t þ 0:6521:
The proportionality coefficient of the latter trend is close to zero, which confirms
that the dynamic of variation of the imaginary constituent b1 of the complex
regression is quite low, which is why we can consider this coefficient almost
constant.
We see that the logarithmic complex-valued model (7.5.2) is not applicable for
modeling the production of this factory, which is confirmed by systematic variation
of one of the model coefficients, but it can be used, with the trends obtained, for
purposes of forecasting and multivariate modeling. Taking into account the linear
variation in time of one of the coefficients, the logarithmic complex-valued model
of the Diatom factory production will have the form
Gt þ iCt ¼ ð0:1051t
0:8496 þ i0:6521Þ lnðKt þ iLt Þ:
Now, setting different alternatives for the economic development of the factory
resources, one can obtain alternative complex production results.
A similar method was used to calculate the coefficients of this model with
respect to the statistical manufacturing data in Russia. To avoid calculating the
224
6 Production Functions of Complex Variables
Table 6.12 Coefficients of logarithmic complex variables production function of Inza Diatom
factory
t
1
2
3
4
5
6
7
8
9
10
11
Table 6.13 Coefficients of
logarithmic production
function of complex variables
for Russian industry
b0
0.09084
0.3334
0.99276
0.125047
0.14192
3.03984
0.19265
1.018661
0.32793
0.084627
1.197226
t
1998
1999
2000
2001
2002
2003
2004
b1
0.22095
1.05475
3.642535
0.568678
3.308633
0.386848
0.67284
0.58261
1.0547
0.177843
2.964839
b0
2.150242
1.284403
0.34767
0.783986
1.11559
0.08035
4.527272
b1
3.502649
3.327341
9.019879
0.1516
1.390457
3.233731
5.703955
free term, we center the gross margin and production costs with reference to their
averages. In the same way we center the principal logarithmic values. The coefficient variation dynamics is given in Table 6.13, which shows clearly that variation
of this dynamics is crucial, without any visual trend of coefficients, which shows
that a logarithmic model could not be applied to this case.
If a complex-value function with complex coefficients is applicable to virtually
any production process with a particular degree of accuracy, the logarithmic
function, as we can see from the above-mentioned example, is more “whimsical”
and not as universal.
An example of a successful application of a logarithmic production function of
complex variables to a company in St. Petersburg is SPb ZPS. The example was
found and calculations made by D. Duhanina. Since the company works under
conditions of unstable market conditions and its production cycle is quite significant, production activity performance in months is unstable. These data are given in
Table 6.14. The company is a small business, which is reflected in a slight variation
of the resources.
Duhanina expressed these data in a nondimensional value and centered them
around their averages, after which she calculated the dynamics of each coefficient
of the complex-valued logarithmic production function.
6.7 Logarithmic Production Function of Complex Variables
225
Table 6.14 Monthly data on production activity of SPb ZPS for 2005–2007 and model
coefficients
Time, t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Revenue
Profit
Costs
Fixed assets
Model coefficients (6.94)
Q
2,231,714
2,015,300
1,769,635
8,539,410
3,670,696
2,884,686
3,004,009
4,024,720
2,098,340
5,340,695
1,709,550
2,923,416
3,247,150
4,750,283
1,838,114
8,742,325
7,783,112
1,748,450
4,445,690
2,935,020
1,865,884
534,629
4,858,625
G
C
2,262,644
2,162,551
1,799,567
9,459,319
4,112,242
2,986,656
3,109,786
3,994,986
1,951,956
4,760,218
1,357,741
2,791,468
3,308,128
4,694,023
1,614,787
7,958,985
6,895,362
1,310,727
4,351,020,
2,568,162
1,410,011
1,206,204
3,737,405
K
3,775,895
3,746,729
3,756,818
3,756,818
3,727,652
3,698,485
3,669,318
3,640,152
3,610,985
3,581,818
3,552,652
3,523,485
3,494,318
3,465,152
3,508,333
3,479,167
3,450,000
3,420,833
3,391,667
3,362,500
3,333,333
3,304,167
3,275,000
b0
0.0541
0.0711
0.0417
0.2092
0.0920
0.0381
0.0378
0.0165
0.0010
0.0429
0.0324
0.0006
0.0404
0.0197
0.0164
0.1553
0.2038
0.1285
0.0234
0.1771
0.3221
0.4863
0.4300
30,930
147,251
29,932
919,909
441,546
101,970
105,777
29,734
146,384
580,477
351,809
131,948
60,978
56,260
223,327
783,340
887,750
437,723
94,670
366,858
455,873
671,575
1,121,220
b1
0.0301
0.0329
0.0190
0.0683
0.0170
0.0090
0.0099
0.0048
0.0000
0.0025
0.0136
0.0003
0.0237
0.0130
0.0090
0.1147
0.1420
0.1000
0.0194
0.1582
0.3457
0.5474
0.5057
It is evident from the table that the complex proportionality coefficient varies
from observation to observation, but this variation does not represent a trend, in
contrast to the case with the Diatom factory. These variations are very slight and
characterize deviations with reference to some average values. The last three
observations are exceptions, as the coefficients calculated for them differ significantly from the whole series.
This means that on a given set of production results, a complex-valued logarithmic model of the production function will satisfactorily describe the production
process. The deviations of the last three observations are due either to random
factors or to changes in the development trends. Since thorough research of the
situation at this small business is not the object of our study, let us focus on a formal
problem – the possibility of modeling production by means of this model.
To build a model on the available statistical data, one can use the values from
Table 6.14. Duhanina used LSM to estimate the coefficients of the complex-valued
logarithmic model of the production function. This model has the form
Gt þ iCt ¼ ð0:0909
i0:1715Þ þ ð0:0436
i0:0389ÞðKt þ iLt Þ:
(6.100)
226
6 Production Functions of Complex Variables
To draw a conclusion regarding the applicability of the model in question to
production modeling, let us compare the properties of this model with those of other
real-variable models. Duhanina used LSM to find, based on the same data,
coefficients of the Cobb-Douglas production function, which has the form
Qt ¼ 1:2603Kt0:3326 Lt0:6674 :
(6.101)
Now we can compare the models’ suitability for describing production
processes.
The logarithmic production function makes it possible to calculate both the gross
margin and production costs. The approximation error was G ¼ 58.07% for the
gross margin. It is evident from Table 6.14 that the profit is subject to a high
variance; this is why this approximation error should not mislead the reader. The
model is much more accurate when it describes production-cost dynamics. The
average approximation error for this parameter in the original series was C ¼ 8.2%.
Since the sum of the gross margin and production costs gives an output value, we
use the logarithmic production function of complex variables to approximate this
parameter too. The average approximation error for Q was 33.14%.
The use of the Cobb-Douglas function for the purposes of modeling gives an
average approximation error of 62.49%, which is almost twice as high as the
approximation error of the complex-value model.
Again, we will not focus here on the problem of selecting the best model for
describing the given production process. It is not of interest here. It is important to
note that from the given example it follows that for some production processes a
logarithmic complex-valued function can be successfully applied.
We will not calculate the elasticity of logarithmic complex-valued production
functions because they do not have a very simple form or as clear an economic
interpretation as in the case with the power production function with real
coefficients. Therefore, the elasticity will not contribute to our understanding of
the model properties.
6.8
Exponential Production Function of Complex Variables
To complete our study of elementary complex-valued functions that may be used as
production function models, let us turn our attention to the properties and features
of applying an exponential production function. This function can have the most
varied bases but the function properties will not change. It is the degree of
complexity of model usage that changes. It is evident that the exponential function,
where e is the base, will be the least problematic since the exponential form of
complex variables is practically always used to understand model properties. This is
why we can use a model of an exponential complex-valued function as an example
of exponential models in general, and it has the following form:
6.8 Exponential Production Function of Complex Variables
Gt þ iCt ¼ ða0 þ ia1 Þeðb0 þib1 ÞðKt þiLt Þ :
227
(6.102)
The simplest variant of this model is one with real coefficients. It represents a
rather simple function, which is of little interest:
Gt þ iCt ¼ a0 eb0 ðKt þiLt Þ :
(6.103)
Indeed, grouping the variables the comprise the model and the polar angle of the
right-hand side of the equality we obtain
Gt þ iCt ¼ a0 eb0 Kt eib0 Lt :
(6.104)
If we turn to the trigonometric form we obtain the following equalities for the
real and imaginary parts:
(
Gt ¼ a0 eb0 Kt cosðb0 Lt Þ;
Ct ¼ a0 eb0 Kt sinðb0 Lt Þ:
(6.105)
This means that for positive b0, an increase in capital from the zero value will be
followed by an increase in the production scale and, therefore, in the gross margin;
this growth will retain the same proportions between these variables. In other
words, the growth in production volumes maintains an unchanged level of profitability. For real production this is possible when the capital-labor ratio is low and its
growth significantly affects productivity. There is also some control over pricing.
Growth in the labor resource from the zero point will be followed by an increase
in the polar angle, the production scale remaining unchanged with unchanged
capital. This means growth in the production costs and reduction in the gross
margin. One should not forget that periodic functions can be both positive and
negative. Therefore, for modeling production situations it is necessary either to
center the original variables, in which case they will take both positive and negative
values, or to impose restrictions on the b0 coefficient, based on the economic
meaning of the problem. This is all that can be said about this function.
If now instead of the real proportionality coefficient in the exponent we consider
a situation where the proportionality coefficient is imaginary, i.e., b0 ¼ 1, b1 6¼ 0,
we will see certain rather “symmetrical” changes in the function’s properties. The
production function in this case will have the form
Gt þ iCt ¼ a0 eib1 ðKt þiLt Þ :
(6.106)
The trigonometric form of this model will be as follows:
(
G t ¼ a0 e
Ct ¼ a0 e
b1 Lt
b1 Lt
cosðb1 Kt Þ;
sinðb1 Kt Þ:
(6.107)
228
6 Production Functions of Complex Variables
Growth in the labor resources definitely leads to a decrease in the production
scale (if b1 > 0) and a decrease in both gross margin and production cost values.
Growth in capital from the zero point will lead to a decrease in the gross margin and
an increase in production costs. On the whole, this model can describe crisissituation production, when to attain the desired production performance it is
necessary to reduce the number of employees and abandon noncore production,
eliminating capital in this area of production.
It is understood that the use of complex coefficients makes it possible to
synthesize these diverse properties into a single complex production relationship.
To understand the influence of production resources on production performance in
the complex coefficient model (6.102) the modulus and the polar angle in the righthand side of the equality should be singled out. For that, we open the brackets in the
exponent and express the complex proportionality coefficient in exponential form.
Then taking account the designations previously introduced we obtain
Gt þ iCt ¼ aeb0 Kt
b1 Lt iðb0 Lt þb1 Kt þaÞ
e
:
(6.108)
Now it is easy to obtain two equalities of real variables that model the real and
imaginary parts of the model, i.e., describe the effect of production resources on the
gross margin and production costs:
(
Gt ¼ aeb0 Kt
Ct ¼ aeb0 Kt
b 1 Lt
b 1 Lt
cosðb0 Lt þ b1 Kt þ aÞ;
sinðb0 Lt þ b1 Kt þ aÞ:
(6.109)
With positive coefficients the model will possess the following properties.
Capital growth entails growth in the gross margin and production costs. However,
this growth is not similar. Thus, the gross margin’s exponential growth induced by
an increase in Kt in the exponent is somehow nullified as the cosine of the polar
angle decreases with an increase in capital, and their product gives complex
nonlinear dynamics.
Costs increase more intensively with the capital increase since with capital
growth, an exponential increase is associated with an increase in the sine. Their
product gives a particular multiplicative effect.
However, this general characteristic is corrected by the argument of the
proportionality coefficient a ¼ arctg aa10 . It characterizes a phase shift of the cosine
and sinusoid. This shift may lead to inverse dependences.
The same complex character in this model characterizes the dependence of the
production performance on labor resources. In the first approximation, the increase
in labor resources is followed by a very marked decrease in the gross margin, as the
decrease of the exponent in the first equality (6.109) is associated with a decrease in
the cosine. Their product reinforces the trend.
The cost behavior is not as unambiguous – growth in the labor resource is
followed by a decrease in the exponential component of the second equality
(6.109); however, its harmonic – sinusoid – component grows.
6.9 Summary
229
Again, this complex character of the dependence is to a large extent corrected by
the argument of the proportionality coefficient – its various values provide a phase
shift in the harmonic factors, and these factors can behave in a different way
compared to the original idea.
This is why an exponential complex-valued function with complex coefficients
is able to describe various production types.
According to the manufacturing data in Russia that have already been used in
this chapter as the basis for verifying the properties of PFCVs, A.M. Chuvazhov
built an exponential model of the production function with coefficients estimated by
LSM. The model has the form
Gt þ iCt ¼ ð1:656 þ i0:534Þeð0:265þi0:015ÞðKt þiLt Þ :
(6.110)
It describes the original data quite accurately – profit approximation errors are
7.06% and those of production costs are 2.64%. Therefore, the exponential
complex-valued model of the production function may rightfully be included in
the range of production function models since there could definitely occur cases
where this model turns out to be the best possible one.
6.9
Summary
If we compare complex-argument models (Chap. 5) and complex-variable models,
we clearly see that the latter are much more interesting for economic researchers
and more universal. In this chapter functions of complex variables have been used
as production function models, and in a number of cases they demonstrate a clear
advantage over models of real variables. Each of the complex-variable functions
has its original properties and can be used as a production function model in various
situations. The researcher investigating a particular production process should
select the best one.
It is very important that models of complex variables, being compact, allow for
modeling several production parameters simultaneously – production volume,
gross margin, and production costs. This is an obvious advantage of the proposed
models.
The complex-valued power production function with real coefficients possesses
the most interesting properties from among the models of production functions, as it
shows very well real production processes. In addition, such important
characteristics as the elasticity of the gross margin, production costs and production
volumes for each of the resources and for a complex resource can be easily
calculated using this function. This means that this function has considerable
analytic potential.
In conclusion, another important property of PFCVs should also be noted.
Complex-variable models used as production models make it possible to solve
230
6 Production Functions of Complex Variables
some problems that are difficult to pose in the area of real variable models. The
following problem is standard: determining the output volume for various resource
combinations. Complex-variable models expand this problem since one can not
only calculate the output but find the values for the gross margin and production
costs for various resource combinations.
There are also more complicated problems in studying production processes and
production planning: for example, understanding how efficient production is, determining how one can attain a particular production volume, reduce production costs,
and achieve the largest profit.
In real-variable models, in solving this problem one should vary the resources
and calculate the results finding the best resource combination for the set criterion.
PFCVs make it possible to solve these problems simply using the inverse
function. By definition, building an inverse function means deriving a relationship
like x ¼ F 1 ðyÞ, for which x ¼ FðxÞ.
The inverse function of a real variable is possible only in the case of a
monofactor dependence. And all production functions, even the simplest ones
like Cobb-Douglas, are multifactorial, and it is not possible to calculate an inverse
function.
In relation to complex-variable or complex-argument models, it is rather a
simple and solvable problem, as we consider the dependence of one complex
variable on another, i.e., monofactor dependence. Therefore, one can apply an
inverse function to these dependences.
For the complex variable production function
G þ iC ¼ FðK þ iLÞ
(6.111)
K þ iL ¼ F 1 ðG þ iCÞ:
(6.112)
let us derive the function
Substituting the required values of the gross margin and costs into the obtained
inverse function it is easy to determine what capital and labor resources are
necessary to obtain the required values of the production performance.
Let us derive the following types of inverse functions for the above-mentioned
models:
1. Power,
2. Logarithmic,
3. Exponential.
The complex-valued power production function (6.17) has the form
Gt þ iCt ¼ ða0 þ ia1 ÞðKt þ iLt Þb0 þib1 :
6.9 Summary
231
The inverse function is found in the following way:
K þ iL ¼
G þ iC
a0 þ ia1
1
b þib
0
1
:
(6.113)
The logarithmic complex-valued production function (6.93) has the following
form:
Gt þ iCt ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðKt þ iLt Þ:
The inverse function (6.112) will have a bit more complex form:
lnðKt þ iLt Þ ¼
Gt þ iCt ða0 þ ia1 Þ
;
b0 þ ib
(6.114)
though if necessary, it may be written as follows:
Kt þ iLt ¼ e
Gt þiCt ða0 þia1 Þ
b0 þib
(6.115)
This form makes it possible to calculate the required values of capital and labor
at the same time.
Exponential complex-valued function (6.102) considered in the last section of
this chapter was given in the following form:
Gt þ iCt ¼ ða0 þ ia1 Þeðb0 þib1 ÞðKt þiLt Þ :
It can be used to obtain the inverse function
Kt þ iLt ¼
þiCt
ln Ga0t þia
1
b0 þ ib1
:
(6.116)
Calculations with any of these functions are easy. First, any econometric method
is used to find coefficients of the original production function of complex variables,
after which they are substituted into the respective inverse function (6.114),
(6.115), or (6.116). Now, setting the desired profit, costs, or production volume
values one can estimate the required capital and labor resource values.
One can also solve a problem relating to the maintence of the same profitability
level, as well as other problems.
We should point out one assumption that we used a priori without specifying it.
The proposed production functions modeled production costs and profit, and the
economic interpretation of the obtained results complies with these indicators.
However, the company or industry gross margin is determined in a market economy
not just by the internal company forces or the number of resources used, but by the
232
6 Production Functions of Complex Variables
market itself. And the latter was not taken into account in the models under
consideration, and it is not incorporated into real-variable models. It is assumed
with this approach that everything that is to be produced in a company will find a
consumer and will be sold with the modeled profitability.
This circumstance should be taken into account for more careful scientific
research.
However, despite that, real examples of participants in economic activity
assured us that complex-valued production functions could be an efficient tool for
the modeling and analysis of production processes and, therefore, could be included
in the arsenal of the production function theory.
Reference
1. Svetunkov SG, Svetunkov IS (2008) Production functions of complex variables. LKI, Moscow
Chapter 7
Multifactor Complex-Valued Models
of Economy
The economy as an object of research is a multifactor system. This is why to
provide adequate modeling, multifactor models should be applied. This chapter
considers multifactor, complex-valued models with respect to production functions.
The production performance is represented in the form of a complex variable where
the gross margin refers to the real part and the production costs to the imaginary
part. The production resource is represented by two complex variables. The first
variable characterizes capital expenses and is represented in complex form. Its real
part includes costs of fixed production assets, and its imaginary part thecosts of
fixed nonproduction assets. The second complex variable characterizes the labor
costs and is also represented as a complex variable. Its real part comprises industrial
production capital, and its imaginary part – nonproduction workers.
These multifactor production functions are considered in linear form, CobbDouglas power function form, and power form. With Russia as an example, it is
shown how to use these models to model the development of the economy taking
into account of its “shadow” part.
7.1
General Provisions of Complex-Valued Model Classification
There is no doubt that models of production functions of complex variables
discussed in the previous chapter extend the instrumental base of the production
function theory and in certain cases provide more adequate modeling results than
the existing real-variable models. However, one should not overestimate the results
of application of certain elements of the complex-variable function theory in the
economy since complex-valued economics is not an alternative to existing models
of real variables; it only extends the arsenal of economic and mathematical methods
and models, supplying economists with new tools. Along with the obvious
advantages, production functions of a complex argument and complex-variable
functions have some disadvantages restricting their sphere of application. Some
of them were shown in their respective chapters. However, there is another
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
233
DOI 10.1007/978-1-4614-5876-0_7, # Springer Science+Business Media New York 2012
234
7 Multifactor Complex-Valued Models of Economy
circumstance that can be considered a drawback of the above-mentioned complexvalued functions.
Indeed, in general form, production functions of complex variables (complexargument functions can be considered as their simplified analog) have the following
form:
Gt þ iCt ¼ FðKt þ iLt Þ:
(7.1)
An important advantage of such a model compared to real-variable models is the
modeling of dependences of two economic variables on two other economic
variables. Economics differs from other natural and exact subjects and sciences in
that a set of variables is determined by the effect of other variables. In this case
model (7.1) in the economy is a step ahead compared to real-variable models.
However, if we try to extend, for example, the number of production resources in
function (7.1) we will face a very complicated problem – how do we do it? A
complex variable, by definition, consists of a pair of real ones. What will we do if
we need to take into account the effect on production of one more resources, for
example, the area of land for agriculture? If we consider it as a simple real variable,
we do not take into account its influence on production costs, and if we take it as the
imaginary part, we miss the influence of the new resource on the gross margin. On
the other hand, it is quite difficult to add one more variable to model (7.1) without
taking specific steps. This is the first essential shortcoming restricting the sphere of
application of models like (7.1).
The second disadvantage is due to the necessity of very accurate work with the
dimension and scale of each of the variables. And if everything is clear about the
production results, since these variables are measured in the same units and have the
same scale, it is not the case for the production resources as the complex-valued
functions under consideration are nonuniform.
Indeed, if we take the power production function of real variables
Qt ¼ aKta Lbt
(7.2)
and change, for example, the capital resource scale by multiplying the resource by
l, we will have
Qt ¼ aðlKt Þa Lbt ¼ ala Kta Lbt :
(7.3)
That is, as a result of the change in the magnitude of the variable, only the aspect
ratio will change; there is no need to adjust the other variables and coefficients.
Complex-valued functions are another matter. The complex resource variable
represented as ðKt þ iLt Þ significantly changes the model in the scale of either of
the two resources. For example, changing the scale of the variable capital share for
the same factor l leads to a substantial change in the most complex variable, as is
easily seen:
lKt þ iLt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lt
ðlKt Þ2 þ L2t eiarctglKt ;
7.1 General Provisions of Complex-Valued Model Classification
235
which means that the complex variable of a production function is not uniform;
therefore, any complex-valued function whose scale was changed will change all its
coefficients and the accuracy of the description of the original variables.
Let us take the power production function with real coefficients as an example:
Gt þ iCt ¼ aðKt þ iLt Þb :
(7.4)
The gross margin is described by this function as follows:
Gt ¼ að
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b
Lt
Kt 2 þ L2t Þ cosðbarctg Þ:
Kt
(7.5)
If we change the capital resource scale l times we will have the following
expression for the gross margin:
Gt ¼ að
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b
Lt
Þ:
l2 Kt 2 þ L2t Þ cosðbarctg
lKt
(7.6)
This means that an adjustment to variable Kt will cause changes in both the
proportionality coefficient a and exponent b, and the accuracy of the description of
the gross margin, which in practice will be influenced by random errors. And since
in the previous chapter we discussed the analytical properties of this very model and
an interpretation of exponent b, it is quite easy to understand how essential it is to
have a cautious stance with respect to the dimension and scale of variables of
production functions with complex variables.
Thus, two essential problems (besides numerous other, less important, ones)
require the development of complex-valued production functions:
1. The need for new economic variables, which is impossible for model of the (7.1)
type,
2. Nullifying the influence of change in the scale of the original variables on
modeling results.
Before we show how to create economic models of complex variables that could
be free from the above-mentioned defects, let us turn our attention to the very
essence of economic indicators, which is best illustrated by the production process.
All the indicators used for modeling any economic process represent a result of
some aggregation (and abstraction, which is taken for granted).
Indeed, the gross margin Gt consists of many components that can be divided
into two parts – one part of the profit is assigned to the state budget in the form of
income tax and the other stays in the company’s coffers.
Production costs Ct also consist of many terms, but they can also be represented
in the form of two components – fixed costs (depreciation, employee compensation,
etc.) and variable costs (raw materials, semifinished products, energy for
operations, etc.).
236
7 Multifactor Complex-Valued Models of Economy
Capital Kt in any form can also be represented as two terms – core and noncore
(for example, core and noncore assets).
In the same way, labor resources Lt consist of two large groups – industrial and
production on the one hand and other personnel on the other hand.
The list could go on, but the idea is that practically each economic indicator, not
only of production, can be represented as the sum of two terms based on some
classification. As always, justification could be found to subdivide these two classes
into active and passive parts depending on their different effects on production.
Indeed, labor resources that were used in the production function theory in
general and in previous chapters in particular reflect, very roughly, the influence
of this indicator on production performance, as an increase in the number of
production and industrial workers has a different effect on production performance
compared to other workers. Their summation nullifies this influence and leads to a
deterioration of the model properties – both analytical and forecasting.
In the same way, the size of the active part of fixed production assets (machines,
equipment, production lines, etc.) has a different effect on the production variant
than their passive parts (buildings, construction, infrastructure, etc.).
Since the classification of practically every economic variable implies a division
into active and passive groups, the natural conclusion is to represent them in a
complex-valued economy in the form of a complex variable where the real part is
associated with the active term and the imaginary part with the passive one. This is
the rule we will stick to henceforth.
Then we can write the capital resource in the following way:
K0 þ iK1 ;
(7.7)
where K0 means fixed production assets and K1 major nonproduction assets.
The labor resource is to be represented as follows:
L0 þ iL1 ;
(7.8)
where L0 is industrial and production personnel and L1 nonproduction personnel.
Output can be represented, as previously, in the form of a complex variable
including the gross margin and production costs, or as some classified variable, for
example, gross output (including sold and unsold products):
Q0 þ iQ1 :
(7.9)
This idea of production modeling easily solves the first problem – that of adding
a new variable. If it is necessary to add to a model a new production resource S, the
latter can be represented as a complex variable consisting of an active and a passive
part:
S0 þ iS1 :
(7.10)
7.2 Linear Classification Production Function
237
For example, to model agricultural production, the land area can be represented
as the area for crops. livestock, etc.
Taking into account all of the foregoing considerations, the general production
model will have the following form:
Q0 þ iQ1 ¼ F½ðK0 þ iK1 Þ; ðL0 þ iL1 Þ; ðS0 þ iS1 Þ:
(7.11)
Extrapolation of this approach to all economic models, not only production ones,
will result in the following general form:
y0 þ iy1 ¼ Fðx0j þ ix1j Þ:
(7.12)
Here j is the number of the economic variable included in the model.
Model (7.12) is a multifactor model. Since this model opens up a new class of
complex-valued models based on the classification of economic variables, we will
call them classification models.
Classification models are free from the above-mentioned defect of nonuniformity of (7.1). Classification variables by definition have the same dimension since
they represent two parts of one whole. This is why if you need to scale some
complex variable, you should multiply both the real and the imaginary part of this
variable by the scaling coefficient l:
lK0t þ ilK1t ¼ l
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K1t
2 eiarctgK0t ¼ lðK þ iK Þ:
Kot 2 þ K1t
0t
1t
We see that this is a uniform variable of the first degree. In this case uniformity
or nonuniformity is determined not by the features of complex variables but by the
peculiarities of the models used.
In this chapter we will discuss economic classification models with respect to
production process modeling problems.
7.2
Linear Classification Production Function
Let us again consider a production function describing the behavior of a complex
production result including the gross margin Gt and the production costs Ct subject
to the use of capital and labor. In this case both capital and labor can be represented
as complex classification variables (7.7) and (7.8):
K0 þ iK1 ;
L0 þ iL1 :
A linear function is the simplest variant of the production classification function:
Gt þ iCt ¼ ða0 þ ia1 Þ þ ðb0 þ ib1 ÞðK0t þ iK1t Þ þ ðc0 þ ic1 ÞðL0t þ iL1t Þ: (7.13)
238
7 Multifactor Complex-Valued Models of Economy
Here K0 means fixed production assets, K1 fixed nonproduction assets, L0 industrial
and production personnel, аnd L1 nonproduction personnel.
If in this function the imaginary parts of complex proportionality coefficients are
equal to zero, then the function can be transformed into a system of elementary
multifactor (two-factor) equations:
(
Gt ¼ a0 þ b0 K0t þ c0 L0t;
Ct ¼ a1 þ b0 K1t þ c0 L1t :
(7.14)
This model will not be made more complex if the real parts of the complex
proportionality coefficients are equal to zero:
(
Gt ¼ a0
b1 K1t
c1 L1t ;
Ct ¼ a1 þ b1 K0t þ c1 L0t :
(7.15)
This is why, it is only model (7.13) that is reasonable to use in economic and
mathematical modeling, and we can transform it into a system of two real
equalities:
(
Gt ¼ a0 þ b0 K0t b1 K1t þ c0 L0t c1 L1t ;
Ct ¼ a1 þ b0 K1t þ b1 K0t þ c0 L1t þ c1 L0t :
(7.16)
Now we can give an interpretation to the processes that can be described by
means of a complex-valued model like (7.13). It follows from the first equality of
(7.16) that an increase in the fixed production assets K1 and the number of
nonproduction personnel L1 leads to a reduction in the gross margin, and from the
second one that growth in these resources is followed by growth in the production
costs. This is the case in most real production processes – growth in noncore assets
deteriorates the production parameters, as does growth in the administrative
apparatus.
Growth in the fixed production assets K0 and industrial and production personnel
L0 results in a linear growth in the gross margin and production costs. The nature of
this growth is determined by the proportionality coefficients of the variables.
Since model (7.13) can be represented in the form of real variables (7.16), one
can ask if it is reasonable to use a complex-valued function as it may be easier to use
two real functions (7.16). The answer comes from a mere comparison of (7.13) and
(7.16). Model (7.13) is compact, and it is easy to find the model coefficients by
LSM. One need simply solve a system of six equations with six unknowns:
7.2 Linear Classification Production Function
8
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X
t
X
t
X
Gt ¼ na0 þ b0
Ct ¼ na1 þ b1
K0t Gt
t
X
t
K0t
X
t
X
t
X
t
K0t þ b0
K0t K1t þ c0 ð
K0t Ct þ
b1
t
K1t Ct ¼ a0
t
2b1
t
X
X
X
X
X
K1t þ c0
t
K1t þ c0
K0t
a1
t
t
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X
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X
K1t þ a1
X
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X
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L1t ;
t
L1t þ c1
K1t þ b0
K1t L1t Þ
t
c1
t
t
K0t L0t
t
K1t Gt ¼ a0
X
239
X
t
c1 ð
K0t þ b1
X
2
ðK0t
X
t
X
t
L0t ;
t
2
K1t
Þ
K0t L1t þ
2
ðK0t
X
t
L0t K1t Þ;
2
K1t
Þ
X
X
>
>
K1t L0t þ c1
L1t K1t ;
K0t K1t þ c0
K0t L1t
K0t L0t þ
þ2b0
>
>
>
>
t
t
t
t
t
>
>
X
X
X
X
X
X
>
>
>
L0t Gt
L1t Ct ¼ a0
L0t a1
L1t þ b0 ð
K0t L0t
K1t L1t Þ
>
>
>
t
t
t
t
t
t
>
>
X
X
X
X
X
>
>
>
b1 ð
K1t L0t þ
K0t L1t Þþc0 ð
L20t
L21t Þ 2c1
L0t L1t ;
>
>
>
>
t
t
t
t
t
>
>
X
X
X
X
X
X
>
>
>
L
C
þ
L
G
¼
a
L
þ
a
L
þ
b
ð
K
L
K0t L1t Þ
0t
t
1t
t
1
0t
0
1t
0
1t
0t
>
>
>
>
t
t
t
t
t
t
>
X
X
X
X
X
>
>
>
>
K0t L0t þ
K1t L1t Þ þ c1 ð
L20t þ
L21t Þ þ 2c0
L0t L1t :
: þ b1 ð
X
t
X
t
X
t
t
t
Here, T is the number of observations, t ¼ 1,2,3,. . .,T.
In the case of model (7.16), which represents a system of two real linear
equations, the situation is much more complicated – using LSM, it is necessary to
estimate the coefficients of the first equation of the system and then those of the
second one.
Application of LSM to the first equality of system (7.16) makes it necessary to
solve a system of five such equations:
X
X
X
X
8X
G
¼
a
T
þ
b
K
b
K
þ
c
L
c
L1t ;
>
t
0
0
0t
1
1t
0
0t
1
>
>
>
t
t
t
t
t
>
X
X
X
X
X
X
>
>
2
>
>
Gt K0t ¼ a0
K0t þ b0
K0t
b1
K0t K1t þ c0
K0t L0t c1
K0t L1t ;
>
>
>
t
t
t
t
t
t
>X
>
X
X
X
X
X
<
2
K1t L1t ;
K1t L0t c1
K1t
þ c0
K0t K1t b1
K1t þ b0
Gt K1t ¼ a0
>
t
t
t
t
t
t
>
X
X
X
X
X
X
>
>
>
>
L1t L0t ;
L20t c1
K1t L0t þ c0
K0t L0t b1
L0t þ b0
Gt L0t ¼ a0
>
>
>
t
t
t
t
t
t
>
>
X
X
X
X
X
X
>
>
>
L21t :
L0t L1t c1
K1t L1t þ c0
K0t L1t b1
L1t þ b0
Gt L1t ¼ a0
:
t
t
t
t
t
t
(7.17)
240
7 Multifactor Complex-Valued Models of Economy
Here one should be sure that the obtained coefficients will correspond to those
obtained in the solution of the other system of five LSM equations determined by
the second equation of system (7.16):
X
X
X
X
8X
L0t ;
L1t þ c1
K0t þ c0
K1t þ b1
Ct ¼ a1 T þ b0
>
>
>
> t
t
t
t
t
>
X
X
X
X
X
X
>
>
2
>
>
K1t L0t ;
K1t L1t þ c1
K0t K1t þ c0
K1t
þ b1
K1t þ b0
Ct K1t ¼ a1
>
>
>
t
t
t
t
t
t
>
>
X
X
X
X
X
<X
2
K0t L0t ;
K0t L1t þ c1
K0t
þ c0
K0t K1t þ b1
K0t þ b0
Ct K0t ¼ a1
>
t
t
t
t
t
t
>
X
X
X
X
X
>X
>
>
>
L1t L0t ;
L21t þ c1
K0t L1t þ c0
K1t L1t þ b1
L1t þ b0
Ct L1t ¼ a1
>
>
>
t
t
t
t
t
t
>
>
X
X
X
X
X
X
>
>
>
Ct L0t ¼ a1
L0t þ b0
K1t L0t þ b1
K0t L0t þ c0
L0t L1t þ c1
L20t :
:
t
t
t
t
t
t
(7.18)
However, we cannot be sure about this. Moreover, the coefficients resulting from
the solution of system (7.17) will vary from those found by solving equation system
(7.18), at least because all the original data represent the result of the implementation of a random process and therefore include random errors, which will lead to
various coefficient values.
Also, from system (7.17) we see that its solution does not depend at all on
changes in production costs, and from (7.18) we see that its solution does not
depend on the nature of the change in the gross margin. In other words, probability
that solving (7.17) and (7.18) will yield equal coefficients is practically equal to
zero. This means that it is impossible to obtain system (7.16) by this method!
Even if we first solve system (7.17) and find coefficients a0, b0, b1,, and c1 by
substituting the obtained values into system (7.18), we see that each equality of this
system from the first to the fifth will give different values of coefficient a1. To find
the balance, specific coordination procedures are needed.
Solving system (7.17) we obtain model coefficient values:
Gt ¼ a0 þ b0 K0t þ b1 K1t þ c0 L0t þ c1 L1t :
(7.19)
Solving system (7.18) we obtain coefficients of another model:
Ct ¼ a1 þ d0 K1t þ d1 K0t þ f0 L1t þ f1 L0t :
(7.20)
Certainly, they can be synthesized in a model like (7.13):
Gt þ iCt ¼ ða0 þ ia1 Þ þ ½b0 K0t þ b1 K1t þ iðd0 K1t þ d1 K0t Þ þ ½c0 L0t
þ c1 L1t þ iðf0 L1t þ f1 L0t Þ;
(7.21)
7.2 Linear Classification Production Function
241
but it will be quite a different model! One can also transform active parts of the
resources into the real part and passive part into the imaginary one. However, here
the coefficients of the synthesized model will vary from those of the complexvalued model. In addition, the bulkiness of the procedure makes it pointless!
Since in this model we can use the procedure of preliminary centering of the
original parameters with respect to their averages, calculation of the complexvalued model coefficients will be much simplified. The model involving centered
parameters will have the following form:
Gt þ iCt ¼ ðb0 þ ib1 ÞðK0t þ iK1t Þ þ ðc0 þ ic1 ÞðL0t þ iL1t Þ
(7.22)
But a system of normal LSM equations for centered variables will contain four
equations:
X
X
X
8X
2
2
K0t Gt
K1t Ct ¼ b0
ðK0t
K1t
Þ 2b1
K0t K1t
>
>
> t
>
t
t
t
>
>
X
X
X
X
>
>
>
L0t K1t Þ;
K0t L1t þ
K1t L1t Þ c1 ð
K0t L0t
þc0 ð
>
>
>
t
t
t
t
>
>X
X
X
X
>
>
2
2
>
K0t K1t
ðK0t
K1t
Þ þ 2b0
K1t Gt ¼ b1
K0t Ct þ
>
>
>
>
t
t
t
> t
>
X
X
X
X
>
>
>
L1t K1t Þ;
K0t L0t þ
K1t L0t Þ þ c1 ð
K0t L1t
>
< þc0 ð
t
t
t
t
X
X
X
X
X
X
>
K0t L1t Þ
K1t L0t þ
K1t L1t Þ b1 ð
K0t L0t
L1t Ct ¼ b0 ð
L0t Gt
>
>
>
>
t
t
t
t
t
t
>
>
X
X
X
>
>
>
þc0 ð
L20t
L21t Þ 2c1
L0t L1t ;
>
>
>
>
t
t
t
>
X
X
X
X
X
X
>
>
>
>
L0t Ct þ
L1t Gt ¼ b0 ð
K1t L0t
K0t L1t Þ þ b1 ð
K0t L0t þ
K1t L1t Þ
>
>
> t
t
t
t
t
t
>
>
>
> þc ðX L2 þ X L2 Þ þ 2c X L L :
>
:
0t 1t
0
1
1t
0t
t
t
t
(7.23)
If one works with software that provides mathematical operations with complex
variables, the task will be even simpler.
If we record model (7.22) with complex variables:
z_t ¼ b_K_ t þ c_L_t
(7.24)
we will easily obtain a system of LSM estimations of the model coefficients:
(X
X
z_t K_ t ¼ b_
z_t L_t ¼ b_
X
X
X
L_t K_ t ;
X
L_t K_ t þ c_
L_2t :
K_ t2 þ c_
(7.25)
242
7 Multifactor Complex-Valued Models of Economy
The solution will allow us to obtain the required complex coefficients of the
multifactor model.
The conclusion that follows is clear – model (7.13) extends the instrumental base
of economic and mathematical modeling since the building of its analog in the
sphere of real numbers requires complex computational procedures that do not
ensure adequate modeling.
Linear complex-valued multifactor models possess all the drawbacks of linear
production function models with real variables. Therefore, understanding that
model (7.13) has the right to exist and practical use let us leave it for modeling
simple production situations.
7.3
Classification Production Function of Cobb-Douglas Type
Linear production classification functions differ from similar real-variable
functions in their meaning and convenience in usage. However, the properties of
complex-valued functions are most prominent for nonlinear functions. Their
analogs in the sphere of real variables are extremely complicated, as was pointed
out in previous chapters. For that reason it is logical to consider nonlinear, complexvalued classification functions. The simplest variant of such a function that comes
to mind is built according to the Cobb-Douglas production function.
Let us call models like
Gt þ iCt ¼ ða0 þ ia1 ÞðK0 þ iK1 Þa ðL0 þ iL1 Þ1
a
(7.26)
Cobb-Douglas classification production functions.
Here, as in Cobb-Douglas functions, the exponent lies within the range
0 a 1:
(7.27)
We know that functions are linearly uniform if the following condition is met:
f ðlx1 ; lx2 ; . . . ; lxi Þ ¼ lf ðx1 ; x2 ; . . . xi Þ:
(7.28)
Let us check if this condition is met for model (7.26):
ða0 þ ia1 ÞðlK0 þ ilKv Þa ðlL0 þ ilLv Þ1
a
¼ la l1 a ða0 þ ia1 Þ ðK0 þ iKv Þa ðL0 þ iLv Þ1
a
¼ lða0 þ ia1 ÞðK0 þ iKv Þ ðL0 þ iLv Þ
This means that the function is linearly uniform.
1 a
:
a
(7.29)
7.3 Classification Production Function of Cobb-Douglas Type
243
If we represent the right-hand side of model (7.26) in exponential form we obtain
ða0 þ ia1 ÞðK0 þ iK1 Þa ðL0 þ iL1 Þ1
a
¼ Ra eiya RaK eiayK R1L a eið1
aÞyL
:
(7.30)
Here we use the following designations:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1
a20 þ a21 ; ya ¼ arctg ;
a0
(7.31)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K1
K02 þ K12 ; yK ¼ arctg ;
K0
(7.32)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L1
L20 þ L21 ; yL ¼ arctg :
L0
(7.33)
Ra ¼
RK ¼
RL ¼
Then model (7.26) can be represented as a system of two real equations: (1) for
the gross margin:
Gt ¼ Ra RaK R1L
a
cosðya þ ayK þ ð1
aÞyL Þ;
(7.34)
aÞyL Þ
(7.35)
(2) for production costs:
Ct ¼ Ra RaK R1L
a
sinðya þ ayK þ ð1
Now the general characteristics of the model are quite clear. If all the other
resources remain unchanged, then:
1. If investments in fixed capital K0 grow, the absolute value of the capital
resources increases (7.32) and their polar angle decreases. The cosine of the
decreasing angle goes up. When the modulus of the capital resource grows, the
gross margin (7.34) grows as well. For production costs in this situation there are
two opposing trends – the factor, the modulus of capital resources (7.35),
increases but the sine of the decreasing angle goes down. That is why it follows
from (7.35) that depending on the values of the exponent a and the complex
proportionality coefficient y there could occur both growth (if a is close to zero)
and decrease (if a is close to one) in production costs. Intervals are possible
where costs remain unchanged.
2. If investments in nonfixed capital grow, the modulus of the capital resources will
grow as well as the polar angle. This means that the gross margin is calculated as
the product of two opposing trends – the increasing modulus of capital resources
and decreasing cosine of the sum of angles. However, depending on the original
data and exponent value, this can in some cases cause an increase in the resulting
indicator, in other cases a decrease in its values, and in other cases stability of the
indicator. The dynamics of the production costs in this situation is determined by
244
7 Multifactor Complex-Valued Models of Economy
multiplying the increasing modulus by the increasing sine of the sum of polar
angles. This means that production costs will uniquely increase.
Similar conclusions can be drawn with respect to the behavior of another
complex variable – labor resources. With an increase in the number of industrial
and production workers the gross margin grows and costs may increase, remain
unchanged, or decrease. With an increase in nonproduction personnel costs will
definitely rise – the profit dynamics being determined by the model coefficients
growth, stability – or decrease.
Thus, we see that model (7.26) covers all the possible alternatives of production
relationships, except perhaps production stagnation.
A thorough study of function (7.26)’s properties was carried out by E.V.
Sirotina. Let us show some of the results of that study.
To find unknown parameters a0, a1, and a of model (7.26), let us use LSM
estimation of nonlinear models of complex variables with complex coefficients
suggested in Chap. 4.
For models of the Cobb-Douglas type, as with Cobb-Douglas production
functions, it is a little easier than for power functions with unlimited exponent
values. For this equation (7.26) should be transformed into linear form by taking the
logarithm the left- and right-hand sides of the natural base:
lnðGt þ iCt Þ ¼ lnða0 þ ia1 Þ þ a lnðK0 þ iK1 Þ þ ð1
aÞ lnðL0 þ iL1 Þ:
(7.36)
Using simple transformations we can present this expression in the following
form:
ln
Gt þ iCt
K0t þ iK1t
¼ lnða0 þ ia1 Þ þ a ln
:
L0 þ iL1
L0t þ iL1t
(7.37)
We see that the model looks like a simple logarithmic equation of two complex
variables. To simplify further our conclusions, let us introduce the following
designations of these variables. The resulting variable under the logarithm sign,
which E.V. Sirotina proposed calling “complex productivity,” can be transformed
into the following form, which is convenient for research:
gt ¼
Gt þ iCt Gt L0t þ Ct L1t
Ct L0t Gt L1t
¼
þi
:
L0 þ iL1
L20 þ L21
L20 þ L21
(7.38)
The complex resource, also under the logarithm sign on the right-hand side, will
be designated in the following way:
kt ¼
K0t þ iK1t K0t L0t þ K1t L1t
K1t L0t K0t L1t
¼
þi
:
L0t þ iL1t
L20t þ L21t
L20t þ L21t
(7.39)
7.3 Classification Production Function of Cobb-Douglas Type
245
Since this variable represents the capital-to- labor ratio, it can just be called that.
These two new economic parameters can provide the economist with an additional characteristic of the production processes taking place, but will not explore
this idea further.
As was stated in the Chap. 1, in this study we use principal logarithmic values,
which is why equality (7.37) can be presented in the following way:
ln Rg þ iyg ¼ A0 þ iA1 þ aðln Rk þ iyk Þ:
(7.40)
Here:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
Gt L0t þ Ct L1t
Ct L0t Gt L1t
Ct L0t Gt L1t
Rg ¼
;
þ
; yg ¼ iarctg
2
2
2
2
G
L0 þ L1
L0 þ L1
t L0t þ Ct L1t
(7.41)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
K0t L0t þ K1t L1t
K1t L0t K0t L1t
; yk
þ
Rk ¼
L20t þ L21t
L20t þ L21t
¼ iarctg
K1t L0t K0t L1t
;
K0t L0t þ K1t L1t
A0 ¼ ln
(7.42)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1
a20 þ a21 ; A1 ¼ arctg :
a2
(7.43)
Now coefficients of this model can be found by LSM. With reference to this
model the problem becomes trivial, since it is necessary to solve a system of
equations with three unknowns:
8X
X
>
ln Rg ¼ TA0 þ a
ln Rk ;
>
>
>
t
t
>
>
X
>
<X
yg ¼ TA1 þ a
yk ;
t
t
>
>
X
>
X
X
X
X
X
>
>
2
>
>
y2k :
y
þa
ln
R
þ
ln
R
þ
A
y
y
¼
A
ln
R
ln
R
þ
k
k
k
1
g k
0
g
k
:
t
t
t
t
t
t
(7.44)
E.V. Sirotina built a Cobb-Douglas-type production classification function for
statistical data for the period from 1999 to 2006 on the company Lenoblgaz plc.
Since these data represent a commercial secret, we will not present them here.
However, in accordance with the statistical data, after reduction of the variables to a
single scale and calculation of all the intermediate variables, a system of LSM
equations (7.44) was obtained, which, with respect to this company, will have the
following form:
246
7 Multifactor Complex-Valued Models of Economy
8
>
< 14:21 ¼ 8A0 þ 19:07a;
7:37 ¼ 8A1 þ 1:59a;
>
:
32:65 ¼ 14:21A0 1:59A1 þ 46:68a;
where it is easy to find the required coefficient values: A0 ¼ 0.084, A1 ¼ 1.062,
a ¼ 0.71. Using the known values of coefficients A0 and A1 one can find the
original values of coefficients a0 and a1:
a0 þ ia1 ¼ e0:084þi1:062 ¼ 0:53 þ i0:95
Then the model of the Cobb-Douglas-type classification production function for
Lenoblgazfor the period under consideration has the following form:
Gt þ iCt ¼ ð0:53 þ i0:95ÞðK0 þ iK1 Þ0:71 ðL0 þ iL1 Þ0:29 :
(7.45)
It is evident that there were no difficulties in building such a function with two
complex variables. Moreover, the Cobb-Douglas-type classification production
function turned out to be easy to apply in practice.
7.4
Elasticity and Other Characteristics of a Classification
Production Complex-Valued Function
According to its properties, the Cobb-Douglas-type classification production function is more suitable for economic reality than the Cobb-Douglas function as it
describes most actually existing production relationships between resources and
production performance. It is obvious that it also describes the relationships that are
unattainable in the sphere of real-variable models.
However, to consciously use this new mathematical tool for economic modeling it
is necessary to examine its properties more carefully. For this we will use the standard
set of tools used by specialists for studying properties of “neoclassical” production
functions and the Cobb-Douglas function as a kind of “neoclassical” production
function. The following conditions will hold for the power production function
model with real variables with exponents lying within a range from zero to one:
1. In the absence of one of the resources, production is impossible in general, which
is expressed mathematically as follows:
Fð0; LÞ ¼ FðK; 0Þ ¼ 0:
(7.46)
7.4 Elasticity and Other Characteristics of a Classification Production. . .
247
2. With unlimited increase in one of the resources the output increases infinitely,
i.e.,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
FðK; 1Þ ¼ Fð1; LÞ ¼ 1 b2 4ac:
(7.47)
dQ
dQ
> 0;
> 0:
dK
dL
(7.48)
3. With growth in resources output grows, which means that the first derivatives are
positive:
4. With an increase in resources the rate of output increase slows down, which
means that the second derivative is negative:
@2Q
@2Q
<
0;
< 0:
@K 2
@L2
(7.49)
These properties follow from the mathematical form of the power “neoclassical”
production function, which is considered by economists as important arguments for
practical applicability of the model.
Since in this section we are studying the properties of one of the simplest
classification production functions – Cobb-Douglas-type of functions with complex
variables (7.26) – let us look for similar properties of this function.
The Cobb-Douglas-type classification production function has the following
form:
Gt þ iCt ¼ ða0 þ ia1 ÞðK0 þ iK1 Þa ðL0 þ iL1 Þ1 a :
Here the exponent lies within the range 0 a 1 , which makes it possible to
consider the function as a certain analog of the well-known Cobb-Douglas function.
It is evident that if at least one of the complex resources is equal to zero, the
output modeled by this function will also be equal to zero, i.e., condition (7.46) is
satisfied.
Theoretically it can happen that one component of a complex resource is equal to
zero and the other is not. In this case (7.46) does not hold, for example, L0 ¼ 0,
L1 > 0. However, these cases do not occur in actual economic practice. It is
impossible to imagine a situation where industrial and production workers are
absent at a company and nonproduction personnel works, or, conversely, where a
company has industrial and production personnel but does not have managers,
accountants, or a general director. The same can be said about the fixed assets of
an enterprise – if there is no path (K0) to the factory floor (K0), the workers will not
get to the floor.
This is why the real economy reflects a situation where both the real and
imaginary components of a complex result are equal or not equal to zero
248
7 Multifactor Complex-Valued Models of Economy
simultaneously. Other variants refer to idealization where the object models are
associated with properties that are not inherent in those models. While building
models, we use abstraction but not idealization.
The second property i obvious: with an unlimited increase in one of the complex
resources – capital or labor –output increases infinitely. Growth in the modulus of
the complex resource in the model is shown by growth in the modulus of the
production result.
This means that the first and second properties inherent in “neoclassical” models
are also inherent in the model under consideration, which makes the analogy
between them more sound.
Now let us find the signs of the first and second derivatives of function (7.26) to
verify the feasibility of conditions (7.48) and (7.49). E.V. Sirotina made certain
calculations, which we can use. First, to simplify calculation of the derivatives, let
us represent the model of production function (7.26) in exponential form:
a
G þ iC ¼ Ra eiya ðRK eiyK Þ ðRL eiyL Þ
1 a
;
(7.50)
where
Ra ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a20 þ a21 ;
RL ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L20 þ L21 ;
a1
; RK ¼
a0
L1
yL ¼ arctg :
L0
ya ¼ arctg
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K02 þ K12 ;
yK ¼ arctg
K1
;
K0
In this form (7.50), the model’s modulus and polar angle are easily grouped:
G þ iC ¼ ðRa RaK R1L a Þeiðya þiayK þ
ið1 aÞyL Þ
:
(7.51)
This is why the model of a Cobb-Douglas-type classification production function
can be represented in trigonometric form:
G þ iC ¼ Ra RaK R1L a ½cosðya þ ayK þ ð1
aÞyL Þ þ i sinðya þ ayK þ ð1
aÞyL Þ:
(7.52)
This makes it possible to calculate the first and second partial derivatives of the
function with respect to resources.
According to the d’Alembert-Euler (Riemann-Cauchy) condition, to find a
derivative of a complex-valued function, it is sufficient to take the derivatives
with respect to their real or imaginary part. The real part of model (7.52) has the
following form:
G þ iC ¼ R cos y ¼ U:
(7.53)
7.4 Elasticity and Other Characteristics of a Classification Production. . .
249
This is why the derivative of function (7.26) with respect to the complex capital
resource, for example, can be found in the following way:
@ðG þ iCÞ @U
¼
@K
@K0
i
@U
@ðR cos yÞ
¼
@K1
@K0
i
@ðR cos yÞ
:
@K1
Let us calculate the first component of derivative (7.54), namely,
derivative of a complex function:
(7.54)
@ðR cos yÞ
,
@K0
@ðR cos yÞ
@R
@ cos y
¼
cos y þ
R:
@K0
@K0
@K0
as a
(7.55)
The first term is
@R
@Ra RaK RL1
cos y ¼
@K0
@K0
Since
a
K0
ffi cos y:
cos y ¼ aRa R1L a RaK 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K02 þ K12
(7.56)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K02 þ K12 ¼ RK , we finally get
@R
@Ra RaK R1L
cos y ¼
@K0
@K0
a
cos y ¼ aRa R1L a RaK 2 K0 cos y:
(7.57)
The second term (7.56) is a derivative of the cosine of the argument with respect
to K0. It can also be found:
@ cos y
@ cosðya þ ayK þ ð1
R¼
@K0
@K0
¼
R sin ya
@ðarctg KK01 Þ
@K0
aÞyL Þ
R¼
R sin y
@ðayK Þ
@K0
;
where we finally obtain
@ cos y
R¼
@K0
¼
aR sin y
K1
K02 ð1 þ
aRa RaK 2 R1L a K1
K12
Þ
K02
¼ aRa RaK R1L
a
sin y
K02
K1
þ K12
(7.58)
sin y:
Substituting (7.57) and (7.58) to (7.55) we obtain a partial derivative of the
classification production function with respect to the fixed capital:
@ðR cos yÞ
¼ aRa R1L a RaK 2 K0 cos y þ aRa RaK 2 R1L a K1 sin y
@K0
¼ aRa R1L a RaK 2 ðK0 cos y þ K1 sin yÞ:
(7.59)
250
7 Multifactor Complex-Valued Models of Economy
In the same way we can find a partial derivative of the Cobb-Douglas-type
cos yÞ
. Since this derivative is a derivative of a
classification production function: @ðR@K
1
complex function, it can be calculated as follows:
@ðR cos yÞ
@R
@ cos y
¼
cos y þ
R:
@K1
@K1
@K1
(7.60)
The first term (7.60) as the modulus derivative will be recorded as follows:
@R
K1
cos y ¼ aRa RL1 a RaK 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos y ¼ aRa R1L a RaK 2 K1 cos y:
@K1
K02 þ K12
(7.61)
The second term (7.60) represents the derivative of the cosine of the argument
with respect to subsidiary capital K1. This derivative also can be derived as follows:
@ cos y
@ cosðya þ ayK þ ð1
R¼
@K1
@K1
aÞyL Þ
R¼
R sin ya
@ðarctg KK01 Þ
@K1
:
After calculating the derivative of the arctangent we finally have
@ cos y
R¼
@K1
¼
aR sin y
1
K0 1 þ
K12
K02
¼
aRa RaK R1L
a
sin y
K02
K0
þ K12
(7.62)
aRa RaK 2 R1L a K0 sin y:
Substituting (7.61) and (7.62) into (7.60) we obtain a partial derivative of the
classification production function with respect to subsidiary capital:
@ðR cos yÞ
¼ aRa R1L a RaK 2 K1 cos y aRa RaK 2 R1L a K0 sin y
@K1
¼ aRa R1L a RaK 2 ðK1 cos y K0 sin yÞ:
(7.63)
Since we have all the terms we need to obtain a derivative of the classification
function with respect to complex capital, we can do it:
@ðG þ iCÞ @U
@U @R cos y @R cos y
¼
i
¼
i
@K
@K0
@K1
@K0
@K1
¼ aRa RL1 a RaK 2 ðK0 cos y þ K1 sin yÞ iaRa R1L a RaK 2 ðK1 cos y
¼ aRa R1L a RaK 2 ½ðK0
iK1 Þ cos y þ ðK1 þ iK0 Þ sin y:
K0 sin yÞ
(7.64)
7.4 Elasticity and Other Characteristics of a Classification Production. . .
251
In the second term of the last factor of the resulting expression we can take the
imaginary unit out and get
@ðG þ iCÞ
¼ aRa R1L a RaK 2 ½ðK0 iK1 Þ cos y þ iðK0 iK1 Þ sin y
@K
¼ aRa R1L a RaK 2 ðK0 iK1 Þðcos y þ i sin yÞ:
(7.65)
Since cos y þ i sin y ¼ eiy , (7.47) can be written in more convenient form:
@ðG þ iCÞ
¼ ða0 þ ia1 ÞðK0 þ iK1 Þa ðL0 þ iL1 Þ1 a ½aRK 2 ðK0
@K
iK1 Þ:
(7.66)
Let us simplify the expression in square brackets:
aRK 2 ðK0
iK1 Þ ¼ a
K0 iK1
a
¼
¼ aðK0 þ iK1 Þ 1 :
2
2
K0 þ iK1
K0 þ K1
(7.67)
Substituting this value into (7.66) we obtain the final formula for the first
derivative of the classification production function with respect to complex capital:
@ðG þ iCÞ
¼ aða0 þ ia1 ÞðK0 þ iK1 Þa 1 ðL0 þ iL1 Þ1 a :
@K
(7.68)
After carrying out similar difficult calculations we obtain the resulting formula
of the first derivative of the classification production function of complex variables
with respect to complex labor:
@ðG þ iCÞ
¼ ð1
@L
aÞða0 þ ia1 ÞðK0 þ iK1 Þa ðL0 þ iL1 Þ a :
(7.69)
Both (7.68) and (7.69) represent a complex number consisting of real and
imaginary parts. And, as is known, complex numbers can be neither positive nor
negative. This requirement (of positivity, for example) can be made either with
respect to the modulus of the complex variable or to its polar angle, to its real or
imaginary part. This means that with reference to this complex-valued production
function, condition (7.48) on its first derivative will not hold.
Therefore, let us consider the essence of the first derivatives of the function and
then draw a conclusion.
Since by definition 0 a 1 and all the production resources lie in the first
quadrant of the complex plane, the modeled result will lie in the first quadrant of the
complex plane of the production result, subject to the nonnegativity of each of the
coefficients of the complex proportionality coefficient:
a0 0; a1 0:
(7.70)
252
7 Multifactor Complex-Valued Models of Economy
One should remember that positivity of the first derivatives of the “neoclassical”
production function testifies to the fact that positive growth in resources leads to
positive growth in production performance.
In the case under consideration, if any of the conditions (7.70) in the CobbDouglas-type classification production function of complex variables is violated,
the modeled result will not fall into the first quadrant. Therefore, the use of some
resource worsens the production result and vice versa – a reduction in the labor or
capital resource has a favorable effect on production performance. Such cases exist
in the economy, which is why one can say that the model of a Cobb-Douglas-type
classification production function of complex-variable models more diverse
alternatives than real-variable models. But if researchers are going to follow a strict
rule – an increase in resources should lead to an increase in production performance, then the Cobb-Douglas-type classification model, besides restrictions on the
exponent 0 a 1, should be supplemented by condition (7.70). But, for CobbDouglas functions the condition of positivity of the proportionality coefficient is
extended in and of itself.
Let us calculate the second derivative with respect to complex resources.
Omitting those calculations that are similar to the preceding ones, we present the
final formulas of the second derivative of the Cobb-Douglas-type of the classification production function of complex variables with respect to its complex resources.
The second derivative of this function with respect to the complex capital will
have the following form:
@ 2 ðG þ iCÞ
¼ aða
@K 2
1Þða0 þ ia1 ÞðK0 þ iK1 Þa 2 ðL0 þ iL1 Þ1 a :
(7.71)
Since the exponent of this function is positive 0 a 1, as the coefficients of
the complex proportionality coefficient are positive (7.70), the second term (7.71)
will be negative, and with all the other factors positive, the modeled result will fall
into the third quadrant of the complex plane, i.e., both the real and imaginary parts
of the second production complex-valued function are negative.
The second derivative of the function under consideration with respect to
complex labor resources will be equal to
@ 2 ðG þ iCÞ
¼
@L2
að1
aÞða0 þ ia1 ÞðK0 þ iK1 Þa ðL0 þ iL1 Þ
a 1
:
(7.72)
For the same reasons, all the factors of (7.72) are positive, and since there is a
minus sign in their product, both the real and imaginary parts of the second
derivative with respect to complex labor will be negative.
Thus, when we set conditions
0 a 1;
a0 0; a1 0;
(7.73)
7.4 Elasticity and Other Characteristics of a Classification Production. . .
253
for the Cobb-Douglas-type classification production function of complex variables,
it acquires properties similar to those of “neoclassical” production functions of real
variables.
Then it is necessary to calculate the elasticity of the Cobb-Douglas-type classification production function of complex variables. This is easy to do since the
calculation of the first derivatives of this function with respect to complex capital
and complex labor was performed previously.
Since the formula of the elasticity in the sphere of real numbers has the following
form:
ex ¼
@y x
;
@x y
it can also be used for models of complex variables, as was done in previous
chapters, if we know the first derivatives of the complex-valued function.
Then the elasticity of the Cobb-Douglas-type classification production function
of complex variables with respect to complex capital will have the following form:
eK ¼
@ðG þ iCÞ K
¼ aða0 þ ia1 ÞðK0 þ iK1 Þa 1 ðL0 þ iL1 Þ1
@K
G þ iC
a
K
G þ iC
(7.74)
or, with obvious reductions,
eK ¼ a:
(7.75)
Similarly, we can find the elasticity with respect to complex labor:
eL ¼
@ðG þ iCÞ L
¼ ð1
@L
G þ iC
aÞða0 þ ia1 ÞðK0 þ iK1 Þa 1 ðL0 þ iL1 Þ
a
L
;
G þ iC
(7.76)
where finally we have
eL ¼ ð1
aÞ:
(7.77)
The conclusion that follows from the foregoing results is clear – in the CobbDouglas-type classification production function of complex variables the exponents
play the same role as the exponents in the Cobb-Douglas production function: they
are parameters of the elasticity of the production result with respect to the appropriate complex resource.
Since the proposed Cobb-Douglas-type classification production function
represents each research as a complex variable, one can say that the function itself,
besides elasticity of the complex resource as a whole, also has elasticity of each of
the resource components, real and imaginary.
254
7 Multifactor Complex-Valued Models of Economy
Let us derive a formula of elasticity of the Cobb-Douglas-type classification
production function with respect to fixed capital. For that, let us first calculate the
first partial derivative of the production complex-valued function with respect to the
fixed capital using the same designations as previously:
@ðG þ iCÞ @ðR cos yÞ
@ðR sin yÞ
¼
þi
:
@K0
@K0
@K0
(7.78)
Previously, in (7.59), for the first term we had
@ðR cos yÞ
¼ aRa R1L a RaK 2 ðK0 cos y þ K1 sin yÞ
@K0
and for the second term (7.78)
@ðR sin yÞ
¼ aRa R1L a RaK 2 ðK0 sin y
@K0
K1 cos yÞ:
(7.79)
Substituting the results obtained into (7.78) we can find the first partial derivative
of the Cobb-Douglas-type classification production function with respect to fixed
capital:
@ðG þ iCÞ
¼ aRa R1L a RaK 2 ½ðK0 cos y þ K1 sin yÞ þ iðK0 sin y
@K0
K1 cos yÞ: (7.80)
Simplifying the expression we have
@ðG þ iCÞ
¼ aða0 þ ia1 ÞðK0 þ iK1 Þa ðL0 þ iL1 Þ1
@K0
a
K0 iK1
;
K02 þ K12
(7.81)
where we can find the elasticity of the Cobb-Douglas-type classification production
function of complex variables with respect to fixed capital:
e K0
@ðG þ iCÞ K0
K02 iK0 K1
K02
¼
¼a
¼a
2
2
2
@K0
G þ iC
K0 þ K1
K0 þ K12
K0 K1
i 2
: (7.82)
K0 þ K12
This is a complex coefficient. Let us give it an economic interpretation.
If the fixed capital increases by one, the real part of the complex production
result, i.e., the gross margin, increases by less than a since the real part of the
complex coefficient is less than unity. With an increase in the fixed capital to
infinity, an increase in the fixed capital by unity leads to an increase in the gross
margin by a.
The imaginary component of the complex coefficient of elasticity of the production result for the fixed capital is negative. This means that the imaginary
7.4 Elasticity and Other Characteristics of a Classification Production. . .
255
component of the production result, i.e., gross costs, decreases by a value lower
than a. When the fixed capital tends to infinity, the costs stop changing.
In a similar way we can calculate the first derivative of the Cobb-Douglas-type
classification production function for subsidiary capital and on the basis of these
calculations determine the formula for calculating the elasticity of this resource.
Omitting intermediary calculations, we can present the final formula for the
elasticity:
e K1 ¼
@ðG þ iCÞ K1
K 2 þ iK0 K1
K12
K0 K1
¼a 12
¼
a
þ
i
: (7.83)
@K1
G þ iC
K02 þ K12
K0 þ K12
K02 þ K12
This can also receive an economic interpretation.
An increase in the subsidiary capital leads to an increase in the gross margin as the
real component and the gross cost as the imaginary component of the production
result. However, with an increase in the subsidiary capital to infinity, the real component of the complex coefficient of elasticity tends to a, and the imaginary one tends to
zero. This means that with high values of the subsidiary capital its growth leads to
growth in the gross margin by a without influencing the production costs.
The sum of the complex coefficient of the elasticity of fixed capital (7.82) and
that of the subsidiary capital (7.83) will give the value of the total elasticity of
complex capital:
eK0 þ eK1 ¼ að
K02
K02 þ K12
i
K0 K1
K12
K0 K1
þi 2
þ
Þ ¼ a:
K02 þ K12 K02 þ K12
K0 þ K12
In a similar way we derive the formulas for elasticity of the real part of complex
labor:
eLo ¼
ð1 aÞ Lo 2
Lo 2 þ L1 2
i
ð1
aÞ Lo L1
Lo 2 þ L1 2
(7.84)
and of the imaginary part:
eL1 ¼
ð1 aÞ L1 2
ð1 aÞ Lo L1
þi
:
2
2
Lo þ L1
L o 2 þ L1 2
(7.85)
Then these coefficients show how an increase in industrial and production
workers or nonproduction workers affect the gross margin and costs. The contribution of each component is determined by both the exponents and particular values
of the hired labor.
The sum of these partial coefficients of elasticity will give us the total elasticity
of complex labor:
eL ¼ eLo þ eL1 ¼
ð1
aÞ Lo 2 þ ð1 aÞ L1 2
¼1
Lo 2 þ L1 2
a:
256
7 Multifactor Complex-Valued Models of Economy
Analysis of partial coefficients of elasticity shows that they vary with variation
in the volumes of resources use, i.e., return on resources varies with the resource
scale, their sum remaining unchanged:
eK0 þ eK1 ¼ a; eLo þ eL1 ¼ 1
a:
We will not give an economic interpretation to this interesting property since
careful study of the economic meaning of model parameters is not the focus of this
work.
7.5
Classification Power Production Function
The artificial restriction imposed on the limits of variation of exponent a in the
Cobb-Douglas-type classification production function also limits the application of
this model. This restriction (7.27) must simplify considerably the problem of
finding model coefficients, but such simplifications almost always worsens the
model approximation properties. This is why actual production and other economic
processes will be more adequately described by models that do not impose any a
priori restrictions on their coefficients, unless it follows from the core of the
modeled economic situation. Removal of limitations on the domain of the
exponents of the power classification function extends its capacities.
Thus, E.V. Sirotina estimated the coefficients of one of the alternatives of the
power classification production function using as an example the company Lengaz
– a function with real exponents – and obtained the following model:
Gt þ iCt ¼ ð2:34 þ i3:65ÞðK0 þ iK1 Þ0:44 ðL0 þ iL1 Þ0:50 :
This model describes the company’s production process more precisely than the
system of power models of real variables describing the gross margin and production costs separately.
We will not concentrate on this type of model since its properties are similar to
those determined for the Cobb-Douglas-type classification production function
considered in the previous section, but we will pay attention to a model of another
type – classification power production functions with complex exponents:
Gt þ iCt ¼ ða0 þ ia1 ÞðK0 þ iK1 Þðb0 þib1 Þ ðL0 þ iL1 Þðc0 þic1 Þ :
(7.86)
Let us verify the uniformity of this function in accordance with rule (7.28):
ða0 þ ia1 ÞðlK0 þ ilKv Þðb0 þib1 Þ ðlL0 þ ilLv Þðc0 þic1 Þ ¼
lðb0 þib1 Þ lðc0 þic1 Þ ða0 þ ia1 ÞðK0 þ iKv Þðb0 þib1 Þ ðL0 þ iLv Þðc0 þic1 Þ ¼
l
ðb0 þib1 Þþ ðc0 þic1 Þ
ða0 þ ia1 ÞðK0 þ iKv Þ
ðb0 þib1 Þ
ðL0 þ iLv Þ
ðc0 þic1 Þ
;
(7.87)
7.5 Classification Power Production Function
257
i.e., this function is nonuniform with reference to the complex exponent
ðb0 þ ib1 Þ þ ðc0 þ ic1 Þ:
(7.88)
Setting various exponent values, one can obtain its most diverse forms. For
example, a Cobb-Douglas-type classification production function will be obtained
if the sum (7.88) is equal to unity and it will become uniform to the first power.
It is logical to consider production functions that are linearly pseudo uniform,
namely, uniform to power i. This can happen if (7.88) gives in sum an imaginary
unity:
ðb0 þ ib1 Þ þ ðc0 þ ic1 Þ ¼ i:
Let us designate the exponent of this function as b. Then the pseudo-uniform
Cobb-Douglas-type classification production function will have another form:
Gt þ iCt ¼ ða0 þ ia1 ÞðK0 þ iK1 Þib ðL0 þ iL1 Þið1
bÞ
(7.89)
with the restrictions
0 b 1:
(7.90)
To understand the core of the behavior of the pseudo-uniform Cobb-Douglastype classification production function, let us represent it in exponential form:
ib ið1 bÞ iyL ið1 bÞ
iyK
Gt þ iCt ¼ Ra eiya Rib
RL
Ke
e
:
(7.91)
Here we use previously introduced designations:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1
a20 þ a21 ; ya ¼ arctg ;
a0
(7.92)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K1
K02 þ K12 ; yK ¼ arctg ;
K0
(7.93)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L1
L20 þ L21 ; yL ¼ arctg :
L0
(7.94)
Ra ¼
RK ¼
RL ¼
If we transform the right-hand side of model (7.91) so that we can calculate the
modulus of the right-hand side of the complex variable and its polar angle, we will
have
Gt þ iCt ¼ Ra e
ið1
byK yL ðb 1Þ ln Rib
ln RL
K
e
e
e
bÞ
eiya :
(7.95)
258
7 Multifactor Complex-Valued Models of Economy
Since
ib
eln RK ¼ eib ln
pffiffiffiffiffiffiffiffiffiffi
ffi
2
2
K0 þK1
;
and
ið1 bÞ
eln RL
¼ eið1
bÞ ln
pffiffiffiffiffiffiffiffiffi
ffi
2
2
L0 þL1
;
the gross margin, according to the pseudo-uniform Cobb-Douglas-type classification production function, will be calculated as follows:
L
G t ¼ Ra e
K
ðb 1ÞarctgL1 barctgK1
0
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos ya þ b ln K02 þ K12 þ ð1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bÞ ln L20 þ L21 :
(7.96)
Accordingly, the production costs will be calculated in compliance with the
formula
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L
K
ðb 1ÞarctgL1 barctgK1
2
2
0
0
sin ya þ b ln K0 þ K1 þ ð1 bÞ ln L20 þ L21 :
Ct ¼ Ra e
(7.97)
Let us now study what aspect of production is being modeled by this pseudouniform Cobb-Douglas-type classification production function. With the assumption of unchanged complex labor resources we will get the following character of
the effect of the complex capital resource on production performance:
1. Growth in investments in fixed capital K0 leads to a decrease in the polar angle of
these resources and an increase in the modulus of the capital resources. The gross
margin in (7.96) represents the multiplication of two varying factors – growing
L
K
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðb 1ÞarctgL1 barctgK1
0
0 and decreasing cosine cosðy þ b ln
exponent e
K02 þ K12 þ ð
a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 bÞ ln L20 þ L21 Þ . Depending on the ratio between the fixed and noncore
capital, as well as on the values of the complex proportionality coefficient and
exponent b, it is possible to model an increase in profits, the stability of its values,
and a decrease in the volumes. However, for the production costs in this situation
two trends are multiplied – the exponent grows with the growing sine of the angle.
This is why it follows from (7.96) that costs will definitely grow.
2. If investments in nonfixed capital grow, the polar angle grows; therefore, the
exponent where this angle is used with the “minus” sign decreases. The argument of the cosine component increases with the increase in the modulus of
capital resources, which means a decrease in this very factor. Thus, two decreasing trends are multiplied. This means that investments that made in nonfixed
capital lead to a decrease in the gross margin. As for production costs, it is shown
that if the exponential component decreases, the sine of the increasing angle
increases. Therefore, multiplication of these two factors can result in different
trends for different coefficients.
The same conclusions can be drawn on the character of influence of the other
complex variable – labor – on production performance. With an increase in the
7.6 The Shadow Economy and Its Modeling by Means of Complex-Valued Functions
259
number of industrial and production workers, costs will grow, but changes in the
gross margin can vary widely but an increase in nonproduction workers will lead to
a decrease in the gross margin and a varied behavior of costs.
Since the relationships modeled by the pseudo-uniform Cobb-Douglas-type
classification production function are in agreement with many actual production
processes, one can say that this function can be used in economic modeling.
However, since in this model a priori assumptions are made about its uniformity
and restrictions on the exponent signs (7.90), application of this model will be
limited. This is why we will not pay too much attention to its properties, as we did
with respect to the pseudo-uniform Cobb-Douglas-type classification production
function in the previous section.
A model where any restrictions on the uniformity of model (7.86) and the
coefficient signs are removed will be considered universal. The application area
of this model is extended though it gets much more complicated and the procedure
for calculating complex coefficients becomes more difficult.
Taking logarithms of the right- and left-hand sides of equality (7.86) we obtain
the following model:
lnðGt þ iCt Þ ¼ lnða0 þ ia1 Þ þ ðb0 þ ib1 Þ lnðK0 þ iK1 Þ þ ðc0 þ ic1 Þ ln ðL0 þ iL1 Þ:
(7.98)
We do not foresee any theoretical problems in connection with the estimation of
coefficients of this model on the basis of statistical data – everything is quite simple.
On the other hand, to do this, let us solve a system of six equations with six
unknowns (if we work with real variables) or a system of three complex-valued
LSM equations. Here, it is necessary to observe the polar angles of the variables
since most modeling programs calculate angles by cutting off their values in a range
of 0–2p, which can cause errors in the estimation of model coefficients.
In any case, if economists end up working with the power classification production function of complex variables, they will achieve more adequate modeling of
production processes and of the influence of the active and passive parts of the
production resources than they will if they use real-variable models.
It is obvious that besides power models other classification types can be used –
exponential, logarithmic, or a combination of the two. It is crucial that the number
of resources used is limited only by the computational capacities of the technology
and problem context.
7.6
The Shadow Economy and Its Modeling by Means
of Complex-Valued Functions
Using the approach of singling out active and passive parts, one can classify
economic processes into two groups – legal and shadow economies. It is evident
that legal economy indicators should be assigned to the real parts of economic
260
7 Multifactor Complex-Valued Models of Economy
variables and shadow economy parameters to the imaginary components. On this
basis we can build diverse models describing the shadow economy. Let us consider
one of them.
Let us introduce the following variables:
K0 – cost of fixed assets shown in statistical reports;
K1 – cost of fixed assets used in illegal production;
L0 – number of workers involved in production;
L1 – number of employees involved in the shadow economy;
Q0 – gross domestic product as shown in official reports;
Q1 – gross domestic product of the shadow economy.
The simplest power model taking into account the shadow business has the
following form:
Q0 þ iQ1 ¼ aðK0 þ iK1 Þa ðL0 þ iL1 Þb ;
(7.99)
where a and b are function exponents without any restrictions on values and signs.
From the point of view of best approximation of socioeconomic phenomena and
taking into account shadow processes, the power production function of complex
variables with complex coefficients will be more precise:
Q0 þ iQ1 ¼ ða0 þ ia1 ÞðK0 þ iK1 Þb0 þib1 ðL0 þ iL1 Þc0 þic1 :
(7.100)
Its coefficients can be found by LSM, as was already mentioned. However, from
the point of view of the economic core of the modeled process, the real coefficient
model (7.99) turns out to be more feasible.
Indeed, if the values for the fixed assets and workers involved in the shadow
economy are equal to zero, the output in the shadow economy will also be equal to
zero. Model (7.99) explains this fact: if K1 ¼ 0 and L1 ¼ 0, then it is evident that
Q1 ¼ 0. And in a model with complex-valued coefficients (7.100), if the shadow
resources are equal to zero, the real variable, put to a complex power, will result in a
complex variable and calculate the volume of shadow products associated with the
imaginary part and not equal to zero. This means that this model is not as precise in
showing real processes. This is the first reason.
The second reason for preferring model (7.99) over model (7.100) stems from
the nature of the original data. Indeed, statistical data on the legal economy, that is,
their order and trends, though full of mistakes, are not very distorted by these
mistakes. However, information on the shadow economy and its components
cannot be taken into account in a similar way by definition. Here we must use
expert evaluations whose accuracy is, of course, not great. Under these conditions,
when the original data are not exact, it is pointless to complicate the model in order
to increase its accuracy. This is another reason why we would prefer the simple
model to model (7.100).
7.6 The Shadow Economy and Its Modeling by Means of Complex-Valued Functions
261
Let us make a preliminary analysis of the properties of the proposed model of
economic dynamics taking into account the shadow economy. In exponential form,
model (7.99) is transformed into
a
b
Q0 þ iQ1 ¼ aðK02 þ K12 Þ2 ðL20 þ L21 Þ2 e
K
L
iðaarctgK1 þbaarctgL1 Þ
0
0
:
(7.101)
Here we can determine how the model takes into account the influence of the
factors on the legal and shadow economies. The volume of officially shown
production is
b
a
K1
L1
Q0 ¼ aðK02 þ K12 Þ2 ðL20 þ L21 Þ2 cos aarctg þ baarctg
:
K0
L0
(7.102)
Similarly, one can determine how the proposed model describes illegal output:
Q1 ¼
aðK02
b
a
þ
K12 Þ2 ðL20
þ
L21 Þ2
K1
L1
sin aarctg þ baarctg
:
K0
L0
(7.103)
We see that the model describes the influence of both legal and illegal resources
on both the official and shadow economies. Does this occur in the real economy?
Indeed, L1 who are employed in the shadow economy produce an illegal product
that is legalized through official sales and shown as a part of real GDP Q0, and those
employed in legal production often produce, without suspecting as much, products
hidden from taxation and included in the turnover in the shadow economy. In
addition, the remuneration of people involved in illegal business cause them to
direct most their earnings to satisfy their demands using legally produced goods.
Similarly, on fixed assets K0 officially shown in balance sheets, workers produce
products for shadow turnover, though in various “backstreet” shops and with the
equipment assigned to the fixed assets of the shadow business K1 they produce goods
distributed in official turnover. Complex-valued power functions show these complex relationships.
Let us see what trends are described by the proposed model. Suppose that the
situation in the region under study leads to an increase in the number of
entrepreneurs involved in the shadow economy and the resources involved in
legal production remain unchanged. This means that both capital K1 used in illegal
production and the number of employees in the shadow economy L1 increase.
a
According to (7.102), this leads to an increase in both factors ðK02 þ K12 Þ2 and
b
ðL20 þ L21 Þ2 and a decrease in the trigonometric factor – the cosine of the increasing
angle. Then, how does strengthening of the shadow economy influence the legal
economy? The answer to this question is determined by a and b exponent values. If
they are less than one, then growth in the power components will occur to a lesser
degree than the cosine growth modeling either stability or a slight decrease in legal
production. The closer these parameters are to zero, the more negative is the
influence of the shadow economy on the legal one.
262
7 Multifactor Complex-Valued Models of Economy
If the exponents are greater than one, the power factors increase faster than the
cosine decreases. Therefore, in this case we model a positive influence of the
shadow economy on the results of the legal economy.
How do the resources involved in the shadow business influence its scope? The
answer can be derived from the second component of model (7.103). The power
factors grow with the growth in resources K1 and L1. The trigonometric factor also
grows – the sine curve increases its values with the growth in the angle. Therefore,
we model the implicit growth in shadow production whether the exponent is close
to zero or greater than one.
Let us now assume again that the production situation makes pointless the
development of the shadow production. Then entrepreneurs will fix it at a particular
level and invest the resources in legal industries. This means that capital K0
involved in legal production and the number of those employed in the economy
L0 increase. How will model (7.99) react to such resource variation?
The dynamics of legal production determined by the product of power factors
and trigonometric factor (7.102) will be as follows. The power factors increase, and
this increase is determined by the exponents. With the growth in K0 and L0 the
cosine argument will decrease. The cosine of the decreasing angle will increase.
Therefore, the scope of legal production will increase in any case.
And what happens with the shadow economy? The power components also
increase, but the trigonometric factor – the sine curve – decreases. Therefore,
depending on the exponent values and the ratio of the resources of the legal to
those of the illegal economy, the scope of illegal production could increase,
decrease, or remain unchanged. If the exponents are less than one, then growth in
the power factors occurs to a smaller extent than the sine curve decrease.
As a result, the model correctly shows the possible alternatives of influence of
the resources of the legal and shadow economies on each other.
According to the available data it is not difficult to build model (7.99) – we
already did it in previous sections. However, the main difficulty we face when
building model (7.99) is the absence of reliable information on the variables
assigned to the shadow economy in Russia and other countries. To overcome this
problem, I.S. Savkov suggested relying on the available expert information on
Russia’s shadow economy. Since we know how many people the experts believe
are involved in Russia’s shadow economy (the average estimate is 41 %) and how
big the shadow economy is compared to the official GDP (the average expert
estimate is 45 %), we can use these data for model building. However, the cost of
the fixed assets (capital) involved in the shadow economy is unknown. We have not
seen expert estimates of this value. To determine it we propose the following
approach. According to the available statistical data on Russia we calculate the
parameters of the power production function of real variables, which has the
following form:
Qt ¼ 0:5241Kt0:569 Lt0:579 :
(7.104)
7.6 The Shadow Economy and Its Modeling by Means of Complex-Valued Functions
263
Since the exponents reflect the contribution of each resource to production
performance, the shadow economy’s technology differs very little from that of
official production, especially since most of the shadow products are manufactured
on the same production lines, we can use these coefficients for model (7.99):
Q0t þ iQ1t ¼ 0:5241ðK0t þ iK1t Þ0:569 ðL0t þ iL1t Þ0:579 ;
(7.105)
where we can obtain an estimation of the fixed assets used in Russia’s shadow
economy. For that let us take the logarithms of the right- and left-hand sides of this
equality (let us use the principal logarithm values):
ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q1t
2 þ K 2 þ iarctg K1t
Q20t þ Q21t þ iarctg
¼ ln 0:5241 þ 0:569 ln K0t
1t
Q0t
K0t
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L1t
þ 0:579 ln L20t þ iL21t þ iarctg
:
L0t
Now we can calculate the unknown value of the capital in Russia’s shadow
economy.
Incidentally, the specific properties of complex variables allow for simplifying
this problem. Since two complex numbers are equal to each other only when their
real and imaginary parts are equal, then with reference to the exponential form, this
means that the moduli of the complex numbers and their polar angles should be
equal to each other. This is why from (7.105) one can obtain formulae that are more
convenient for calculation:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q
0:569
0:579
iarctgQ1t
2
2 2
0t ¼ 0:5241ðK
Q20t þ Q21t e
ðL20t þ L21t Þ 2
0t þ K1t Þ
(7.106)
or
arctg
Q1t
K1t
L1t
¼ 0:569arctg
þ 0:579arctg :
Q0t
K0t
L0t
(7.107)
Solving any of the foregoing equations with respect to the fixed assets of the
shadow economy K0t, we see that lately they have made up 37.5% of the official
value of the fixed assets of Russia. Thus, the model shows that about one-third of
the fixed assets of the Russian economy exists in the shadow economy.
Indeed, the calculation results for the model obtained are mainly conditional
since they are based on a number of simplifications and assumptions; and should
relatively reliable variable values assigned to the shadow economy emerge, the
model coefficients (7.105) should be recalculated and will certainly change. However, what is really important is that the model for classifying a complex-valued
function can be used for economic modeling taking into account its shadow
component. Models of real variables represent in this case a less fine tool for
research and analysis.
264
7.7
7 Multifactor Complex-Valued Models of Economy
Formation of Complex, Multifactor Models
of Complex Variables
Classification models of complex variables expand considerably the capacities of a
complex-valued economy. So far we have been looking at models of one type –
linear and power models. However, in the real economy factors that affects
performance need not follow only one particular law. This is why in real-variable
models we see complex multifactor models such as, for example,
yt ¼ axa1t xb2t þ bx3t :
It is difficult to apply LSM directly to this model since we would obtain a system
of four nonlinear equations with four unknown coefficients. This system can be
solved by numeric methods, but it is easier, using the same numeric methods
available from software packages, to find, by direct iteration, satisfactory
coefficients corresponding to an LSM criterion. For that, for example, there is the
“Search for Solution” function in Microsoft Excel.
However, it is not possible to use this function or a built-in algorithm in the
software to search for coefficient values of a similar complex-valued model:
yrt þ iyit ¼ ða0 þ ia1 Þðx1rt þ ix1it Þa ðx2rt þ ix2it Þb þ ðb0 þ ib1 Þðx3rt þ ix3it Þ:
There is a need for building complex, nonlinear, multifactor, complex-valued
models. For example, the volume of produced crops and livestock in agriculture is
determined by the number of workers in these industries, land area allocated for
land and pastures, number of machines for working the land, stock breeding, and
other factors. To correctly model agricultural production it is necessary to build
multifactor, complex-valued models, which will definitely not be linear.
In the fullness of time, a method of synthesizing monofactor models into
multifactor ones was proposed to provide solutions to similar problems in the
field of real variables.
Let us assume that a researcher has used some method, for example LSM, to
build several monofactor dependences of parameter y on factors k ¼ 1,2,3,. . .n,
y ¼ fk ðxk Þ þ ek :
(7.108)
Each of these models describes the behavior of the parameter with variance:
s2k ¼
1
n
1
n
X
k¼1
ðyk
fk ðxk ÞÞ2 :
Since in all n models (7.108) the left-hand sides of the equalities are equal to
each other and equal to y, let us sum up their left- and right-hand sides n times:
7.7 Formation of Complex, Multifactor Models of Complex Variables
ny ¼
n
X
k¼1
n
X
fk ðxk Þ þ
ek ;
265
(7.109)
k¼1
where
y¼
n
n
1X
1X
fk ðxk Þ þ
ek :
n k¼1
n k¼1
(7.110)
We have a multifactor model as a synthesis of monofactor ones. From the record
obtained it follows that each monofactor model is incorporated into a multifactor
one with the same weight equal to 1/n. Since the variances of each model differ
from each other, to minimize the total approximation and variance error following
synthesis into a common multifactor one, we introduce the weight of each model vi.
Here it is obvious that the higher the variance of monofactor models, the less weight
it should have when incorporated into a common multifactor one.
Taking into account the foregoing discussion, the multifactor model as a synthesis of monofactor models will have the following form:
y¼
n
X
k¼1
vk fk ðxk Þ þ
n
X
vk ek ;
(7.111)
k¼1
where
vl ¼
1
n
n
P
sk
sl
k¼1
1
n
P
:
(7.112)
sk
i¼1
We see that the sum of weights given by formula (7.112) is equal to one. There
could be other methods of establishing the weights of each monofactor model in a
multifactor one.
Let us use this method to build multifactor, complex-valued models. Let us
assume the construction of several monofactor, complex-valued dependences of
some complex parameter (yr + iyi) on complex factors (xrk + ixik), k ¼ 1,2,3,. . .l,
. . .n, each of the models describing the behavior of a complex parameter with the
average approximation error (erk + ieik):
yr þ iyi ¼ fi ðxrk þ ixik Þ þ ðerk þ ieik Þ:
(7.113)
Let us synthesize these monofactor models into a multifactor one with the
corresponding weights:
yr þ iyi ¼
n
X
k¼1
vk fk ðxrk þ ixik Þ þ
n
X
k¼1
vk ðerk þ ieik Þ;
(7.114)
266
7 Multifactor Complex-Valued Models of Economy
where
vl ¼
1
n
n
P
k¼1
1
ðsrk þ isik Þ
n
P
k¼1
ðsrl þ isil Þ
(7.115)
ðsrk þ isik Þ
is the complex weight of each factor in the common multifactor, complex-valued
model.
Where it is assumed an additive multifactor model will be built, the proposed
method significantly simplifies the process forming multifactor, complex-valued
models and reduces its labor intensiveness.
It is a bit more complicated when researchers propose building a multifactor
multiplicative model. In this case each monofactor model should be considered
through the multiplicative approximation error:
y
:
fk ðxk Þ
mk ¼
(7.116)
If the model uniquely describes a parameter, the multiplicative approximation
error will always be equal to one. If the description provides some variance, the
multiplicative approximation error will vary around one, and the worse the model
describes the parameter, the stronger the variation of the error. Therefore, the error
can estimate the model accuracy:
ek ¼ 1
mk :
(7.117)
Here the variance can be calculated easily.
Again the following rule is valid: the higher the variance, the lower the weight
the model should have. However, if in the additive case the sum of weights equals
one, then in the multiplicative case their product should be equal to one. Then each
weight can be found by the formula
vl ¼
n
Q
sk
k¼1
sl
1n
:
(7.118)
Taking into account the foregoing discussion, the multifactor multiplicative
model will have the following form:
y¼
Y
n
k¼1
vk fk ðxk Þ
1n
:
(7.119)
7.8 Summary
267
The same method can be used to build a multifactor, complex-valued model:
yr þ iyi ¼
Y
n
k¼1
1n
vk fk ðxrk þ ixik Þ ;
(7.120)
where complex weights are calculated by the formula
vl ¼
n
Q
k¼1
ðsrk þ isik Þ
srl þ isil
1n
:
(7.121)
Thus, one can avoid serious computational difficulties and build a complexvalued model of a given accuracy. In the final chapter this method will be used to
build complex, multifactor, complex-valued models.
Of course, one should carefully watch the limits of the parameter and coefficient
variations since many complex-valued functions are multivalent and periodic, and
as a result the pursuit of complexity of a model can produce a meaningless model.
7.8
Summary
The TFCV is limited to the study of monofactor, complex variables. The economy
represents a complex object, and attempts to maximally take into account the actual
complexity of the relationships involved in modeling will inevitably lead to the
building of multifactor dependences. In this chapter we have shown that multifactor, complex-valued models do expand researchers’ mathematical apparatus. Using
this apparatus, it is possible to solve problems that could never even be posed in the
domain of real variables. Classification models where each complex variable
represents two parts of one whole economic indicator or factor make it possible
to model various contributions of these components to the performance. The
efficiency of these models was demonstrated using production functions of an
actually operating economic entity and a complex model of the Russian economy
taking into account its illegal part.
Bearing in mind the highly promising nature of multifactor, complex-valued
modeling let us nevertheless point out that building these models involves rather
labor-intensive problems for estimating model parameters since the number of
unknown coefficients in these models is large.
In this chapter we mainly looked at power multifactor, complex-valued models,
as linear and power multiplicative multifactor models prevail in economic analysis
with real variables.
268
7 Multifactor Complex-Valued Models of Economy
The last section showed how to build complex, nonlinear, multifactor, complexvariable models by synthesizing monofactor complex-valued ones.
The richness of the TFCV (and this chapter shows that it is possible to work with
models of not one but several complex variables) opens up multidimensional
perspectives for practical usage of complex-valued models in economic modeling.
Chapter 8
Modeling Economic Conditions of the Stock
Market
The models and methods of a complex-valued economy can be used not only in the
modeling of production in the form of production functions. One area of application
of the TFCV is the stock market. This chapter shows how to build a complex index
and compare it with the index of real variables. An important result was obtained
from an analysis of the stock market in the phase plane of a complex index modulus
and the polar angle of the complex index. In this case it is possible to reveal and
describe a complex nonlinear relationship between stock sales volume and stock
prices. We called this relationship the “K-pattern.” Methods for determining
K-patterns and for modeling them are given in this chapter. The main results of
the research are demonstrated for the Russian stock market.
8.1
Stock Market Indexes
One of the directions specified in early studies aimed at using the TFCV in
economic was for modeling economic conditions in stock markets [1]. Economics
has been working on this problem for over 100 years. Considerable progress has
been made, and economists everywhere are using some standard set of methods to
move forward.
Two approaches are used to model economic conditions: the first one determines
the parameters of economic conditions as a certain dependence on the conditioncreating factors, and the second one implies aggregation of the parameters into
some general value – index – which shows the status of the conditions.
The basis for using complex-valued models with the first approach are stated in
previous chapters. Using complex-variable models it is possible to build more
complex and probably more adequate models of dependence of the parameters of
economic conditions on the condition-creating factors. We do not expect methodological difficulties here. As for the use of complex-valued structures with the
second approach, i.e., for building indexes for economic conditions, there are no
ready-made solutions.
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
269
DOI 10.1007/978-1-4614-5876-0_8, # Springer Science+Business Media New York 2012
270
8 Modeling Economic Conditions of the Stock Market
In order to be able to judge the behavior of some entity on the whole, one should
use not an aggregate of parameters but some generalized one that includes the
diagnostic properties of the aggregate of the parameters. This is clear as it is
difficult to compare a set of parameters of one entity with a set of parameters of
another entity, or the same one but in the prior time period, and draw some general
conclusion. More often, certain parameters of one set testify to an entity’s
advantages, while others demonstrate the advantages of the other one. A
generalized parameter involves the main features of the majority of parameters
showing the entity’s average status.
Since the state of economic conditions of any market is reflected by a set of
possible parameters, an index appears to be more preferable here as it aggregates
varied information. In this chapter we will not consider the modeling of market
conditions in general, focusing our attention on one type of market – the stock or
securities market.
Every major national security market usually has its own stock index or several
indexes. Stock traders are guided in their activities by their knowledge of those
markets. A stock market index is a number that characterizes its state. As a rule, the
value of this number bears no crucial information. It is not the mere value of this
number that is important, but its correlation with its previous values. The index can
characterize the stock market in general, the market of groups of securities (state
securities, bonds, stocks, etc.), the market of securities of certain industries (oil and
gas, telecommunications, transport, banks, etc.), and others.
Comparing the dynamics of the behavior of these indexes we see how the state of
an industry changes with respect to the market on the whole and therefore market
conditions or the dynamics of changes taking place there.
The index theory has clear logical parallels with indifference curves and surfaces
known in economic theory. In accordance with the conclusions of the latter, the
sums of the cost of goods in a closed system given various prices with other
conditions unchanged, will remain the same (constant consumption level):
X
j
P j Q j ¼ const:
If the situation changes in this closed system over time, the aggregate cost will
also change. An index shows the variation in this aggregate cost over time.
Therefore, the aggregate cost of all the purchases in a given securities market or
ratio of this cost to the same value at the previous moment of time will be used as
the generalizing value at each moment t:
It ¼
m
P
j¼1
m
P
j¼1
Ptj Qtj
Ptj 1 Qtj
;
1
(8.1)
8.1 Stock Market Indexes
271
where Ptj is the price for the jth stock sold on the market; Qtj is the volume of the jth
stock sold on the market; j is the stock number (or that of the enterprise selling the
goods) on the market, j ¼ 1, 2, 3, . . . m; and t is time.
The index helps to compare aggregate costs at a given moment t with the
aggregate cost at the previous moment (t l). If economic conditions in the market
have improved compared to the previous moment, then business activity has
increased and the number of deals has grown compared to the previous moment.
Therefore, the aggregate cost of sales has grown as well, and index (8.1) is greater
than one. If market conditions have deteriorated, then the activities of market
participants have decreased, and the number of deals and sales volume has fallen
and aggregate costs have decreased as well. Therefore, the numerator in (8.1) is
lower than the denominator, the index being less than one. If the market conditions
do not change, then the index is equal to one.
In this way, various values of index (8.1) allow for interpreting the market
conditions at one moment compared to the previous one. Market condition indexes
have the great advantage of generalizing massive amounts of data since their
numerator and denominator represent the sums of products of prices for goods
and their sales volumes. It is not difficult to add another term to this sum; therefore,
an index can take into account and generalize information on changes in the costs of
all goods sold and bought on this market. Therefore, the index gives a unique
opportunity to use all the information at the researcher’s disposal. However, the
possibility of generalizing a great amount of data is in turn a condition of the
index’s existence as the principle of index calculation implies a generalization of
large amounts of data. This objective necessity represents the disadvantages of an
index – a general index is not able to signal system disproportions whose trends are
gaining strength in the market nowadays.
Indeed, due to the fact that the numerator and denominator of (8.1) comprise the
sums of products, there could be cases where a decrease in one parameter in the sum
will be compensated by an increase of another one, for example, halving of the
stock price (“goods” means stocks) will be compensated by doubling of the sales
volume of these securities. A sharp decrease in the price of a firm’s stock might be
the sign of a crisis at the company, and if it is part of a chain of economic
interconnections, it may result in further collapse of the entire chain. This may
lead to various consequences for the market, even the collapse of the entire market.
There could be another case where a decrease in the price and sales volumes of a
stock in the whole aggregate will be compensated by an increase in price and sales
volumes of another stock. In general, the index itself does not change. This change
may lead to a number of unfavorable circumstances – we know of several “dark
days,” which means sudden collapses at stock exchanges, though the indexes of
economic circumstances provided no hint of collapse. That is why attempts are
made to limit the number of parameters included in the index to the most important
ones.
Lately, practicing economists tend not to use an index as a ratio of the amount of
sales at the given moment to that at the previous moment. They use the sales
272
8 Modeling Economic Conditions of the Stock Market
volume of some selected set of stocks and analyze how it varies over time. Similar
stock indexes are calculated by various information agencies, stock exchanges,
rating agencies, etc. They are calculated as the average of stock prices of companies
included in the selection. Index developers apply various approaches – for example,
simple average, geometric mean, weighted average.
The general formula for calculating an index’s average has the following form:
I¼
m
P
Pj
j¼1
m
;
(8.2)
where I is the index, Pj is the price of stock j sold on the market, and m is the number
of companies.
The index’s geometric mean has a more complex calculation:
I¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
m
I1 I2 . . . Im ;
(8.3)
where I is the composite index; I1 ; I2 ; I3 ; :::::In are individual indexes of
companies, and m is the number of companies in the selection.
The weighted-average method is the best known among index calculations. It
takes into account the company size and scale of operations in the stock market. As
a rule, the market capitalization of a company, i.e., the total market value of a
company’s stocks, is taken as the weight.
According to the weighted-average method, index calculation has the following
form:
I¼
m
P
j¼1
m
P
j¼1
Ptj Qtj
P0j Q0j
I0 ;
(8.4)
where P0j and Ptj are stock prices of company i in the basic and reporting periods;
Q0j and Qtj are the number of stocks in circulation in the basic and reporting periods;
i ¼ 1, 2,. . ., m is number of companies in the selection; and
I0 is the index’s basic value.
The variety of approaches to index calculation is due to the fact that none of
them possesses sufficient diagnostic properties. This is why analysts improve the
indexes using the most varied methods.
Let us see how the general principles and approaches to a complex-valued
economy that are stated in this study can be used to develop tools for stock market
analysis.
8.1 Stock Market Indexes
273
As a product, a stock has two components: consumer properties inherent in
goods and price, which is a monetary assessment of consumer properties of this
product by a particular consumer. According to our study, these two aspects
characterize the product, and therefore these two parameters characterize the
product in general, and they should not be considered separately from each other
but in tandem, i.e., as a complex variable:
zjt ¼ qjt þ ipjt :
(8.5)
Here
q is the sales volume,
p is the unit price, and
j is the stock number.
The original values of the price and volume must be represented as dimensionless quantities; otherwise a complex variable cannot be created.
The above-mentioned expression (8.5) makes it possible to fully describe the
properties of a particular stock and mathematically correctly work both with each of
the components and with the aggregate in general if the original variables are
brought to one scale and dimension.
Complex variable (8.5) is characterized by its modulus and polar angle:
Rjt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pjt
q2jt þ p2jt ; yjt ¼ arctg :
qjt
(8.6)
Therefore, it can be represented in both arithmetic (8.5) and trigonometric and
exponential forms. Later on we will need the exponential representation of this
complex variable:
zjt ¼ Rjt eiyjt :
(8.7)
If a researcher possesses data on stock sales at moment t and the previous
moment t 1, then by comparison one can judge the change in the market for
these stocks. In our case a comparison can be carried out either by subtraction from
zjt of its previous value or by division of zjt by zjt 1.
In the first case the real part of the obtained difference will characterize the
change in the sales volume and the imaginary one the change in the unit price. The
quotient of the two complex numbers will be characterized by the ratio of the
modules and angles of the two complex variables. The modulus will be less or
greater than one depending on whether the modulus of the variable has increased or
decreased, and the polar angle will characterize variation in the price with respect to
the volume change. Since information on each stock is in its complex model (8.5),
then, besides activity with individual stocks one can create a parameter generalizing
information for all stocks, in other words, an index. This index should contain
274
8 Modeling Economic Conditions of the Stock Market
information on all the relevant sales activity in the market, i.e., for this it is
necessary to use m complex variables (8.7).
A simple sum of complex variables (8.5) would be meaningless since we would
sum up the real and imaginary parts separately and the resulting complex number
would not have any crucial significance. However, it is obvious that the properties
of each individual stock will get lost, with no new properties appearing for the
complex value as a result of this generalization. Multiplication of the complex sales
variables of all m stocks in the given market will have the following meaning:
Zt ¼
m
Y
j¼1
zjt ¼
m
Y
j¼1
i
iyjt
ðRjt e Þ ¼ e
m
P
m
Y
yjt
j¼1
Rjt :
(8.8)
j¼1
Similarly, one can find the product of the complex sales variables in the same
market at the previous moment t 1:
Zt
1
¼
m
Y
zjt
1
j¼1
m
Y
¼
j¼1
ðRjt 1 e
iyjt
i
1
Þ¼e
m
P
yjt
1
j¼1
m
Y
Rjt 1 :
(8.9)
j¼1
The ratio of (8.8) to (8.9) will also be a complex variable and will characterize
the situation in the market, i.e., it will be some index:
ISt ¼
m
Q
j¼1
m
Q
1
m
zjt
zjt
¼e
1
im1
m
P
yjt
j¼1
m
P
yjt
1
j¼1
m
Q
j¼1
j¼1
m
Q
1
m
Rjt
Rjt
:
(8.10)
1
j¼1
According to complex-variable properties, this very index IS is a complex
variable with real and imaginary parts. It also can be presented in exponential
form using modulus Rz and polar angle ’z.
The polar angle of index IS is found as the exponent in (8.10):
e
im1
m
P
yjt
j¼1
m
P
j¼1
yjt
1
;
(8.11)
i.e., it is equal to the difference
1
’z ¼
m
X
m
j¼1
yjt
m
X
j¼1
yjt
1
:
(8.12)
8.1 Stock Market Indexes
275
p
zt
pt
Rt
zt-1
pt-1
Rt-1
qt
qt-1
q
qt
0
qt-1
Fig. 8.1 Stock behavior shown in the complex plane
The modulus of index IS is determined according to the formula
Rzt ¼
m
Q
j¼1
m
Q
1
m
Rjt
Rjt
1
j¼1
¼
Y
m
Rjt
R
j¼1 jt 1
m1
:
(8.13)
To understand the meaning of the modulus and polar angle of the index, let us
turn to its graphic interpretation in a complex plane. By definition, the coordinate
axes will be the scaled unit price and stock sales volume.
Let us first consider the situation where m ¼ 1, i.e., where we study the behavior
of only one stock. Figure 8.1 shows the position of the complex variable at moment
t and at the previous moment t 1.
In the figure the complex variable zt is shown in circumstances where the sales
volume of this stock qt fell somewhat compared to the previous moment qt 1, but
the unit price pt considerably increased: pt > pt 1. This is shown in the model by
the fact that the modulus of the complex variable has increased: Rt > Rt 1, and the
polar angle, yt > yt 1, has increased as well.
It is evident that for the polar angle of any security lies within a range of 0 to p/2.
If now we relate the complex variable at moment t to its values at the previous
moment, we obtain a new complex variable (private index):
It ¼
zt
Rt iðyt
¼
e
z t 1 Rt 1
yt 1 Þ
¼
Rt
cosðyt
Rt 1
yt 1 Þ þ i
Rt
sinðyt
Rt 1
yt 1 Þ: (8.14)
276
8 Modeling Economic Conditions of the Stock Market
Since the polar angle of this complex index is equal to
yIt ¼ yt
yt
1
¼ arctg
pt
qt
arctg
pt
qt
1
;
(8.15)
1
and the modulus to
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q2t þ p2t
RIt ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
;
q2t 1 þ p2t 1
(8.16)
one can make a preliminary conclusion on the variations in the complex index’s
polar angle (8.14) and its modulus.
First of all it should be noted that the complex index’s polar angle lies within the
limits
p=2 < yIt < p=2:
(8.17)
Polar angle (8.15) of partial index (8.14) is equal to zero if the proportion
between the price and sales volume has not changed; this follows from (8.15). A
proportionate increase in price with an equivalent concomitant growth in sales
volume means increasing interest in stocks and buoyancy of the market. A proportionate decrease in price with an equivalent concomitant decrease in the sales
volume means stockholders’ tendency to hold onto stocks with falling prices.
Therefore, the fact that the polar angle of the private index (8.14) is equal to zero
suggests a certain stability of this stock in the market – an increase in its price leads
to an increase in trading in these securities, while a decrease in the price leads to a
decrease in trading.
With reference to the arithmetic form of the complex index, its imaginary part is
equal to zero.
The polar angle (8.15) is greater than zero if the stock price has grown but the
sales volume has remained unchanged, or if the price has not changed but the sales
volume has fallen. The first suggests a growth in interest in these securities and a
reluctance on the part of a number of shareholders to get rid of them, while the
second suggests that this price does not satisfy shareholders and they refrain from
selling. This is a situation of expectations where market participants consider these
securities promising.
The positive character of the polar angle means that both cosine and sine are
positive, as are the real and imaginary parts of index (8.14). The greater the
imaginary part of this index, the more expectations the market has with respect to
future changes in the stock’s behavior, meaning price increases, which is why stock
prices have grown compared to the previous observation. The positive character of
the polar angle of the private index reflects an increase in the interest in the
securities on the market.
8.1 Stock Market Indexes
277
If the polar angle of the private index is less than zero, this means that at fixed
prices the sales volume of this stock is increasing or that at fixed sales volumes the
unit price falls. The first means that shareholders are in a hurry to get rid of this
stock, expecting a deterioration in its positions. The second means that the price is
starting to decrease but the stock is not held onto because demand for it is not high.
Thus, a negative polar angle of the private index reflects a decrease in interest in this
stock.
The arithmetic form means that the imaginary part of the complex private index
is negative, and the higher the modulus, the worse the positions of this stock in the
market.
Now let us consider possible dynamics of modulus (8.16) of the private complex
index. It can remain stable, increase, or decrease. Figure 8.1 shows a dotted circle
with radius Rt. At any point of this circle the radius will be the same – for an
increasing polar angle (increase in price and decrease in sales volume) and a
decreasing polar angle (decrease in price and increase in the sales volume).
Therefore, this index does not show the expectations of market participants. It
shows only the scale (not volume!) of activity with this stock.
The sales volume of this stock represents the product of the price and volume or,
in the case under consideration, the product of the real part of the private complex
index and the real component of its imaginary part, i.e.,
1
pt qt ¼ Rt cos yt Rt sin yt ¼ R2t sinð2yt Þ:
2
(8.18)
A decrease in the modulus of the private complex index with permanent polar
angle reflects a decrease in sales volume, and its increase an increase in sales
volume. Simultaneous change in both the polar angle and the modulus can lead to
constant sales volume but a dramatic change in the attitude of market participants
toward this stock (Fig. 8.2).
Sales volume can remain constant if, for example, there is an increase in the
modulus of the complex variable but its polar angle decreases. Figure 8.2 show that
this corresponds only to a situation where the price falls and the sales volume
increases.
Can this describe a situation where stock owners dump a stock? For this reason,
sales volume conveys little information about market conditions, though a change
in the polar angle and the imaginary part of the complex variable are more
informative.
Let us summarize the information with reference to the private complex index:
1. Proximity to zero of the imaginary part of an index reflects a stable market. An
increase in the real part indicates growing interest in the securities, while a
decrease means that interest has fallen off.
2. If the imaginary part of the complex private index is positive, this reflects the
fact that the stock has generated some interest. If the imaginary part is positive
and far from zero values, this means that there is a demand for the stock in the
278
8 Modeling Economic Conditions of the Stock Market
p
zt-1
pt-1
Rt-1
zt
pt
Rt
qt-1
qt
q
qt-1
0
qt
Fig. 8.2 Stability of sales volumes with changing market conditions for a given stock
Table 8.1 Conditional example for index calculation (dimensionless units)
Stock 1
t
t–1
Price, p
11
10
Stock 2
Sales
volume, q
10
11
Price, p
10
8
Stock 3
Sales
volume, q
8
10
Price, p
7
5
Stock 4
Sales
volume, q
5
7
Price, p
15
10
Sales
volume, q
10
15
market that is close to exuberance. If the real part is high, then we observe a
boom; if the real part is low, then sales volume has decreased in the expectation
of further price increases of prices.
3. If the imaginary part of the index is negative, this means a decrease in interest in
this stock, and the greater its modulus, the sharper the decrease in interest. A
high value of the real part will reflect the fact that the sales volume has increased
–market participants are dumping the stock; if not, market participants are
holding this stock in the expectation of improved conditions.
Since the generalizing index (8.10) represents a product of private indexes, the
above-mentioned conclusions also extend to its properties. Table 8.1 presents a
conditional example demonstrating properties of the index of complex variables
compared to the standard approach.
The stock prices and volume are selected to have the classic index (8.1) equal to
1, showing stability in the market. One can easily see that
It ¼
11 10 þ 10 8 þ 7 5 þ 15 10
¼ 1:
10 11 þ 8 10 þ 5 7 þ 10 15
(8.19)
8.1 Stock Market Indexes
279
Table 8.2 Calculated values
of stock moduli for each time
moment
Stock 1
t
t–1
Table 8.3 Calculated values
of polar angle (in radians)
t
t–1
Stock 2
Stock 3
Stock 4
R1
R2
R3
R4
14.87
14.87
12.80
12.80
8.60
8.60
18.03
18.03
Stock 1
Stock 2
Stock 3
Stock 4
y1
y2
y3
y4
0.83
0.74
0.90
0.67
0.95
0.62
0.98
0.59
Now for each stock and at each moment of time, one can calculate the modulus
of the complex variable (Table 8.2).
Then we can determine the polar angles of each variable at various moments of
time (Table 8.3).
Let us now consider the values of private complex indexes.
The first one is
I1t ¼
14:87 ið0:83
e
14:87
0:74Þ
¼ 1 cos 0:09 þ i1 sin 0:09 ¼ 0:995952733 þ i0:089878549;
the second one is
I2t ¼
12:80 ið0:90
e
12:80
0:67Þ
¼ cos 0:23 þ i sin 0:23 ¼ 0:973666395 þ i0:227977524;
the third one
I3t ¼
8:60 ið0:95
e
8:60
0:62Þ
¼ cos 0:33 þ i sin 0:33 ¼ 0:946042344 þ i0:324043028;
and the fourth and last one is
I4t ¼
18:03 ið0:98
e
18:03
0:59Þ
¼ cos 0:39 þ i sin 0:39 ¼ 0:92490906 þ i0:380188415:
The imaginary parts of all the indexes are positive, which reflects a growing
interest in the stocks. The imaginary part of the first index is close to zero, which is
why interest in this stock can be considered quite normal. However, the fourth stock
has a considerable imaginary part – 0.380188415. This reflects surge in the stock.
Indeed, we see from Table 8.1 that the price for this stock grew by 50%.
Now let us calculate the summarizing complex index (8.10) characterizing the
market in general. The index modulus is
280
8 Modeling Economic Conditions of the Stock Market
Rt ¼
14:87 12:80 8:6 18:03
14:87 12:80 8:6 18:03
14
¼
29:513
29:513
14
¼ 1:
(8.20)
Its polar angle is
1
yt ¼ ½ð0:83
4
0:74Þ þ ð0:9
0:67Þ þ ð0:95
0:62Þ þ ð0:98
0:59Þ ¼ 0:26:
(8.21)
Then the arithmetic form of the complex index is
ISt ¼ 0:966389978 þ i0:257080552:
(8.22)
The imaginary part of this coefficient is positive and greater than zero, which
reflects a favorable economic situation and buoyancy of the market.
The classic index for the conditional example under study is equal to one, which
means a stable situation, though we can see a clear change. Again, we see that usage
of complex variables makes it possible to model the economic entity in a different
way and draw different conclusions compared to those obtained by models and
indexes of real variables.
Very often, to show market conditions, economists take into account some
“summary” index representing the sum of sales volumes of a certain set of stocks
(“blue chips”):
It ¼
X
Ptj Qtj ¼ const:
(8.23)
j
To analyze the situation in the market, we do not divide the current index value
by the previous one but just analyze the index dynamics. It goes without saying that
the dynamic range of this index makes it possible to compare both neighboring and
remote indexes with each other.
If by analogy to (8.23) we use index (8.8),
Zt ¼
m
Y
j¼1
zjt ¼
m
Y
i
iyjt
ðRjt e Þ ¼ e
j¼1
m
P
j¼1
yjt
m
Y
Rjt ;
j¼1
then we could face a problem – as a complex number this index increases its
modulus with an increase in the number of data used and increases the polar
angle, which is equal to the sum of polar angles, each of them being positive:
y¼
m
X
j¼1
yjt :
8.1 Stock Market Indexes
281
This is why this index is a periodic value, since for various combinations of the
stocks comprising the index one can obtain polar angles that differ from each other
by 2pk; therefore, the ratio of their real and imaginary parts will be similar, though
the processes are different. Index (8.10), which includes the quotient of the complex
indexes, is free from this shortcoming since the polar angles of the same stocks are
subtracted from each other. The question is whether it is possible to obtain a certain
analog of the consolidated index (8.23) in the sphere of complex variables.
The geometric mean can meet these requirements (8.8):
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u P
m
P
u i m yjt Y
m
yjt
im1
m
u
It ¼ te j¼1
Rjt ¼ e j¼1
j¼1
vffiffiffiffiffiffiffiffiffiffiffiffi
uY
m
u
m
t
Rjt :
(8.24)
j¼1
Calculation of this index will allow us to analyze separately the dynamics of its
four characteristics:
1. Polar angle:
’t ¼
m
1 X
yjt ;
m j¼1
2. Modulus:
vffiffiffiffiffiffiffiffiffiffiffiffi
uY
m
u
m
Rt ¼ t Rjt ;
j¼1
3. Real part:
Irt ¼ Rt cos ’t ;
4. Imaginary part:
Iit ¼ Rt sin ’t :
Looking at the complex index as a supplementary tool of stock market analysis,
but not an alternative to real-variable tools, we have every reason to assert that such
a development of the instrumental base of the economy is quite productive.
282
8.2
8 Modeling Economic Conditions of the Stock Market
Phase Plane and K-patterns
The stock market is a special market where one will not find vendors and end users.
Its goods do not meet immediate needs, vendors quickly become buyers, and vice
versa. Limiting usefulness or labor value theories fail to explain their behavior.
Models and methods of economic theory applied to this market do not work – the
stock market operates by other laws, other interrelations, other behavior styles. This
market more than others is governed by psychological and even sociological laws.
Mass panic in the market leads to its collapse, and expectations represent the main
driving force. Much effort is expended on forecasting price movements with little to
show for it.
Stock market theory is fully completed. One should remember that the goal of
any theory is to explain the phenomenon under study including the models
involved.
Nowadays there are two approaches to studying the stock market. The first is
known as fundamental analysis and represents verbal and graphic models to
describe complex relationships of causes and factors affecting the market. The
second one is called “technical analysis” and involves various mathematical models
trying to describe and forecast numerical characteristics of the market without
getting into the system of cause-and-effect relationships and without taking into
account the influence of factors measured on a nominal or ordinal scale. Synthesis
of these two directions makes up a kind of stock market theory.
Since in this market the unit price for one share of stock and the sales volume
thereof are the main indicators studied by stock market theory, it is quite natural to
determine and understand the relationships between these two economic
parameters. However, so far their use in stock market theory has been limited to
the calculation of various indexes like (8.1) or (8.2). Nothing more has been
accomplished.
However, economists understand that between the unit price and the sales
volume of a given product, even a stock in the stock market, there is an economic
relationship that is intuitively felt but not thus far inexplicable. This is why
nowadays stock market theory recommends analyzing visually the dynamics of
price pt and sales volume qt, placing one graph under the other and correlating the
time axis scale. This combined positioning of the dynamics of economic parameters
gives economists some intuitive impression on the process under study. There is
also a certain understanding of the mutual influence of these two parameters,
though no model is capable of describing either the direction or the core significance of this influence.
Since both the stock price and the sales volume are interrelated and distributed in
time, one could combine them in one graph representing what we know from the
natural sciences as a phase plane, as each value of the price and sales volume has an
index – time – providing their precise identification. This is why, if we place price
on one axis and sales volume on the other, we will obtain a phase plane whereon we
can plot a set of points, which often looks like a law and is therefore called a “phase
8.2 Phase Plane and K-patterns
283
Fig. 8.3 Typical phase
portraits of a stock
on the stock market
pt
qt
0
portrait.” The “Hysteresis loop,” well-known in physics, is the most vivid example
of the phase portrait.
Attempts to build phase portraits of securities listed on various stock markets
fail. We obtain not a portrait but a chaotic accumulation of points and their
connecting lines. Figure 8.3 gives an example of the phase plane of a certain stock.
Therefore, we should acknowledge that using real-variable instruments and
well-known approaches it will not be possible to solve the problem of determining
and describing relationships between the unit price of this stock and its circulation
volume in the stock market. This means that science is not able to solve the main
task of revealing the market’s driving force as demonstrated by its laws. As a matter
of fact, nobody argues that modern stock market theory has developed a large body
of empirical materials and discovered its numerous dynamic laws. However, the
empirical-deductive method used is not able to reach a level of theoretical
generalizations. The hypothesis of existence of a relationship between price and
volume in this market has not been confirmed so far.
Since this research is aimed at demonstrating only a small part of those rich
possibilities that open up for economists when using the TFCV, let us show how to
solve the set problem and test the hypothesis on the existence of interrelations
between stock prices and sales volume in the stock market. For that let us use the
previously introduced variable (8.5):
zjt ¼ qjt þ ipjt :
This complex variable can be written both in exponential and trigonometric
forms for which we calculate its modulus and polar angle:
Rjt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pjt
q2jt þ p2jt ; yjt ¼ arctg :
qjt
(8.25)
Both the modulus of the complex variable and its polar angle characterizing the
stock (price and sales volume) undergo changes in time since the original complex
284
8 Modeling Economic Conditions of the Stock Market
Fig. 8.4 Typical phase
portrait showing behavior
of polar coordinates of
securities on stock market
Rjt
0
jt
variable is dynamic. This is why graphs of change in time of both the polar angle
and the modulus can give additional characteristics for the stock. The economic
meaning of an increase or decrease in these parameters was stated in the previous
section (Figs. 8.1 and 8.2). This is why, on the one hand, in building these graphs
economists obtain additional information on changes in the stock with time.
On the other hand, it definitely follows from (8.25) that these two variables are
related to each other since the same pair – price and volume – are used to calculate
the modulus and the polar angle. This means that the dynamic variables (8.25) can
be placed on one graph – phase plane graph – the relationship between them, if any,
characterizing the phase portrait of the process under study.
T.V. Koretskaya built several tens of these securities listed on MICEX from
2007 to 2009. In the phase planes of all the studied securities one can clearly see the
phase portrait of a typical shape (Fig. 8.4).
In the phase portrait we see that at a certain period of time, the points of the
securities’ modulus and polar angle lie on a concave curve (in Fig. 8.4 we see these
curves as a bold dotted line). When considered in their dynamics, the position of
these points represents a swinging movement along the bold line up and down.
Then, when the market dynamics changes, the phase portrait points “break off”
these stable lines and move to another, lower level, where they stay for a rather long
period of time on a similar concave line. Since T.V. Koretskaya considered only the
period characterized by a global economic crisis, resulting in a collapse of stock
markets and withdrawal from the market of the funds invested in securities by
financial companies , the transition from one position in the phase plane to another
has a downward trend. I am quite sure that with the first signs of improvement in the
global economy and the available resources reappear in the stock market, the phase
planes of securities will demonstrate a transition from one phase state to another
with an upward trend.
We call the phase portrait lines K-patterns patterns [2]. Graphic analysis of the
phase state of a security shown in the graph as a K-pattern made it possible to
formulate a hypothesis on parallel K-patterns of the same securities. To test this
hypothesis, Koretskaya proposed a simple but very productive approach – taking
the logarithm of the securities’ moduli and polar angles and further building the
8.2 Phase Plane and K-patterns
285
Fig. 8.5 Phase portrait
of security in logarithmic
phase plane
lnRjt
0
lnqjt
phase plane in logarithms. This transformation of the complex variables and the
phase plane yielded phase portraits in the form of parallel straight lines (Fig. 8.5).
Of course, since graphic models are built according to sample parameter values,
this happens under the influence of numerous arbitrary factors that allow some lines
to deviate from the general location in the phase plane. However, the parallel
nature, on average, of the straight lines in a logarithmic phase plane under visual
analysis of tens of security graphs is evident. This was confirmed by statistical
analysis when the slope angle of linear models fluctuated around some average
value.
Having solved the problem of determining the K-pattern relationship of the unit
price and the sales volume of a security in the stock market it is necessary to give an
economic interpretation to the results obtained.
The relationship between price and volume is complex and nonlinear since the
K-pattern represents a nonlinear relationship of the modulus of a complex variable (square root of the sum of squares of price and volume) and its polar angle
(arctangent of price-to-volume ratio). The complex form of this relationship does
not provide an unambiguous interpretation like, for example, that a price increase
leads to a particular trend. These explicit conclusions do not suggest themselves;
others do. K-patterns as an aggregation of points of a phase plane may have
something in common that determines their particular position in the plane. But
what is the nature of these causes that are reflected in the same position of phase
portrait points and why are phase transitions from one state to another observed?
Is the assumption true that during periods of stability in the stock market stocks
are arranged in K-patterns, and in the times of instability they “break away from”
these patterns?
To answer this question, let us turn our attention to the processes taking place in
the stock market. Let us consider the MICEX stock market of 2008 as an example.
On January 21, 2008 auctions were not held in the USA. However, on that very
day Asian stock exchanges collapsed along with the Russian stock market. In that
crisis situation of instability, many market participants hurried to unload their
securities and withdraw funds from the market, leading to a considerable decrease
in the volume of funds circulating in the market. This situation in the MICEX
286
8 Modeling Economic Conditions of the Stock Market
market did not change for 3 and a half months up to May 4, 2008. If we build a
phase portrait of any of the MICEX “blue chips,” we see that this period of
“stability” manifests itself in the phase plane of this security by a K-pattern. Of
course, there are obvious deviations from the line induced by random factors;
however, this dispersion is not critical. The period shown by the K-pattern is
characterized by a similar attitude of market participants toward the securities
and – what is important – by the relative stability in the market in general.
Beginning on May 5 the points of the phase plane started a gradual transition to a
new state. They started going lower and lower than the K-pattern. This decreasing
trend coincided with the events of August 18, 2008, which was marked by a new fall
of Russian stocks against the background of the Georgia-South Ossetia conflict. On
the day of the summit of NATO ministers of foreign affairs in Brussels on August
19, 2008, when the question of how to punish Russia for its prolonged occupation of
Georgia, the Russian stock market collapsed, which is shown in the phase plane by
a chaotic fluctuation of the points with a clear tendency to a new level.
On September 16, 2008, influenced by the bankruptcy of the American investment bank Leman Brothers, the Russian stock market underwent another collapse.
This period saw an active withdrawal of funds from deposit accounts in Russia, this
instability being demonstrated by a downward trend of the polar angle and
modulus.
On Monday, September 29, the Russian stock market again collapsed.
Nervousness reached a fever pitch, which showed on October 6, 2008, that adoption
of [U.S. Treasury Secretary Hank] Paulson’s plan did not relieve the tension from
the American stock market and did not dispel doubts about its efficiency. As a
result, on Monday, October 6, 2008, the market collapsed again.
After the collapse, a measure of stability returned to the market from October 29,
when the U.S. Federal Reserve announced a 0.5 % decrease in the discount rate.
This pushed the Asian and European markets up on October 30, and the Russian
stock market also showed some upward movement. The volume of the funds
circulating in the market was fixed at some stable level. From that day until the
end of 2008, the points of the phase plane accurately arranged themselves into a
new K-pattern.
Analysis of other phase portraits shows that the periods of time characterized by
stable markets show all points in one K-pattern. As soon as there is a change in the
stock market, the points of the phase portrait start moving toward another K-pattern,
which takes shape in the subsequent period of stock market stability.
It is interesting to note that in the early 2000s Lukoil stock was arranged in a
K-pattern, and beginning in 2005 stopped behaving in this way. A.M. Chuvazhov,
who discovered this phenomenon, assumed that the absence of K-patterns in this
period was due to certain political and economic causes, including the acquisition
of stocks by American companies, as a result of which changes in the Russian stock
market stopped influencing the behavior of the Lukoil’s stock.
8.3 Mathematical Models of K-Patterns
8.3
287
Mathematical Models of K-Patterns
The linear character of K-patterns in the logarithmic phase plane predetermines the
type of mathematical model of the K-pattern. In logarithms it has the following
form:
ln Rjt ¼ a0 þ a1 lnðarctgyjt Þ:
(8.26)
Moving from the logarithmic to the multiplicative form we have
Rjt ¼ ea0 arctgyjt a1 :
(8.27)
If now we substitute price and volume for the modulus and polar angle, we have
the following K-pattern equation:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pjt a1
a0
2
2
pjt þ qjt ¼ e artg
:
qjt
(8.28)
It is difficult to transform this model into an explicitly given dependence of price
on volume or volume on price. However, since for the given polar angle we
calculate the complex-variable modulus according to (8.27), with these two
parameters it will be quite possible to calculate the sales volume as the real part
of the complex variable:
qjt ¼ ea0 arctgyjt a1 cos yjt
(8.29)
and the unit stock price as the imaginary part of the complex variable:
pjt ¼ ea0 arctgyjt a1 sin yjt :
(8.30)
There is an obvious dependence between the unit stock price and sales volume,
which follows from complex-variable properties:
pjt ¼ qjt tgyjt :
(8.31)
Now one can propose an algorithm of stock market modeling. If there are
grounds to consider the polar angle equal to some set (for example, forecasted)
value, then under stable market conditions, when K-patterns are observed, it is
possible to calculate the sales volume according to (8.29) and the unit stock price
according to (8.30).
For practical purposes model (8.26), or its analog (8.27), is quite suitable since
this model’s coefficients are easily found from statistical data using any method of
statistical estimation, for example, LSM.
288
8 Modeling Economic Conditions of the Stock Market
According to the data on changes in quotations of Aeroflot at MICEX for 2008,
T.V. Koretskaya found several K-patterns using LSM equations. Let us consider
two of them. The first K-pattern corresponds to a relatively stable period from
January 21 to May 4, 2008, the second one from October 29, 2008 to the end of that
year.
The model Eq. (8.27) for the first K-pattern has the following form:
ln R^t ¼ 0:2366
0:8261 ln yt :
(8.32)
The model determination coefficient with factual data is 0.955.
The same equation for the second K-pattern is
ln R^t ¼
0:9945
0:9396 ln yt :
(8.33)
This model describes the original data just as well since the determination
coefficient with factual data is 0.9613.
It is easy to see that the free term decreased and became negative, which
indicates a downward shift in the K-pattern. The model proportionality coefficient
also changed, which is due to the influence of random factors, but the scale and sign
did not change. The regression equations of all the K-patterns are statistically
significant.
How can we use K-pattern equations calculated by LSM in practice? Assume
that we know that in December 2008 the market situation is stable and we know that
one of the payers plans to sell a large stake Qt+1 ¼ 100,000 of Aeroflot. We have a
K-pattern equation at our disposal with coefficients calculated by LSM (8.33).
Expressing this volume as a basic value we get, for example, qt+1 ¼ 1.28.
To make the right decision we should forecast a possible price for a share of
Aeroflot stock that will take shape as a result of the auctions given the volume of
stock up for sale. The market does not change; therefore, the K-pattern equation
does not change either, but all the K-pattern points are characterized by a changed
polar angle and modulus. This is why the calculation algorithm (8.29)–(8.30)
cannot be applied in this case.
To solve this problem, it is necessary to use a mathematical equation of
a K-pattern. Substituting the volume of stocks put on sale into the equation of the
K-pattern model we obtain
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p2tþ1 þ 1:282 ¼ e
0:9945
artg
ptþ1
1:28
0:9396
ptþ1
¼ 2:703 artg
1:28
0:9396
It is impossible to derive the value of the forecasted price and calculate it from
the preceding equation, but it is not impossible to solve the given problem since the
equation
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p2tþ1 þ 1:282
ptþ1
2:703 artg
1:28
0:9396
¼ 0;
8.3 Mathematical Models of K-Patterns
K-patternI
289
K-patternII
K-patternIII
K-patternIV
P
1,20
1,10
1,00
0,90
0,80
0,70
0,60
0,50
0,40
0,30
0,20
0,10
Q
0,00
0,00
0,50
1,00
1,50
2,00
2,50
3,00
3,50
4,00
Fig. 8.6 K-patterns for the Aeroflot shares
with respect to the only unknown pt+1, can be solved using one of the numerical
methods.
The obtained stock price value can be used as the forecasted one if the stake
is sold.
According to the results stated in this chapter, we could draw the conclusion that
the complex index (8.10) used along with other indexes makes it possible for
researchers to obtain additional information about stock market conditions. New
information on the stock market can also be obtained calculating the complex index
(8.24) and analyzing its main characteristics.
Identification of a K-pattern and justification of the form of its mathematical
model (8.26) allows for developing theoretical ideas about the stock market since it
represents a mathematical model of the relationship of a unit stock price and sales
volume, which is quite complex and unidentifiable using real-variable models. The
general idea of this relationship is given by Fig. 8.6
The transition process from one K-pattern to another is of particular interest.
Since K-patterns reflect periods of stable markets, it is most interesting to forecast
instability – the transition from one state to another. Perhaps combining K-patterns
with market cycles would make it possible to obtain the desired results – but that is
a task for the future. The most important thing is that has become possible to
identify these K-patterns and show that they correspond to periods of stability in the
markets.
290
8 Modeling Economic Conditions of the Stock Market
References
1. Svetunkov SG (2006) Complex variables in the index theory. Theory of function of complex
variable in economic and mathematical modeling: materials of All-Russian Scientific Seminar
(19 December 2005). SUEF, St. Petersburg, pp 15–37
2. Svetunkov SG, Koretsksya TV (2009) Comparative research of classical index and complex
variable index for stocks dynamics at MICEX. Bulletin of Orenburg State University
5(2009):78–81
Chapter 9
Modeling and Forecasting of Economic
Dynamics by Complex-Valued Models
This chapter is devoted to one goal – modeling economic dynamics using models of
complex variables. The chapter opens with the model of Ivan Svetunkov, which
represents a complex-valued analog of the model of short-term forecasting of
exponential smoothing (Brown’s model). This model possesses remarkable
properties and ushers in a new class of models of short-term economic forecasting.
It shows how to use a complex-valued analog of Solow’s model of economic
dynamics, and its particular properties are demonstrated. The chapter and book
conclude with a section devoted to diagnostics of regional socioeconomic
development.
9.1
Ivan Svetunkov’s Model for Short-Term Forecasting
Ivan Svetunkov’s short-term forecasting model is one of the remarkable results of
the application of the methods of complex-valued economic modeling. It successfully combines the advantages of Brown’s short-term forecasting model and
properties of complex-valued models. The core of the model is as follows [1].
In 80 % of the cases of practical application, the problem of forecasting
socioeconomic dynamics in the short term is solved by means of Brown’s method
(also known as the exponential smoothing method). The main idea is that the
forecasted value is defined as the weighted average of the previous range, which
in compact form is written as follows:
Y^tþ1 ¼ aYt þ ð1
aÞY^t :
(9.1)
The properties that made the model rather popular among forecasters are evident
for such a grouping:
Y^tþ1 ¼ Y^t þ a Yt
Y^t :
(9.2)
S. Svetunkov, Complex-Valued Modeling in Economics and Finance,
291
DOI 10.1007/978-1-4614-5876-0_9, # Springer Science+Business Media New York 2012
292
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
From this way of writing BrownBrown’s model we see that the forecasted value
is calculated using the previously forecasted one but corrected by the deviation of
fact from forecast.
The accuracy of Brown’s model for forecasting problems is determined by
values of coefficient a, called the smoothing constant. Smoothing constant a values
lie within the range
0 < a < 2:
(9.3)
Since numerous values of Brown’s model lying within
1 a<2
(9.4)
have been ignored by economists up till now, and as early as 1997 it was called the
transcendental set of Brown’s method [2], we will not dwell on it. We merely
mention that situations with transcendental cases of Brown’s method are quite
frequent in modern economics, especially when the researcher is faced with the
problem of forecasting irreversible processes.
Ivan Svetunkov developed the logic of Brown’s method and proposed the use of
features of complex-valued economics to forecasting of two parameters simultaneously: parameter Yt values and its deviation from the actual one et:
et ¼ Y t
Y^t :
To do this the forecasted parameter and deviation from it can be represented in
the form of a single complex variable:
Yt þ iet ;
(9.5)
which is considered to be the subject of short-term forecasting by Brown’s method.
Since both the parameter and the deviation have the same dimension and show
various sides of the same process, this variable has a right to exist. Let us designate
the calculated forecasting value of this complex variable as
Y^t þ i^et :
Then, similarly to Brown’s model (9.1), we can obtain the following forecasted
model:
Y^tþ1 þ i^etþ1 ¼ ða0 þ ia1 ÞðYt þ iet Þ þ ðð1 þ iÞ
ða0 þ ia1 ÞÞ Y^t þ i^et :
(9.6)
Unlike Brown’s model, in the case under consideration we use the complex
smoothing constant:
a0 þ ia1 :
(9.7)
9.1 Ivan Svetunkov’s Model for Short-Term Forecasting
293
However, the variation limits of this smoothing constant need not coincide with
those determined for the original Brown’s model (9.3). The boundaries will be
examined later, but for now we will study the properties of Ivan Svetunkov’s shortterm forecasting model.
Taking into account properties of complex variables, namely, their real and
imaginary parts, the complex-valued model (9.6) can be reduced to the following
system of real equations:
(
Y^tþ1 ¼ a0 Yt þ ð1
^etþ1 ¼ ða0 et þ ð1
a0 ÞY^t
ð a1 e t þ ð 1
a0 Þ^et Þ þ a1 Yt þ ð1
1
^et ;
a1 Þ^et Þ ¼ Y^t0
0 1
^
a1 ÞY t ¼ ^e þ Y^ :
t
(9.8)
t
It is seen from (9.8) that the forecasted value Y^tþ1 is determined as some
forecasted value Y^t0 found by Brown’s method and corrected for some value ^e1t ,
also forecasted by Brown’s method ^e1t : For its part, the forecasted value of the
correction index ^etþ1 is also determined by two constituents found by the same
Brown’s method, only by addition: the forecasted correction index ^e0t and the
forecasted value Y^t1 . Here, the top indexes “0” and “1” show which value of the
two is used to calculate these values (a0 or a1).
Obviously, in model (9.6) the forecasted values are created by the previous
factual ones with some complex weights set by the algorithm, which is similar to
exponential smoothing in Brown’s method but somewhat more complicated. Let us
represent in formula (9.6) the calculated value Y^t þ i^et via the previous actual Yt 1
þiet 1 :
Y^tþ1 þi^etþ1 ¼ ða0 þia1 ÞðYt þiet Þþða0 þia1 Þ½ð1 a0 Þþið1 a1 ÞððYt
1 þiet 2 Þþ...
Then it is evident that this case also involves the process of calculating the
weighted average, but using complex numbers and variables. Complex variables of
(9.5) are multiplied by weight coefficients, which in this case are complex. The
series of these complex weights from the above-mentioned equality can be
represented as follows:
ða0 þ ia1 Þ; ða0 þ ia1 Þ½ð1
a0 Þ þ i ð 1
a1 Þ; ða0 þ ia1 Þ½ð1
a0 Þ þ ið1
a1 Þ2 ; . . .
(9.9)
which is simply the geometric progression series of complex numbers.
With this range we can determine the variation boundaries of the complex
smoothing variable (9.7). By definition this series should converge since otherwise
older observations would carry more weight than the new ones, which could result
in inaccuracy of the model forecast.
This means that for the complex series of weights (9.9), the following equality
should hold:
lim ða0 þ ia1 Þ½ð1
n!1
a0 Þ þ ið1
a1 Þn ¼ 0:
(9.10)
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9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
2,0
1,8
1,6
1,4
a1
1,2
1,0
0,8
0,6
0,4
0,2
0,0
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
a0
Fig. 9.1 Domain of complex smoothing constant
With constant values of the complex smoothing constant and obviously existing
inequality
ða0 þ ia1 Þ 6¼ 0;
equality (9.10), as well as for the series of the geometric progression of real
numbers, converges to a certain number when the following condition is met:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 a0 Þ2 þ ð1 a1 Þ2 < 1
(9.11)
In the a0 and a1 plane inequality (9.11) represents a circle with its center in the
point (1;1) and unit radius (Fig. 9.1). If the value of the complex smoothing constant
is inside the circle, then series (9.9) converges to a complex number. If the value
appears to be on the boundary or goes beyond the circle, then series (9.9) diverges.
From (9.11), by elementary transformations one can obtain the boundaries
within which a1 should lie for the series to converge:
1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ð1 a0 Þ2 < a1 < 1 þ 1 ð1 a0 Þ2 :
9.1 Ivan Svetunkov’s Model for Short-Term Forecasting
295
For its part, this term is clearly true only when the radicand is positive (the
situation where a1 is a complex number is not considered):
1
ð1
a0 Þ2 > 0:
(9.12)
From the restriction (9.12) it is easy to derive the boundaries within which a0 and
a1 should lie:
8
0 < a0 < 2;
>
>
<
0 < a1 < 2;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
>
:
1
1 ð1 a0 Þ2 < a1 < 1 þ 1 ð1 a0 Þ2 :
(9.13)
Thus, we have the boundaries within which the real and imaginary parts of the
complex smoothing coefficient should lie for the series of complex weights (9.9) to
converge to a certain number. In addition, Ivan Svetunkov calculated what this
number is. It is calculated according to the formula
S¼
a20
a1 þ a21 þ iða0 Þ
a20 þ ð1
a1 Þ 2
:
(9.14)
One can see that in (9.4) the real part can be represented by any real number
(positive or negative) but the imaginary part can be represented only by a positive
real number. This means that in model (9.6) the series of weights (9.9) has a more
complex meaning than in Brown’s model, where for the model to exist the series of
weights should definitely converge to 1.
The specific properties of model (9.6) also appear in the fact that the series of
weights (9.9) converge to (9.14) with varying rate depending on the constant
smoothing value. Research has shown that the closer the constant smoothing
value is to the circle edge (9.11), the slower is the convergence. The modulus of
the number is in this case a certain characteristic of the convergence rate:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v ¼ ð 1 a0 Þ 2 þ ð 1 a1 Þ 2 :
(9.15)
According to (9.11) v lies within a range of 0 to 1, and the closer it is to 1, the
more inert is the model, i.e., the forecast is influenced by obsolete data, and the
closer it is to 0, the less important are the old data in the creation of the forecast, and
the model becomes more adaptive.
We have data on the magnitude of electric power generation by windmills in one
U.S. states. This time series consists of 20,000 observations taken every half hour.
Since these data are intended exclusively for official use, it was agreed that the
American side would not provide even selective values of this series but demonstrate only the results of the calculations.
296
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
If one uses Brown’s method directly for this series, the optimal value of a is 0.28,
which shows that the model adapts slowly to incoming information. The mean
relative approximation error in this case was 7.54 %, the determination coefficient
was 0.1485, and the compliance coefficient was 94.22 %. These parameters show
that the model does not describe and forecast wind generation series very well.
If we apply to the same series Ivan Svetunkov’s model (9.6), we can also find the
optimal value of the complex smoothing, which is equal to
a0 þ ia1 ¼ 0:59 þ 1:00i
The mean relative approximation error is 5.06 %, the determination coefficient is
0.6051, and the compliance coefficient is 95.68 %. According to these parameters,
the results forecast by the correction method appear to be more accurate than was
possible with Brown’s method or its identifications.
It should be noted that Ivan Svetunkov’s model differs from all existing shortterm forecasting models by its ability to detect trends in series data that make it
possible to describe them. The correction model “predicts” the series dynamics
while Brown’s model always “procrastinates.”
Ivan Svetunkov made a comparative analysis of the proposed model with
various modifications to Brown’s model (Holt method, Holt modification method,
etc.). This analysis showed that in a series of cases model (9.6) gives much more
accurate results than these modifications. This reflects the fact that Ivan
Svetunkov’s model can be included in the arsenal of tools for forecasting socioeconomic activity along with the existing models.
In conclusion, it should be noted that additional research by E. Tsedyakova
showed that Ivan Svetunkov’s model gives better forecasts in data series with
chaotic dynamics for which the optimal smoothing coefficient, according to
Brown’s, model is close to 0.
9.2
Complex-Valued Autoregression Models
If application of Brown’s model of complex variables leads to such interesting
results, it is useful to consider other forecasting models to which complex variables
could be applied.
The complex-valued production functions considered in Chaps. 5 and 6 can well
be used for these purposes as forecasting models – linear and nonlinear. These
models’ properties were studied earlier and there is no sense in repeating them here.
From these points of view autoregression models seem more interesting if one
considers autoregression of a complex variable.
The first-order autoregression model of a complex variable can be written as
follows:
yrt þ iyit ¼ ða0 þ ia1 Þðyrt
1
þ iyit 1 Þ; t ¼ 1; 2; 3; . . .
(9.16)
9.2 Complex-Valued Autoregression Models
297
Fig. 9.2 Complex-valued autoregression model at power base with modulus higher than 1 (9.20)
It is important to recall here that the first-order autoregression model of a real
variable is written as follows:
yt ¼ ayt
(9.17)
1
and ultimately represents the following function:
y t ¼ at y 0 :
(9.18)
The dynamic of variation in time of this exponential function is completely
determined by the function base – coefficient a.
It is easy to show that in a similar way the complex-valued autoregression model
(9.16) can be represented as a power function:
yrt þ iyit ¼ ða0 þ ia1 Þt ðyr0 þ iyi0 Þ:
(9.19)
It is known that the complex-valued power function is periodic and diverges in
the form of a spiral with increases in the exponent if the modulus of the base is
greater than one and converges along the spiral to zero if the complex-valued
modulus in the power base is less than one.
The graph in Fig 9.2 shows the variation in the autoregression function:
yrtþ1 þ iyitþ1 ¼ ð0:7 þ i0:8Þðyro þ iyi0 Þ;
yro ¼ 1; yi0 ¼ 1:
(9.20)
The complex coefficient autoregression modulus is equal to 1.063 > 1. Therefore, the graph shows how the parameter modeled by autoregression diverges from
298
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Imaginary part of model
Real part of model
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
0
10
20
30
30
40
-0.5
-1
-1
-1.5
-1.5
0
10
20
30
40
Fig. 9.3 Variation in time of real and imaginary parts of complex-valued autoregression model (9.20)
the initial point in the complex plane with coordinates (1;1) along a spiral. It is
evident that one can set any initial point and the character of the modeled relationship will not change, except for the initial point’s location.
Since in economics phase planes like the one shown in Fig. 9.2 are not practically applied, using mainly time variations of the parameters, Fig. 9.3 shows the
variation in time of the real and imaginary parts of the autoregression model (9.20).
Their dynamics have a fluctuating character that diverges with time.
If in the complex-valued autoregression model the modulus of the complex
coefficient of autoregression is less than one, then the model will generate a process
converging to the zero point along a spiral. Thus, for the model
yrt þ iyit ¼ ð0:7 þ i0:6Þðyrt
1
þ iyit 1 Þ;
yro ¼ 1; yi0 ¼ 1;
(9.21)
the modulus of the complex-valued proportionality coefficient is equal to
0.922 < 1.
The autoregression model generates a series converging to zero from the initial
point with coordinates (1;1) in the complex plane (Fig. 9.4).
Variation in time of the real and imaginary parts of the autoregression model
(9.21) will have an oscillatory and damped character, and over time, both parts tend
to zero.
Second- and higher-order autoregression models are more complicated. Let us
consider the second-order autoregression model of a real variable:
yt ¼ ayt 2 ; t ¼ 2; 3; 4; . . .
(9.22)
For t ¼ 2 it is easy to see that y2 ¼ ay0. For t ¼ 3, y3 ¼ ay1 is calculated. We see
that these two calculated values do not depend on each other. Further, we see that
for t ¼ 4 y4 ¼ ay2 ¼ a2 y0 , for t ¼ 5 y5 ¼ ay3 ¼ a2 y1 are calculated.
In general, for the second-order autoregression model (9.22):
(
t
y t ¼ a2 y 0 ;
t 1
2
if t is an even number,
yt ¼ a y1 ; if t is odd:
(9.23)
9.2 Complex-Valued Autoregression Models
299
Fig. 9.4 Complex-valued autoregression model at power base with modulus less than one
Fig. 9.5 Second-order autoregression model yt ¼ 1:3yt 2 ; y0 ¼ 3; y1 ¼ 1
Then, if the modulus of the autoregression coefficient is higher than one, the
model generates an oscillatory process with increasing amplitude, as shown in
Fig. 9.5, and if the modulus is less than one, the model will generate a damped
oscillatory process.
The character of the modeled process for second-order autoregression is determined by the autoregression coefficient and the first two values of the modeled
series.
Similarly, the second-order autoregression model for the complex series
yrt þ iyit ¼ ða0 þ ia1 Þðyrt
2
þ iyit 2 Þ; t ¼ 2; 3; . . .
(9.24)
300
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Fig. 9.6 Second-order
complex-valued
autoregression model (9.25)
will have a more complicated dynamic than that of the first-order model. The
character of the dynamics is determined by both values of the complex auto
regression coefficient and the first two values of the complex-valued series.
As an example, let us consider the dynamics of the second-order complexvalued autoregression model
yrt þ iyit ¼ ð0:7 þ i0:6Þðyrt
2
þ iyit 2 Þ;
yro ¼ 1; yi0 ¼ 1;
yr1 ¼ 0:4; yi1 ¼ 2:1:
(9.25)
The phase portrait of this autoregression model is given in Fig. 9.6.
Since the modulus of the autoregression coefficient is less than one, the model
converges to zero. If it is higher than one, the model will diverge in the complex
plane.
Figure 9.7 demonstrates the variation in time of the real and imaginary parts of
this second-order complex-valued autoregression model.
The figure demonstrates that both parts of the model show a damped process
with a complex oscillatory structure. It is evident that their oscillation period is the
same and they are shifted with respect to each other by the same lag.
Autoregression models of a more complex order will generate more complex
variants of the dynamics.
This dynamic can be demonstrated by the autoregression model with lags
distributed for one and two observations:
yrt þ iyit ¼ ð 0:3
i0:4Þðyrt
yro ¼ 1; yi0 ¼ 1;
1
þ iyit 1 Þ þ ð0:6
yr1 ¼ 1:2; yi1 ¼ 0:8
i0:8Þðyrt
2
þ iyit 2 Þ;
(9.26)
The dynamics of variation of the real and imaginary parts of these parameters are
given in Fig. 9.8.
9.3 Solow’s Model of Economic Dynamics and Its Complex-Valued Analog
Imaginary part of model
Real part of model
1.5
1
0.5
0
-0.5
0
10
20
301
30
30
40
-1
-1.5
-2
2.5
2
1.5
1
0.5
0
-0.5 0
-1
-1.5
-2
10
20
30
40
Fig. 9.7 Real and imaginary parts of second-order complex-valued autoregression model (9.25)
Real part of model
Imaginary part of model
2
2
1.5
1.5
1
1
0.5
0.5
0
-0.5
0
10
20
30
40
0
-0.5
-1
-1
-1.5
-1.5
-2
-2
0
10
20
30
40
Fig. 9.8 Real and imaginary parts of complex-valued autoregression model with distributed
lags (9.26)
The give graph of complex-valued autoregression models show that they can be
applied to the modeling of many processes with a seasonal component or, for
example, in stock markets. This means that complex-valued autoregression models
can take their rightful place in the range of socioeconomic dynamic forecasting
models. Coefficients of these models for the available statistical data can be found
using LSM, especially as described in Chap. 4.
9.3
Solow’s Model of Economic Dynamics and Its ComplexValued Analog
Solow’s model underlies numerous modern economic dynamics models. It is not
necessary to delve deep into the specifics of the economic dynamics modeling
apparatus. We will only show how complex-valued economics can enrich the
apparatus of economic and mathematical modeling using Solow’s fundamental
model as an example.
302
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Solow’s model has the following form:
In the model, the final product is determined the by Cobbs-Duglas function:
Yt ¼ aKta L1t a :
(9.27)
In discrete time t the final product Yt is distributed to gross investments It and
consumption Сt:
Yt ¼ It þ Ct :
(9.28)
It is assumed that the part of the final product that falls on investments is set in
the form of the rate of accumulation r:
It ¼ rYt :
(9.29)
Obviously, investments contribute to the growth of the fixed production assets of
the coming year Kt+1 and are expressed via the obsolete funds Kt taking into
account the share of the depleted fixed production assets m:
Ktþ1 ¼ ð1
mÞKt þ It :
(9.30)
The number of those employed in the economy Lt+1 is determined via the
number of employed in the current year Lt taking into account the annual growth
rate of the number of employees n:
Ltþ1 ¼ ð1 þ nÞLt :
(9.31)
The interrelated (9.27), (9.28), (9.29), (9.30), and (9.31) represent Solow’s
model, and they can be used to model economic growth of some idealized system.
Instead of the Cobbs-Douglas production function, let us use in the economic
dynamics model the complex-valued power production function with real
coefficients that was considered in detail in Chap. 6. Then the economic dynamics
model will have the following form:
It þ iCt ¼ aðKt þ iLt Þb ;
(9.32)
Ktþ1 ¼ ð1
(9.33)
mÞKt þ It ;
Ltþ1 ¼ ð1 þ nÞLt :
(9.34)
We see that the model obtained involves only three equalities since the production function immediately calculates accumulation It, which is to be used later on,
and consumption Ct. We see that the rate of accumulation is not specified. If
necessary, it can be calculated separately:
9.3 Solow’s Model of Economic Dynamics and Its Complex-Valued Analog
r¼
303
It
:
It þ C t
It is obvious that the exponent of model (9.32) fulfills this role because
r ¼ f ðbÞ:
If we use the other complex-valued production functions studied in Chap. 6
instead of the complex-valued power production function, we obtain completely
different models of economic dynamics and other development trajectories. The
model can be expanded by replacing other equations with complex-variable
models, thereby developing the instrumental base of economists who model economic dynamics.
Ilyas Abdullayev, a lecturer at Khorezm National University, used the basic
conditions of a complex-valued economy to build a system of models of economic
dynamics of particular branches of the Republic of Uzbekistan and Khorezm
Region. Let us use only one result to demonstrate how to apply complex-valued
economics to the modeling of real economic dynamics.
For the manufacturing dynamics of the Republic of Uzbekistan from 1995 to
2008 using Solow’s economic dynamics model, Abdullayev set out to build models
with the following production functions:
Cobb-Douglas production function,
Neoclassical production function,
Power production function and
Complex-valued production function
The Cobb-Douglas function is not defined on this set – the exponent with labor
resources is negative; in addition, the “neoclassical” production function is undefined as well. LSM makes it possible to find on this set only the power production
function, which has the form
Q ¼ 0:13K 1:2 L5:8 :
(9.35)
This model poorly approximates the original data. The mean absolute approximation error is 132 %. Therefore, this function should not be used in economic
dynamics models.
Using the method of synthesizing monofactor complex-valued models into one
multifactor complex-valued model described in Sect. 6.7 Abdullayev obtained the
following production function model of Uzbekistan manufacturing:
Q ¼ 0:832eð
0:53 i2:7Þ
K ð0:37þi0:121Þ ðLnnt þ iLit Þ3:03 :
(9.36)
Here, Lппt is the number of manufacturing and production workers, and Lit is the
number of nonproduction workers in Uzbekistan manufacturing (in dimensionless
relative values).
304
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Model (9.36) describes the dynamics of Uzbekistan’s manufacturing with an
average absolute approximation error of 32.8 %, which is almost five times more
accurate than real-variable models.
9.4
Modeling Regional Socioeconomic Development
The socioeconomic development of a region is a very complex subject in scientific
research because the level of this development results from many factors and
conditions.1 Any region of a given country represents a complex system of
interrelating economic, social, political, historic, cultural, and other subsystems.
Uneven regional socioeconomic status, the objective result of a great variety of
development conditions, leads to the need for redistributing income at the state
level. The general scheme of this redistribution is similar all over the world – each
region pays a legally stipulated part of the collected taxes to the state treasury,
leaving the other part at the disposal of the regional budgets.
The regions, called “donors,” collect enough taxes so that the remaining funds
exceed the sum necessary to support subsistence living. This surplus allows
regional authorities to pass more active social and economic policies in the region.
As a rule, they receive no return on the funds they transfer to the state budget
(except for some state target programs).
Economically underdeveloped regions transfer to the treasury part of the collected taxes in the same proportion to the funds retained for their use as donor
regions. But the funds remaining in the regional budget do not allows the region to
work at a breakeven level: the authorities cannot “feed” themselves, much less
implement planned social programs. For these regions subsidies come from the
state budget at the expense of the funds coming from donor regions. These subsidies
make it possible to solve the set social problems and to a certain extent level out
social development of the regions.
In this situation where the level of social development is subject to thorough
regulation by state authorities, and it is quite similar in state regions but the level of
economic development is different, it is not very correct to speak about socioeconomic development of regions as a single parameter of regional authorities’ activity. It makes sense to talk about two components – the social development of a
region and its economic development. Modern economics does not do this – all
approaches to studying regional development are based on the principle of combined estimation of regional social and economic development. It is for this reason
that the estimation process involves numerous development indexes representing a
convolution of an aggregate of various parameters – economic and social – in one
1
This work performed under the International grant Russian Foundation for Humanities – The
National Academy of Sciences of Ukraine № 10-02-00716/U “Valuation models are uneven and
cyclical dynamics of the socio-economic Development of Regions of Ukraine and Russia”.
9.4 Modeling Regional Socioeconomic Development
305
index, though they are measured on various scales. Constructive critical analysis of
these indexes shows that they differ in either the convolution method or the
convolution filling. In any case these indices are obtained as the results of the
summation of two different parts.
As it is not allowed to add the demand price and the volume of demand to
estimate the demand, so similarly, it is senseless to combine the characteristics of
the social and economic development of a region into one parameter to estimate the
level of its development. It was shown earlier that the former are the result of state
policy, the latter of regional policy.
This is why a complex-valued indicator of socioeconomic development was
proposed that was free of this defect [3]. This complex-valued parameter Z includes
a real part as a ratio of the average per capita income C to the subsistence minimum
LV, which can be called “wealth level” d, and an imaginary part as the ratio of paid
services to the population PS to the amount of the total commodity turnover of the
region CC, which can be called “social satisfaction level” s:
Z¼
C
PS
þi
¼ d þ is:
LV
CC
(9.37)
The real part of this complex-valued parameter characterizes the income level of
the residents of this region and therefore is a generalizing indicator of the level of its
economic development. Indeed, the average income per capita C characterizes the
entire aggregate of cash funds received by the average resident of this region from
various sources, primarily from the earnings at production enterprises in the region.
It is one of the rather accurate and broad indicators of the level of economic
development of a region. The subsistence minimum LV characterizes the development of market relations in the region – the more competition there is in the
regional market, the more participants compete in the market, the lower the prices
for the goods included in the subsistence minimum, the lower the subsistence
minimum. Then it is obvious that the ratio of the average per capita income to
the subsistence minimum, called the wealth level, shows the average economic
well-being of the region. Table 9.1 shows data on the change in the real part of the
complex-valued parameter of regional development for Northwest Russia.
One can see from the table that with respect to the dynamics of economic
development, the leader in 2007 was the Nenets autonomous district, followed by
Saint-Petersburg. It is interesting that the gross regional product per capita was
1.156 million rubles per person in the Nenets autonomous district in 2007 and 0.243
million rubles per person in Saint-Petersburg.
The Republic of Karelia, Archangelsk region, Vologda region, Kaliningrad
region, Leningrad region, Murmansk region, Novgorod region and Pskov region
are significantly behind the leaders – the average wealth level for them in 2007 was
2.57, which is 2.3 times less than the leading region.
The dynamics of the imaginary part of this parameter – the level of social
satisfaction of residents in the region –is also interesting. It is given in Table 9.2.
306
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Table 9.1 Dynamics of wealth level in certain regions of Russia
Number
1
2
3
4
5
6
7
8
9
10
11
Region
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Saint-Petersburg
2001
–
2.53
2.33
2.27
–
1.85
–
2.13
–
1.43
–
2002
2.11
2.71
1.86
2.65
1.91
1.41
1.27
2.14
1.72
1.78
2.03
2003
2.09
2.80
2.01
3.62
2.18
1.69
1.43
2.19
1.83
2.02
2.67
2004
2.15
2.97
2.23
4.52
2.33
1.78
1.86
2.27
1.86
2.16
3.25
2005
2.25
3.10
2.39
4.27
2.27
2.05
2.10
2.27
2.00
2.09
3.92
2006
2.43
3.13
2.40
4.70
2.51
2.50
2.55
2.39
2.29
2.21
4.08
2007
2.29
3.26
2.52
5.97
2.70
2.89
2.87
2.68
2.26
2.36
4.31
Table 9.2 Dynamics of rate of social satisfaction in certain regions of Russia
Number
1
2
3
4
5
6
7
8
9
10
11
Region
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Saint-Petersburg
2000
0.22
0.19
0.24
0.16
0.27
0.23
0.20
0.21
0.22
0.19
0.36
2001
0.22
0.19
0.22
0.13
0.29
0.25
0.21
0.25
0.23
0.20
0.38
2002
0.23
0.20
0.26
0.14
0.32
0.30
0.25
0.31
0.26
0.21
0.41
2003
0.28
0.23
0.26
0.16
0.33
0.32
0.28
0.36
0.31
0.23
0.47
2004
0.30
0.24
0.31
0.17
0.37
0.32
0.23
0.41
0.35
0.24
0.47
2005
0.31
0.24
0.34
0.15
0.45
0.33
0.24
0.50
0.36
0.25
0.45
2006
0.31
0.25
0.37
0.21
0.46
0.35
0.23
0.52
0.36
0.27
0.42
2007
0.32
0.25
0.42
0.24
0.43
0.34
0.21
0.48
0.34
0.27
0.42
An interesting fact is that practically all the regions, districts, and the Komi
Republic are characterized by an increase in the level of social satisfaction, except
for the Leningrad region.
It is surprising that the Nenets autonomous district, which is the leader with
respect to wealth level (Table 9.1), is an outsider, along with the Leningrad region,
with respect to the level of social satisfaction (Table 9.2).
This can be explained by the fact that many workers in the region are not
permanent residents. These “shift workers” agree to refuse social services in
exchange for higher pay and subsequent satisfaction of their needs and demands
in other regions of the country.
The low position of the Leningrad region in the social satisfaction rating is quite
clear – some residents in the Leningrad region live in the suburbs of SaintPetersburg, where they work and satisfy their social needs (salons, clinics, clubs,
theaters, museums, etc.).
It is obvious that consideration of the real and imaginary parts of the complex
parameter of socioeconomic development provides the researcher with important
information that makes it possible to estimate the level of the results obtained.
9.4 Modeling Regional Socioeconomic Development
307
However, it is not the two real variables that are of interest, but one complex one.
Simple consideration of these complex-valued variables does not tell the researcher
much (Table 9.3).
Since a complex variable can be written not only in arithmetic but also in
exponential and trigonometric form
Z¼
C
PS
þi
¼ d þ is ¼ Reiy ¼ R cos y þ iR sin y;
LV
CC
(9.38)
its additional characteristics are the modulus of the complex variable
R¼
and its polar angle
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d 2 þ s2
s
y ¼ arctg :
d
(9.39)
(9.40)
It is easy to see that a decrease in the modulus values (9.39) can occur only in the
event of a decrease in one of its constituents characterizing either economic or
social development. The constant nature of the modulus reflects both the growth in
one parameter and a decrease in the other. If the polar angle decreases, the
numerator of fraction (9.40) decreases compared to the denominator, i.e., the
level of social development is lower than that of economic development.
Table 9.4 shows the dynamics of the modulus of the complex-valued indicator of
socioeconomic development of the same regions of Northwest Russia. Analysis of
this parameter makes it possible to draw some conclusions. In the last year under
observation socioeconomic development of such regions as the Republic of Karelia
and the Vologda and Novgorod regions has slowed down.
Modules of complex-valued parameters of other regions have positive dynamics,
which reflects the growth in the level of socioeconomic development of each region
as the whole. Especially high rates are characteristic for the Nenets autonomous
district. Previously, we looked at the fact that this region is characterized by a high
rate of economic development and a low rate of the social component. Therefore,
additional information on the correlation of the economic and social constituents
and their dynamics can be obtained from an analysis of changes in the polar angle of
the complex-valued parameter. Table 9.5 gives this dynamic for the regions under
consideration.
It follows from (9.40) that the polar angle will grow if growth of the social
component exceeds the growth of the economic one, and the polar angle will
decrease if there is a more intensive growth of the economic component and less
noticeable growth of the social one.
The polar angle may remain unchanged when both the economic and social
components characterizing the region’s condition have similar growth rates.
308
Number
1
2
3
4
5
6
7
8
9
10
11
Region
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Saint-Petersburg
2001
–
2.53 +
2.33 +
2.27 +
–
1.85 +
–
2.13 +
–
1.43 +
–
0.19i
0.22i
0.13i
0.25i
0.25i
0.2i
2002
2.11 +
2.71 +
1.86 +
2.65 +
1.91 +
1.41 +
1.27 +
2.14 +
1.72 +
1.78 +
2.03 +
0.23i
0.2i
0.26i
0.14i
0.32i
0.3i
0.25i
0.31i
0.26i
0.21i
0.41i
2003
2.09 + 0.28i
2.8 + 0.23i
2.01 + 0.26i
3.62 + 0.16i
2.18 + 0.33i
1.69 + 0.32i
1.43 + 0.28i
2.19 + 0.36i
1.83 + 0.31i
2.02 + 0.23i
2.67 + 0.47i
2004
2.15 +
2.97 +
2.23 +
4.52 +
2.33 +
1.78 +
1.86 +
2.27 +
1.86 +
2.16 +
3.25 +
0.3i
0.24i
0.31i
0.17i
0.37i
0.32i
0.23i
0.41i
0.35i
0.24i
0.47i
2005
2.25 + 0.31i
3.1 + 0.24i
2.39 + 0.34i
4.27 + 0.15i
2.27 + 0.45i
2.05 + 0.33i
2.1 + 0.24i
2.27 + 0.5i
2 + 0.36i
2.09 + 0.25i
3.92 + 0.45i
2006
2.43 + 0.31i
3.13 + 0.25i
2.4 + 0.37i
4.7 + 0.21i
2.51 + 0.46i
2.5 + 0.35i
2.55 + 0.23i
2.39 + 0.52i
2.29 + 0.36i
2.21 + 0.27i
4.08 + 0.42i
2007
2.29 + 0.32i
3.26 + 0.25i
2.52 + 0.42i
5.97 + 0.24i
2.7 + 0.43i
2.89 + 0.34i
2.87 + 0.21i
2.68 + 0.48i
2.26 + 0.34i
2.36 + 0.27i
4.31 + 0.42i
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Table 9.3 Dynamics of complex-valued parameter of socioeconomic development in certain regions of Russia
9.4 Modeling Regional Socioeconomic Development
309
Table 9.4 Dynamics of modulus of complex-valued indicator of socioeconomic development
of some Northwest regions of Russia
Number
1
2
3
4
5
6
7
8
9
10
11
Region
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Saint-Petersburg
2001
–
3.152
3.143
2.606
–
2.983
–
3.164
–
2.459
–
2002
3.027
3.368
3.013
2.992
3.406
3.037
2.680
3.461
2.961
2.685
3.951
2003
3.320
3.542
3.138
3.962
3.596
3.288
2.897
3.755
3.320
2.979
4.587
2004
3.420
3.763
3.541
4.822
3.878
3.335
2.866
4.072
3.564
3.141
4.925
2005
3.576
3.848
3.773
4.516
4.247
3.543
3.041
4.487
3.681
3.152
5.248
2006
3.692
3.902
3.936
5.066
4.397
3.913
3.348
4.625
3.806
3.329
5.330
2007
3.633
4.007
4.240
6.329
4.389
4.115
3.510
4.634
3.708
3.423
5.496
Table 9.5 Dynamics of polar angle (in radians) of complex-valued rate of socioeconomic
development in certain regions of Russia
Number
1
2
3
4
5
6
7
8
9
10
11
Region
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Saint-Petersburg
2001
–
0.639
0.736
0.513
–
0.902
–
0.832
–
0.950
–
2002
0.799
0.636
0.905
0.483
0.975
1.088
1.077
0.904
0.951
0.846
1.031
2003
0.889
0.659
0.876
0.418
0.919
1.031
1.055
0.948
0.987
0.826
0.949
2004
0.891
0.661
0.889
0.356
0.926
1.008
0.864
0.979
1.022
0.812
0.850
2005
0.891
0.634
0.885
0.332
1.006
0.954
0.809
1.04
0.996
0.846
0.727
2006
0.852
0.639
0.915
0.382
0.963
0.878
0.705
1.028
0.925
0.845
0.699
2007
0.889
0.621
0.934
0.338
0.908
0.792
0.613
0.954
0.915
0.810
0.669
Table 9.5 shows that the polar angle is growing only for the Archangelsk region.
Earlier, separate analysis of the real and imaginary parts of the complex-valued
parameter (9.37) and analysis of the modulus dynamics did not distinguish the
Archangelsk region in any way. Now we have discovered that it actively
implements social programs and does everything for the residents to live more
comfortably in their region.
Regions where the dynamics of the polar angle are approximately constant are
the Republic of Karelia and Pskov region. The low modulus shows that no active
state programs – either economic or social – are implemented there. These regions
are developed independently, without active state support.
All the other regions of the Northwest are characterized by nonlinear dynamics
with a decrease in the polar angle with time. This means that improvement of the
economic conditions of these regions exceeds the satisfaction of the social needs
and demands of their residents.
310
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Since the complex-valued indicator of the regions’ condition made it possible to
reveal trends of socioeconomic development of Northwest regions of Russia, a
natural question arises as to whether these trends can be described by some law.
This is crucial because when analyzing the socioeconomic development of a region,
one should judge the trends in general but not separate their components, i.e., it is
necessary to aggregate the information.
There is a simple way to do this –build a trend of development of the economic
component d of the complex parameter (9.37) showing the socioeconomic development of a region:
d ¼ fd ðtÞ
(9.41)
s ¼ fs ðtÞ:
(9.42)
and a trend of its social constituent:
After that it is rather easy to make a forecast for each of the components for a
certain period. However, this approach does not take into account the interrelation
between the economic and social components of the common complex-valued
parameter, which definitely exists. Therefore, it is more correct to build a
complex-valued trend where the complex parameter (9.37) is represented as a
certain complex-valued function of time that is represented as a real discrete
variable:
d ¼ is ¼ f ðtÞ:
(9.43)
Various functions that may be used as these models are considered in detail in
the Chap. 2. The power function is the most suitable for our problem. With
reference to the class of models under consideration it will have the form
yrt þ iyit ¼ ða0 þ ia1 Þtðb0 þib1 Þ :
(9.44)
The complex proportionality coefficient can be represented in exponential form:
a0 þ ia1 ¼ Ra ea :
Then (9.44) can be represented in a form that is convenient for analyzing the
model properties:
yrt þ iyit ¼ Ra ea tb0 eib1 ln t ¼ Ra tb0 eaþib1 ln t :
(9.45)
It is obvious from the formula that the modulus of the complex result is
Ryt ¼ Ra tb0
(9.46)
9.4 Modeling Regional Socioeconomic Development
311
and the modeled polar angle is
y ¼ a þ b1 ln t:
(9.47)
It is easy to see that each constituent of the complex exponent influences the
complex-valued result modeled by the trend. With a positive real component of
the complex exponent b0 the trend will be characterized by a growing modulus
of the complex result, and with negative values it will be characterized by a
decreasing modulus and all its component (economic and social). If this coefficient is equal to zero, then the modulus of the complex result does not change but
the proportions between the economic and social result (on the circle) do. This is
certainly an incredible case.
The imaginary part of the complex exponent has another influence on the result
since it affects the polar angle of the modeled result. If coefficient b1 is positive,
then the polar angle increases, which is possible when the growing trend of the
social component of the region development dominates over the growing trend of
the economic one. If the imaginary component is less than zero, then the modeled
trend represents a sharper increase in the economic component over the social one.
If this coefficient is close to zero, the polar angle of the modeled parameter is
constant and equal to a. This is possible when the proportions between the rates of
social and economic development of a region do not vary with time.
It is evident now that the trend’s complex parameter (9.44) operates as a certain
diagnosing coefficient, where the real part characterizes rates of the region’s
development and the imaginary one shows which part is developing more actively
– the economic or the social one. Knowing these specifics, one can assume that for
regions developing in postindustrial society the real and imaginary parts of the
complex exponent will be positive.
In order to use trend (9.44) for purposes of forecasting socioeconomic development of regions, it is necessary to estimate the coefficients of the trend models using
statistical data. The problem of estimating complex-valued model coefficients by
LSM has already been solved; thus, let us show how to do it with respect to the trend
under consideration.
Since the trend is nonlinear, it should be linearized by taking logarithms of the
left- and right-hand side of equality (9.44):
lnðyrt þ iyit Þ ¼ lnða0 þ ia1 Þ þ ðb0 þ ib1 Þ ln t:
(9.48)
To avoid solving huge systems of normal equations to estimate four real
coefficients, let us center the variables of the linear model around their averages:
Yrt þ iYit ¼ lnðyrt þ iyit Þ
y; T ¼ ln t
t;
where
y ¼
X lnðyrt þ iyit Þ
T
; t ¼
X ln t
T
:
(9.49)
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9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Table 9.6 Trends in socioeconomic development in certain regions of Russia
Number
1
2
3
4
5
6
7
8
9
10
11
Region
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Saint-Petersburg
Trend in socioeconomic development
yrt þ iyit
yrt þ iyit
yrt þ iyit
yrt þ iyit
yrt þ iyit
yrt þ iyit
yrt þ iyit
yrt þ iyit
yrt þ iyit
yrt þ iyit
yrt þ iyit
¼ ð2:055 þ i0:240Þt0:071þi0:013
¼ ð2:492 þ i0:188Þt0:130þi0:002
¼ ð2:041 þ i0:206Þt0:079þi0:030
¼ ð2:129 þ i0:120Þt0:479 i0:010
¼ ð1:907 þ i0:307Þt0:173þi0:008
¼ ð1:467 þ i0:263Þt0:241 i0:014
¼ ð1:157 þ i0:249Þt0:456 i0:073
¼ ð2:042 þ i0:240Þt0:101þi0:046
¼ ð1:660 þ i0:273Þt0:162þi0:002
¼ ð1:481 þ i0:195Þt0:239 i0:009
¼ ð2:018 þ i0:422Þt0:429 i0:062
Then trend (9.44) with reference to the case under consideration will look like a
simple linear complex-valued function of a real argument:
Yrt þ iYit ¼ ðb0 þ ib1 Þ T:
(9.50)
Now, one can use LSM to find the complex exponent values:
b0 þ ib1 ¼
P
ðYrt þ iYit ÞT
P 2
:
T
(9.51)
We used (9.51) and the data of Table 9.3 to build regression models of trends of
each of the regions under consideration (Table 9.6):
|Here in this section we are not looking to determine confidence limits of LSM
estimations of trend coefficients; for our purposes this problem is not crucial. Our
target is values of the complex exponent of each trend.
The maximum value of the real part of this complex exponent falls on the Nenets
autonomous district (0.479), Leningrad region (0.456) and Saint-Petersburg
(0.429). This means that during the period under consideration, these three regions
were developing most dynamically. The lowest dynamic of socioeconomic development belongs to Archangelsk region (0.079) and the Republic of Karelia (0.071).
According to the imaginary part of the complex-valued trend, one can judge the
type of regional development – social, economic, or socioeconomic.
The imaginary components of the complex exponent are close to zero in the
Nenets autonomous district ( 0.010), Kaliningrad region ( 0.014), Pskov region
( 0.009), Republic of Karelia (0.013), Komi Republic (0.002), Vologda region
(0.008), and Novgorod region (0.002). This means that the proportions of development of the economic and social components in these regions remain stable and one
can speak of balanced socioeconomic development.
Negative values of the imaginary constituent of the complex exponent are
assigned to the Leningrad region ( 0.073) and Saint-Petersburg ( 0.062). This
9.4 Modeling Regional Socioeconomic Development
313
means that in these regions the economic component is more active than the social
one, Saint-Petersburg being the leader with its rather huge real component.
Regions with a social type of development for the period under consideration are
those where the imaginary component of the complex exponent is much greater
than zero. These are two northern regions – Archangelsk region (0.030) and
Murmansk region (0.046).
It should be noted that instead of numerous and various parameters of socioeconomic development of a region (like per capita Gross Regional Product, per capita
investments in fixed capital, per capita volume of foreign trade turnover, monthly
average income, per capita number of investments, local budget deficit, number of
beds in hospitals, etc.) we used only four – average per capita income, subsistence
level, social services, and total turnover in region. These four parameters help to
make a comparative analysis of the socioeconomic development of the regions and
draw certain conclusions. Certainly, studying the rates of regional development in
full detail requires additional analysis of numerous parameters characterizing the
particular features of each region. Nevertheless, to form a general picture of the
regional development in the country the proposed approach seems quite suitable.
However, research is aimed not so much at estimating the state of an entity as at
the possibility of governing this entity to reach some optimum state. With reference
to the problem of regional management this goal implies finding cause-effect
relationships between the parameters showing the level of socioeconomic development and factors predetermining the regional development.
It is evident that there are many factors influencing the level of regional
development. This influence can be direct or indirect, immediate or delayed, strong
or weak, etc. It is impossible for a researcher to take into account all these factors,
so they restrict themselves to the most important ones. Formally, correlation
analysis handles this with a certain degree of success. With reference to the
complex-valued parameter (9.37), correlation analysis is inapplicable as it was
developed to determine the degree of approximation of the relationship between
random factors to the linear one, provided the factors themselves are real variables.
In Chap. 4 we proposed a specially developed apparatus of complex-valued econometrics, one part of which is the correlation analysis of random complex variables.
In that chapter we obtained a formula for the pair correlation coefficient between
two random complex variables:
rXY
P
ðyrt þ iyit Þðxrt þ ixit Þ
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
P
P
ðxrt þ ixit Þ2 ðyrt þ iyit Þ2
(9.52)
This coefficient is complex. Its real part, as in the case of the pair correlation
coefficient of real random variables, characterizes the degree of approximation
between two random complex variables to the linear form and the imaginary part is
the degree of scattering of actual points with respect to linear regression
dependence.
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9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Table 9.7 Coefficients of complex correlation between socioeconomic development in regions
of Russia and first complex factor
Northwest region of Russia
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Saint-Petersburg
Per capita monthly average cash income (rubls)
Per capita rubles/monthly average cash expenses (rubles)
0.84041 + i0.02443
0.98036 i0.00815
0.96443 + i1.49141
0.98504 + i0.00970
1.02798 + i0.04992
1.00522 i0.00513
0.99044 + i0.00703
1.04353 + i0.07393
0.96741 0.02843
0.96027 + i0.03651
0.97921 i0.02597
While creating complex variables to describe a certain economic process it
should be noted that complex variables represent a convenient form of recording
two real variables. This is why one can combine in a complex variable such pairs of
socioeconomic parameters that above all reflect various sides of the same phenomenon or entity but have similar scales and dimensions.
With reference to the problem of finding factors determining socioeconomic
dynamics of regional development, we created several complex variables that,
according to economic analysis, imply an influence on regional development:
1. Per capita monthly average cash income of the population x1r and per capita
monthly average cash income x1i of the population x1r + ix1i;
2. Gross regional product x2r and agricultural produce x2i of the region x2r + ix2i;
3. Crops x3r and livestock x3i of the region x3r + ix3i.
The complex pair correlation coefficients between the complex parameter of
socioeconomic development of each of the regions of the Northwest of Russia
(Table 9.3) and the first of the three factors are given in Table 9.7.
As follows from the analysis of values of the calculated complex pair correlation
coefficients, for all the regions of the Northwest of Russia, one can use the
following linear complex-valued model:
dt þ ist ¼ ða01 þ ia11 Þ þ ðb01 þ ib11 Þðx1rt þ ix2it Þ:
(9.53)
The Republic of Karelia is an exception. Its real part of the complex pair
correlation coefficient is relatively far from one; therefore in this case it is
recommended to use a nonlinear function.
The degree of influence of the second complex variable – gross regional product
x2r and agricultural produce x2i of the region – on the complex parameter of
socioeconomic development Zt is shown in Table 9.8, which contains the results
of the complex-valued correlation analysis.
9.4 Modeling Regional Socioeconomic Development
315
Table 9.8 Coefficients of complex correlation between indicator of socioeconomic development
in regions of Russia and second complex factor
Northwest region of Russia
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Gross regional product, millions of rubles/agric.
produce – total, millions of rubles
0.874985 + i0.05414
0.94775 i0.00561
0.96857 + i0.017192
0.90402 + i0.00096
0.91992 + i0.00571
0.99416 + i0.00020
0.99332 + i0/00012
0.96576 + i0.11550
0.96684 i0.00482
0.91702 + i0.02245
Some commentary on this table is warranted. First, it should be mentioned that
the line for the Saint-Petersburg complex correlation coefficient is not filled
because there are no livestock or crops as a branch of the regional economy.
Practically for all the regions except the Republic of Karelia one can use the
linear complex-valued model:
dt þ ist ¼ ða02 þ ia12 Þ þ ðb02 þ ib12 Þðx2rt þ ix2it Þ:
(9.54)
However, for the Republic of Karelia, the real part of the complex pair correlation coefficient is close to one, but it is possible that the model of the complexvalued power function will be more accurate for approximation and for forecasting
dynamics.
Small values of the imaginary part of the complex correlation coefficient for
practically all regions reflect the small variance of the linear models. The
Murmansk region is an exception. Here, the imaginary part of the complex pair
correlation coefficient is 0.11550, which is much greater than in other regions. This
means that the linear regression for this region will be accompanied by a higher
variance in the actual values of the complex socioeconomic parameter compared to
the calculated values in other regions. The multifactor relationship in this case is
preferable.
The role and influence of the last of the considered complex factors on the rate of
socioeconomic development of regions, i.e., crops x3r and livestock x3i, can be
determined from Table 9.9. This factor also excludes Saint-Petersburg from consideration as a city without such a branch.
It should also be noted that for the Republic of Karelia the real part of the
complex correlation coefficient is not that close to one. The relatively high value of
the imaginary part of this coefficient must also mean that the influence of local
agricultural produce in the republic does not prevail and this complex factor can be
neglected for the purpose of building regional development models.
316
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
Table 9.9 Coefficients of complex correlation between indicator of socioeconomic development
in regions of Russia and third complex factor
Northwest region of Russia
Republic of Karelia
Komi Republic
Archangelsk region
Nenets autonomous district
Vologda region
Kaliningrad region
Leningrad region
Murmansk region
Novgorod region
Pskov region
Agricultural produce (crops), millions of rubles/agric.
produce (stock breeding), millions of rubles
0.85823 i0.10777
0.99021 + i0.00773
1.05721 + i0.15713
0.81765 + i0.00932
0.93737 + i0.07753
0.99772 + i0.01498
0.99477 + i0.01988
1.00994 + i0.00782
0.98000 i0.01741
0.95432 i0.00634
A similar situation is observed in the Nenets autonomous district. This region
gets its supply of agricultural products from other territories.
For other territories agriculture is an important part of the regional socioeconomic system; therefore, the level of their socioeconomic development can be
described by a linear complex-valued model:
dt þ ist ¼ ða03 þ ia13 Þ þ ðb03 þ ib13 Þðx3rt þ ix3it Þ:
(9.55)
Thus, modeling the regional development of practically all the territories of
Northwest Russia can involve linear complex-valued models and with all three
complex factors. Since the three considered factors influence one complex socioeconomic parameter, it is advisable to use a multifactor linear model of the
following type:
dt þ ist ¼ ða0 þ ia1 Þ þ ðb01 þ ib11 Þðx1rt þ ix1it Þ þ ðb02 þ ib12 Þðx2rt þ ix2it Þ
þ ðb03 þ ib13 Þðx3rt þ ix3it Þ:
(9.56)
We see that the apparatus of a complex-valued economy can indeed be used for
modeling regional development.
9.5
Conclusion
In December 2004 when it was first proposed to use complex numbers as a form of
representing socioeconomic variables, it was clear how to develop this proposal but
difficult to predict that complex variable models would provide researchers with not
so much alternative results as new ones.
9.5 Conclusion
317
Six years of hard and intense work by a team of authors made it possible to
formulate the principles of the new scientific direction called “complex-valued
economics.” The neglect of education and science by Russian state authorities
that prevailed in the country from the 1990s made it impossible for us us to focus
on this problem – all those involved in complex-valued economics had to think first
about how to earn a living – and only then turn to scientific research.
In these circumstances the support rendered to us by the Russian Foundation of
Fundamental Research turned out to be invaluable. Under the aegis of this foundation we conducted work within the framework of Grant No. 07-06-00151, “Development of principles of economic and mathematical modeling using complex
variables,” which lasted for 3 years from 2007 till 2009. In addition, with the
material support of the RFFR a study, “Production functions of complex variables”
(Grant No. 07-06-07030-d), was issued; it was the first serious work on complexvalued economics presented to a broad scientific society.
The present study shows that it represents a completed work that provides
economists with the principles of complex-valued economics – from the general
idea of complex-valued economic and mathematical models to particular methods
of application of complex-valued models. On the basis of the results obtained one
can further develop this scientific direction by uncovering an increasing number of
practical economic problems where complex-valued economics would help to
deepen one’s understanding of the processes and complement the arsenal of economic and mathematical methods and models with a new more “fine-tuned”
research instrument.
Due to a lack of time and strength, many provisions of the study are stated briefly
without delving deeper into the given problem.
Econometrics of complex variables serves as a good example. It comprises a
thorough description of the application of LSM and shows that assumptions
adopted in modern mathematical statistical theory about random complex variables
are not true – assumptions about how fundamental characteristics such as variance,
correlation moment, and covariance should be real since they characterize the
measure of variability further lead to a dead end. Elimination of this assumption
and laying out of a hypothesis on the possibility of the existence of complex
variance and correlation moment immediately eliminate the imbalance of further
econometric structures and make it possible to obtain the required statistical
estimations.
Since the concept of “function minimum” does not exist in the TFCV, a certain
criterion was stated when LSM estimations were obtained that made it possible to
determine the required model coefficients. The first structures of this type were
estimated by means of a Hessian matrix for the correspondence of the estimations to
the minimum of the formed criterion. However, we would like to obtain such a
conclusion in a universal form – to prove that in all cases these estimations
correspond to this minimum. A lack of time and strength made it impossible for
us to solve this problem completely.
We were able to state the approach and method of interval estimation of
coefficients of an econometric linear complex-valued model, but we would like to
318
9 Modeling and Forecasting of Economic Dynamics by Complex-Valued Models
extrapolate it to other statistical characteristics, for example, to the value of a
complex pair correlation coefficient.
We studied quite fully the properties of complex-valued production functions –
complex argument functions, complex variable functions, and even functions of
several complex arguments; however, so far we have not had sufficient time or
strength to include such an important factor as scientific–technical progress
(innovations) for a consideration of this part of the production function theory.
New results were obtained in stock market theory; however, they also require a
more thorough development and explanation. The cyclic character of the market as
an objective property can be best shown using complex-variable models; however,
this direction is also spotlighted quite briefly.
A new direction of scientific research was set out in the part regarding the
building of two-factor models under conditions of multicollinearity; however, so
far, it has not been possible to extrapolate this approach to the problem of building
models with a large number of factors – the hypotheses based on this model need
verification and substantiation.
This is why the materials stated in this study represent only the foundation for
the models and methods of complex-valued economics – the formation of complexvalued economics as a complete and balanced subdivision of economics is yet to
come.
References
1. Svetunkov IS (2011) Short-term forecasting of socio-economic processes with the use of model
with correction. Bus Inf 1(5):109–112
2. Svetunkov SG, Butukhanov AV, Svetunkov IS (2006) Zapredelnye sluchai metoda Brauna v
ekonomicheskom prognozirovanii. SPbGUEF Publishers, St. Petersburg (in Russian)
3. Svetunkov SG, Svetunkov IS (2011) Power production functions of complex variables. Econ
Math Methods 48:67–79