Mathematical methods in economics are an important tool for analysis. They are used in the construction of theoretical models that allow us to display existing connections in everyday life. Also, using these methods, the behavior of business entities and the dynamics of economic indicators in the country are quite accurately predicted.

I would like to dwell in more detail on forecasting the indicators of economic objects, which is a tool of decision-making theory. Forecasts of socio-economic development of any country are based on certain indicators (inflation dynamics, gross domestic product, etc.). The formation of expected indicators is carried out using such methods of applied statistics and econometrics as regression and correlation analysis.

The field of research “Economics and mathematical methods” has always been quite interesting for scientists in this field. Thus, Academician Nemchinov identified five mathematical ones in planning and forecasting:

Method of mathematical modeling;

Vector-matrix method;

Successive approximation method;

Method of optimal social assessments.

Another academician, Kantorovich, divided mathematical methods into four groups:

Models of interaction between economic units;

Macroeconomic models, including demand models and the balance sheet method;

Optimization models;

Linear modeling.

The system is used to make effective and correct decisions in the economic sphere. In this case, modern computer technology is mainly used.

The modeling process itself should be carried out in the following order:

1. Statement of the problem. It is necessary to clearly formulate the problem, determine the objects related to the problem being solved, and the situation realized as a result of its solution. It is at this stage that the quantitative and subjects, objects and situations related to them are made.

2. System analysis of the problem. All objects must be divided into elements with a definition of the relationship between them. It is at this stage that it is best to use mathematical methods in economics, with the help of which a quantitative and qualitative analysis of the properties of newly formed elements is carried out and as a result of which certain inequalities and equations are derived. In other words, a system of indicators is obtained.

3. System synthesis is a mathematical formulation of a problem, during the organization of which a mathematical model of an object is formed and algorithms for solving the problem are determined. At this stage, there is a possibility that the accepted models of the previous stages may turn out to be incorrect, and to obtain the correct result you will have to go back one, or even two steps.

Once the mathematical model is formed, you can proceed to developing a program to solve the problem on a computer. If you have a fairly complex object that consists of a large number of elements, you will need to create a database and available tools to work with it.

If the problem takes a standard form, then any suitable mathematical methods in economics and a ready-made software product are used.

The final stage is the direct operation of the formed model and obtaining the correct results.

Mathematical methods in economics must be used in a certain sequence and with the use of modern information and computing technologies. Only in this order does it become possible to exclude subjective volitional decisions based on personal interest and emotions.

Mathematical Economics. Kolemaev V.A.

2nd ed., revised. and additional - M.: 2002. - 399 p.

A systematic view of the economy is given using mathematical models of both macro- and microeconomics, as well as the production and financial-credit subsystems of the economy.

The textbook consists of sections: “Mathematical models of macroeconomics”, “Mathematical models of microeconomics” and “Models of analysis, forecasting and regulation of the economy”. The functional structure of the economy is reflected by modeling pricing, taxation, etc. The most important results obtained by domestic and foreign schools of mathematical economics in the 20th century are reflected, as well as new results obtained by the author (1st ed. - UNITI, 1998).

Questions and tasks for independent solution are given.

For undergraduates, graduate students and teachers of economic universities, as well as researchers.

Format: djvu

Size: 26.1 MB

Download: yandex.disk

Content
Preface 3
Introduction. Economy as an object of mathematical modeling 4
PART 1. MATHEMATICAL MODELS OF MACROECONOMICS 14
Chapter 1. Static models of macroeconomics 15
1.1. Macroeconomic production functions 16
1.2. Leontiev model 28
Chapter 2. Linear dynamic models of macroeconomics with discrete time 35
2.1. Economy as a dynamic system 36
Keynes' dynamic model 38
Samuelson-Hicks model 40
2.2. Dynamic model of Leontiev 44
2.3. Neumann model 46
Chapter 3. Linear dynamic models of macroeconomics with continuous time 52
3.1. Mathematical methods for studying economic dynamic systems 53
3.1.1. Linear dynamic element 54
3.1.2. Multiplier 55
3.1.3. Accelerator 56
3.1.4. Inertial link 57
3.1.5. Economy in the form of Keynes's dynamic model as an inertial link 59
3.1.6. Transfer function 60
3.1. 7. Oscillatory link 62
3.1.8. Economics in the form of the Samuelson-Hicks model as a linear dynamic link of the second order 67
3.1.9. Characteristics of the dynamic link 68
3.2. Analysis and synthesis of dynamic systems, transient processes in them 72
3.2.1. Transfer function of serial connection 74
3.2.2. Parallel connection transfer function 75
3.2.3. Closed-loop transfer function with feedback 76
3.2.4. Introduction of the multiplier into the feedback loop with Keynes's dynamic model 77
3.2.5. Introducing an accelerator into a positive feedback loop with Keynes's dynamic model 80
3.2.5. Stability of linear dynamic systems 82
3.2. 7. Conditions for economic stability in the form of the Samuelson-Hicks model 84
3.3. Linear multiply connected dynamic systems 85
Economy in the form of a dynamic inter-sectoral balance as a multiply connected linear dynamic system 88
3.4. Nonlinear dynamic systems. Market cycles in the economy 90
3.4.1. Nonlinear dynamic model of Keynes 92
3.4.2. Market cycles in the economy 94
3.5. Optimal control of dynamic systems 98
3.5.1. Pontryagin's maximum principle 99
3.5.2. Necessary conditions for optimality (maximum principle) 101
Chapter 4. Small-sector nonlinear dynamic models of macroeconomics 103
4.1. Model Solow 105
4.1.1. Transition regime in the Solow 108 model
4.1.2. The Golden Rule of VP Accumulation
4.1.3. Gain, in current consumption - loss, in the near future 111
4.2. Accounting for delays when entering funds 112
4.3. Single-sector model of optimal economic growth 116
4.4. Three-sector economic model 122
4.5. Production functions of sectors of the Russian economy 126
4.6. Modeling stagnation and balanced economic growth 130
4.6.1. Stagnation 131
4.6.2. Balanced economic growth 134
4.7. Study of balanced steady states 147
4.7.1. The golden rule for the distribution of labor and investment between sectors 149
4.7.3. An alternative way to determine the technological optimum 157
PART II. MATHEMATICAL MODELS OF MICROECONOMICS 163
Chapter 5. Patterns of consumer behavior 164
5.1. Consumer preferences and his utility function 165
Consumer behavior model 167
5.2. Slutsky equation 168
5.2.1. Change in demand with an increase in price with compensation 169
5.2.2. Change in demand when income changes 170
Chapter 6. Models of behavior of producers 173
6.1. Company model 174
6.1. 1 Manufacturer’s reaction to a change in output price 180
6.1.2. Manufacturer's response to changes in resource prices 181
6.2. Behavior of firms in competitive markets 185
6.2.1. Cournot equilibrium 187
Chapter 7. Models of interaction between consumers and producers 191
7.1. Models for establishing equilibrium prices 192
7.1.1. Web model 193
7.1. 2. Evans Model 195
7.2. Walras model 197
PART III. MODELS OF ANALYSIS, FORECASTING AND ECONOMIC REGULATION 201
Chapter 8. Mathematical models of market economies 202
8.1. Classical model of a market economy 203
8.1.1. Labor market 204
8.1.2. Money market 206
8.2. Keynes model 208
8.3. Mathematical models of the financial market 212
8.3.1. Financial transactions 213
8.3.2. Financial risk 217
8.3.3. Equilibrium in the securities market 230
8.4. Forecasting currency crises and financial risks 232
8.4.1. Model for forecasting financial risks 233
8.4.2. Forecasting currency crises 236
Chapter 9. Modeling inflation 239
9.1. The essence of inflation 240
9.2. Study of inflation using a three-sector economic model 244
9.2.1. The first half-turn of inflation 246
9.2.2. Second half-round of inflation 247
9.3. Conditions for the emergence and self-sustaining of inflation 249
9.4. The impact of inflation on production 250
Chapter 10. Mathematical models of state regulation of the economy 260
10.1. The role and functions of taxes in society 261
10.2. Taxes in a three-sector economy 266
10.3. The impact of tax increases on production and consumption 274
Chapter 11. Modeling foreign trade 280
11.1. Model of an open three-sector economy 281
11.2. Conditions for the possibility and feasibility of the national economy entering the world market 285
11.2.1. Entering the world market while fixing the shares of resources entering the fund-creating sector 287
11.3. Golden Rule of Foreign Trade 292
11.3.1. The Golden Rule of Resource Allocation 295
11.4. The influence of foreign trade on the national economy 300
11.4.1. Redistribution of resources between the material and consumer sectors 301
11.4.2. Redistribution of resources between the material and asset-creating sectors 305
Chapter 12. Modeling the goal of social development 308
12.1. Mathematical Theory of Public Choice 311
12.2. Models of cooperation and competition 327
12.2.1. Co-op games 328
12.2.2. Cooperation and competition in a three-sector economy* 332
12.3. Modeling scientific and technological progress 337
12.3.1. Evolutionary models of scientific and technological progress 338
12.3.2. Model of technological change 339
12.3.3. Model of rearmament of a three-sector economy 344
Applications 349
Appendix 1. Properties of an indecomposable direct cost matrix 350
Appendix 2. Linear differential equations and systems of linear differential equations with constant coefficients 353
Appendix 3. Study of expressions that determine the behavior of a three-sector economy in a stationary state 358
Appendix 4. Optimal balanced growth in a three-sector economy 364
Appendix 5. Kuhn-Tucker conditions 382
Literature 386

Magnitogorsk 2005

Collection of problems for the course “Mathematical Economics”. - Magnitogorsk: MaSU, 2005. – 184 p.

The collection provides an overview of the key categories and provisions used in the Mathematical Economics course. Examples of solving typical problems are presented, and questions for self-test on the material being studied are given. The materials of the manual can be used in the courses “Financial Mathematics”, “Mathematical Methods of Financial Analysis”, “Financial Management”, “Financial Analysis”, etc.

The work is aimed at teachers, graduate students and full-time and part-time students, researchers and practitioners specializing in the field of financial management and investment projects, the application of mathematical methods and models in the study of economic systems and phenomena.

Compilers. G.N. Chusavitina,

V.B. Lapshina.

 Chusavitina G.N., Lapshina V.B. 2005

 Magnitogorsk State University, 2005

INTRODUCTION 5

Chapter 1 simple interest 7

1.1. Determining rates and calculating interest 7

1.2. Simple interest rate 10

1.3. Simple discount rate 21

1.4. Loan repayment and depreciation charges 32

1.5. Calculating averages 41

1.6. Currency payments 48

1.7. Income tax 53

1.8. Inflation 56

1.9. Replacement and consolidation of payments 64

Chapter 2 COMPOUND INTEREST 73

2.1. Compound interest rate 73

2.2. Complex discount rate 91

2.3. Continuous rate 101

2.4. Rate equivalence 107

2.5. Inflation and compounding and continuous interest 112

2.6. Replacement of payments and terms of their payments 125

Chapter 3 ANNUITIES 132

3.1. Permanent annuity 132

3.2. Continuous and variable annuities 148

3.3. Valuation of an annuity with a period of more than a year 157

INTRODUCTION

"Mathematical economics" is the name of the discipline invented by mathematicians. Economists prefer another name - “Economic and mathematical models and methods”. This name is often found in the curricula and standards of economics departments. In our opinion, these two names equally accurately convey the internal content of the subject, where economic and mathematical aspects are harmoniously combined. Unfortunately, in practice, the EMM&M course program is often compiled entirely from separate sections of “Operations Research and Mathematical Programming”, which, firstly, have already been completed before this course, and secondly, contain mathematical models of decision making and optimization, and not economic -mathematical models as such.

Mathematical economics is a science that uses mathematics as a method for studying economic systems and phenomena.

Thus, the object of study (or subject area) of mathematical economics is economics - as a part of being or part of a vast area of human activity.

Like other sciences that study economics as a whole or its components, mathematical economics uses a certain methodology and has its own specifics. The specificity of mathematical economics, its methodological feature, is that it studies not the economic objects and phenomena themselves, but their mathematical models. Its goal is to obtain objective economic information and develop recommendations of important practical importance. Formally, mathematical economics can be classified as both economic and mathematical sciences. In the first case, it should be understood as that section of economics that studies quantitative and qualitative categories, as well as behavioral aspects of economic entities. Considering mathematical economics to be one of the areas of mathematics, we can attribute it to those sections of applied mathematics that deal with optimization problems and decision-making problems

By its nature, economics is the social science closest to mathematics. Already in the definition of the very concept of economics, its main tasks, one can see mathematical concepts and terminology.

Indeed, economics is the social science of using limited resources in order to maximize the satisfaction of the unlimited material needs of the population. The central problems of economic science - rational management of the economy, optimal distribution of limited resources, the study of economic management mechanisms, development of methods of economic calculations - are essentially problems solved within the framework of mathematical sciences. Quantitative and qualitative methods of mathematics are the best auxiliary apparatus for obtaining answers to the basic questions of economics:

What should be produced (i.e., what goods and services should be produced and in what quantities)?

How will goods be produced (i.e. by whom and with what resources and what technology)?

Who are these goods intended for (i.e., by whom and how will these goods be consumed)?

Finally, the task of economic theory associated with systematizing, interpreting and generalizing the behavior of economic participants in the process of production, exchange and consumption goes back to mathematical problems of optimization and decision making.

Taking into account the above, we can talk about the following main tasks facing mathematical economics:

development of mathematical models of economic objects, systems and phenomena (general and specific problems of economics under various conditions, prerequisites and at various levels);

studying the behavior of economic participants (conditions for the existence of optimal solutions and their characteristics, as well as methods for calculating them in models of consumption, the firm, perfect and imperfect competition, etc.);

study of descriptive models of the economy (planning models, input-output, expanding economy, economics of welfare and growth, etc.);

analysis of economic values and statistical data (elasticity, average and marginal values, regression and correlation analysis and forecasting of economic factors and indicators).

MATHEMATICAL ECONOMICS

A mathematical discipline whose subject is economic models. objects and processes and methods of their research. However, concepts, results, methods of M. e. it is convenient and customary to present them in close connection with their economics. origin, interpretation and practicality. applications. The connection with economics is especially significant. science and practice.

M. e. as a part of mathematics began to develop only in the 20th century. Previously there were only episodes. research that cannot, in a strict sense, be classified as mathematics.

Features of economic and mathematical modeling. Economical feature modeling lies in the exceptional diversity and heterogeneity of the subject of modeling. The economy contains elements of controllability and spontaneity, rigid certainty and significant ambiguity and freedom of choice, technical processes. character and social processes, where human behavior comes to the fore. Different levels of the economy (eg, workshop and national economy) require significantly different descriptions. All this leads to great heterogeneity of mathematical models. apparatus. A subtle issue is the reflection of the type of socio-economic. systems, edges are modeled, taking into account the social system. It often turns out that abstract mathematics. one or another economic object or process can be successfully applied to both capitalist and socialist economies. It's all about the method of use and interpretation of the analysis results.

Production, efficient production. Economics deals with goods, or products, which are understood in economics. extremely wide. The general term ingredients is used for them. The ingredients are services, natural resources, environmental factors negatively affecting humans, comfort from the existing security system, etc. It is usually believed that there are of course ingredients and products - a Euclidean space where l - number of ingredients. Point z from under the right conditions can be considered a "production" mode, the positive components indicating the production volumes of the relevant ingredients, and the negative components indicating the costs. The word "production" is in quotation marks because production is understood in its broadest sense. The set of available (given, existing) production possibilities is. A production method is effective if there is no such thing that strict . The task of identifying effective methods is one of the most important in economics. It is usually assumed, and in many cases this agrees well with reality, that Z- convex By expanding the product space, the problem of analyzing efficient methods can be reduced to the case when Z- convex closed

A typical task of identifying an effective method is the main task of production planning. Given production methods and a vector of needs and resource limitations. It is required to find a way such that for everyone If Z- convex closed cone, then this is a general problem convex programming. If Z is given by a finite number of generators (so-called basic methods), then this is a general problem linear programming. Solution lies on the border Z. Let p be the coefficients of the support hyperplane for Z at the point i.e. for all and The main convex programming finds the conditions under which p l>0. For example, a sufficient condition: there is a vector (the so-called Slater condition). The coefficients I, characterizing the effective method, have important economic implications. meaning. They are interpreted as prices that measure the cost-effectiveness and production of individual ingredients. The method is effective if and only if the cost of output is equal to the cost of inputs. This effective methods of production and their characterization with the help of p had a revolutionary impact on the theory and practice of socialist planning. economy. It formed the basis for objective quantitative methods for determining prices and public assessments of resources, making it possible to select the most effective economic ones. decisions in socialist conditions. farms. The theory naturally generalizes to an infinite number of ingredients. Then the ingredient space turns out to be a suitably chosen function space.

Efficient growth. Ingredients belonging to different moments or time intervals can formally be considered different. Therefore, the description of production in dynamics, in principle, fits into the above scheme, consisting of objects (X,Z, b), Where X- ingredient space, Z- many production possibilities, b- setting requirements and restrictions on the economy. However, the study itself is dynamic. aspect of production requires more special forms of describing production capabilities.

The production capabilities of a fairly general economic model. speakers are specified using a point-set mapping (multivalued function) Here is the (phase) space of the economy, interpreted as the state of the economy at one time or another, where x k - quantity of product k available at this moment. The set a(x) consists of all states of the economy, in which it can go from state to X. We will call

display graph a. Points ( x, y).- permissible production processes.

Various options for setting possible trajectories of economic development are considered. In particular, the consumption of the population is taken into account either in the display itself, or is highlighted explicitly. For example, in the second case, an admissible trajectory is such that

For all t. Various concepts of trajectory efficiency are studied. A trajectory is consumption efficient if there is no other feasible trajectory ( X, C), leaving the same initial state, for which a trajectory is internally effective if there is no other admissible trajectory (X, C) leaving the same initial state, time t 0 and number l>1, such that

The optimality of a trajectory is usually determined depending on the utility function and the coefficient of reduction of utility over time (see below for the utility function). The trajectory is called (u, m)-about ptpmal if

for any admissible trajectory ( X, C), emerging from the same initial state. There are quite general existence theorems for the corresponding trajectories.

Trajectories that are effective in various senses are characterized by a sequence of prices in the same way that an effective method was characterized by prices (coefficients of the reference hyperplane) P. That is, if for the efficient method the cost of inputs is equal to the cost of output at optimal prices, then on the efficient trajectory the cost of states is constant and maximum, and on all other admissible trajectories it cannot increase.

All the above definitions are easily generalized to the case when production a, function u and m depend on time. Time itself can be continuous, or in general the parameter t can run through a set of a rather arbitrary form.

With economical From a point of view, the trajectories that are of interest are those that achieve the maximum possible rate of economic growth, which it can sustain for an indefinitely long time. It turns out that when a and and are constant in time, such trajectories are stationary, i.e., they have

where a is the growth (expansion) rate of the economy. Stationary effective in one sense or another, as well as stationary optimal trajectories are called. highways.

Under very broad assumptions, the theorems about the highway take place, stating that any effective , regardless of the initial state, approaches the highway over time. There are a large number of different theorems about the highway, differing in the definition of efficiency or optimality, the method of measuring the distance to the highway, the type of convergence, and finally, the finite or infinite time interval.

Economical model dynamics, whose production capabilities are set by a multifaceted convex cone, called. Neumann model. A special case of the Neumann model is the closed Leontief model, or (in other terminology) a closed dynamic inter-industry balance (the term “closed” is used here as a characteristic of the property of the economy, which consists in the absence of irreproducible products), which is specified by three matrices with non-negative elements Ф, Ау Ordered Process if and only if there are vectors v, such that the following inequalities are satisfied:

The input-output balance model has become widespread due to the convenience of obtaining initial information for its construction.

Economy models dynamics are also considered in continuous time. Continuous-time models were among the first to be studied. In particular, a number of works were devoted to the simplest single-product model given by the equation

Where X - volume of funds per unit of labor resources, c - consumption per capita, f- production function (increasing, concave). Non-negative functions satisfying this equation characterize the admissible trajectory. For a given utility function and discount factor m is determined. Optimal trajectories (and only they) satisfy an analogue of the Euler equation

where is the maximum number that satisfies the condition f(x) -c=x.

Leontief's model was also first formulated in continuous time as a system of differential equations

Where X- product flows, AI IN - matrices of current and capital costs, respectively, WITH - final consumption flows.

Efficient and optimal trajectories in continuous-time models are studied using methods of the calculus of variations, optimal control, and mathematics. programming in infinite-dimensional spaces. Models are also considered in which admissible trajectories are specified by differential inclusions of the form (x) , Where A - production display.

Rational consumer behavior. The tastes and goals of consumers, which determine their rational behavior, are given in the form of a certain system of preferences in the space of products. Namely, for each consumer i a point-set mapping is defined where Z- a certain space of situations in which the consumer may find himself in the selection process, X- the set of vectors available to the consumer. In particular, Z may include as a subspace the content-rich set consists of all vectors that are (strictly) preferred to the vector x in the situation z. For example, display P i can be specified as a utility function And, where u(x) shows the utility from consuming a set of products X. Then

Let the description of situation z include prices p . for all products and consumer cash income d. Then there are many sets that the consumer can purchase in a situation z. This is a lot of names. budgetary. The rationality of consumer behavior lies in the fact that he chooses such sets of xyz B i(z) , for which Let D(z) be the set of sets of products chosen by fighter r in situation z; D i called displayed by i-e m (or function in the case when D i(z) consists of one demand point. There are a number of studies devoted to elucidating the properties of mappings Р i, В i, Di. In particular, the case when the mappings P i can be specified as functions. Conditions have been determined under which the mappings In i And D i are continuous. Of particular interest is the study of the properties of the demand function D i. The fact is that sometimes it is more convenient to consider the demand functions as primary D i, not preferences P i, since they are easier to construct based on available information about consumer behavior. For example, in economics (trading) there can be values that approximately estimate the partial derivatives

where R is the price of product p, d- income.

Adjacent to the theory of rational consumer behavior is the theory of group choice, which usually deals with discrete options. It is usually assumed that there are a finite number of group members and a finite number of, for example, alternative options. The problem is to make a group decision about choosing one of the options given the preference relations between the options for each participant. Group choice provides various voting schemes, and axiomatic and game-theoretic approaches are also considered.

Coordination of interests. The bearers of interests are individual parts of the economy. systems, as well as society as a whole. Such parts are consumers (consumer groups): enterprises, ministries, territorial government bodies, planning and financial authorities, etc. There are two mutually intertwined approaches to the problem of reconciling interests - analytical, or constructive, and synthetic, or descriptive. According to the first approach, the global optimality criterion (formalization of the interests of society as a whole) is taken as the initial one. The task is to derive local (private) criteria from the general one, taking into account private interests. In the second approach, the initial ones are precisely private interests and the task is to combine them into a single consistent system, the functioning of which leads to results that are satisfactory from the point of view of society as a whole.

The first approach directly includes decomposition methods of mathematics. programming. Suppose, for example, that there is a productivity in the economy and each producer j is given by the set of production possibilities Yj, where and is a convex compact set. Given V of the entire society as a whole, where - concave function. The economy must be organized in such a way that the convex programming problem is solved: find from the conditions

According to the theorem about the characteristics of efficient production methods, there are prices such that

for all j,

The value y (j) p is interpreted as the profit of the jth producer at prices R. It follows that the profit maximization criterion for each of the producers does not contradict the overall goal if the current prices are determined accordingly. Schemes related to the second approach have received great development within the framework of economic models. balance.

Economic equilibrium. It is assumed that the economy consists of separate parts that are carriers of their own interests: producers, numbered with indices j = 1, ..., T, and consumers numbered with indices i=1, ..., P. Producer j is described by the production possibilities set and the mapping defining his system of preferences. Here Z- a set of possible states of the economy, specified below. Consumer r is described by the set of possible sets of products available for consumption, the initial stock of products, and preferences and, finally, by the income distribution function, where a i(z) shows the amount of money flowing to consumer i in state z. There are many possible prices in the economy Q. Then the set of possible states is Budget display B i is defined here like this:

The equilibrium state of the described economy is one that satisfies the conditions

Essentially, the equilibrium state of the economy coincides with the definition of the solution non-cooperative game many persons in the Neumann-Nash sense with the additional condition that a balance be satisfied for all products. The existence of an equilibrium state has been proven under very general conditions for the original economy. Much more stringent conditions must be imposed in order for the equilibrium state to be optimal, i.e., to achieve a certain global optimization problem with an objective function depending on the interests of consumers. For example, let P i given by a concave continuous function a Fj given by the function

Where Y j , X i - convex compacts,

Any subset S=(i 1 , ..., i r ) consumer indices forms a sub-economy of the original economy, in which each consumer i s from S there corresponds (one and only one) producer, the set of production possibilities of which exists

The income distribution functions in this case have the form

State of the name balanced if

They say that a balanced state z the original economy is blocked by a coalition of consumers S, if in a sub-economy determined by the coalition S, there is such a balanced state that For s= 1, ..., r and for at least one index there is a strict inequality. The core of the economy is called. the set of all balanced states that are not blocked by any coalition of consumers. For an economy with the described properties, the theorem holds: every equilibrium state belongs to the core. The converse is not true, but a number of sufficient conditions have been found under which many equilibrium states are close to each other or even coincide. In particular, if the number of consumers tends to infinity and the influence of each consumer on the state of the economy becomes increasingly small, then the set of equilibrium states tends to the core. The coincidence of the core and the set of equilibrium states occurs in an economy with an infinite (continuous) number of consumers (Aumann’s theorem).

Let the economy be a market model (i.e., there are no producers), the set of participants (consumers) is a closed single segment , hereinafter denoted T. The state of the economy is z=(x,p), where is the function displaying TV R + l, each component is Lebesgue integrable on the interval T. The initial products between participants are specified by the function w,. thus the balanced state z is such that the Coalition of participants is a Lebesgue measurable subset of the set T. If a subset has measure 0, then the corresponding is called. null. The core is the set of all balanced states that are not blocked by any non-zero coalition. A state is an equilibrium if for almost every participant i

Aumann's theorem states that in the described economy and the set of equilibrium states coincide. Of interest is the question of the structure of the set of equilibrium states, in particular when this set is finite or consists of one point. Debreu's theorem applies here. Let there be many market models where are the initial inventories of products for participant i, the vector is a parameter that defines a specific model from the set The display represents the demand function for the i-th participant. Functions D 1, ..., Dn are given (do not change) for the entire set of economies W. Let W 0 , - a set of economies in which the set of equilibrium states is infinite. Debreu's theorem states that if functions D 1, ..., Dn are continuously differentiable and there are no saturation points for at least one of the participants, then W 0 has (Lebesgue) measure in space W.

About numerical methods. M. e. has a close connection with computational mathematics. Linear, linear economic. models have had a major influence on computational methods in linear algebra. Essentially thanks to linear programming, inequalities in computational mathematics have become as common as equations.

A difficult and multifaceted issue is the calculation of economics. balance. For example, many works are devoted to the conditions for convergence to equilibrium of a system of differential equations

Where R - price vector, F- excess demand function, i.e., supply and demand functions. Equilibrium prices, by definition, ensure equality of supply and demand:

The excess demand function F is specified either directly or through more primary concepts of the corresponding equilibrium model. S. Smale studied a significantly more general dynamic. system than (*), in relation to the market model; along with changes in prices over time R a change in state x is considered; in this case, the permissible trajectory satisfies certain differential inclusions of the form where K(p). and C(p) - set of possible directions of change X, determined through a market model.

Economical an equilibrium, a solution to a game, a solution to one or another extremal problem can be defined as fixed points of a suitably formulated point-set mapping. As part of research on M. e. Numerical methods for searching for fixed points of different classes of mappings are being developed. The most famous is Scarf's method, which is a combination of the ideas of Sperner's lemma and the simplex method for solving linear programming problems.

Related issues. M. e. is closely related to many mathematical fields. disciplines. Sometimes it is difficult to determine where the boundaries between M. e. and mathematical statistics or convex analysis, functional analysis, topology, etc. One can point out, for example, the development of the theory of positive matrices, positive linear (and homogeneous) operators, and the spectral properties of superlinear point-set mappings under the influence of the needs of mathematical economics.

Lit.: Neumann J., Morgenstern O., Game Theory and Economic Behavior, trans. from English, M., 1970; K a n t o r o v i h L. V., Economic calculation of the best use of resources, M., 1959; Nikaido X., Convex structures and mathematical economics, trans. from English, M., 1972; M a k a r o v V. L., Rubinov A. M., Mathematical theory of economic dynamics and equilibrium, M., 1973; M i r k i n B. G., The problem of group choice [information], M., 1974; Scarf H., The Computation of Economic Equilibria, L., 1973; Dantzig J., Linear programming, its applications and generalizations, trans. from English, M., 1966; Smale S., "J. math. Economics", 1976, No. 2, p. 107-20. L.V. Kantorovich, V. L. Makarov.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

Economic dictionary

Subject and methods of economic theory

Economic relations permeate all spheres of human life. The study of their patterns has occupied the minds of philosophers since ancient times. The gradual development of agriculture and the emergence of private property contributed to the complication of economic relations and the construction of the first economic systems. Scientific and technological progress, which determined the transition from manual labor to machine labor, gave a strong impetus to the consolidation of production, and therefore to the expansion of economic ties and structures. In the modern world, economics is increasingly considered in conjunction with other related social sciences. Namely, at the junction of two directions there are various solutions that can be applied in practice.

The fundamental direction towards economics itself took shape only by the middle of the nineteenth century, although scientists in many countries over the centuries created special schools that studied the patterns of people's economic life. Only at this time, in addition to a qualitative assessment of what was happening, scientists began to study and compare actual events in the economy. The development of classical economics contributed to the formation of applied disciplines that study narrower areas of economic systems.

The main subject of studying economic theory is the search for optimal solutions for economies at various levels of organization in terms of meeting increasing demand, subject to limited resources. Economists use various methods in their research. Among them, the most frequently used are the following:

Methods that allow you to evaluate general elements or generalize individual structures. They are called methods of analysis and synthesis.
Induction and deduction make it possible to consider the dynamics of processes from the particular to the general and vice versa.
The systems approach helps to see a separate element of the economy as a structure and analyze it.
In practice, the abstraction method is widely used. It allows you to separate the object or phenomenon being studied from its relationships and external factors.
As in other sciences, the language of mathematics is often used in economics, which helps to visually display the elements of the economy under study, as well as carry out an analysis or form the necessary forecast of trends.

The essence of mathematical economics

Modern economics is distinguished by the complexity of the systems it studies. As a rule, one economic agent enters into many relationships at once, and every day. If we are talking about an enterprise, then the number of its internal and external interactions increases thousands of times. To facilitate the research and analytical tasks facing economists and scientists, the language of mathematics is used. The development of mathematical tools makes it possible to solve problems that are beyond the power of other methods used in economic theory.

Mathematical economics is an applied branch of economic theory. Its main essence lies in the use of mathematical methods, means and tools to describe, study and analyze economic systems. However, this discipline has its own specifics. It does not study economic phenomena as such, but deals with calculations associated with mathematical models.

Note 1

The goal of mathematical economics, like most applied areas, can be called the formation of objective information and the search for solutions to practical problems. It studies, first of all, quantitative and qualitative indicators, as well as the behavior of economic agents in dynamics.

The challenges facing mathematical economics are as follows:

Construction of mathematical models describing processes and phenomena in economic systems.
Study of the behavior of various subjects of economic relations.
Providing assistance in constructing and evaluating plans, forecasts, and various types of events over time.
Carrying out analysis of mathematical and statistical quantities.

Applied mathematics in economics

Mathematical economics in its social significance is quite close to mathematics. If we consider this discipline from the perspective of mathematical science, then for it it is an applied direction. Applied mathematics makes it possible to consider and analyze individual elements of complex economic systems, since it has broad functionality based on fundamental mathematical knowledge. Such possibilities of mathematics contributed to the emergence of mathematical ecology, sociology, linguistics, and financial mathematics.

Let's consider the most important mathematical methods used in the study of economic systems:

Operations research deals with the study of processes and phenomena in systems. This includes analytical work and optimization of the practical application of the results obtained.
Mathematical modeling includes a wide range of methods and tools that make it possible to solve problems facing scientists and economists. The most commonly used are game theory, service theory, schedule theory, and inventory theory.
Optimization in mathematics deals with the search for extreme values, both maximum and minimum. Function graphs are usually used for these purposes.

The methods of mathematics listed above make it possible to study statistical situations in the economy, or processes in short-term periods. As is known, currently the main goal of economic entities is to find long-term equilibrium. An important factor in these studies is the time factor, which can be taken into account by using probability theory and the theory of optimal solutions for calculations.

Note 2

Thus, mathematics and economics are closely related to each other. It is customary to dress up the dynamics of economic structures in mathematical models, which can then be divided into separate subtasks and all possible methods of economic analysis, as well as mathematical calculations, can be applied. Decision-making in the economic sphere is a rather complex action, since it is associated with the imperfection and incompleteness of available information. The use of mathematical modeling makes it possible to reduce the riskiness of management decisions.