Foundations of Mathematical Economics. Mathematical economics. Goals and objectives of studying the topic

MATHEMATICAL ECONOMY

Mathematical discipline, the subject of a cut are models of economics. objects and processes and methods of their research. However, the concepts, results, methods of M. e. convenient and customary to present in close connection with their economical. origin, interpretation and practical. applications. The connection with economics is especially important. science and practice.

M. e. as part of mathematics, it began to develop only in the 20th century. Previously, there were only episodes. research, to-rye. cannot in the strict sense be attributed to mathematics.

Features of economic and mathematical modeling. Feature economical modeling consists in the exceptional diversity and heterogeneity of the subject of modeling. In the economy there are elements of controllability and spontaneity, rigid certainty and significant ambiguity and freedom of choice, technical processes. character and social processes, where human behavior is highlighted. Different levels of the economy (eg, shop and national economy) require significantly different descriptions. All this leads to a great heterogeneity of mathematical models. apparatus. The 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 an abstract mathematical. one or another economical. object or process can be successfully applied to both capitalist and socialist economies. It's all about the way of using, interpreting the analysis results.

Manufacturing, efficient production. Economics deals with goods, or products, to-rye are understood in M. e. extremely wide. The general term ingredients are used for them. Ingredients are services, natural resources, environmental factors that negatively affect a person, comfort from the existing security system, etc. It is usually believed that there are of course ingredients and products - Euclidean space, where l - number of ingredients. The z-point of, under the right conditions, can be viewed as a "production" method, with positive components indicating the output of the respective ingredients, and negative components indicating costs. The word "production" is put in quotation marks, since production is understood in the broadest sense. The set of available (given, existing) production possibilities is. A method of production is efficient if there is no such that a strict one is fulfilled for at least one component. The task of identifying effective methods is one of the most important in economics. It is usually assumed, and in many cases this is in good agreement with reality, that Z - convex. By expanding the product space, the problem of analyzing effective methods can be reduced to the case when Z - convex closed.

The typical task of identifying an effective method is the main task of production planning. Production methods and the vector of needs and resource constraints are given.It is required to find a way such that for all If Z - closed convex cone, then this is a general problem convex programming. If Z is given by a finite number of generators (the 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 a point, i.e., for all and The main convex programming finds conditions for which p l> 0. For example, the condition is sufficient: there is a vector (the so-called Slater condition). The coefficients I characterizing the effective way are of an important economical. meaning. They are interpreted as prices that are commensurate with the cost and production efficiency of the individual ingredients. The method is effective if and only if the cost of output is equal to the cost of inputs. Given effective modes of production and their characterization with p had a revolutionary impact on the theory and practice of socialist planning. economy. It formed the basis of objective quantitative methods for determining prices and public assessments of resources, making it possible to choose the most effective economic. decisions in the conditions of socialist. farms. The theory naturally generalizes to an infinite number of ingredients. The ingredient space then turns out to be a suitably chosen functional space.

Effective growth. Ingredients related to different moments or intervals of time 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 constraints on the economy. However, the study itself is dynamic. aspect of production requires more special forms of description of production capabilities.

The production capabilities of a fairly general economic model. dynamics are set using point-multiple display (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 - the quantity of product k available at that moment. The set a (x). Consists of all states of the economy, into which it can pass in a single time from the state NS. We will call

graph display a. Points ( x, y) .- acceptable manufacturing processes.

Various options for specifying possible trajectories of economic development are considered. In particular, the consumption of the population is taken into account either in the self-mapping itself, or it is singled out explicitly. For example, in the second case, an admissible trajectory is such that

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

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

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

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

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

With economical From the point of view, trajectories are of interest, on which the maximum possible growth rate of the economy is achieved, to-ry it can withstand as long as you like. It turns out that for constant a and and such trajectories are stationary in time, that is, 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, there are backbone theorems, which state that every efficient one, regardless of the initial state, approaches the backbone over time. There are a large number of different theorems about the backbone, differing in the definition of efficiency or optimality, the method of measuring the distance to the backbone, the type of convergence, and finally, a finite or infinite time interval.

Economic model dynamics, in which production capabilities are given by a polyhedral convex cone, called. Neumann's model. A special case of the Neumann model is the closed Leontief model, or (in another terminology) a closed dynamic input-output balance (the term "closed" is used here as a characteristic of the property of the economy, consisting in the absence of non-reproducible products), which is specified by three matrices with non-negative elements Ф, А, and Order Process if and only if there are vectors v, such that the following inequalities hold:

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

Economical models dynamics are also considered in continuous time. One of the first to study exactly models with continuous time. In particular, a number of works were devoted to the simplest single-product model given by the equation

where NS - the 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 the discount coefficient m is determined. Optimal trajectories (and only they) satisfy the analogue of the Euler equation

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

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

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

Effective and optimal trajectories in models with continuous time are studied using the methods of calculus of variations, optimal control, mathematical. programming in infinite-dimensional spaces. Models are also considered in which admissible trajectories are given by differential inclusions of the form (x) , where a - production display.

Rational consumer behavior. Tastes and goals of consumers, to-rye determine their rational behavior, are given in the form of a certain system of preferences in the product space. Namely, for each consumer i, a point-to-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 can include as a subspace. In essence, the set consists of all vectors that are (strictly) preferred to the vector xv of the situation z. For example, the mapping P i can be specified as a utility function and, where u (x). shows the utility from the consumption of a set of products NS. 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 set is called. budgetary. The rationality of the consumer's behavior lies in the fact that he chooses such sets х from B i(z) , for which Let D (z) be the set of sets of products chosen by the fighter r in the situation z; D i called displayed by i-e m (or by a function in the case when D i(z) consists of one point) demand. There are a number of studies devoted to clarifying the properties of the mappings Р i, В i, Di. In particular, the case where the mappings P i can be specified as functions. Conditions are determined for 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 ones D i rather than preferences P i because they are easier to construct from the available information on consumer behavior. For example, in the economy (trade), values ​​can be observed that approximate the partial derivatives

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

The theory of rational consumer behavior is adjacent to the theory of group choice, which deals, as a rule, with discrete options. It is usually assumed that there are a finite number of group members and a finite number, for example, of alternatives. The task is to select a group decision on the choice of one of the options for a given preference relationship between options for each participant. Group choice provides various voting schemes; axiomatic and game-theoretic approaches are also considered.

Harmonization of interests. The bearers of interests are the individual parts of the economics. systems, as well as society as a whole. Such parts are consumers (consumer groups): enterprises, ministries, territorial government bodies, planning and financial bodies, etc. There are two mutually intertwining approaches to the problem of reconciling interests - analytical, or constructive, and synthetic, or descriptive. According to the first approach, the global criterion of optimality (formalization of the interests of the whole society as a whole) is taken as the initial one. The task is to derive local (private) criteria from the general, while taking into account private interests. In the second approach, private interests are the initial ones 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 the whole society as a whole.

The first approach is directly related to the decomposition methods of mathematical. programming. Let, for example, in the economy there is m productive and each producer j is given by a set of production possibilities Y j, where and is a convex compact set. The V of the whole society as a whole is given, where - concave function. The economy should be organized in such a way that the convex programming problem is solved: find from the conditions

According to the theorem on the characteristic of efficient production methods, there are prices such that

The value y (j) p is interpreted as the profit of the j-th producer at prices R. Hence it follows that the criterion of profit maximization for each of the producers does not contradict the general goal if the current prices are determined accordingly. Schemes related to the second approach have been extensively developed 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 j = 1, ..., T, and consumers, numbered by indices i = 1, ..., NS. Manufacturer j is described by a variety of production possibilities and display setting his system of preferences. Here Z - the set of possible states of the economy, elaborated below. Consumer r is described by a set of possible sets of products available for consumption, an initial stock of products, a preference and, finally, the income distribution function, where a i(z) shows the amount of money going 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 defined here like this:

The equilibrium state of the described economy is satisfying the conditions

Essentially, the equilibrium state of the economy coincides with the definition of a solution a coalition-free game many persons in the sense of Neumann - Nash with the additional condition that the balance for all products is fulfilled. The existence of a state of equilibrium 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, that is, to deliver a certain global optimization problem with an objective function that depends on the interests of consumers. For example, let Р i given by a concave continuous function a F j given by function

where Y j, X i - convex compact sets,

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 is

In this case, the income distribution functions have the form

Condition called balanced if

It is said that a balanced state z the original economy is blocked by a coalition of consumers S, if in a coalition-driven economy S, there is such a balanced state that for s = 1, ..., r and a strict inequality holds for at least one index. The core of the economy is called. the set of all balanced states, which are not blocked by any coalition of consumers. For an economy with the described properties, the following theorem holds: every equilibrium state belongs to the core. The converse is not true, but a number of sufficient conditions have been found for which the set of equilibrium states and 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 ever smaller, then the set of equilibrium states tends to the core. The coincidence of the core and the set of equilibrium states takes place in an economy with an infinite (continuous) number of consumers (Auman's theorem).

Let the economy be a market model (i.e., there are no producers), the set of participants (consumers) is a closed unit segment , denoted below T. The state of the economy is z =(x, p), where there is a function representing TV R + l, each component of which is Lebesgue integrable on the interval T. The initial products between the participants are given 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, which are not blocked by any nonzero 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 holds here. Let the set of market models where is the initial stocks of products for participant i, the vector is a parameter that determines a specific model from the set The mapping is the demand function for the i-th participant. Functions D 1, ..., D n set (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, ..., D n are continuously differentiable and there are no saturation points for at least one of the participants, then W 0 has (Lebesgue) measure in the space W.

On numerical methods. M. e. has a close relationship with computational mathematics. Linear, linear economical the models have had a great influence on the computational methods of linear algebra. Essentially, linear programming has made inequalities as common in computational mathematics as equations.

Computing economics is a difficult and multidimensional issue. balance. For example, many papers are devoted to conditions for the 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 much more general dynamic. system than (*), applied to the market model; along with price changes over time R the change in the state x is considered; in this case, the admissible trajectory satisfies certain differential inclusions of the form where K (p) and C (p) - many possible directions of change of ri NS, defined through a market model.

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

Related questions. M. e. closely related to many mathematical. disciplines. Sometimes it is difficult to determine where the boundaries between M. e. and math. 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-to-set mappings under the influence of the needs of M. e.

Lit.: Neumann J., Morgenstern O., Game theory and economic behavior, trans. from English., M., 1970; K and t about r about in and the p LV, Economic calculation of the best use of resources, M., 1959; Nikaido X., Convex structures and mathematical economics, trans. from English, M., 1972; M and to and r about in V. L., Rubinov A. M., Mathematical theory of economic dynamics and equilibrium, M., 1973; M and r to and B. G. N., The problem of group choice [information], M., 1974; Scarf H., The Computation of Economic Equilibria, L. 1973; Danzig 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.

Encyclopedia of Mathematics. - M .: Soviet encyclopedia... I. M. Vinogradov. 1977-1985.

• Economic Dictionary
• 
  

Mathematical economics is a theoretical and applied science, the subject of which is mathematical models of economic objects and processes and methods of their research.

The emergence of the mathematical sciences was undoubtedly associated with the needs of the economy. It was required, for example, to find out how much land to sow with grain in order to feed the family, how to measure the sown field and estimate the future harvest.

With the development of production and its complication, the needs of the economy in mathematical calculations also grew. Modern production is a strictly balanced work of many enterprises, which is ensured by the solution of a huge number of mathematical problems. A huge army of economists, planners and accountants is engaged in this work, and thousands of electronic computers carry out calculations. Among such tasks are the calculation of production plans, and the determination of the most advantageous location of construction projects, and the choice of the most economical transportation routes, etc. Mathematical economics is also engaged in a formalized mathematical description of already known economic phenomena, testing various hypotheses on economic systems described by some mathematical relations.

Let's consider two simple examples that demonstrate the use of mathematical models for this purpose.

Let the demand and supply of the goods depend on the price. For equilibrium, the price in the market must be such that the product is sold out and there is no surplus:

. (1)

But if, for example, the proposal is late by one time interval, then, as shown in Fig. 1 (which shows the curves of supply and demand as functions of price), when the price, demand exceeds supply. And since the supply is less than the demand, the price rises and the goods are bought up at the price. At this price, the supply rises to a value; now the supply is higher than the demand and the producers are forced to sell the goods at the price, after which the supply falls and the process repeats. The result is a simple business cycle model. Gradually, the market comes to equilibrium: demand, price and supply are set at the level.

Rice. 1 corresponds to the solution of equation (1) by the method of successive approximations, which determines the root of this equation, i.e. equilibrium price and the corresponding value of supply and demand.

Let's consider a more complex example - the “golden rule” of accumulation. The amount of output by the enterprise (in rubles) of the final product at a time is determined by labor costs, the productivity of which depends on the ratio of the degree of its saturation with equipment to labor costs. The mathematical notation for this is:

. (2)

The final product is allocated to consumption and storage of equipment. If we denote the share of output going to accumulation through, then

In economics, it is called the rate of accumulation. Its value is enclosed between zero and one.

Per unit of time, the volume of equipment changes by the amount of accumulation

. (4)

With balanced economic growth, all of its components grow with the same growth rate. Using the compound interest formula, we get:

, , , .

If we introduce the values ​​characterizing consumption, equipment volume and output per employee, then the system of relations (2) - (4) will go into the system

, , . (5)

The second of these ratios, at a given growth rate and consumption, will determine the capital-labor ratio as the intersection point of the curve and the straight line in Fig. 2. These lines will necessarily intersect, since the function, albeit monotonously, grows, which means an increase in output with an increase in the number of workers, but more and more gently, i.e. it is a concave function. The latter circumstance reflects the fact that an additional increase in equipment per worker, due to the increase in its load, becomes less and less effective ("the law of diminishing utility"). A family of curves corresponds to different values ​​of the accumulation rate. The length of the segment, as follows from formula (5), is equal to consumption. At (point in Fig. 2) there is no consumption at all - all production goes to the accumulation of equipment. Let us now reduce the rate of accumulation. Then consumption (length) will no longer be zero, although the growth rate of the economy (the slope of the straight line) remains the same. At the point with the ordinate, for which the tangent to the curve is parallel to the straight line, consumption is maximum. It corresponds to the curve of a family with a certain rate of accumulation, called the "golden rate of accumulation."

After the Olympiad, it is interesting to discuss problem solutions.

Comparison of theory and practice is not an easy problem in mathematical economics: it is extremely difficult to measure economic indicators - they are not measured on laboratory facilities, observations can be carried out extremely rarely (remember the censuses!), They are carried out in different conditions and contain a lot of inaccuracies. Therefore, it is difficult here to use the measurement experience gained in other sciences, and the development of special methods is required.

The development of mathematical economics has caused the emergence of many mathematical theories, united by the name "mathematical programming" (you can read about linear programming in the article "Geometry").

The problems of applying mathematical methods in economics were developed in the works of the Soviet mathematician L.V. Kantorovich, who were awarded the Lenin and Nobel Prizes.

First of all, it is necessary to consider the formulas for economics, which relate to supply and demand. The demand function equation can be represented as the following formula:

y = k * x + b

The demand function itself looks like this:

QD = k * P + b

Suggestion function:

Qs = k * P + b

If we consider the elasticity indicators, then we can distinguish formulas for economics that determine the price elasticity of demand:

EDP ​​= Δ QD (%): Δ P (%)

EDP ​​= (Q2 –Q1) / (Q2 + Q1): (P2 –P1) / (P2 + P1)

The second formula is the calculation of the midpoint, here the value of P1 is the price of the product before the change, P2 is the price of the product after the change, Q1 is the demand before the price change, Q2 is the demand after the price change.

The formula for the elasticity of demand in general:

EDI = (Q2 –Q1) / Q1: (P2 –P1) / P1

Macroeconomic formulas

Formulas for economics include formulas for microeconomics (supply and demand, firm costs, etc.), as well as formulas for macroeconomics. An important formula for macroeconomics is the formula for calculating the amount of money required in circulation:

KD = ∑ CG - K + SP - VP / CO

КД - the amount of money in circulation,

CG - the sum of prices for goods;

K - goods sold on credit;

SP - urgent payments;

VP - mutually redeemable payments under barter transactions;

CO is the annual rate of turnover of the monetary unit.

In order to determine the money supply in circulation, you must use the following formula:

M = P * Q / V

Here M is the money supply in circulation;

V is the velocity of money circulation;

Р - average prices for products;

Q is the number of products manufactured at constant prices.

The exchange equation can be represented by the following equality:

M * V = P * Q

This equation reflects the equality of total expenditures in monetary terms and the value of all goods and services that are produced in the state.

Other macroeconomic formulas

Consider a few more formulas in economics, among which the formula for calculating real income occupies an important place:

RD = ND / CPI * 100%

Here RD is real income,

ND - nominal income,

CPI is an indicator of the consumer price index.

The formula for calculating the consumer price index is represented by the following expression:

CPI = STTG / STBG

STTG - the cost of the consumer basket in the current year,

STBG - in the base year.

In accordance with the indicator of price indices, the inflation rate can be determined using the appropriate formula:

TI = (CPI1 - CPI0) / CPI0 * 100%

In accordance with the rate of inflation, several types can be distinguished:

1. Creeping inflation with a rise in prices up to 5% per annum,

2. Moderate inflation up to 10% per annum,

3. Galloping inflation with a rise in prices of 20-200% per annum,

4. Hyperinflation with catastrophic price increases of more than 200% per year.

Interest formulas

Economic calculations often require the calculation of interest. Formulas for economics include the calculation of both compound and simple interest. The formula for calculating simple interest is presented as follows:

C = P * (1 + in / 360)

Here P is the amount of debt, including interest;

С - the total amount of the loan;

n is the number of days;

i - annual percentage in shares.

The formula for calculating compound interest looks like this:

C = P (1 + in / 360) k

K is the number of years.

The formula for calculating compound interest, which is calculated over several years:

С = Р (1 + i) k

Formula of unemployment, employment and GNP

UB = Number of unemployed / HR * 100%

Here HR is the size of the labor force.

The formula for calculating the employment rate is as follows:

UZ = Number of employed / HR * 100%

The formula for calculating the gross national product is calculated as follows:

GNP =% + ZP + Tr + KNal - ChS + R + Am + DS

Here Tr are corporations,

Knal - indirect taxes,

PC - net subsidies,

Р - rent,

Am is the amount of depreciation,

DS - property income.

The formula for calculating GNP in accordance with costs:

GNP = LPR + GZ + VChVI - CHI

Calculation of revenue, profit and costs

Formulas for economics when calculating revenue and profit:

TR = P * Q

Profit = TR - TC

The formula for calculating average total costs looks like this:

AC = AFC + AVC or

AC = TC / Q

TC = TFC + TVC

Formula for calculating average fixed costs.

Subject and methods of economic theory

Economic relations permeate all spheres of human life. The study of their regularities occupied the minds of philosophers even in antiquity. The gradual development of agriculture, the emergence of private property contributed to the complication of economic relations and the construction of the first economic systems. Scientific and technical 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 the two directions, there are various solutions that can be applied in practice.

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

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

1. Methods for evaluating elements of the general, or generalizing individual structures. They are called methods of analysis and synthesis.
2. Induction and deduction make it possible to consider the dynamics of processes from the particular to the general and vice versa.
3. The systems approach helps to see a separate element of the economy, as a structure, and to analyze it.
4. In practice, the abstraction method is widely used. It allows you to separate the studied object or phenomenon from its interrelationships and external factors.
5. As in other sciences, economics often uses the language of mathematics, which helps to visually display the studied elements of the economy, as well as to analyze 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 direction of economic theory. Its main essence lies in the application of mathematical methods, means and tools for the description, study and analysis of economic systems. However, this discipline has its own specifics. She does not study economic phenomena as such, but deals with calculations related to mathematical models.

Remark 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. She studies, first of all, quantitative, 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 the construction and assessment of plans, forecasts, various kinds of events in dynamics.
• Analysis of mathematical and statistical values.

Applied Mathematics in Economics

In terms of its social significance, mathematical economics is close enough to mathematics. If we consider this discipline from the side of mathematical science, then for it it is an applied direction. Applied mathematics makes it possible to consider and analyze individual elements of the most complex economic systems, since it has a wide functionality based on fundamental mathematical knowledge. Such capabilities of mathematics contributed to the emergence of mathematical ecology, sociology, linguistics, financial mathematics.

Consider the most important mathematical methods used in the study of economic systems:

1. Operational 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.
2. Mathematical modeling includes a wide range of methods and tools that make it possible to solve the problems facing scientists and economists. The most commonly used are game theory, service theory, scheduling theory, and inventory theory.
3. Optimization in mathematics deals with the search for extreme values, both maximum and minimum. For these purposes, graphs of functions are usually used.

The above methods of mathematics make it possible to study statistical situations in the economy, or processes in short-term periods. As you know, at present, the main goal of economic agents is to find a long-term balance. An important factor in these studies is the time factor, which can be taken into account by applying the theory of probability, the theory of optimal solutions for calculations.

Remark 2

Thus, mathematics and economics are closely related to each other. It is customary to clothe the dynamics of economic structures in mathematical models, which can then be broken down 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 difficult action, since it is associated with the imperfection and incompleteness of the available information. The use of mathematical modeling makes it possible to reduce the riskiness of the management decisions taken.

Federal Agency for Education

State educational institution of higher professional education

Vladimir State University

A.A. Galkin

MATHEMATICAL

ECONOMY

Approved by the Ministry of Education and Science of the Russian Federation as a textbook

for students of higher educational institutions studying in the specialty "Applied Informatics (in Economics)"

Vladimir 2006

UDC 330.45: 519.85 BBK 65 V 631

Reviewers:

Doctor of Technical Sciences, Professor Head. Department of Automated Information and Control Systems, Tula State University

V.A. Fatuev

Doctor of Technical Sciences, Professor Head. Department of Information Systems

Tver State Technical University

B.V. Palukh

Doctor of Economics, Professor Head. Department of Economics and Management at Enterprises

Vladimir State University

V.F. Arkhipova

Doctor of Physical and Mathematical Sciences, Professor Head. Department of Algebra and Geometry, Vladimir State University

N.I. Dubrovin

Reprinted by the decision of the Editorial and Publishing Council of Vladimir State University

Galkin, A.A.

G16 Mathematical economics: textbook / A. A. Galkin; Vladim. state un-t. - Vladimir: Publishing house Vladim. state University, 2006 .-- 304 p. - ISBN 5-89368-624-1.

A wide range of typical optimization problems arising in economics and algorithms that allow solving these problems are considered. A methodology for formalizing these tasks and their classification are given. Methods for solving deterministic problems of static and dynamic optimization are presented. For each type of problem and algorithm, examples are given that demonstrate the technique of practical use of these algorithms, as well as a set of problems for independent solution.

Designed for university students studying in the specialty 080801 - applied informatics (in economics), as well as students, undergraduates and postgraduates of related specialties full-time, part-time education, persons receiving a second higher education, as well as practitioners.

Tab. 80. Ill. 60. Bibliography: 39 titles.

§ 6.8. The problem of the optimal phased distribution of allocated funds between enterprises during

§ 8.6. The problem of control of a linear dynamic system

with by minimizing the generalized quadratic integral

§ 9.2. Control of a linear discrete system of arbitrary order with optimization of the total generalized

List of accepted abbreviations

CF - target function ODR - area of ​​feasible solutions

LP - linear programming LPP - LP problem CLP - canonical LPP

TZ - transport task PO - points of departure, PN - points of destination in TZ

ISZU - method of the northwest corner ISM - method of the golden section DP - dynamic programming VI - calculus of variations PM - maximum principle; DE - differential equation

FOREWORD

V the preparation of students of various technical and economic specialties and directions takes a significant place in the study of mathematical models and methods typical for the corresponding subject area, which allow, operating with these models, to explain the behavior of the systems under consideration, evaluate their characteristics, reasonably make constructive, technological, economic, organizational and other decisions ...

The mastery of these models and methods builds on the foundation laid in a fairly universal classical discipline, usually called "Higher Mathematics". The mathematical apparatus that makes it possible to solve typical and most important problems for the corresponding field of applications is studied in special disciplines.

For students studying in the specialty "Applied Informatics (in Economics)", one of these disciplines is "Mathematical Economics". In accordance with the current state educational standard (SES), the program of this discipline includes a large amount of educational material related to mathematical calculations in the field of economics. This material is divided into two parts.

V the first part examines the problems of financial analysis, which were considered in the SES of the previous generation in a special discipline - "Financial mathematics".

The second part of the program contains, from the point of view of mathematics, more complex problems and methods related to finding the best ones, i.e. optimal solutions to various problems encountered in the field of applied economics. Previously, students mastered this material while studying the discipline "Theory of optimal control in economic systems."

The curriculum of the discipline "Mathematical Economics" contains a wide range of rather difficult questions to study. Since the amount of time allocated for classroom studies in this discipline is quite small, the independent work of students with educational literature is of particular importance.

It should be noted that over the past 30 years, many different monographs, textbooks and teaching aids on mathematical methods used in economics have been published in our country. However, when working with them, students have serious difficulties. First, many of these books are now practically inaccessible to students, since they are either absent in the libraries of universities, or are available in single copies. Secondly, to study all the material provided by the program, one textbook is not enough, and different books, as a rule, use a different style of presentation, different designations. Quite often the level of presentation of the material is not available to a “real” student. Thirdly, when organizing the educational process in disciplines of a mathematical nature, it is fundamentally important for students to acquire practical skills in using the methods studied, and this requires tasks for independent solution. Most of the textbooks on the topic under consideration contain examples and tasks to illustrate the technique of applying the methods outlined, but they are not enough to give all students of a regular study group individual tasks.

The proposed textbook is intended to study the second, more complex part of the discipline "Mathematical Economics", which considers optimization problems arising in economics and algorithms for their solution. It has been prepared in the light of the above circumstances.

The book contains the formulations of typical optimization problems arising in the economic sphere, their formalization is carried out, the essence of methods and algorithms is set forth, which allow performing a solution with an illustration of the technique of these algorithms using specific examples. In addition, for each topic, a sufficiently large set of tasks for independent solution is presented, which allows each student to give his own individual task.

From a huge variety of possible optimization problems and methods proposed by modern science, deterministic problems and algorithms for static and dynamic optimization are selected for inclusion in this textbook. Due to the limited volume of the book, optimization problems with uncertainties, including probabilistic-statistical, interval, fuzzy and other problems and models, as well as vector optimization problems, are not considered.

The book includes nine chapters. The first section gives examples of optimization problems of an economic nature, which demonstrate the formalization technique, i.e. obtaining a mathematical model of the problem being solved, the classification of optimization problems is given.

Chapters two, three and four are devoted to linear static optimization problems. In the second chapter, problems and methods of linear programming are presented, separately in the third, transport problems are considered, and in the fourth, optimization problems are interpreted on graphs. For each problem, the most effective method (algorithm) for solving is presented and an example is given that demonstrates the technique of practical use of this algorithm. The fifth chapter presents analytical and numerical methods for solving nonlinear static optimization problems in the absence and presence of constraints.

Dynamic optimization problems, commonly called optimal control problems, are discussed in chapters six through nine. In the sixth chapter, a general idea of ​​dynamical systems of continuous and discrete type is given, the classical problem of optimal control and dynamic programming (DP) is formulated, the essence of DP is stated, and the technique of its practical application is shown using various examples of an economic nature. The seventh chapter sets out the foundations of the calculus of variations, the eighth - the maximum principle for continuous systems, and the ninth - for discrete systems. In each of these chapters, much attention is paid to the analysis of various particular problems and examples that illustrate the methodology for the practical use of calculated ratios.

At the end of each of the chapters from the first to the sixth there are tasks for independent solution. At the end of the ninth chapter, problems are given for independent solution, devoted to the methods of optimal dynamic control.

A special problem, for the solution of which the author took considerable efforts in the process of working on the book, was the fact that some methods and algorithms in the original literature are presented in such a way that it is rather difficult for students of a non-mathematical and information-economic profile to understand them. Therefore, it was necessary to find opportunities for adapting the relevant theoretical material to the real level of training of the students to whom the book is oriented.

In addition, the author strove, when presenting a large number of significantly different tasks and methods, to maintain as much as possible a single style, character, system of presentation of the material. I would like to hope that this has been accomplished to a certain extent.

In preparing the textbook, the material of lectures and practical classes on the disciplines "Optimization Methods", "Control Theory", "Theory of Optimal Control in Economic Systems" and "Mathematical Economics" was used, which the author taught for 25 years at the Vladimir State University (VlSU) ... In these lessons, most of the theoretical material and tasks for independent solution were tested. The electronic version of the textbook is included in the information resources of the VlSU electronic library.

Despite the fact that the textbook was prepared for students of the specialty "Applied Informatics (in Economics)", undoubtedly, it can be useful for students, undergraduates, graduate students and specialists of other profiles, since optimization problems arise everywhere. It is no accident that they say that "there is nothing in nature in which it would be impossible to see the meaning of any maximum or minimum."

He will be grateful to all those who take advantage of the book and give their opinion on its content, possibly about shortcomings or inaccuracies. To do this, you can use e_mail: [email protected].

The work on the book, with some interruptions, was carried out for about 10 years, but it could drag on indefinitely, if not for the prompt and highly qualified assistance in the work on the manuscript, which was provided by graduate student I.V. Camp. For this, the author expresses special gratitude to her.