Subject and Methods economic theory

Economic relations permeate all spheres of human life. The study of their patterns occupied the minds of philosophers in antiquity. gradual development Agriculture, the emergence of private property contributed to the complication of economic relations and the construction of the first economic systems. Scientific - technical progress, which determined the transition from manual labor to machine labor, gave a strong impetus to the consolidation of production, and hence 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 trend towards economics 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 investigate 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 study of economic theory is the search for optimal solutions for economies of various levels of organization in terms of meeting the growing demand under the condition of limited resources. Economists use a variety of methods in their research. Among them, the most commonly used are the following:

1. Methods that allow you to evaluate the elements of the general, or generalize 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. A systematic approach helps to see a separate element of the economy as a structure and analyze it.
4. In practice, the abstraction method is widely used. It allows you to separate the studied object or phenomenon from its relationships and external factors.
5. As in other sciences, the language of mathematics is often used in economics, which helps to visualize the studied elements of the economy, as well as to analyze or form the necessary forecast of trends.

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 area of ​​economic theory. Its main essence lies in the application of mathematical methods, means and tools for describing, studying and analyzing economic systems. However, this discipline has its own specifics. She doesn't 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 for practical problems. It studies, first of all, quantitative, qualitative indicators, as well as the behavior of economic agents in dynamics.

Challenges ahead mathematical economics, are as follows:

• Construction of mathematical models describing processes and phenomena in economic systems Oh.
• Study of the behavior of various subjects of economic relations.
• Implementation of assistance in the construction and evaluation of plans, forecasts, various kinds of events in dynamics.
• Carrying out the analysis of mathematical and statistical values.

Applied Mathematics in Economics

Mathematical economics in its social significance 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 possibilities of mathematics contributed to the emergence of mathematical ecology, sociology, linguistics, and 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 application in practice 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, function graphs are usually used.

The methods of mathematics listed above 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 entities is to find a long-term equilibrium. Important in these studies is the time factor, which can be taken into account by applying probability theory, the theory of optimal solution, for calculations.

Remark 2

Thus, mathematics and economics are closely related to each other. It is customary to dress the dynamics of economic structures in mathematical models, which can then be divided into separate subtasks and all possible methods can be applied. economic analysis, as well as mathematical calculations. Decision-making in the economic sphere is a rather complicated 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 managerial decisions.

FEDERAL AGENCY FOR EDUCATION

STATE TECHNICAL UNIVERSITY

__________________________________________________________________

Department of Information Systems

MATHEMATICAL ECONOMICS

Lecture notes

For third-year students of the specialty

"Applied Informatics (in Economics)"

Tver 2009

1. Assessment methods investment projects

At present, in developed countries market economy in the analysis of investment projects, the discounting technique based on the logic of compound interest began to be widely used. Therefore, this section provides the essence and advantages of using these methods.

^ 1.1 Net present value method

Net present value is calculated as the difference
discounted to one point in time streams of income and expenses
by project:

where CF INt - cash inflow for period t;

CF OFt - cash outflow for period t;

R - discount rate;

N - project life cycle.

In cases where the investment is a one-time investment in the initial period, the NPV calculation formula will look like this:

where C 0 - capital investments in the zero period.

Using this criterion when making decisions is quite simple. A positive NPV value indicates the amount of income that the investor will receive in excess of the required level. In the case when the NPV is equal to zero, the investor not only returns his capital, but also increases it by the amount specified by the discount rate. The resulting negative NPV indicates that the project should be rejected.

It should be noted that NPV is additive over time. This property allows you to summarize today's net values various projects, which is very important when analyzing the optimality of the investment portfolio.

^ 1.2 Method for calculating the return on investment index

The profitability index is the ratio of discounted profits and project costs. That is, in relation, for example, to one-time investments, the calculation is made according to the formula:

In the case when the value of PI>1, the project is profitable. If PI<1, то от инвестирования следует отказаться. Значение индекса рентабельности, равное единице, говорит о том, что проект и ни прибыльный, и ни убыточный.

The advantage of this indicator over the NPV indicator is that it is relative. Therefore, it is easy to use when it is necessary to select one project from a number of alternative ones with approximately the same NPV values, as well as when forming an investment portfolio with the maximum total NPV value.

Such a task arises when there are several attractive investment projects to choose from, but due to limited financial resources, the investor cannot participate in all projects at the same time. Then PI is calculated for each project and the projects are ranked in descending order of PI. The investment portfolio includes the first m-projects, which in total can be fully financed.

If the next project lends itself to splitting, then it is also included in the portfolio in that part of it that can be financed.

^ 1.3 Method for calculating the rate of return on an investment

The rate of return (internal rate of return) is the value of the interest rate at which the net present value of the project is zero:

where IRR is the rate of return (internal rate of return).

The IRR value shows the maximum allowable relative level of costs that in one way or another can be associated with the project in question. For example, if the project is fully financed by a loan, then the IRR value will show the upper limit of the bank interest rate, the excess of which will make the project unprofitable.

To determine the IRR, either calculation or calculation-graphic methods are used. In the first case, the annual cash flows (taking into account the necessary capital investments) are discounted at various trial discount rates in increments of one percent. This will produce a set of corresponding net present values, the smallest positive value of which will indicate the exact rate of return to be taken into account.

The use of the calculation and graphical method boils down to the fact that rates of return are plotted along the vertical axis on the coordinate system, and net today's values ​​are plotted along the horizontal axis. Then two NPV values ​​are calculated corresponding to any two rates of return. A straight line is drawn between these two points, the point of intersection of which with the vertical axis is the estimated internal rate of return. However, it should be noted that the obtained value must be checked for zero, and if necessary, an adjustment should be made.

^ 1.4 Method for determining the discounted payback period

The discounted payback period is the period of time during which the investor fully returns his initial costs, while ensuring the required level of profitability:

where T is the discounted payback period;

PV is the present value of the investment.

This method is one of the simplest and most widely used, but is usually used to obtain additional information about the project in cases where the main thing is that the investment pays off as soon as possible. In addition, the method is also convenient when analyzing projects with a high degree of risk, since the shorter the payback period, the less risky the project is.

^ 2. Features of the application of methods for evaluating investment projects

The methods described above are fair in their entirety when analyzing independent investment projects. That is, the criteria of these methods only then will not conflict with each other.

When analyzing competing projects, a different situation arises, the importance of considering which is due to the desire to increase competition between enterprises in order to reduce the cost of projects through the use of internal reserves of companies. In addition, such a situation may arise under severe financial constraints.

Consider two projects competing with each other. Calculate the net present value of projects, as well as their internal rate of return, provided that the discount rate is 11%.

Table 1

PROJECT	СF by years (million rubles)					NPV at r=11%	IRR
	0	1	2	3	4
X1	-50	0	0	15	110	33,5	26,7%
X2	-50	40	15	15	20	22,4	35,0%

As can be seen from Table 1, the NPV of the X1 project will be 33.5 million rubles, which is clearly preferable to the NPV of the X2 project - 22.4 million rubles. However, if we focus on the internal rate of return, then preference should be given to the X2 project with IRR = 35% versus 26.7% for the X1 project. Thus, the NPV and IRR criteria are in conflict with each other, despite the fact that both methods are based on the same formula.

The problem that has arisen is easily solved if we consider in more detail the essence of the IRR criterion, the calculation of which provides for the possibility of reinvesting the intermediate income of the project, providing a return equal to IRR. But is it realistic to ensure such a return if the return on reinvestment is less than the IRR? As further consideration of the example will show, no.

Let's calculate the absolute value of the investor's income at the end of the fourth year, or, in other words, the future value of projects (future value), provided that the reinvestment rate is 11%:

FV (X1) \u003d 110 + 15 * (1 + 0.11) \u003d 126.65 million rubles,

FV (X2) \u003d 20 + 15 * (1 + 0.11) + 15 * (1 + 0.11) 2 + 40 * (1 + 0.11) 3 \u003d 109.84 million rubles.

Let us determine the profitability of this operation, based on the following dependence:

A number of researchers, taking into account the shortcomings of the IRR criterion, have proposed using another criterion instead - MIRR (modified IRR). MIRR is the expected return, provided that all intermediate income of the project is reinvested at a given rate of return.

table 2

As can be seen from Table 2, the use of the MIRR criterion removes the contradiction between the absolute and relative indicators of the result of the project implementation. Now the question is removed: preference should be given to the X1 project. In addition, in the future, when comparing two competing projects, NPV should be considered the best criterion.

The examples given were based on the contradiction between the NPV and IRR criteria when analyzing projects with the same amount of capital investment. Therefore, it is also necessary to consider an example of the analysis of competing projects with different investment volumes.

Table 3

PROJECT	СF by years (million rubles)					NPV (r=11%)	IRR	MIRR (r=11%)
	0	1	2	3	4
X3	-5	4,5	2,2	2,5	2,5	4,3	54%	29,82%
X2	-50	40	15	15	20	22,4	35%	21,74%

An analysis of the data presented in Table 3 shows that the IRR and MIRR criteria point to the X3 project, while the NPV criterion, which is taken as the main one in the previous example, is clearly on the side of the X2 project. That is, in this situation, the problem of disproportionate projects arose (the problem of scale). Therefore, the final decision here can be made only after analyzing the possible embedding of the difference between CFo (X3) and CFo (X2). In our example, this difference is 45 million rubles.

Suppose we have the opportunity to invest these funds in the following way:

Table 4

PROJECT	СF by years (million rubles)					NPV (r=11%)	IRR	MIRR (r=11%)
	0	1	2	3	4
X4	-45	36	13	13	18	19,3	34%	21,38%

Now it is necessary to find out what is preferable - the X3 and X4 projects or the X2 project?

Table 5

PROJECT	СF by years (million rubles)					NPV (r=11%)	IRR	MIRR (r=11%)
	0	1	2	3	4
X3+X4	-50	40,5	15,2	15,5	20,5	23,7	36%	22,30%
X2	-50	40	15	15	20	22,3	35%	21,74%

Considering the results reflected in Table 5, it becomes quite clear that the investor will reject the X2 project in favor of the implementation of the two projects X3 and X4. At the same time, it should be noted that the final choice will still be the X1 project:

Table 6

PROJECT	СF by years (million rubles)					NPV (r=11%)	IRR	MIRR (r=11%)
	0	1	2	3	4
X3+X4	-50	40,5	15,2	15,5	20,5	23,7	36%	22,30%
X1	-50	0	0	15	110	33,5	26,7%	26,16%

However, there may be situations when, apart from the X3 and X4 projects, there are no more projects with a positive NPV. In this case, it is necessary to focus not on the rate of return, but on NPV.

It should be noted that the problem of scale can also arise in the case of NPV - PI. In this case, the solution method will be similar.

Thus, we can draw the following conclusion: it is desirable to analyze investment projects by several methods at once, which will allow obtaining additional important information about them.

^ 3. Accounting for inflation in the analysis of projects

The effect of inflation can be taken into account by adjusting either future receipts or a discount rate for its index. In this case, it is advisable to use the following dependence:

Where r nom is the nominal interest rate;

R real - real interest rate;

λ is the general level of inflation.

For small values r And λ formula (7) can be written as follows:

R nom ≈ r eal + λ (8)

Both nominal and real interest rates can be used as the discount rate. The choice depends on how the project's cash flow is measured. If the cash flow is presented in real terms (at constant prices), then the real interest rate should be used for discounting.

However, using real interest rates and calculating cash flows at constant prices does not allow for structural inflation. In such cases, the calculation must be carried out at current prices:

In the latter case, however, the ability to predict price increases is required.

^ 4. Accounting for risk in the analysis of a single project

A risk-based analysis of a single project is carried out only if the investment project is independent. In this case, it is quite sufficient to use two indicators: the expected return and the standard deviation (RMS) of the return, which completely determine the normal distribution.

The expected return is calculated as follows:

(11)

where R i - yield on the i-th scenario;

P i - probability of development of events according to the i-th option;

N is the number of options considered.

Thus, it is clear that the expected return is the most likely return on the project, while the standard deviation, which measures the variance of the expected return, is an indicator of the risk of the project:

When comparing risks for assets with different expected returns, it is advisable to use the coefficient of variation (that is, a measure of relative dispersion):

(13)

Obviously, the higher the SD and CV, the higher the risk. As an example, consider the random sample data presented in Table 7:

Table 7

Project	R		CV
X1	12,5%	3,12	0,25
x2	11,0%	3,32	0,30
X3	12,2%	2,68	0,22

In this example, the X2 project is the least profitable and at the same time the most risky, therefore, it should be rejected immediately, and the further choice will depend on the investor's attitude to risk. If it is negative, the XZ project will be implemented. If the investor is risk averse, XI will be preferred.

Practice shows that investors at the level of municipal officials try to choose the minimum risk. Thus, in our case, the KhZ project will be accepted for investment.

^ 5. Accounting for risk in portfolio analysis

Usually, in order to reduce the non-systematic part of the risk, diversification is used, which is based on the creation of an effective portfolio by analyzing the correlation of its assets. At the same time, it should be noted that each new investment here should be considered taking into account the current portfolio.

Let us consider the methodology for calculating the risk of a portfolio consisting of three projects, using the data presented in Table 7 as an example, and also under the condition that each project receives a third of the invested amount.

The portfolio return will be determined as follows:

(14)

Where R k is the expected profitability of the k-th project;

X k - share of funds invested in the k-th project;

M - the number of projects in the portfolio.

In our example:

R portfolio = 12,5 1 / 3 + 11 1 / 3 + 12,2 1 / 3 = 11,9%.

In our example:

cov 12 = 7.34 and cov 13 = – 8,12.

Thus, it is obvious that the returns of projects X1 and X2 change in the same direction, and the returns of projects X1 and X3, as well as X2 and X3 - in the opposite direction. However, since the absolute value of covariance is difficult to interpret, the degree of interdependence between indicators is calculated using the correlation coefficient:

At r = +1, the indicators change over time in exactly the same way, at r = -1 there is a completely negative correlation, zero indicates the absence of a relationship.

In this example:

r 12 = 0.71, r 13 = -0.96 and r 23 = -0.6.

Obviously, in order to reduce the risk, a combination of a portfolio of projects X1 and X3 would be most appropriate. At the same time, however, it is necessary to calculate the portfolio risk itself, taking into account the correlation between projects:

Calculate the portfolio risk (X1, X3) under the condition of equal equity investment:

.

Thus, the risk of our portfolio is significantly lower than the risks of its constituent projects, and at r< 0 диверсификация всегда будет приводить к подобным результатам. Однако при 0 < r < 1 также можно сократить риск, причем при определенных значениях r риск портфеля может оказаться ниже самого рискованного его актива.

The methodology for compiling a portfolio of multiple projects is the same as for compiling a two-asset portfolio.

From the entire set of portfolios indicated by the area in Fig. 1, it is necessary to select those portfolios that are on the AB line - they are the ones that provide the minimum risk with the highest expected return. In this case, the specific choice among them depends on our attitude to risk. Graphically, the choice between risk and return is expressed by indifference curves, a unique set of which exists for each individual in terms of that person's preferences for risk and return.

Fig.1 The problem of choosing the optimal portfolio.

The straight line from the return point on a risk-free asset through the tangent point of the possible portfolio curve AB is called the Capital Market Line (CML) and reflects the choice in the risk-return system. Point C in fig. 1 thus reflects the risk and return of the market portfolio. The highest level of utility is achieved by the investor at the point where his indifference curve to risk and return touches the line of the capital market. If the investor prefers certainty, then this point will be located to the left of the market portfolio (to the left of C); the investor invests in both risk-free and risky assets, and his portfolio, as a result, has a low risk and low return. If the investor is more risk averse, the touch point will be to the right of the market portfolio (to the right of C); funds are invested in riskier assets and the portfolio has more risk and higher returns.

The problem of finding an optimal portfolio consisting of many assets can, in principle, be solved by the selection procedure - we are looking for a portfolio with the highest expected return for a given risk level. However, in practice, it is expedient to solve the problem of capital allocation using a quadratic version of linear programming.

Let's determine the share of the i-th asset in the portfolio by costs:

where CF OFt max is the maximum allowable size of the investment program for period t.

Consider the summary risk indicator:

The objective function (20), which minimizes the risk of the final portfolio, where the binary variable X i acts as a criterion for participation in the portfolio, the unit value of which indicates the entry of the i-th project into the portfolio, and the zero value indicates the refusal of the i-th project to invest, looks like in the following way:

with restrictions:

where NPV min is the size of the minimum acceptable net present value of the portfolio;

T n - the initial period of the investment program;

T to - the final period of the investment program;

V k - vector of competing projects;

V - set of vectors of competing projects;

N l - the number of projects of the previous portfolio, T to which exceeds T n of the portfolio being compiled.

Obviously, when calculating the objective function (20), only that part of the variance-covariance matrix (19) is used, which is located on and below the main diagonal, which is caused by the application of the restrictive condition in the nested loop but to the columns, while, since there are two covariances for for each possible pair of projects, a doubling factor is introduced for the values ​​of the nested loop.

Thus, the task of optimization is to determine which projects should be accepted for investment so that the amount of expected income and the level of risk optimally correspond to the goals of the investor, which are determined by the direction of the objective function and a set of restrictions:

1. The risk measured by dispersion (RMS) of the portfolio is minimized.

2. The income from the portfolio, equal to the additive indicator of the expected net present value of the accepted projects, must not be less than the required amount, given by the value discounted to the initial investment period.

3. The total volume of annual investments cannot exceed the limits of available (allocated) funds established for a given period of time separately for each year of the investment program.

4. Only one of the projects representing the same group of competing projects can be included in the portfolio.

5. The compilation of a new portfolio is carried out taking into account the mandatory inclusion in its composition of those projects of the previous portfolio, the completion period of the investment program for which exceeds the period of the start of the investment program of the new portfolio.

6. The considered projects are not subject to crushing.

The described problem includes a number of restrictions in the form of inequalities, which mainly set limits for investing in certain areas. Otherwise, it is impossible to guarantee that the resulting solution will be on the efficiency frontier. In doing so, we may end up with a riskier portfolio, but we will not need to use all of our money and/or we will be able to earn a higher return.

Calculation and issuance of the resulting characteristics of the portfolio:

Many selected projects:

Expected net present value of the portfolio:

Expected portfolio return:

Project portfolio risk:

Saving financial resources:

There are various definitions of the concept of "risk", therefore, summarizing the above, we will understand the risk as a situation where there are several possible results of certain actions, and there are also necessary data from past periods that make it possible to calculate some dependencies to predict possible future results.

The CAPM (capital asset pricing model), widely used for portfolio compilation, developed by W. Sharp, proceeds from the fact that it is important to take into account only the systematic risk of each individual asset. However, the works of G. Markowitz proved the importance of taking into account the overall risk as a whole. Therefore, the previous reasoning was based precisely on this premise.

Systematic risk is caused by factors such as inflation. economic crisis, other general market factors.

The presence of non-systematic risk is associated with random events that affect specific assets or companies.

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Software and information support

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   Microsoft Office 2000: Microsoft Excel.
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Magnitogorsk 2005

Collection of tasks for the course "Mathematical Economics". - Magnitogorsk: MaGU, 2005. - 184 p.

The collection provides an overview of the key categories and provisions used in the course "Mathematical Economics". Examples of solving typical problems are presented, questions for self-examination on the studied material 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, scientific and practical workers 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. Determination of rates and calculation of interest 7

1.2. Simple interest rate 10

1.3. Simple discount rate 21

1.4. Loan repayment and depreciation 32

1.5. Calculating Averages 41

1.6. Currency settlements 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 73

2.2. Compound discount rate 91

2.3. Continuous rate 101

2.4. Equivalence rates 107

2.5. Inflation and Compound 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. Annuity assessment with a period of more than a year 157

INTRODUCTION

"Mathematical economics" is the name of the discipline coined by mathematicians. Economists prefer another name - "Economic and Mathematical Models and Methods". This name is often found in the curricula and standards of economic faculties. In our opinion, these two names equally accurately convey the inner content of the subject, where economic and mathematical aspects are harmoniously combined. Unfortunately, in practice, the EMMM course program is often composed entirely of separate sections "Operations Research and Mathematical Programming", which, firstly, have already been completed before this course, and secondly, contain mathematical models of decision making and optimization, rather than economics. - mathematical models as such.

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

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

Like other sciences that study the economy 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 lies in the fact that it studies not the economic objects and phenomena themselves as such, but their mathematical models. Its goal is to obtain objective economic information and develop recommendations of great practical importance. Formally, mathematical economics can be attributed to both economic and mathematical sciences. In the first case, it should be understood as that section of the economy that studies quantitative and qualitative categories, as well as the behavioral aspects of economic entities. Considering mathematical economics as one of the areas of mathematics, it can be attributed to those sections of applied mathematics that deal with optimization problems and decision-making problems.

By its nature, economics is the closest social science 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 a social science about the use of limited resources in order to maximize the satisfaction of the unlimited material needs of the population. The central problems of economic science - the rational management of the economy, the optimal distribution of limited resources, the study of economic management mechanisms, the 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 main questions of economics:

what should be produced (i.e. what goods and services and in what quantity should be produced)?

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

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

Finally, the task of economic theory, connected with bringing into the system, interpretation and generalization of the behavior of economic participants in the process of production, exchange and consumption, goes back to the mathematical problems of optimization and decision making.

In view of 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 particular problems of the economy under various conditions, prerequisites and at various levels);

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

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

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

The collection provides an overview of the key categories and provisions used in the course "Mathematical Economics". Examples of solving typical problems are presented, questions for self-examination on the studied material 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, scientific and practical workers 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.

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

The emergence of the mathematical sciences was undoubtedly connected 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 provided by the solution of a huge number of mathematical problems. This work is occupied by a huge army of economists, planners and accountants, and the calculations are carried out by thousands of electronic computers. Among such tasks are the calculation of production plans, and the determination of the most advantageous location of construction sites, 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 certain mathematical relationships.

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

Let the demand and supply of goods depend on the price. For equilibrium, the market price 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 (where demand and supply curves are shown as functions of price), at price demand exceeds supply. And since supply is less than demand, the price rises and the goods are bought up at a price of . At this price, supply rises to ; now supply is higher than demand and producers are forced to sell the goods at a price, after which supply falls and the process repeats. The result is a simple business cycle model. Gradually, the market comes into balance: 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.

Consider a more complex example - the "golden rule" of accumulation. The value of the enterprise's output (in rubles) of final products at a point in time is determined by labor costs, the productivity of which depends on the ratio of the degree of saturation of its equipment to labor costs. The mathematical notation for this is:

. (2)

The final product is distributed to the consumption and accumulation of equipment. If we denote the share of output that goes to accumulation through , then

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

For a unit of time, the volume of equipment changes by the amount of accumulation

. (4)

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

, , , .

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

, , . (5)

The second of these ratios, given growth rates and consumption, will determine the capital-labor ratio as the point of intersection of the curve and the straight line in Fig. 2. These lines will definitely intersect, since the function , albeit monotonously, grows, which means an increase in output with an increase in the labor force , but more gently, i.e. it's a concave function. The latter circumstance reflects the fact that an additional increase in equipment per worker, due to an increase in its workload, becomes less and less effective (“the law of diminishing utility”). The 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 already be non-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, the consumption is maximum. It corresponds to a family curve with a certain accumulation rate, called the "golden accumulation rate".

LEONID VITALIEVICH KANTOROVICH (1912-1986) L. V. Kantorovich - Soviet mathematician and economist, creator of linear programming and the theory of optimal planning of the socialist economy, academician, Nobel Prize winner. L. V. Kantorovich was born in St. Petersburg, in the family of a doctor. His abilities manifested unusually early. Already at 4 goals, he freely operated with multi-digit numbers, at the age of seven he mastered a chemistry course according to his older brother's textbook. At the age of 14, he became a student at St. Petersburg University. By the time he graduated from the university, in 1930, L. V. Kantorovich was already a well-known scientist, the author of a dozen papers published in leading international mathematical journals, and at the age of 20 he was a professor of mathematics. In 1935, the scientist introduced and studied a class of function spaces in which an order relation is defined for a certain set of their elements. The theory of such spaces, they are called Kantorovich spaces, or -spaces, is one of the main sections of functional analysis. Recent work related to the solution of the continuum problem has determined the place of -spaces among the most fundamental mathematical structures. L. V. Kantorovich was distinguished by his amazing ability to see the core of the problem in a particular problem and, having created a theory, to give a general method for solving a wide class of similar problems. This was revealed especially clearly in his works on computational mathematics and mathematical economics. In the early 30s. L. V. Kantorovich was one of the first prominent scientists to become interested in computational mathematics. The modern appearance of this science was largely determined by his works. Among them are the fundamental and classic monograph "Approximate Methods of Higher Analysis"; computational methods bearing his name; general theory of approximate methods, built on the basis of functional analysis (State Prize, 1949); works on automatic programming, performed at the dawn of the computer era and anticipating many modern ideas, and finally, a number of inventions in the field of computer technology. In 1939, a small pamphlet "Mathematical Methods of Organization and Planning of Production" was published in Leningrad, which actually contained a new section of applied mathematics, later called linear programming (see Geometry). The reason for writing it was a specific production task. Realizing the key importance of the concepts of variance and optimality in a socialist economy, such important indicators as price, rent, efficiency, he proceeds to develop the theory of optimal planning, which was subsequently awarded the Lenin (1965) and Nobel (1975) prizes. The book "Economic Calculation of the Best Use of Resources", expounding this theory, was written under the conditions of the Leningrad blockade and completed already in 1942. Understanding the exceptional importance of these studies, the scientist persistently pursued the practical use of their results. However, the work was not published until 1959 and even then was attacked by orthodox political economists. L. V. Kantorovich's book shaped the views of a whole generation of Soviet economists. Many of the ideas first expressed there are being implemented in the course of perestroika.

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

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

The development of mathematical economics caused the emergence of many mathematical theories, united by the name "mathematical programming" (for linear programming, see the article "Geometry").

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

The main purpose of the economy- providing society with consumer goods. There are stable quantitative patterns in the economy, so their formalized mathematical description is possible.

An object studying the discipline - economics and its divisions.

Subject - mathematical models of economic objects.

Method - system analysis of the economy as a complex dynamic system.

Model - this is an object that replaces the original, reflects the most important features and properties of the original for this study.

A model that is a set of mathematical relationships is called mathematical.

SIMULATION ELEMENTS

System is a set of interrelated elements that jointly realize certain goals.

Supersystem - the environment surrounding the system in which the system operates.

Subsystem - a subset of elements that implement goals consistent with the goals of the system (a subsystem can implement part of the goals of the system).

The economic system: allocates resources, produces goods, distributes commodities, and accumulates.

Supersystem of the national economy- nature, world economy and society.

The main subsystems of the economy- production and financial-credit.

FEATURES OF THE ECONOMY AS A MODELING OBJECT

In economics, models similar to technical ones are impossible, because it is impossible to build an exact copy of the economy and work out options for economic policy on this copy.

Experimentation is limited in the economy, since all its parts are tightly interconnected with each other.

Direct experiments with the economy have both positive and negative sides.

Positive side- the short-term results of the economic policy pursued are immediately visible.

Negative side- it is impossible to directly predict the medium and long-term consequences of the decisions made.

Thus, in order to develop correct economic decisions, it is necessary to take into account both all past experience and the results obtained in calculations using mathematical models that are adequate to the given economic situation.

The development of mathematical models is laborious, but very promising. Thus, Keynes's model, which reflects the ability of a market economy to adapt to disturbing influences, was built under the impression of the crisis of 1929-1933. However, the application of this model to overcome the post-war crisis in Germany and Japan was very successful and was called the "economic miracle".

LET'S CONSIDER THE STRUCTURE OF THE ECONOMY AS AN OBJECT OF MATHEMATICAL MODELING

The economy is a complex system consisting of production and non-production (financial) cells (economic units) that are in production, technological and (or) organizational and economic relations with each other.

In relation to the economic system, each member of society plays a dual role: on the one hand, as a consumer, and on the other, as a worker.

In addition to labor, material resources are natural resources and means of production.

All branches of material production create a gross domestic product (GDP).

IN natural-real form of GDP - these are the means of labor and consumer goods,

In value form - the fund for compensation for the retirement of fixed assets (depreciation fund) and the newly created value (national income).

In the process of creating GDP, an intermediate product is produced and re-consumed.

By tangible the composition of the intermediate product - these are the objects of labor used for current production consumption, their value is entirely transferred to the cost of the means of labor or commodities included in GDP.

USE OF MATHEMATICS IN ECONOMY ALLOWS:

1. highlight and formally describe the most important relationships of economic variables and objects;

2. gain new knowledge about the object;

3. evaluate the type of dependencies of factors and parameters of variables, draw conclusions.

WHAT IS AN ECONOMIC AND MATHEMATICAL MODEL?

This is a simplified formal description of economic phenomena.

The mathematical model of an economic object is its display in the form of a set of equations, inequalities, logical relations, graphs.

Models make it possible to identify the features of the functioning of an economic object and, on this basis, predict the behavior of an object in the future when parameters change.

MODEL BUILDING STAGES:

1. the subject and goals of the study are formulated;

2. in the economic system, structural or functional elements corresponding to this goal are distinguished;

3. the most important qualitative characteristics of these elements are revealed;

4. verbally, qualitatively describes the relationship between the elements;

5. symbolic designations are introduced for the characteristics of an economic object and the relationships between them are formulated;

6. calculations are carried out according to the model and the results obtained are analyzed;

MODEL STRUCTURE:

To build a model, it is necessary to define exogenous and endogenous variables and parameters.

exogenous variables– are set outside the model, i.e. known at the time of calculation.

endogenous variables– are determined in the course of calculations according to the model.

Parameters are the coefficients of the equations.

CLASSES OF ECONOMIC AND MATHEMATICAL MODELS

Economic and mathematical models are divided into the following classes:

1. By level of generalization

a. Macroeconomic - describe the economy as a whole, link aggregated indicators: GDP, consumption, investment, employment. Macromodels reflect the functioning and development of the entire economic system or its fairly large subsystems. In macromodels, economic cells are considered indivisible.

b. Microeconomic - describe the interaction of structural and functional components of the economy. Micromodels - the functioning of economic units and their associations. In micromodels, an economic unit can be considered as a complex system.

2. By abstraction level

a. Theoretical - allow you to study the general properties of the economy by deriving from formal premises. Used to study the general properties of the economy and its elements (supply and demand models)

b. Applied - make it possible to evaluate the parameters of the functioning of a particular economic object and develop recommendations for decision-making. Used to assess the parameters of specific economic objects. This includes econometric models that apply the methods of mathematical statistics.

3. Models of equilibrium and growth

a. Equilibrium - descriptive (descriptive) models. They describe such a state of the economy, when the resultant of all forces seeking to bring the economy out of this state is equal to zero. An example is the Leontief model (input-output),

b. Growth models are designed to determine how the economy should develop under certain criteria. Example - Solow, Samuelson-Hicks Model

4. By taking into account the time factor.

a. Static - describe the state of an object at a particular moment or period of time.

b. Dynamic - include the relationship of variables over time. Usually use the apparatus of differential equations.

5. By taking into account the factor of chance.

a. Deterministic - imply rigid functional relationships between model variables.

b. Stochastic - allow random effects on indicators and use probability theory and mathematical statistics.

TEST QUESTIONS

1. What is economic and mathematical modeling? Its place in economic analysis and forecasting.

2. Stages of modeling. model factors.

3. Classes of economic and mathematical models.