Such a concept as a production function is given. Cobb-Douglas production function: solution examples

production called any human activity on the transformation of limited resources - material, labor, natural - into finished products. The production function characterizes the relationship between the amount of resources used (factors of production) and the maximum possible output that can be achieved, provided that all available resources are used in the most rational way.

The production function has the following properties:

1 There is a limit to the increase in production that can be reached by increasing one resource and keeping other resources constant. If, for example, in agriculture increase the amount of labor with constant amounts of capital and land, then sooner or later there comes a point when output stops growing.

2 Resources complement each other, but within certain limits, their interchangeability without reducing output is also possible. Manual labor, for example, may be replaced by the use of more machines, and vice versa.

Manufacturing cannot create products out of nothing. The production process is associated with the consumption of various resources. The number of resources includes everything that is necessary for production activities - raw materials, energy, labor, equipment, and space.

In order to describe the behavior of a firm, it is necessary to know how much of a product it can produce using resources in various volumes. We will proceed from the assumption that the company produces a homogeneous product, the amount of which is measured in natural units - tons, pieces, meters, etc. The dependence of the amount of product that a company can produce on the volume of resource costs is called production function.

But an enterprise can carry out the production process in different ways, using different technological methods, different options for organizing production, so that the amount of product obtained with the same resource costs can be different. Firm managers should reject production options that give a lower output of the product if, for the same input of each type of resource, a higher output can be obtained. Similarly, they should reject options that require more input of at least one resource without increasing the yield of the product and reducing the cost of other resources. Options rejected for these reasons are called technically inefficient.

Let's say your company manufactures refrigerators. For the manufacture of the case, you need to cut sheet metal. Depending on how the standard sheet of iron is marked and cut, more or less parts can be cut out of it; accordingly, for the manufacture of a certain number of refrigerators, less or more standard iron sheets will be required. At the same time, the consumption of all other materials, labor, equipment, electricity will remain unchanged. Such a production option, which can be improved by more rational cutting of iron, should be recognized as technically inefficient and rejected.

technically efficient are called production options that cannot be improved either by increasing the production of a product without increasing the consumption of resources, or by reducing the costs of a resource without reducing output and without increasing the costs of other resources. The production function takes into account only technically efficient options. Its meaning is greatest the quantity of a product that an enterprise can produce given the volume of resource consumption.

Consider first the simplest case: an enterprise produces a single type of product and consumes a single type of resource. An example of such production is quite difficult to find in reality. Even if we consider an enterprise providing services at customers' homes without the use of any equipment and materials (massage, tutoring) and spending only the labor of workers, we would have to assume that workers go around customers on foot (without using transport services) and negotiate with customers without the help of mail and telephone.

production function- shows the dependence of the amount of product that a firm can produce on the amount of costs of the factors used

Q= f(x1, x2…xn)

Q= f(K, L),

where Q- output volume

x1, x2…xn– volumes of applied factors

K- volume of the capital factor

L- volume of labor factor

So, the enterprise, spending a resource in the amount X, can produce a product in quantity q. production function

Federal Agency for Education of the Russian Federation

State educational institution of higher professional education

"South Ural State University"

Faculty of Mechanics and Mathematics

Department of Applied Mathematics and Informatics

Production function of the firm: essence, types, application.

EXPLANATORY NOTE TO THE COURSE WORK (PROJECT)

in the discipline (specialization) "Microeconomics"

SUSU–080116 . 2010.705.PZ KR

Head, Associate Professor

V.P. Borodkin

Student group MM-140

N.N. Basalaeva

2010

Work (project) is protected

with an assessment (in words, numbers)

___________________________

2010

Chelyabinsk 2010

INTRODUCTION……………………………………………………………………..3

THE CONCEPT OF PRODUCTION AND PRODUCTION FUNCTIONS ... ..7

2.1. Cobb-Douglas production function……………………………..13

2.2. CES production function……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

2.3. Production function with fixed proportions………...14

2.4. Cost-output production function (Leontief function)……14

2.5. Production function of the analysis of methods of production activity…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

2.6. Linear production function………………………………………15

2.7. Isoquant and its types………………………………………………………….16

PRACTICAL APPLICATION OF THE PRODUCTION FUNCTION.

3.1 Modeling the costs and profits of an enterprise (firm)…………...21

3.2 Methods of accounting for scientific and technological progress…………………………..28

CONCLUSION………………………………………………………………...34

Bibliographic list……………………………………………………35

INTRODUCTION

Economic activity can be carried out by various entities - individuals, family, state, etc., but the main productive functions in the economy belong to an enterprise or firm. On the one hand, a firm is a complex material, technological and social system that ensures the production of economic benefits. On the other hand, this is the very activity of organizing the production of various goods and services. As a system that produces economic goods, the firm is integral and acts as an independent reproductive link, relatively isolated from other links. The company independently carries out its activities, disposes of the released products and the profit received, remaining after paying taxes and other payments.

So what is a production function? Let's look at the dictionary and get the following:

PRODUCTION FUNCTION - an economic-mathematical equation that connects variable costs (resources) with production (output) values. Production functions are used to analyze the influence of various combinations of factors on the volume of output at a certain point in time (static version of the production function) and to analyze and predict the ratio of the volumes of factors and output at different points in time (dynamic version of the production function) at various levels of the economy - from a firm (enterprise) to National economy as a whole (aggregate production function, in which the output is an indicator of the total social product or national income, etc.). In an individual firm, corporation, etc., the production function describes the maximum amount of output that they are able to produce for each combination of factors of production used. It can be represented by many isoquants associated with different levels of output.

This type of production function, when an explicit dependence of the volume of production on the availability or consumption of resources is established, is called the output function.

In particular, output functions are widely used in agriculture, where they are used to study the impact on yields of such factors as, for example, different types and compositions of fertilizers, tillage methods. Along with similar production functions, the inverse functions of production costs are used. They characterize the dependence of resource costs on output volumes (strictly speaking, they are inverse only to production functions with interchangeable resources). Special cases of production functions can be considered the cost function (the relationship between the volume of production and production costs), the investment function (the dependence of the required investment on production capacity future enterprise), etc.

Mathematically, production functions can be represented in various forms- from such simple ones as the linear dependence of the result of production on one factor under study, to very complex systems of equations, including recurrence relations that connect the states of the object under study in different periods of time.

The most widely used are multiplicative-power forms of representation of production functions. Their peculiarity is as follows: if one of the factors is equal to zero, then the result vanishes. It is easy to see that this realistically reflects the fact that in most cases all analyzed primary resources are involved in production, and without any of them, production is impossible. In its most general form (it is called canonical), this function is written as follows:

Or

Here, the coefficient A in front of the multiplication sign takes into account the dimension, it depends on the chosen unit of measurement of costs and output. Factors from the first to the nth can have different content depending on what factors influence the overall result (output). For example, in a production function that is used to study the economy as a whole, one can take the volume of the final product as a performance indicator, and the factors - the number of employed people x 1, the sum of fixed and working capital x 2, the area of ​​land used x 3. There are only two factors in the Cobb-Douglas function, with the help of which an attempt was made to assess the relationship of factors such as labor and capital with the growth of US national income in the 20-30s. XX century:

N = A L α K β ,

where N is the national income; L and K are the volumes of applied labor and capital, respectively.

The power coefficients (parameters) of the multiplicative power production function show the share in the percentage increase in the final product that each of the factors contributes (or by what percentage the product will increase if the costs of the corresponding resource are increased by one percent); they are coefficients of elasticity of production with respect to the costs of the corresponding resource. If the sum of the coefficients is 1, this means that the function is homogeneous: it increases in proportion to the increase in the amount of resources. But such cases are also possible when the sum of the parameters is greater or less than unity; this shows that an increase in costs leads to a disproportionately large or disproportionately small increase in output (Effects of scale).

In the dynamic version, apply different forms production functions. For example, (in the 2-factor case): Y(t) = A(t) L α (t) K β (t), where the factor A(t) usually increases over time, reflecting the overall increase in the efficiency of production factors over time.

By taking a logarithm and then differentiating this function with respect to t, one can obtain the ratio between the growth rates of the final product (national income) and the growth of production factors (the growth rates of variables are usually described here as a percentage).

Further “dynamization” of production functions may be to use variable coefficients elasticity.

The ratios described by the production function are of a statistical nature, i.e., they appear only on average, in big mass observations, since in reality the result of production is affected not only by the analyzed factors, but also by many unaccounted ones. In addition, the applied indicators of both costs and results are inevitably products of complex aggregation (for example, a generalized indicator of labor costs in a macroeconomic function includes labor costs of different productivity, intensity, qualifications, etc.).

A special problem is taking into account the factor of technical progress in macroeconomic production functions. With the help of production functions, we also study the equivalent interchangeability of factors of production, which can be either constant or variable (that is, dependent on the volume of resources). Accordingly, functions are divided into two types: with constant elasticity of substitution (CES - Constant Elasticity of Substitution) and with variable (VES - Variable Elasticity of Substitution).

In practice, three main methods are used to determine the parameters of macroeconomic production functions: based on the processing of time series, based on data on the structural elements of aggregates, and on the distribution of national income. The last method is called distribution.

When constructing production functions, it is necessary to get rid of the phenomena of multicollinearity of parameters and autocorrelation - otherwise gross errors are inevitable.

Here are some important production functions

Linear production function:

P = a 1 x 1 + ... + a n x n ,

where a 1 , ..., a n are the estimated parameters of the model: here the factors of production are substituted in any proportions.

CES Feature:

P \u003d A [(1 - α) K - b + αL - b] - c / b,

in this case, the elasticity of resource substitution does not depend on either K or L and, therefore, is constant:

This is where the name of the function comes from.

The CES function, like the Cobb-Douglas function, assumes a constant decrease in the marginal rate of substitution of the resources used. Meanwhile, the elasticity of the replacement of capital by labor and, conversely, of labor by capital in the Cobb-Douglas function, which is equal to one, here can take on different values ​​that are not equal to one, although it is constant. Finally, unlike the Cobb-Douglas function, taking the logarithm of the CES function does not lead it to linear form, which forces us to use more complex methods of nonlinear regression analysis to estimate the parameters.

1. THE CONCEPT OF PRODUCTION AND PRODUCTION FUNCTIONS.

Production is understood as any activity for the use of natural, material, technical and intellectual resources to obtain both tangible and intangible benefits.

With the development of human society, the nature of production is changing. In the early stages of human development, natural, natural, naturally occurring elements of the productive forces dominated. And man himself at that time was more a product of nature. Production during this period was called natural.

With the development of the means of production, the historically created material and technical elements of the productive forces begin to predominate. This is the age of capital. At present, knowledge, technology, and the intellectual resources of the person himself are of decisive importance. Our era is the era of informatization, the era of the dominance of scientific and technical elements of the productive forces. Possession of knowledge, new technologies is crucial for production. In many developed countries, the task of universal informatization of society is set. The worldwide computer network Internet is developing at a tremendous pace.

Traditionally, the role of the general theory of production is played by the theory of material production, understood as the process of transforming production resources into a product. The main production resources are labor ( L) and capital ( K). The modes of production or existing production technologies determine how much output is produced with given amounts of labor and capital. Mathematically existing technologies are expressed through production function. If we denote the volume of output by Y, then the production function can be written

Y= f(K, L).

This expression means that the volume of output is a function of the amount of capital and the amount of labor. The production function describes the set of existing this moment technologies. If a better technology is invented, then with the same expenditure of labor and capital, output increases. Consequently, changes in technology also change the production function. Methodologically, the theory of production is largely symmetrical to the theory of consumption. However, if in the theory of consumption the main categories are measured only subjectively or are not yet subject to measurement at all, then the main categories of the theory of production have an objective basis and can be measured in certain physical or value units.

Although the concept of production may seem very broad, vague and even vague, since in real life production is understood as an enterprise, and construction, and an agricultural farm, and a transport enterprise, and a very large organization such as a branch of the national economy, however, economic and mathematical modeling highlights something in common that is inherent in all these objects. This common is the process of converting primary resources (production factors) into the final results of the process. Therefore, the main initial concept in the description of an economic object is the technological method, which is usually represented as a vector of production costs v, which includes the enumeration of the volumes of expended resources (vector x) and information about the results of their transformation into final products or other characteristics (profit, profitability, etc.) (vector y):

v= (x; y).

Dimension of vectors x and y, as well as methods of their measurement (in natural or cost units) significantly depend on the problem under study, on the levels at which certain tasks of economic planning and management are set. The set of vectors of technological methods that can serve as a description (from an acceptable point of view of the researcher with accuracy) of the production process that is actually feasible at some object is called the technological set V this object. For definiteness, we will assume that the dimension of the cost vector x is equal to N, and the output vector y respectively M. Thus, the technological v is a vector of dimension ( M+ N), and the technological set VCR +   M + N. Among all the technological methods implemented at the facility, a special place is occupied by methods that compare favorably with all others in that they require either lower costs for the same output, or correspond to a larger output for the same costs. Those of them that occupy in a certain sense the limiting position in the set V, are of particular interest because they are a description of a feasible and marginally profitable real production process.

Let's say that the vector ν (1) =(x (1) ;y (1) ) preferred over vector ν (2) =(x (2) ;y (2) ) with the designation ν (1) > ν (2) if the following conditions are met:

1) at i (1) ≥ y i (2) (i=1,…,M);

2) x j (1) ≤ x j (2) (j=1,…M);

and at least one of the following occurs:

a) there is such a number i 0 that at i 0 (1) > y i 0 (2)

b) there is such a number j 0 that x j 0 (1) x j 0 (2)

A technological method ۷ is called effective if it belongs to the technological set V and there is no other vector ν Є V that would be preferable to ۷. The above definition means that those methods are considered effective that cannot be improved in any cost component, in any position of the product, without ceasing to be acceptable. The set of all technologically efficient methods will be denoted by V*. It is a subset of the technological set V or matches it. In essence, the task of planning the economic activity of a production facility can be interpreted as the task of choosing an effective technological method that best suits some external conditions. When solving such a problem of choice, the idea of ​​the very nature of the technological set turns out to be quite significant V, as well as its effective subset V*.

In a number of cases, it turns out to be possible to admit, within the framework of fixed production, the possibility of interchangeability of certain resources (various types of fuel, machines and workers, etc.). At the same time, the mathematical analysis of such productions is based on the premise of the continual nature of the set V, and consequently, on the fundamental possibility of representing variants of mutual replacement using continuous and even differentiable functions defined on V. This approach has received its greatest development in the theory of production functions.

With the help of the concept of an effective technological set, a production function can be defined as a mapping

y= f(x),

where ν \u003d (x; y) ЄV*.

This mapping is, generally speaking, multi-valued, i.e. a bunch of f(x) contains more than one point. However, for many realistic situations, production functions turn out to be single-valued and even, as mentioned above, differentiable. In the simplest case, the production function is the scalar function N arguments:

y = f(x 1 ,…, x N ).

Here the value y has, as a rule, a cost character, expressing the volume of production in monetary terms. The arguments are the volumes of resources expended in the implementation of the corresponding effective technological method. Thus, the above relation describes the boundary of the technological set V, because at given vector costs ( x 1 , ..., x N) to produce products in quantities greater than y, is impossible, and the production of products in quantities less than specified corresponds to an inefficient technological method. The expression for the production function can be used to evaluate the effectiveness of the management method adopted at a given enterprise. Indeed, for a given set of resources, one can determine the actual output and compare it with that calculated from the production function. The resulting difference provides useful material for evaluating efficiency in absolute and relative terms.

The production function is a very useful tool for planning calculations, and therefore a statistical approach has now been developed to construct production functions for specific economic units. In this case, a certain standard set of algebraic expressions is usually used, the parameters of which are found using the methods of mathematical statistics. This approach means, in essence, estimating the production function based on the implicit assumption that the observed production processes are efficient. Among the various types of production functions, linear functions of the form

since for them the problem of estimating coefficients from statistical data is easily solved, as well as power functions

for which the problem of finding the parameters is reduced to estimating the linear form by passing to logarithms.

Under the assumption that the production function is differentiable at each point of the set X possible combinations of inputs, it is useful to consider some quantities associated with the production function.

In particular, the differential

represents the change in the cost of output when moving from the cost of a set of resources x=(x 1 , ..., x N) to the set x+dx=(x 1 +dx 1 ,..., x N +dx N) provided that the properties of the efficiency of the corresponding technological methods are preserved. Then the value of the partial derivative

can be interpreted as the marginal (differential) resource return or, in other words, the marginal productivity coefficient, which shows how much the output will increase due to the increase in the cost of the resource with the number j for a small unit. The value of the marginal productivity of the resource can be interpreted as the upper limit of the price p j, which the production facility can pay for an additional unit j-that resource in order not to be at a loss after its acquisition and use. Indeed, the expected increase in output in this case will be

and hence the ratio

will generate additional profit.

In the short run, when one resource is treated as fixed and the other as variable, most production functions have the property of diminishing marginal product. The marginal product of a variable resource is the increase in the total product due to the increase in the use of this variable resource per unit.

The marginal product of labor can be written as the difference

MPL= F(K, L+ 1) - F(K, L),

where MPL marginal product of labor.

The marginal product of capital can also be written as the difference

MPK= F(K+ 1, L) - F(K, L),

where MPK marginal product of capital.

A characteristic of a production facility is also the value of the average resource return (productivity of the production factor)

having a clear economic sense the quantity of products produced per unit of resource used (production factor). The reciprocal of the resource return

commonly referred to as resource intensity because it expresses the amount of a resource j required to produce one unit of output in value terms. Very common and understandable are terms such as capital intensity, material intensity, energy intensity, labor intensity, the growth of which is usually associated with a deterioration in the state of the economy, and their decline is regarded as a favorable result.

The quotient of dividing the differential productivity by the average

is called the coefficient of elasticity of production by the production factor j and gives an expression for the relative increase in production (in percent) with a relative increase in the cost of the factor by 1%. If a E j 0, then there is an absolute decrease in output with an increase in the consumption of the factor j; this situation may occur when technologically unsuitable products or modes are used. For example, excessive consumption of fuel will lead to an excessive increase in temperature and the chemical reaction necessary for the production of the product will not take place. If 0 E j 1, then each subsequent additional unit of the expended resource causes a smaller additional increase in production than the previous one.

If a E j> 1, then the value of the incremental (differential) productivity exceeds the average productivity. Thus, an additional unit of resource increases not only the volume of output, but also the average resource return characteristic. This is how the process of increasing the return on assets occurs when highly progressive, efficient machines and devices are put into operation. For a linear production function, the coefficient a j numerically equal to the value of differential productivity j-th factor, and for a power function, the exponent a j has the meaning of the coefficient of elasticity in terms of j-that resource.

2. TYPES OF PRODUCTION FUNCTIONS.

2.1. Cobb-Douglas production function.

The first successful experience in constructing a production function as a regression equation based on statistical data was obtained by American scientists - mathematician D. Cobb and economist P. Douglas in 1928. The function they proposed originally looked like this:

where Y is the volume of output, K is the value of production assets (capital), L is labor costs, - numerical parameters (scale number and elasticity index). Due to its simplicity and rationality, this function is still widely used today, and has received further generalizations in various directions. The Cobb-Douglas function will sometimes be written as

It is easy to check that and

In addition, function (1) is linearly homogeneous:

Thus, the Cobb-Douglas function (1) has all the above properties.

For multifactorial production, the Cobb-Douglas function has the form:

To take into account technical progress, a special multiplier (technical progress) is introduced into the Cobb-Douglas function, where t is the time parameter, is a constant number characterizing the rate of development. As a result, the function takes a "dynamic" form:

where not required. As will be shown in the next section, the exponents in function (1) have the meaning of the elasticity of output with respect to capital and labor.

2.2. production functionCES(with constant elasticity of substitution)

Looks like:

Where is the scale coefficient, is the distribution coefficient, is the replacement coefficient, is the degree of homogeneity. If the conditions are met:

then function (2) satisfies the inequalities and . Taking into account technological progress, the CES function is written:

The name of this function follows from the fact that for it the elasticity of substitution is constant.

2.3. Production function with fixed proportions. This function is obtained from (2) at and has the form:

2.4. Cost-output production function (Leontief function) is obtained from (3) when :

Here, is the amount of costs of type k required to produce one unit of output, and y is output.

2.5. The production function of the analysis of the methods of production activity.

This function generalizes the input-output production function to the case when there is a certain number (r) of basic processes (modes of production activity), each of which can proceed with any non-negative intensity. It has the form of an "optimization problem"

Where (5)

Here, is the output at a unit intensity of the j-th basic process, is the level of intensity, is the amount of costs of the type k required at a unit intensity of the method j. As can be seen from (5), if the output produced at a unit intensity and the costs required per unit of intensity are known, then the total output and total costs are found by adding the output and costs, respectively, for each basic process at the selected intensities. Note that the problem of maximizing the function f in (5) under given inequality constraints is a model for the analysis of production activity (maximization of output with limited resources).

2.6. Linear production function(resource substitution function)

It is used in the presence of a linear dependence of output on costs:

Where is the cost rate of the kth type for the production of a unit of output (marginal physical cost product).

Among the production functions given here, the most common is the CES function.

To analyze the production process and its various indicators along with marginal products,

(upper dashes indicate fixed values ​​of variables), showing the amount of additional income obtained by using additional quantities of costs, the concepts of average products are applied.

The average product for the k-th type of costs is the volume of output per unit of costs of the k-th type at a fixed level of costs of other types:

Let us fix the costs of the second type at a certain level and compare the graphs of the three functions:

Fig.1. release curves.

Let the function graph have three critical points(as shown in Fig. 1): - inflection point, - point of contact with the ray from the origin, - maximum point. These points correspond to the three stages of production. The first stage corresponds to the segment and is characterized by the superiority of the marginal product over the average: Therefore, at this stage, the implementation of additional costs is advisable. The second stage corresponds to the segment and is characterized by the superiority of the average product over the marginal one: (Additional costs are not reasonable). In the third stage and additional costs lead to the opposite effect. This is explained by the fact that is the optimal amount of costs and their further increase is unreasonable.

For specific names of resources, average and marginal values ​​acquire the meaning of specific economic indicators. Consider, for example, the Cobb-Douglas function (1) , where is capital and is labor. Medium Products

make sense, respectively, of the average productivity of labor and the average productivity of capital (average return on assets). It can be seen that the average productivity of labor decreases with increasing labor resources. This is understandable, since the production assets (K) remain unchanged, and therefore the newly attracted labor force is not provided with additional means of production, which leads to a decrease in labor productivity. A similar reasoning is true for capital productivity as a function of capital.

For function (1) marginal products

make sense, respectively, of the marginal productivity of labor and the marginal productivity of capital (marginal return on assets). In the microeconomic theory of production, it is believed that the marginal productivity of labor is equal to wages (the price of labor), and the marginal productivity of capital is equal to rent payments (the price of services of capital goods). It follows from the condition that with constant fixed assets (labor costs), an increase in the number of employees (the volume of fixed assets) leads to a drop in the marginal productivity of labor (marginal return on assets). It can be seen that for the Cobb-Douglas function, the marginal products are proportional to the average products and less than them.

2.7. Isoquant and its types

When modeling consumer demand, the same level of utility of various combinations of consumer goods is graphically displayed using an indifference curve.

In economic and mathematical models of production, each technology can be graphically represented by a point, the coordinates of which reflect the minimum necessary costs of resources K and L for the production of a given volume of output. Many such points form a line of equal output, or an isoquant. Thus, the production function is graphically represented by a family of isoquants. The further the isoquant is located from the origin, the greater the volume of production it reflects. Unlike an indifference curve, each isoquant characterizes a quantified amount of output.

Fig.2. Isoquants Corresponding to Different Volumes of Production

On fig. 2 shows three isoquants corresponding to a production volume of 200, 300 and 400 units. It can be said that for the production of 300 units of production, K 1 units of capital and L 1 units of labor or K 2 units of capital and L 2 units of labor are needed, or any other combination of them from the set represented by the isoquant Y 2 = 300.

In the general case, in the set X of feasible sets of production factors, a subset is allocated, called the isoquant of the production function, which is characterized by the fact that for any vector the equality

Thus, for all sets of resources corresponding to the isoquant, the volumes of output are equal. Essentially, an isoquant is a description of the possibility of mutual substitution of factors in the process of production of goods, providing a constant volume of production. In this regard, it is possible to determine the coefficient of mutual replacement of resources, using the differential relation along any isoquant

Hence, the coefficient of equivalent replacement of a pair of factors j and k is equal to:

The obtained ratio shows that if production resources are replaced in a ratio equal to the ratio of incremental productivity, then the amount of output remains unchanged. It must be said that knowledge of the production function makes it possible to characterize the extent of the possibility to carry out the mutual replacement of resources in efficient technological methods. To achieve this goal, the coefficient of elasticity of the replacement of resources for products is used.

which is calculated along the isoquant at a constant level of costs of other production factors. The value s jk is a characteristic of the relative change in the coefficient of mutual replacement of resources when the ratio between them changes. If the ratio of interchangeable resources changes by s jk percent, then the mutual replacement ratio sjk will change by one percent. In the case of a linear production function, the mutual substitution coefficient remains unchanged for any ratio of resources used, and therefore we can assume that the elasticity s jk = 1. Accordingly, large values ​​of s jk indicate that greater freedom is possible in replacing production factors along the isoquant and, at the same time, the main the characteristics of the production function (productivity, interchange ratio) will change very little.

For power-law production functions for any pair of interchangeable resources, the equality s jk = 1 is true. In the practice of forecasting and preplanning calculations, constant elasticity of substitution (CES) functions are often used, which look like:

For such a function, the resource replacement elasticity coefficient

and does not change depending on the volume and ratio of resources expended. For small values ​​of s jk, resources can replace each other only to a small extent, and in the limit at s jk = 0, they lose their interchangeability property and appear in the production process only in a constant ratio, i.e. are complementary. An example of a production function that describes production under the conditions of the use of complementary resources is the cost release function, which has the form

where a j is a constant coefficient of resource return of the j -th production factor. It is easy to see that a production function of this type determines the bottleneck output on the set of production factors used. Different cases of the behavior of isoquants of production functions for different values ​​of the elasticity coefficients of substitution are shown in the graph (Fig. 3).

The representation of an effective technological set using a scalar production function is insufficient in cases where it is impossible to manage with a single indicator describing the results of the production facility, but it is necessary to use several (M) output indicators. Under these conditions, one can use the vector production function

Rice. 3. Various cases of behavior of isoquants

The important concept of marginal (differential) productivity is introduced by the relation

All other main characteristics of scalar production functions admit a similar generalization.

Like indifference curves, isoquants are also classified into different types.

For a linear production function of the form

where Y is the volume of production; A , b 1 , b 2 parameters; K , L costs of capital and labor, and the complete replacement of one resource by another isoquant will have a linear form (Fig. 4).

For the power production function

isoquants will look like curves (Fig. 5).

If the isoquant reflects only one technological method for the production of a given product, then labor and capital are combined in the only possible combination (Fig. 6).

Rice. 6. Isoquants under strict complementarity of resources

Rice. 7. Broken isoquants

Such isoquants are sometimes called Leontief-type isoquants after the American economist W.V. Leontiev, who put this type of isoquant as the basis of the inputoutput method he developed.

The broken isoquant implies the presence of a limited number of technologies F (Fig. 7).

Isoquants of this configuration are used in linear programming to substantiate the theory of optimal resource allocation. Broken isoquants most realistically represent the technological capabilities of many production facilities. However, in economic theory Traditionally, isoquant curves are mainly used, which are obtained from broken lines with an increase in the number of technologies and an increase in breakpoints, respectively.

3. PRACTICAL APPLICATION OF THE PRODUCTION FUNCTION.

3.1 Modeling the costs and profits of an enterprise (firm)

At the heart of the construction of models of behavior of the manufacturer (individual enterprise or firm; association or industry) is the idea that the manufacturer seeks to achieve a state in which he would be provided with the greatest profit under the prevailing market conditions, i.e. First of all, with the existing price system.

The simplest model of the optimal behavior of a producer under conditions of perfect competition has the following form: let an enterprise (firm) produce one product in the quantity y physical units. If a p exogenously given price of this product and the firm sells its output in full, then it receives a gross income (revenue) in the amount of

In the process of creating this quantity of product, the firm incurs production costs in the amount of C(y). At the same time, it is natural to assume that C"(y) > 0, i.e. costs increase with the volume of production. It is also commonly assumed that C""(y) > 0. This means that the additional (marginal) cost of producing each additional unit of output increases as the volume of production increases. This assumption is due to the fact that in a rationally organized production, with small volumes, the best machines and highly skilled workers can be used, which will no longer be at the disposal of the company when the volume of production increases. Production costs consist of the following components:

1) material costs C m, which include the cost of raw materials, materials, semi-finished products, etc.

The difference between gross income and material costs is called added value(conditionally pure products):

2) labor costs C L ;

Rice. 8. Lines of revenue and costs of the enterprise

3) expenses associated with the use, repair of machinery and equipment, depreciation, the so-called payment for capital services C k ;

4) additional costs C r associated with the expansion of production, the construction of new buildings, access roads, communication lines, etc.

Total production costs:

As noted above,

however, this dependence on the volume of output ( at) is different for different types of costs. Namely, there are:

a) fixed costs C 0 , which are practically independent of y, incl. payment of administrative personnel, rent and maintenance of buildings and premises, depreciation, interest on loans, communication services, etc.;

b) proportional to the volume of output (linear) costs C 1, this includes material costs C m, remuneration of production personnel (part of C L), expenses for the maintenance of existing equipment and machinery (part C k) etc.:

where a a generalized indicator of the costs of these types per one product;

c) super-proportional (non-linear) costs With 2 , which include the acquisition of new machines and technologies (i.e., costs such as With r), overtime pay, etc. For a mathematical description of this type of cost, a power law is usually used

Thus, to represent total costs, one can use the model

(Note that the conditions C"(y) > 0, C""(y) > 0 are satisfied for this function.)

Consider possible options for the behavior of an enterprise (firm) for two cases:

1. The enterprise has a sufficiently large reserve of production capacities and does not seek to expand production, so we can assume that C 2 = 0 and total costs are a linear function of output:

The profit will be

It is clear that for small production volumes

The firm is making a loss because

Here y w break-even point (profitability threshold), determined by the ratio

If a y> y w, then the firm makes a profit, and the final decision on the volume of output depends on the state of the market for the sale of manufactured products (see Fig. 8).

2. In a more general case, when With 2 0, there are two break-even points and, moreover, the firm will receive a positive profit if the output y satisfies the condition

On this segment, at the point, the greatest value of profit is achieved. Thus, there is optimal solution profit maximization problem. At the point BUT, corresponding to the costs at optimal output, tangent to the cost curve With parallel to the straight line of income R.

It should be noted that the final decision of the firm also depends on the state of the market, but from the point of view of observing economic interests, it should recommend the optimizing value of output (Fig. 9).

Rice. 9. Optimal output

By definition, profit is the value

The break-even points and are determined from the condition of equality of profit to zero, and its maximum value is reached at the point that satisfies the equation

Thus, the optimal volume of production is characterized by the fact that in this state the marginal gross income ( R(y)) is exactly equal to the marginal cost C(y).

Indeed, if y R( y) > C(y), and then output should be increased, since the expected additional income will exceed the expected additional costs. If y> , then R(y) C ( y), and any increase in volume will reduce profits, so it is natural to recommend reducing the volume of production and come to a state y= (Fig. 10).

It is easy to see that as the price increases ( R) the optimal output as well as the profit increase, i.e.

This is also true in the general case, since

Example. The company produces agricultural machines in the amount at pieces, and the volume of production, in principle, can vary from 50 to 220 pieces per month. At the same time, naturally, an increase in production volume will require an increase in costs, both proportional and superproportional (nonlinear), since it will be necessary to purchase new equipment and expand production areas.

In a specific example, we will proceed from the fact that the total costs (cost) for the production of products in the amount at products are expressed by the formula

C(y) = 1000 + 20 y+ 0,1 y 2 (thousand rubles).

This means that fixed costs

C 0 = 1000 (tons of rubles),

proportional costs

C 1 = 20 y,

those. the generalized indicator of these costs per product is equal to: a= 20 thousand rubles, and the non-linear costs will be C 2 = 0,1 y 2 (b= 0,1).

The above formula for costs is a special case general formula, where the exponent h= 2.

To find the optimal volume of production, we use the formula for the maximum profit point (*), according to which we have:

It is quite obvious that the volume of production at which the maximum profit is achieved is very significantly determined by the market price of the product. p.

In table. 1 shows the results of calculating the optimal volumes for various price values ​​from 40 to 60 thousand rubles per product.

The first column of the table contains possible output volumes at, the second column contains data on total costs With(at), the third column shows the cost per one product:

Table 1

Data on output volumes, costs and profits

The fourth column characterizes the values ​​of the above marginal costs MS, which show how much it costs to produce one additional product in a given situation. It is easy to see that marginal costs increase as production increases, which is in good agreement with the position expressed at the beginning of this paragraph. When considering the table, you should pay attention to the fact that the optimal volumes are exactly at the intersection of the line (marginal costs MS) and column (price p) with their equal values, which quite neatly correlates with the optimality rule established above.

The above analysis refers to a situation of perfect competition, when the producer cannot influence the price system by his actions, and therefore the price p for goods y acts in the manufacturer's model as an exogenous value.

In the case of imperfect competition, the producer can directly influence the price. In particular, this applies to the monopoly producer of goods, which forms the price for reasons of reasonable profitability.

Consider a firm with a linear cost function that sets its price in such a way that the profit is a certain percentage (a fraction of 0

Hence we have

Gross income

and production breaks even, starting with the smallest volumes of production ( y w 0). It is easy to see that the price depends on the volume, i.e. p= p(y), and with an increase in production volume ( at) the price of the good decreases, i.e. p"(y)

The profit maximization requirement for a monopolist has the form

Assuming still that >0, we have an equation for finding the optimal output ():

It is useful to note that the optimal output of a monopolist () is usually not greater than the optimal output of a competitive producer in the formula marked with an asterisk.

A more realistic (but also simpler) model of the firm is used to take into account the resource constraints that play a very large role in the economic activities of producers. The model identifies one most scarce resource (labor, fixed assets, rare material, energy, etc.) and assumes that the firm can use it in no more than Q. The firm can produce n various products. Let be y 1 , ..., y j , ..., y n the desired volumes of production of these products; p 1 , ..., p j , ..., p n their prices. Let also q unit price of a scarce resource. Then the gross income of the firm is

and the profit will be

It is easy to see that for fixed q and Q the profit maximization problem is transformed into the gross income maximization problem.

Suppose further that the resource cost function for each product C j (y j) has the same properties that were stated above for the function With(at). Thus, C j " (y j) > 0 and C j "" (y j) > 0.

In its final form, the model of the optimal behavior of a firm with one limited resource is as follows:

It is easy to see that in a fairly general case, the solution to this optimization problem is found by studying the system of equations:

Note that the optimal choice of the firm depends on the entire set of product prices ( p 1 , ..., p n), and this choice is a homogeneous function of the price system, i.e. when prices change by the same number of times, optimal outputs do not change. It is also easy to see that from the equations marked with asterisks (***) it follows that with an increase in the price of the product n(at constant prices for other products), its output should be increased in order to maximize profits, since

and the production of other goods will decrease, since

These ratios together show that in this model, all products are competing. Formula (***) also implies the obvious relation

those. with an increase in the volume of a resource (capital investment, labor, etc.), optimal outputs increase.

A number of simple examples can be given to help you better understand the rule of optimal firm selection based on the principle of maximum profit:

1) let n = 2; p 1 = p 2 = 1; a 1 = a 2 = 1; Q = 0,5; q = 0,5.

Then from (***) we have:

0.5; = 0.5; P = 0.75; = 1;

2) let now all the conditions remain the same, but the price of the first product has doubled: p 1 = 2.

Then the firm's optimal profit plan: = 0.6325; = 0.3162.

The expected maximum profit increases markedly: P = 1.3312; = 1.58;

3) note that in the previous example 2, the firm must change the volume of production, increasing the production of the first and decreasing the production of the second product. Suppose, however, that the firm is not pursuing maximum profit and will not change the established production, i.e. choose a program y 1 = 0,5; y 2 = 0,5.

It turns out that in this case the profit will be P = 1.25. This means that when prices rise in the market, the firm can get a significant increase in profits without changing the output plan.

3.2 Methods of accounting for scientific and technological progress

It should be considered generally accepted that over time at an enterprise that maintains a fixed number of employees and a constant volume of fixed assets, output increases. This means that in addition to the usual production factors associated with the cost of resources, there is a factor that is usually called scientific and technological progress (NTP). This factor can be viewed as a synthetic characteristic that reflects the combined impact on economic growth of many significant phenomena, among which the following should be noted:

a) improvement over time in the quality of the labor force due to the improvement of the skills of workers and the development of methods for using more advanced technology;

b) improvement in the quality of machinery and equipment leads to the fact that a certain amount of capital investment (at constant prices) makes it possible, over time, to acquire a more efficient machine;

c) improvement of many aspects of the organization of production, including supply and marketing, banking operations and other mutual settlements, the development of an information base, the formation of various kinds of associations, the development of international specialization and trade, etc.

In this regard, the term scientific and technological progress can be interpreted as a set of all phenomena that, with a fixed amount of input production factors, make it possible to increase the output of high-quality, competitive products. The very vague nature of such a definition leads to the fact that the study of the influence of scientific and technical progress is carried out only as an analysis of that additional increase in production, which cannot be explained by a purely quantitative increase in production factors. The main approach to accounting for scientific and technical progress is that time is introduced into the totality of output or cost characteristics ( t) as an independent production factor and considers the transformation in time of either a production function or a technological set.

Let us dwell on the methods of accounting for scientific and technical progress by transforming the production function, and we will take the two-factor production function as a basis:

where the factors of production are capital ( To) and labor ( L). The modified production function in the general case has the form

and the condition

which reflects the fact of the growth of production over time at fixed costs of labor and capital.

When developing specific modified production functions, they usually seek to reflect the nature of the scientific and technical progress in the observed situation. There are four cases:

a) a significant improvement over time in the quality of the workforce allows you to achieve the same results with fewer people employed; this type of STP is often called labor-saving. The modified production function has the form where is the monotonic function l(t) characterizes the growth of labor productivity;

Rice. 11. Growth of production over time with fixed costs of labor and capital

b) the predominant improvement in the quality of machinery and equipment increases the return on assets, there is a capital-saving scientific and technical progress and the corresponding production function:

where is the increasing function k(t) reflects the change in capital productivity;

c) if there is a significant influence of both mentioned phenomena, then the production function is used in the form

d) if it is not possible to identify the influence of scientific and technical progress on production factors, then the production function is used in the form

where a(t) an increasing function that expresses the growth of production at constant values ​​of the costs of factors. To study the properties and features of scientific and technical progress, some correlations between the results of production and the costs of factors are used. These include:

a) average labor productivity

B) average return on assets

c) employee's capital-labor ratio

d) equality between the level of wages and the marginal (marginal) productivity of labor

e) equality between the marginal return on assets and the rate of bank interest

An NTP is said to be neutral if it does not change certain relationships between given quantities over time.

1) progress is called Hicks-neutral if the ratio between the capital-labor ratio ( x) and the marginal rate of replacement of factors ( w/r). In particular, if w/r= const, then the replacement of labor for capital and vice versa will not bring any benefit and capital-labor ratio x=K/L will also remain constant. It can be shown that in this case the modified production function has the form

and Hicks neutrality is equivalent to the impact of scientific and technical progress directly on output discussed above. In the situation under consideration, the isoquant shifts to the left down over time by means of a similarity transformation, i.e. remains exactly the same shape as in the original position;

2) progress is called Harrod-neutral if, during the period under consideration, the rate of bank interest ( r) depends only on the return on assets ( k), i.e. it is not affected by NTP. This means that the marginal return on assets is set at the level of the rate of interest and a further increase in capital is not advisable. It can be shown that this type of STP corresponds to the production function

those. technological progress is labor-saving;

3) progress is Solow-neutral if the equality between wage levels ( w) and the marginal productivity of labor and a further increase in labor costs is unprofitable. It can be shown that in this case the production function has the form

those. NTP turns out to be fund-saving. Let's give a graphical representation of three types of scientific and technological progress using the example of a linear production function

In the case of Hicks neutrality, we have a modified production function

where a(t) increasing function t. This means that over time the isoquant Q(line segment AB) is shifted to the origin by parallel translation (Fig. 12) to the position A 1 B 1 .

In the case of Harrod neutrality, the modified production function has the form

where l(t) is an increasing function.

Obviously, over time, the point BUT remains in place and the isoquant is shifted to the origin by rotating to the position AB 1 (Fig. 13).

For Solow-neutral progress, the corresponding modified production function

where k(t) is an increasing function. The isoquant shifts to the origin, but the point AT does not move and rotates to position A 1 B(Fig. 14).

When constructing production models taking into account scientific and technical progress, the following approaches are mainly used:

a) the idea of ​​exogenous (or autonomous) technical progress, which also exists when the main production factors do not change. A special case of such an NTP is Hicks-neutral progress, which is usually taken into account using an exponential factor, for example:

Here l > 0, characterizes the rate of STP. It is easy to see that time here acts as an independent factor in the growth of production, but at the same time it seems that scientific and technical progress occurs on its own, without requiring additional labor and capital investments;

b) the idea of ​​technical progress embodied in capital connects the growth of the influence of scientific and technical progress with the growth of capital investments. To formalize this approach, the Solow-neutral progress model is taken as a basis:

which is written as

where K 0 fixed assets at the beginning of the period, D K accumulation of capital over a period equal to the amount of investment.

Obviously, if no investment is made, then D K= 0, and there is no increase in output due to scientific and technical progress;

c) the above approaches to modeling scientific and technical progress have a common feature: progress acts as an exogenously given value that affects labor productivity or capital productivity and thereby affects economic growth.

However, in the long run, STP is both the result of development and, to a large extent, its cause. Since it is economic development that allows wealthy societies to finance the creation of new models of technology, and then reap the fruits of the scientific and technological revolution. Therefore, it is quite legitimate to approach STP as an endogenous phenomenon caused (induced) by economic growth.

There are two main directions of modeling scientific and technical progress:

1) the induced progress model is based on the formula

moreover, it is assumed that the society can distribute the investments intended for scientific and technical progress between its various directions. For example, between the growth of capital productivity ( k(t)) (improving the quality of machines) and the growth of labor productivity ( l(t)) (training of employees) or the choice of the best (optimal) direction of technical development with a given volume of allocated capital investments;

2) the model of the learning process in the course of production, proposed by K. Arrow, is based on the observed fact of the mutual influence of labor productivity growth and the number of new inventions. In the course of production, workers gain experience, and the time to manufacture a product decreases, i.e. labor productivity and the labor contribution itself depend on the volume of production

In turn, the growth of the labor factor, according to the production function

leads to an increase in production. In the simplest version of the model, the following formulas are used:

those. return on investment increases.

CONCLUSION

Thus, in this term paper I have considered many important and interesting facts from my point of view. It was found, for example, that the production function is a mathematical relationship between the maximum output per unit of time and the combination of factors that create it, given the current level of knowledge and technology. In production theory, they mainly use a two-factor production function, which in general view looks like this: Q = f(K,L), where Q is the volume of production; K - capital; L - labor. The question of the ratio of costs of factors of production replacing each other is solved with the help of such a concept as the elasticity of substitution of factors of production. The elasticity of substitution is the ratio of the costs of substituting factors of production at a constant output. This is a kind of coefficient that shows the degree of efficiency in replacing one factor of production with another. The measure of the interchangeability of factors of production is the marginal rate technical substitution MRTS, which shows how many units one of the factors can be reduced while increasing the other factor by one, keeping the output unchanged. The marginal rate of technical substitution is characterized by the slope of isoquants. MRTS is expressed by the formula: Isoquant - a curve representing all possible combinations of two costs that provide a given constant volume of production. Funding is usually limited. Thus, the optimal combination of factors for a particular enterprise is the general solution of the isoquant equations.

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Milgrom D.A. Evaluation of the competitiveness of economic technologies // Marketing in Russia and abroad, 1999, No. 2. - p. 44-57 production function firms is an isoquant map with different levels...

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production function- dependence of production volumes on the quantity and quality of available production factors, expressed using mathematical model. The production function makes it possible to determine the optimal amount of costs required to produce a certain portion of goods. At the same time, the function is always intended for a specific technology - the integration of new developments entails the need to revise the dependence.

Production function: general appearance and properties

Production functions have the following properties:

• An increase in output due to one production factor is always limiting (for example, a limited number of specialists can work in one room).

• Factors of production are interchangeable (human resources are replaced by robots) and complementary (workers need tools and machines).

In general, the production function looks like this:

Q = f (K, M, L, T, N),

Each company, undertaking the production of a particular product, seeks to achieve maximum profit. The problems associated with the production of products can be divided into three levels:

1. An entrepreneur may be faced with the question of how to produce a given quantity of products in a particular enterprise. These problems relate to the issues of short-term minimization of production costs;
2. the entrepreneur can decide on the production of the optimal, i.e. bringing a large amount of products at a particular enterprise. These questions are about long-term profit maximization;
3. the entrepreneur may be faced with finding out the most optimal size of the enterprise. Similar questions pertain to long-term profit maximization.

You can find the optimal solution based on an analysis of the relationship between costs and production volume (output). After all, profit is determined by the difference between the proceeds from the sale of products and all costs. Both revenue and costs depend on the volume of production. Economic theory uses the production function as a tool for analyzing this dependence.

The production function determines the maximum amount of output for each given amount of resources. This function describes the relationship between resource input and output, allowing you to determine the maximum possible output for each given amount of resources, or the minimum possible amount of resources to provide a given output. The production function summarizes only technologically efficient methods of combining resources to ensure maximum output. Any improvement in production technology that contributes to an increase in labor productivity leads to a new production function.

PRODUCTION FUNCTION - a function that displays the relationship between the maximum volume of the product produced and the physical volume of production factors at a given level of technical knowledge.

Since the volume of production depends on the volume of resources used, the relationship between them can be expressed as the following functional notation:

Q = f(L,K,M),

where Q is the maximum volume of products produced with a given technology and certain production factors;
L - labor; K - capital; M - materials; f is a function.

The production function with this technology has properties that determine the relationship between the volume of production and the number of factors used. For different types of production, production functions are different, however? they all have general properties. Two main properties can be distinguished.

1. There is a limit to the growth in output that can be achieved by increasing the cost of one resource, other things being equal. So, in a firm with a fixed number of machines and production facilities, there is a limit to the growth of output by increasing additional workers, since it will not be provided with machines for work.
2. There is a certain complementarity (completeness) of factors of production, however, without a decrease in the volume of output, a certain interchangeability of these factors of production is also likely. Thus, various combinations of resources can be used to produce a good; it is possible to produce this good by using less capital and more labor, and vice versa. In the first case, production is considered technically efficient in comparison with the second case. However, there is a limit to how much labor can be replaced by more capital without reducing production. On the other hand, there is a limit to the use of manual labor without the use of machines.

In graphical form, each type of production can be represented by a point, the coordinates of which characterize the minimum resources necessary for the production of a given volume of output, and the production function - by an isoquant line.

Having considered the production function of the firm, let's move on to characterizing the following three important concepts: total (cumulative), average and marginal product.

Rice. a) Curve of the total product (TR); b) curve of average product (AP) and marginal product (MP)

On fig. the curve of the total product (TP) is shown, which varies depending on the value of the variable factor X. Three points are marked on the TP curve: B is the inflection point, C is the point that belongs to the tangent coinciding with the line connecting given point with the origin, D is the point of maximum TP value. Point A moves along the TP curve. Connecting point A to the origin, we get the line OA. Dropping the perpendicular from point A to the abscissa axis, we get the triangle OAM, where tg a is the ratio of the side AM to OM, i.e., the expression for the average product (AP).

Drawing a tangent through point A, we get the angle P, the tangent of which will express the marginal product MP. Comparing the triangles LAM and OAM, we find that up to a certain point the tangent P is greater than tg a. Thus, marginal product (MP) is greater than average product (AR). In the case when point A coincides with point B, the tangent P takes on a maximum value and, therefore, the marginal product (MP) reaches the largest volume. If point A coincides with point C, then the value of the average and marginal product are equal. The marginal product (MP), having reached its maximum value at point B (Fig. 22, b), begins to decline and at point C it intersects with the graph of the average product (AP), which at this point reaches its maximum value. Then both the marginal product and the average product decrease, but the marginal product decreases at a faster rate. At the point of maximum total product (TP), marginal product MP = 0.

We see that the most effective change the variable factor X is observed on the segment from point B to point C. Here, the marginal product (MP), having reached its maximum value, begins to decrease, the average product (AR) still increases, and the total product (TR) receives the greatest increase.

Thus, the production function is a function that allows you to determine the maximum possible volume of output for various combinations and quantities of resources.

In production theory, a two-factor production function is traditionally used, in which the volume of production is a function of the use of labor and capital resources:

Q = f(L, K).

It can be presented as a graph or curve. In the theory of the behavior of producers, under certain assumptions, there is a unique combination of resources that minimizes the cost of resources for a given volume of production.

The calculation of the firm's production function is the search for the optimum, among many options involving various combinations of factors of production, one that gives the maximum possible output. In the face of rising prices and cash costs, the firm, i.e. the cost of acquiring factors of production, the calculation of the production function is focused on finding such an option that would maximize profits at the lowest cost.

The calculation of the firm's production function, seeking to achieve an equilibrium between marginal cost and marginal revenue, will focus on finding such a variant that will provide the required output at minimum production costs. The minimum costs are determined at the stage of calculating the production function by the method of substitution, the displacement of expensive or increased in price factors of production by alternative, cheaper ones. Substitution is carried out using a comparative economic analysis interchangeable and complementary factors of production at their market prices. A satisfactory option would be one in which the combination of factors of production and a given volume of output meets the criterion of the lowest production costs.

There are several types of production function. The main ones are:

1. Nonlinear PF;
2. Linear PF;
3. Multiplicative PF;
4. PF "input-output".

Production function and selection of the optimal production size

A production function is the relationship between a set of factors of production and the maximum possible amount of product produced by this set of factors.

The production function is always concrete, i.e. intended for this technology. New technology - new productive function.

The production function determines the minimum amount of input needed to produce a given amount of product.

Production functions, no matter what kind of production they express, have the following general properties:

1. An increase in production due to an increase in costs for only one resource has a limit (you cannot hire many workers in one room - not everyone will have places).
2. Factors of production can be complementary (workers and tools) and interchangeable (production automation).

In its most general form, the production function looks like this:

Q = f(K,L,M,T,N),

where L is the volume of output;
K - capital (equipment);
M - raw materials, materials;
T - technology;
N - entrepreneurial abilities.

The simplest is the two-factor model of the Cobb-Douglas production function, which reveals the relationship between labor (L) and capital (K). These factors are interchangeable and complementary.

Q = AK α * L β ,

where A is a production coefficient showing the proportionality of all functions and changes when the basic technology changes (in 30-40 years);
K, L - capital and labor;
α, β are the elasticity coefficients of the volume of production in terms of capital and labor costs.

If = 0.25, then a 1% increase in capital costs increases output by 0.25%.

Based on the analysis of elasticity coefficients in the Cobb-Douglas production function, we can distinguish:

1. a proportionally increasing production function when α + β = 1 (Q = K 0.5 * L 0.2).
2. disproportionately - increasing α + β > 1 (Q = K 0.9 * L 0.8);
3. decreasing α + β< 1 (Q = K 0,4 * L 0,2).

The optimal sizes of enterprises are not absolute in nature, and therefore cannot be established outside of time and outside the location, since they are different for different periods and economic regions.

The optimal size of the projected enterprise should provide a minimum of costs or a maximum of profit, calculated by the formulas:

Ts + S + Tp + K * En_ - minimum, P - maximum,

where Tc - the cost of delivery of raw materials and materials;
C - production costs, i.e. production cost;
Tp - the cost of delivering finished products to consumers;
K - capital costs;
En is the normative coefficient of efficiency;
P is the profit of the enterprise.

In other words, the optimal sizes of enterprises are understood as those that ensure the fulfillment of the tasks of the plan for output and increase in production capacity minus the reduced costs (taking into account capital investments in related industries) and the maximum possible economic efficiency.

The problem of optimizing production and, accordingly, answering the question of what should be the optimal size of the enterprise, with all its severity, also confronted Western entrepreneurs, presidents of companies and firms.

Those who failed to achieve the necessary scale found themselves in the unenviable position of high-cost producers, doomed to exist on the brink of ruin and ultimately bankruptcy.

Today, however, those US companies that are still striving to compete by saving on concentration are gaining rather than losing. In modern conditions, this approach initially leads to a decrease not only in flexibility, but also in production efficiency.

In addition, entrepreneurs remember that small businesses mean less investment and therefore less financial risk. As for the purely managerial side of the problem, American researchers note that enterprises with more than 500 employees become poorly managed, clumsy and poorly responsive to emerging problems.

Therefore, a number of American companies in the 60s went to the downsizing of their branches and enterprises in order to significantly reduce the size of the primary production links.

In addition to the simple mechanical disaggregation of enterprises, the organizers of production carry out a radical reorganization within enterprises, forming command and brigade org. structures instead of linear-functional ones.

When determining optimal size the enterprises of the firm use the concept of the minimum effective size. It is simply the lowest level of output at which a firm can minimize its long-run average cost.

Production function and the choice of the optimal production size.

Production is any human activity to transform limited resources - material, labor, natural - into finished products. The production function characterizes the relationship between the amount of resources used (factors of production) and the maximum possible output that can be achieved, provided that all available resources are used in the most rational way.

The production function has the following properties:

1. There is a limit to the increase in production that can be reached by increasing one resource and keeping other resources constant. If, for example, the amount of labor in agriculture is increased with constant amounts of capital and land, then sooner or later there comes a point when output stops growing.
2. Resources complement each other, but within certain limits, their interchangeability is also possible without reducing output. Manual labor, for example, may be replaced by the use of more machines, and vice versa.
3. The longer the time period, the more resources can be reviewed. In this regard, there are instant, short and long periods. Instant period - the period when all resources are fixed. A short period is a period when at least one resource is fixed. The long period is the period when all resources are variable.

Usually in microeconomics, a two-factor production function is analyzed, reflecting the dependence of output (q) on the amount of labor used ( L) and capital ( K). Recall that capital refers to the means of production, i.e. the number of machines and equipment used in production, measured in machine hours. In turn, the amount of labor is measured in man-hours.

As a rule, the considered production function looks like this:

q = AK α L β

A, α, β - given parameters. Parameter A is the coefficient of total productivity of production factors. It reflects the impact of technological progress on production: if the manufacturer introduces advanced technologies, the value of A increases, i.e., output increases with the same amount of labor and capital. The parameters α and β are the elasticity coefficients of output with respect to capital and labor, respectively. In other words, they show the percentage change in output when capital (labor) changes by one percent. These coefficients are positive, but less than unity. The latter means that with the growth of labor with constant capital (or capital with constant labor) by one percent, production increases to a lesser extent.

Building an isoquant

The above production function says that the producer can replace labor with capital and capital with labor, leaving the output unchanged. For example, in agriculture in developed countries, labor is highly mechanized, i.e. there are many machines (capital) for one worker. On the contrary, in developing countries the same output is achieved through a large amount of labor with little capital. This allows you to build an isoquant (Fig. 8.1).

The isoquant (line of equal product) reflects all combinations of two factors of production (labor and capital) in which output remains unchanged. On fig. 8.1 next to the isoquant is the release corresponding to it. Yes, release q 1, achievable using L1 labor and K1 capital or using L 2 labor and K 2 capital.

Rice. 8.1. isoquant

Other combinations of the amounts of labor and capital required to achieve a given output are also possible.

All combinations of resources corresponding to a given isoquant reflect technically effective ways production. Production method A is technically efficient compared to method B if it requires the use of at least one resource in a smaller amount, and all the others not in large quantities compared to method B. Accordingly, method B is technically inefficient compared to A. Technically inefficient modes of production are not used by rational entrepreneurs and do not belong to the production function.

It follows from the above that an isoquant cannot have a positive slope, as shown in Fig. 8.2.

The segment marked with a dotted line reflects all technically inefficient methods of production. In particular, in comparison with method A, method B to ensure the same output ( q 1) requires the same amount of capital but more labor. It is obvious, therefore, that way B is not rational and cannot be taken into account.

Based on the isoquant, it is possible to determine the marginal rate of technical replacement.

The marginal rate of technical replacement of factor Y by factor X (MRTS XY) is the amount of the factor Y(for example, capital), which can be abandoned by increasing the factor X(for example, labor) by 1 unit so that the output does not change (we remain on the same isoquant).

Rice. 8.2. Technically efficient and inefficient production

Consequently, the marginal rate of technical replacement of capital by labor is calculated by the formula
For infinitely small changes in L and K, it is
Thus, the marginal rate of technical replacement is the derivative of the isoquant function at a given point. Geometrically, it is the slope of the isoquant (Fig. 8.3).

Rice. 8.3. Marginal rate of technical replacement

When moving from top to bottom along the isoquant, the marginal rate of technical replacement decreases all the time, as evidenced by the decreasing slope of the isoquant.

If the producer increases both labor and capital, then this allows him to achieve a higher output, i.e. move to a higher isoquant (q2). An isoquant located to the right and above the previous one corresponds to a larger output. The set of isoquants forms an isoquant map (Fig. 8.4).

Rice. 8.4. Isoquant map

Special cases of isoquants

Recall that the given isoquants correspond to a production function of the form q = AK α L β. But there are other production functions. Let us consider the case when there is a perfect substitution of factors of production. Let us assume, for example, that skilled and unskilled loaders can be used in warehouse work, and the productivity of a skilled loader is N times higher than that of an unskilled one. This means that we can replace any number of skilled movers with unskilled ones at a ratio of N to one. Conversely, one can replace N unskilled loaders with one qualified one.

The production function then looks like: q = ax + by, where x- the number of skilled workers, y- the number of unskilled workers, a and b- constant parameters reflecting the productivity of one skilled and one unskilled worker, respectively. The ratio of the coefficients a and b is the marginal rate of technical replacement of unskilled movers by qualified ones. It is constant and equal to N: MRTSxy=a/b=N.

Let, for example, a qualified loader be able to process 3 tons of cargo per unit time (this will be the coefficient a in the production function), and an unskilled one - only 1 ton (coefficient b). This means that the employer can refuse three unskilled loaders, additionally hiring one qualified loader, so that the output (total weight of the handled load) remains the same.

The isoquant in this case is linear (Fig. 8.5).

Rice. 8.5. Isoquant under perfect substitution of factors

The tangent of the slope of the isoquant is equal to the marginal rate of technical replacement of unskilled movers by qualified ones.

Another production function is the Leontief function. It assumes a rigid complementarity of factors of production. This means that the factors can only be used in a strictly defined proportion, the violation of which is technologically impossible. For example, an air flight can normally be operated with at least one aircraft and five crew members. At the same time, it is impossible to increase aircraft-hours (capital) while simultaneously reducing man-hours (labor), and vice versa, and to keep output unchanged. Isoquants in this case have the form of right angles, i.e. the marginal rates of technical replacement are zero (Fig. 8.6). At the same time, it is possible to increase output (the number of flights) by increasing both labor and capital in the same proportion. Graphically, this means moving to a higher isoquant.

Rice. 8.6. Isoquants in the case of rigid complementarity of factors of production

Analytically, such a production function has the form: q = min (aK; bL), where a and b are constant coefficients reflecting the productivity of capital and labor, respectively. The ratio of these coefficients determines the proportion of the use of capital and labor.

In our flight example, the production function looks like this: q = min(1K; 0.2L). The fact is that the productivity of capital here is one flight for one aircraft, and the productivity of labor is one flight for five people, or 0.2 flights for one person. If an airline has a fleet of 10 aircraft and 40 flight personnel, then its maximum output will be: q = min( 1 x 8; 0.2 x 40) = 8 flights. At the same time, two aircraft will be idle on the ground due to a lack of personnel.

Let us finally look at the production function, which assumes the existence of a limited number of production technologies for the production of a given amount of output. Each of them corresponds to a certain state of labor and capital. As a result, we have a number of reference points in the “labor-capital” space, connecting which, we get a broken isoquant (Fig. 8.7).

Rice. 8.7. Broken isoquants in the presence of a limited number of production methods

The figure shows that output in volume q1 can be obtained with four combinations of labor and capital, corresponding to points A, B, C and D. Intermediate combinations are also possible, achievable in cases where an enterprise jointly uses two technologies to obtain a certain total release. As always, by increasing the amount of labor and capital, we move to a higher isoquant.

Production is the main area of ​​activity of the company. Firms use factors of production, which are also called input (input) factors of production.

A production function is the relationship between a set of factors of production and the maximum possible amount of product produced by a given set of factors.

A production function can be represented by many isoquants associated with different levels of output. This type of function, when an explicit dependence of the volume of production on the availability or consumption of resources is established, is called the output function.

In particular, output functions are widely used in agriculture, where they are used to study the impact on yields of such factors as, for example, different types and compositions of fertilizers, tillage methods. Along with similar production functions, the inverse functions of production costs are used. They characterize the dependence of resource costs on output volumes (strictly speaking, they are inverse only to PF with interchangeable resources). Special cases of PF can be considered a cost function (connection between the volume of production and production costs), an investment function: the dependence of the required investment on the production capacity of the future enterprise.

There is a wide variety of algebraic expressions that can be used to represent production functions. The simplest model is a special case of the general production analysis model. If only one activity is available to the firm, then the production function can be represented by rectangular isoquants with constant returns to scale. There is no ability to change the ratio of factors of production, and the elasticity of substitution is certainly zero. This is a highly specialized manufacturing function, but its simplicity explains its widespread use in many models.

Mathematically, production functions can be represented in various forms - from such simple ones as a linear dependence of the result of production on one factor under study, to very complex systems of equations, including recurrence relations that connect the states of the object under study in different periods of time..

The production function is graphically represented by a family of isoquants. The further the isoquant is located from the origin, the greater the volume of production it reflects. Unlike an indifference curve, each isoquant characterizes a quantified amount of output.

Figure 2 _ Isoquants corresponding to different production volumes

On fig. 1 shows three isoquants corresponding to a production volume of 200, 300 and 400 units. It can be said that for the production of 300 units of production, K 1 units of capital and L 1 units of labor or K 2 units of capital and L 2 units of labor are needed, or any other combination of them from the set represented by the isoquant Y 2 = 300.

In the general case, in the set X of admissible sets of production factors, a subset X c is allocated, called the isoquant of the production function, which is characterized by the fact that for any vector the equality

Thus, for all sets of resources corresponding to the isoquant, the volumes of output are equal. Essentially, an isoquant is a description of the possibility of mutual substitution of factors in the process of production of goods, providing a constant volume of production. In this regard, it is possible to determine the coefficient of mutual replacement of resources, using the differential relation along any isoquant

Hence, the coefficient of equivalent replacement of a pair of factors j and k is equal to:

The obtained ratio shows that if production resources are replaced in a ratio equal to the ratio of incremental productivity, then the amount of output remains unchanged. It must be said that knowledge of the production function makes it possible to characterize the extent of the possibility to carry out the mutual replacement of resources in efficient technological methods. To achieve this goal, the coefficient of elasticity of the replacement of resources for products is used.

which is calculated along the isoquant at a constant level of costs of other production factors. The value sjk is a characteristic of the relative change in the coefficient of mutual replacement of resources when the ratio between them changes. If the ratio of interchangeable resources changes by sjk percent, then the mutual replacement ratio sjk will change by one percent. In the case of a linear production function, the mutual substitution coefficient remains unchanged for any ratio of resources used, and therefore we can assume that the elasticity s jk = 1. Accordingly, large values ​​of sjk indicate that greater freedom is possible in replacing production factors along the isoquant and, at the same time, the main characteristics production function (productivity, interchange factor) will change very little.

For power production functions for any pair of interchangeable resources, the equality s jk = 1 is true.

The representation of an effective technological set using a scalar production function turns out to be insufficient in cases where it is impossible to manage with a single indicator describing the results of the production facility, but it is necessary to use several (M) output indicators (Figure 3).

Figure 3 _ Various behaviors of isoquants

Under these conditions, one can use the vector production function

The important concept of marginal (differential) productivity is introduced by the relation

All other main characteristics of scalar PFs admit a similar generalization.

Like indifference curves, isoquants are also classified into different types.

For a linear production function of the form

where Y is the volume of production; A , b 1 , b 2 parameters; K , L costs of capital and labor, and the complete replacement of one resource by another isoquant will have a linear form (Figure 4, a).

For the power production function

Then the isoquants will look like curves (Figure 4, b).

If the isoquant reflects only one technological method for the production of a given product, then labor and capital are combined in the only possible combination (Figure 4, c).

d) Broken isoquants

Figure 4 - Different variants of isoquants

Such isoquants are sometimes called Leontief-type isoquants after the American economist W.V. Leontiev, who put this type of isoquant as the basis of the inputoutput method he developed.

The broken isoquant implies the presence of a limited number of technologies F (Figure 4, d).

Isoquants of this configuration are used in linear programming to substantiate the theory of optimal resource allocation. Broken isoquants most realistically represent the technological capabilities of many production facilities. However, in economic theory, isoquant curves are traditionally used, which are obtained from broken lines with an increase in the number of technologies and an increase in breakpoints, respectively.

The most widely used are multiplicative-power forms of representation of production functions. Their peculiarity is as follows: if one of the factors is equal to zero, then the result vanishes. It is easy to see that this realistically reflects the fact that in most cases all analyzed primary resources are involved in production, and without any of them, production is impossible. In its most general form (it is called canonical), this function is written as follows:

Here, the coefficient A in front of the multiplication sign takes into account the dimension, it depends on the chosen unit of measurement of costs and output. Factors from the first to the nth can have different content depending on what factors influence the overall result (output). For example, in the PF, which is used to study the economy as a whole, it is possible to take the volume of the final product as a performance indicator, and the factors - the number of employed people x1, the sum of fixed and working capital x2, the area of ​​land used x3. There are only two factors in the Cobb-Douglas function, with the help of which an attempt was made to assess the relationship of factors such as labor and capital with the growth of US national income in the 20-30s. XX century:

N = A Lb Kv,

where N is the national income; L and K - respectively, the volume of applied labor and capital (for details, see; Cobb-Douglas function).

The power coefficients (parameters) of the multiplicative power production function show the share in the percentage increase in the final product that each of the factors contributes (or by what percentage the product will increase if the costs of the corresponding resource are increased by one percent); they are coefficients of elasticity of production with respect to the costs of the corresponding resource. If the sum of the coefficients is 1, this means that the function is homogeneous: it increases in proportion to the increase in the amount of resources. But such cases are also possible when the sum of the parameters is greater or less than unity; this shows that an increase in costs leads to a disproportionately large or disproportionately small increase in output - economies of scale.

In the dynamic version, different forms of the production function are used. For example, in the 2-factor case: Y(t) = A(t) Lb(t) Kv(t), where the factor A(t) usually increases over time, reflecting the overall increase in the efficiency of production factors in dynamics.

Taking the logarithm and then differentiating this function with respect to t, one can obtain the ratio between the growth rates of the final product (national income) and the growth of production factors (the growth rates of variables are usually described here as a percentage).

Further “dynamization” of the PF may consist in the use of variable elasticity coefficients.

The described PF relationships are of a statistical nature, i.e., they appear only on average, in a large number of observations, since not only the analyzed factors, but also many unaccounted ones, actually affect the result of production. In addition, the applied indicators of both costs and results are inevitably products of complex aggregation (for example, a generalized indicator of labor costs in a macroeconomic function includes labor costs of different productivity, intensity, qualifications, etc.).

A special problem is taking into account the factor of technical progress in macroeconomic PFs (for more details, see the article “Scientific and technical progress”). With the help of PF, the equivalent interchangeability of factors of production is also studied (see Elasticity of substitution of resources), which can be either constant or variable (that is, dependent on the volume of resources). Accordingly, functions are divided into two types: with constant elasticity of substitution (CES - Constant Elasticity of Substitution) and with variable (VES - Variable Elasticity of Substitution) (see below).

In practice, three main methods are used to determine the parameters of macroeconomic PFs: based on the processing of time series, based on data on the structural elements of aggregates, and on the distribution of national income. The last method is called distribution.

When constructing a production function, it is necessary to get rid of the phenomena of multicollinearity of parameters and autocorrelation - otherwise gross errors are inevitable.

Here are some important production functions.

Linear production function:

P = a1x1 + ... + anxn,

where a1, ..., an are the estimated parameters of the model: here the factors of production are substituted in any proportions.

CES Feature:

P \u003d A [(1 - b) K-b + bL-b] -c / b,

in this case, the elasticity of resource substitution does not depend on either K or L and, therefore, is constant:

This is where the name of the function comes from.

The CES function, like the Cobb-Douglas function, is based on the assumption of a constant decrease marginal norm replacement of used resources. Meanwhile, the elasticity of the replacement of capital by labor and, conversely, of labor by capital in the Cobb-Douglas function, which is equal to one, here can take on different values ​​that are not equal to one, although it is constant. Finally, unlike the Cobb-Douglas function, the logarithm of the CES function does not lead it to a linear form, which forces us to use more complex methods of nonlinear regression analysis.

The production function is always concrete, i.e. intended for this technology. New technology - new productive function. The production function determines the minimum amount of input needed to produce a given amount of product.

Production functions, no matter what kind of production they express, have the following general properties:

• 1) An increase in production due to an increase in costs for only one resource has a limit (you cannot hire many workers in one room - not everyone will have places).
• 2) Factors of production can be complementary (workers and tools) and interchangeable (production automation).

In its most general form, the production function looks like this:

where is the volume of output;

K- capital (equipment);

M - raw materials, materials;

T - technology;

N - entrepreneurial abilities.

The simplest is the two-factor model of the Cobb-Douglas production function, which reveals the relationship between labor (L) and capital (K).

These factors are interchangeable and complementary. Back in 1928, American scientists - economist P. Douglas and mathematician C. Cobb - created a macroeconomic model that allows you to evaluate the contribution of various factors of production to an increase in production or national income. This function has the following form:

where A is a production coefficient showing the proportionality of all functions and changes when the basic technology changes (in 30-40 years);

K, L- capital and labor;

b, c - coefficients of elasticity of the volume of production for capital and labor costs.

If b = 0.25, then a 1% increase in capital costs increases output by 0.25%.

Based on the analysis of the coefficients of elasticity in the Cobb-Douglas production function, we can distinguish:

1) a proportionally increasing production function, when

2) disproportionately - increasing

3) decreasing

Let us consider a short period of a firm's activity, in which labor is the variable of two factors. In such a situation, the firm can increase production by using more labor resources (Figure 5).

Figure 5_ Dynamics and relationship of total average and marginal products

Figure 5 shows a graph of the Cobb-Douglas production function with one variable is shown - the TRn curve.

The Cobb-Douglas function had a long and successful life without serious rivals, but recently she has been a strong competitor new feature Arrow, Chenery, Minhasa and Solow, which we will refer to as SMAC for short. (Brown and De Cani also developed this feature independently). The main difference of the SMAC function is that the elasticity of substitution constant y is introduced, which is different from one (as in the Cobb-Douglas function) and zero: as in the input-output model.

The diversity of market and technological conditions that exists in today's economy suggests the impossibility of satisfying the basic requirements of reasonable aggregation, except perhaps for individual firms in the same industry or limited sectors of the economy.

Thus, in the economic and mathematical models of production, each technology can be graphically represented by a point, the coordinates of which reflect the minimum necessary costs resources K and L for the production of a given output. Many such points form a line of equal output, or an isoquant. That is, the production function is graphically represented by a family of isoquants. The further the isoquant is located from the origin, the greater the volume of production it reflects. Unlike an indifference curve, each isoquant characterizes a quantified amount of output. Usually in microeconomics, a two-factor production function is analyzed, reflecting the dependence of output on the amount of labor and capital used.