I. ECONOMIC THEORY

10. Production function. Law of diminishing returns. scale effect

production function is the relationship between a set of factors of production and the maximum possible volume of product produced using 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:

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.

,

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;

Elasticity coefficients of output for capital and labor inputs.

If = 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. The graph of the Cobb-Douglas production function with one variable is shown in Fig. 10.1 (curve TP n).

In the short run, the law of diminishing marginal productivity applies.

The law of diminishing marginal productivity operates in the short run when one factor of production remains unchanged. The operation of the law assumes an unchanged state of technology and production technology, if the latest inventions and other technical improvements are applied in the production process, then an increase in output can be achieved using the same production factors. That is, technological progress can change the boundaries of the law.

If capital is a fixed factor and labor is a variable factor, then the firm can increase production by employing more labor. But on the law of diminishing marginal productivity, a consistent increase in a variable resource, while the others remain unchanged, leads to diminishing returns of this factor, that is, to a decrease in the marginal product or marginal productivity of labor. If the hiring of workers continues, then in the end, they will interfere with each other (marginal productivity will become negative) and output will decrease.

The marginal productivity of labor (marginal product of labor - MP L) is the increase in output from each subsequent unit of labor

those. productivity gain to total product (TP L)

The marginal capital product MP K is defined similarly.

Based on the law of diminishing productivity, let's analyze the relationship between total (TP L), average (AP L) and marginal products (MP L) (Fig. 10.1).

There are three stages in the movement of the total product (TP) curve. At stage 1, it rises at an accelerating pace, since the marginal product (MP) increases (each new worker brings more production than the previous one) and reaches a maximum at point A, that is, the growth rate of the function is maximum. After point A (stage 2), due to the law of diminishing returns, the MP curve falls, that is, each hired worker gives a smaller increment in the total product compared to the previous one, so the growth rate of TP after TS slows down. But as long as MP is positive, TP will still increase and peak at MP=0.

Rice. 10.1. Dynamics and relationship of the total average and marginal products

At stage 3, when the number of workers becomes redundant in relation to fixed capital (machines), MR becomes negative, so TP begins to decline.

The configuration of the average product curve AR is also determined by the dynamics of the MP curve. At stage 1, both curves grow until the increment in output from newly hired workers is greater than the average productivity (AP L) of previously hired workers. But after point A (max MP), when the fourth worker adds less to the total product (TP) than the third, MP decreases, so the average output of four workers also decreases.

scale effect

1. Manifested in a change in long-term average production costs (LATC).

2. The LATC curve is the envelope of the firm's minimum short-term average cost per unit of output (Figure 10.2).

3. The long-term period in the company's activity is characterized by a change in the number of all production factors used.

Rice. 10.2. Curve of long-run and average costs of the firm

The reaction of LATC to a change in the parameters (scale) of a firm can be different (Fig. 10.3).

Rice. 10.3. Dynamics of long-term average costs

Stage I: positive effect of scale	An increase in output is accompanied by a decrease in LATC, which is explained by the effect of savings (for example, due to the deepening of the specialization of labor, the use of new technologies, the efficient use of waste).
Stage II: constant returns to scale	When the volume changes, the costs remain unchanged, that is, an increase in the amount of resources used by 10% caused an increase in production volumes also by 10%.
Stage III: negative scale effect	An increase in production (for example, by 7%) causes an increase in LATC (by 10%). The reason for the damage from the scale can be technical factors (unjustified gigantic size of the enterprise), organizational reasons (growth and inflexibility of the administrative and management apparatus).

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 more profit, the number of products at a particular enterprise. These questions are about long-term profit maximization;
3. the entrepreneur may be faced with the task of 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 costs 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 for 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 common 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 the worker 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 this point with the origin, D – 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 in the variable factor X is observed in 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, the total product (TR) receives the largest growth.

Thus, the production function is a function that allows you to determine the maximum possible 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 calculation of 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 with the help of a comparative economic analysis of 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 size of enterprises is understood as those that provide the targets for 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 the optimal size of the enterprise, firms 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 called any human 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, 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 this isoquant reflect technically efficient methods of 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 the output of 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 two technologies are used together to obtain a certain total output . As always, by increasing the amount of labor and capital, we move to a higher isoquant.

production functions are called economic-mathematical models that link variable costs with output values. The concepts of "costs" and "output" are related, as a rule, to the production process; this explains the origin of the name of this type of models. If the economy of a region or a country as a whole is considered, then aggregated production functions are developed, in which the output is an indicator of the total social product. Particular cases of production functions are release features (dependence of production volume on the availability or consumption of resources), cost functions (the relationship between the volume of production and production costs), capital cost functions (dependence of capital investments on the production capacity of the enterprises being created), etc.

Multiplicative forms of representation of production functions are widely used. In its most general form, the multiplicative production function is written as follows:

Here the coefficient BUT determines the dimension of quantities and depends on the chosen units of measurement of costs and output. Factors X i represent influencing factors and may have different economic content depending on what factors affect the output R. The power parameters α, β, ..., γ show the share in the growth of the final product that each of the factors contributes; they're called coefficients of elasticity of production with respect to costs of the corresponding resource and show by what percentage the output increases with an increase in the cost of this resource by one percent.

The sum of elasticity coefficients is important for characterizing the properties of the production function. Suppose that the costs of all types of resources increase in k once. Then the value of output in accordance with (7.16) will be

Therefore, if , then with an increase in costs in to times the output also increases in k once; the production function in this case is linearly homogeneous. At E > 1 the same increase in costs will lead to an increase in output by more than to times, and at E < 1 – менее чем в to times (the so-called scale effect).

An example of multiplicative production functions is the well-known Cobb-Douglas production function:

N - national income;

BUT – coefficient of dimension;

L, K - the volume of applied labor and fixed capital, respectively;

α and β are coefficients of elasticity of national income to labor L and capital TO.

This function was used by American researchers in the analysis of the development of the US economy in the 30s of the last century.

The efficiency of resource use is characterized by two main indicators: average (absolute ) efficiency resource

and marginal efficiency resource

The economic meaning of μi is obvious; depending on the type of resource, it characterizes such indicators as labor productivity, capital productivity, etc. The value v i shows the marginal increase in product output with an increase in the cost of the i-th resource by a "small unit" (by 1 ruble, by 1 standard hour, etc.).

Many points n -dimensional space of production factors (resources) that satisfy the condition of output constancy R (X ) = C, called isoquant. The most important properties of isoquants are the following: isoquants do not intersect with each other; a larger output value corresponds to an isoquant that is more distant from the origin of coordinates; if all resources are absolutely necessary for production, then isoquants have no common points with coordinate hyperplanes and coordinate axes.

In material production, the concept of interchangeability of resources. In the theory of production functions, the possibilities of substitution of resources characterize the production function in terms of various combinations of inputs of resources that lead to the same level of output. Let's explain this with a hypothetical example. Let the production of a certain amount of agricultural products require 10 workers and 2 tons of fertilizer, and if only 1 ton of fertilizer is applied to the soil, 12 workers will be required to obtain the same crop. Here, 1 ton of fertilizer (the first resource) is replaced by the labor of two workers (the second resource).

The conditions for equivalent interchangeability of resources at some point follow from the equality dP = 0:

From here marginal rate of substitution (equivalent substitutability) of any two resources k and l given by the formula

(7.20)

The marginal rate of substitution as an indicator of the production function characterizes the relative efficiency of interchangeable factors of production when moving along the isoquant. For example, for the Cobb-Douglas function, the marginal rate of replacement of labor costs by capital costs, i.e. production assets has the form

(7.21)

The minus sign in the right parts of formulas (7.20) and (7.21) means that at a fixed volume of production, an increase in one of the interchangeable resources corresponds to a decrease in the other.

Example 7.1. Consider an example of the Cobb-Douglas production function, for which the coefficients of output elasticity for labor and capital are known: α = 0.3; β = 0.7, as well as labor and capital costs: L = 30 thousand people; To = 490 million rubles. Under these conditions, the marginal rate of replacement of production assets by labor costs is equal to

Thus, in this conditional example, at those points of the two-dimensional space ( L, K ), where labor and capital resources are interchangeable, a decrease in production assets by 7 thousand rubles. can be offset by an increase in labor costs per person, and vice versa.

Related to the concept of the marginal rate of substitution is the concept elasticity of resource substitution. The coefficient of elasticity of substitution characterizes the ratio of the relative change in the ratio of resource costs k and l to a relative change in the marginal rate of substitution of these resources:

This coefficient shows by what percentage the ratio between interchangeable resources must change in order for the marginal rate of replacement of these resources to change by 1%. The higher the elasticity of resource substitution, the more widely they can replace each other. With infinite elasticity () there are no boundaries for the interchangeability of resources. With zero elasticity of substitution () there is no possibility of replacement; in this case, the resources complement each other and must be used in a certain ratio.

Consider, in addition to the Cobb-Douglas function, some other production functions widely used as econometric models. Linear production function has the form

are the estimated parameters of the model;

, - factors of production, mutually substitutable in any proportions (elasticity of substitution ).

The isoquants of this production function form a family of parallel hyperplanes in a non-negative orthant n -dimensional space of factors.

Many studies use production functions with constant elasticity of substitution.

(7.23)

The production function (7.23) is a homogeneous function of the degree P. All elasticities of substitution of resources are equal to each other:

Therefore, this function is called function with constant elasticity of substitution (CES function ). If , the elasticity of substitution is less than one; if , the value is greater than one; when , the CES function is transformed into a multiplicative power production function (7.16).

Two factor function CES has the form

At n = 1 and p = 0, this function is transformed into a function of the type of the Cobb-Douglas function (7.17).

In addition to production functions with constant coefficients of elasticity of output from resources and constant elasticity of resource substitution, more general functions are also used in economic analysis and forecasting. An example is the function

This function differs from the Cobb-Douglas function by the factor , where z = K/L- capital-labor ratio (capital-labor ratio) of labor, and in it the elasticity of substitution takes on different values ​​depending on the level of capital-labor ratio. In this regard, this function belongs to the type production functions with variable elasticity of substitution (VES functions ).

Let us turn to the consideration of a number of issues of the practical use of production functions in the economy.

chemical analysis. Macroeconomic production functions are used as a tool for forecasting the volume of gross output, final product and national income, to analyze the comparative efficiency of production factors. Thus, an important condition for the growth of production and labor productivity is an increase in the capital-labor ratio of labor. If for the Cobb-Douglas function

set the condition of linear homogeneity , then from the ratio between labor productivity ( P/L ) and capital-labor ratio ( K/L )

(7.24)

it follows that labor productivity grows more slowly than the capital-labor ratio, since . This conclusion, like many other results of analysis based on production functions, is always valid for static production functions that do not take into account the improvement of technical means of labor and the qualitative characteristics of the resources used, i.e. regardless of technological progress. To estimate the parameters of the model (7.24), it is linearized by taking a logarithm:

Along with a quantitative increase in the amount of resources used (labor resources, production assets, etc.), the most important factor in the growth of production is scientific and technological progress, which consists in improving technical means and technology, improving the skills of workers, and improving the organization of production management. Static econometric models, including static production functions, do not take into account the factor of technical progress; therefore, dynamic macroeconomic production functions are used, the parameters of which are determined by processing time series. Technological progress is usually reflected in production functions in the form of a time-dependent trend in the development of production.

For example, the Cobb-Douglas function, taking into account the technological progress factor, takes the following form:

In model (7.25), the factor reflects the trend in the development of production associated with scientific and technological progress. In this multiplier t - time, and λ - the rate of growth of output due to technical progress. In the practical use of model (7.25), to estimate its parameters, linearization is carried out by taking logarithms, similarly to model (7.24):

It should be specially noted that when constructing production functions, as for all multifactorial econometric models, a very important point is the correct selection of influencing factors. In particular, it is necessary to get rid of the phenomena of multicollinearity of factors and the phenomena of autocorrelation within each of them. This issue is described in detail in paragraph 7.1 of this chapter. When estimating the parameters of production functions based on statistical observations, including time series, the main method is the least squares method.

Consider the application of production functions for economic analysis and forecasting on a conditional example from the field of labor economics.

Example 7.2. Let the output of the industry be characterized by a production function of the Cobb-Douglas type:

R - the volume of output (million rubles);

T - the number of industry employees (thousand people);

F - the average annual cost of fixed production assets (million rubles).

Suppose the parameters of this production function are known and equal: a = 0.3; β = 0.7; dimension factor A = = 0.6 (thousand rubles/person) 0.3. The value of the average annual cost of fixed production assets is also known F = 900 million rubles. These conditions require:

• 1) determine the number of industry employees required to produce products in the amount of 300 million rubles;
• 2) find out how output will change with an increase in the number of employees by 1% and the same volumes of production assets;
• 3) evaluate the interchangeability of material and labor resources.

To answer the question of the first task, we linearize this production function by taking the logarithm in natural base;

whence it follows that

Substituting the initial data, we get

Hence (thousand people).

Let's consider the second task. Since , this production function is linearly homogeneous; in accordance with this, the AIR coefficients are the coefficients of output elasticity for labor and funds, respectively. Consequently, an increase in the number of employees in the industry by 1% with a constant volume of production assets will lead to an increase in output by 0.3%, i.e. the issue will amount to 300.9 million rubles.

Turning to the third task, we calculate the marginal rate of replacement of production assets with labor resources. According to formula (7.21)

Thus, subject to the interchangeability of resources to ensure the constancy of output (i.e., when moving along the isoquant), a decrease in the production assets of the industry by 3.08 thousand rubles. can be compensated by an increase in labor resources by 1 person, and vice versa.

Characterizes the relationship between the amount of resources used () 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. short period— the period when at least one resource is fixed. A long period - 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 () and capital () used. Recall that capital refers to the means of production, i.e. the number of machines and equipment used in production and measured in machine hours (topic 2, paragraph 2.2). In turn, the amount of labor is measured in man-hours.

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

A, α, β are given parameters. Parameter BUT is the coefficient of total factor productivity. It reflects the impact of technological progress on production: if the manufacturer introduces advanced technologies, the value BUT increases, i.e. output increases with the same amount of labor and capital. Options 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 given production function says that the producer can replace labor by captain and capital by 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).

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. Thus, output , is achievable using labor and capital, or using labor and captain.

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 efficient production methods. Mode of production A is technically efficient in comparison with the method AT, if it requires the use of at least one resource in a smaller amount, and all the others not in large quantities in comparison with the method AT. Accordingly, the method AT is technically inefficient compared to BUT. 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 the method BUT way AT to ensure the same output () requires the same amount of capital, but more labor. It is obvious, therefore, that the 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.

Marginal Rate of Technical Replacement of Factor Y by Factor X (MRTS XY)- this is the amount of a factor (for example, capital), which can be abandoned when the factor (for example, labor) is increased 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

With infinitesimal changes L and K she 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 (q 2). An isoquant located to the right and above the previous one corresponds to a larger output. The set of isoquants forms isoquant map(Fig. 8.4).

Rice. 8.4. Isoquant map

Special cases of isoquants

Recall that the given ones correspond to a production function of the form . 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 in N times higher than the unskilled. This means that we can replace any number of qualified movers with unskilled ones in the ratio N to one. Conversely, one can replace N unskilled loaders with one qualified one.

In this case, the production function has the form: where is the number of skilled workers, is the number of unskilled workers, a and b- constant parameters reflecting the productivity of one skilled and one unskilled worker, respectively. Coefficient ratio a and b- the marginal rate of technical replacement of unskilled loaders by qualified ones. It is constant and equal 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 plane, 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 the output in the volume q 1 can be obtained with four combinations of labor and capital corresponding to the points A, B, C and D. Intermediate combinations are also possible, achievable when an enterprise uses two technologies together to obtain a certain aggregate output. As always, by increasing the amount of labor and capital, we move to a higher isoquant.

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.

Consideration of the concept of "production function" will begin with the simplest case, when production is due to only one factor. In this case, the production function - this is a function, the independent variable of which takes the values ​​of the resource used (factor of production), and the dependent variable - the values ​​of the volume of output y=f(x).

In this formula, y is a function of one variable x. In this regard, the production function (PF) is called one-resource or one-factor. Its domain of definition is the set of non-negative real numbers. The symbol f is a characteristic of the production system that converts a resource into an output.

Example 1. Take the production function f in the form f(x)=ax b , where x is the value of the resource expended (for example, working hours), f(x) is the volume of output (for example, the number of refrigerators ready for shipment). The values ​​a and b are parameters of the production function f. Here a and b are positive numbers and the number b1, the parameter vector is a two-dimensional vector (a,b). The production function y=ax b is a typical representative of a wide class of one-factor PFs.

Rice. one.

The graph shows that with the increase in the value of the resource expended, y grows. However, at the same time, each additional unit of the resource gives an ever smaller increase in the volume y of output. The noted circumstance (an increase in the volume of y and a decrease in the increase in the volume of y with an increase in the value of x) reflects the fundamental position of economic theory (well confirmed by practice), called the law of diminishing efficiency (diminishing productivity or diminishing returns).

PFs can have different areas of use. The input-output principle can be implemented both at the micro- and macroeconomic levels. Let's focus on the microeconomic level first. PF y=ax b , discussed above, can be used to describe the relationship between the value of the spent or used resource x during the year at a separate enterprise (firm) and the annual output of this enterprise (firm). The role of the production system here is played by a separate enterprise (firm) - we have a microeconomic PF (MIPF). At the microeconomic level, an industry, an intersectoral production complex, can also act as a production system. MIPF are built and used mainly for solving problems of analysis and planning, as well as forecasting problems.

The PF can be used to describe the relationship between the annual labor input of a region or country as a whole and the annual final output (or income) of that region or country as a whole. Here, a region or a country as a whole acts as a production system - we have a macroeconomic level and a macroeconomic PF (MAPF). MAFFs are built and actively used to solve all three types of problems (analysis, planning and forecasting).

We now turn to the consideration of production functions of several variables.

Production function of several variables is a function whose independent variables take the values ​​of the volumes of resources spent or used (the number of variables n is equal to the number of resources), and the value of the function has the meaning of the values ​​of output volumes:

y=f(x)=f(x 1 ,…,х n).

In the formula, y (y0) is a scalar, and x is a vector quantity, x 1 ,…,x n are the coordinates of the vector x, that is, f(x 1 ,…,x n) is a numerical function of several variables x 1 ,…,x n. In this regard, the PF f(x 1 ,…,х n) is called multi-resource or multi-factorial. More correct is such a symbolism f(x 1 ,…, x n ,a), where a is the vector of PF parameters.

According to the economic sense, all variables of this function are non-negative, therefore, the domain of definition of the multifactorial PF is the set of n-dimensional vectors x, all coordinates x 1 ,…, x n of which are non-negative numbers.

A graph of a function of two variables cannot be drawn in a plane. The production function of several variables can be represented in a three-dimensional Cartesian space, two coordinates of which (x1 and x2) are plotted on the horizontal axes and correspond to resource costs, and the third (q) is plotted on the vertical axis and corresponds to the output of the product (Fig. 2). The graph of the production function is the surface of the "hill", rising with the growth of each of the coordinates x1 and x2.

For a separate enterprise (firm) producing a homogeneous product, the PF f(x 1 ,…,х n) can link the volume of output with the cost of working time for various types of labor activity, various types of raw materials, components, energy, fixed capital. PF of this type characterize the current technology of the enterprise (firm).

When constructing the PF for a region or country as a whole, the aggregate product (income) of the region or country, usually calculated at constant rather than current prices, is often taken as the value of annual output Y, fixed capital is considered as resources (x 1 (= K) - the volume of fixed capital used during the year) and live labor (x 2 (= L) - the number of units of living labor expended during the year), usually calculated in value terms. Thus, a two-factor PF Y=f(K,L) is built. From two-factor PF are moving to three-factor. In addition, if the PF is constructed from time series data, then technological progress can be included as a special factor in production growth.

PF y=f(x 1 ,x 2) is called static, if its parameters and its characteristic f do not depend on time t, although the volume of resources and the volume of output may depend on time t, that is, they can be represented in the form of time series: x 1 (0), x 1 (1),…, x 1 (T); x 2 (0), x 2 (1), ..., x 2 (T); y(0), y(1),…,y(T); y(t)=f(x 1 (t), x 2 (t)). Here t is the number of the year, t=0.1,…,Т; t= 0 - base year of the time interval covering years 1,2,…,T.

Example2. To model a particular region or country as a whole (that is, to solve problems at the macroeconomic, as well as at the microeconomic level), the PF of the form y= is often used, where a 0 , a 1 , and 2 are the parameters of the PF. These are positive constants (often a 1 and a 2 are such that a 1 + a 2 =1). The PF of the form just given is called the Cobb-Douglas PF (CPKD) after the two American economists who proposed its use in 1929.

PPCD is actively used to solve various theoretical and applied problems due to its structural simplicity. PFKD belongs to the class of so-called multiplicative PFs (MPFs). In applications, PFKD x 1 = K is equal to the volume of fixed capital used (the volume of fixed assets used - in domestic terminology), - the cost of living labor, then PFKD takes on the form often used in the literature:

Example3. Linear PF (LPF) has the form: (two-factor) and (multifactor). PSF belongs to the class of so-called additive PF (APF). The transition from the multiplicative PF to the additive one is carried out using the logarithm operation. For a two-factor multiplicative PF

this transition looks like: . Introducing the appropriate substitution, we obtain an additive PF.

For the production of a particular product, a combination of various factors is required. Despite this, various production functions share a number of common properties.

For definiteness, we confine ourselves to production functions of two variables. First of all, it should be noted that such a production function is defined in a non-negative orthant of the two-dimensional plane, that is, at. The PF satisfies the following set of properties:

• 1) there is no output without resources, i.e. f(0,0,a)=0;
• 2) in the absence of at least one of the resources, there is no output, i.e. ;
• 3) with an increase in the cost of at least one resource, the volume of output increases;

4) with an increase in the cost of one resource with a constant amount of another resource, the volume of output increases, i.e. if x>0, then;

5) with an increase in the cost of one resource with a constant amount of another resource, the value of the increase in output for each additional unit of the i-th resource does not increase (the law of diminishing efficiency), i.e. if then;

• 6) with the growth of one resource, the marginal efficiency of another resource increases, i.e. if x>0, then;
• 7) PF is a homogeneous function, i.e. ; at p>1 we have an increase in production efficiency due to the increase in the scale of production; at p

Production functions allow us to quantitatively analyze the most important economic dependencies in the sphere of production. They make it possible to estimate the average and marginal efficiency of various production resources, the elasticity of output for various resources, the marginal rates of substitution of resources, the effect of production scale, and much more.

Task 1. Let a production function be given that relates the volume of output of an enterprise with the number of workers, production assets and the volume of machine hours used

It is necessary to determine the maximum output under restrictions

Solution. To solve the problem, we compose the Lagrange function

we differentiate it with respect to variables, and equate the resulting expressions to zero:

It follows from the first and third equations that, therefore,

whence we obtain a solution for which y=2. Since, for example, the point (0,2,0) belongs to the admissible region and y=0 in it, we conclude that the point (1,1,1) is the global maximum point. The economic implications of the resulting solution are obvious.

It should also be noted that the production function describes a set of technically efficient production methods (technologies). Each technology is characterized by a certain combination of resources required to obtain a unit of output. Although production functions are different for different types of production, they all have common properties:

• 1. There is a limit to the increase in production that can be achieved by increasing the cost of one resource, all other things being equal. This means that in a firm with a given number of machines and production facilities, there is a limit to increasing production by attracting more workers. The increase in output with an increase in the number of employed will approach zero.
• 2. There is a certain complementarity (complementarity) of production factors, but without a reduction in production volumes, a certain interrelation of these factors is also possible. For example, the work of workers is effective if they are provided with all the necessary tools. In the absence of such tools, the volume can be reduced or increased with an increase in the number of employees. In this case, one resource is replaced by another.
• 3. Production method BUT considered technically more efficient than B, if it involves the use of at least one resource in less, and all the others - not in more than the method B. Technically inefficient methods are not used by rational producers.
• 4. If way BUT involves the use of some resources in more, and others - in a smaller amount than the method B, these methods are incomparable in terms of technical efficiency. In this case, both methods are considered technically efficient and are included in the production function. Which one to choose depends on the price ratio of the resources used. This choice is based on cost-effectiveness criteria. Therefore, technical efficiency is not identical to economic efficiency.

Technical efficiency is the maximum possible volume of production achieved as a result of the use of available resources. Economic efficiency is the production of a given volume of output at minimum cost. 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:

Graphically, each production method (technology) can be represented by a point that characterizes the minimum required set of two factors needed to produce a given volume of output (Fig. 3).

The figure shows various methods of production (technology): T 1 , T 2 , T 3 , characterized by different ratios in the use of labor and capital: T 1 = L 1 K 1 ; T 2 = L 2 K 2 ; T 3 = L 3 K 3 . the slope of the beam shows the size of the application of various resources. The higher the angle of inclination of the beam, the greater the cost of capital and the lower the cost of labor. Technology T 1 is more capital intensive than technology T 2 .

Rice. 3.

If you connect different technologies with a line, you get an image of the production function (line of equal output), which is called isoquants. The figure shows that the volume of production Q can be achieved with different combinations of factors of production (T 1, T 2, T 3, etc.). The upper part of the isoquant reflects capital-intensive technologies, while the lower part reflects labor-intensive technologies.

An isoquant map is a set of isoquants that reflect the maximum achievable level of output for any given set of production factors. The farther the isoquant is from the origin, the greater the output. Isoquants can pass through any point in space where there are two factors of production. The meaning of the isoquant map is similar to the meaning of the indifference curve map for consumers.

Fig.4.

Isoquants have the following properties:

• 1. Isoquants do not intersect.
• 2. The greater distance of the isoquant from the origin corresponds to a greater level of output.
• 3. Isoquants - descending curves, have a negative slope.

Isoquants are similar to indifference curves with the only difference that they reflect the situation not in the sphere of consumption, but in the sphere of production.

The negative slope of the isoquants is explained by the fact that an increase in the use of one factor at a certain volume of output of the product will always be accompanied by a decrease in the amount of another factor.

Consider possible isoquant maps

On fig. Figure 5 shows some isoquant maps that characterize various situations that arise when two resources are consumed in production. Rice. 5a corresponds to the absolute mutual substitution of resources. In the case shown in Fig. 5b, the first resource can be completely replaced by the second: the isoquant points located on the x2 axis show the amount of the second resource, which makes it possible to obtain one or another output of the product without using the first resource. The use of the first resource reduces the cost of the second, but it is impossible to completely replace the second resource with the first. Rice. 5c depicts a situation in which both resources are needed and neither can be completely replaced by the other. Finally, the case shown in Fig. 5d is characterized by absolute complementarity of resources.

Rice. 5. Examples of isoquant maps

To explain the production function, the concept of costs is introduced.

In the most general form, costs can be defined as a set of costs that a manufacturer incurs when producing a certain volume of output.

There is their classification according to time periods during which the company makes a particular production decision. To change the volume of production, the firm has to adjust the amount and composition of its costs. Some costs can be changed fairly quickly, while others require a certain amount of time.

The short-term period is a time interval that is insufficient for the modernization or commissioning of new production capacities of the enterprise. However, during this period, the company can increase output by increasing the intensity of the use of existing production capacities (for example, hire additional workers, purchase more raw materials, increase the equipment maintenance shift ratio, etc.). It follows that in the short run costs can be either fixed or variable.

Fixed costs (TFC) are the sum of costs that do not depend on changes in the volume of production. Fixed costs are associated with the very existence of the firm and must be paid even if the firm does not produce anything. They include depreciation charges on buildings and equipment; property tax; insurance payments; repair and maintenance costs; bond payments; salaries of senior management personnel, etc.

Variable cost (TVC) is the cost of resources that are used directly to produce a given output. Elements of variable costs are the costs of raw materials, fuel, energy; payment for transport services; payment for most of the labor resources (wages). Unlike fixed costs, variable costs depend on the volume of output. However, it should be noted that the increase in the amount of variable costs associated with an increase in production by 1 unit is not constant.

At the beginning of the process of increasing production, variable costs will increase for some time at a decreasing rate; and so it will continue until a specific value of the volume of production. Then variable costs will begin to increase at an increasing rate per each subsequent unit of output. This behavior of variable costs is determined by the law of diminishing returns. An increase in marginal product over time will cause smaller and smaller increments of variable resources to produce each additional unit of output.

And since all units of variable resources are purchased at the same price, this means that the sum of variable costs will increase at a decreasing rate. But as marginal productivity begins to fall in accordance with the law of diminishing returns, more and more additional variable resources will have to be used to produce each successive unit of output. The sum of variable costs will thus increase at an accelerating rate.

The sum of fixed and variable costs associated with the production of a certain amount of output is called total cost (TC). Thus, we get the following equality:

TC - TFC + TVC.

In conclusion, we note that production functions can be used to extrapolate the economic effect of production in a given period of the future. As in the case of conventional econometric models, an economic forecast begins with an assessment of the predicted values ​​of production factors. In this case, the method of economic forecasting that is most suitable in each individual case can be used.