production function. The concept of a production function

10.10.2019

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, release functions are widely used in agriculture, where they are used to study the effect on the yield of such factors as, for example, different types and compositions of fertilizers, methods of tillage. 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 (the relationship 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 as simple 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 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 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 replacement coefficient remains unchanged for any ratio of resources used, and therefore we can assume that the elasticity s jk = 1. Accordingly big values sjk indicate that greater freedom is possible in the replacement of production factors along the isoquant, and at the same time the main characteristics of the 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 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 isoquant

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 population x1, the sum of the main and revolving funds x2, 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 the homogeneity of the function: 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, apply different forms production function. 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 measures of both inputs and outputs are inevitably products of complex aggregation (e.g., the aggregated indicator labor costs in the macroeconomic function, it 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, 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 various meanings, 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 the use of more complex methods of non-linear regression analysis to estimate the parameters.

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

Production functions, regardless of 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 the most general view 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 with a change in the basic technology (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 it has been strongly competed by a new feature of Arrow, Chenery, Minhas and Solow, which we will call 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 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. 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.

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:

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;
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;
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. As a tool for analyzing this dependence economic theory uses a production function.

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 contributing to the growth of labor productivity, causes 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 production production functions are different, however? they all have common properties. Two main properties can be distinguished.

There is a limit to the growth in output that can be achieved by increasing the cost of one resource, other things being equal. Thus, in a firm with a fixed number of machines and industrial premises there is a limit to the growth of output by increasing additional workers, since it will not be provided with machines for work.
There is a certain complementarity (complementarity) 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 can be represented 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 (AR).

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.

Calculation of the firm's production function is a 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 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:

Nonlinear PF;
Linear PF;
Multiplicative PF;
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:

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).
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:

a proportionally increasing production function when α + β = 1 (Q = K 0.5 * L 0.2).
disproportionately - increasing α + β > 1 (Q = K 0.9 * L 0.8);
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 acuteness, 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: small size businesses means 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, the series American companies in the 1960s, he went on to break down his branches and enterprises in order to significantly reduce the size of primary production units.

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:

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

It 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 A is the coefficient of total factor productivity. It reflects the impact of technological progress on production: if the manufacturer introduces advanced technologies, the value A increases, i.e. output increases with the same amount of labor and capital. 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 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 V, 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 V. Accordingly, the method V 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 the method A way V 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.

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.

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.

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.

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 in manufacturing process If the latest inventions and other technical improvements are applied, an increase in output can be achieved using the same factors of production. 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).

production function- this is the relationship between the amount and structure of the resources used (L-labor, K-capital) and the maximum possible amount of output (Q) that the company is able to produce within a certain period of time.

The production function characterizes this technology. The improvement of technology, which provides a new achieved volume of output for any combination of factors, is reflected in a new production function.

A set of production factors or resources can be represented as the cost of labor, capital (tools and materials), then the production function can be described as follows:

Q = f(L, K),

where Q is the maximum volume of products produced with a given technology and a given ratio of labor - L, capital - K.

2.2.Properties of the production function

All production functions have common properties:

There are limits to the growth in output that can be achieved by increasing the cost of one resource while other resources remain unchanged.

A certain mutual complementarity (complementarity) of factors of production is possible, but without a decrease in the volume of production, a certain interchangeability of these factors is also possible.

Changes in the use of factors of production are more elastic over a long period of time than over a short period of a firm's activity.

Short period of time- this is the period of production during which all resources except one are unchanged, then the entire increase in production is associated with an increase in the use of this particular factor.

Long term time- this is the period during which the manufacturer can change all the factors of production of this product. In theory, a long period of time is considered as successive short periods.

Total product of a variable factor of production (TR)- This is the amount of output produced with a certain amount of this factor and with other factors of production unchanged.

Average product of a variable factor of production is the ratio of the total product of a variable factor to the amount of that factor used. For example, the average product of labor AP(L) is the total product of labor TP(L) divided by the number of hours of labor (L):

The value shown is labor productivity or the amount of output for each hour of labor.

Average product of capital:

marginal product of a variable factor of production is the change in the total product of that factor (for example, TR L) when the factor used changes per unit (for example, the labor factor (L) changes by one, and capital does not change).

where F is the factor of production (L or K).

Law of diminishing returns(marginal productivity of factors of production):

In the context of the implementation of production activities, the company must use the main factors of production in a certain proportion between fixed and variable resources. If the enterprise increases only the number of variable factors without changing the constant factor, then in this case, law of diminishing returns.

Law of diminishing marginal productivity of factors of production states that if a firm increases the use of only some or one of the factors of production, then the increase in output brought by additional volumes of these factors will eventually begin to decline.

According to the law, a continuous increase in the use of one variable resource, combined with an unchanged amount of other resources, at a certain stage will lead to the cessation of the growth of returns, and then to its decrease. It should be noted that quite often the operation of the law assumes the constancy of the technological level of production, and therefore the transition to a more advanced technology can increase the return, regardless of the ratio of constant and variable factors.

Consider the following example. How will the return on the variable factor change in the enterprise in the short run if part of the resources or factors of production remains constant. In the short term, the enterprise is not able to introduce new workshops, install new equipment, etc.

Assume that an enterprise in its activities uses only one variable resource - labor, the return of which is productivity. It is necessary to determine how the firm's costs will change with a gradual increase in the variable resource (number of workers).

In a small workshop for 3 pieces of equipment, one worker makes 5 items per shift. With the involvement of the second worker, together they will make 12 products per shift, the third - 20, with the fourth - 25, with the fifth - also 25, with the sixth - 20. The addition of the second worker gives an increase of 7 units, the third - 8 units, the fourth - 5 units, the fifth - does not give an increase at all. Thus, already from the fourth unit of the variable factor, we fix diminishing returns. The same is observed in the case of the average value of output. One worker - 5 products, two - 6 each, three - 6.7 each, four - 6.2 each, five - 5 each, six - 3.3. The question arises why the return drops so sharply? Because with the same production capacity (three machines), the fifth and sixth workers are no longer just superfluous, they interfere with the rational production process.

Table 5.3

Number of workers (L)	Overall Performance (TP)	Ultimate Performance (MP)	Average performance (AP)

Let's write down the given data in tab. 5.3 and construct the corresponding graphs 5.6 and 5.7.

The data of the table and the graphs built on them indicate that, starting from a certain moment, both the total, the marginal, and the average productivity decrease. This shows the essence law of diminishing returns.

scale effect

It is possible to eliminate the effect of the law of diminishing returns if the firm opens additional production, that is, new production capacities are put into operation. In fact, there will be an increase in production potential - a permanent resource (long-term period)

In the long run, the use of factors of production (L and K) must be considered as variables. This is due to the fact that the firm can actively change the attracted production resources. In this case, all costs of the enterprise will act as variables.

The relationship between the increase in factors of production and the volume of output is characterized by scale effect:

scale effect
Recoil state	Ratio of production rates and costs	State of the cost
Increasing returns to scale (positive returns to scale)	Output grows faster than costs	Average costs are falling
Diminishing returns to scale (negative returns to scale)	Output grows faster than costs	Average costs rise
Constant returns to scale	Output and costs rise at the same rate	Average costs do not change

The economies of scale will be positive if the average gross costs decrease with an increase in production volumes, and negative if they increase.

An analysis of the company's costs in the short and long term is a necessary but not sufficient condition for planning output in the near future and in the future. Cost minimization is not an end in itself, but only a means of increasing profits or reducing losses, and ultimately ensuring the stability and sustainability of the company's position in the market.

Thus, if in the short term it is important for the company to find the optimal ratio of factors of production (K , L), then in the long term the company solves the problem of choosing the required scale of the company's activities.