Each company, having undertaken the production of a specific product, strives to achieve maximum profit. Problems associated with product production can be divided into three levels:

1. An entrepreneur may be faced with the question of how to produce a given quantity of products at a certain enterprise. These problems relate to issues of short-term minimization of production costs;
2. the entrepreneur can solve questions about the production of the optimal, i.e. bringing greater profit, the amount of production at a particular enterprise. These questions concern long-term profit maximization;
3. An entrepreneur may be faced with the task of determining the most optimal size of an enterprise. Similar questions relate to long-term profit maximization.

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

The production function determines the maximum volume of output for each given amount of input. This function describes the relationship between resource costs and output, allowing you to determine the maximum possible volume of output for each given amount of resources, or the minimum possible amount of resources to ensure a given volume of 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 determines a new production function.

PRODUCTION FUNCTION - a function that reflects the relationship between the maximum volume of a product produced and the physical volume of factors of production 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 using a given technology and certain factors of production;
L – labor; K – capital; M – materials; f – function.

The production function for a given 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 of output that can be achieved by increasing the costs of one resource, all other things being equal. Thus, 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 mutual complementarity (completeness) of production factors, however, without a decrease in output, a certain interchangeability of these production factors is also likely. Thus, various combinations of resources can be used to produce a good; it is possible to produce this good 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 required to produce a given volume of output, and the production function - by an isoquant line.

Having considered the production function of the company, we move on to characterize the following three important concepts: total (total), average and marginal product.

Rice. a) Total product (TP) curve; b) curve of average product (AP) and marginal product (MP)

In Fig. shows the total product (TP) curve, which varies depending on the value of the variable factor X. Three points are marked on the TP curve: B – inflection point, C – point that belongs to the tangent coinciding with the line connecting this point to the origin, D – point of maximum TP value. Point A moves along the TP curve. By connecting point A to the origin of coordinates, we obtain line OA. Dropping the perpendicular from point A to the x-axis, we obtain a triangle OAM, where tg a is the ratio of the side AM to OM, i.e., the expression of the average product (AP).

Drawing a tangent through point A, we obtain an angle P, the tangent of which will express the limiting product MP. Comparing the triangles LAM and OAM, we find that up to a certain point the tangent P is greater than tan a. Thus, marginal product (MP) is greater than average product (AP). In the case when point A coincides with point B, the tangent P takes on its maximum value and, therefore, the marginal product (MP) reaches its greatest volume. If point A coincides with point C, then the values ​​of the average and marginal products are equal. The marginal product (MP), having reached its maximum value at point B (Fig. 22, b), begins to contract 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 and average product decrease, but the marginal product decreases at a faster pace. At the point of maximum total product (TP), the marginal product MP = 0.

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

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

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

Q = f (L, K).

It can be presented in the form of a graph or curve. In the theory of producer behavior, under certain assumptions, there is a single combination of resources that minimizes resource costs for a given volume of production.

Calculation of a firm's production function is a search for the optimum, among many options involving various combinations of production factors, one that gives the maximum possible volume of output. In an environment of rising prices and cash costs, the firm, i.e. costs of purchasing factors of production, the calculation of the production function is focused on searching for an option that would maximize profits at the lowest costs.

The calculation of the firm's production function, seeking to achieve a balance between marginal costs and marginal revenue, will focus on finding an option that will provide the required output at minimal production costs. Minimum costs are determined at the stage of calculations of the production function by the method of substitution, displacing expensive or increased in price factors of production with alternative, cheaper ones. Substitution is carried out using a comparative economic analysis of interchangeable and complementary factors of production at their market prices. A satisfactory option will be one in which the combination of production factors and a given volume of output meets the criterion of 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 choice of optimal production size

A production function is the relationship between a set of factors of production and the maximum possible output produced by that set of factors.

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

Using the production function, the minimum amount of input required to produce a given volume of product is determined.

Production functions, regardless of what type of production they express, have the following general properties:

1. Increasing production volume due to increasing costs for only one resource has a limit (you cannot hire many workers in one room - not everyone will have space).
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 Cobb-Douglas production function model, which reveals the relationship between labor (L) and capital (K). These factors are interchangeable and complementary

Q = AK α * L β,

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

If = 0.25, then an increase in capital costs by 1% increases production volume by 0.25%.

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

1. 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 size of enterprises is not absolute in nature, and therefore cannot be established outside of time and outside the area of ​​location, since they are different for different periods and economic regions.

The optimal size of the designed enterprise should ensure a minimum of costs or a maximum of profits, calculated using the formulas:

Тс+С+Тп+К*En_ – minimum, П – maximum,

where Тс – costs of delivery of raw materials;
C – production costs, i.e. production cost;
Тп – costs of delivering finished products to consumers;
K – capital costs;
En – standard efficiency coefficient;
P – enterprise profit.

Sl., the optimal size of enterprises is understood as those that provide the plan targets for product output and the increase in production capacity minus the reduced costs (taking into account capital investments in related industries) and the highest possible economic efficiency.

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

Those that failed to achieve the required scale found themselves in the unenviable position of high-cost producers, condemned to an existence on the brink of ruin and eventual bankruptcy.

Today, however, those American companies that still strive to succeed in the competitive struggle through economies of concentration of production are not winning as much as they are losing. In modern conditions, this approach initially leads to a decrease in not only flexibility, but also production efficiency.

In addition, entrepreneurs remember: small enterprise size 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, slow and poorly responsive to emerging problems.

Therefore, a number of American companies in the 60s decided to disaggregate their branches and enterprises in order to significantly reduce the size of the primary production units.

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

When determining the optimal enterprise size, firms use the concept of minimum efficient size. It is simply the smallest level of production at which the firm can minimize its long-run average cost.

Production function and selection of optimal production size.

Production is any human activity involving the transformation of limited resources - material, labor, natural - into finished products. The production function characterizes the relationship between the amount of resources used (factors of production) and the maximum possible volume of 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 achieved by increasing one resource and holding other resources constant. If, for example, in agriculture we increase the amount of labor with constant amounts of capital and land, then sooner or later a moment comes when output stops growing.
2. Resources complement each other, but within certain limits their interchangeability is possible without reducing output. Manual labor, for example, can be replaced by the use of more machines, and vice versa.
3. The longer the time period, the more resources can be revised. In this regard, instantaneous, short and long periods are distinguished. An instantaneous period is a period when all resources are fixed. Short period - a period when at least one resource is fixed. A long period is a 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). Let us 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. In turn, the amount of labor is measured in man-hours.

Typically, the production function in question looks like this:

q = AK α L β

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

Construction of an isoquant

The given production function suggests that the producer can replace labor with capital and capital with labor, leaving output unchanged. For example, in agriculture in developed countries, labor is highly mechanized, i.e. There are many machines (capital) per 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 construct an isoquant (Fig. 8.1).

An isoquant (line of equal product) reflects all combinations of two factors of production (labor and capital) at which output remains unchanged. In Fig. 8.1 next to the isoquant the corresponding release is indicated. Yes, release q 1, achievable by using L 1 labor and K 1 capital or using L 2 labor and K 2 capital.

Rice. 8.1. Isoquant

Other combinations of labor and capital volumes are possible, the minimum required to achieve a given output.

All combinations of resources corresponding to a given isoquant reflect technically efficient methods of production. Production method A is technically efficient in comparison with method B if it requires the use of at least one resource in smaller quantities, and all others in smaller quantities, in comparison with method B. Accordingly, method B is technically ineffective in comparison with A. Technically ineffective production methods are not used by rational entrepreneurs and are not part of the production function.

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

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

Based on the isoquant, the marginal rate of technical substitution can be determined.

The marginal rate of technical replacement of factor Y by factor X (MRTS XY) is the amount of factor Y(for example, capital), which can be abandoned when the factor increases X(for example, labor) by 1 unit so that output does not change (we remain at 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 infinitesimal changes in L and K, it is
Thus, the marginal rate of technical substitution is the derivative of the isoquant function at a given point. Geometrically, it represents the slope of the isoquant (Fig. 8.3).

Rice. 8.3. Limit rate of technical replacement

When moving from top to bottom along an 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 greater output, i.e. move to a higher isoquant (q2). An isoquant located to the right and above the previous one corresponds to a larger volume of output. The set of isoquants forms an isoquant map (Fig. 8.4).

Rice. 8.4. Isoquant map

Special cases of isoquants

Let us recall that the given isoquants correspond to the production function of the form q = AK α L β. But there are other production functions. Let us consider the case when there is perfect substitutability 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 qualified loader is N times higher than that of an unskilled loader. This means that we can replace any number of qualified movers with unqualified movers at a ratio of N to one. Conversely, you can replace N unqualified loaders with one qualified one.

The production function then has the form: q = ax + by, Where x- number of qualified workers, y- number of unskilled workers, A And b- constant parameters reflecting the productivity of one skilled and one unskilled worker, respectively. The ratio of coefficients a and b is the maximum rate of technical replacement of unskilled loaders with 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 coefficient a in the production function), and an unskilled loader - only 1 ton (coefficient b). This means that the employer can refuse three unqualified loaders, additionally hiring one qualified loader, so that the output (total weight of the processed cargo) remains the same.

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

Rice. 8.5. Isoquant with perfect substitutability of factors

The tangent of the isoquant slope is equal to the maximum rate of technical replacement of unskilled loaders with qualified ones.

Another production function is the Leontief function. It assumes strict complementarity of production factors. This means that factors can only be used in a strictly defined proportion, violation of which is technologically impossible. For example, an airline flight can be carried out normally 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 keep output constant. Isoquants in this case have the form of right angles, i.e. the maximum rates of technical replacement are equal to 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 strict complementarity of production factors

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 use of capital and labor.

In our flight example, the production function looks like this: q = min(1K; 0.2L). The fact is that capital productivity here is one flight per plane, and labor productivity is one flight per five people or 0.2 flights per person. If an airline has an aircraft fleet of 10 aircraft and has 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 that there are a limited number of production technologies to produce a given quantity 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 obtain a broken isoquant (Fig. 8.7).

Rice. 8.7. Broken isoquants with 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 two technologies are used together to obtain a certain total output . As always, by increasing the quantities of labor and capital, we move to a higher isoquant.

Production function

The relationship between input factors and final output is described by a production function. It is the starting point in the microeconomic calculations of the company, allowing you to find the optimal option for using production capabilities.

Production function shows the possible maximum output (Q) for a certain combination of production factors and selected technology.

Each production technology has its own special function. In its most general form it is written:

where Q is production volume,

K – capital

M – natural resources

Rice. 1 Production function

The production function is characterized by certain properties :

There is a limit to the increase in output that can be achieved by increasing the use of one factor, provided that other factors of production do not change. This property is called law of diminishing returns of a factor of production . It works in the short term.

There is a certain complementarity of factors of production, but without a reduction in 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.

The production function can be considered as single-factor and multi-factor. One-factor assumes that, other things being equal, only the factor of production changes. Multifactorial involves changing all factors of production.

For the short-term period, single-factor is used, and for the long-term, multi-factor.

Short term – This is a period during which at least one factor remains unchanged.

Long term – it is a period of time during which all factors of production change.

When analyzing production, concepts such as total product (TP) – the volume of goods and services produced over a certain period of time.

Average Product (AP) characterizes the amount of output per unit of production factor used. It characterizes the productivity of a production factor and is calculated using the formula:

Marginal product (MP) - additional output produced by an additional unit of a factor of production. MP characterizes the productivity of an additionally hired unit of production factor.

Table 1 - Production results in the short term

Analysis of the data in Table 1 allows us to identify a number of patterns of behavior total, average and marginal product. At the point of maximum total product (TP), the marginal product (MP) is equal to 0. If, with an increase in the volume of labor used in production, the marginal product of labor is greater than the average, then the value of the average product increases and this indicates that the ratio of labor to capital is far from optimal and Some equipment is not used due to labor shortages. If, as the volume of labor increases, the marginal product of labor is less than the average product, then the average product of labor will decrease.

Law of substitution of factors of production.

Equilibrium position of the firm

The same maximum output of a firm can be achieved through different combinations of factors of production. This is due to the ability of one resource to be replaced by another without compromising production results. This ability is called interchangeability of factors of production.

Thus, if the volume of the labor resource increases, then the use of capital may decrease. In this case, we resort to a labor-intensive production option. If, on the contrary, the volume of capital used increases and labor is displaced, then we are talking about a capital-intensive production option. For example, wine can be produced using a labor-intensive manual method or a capital-intensive method using machinery to squeeze grapes.

Production technology Firms are a way of combining factors of production to produce products, based on a certain level of knowledge. As technology develops, a firm is able to produce the same or greater volume of output with a constant set of production factors.

The quantitative ratio of interchangeable factors allows us to estimate the coefficient called the marginal technological rate of substitution (MRTS).

Limit rate of technological substitution labor by capital is the amount by which capital can be reduced by using an additional unit of labor without changing output. Mathematically this can be expressed as follows:

MRTS L.K. = - dK / dL = - ΔK / ΔL

Where ΔK - change in the amount of capital used;

ΔL change in labor costs per unit of production.

Let's consider the option of calculating the production function and substitution of production factors for a hypothetical company X.

Let us assume that this firm can change the volume of production factors, labor and capital from 1 to 5 units. Changes in output volumes associated with this can be presented in the form of a table called “Production grid” (Table 2).

table 2

The company's production networkX

For each combination of main factors, we determined the maximum possible output, i.e., the values ​​of the production function. Let us pay attention to the fact that, say, an output of 75 units is achieved with four different combinations of labor and capital, an output of 90 units with three combinations, 100 with two, etc.

By representing the production grid graphically, we obtain curves that are another variant of the production function model previously fixed in the form of an algebraic formula. To do this, we will connect the dots that correspond to combinations of labor and capital that allow us to obtain the same volume of output (Fig. 1).

Rice. 1. Isoquant map.

The created graphical model is called isoquant. A set of isoquants - an isoquant map.

So, isoquant- this is a curve, each point of which corresponds to combinations of production factors that provide a certain maximum volume of output of the company.

In order to obtain the same volume of output, we can combine factors, moving in search of options along the isoquant. An upward movement along an isoquant means that the firm gives preference to capital-intensive production, increasing the number of machine tools, the power of electric motors, the number of computers, etc. A downward movement reflects the firm's preference for labor-intensive production.

The choice of a firm in favor of a labor-intensive or capital-intensive version of the production process depends on the conditions of business: the total amount of monetary capital that the firm has, the ratio of prices for factors of production, the productivity of factors, and so on.

If D - money capital; R K - price of capital; R L - the price of labor, the amount of factors that a firm can acquire by completely spending money capital, TO - amount of capital L– the amount of labor will be determined by the formula:

D=P K K+P L L

This is the equation of a straight line, all points of which correspond to the full use of the firm's monetary capital. This curve is called isocost or budget line.

A

Rice. 2. Producer equilibrium.

In Fig. 2 we combined the line of the company's budget constraint, isocost (AB) with an isoquant map, i.e. a set of alternatives to the production function (Q 1, Q 2, Q 3) to show the producer’s equilibrium point (E).

Producer Equilibrium- this is the position of the company, which is characterized by the full use of monetary capital and at the same time achieving the maximum possible volume of output for a given amount of resources.

At the point E isoquant and isocost have an equal slope angle, the value of which is determined by the indicator of the marginal rate of technological substitution (MRTS).

Dynamics of the indicator MRTS (it increases as you move upward along the isoquant) shows that there are limits to the mutual substitution of factors due to the fact that the efficiency of using production factors is limited. The more labor is used to displace capital from the production process, the lower the productivity of labor. Likewise, replacing labor with more and more capital reduces the return of capital.

Production requires a balanced combination of both production factors for their best use. An entrepreneurial firm is willing to substitute one factor for another provided there is a gain, or at least an equality of loss and gain in productivity.

But in the factor market it is important to take into account not only their productivity, but also their prices.

The best use of the firm's monetary capital, or the producer's equilibrium position, is subject to the following criterion: the producer's equilibrium position is achieved when the marginal rate of technological substitution of factors of production is equal to the ratio of prices for these factors. Algebraically, this can be expressed as follows:

- P L / P K = - dK / dL = MRTS

Where P L , P K - prices of labor and capital; dK, dL - changes in the amount of capital and labor; MTRS - marginal rate of technological substitution.

Analysis of the technological aspects of the production of a profit-maximizing company is of interest only from the point of view of achieving the best final results, i.e., the product. After all, investments in resources for an entrepreneur are only costs that must be borne in order to obtain a product that is sold on the market and generates income. Costs have to be compared with results. Result or product indicators therefore acquire special significance.

Characterizes the relationship between the amount of resources used () and the maximum possible volume of 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 achieved by increasing one resource and keeping other resources constant. If, for example, in agriculture we increase the amount of labor with constant amounts of capital and land, then sooner or later a moment comes when output stops growing.

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

3. The longer the time period, the more resources can be revised. In this regard, instantaneous, short and long periods are distinguished. Instantaneous period - a period when all resources are fixed. Short period- a period when at least one resource is fixed. A long period - a period when all resources are variable.

Typically in microeconomics, a two-factor production function is analyzed, reflecting the dependence of output (q) on the amount of labor () and capital () used. Let us 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, clause 2.2). In turn, the amount of labor is measured in man-hours.

Typically, the production function in question looks like this:

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

Construction of an isoquant

The given production function suggests that the producer can replace labor with capital and capital with labor, leaving output unchanged. For example, in agriculture in developed countries, labor is highly mechanized, i.e. There are many machines (capital) per 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 construct an isoquant (Fig. 8.1).

Isoquant(equal product line) reflects all combinations of two factors of production (labor and capital) for which output remains unchanged. In Fig. 8.1 next to the isoquant the corresponding release is indicated. Thus, output is achievable using labor and capital or using labor and capital.

Rice. 8.1. Isoquant

Other combinations of labor and capital volumes are possible, the minimum required to achieve a given output.

All combinations of resources corresponding to a given isoquant reflect technically efficient production methods. Mode of production A is technically effective in comparison with the method IN, if it requires the use of at least one resource in smaller quantities, and all others not in large quantities in comparison with the method IN. Accordingly, the method IN is technically ineffective compared to A. Technically inefficient production methods are not used by rational entrepreneurs and are not part of the production function.

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

The dotted line reflects all technically inefficient production methods. In particular, in comparison with the method A way IN 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, the marginal rate of technical substitution can be determined.

Marginal rate of technical replacement of factor Y by factor X (MRTS XY)- this is the amount of a factor (for example, capital) that can be abandoned when the factor (for example, labor) increases by 1 unit, so that output does not change (we remain at 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 infinitesimal changes L And K it amounts to

Thus, the marginal rate of technical substitution is the derivative of the isoquant function at a given point. Geometrically, it represents the slope of the isoquant (Fig. 8.3).

Rice. 8.3. Limit rate of technical replacement

When moving from top to bottom along an 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 greater 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 volume of output. The set of isoquants forms isoquant map(Fig. 8.4).

Rice. 8.4. Isoquant map

Special cases of isoquants

Let us recall that these correspond to a production function of the form . But there are other production functions. Let us consider the case when there is perfect substitutability 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 qualified loader is N times higher than unskilled. This means that we can replace any number of qualified movers with unqualified ones in the ratio N to one. Conversely, you can replace N unqualified loaders with one qualified one.

The production function then 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 maximum rate of technical replacement of unqualified loaders with 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 coefficient a in the production function), and an unskilled loader - only 1 ton (coefficient b). This means that the employer can refuse three unqualified loaders, additionally hiring one qualified loader, so that the output (total weight of the processed cargo) remains the same.

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

Rice. 8.5. Isoquant with perfect substitutability of factors

The tangent of the isoquant slope is equal to the maximum rate of technical replacement of unskilled loaders with qualified ones.

Another production function is the Leontief function. It assumes strict complementarity of production factors. This means that factors can only be used in a strictly defined proportion, violation of which is technologically impossible. For example, an airline flight can be carried out normally 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 keep output constant. Isoquants in this case have the form of right angles, i.e. the maximum rates of technical replacement are equal to 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 strict complementarity of production factors

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

In our airline flight example, the production function looks like this: q = min(1K; 0.2L). The fact is that capital productivity here is one flight per plane, and labor productivity is one flight per five people or 0.2 flights per person. If an airline has an aircraft fleet of 10 aircraft and has 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 that there are a limited number of production technologies to produce a given quantity 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 obtain a broken isoquant (Fig. 8.7).

Rice. 8.7. Broken isoquants with a limited number of production methods

The figure shows that product output in the amount of 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 in cases where an enterprise jointly uses two technologies to obtain a certain total output. As always, by increasing the quantities of labor and capital, we move to a higher isoquant.

The dependence of the quantity of goods produced on the corresponding factors of production with the help of which it is produced. Let's look at this concept in more detail.

A production function always has a specific form, since it is intended for a specific technology. The introduction of new technological developments entails a change or the creation of a new type of dependence.

This function is used to find the optimal (minimum) amount of costs that are necessary to produce a certain number of goods. All production functions, regardless of what they express, are characterized by the following general properties:

The growth in the volume of goods produced due to only one factor (resource) has a finite limit (only a certain number of workers can work normally in one room, since the number of places is limited by area);

Factors of production can be interchangeable and complementary (workers and tools).

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

Q = f (K, L, M, T, N), in this formula

Q is the volume of goods produced;

K - equipment (capital);

M - costs of materials and raw materials;

T - technologies used;

N - entrepreneurial abilities.

Types of production functions

There are many types of this dependence, which take into account the influence of one or several of the most important factors. However, two main types of production functions are most famous: a two-factor model of the form Q = f (L; K) and the Cobb-Douglas function.

Two-factor model Q = f (L; K)

This model considers the dependence of output (Q) on (L) and capital (L). Quite often, a group of isoquants is used to analyze this model. An isoquant is a curve that connects all possible combination points that allow the production of a specific volume of goods. The X-axis usually shows labor costs, and the Y-axis usually shows capital costs. Several isoquants are drawn on the same graph, each of which corresponds to a certain volume of production when using a specific technology. The result is a map of isoquants with different quantities of goods produced. It will be the production function for this enterprise.

Isoquants have the following general properties:

The concave and downward type of isoquant is due to the fact that a decrease in the use of capital with a stable volume of goods produced causes an increase in labor costs;

The concave shape of the isoquant curve depends on the maximum permissible rate of technological substitution (the amount of capital that can replace 1 additional unit of labor).

Cobb-Douglas function

This production function, named after two American discoverers, where the total volume of output Y depends on the resources used in the production process, for example, labor L and capital K. Its formula is:

where α and b are constants (α>0 and b>0);

K and L are capital and labor, respectively.

If the sum of the constants α and b is equal to one, then it is generally accepted that such a function has a production constant. If parameters K and L are multiplied by any coefficient, then Y must also be multiplied by the same coefficient.

The Cobb-Douglas model can be applied to any individual company. In this case, α is the share of total costs going to capital, and β is the share going to labor. Cobb-Douglas models can also contain more than two variables. For example, if N is then the production function takes the form Y=AKαLβNγ, where γ is a constant (γ>0), and α + β +γ = 1.

I. ECONOMIC THEORY

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

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

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

Using the production function, the minimum amount of input required to produce a given volume of product is determined.

Production functions, regardless of what type of production they express, have the following general properties:

1) Increasing production volume due to increasing costs for only one resource has a limit (you cannot hire many workers in one room - not everyone will have space).

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

,

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

K, L - capital and labor;

Elasticity coefficients of production volume with respect to capital and labor costs.

If = 0.25, then an increase in capital costs by 1% increases production volume by 0.25%.

Based on the analysis of elasticity coefficients in the Cobb-Douglas production function, we can distinguish:
1) proportionally increasing production function, when ( ).
2) disproportionately – increasing);
3) decreasing.

Consider a short period of a firm's activity in which labor is the variable of the 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 (TP n curve).

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

The law of diminishing marginal productivity operates in the short term when one factor of production remains constant. The effect of the law assumes the 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 scope of the law.

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

Marginal productivity of labor (marginal product of labor - MP L) is the increase in production volume from each subsequent unit of labor

those. productivity gain to total product (TP L)

The marginal product of capital MP K is determined similarly.

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

The movement of the total product (TP) curve can be divided into three stages. At stage 1, it rises upward at an accelerating pace, as the marginal product (MP) increases (each new worker brings more output than the previous one) and reaches a maximum at point A, that is, the rate of growth 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 increase in the total product compared to the previous one, therefore the growth rate of TR after the TS slows down. But as long as MR is positive, TP will still increase and reach a maximum at MR=0.

Rice. 10.1. Dynamics and relationship between the general average and marginal products

At stage 3, when the number of workers becomes excessive in relation to the fixed capital (machines), MP becomes negative, so TR begins to decline.

The configuration of the average product curve AP 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 the four workers also decreases.

Economies of scale

1. Manifests itself in changes 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 activities is characterized by a change in the quantity of all production factors used.

Rice. 10.2. The firm's long-run and average cost curve

The reaction of LATC to changes in the parameters (scale) of the company can be different (Fig. 10.3).

Rice. 10.3. Dynamics of long-term average costs