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

Production function: general form and properties

Production functions are characterized by the following properties:

• Increasing output volumes due to one production factor always to the limit (for example, a limited number of specialists can work in one room).

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

IN general view The production function looks like this:

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

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 the production factor and is calculated by 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 amount of capital employed increases and labor is displaced, then we're talking about 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).

K

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.

K

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.

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 a large amount of products to a particular enterprise. These questions concern long-term profit maximization;
3. An entrepreneur may be faced with 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 this technology and certain factors 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 production production functions are different, however? they all have general properties. Two main properties can be distinguished.

1. There is a limit to the growth 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 it 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 is the inflection point, C is the point that belongs to the tangent coinciding with the line connecting this point with the origin, D is the 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 factors of production, those that produce 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 comparative economic analysis 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 ensure the fulfillment of plan targets for product output and growth production capacity with a minus of 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 optimal size The company's enterprises 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 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 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 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 using more cars, 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. In contrast, in developing countries the same output is achieved by large quantity 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 an enterprise jointly uses two technologies to obtain a certain total release. As always, by increasing the quantities of labor and capital, we move to a higher isoquant.

Production refers to 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 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.

Manufacturing cannot create products out of nothing. The production process involves the consumption of various resources. Resources include everything needed to production activities, – raw materials, energy, labor, equipment, and space.

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

But an enterprise can implement it in different ways manufacturing process using different technological methods, different variants organization of production, so that the amount of product obtained with the same expenditure of resources may be different. Firm managers should reject production options that give lower output if a higher output can be obtained with the same costs of each type of resource. Likewise, they should reject options that require more input from at least one input without increasing yield or reducing the input of other inputs. Options rejected for these reasons are called technically ineffective.

Let's say your company produces refrigerators. To make the body, you need to cut sheet iron. Depending on how a standard sheet of iron is marked and cut, more or fewer parts can be cut out of it; Accordingly, to manufacture a certain number of refrigerators, less or more standard sheets of iron will be required. At the same time, the consumption of all other materials, labor, equipment, and electricity will remain unchanged. This production option, which could be improved by more rational cutting of iron, should be considered technically ineffective and rejected.

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

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

Production function– shows the dependence of the amount of product that a company can produce on the volume of costs of the factors used

Q = f(x1, x2…xn)

Q = f(K, L),

Where Q- volume of output

x1, x2…xn– volumes of applied factors

K- volume of capital factor

L- volume of labor factor

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

Another type of production function is the linear production function, which has the following form:

Q(L,K) = aL + bK

This production function is homogeneous of the first degree, therefore, it has constant returns to scale of production. Graphically, this function is presented in Figure 1.2, a.

The economic meaning of a linear production function is that it describes production in which factors are interchangeable, that is, it does not matter whether you use only labor or only capital. But in real life such a situation is practically impossible, since any machine is still serviced by a person.

The coefficients a and b of the function, which are found under the variables L and K, show the proportions in which one factor can be replaced by another. For example, if a=b=1, then this means that 1 hour of labor can be replaced by 1 hour of machine time in order to produce the same volume of output.

It should be noted that in some types of economic activity, labor and capital cannot replace each other at all and must be used in a fixed proportion: 1 worker - 2 machines, 1 bus - 1 driver. In this case, the elasticity of factor substitution is zero, and the production technology is reflected by the Leontief production function:

Q(L,K) = min(; ),

If, for example, each long-distance bus must have two drivers, then if there are 50 buses and 90 drivers in the bus fleet, only 45 routes can be served simultaneously:
min(90/2;50/1) = 45.

Application

Examples of solving problems using production functions

Problem 1

A firm engaged in river transportation uses carrier labor (L) and ferries (K). The production function has the form . The price per unit of capital is 20, the price per unit of labor is 20. What will be the slope of the isocost? How much labor and capital must the firm attract to carry out 100 shipments?

Solution

The isocost is given by the equation:

where C is the value of total costs (some constant). From here:

,

those. the slope of this line is -1.

The optimal amount of labor and capital for 100 transportations is determined as the point of tangency of the isoquant and isocosts at some C. Solving the isoquant equation we get:

√(L×K) = 100/10 = 10, then .

Then . Because total costs must be minimal, then by minimizing C over L, we find the amount of labor L: And . We will find the amount of capital using the formula.

Answer: To carry out 100 shipments, a firm must attract 10 units of labor and 10 units of capital.

Problem 2

The production function has the form , where Y- quantity of products per day, L- hours of labor, K- machine operating hours. Let us assume that 9 hours of labor and 9 hours of machinery are spent per day.

What's it like maximum amount products produced per day? Suppose the firm doubles the costs of both factors. Determine the economies of scale in production.

Solution

In the conditions of the task per day it is produced units of production. If the inputs of both factors double, then output becomes equal , i.e. also doubles. Then the effect of changes in the scale of production, determined from the condition , is equal to one.

Problem 3

In the short term, the firm's production function has the form: , where L is the number of workers. At what level of employment will total output be maximum?

Solution

To answer the question of the problem, you need to find the maximum point of the function Y(L) . Let's differentiate it with respect to L and equate the derivative to zero: . We get quadratic equation, whose discriminant is , and the roots are . Since one of the roots is negative, we take . The number of workers is an integer, therefore, rounding, we get .

Conclusion

Resources in the economy act as factors of production, which include:

2. land (natural resources);

3. capital;

4. entrepreneurial ability;

5. scientific and technological progress.

All these factors are closely interrelated.

A production function is a mathematical relationship between the maximum volume of output per unit of time and the combination of factors that create it, given the existing level of knowledge and technology. Moreover, the main task of mathematical economics from a practical point of view is to identify this dependence, that is, to construct a production function for a specific industry or a specific enterprise.

In production theory, they mainly use a two-factor production function, which in general looks like this:

Q = f(K, L), where Q is production volume; K - capital; L – labor.

The issue of the relationship between the costs of substituting factors of production is solved using such a concept as elasticity of substitution of production factors.

The elasticity of substitution is the ratio of the costs of factors of production that replace each other with a constant volume of output. This is a kind of coefficient that shows the degree of efficiency of replacing one factor of production with another.

A measure of the interchangeability of production factors is the marginal rate of technical substitution MRTS, which shows how many units one of the factors can be reduced by increasing another factor by one, keeping output unchanged.

An isoquant is a curve representing all possible combinations of two costs that provide a given constant volume of production.

Funds are usually limited. A line formed by many points showing how many combined factors of production or resources can be purchased with available funds is called an isocost. Thus, the optimal combination of factors for a particular enterprise is the general solution of the isocost and isoquant equations. Graphically, this is the point of tangency between the isocost and isoquant lines.

The production function can be written in a variety of algebraic forms. Typically, economists work with linearly homogeneous production functions.

The work also considered specific examples solving problems using production functions, which allowed us to conclude that they are of great practical importance in economic activity any enterprise.

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