Theory of a firm, Production function and its properties - Microeconomics

Theory of the firm

As a result of studying this chapter, the student must:


• The concept of the production function and its properties, the relationship between the output volume, the quantity and prices of the factors of production, the cost function and its properties, the laws of the firm's behavior when prices are set or the firm has the opportunity to set prices;

be able to

solve the optimization problem of maximizing the company's profit, describe the firm's reaction to market incentives in the short and long terms;


• methods of solving the problems of minimizing costs for a given level of output and finding the volume of output that maximizes profit.

Production function and its properties

Production is the process of converting production factors into products. The reality with which the firm is facing is the technological problem -

Sti. Technology determines and limits the ability to combine factors of production for output.

Many production opportunities are the most common way of describing a company's technology. The production function characterizes the maximum possible output that can be obtained by using this combination of resources.

If only one product is produced with the help of many factors (and in the future we will consider firms producing a single product from many factors), we will designate the output volume through y, and the volume of the i-th factor will be denoted by xi, so that for n factors the entire vector of factors will be denoted as . It is required that and

The production function is built for this technology. Improvement of technology, increasing the maximum possible output of products with any combination of factors, is displayed by a new production function.

Although production functions are different for different types of productions, they all have common properties.

The production function is continuous, strictly increasing , and strictly quasi-concave. Continuity ensures that small changes in the costs of factors of production lead to small changes in output. The condition of strict increase provides an increase in the volume of output with increasing costs of any of the factors. Strict quasi-concavity ensures the complementarity of factors of production, which means that the production of this product can not be carried out without any input factor costs.

The listed (desirable) properties of the production function are completely consistent with its definition, since they only concern the cost-output relationship.

Let's give examples of the most successfully constructed and therefore often used in practice production functions. For simplicity, we will consider a two-factor one-product production function of the form

1. Cobb-Douglas production function.

The first successful experience of constructing a production function as a regression equation on the basis of statistical data was obtained by American scientists - mathematician D. Cobb and economist P. Douglas in 1928. The function they proposed initially had the form

( 4 . 1 )

where Y is the output volume; K - the value of production assets (capital); L - labor costs; - number

Parameters (scale number and elasticity index). Due to its simplicity and rationality, this function is widely used until now and has received further generalizations in various directions. The Cobb-Douglas function can sometimes be written in the form

It is easy to verify that V (0,0) = 0 and

In addition, the function (4.1) is linearly homogeneous:

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

To account for technological progress, a special multiplier (technical progress) is introduced in the Cobb-Douglas function , where t is the time parameter; v is a constant number characterizing the rate of development. As a result, the function accepts dynamic view:

where optional

2. The production function CES (with a constant elasticity of substitution) has the form

( 4 . 2 )

where - scale factor; - distribution ratio; - factor substitution; - the degree of homogeneity. If the conditions are satisfied, then the function (4.2) satisfies the inequalities and

Given the technical progress, the CES function is written

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

3. Production function with fixed proportions. This function is obtained from (4.2) with and has the form


4. The production cost-output function (the Leontief function) is obtained from (4.3) with

In essence, this function sets the proportion by which the amount of costs of each type is determined, which is necessary for the production of one unit of output. Therefore, in the literature there are often other forms of writing:

( 4 . 4 )


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

5. The production function of analyzing the methods of production activity. This function generalizes the production cost-output function to the case where there are a number of basic processes (methods of production activity), each of which can occur with any nonnegative intensity. It has the form of an optimization task


where - output at a single intensity of the j-th base process; - intensity level; - the amount of costs required for the j type k, required for a single intensity of the method.

As we see from (4.5), if the output produced at a unit intensity and the costs necessary per unit intensity are known, then the total output and total costs are found by adding the output and costs, respectively, for each base process at the selected intensities. Note that the task of maximizing the function img src="images/image326.jpg"> in (4.5) for given constraints-inequalities is a model for analyzing production activity (maximizing output with limited resources).

6. The linear production function (function with interchange of resources) is applied in the presence of a linear dependence of release on costs:


The volume of output with increasing use in the production of the variable factor will increase, but this growth has certain limits within the given technology. If the production function is differentiated, then its partial derivative is called the limiting product of the i-th production factor and shows the output change when using an additional unit of the i-th factor.

Graphical representation of the production function is isoquant - a curve representing an infinite set of combinations of factors of production that ensure the same output. We denote this set by . For a given production factor vector x , the isoquant passing through the point x, is the set of production factor vectors, each of which allows producing the same output volume as x , namely

The analogue of the marginal rate of substitution in consumption theory in firm theory is the marginal rate of technical substitution (MRTS). It measures the extent to which one factor can be replaced by another without changing the output volume. Formally, MRTS is defined as the ratio of marginal products


MRTS of any two factors of production, generally speaking, depends on the number of all factors used. In empirical studies, however, it is often assumed that the factors of production can be broken down into a relatively small number of types, and the degree of substitution between factors of one type differs from the degree of substitution between factors of different types. Production functions that have this property are called separable, and there are at least two basic types of separability.

Let - the number of factors of production and assume that this set can be divided into disjoint subsets . The production function is called non-strictly separable if MRTS between two factors of the same group does not depend on factors belonging to another group:

for all ,

where and are the limiting products i and j factors.

The production function is called strictly separable if the MRTS between two factors from different groups does not depend on factors outside of these groups:

for all

Isoquants (as well as indifference curves) can have a different configuration (Figure 4.1).

Isoquant species: a is isoquantum when the factors are completely interchangeable, b is an isoquant for which substitution is incomplete, and c is an isoquantum in which factors are not interchangeable

Fig. 4.1. Species isoquant: a - isoquantum, when the factors are completely interchangeable; b is an isoquant for which substitution is incomplete; c - isoquantum, in which factors are not interchangeable

The shape of the isoquant is given by the elasticity of substitution. For the production function f (x), the replacement elasticity between the t-th and j-th factors at the point x is defined as

where and are the limiting products of the i-th and j factors.

If the production function is quasi-concave, then the elasticity of substitution can not be negative. The closer it is to zero, the harder there will be a substitution of factors; the more it is - the easier it is substitution between them. In Fig. 4.1a the elasticity is infinite; in Fig. 4.16 the elasticity is finite, but greater than zero; in Fig. 4.Ie elasticity is zero.

All production functions with constant elasticity of substitution (including the Cobb-Douglas and Leontief functions) are included in the class of homogeneous first-degree production functions, which plays an important role in theoretical and applied research. Homogeneity of the first degree further structure the production function; homogeneous first-degree functions are always concave (Shepard's theorem).

Example 4.1

Consider a production function with constant elasticity of substitution y = (Χχρ X2pI17p for 0 ≠ p <.) To calculate the elasticity of substitution, we note that In (X2Zx1) = In (X2) In (X1 ) 1, therefore, if we take the total differential of the numerator σ, we obtain

Having calculated the partial derivatives of a function, dividing them into each other and taking logarithms, we obtain

Taking the total differential, we find the denominator σ:

By dividing (a) by (b), we get the elasticity of substitution , which is a constant, hence the abbreviation

CES is the constant elasticity of substitution.

For CES functions, the degree of substitution between factors is always the same regardless of the release level or the ratio of factors. This limits the range of technologies described by such functions. At the same time, with the help of various values ​​of the parameter p, and hence of the various values ​​of the parameter σ, it is possible to specify technologies with a very different (but everywhere constant) elasticity of substitution of the factors. The closer p is to unity, the greater σ; if p = 1, then σ is infinite, the production function is linear, and its isoquants are similar to those shown in Fig. 4.1 a. Other popular production functions can also be considered as special cases of some CES functions.

thematic pictures

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