Models of the company (manufacturer)

The company’s costs for the production of products, the task of their minimization

Cost classification. Any production is associated with the costs of raw materials, electricity, labor, equipment, land and so on. Without the use of the necessary resources, it is impossible to create new goods.

Production costs are a set of costs incurred by an entrepreneur in ensuring a particular volume of production and its subsequent sale in a certain period of time. Costs can be classified according to many characteristics.

Fixed costs (FC) are costs that do not change in the short term with an increase or decrease in production volume. Fixed costs include costs associated with the use of buildings and structures, machinery and production equipment, rent, major repairs, as well as administrative costs. They are paid even when the products are not produced at all.

Variable costs (VC) are costs that vary with the increase or decrease in output. These include the cost of raw materials, electricity, auxiliary materials, labor costs.

Total costs (TC) are a set of fixed and variable costs of the firm in connection with the production of products in the short term.

Every producer needs to know the average costs (i.e. the cost of producing a unit of output), since they are comparable to the price. They are important in determining the profitability and unprofitability of production.

Average Fixed Cost (AFC) is the fixed cost of producing a unit of output (FC/y). As total revenue increases as output increases, average fixed costs are smaller and smaller.

Average Variable Costs (AVC) – Variable Unit Production Costs (VC/y). They reach their minimum when they reach the technologically optimal size of the enterprise. The concept of average variable costs is necessary to determine the efficiency of the firm’s management, the equilibrium position and determine the immediate prospects for development – expansion, reduction of production or withdrawal from the industry.

Average total costs (PBX) – the ratio of total costs to the volume of output (TC / y).

Marginal cost (MS) is the increment of total costs caused by an infinitesimal increase in production (ΔTC/Δu). When the MS<ATS, the average cost curve goes down: the production of each new unit of output reduces the average cost. When the MS>ATS, the average cost curve goes up: the production of a new unit of output increases the average cost. When PBX=min, THEN MS=PBX. The marginal cost curve intersects the average variable cost curve and the average total cost curve at the points of their minimum value.

Let’s define the cost function. The cost function expresses the dependence between the volume of products produced and the minimum necessary costs of its production:

C=C (y).                                        (6-1)

To quantitatively characterize the dependence of total costs on the volume of output, the coefficient of elasticity of costs from output (eC, Q) is used. It shows how much the total cost will change if you change output by 1%:

eC,Q = (ΔTC/Δy)*(y/TC) = MC/AC                (6-2)

To quantify costs, you need to know the prices of factor services. In general, dependence:

x=x(y)                                                (6-3)

it is the dependence of the volume of resources on the volume of output. Such a function is called the function of production inputs of resources, and the costs themselves are called

Z(y)=q1*x1(y)+q2*x2(y)+…+qn*xn(y)                (6-4)

Consider the case when there are two factors of production: labor and capital. Let’s denote the wage rate (labor price) – rL, and the rent for the use of capital per unit of time – rK. Then the total costs (CU) of the output of a certain number of products are equal to:

TC = rL*L+rK*K                                        (6-5)

The volumes of factors of production used at a given output are predetermined by the technology represented by the production function y = y (L, K). Therefore, L=L(y), K=K(y), and therefore TC=TC(y). Suppose that the production technology is characterized by the Cobb-Douglas production function:

y=A*x1a1*x2a2*…*xnan,                                (6-6)

where A>0, 0<aj<1, j=1,…,n (in our case y=LaK1-a ) (6-7)

(the different types of production functions will be discussed in more detail in chapter II)

In the short term, the amount of capital is fixed, and the production function contains only the amount of labor used as an argument. To produce at a given technology in units of production, L=(y*K1-a)1/a units of labor are required. Let’s substitute this value into the formula (6-5):

TC=rL*(y*K1-a)1/a+rK*K=TC (y)                (6-8)

The marginal cost of this technology, characterized by the Cobb-Douglas function, is:

MC=rL/α*(y/K)(1-α)/α                                (6-9)

Average fixed costs, average variables and average total costs are rK*K/y, rL*(y/K)(1-α)/α and rK*K/y+rL*(y/K)(1-α)/α, respectively.

Since, by definition, the cost function expresses the dependence between the output of products and the minimum costs of its production, it is first necessary to find such a combination of labor and capital that provides the minimum costs for a given output. With a given sum of the total production costs of the CU, the set of possible combinations of labor and capital is determined by equation (6 – 5) Let’s solve it with respect to K:

K=TC/rK-(rL/rK)*L                                        (6-10)

Equation (6-10) is the isocoste equation. The tangent of the angle of inclination of the isocoste is equal to the ratio of prices for factors, and its distance from the origin is determined by the volume of production costs. All combinations of the volumes of labor and capital corresponding to the points on the isocoste and below it are “affordable” for the producer, and all combinations of both factors, marked by points above the isocoste, are not available to him. The trajectory of the isocquant touch points (graphs of production functions) and isocoste indicates a combination of resources at which the costs required for each of the outputs are minimal. At the point of contact, the slope of the isoquant coincides with the slope of the isocoste. Thus, the problem of minimizing costs boils down to the following: it is necessary to find such an isocoste that would be tangential to a given isoquant, that is, it is necessary to find the point of contact of the isocost with the most distant isoquant. Touch point x* is the optimal solution. We will solve this problem by the method of Lagrange multipliers.

The task of minimizing costs

For the problem of minimizing costs, the Lagrange function is:

F(x1,x2,…,xn,λ1)=q1*x1+q2*x2+…+qn*xn+λ1*(y0-f(x1,x2,…,xn)), (6-11)

where q1,q2,…,qn are the resource prices x1,x2,…,xn, respectively


x1>=0, x2>=0, …, xn>=0

Next, we get a system of equations:

F/  xj=qj-λ1*  f/  xj=0,  j=1,2,…,n                                         (6-12)

F/  λ1=y0-f(x1,x2,…,xn)=0  or

f/  xj=1/λ1*qj,  j=1,2,…,n;  f(x1,x2,…,xn)=y0                        (6-13)

At the minimum point, we get:

The marginal productivity of resources is proportional to their prices (the proportionality coefficient is 1/λ*1), i.e.:

f / xj=(1/λ*1)*qj;  j=1,2,…,n;                                                (6-14)

the ratio of marginal productivity of resources is equal to the ratio of their prices, i.e.:

(  f/  xj):( f/  xi)=qj/qi;  j,i=1,2,…,n;  j≠i;                                (6-15)

the ratio of the marginal productivity of resources to their prices is equal to each other, i.e.:

(  f/  xj):qj=1/λ*1,  j=1,2,…,n                                                (6-16)

The obtained relations form the basis of the theory of marginal productivity of factors of production as a theory of value, namely: the prices of resources are proportional to the marginal productivity of resources, in particular for labor we have that it is estimated in accordance with its marginal productivity.

Let’s give an interpretation of the Lagrange multiplier. Have:

dZ=q1*dx1+q2*dx2+…+qn*dxn .

At the  minimum point qj=(λ*1)*(  f/  xj), j=1,2,…,n, hence

dZ=λ*1((  f/  x1)*dx1+(  f/  x2)*dx2+…+(  f/  xn)*dxn)=(λ*1)*dy.  (6-17)


λ*1=dZ/dy,                                                                        (6-18)

i.e. λ*1 is the total marginal cost per unit of additional output.

Let’s look at some types of resource cost and cost functions.

Let be a linear heterogeneous resource cost function specified

xj=αj*y+bj,  j=1,2,…,n,  αj>0, bj>0,  j=1,2,…,n.

Then the cost function is as follows:


where a=∑qj*αj,  b=∑qj*bj.

If the nonlinear resource cost function xj=φj(y),  j=1,2,…,n is specified, then


The challenge of maximizing output

The task of maximizing the volume of production is to determine the maximum volume of output at a given cost of resources. Mathematically, it is formulated as follows:

Y=f(x1,x2,…,xn)→max                                (6-19)

under conditions

q1*x1+q2*x2+…+qn*xn=C,                        (6-20)

x1>=0, x2>=0,…,xn>=0                                (6-21)

Geometrically, this means finding an isoquant of a production function y=f(x1,x2) that would relate to a given isocost q1*x1+q2*x2=C.

Each isoquant is characterized by the following properties:

an isoquant lying above and to the right of another corresponds to a greater number of products produced; isocquants do not intersect; in the economic field, the isocquants have a negative slope, i.e. they are facing the origin with a convexity.

Consider some of the manufacturing functions that are often used when analyzing the behavior of the manufacturer:

A linear production function is as follows:


for which a1>0, a2>0,…, an>0, i.e. a linear dependence of output on the costs of factors of production is assumed;

Production function with constant parameters. This function is given by the ratio:

y=min(x1/a1,x2/a2,…,xn/an),  where a1>0, a2>0,…, an>0 ;

Cobb-Douglas production function. The function is y=A*x1α1*x2α2*…*xnαn, where A>0, 0<αj<1,  j=1,2,…,n (this type of production function will be considered as an example in my term paper); Production function with constant replacement elasticity (CES). This function is as follows:

y=A*(B1*x1-p+…+Bn*xn-p)-γ/p,  for which A>0, 0<Bj<1,  j=1,2,…,n. .

The problem of maximizing the volume of output will also be solved by the method of Lagrange multipliers.

The Lagrange function of this task will look like this:


Optimality conditions will be:

F/  xj=  f/  xj-λ2*qj=0,  j=1,2,…,n;

F/  λ2=C-∑qi*xi=0


f/  xj=λ2*qj,  j=1,2,…,n;  ∑qi*xi=C                        (6-22)

At the maximum point x* there will be ratios similar to the corresponding ratios in the problem of minimizing costs, namely:

The marginal productivity of resources is proportional to their prices with a proportionality factor of λ*2, i.e.:

f/  xj=(λ*2)*qj,  j=1,2,…,n;

the ratio of the marginal productivity of resources is equal to the ratio of their prices, i.e., the ratio of the marginal productivity of resources.

(  f/  xj): ( f/  xi)=qj:qi;  j,i=1,2,…,n,  j≠i;

the ratio of the marginal productivity of resources to their prices is equal to each other, i.e.

(  f/  xj)=λ*2,  j=1,2,…,n .

Let’s define the economic meaning of the multiplier λ*2. The complete differential of the production function will be:

dy=(  f/  x1)*dx1+(  f/  x2)*dx2+…+(  f/  xn)*dxn.                        (6-23)

Since at the point of maximum x* there is a ratio:

f/  xj=(λ*2)*qj (j=1,2,…,n), then

dy=(λ*2)*(q1*dx1+q2*dx2+…+qn*dxn)=(λ*2)*dZ=(λ*2)*dC.  (6-24)

From here we get  dy/dZ=dy/dC=λ*2.                                (6-25)

Thus, λ*2 expresses the additional output per unit of total costs, i.e. it expresses the total marginal productivity of resources.

Thus, it can be concluded that λ*1 and λ*2 for the problems under consideration are mutually opposite in meaning. Therefore, these two tasks are called mutual tasks for the manufacturer.

If we take Z*=Zmin obtained in the problem of minimizing costs at y=u0 as C in the problem of maximizing output, then the maximum output volume is y*=u0, λ*2=1/λ*1, and the optimum points coincide.


The technological relationship between output and input is given by the function y=f(x)=f(x1,x2,…,xn), which depends on n variables, which is called the production function. And the function C(y)=Zmin(y)=∑qi*x*i(y) is called a cost function.

The task of minimizing the cost of production:


and the task of maximizing the volume of output:


are mutual tasks for the manufacturer.

Moreover, at the point of optimum of both costs and output, the following ratios are observed:

the marginal productivity of resources is proportional to their prices; the ratio of marginal productivity of resources is equal to the ratio of their prices; the ratio of the marginal performance of resources to their prices is equal to each other.

The geometric solution to the problem of determining the maximum possible output with the money available to the manufacturer, represented by isocost, and a given production function, represented by the isocquant family, is as follows: you need to find the point of contact of the isocoste with the most distant isoquant.