On the theory of optimization of transaction costs of multifactor manufacturing enterprises

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Introduction

One of the important and relevant areas of modern economic theory is the development of economic and mathematical methods for predicting the economic performance of an enterprise, taking into account the interaction of economic entities with the surrounding social environment. The activity of any industrial enterprise is carried out in the sphere of material production and in a certain social environment. The result of the impact of the enterprise on its own production are certain transformations, which are accompanied by output, production costs and profits. The interaction of the enterprise with the social sphere is carried out in the form of a set of transactions that generate transaction costs and redistribution of profits. The production activity of the enterprise forms its production transformation costs, and the interaction of the enterprise with the social environment generates non-production transaction costs. [12].

Transaction costs can include the cost of finding economic information, the cost of measuring the parameters of various goods, the cost of negotiating and negotiating contracts, the cost of creating specifications and protecting property rights, the cost of managing opportunistic behavior, and so on. In some cases, transaction costs are the result of strengthening measures taken by management to improve the quality of products with new enhanced consumer properties. A significant part of transaction costs can be spent by management on social programs for staff, on professional development programs for employees, on the environment, on scientific and charitable projects, etc. [3–13].

The implementation of such programs contributes to the growth of quality products and sales volumes, develops the innovative component of the enterprise, and attracts new volumes of investment. At the same time, the implementation of these programs generates significant transaction costs and reduces the profit of the enterprise. The presence of transaction costs forces the management of the enterprise to maximize not the profit function, but a certain transactional utility function, which takes into account the opportunistic interests of the management and the outflow of part of the enterprise's profits for non-production needs. These circumstances prevent the achievement of the maximum possible profit of the enterprise, instead of which it is necessary to restrict itself to its optimal value [14–21].

The aim of the published work is to develop new economic and mathematical models for calculating the optimal profit of multifactorial manufacturing enterprises based on the optimization of transaction costs.

The scientific novelty of the obtained models lies in the fact that they take into account the influence of the interaction of transformational and transactional costs of manufacturing enterprises with an arbitrary set of production factors on their economic performance.

 

1. Statement of the problem

The set of resources that the enterprise provides for the production and release of its products can be presented in the form of a  – dimensional vector of the space  of volumes of production factors

,

where  are the components that represent the main and labor resources involved in production;  are resources that provide non-productive transactional activities of the enterprise.

It should be noted that factors of production  are only sources of production or transformation costs, and resources  are sources of both production and transaction costs.

The general view of an arbitrary multifactorial production function is given by the expression

,                                                   (1.1)

 

where  is the volume of products manufactured by the manufacturing enterprise.

We will assume that, within the framework of our model, product output is provided by the Cobb – Douglas multiplicative production function

,                                                             (1.2)

 

where  is the cost of products produced for unit volumes of resources, and the power-law indicators of the production function  and  are the elasticities of output with respect to the corresponding resources .

The proportional costs of an enterprise with such resources are

,                                               (1.3)

 

where  are the cost of costs for unit volumes of resources, respectively,  are the fixed costs of the enterprise.

The expression for the profit of the enterprise  is expressed by the difference of formulas (1.2) and (1.3)

 

где  – стоимости затрат на единичные объемы ресурсов, соответственно,  – постоянные затраты предприятия.

.                                  (1.4)

 

To calculate the highest income, the profit function (1.4) should be maximized. However, in practice, the enterprise has to maximize the target transactional utility function, which takes into account the redistribution of profits in the interests of the enterprise management and for the implementation of socially oriented programs. Here the utility function is assumed to be linear

,                                     (1.5)

 

where  are the coefficients of the utility function (1.5), which determine the level of redistribution of funds between profit and non-productive resources. It should be noted that all the coefficients of the utility function (1.5) are non-negative

The influence of the target transactional utility function (1.5) of the enterprise on the redistribution of profits in the interests of the enterprise management and for the implementation of socially oriented programs is completely determined by a set of coefficients . These coefficients have lower and upper bounds .

The values  of the parameters correspond to a situation in which the enterprise does not finance any social or non-production programs at all and the transactional utility function (1.5) coincides with the profit function (1.4).

The values  correspond to a situation in which the enterprise begins to spend all its profits on social or non-production programs . By controlling the choice of coefficients , it is possible to choose such a mode of operation of the enterprise, in which both the economic component of production and its social orientation will remain quite effective.

The maximum possible profit of the enterprise  and the resources corresponding to it  and  are found from the conditions

                                                         (1.6)

The system of equations (1.6) can be written in the form

                                               (1.7)

where  , .

The structure of the system of equations (1.7) shows that the resources of the enterprise  and  are related by the ratios

        (1.8)

Substituting relations (1.8) into the system of equations (1.7), we obtain

                     (1.9)

Solving the system of equations (1.9) with respect to the quantities  and , we find the values of the volumes of resources  and , corresponding to the maximum profit

                         (1.10)

The expression for the maximum profit takes the form

                     (11)

 

Since in real conditions, in addition to the profit function, it is necessary to take into account the target transactional utility function, formulas (1.10) and (1.11) express the maximum values of the enterprise's profit and the corresponding values of resources unattainable in practice.

To obtain real optimal values of the enterprise's work, along with the maximization of the profit function (1.4), it is necessary to maximize the target transactional utility function (1.5).

Substitution of the profit function (1.4) into the transactional utility function (1.5) gives

                       (1.12)

 

The optimal profit  and the resources corresponding to it  and  are found from the conditions

                                                             (1.13)

or

                                        (1.14)

where  .

The structure of equations (1.14) shows that in this case the resources of the enterprise  and  are related by the relations

                                  (1.15)

Substituting relations (1.15) into the system of equations (1.14), we obtain

                             (1.16)

Solving the system of equations (1.16) with respect to the quantities  and , we find the values of the volumes of resources  and , corresponding to the maximum profit

                            (1.17)

The expression for the optimal profit takes the form

                   (1.18)

Let us now apply the developed method for calculating the maximum profit, optimal profit and transaction costs (1.6) – (1.18) for some variants of enterprise models.

 

2. Calculation of the maximum and optimal profit for two-factor manufacturing enterprises

Let us first consider the case when the output of an enterprise is provided by one production factor  and one non-production resource . Then the formulas for the production function, costs, profit and utility function (1.2) – (1.5) take the form

,                                                                (2.1)

,                                                     (2.2)

,                                            (2.3)

.                                                                (2.4)

The influence of the target transactional utility function (2.4) of the enterprise on the redistribution of profits in the interests of the enterprise management and for the implementation of socially oriented programs is completely determined by the parameter , .

The value of the parameter  corresponds to a situation in which the enterprise does not finance any social or non-production programs at all and the transaction utility function (2.4) coincides with the profit function (2.3).

The values  correspond to a situation in which the enterprise begins to spend all its profits on social or non-production programs. By controlling the choice of the coefficient, it is possible to select such a mode of operation of the enterprise, in which both the economic component of production and its social orientation will remain quite effective.

The peculiarities of the enterprise's work differ significantly in the short-term and long-term periods of activity. When constructing a mathematical model of an enterprise with a short-term period of work, changes in basic and labor resources can be neglected .

Obviously, the maximum possible value of the profit function (2.3) can be obtained only if the target transactional utility function (2.4) is not fully taken into account. The equation for calculating this value is

,                                                  (2.5)

where .

The resource value  corresponding to the maximum profit of the enterprise is found from the solution of equation (2.5)

.                                                                       (2.6)

The maximum value of the profit of the enterprise is expressed by the ratio

,                                     (2.7)

or

.                         (2.8)

Formulas (2.6) – (2.8) show the unattainable in practice maximum values of the profit of the enterprise  and the corresponding value of the resource , since in real conditions it is necessary to take into account the target transactional utility function. For real-optimal values of the enterprise, along with maximization of profit function (2.3) is necessary to maximize the target transaction utility function (2.4). Substitution profit function (2.3) in the transaction utility function (2.4) gives

.                                 (2.9)

The optimal value of the profit function (2.3), taking into account the influence of the target transactional utility function, is determined from the condition

.                                        (2.10)

The solution to equation (2.10) gives the optimal resource value

,                                                            (2.11)

where .

It should be noted that there are obvious inequalities  and . Comparison of the right-hand sides of formulas (2.6) and (2.11) gives

.                                                                       (2.12)

The optimal value of the profit of the enterprise is expressed by the ratio

,                           (2.13)

or

.             (2.14)

It follows from relations (2.6) and (2.8), (2.11) and (2.14) that

.                                                                 (2.15)

Let's apply the obtained formulas to calculate the maximum possible value of the profit function, and the optimal value of the profit function.

Figure 1 shows the curve of the profit function , the line of indifference of the target transactional utility function  and the lines of indifference of the target transactional utility function, corresponding to the boundary values of the parameter  and . Points of tangency ,  curve  and lines  are calculated by formulas (24), (26), (29), (32). The parameter value  is equal to the absolute value of the tangent of the slope of the tangent curve of profit at the point of its intersection with the abscissa axis

,

where  is the value of the resource  at which the profit of the enterprise vanishes .

Figure 1 shows that the indifference lines are a one-parameter family of straight lines with a parameter  for which the profit curve is an envelope line.

Suppose now the period of operation of the enterprise is long-term, and the production factor  is a variable. The enterprise can achieve the maximum possible value of the profit function (8) only if the target transactional utility function (2.9) is not fully taken into account. It is determined from the conditions

                                                              (2.16)

 

Figure 1 – Curve of the profit function  (solid line) and indifference lines of the target transactional utility function  and indifference lines of the target transactional utility function, corresponding to the boundary values of the parameter  and  (dashed lines). Touch points ,  and . Calculated values: , , , , , , , , , . Touch points: , , .

 

The system of equations (2.16) can be written in the form

                                                (2.17)

where .

 

Equations (2.17) show that the quantities  and  are related by the relation

.                                                              (2.18)

Substituting formula (2.18) into the first equation of system (2.17), we find the values of the resources at which the profit of the enterprise takes the maximum value

                 (2.19)

The maximum profit value is calculated by the formula

    (2.20)

Let us now calculate the optimal profit of the enterprise, taking into account the target transactional utility function (2.4). To do this, it is necessary to jointly maximize the profit function (2.3) and the target transactional utility function (2.4). Obviously, the optimal values of the resources, the profit function and the transactional utility function are found from the conditions

                                (2.21)

Figure 2 – Graphs of the surface of the profit function  and the plane of indifference of the target transactional utility function , constructed using formulas (21) and (22) for the parameter . The point of contact between the surface and the plane  is calculated using formulas (40) and (42). Calculated values: ; ; ; ; ; . Calculated parameter values: ; ; ; ; ;

 

The solution to equation (2.21) has the form

.                           (2.22)

Comparison of formulas (2.19) and (2.22) shows that the inequalities

.                                     (2.23)

The optimal value of the profit of the enterprise is expressed by the ratio

           (2.24)

It follows from relations (2.7), (2.10), and (2.11) that

.                                               (2.25)

Let's apply the obtained formulas to calculate the maximum possible value of the profit function, and the optimal value of the profit function.

Figure 2 shows the graphs of the surface of the profit function  and the plane of indifference of the target transactional utility function , constructed using formulas (2.3) and (2.4) for the parameter .

 

3. Calculation of the maximum and optimal profit for three-factor manufacturing enterprises.

If the output of the enterprise is provided by two production factors  and one non-production resource , then the formulas for the production function, costs, profit and utility function take the form

,                                                                     (3.1)

,                                                   (3.2)

,                                     (3.3)

,                                                                           (3.4)

where  is the fixed capital (production assets),  is the labor resources involved in production,  is the cost of products produced per unit volume of resources,  are the elasticity of output for the corresponding resources,  are the cost of costs per unit amount of resources,  is the fixed costs of the enterprise.

The maximum possible value of the profit function (3.3) can be obtained only if the target transactional utility function (3.4) is not fully taken into account. The system of equations (1.7) for finding this value takes the form

                                                        (3.5)

The solution of the system of equations (3.5) for the maximum values of the volumes of resources , ,  is written in the form

                                        (3.6)

The expression for the maximum profit takes the form

.               (3.7)

To calculate the optimal profit of the enterprise, it is necessary to jointly maximize the profit function (3.3) and the target transactional utility function (3.4). The system of equations (1.14) for finding the optimal values of resources, profit function and transaction utility function takes the form

                                        (3.8)

The solution of the system of equations (3.7) for the optimal values of the volumes of resources , ,  is written in the form

                                (3.9)

The expression for the optimal profit takes the form

.             (3.10)

Since in this case the profit function and the target transactional utility function are functions of three variables, graphical interpretation is impossible.

Using formulas (3.6), (3.7), (3.9) and (3.10), the maximum possible values of the profit function are calculated, not taking into account the transaction utility function and the optimal values of the profit function, taking into account the transaction utility function ; ; ; ; ; ; ; . Calculated values: ; ; ; ; ; ; ; ; .

 

Conclusion

New economic and mathematical models have been developed for optimizing transaction costs of multifactorial manufacturing enterprises.

It was found that transaction costs force to maximize not only the profit function of the enterprise, but also the target transactional utility function, which takes into account the redistribution of profits between the production area of ​​the enterprise and the social environment surrounding it.

The influence of production and transaction costs on the size of the maximum and optimal profit in the short and long term of the enterprise has been investigated.

It is shown that an enterprise bearing transaction costs can only achieve the optimal amount of profit, the value of which is less than the maximum possible amount of profit.

About the authors

Elena A. Ilyina

Samara National Research University

Author for correspondence.
Email: elenaalex.ilyina@yandex.ru
ORCID iD: 0000-0002-2590-6138

Candidate of Physical and Mathematical Sciences, associate professor of the Department of Mathematics and Business Informatics

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russian Federation

Leonid A. Saraev

Samara National Research University

Email: saraev_leo@mail.ru
ORCID iD: 0000-0003-3625-5921

Doctor of Physical and Mathematical Sciences, professor, head of the Department of Mathematics and Business Informatics

Russian Federation, 34, Moskovskoye shosse, Samara, 443086, Russian Federation

References

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