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DOI: 10.18287/2542-0461-2021-12-4-182-194 ЮУ
scientific article
Submitted: 21.08.2021 Revised: 25.09.2021 Accepted: 26.11.2021
On the theory of optimization of transaction costs of multifactor manufacturing enterprises
E.A. Ilyina
Samara National Research University, Samara, Russian Federation E-mail: elenaalex.ilyina@yandex.ru. ORCID: http://orcid.org/0000-0002-2590-6138
L.A. Saraev
Samara National Research University, Samara, Russian Federation E-mail: saraev_leo@mail.ru. ORCID: http://orcid.org/0000-0003-3625-5921
Abstract: The published article proposes mathematical models for calculating the optimal profit of multifactorial manufacturing enterprises that incur both production (transformational) and certain non-production (transactional) costs, the sources of which may be forced costs of searching for economic information, measuring the parameters of various goods, negotiating and the conclusion of contracts, for the development of specifications and the protection of property rights, for the opportunistic behavior of employees and managers of the enterprise, etc. A numerical analysis of the presented models for calculating the optimal profit of multifactor enterprises that bear transaction costs shows the unattainability of the maximum possible profit values, since in practice the enterprise management maximizes not the profit itself, but its utility, expressed in the form of the corresponding transaction function.
Key words: resources, factors of production, production function, profit, transformation costs, transaction costs, transactional utility function.
Citation. Ilyina E.A., Saraev L.A. On the theory of optimization of transaction costs of multifactor manufacturing enterprises. Vestnik Samarskogo universiteta. Ekonomika i upravlenie = Vestnik of Samara University. Economics and Management, 2021, vol. 12, no. 4, pp. 182-194. DOI: http://doi.org/10.18287/2542-0461-2021-12-4-182-194. (In Russ.)
Information on the conflict of interest: authors declare no conflict of interest. © Ilyina E.A., Saraev L.A., 2021
Elena A . Ilyina - Candidate of Physical and Mathematical Sciences, associate professor of the Department of Mathematics and Business Informatics, Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russian Federation.
Leonid A. Saraev - Doctor of Physical and Mathematical Sciences, professor, head of the Department of Mathematics and Business Informatics, Samara National Research University, 34, Moskovskoye shosse, Samara, 443086, Russian Federation.
научная статья
УДК 330.42
Дата поступления: 21.08.2021 рецензирования: 25.09.2021 принятия: 26.11.2021
К теории оптимизации транзакционных издержек многофакторных производственных предприятий
Е.А. Ильина
Самарский национальный исследовательский университетимени академика С.П. Королева,
г. Самара, Российская Федерация E-mail: elenaalex.ilyina@yandex.ru. ORCID: http://orcid.org/0000-0002-2590-6138
Л.А. Сараев
Самарский национальный исследовательский университетимени академика С.П. Королева,
г. Самара, Российская Федерация E-mail: saraev_leo@mail.ru. ORCID: http://orcid.org/0000-0003-3625-5921
Аннотация: В публикуемой статье предложены математические модели расчета оптимальной прибыли многофакторных производственных предприятий, несущих как производственные (трансформационные), так и определенные непроизводственные (транзакционные) издержки, источниками которых могут быть вынужденные расходы на поиск экономической информации, на измерения параметров различных благ, на ведение переговоров и заключение контрактов, на разработку спецификаций и защиту прав собственности, на оппортунистическое поведение сотрудников и руководителей предприятия и т.д. Численный анализ представленных моделей расчета оптимальной прибыли многофакторных предприятий, несущих транзакционные издержки, показывает недостижимость максимально возможных значений прибыли, поскольку на практике руководство предприятия максимизирует не саму прибыль, а свою полезность, выраженную в виде соответствующей транзакционной функции.
Ключевые слова: ресурсы; факторы производства; производственная функция; прибыль; трансформационные издержки; транзакционные издержки; транзакционная функция полезности. Цитирование. Ilyina E.A., Saraev L.A. On the theory of optimization of transaction costs of multifactor manufacturing enterprises // Вестник Самарского университета. Экономика и управление. 2021. Т. 12, № 4. С. 182-194. DOI: http://doi.org/10.18287/2542-0461-2021-12-4-182-194.
Информация о конфликте интересов: авторы заявляют об отсутствии конфликта интересов.
© Ильина Е.А., Сараев Л.А.
Елена Алексеевна Ильина - кандидат физико-математических наук, доцент кафедры математики и бизнес- информатики, Самарский национальный исследовательский университет имени академика С.П. Королева, 443086, Российская Федерация, г. Самара, Московское шоссе, 34.
Леонид Александрович Сараев - доктор физико-математических наук, профессор, заведующий кафедрой математики и бизнес-информатики, Самарский национальный исследовательский университет имени академика С.П. Королева, 443086, Российская Федерация, г. Самара, Московское шоссе, 34.
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 n - dimensional vector of the space Rn of volumes of production factors
Q = ( 0!,a, s1, S2,..., S„),
where Qt are the components that represent the main and labor resources involved in production; Sj are resources that provide non-productive transactional activities of the enterprise.
It should be noted that factors of production Qt are only sources of production or transformation costs, and resources Sj are sources of both production and transaction costs.
The general view of an arbitrary multifactorial production function is given by the expression
V = V (Qi, 02,..., Qm, Sj, S2,..., Sn) , (1.1)
where V 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
m n
V = P -П Qa П Sp, (12)
s=1 p=1
where P is the cost of products produced for unit volumes of resources, and the power-law indicators of the production function as and care the elasticities of output with respect to the corresponding resources
(0 < as < 1,0 < Cp < 1).
The proportional costs of an enterprise with such resources are
m n
TC = £AQ • Qs + E A • Sp + TFC, (1.3)
s=1 p=1
where AQ , ASP are the cost of costs for unit volumes of resources, respectively, TFC are the fixed costs of the enterprise.
The expression for the profit of the enterprise PR = TR — TC is expressed by the difference of formulas (1.2) and (1.3)
где AS Ap - стоимости затрат на единичные объемы ресурсов, соответственно, TFC - постоянные затраты предприятия.
PR = P - П Qa - П Spp —E ÄQ - Qs —E Aps - Sp — TFC • (14
s
's 1 l^p /Li^Q ^s /
s=1 p =1 s=1 p =1
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
U = U(PR,Sl3S2,...,S„) = PR + £qp ■ Sp
(1.5)
p=1
where qare 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 (Vp : qp > 0)
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 qp. These coefficients have lower and upper bounds
q0 < q < q00.
lp lp -i 0
The values qp = q p = 0 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 qp = qrp correspond to a situation in which the enterprise begins to spend all its profits on social or non-production programs PR = 0. By controlling the choice of coefficients qp, 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 PRmax and the resources corresponding to it Qmax and Smax are found from the conditions
ÔPR
= 0
ÔPR
= 0.
8Q, dSj
The system of equations (1.6) can be written in the form
m n
P-n-nS? ="Q ■ Q, (i=i.m),
(1.6)
p=1
(1.7)
p ■n q- ■n s?=ai ■ s, ( j=i-n ),
p=1
where aQ = —° > al
a,■
Aj
AS
The structure of the system of equations (1.7) shows that the resources of the enterprise Q Q and S1,S2,...,Sn are related by the ratios
Qs = Q,
Substituting relations (1.8) into the system of equations (1.7), we obtain
g, 0s=Sp G, s,=s-a, Sp=Q, ■a.
gCQ aQ as as
-, 0
(1.8)
m n , .
Z a Z cp ™ С a Л
p ■ or ■ or ■n
s=1
"0 g j
■n
p=1
С ai
uo ap
yus J
= aQ ■ 0г
(1.9)
m
Z as Z ' T-r
P■ j ■ j ■n
m с G л
s
s=1
\ao J
n
p=1
í jACp
G ap
V" s
= al ■ S,. .
Solving the system of equations (1.9) with respect to the quantities Q and S ., we find the values of the
1 j
volumes of resources Q,max and Smax, corresponding to the maximum profit
1
s
1
s
c
a
a
Qmax _
S m ax _
—-П — -П
a 11 as 11
"Q s_l V Q У
( „i Y
( ~ i yp ^
ap
V S У
1—E as—E cp
f \as f
P-П a -Пía
V aS s_1 V aQ У p_1 V aS , У
1—E as—E cp
The expression for the maximum profit takes the form
m n m n
PRmax _ P - П( Qsmax )as - П ( S7x Г—E A - Qsmax—E ÄSp - Spmax—TFC •
p_1
p_1
(1.10)
(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
U _P-ПQa-ПS? —EAQ -Qs —EAsp - Sp — TFC + Eqp - Sp.
^ Ц p Q ^s ^ S p
s_1 s_1
(1.12)
The optimal profit PRopt and the resources corresponding to it Qopt and Sopt are found from the condi-
tions
or
ôU Л ôU Л
_ 0,-_ 0.
(1.1З)
ôQ
ôS..
P-ПQaa-ПSj _ aQ -Qi , (i _ 1..m)
s_1
p_1
(1.14)
p-П Qaa-П sj - s., ( j _ l..n ),
s_1
p_i
q
where -J _aj — — •
The structure of equations (1.14) shows that in this case the resources of the enterprise Qx,Q2,...,Qm and Sx,S2,...,Sn are related by the relations
a
Q -j -S aQ
Q _ Q - — , Q _ S -—, S _ S.- — , S _ Q.- — ^-s ' s p as p j p p i p
a
*Q aQ ^S VS
Substituting relations (1.15) into the system of equations (1.14), we obtain
eas _ec> ^( aQ ys
P - Qr1 - Qr1 -П
s
v aQ у
P-J -^1 -П
cp m ( i \ 'IS
s
v aQ у
v Q у
-П
p_1
n
-П
p_1
(a y'
Q
-j У
_aQ-Qi,
(j yp
qps
_qi- s¡ •
(1.15)
(1.16)
Solving the system of equations (1.16) with respect to the quantities Q and S ., we find the values of the
1 j
volumes of resources Q°pt and Sopt, corresponding to the maximum profit
s_1
s_1
s_1
s_1
c
s_1
a
a
s_1
Q,opt =
Sopt =
p
a
n
o s=1
с a \as aQ
V Q J
■П
p=1
Ca ^ ^
m
-Z a-Z cp
s=1 p=1
(1.17)
P
n
( ] \a
Us s=1
V Q J
■n
p=1
vS
m
1-Z as -Z cp
The expression for the optimal profit takes the form
m n m n
PRopt = P-n(QTT -n(spptf -IAQ ■ Qsopt-XA ■ Sopt-TFC.
(1.18)
s=1 p=1 s=1 p=1
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 Q and
one non-production resource S. Then the formulas for the production function, costs, profit and utility function (1.2) - (1.5) take the form
V = P • Qa • Sc , (2.1)
TC = Aq • Q + AS • S + TFC , (2.2)
PR = p ■ Qa ■ Sc - A0 ■ Q - As ■ S - TFC ,
(2.3)
U = PR + q • S. (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 q, (q0 < q < q j}.
The value of the parameter q = q0 = 0 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 q = qœ 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 Q = const.
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
dPR ~dS
c■(P■ Qa ■ Sc1 -as) = 0
(2.5)
where a„
As
The resource value S corresponding to the maximum profit of the enterprise is found from the solution of equation (2.5)
f
S„„ =
P ■ 00
■V
V as J
The maximum value of the profit of the enterprise is expressed by the ratio
PRmax = P ^ Qa ^ Sm ax " Aq • Q - A, • S^ - TFC ,
(2.6) (2.7)
or
s=1 p=\
c
c
PRmax = P • Q0
r P • Qa Л1—c
а
— Aq • Q — A,
S У
P • Q
1
a ЛИ—е
а
— TFC .
(2.8)
Formulas (2.6) - (2.8) show the unattainable in practice maximum values of the profit of the enterprise PRmax and the corresponding value of the resource Smax , 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
U = (P • Qa • Sc - Aq • Q - AS • S - TFC) + q • S . (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
dU dS
P . c. Qa
S
1—c
— AS
+ q = 0
(2.10)
The solution to equation (2.10) gives the optimal resource value S
opt
Sopt =
f P . Qa Л
-S
1—c
(2.11)
where -S = aS
q
It should be noted that there are obvious inequalities c > 0, q > 0 and rjs <as . Comparison of the right-hand sides of formulas (2.6) and (2.11) gives
(2.12)
Sopt > Smax .
The optimal value of the profit of the enterprise is expressed by the ratio
PRopt = P. Qaa. Socpt — Aq . Q — A, . Sopt — TFC
opt
-S "opt
or
PRopt = P • Qa
c
r P • Qa Л^
-S
— Aq • Q — Aj
rP • Qa Л
-S
— TFC .
(2.13)
(2.14)
It follows from relations (2.6) and (2.8), (2.11) and (2.14) that
PR ( Sopt )< PR ( Smax ) . (215)
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 PR = PR ( S ), the line of indifference of the target transactional utility function U ( PR, S ) = U ( PRopt, Sopt ) = Uopt and the lines of indifference of the target transactional utility function, corresponding to the boundary values of the parameter (q = 0) and (q = qœ) . Points oftangency (PR^,Smax), (PRopt,Sopt) curve PR = PR(S) and Unes U(PR,S) = U(PR^,) = U^ are calculated by formulas (24), (26), (29), (32). The parameter value ( q = qœ) 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
OPR ( Sf )
qœ =-
öS
where SF is the value of the resource S at which the profit of the enterprise vanishes (PRF = 0).
Figure 1 shows that the indifference lines are a one-parameter family of straight lines with a parameter q0 < q < qœ for which the profit curve is an envelope line.
c
1
1
c
Suppose now the period of operation of the enterprise is long-term, and the production factor Q 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
dPR=0, dIR=0. (2.16)
SQ PR ,U
SS
12
\ \ PRmax rr\ PRopt / ! 1 \ \ 4
/ / 1 1 / > > / < 1 / 1 1 1 1 i I i Y \ \ \ч-
S
SN Smax Sopt SF 30
Figure 1 - Curve of the profit function PR = PR (S) (solid line) and indifference lines of the target transactional utility function U (PR, S) = U (PRopt, Sopt ) = Uopt and indifference lines of the target transactional utility function, corresponding to the boundary values of the parameter (q = 0) and (q = qœ) (dashed lines). Touch points (PRmax ,S^ ) , (PR0Vt, Sopt ) and (PRf , Sf ) . Calculated values: P = 20, a = 0,25, c = 0,33, AQ = 4, AS = 1,7, TFC = 20, Q = 1,5, q0 = 0, q = 0,45, qF = 0,8783. Touch points: (PR^ = 9,8062;S^ = 8,8095), (pr^ = 8,5051;S^ = 13,9400), (PRf = 0 ; SF = 26,0713) .
The system of equations (2.16) can be written in the form
P ■ Qa ■ Sc =aQ ■ Q , P ■ Qa ■ Sc =ap ■ S,
(2.17)
where a = —^.
Q
a
Equations (2.17) show that the quantities Smax and Qmax are related by the relation
a
S = ■ Q
max ¿^m
(2.18)
a
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
(
Qmax
P
1—c c \aQ aS J
\l—a—c (
, S„„„ =
P
aa ■a1'"
Q S J
1—a —c
(2.19)
The maximum profit value is calculated by the formula
6
0
pRmax = p • Qmax • S^ — А • Qnax — As • S_ — TFC =
=p
pa
ac VaQ ^aS у
— aq •
p
^1—a—c
1 —c c
a •a
V" Q
— As •
p
(2.20)
vaQ
a
— TFC.
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
dU dQ dU dS
= a
( p • Qa—1 • Sc — aQ ) = 0, c •( p • Qa • Sc—1 —as ) + q = 0.
(2.21)
<pr ,U
Q
S
Figure 2 - Graphs of the surface of the profit function PR = PR (Q, S) and the plane of indifference of the target transactional utility function U (PR, S) = U (PRopt, Sopt) = Uopt, constructed using formulas (21) and (22) for the parameter q = 0,45. The point of contact between the surface and the plane (PRopt, Qopt, Sopt) is calculated using formulas (40) and (42). Calculated values: P = 20; a = 0,25;
c = 0,33; Aq = 2,4
As = 2,7:
TFC = 20 . Calculated parameter values: Qmax = 6,5087 ;
S max = 7,6369 ; pRmax = 6,24311; Qopt = 7,5112 ; ^ = 10,5758; pRoat = 5,5260
opt
opt
The solution to equation (2.21) has the form
Qopt
p
1 —c c
aQ -Vs
Q
1 —a—c
Sopt =
p
a 1 —a
aQ Vs
1 — a —c
Comparison of formulas (2.19) and (2.22) shows that the inequalities
q > q s > S
¿^opt >^max ' opt max
The optimal value of the profit of the enterprise is expressed by the ratio
pRopt = p • Qopt • Socpt — Aq • Qopt — As • Sopt — TFC =
= p •
pa
^1— a— c
opt
c
opt
aQ •Vs у
— Аг
p
opt 1
^1—a—c
1—c c
a; • Vs
V- Q
opt (
— As •
p
^1—a—c
aQ
VQ
— TFC.
It follows from relations (2.7), (2.10), and (2.11) that
(2.22)
(2.23)
(2.24)
PR(Qopt, Sopt )< PR(Qmax, Smax ) . (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 PR = PR (Q, S) and the plane of indifference of the target transactional utility function U (PR, S) = U (PRopt, Sopt) = Uopt, constructed using formulas (2.3) and (2.4) for the parameter q = 0,45.
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 Q1 = K, Q2 = L and one non-
production resource S1 = S, then the formulas for the production function, costs, profit and utility function take the form
V = P • Ka • Lb • Sc , (3.1)
TC = AK • K + Al • L + AS • S + TFC , (3.2)
PR = P • Ka • Lb • Sc - AK • K - Al • L - AS • S - TFC , (3.3)
U = PR + q • S , (3.4)
where K is the fixed capital (production assets), L is the labor resources involved in production, P is the cost of products produced per unit volume of resources, a, b, c are the elasticity of output for the corresponding resources, AK, AL, AS are the cost of costs per unit amount of resources, TFC 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 Km L , S is written in the form
max ' max
P ка max Lb max • Sc max и Kmax
P к a max Lb max •Sc max и T max
P к a max Lb max •Sc max и a S max
Kmax =
Lmax
Smax =
P
aK
f Y a
VaL J
f Y Y-a-b-'-'
a
V as J
r P a c Л
aT V L VaK J VaS J J
f P f aл a aл b Л
aS V S VaK J VaL J J
1-a-b-c
1-a-b-c
(3.6)
The expression for the maximum profit takes the form
PR = P • K" • L? • SI., - A^ • K„ - A • L - Aq • S - TFC .
„ ^ „„ w ^ . (3.7)
-________-____ -____ -____ K max L max S max v '
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
p •k" • Tb •Sc =a • K
r ^opt opt °opt _ UK ^opt ■
p • Ka • Tb • Sc =a • T
r ^opt opt °opt _ UL opt 5
(3.8)
p^K:
• L • Sc = n • S
"opt %pt opt 'Is °opt •
The solution of the system of equations (3.7) for the optimal values of the volumes of resources K
opt
L t, S t is written in the form
opt opt
Kopt =
p a
V
K
f V a
VaL у
( Y ^1— a—b—,
a
Vs
Lopt =
Sopt =
p_
aL p
Vs
( y ( y \l—a—b—,
a
Vaк у
a
as
Vaк у
a
KVsy (a >
(3.9)
b
VaL у
1—a—b—c
The expression for the optimal profit takes the form
Ptf = p• Kl • ZL • S! - AK • Kopt - Ar • Lopt - AS • Sopt - TFC .
opt
opt opt opt
K opt L opt
S opt
(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 Kmax = 16,3764; Lmax = 24,3427; Smax = 13,3922.
<—' max J j max 1 j max 1
PRmax = 24,0197; Kopt = 19,0222; L^ = 28,2755; Sopt = 18,6671; PR^ = 22,7314. Calculated values^ P = 20; a = 0,245; b = 0,24; c = 0,23; AK = 2,4 ; al = 1,55 ; AS = 2,7 ; TFC = 20; q = 0,45 .
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.
References
1 Coase R.H. The nature of the firm. Economica. New Series, 1937, vol. 4, no. 16, pp. 386-405. Available at: http://links.jstor.org/sici?sici=0013-0427%28193711%292%3A4%3A16%3C386%3ATN0TF%3E2.0.C0%3B2-B.
2 Coase R.H. The problem of social cost. Journal of Law and Economics, 1960, vol. 3, no. 3, pp. 1-44. Available at: http://wwwjstor.org/stable/724810?origin=JST0R-pdf.
3 Williamson O.E. Transaction-cost economics: The governance of contractual relations. Journal of Law and Economics, 1979, vol. 22, no. 2, pp. 233-261. DOI: http://dx.doi.org/10.1086/466942.
4 Williamson O.E. Comparative economic organization: The analysis of discrete structural alternatives. Administrative Science Quarterly, 1991, vol. 36, no. 2, pp. 269-296. DOI: http://doi.org/10.2307/2393356.
5 Williamson O.E. Opportunism and its critics. Managerial and Decision Economics, 1993, vol. 14, issue 2, pp. 97-107. DOI: https://doi.org/10.1002/mde.4090140203.
6 Williamson O.E. The Economic Institutions of Capitalism: Firms, Markets, Relational Contracting. Detroit: Free Press, 1998, 450 p. Available at: http://www.bookre.org/reader?file=1066388.
7 Williamson O.E. Strategy research: Governance and competence perspectives. Strategic Management Journal, 1999, vol. 20, no. 12, pp. 1087-1108. DOI: http://doi.org/10.1002/%28SICI%291097-0266%28199912%2920%3A120/o3C10870/o3A0/o3AAro-SMJ710/o3E3.0.CO0/o3B2-Z.
8 Williamson O.E. The new institutional economics: Taking stock, looking ahead. Journal of Economic Literature, 2000, vol. 38, no. 3, pp. 595-613. DOI: http://dx.doi.org/10.1257/jel.38.3.595.
9 Williamson O.E. Transaction cost economics: The origins. Journal of Retailing, 2010, vol. 86, no. 3, pp. 227-231. DOI: https://doi.org/10.1016/jjretai.2010.07.006.
10 Williamson O.E. The Economic Institutions of Capitalism Firms, Markets, Relational Contracting. Saint Petersburg: Lenizdat, SEVPress, 1996, 702 p. Available at: https://knigogid.ru/books/109125-ekonomicheskie-instituty-kapitalizma-firmy-rynki-otnoshencheskaya-kontraktaciya/toread. (In Russ.)
11 Benkler Y. Coase's penguin, or, Linux and the nature of the firm. Yale Law Journal, 2002, vol. 112, no. 3, pp. 369-446. DOI: http://dx.doi.org/10.2307/1562247.
12 Benkler Y. The wealth of networks. New Haven: Yale University Press, 2006.
13 Benkler Y. Peer production, the commons and the future of the firm. Strategic Organization, 2017, vol. 15, no. 2, pp. 264-274. DOI: http://doi.org/10.1177/1476127016652606.
14 Furubotn E.G., Richter R. Institutions and economic theory: The contribution of the new institutional economics. Ann Arbor: University of Michigan Press, 1997, 542 p. DOI: https://doi.org/10.3998/mpub.6715.
15 Furubotn E.G., Richter R. Institutions and economic theory: The contribution of the new institutional economics. Saint Petersburg: Izd. dom Sankt-Peterburgskogo gos. un-ta, 2005, 702 p. Available at: https://knigogid.ru/books/1178725-instituty-i-ekonomicheskaya-teoriya-dostizheniya-novoy-institucionalnoy-ekonomicheskoy-teorii/toread. (In Russ.)
16 Popov E.V., Konovalov A.A. A model of information retrieval costs optimization. Problemy Upravleniya = Control Sciences, 2008, no. 3, pp. 69-72. Available at: http://mi.mathnet.ru/pu160. (In Russ.)
17 Mantulenko A.V., Saraev A.L., Saraev L.A. On the theory of optimal allocation of production factors and transaction costs. Vestnik of Samara State University, 2013, no. 7 (108), pp. 117-126. Available at: https://elibrary.ru/item.asp?id=20886447; https://journals.ssau.ru/index.php/eco/article/view/5372. (In Russ.)
18 Saraev A.L., Saraev L.A. Optimization model of profit of organizations, considering superproportionally production and transaction costs. Vestnik of Samara State University, 2013, no. 10 (111), pp. 230-237. Available at: http://vestnikoldsamgu.ssau.ru/articles/111_35.pdf. (In Russ.)
19 Ilyina E.A. The model of formation of the optimal profit of the enterprise, taking into account the interaction of transformational and transactional costs. Journal of Economy and entrepreneurship, 2018, no. 12 (101), 2018, pp. 1191-1999. Available at: https://elibrary.ru/item.asp?id=36722316. (In Russ.)
20 Ilyina E.A. To the calculation of the optimal profit of the enterprise, bearing production and transaction costs. Journal of Economy and entrepreneurship, 2019, no. 8 (111), pp. 842-849. Available at: https://elibrary.ru/item.asp?id=41482468. (In Russ.)
21 Ilyina E.A. Influence of transaction costs of a production enterprise on the formation of its profit. Vestnik Samarskogo universiteta. Ekonomika i upravlenie =Vestnik of Samara University. Economics and Management, 2020, vol. 11, no. 1, pp. 144-152. Available at: https://elibrary.ru/item.asp?id=42834925. (In Russ.)
Библиографический список
1 Coase R. H. The nature of the firm. Economica // New Series. 1937. No. 16. Vol. 4. P. 386-405. URL: http://links.jstor.org/sici?sici=0013-0427%28193711%292%3A4%3A16%3C386%3ATNOTF%3E2.0.CO% 3B2-B.
2 Coase R.H. The problem of social cost // Journal of Law and Economics. 1960. No 3. Vol. 3. P. 1-44. URL: http://www.jstor.org/stable/724810?origin=JSTOR-pdf.
3 Williamson O. E. Transaction-cost economics: The governance of contractual relations // Journal of Law and Economics. 1979. No 22 (2). P. 233-261. DOI: http://dx.doi.org/10.1086/466942.
4 Williamson O.E. Comparative economic organization: The analysis of discrete structural alternatives // Administrative Science Quarterly. 1991. Vol. 36, no. 2. Р. 269-296. DOI: http://doi.org/10.2307/2393356.
5 Williamson O.E. Opportunism and its critics // Managerial and Decision Economics. 1993. Vol. 14, issue 2. Р. 97-107. DOI: http://doi.org/10.1002/mde.4090140203.
6 Williamson O.E. The Economic Institutions of Capitalism: Firms, Markets, Relational Contracting. Detroit: Free Press, 1998, 450 p. URL: http://www.bookre.org/reader?file=1066388.
7 Williamson O.E. Strategy research: Governance and competence perspectives // Strategic Management Journal. 1999. Vol. 20, no. 12. Р. 1087-1108. DOI: https://doi.org/10.1002/%28SICI%291097-0266%28199912%2920%3A12°/o3C10870/o3A0/o3AAro-SMJ710/o3E3.0.TO°/o3B2-Z.
8 Williamson O.E. The new institutional economics: Taking stock, looking ahead // Journal of Economic Literature. 2000. Vol. 38, no. 3. Р. 595-613.
9 Williamson O. E. Transaction cost economics: The origins // Journal of Retailing, 2010, vol. 86, no. 3, pp 227-231. DOI: htts://doi.org/10.1016/jjretai.2010.07.006.
10 Уильямсон О.И. Экономические институты капитализма. Фирмы, рынки, отношенческая контрактация. Санкт-Петербург: Лениздат, SEVPress, 1996. 702 с. URL: https://knigogid.ru/books/109125-ekonomicheskie-instituty-kapitalizma-firmy-rynki-otnoshencheskaya-kontraktaciya/toread.
11 Benkler, Y. Coase's penguin, or, Linux and the nature of the firm // Yale Law Journal. 2002, vol. 112, no. 3, pp. 369-446. DOI: http://dx.doi.org/10.2307/1562247.
12 Benkler Y. The wealth of networks. New Haven: Yale University Press, 2006.
13 Benkler, Y. Peer production, the commons and the future of the firm // Strategic Organization, 2017, vol. 15, no. 2, pp. 264-274. DOI: https://doi.org/10.1177/1476127016652606.
14 Furubotn E.G., Richter R. Institutions and economic theory: The contribution of the new institutional economics. Ann Arbor: University of Michigan Press, 1997. 542 p. DOI: https://doi.org/10.3998/mpub.6715.
15 Фуруботн Э.Г., Рихтер Р. Институты и экономическая теория. Достижения новой институциональной экономической теории. Санкт-Петербург: Изд. дом Санкт-Петербургского гос. ун-та, 2005, 702 с. URL: https://knigogid.ru/books/1178725-instituty-i-ekonomicheskaya-teoriya-dostizheniya-novoy-institucionalnoy-ekonomicheskoy-teorii/toread.
16 Попов Е.В., Коновалов А.А. Модель оптимизации издержек поиска информации // Проблемы управления. 2008. № 3. С. 69-72. URL: http://mi.mathnet.ru/pu160.
17 Мантуленко А.В., Сараев А.Л., Сараев Л.А. К теории оптимального распределения факторов производства, производственных и трансакционных издержек // Вестник Самарского государственного университета. 2013. № 7 (108). С. 117-126. URL: https://elibrary.ru/item.asp?id=20886447; https://journals.ssau.ru/
index.php/eco/article/view/5372.
18 Сараев А.Л., Сараев Л.А. Модель оптимизации прибыли предприятия, учитывающая сверхпропорциональные производственные и трансакционные затраты // Вестник Самарского государственного университета. 2013. № 10 (111). С. 230-237. URL: http://vestnikoldsamgu.ssau.ru/ articles/111_35.pdf.
19 Ильина, Е.А. Модель формирования оптимальной прибыли предприятия, учитывающая взаимодействие трансформационных и трансакционных издержек // Экономика и предпринимательство. 2018. № 12 (101). С. 1191-1199. URL: https://elibrary.ru/item.asp?id=36722316.
20 Ильина, Е.А. К расчету оптимальной прибыли предприятия, несущего производственные и трансакционные издержки // Экономика и предпринимательство. 2019. № 8 (109). С. 842-849.
21 Ильина, Е.А. Влияние трансакционных издержек производственного предприятия на формирование его прибыли // Вестник Самарского университета. Экономика и управление. 2020. Т. 11, № 1. С. 144-152. URL: https://elibrary.ru/item.asp?id=42834925
