Empirical investigation of the theory of production function, with the data of alloy production in Ukraine
In this research, a mathematical form of production function is investigated, which is a concept of microeconomics theory, with the actual data from the factory in Dnepropetrovsk Region of Ukraine, which produces the alloys from several input materials. A linear form of the production function was s...
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Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
2014
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| Cite this: | Empirical investigation of the theory of production function, with the data of alloy production in Ukraine / Y. Matsuki, P. Bidyuk, V. Kozyrev // Системні дослідження та інформаційні технології. — 2014. — № 2. — С. 29-39. — Бібліогр.: 1 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860249782336880640 |
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| author | Matsuki, Y. Bidyuk, P. Kozyrev, V. |
| author_facet | Matsuki, Y. Bidyuk, P. Kozyrev, V. |
| citation_txt | Empirical investigation of the theory of production function, with the data of alloy production in Ukraine / Y. Matsuki, P. Bidyuk, V. Kozyrev // Системні дослідження та інформаційні технології. — 2014. — № 2. — С. 29-39. — Бібліогр.: 1 назв. — англ. |
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| description | In this research, a mathematical form of production function is investigated, which is a concept of microeconomics theory, with the actual data from the factory in Dnepropetrovsk Region of Ukraine, which produces the alloys from several input materials. A linear form of the production function was selected as the model, which consists of the variables that represent input materials together with their weighting factors, then the Lagrangean multiplier technique was used to transform this model in order to find the conditions for maximizing the output of the production, under a given cost constraint. The obtained conditions present the mathematical relations between the prices and the quantities of the input materials, which include unknown weighting factors. In order to get the values of the weighting factors, statistical analysis is made with the actual data. The result shows statistical significance of the model, therefore it is concluded that the selected linear function can be the production function.
Проаналізовано математичну форму виробничої функції, яка є концепцією мікроекономічної теорії, з використанням фактичних даних, отриманих від підприємства в Дніпропетровській області України, яке виробляє сплави з декількох вхідних матеріалів. Лінійну форму виробничої функції було обрано в якості моделі, яка включає змінні, що представляють потоки вхідних матеріалів разом із своїми ваговими коефіцієнтами. Для розв’язання оптимізаційної задачі потім було застосовано метод множників Лагранжа для трансформації цієї моделі з метою визначення умови для максимізації об’єму вихідної продукції при обмеженні на видатки. Отримані умови представляють математичні співвідношення між ціною та об’ємом вхідних матеріалів, у тому числі невідомих вагових коефіцієнтів. Для того, щоб отримати значення вагових коефіцієнтів, виконано статистичний аналіз наявних фактичних даних. Отриманий результат свідчить про статистичну значущість моделі, а тому можна зробити висновок, що обрана лінійна функція може бути виробничою функцією.
Проанализирована математическая форма производственной функции, которая является концепцией микроэкономической теории, с использованием фактических данных от предприятия в Днепропетровской области Украины, которые производит сплавы из нескольких входящих материалов. Линейная форма производственной функции была выбрана в качестве модели, состоящей из переменных, представляющих входящие материалы вместе со своими весовыми коэффициентами. Для решения оптимизационной задачи затем был применен метод множителей Лагранжа для трансформации этой модели с целью определения условий для максимизации объема выходной продукции при ограничениях на ресурсы. Полученные условия представляют математические соотношения между ценой и количеством входящих материалов, в том числе неизвестных весовых факторов. Для того, чтобы получить значения весовых коэффициентов, выполнен статистический анализ имеющихся фактических данных. Полученный результат свидетельствует о статистической значимости модели, поэтому можно сделать вывод, что выбранная линейная функция может быть производственной функцией.
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| first_indexed | 2025-12-07T18:41:44Z |
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Y. Matsuki, P. Bidyuk, V. Kozyrev, 2014
Системні дослідження та інформаційні технології, 2014, № 2 29
UDC 519.004.942
EMPIRICAL INVESTIGATION OF THE THEORY
OF PRODUCTION FUNCTION, WITH THE DATA OF ALLOY
PRODUCTION IN UKRAINE
Y. MATSUKI, P. BIDYUK, V. KOZYREV
In this research, a mathematical form of production function is investigated, which is
a concept of microeconomics theory, with the actual data from the factory in Dne-
propetrovsk Region of Ukraine, which produces the alloys from several input mate-
rials. A linear form of the production function was selected as the model, which con-
sists of the variables that represent input materials together with their weighting
factors, then the Lagrangean multiplier technique was used to transform this model
in order to find the conditions for maximizing the output of the production, under
a given cost constraint. The obtained conditions present the mathematical relations
between the prices and the quantities of the input materials, which include unknown
weighting factors. In order to get the values of the weighting factors, statistical
analysis is made with the actual data. The result shows statistical significance of the
model, therefore it is concluded that the selected linear function can be the produc-
tion function.
INTRODUCTION
Production function of the microeconomics theory [1] gives the information for
decision-making in producing industrial materials. In the theory, the production
function defines the optimal combination of input materials with their weighting
factors. In order to specify the weighting factors, the Lagurangean multiplier
technique [1] is used under the conditions for maximizing the production, which
is given by cost constraint that is made of the prices of the materials together with
their quantities.
The mathematical forms of production function are given in the literatures of
microeconomics, and Cobb-Douglas function [1] is known as an example in non-
linear form. The procedure, the Lagrangean multiplier technique, of finding the
conditions for maximizing the production under cost constraintis obtained from
those literatures. In this research, a linear form of production function is selected,
and the appropriateness of this form is tested with the data taken from the produc-
tion system of alloy at a factory in Dnepropetrovsk of Ukraine.
The data that are used in this analysis include quantities and the prices of the
input materials, i.e., lime, bentonite, ore, gas, electricity as well as the quantity of
the final product, iron ore and pellets.
The descriptive statistics of those input materials are shown in Table 1 and 2.
Correlations between the variables selected are given in Table 3. Figures 1–3
show time histories for the quantity of final products, prices of gas, electricity,
ore, bentonite, and lime for 36 months. Figures 4–5 illustrate quantities dynamics
for gas, bentonite, lime, electricity, and iron, ore also for 36 months.
Y. Matsuki, P. Bidyuk, V. Kozyrev
ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 30
T a b l e 1 . Descriptive statistics of the prices of gas, electricity, ore, lime and
bentonite
Statistics
Gas price
(UAH/m3)
Electricity
price
(UAH/kWh)
Ore price
(UAH/ton)
Lime price
(UAH/ton)
Bentonite
price
(UAH/ton)
Mean 1.9167 0.3667 9.0667 700.00 566.67
Median 1.9100 0.4000 7.9000 700.00 550.00
Max. 2.1600 0.4300 11.600 700.00 600.00
Min. 1.6800 0.2700 7.7000 700.00 550.00
Std.Dev 0.1988 0.0704 1.8186 0.0000 23.905
Skewness 0.0510 –0.6094 0.7005 NA 0.7071
Kurtosis 1.5000 1.5000 1.5000 NA 1.5000
Obs. 36 36 36 36 36
Note Max. — maximum value; Min. — minimum value; Std. Dev. — stan-
dard deviation; Obs. — number of observations; NA — not available, because the
lime price doesn’t change over 36 months in the obtained database. UAH —
Ukrainian currency (hryvnya); kWh — kilo watt-hour.
T a b l e 2 . Descriptive statistics of quantities of gas, electricity, ore, lime, ben-
tonite and the final product
Statistics
Gas
quantity
Electricity
quantity
Ore
quantity
Lime
quantity
Bentonite
quantity
Final product
quantity
Mean 719570.6 18107646 3028497 44140.3 50020.2 1007887
Median 720530.0 18344268 3066452 43507.5 49443.5 998094.0
Max. 826160.0 20469762 3441243 50690.0 56923.0 1146490
Min. 620210.0 15400996 2568431 38676.0 42735.0 862363.0
Std.Dev 64555.82 1382721 253399.9 3807.67 4284.58 84840.57
Skewness –0.056677 –0.281579 –0.0740 0.2199 –0.0296 0.1906
Kurtosis 1.834927 2.2615 1.9633 1.8112 1.6549 2.1655
Obs. 36 36 36 36 36 36
F
in
al
p
ro
du
ct
v
ol
um
e
Date
Fig. 1. Quantity of final products for 36 months from January 2008 (tons)
Empirical investigation of the theory of production function, with the data of alloy production …
Системні дослідження та інформаційні технології, 2014, № 2 31
T a b l e 3 . Correlations of quantities and/or prices of the final product and input
materials
Variables Final
product
Gas
price
Electricity
price
Ore
price
Bentonite
price
Gas
quantity
Electricity
quantity
Ore
quantity
Lime
quantity
Bentonite
quantity
Final
product 1
Gas
price
–0.2193 1
Electricity
price
–0.1896 0.9322 1
Ore
price 0.1589 –0.8292 –0.9752 –1
Bentonite
price
–0.2121 0.8778 0.6449 –0.4600 1
Gas
quantity
–0.2175 0.2983 0.2213 –0.1596 0.3370 1
Electricity
quantity 0.0719 0.1548 0.1773 –0.1794 0.0920 0.2842 1
Ore
quantity 0.1454 –0.2165 –0.1238 0.0590 –0.2933 –0.1523 –0.0819 1
Lime
quantity
–0.0558 –0.0401 0.0268 –0.0659 –0.1202 –0.0736 0.1100 0.2609 1
Bentonite
quantity 0.0325 0.0825 0.0868 –0.0837 0.0593 0.0632 0.2326 –0.1484 –0.3286 1
Fig. 2. Prices of gas, electricity, and ore for
36 months
Fig. 3. Prices of bentonite and lime for
36 months
Fig. 4. Quantities of gas, bentonite and lime
for 36 months
Fig. 5. Quantities of electricity and ore for
36 months
Y. Matsuki, P. Bidyuk, V. Kozyrev
ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 32
Note: Lime price is omitted from this table because the lime price doesn’t
change over the given 36 months as shown in Fig. 3, therefore it doesn’t have any
correlation with other variables.
METHODOLOGY
Production function is a theory to indicate the levels of production of industrial
materials with various input materials, ,iX where ,,....,2,1 ni such as raw mate-
rials,electricity and gas. The producers and/or sellers wish higher level of produc-
tion, ),,...,,,( 321 nXXXXQ but the constraints are given by the total cost or
budget, ,oC together with the prices
ixP for different kinds of input materials iX
respectively, where
.
1
i
n
i
x
o XPC
i
(1)
Under this constraint, the condition for obtaining the maximum production is
to be found, using the Lagrangean multiplier technique, as shown below: at first,
the Lagrangean is defined as the follows:
),(),....,,(
1
321
n
i
ix
o
n XPCXXXXQZ
i
(2)
here, is an unknown variable, which is called the “Lagrangean multiplier”.
The first order condition to get the maximum production,
),,...,,,( 321 nXXXXQ is that the partial derivatives of Z by each of
nXXXX ,...,,, 321 and are equal to zero, i.e.,
,0
ix
ii
PX
Q
X
Z (3)
.0
1
n
i
ix
o XPCZ
i (4)
For example, by dividing i-th equation by )1( i -th equation of the above
(1)–(4), we get the following:
,
j
i
X
X
j
i
P
P
X
Q
X
Q
(5)
where, .ji
The above equation (5) means that the ratio of marginal production of inputs
(the ratio of these two partial derivatives of production function by iX and jX )
should be equal to the ratio of the prices of these iX and jX in order to get the
maximum production [1]. In other words, although producers and/or sellers wish
to achieve the higher/larger production, the maximum production is always con-
Empirical investigation of the theory of production function, with the data of alloy production …
Системні дослідження та інформаційні технології, 2014, № 2 33
strained by the total cost or total budget and the prices, and the maximum produc-
tion is obtained only where and/or when the ratio of marginal productions,
j
i
X
Q
X
Q
, and the ratio of the corresponding two prices,
j
i
X
X
P
P
, are equal. This
point is the equilibrium to achieve the maximum production, which is given under
the total cost constraint (equation (1)). In other words, the production is at the
maximum, and there is enough amount of budget when equation (5) is satisfied.
The mathematical model of the production function needs to be found. In
this research, a linear model (equation (6)) is assumed, and then empirical analy-
sis is made for testing the fitting of the model to the actual data:
,
1
i
n
i
i XaQ
(6)
where
,1
1
n
i
ia (7)
here ia is a weighting factor to combine various input materials, iX , to make up
a production function .Q
In order to make the statistical test, the variables included in the equation (6)
are not enough because the actual value of .Q is unknown, therefore this model
needs to be transformed to the other linear equations, with the Lagrangean multi-
plier technique as shown below, with which each quantity of input material, iX ,
can be mathematically indicated as the function of the total cost, ,oC and the
prices of various input materials,
ns xxxx PPPP ,...,,,
31
, together with rest of the
other input materials, jX , where ,ji which are available in the actual database.
Then, the linear regression analysis can be carried out for the statistical test.
For the linear model, ,
1
i
n
i
i XaQ
the Lagrangean is:
).(.
11
i
n
i
X
o
i
n
i
i XPCXaZ
i
(8)
Given the cost constraint, the first order condition for maximizing the pro-
duction, ,
1
i
n
i
i Xa
is that the partial derivatives of Z by each of
nXXXX ,...,,, 321 and are equal to zero, i.e.,
,0
iXi
i
PaX
Z (9)
,0
1
i
n
i
x
o XPCZ
i (10)
where, .,...,2,1 ni
Y. Matsuki, P. Bidyuk, V. Kozyrev
ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 34
From (9)
.
i
X
a
P
i
(11)
From (10)
.
1
i
n
i
X
o XPC
i
(12)
Then, replace
jXP of (12) by (11) to get:
,
1
1
n
i
j
j
iX
o X
a
XPC
i
(13)
where .ji
From (11)
.
1
i
X
a
P
i
(14)
Then, replace
1
of (13) by (14) to get:
.
1
1
j
n
j i
j
X
o
i X
a
a
P
C
X
i
(15)
The next step is to test if this model statistically fits in the actual data, upon
the mathematical model shown in the equation (15).
RESULTS
For the statistical test, one more variable, the total cost, ,oC was calculated upon
the equation (1), in addition to the variables shown in Table 1 and 2. Then, in or-
der to get the coefficients of the production function, shown in the equation (6),
the equation (15) was made up with combinations of the input materials. In Table 3,
various combinations of the variables for input materials are shown. Then, the
statistical test was made with the data. Also in Table 3, the value of R2 is shown
on each combination of the input materials, which indicates how each model fits
in the data.
As the result, the model of the production with lime and bentonite shows the
best values of .2R As shown in the model № 17 of Table 4, 2R of the model for
the equation (15) with the quantity of lime as the dependent variable is 0,8238,
and 2R of the model with the quantity of bentonite as the dependent variable is
0,7874, both of which satisfactory show the statistical fitting of the data on the
mathematical model. More details of the statistical check of the model № 17 of
Table 4 is shown in Table 5.
Empirical investigation of the theory of production function, with the data of alloy production …
Системні дослідження та інформаційні технології, 2014, № 2 35
T a b l e 4 . R2 of the linear functions
№ Model of
equation (6) Model of equation (15) R2
Xlime=α1+α2*Co/Plime+α3*Xbentonite+α4*Xelectricity+α5*Xore+
+α6*Xgas
0.3645
Xbentonite=α1+α2*Co/Pbentonite+α3*Xlime+α4*Xelectricity+
+α5*Xore+α6*Xgas
0.2611
Xelectricity=α1+α2* Co/Pelectricity+α3*Xbentonite+α4*Xlime+
+α5*Xore+α6*Xgas
0.1801
Xore=α1+α2* Co/Pore+α3*Xbentonite+α4*Xelectricity+
+α5*Xlime+α6*Xgas
0.1015
1
gas*5
ore*4
yelectricit*3
bentonite*2
lime*1
Xa
Xa
Xa
Xa
XaQ
Xgas=α1+α2* Co/Pgas+α3*Xbentonite+α4*Xelectricity+
+α5*Xlime+α6*Xore
0.1364
Xlime=α1+α2* Co/Plime+α3*Xbentonite+α4*Xelectricity +α5*Xore 0.3559
Xbentonite=α1+α2* Co/Pbentonite+α3*Xlime+α4*Xelectricity +α5*Xore 0.2582
Xelectricity=α1+α2* Co/Pelectricity+α3*Xlime+α4*Xbentonite +α5*Xore 0.1150
2
ore*4
yelectricit*3
bentonite*2
lime*1
Xa
Xa
Xa
XaQ
Xore=α1+α2* Co/Pore+α3*Xbentonite+α4*Xelectricity +α5*Xlime 0.0880
Xlime=α1+α2* Co/Plime+α3*Xbentonite+α4*Xgas +α5*Xore 0.2525
Xbentonite=α1+α2* Co/Pbentonite+α3*Xlime+α4*Xgas +α5*Xore 0.1638
Xgas=α1+α2* Co/Pgas+α3*Xbentonite+α4*Xlime +α5*Xore 0.0691
3
ore*4
gas*3
bentonite*2
lime*1
Xa
Xa
Xa
XaQ
Xore=α1+α2* Co/Pore+α3*Xbentonite+α4*Xgas +α5*Xlime 0.1034
Xlime=α1+α2* Co/Plime+α3*Xbentonite+α4*Xelectricity +α5*Xgas 0.6109
Xbentonite=α1+α2* Co/Pbentonite+α3*Xlime+α4*Xelectricity +α5*Xgas 0.8124
Xelectricity=α1+α2* Co/Pelectricity+α3*Xlime+α4*Xbentonite +α5*Xgas 0.1753
4
gas*4
yelectricit*3
bentonite*2
lime*1
Xa
Xa
Xa
XaQ
Xgas=α1+α2*Co/Pgas+α3*Xbentonite+α4*Xelectricity +α5*Xlime 0.1413
Xelectricity=α1+α2*Co/Pelectricity+α3*Xbentonite+α4*Xgas +α5*Xore 0.1343
Xbentonite=α1+α2*Co/Pbentonite+α3*Xelectricity+α4*Xgas +α5*Xore 0.2153
Xgas=α1+α2*Co/Pgas+α3*Xbentonite+α4*Xelectricity +α5*Xore 0.1211
5
ore*4
gas*3
bentonite*2
yelecteicit*1
Xa
Xa
Xa
XaQ
Xore=α1+α2*Co/Pore+α3*Xbentonite+α4*Xgas +α5*Xelectricity 0.0855
Xelectricity=α1+α2*Co/Pelectricity+α3*Xlime+α4*Xgas +α5*Xore 0.1116
Xlime=α1+α2*Co/Plime+α3*Xelectricity+α4*Xgas +α5*Xore 0.2986
Xgas=α1+α2*Co/Pgas+α3*Xlime+α4*Xelectricity +α5*Xore 0.1323
6
ore*4
gas*3
lime*2
yelecteicit*1
Xa
Xa
Xa
XaQ
Xore=α1+α2*Co/Pore+α3*Xlime+α4*Xgas +α5*Xelectricity 0.1302
Xlime=α1+α2*Co/Plime+α3*Xbentonite+α4*Xelectricity 0.8239
Xbentonite=α1+α2*Co/Pbentonite+α3*Xlime+α4*Xelectricity 0.6287 7
yelectricit*3
bentonite*2
lime*1
Xa
Xa
XaQ
Xelectricity=α1+α2*Co/Pelectricity+α3*Xlime+α4*Xbentonite 0.0965
Xlime=α1+α2*Co/Plime+α3*Xbentonite+α4*Xore 0.2488
Xbentonite=α1+α2*Co/Pbentonite+α3*Xlime+α4*Xore 0.1561 8
ore*3
bentonite*2
lime*1
Xa
Xa
XaQ
Xore=α1+α2*Co/Pore+α3*Xlime+α4*Xbentonite 0.0826
Xbentonite =α1+α2*Co/Pbentonite+α3*Xgas+α4*Xelectricity 0.8159
Xelectricity=α1+α2*Co/Pelectricity+α3*Xbentonite+α4* Xgas 0.1285 9
gas*3
yelectricit*2
bentonite*1
Xa
Xa
XaQ
Xgas=α1+α2*Co/Pgas+α3*Xelectricity+α4*Xbentonite 0.1050
Xore =α1+α2*Co/Pore+α3*Xgas+α4*Xelectricity 0.4047
Xelectricity=α1+α2*Co/Pelectricity+α3*Xore+α4* Xgas 0.0990 10
gas*3
yelectricit*2
ore*1
Xa
Xa
XaQ
Xgas=α1+α2*Co/Pgas+α3*Xelectricity+α4*Xore 0.1183
Xlime =α1+α2*Co/Plime+α3*Xgas+α4*Xelectricity 0.8002
Xelectricity=α1+α2*Co/Pelectricity+α3*Xlime+α4* Xgas 0.1010 11
gas*3
yelectricit*2
lime*1
Xa
Xa
XaQ
Xgas=α1+α2*Co/Pgas+α3*Xelectricity+α4*Xlime 0.1418
Y. Matsuki, P. Bidyuk, V. Kozyrev
ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 36
Continue of table 4
Xore =α1+α2*Co/Pore+α3*Xgas+α4*Xbentonite 0.1309
Xbentonite=α1+α2*Co/Pbentonite+α3*Xore+α4* Xgas 0.1160 12
gas*3
bentonite*2
ore*1
Xa
Xa
XaQ
Xgas=α1+α2*Co/Pgas+α3*Xbentonite+α4*Xore 0.0599
Xore =α1+α2*Co/Pore+α3*Xelectricity+α4*Xbentonite 0.0637
Xbentonite=α1+α2*Co/Pbentonite+α3*Xore+α4* Xelectricity 0.2088 13
yelectricit*3
bentonite*2
ore*1
Xa
Xa
XaQ
Xelectricity=α1+α2*Co/Pelectricity+α3*Xbentonite+α4*Xore 0.0727
Xore =α1+α2*Co/Pore+α3*Xgas+α4*Xlime 0.1665
Xlime=α1+α2*Co/Plime+α3*Xore+α4* Xgas 0.1939 14
gas*3
lime*2
ore*1
Xa
Xa
XaQ
Xgas=α1+α2*Co/Pgas+α3*Xlime+α4*Xore 0.0680
Xore =α1+α2*Co/Pore+α3*Xelectricity+α4*Xlime 0.1192
Xlime=α1+α2*Co/Plime+α3*Xore+α4* Xelectricity 0.2903 15
lime*3
yelectricit*2
ore*1
Xa
Xa
XaQ
Xelectricity=α1+α2*Co/Pelectricity+α3*Xlime+α4*Xore 0.0443
Xore =α1+α2*Co/Pore+α3*Xbentonite+α4*Xlime 0.0826
Xlime=α1+α2*Co/Plime+α3*Xore+α4* Xbentonite 0.2488 16
lime*3
bentonite*2
ore*1
Xa
Xa
XaQ
Xbentonite=α1+α2*Co/Pbentonite+α3*Xlime+α4*Xore 0.1561
Xlime=α1+α2*Co/Plime+α3*Xbentonite 0.8238
17
bentonite*2
lime*1
Xa
XaQ
Xbentonite =α1+α2*Co/Pbentonite+α3*Xlime 0.7874
Xore=α1+α2*Co/Pore+α3*Xbentonite 0.1071
18
bentonite*2
ore*1
Xa
XaQ
Xbetonie=α1+α2*Co/Pbentonite+α3*Xore 0.1045
Xore=α1+α2*Co/Pore+α3*Xlime 0.1517
19
lime*2
ore*1
Xa
XaQ
Xlime=α1+α2*Co/Plime+α3*Xore 0.1889
Xore=α1+α2*Co/Pore+α3*Xelectricity 0.4564
20
bentonite*2
ore*1
Xa
XaQ
Xelectricity=α1+α2*Co/Pelectricity+α3*Xore 0.0240
Xore=α1+α2*Co/Pore+α3*Xgas 0.9760
21
gas*2
ore*1
Xa
XaQ
Xgas=α1+α2*Co/Pgas+α3*Xore 0.0564
Xlime=α1+α2*Co/Plime+α3*Xelectricity 0.8213
22
yelectricit*2
lime*1
Xa
XaQ
Xelectricity =α1+α2*Co/Pelectricity+α3*Xlime 0.0239
Xlime=α1+α2*Co/Plime+α3*Xgas 0.9973
23
gas*2
lime*1
Xa
XaQ
Xgas =α1+α2*Co/Pgas+α3*Xlime 0.0671
Xbentonite=α1+α2*Co/Pbentonite+α3*Xelectricity 0.8347
24
yelectricit*2
bentonite*1
Xa
XaQ
Xelectricity =α1+α2*Co/Pelectricity+α3*Xbentonite 0.0597
Xbentonite=α1+α2*Co/Pbentonite+α3*Xgas 0.9985
25
gas*2
bentonite*1
Xa
XaQ
Xgas =α1+α2*Co/Pgas+α3*Xbentonite 0.0340
Xelectricity=α1+α2*Co/Pelectricity+α3*Xgas 0.8974
26
gas*2
yelectricit*1
Xa
XaQ
Xgas =α1+α2*Co/Pgas+α3*Xelectricity 0.1321
In Table 5, the T-statistics of each independent variable, the Akaike Informa-
tion Criterion (AIC) and Shwartz Criterion don’t show sufficient statistical fitting.
According to the mathematical model of the equation (16), the coefficient,
,
iX
o PC should be 1.0, but in Table 5, the coefficients of
ie
o PC lim and
bentonitPCo are 0,8368 and 0,7794. In this analysis, approximation is taken for
the further steps of the analysis, and they are both assumed to be 1.0.
Empirical investigation of the theory of production function, with the data of alloy production …
Системні дослідження та інформаційні технології, 2014, № 2 37
T a b l e 5 . Statistical test on the linear model of production function with lime
and bentonite
Model
Depen-
dent
Variable
Independent
Variable
Coeffi-
cient
α1, α2 , …
T-
Statistics
R2 AIC Schwartz
Interception 10600 2.0064
Total cost
(C0)/Lime price
(Plime)
0.8368 11.581 Xlime=α1+α2*Co/
Plime+α3*Xbentonite
Quantity
of Lime
(Xlime)
Quantity of Ben-
tonite (Xbentonite)
–0.7455 –9.8316
0.8238 17.730 17.861
Interception 17038 2.7269
Total cost
(C0)/Bentonite
price (Pbentonite)
0.7794 10.271
Xbentonite=
=α1++α2*Co/
Pbentonite+α3*Xlime
Quantity
of Ben-
tonite
(Xbentonite) Quantity of Lime
(Xlime)
–1.100869 –9.573509
0.7874 18.153 18.285
The next step is to estimate the weighting factors, which are indicated as the
coefficients ia , where ni ,...,2,1 of the equation (6).
When
,ij
i
j
a
a
(16)
where, ij is the observed value of the coefficient that is obtained by the linear
regression analysis, as shown in Table 5.
From (15) and (16)
,
1
1
j
n
j
ij
X
o
i X
P
C
X
i
(17)
where
.
1
1
1
n
j
ij
i
n
j
j
a
a
(18)
From (7)
.1
1
11
n
j
ji
n
i
i aaa (19)
Then, from (18) and (19)
,
1 1
1
n
j
ij
i
i
a
a
(20)
,1
1
1
n
j
ijii aa (21)
Y. Matsuki, P. Bidyuk, V. Kozyrev
ISSN 1681–6048 System Research & Information Technologies, 2014, № 2 38
.11
1
1
n
j
ijia (22)
Therefore
.
1
1
1
1
n
j
ij
ia
(23)
From the equation (17) and the values of the coefficients of lime and ben-
tonite in Table 4, the following 2 equations are obtained:
,74546,0 bentonit
lim
lime X
P
C
X
e
o
(24)
.10087,1 lime
bentonit
bentonit X
P
C
X
o
(25)
With the equation (23) and the values of the coefficients in the equations
(24) and (25), the following production function is obtained:
.4760,05729,0 bentonitlime XXQ (26)
The correlation between the quantity of the final product and the calculated
values upon the equation (26) is shown in Table 6. With data of 36 months from
January 2008 to December 2010, the statistical values don’t show any fitting of
the calculated value in the actual data. However, with the data of 12 months from
January to December 2008, the statistical indicators show the improvement. The
actual value of the final product quantity is 26,88 times larger than the calculated
value, but the behavior in time series over 12 months show proportional rise and
fall of the product, and therefore it shows a predictability of the final product
upon quantity of bentonite and lime, as shown in Fig. 6. In this period, the first 12
months, the most of the prices of the input materials are stable as shown in Fig. 2
and Fig. 3, and it shows that the stable prices improved the predictability by the
obtained production function in the equation (26).
T a b l e 6 . Correlation between the final product quantity and the calculated
value
№
Dependent
Variable
Independent
Variable
Coefficient
T-
Statistics
R2 AIC Schwartz
Durbin-
Watson
Interception 1053544 3.8250
1
Final product
quantity Calculated Q –0.7414 –0.1323
0.0005 25.500 25.588 2.0003
Interception –272693.4 –0.4470
2
Final product
quantity Calculated Q 26.8764 2.1457
0.3153 24.989 25.070 1.4704
In Table 5 data is from January 2008 to December 2010. In Table 6 data is
from January 2008 to December 2008.
Empirical investigation of the theory of production function, with the data of alloy production …
Системні дослідження та інформаційні технології, 2014, № 2 39
CONCLUSIONS AND RECOMMENDATIONS
Upon the analysis of the given data of the alloy production in Dnepropetrovsk, it
is concluded that the productivity of the manufacturing process can be predicted
by the linear form of the production function, as long as the prices of the input
materials are stable.
Fewer numbers of input variables can predict the quantity of the final prod-
ucts. In this analysis, only the quantities of bentonite and lime are the input vari-
ables of the production function, given that the prices are stable; and, the other
input materials and utilities, ore, electricity and gas were not used.
On this analysis, the obtained quantity of the final product by the obtained
utility function needs to be multiplied by the factor of about 27, because of the
fewer input variables included in the production function.
Further research and analysis are needed for different production systems
and products, to compare the results with this analysis.
REFERENCE
1. Browning E.K., Browning J.M. Microeconomic Theory and Application, Third Edi-
tion. — Glenview: Scott, Foresman and Company, 1989. — 637 p.
Received 08.09.2013
F
in
al
p
ro
d
u
ct
vo
lu
m
e
C
al
cu
la
te
Рис. 6. Comparison of the quantity of final product and the calculated value in 2008
|
| id | nasplib_isofts_kiev_ua-123456789-85496 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1681–6048 |
| language | English |
| last_indexed | 2025-12-07T18:41:44Z |
| publishDate | 2014 |
| publisher | Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
| record_format | dspace |
| spelling | Matsuki, Y. Bidyuk, P. Kozyrev, V. 2015-08-06T19:28:29Z 2015-08-06T19:28:29Z 2014 Empirical investigation of the theory of production function, with the data of alloy production in Ukraine / Y. Matsuki, P. Bidyuk, V. Kozyrev // Системні дослідження та інформаційні технології. — 2014. — № 2. — С. 29-39. — Бібліогр.: 1 назв. — англ. 1681–6048 https://nasplib.isofts.kiev.ua/handle/123456789/85496 519.004.942 In this research, a mathematical form of production function is investigated, which is a concept of microeconomics theory, with the actual data from the factory in Dnepropetrovsk Region of Ukraine, which produces the alloys from several input materials. A linear form of the production function was selected as the model, which consists of the variables that represent input materials together with their weighting factors, then the Lagrangean multiplier technique was used to transform this model in order to find the conditions for maximizing the output of the production, under a given cost constraint. The obtained conditions present the mathematical relations between the prices and the quantities of the input materials, which include unknown weighting factors. In order to get the values of the weighting factors, statistical analysis is made with the actual data. The result shows statistical significance of the model, therefore it is concluded that the selected linear function can be the production function. Проаналізовано математичну форму виробничої функції, яка є концепцією мікроекономічної теорії, з використанням фактичних даних, отриманих від підприємства в Дніпропетровській області України, яке виробляє сплави з декількох вхідних матеріалів. Лінійну форму виробничої функції було обрано в якості моделі, яка включає змінні, що представляють потоки вхідних матеріалів разом із своїми ваговими коефіцієнтами. Для розв’язання оптимізаційної задачі потім було застосовано метод множників Лагранжа для трансформації цієї моделі з метою визначення умови для максимізації об’єму вихідної продукції при обмеженні на видатки. Отримані умови представляють математичні співвідношення між ціною та об’ємом вхідних матеріалів, у тому числі невідомих вагових коефіцієнтів. Для того, щоб отримати значення вагових коефіцієнтів, виконано статистичний аналіз наявних фактичних даних. Отриманий результат свідчить про статистичну значущість моделі, а тому можна зробити висновок, що обрана лінійна функція може бути виробничою функцією. Проанализирована математическая форма производственной функции, которая является концепцией микроэкономической теории, с использованием фактических данных от предприятия в Днепропетровской области Украины, которые производит сплавы из нескольких входящих материалов. Линейная форма производственной функции была выбрана в качестве модели, состоящей из переменных, представляющих входящие материалы вместе со своими весовыми коэффициентами. Для решения оптимизационной задачи затем был применен метод множителей Лагранжа для трансформации этой модели с целью определения условий для максимизации объема выходной продукции при ограничениях на ресурсы. Полученные условия представляют математические соотношения между ценой и количеством входящих материалов, в том числе неизвестных весовых факторов. Для того, чтобы получить значения весовых коэффициентов, выполнен статистический анализ имеющихся фактических данных. Полученный результат свидетельствует о статистической значимости модели, поэтому можно сделать вывод, что выбранная линейная функция может быть производственной функцией. en Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України Системні дослідження та інформаційні технології Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи Empirical investigation of the theory of production function, with the data of alloy production in Ukraine Емпіричні дослідження теорії виробничої функції за даними стосовно виробництва сплавів в Україні Эмпирические исследования теории производственной функции с данным производства сплавов в Украине Article published earlier |
| spellingShingle | Empirical investigation of the theory of production function, with the data of alloy production in Ukraine Matsuki, Y. Bidyuk, P. Kozyrev, V. Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи |
| title | Empirical investigation of the theory of production function, with the data of alloy production in Ukraine |
| title_alt | Емпіричні дослідження теорії виробничої функції за даними стосовно виробництва сплавів в Україні Эмпирические исследования теории производственной функции с данным производства сплавов в Украине |
| title_full | Empirical investigation of the theory of production function, with the data of alloy production in Ukraine |
| title_fullStr | Empirical investigation of the theory of production function, with the data of alloy production in Ukraine |
| title_full_unstemmed | Empirical investigation of the theory of production function, with the data of alloy production in Ukraine |
| title_short | Empirical investigation of the theory of production function, with the data of alloy production in Ukraine |
| title_sort | empirical investigation of the theory of production function, with the data of alloy production in ukraine |
| topic | Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи |
| topic_facet | Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/85496 |
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