Sergey A. Karganov
About the fallacy of using V. Leontiev’s economic – mathematical model
and ‘Input-Output’ intersectoral balance in economic planning.
Abstract
Since the publication of the first V. Leontiev’s intersectoral ‘InputOutput’ balance (1936), the world’s economic community has been
constantly pinning its hopes on the use of such tables and the economic
and mathematical models of their building for planning the development
of the economies of their countries.
This paper presents the examples of nine structural flaws of V. Leontiev’s
economic and mathematical model and his model of inter-sectored
balance ‘Input-Output’. The presence of these defects explains the futility
and unfounded use of V. Leontiev’s models not only for the purposes of
long-term intersectoral planning and forecasting, but also in the
development of:
- balanced models of inner planning;
- different types of product – labour models;
- dynamic models of interdisciplinary balance.
Key words: Leontiev’s economic–mathematical model, intersectoral
balance, gross domestic product (GDP), value added, errors and
ambiguity of calculations.
Introduction
The contemporary economic theory recommends intersectoral balance
‘Input – Output’ to be used in macro planning. ‘Input – Output’ model
presents in a form of a chessboard (Table 1) relationships between the
volume of production outlays (in respective sectors) and the volume of
production manufactured by different sectors of businesses.
Table 1
Model of intersectoral „Input–Output” balance
Producers
- sectors
1
2
...
N
Consumers - sectors
1
2
…
n
X11 X12
...
X1n
X21 X22
...
X2n
...
...
...
...
Xn1 Xn2
...
Xnn
Final
product
Global
product
Y1
Y2
...
Yn
X1
X2
...
Xn
Value
added
Z1
Z2
...
Zn
Global
Product
X1
X2
...
Xn
n
n
j 1
i 1
 Z j   Yi
n
n
X X
i 1
i
j 1
j
The following indices have been used to create and calculate the In-Out
inter-sector balance:
 Xij,- production volume of an industry sector i, used in sector j. In
In-Out inter-sector balance convention i – is the number of a line
i.e. the number of a producer and j – is the number of a column, i.e.
the number of a business sector that uses;
 Xi – total volume of production of a given sector i for a given
period, defined by the sum of needs in the production of a given
sector:
n
Xi =
X
j 1
ij
 Yi ,
i = 1,2,…,n
(1)
 Xj – volume of demand of j-th sector in the production of i-th
sectors and other production factors, defined by the following
equation:
n
Xj =
X
i 1
ij
Zj
(2)
 Yi – volume of utilization of the final product of sector I for the
purposes of consumption of households, the country’s
administration, commercial and non-profit organizations,
investment, changes in reserves, pure accumulation of goods,
creating export – import balance;
 Zj – value added (symbolically pure production), which includes
payment for work, profit and depreciation.
A technological matrix of coefficients of simple material outlays A(aij)
constitutes the core of the Input – Output model. Coefficients of simple
material outlays aij can be calculated from the following formula:
aij  X ij / X j , i, j = 1, 2,…, n.
(3)
On the one hand it allows to estimate the necessary volume of
production of sector i ensuring production of sector j , and on the other it
shows what part of production of sector j needs to be produced to ensure
maintaining production of i-th sector with Xij size.
Taking into consideration (3), Equation (1) will take the following
form:
n
Xi =
 aij X j  Yi
j 1
(4)
or in the form of a matrix:
X = AX + Y
According to the current methodology Equation (4) can be used to
perform the following calculations:
1. Assuming for every sector the volume of global production (Xi), it is
possible to find the volume of the final production for every sector (Yi):
Y = (E – A)X
(5)
1.2. Assuming the volume of final production of i – th sector Yi, it is
possible to find the volume of global production of every sector (Xi):
X = (E – A)-1Y =BY
(6)
In (5) and (6) symbol E denotes an identity matrix of an order n, and
B = (E – A)-1 – a matrix inverse to matrix (E – A). Elements of matrix B
are commonly referred to as coefficients of complete material
expenditures.
To illustrate how accurate are results obtained using Input – Output
model and examples first put forward by I. W. Orlova [3] is presented.
The example gives the following values of coefficients of simple
expenditure aij and final production Yi for sectors in a symbolic economic
system:
 0,3 0,1 0,4 


А   0,2 0,5 0  ,
 0,3 0,1 0,2 


 200 


Y   100  .
 300 


Then, according to the Input – Output model the matrix B of coefficients
of complete expenditures and the vector X of global production will take
the following form:
B = (E – A)
-1
 2,040816 0,612245 1,020408 


=  0,816327 2,244898 0,408163  ;
 0,867347 0,510204 1,683673 


 775,51 


X =  510,20  .
 729,59 


Knowing parameters of vector X, we will find the volume of inter-sector
supplies Xij, which are necessary to create a table of inter-sector balance
(Table 1). From Equation (3) we have:
 232,6531 51,0204 291,8367 


X ij  aij X j   155,1020 255,1020
0
 .
 232,6531 51,0204 145,9184 


Taking into consideration the above calculations, the Input – Output
model will take the following form Table 2.
Table 2.
Example intersectoral „Input–Output” balance, in monetary unit (m. u.).
Consumers - sectors
1
2
3
232,7
51,05
291,8
155,1
255,0
0
232,7
51,05
145,9
Producers - sectors
1
2
3
Value added
Global product
155,0
153,1
291,9
775,5
510,2
729,6
Final
product
Global
product
200
100
300
775,5
510,2
729,6
600
2015,3
The balance character of the Input – Output model is maintained when
the following equations are satisfied:
n
n
i 1
j 1
 Xi   X j
i
n
n
i 1
j 1
Yi   Z j
(7)
which allows to use the model in a broad range of applications, including
economic planning, models of consumption, human resources,
international trade (linear models of exchange) and dynamic models of
inter-sector balance.
However, the above mentioned objectives cannot be achieved because
of:
 perversions in the use of economic notions and definitions of
parameters assessed using the model;
 the assumed system of estimation coefficients and rules according
to which the models functions;
 indeterminacy which arises while constructing production sectors
of the model, i.e. whether or not they can be described as the socalled ‘pure’ or ‘economic’ sectors;
 false interpretation of economic procedures of calculations in W.
Leontiev’s model.
Let us provide proof for the existence of the above mentioned defects of
the model.
1. Perversions in the use of economic notions and definitions of
parameters assessed using the model.
The main idea of the model is to connect in one table two basic
documents:
1) calculations of expenditure on production (with detailed
specification of all kinds of expenditure);
2) private material balances.
The first document is shown in columns of a table whereas the second
is presented in its lines. The member of the Academy of Sciences N. P.
Fiedorienko stresses the fact that this method allows to analyze every
index of the table’s first square from two different points of view: as an
element of material expenditure on production and as production supply
by the sector to a consumer1.
It was stressed that from the point of view of economic theory it is not
justified to refer inter-sector supplies to the category of material
expenditure, because production supplied by sectors can only be
industrial services and works.
Identifying volume of production in a sector Xij, when i = j, with
material expenditure on creating this production makes, according to the
author, all the indices calculated with the model meaningless in the
economic sense. It is true that:
- if ‘indirect production’ (in the terminology of the authors of the
model) is not finished production of a company, then it makes no
economic sense to add its volume to ‘final production’ of sectors as
is the case when calculating results according to the lines of the
model;
- if ‘indirect production’, however, according to the register of
expenditure equals the volume of ‘final production’ of a sector,
then it makes no economic sense to add its volume of production to
the value of ‘symbolically pure production’ in the sector as is the
case when calculating results according to the lines of the model.
As a result of similar methodological imprecision errors are made
both in calculating the volume of ‘symbolically pure production’ in a
sector2 ( Z j ) and in determining the volume of ‘final production’ ( Yi ).
Given these conditions the only possibility to avoid imprecision and
errors connected with creating the vector of ‘final production’ and
determining the volume of ‘symbolically pure production’ could be a
change in the structure of ‘Input – Output’ model. The change would
involve excluding from the matrix inter-sector supplies to volumes of
1
N. P. Fiedorienko, Economics and mathematics, Izdatielstvo Znanije, Moskva 1967, p. 43.
In the current methodology of the model the volume of conditionally pure production of j –th sector is
calculated using another property: Z  X  X
 ij
j
j
2
i
production supplies within sectors (coefficients X ij , when i = j) and
including them in ‘final production’ of a sector.
To illustrate these changes in the model, it is necessary to calculate
components of ‘final production’ vector according to:
Yi*  Yi  X ij
przy i = j
(8)
and to introduce appropriate changes into the matrix of coefficients of
simple material expenditures A(aij) - the coefficient aij equals zero when
i = j.
Let us illustrate a similar situation using the data of the investigated
case. Using the data from Table 2 we will create a new vector of final
production Y* which consists of components calculated using (8).
At the same time, coefficients aij, for i = j wil be excluded from the
matrix of coefficients of simple material expenditures A(aij). The new
initial data to calculate inter-sector balance will take the following form:
 0 0,1 0,4 


А   0,2 0
0 ,
 0,3 0,1 0 


 432,7 


Y   355,0 
 445,9 


Then, according to the model, the matrix B of coefficients of complete
expenditures will take the following form:
B = (E – A)
-1
 1,173709 0,164319 0,469484 


=  0,234742 1,032864 0,093897  .
 0,375587 0,152582 1,150235 


The results of calculations in the model are presented in Table 3.
Table 3. Iinput–Output” calculations results taking account of internal
sector supplies in the „final production” indexes, in m. u.
Producers - sectors
1
2
3
Value added
Global product
Users - sectors
1
2
3
0
51,05
291,8
155,1
0
0
232,7
51,05
0
387,7
408,1
437,8
775,5
510,2
729,6
Final
product
Global
product
432,7
355,0
445,9
775,5
510,2
729,6
1233,6
2015,3
The data in Table 3 show that although the changes introduced into the
model did not lead to a change in the value of sector production (Xi), they
allowed to properly assess the volume of production manufactured by the
sector of symbolically pure production.
Indeed, if from the figure of total production volume manufactured in a
sector we subtract the production of other sectors used for its
manufacturing, the obtained difference illustrates the contribution of a
given sector into increasing the socially necessary production. The
coefficient of symbolically pure production will be the measure of this
contribution. On the basis of the data from Table 2 the values of
symbolically pure production will amount to:
- for the first sector 775.5 – 155.1 – 232.7 = 387.7 m. u.;
- for the second sector 510.2 – 51.05 – 51.05 = 408.1 m. u.;
- for the third sector 729.6 – 291.8 = 437.8 m. u..
The results of calculating symbolically pure production, quoted in
Table 3, show that such changes will take place if the proposed changes
in the structure of the model will in fact be introduced.
Symbolically pure production increased and reached 232.7 m. u. (387.7
– 150.0). This is why we can conclude that in fact 232.7 U m. u. were not
material outlays.
The proposed changes of the structure of the model make the
coefficients once again meaningful in the economic sense and they also
allow to assess the degree of discrepancy between the volume of
symbolically pure production calculated using the model and the
expected volume of this production. It is going to be explained a little bit
later.
2. The inadequacy of indices calculated according to ‘Input –
Output’ model compared with their computed values
It is a well known fact that one criterion of a well built ‘Input – Output’
model is the social equality of necessary costs of production and the value
of its manufacturing:
Xi = Xj
for i = j
The equation can also be expressed as:
n
a
j 1
ij
X j  Yi =
n
X ij  Z j

i 1
(9)
It should be stressed that the condition mentioned for Equation (9) is
formally satisfied when preparing practically all currently published intersector balances. It is also satisfied in the quoted example (Table 2).
According to Equation (9) ‘final production’ Yi is an independent part
of total volume of production (Xi) in sector i.. This is why its possible
increase by 10% should also lead to an increase of index Xi and an
increase of necessary production outlays (Xj) only by 0.1 ·Yi and no
more. However, calculations performed on the basis of W. Leontiev’s
model show that in a similar situation the increase of production outlays
and the volume of production supplies to the market are significantly
higher than volumes that triggered their causes.
We can see the evidence of that in the initial data of ‘Input – Output’
model presented by I. W. Orlova if the absolute meanings Yi of the ‘final
production’ vector Y are increased by 10%. Then, in line with the
introduced changes and current methodology, the results of calculations
will have the values presented in Table 4.
Table 4. „Input–Output” calculation results taking account of internal
sector supplies in the „final production” indexes increased by 10%
Producers sectors
1
2
3
Value added
Global product
Consumers - sectors
1
2
3
255,9
56,2
321,0
170,6
280,6
0
255,9
56,1
160,6
170,7
168,3
321,0
853,1
561,2
802,6
Final
product
220
110
330
Global
product
853,1
561,2
802,6
660
2216,9
The data in Tables 2 and show that an increase of final production by 60
UMUs (i.e. by 10%) will require, according to W. Leontiev’s algorithm,
an increase of global production by 201.6 m. u. (i.e. more than threefold). The obtained result cannot be explained on the grounds of real
economics and the labour theory of value according to which the value of
a commodity is related to the labour needed to produce or obtain that
commodity.
Particular attention should be paid to the problem of precise definitions
of the initial values of ‘final production’. The quoted example shows that
when W. Leontiev’s algorithm is used in calculations, errors made in
defining the components of ‘final production’ vector will be reflected in
‘global product’ index. An additional proof of that are data presented in
Table 5.
Table 5. Size of changes in global production with various corrections of
„final production” vector
Secto
rs
1
І
ІІ
ІІІ
Total
Possibile errors of defining parameters of ‘final production’ vector in %
and their effects
1%
5%
10%
growth
for
for grow for
for
for
for
growth
„K.p.” „G.p.” th „K.p.” „G.p.”
„K.p.” „G.p.”
2
2,0
1,0
3,0
6,0
3
7,4
3,4
9,3
20,1
4
3,7
3,4
3,1
3,36
5
10,0
5,0
15,0
30,0
6
37,3
16,8
46,6
100,7
7
3,7
3,4
3,1
3,36
8
20,0
10,0
30,0
60,0
9
74,4
33,7
93,3
201,4
The data in Table 5 demonstrate that the effect of error multiplication
occurs as a result of any mistakes made when defining parameters of
‘final production’ vector. For the presented model, error multiplication
Km, given any precision degree of defining parameters of ‘final
production’ vector, can be given by:
Km 
X
Y
i
i
i
i
Therefore, for the investigated case and using the data from Table 2,
Km = 2015,3:600 = 3.36, which is fully confirmed by the total values of
the data in lines 4, 7 and 10 of Table 5.
The quoted examples also confirm the fact that when final production
vector is corrected, in W. Leontiev’s model the meanings of assessed
indices do not make economic sense and pervert the understanding of
processes which take place in an economic system.
At this point, we should also mention the flaws of the model that were
pointed out by a team of scientists of The Chair of General Economic
Theory of the State University of Economics and Finances of Sankt
Petersburg. They stated [3] that ‘it should be stressed that while
developing the methodology of building inter-sector balances in this
country, economic indices were stripped of any market meaning.’ And so:
- indices of II quadrant of the balance were interpreted as elements
of divided final product and not as structural components of
demand at the macro-level.
- global indices of I and III quadrants were interpreted as indices that
characterize the value structure of the global social product and not
10
3,7
3,4
3,1
3,36
the structure of its distribution into profits of different economic
entities, etc.
As a result, the model has become a tool of assessing production and
production’s distribution and not a tool that could assess structural
parameters of macro-economic stability in ‘Input – Output’ model.
3. The uselessness of ‘Input – Output’ model to determine the
final volume of expenditure and production results.
According to labour theory of value, the value of labour, commodity or
service is related to the labour needed to produce or obtain that
commodity. In a fair and just competition system this can be reflected in
market prices.
Even Aristotle thought that without fair exchange social life would not
be possible. He claimed that an exchange is fair if the interdependence of
the parties involved reflects the interdependence of their labour i.e. the
equivalence of labour outlays. The total measure of labour outlays,
according to Aristotle, was necessity. An exchange took place if the
mutual needs of sellers and buyers were equal or if labour outlays on the
exchanged products were equal.
‘Input – Output’ model can fully present only the needs of production
producers in two aspects: production outlays Xj that have to be
compensated and manufactured production Xi (for i = j) which has to be
sold.
It also follows from the model that all the needs of j –th sector in the
production of i –th sector will be fully satisfied with the cost of intersector supplies  X ij .
i
Therefore, ‘final production’ manufactured by i – th sector with the size
defined by:
Yi = Xi –  X ij
j
is not produced in order to satisfy needs of j – th sector – producers in i –
th sector, but to satisfy the needs in this production of other sectors –
producers.
However, the data of the model do not include any information about:
- names of ‘other sectors – producers’, their composition and the
volume of production exchanged into the production of i – th
sector;
- production outlays in other sectors – producers and general volume
of their production.
Only one thing seems to be obvious; according to labour theory of value
total production manufactured by other sectors – producers should equal
the total volume of final production which was listed in the model.
As a result of that, as can be seen in the investigated example which
was taken from I. W. Orlova’s handbook, the total volume of production
 n

  X i  2015.3m.u.  in our example does not include the volume of
 i 1

production of ‘other sectors – producers’, which can be assumed to be


equal to the total value of ‘final production’ ”   Yi  600 m.u.  of all the

i

sectors and amounts to no less than 2615.3 m. u. (2015.3 + 600)
according to the equation:
n

n
 X  Y
i
i 1
i
(10)
i
According to (10) the socially necessary outlays for production in a
symbolic economic system will practically amount to the value of Ψ =
2615.3 m. u. which was determined from another equation:

n
n
 X  Y
j 1
j
i
(11)
i
However, given the current methodology of the model, calculations
performed using (10) and (11) cannot give a complete picture of the
volume of production manufactured in an economic system. This is
confirmed by several reasons:
1. There are no guarantees that ‘final production’ vector calculates
production of all the ‘other sectors – producers’ that participate in
the process of social production.
2. There are no guarantees that total volumes of production of ‘other
sectors – producers’ equal
n
Y .
i
i
3. It is not possible to obtain even estimated value of inter-sector
supplies in the group of ‘other sectors – producers’.
To confirm these arguments we referred to the most complete
description of methodology of constructing quadrants of inter-sector
balance provided in [4], a handbook for lecturers and students of
economics, specialist working in the field of economics and accountancy
as well as for those employed in legislature and executive government. In
Chapter 12.3 of the handbook entitled ‘Methodology of constructing the
second quadrant of inter-sector balance’ three directions of utilizing ‘final
production’ are defined: final consumption, global reserves and export.
According to the handbook, final users are:
- households,
- organs of national administration
- non-profit organization.
In practical terms, statistical data about the volume of production of
these ‘sectors’ and the level of needs in this production are not possible to
determine, because the quoted consumers are not included in the current
economical classification of the Russian Federation.
It is obvious that the names of sectors can be redefined and the list of
sectors can be expanded or narrowed down depending on the list of
sectors provided for in the first quadrant of ‘Input – Output’ model and
the adopted degree of aggregation of economic sectors.
Most authors, including the earlier quoted I. W. Orlova do not reveal
any names of sectors – consumers of ‘final production’. They directly
calculate in the model only the sectors of material production. If we make
that assumption, all the sectors that were not taken into account in the
first quadrant of the model should be included in the group of final
consumers. A list of such sectors should include most of all the sectors of
non-material production and the social sphere. According to the
classification applied in the Russian Federation the list of final products
and services should also include the production of the following sectors:
- construction industry;
- public utility and services industry;
- health care system;
- physical culture;
- social security;
- education;
- culture and art;
- science and scientific services;
- finances, loans, securities and pension schemes;
- social organizations.
It is without doubt that when the nomenclature of sectors utilizing ‘final
production’ increases, the possibility of precisely determining the
components of ‘final production’ vector and results of calculations using
the model decreases.
The authors of the handbook consider using ‘final production’ in other
character, to use it for ‘global reserves and for export’.
While we mention ‘global accumulation’, it should be explained that it
can be used only for goods paid for and manufactured in the sector of
material production. The illustration in ‘final production’ vector of
outlays for ‘global accumulation’ means that in the model the values of
production of material manufacturers (Xij) calculated using the model do
not include production meant to be a reserve. This is why in the list of
‘final production’ consumers, apart from sectors of non-material
production and the social sphere, you also have to calculate sectors of
material production which participate in production for the purposes of
‘global accumulation’. As it is impossible to determine directions of
utilizing ‘global reserves’ to construct ‘Input – Output’ inter-sector
balance, any attempts of implementing such recommendations will
always be one of the reasons for mistakes in calculations performed using
the model.
Utilization of ‘final production’ for export supplies also needs to be
further explained. The existence of export supplies proves the existence
of sectors – consumers that were not taken account of in the model. The
presence of such sectors – consumers (importers of national production)
confirms the fact that disproportions of the model when the volume of
production exceeds the value of consumption of national industries has
the character of objective positive deviation.
When the country’s needs in some kind of import production are
satisfied, the volume of production consumption exceeds the volume of
its production. In this particular case, assessment of deviation between
production volume and consumption volume, measured in market prices,
will have the character of objective negative deviation.
Naturally, the index of gross domestic product (GDP) should reflect
internally (domestically) balanced production capabilities. Otherwise,
GDP is not useful as an index. It is in every country’s interest that GDP
should be balanced with that country’s needs.
On this basis a conclusion can be drawn that recommendations, after
calculating in the list of ‘final production’ export supplies of production,
and particularly the balance of export – import production supplies are
one of the reasons for disturbing the balance of manufacturing and
consuming production.
The more possible directions are defined by the components of ‘final
production’ vector, the less justifiable it is to calculate using the model
the assessment of global production, symbolically pure production and
newly created value.
It should also be stressed that it would be a mistake to consider ‘final
production’ as if it were ‘earlier paid for by future consumers’ and means
necessary to buy it should not be listed in the catalogue of total outlays
for production of sectors illustrated in the model.
These suppositions could be made as in the model general, socially
necessary outlays of labour equal general market prices of production.
This is why, theoretically, such components of general outlays as outlays
for labour and profit of sectors – producers, could be fully sources of
payment when purchasing ‘final production’ of its own sector or
production of other sectors.
However, such a speculation cannot be considered as justified for two
reasons:
First, remuneration for labour and profit will appear at producers only
when they complete their own production or will have it exchanged into
another production. Consequently, the acquisition of ‘final production’
can only be performed by ‘other sectors – producers’ only through
exchange of their production into production of sector i or into money
obtained for the production of ‘other sectors – producers’;
Second, if it is acknowledged that given production is utilized not by
‘other sectors – producers’ but by the same sectors i , then it s quite
difficult to understand what sectors and to which degree utilize this
production.
These remarks demonstrate the fact that ‘Input – Output model cannot
be used to determine total volume of outlays.
4. ‘Input – Output’ model cannot be used to calculate the volume
of production and the country’s demand for this production
Balancing demand for production with its volume must be considered
as one of the fundamental properties of the optimal market economy.
However, W. Leontiev’s model cannot illustrate this kind of balance.
In its classical form, the model:
X = (E – A)-1Y =BY
does not allow to draw such conclusions.
Everything becomes obvious when the model is presented in a
simplified form: (E – A)X = Y. Taking into consideration I. W. Orlova’s
example and earlier assumptions, the model takes the following form:
0,7 ·Х1 – 0,1 ·Х2 – 0,4 ·Х3 = 200
–0,2 ·Х1 + 0,5 ·Х2
= 100
–0,3 ·Х1 – 0,1 ·Х2 + 0,8 ·Х3 = 300
The aim of solving this system of equations is to find such values of
production Xi that would ensure starting ‘final production’ according to
the adopted plan. Unofficially it is assumed that if demand for ‘final
production’ is set properly, then the volume of production Xi calculated
with the model would reflect global demand of the country – Θj for the
production of sectors i .
However, none of the planned solutions obtained using the model
cannot produce a result that would satisfy the equation: Xi = Θj, even if
we eliminate the earlier mentioned defects of the model.
To illustrate the presented assumptions, let us use I. W. Orlova’s
perfectly calculated indices of simple material outlays (indices of matrix
A) and values of ‘final production’ vector (Y). Given this condition, it is
possible to balance the volume of production of sectors (Xi) calculated
with the model with the demand for the production of these sectors only
when the volume of inter-sector supplies of ‘indirect production’ is
balanced with. The condition of balancing the volume of inter-sector
supplies of ‘indirect production’ for j-th sector k can be given by:
n
n
j 1
i 1
Dk   X jk   X ki  0 ,
for i = 1,2,..,k,..,n; j = 1,2,...,k,...,n
(12)
Where:
n
X
j 1
jk
- total demand in indirect production of sector k;
ki
- total demand in the production of k-th sector.
n
X
i 1
The assessment of how the volume of ‘indirect production’ has been
balanced with Equation (12) for ‘Input – Output’ model from Table 2 is
presented in Table 6.
Table 6
Inter - sector demand/supply balance reflecting data on „indirect
production” of Table 2
Producers sectors
(i)
1
X1j
X2j
X3j
Consumption of production
Xi1
Xi2
Xi3
2
232,7
155,1
232,7
3
51,0
255,1
51,0
4
291,8
0
145,9

ij
Di
j
5
575,5
410,2
429,6
6
+45,0
–53,1
+8,1

ij
620,5
i
357,1
437,7
1415,3
D
i
0
i
As can be seen from the data in line 6 of Table 6 the total demand in the
production of the first sector was higher by 45 m. u. than the planned
possibility of its manufacturing. On the other hand, the demand in the
production of the third sector was higher by 8.1 m. u. than the production
limit. At the same time companies in the second sector produce 53.1 m. u.
(i.e. 14.9%) more than the demand for this production.
Let us stress that both unsatisfied demand for production (data in line 6
of Table 6 with the + sign) and surplus production (data in line 6 of Table
6 with the – sign) are sources of unemployment and inflation in the
country.
Unemployment will be due to decreased production output of
companies for whom their production exceeds demand and inflation will
be due to too low production level relative to demand.
As follows from the data in line 6 of Table 6 the size of phenomena that
can trigger unemployment and inflation is the same. Therefore, the
forecast relative increase of unemployment ( Ibezrobocie ) and inflation ( I inflacja )
in the country due to unbalanced production and consumption of
production will also be equal and their sizes can be found using the
following equation:
I bezrobocie  I inflacja 
 
2  
i
(13)
i
i
i
Using the data from the case in question, the forecast values of the
estimated indices will amount to:
I bezrobocie.  I inflacja 
 

2 
i

i
i
45   53.1  8.1
2  2015.3

106.2
 0.0263 or 2.63%
4030.6
i
On the grounds of these calculations a conclusion can be drawn that the
current methodology in case of Xi  Θj does not guarantee satisfying
consumers’ demand. However, if Xi = Θj using Leontiev’s model does
not make any sense.
It is therefore obvious that planned disproportions between sectors and
an increase of unemployment and inflation in the country are unavoidable
if W. Leontiev’s model is to be used.
5. The impossibility of using the economic – mathematical ‘Input
– Output’ model to assess the results of implementing means of
scientific – technological progress
It should additionally be pointed out that the ‘Input – Output’ model is
useless if we wanted to use it to assess the results of implementing some
achievement of scientific – technological progress. It can be explained
using the example of calculations presented in Table 2.
Let us assume that as a result of implementing an innovation in
companies of the second sector the structure of technological outlays in
production has changed:
- the demand of the second sector in the production supplies to
companies of the first sector increased by 50 m. u. (from 51.05 to
101.05);
- the demand of the consumption of the second sector’s own
production decreased by 100 m. u. (from 255.1 to 155.1).
There are two possible alternative variants of using the innovation:
1) to make profit through decreased outlays for production, given the
same level of production;
2) to make profit through increase level of production, given the same
outlays for production.
To assess the realization of the first variant using W. Leontiev’s
model some necessary changes must be introduced into the matrix of
coefficients of simple outlays A(aij).
Following the implementation of the innovation, new coefficients of
outlays for the second sector will take the following form:
a12 = 101.05:510.2 = 0.198
a22 = 155.0:510.2 = 0.304
a32 = 0.1 (it has remained unchanged).
As follows from the conditions of the first variant of implementing
innovation, the total demand for total production manufactured by the
second sector has not changed and the remaining coefficients of the
matrix of simple outlays and ‘final production’ volume have not changed,
either. The matrix A and vector Y will take the following form:
 0,3 0,198 0,4 


А   0,2 0,304 0  ,
 0,3 0,1 0,2 


 200,0 


Y   100,0  .
 300,0 


Please, note that owing to an increase in the profit by 50.0 m. u. (100 –
50), the volume of ‘symbolically pure production’ in the second sector
should also increase.
The results of calculations using the model for the presented initial data
are shown in Table 7. On the basis of these results it was found that the
results produced using W. Leontiev’s algorithm directly contradict
expectations:
- the volume of global production of the first sector increased not by
50.0 m. u.s but only by 30 m. u.;
- the global production of the second sector did not maintain the
present level and decreased by 135 m. u.;
- the volume of ‘symbolically pure production’(Value added) of the
second sector did not increase by 50.0 m. u., either. Instead, it fell
by 3.8 m. u.;
- the total planned volume of global production was not maintained,
but it fell by 110.6 m. u. (2015.3 – 1904.7).
Table 7
„Input–Output” balance reflecting the implementation of the first
variant of innovation
Producers
- sectors
1
2
3
Value
added
Consumers - sectors
1
2
3
241,6 74,3
289,6
161,1 114,1
0
241,6 37,5
144,9
161,2 149,3
289,5
Global
805,5 375,2
production
724,0
Final product
200
100
300
Global product
805,5
375,2
724,0
600
1904,7
In the implementation of the second variant of innovation, coefficients
of simple outlays matrix calculated for the first variant and reflecting
technological changes in production maintain their values. Any changes
of the volume of production as a result of implementing innovation can
only be reflected in indices of ‘final production’. It should be reminded
that according to the labour theory of value an increase of ‘final
production’ of the second sector can only take place in a value equal to
the increase of profit (i.e. by 50.0 m. u.. Then, its value will be 150.0 m.
u.. As a result, the initial data to calculate the balance will take the
following form:
 0,3 0,198 0,4 


А   0,2 0,304 0 
 0,3 0,1 0,2 


i
 200,0 


Y   150,0  ,
 300,0 


The results of these calculations are presented in Table 8.
A comparison of the data presented in Tables 2 and 8 shows that:
- the global production of companies of the first sector increased by 67.2
m. u., although its forecast increase was to amount to 50.0 m. u. only;
- the global production of companies of the second sector decreased by
52.5 m. u. although the forecast predicted its increase following
innovation by 50.0 m. u.;
- the total planned volume of global production increased not by 50.0 m.
u. but by 33.3 m. u. (2048.6 – 2015.3).
It follows from the analysis that using W. Leontiev’s model does not
guarantee obtaining predictable results.
Table 8
„Input–Output” balance reflecting the implementation of the second
variant of innovation
Producers
- sectors
1
2
3
Value
added
Global
product
Consumers - sectors
1
2
3
252,8
90,6
299,3
168,5
139,2
0
252,8
45,6
149,8
168,66
182,33
299,11
842,7
457,7
748,2
Final
product
200
150
300
Global product
842,7
457,7
748,2
6500
2048,6
It could also have been predicted that the differing results presented in
Tables 7 and 8 are due to the fact that the use of Leontiev’s model allows
to consider all the complex mechanism of inter-sector production supplies
and not only the visible results of implementing innovation. However,
these predictions are apparently unjustified as using the model leads to:
- results that quite contradict those that could be expected as a result
of which obtained results cannot be considered as reliable values of
indices;
- obtained results cannot be considered as reliable, because both the
results and the results presented in Table 2 do not ensure balanced
inter-sector production supplies. On the basis of the assessment of
the degree of balancing inter-sector supplies presented in point 4 of
the present paper an assessment of disproportions in different
sectors can be conducted (Table 9).
Let us also highlight the fact that the results of the analysis quoted in
points 2,3 and 4 show that there is no perspective of using ‘Input –
Output’ model to predict changes of the global production through a
change of the coefficients of simple material expenditures (aij) or the
volume of final production (Yi).
Table 9
Results of planned production volume estimation in the sectors (“+”
surplus, “–“ insufficiency) obtained on the basis of Leontiev’s
economic-mathematical model
Global
production in
sectors
X1
X2
X3
 
j
j
ΔXj in UMUs for data from Table
nr 2
nr 6
nr 7
+45,0
–53,1
+8,1
+38,8
–49,3
+10,5
+31,4
–32,3
+0,9
0
0
0
6. The uselessness of Leontiev’s model for planning and
assessment of forecast volume in symbolically pure production
W. Leontiev’s economic – mathematical model contains several gross
mistakes due to which it is impossible to use the model to assess planned
volume of symbolically pure production in the third quadrant of the
model.
Firstly, the components of symbolically pure production (appreciation,
remuneration, profit and taxes) cannot be defined using the model. Highly
qualified specialists are responsible for defining these indices using
methods which are known only to themselves.
Secondly, the calculation, using the model, of the volume of total
symbolically pure production for each sector is done by subtracting from
calculated figures of global production the data of indirect production
with the volume that is used by that particular sector.
Indices obtained in this way cannot be defined as indices of
symbolically pure production, because according to the economic theory
symbolically pure production is defined by subtracting the value of used
material outlays from the volume of production. However, if the
production of other sectors used in the process of manufacturing only in
part and not always3 can be referred to outlays, then the sector’s own
production does not constitute any part of such outlays.
Additionally, material outlays which constitute a part of the production
process can only be the result of supplies of other producers. Therefore,
when the model reflects a sector’s own production in the list of indirect
production it leads to falsified assessment of symbolically pure
production and to double records of production from other producers.
Thirdly, symbolically pure production constitutes a different part of any
kind of production generated by sector j for sector i.4 The same applies
for a sector’s own production. This means that for kind of production
with the size Xij, and also for Xi,j+1= Уi, there is an appropriate size of
symbolically pure production. After changes is inter-sector supplies, the
volume of symbolically pure production should also be changed.
However, Leontiev’s model does not provide a possibility to
accommodate such changes in a detailed record and so therefore it does
not allow to keep a record of changes in the volume of symbolically pure
production.
It is easy to understand it by comparing the results of calculations
presented in Tables 7 and 8. The data in these two table differ in so much
as that the volume of final production of the second sector increased one
and a half times. However, this increase did not lead to any changes in
symbolically pure production which remained at an unchanged level
0.3979 (149.3:375.2).
The size of this production could not change because:
- the matrix of coefficients of symbolically pure production was not
taken into consideration while conducting calculations using the
algorithm;
- an algorithm using the matrix of coefficients of symbolically pure
production has not been developed;
- it is impossible to construct a matrix of coefficients of symbolically
pure production without a record of differences in requirements
3
For example, production of non-material production sectors.
This is due to differences in material consumption indices, necessary funds and profitability of
production.
4
consumers have about final production, no data are known about
concrete consumers.
7. Flaws in the information security conditioned by the W.
Leontiev’s model’s system of estimated indices and rules of
functioning of the system
First of all, it should be noted that the assessed indices of total volume
of production (indices of ‘global production’ and their sum – ‘the total
(global) national product’) in the economic sense do not correspond to
any current estimation indices of how economic systems function. Their
very well known flaws include: large size of material outlays while
summing up global production of respective companies and limitation of
the assessed object to the sphere of material production.
However, currently the whole world and also Russia for the purposes of
general macroeconomic assessment use only estimated of final
production which includes all the spheres of social production. These
include GDP (gross domestic product) and GNP (gross national product).
GNP equals the sum of GDP and net income from ownership rights
abroad.
GDP refers to the market value of all final goods and services
produced within a country in a given period.
GDP can be calculated using three methods:
- production approach also known as gross value added approach in
which all the outputs i.e. results of the process of social production
are added. The gross value added is calculated by subtracting from
the value of goods produced in a company the sum of material
costs of production factors used while manufacturing these goods;
- expenditure approach where we sum up all expenditure on final
goods manufactured by a country’s companies. These include:
expenditure on consumption goods (i.e. products and services)
produced in the country, expenditure on domestic investment
goods, government’s expenditure on domestically produced final
products and services excluding transfer payments, foreign
expenditure on domestic exported goods;
- income approach in which incomes of all economic entities sum up
(remuneration, interest rates, sickness benefit, profit), appreciation
and the country’s income (taxes).
Knowing that these three methods are used it is obvious that the volume
of ‘global production’ calculated using W. Leontiev’s ‘Input – Output’
model does not correspond to published indices of GDP calculated
according to expenditure or according to results of production.
Indices of ‘global production’ calculated with the model for the same
reasons cannot change GNP indices particularly bearing in mind the fact
that differences between GDP and GNP are not productive but ‘rental’ in
character.
Other flaws of information security of the model which distort results of
calculations are two adopted postulates of the model’s functioning:
1) there is linear dependence between the volume of ‘indirect
production’ and the volume of ‘final production’;
2) stability of coefficients of simple expenditures aij of matrix A given
that the volume of production has changed.
First postulate. It follows from the first postulate that the volume of
‘global production’ is calculated using Equation (6). At the same time, it
is obvious that given increased demand for ‘final production’ (Yi) of
every i -th sector by 1%, the volume of ‘global production’ (Xi) generated
by every sector should increase no more than 0.01 ·Yi and it should
amount to:
X i(1)  X i( 0 )  0,01  Yi ,
where: X i(1) and X i(0) are volumes of ‘global production’ of i –th sectors
‘after’ and ‘before’ an increase in demand for ‘final production’
of these sectors.
However, when the calculation is done using Equation (6), the new
estimation of the volume of ‘global production’ of i –th sector amounts
to:
X i(1)  1,01  X i( 0) ,
It is a number much higher than the increase of index Yi, which means
that the increase of indirect production is not reliable.
Second postulate was introduced to ensure stability and
comparability of calculations made using the model. However, given the
present structure of the model, the achievement of set goals is impossible
because of two reasons.
Firstly, all the needs of participants of the process of social production
are divided into sectors. Total figures of needs are calculated for each
sector in order to be fully satisfied. In the model, the process of satisfying
consumer needs in one sector for the production of other sectors is
reflected in the volume of their inter-sector supplies Xij. It means that the
consumption of j –th sector of the production of i –th sector in the volume
of Xij aims not only at satisfying technological needs of j –th sector, but
also ecological, social, spiritual and other needs. Naturally, the demand of
j-th sector for the production of i –th sector to satisfy the above
mentioned needs does not change proportionally to changes in the volume
of production in j –th sector.
Secondly, a requirement of the stability of coefficients aij, calculated
using Equation (3) no longer makes sense (at a given level of technology)
together with changes of vector Yi (‘final production’), because different
meanings of the coefficients of vector Yi predict different needs of j –th
sector in the production of i –th sector.
This is why the postulate of the stability of coefficients aij, which in
the nomenclature of the model are called ‘coefficients of simple material
expenditures’ are which are calculated using Equation (3) is, according to
the author, the reason of addition distortions of ‘Input – Output’ model.
8. Flaws in the information security which arise while defining
the structure of sectors in ‘Input – Output’ model
The model is constructed using pure sectors. Pure (or technological)
sector is a specific notion that belongs to the current theory of inter-sector
balances. A pure sector unites companies independently from the ministry
they belong to according to what the majority of their main production is.
The evidence of pure sectors is done on the basis of reorganization of
data collected in current statistical evidence of sectors of the country’s
industry. Respective companies are included in a respective pure sector
taking into account their total production.
It was stressed in [5] that ‘It is a kind of abstract entity, as in fact pure
sectors do not exist. Their artificial character is immediately revealed in
planning as many products are manufactured simultaneously by different
branches of industry. This is why special arrangements are made to make
the model compatible with the system of indices of the national economic
plan. As a result, a balance is produced which is constructed not
according to pure, but according to economic (administrative) sectors’.
This system of processing information deprives the model of any
efficiency and precision as it leads to mistakes in calculating indices not
only for pure sectors, but also for economic branches.
9. Flaws connected with the economic interpretation of matrix B
(matrix of coefficients of complete expenditures) of ‘Input –
Output’ model
The model assumes that coefficients bij of matrix (B) of coefficients of
complete material outlays show the need of global start of production of
the sector i for manufacturing one unit of ‘final production’ of sector j.
They consist of simple outlays of each sector for a given product and
indirect outlays. Indirect outlays should be understood as not direct
simple outlays for a given product, but outlays of connected sectors.
Examples found in the literature usually mention the problem of
defining complete outlays of electricity for manufacturing any
production. To manufacture steel one needs to use a certain amount of
electricity and to manufacture this electricity one needs a certain amount
of steel. It should also be stressed that an increase in steel production
would require an increase of electricity outlays for obtaining metal ore
from which steel is produced, etc.
It is assumed that coefficients of complete expenditures take into
account these mutual relationships contrary to coefficients of simple
expenditures and that a solution of W. Leontjev’s equations allows to
express coefficients of complete expenditures through coefficients of
simple expenditures.
It is stressed that coefficients of complete expenditures often broaden,
in comparison with coefficients of simple expenditures, the range of
resources taken into consideration, e.g. crude oil is not directly used to
produce cast iron (coefficient of simple expenditures equals zero), but it
is to be found in the evidence of complete expenditures (through the use
of electricity in transportation).
It is thought that the use of the system of coefficients of complete
expenditures allows to perform a quick assessment what corrections
should be introduced to material resources in order to ensure a good
balance of the plan of national economy and also to assess the influence
of changes in the inter-sector proportions on the effectiveness of
manufacturing production.
All the above mentioned interpretations of the mutual relationship
between coefficients of simple and complete expenditures and all the
attempts to give coefficients of complete expenditures calculated with W.
Leontiev’s model do not have any scientific basis.
Firstly, complete material expenditures defined as the sum of simple
expenditures of each sector for a given product and indirect outlays in the
whole chain of interrelated goods is an economically abstract category.
Determining such outlays is methodologically groundless and makes no
sense as it serves no practical purpose. Coefficients bij of matrix B, which
is called ‘matrix of coefficients of complete material expenditures’
actually do no affect complete material expenditures in the light of such
definition.
The calculation of coefficients of complete material expenditures in
‘Input – Output’ model is conducted on the basis of matrix A of
coefficients of simple material outlays by means of an operation called
matrix inversion, according to the formula B = (E – A)-1, which can be
given by:
B=(E – A)-1 =E + A + A2 + A3 + ...=


k
(14)
k 1
Incidentally, the precision of calculations conducted using Equation (14)
depends on the number of approximations k.
It is obvious that the secret of transforming coefficients of simple
outlays aij into coefficients of complete expenditures bij begins from the
third component of Equation (9). We shall investigate the process on an
example of a three-dimensional matrix A, similar to the one found in the
quoted example. In this case:
  11 12 13    11 12 13 

 

A    21 22 23     21 22 23  
       
 31 32 33   31 32 33 
 11 11   12 21   13 31  11 12   12 22   13 32  11 13   12 23   13 33  
                        
22 21
23 31
21 12
22 22
23 32
21 13
22 23
23 33 
 21 11
 31 11   32 21   33 31  31 12   32 22   33 32  31 13   32 23   33 33  
2
Then, in the second approximation the meaning of the coefficient of
complete material outlays, e.g. b12, defined with Equations (14) and (3)
takes the following form:
b12= 12  1112  12 22  13 32  =
12 1112 12  22 13  32



2
1  2
22
3 2
(15)
It follows from Equation (15) that the calculation of these coefficients
requires adding and multiplying coefficients calculated on different bases.
Adding and multiplying such coefficients is in itself a pointless economic
task and even more so if these bases are different from that for which a
given coefficient of complete material expenditures is determined.
Therefore, the claim held by many economists in the literature that
coefficients of complete material expenditures bij have a profound
economic sense is but a fairy tale for trustful economists.
Secondly, the matrix of coefficients of complete material expenditures
B has a structure which is different from that of matrix A. By the
structure of matrix we understand mutual relationships of its elements in
column.
This argument is not only received with surprise, as according to W.
Leontiev’s algorithm matrix B is created on the basis of matrix A, but a
question also arises how these structural differences should be calculated
and what appropriate procedure of calculations should be applied.
The investigated example provided by I. W. Orlova lacks simple
material outlays for the production of the second sector necessary to
generate production of the third sector (the appropriate coefficient of
simple material expenditures equals zero). At the same time the
coefficient of complete material outlays for the production of the second
sector for one unit of final production of the third sector is 0.408 which
requires a further increase of production in the second sector by
0.408 · Y3 = 0.408 · 300 = 122.4 m. u.. Consequently, the following
questions remain unanswered: should in fact production supplies of the
second sector to the third sector in the amount of 122.4 m. u. be taken
into consideration? And if so, why such supplies are not to be found in
‘Input – Output’ model? If not, then to what sector and in what amount
such production supplies should be included in the model?
All these questions will always remain unanswered particularly
when we consider the fact that the same results can be obtained by
resigning from coefficients bij.
According to the author, the same results of global production in
Leontiev’s model can be obtained using Equation (16):
X  UY  Y  ( E  U )Y
(16)
where matrix U is constructed by exchanging coefficients aij of matrix A
into coefficients uij calculated using the following equation:
u ij 
 ij
Yi
Then, on the basis of the investigated I.W. Orlova’s initial ‘Input
– Output’ model, coefficients of matrix U and of vector Y take the
following form:
 1,16325 0,5102 0,9728 


U   0,77550 2,5510
0 
 1,16325 0,5102 0,4864 


 200 


и Y   100  ,
 300 


and global production vector X calculated using Equation (11) will take
the following form:
 775,5 


X   510,2 
 729,6 


which is fully in line with the calculations conducted using W. Leontiev’s
algorithm.
The existence of two, different interchangeable systems of coefficients
(B and U) also shows that they are unreliable and that their economic
interpretation does not make any sense at all.
Conclusions
The earlier mentioned flaws of ‘Input – Output’ model of inter-sector
balance and W. Leontiev’s economic – mathematical model prove that
they cannot be used to establish, optimize and maintain planned
proportions in the economy.
Improvements of the state system of inter-sector planning should
include:
1.
Changes of rules according to which inter-sector balance is
constructed. Such changes should particularly reflect the
influence of all categories of participants in the process of social
production as well as a change of the ‘expenditure – production’
method of constructing ‘Input – Output’ inter-sector balance into
‘consumption – distribution’ method. The ‘consumption –
distribution’ method of constructing the balance means that the
balance should reflect the volume of demand for the socially
necessary production and the expected volume of its production.
According to the author, such balances are called supply –
demand inter-sector balances and present a square matrix which
represent according to its line the demand of producers – sectors
for the production of other sectors to conduct their professional
activities. The columns, on the other hand, reflect the demand of
other sectors for the results of production of a given sector –
producer.
2.
A development of methods of optimizing the system of mutual
needs in supply - demand model. Constructing balanced supply –
demand models will allow to both reduce unemployment and
inflation in the economy and to increase the effectiveness of
estimating and assessing the implementation of achievements of
scientific – technological progress.
The most pressing tasks include informing economists through the
publication in economic journals and through lectures at scientific
conferences about the flaws of ‘Input – Output’ balance and W.
Leontiev’s economic – mathematical model. Apart from that, in order to
avoid any further useless waste of time and money, all the attempts of
using these models for constructing:
- balanced models of one’s own planning,
- various kinds of consumer – employee models,
- models of international trade (linear models of exchange)
- dynamic models of inter-sector balance
should be immediately stopped.
References:
1. Fiedorienko N.P.: Ekonomika i matematika, Izdatelstvo Znanije,
Moskva 1967.
2. Łopatnikow L.I.: Ekonomiko-matematičeskij slovar. ANZSRR,
Izdatelstvo Nauka, Moskva 1987.
3. Orłowa I.W.: Ekonomiko-matematičeskoje. Praktičeskoje posobije
po rešeniju zadač, Vuzovskij učebnik, Moskva 2005.