Computational method of finding optimal structural change in economic systems: an input-output projected-gradient approach Alexander Vaninsky∗ Mathematics Department Hostos Community College The City University of New York 500 Grand Concourse Av., Room B 409 Bronx, NY 10451 U.S.A. Abstract A method of finding locally optimal structural change in economic systems described by input-output models and aimed to maximization of final product per capita is suggested. The method is based on a projected-gradient. Both projected-gradient and corresponding locally optimal structural change are obtained in explicit form. Numerical examples are provided. Keywords : Projected gradient method, optimal-structural change, input output models.
1.
Introduction
In this paper a computational method of finding locally optimal structural change in economic systems described by input-output models is suggested. Input-output model, see e.g. Gregory and Stuart (2004), is combined with a special method of factor decomposition developed in Vaninsky (1984) and a projected-gradient method. Suggested method uses a known fact that a projection of a gradient on a plane tangent to a surface is a gradient of the limitation of the original function on the ∗ E-mail:
[email protected]
——————————– Journal of Interdisciplinary Mathematics Vol. 9 (2006), No. 1, pp. 61–76 c Taru Publications °
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A. VANINSKY
surface, see e.g. Maital and Vaninsky (1999) for detail. Input-output model is transformed to a structured form, with all its elements expressed as shares of sector gross output and of total gross output, see Ghosh (1964) for detail. Final product per capita is considered as an objective function, and is transformed to the function of structured variables and one quantitative variable that does not appear in the final result. As known, movement along gradient line provides maximal increase in objective function per unit of distance. This observation is used in the suggested method for obtaining locally optimal structural change. Simple analytical form of the obtained result allows its further application to Economic Analysis problems. Structured form of input-output model allows elimination of inflation issues from consideration. It is shown that the movement along the projected-gradient line leads an economic system either to stable one-sector state or to unstable state with equal shares of all sectors in gross output. The latter is referred to below as an even-sector state. As economic system with one-sector is unlikely practical, external forces are needed for correction of undesired structural change. The nature of these forces is beyond the scope of this paper. The paper is organized as follows. In Section 2 transformation of input-output model and objective function to the structured form is given. Section 3 contains mathematical solution of the problem based on projected-gradient; the solution is presented in Section 4 in explicit analytical form. In Section 5 economic interpretation is discussed, and in Section 6 numerical example is given. 2.
Mathematical statement of the problem
Consider an economic system described by input-output model with final product per capita as an objective function. The input-output model is this: X = AX + Y ,
(2.1)
where X = ( Xi ) stands for a vector of gross output, Y = (Yi ) , for a vector of final product, and A = ( Ai j ) , for a matrix of intermediate (technological) use; lower symbols i and j stand for the sectors of the economic system. This system may be transformed to structured form as follows.
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OPTIMAL STRUCTURAL CHANGE
Dividing each equation of (2.1) by Xi , we get: n
1=
∑ Ci j + Ui ,
i = 1, . . . , n
(2.2)
j=1
where n stands for the number of sectors in the economic system, Ci j = ( Ai j X j )/ Xi , for a share of gross output of i -sector used technologically in j -sector, and Ui = Yi / Xi , for final product of i -sector as a share of i -sector gross output. Each equation in (2.2) corresponds to a particular sector. A set of variables Di , the shares of i -sector gross output in total, combines them into a system of equations. Di = Xi / X .
(2.3)
As follows from the definition, n
∑ Di = 1 .
(2.4)
i =1
Input-output model, given by equations (2.2) and (2.4), is referred to below as a structured model. At the next step we transform final product per capita, an objective function, to a function of structural variables. By definition, final product per capita is Z = Y / N,
(2.5)
where Y stands for total final product, N , for population. Recall that in accordance with the definitions given above, Yi = Xi Ui = ( XDi )Ui ;
Y=
n
n
i =1
i =1
∑ Yi = ∑ (XDi )Ui ,
so that after the substitution of these expressions into (2.5) we get: µ ¶ N n X e ∑ Di Ui , Z= Di Ui = X ∑ N i =1 i =1
(2.6)
(2.7)
e = X / N stands for total gross output per capita. where X The objective function (2.7) does not depend on structural variables of intermediate use Ci j explicitly. They will be included into consideration in the operation over projected gradient below.
64 3.
A. VANINSKY
Solution of the problem
Consider objective function given by (2.7) as a function of all variables: e Ui , Di , Ci j ), Z = Z ( X,
i, j = 1, . . . , n .
(3.1)
The components of its gradient are as follows: n ∂Z = ∑ Di Ui , e ∂X i =1
∂Z e i, = XU ∂Di
∂Z e i, = XD ∂Ui
∂Z = 0. ∂Ci j
(3.2)
Note, that the components corresponding to variables Ci j are equal to zero, because these variables do not appear in the function explicitly. To find a gradient projection on a plane defined by equations (2.2) and (2.4), we first rewrite them in a form: n
∑ Ci j + Ui − 1 = 0 ,
i = 1, . . . , n
(3.3)
j=1 n
∑ Di − 1 = 0 ,
(3.4)
i =1
e , i, j = 1, . . . , n . and put variables in the following order Di , Ci j , Ui , X The total number of the variables is, correspondingly, n + n2 + n + 1 = (n2 + 2n + 1) . Let H = ( Hkl ) be a transposed Jacobian matrix of the left-hand side e , i, j = 1, . . . , n , of the equations (3.3), (3.4) and variables Di , Ci j , Ui , X see e.g. Marsden and Hoffman (2001), so that each column of the matrix corresponds to a particular equation from (3.3), (3.4). Then the number of columns is (n + 1) , the same as that of equations. Each row of the matrix corresponds to one variable, with Hkl being a partial derivative e, of the left-hand side of k -th equation by l -th variable Di , Ci j , Ui , and X correspondingly. Thus, we get Hkl =
∂(left-hand side of equation number l) , ∂(variable number k) k = 1, . . . , (n2 + 2n + 1);
l = 1, . . . , (n + 1) ,
(3.5)
with first n columns corresponding to equations (3.1) and last column, to equation (3.4). First n columns are similar in structure due to the similarity in the equations (3.3). Each column contains (n2 + 2n + 1) elements, corre-
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OPTIMAL STRUCTURAL CHANGE
sponding to one variable each. For the column corresponding to the k -th equation in (3.3) we have (n + 1) non-zero elements only:
• an element corresponding to Ui for i = k : ∂(left-hand side of equation number k) = 1, ∂(Uk )
k = 1, . . . , n , (3.6)
and
• n elements corresponding to Ci j for i = k : ∂(left-hand side of equation number k) =1 , ∂(Ck j )
k = 1, . . . , n ,
(3.7)
For a column corresponding to equation (3.4) we have n non-zero elements corresponding to variables Di : ∂(left-hand side of equation number (n + 1)) =1 , ∂ ( Di )
k = 1, . . . , n ,
(3.8)
All the rest elements of matrix H are equal to zero. As known, columns of the matrix H are vectors normal to the subspace defined by equation (3.3), (3.4). As shown in Albert (1972), a projection of a gradient-vector on the plane is this Pr H (grad Z ) = (grad Z )( E − H H + )
(3.9)
where grad Z stands for a gradient row-vector, E , for a unit matrix of the size (n2 + 2n + 1) , H , for a (n2 + 2n + 1) × (n + 1) matrix given by (3.6)(3.8), H + , for a general inverse of matrix H , and Pr H , for the projectedgradient. In our case, the columns of the matrix H are linearly independent, so that matrix H + may be rewritten as H + = ( H T H )−1 H T ,
(3.10)
where upper symbol “ T ” stands for matrix transposition. Substituting (3.10) to (3.9) we get: Pr H (grad Z ) = (grad Z )( E − H H + )
= (grad Z )( E − H ( H T H )−1 H T ) .
(3.11)
Formula (3.11) defines components of the projected-gradient of the objective function (2.7) on a subspace of structural variables defined by
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A. VANINSKY
equations (3.3), (3.4). The components of projected-gradient determine locally optimal change in structural variables. 4.
Presentation of the solution in explicit form
In this section it is shown how projected-gradient (3.11) may be represented explicitly. For simplicity we start with a two-sector numerical example of input-output model (2.1), and after that extend the obtained result for a general case. Let input-output model be this: X 100 60
A =
30 20
20 20
=
Y 50 20
Transform it to the structured form (2.2), (2.4) by division of each row by the sector gross output: C 1 1
=
0.3000 0.3333
0.2000 0.3333
=
U 0.5000 0.3334
Total gross output is 100 + 60 = 160 , so that values of Di , the shares of a sector gross output in total gross output, are as follows: D1 = 100/160 = 0.6250 , D2 = 60/160 = 0.3750 , see (2.3). Put the variables in order as given above: X1 = D1 , X2 = D2 , X3 = C11 , X4 = C12 , X5 = C21 , e . Then objective function (2.7) and X6 = C22 , X7 = U1 , X8 = U2 , X9 = X its gradient (3.2) are as follows: Z= ∂Z ∂X1 ∂Z ∂X2 ∂Z ∂X7 ∂Z ∂X8 ∂Z ∂X9
X9 ( X1 X7 + X2 X8 ), ∂Z = = X7 X9 , ∂D1 ∂Z = = X8 X9 , ∂D2 ∂Z = = X1 X9 , ∂U1 ∂Z = = X2 X9 , ∂U2 ∂Z = = ( X1 X7 + X2 X8 ) . e ∂X
All the rest components of the gradient vector are equal to zero.
(4.1)
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OPTIMAL STRUCTURAL CHANGE
Recall equations of the structural variables interconnection (3.3), (3.4) for n = 2 :
( X3 + X4 ) + X7 − 1 = 0 ,
( X5 + X6 ) + X8 − 1 = 0 ,
( X1 + X2 ) − 1 = 0 .
(4.2)
Using formulae (3.5)-(3.8), we obtain matrix H T shown below where empty cells stand for zero values. As we are interested in structural variables only, the problem is considered in the orthogonal complement e = X9 . This to the coordinate vector of the quantitative variable X observation results in the appearance of an additional row in the matrix e = X9 . H T containing 1 in the last column corresponding to X
Matrix products in formulae (3.10), (3.11) are as follows: 3 HT H =
3 2 1 1/3
( H T H )−1 =
1/3 1/2 1
H + = ( H T H )−1 H T
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A. VANINSKY
And finally,
From formula (4.1), grad Z = ( X7 X9 , X8 X9 , 0, 0, 0, 0, X1 X9 , X2 X9 , ( X1 X7 + X2 X8 ))T e U2 X, e 0, 0, 0, 0, D1 X, e D2 X, e ( D1 U1 + D2 U2 ))T . (4.3) = (U1 X, Substituting this vector and matrix E − H H + given above into (3.11), we get: µ e U1 − U1 + U2 , U2 − U1 + U2 , − D1 , − D1 , Pr H (grad Z ) = X 2 2 3 3 ¶ D2 D2 2D1 2D2 , ,0 . (4.4) − ,− , 3 3 3 3 Formula (4.4) has block structure with each block corresponding to one structural variable Di , Ci j , and Ui , respectively. Two elements are in each
69
OPTIMAL STRUCTURAL CHANGE
block in 2-case; n elements, in general case. Components of the first block corresponding to sectors’ shares in gross output may be either positive or negative. The sign depends on the share of sector final product in sector gross output Ui . If it is greater than average, then corresponding component is positive and vice versa. Components corresponding to intermediate use form two blocks, each containing two variables: X3 , X4 , and X5 , X6 , respectively. These components are always negative and proportional to the share of sector gross output in total. The coefficient of proportionality is −1/3 , in general case, −1/(n + 1) . Next block corresponds to final product variables Ui . Its components are always positive and proportional to the share of sector final product in sector gross output. Coefficient of proportionality equals to 2/3 , in general case, n/(n + 1) . The last block contains only one component corresponding to quantie and is equal to zero, because this variable keeps constant tative variable X value on a plane of structural change defined by (2.2), (2.4). To extend the result obtained for n = 2 to a general case, we retain the same block structure while increasing the sizes of the blocks. Doing so, we obtain matrix E − H H + and projected-gradient (4.4) as follows:
where M1 (m, k) , M2 (m, p, k) , M3 (m, p, k) , and M4 (m, k) are k × k matrices of the following structure: 1 − 1/m M1 (m, k) =
−1/m ... −1/m
−1/m 1 − 1/m ... −1/m
... ... ... ...
−1/m −1/m ... 1 − 1/m
70
A. VANINSKY
M2 (m, p, k) = M3T (m, p, k) m-th
=
0
...
0
...
0
...
0
...
−1/ p −1/ p −1/ p −1/ p
0
...
0
...
0
...
0
...
M4 (m, k) = diag(1 − 1/m) 1 − 1/m 1 − 1/m
=
... 1 − 1/m
The last column of matrix E − H H + corresponding to quantitative e is zero. variable X Formula of the projected-gradient (4.4) is generalized in a similar way:
µ e U1 − U, ¯ U2 − U, ¯ . . . , Un − U, ¯ − D1 , . . . , − D1 , − D3 , Pr H (grad Z ) = X n+1 n+1 n+1 ¶ D2 Dn Dn nD1 nD1 . . .− ,...,− , . . .− , ,..., , 0 (4.5) n+1 n+1 n+1 n+1 n+1
where U¯ = (U1 + . . . + Un )/n stands for an average share of final product in sector gross output. As follows from (2.2) and (2.4), shares of sectors’ gross outputs in total add up to one. The same is true for the sums of shares of intermediate use and final product in each sector. From this observation follows that the components of projected-gradient corresponding to (a) gross output and (b) intermediate and final product of any sector add up to zero: n
n
∑ (Ui − U¯ ) = ∑ Ui − nU¯ = 0 ;
i =1 n
∑
j=1
µ
D − i n+1
¶
i =1
+
nDi = 0; n+1
i = 1, . . . , n .
(4.6)
It may be noted that in the formula of projected-gradient (4.4), total gross e , the quantitative variable, is separated from structural output per capita X variables. This observation allows computation of the contributions of
OPTIMAL STRUCTURAL CHANGE
71
each structural variable in the change in objective function on a finite interval, see Vaninsky (1984) for detail. 5.
Economic applications
A comprehensive discussion of possibilities provided by suggested approach is beyond the scope of this paper. In this section below we merely outline some of them that seem important to us. Projected-gradient approach allows shed light on economic dynamics. From formula (4.5), it follows that objective function increases locally together with a sector share in total gross output, if the sector has higher than average share in final product, Ui − U¯ > 0 . Increase in a sector share along a gradient line leads eventually to one-sector economic system, a stable equilibrium state. This situation may be considered as an extension of the Ricardian model of international economics that states that international trade leads national economies to specialization in a single product in which they have comparative advantage, Krugman and Obstfeld (2002). In our case, greater share in final product may be considered as comparative advantage of a sector that leads to onesector economy. There is another equilibrium state corresponding to equal sectors’ shares in final product, Ui − U¯ = 0 , or Ui = U¯ , i = 1, . . . , n . In this case, all components of projected-gradient corresponding to the shares of sectors’ gross output are equal to zero. Such economy keeps its gross output structure constant. This equilibrium state is unstable. Any small change in one sector’s share in final product will result in either its disappearance or in domination over other sectors and eventually in one sector economic system. Projected-gradient approach may be of practical interest in situations, where impact on structural change is possible. The suggested approach allows determination of the sectors that deserve accelerated expansion. First of all, national economies in transition should be mentioned, like those of China, Eastern Europe countries, and Russia. In transition process, some of them choose “shock therapy”, or “Big Bang”, while other ones pursue “gradualism”, see Gregory and Stuart (2004) for detail. In any case, it is of principal importance for a government to demonstrate quickly the advantages of the chosen variant of socio-economic development. Structural change aimed to maximal increase in final product per capita may serve as one of the macroeconomic tools. Another area of practical application of the suggested approach is
72
A. VANINSKY
indicative planning, when market determines resource allocation, while plan is used to guide decision-making. Planners project sectoral trends and provide information that is voluntary used by market participants, Gregory and Stuart (2004). Situations requiring structural adjustment may arise in developed economies as well, for instance as a tool of battling deflation, see e.g. Zaun (2005) for brief summary of current macroeconomic problems of Japan. 6.
Examples
6.1 Computation of optimal structural change In this section, we continue consideration of the two-sector model and demonstrate how locally optimal structural change may be found. In section 4 above we have obtained the following values of structural variables: D1 = 0.6250 , D2 = 0.3750 , C11 = 0.3000 , C12 = 0.2000 , C21 = 0.3333 , C22 = 0.3333 , U1 = 0.5000 , U2 = 0.3334 . Average of sectors’ shares in sectors final product is U¯ = (U1 + U2 )/2 = 0.4167 . The e is a multiplier in formula (4.5) that value of gross output per capita X keeps constant value with change in structural variables. For simplicity, it e = 1 below. is assumed X From formula (4.5), projected-gradient is: Pr H (grad Z ) = (0.0833 , −0.0833, −0.2083, 0.2083, 0.1250, 0.1250, 0.4167, 0.2500) . Denote as “ s ” the length of a vector directed along the projected-gradient. Then the new (locally optimal) values of the structural variables are as follows: D1 = 0.6250 + 0.0833 × s , D2 = 0.3750 − 0.0833 × s , C11 = 0.3000 − 0.2083 × s , C12 = 0.2000 − 0.2083 × s , C21 = 0.3333 − 0.1250 × s , C22 = 0.3333 − 0.1250 × s , U1 = 0.5000 + 0.4167 × s , U2 = 0.3334 + 0.2500 × s . To determine reasonable value of s , we assume, for simplicity, that it corresponds to relatively small, about 5%-10%, change in the length of initial vector of structural variables:
kPr H (grad Z )k × s q 2 + C2 + C2 + C2 +U 2 +U 2 , ≤ (5 − 10%) D12 + D22 + C11 12 22 22 1 2
(6.1.1)
where k • k stands for Euclidean norm of a vector, a square root of sum of squares of its components. Given the values of = 0.6067 and q our example, we get kPr H (grad Z )k 2 + C2 + C2 + C2 + U 2 + U 2 D12 + D22 + C11 = 1.1156 , so that 12 22 22 1 2 s ≤ (5 − 10%) × 1.1156/0.6067 . This implies 0.0919 ≤ s ≤ 0.1838 ,
OPTIMAL STRUCTURAL CHANGE
73
and we assume below s = 0.1 . Given this assumption and using formulae given above, the following set of new values of the structural variables are obtained: D1 = 0.6250 + 0.0833 × 0.1 = 0.6333 , D2 = 0.3750 − 0.0833 × 0.1 = 0.3667 , C11 = 0.3000 − 0.2083 × 0.1 = 0.2792 , C12 = 0.2000 − 0.2083 × 0.1 = 0.1792 , C21 = 0.3333 − 0.1250 × 0.1 = 0.3208 , C22 = 0.3333 − 0.1250 × 0.1 = 0.3208 , U1 = 0.5000 + 0.4167 × 0.1 = 0.5416 , U2 = 0.3334 + 0.2500 × 0.1 = 0.3584 . To validate this result, check that the new values of structural variables satisfy formulae (3.3) and (3.4): (0.2792 + 0.1792) + 0.5416 = 1.0000 , (0.3208 + 0.3208) + 0.3584 = 1.0000 , 0.6333 + 0.3667 = 1.0000 . n
Term ∑ Di Ui in formula (2.7), may be considered as a share of structural i =1
variables in the value of the objective function, final product per capita. Check, that new values provide greater value of the objective function than initial ones. With initial values, we have n
∑ Di Ui = 0.6250 × 0.5000 + 0.3750 × 0.3334 = 0.4375,
i =1
while with optimized ones, n
∑ Di Ui = 0.6333 × 0.5416 + 0.3667 × 0.3584 = 0.4744 .
i =1
The increase constitutes (0.4744 − 0.4375)/0.4375 × 100% = 8.43% . The reason of the increase is a greater share of more efficient first sector in total gross output, provided by new values of structural variables. 6.2 Case study: Japan 2000 In this section we apply suggested approach to actual statistical information of the economy of Japan. The objective is to derive conclusions regarding structural change in the economy. Data of 2000, the latest available by the time of writing the paper, are posted on the web site of the Statistics Bureau of the Ministry of Internal Affairs and Communication of Japan http://www.stat.go.jp/english/data/io/. As known, contemporary economy of Japan is suffering from deflation for last seven years, with decrease in consumer spending being one of the main factors. In this situation, government or central bank intervention may be useful. Analysis of structured input-output model may help determination of the sectors of economy that have greatest potential to increase final demand per capita and thus, consumer spending.
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A. VANINSKY
For analysis we used 13-sector input-output table valued at purchasers’ prices; sectors were aggregated into a standard 6-sector model for better readability. Data in yens are given in Table 1, structure input-output model (3.3), (3.4), in Table 2, and components of projected gradient, in Table 3. Data in tables 2 and 3 are given in percent. As above, we assumed e = 1 , because only ratio of components matters. X Table 1 Statistical data
Japan 2000, purchasers’ prices, million yen Total demand A I C W T S
22537651 454872468 77310529 27006396 60128230 371192363
A
I
C
W
T
S
Final demand
1626403 10440101 244599 0 3015 2091571 8131962 3311929 149267432 28805945 4414152 8481546 46584278 214007186 80907 1296210 199012 1258735 653064 5491288 68331313 91925 6381447 539282 1623262 1206623 8259817 8904040 376952 4988307 3214725 530854 8375268 18555198 24086926 806739 29878922 7848730 3888582 15143297 63444451 250181642
N OTES : Source: http://www.stat.go.jp/english/datalio/ Sectors: A : Agriculture, forestry, and fishing industry, sector 01 I : Industry, combines sectors 02 and 03 C : Construction, sector 04 W : Water and electricity, sector 05 T : Transport and communication, combines sectors 09 and 10 S : Commerce and services, combines sectors 06, 07, 08, 11, 12 and 13
Table 2 Structured input-output model
% Total demand A I C W T S
2.22 44.90 7.63 2.67 5.94 36.64
A 7.22 0.73 0.10 0.34 0.63 0.22
I 46.32 32.82 1.68 23.63 8.30 8.05
C
W
T
1.09 6.33 0.26 2.00 5.35 2.11
0.00 1.86 0.84 4.47 13.93 4.08
0.01 1.86 0.84 4.47 13.93 4.08
S 9.28 10.24 7.10 30.58 30.86 17.09
Final demand 36.08 47.05 88.39 32.97 40.06 67.40
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OPTIMAL STRUCTURAL CHANGE
Table 3 Projected-gradient
%
A I C W T S
Total demand –15.91 –4.94 36.39 –19.02 –11.93 15.41
A –0.32 –6.41 –1.09 –0.38 –0.85 –5.23
I –0.32 –6.41 –1.09 –0.38 –0.85 –5.23
C
W
T
–0.32 –6.41 –1.09 –0.38 –0.85 –5.23
–0.32 –6.41 –1.09 –0.38 –0.85 –5.23
–0.32 –6.41 –1.09 –0.38 –0.85 –5.23
S –0.32 –6.41 –1.09 –0.38 –0.85 –5.23
Final demand 1.91 38.49 6.54 2.29 5.09 31.41
N OTE: Total demand per capita is set up to 1
As follows from Table 3, there are two sectors with positive values of the components corresponding to the sectors’ shares in total demand (total demand stands for total gross output in previous considerations): Construction sector (36.39), and Commerce and Services sector (15.41). Increase in shares of those sectors in total gross output will lead, in accordance with the suggested model, to the increase in final demand. So, namely these sectors may be suggested as main targets of central bank and government intervention. It may be noted, that the sectors are not equal in there abilities: Construction sector is more than twice influential than Commerce and Services sector. 7.
Conclusions 1. In this paper a problem of finding locally optimal structural change in economic systems is considered. A system is described by an input-output model, final product per capita serves as an objective function. A method of projected-gradient is applied. Vector of locally optimal structural change is obtained in a simple analytical form. Two feasible equilibrium states are found: one-sector economic system (stable state) and even-sector economic system (unstable state). 2. Possible applications of the suggested approach are discussed. They include economies in transition, developed economies battling deflation, and indicative planning problems. 3. Numerical examples are provided. A theoretical example demonstrates computational aspects; case study of Japanese economy of 2000 serves as an example of economic analysis aimed to determination of critical sectors of national economy.
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A. VANINSKY
References [1] A. Albert (1972), Regression and Moore-Penrose Pseudoinverse, New York, Academic Press. [2] A. Ghosh (1964), Experiments with Input-Output Models, Cambridge University Press, Cambridge. [3] P. Gregory and R. Stuart (2004), Comparing Economic Systems in the Twenty-First Century, 7th edn., Houghton Mifflin, New York. [4] http://www.stat.go.jp/english/data/io/, web site of the Statistics Bureau of the Ministry of Internal Affairs and Communications of Japan. [5] P. Krugman and M. Obstfeld (2002), International Economics, Theory and Policy, 6th edn., Addison-Wesley. [6] S. Maital and A. Vaninsky (1999), Data envelopment analysis with a single DMU: a graphic projected-gradient approach, European Journal of Operational Research, Vol. 115, pp. 518–528. [7] J. Marsden and M. Hoffman (2001), Elementary Classical Analysis, 7th edn., W.H. Freeman & Co. [8] A. Vaninsky (1984), Generalization of the integral method of economic analysis to interconnected and derived factors, Automation and Remote Control, Vol. 44 (8) (2), pp. 1074–1083. [9] T. Zaun (2005), Japan Consumer Price is Steepest Since ’03, The New York Times, March 26, 2005, C3.
Received April, 2005
