Prof Author Louis De Mesnard
email: louis.de-mesnard@u-bourgogne.fr

On the consistency of the commodity-based technology in make-use model of production

Keywords: KEYW ORDS. Matrices, input-output, demand, supply

In the Stone make-use model, the industry-based technology is consistent because his solution can be explained in variational terms inside a circuit. However, the alternative model, the commodity-based technology, is not economically realistic: it never corresponds to a circuit, even if an exact solution can be found when the number of commodities is equal to the number of industries. This model hesitates between a supply-driven and a demand-driven model but when it is converted into a true supply-driven one, it retrieves a consistency.

Introduction

The input-output model has a variant, developed by Stone (1961), where two rectangular homogenous matrices are considered:

• The Use matrix, denoted U, with industries as columns and commodities as rows and with final demand as supplementary column and added value as supplementary row; it indicates how much of each commodity each industry buys to produce.

Wil U 12 e\ U2\ U22 e2 9 2 W 31 U32 es 9 3 W i W2
Xi X 2

• The Make matrix, denoted V, with industries as rows and commodities as columns, indicating how many of each commodity an industry produces.

V ll V12 V13 V21 V22 V23 q\ <J2 #3

XI

X2

One has four accounting identities: x, =Xj=i vu f°r aU h = uy + wj for all j and qt = XJLi utj + ei for all i, qj = X"=i vy for all j, where Wj is the value added of industry j and e, is the amount of commodity i sold to final demand; that is:

(1)

(2)

(3)

(4) x = V s x = U/ s+ w q = U s + e q = V 's
These identities require homogeneity by row and columns, so all terms are expressed in value.

Are defined technical coefficients: a" = or A" = U x-1. In this model, one can pose two alternative hypotheses about matrix V:

• The total output qj of a commodity j is supplied by any industry i in fixed proportions, i.e.,

V/i

the commodity-output proportion dy = -rf is fixed (case referred as technology based on ■ ij industries), that is:

(5) D = V q-1
• The total output x, of any industry i is composed of commodities in fixed proportions, i.e., the industry output proportion cy = j j is fixed (case referred as technology based on commodities), that is:

(6) C = x " 'V
In both cases, the model has a solution, if the number of commodities is equal to the number of industries (Miller and Blair, 1985, pp. 174). In this short note, I will show that the economic interpretation of the commodity based technology is problematic, unless the model is assumed to be supply-driven.

II. The circuit in the traditional Leontief model

The central equation of the traditional input-output economics (Leontief, 1936) is:

(7) x = A x + f
where a a = -is the technical coefficient, x, is the output of sector j, ft is the final demand of Xj commodity i, zy indicates how much of commodity i is bought by sector j , that is the flow from i to j , and v7 is the added value of sector j, with the unique matrix:

Z11 Z12 Z13 Z21 Z22 Z23 Z 31 Z32 Z33 Vi V2 V3 *1 X2 *3 f \ X\ f l X2 h *3
The model can be solved by simply obtaining from equation ( 7):

(8) x = (I -A )" 1f but it must also support a reasoning based on impulsions of demand, that is a differentiation of equation ( 7

): A/)0) -> Axj0) = A/)0) -> Ax[1} = Az^ = ay Axf}. So, the total increase o f the output of sector i is: =X^=i ay Ax]0)
. This continues at steps 2, ..., etc., and at step k:

Ax\k) = a ij that is Ax(i) = A = A* Ax(0) = A k
Af, so the total increase of output is given by: Ax = X ^i AxW = (S^=i A4 j A f ^ ( I -A ) " ' f and equation ( 8) is retrieved. This is well known but it must be reminded: if the second type of solution is impossible, the first cannot be interpreted and it is only a vain exercise without economical meaning. Now, this second reasoning describes a circular process: a production of a sector generates a demand of some intermediate commodities described by the technical coefficients, what generates at its turn a production of the concerned sectors (remembering that the bijective correspondence sector-products is assumed).

III. The circuit with the technological hypotheses of the make-use

For the make-use model, all is in the plausibility of the circular process as described by both alternative hypotheses. Either the process is plausible and the solution of the model has an economic meaning, either it is not.

With the industry-based hypotheses, commodities are produced by industries following the commodity-output-coefficients dy found from the Make matrix, and industries demand products by the mean of technical coefficients a" found from the Use matrix, that is they are found by the product of two matrices, the matrix of technical coefficients, A", and the Markovian matrix commodity-output-proportion matrix D, following the model q = A" D q + e (Miller and Blair, 1985, p. 167) (9) A" D = (U i" 1) (V q"1) = U <V s>_1 V <V' s>_1

A final demand Aej0) of commodity j is an equal need for commodity j: Aqf* = Aej0). This generates an increase in the production of industry i: Axj^ = dy Aej0); so, to the total, industry i has to produce: Axj1* = dy Aej1*. Then, the additional production of industry i generates the need of intermediate goods, that is for commodity /: Aq]l) = Am^ = a" , Ax)!). To the total, the need for the intermediate commodity / is: A#; (1) = £"=1 au a . And the circuit is closed and starts again with this demand Aqf* of commodity I. At step k, one has: = X " i dy Aef~l), that is Ax(i) = D Ae(*-1), and Aqf} = Z"=i au a Axf], that is 1 This works even for rectangular matrices. However, most formulae of the model as exposed in Miller and Blair (1985) require that the number of commodity is equal to the number of industries: matrices Make and Use must be square, what is much restricting.

Aq(i) = A" Axw . Finally, Aq(i) = A" D Ae(k !) and equation ( 9) is retrieved. Obviously, the process could begin by a demand to an industry instead of a demand of a commodity.

With the commodity-based technology, industries demand commodities -as above -by the technical coefficients, but these commodities are assumed to be produced by industries following the industry output proportions, cy. If we want to translate it in terms of circuit, the process could begin by a final demand of commodity j: A e]0) -» A qf* = Aej0). Unlike D, the C matrix does not indicate what industry will produce this commodity; but if it is a particular A (0) industry i, arbitrary chosen, that produces this commodity j, that is Ax, = --, then i will u have to increase its production of a ll the other commodities I that it produces usually:

= cu Axj1*. All this is highly unrealistic! If the process begins by the demand Axf* to an industry z, then this generates a demand of intermediate commodities j by the mean of the technical coefficients ajj: Aqf* = a" Ax f*; but after this, the above problem of allocation toward the industries occurs again.

Alternately, this could mean that an industry i, that produces commodities following coefficients cy, also demands intermediate commodities following the technical coefficients.

So, a demand Ax-0) to an industry i implies that this industry increases its production of a ll commodities following Aq~p = cy Ax-0) and, simultaneously, it demands some intermediate commodities following the technical coefficients: Aqf^ = a\ Ax f*. But this is not a circular process because after this demand of intermediate commodities, there is no continuation.

The difficulty met with both interpretations is linked to the fact that one has to inverse matrix C to deduce the commodity-by-commodity balance accounting identity with the commodity-based technology: q = A" C"1 q + e (formula 5-18 of Miller and Blair (1985, pp. 169), what requires again all matrices to be square. In any case, this behavior corresponds to a push-process, similar to those assumed by Ghosh with its hypothesis of a supply-driven model in the traditional input-output economics. It appears to be contradictory with the existence of technical coefficients in the Use matrix, i.e., a demand-driven hypothesis. However, it is possible to reverse the perspective. Replace the technical coefficient matrix A" by a matrix of allocation coefficients, by = ~, that is:

(10)

B " = q-' U From ( 6), it follows V = x C, what reported in (4) gives:

(11) q = C x From (10), it comes q B " = U and by reporting it in (2) it is obtained x = Bu/ q s + w = Bu/ q + w; so, the equation of the model is obtained by reporting (11) in this last equation:

(12) x = B"/ C/ x + w Now, the push-process is set in complete conformity with a supply-driven model and it is circular. A supply of a commodity j generates an output of all industries as indicated by Bu, in the Ghosh way, then the industries sell commodities following the proportions indicated by the coefficients cy. The initial increase of the value added v, of an industry i generates an equal increase in the output of this industry: Ax-0) = Av-0). By the matrix C, this generates an increase in the supply of all commodities: Ag^ = cy Axf), that is, to the total the increase in the supply of commodity j is: Aq f = 2 " =1 cy Axj0). This supplementary supply of a commodity j induces an increase of the output of all industries I following Bu:

A q f -> Ax;1} = b" i A q f , so, to the total, the industry I increases its output of Ax/(1) = X ;, bj! A qf*. At step k, , one has: A qf* = X " =1 cy Axf-1) and Axf* = X " i b", A q f , that is Aqw = C' Ax(*_1) and Axw = Bu/ Aq®, so Ax(*> = Bu/ C' Ax(i_1). This is in conformity with the corresponding model ( 12): the supply-driven commodity-based-technology model is consistent2.

IV. Conclusion

In the Stone make-use model, the industry-based technology is consistent because his solution can be explained in variational terms inside a circuit. However, the alternative model, the commodity-based technology, is not economically realistic: it never corresponds to a circuit, even if an exact solution can be found when the number of commodities is equal to the number of industries. This model hesitates between a supply-driven and a demand-driven model but when it is converted into a true supply-driven one, it retrieves a consistency. All this is irrespective of the discussion about the artificial character of a supply-driven model: it is just to demonstrate that the commodity-based technology is inconsistent, hesitating between a supply-driven and a demand-driven-model.

V. Bibliographical references

Review of Economics and Statistics 18

  LEONTIEF, Wassily 1936. "Quantitative Input-Output Relations in the Economic System of the United States," , 3, 105-125. MILLER, Ronald E. and Peter D. BLAIR 1985. Input-output analysis: foundations and extensions. Prentice-Hall, Englewood Cliffs, New-Jersey.

STONE, Richard 1961. Input-output and national accounts. Organisation for European Economic Cooperation, Paris.

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