NOTE ON THE CONSTRUCTION OF SRAFFA'S STANDARD SYSTEM
Emmenegger J.-F., Chable D.
Emmenegger J.-F., Chable D. Fribourg, Switzerland Jean-Francois.Emmenegger@unifr.ch NOTE
ON THE CONSTRUCTION OF SRAFFA'S STANDARD SYSTEM
Sraffa‘s seminal book Production of Commodities by means of Commodities (PCMC, 1960) treats a circularproduction economy. He considers n different industries, each industry producing exactly one commodity. This gives rise to Input-Output tables, whose kernel consists in a generally non-negative and irreducible (n x n)-matrix S, describing the flow of the commodities. Each commodity occupies one row and each industry occupies one columnin matrix S=(sij), i,j=1,…,n. The entries sijgive the quantity of commodity i needed to produce commodity j. Sraffa proposes a constant profits rate r for the entrepreneurs and a constant wage rate w for the workers. He then takes as currency one of 57 the produced commodities, called numéraire. He proposes a price model to calculate the prices pi of each of the n commodities i, expressed by that numéraire,collected in a vector p=[p1,…, pn]‘. How have prices of commodities, the constant profits rate r and the constant wage rate w to behave, in order to guarantee a reproduction with given surplusd0?Technically, consider the vector of ones e=[1,…,1]‘, calculate the vector of total output q= Se +d,diagonalize it to get the diagonal matrix , the elementsqi in the diagonal.Given a labor vector L=[L1,…,Ln]‘ and the wage rate w, calculate the inputcoefficients matrix C=S -1 , its Frobenius number лC=1/(1+R)<1with standard ratio R. The rate of profits r is chosen in the interval [0,R],0≤r≤R. Sraffa proposes theprice model C’p(1+r) + w -1 L= p.(1) In PCMC (Par. 23 and 24), Sraffaasks the question: "The necessity of having to express the price of one commodity in terms of another, which is arbitrarily chosen as numéraire, complicates the study of price movements which accompany a change in distribution between wages and profits. It is impossible to tell of any particular price fluctuation whether it arises from the peculiarities of the commodity which is being measured or from those of the measuring standard". Sraffa then continues (PCMC, Par. 24): "It is not likely that an individual commodity could be found which possesses even approximately the necessary requisites. A mixture of commodities, however, or a composite commodity, would do equally well;.....". Then Sraffa continues (PCMC, Par. 26): We shall call a mixture of this type the … Standard commodity; and the set of equations (or industries), taken in the proportions that produce the Standard commodity, the Standard system.‖ Then comes the definition of the Standard system (PCMC, Par. 25): ―…the various commodities are represented among its aggregate means of production in the same proportions as they are among the products.‖ In this note, we present the mathematics of the Standard system: First, a commodity is basic, when it enters directly or indirectly into the production of all other commodities. Second, a Standard system is composed of a subsystem of k≤nbasic industries taken among the n industries. Thekcommoditiesproduced by this Standard system form together aStandard commodity. The required proportionality means that we have for the k-vectors , the parallelism q || q-d=Se || d. (2) We will also show that the proportionality factor is the standard ratioR, giving q=(1+R)Se . (3) We have to show how a standard system, designed by ,is constructed from a non-standard system . Sraffa only gives an arithmetic explanation for this construction. He multiples each line i, i=1,..,k, of the subsystem S, L, q, by multipliers i , composing the vector = [1 ,.., k]. We have identified this operation as an orthogonal Euler-map, operated with the diagonal matrix on the matrices S, L,q . (4) In this note we will also present a numerical example, how to calculate a standard system from a non-standard system. One shows also how standard systems answer Sraffa‘s question of price fluctuations for such a system.