# A mathematical study of the Sraffa model.

 Autor Hall, Luis J. Cargo Texto en ingl Páginas 77(12)

INTRODUCTION

It is unknown whether Piero Sraffa was aware of the Perron-Frobenius's work (1907 and 1912) when he started to work on his book "Production of Commodities by Means of Comodities " in the late forties. Nevertheless, his book reflects a clear demonstration of the existence of a maximum positive eigenvalue associated with a unique, up to a scalar, positive eigenvector when working with an irreducible non-negative matrix.

It is through this mathematical result that the concept of the standard commodity developed by Sraffa acquires full generality; the Sraffian-Price system PA(1+r) + [a.sub.n]l = P can use as its numeraire r = [rho](A)(1-w), a numeraire developed by using the standard product. Sraffa employed this result to solve the problem posed by Ricardo  of finding a commodity whose capital and labor proportions were such that those price fluctuations created by changes in distribution, say changes in the rate of profit or wages, did not affect this price. Employing the price of this composite commodity as numeraire, one could generalize Ricardo's result of the corn model and establish a linear relation between the real wage and the rate of profit.

One must add from the start that this standard commodity does not imply that fluctuations of prices are not obtained from changes in distribution as Ricardo had already recognized; rather, the standard commodity is a clever constructed numeraire which does not fluctuate to changes in distribution so that one can understand and measure fluctuations in prices obtained from changes in distribution.

Furthermore, it brings enough criticism by itself on any definition of the term "capital". It is immediate that any fluctuation in distribution brings about movements in relative prices. Then, defining capital as a bundle of commodities whose value is determined by supply and demand becomes circular and indeterminate. The very idea of a demand curve for a factor of production implies that changes in distributional rates occurs, the vertical axis, and hence fluctuations in the value of those defined "capital goods", the horizontal axis, changes in an indeterminate manner.

This paper has two primary objects. on the one hand, it presents a rigorous derivation of the Sraffa model recurring to some very abstract mathematical theorems discovered by Frobenius and Perron at the beginning of the century. This presentation is used to exploit the concept of productive economies or surplus economies, and also to provide a clear derivation of the standard commodity. The second goal of the paper is to sketch von Neumann's balanced growth model and show how Sraffa's model is related to Neumann's model. By studying the set of assumptions underlying both models, one can study the relationship between both models and its implications.

The paper is organized as follows. First it introduces the theorems of Frobenius and Perron and analyze subsistence economies. Next it introduces productive economies and study the solution of these economies and the structure of these systems proposed by Sraffa. Next, the concept of the Standard Commodity is derived and analyzed. The paper follows with the sketch of the model of von Neumann and its properties. Then the relation of Neumann's model to the model of Sraffa is studied. Conclusion remarks follows.

1. THE STANDARD COMMODITY IN THE MODEL OF SRAFFA

First, consider the fundamental theorems of Frobenius on nonnegative matrices which extend the results obtained by Perron for positive matrices

Theorem 1

2. (Frobenius non-negative) Let A [mayor que o igual a] 0 be an nXn matrix then [rho](A) [mayor que o igual a] 0 is an eigenvalue of A and there is a nonnegative vector x [mayor que o igual a] 0; x [desigual a] 0, such that Ax = [rho](A)x. (2)

3. A irreducible iff [(l + A).sup.n-1] > 0

4. (Frobenius nonnegative and irreducible) If A [mayor que o igual a] 0 and irreducible then [rho](A) is a real, multiplicity one eigenvalue. The left and right eigenvectors are positive and unique up to a scalar. Moreover p > 0 (Frobenius) and for any other [lambda] [menor que o igual a] [rho](A),[p.sub.[lambda],i]

For a proof see Horn  and Pasinetti . (3)

Let represent the technical coefficient of the economy. We avoid any discussion on how these coefficients are determined a fact that has actually determined different schools in economic thought. For instance, one can have some so-called Neo-Ricardians who tend to find on a kind of technology as such. In a line close to some thinkers of the "second international socialist", these authors deliberately emphasize on the technical side and reduce the social configuration of such structure to its minimum. Moreover, it is a logical implication from this attitude their recognized neglecting of the Labor Theory of Value. For our present purpose, is an non-negative matrix for a single product and denominated "an economy."

2.0.1 Subsistence system and prices

Consider first the case denominated by Sraffa "Production for Subsistence." Sraffa proposes to find out a set of exchange values that allow economic agents to exchange all quantities in such a way that production can take place next period and there is no surplus created after production. This exchange values, he adds, spring directly from the methods of production. In matrix notation he argues that pA = p.

For pA = p = [lambda]p to hold, one argues that [lambda](A)=1 where [lambda](A) is an eigenvalue of A. There are some attributes associated to this type of economy that deserves to be emphasized and will lead us into the role of basic and non-basic commodities.

We just assume that A is non-negative. one interesting issue here is to see what other restrictions are required to generate economic meaning for this type of model. From theorem (1), if A [mayor que o igual a] 0 then one knows that [rho](A) [mayor que o igual a] 0 and the price vector associated is p [mayor que o igual a] 0. The spectral radius is an eigenvalue not unique and hence the eigenvector associated is not unique either.

The economy is denominated a reducible economy in the sense that not all its sectors are somehow connected. Mathematically speaking, this means that there is a permutation matrix P which allows one to re-write A as

[EXPRESIÓN MATEMÁTICA IRREPRODUCIBLE EN ASCII]

Matrix B and D are square and irreducible matrices (4). Furthermore, this normal form of matrix A is unique up to a permutation of the diagonal (Gantmacher ).

Let us spell out these properties (5). First, as claimed before, [rho](A) = p has a positive solution for the eigenvalue [lambda] = p(A). Note that we work with economies which are only nonnegative but we restrict this set of economies to only those with [rho](A) [menor que o igual a] 1. (6)

The case for [rho](A) = 0 is equivalent, not with a A = 0, but rather with [A.sup.m] = 0 for some m integer. Then, although the economy could have initially some technical positive values, they will eventually become null. These eventually null economies shall be disregarded as having no economic meaning.

Second, the vector of prices can have zero elements. I return to this point below.

Third, one can write the A matrix on its normal form. In economic terms, this means that we can partition the economy in groups. Matrix B and D would be square matrix that are actually irreducible ones and represent different subsectors of the economy. Moreover, if C contains some nonzero elements, then one has that D is related to B but not viceversa. Sraffa denominated subsector B of A as a sector composed of basic commodities and the rest sectors as nonbasic commodities. It is direct that nonbasic commodities depend upon basic ones for its production and since this partition is unique, one has that it is not possible to modify this partition of commodities.

Dividing the vector of prices for each group, one gets

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One has that prices for sector B do not depend upon the rest of the economy-prices while viceversa is not the case. Additionally, it can easily be shown that the rate of expansion of the economy or in mathematical terms the maximum eigenvalue of economy A is the same that the maximum eigenvalue for B. Thus, permutations within the diagonal are always possible, the partition is unique and [rho](A) = [rho](B). Sraffa employed these important properties to construct the standard commodity as shown below. This last property let Sraffa take subsystem B instead of A which has a positive Frobenius root and a positive left and right eigenvector associated to work out the Standard Commodity.

Fourth, there is the fundamental issue on prices which must be closely studied when working with a nonnegative matrix. The nonnegative economy allows to have a decomposable system that can be written as the equation above and hence allows for the very important notion of basics and nonbasics. This main notion lets one distinguish between some sectors of the economy that are totally independent (7) from those dependent ones. This distinction is a very important one for...

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