I am not sure about the economic logic in this post. Maybe somebody like D'Agata or Zambelli could do something with this. These ideas were suggested to me by email with a sometime commentator.

I start out with notation for Sraffa's price system, modified in an unusual way to allow for persistent variations in the rate of profits among industries:

**a**_{0}is a row vector of labor coefficients in each of*n*industries.**A**is a Leontief input-output matrix, where*a*_{i, j}is the quantity of the*i*th commodity needed as input to produce one unit of the*j*th commodity.**S**is a diagonal matrix, where all off-diagonal elements are zero.*s*_{j, j}is the markup on non-labor costs in the*j*th industry.**p**is a row vector of prices.*w*is the wage.*r*is the scale factor for the rate of profits.

The coefficients of production, as expressed in the labor coefficients and the Leontief matrix are given parameters. Relative markups are also taken as given. Prices, the wage, and the scale factor for the rate of profits are the unknowns to be determined. My problem is to find a numeraire such that the wage and the scale factor for the rate of profits trade off in a straight-line relationship, at least when labor is advanced and wages are paid out of the net product:

r=R(1 -w)

I assume all elements of **A** are non-negative and that all elements of **a**_{0}
and all diagonal elements of **S** are positive. The economy is assumed to be viable,
that is, as capable of producing a surplus product. For simplicity, assume that the Leontief
matrix is indecomposable. More generally, I need **A** **S** to be a Sraffa matrix.

For my purposes here, I formulate price equations as so:

pAS(1 +r) +a_{0}w=p

Consider the case when wages are zero and the scale factor for the rate of profits is at its
maximum *R*:

pAS(1 +R) =p

Or:

pAS= (1/(1 +R))p

I observe that prices are a left-hand eigenvector of the matrix **A** **S**,
with (1/(1 + *R*)) the corresponding eigenvalue. To ensure that prices are
positive, of the *n* eigenvalues, choose the maximum. The maximum eigenvalue
is also known as the Perron-Frobenius root of **A** **S**.

Let **y ^{*}** be a right-hand eigenvector of

**A**

**S**corresponding to its Perron-Frobenius root. Let

**q**be gross output such that the net output is

^{*}**y**:

^{*}y=^{*}q-^{*}Aq^{*}

These quantities flow define the standard system here, when scaled so as employ a unit quantity of labor:

a_{0}q= 1^{*}

The net output of the standard system is the desired numeraire:

py= 1^{*}

With this definition of the standard system, the ratio of physical
gross outputs to circulating capital inputs varies among commodities.
This result contrasts with Sraffa's standard system. I suppose
I could restore this property by choosing **q ^{*}**,
not

**y**, to be an eigenvector. Either way, the ratio of net outputs to circulating capital inputs varies among industries. Either way, the relative ratios of commodities in the standard industry depends on relative markups.

^{*}Do Marx's invariants hold with the above definition of the standard system? I expect not. Nevertheless, does this mathematics provide some insight into classical or Marxist political economy?

## 2 comments:

I was just thinking about a similar problem over the weekend (my puzzle: what if we allowed each industry to have separate rates of profit).

Out of curiosity, when you write that you "need

A Sto be a Sraffa matrix", what is the definition of a "Sraffa matrix"? I assume it's related to your notes on irreducible matrices?Your approach is interesting, but my intuition tells me there may be some "projective" (as in projective geometry) concerns: what's preventing me from considering some constant multiple

S= λS'for λ positive, and (1 +R')/λ = (1 +R) so thatS'(1 +R') =S(1 +R), we have some unconstrained degree of freedom, don't we? Maybe there's a trick here I'm missing, or I'm thinking about things the wrong way...I've decided that this post is misdirected, though it will take me weeks to develop why.

A Sraffa matrix is defined here: Kurz and Salvadori, 1995: 123-124. The definition does relate to my notes on irreducible matrices and to the Perron-Frobenius theorems. Consider a Leontief input-output matrix. Sraffa assumes that there exists at least one basic commodity, that is, a commodity that enters directory or indirectly into the production of all commodities. He assumes that the economy is at least viable, that is, that the capital goods used up in producing gross output can be reproduced. If the economy is more than viable, some positive net output is left over. He also rules out non-basic goods like the beans in the appendix in his book. Some non-basic goods, like, say, ostrich eggs enter into their own production. One needs the maximum rate of profits for non-basics, so to speak, to not exceed the maximum rate of profits for the submatrix formed only out the basic commodities.

I agree that I need some sort of normalization condition there.

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