**Summary:**

1.0 Introduction One inspiration for this post is stumbling across this abstract 2.0 Marx Start with labor coefficients and a Leontief input/output matrix, in physical terms. You can construct this from make and use tables for your country, given price indices by sectors. For any existing capitalist economy, I expect that matrix to characterize a more than viable economy. After all the capital goods used up in producing the final demand in, say, a year are reproduced, some commodities will remain for consuming and investing. As with any other matrix, a set of n Eigenvalues and corresponding Eigenvectors can be found for that matrix. And, if I recall correctly, the maximum Eigenvalue has an associated Eigenvector in which all elements are non-negative. The components for Sraffa's

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Robert Vienneau considers the following as important: Example in Mathematical Economics, Karl Marx

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**1.0 Introduction**

One inspiration for this post is stumbling across this abstract

**2.0 Marx**

Start with labor coefficients and a Leontief input/output matrix, in physical terms. You can construct this from make and use tables for your country, given price indices by sectors.

For any existing capitalist economy, I expect that matrix to characterize a more than viable economy. After all the capital goods used up in producing the final demand in, say, a year are reproduced, some commodities will remain for consuming and investing.

As with any other matrix, a set of *n* Eigenvalues and corresponding Eigenvectors can be found for
that matrix. And, if I recall correctly, the maximum Eigenvalue has an associated Eigenvector in which
all elements are non-negative. The components for Sraffa's basic commodities are all strictly positive.
And they are in the proportions of Sraffa's standard commodity. If wages are zero
(the worker's live on air) and the final demand is all invested, the final demand will be in proprortions
of Sraffa's standard economy and the economy will expand at a rate of growth related to this eigenvalue.
One can read
von Neumann (1945) as describing
this model of uniform growth.

So given the technique in use in a capitalist economy, a composite commodity of average capital composition is picked out. Georg von Charasoff called this composite commodity 'Urkapital' which, I guess, is translated as "original capital". In some sense, this urkapital is a commodity of average organic composition of capital. This numeraire is picked out by the technique in use. In the production of this numeraire:

- The rate of profits, as calculated in the system of embodied labor values, is identically equal to the rate of profits, as calculated in the system of prices of production.
- Gross output, as evaluated at embodied labor values, is equal to gross outputs, as evaluated at prices of production.
- Net output, as evaluated at embodied labor values, is equal to net output, as evaluated at prices of production.
- The division of net output between workers and capitalist makes no difference to the above invariants.

So Volumes 1 and 3 of Marx's *Capital*
are
consistent.

**3.0 Marginalist Economics**

If one is ill-informed and malicious these days, one could insist that Sraffa's model is a special case of a neoclassical model of intertemporal equilibrium. Consumers maximize utility, and managers of firms maximize profits. Initial endowments just happen to be so that the economy expands along a steady state growth path.

**4.0 Conclusions**

But, of course, utility-maximization is nonsense, especially inter-temporally. Furthermore, the von Neumann ray has saddle-point instability. As Joan Robinson showed with her models of metallic ages, one need not assume that the workforce is fully employed in the long-run.

The model that has no need of restrictive assumptions is the more general.