**1.0 Introduction**

This post outlines an unoriginal argument against Marx's version of the Labor Theory of Value (LTV), if that is the right name. Somehow, this post was obliquely inspired by Fabio Petri's recent working paper.

Suppose technology, net output, and the real wage are given. Then the rate of profits and prices of production are determined.

Suppose that the technology provides a choice of technique. Then the determination of the choice of technique requires an analysis at the level of prices of production. One can do labor value accounting only after the determination of the technique in use.

Given the technique and the level of operation of each process, one can then determine the labor value embodied in the output of each industry. One can then use this labor value accounting in the overall system of labor values to calculate a supposed rate of profits. In general, this rate of profits is unequal to the rate of profits in the system of prices of production.

One can also use the technique in use to identify a commodity of an average organic composition of capital, in some sense. And one can calculate the rate of profits in the system of labor values for the industry producing this average commodity. And all of Marx's other volume 3 invariants hold in the production of Sraffa's standard commodity.

But the composition of the standard commodity also varies with the technique in use and, thus, depends on the real wage and an analysis at the level of prices of production.

Supply and demand, as conceptualized in marginalist economic theory, however, remains nonsense, not even wrong.

**2.0 Givens**

Suppose a capitalist economy is observed at a given point in time. The net output of the economy
consists of a column vector **d**, in which each element is measured in physical units (kilotons, bushels, etc.)

Suppose the capitalists know of the processes comprising two techniques, Alpha and Beta, for producing
the given net output. Each technique is characterized by a row vector of direct labor
coefficients, **a**_{0}(α) and **a**_{0}(β)
and a Leontief input-output matrix, **A**(α) and **A**(β).
These vectors and matricies are given in physical units.

I assume constant returns to scale and, here, that all advanced capital is circulating capital. Both techniques must be able to produce the given net output, but different intermediate commodities may be produced. The economy must hang together, in some sense. That is at least one basic commodity exists in each technique, although the which commodities are basic may vary with the technique. Also, nothing like Sraffa's 'beans', in Appendix B of his book, exists in either technique.

One could articulate these assumptions more rigorously.

**3.0 Prices of Production**

Prices of production, for a competitive capitalist economy, are such that the same rate of (accounting) profits is obtained in each operated process. For the Alpha technique, prices of production must satisfy the following system of equations:

p(w, α)A(α) (1 +r(w, α)) +wa_{0}(α) =p(w, α)

If one takes net output as the numeraire, prices of production must be such that:

p(w, α)d= 1

Prices of production and the rate of profits can be found for a non-negative wage up to a certain maximum. Prices of production and the corresponding rate of profits can be found for each technique.

**4.0 Choice of Technique**

The determination of prices of production for each technique, at the given wage, allows for an analysis of the choice of technique.

If Alpha is the cost-minimizing technique at the wage *w*, supernormal profits
cannot be obtained by operating Beta.
The cost of operating each process in the Beta technique at Alpha prices cannot fall
below the revenue obtained.
The following must hold:

p(w, α)A(β) (1 +r(w, α)) +wa_{0}(β) ≥p(w, α)

If the wage is not that at a switch point, a strict inequality must hold for at least one process in Beta. I suppose that prices of commodities only produced in Beta are zero for the above display and that prices for commodities only produced under Alpha do not appear in the price vector.

For the case of circulating capital, the above is equivalent to the rate of profits for Alpha exceeding the rate of profits for Beta:

r(w, α) ≥r(w, β)

If Beta is the cost-minimizing technique, these conditions are reversed. In any case, which technique is cost-minimizing may vary with the wage.

**5.0 Labor Values**

Let **a**_{0}(*w*) be the row vector of direct labor coefficients
and **A**(*w*) be the Leontief input-output matrix of the cost-minimizing
technique at the wage *w*.

The labor embodied in each commodity produced for the cost-minimizing technique is found from the vector of direct labor coefficients and the Leontief inverse:

v=a_{0}(w) (I-A(w)^{-1}

One could go on to perform further calculations, including with the dominant eigenvalue
and corresponding eigenvector for the input-output matrix **A**(*w*).
As this notation emphasizes, the data for labor value accounting depends on the wage.
The data also depend on the above analysis of prices of production.

**6.0 Conclusion**

One could respond to this argument by asserting that typically a single technique is dominant for any distribution of income. New techiques come about by innovation, replacing existing techniques.

This rebuttal does not work for, at least, some aspects of joint production. Extensive rent or differential rent of the first kind, for example, requires an analysis to identify which type of land pays no rent and is nevertheless cultivated. It is only after such an analysis at the level of prices of production that labor values can be calculated.

Supply and demand curves have not been drawn above. Nor has anything been said about utility maximization. As Laplace told Napoleon in a different context,"I have no need of that hypothesis."

**References**

- Fabio Petri. 2024. What remains valid of the first chapter of Marx's
*Capital*?, Centro Sraffa Working Paper 65. - Paul A. Samuelson. 1989. Revisionist findings on Sraffa.
*Essays in Honour of Piero Sraffa*(ed. by K. Bharadwaj and B. Shefold).

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