John Roemer’s Reproducible Solution

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Robert Vienneau
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Can I adapt Roemer's work, suitably taking into account later work by D'Agata and Zambelli, to found this approach to markup pricing? As a start, I here quote Roemer on a reproducible solution (RS), before he takes into account unequal rates of profits and a choice of technique. Given the role of endowments, is this a neoclassical approach, like Hahn's 1984 CJE paper? Even so, is it a valid justification for Sraffa's price equations? Notice there are no subscripts for time below.

"There are N capitalists; the νth one is endowed with a vector of produced commodity endowments ω^ν ... Capitalist ν starts with capital ω^ν, which he seeks to turn in more wealth at the highest rate of return. Thus the program of capitalist ν is

Facing prices p, to

choose x^ν ≥ 0 to

max (p - (p A + L)) x^ν

s.t. (p A + L) x^ν ≤ p ω^ν

(The constraint says that the inputs costs can be covered by current capital.) Let us call A^ν(p) the set of solution vectors to this program." -- Roemer (1981: 18-19, I made changes for typesetting mathematics).

Roemer defines a RS:

"Definition 1.1: A price vector p is a reproducible solution for the economy {A, L; b; ω¹, ..., ω^N} if:

• For all ν, there exists x^ν in A^ν(p), such that (profit maximization)
• x = Σ x^ν and x ≥ A x + (L x) b (reproducibility)
• p b = 1 (subsistence wage)
• A x + (L x) ≤ ω = Σ ω^ν (feasibility)

We shall also refer to the entire set {p, x¹, ..., x^N} as a reproducible solution." -- Roemer (1981: 19-20, with for math).

A RS can only exist if the elements of the endowment vector are in certain proportions:

"Theorem 1.2: Let the model {A, L, b} be given with A productive and indecomposable, and the rate of exploitation e > 0. Let {p, x¹, ..., x^N} be a nontrivial RS. (i.e., Σ x^ν = x ≠ 0). Then the vector of prices p is the E[qual] P[rofit] R[ate] vector p^*. Furthermore, a RS exists if and only if omega is an element of C^*, where C^* is a particular convex cone in [the space of n-dimensional real vectors] containing the balanced growth path of {A, L, b}. (C^* is specified precisely below.)" -- Roemer (1981: 20, with changes for math).

Even though endoments are taken as given in defining the firm's LP, endowments are endogenous in the sense that they must lie close to those on a balanced growth path. I like to have labor advanced and wages paid out of the surplus, instead of vice versa as above. The above does not allow for a choice of technique. Roemer has at least some of this in later chapters.

• John E. Roemer. 1981. Analytical Foundations of Marxian Economic Theory. Cambridge University Press.