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John Roemer’s Reproducible Solution

Summary:
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 ων ... Capitalist ν starts with capital ων, 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 xν ≥ 0 to max (p - (p A + L)) xν

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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 ων ... Capitalist ν starts with capital ων, 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 xν0 to
max (p - (p A + L)) xν
s.t. (p A + L) xνp ων
(The constraint says that the inputs costs can be covered by current capital.) Let us call Aν(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, ..., ωN} if:
  • For all ν, there exists xν in Aν(p), such that (profit maximization)
  • x = Σ xν and xA x + (L x) b (reproducibility)
  • p b = 1 (subsistence wage)
  • A x + (L x) ≤ ω = Σ ων (feasibility)
We shall also refer to the entire set {p, x1, ..., xN} 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, ..., xN} be a nontrivial RS. (i.e., Σ xν = x0). 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.

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

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