# [OPE-L:1918] [GIL] Re: urgent request

glevy@acnet.pratt.edu (glevy@acnet.pratt.edu)
Tue, 23 Apr 1996 15:29:48 -0700

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---------- Forwarded message ----------
Date: Tue, 23 Apr 1996 17:58:26 EST
From: Gilbert Skillman <gskillman@wesleyan.edu>
To: ope-l@anthrax.ecst.csuchico.edu
Subject: Re: [OPE-L:1914] urgent request

> Would someone please send me quickly the basic price and profit rate
> equations that Roemer uses to "generalize" the Okishio Theorem to fixed
> capital for his two special cases:
>
> 1. non-depreciating fixed capital
>
> 2. the von-Neuman rate of profit with depreciating fixed capital
>
> In other words: what replaces the basic equation for circulating capital only?
>
> p = (pA + pbL)(1 + r)

For the non-depreciating fixed capital model:

p = rpT + (1+r)(pA + L) (Roemer uses pi for r and theta for T),

where pA + L represents "cost of materials used", r(pA +L) represents
"markup on materials used", and rpT represents "markup on fixed
capital used" (T is "the nXn matrix of fixed capital coefficients"),
and assuming wage normalization 1=pb, where b is a vector of
subsistence wage goods.

For the Von Neumann model with depreciating capital: Define B as an
n x m matrix of output coefficients, A as an
n x m matrix of input coefficients, L as an m-row vector of direct
labor inputs, b as an n-column vector of subsistence wage goods, M =
A + bL, and T is defined as before.

Roemer writes: "An equilibrium price vector and profit rate (p, r)
must satisfy pB _<_ (1+r)pM. We shall call the von Neumann profit
rate the minimum r for which there exists a price vector p _>_ 0
satisfying this inequality.....

..Now, by the Frobenius-Perron theorem, it is known that the minimum
r for which there exists a nonnegative vector p such that (the
foregoing inequality) holds is, in fact, the value r such that

p = p[M + r(M + T)].

If the matrix M is indecomposable, then the value r is unique, as is
the price vector p, and in fact (p, r) is the equilibrium discussed
in (the section on fixed nondepreciating capital)."

> I need this for my class at UNAM tomorrow (Wednesday!) and don't have the
> Roemer articles with me, so a prompt response would be very much
> appreciated. If you want, you can send to me directly: fmoseley@laneta.apc.org

Voila.

In solidarity, Gil