I thank Jerry for highlighting and responding to my post about moral
depreciation and the Okishio theorem.
Responses to his questions: no, this wasn't an "ope-l original," but is
based on Andrew Kliman, "The Profit Rate Under Continuous Technological
Change," RRPE 20:2-3 (if I remember), dated 1988; and Andrew Kliman, "A
Value-theoretic Critique of the Okishio Theorem," in _Marx and Non-
Equilibrium Economics, edited by the illustrious ope-l members Alan Freeman
and Guglielmo Carchedi. JUst published (1995) by Edward Elgar.
The results--the refutation of Okishio--does indeed hold up under the
other assumptions Jerry noted. Simultaneous production is not necessary,
and in the above papers, in fact, production takes time (I thus use
difference equations and not differential equations). Physically
non-depreciating fixed capital isn't necessary, because fixed capital
itself isn't necessary--John Ernst, "Simultaneous Valuation Extirpated,"
RRPE 14:2 (if I remember), 1982, was the first to show this. If input
prices exceed output prices, then, even when Okishio's profit rate--
in which the input and output prices are equated (the capital is
retroactively revalued, without capital losses being charged!) rises, the
profit rate based on actual prices can fall.
Nor do things change in a multisector model, substantially. Here's
another toy model. Assume extraction of living labor is constant in
each period. Assume workers live on air (note that the real wage is
constant!). Assume no material inputs are used anywhere except
physically nondepreciating fixed capital. Thus, total value equals
surplus-value.
Now, on the basis of Marx's theory, under these assumptions, total value
in period t is TV(t) = L, where L is living labor. And Marx shows, based
on his value theory, that total value equals total price; hence, TP(t) = L
(assuming no change in the monetary expression of value).
Now, assume that the capitalists reinvest all surplus-value. Then the
value (= price) of the capital stock, K, is given as follows
K(t+1) = K(t) + L.
The solution to this equation is K(t) = Ko + t x L.
The profit rate is
r(t) = TP(t)/K(t) = L/(Ko + t x L) = 1/(Ko/L + t).
The profit rate continuously falls and, as t approaches infinity, r
approaches zero from above.
The above is not a one-sector model, but is applicable to any number of
sectors. K, TV, and TP are aggregates for the whole economy. Note also
that no particular technological assumptions are made except for the
constancy of L. The whole thing really relies only on (a) the determination
of value by labor-time and (b) Marx's demonstration that TP = TV.
As to Jerry's last question--does the lack of response indicate that
everyone else on this list agrees that the Okishio theorem is wrong [for
the reasons I've adumbrated]?--I don't know. But of course I posted my
post to find out; I too would like to know.
Ciao--Andrew