# [OPE-L:648] Re: Forms of Tech. Change [digression]

akliman@acl.nyit.edu (akliman@acl.nyit.edu)
Mon, 4 Dec 1995 14:51:29 -0800

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Andrew here. John keeps making an excellent point about Marx's law of the
falling rate of profit not needing to presume that the "capital/output"
ratio rises in physical terms. Eveyone who has looked at the theory through
the lenses of static equilibrium notions of value has come to the opposite
conclusion--that the output/output ratio must rise or a fall in the profit
rate isn't possible. But the problem is, the static equilibrium view
sweeps moral depreciation under the rug. It assumes that old fixed
capital is RETROACTIVELY revalued each moment so that it has the same
price as new output. In the Okishio theorem, the capital losses are not
even charged against profits--and this is THE reason, T H E reason, Okishio
purportedly "refuted" Marx's law. As if an accountant's profit rate
rises because he buys a computer for \$6000 that's now worth only \$1200.
No amount of babbling about the cheapening of the elements of constant
capital changes the fact that \$6000 was sunk.

Once one understands the moral depreciation factor, it is simple to show
that the rate of profit can fall even if the "capital/output" ratio FALLS.
Here's how you can do it at home in your spare time:

Assume a single sector economy, producing output (X) by means of living
labor (L) and physically nondepreciating fixed capital (F) only. Assume
also that production is *instantaneous*, so that the input and output
price are the same. Now, let the physical quantities grow as follows:

X(t) = Xo*exp(at)

L(t) = Lo*exp(bt)

F(t) = Fo*(c/a)*exp(at) + Fo*(1- c/a).

Now, assume that a > c > b. Thus, X/L and F/L rise, but F/X, the "capital/
output ratio" falls.

Now, the determination of value by labor-time, given no quantitative
price/value deviations (because a single sector has been assumed), implies
that

P(t) = V(t) = L(t)/X(t).

Now, at any moment, investment (in value/price terms) is

P(t)*(dF/dt) = (Lo*Fo/Xo)*c*exp(bt)

and, integrating this, you'll get the value/price of the capital stock:

K(t) = (Lo*Fo/Xo)*(c/b)*exp(bt) + (Lo*Fo/Xo)*(1 - c/b),

(because Ko = (Lo*Fo/Xo)).

Note that K(t) exceeds the "replacement cost" of the capital stock, i.e.,

P(t)*F(t) = (Lo*Fo/Xo)*(c/a)*exp(bt) + (Lo*Fo/Xo)*(1 - c/a)*exp([b-a]t)

for all t > 0, because a > b. Economically, a > b implies that the
unit value = price is falling, due to rising labor productivity, and
because almost all of the capital stock was acquired when values were higher
than they are "now," the historical cost of capital exceeds its replacement
cost.

Now for the really fun part. Plug all of this into the profit rate
formula:

r(t) = P(t)X(t) - P(t)wL(t)
____________________
K(t) + P(t)wL(t)

where w is the real wage (constant if you want to refute Okishio, Roemer, et
al.), and it is certainly possible to find cases where r falls. Try, e.g.,
Xo = 1, Lo = 1, w = 1/7, Fo = 2, a = .04, b = .01, and c = .02. The profit
rate declines continuously, from an inital value of 40% to a limit of 25%.
The "capital/output" ratio falls from an initial value of 2 to a limit of
1.

Now, the static equilibrium profit rate used in the Okishio theorem is
just like that above, except that the "replacement cost" of the fixed
capital is used instead of K(t), the actual value invested. Using the
above data, Okishio's profit rate rises from 40% to a limit of 100% as
time approaches infinity.

Have fun refuting the experts!

Andrew