[OPE-L:4800] Sraffa's Non-Proof

andrew kliman (Andrew_Kliman@msn.com)
Wed, 16 Apr 1997 10:59:43 -0700 (PDT)

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A response to one aspect of Ajit's ope-l 4677. I may have time to
address the rest soon.

I had written "Sraffa fails to prove that, in an economy with a
surplus, there is only one set of exchange ratios that permits simple
reproduction to take place together with equalized rates of return on
capital advanced. Nor does he prove that there is a unique uniform
profit rate in such a case. I demonstrated this in my paper 'The
Okishio Theorem: An Obituary,' which I know that Ajit knows, because
he was a presenter on the same panel at the ASSA in January at which
I presented it."

He responded: "I haven't read your paper. But your claim sounds

Here's what I wrote. Whether or not is it "outlandish," let us see
you -- or anyone else -- disprove it.

* * * * * * * * *

Yet, contrary to the apparent belief of the proponents of the Okishio
theorem (and most readers of Dmitriev and Sraffa), it is simply not
true that the magnitude of the uniform profit rate is uniquely
determined by physical input/output coefficients. To demonstrate this
crucial point, let us turn to the example in which Sraffa (1960:7)
supposedly proves that the uniform profit rate is uniquely determined.
He posits the following input/output relations:

280qr. wheat + 12 t. iron --> 575 qr. wheat
120qr. wheat + 8 t. iron --> 20 t. iron

and immediately concludes:

"The exchange-ratio which enables the advances [on the left-hand sides]
to be replaced and the profits to be distributed to both industries in
proportion to their advances [so that the rate of return on capital
advanced is equalized] is 15 qr. of wheat for 1 t. of iron; and the
corresponding rate of profits in each industry is 25%."

To obtain 15:1 as the unique exchange ratio, Sraffa implicitly assumes
that the exchange ratio between wheat and iron at the time of input is
equal to the exchange ratio at the time of output. This will not
necessarily be the case, but even if it is, the uniform profit rate is
not uniquely determined by the data above, as we shall see presently.

Letting Pw[t] and Pi[t] be the input prices, and Pw[t+1] and Pi[t+1]
be the output prices, of wheat and iron, respectively, Sraffa's
exchange ratio implies that Pi[t] = 15 Pw[t] and that Pi[t+1] =
15 Pw[t+1]. The rate of return on capital advanced in the wheat
sector is therefore

(575 Pw[t+1])/(280 Pw[t] + 12 Pi[t]) - 1 =

(575 Pw[t+1])/(280 Pw[t] + 180 Pw[t]) - 1 =

1.25(Pw[t+1]/Pw[t]) - 1

and the rate of return on capital advanced in the iron sector is

(20 Pi[t+1])/(120 Pw[t] + 8 Pi[t]) - 1 =

(300 Pw[t+1])/(120 Pw[t] + 120 Pw[t]) - 1 =

1.25(Pw[t+1]/Pw[t]) - 1.

The constant exchange ratio of 15:1 thus guarantees that the two rates
of profit are equal. Yet the level of the equilibrium profit rate is
still not determined. It depends as well on the ratio of the output
to the input price of wheat, and it can vary, theoretically, from
-100% to + infinity.

* * * * * * * * *

Andrew Kliman (AX)