[OPE-L:4428] "A Contribution ...," Part II

andrew klima (Andrew_Kliman@msn.com)
Tue, 18 Mar 1997 11:07:20 -0800 (PST)

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TSS Refutations of the Okishio Theorem

Professor Foley dismisses my claim to have refuted the Okishio theorem
because, when the living labor figures I take as data are "corrected" so that
they equal the VNP, "the resulting price and profit rate paths do not
contradict the Okishio theorem" (p. 28). In other words, he dismisses my
claim because he *disagrees* with the TSS interpretation of Marx's value
theory, according to which living labor and the VNP differ. This is hardly
the same thing as having identified an *error* that needs to be *corrected*.
Nor does Professor Foley disprove the authenticity of the TSS interpretation
or even provide any textual evidence or reasoned argument for his own claim
that Marx's concept of value added is identical to the VNP.

What we have, then, are not an error and a correction, but two opposed
interpretations. And the very fact that there does exist an interpretation of
Marx's value theory which succeeds in showing that viable labor-saving
innovation itself can lead to a fall in the general profit rate without a rise
in real wages, an interpretation that has not been shown to be either
internally inconsistent or contrary to the texts, is *itself * a refutation of
the claims that his law of the tendential fall in the profit rate has been
demonstrated to be logically incoherent. Its very existence also refutes the
claims that the Okishio theorem has *proven* that viable labor-saving
innovation at constant real wages cannot lead to a fall in the general profit
rate.

When one reads Marx in one way, one concludes that he made a mistake and that
the Okishio theorem shows it to be a mistake; when one reads Marx in a
different way, e.g., without presupposing that his concept of value added
equals the VNP, one concludes that the law of the falling profit rate flows
inexorably from his value theory and that the Okishio theorem has failed to
show that the opposite is the case. Thus, what were heretofore regarded as
*internal* contradictions within Marx's value theory have now been shown to be
*external* contradictions, contradictions between his actual texts and
interpretations of the texts that fail to make sense of them, and
contradictions between an interpretation that is able to make sense of the
texts and those that fail to do so.

The most peculiar aspect of the current controversy over Marx's value theory
is that those whose interpretation of Marx's value theory replicates his
theoretical results, are told by those whose interpretations fail to do so
that the former interpretation is contrary to Marx's value theory. It is as
if someone who could not succeed in making pudding were to tell someone who
had just done so that s/he was not reading the recipe correctly! Thus, when
it is claimed that the discrepancy between living labor and the VNP in my work
somehow suggests that I've failed to vindicate the logical coherence of the
law of the falling rate of profit, the most conclusive reason for rejecting
this claim is *precisely* that the identification of the VNP with living labor
leads to Okishio's results, against those of Marx, while the TSS
interpretation of value added leads to the refutation of the theorem and to a
vindication of the logical coherence of the law that Marx (1973, p. 748)
considered to be "in every respect the most important law in modern political
economy, and the most essential for understanding the most difficult
relations." The proof of the pudding is in the eating.

In any case, the authenticity of the TSS interpretation of Marx's value
theory simply has no bearing on whether we have *refuted the Okishio theorem*,
in the mathematical and logical sense of the term refutation. It seems that
Professor Foley has forgotten that, to refute the Okishio theorem, it is not
*necessary* that a counterexample conform to Marx's value theory. All that is
*necessary* is that one demonstrate that the theorem's conclusion fails to
follow necessarily from its explicitly stated premises, by, for instance,
producing a counterexample which satisfies all of the explicitly stated
premises of the theorem but arrives at a contrary conclusion.

In order to refute the Okishio theorem, therefore, it is irrelevant whether
the path of prices is that which Marx himself would have generated. One
merely needs to produce *some conceivable* set of prices and quantities that
is not ruled out by the theorem's explicitly stated premises, yet results in a
falling rate of profit. That I have surely done. Even if, for the sake of
argument, I were to say that my set of prices differs from that which Marx
would have generated, it is still a conceivable set of prices, and it yields a
falling rate of profit. Professor Foley himself acknowledges this fact: "the
monetary expression of labor in the New Interpretation sense is not constant,
and *the falling monetary rate of profit* in [Kliman's] examples reflects this
changing monetary expression of labor" (p. 28, emphasis added). No mention is
made of any discrepancy between the example and the explicitly stated premises
of the theorem, because there are none. **I have therefore refuted the
Okishio theorem, as Professor Foley's own statement implies**.

The paper I am presenting at this Convention, "The Okishio Theorem: An
Obituary" (Kliman 1997) spells out in considerable detail how TSS work has
refuted the Okishio theorem. Here I will merely adumbrate the main points.
Readers of Sraffa have often thought that the establishment of a uniform rate
of return on capital advanced requires that input and output prices be equal
(something which is not an explicitly stated premise of the theorem). This is
incorrect. Therefore, given a set of technical coefficients and a uniform
real wage rate, there is not just one, but an infinite number of possible
uniform profit rates. What the Okishio theorem actually shows is that *one* of
these profit rates cannot fall due to labor-saving technical change the
hypothetical profit rate which would exist were output prices to equal input
prices. Yet if output prices fall below input prices, the actual uniform
profit rate will be lower than that hypothetical rate, and if output prices
are low enough, the actual profit rate can be lower than the pre-innovation
rate. This is *sufficient* to refute the Okishio theorem.

For an example, note that in a single-sector economy with identical
producers, the profit rate is necessarily uniform. Imagine such an economy.
Assume that 3 bu. of corn are laid out for seed and 1 bu. for wages, and that
5 bu. of corn are produced. Assume that both the input and output prices of
corn equal 10. The profit rate thus equals 25%. Assume that in the next
period, the same outlays are made, and the labor extracted and thus the real
wage are the same as in period 1, but that 6 bu. of corn are produced. This
is an Okishio-viable technical change, since it reduces unit costs at the
current price of 10. But imagine that the output price of corn falls to 8.
The uniform profit rate has fallen to 20%. This is not a very exciting
refutation of the Okishio theorem, to be sure, but it is a *sufficient* one.


"Vintages" of Labor

In addition to the VNP argument, Professor Foley's paper presents a more
profound objection to the TSS interpretation of Marx's value theory. He
suggests (p. 27) that we do something wrong when we count an hour of labor
expended this year as equivalent to an hour of labor expended last year:

"the gross product contains means of production of various vintages, produced
under different technical conditions, and embodying labor of different
vintages and different productivities. Thus the labor embodied in the gross
product is under these circumstances [technical change] a *vector* of labor of
different vintages. The definition of the monetary expression of labor as the
ratio of a scalar (the contemporary value of the gross product at market
prices) to a vector (the labor embodied in the gross product) is incoherent
mathematically."

Here, Professor Foley seems to suggest that there is a connection between
distinguishing among labors by their "vintages" of labor and the different
productivities of these labors. He does not, however, say what the precise
connection is. In order to demonstrate that the vintage labor notion has the
effect of obliterating completely the antagonism between value and use-value,
I want to first establish the exact connection between different vintages of
labor and different productivities of labor.

For simplicity, I will begin with the case in which only a single commodity
exists, and then generalize the results. Let Kt and Lt indicate the amounts
of dead labor (embodied in means of production) and living labor,
respectively, that are needed to produce a one unit of the commodity in period
t, and let Tt be the total labor embodied. According to the vintage labor
notion, one cannot add Lt and Kt. The labor embodied in the means of
production is last period's labor, a unit of which, he contends, does not
yield the same amount of value as a unit of living labor. To commensurate
labors of different vintages, dead and living labor must be *weighted*
differently.

Letting Wt be the weighting factor for period t's labor, WtLt is the value
added and Wt-1Kt is the value transferred to the product of period t.
Likewise, I think Professor Foley would agree that, according to Marx, the
unit value of the commodity is

WtTt = Wt-1Kt + WtLt (1)

since this equation simply says that the unit value of the commodity is the
sum of constant capital and new value added by living labor.

Wt has the dimension (labor of vintage 0/labor of vintage t), so that the unit
value of the commodity has the dimension (labor of vintage 0/output). In
other words, labor of, say, vintage 0 is the "base year" labor, and the
weights convert labors of other periods into equivalents of base year labor.
[5]

At the end of period t, a certain fraction of output (Xt) becomes the means
of production of period t+1 (At+1). Clearly, in the single-commodity case,
and when labor-times are appropriately weighted, the value of the constant
capital of period t+1 (WtKt+1Xt+1) divided by the value of the output of
period t (WtTtXt) must be equal to the means of production of period t+1
divided by the output of period t:

(WtKt+1Xt+1)/(WtTtXt) = At+1/Xt. (2)

Now, as we know, Professor Foley accepts the Okishio theorem, and he thinks
that when labor-times are weighted correctly, Marx's value theory implies that
the labor-time value rate of profit is of the same magnitude as the profit
rate of the Okishio theorem. In the single commodity case, Okishio's maximum
rate of profit in period t is (xt - At)/At, which, defining at = At/xt, can be
written as (1 - at)/at. The maximum labor-time value rate of profit is the
ratio of the value added by living labor (WtLt) divided by the value of
constant capital (Wt-1Kt), so that

(WtLt)/(Wt-1Kt) = (1 - at)/at. (3)

>From (3), we see that (Wt-1Kt) = (WtLtat)/(1 - at), which, when inserted into
(1), yields

WtTt = (WtLt)/(1 - at) (1')

which is the unit value of the commodity. Substituting this result into (2)
yields

(WtKt+1Xt+1[1 - at])/(WtLtXt) = At+1/xt.

But since (WtKt+1) = (Wt+1Lt+1at+1)/(1 - at+1), and At+1/xt+1 = at+1,

Wt+1/Wt = [(1 - at+1)/Lt+1]/[(1 - at)/Lt,[6]

or, letting Ht = (1 - at)/Lt

Wt+1/Wt = Ht+1/Ht (4)

Ht = (1 - at)/Lt is the ratio of physical net product to living labor in
period t. It is a measure of the physical (use-value) productivity of labor
in that period. Hence, (4) demonstrates that the weights reflect differential
productivities alone. If, for instance, a unit of labor expended in one
period produces only half as much *use-value* as does a unit of labor in
another period, the first unit produces only half as much *value*. This
contradicts Marx's texts, as we shall see below, but it does demonstrate what
Professor Foley seems to imply, that his labors of different vintages are
simply labors of different productivities.

The "Vintage" Labor Notion Makes Simultaneism Consistent

The vintage labor notion has one marked advantage over other simultaneist
interpretations of Marx's value theory (that it is indeed a simultaneist
interpretation will be demonstrated presently). It is internally consistent,
whereas traditional simultaneist interpretations are internally inconsistent
in that they permit a commodity to have two different (social) values at one
and the same time, one as the output of one period, and another as the input
of the next period. This inconsistency leads immediately to other absurd
conclusions. For instance, they imply that there can be a constant, positive
profit rate throughout all time, and all profit can be reinvested, yet the
rate of accumulation of value and the rate of growth of the value of output
are both always zero. The vintage labor notion avoids these logical problems.

Consider the following very unrealistic example, which, however, makes the
analytical issues strikingly clear. There is a one-sector,
circulating-capital economy, in which workers live on air. Measured in
labor-time, surplus-value (profit) therefore equals living labor extracted.
Technical changes occur in the following manner. In the initial period,
period 0, 256 bu. of corn and 128 hrs. of living labor are used to produce 320
bu. In each subsequent period, exactly the same amount of living labor is
extracted, but both the corn input and corn output increase by 25 0.000000e+00ach
period. Through period 3, the physical quantities are thus:

Period Corn Input Living Labor Corn Output
0 256 128 320
1 320 128 400
2 400 128 500
3 500 128 625

Note that all corn output of a period is reinvested as the corn input in the
next period.

According to traditional simultaneist interpretations of Marx's value theory,
in which labors of different productivities are weighted equally, the unit
value of the commodity in any period, both as input and as output, measured in
labor-time, equals the ratio of the living labor extracted to the net product
(corn output minus corn input). These interpretations would thus compute the
following value and profit rate magnitudes:

Table 1
TRADITIONAL SIMULTANEIST VALUATION

Period Unit Value Constant Capital Living Labor Total Value Profit Rate
0 2.000 512 128
640 25%
1 1.600 512 128
640 25%
2 1.280 512 128
640 25%
3 1.024 512 128
640 25%

In this, and the following tables, one adds constant capital and living labor
to obtain total value, and the profit rate equals living labor divided by
constant capital, since variable capital equals zero.

We see that although traditional simultaneism holds that there is a constant
profit rate of 25% (in accordance with the Okishio theorem), and although all
profit is reinvested since the whole of the corn output of one period becomes
the corn input of the next, the total value of output remains stagnant
throughout time, and there is no accumulation of capital in value terms.
Moreover, according to the traditional simultaneist interpretations, although
all corn output is reinvested and becomes the means of production of the next
period, the total value of output in any period does not equal the constant
capital advanced in the next period. Finally, note that traditional
simultaneism violates the well known *identity* between the rate of
accumulation (investment relative to capital), on the one hand, and the profit
rate (profit divided by capital) times the share of profit that is reinvested
on the other:

(Investment/Capital) = (Profit/Capital) x (Investment/Profit) (5)

since (profit/capital) is held to equal 25% and (investment/profit) equals 1
in this example, but (investment/capital) = 0.

These logical problems disappear when labor is weighted according to its
use-value productivity, as the vintage labor notion suggests:

Table 2

"VINTAGE" LABOR VALUATION


Period Con. Cap Prod'y Weight Val. Add Tot. Val Prof. Rate
0 512 0.500 1.00 128 640
25%
1 640 0.625 1.25 160 800
25%
2 800 0.781 1.56 200 1000 25%
3 1000 0.976 1.95 250 1250 25%

The initial value of constant capital is set exogenously, as is the period 0
weight. Then because, for example, productivity in period 1 is 25 0gher
than in period 0 , the weight attached to living labor is also 25 0gher, so
that the same 128 hours of labor yield 25% more value, 160 as against 128.

Table 2 shows that the vintage labor notion avoids the obvious
inconsistencies of prior attempts to reconcile Okishio-type results with
Marx's value theory. The whole of the output of one period is reinvested as
corn input in the next and, accordingly, the value of the output of one period
does equal the value of constant capital in the next. Moreover, with the
profit rate of 25%, and all profit being reinvested, total value increases by
25 0.000000e+00ach period, as it should, and the rate of capital accumulation is
likewise, and appropriately, 25%. Thus identity (5) is satisfied, since the
rate of accumulation of 25 0oes indeed equal the profit rate of 25% times the
share of profit that is reinvested, 1.

For the sake of comparison, Table 3 presents the determination of value
according to the TSS interpretation. It weights the labor of each period as
equal, but like the vintage labor notion, it avoids the internal inconsistency
of the traditional simultaneist interpretations by making the value of
constant capital in one period equal the value of output of the prior period
(given the assumptions of this example). The initial figure for constant
capital is again set exogenously. Adding the living labor gives the total
value of period 0, which becomes the value of constant capital of period 1, to
which the living labor is added to obtain the total value of period 1, and so
forth:

Table 3

CONSISTENT VALUATION, WITH VALUE DETERMINED BY ACTUAL LABOR-TIME

Period Con. Cap Prod'y Weight Val. Add Tot. Val Prof. Rate

0 512 0.500 1.00 128 640
25.0%
1 640 0.625 1.00 128 768
20.0%
2 768 0.781 1.00 128 896
16.7%
3 896 0.976 1.00 128 1024 14.3%

As was the case with the vintage labor figures, here the value of the output
of one period does equal the value of constant capital in the next, which must
be right, since all of the output of each period becomes the corn input of
the next. Moreover, the rates of increase in the total value of output, and
the rate of capital accumulation, conform to the identity that the rate of
accumulation equals the profit rate times the share of profit reinvested.
Since, in this example, all profit is reinvested, the rate of capital
accumulation should equal the profit rate of the preceding period, and the
rate of increase in total value should equal the profit rate of the current
period. These equalities do hold. Constant capital increases by 25%, 20%,
and 16.7%, between periods 0 and 1, 1 and 2, and 2 and 3, respectively, and
total value increases by 20%, 16.7%, and 14.3%.

Unlike the others, Table 3 also exhibits a falling rate of profit under
conditions in which the Okishian profit rate is a constant 25%! The falling
rate of profit under these conditions results from respecting both the simple
facts of economic dynamics and Marx's theory that value is determined by
(real) labor-time.

Notes to Part II:

5. It is also important to point at that the issue at hand concerns the
commensuration of labors of different times, not their monetary expression.
Professor Foley is contending, not that labors of different vintages are
expressed as different money sums, but that they yield different amounts of
value as measured in labor-time. Thus, the weights do not convert labor-time
value into money-value, but rather convert labors of different vintages into
amounts of base year labor-time value.

6. This result can be obtained by several other methods. For instance,
instead of employing (1), (2), and (3), one can replace (3) with the
stipulation that Tt = Ltt/(1 - at), which is the usual simultaneist
definition of the labor embodied per unit of output in period t. Or,
analogously, one could replace (3) with the stipulation that Kt = at[Lt-1/(1
- at-1)]. Also, one could obtain the same result by using only (1) together
with these expressions for Tt and Kt.