[OPE-L:2814] Okishio/Kliman Part 3: "Examples"

Duncan K Foley (dkf2@columbia.edu)
Fri, 9 Aug 1996 13:14:46 -0700 (PDT)

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It is true that Roemer specifies the internal rate of return as his profit
rate. But the internal rate of return under my premises can easily be
to fall. Our earlier discussion of this ran into a lot of complications,
let me try something simpler, which doesn't rely on obtaining an exact
of the IRR. Think of the economy as belonging to a *single* capitalist,
producing a single commodity, and imagine that workers live on air, no
material inputs are used, fixed capital is physically nondepreciating, and
physical capital output ratio is constant. Roemer's profit rate = IRR is
constant. Mine will be, too, if labor productivity is constant. And,
importantly, the two rates will be *equal.* But imagine in the 2nd period
(initial period is the 1st), productivity rises. If this is the only
in which technical change occurs, and production continues with the same
coefficients thereafter, the stream of returns on both investments (which
otherwise would be equal to each other and to Roemer's stream of returns)
be lowered, and thus the IRR's on both investments must be lowered, which
makes them lower than Roemer's.

Duncan now:
Well, I still don't seem to come out at the same place. Here's how I
interpret your model. Consider first the case where there is no technical
change at all. In each period capitalists can invest b units of output and
combine it with a units of labor to yield 1 unit of output per period.
Since the wage is zero, their profit rate with output as numeraire and no
technical change would be 1/b. If we take labor as numeraire, the price of
output in terms of labor will be a, the factory will cost ab, and the
labor numeraire profit rate will be a/ab = 1/b, the same as with output as
numeraire, still assuming no technical change.

Now suppose that in period 2 productivity has a one-shot increase,
corresponding to a fall in the labor coefficient to a', and a fall in the
labor price of output to a'. Investors in period 2 buy a factory for a'b
and their profit rate is a'/a'b = 1/b, on the assumption of no further
technical change. The story for vintages before period 1 is a bit more
complicated, due to the extreme assumption of infinitely-lived capital,
which means that there is no vintage of investment before period 1 which
can be viewed as if it were confronting stationary prices, since every
pre-period 2 vintage will face the fall in prices after period 2. In order
to calculate the rate of profit on pre-period 1 vintages we have to have
some theory of prices in these periods. Suppose that the technical change
was completely unanticipated, so that pre-period 1 investors thought that
the price of output would stay at a forever. Then they would have
anticipated a profit rate of 1/b, which I would take as the equivalent of
the pre-technical change profit rate in the Okishio models. The pre- and
post-innovation profit rates in this case would be the same, which is
consistent with the Okishio theorem. Another interpretation would be to
identify the pre-period 1 stationary profit rate with the IRR of very old
vintages (- infinity, if you will), whose IRR will be very close to 1/b as
well. If the technical change was anticipated and capitalists had the
choice of delaying their investments, then the price of output in periods
before period 2 would rise above a in order to equalize the rate of profit
on investments before and after the technical change. The capitalists who
bought factories in period 1, for example, have a money stream of returns
{a",a',a',...} on a money investment of a"b, where a" is the price of
output in period 1. If a" = a, because they did not anticipate the
technical change, their IRR is going to be lower than 1/b. This makes it
look as though the IRR rises between vintage 1 and vintage 2 capital. If
they correctly anticipated the technical change, and could have waited
until period 2 to invest, the a", the price of output in period 1,
would have to rise above a enough to raise the vintage 1 IRR to 1/b.

I just don't see a falling rate of profit in this example, except in the
sense that the profit rate realized by pre-vintage 1 investments in the
case where the technical change was unanticipated falls below the
anticipated profit rate on the assumption of no technical change.

It would be clearer to work the example out for finite-lived capital, say
2 periods. Then all the investments up to period 1 itself would be
confront stationary prices over their lifetimes, and you could compare the
period 0 IRR to the period 2 IRR (they would be the same). This would
focus attention on the transitional capital, represented by vintage 1
investments, and on the theory of expectations and price formation.

But imagine a productivity rise in the 3rd
period. This lowers the stream of returns on all 3 investments. Etc. So
each new productivity rise lowers each IRR. All IRRs would be equal to
other and equal to Roemer's in the absence of the productivity rise, so
continuous rises in productivity reduce each and every IRR and thus the
economy-wide IRR.

Duncan now:
Since I don't come to the conclusions you adopt as a premise for this part
of the argument, I don't follow you to this conclusion.

I had written: "my examples model precisely the phenomena that Marx said
would give rise to
a FRP--rising productivity and rising organic/technical compositions of
capital--and show that the profit rate can fall on this basis."

Duncan replied: "The example we worked through has a falling ratio of
capital to output."

Andrew: Yes, but also rising productivity (output to living labor) and a
rising technical composition (fixed capital to living labor).

In reference to Duncan's suggestion that the Okishio theorem shows that
"prospective" rate of return on investment rises, I had written: "But if
"prospective" means ex ante, then the theorem shows nothing more than that
viable technical change cannot lower the rate of return capitalists think
will get (if real wages are constant, etc., and if the capitalists are

Duncan replied:
" think this is a recognizable statement of Okishio's theorem. The phrase
"capitalists think they will get" is taken to refer to future capitalists
correctly forecasting their revenues once relative prices have adjusted to
technical change."

Duncan now:
Perhaps the discussion of the quasi-model above clarifies these points.

This implies that the theorem refers to the ex ante rate only because
it is equal to the actual rate. But I've shown that it needn't be equal,
capitalists' expectations are what the *theorem's own premises* state them
be (techniques costed up at current prices, implying expectations of
stationary prices for all time). They don't correctly forecast their
revenues, because the unit price continuously falls.

Duncan now:
The theorem's assumption of stationary prices before and after the
technical change is motivated in part by the hope of avoiding the thorny
problem of putting forward a theory of expectations. When you posit an
environment of constant technical change and constantly changing prices,
which is certainly a more "realistic" representation of capitalist
reality, you have to correspondingly posit a system of price adjustment
and capitalist expectations in order to close the model. Of course the
naive static expectations of the stationary price model won't work very
well in this new set of hypothesized circumstances. This point, however,
raises two difficult analytical problems: 1) what system of expectations
will capitalists use in costing out potential technical innovations? 2)
what prices will competition enforce under these more general
circumstances? It's not obvious that competition will force the price of
output to proportionality with labor input in an environment of constantly
falling prices which everyone recognizes.

The "once relative
prices have adjusted," as I've noted above, is precisely what the theorem
needs to prove, but merely assumes. Without proving the adjustment to a
stationary price scenario, there is no proof that the capitalists'
are correct.

Duncan wrote: "Under conditions of continuing technical change, with a
finite life
of capital, there will always be a gap between historical and replacement
rate of profit. In a steady state, however, this gap will be constant and
could not lead to a situation where one of the profit rates
was rising while the other was falling."

Andrew: What is meant by "steady state"? Why is this case of particular
interest, instead of one in which the gap widens? Also, the result is not
obvious to me; could you explain it?

Duncan now:
The model outlined above with 2-period lifetimes for capital and a single
one-shot technical change would make the point clear, I think. Up to
period 0 and after period 2 all the capital sees stationary prices over
its lifetime, and there will be no discrepancy between historic and
replacement cost. The period 1 vintage investment will have this gap. With
constant technical change and constantly falling prices at a steady rate,
but with a finite lifetime for capital, the gap will stabilize. To make
any progress on this we need to posit a complete model within which the
relevant variables can be calculated.

I had written: "As I understand Marx's value theory, the constant capital
transferred to the value of the product, in the case of "machines" (and
capital generally), is a fractional part of the
*current* cost of the machine (the pre-production reproduction cost), not
historical cost. See, e.g., the next to last page of Ch. 8 of Vol. I.

"For example, if the machine lasts 2 periods, and has a value of 10 at the
beginning of the 1st period, and a value of 6 the beginning of the 2d, I
this means that (1/2)*10 + (1/2)*6 = 8 is transferred."

Duncan replied: "I'm in a summer house and don't have instant access to
Capital, but this strikes
me as wrong. In the absence of technical change, Marx uses an accounting
scheme where the whole initial cost will be written off as part of
capital over the life of the machine."

Andrew: I was discussing what happens when there's technical change, not
it is absent. (I do think that Marx understood the effect of price
changes on
value transferred to be the same whatever the cause, but let me leave that
aside.) So the whole value of the machine is not recovered through sale
products when there's technical change, which lessens the gap between
rates when capital has a finite life and when it has an infinite life.

Duncan now:
It seems to me that this question of depreciation is closely bound up with
assumptions about expectations. If capitalists expect falling prices they
will allow for this in their depreciation schedules. I don't see how we
can make much progress on this front without a theory of expectations.

I don't think I have much to add to what I said already vis-a-vis the
points made at the end of Andrew's reply.

The related issues of price formation, competition, induced technical
change and the rate of profit in a context of continuing nonstationary
technical change, seem to me to well worth exploring.