[OPE-L:2273] Re: Great LeapS Forward

akliman@acl.nyit.edu (akliman@acl.nyit.edu)
Sat, 18 May 1996 11:22:43 -0700

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A reply to Duncan's ope-l 2263.

Among other things, Duncan writes that

"From the point of view of the capitalist thinking about undertaking a
new investment, the issue is the total anticipated stream of profits
over the life of the investment, summarized, for example, in the
Internal Rate of Return. In the examples I've looked at, the IRR does
not fall with the type of technical change you [John] describe (labor
augmenting and capital saving) and a constant real wage."

I agree with Duncan 100% that the expected IRR is the relevant profit
rate for investment decisions. I also *think* we agree that evaluation
of the tendency of the profit rate over time should be based on the
actual (realized) IRR.

Among the examples Duncan has looked at is one of mine. He is right that
it is not a capital saving example. I want to present one now, and to
argue that the IRR does indeed fall over time. Duncan and I have
discussed this issue off-list, and I suspect he might not agree with
my conclusion. Yet I think some of the difference between us is due to
a lack of clarity on my part in explaining how my profit rate measure
translates into the IRR. So I think it is worth another shot. In any
case, feedback from others on the list, especially in regard to
whether I'm calculating the IRR right, would be most appreciated.

For simplicity, I'll assume a one-sector capitalist economy, in which
output (X) is produced with fixed capital (F) that lasts forever--this
simplifies IRR computations immensely--and living labor (N). Workers
will be assumed to live on air, so the real wage is 0, a constant. No
material inputs are used in production. I'll also assume a constant
expression of labor-time in money; it doesn't matter to the results what
this constant is, but I'll make 1 labor unit = $1, so I can ignore the
conversion factor entirely.

Now, I will treat the issue of new investment in a moment, but let me
first write the equations for the economy-wide aggregates. They grow
as follows:

X(t) = Xob^t

F(t) = Fo[(c-1)/(b-1)]b^t + Fo[(b-c)/(b-1)]

N(t) = Nod^t

I will assume that b > c > d and that d is greater than or equal to 1.
Hence, over time, the capital output ratio falls. The technical
composition of capital (F/N) rises, as does labor productivity (X/L).
This is the scenario that John envisions, I think.

Now, to look at the path of the social unit price, the monetary expression
of the commodity's social unit value. By "social," I mean the (weighted)
average, i.e., average labor-time needed to produce the commodity. By
Marx's theory, given no physical depreciation of the fixed capital and
no material inputs used, the total value of output of period t will
equal the living labor extracted in that period. Given that each labor
unit = $1, then, if we let p(t+1) be the unit *output* price of period
t (= the unit *input* price of period t+1), we have

p(t+1)X(t) = N(t) so that p(t+1) = (No/Xo)(d/b)^t .

To begin from an initial stationary price situation, assume also that
p(o) = (No/Xo)--which equals p(1).

Now I want to examine *new investment* in order to get at the IRR, which
looks at the rate of return on each particular investment project. But
I don't want to look at it from the individual firm's perspective--yet.
Rather, I want to do a macroeconomic analysis, and examine the following.
In each period, there is technical change in the following sense: the
*marginal* technical coefficients differ from the existing average
coefficients (through the prior period). At least the marginal technical
composition of capital and the marginal output/labor ratio change. So
I want to ask what the *incremental* rate of return is on the new
investment of this new period. Putting the same thing differently, due
to the (physical) investment, the change in the fixed capital stock
undertaken at this moment, there is a particular increase in social
output *and* a particular increase in the amount of social labor extracted.
So, my questions are: by how much does this new investment increase
surplus value and by how much does it increase constant capital.

Because the fixed capital lasts forever, the stream of returns on the
new investment lasts forever. Whatever the stream of returns *received*
by the innovating firm(s), the *social* return to this new investment is
the amount by which undertaking this investment increases capital's
extraction of living labor. E.g., through last period, the capital
stock was 100 units and 40 hours of living labor were extracted each
period. Now, imagine (for simplicity--there are other ways of thinking
about this that are equivalent) that the *existing* firms do not grow
and do not innovate. Then, were there no new entrants, the capital
stock would remain 100 units and the labor extracted would be 40 units
throughot all time. But now imagine a new entrant who invests in 3
units of fixed capital and extracts 1 labor-hour each period. (Thus,
s/he, like the others, stick with their technology and scale of
production once s/he has entered). What will be the *social* rate of
return on this new investment? And then let's imagine a series of
new entrants, 1 in each new period, and ask the same question.

I realize this is a somethat convoluted, not to mention unrealistic,
story. I'm employing it only to try to get clear what the correct
measure of the IRR on each new investment is.

OK. Since the fixed capital lasts forever, the *incremental* IRR
is exceedingly easy to compute.

Incremental IRR = (return per period)/(initial $ investment).

But the social return per period is the increment to labor extraction,
N(t+1) - N(t) = No(d-1)d^t for the entrant of period t+1. And the
investment, undertaken once and for all when the new entrant
comes in, is the social unit price of the increment to the
capital stock times the increment itself. I.e., P(t+1)*[F(t+1) -
F(t)] = (No/Xo)(d/b)^t * Fo(c-1)b^t = (Fo/Xo)No(c-1)d^t.

This gives a *constant* incremental IRR for each new investment:

Incremental IRR = [Xo(d-1)]/[Fo(b-1)].

Yet, this does NOT mean that the economy-wide profit rate is constant,
because we haven't let looked at the initital, period 0, investment.
The period 0 return per period is No, and the investment is p(o)*Fo =
(No/Xo)*Fo. Thus

Period 0 Incremental IRR = Xo/Fo.

Hence, the next period's investment lowers the social average profit
rate (aggregate IRR). And each new incremental IRR is lower than the
social average, so each continues to lower the average. At time
approaches infinity, therefore, the average profit rate approaches
the Incremental IRR (of all periods but 0) as its limit. (The same
answer for the limit of the economy-wide rate can be found by using
the economy-wide X, F, and N, and computing r(t) = profit/Kt, where
Kt is the value of the aggregate capital stock, given by
K(t+1) = K(t) + I(t), where I(t) is the new investment, computed above.)

So the economy-wide profit rate falls because the marginal rate is always
lower than the average rate.

Why would anyone invest in such new techniques? Simple. The above
Incremental IRRs refer to the return *produced* compared to the
investment. But the individual innovator cares about his/her return
*received*, which will be higher because (again using Marx's theory)
his/her technology is more productive than average, and so s/he will
rake in superprofit. (I can show that each innovator will get the
highest profit rate in the economy, when s/he enters, if s/he sells at
the social price. This BTW is very similar to Geert's "stratification"

It is instructive to compare the above to the results that the
Okishio theorem, or any other simultaneist calculation would give.
The return per period on new investment would be p*[X(t+1) - X(t)]
where p is the stationary unit price, and the new investment would
be p*[F(t+1) - F(t)], so that the Incremental IRR would be

Simultaneist Incremental IRR = [Xo(b-1)]/[Fo(c-1)],

except in period 0, when it would = Xo/Fo.

So it would *rise* over time, since b > c. It begins at the same level
as "my" rate. But mine continuously falls while this continously
rises. Hence, this is another example refuting the Okishio theorem.

P.S. With respect to "my" profit rate: Not only will the new innovator
get the highest profit rate in the economy when s/he enters, but each
new entrant's rate when entering will be equal. If I were to put a
positive (constnat) real wage into the model, each new entrant's rate
when entering would be greater than the prior new entrant's.

Andrew Kliman