Re: Marx's law(?) of the falling profit rate

[Trond Andresen <trond.andresen@xxxxxxxxxxx>]

I have in another message contested whether the profit rate must
neccessarily fall, while not going into the logic of Marx as presented by
Andrew Kliman here on pkt.

But now I want to discuss Kliman's stylized corn economy example, which he
used to demonstrate that the "Temporal single system" interpretation of Marx
*neccessarily* implies a falling profit rate.

At  21.06.99, Andrew described a stylized "corn economy" as follows:

>Assume a one-sector (corn) economy, with no fixed capital, and let us study
>the maximum profit rate (when wages are zero).
[he defines:]
>seed-corn (SC).....corn output (CO)....
>..
> ...(snip)...
>...
>The temporalist maximum profit rate is
>
>        price of CO  -  original cost of SC
>TMPR = ------------------------------------
>              original cost of SC
>
>Let's measure this in terms of labor-time, since that (and not the monetary
>figures) is what is under dispute.  According to Marx's value theory, as
>interpreted by the TSS interpretation thereof, the price of CO here equals
>the value of CO, since there are no price/value discrepancies in a
>one-sector economy.  And the value of C0 equals the value transferred from
>the SC (its original, start-of-period cost) plus the value added by living
>labor.
>
>
>Now, assume that, in labor-time terms, the value of the 256 units of
>seed-corn invested in year 1 totals 256 labor-hours.  In labor-time terms,
>the value added by living labor is simply the amount of living labor
>itself.  Assume it to be 64 labor-hours, and assume that this is *constant*
>over time (technological advance is thus taking place; more and more net
>output is being produced with the same amount of labor).
>
>Thus, in year 2
>
>Price of CO = 256 + 64 = 320.
>
>Remember that, by assumption, all output is reinvested.  (As the table
>above indicates, for instance, the whole 320 units of CO of year 1 become
>the seed-corn invested in year 2.)  Hence, the total price of corn output
>in one year becomes the original cost of seed-corn in the next year (since
>time is continuous; the end of one year *is* the beginning of the next).
>So the original cost of the 320 units of seed-corn invested at the start of
>year 2 is 320 labor-hours.  To this, 64 hours of living labor are added.
>Thus, in year 2:
>
>Price of CO = 320 + 64 = 384.
>
>
>Repeating this process, we arrive at the following figures:
>
>      orig.
>      cost   value  price
>Year  of SC  added  of CO
>----  -----  -----  -----
>  1    256     64    320
>  2    320     64    384
>  3    384     64    448
>  4    448     64    512
>
>
>And, inputting these figures into the formula for the temporalist maximum
>profit rate, we obtain
>
>Year  TMPR
>----  -----
>  1   25.0%
>  2   20.0
>  3   16.7
>  4   14.3
>
>So, in contrast to the simultaneist profit rate, the temporalist rate falls
>continually, as Marx claimed and as the Okishio theorem denies.  If the
>example were to be carried out ad infinitum, the SMPR would remain at 25%
>forever, while the TMPR would approach zero.


First: I agree with Andrew that the profit rate should be calculated on the
basis of actual expenses, not on the basis of ("simultaneist" view)
replacement costs.

But I now want to make the point that Andrew's TMPR only decreases towards
zero if capitalists reinvest 100% of output, i.e. nothing is consumed,
neither by workers nor capitalists. If instead a certain fraction of output
is consumed by workers or capitalists, regardless of how small, the system
will reach a steady state with a constant, positive profit rate.

For simplicity, we will first assume that only capitalists consume out of
output (the case with workers also consuming is considered further below).

Let
R[k] = corn output at the end of year k
L[k] = labor done during year k
a    =  fraction of R[k] reinvested at the end of year k (=start of year k+1)
r[k]  = (R[k] - a*R[k-1])/(a*R[k-1])  , profit rate at the end of year k.

All R's and L's are denominated in labor-hours.

We start with an input of seed corn R[0], and have, for k = 1,2,3,....

R[k] = a*R[k-1] + L[k]                                                     (1)

Andrew's  example corresponds to the special case a = 1, and eq. (1) then
becomes the equation for a discrete integrator with a constant input; the
output increases linearly with time.

Assume from now on, however, that 0 < a < 1, i.e. at least some of the
output, (1-a)*R[k-1], is consumed during year k-1. Then the system has the
following dynamics:

R[1] = a*R[0] + L[1]
R[2] = a*R[1] + L[2] = a*(a*R[0] + L[1]) + L[2] = a^2*R[0] + a*L[1] + L[2]
....
....
R[k] =a*R[k-1] + L[k]
= a^k*R[0] + a^(k-1)*L[1] + a^(k-2)*L[2] + .... +a*L[k-1] + L[k]        (2)

Since all L[k] are assumed equal in Andrew's example, we write L[k] = L,
and (2) may be written as

R[k] = a^k*R[0] +   (a^(k-1) + a^(k-2) + .... +a + 1) * L               (3)

Letting k tend to infinity to check whether a steady state can be reached,
the first term in (3) tends to zero since 0 < a < 1. And the sum in
parentheses is a geometric progression. We then get a stationary
R[k] = R, with

R =  (1+ a + a^2+.....+a^(k) +.....) * L =    L / (1- a)        (4)

The profit rate as defined by Andrew, and by me above, is

r[k]  = (R[k] - a*R[k-1])/(a*R[k-1])                                       (5)

which now becomes constant in the long run

r = (1-a)*R/(a*R) = (1-a) / a                                                    (6)

We note that for Andrew's special case with no consumption, r tends
to zero. We also note that the sustainable profit rate is higher the more
capitalists consume out of output and the less they reinvest!

*****
Let us now consider the slightly more complicated (and fairly reasonable)
case of workers also consuming some of output. To handle this case, it is
convenient to (re)define some entities.

Let
w =  workers' share of output R[k],
   then (1-w) = capitalist's share of output R[k]
s = capitalists' savings rate (investing in corn) out of capitalists' share
of  R[k]

Then the parameter a, defined earlier, becomes

a = 1-w-(1-w)*(1-s) = s*(1-w)                                           (7)

and the profit rate equation (5) becomes

r[k]  = ((1-w)*R[k] - a*R[k-1])/(a*R[k-1])                         (8)

Using (7) and (8), and with k tending to infinity, we get

r = ((1-w) -a)*R/(a*R) = (1-s)/s                                         (9)

Same story: Save less, and the profit rate remains high.
To the degree some part of output is consumed by workers or
capitalists, the profit rate will not go to zero.

But I think it is insufficient to examine this issue through a single
sector example. It must be examined for an entire macroeconomy.
More (hopefully) on that issue later.

Final remark: I may possibly have made some errors in the derivations
here. I am thankful to anyone pointing them out.

Trond Andresen