----------
From: owner-ope-l@galaxy.csuchico.edu on behalf of aramos@aramos.bo
Sent: Friday, February 06, 1998 7:43 PM
To: ope-l@galaxy.csuchico.edu
Cc: Multiple recipients of list
Subject: Re: Addendum, re Marx and historical costs
----------
From: owner-ope-l@galaxy.csuchico.edu on behalf of aramos@aramos.bo
Sent: Saturday, February 07, 1998 6:32 AM
To: ope-l@galaxy.csuchico.edu
Cc: Multiple recipients of list
Subject: Re: Addendum, re Marx and historical costs
"The profit rate obtained by the capitalist without considering the
moral depreciation is:
Lo Lo 1
r[t] = ---- = ------ = --- "
K[t] Lo*a a
As Ale's 2nd post notes, if "without considering" means that the firms' books
fail to reflect the changing values of their assets, then the above isn't
right. But I originally thought "without considering" means that the
(simultaneist) theorist fails to consider that moral depreciation reduces
realized profit. Then the above is right, i.e., it is the simultaneist profit
rate.
Ale is also right that, when the firms' books fail to reflect the changing
values of their assets, they will have as capital advanced
K[t] = v[o]*f[o] + v[o]*f[1] + v[1]*f[2] + ... + v[t-1]*f[t].
"where the "f" are the *additions* to fixed capital carried out at the
beginning of each year, "valued" at the *output* prices of the
*preceding* circuit." (This of course assumes no circulating capital).
Ale asks how to solve this. In general, it doesn't have a closed-form
(analytical) solution, but in the special case Ale is considering, it does. v
and f are scalar. Writing
F[t] = Fo*a^t
X[t] = Xo*a^t
L[t] = Lo*b^t,
v[t] = (Lo/Xo)(b/a)^t
and f[t] = Fo(a-1)a^(t-1), except that f[0] = Fo.
Hence, v[o]*f[o] = (Lo/Xo)Fo, and each subsequent sum of value invested is
v[t-1]*f[t] = [(Lo/Xo)(b/a)^(t-1)]*[Fo(a-1)a^(t-1)] = [Lo*Fo*(a-1)/Xo]*b^(t-1)
Note that this can also be written as
v[t-1]*f[t] = (1/b)*[Lo*Fo*(a-1)/Xo]*b^t, or, for short, H*b^t.
Then
K[t] = (Lo/Xo)Fo + H*{b + b^2 + ... + b^t}
In the numerical example Ale is working with, b = 1, and thus K[t] = (Lo/Xo)Fo
+ H*t. Further, (Lo/Xo)Fo = 8 and H = 8. Thus, as I noted, the capital
advanced is the series 8, 16, 24, 32 ....
Let me anticipate one additional point. If wages are zero and there's no
circulating constant capital, then profit in period t is
Pr[t] = v[t]*X[t] = L[t].
Given L[t] = Lo*b^t, then the profit rate is
r[t] = Lo*b^t/[(Lo/Xo)Fo + H*{b + b^2 + ... + b^t}]
= 1/[(Fo/Xo)*b^(-t) + (Fo*(a-1)/Xo)*{(1/b) + (1/b)^2 + ... + (1/b)^t}]
If b < 1, the series {.} in this last equation increases without limit, so the
profit rate falls to zero as time increases. If b = 1, {.} = t, so it also
increases without limit, and again the profit rate falls to zero as time
increases. If b > 1, a well known procedure gives the limiting value of {.}
as time increases:
(1/b)*{.} = {(1/b)^2 + (1/b)^3+ ... + (1/b)^t + (1/b)^(t+1)}.
Hence {.} - (1/b)*{.} = (1/b) - (1/b)^(t+1)
because all other terms in {.} and (1/b)*{.}, i.e., (1/b)^2 + (1/b)^3+ ... +
(1/b)^t, are the same.
Rewriting the LHS, we have:
(1 - (1/b))*{.} = (1/b) - (1/b)^(t+1)
or {.} = [(1/b) - (1/b)^(t+1)]/(1 - (1/b)).
Now, if b > 1, (1/b)^(t+1) vanishes as t gets large, so the limiting value of
{.} is
lim {.} = (1/b)/(1 - (1/b)) = 1/(b-1)
and thus
lim r[t] = 1/[(Fo/Xo)*b^(-t) + (Fo*(a-1)/[(b-1)Xo]
but since b^(-t) also vanishes as t gets large,
lim r[t] = 1/{(Fo/Xo)*[(a-1)/(b-1)]}.
This can be compared to my result in M&NE. The notation differs, but the
result is the same, given no circulating constant capital. Since 1/(Fo/Xo) is
the limit of the simultaneist, or "material," profit rate, the value/price
rate must be lower than it if a > b. If a = b (no tech. change), then the
two are the same. If a < b (labor-using, productivity-reducing tech. change),
then the value/price rate exceeds the simultaneist rate.
"The formula in point 3.6 makes it clear that, in Andrew''s example, the
profit rate = 0 from year 1 on, only because he assumed a growth rate g = 100%
per year."
This is absolutely correct. That is why my term was "example," NOT model.
"Note that g > 1 yields R[t] < 0 (overaccumlation?)."
g > 1 implies that g*X[t] > X[t], or (a - 1)*X[t] > X[t]. Under the
conditions of the example, in which F[t] = X[t], this implies that (a -
1)*F[t] > X[t], or F[t+1] - F[t] > X[t]. The increase in fixed capital
exceeds output. Without an outside source of supply, this cannot occur.
"Is my "reworking" correct?"
Yes!
Andrew Kliman