# Re: Addendum, re Marx and historical costs

aramos@aramos.bo
Fri, 6 Feb 1998 20:24:50

A comment on Andrew K''s PIAF:

> Date: Tue, 3 Feb 98 16:55:44 UT
> From: "andrew kliman" <Andrew_Kliman@CLASSIC.MSN.COM>
> To: ope-l@galaxy.csuchico.edu
> Subject: Addendum, re Marx and historical costs

1. TEMPORAL DEFINITION OF CAPITAL CIRCUIT

In the spirit of the passage of Capital II, p. 200 (Penguin) that I
quoted in my post of Mon, 2 Feb 1998, 21:58:02, I''ll define the
money-capital circuit "temporally", i.e. I''ll *date* precisely each
successive phase in the circuit:

Table 1: Dated money-capital circuit
---------------------------------------------------------------
-------------> Successive Phases --------------->
---------------------------------------------------------------
1. Circulation = Circulation Production Circulation
2. C''-M'' M-C ... P ... C''-M''
3. Dec. 31[t-1] Jan. 1[t] Jan.2-Dec. 30[t] Dec. 31[t]
4. Output prices = Input prices ....... Out. prices
[t-1] [t] [t]
----------------------------------------------------------------

Year [t-1] *output* is sold on December 31 (C''-M''). The *output*
prices fixed this day are the same as the *input* prices prevailing
on January 1, year [t]. Therefore, the *advanced capital* (M-C)
depends on the *input prices*[t] = *output prices*[t-1].

Subsequently, we see that year [t] production time spans from January
2 to December 30, determining the *output prices* coming out on
December 31[t].

The profit rate and the moral depreciation is obtained through a
comparison between magnitudes prevailing at the end and at the
beginning of year [t], i.e. between the *output* and the *input*
prices each year.

2. NUMERICAL EXAMPLE

I think the following table contains Andrew''s example:

Table 2: "Corn" model with moral depreciation
---------------------------------------------------------------------
Year F L(=Gr.Pr.) X v=L/X K S NP
---------------------------------------------------------------------
0 1 8 1 8 8 0 8
1 2 8 2 4 16 -8 0
2 4 8 4 2 16 -8 0
3 8 8 8 1 16 -8 0
----------------------------------------------------------------------
Notation:

F: Means of production or non depreciating fixed capital
L: Living labor (= Gross profit, because wages = 0)
X: Output
L/X: Unit value
K: Advanced Capital (value magnitude)
S: Loss or moral depreciation
NP: Net Profit

This example strikes me because, from year 1 on, we have that net
profit = 0, as well as the profit rate after the moral depreciation
is taken into account. So that, actually, the process of accumulation
could''nt follow. However, this is a result only of a particular
assumption Andrew does. To see this, let''s consider Andrew''s model.

3. THE MODEL

The model behind Table''s 2 example can be summarized as follows:

3.1 BASIC VARIABLES

F[t] = Fo*a^t

X[t] = Xo*a^t

L[t] = Lo

Lo Lo
V[t] = --------- = ---------
X[t] Xo*a^t

a = (1+g), i.e. "a" is a growth factor and "g" is the rate of growth
of the variable. Andrew assumes that fixed (non depreciating) capital
(Ft) and output (Xt) grow at the same rate, "g", so the ratio capital/
output is a constant Fo/Xo, assumed = 1. Living labor, Lt, is
constant, Lo, which is also equal to the "gross" profit (wages =
0). "Gross" profit is that calculated before fixed-capital moral
depreciation is taken into account. Unit value is the ratio between
living labor and output.

3.2 ADVANCED CAPITAL

Advanced capital is defined as:

K[t] = F[t]*v[t-1]

which, after manipulation and recalling that Fo/Xo = 1, is:

K[t] = a*Lo

In period 0, we have

K[o] = F[o]*v[-1]

but, it is assumed that v[0] = v[-1], so that

K[o] = F[o]*v[o]

Note that v[o] = Lo/Xo.

The specification of the *advanced capital* is central in the model.
Advanced capital at the beginning of period t is determined by the
prices prevailing at the end of the preceding period, t-1 (see Table
1).

3.3 LOSS OR MORAL DEPRECIATION

Moral depreciation in period t is given by the amount of existing
fixed capital AND the difference between *output* prices and *input*
prices of this period:

S[t] = F[t]*(v[t] - v[t-1])

v[t-1] is the *input* price of period t, and v[t] is the *output*
price of the same period. As unit values are falling, v[t] < v[t-1],
which implies that S[t] < 0. (F[t] > 0)

After manipulation S[t] is expressed as:

S[t] = Lo*(1 - a)

As v[-1] = v[0], in period 0, we have:

S[0] = F[0]*0 = 0

3.4 THE PROFIT RATE BEFORE "WRITING DOWN" LOSSES: r[t]

The profit rate obtained by the capitalist without considering the
moral depreciation is:

Lo Lo 1
r[t] = ---- = ------ = ---
K[t] Lo*a a

In period 0, we have

Lo Lo
r[o] = ---- = ------ = 100%
Ko Fo*vo

(Remember that Fo/X0 is assumed = 1.)

3.5 NET PROFIT

Net profit is obtained as the sum of gross profit (Lo) and moral
depreciation S[t]:

NP[t] = Lo + S[t]

= Lo + Lo*(1 - a)

= Lo*(2 - a)

As S[o] = 0, NP[o] = Lo

3.6 THE PROFIT RATE AFTER "WRITING DOWN" THE LOSSES: R[t]

This is the ratio between Net Profit and advanced capital

NP[t] Lo*(2 - a) (2 - a) 1 - g
R[t] = ------- = ------------ = -------- = ------
K[t] Lo*a a 1 + g

4. CONCLUSION

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. If we assume 0 < g < 1 (e.g., g = 50%), we
have R[t] > 0.

Note that g > 1 yields R[t] < 0 (overaccumlation?).

A couple of questions for Andrew:

a) Do you have a "more general" model, in which "moral depreciation"
is taken into account, and L[t] is not a constant but, for example

L[t] = Lo*b^t

where "b" can be either b < a or b > a? I think this will imply that
the profit rates r[t] and R[t] won''t be constants but variables.

b) Is my "reworking" correct?

Alejandro Ramos