# [OPE-L:3827] Redefinition of value in V.3

aramos@aramos.b (aramos@aramos.bo)
Sat, 14 Dec 1996 19:41:33 -0800 (PST)

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1. In OPE-L [3814] Paul Cockshott wrote:

...it has implications for the single system people too,
since their approach involves a redefinition of value
with respect to its original volume 1 definition.

In OPE-L [3816] I said that there is no such a
"redefinition of value" in the single-system approach.
The definition of value in V.1 is a particular case of
that given in V.3, Ch.9. In this post I show this in
algebraical terms:

2. Let us suppose a "scenario" similar to that of
Tugan/Bortkiewicz: There is no technical change, no fixed
capital and no joint production and n commodities. The
following matrices and vectors can be defined:

P: (1xn) vector of unit production prices, in money terms.
v: (1xn) vector of unit values, in money terms.
X: (nx1) vector of gross physical production.
A: (nxn) matrix of physical inputs used per unit of output.
B: (nx1) vector of real wage per unit of living labor.
L: (1xn) vector of living labor used per unit of output.

M = A+BL is the (nxn) matrix the total inputs per unit of
output.

PM is the (1xn) vector of cost-price (in money terms).

P[I-M]X is total profit (a scalar measured in money).

alfa = L/LX is a (1xn) vector of the proportion of living
labor used in each commodity into the total living
labor used.

beta = PM/PMX is a (1xn) vector of the proportion of cost
price invested in each commodity into the total cost
price.

By means of these matrices and vectors, unit values and unit
production prices could be defined as:

v = PM + alfa*P[I-M]X [1]

P = PM + beta*P[I-M]X [2]

Unit values show how the total surplus-value (=total
profit) is produced, exploited: in proportion "alfa" to the
living labor used by each capital. This is the vector
showing the "objectification" of social labor.

Unit production prices show how the total profit (= total
surplus-value) is distributed: in proportion "beta" to the
cost price invested. This is the vector showing the
"appropriation" of social labor.

Therefore, the only difference between unit values and unit
production prices is the difference between "Mehrwert" and
"Profit", as Marx says in the "main manuscript":

Values: W = K + m
Production prices: P = K + p*

(On this, ssee my OPE-L [3590]; the (easy) derivation of
the twin equalities in the conditions assumed in V.3,
Ch.9 is in my OPE-L [3585].)

Equations [1] and [2] require only one normalization, for
instance, to assume that the price of a certain commodity
("gold") = 1.

3. Now, focus on equation [2], the vector of unit
production prices:

Let us suppose an special condition regarding the structure
of PM = PA + PBL, the unit cost-prices vector:

PA = PBLg [3]

where g is an scalar.

This means that vectors PA and PBL are proportional and
that g is composition of capital, equal for all capitals.

Now then, what is the effect of this particular condition
on equation [2]?

Given condition [3], vector beta = PM/PMX becomes a
particular vector beta*: Writing gPBL instead of PA in
the vector of unit cost-prices, it becomes:

(PM)* = PBL(1+g) [4]

Consequently, (PMX)* is:

(PMX)* = PBLX(1+g) [5]

A particular vector beta* is obtained cancelling PB and
(1+g):

beta* = (PM)*/(PMX)* = L/LX = alfa [6]

Therefore, condition [3] (PA = PBLg) means that there is no
difference between the proportion in which the profit is
appropriated (beta*), and the proportion in which surplus-
value is exploited (alfa). When PA is not equal to PBLg,
alfa is not equal to beta, and when PA = PBLg, alfa = beta*.

In such a case the vectors representing the labor time
objectified (value, equation [1]) and the labor time
appropriated (price, equation [2]) "COLLAPSE"; they are the
same vector. There is only one system which represents, at
the same time, prices and values: prices = values. So,
the vector of unit prices and unit values is:

P = v = PM + alfa*P[I-M]X [7]

Condition [3] implies that the solution for P coincides
with that for v; the system of production prices
is reduced to that of values. When capital compositions are
uniform, production and distribution of surplus-value
coincides at the scale of individual capital, not only at
the scale of social capital. This is the theoretical
premise maintained before V.3, Ch. 9.

It is clear that in equation [7], it is not correct to
call the vector of prices as "P", because this vector is
a result of a particular condition (that assumed in [3]).
So, it is convinient to designate this particular VECTOR OF
PRICES with a special symbol. Let us call v* the particular
vector of prices arising from condition [3]:

v* = v*M + alfa*v*[I-M]X [8]

It is important to stress 3 things:

a) v* is only a particular case of P existing, if and
only if, condition [3] is fulfilled. Capitals having
different organic composition implies that unit values no
longer can be calculated by means of [8] but, by means of
the more general definition provided by [1].

b) System [8] does not imply that values are
"independent" of prices. In particular, cost-price is given
by the prices which capitalists pay for their "inputs" (as
happens in V.3, Ch.9). The only difference is that these
MONEY PRICES are equal to values. In V.3, Ch. 9, these
prices correspond to production prices.

c) Equation [8] has exactly the SAME "structure" of
equation [1]: W = K + m, values = cost-price + surplus
value, where surplus-value is a portion of total profit
given by the share of the individual capital in the total
living labor exploited (i.e. vector alfa).

4. This "derivation" shows that, "for the single system
people" there is no "a redefinition of value with respect
to its original volume 1 definition", as claimed by Paul C.
It is the same definition. The only difference is that in
V.1 it is considered under very restrictive premises.

So, my question to Paul is: Why do you consider that in the
single-system approach is there some "redefinition of
value" in V.3, Ch. 9 with respect to the V.1 definition?

5. An important note: The above derivation is ONLY valid
for the abstract situation in which the "transformation
problem" has been traditionally set. It cannot take into
account more complex situations.
It is useful, for instance, to solve correctly the
numerical examples put forward by Tugan and Bortkiewicz
and, consequently, to show that these authors ARE WRONG,
precisely because they were unable to understand the
definition of value proposed in the draft of V.3, Ch.9.

Alejandro Ramos M.
14.12.96