Re: measurement of abstract labor

From: Ian Wright (iwright@GMAIL.COM)
Date: Tue Jun 08 2004 - 13:52:17 EDT

Hi Fred, Ajit and others

I wanted to comment on:

> 2. Marx assumed further that, in the production of value,
> one hour of skilled labor is equivalent to (or counts as)
> a multiple of one hour of simple, unskilled labor (with
> different multiples for different kinds of skilled labor). (Fred)


> Foley simply adds up various kinds of
> concrete labors such as the labor of carpenters and
> masons etc. to get his total direct labor, which is
> what he places against the total money value of the
> net output to derive the labor value of one unit of
> money. It does not solve your problem of abstract
> labor since the value of the money commodity is simply
> based on adding up concrete labors. So I think you
> need to rethink on this problem. (Ajit)


> > Surely by adding them up one abstracts from their
> > concrete type, and adds up all that remains -
> > expenditure of human time. (Paul C.)
> Exactly! So the abstraction has been done by the
> theorist before money comes into the picture. (Ajit)

I am thinking aloud, so I'm ready and willing to recognise flaws in
what I write below. I also may be teaching various grandmothers to
suck eggs ... apologies if so.

The summary point is that the MELT is a theoretical approximation not
unlike a mean-field approximation in physics. The approximation error
can be directly related to the matrix of labour reduction
coefficients. In the special case of homogenous reduction coefficients
the MELT is an error-free measure of the value of money. The key point
is to distinguish the macro-abstractions over concrete labour-types
performed by the theorist and the micro-abstractions over concrete
labour-types performed by the market, and the relations between them.

My partial response to Ajit, therefore, is that the MELT theoretically
summarises a set of micro-abstractions performed in the marketplace.
The micro-abstractions are real. The question then is not whether it
is legitimate to "add up concrete labours" to derive the MELT (I think
it is for reasons not given), but whether the MELT is a useful
representation of the average value of money. For example, the
temperature of a gas is a macro-variable that summarises the energy
distribution of the constituent molecules. It is a useful theoretical
average -- just like the MELT. But it has definite relations to the
real abstractions over concrete labour types that market exchanges

Consider a C-M-C simple commodity economy with non-commodity money,
i.e. non-produced tokens. Assume all commodities are basic and do not
require other commodities as inputs. No replacement costs, so no
distinction between gross and net product. Consider that during a
market period there is a set of market exchanges
E = { e_{1}, e_{2}, ..., e_{n} }
where each e_{i} is a pair
(c_{i}, m_{i})
where c_{i} is a commodity of type 1 ... k (i.e. there are k commodity
types) and m_{i} is an amount of money m_{i}>0 for all i=1...n (i.e.
there are n market exchanges in this period). Each exchange pair
represents an event of exchanging an instance of commodity c_{i} for
an amount of money m_{i}.

To simplify assume the "law" of one price, that is if c_{i}=w then for
all c_{j}=w (i != j) we have m_{i} = m_{j}, for all i=1...n and for
all w=1...k. In other words, instances of the commodity type w
exchange for the same money price in all transactions. Let's label
that price p_{w}.

To simplify assume that for each commodity type w (1..k) there is at
least one exchange e_{i} such that c_{i}=w. In other words, there is
at least one market exchange for each commodity type.

To simplify assume uniform productivity, so that the time required to
produce an instance of commodity type w is the same for all producers
of that commodity, and label it t_{w}.

Define the exchange-value of commodity w, which we label v_{w}, as
v_{w} = p_{w}/t_{w}
which is measured in money-unit per concrete labour-time unit, e.g.
dollars per baker-hour, dollars per tailor-hour, etc.

Note that each exchange-value, v_{w}, can be considered as a
"micro-MELT" because it is the monetary expression of a particular
kind of concrete labour. In an economy of k commodity types there are
k micro-MELTs within any market period.

So at the micro-level we have:
v_{1} dollars are worth 1 hour of labour type 1 (micro-MELT_{1})
v_{2} dollars are worth 1 hour of labour type 2 (micro-MELT_{2})
v_{k} dollars are worth 1 hour of labour type k (micro-MELT_{k})

Now dollars are commensurable but concrete labour types are not. But
the commensurability of money induces an equivalence relation on the
concrete labour types (e.g., Krause, "Money and Abstract Labour").

Let r_{i,j} be the reduction coefficient of concrete labour type i to
type j, such that 1 unit of labour type i is equivalent to r_{i,j}
units of labour type j. The equivalence relation is induced by market
exchanges against money, i.e. in virtue of the respective price of the
product of each labour type 1...k.

In fact
r_{i,j} = v_{i}/v_{j}
i.e., the reduction coefficient of labour type i to j is the ratio of
the respective exchange-values of their products. From the assumption
of uniform productivity and one price, then r_{i,i}=1 for all i=1...k.
Given k commodity types there are k^{2} reduction coefficients.

For example, if the exchange-value of baking is 1$ per baker-hour, and
the exchange-value of tailoring is 2$ per tailor-hour, then the
reduction coefficient of baking to tailoring is 1/2. A baker-hour is
valued 1/2 as much as a tailor-hour in the market etc.

Note also that the reduction coefficients are the ratios of the micro-MELTs.

The key point is that the abstraction from different concrete labours
and their relation to a common thing, i.e. money, has been performed
by the market, not by the theorist.

This preamble allows us to provide a micro foundation to the MELT and
maybe address some of Ajit's questions and criticisms. Note that so
far there has been no "adding up of concrete labours".

What is the value of money in this market period?

There is already one answer -- it is the vector of micro-MELTs
V = ( v_{1} ... v_{k} )
that define the value of money in terms of the k types of concrete
labours. If we are content to view the economy as only a set of market
exchanges at one point in time then there is not much more to say.
Certainly in a C-M-C economy there is never an exchange between an
amount of money and a product of abstract labour, for that does not
make sense. There are only exchanges between money and products of
concrete labours. So the vector of micro-MELTs is the precise
representation of the value of money in this market period.

But the MELT is introduced as a theoretical abstraction to construct a
measure of the average value of money. This abstraction is useful if
we want to know how much concrete labour-time of any type 1$ can on
average buy.

Under the assumptions, in particular no distinction between gross and
net output, the MELT is the total market price divided by total labour

       sum_{i=1...n} m_{i}
MELT = -----------------------
      sum_{i=1...n} t_{c_{i}}

and has units money-unit per time-unit, e.g. dollars per abstract
hour. (If you're using a viewer with proportional fonts the fraction
may not format correctly).

In this formula the denominator does indeed add up concrete
labour-times. But this theoretical abstraction has well-defined
relations to the real abstractions that take place in the market.

For example, if I have 1$ in my pocket then the MELT implies that I'll
be able to buy 1/MELT abstract hours. But if in fact I spend the 1$ on
commodity type i then I buy 1/micro-MELT_{i} concrete hours of labour
type i. This is where the error comes in, because in general:

1/MELT is not equal to 1/micro-MELT_{i} for all i=1...k

The MELT equals all the micro-MELTs only in the special case that all
the reduction coefficients are homogenous, in which case the value of
money can be represented without error as a scalar, rather than a

Let n_{i} be the number of market exchanges in E that satisfy c_{j}=i
where j=1...n. For example, if 10 market exchanges involve commodity
#2 then n_{2}=10. Then the MELT can also be written as:

      sum_{i=1...k} p_{i} n_{i}
MELT = -------------------------
      sum_{i=1...k} t_{i} n_{i}

Homogenous reduction coefficients obtain when
r_{i,j} = v_{i}/v_{j} = 1 for all i and j
in which case
p_{i}/t_{i} = a constant for all i and j (i.e., all the micro-MELTs
have the same value). Call that constant C. Therefore, every p_{i} may
be written as:
p_{i} = C t_{i}
Substitute this into the MELT equation gives:

      sum_{i=1...k} C t_{i} n_{i}
MELT = -------------------------
      sum_{i=1...k} t_{i} n_{i}

Which simplifies to
and therefore all the micro-MELTs and the aggregate MELT coincide. In
this special case, the value of money can be represented as a scalar
not a vector.

So when the reduction coefficients are heterogeneous the MELT will be
an approximation to the value of money for any set of concrete
exchanges. That error can be measured for any commodity bundle -- the
bigger and more varied the bundle the greater the sampling of the
different concrete labours and -- here I wave my hands -- the smaller
the error. The aggregate MELT simply considers *all* the exchanges
that occur in a market period as one huge commodity bundle, and
thereby provides a measure of the average value of money for the
market period. The MELT is a macro-approximation to all the

> It does not solve your problem of abstract
> labor since the value of the money commodity is simply
> based on adding up concrete labors. So I think you
> need to rethink on this problem. (Ajit)

But the value of money is here defined by the market exchanges, and it
is the market that compares concrete labours, not the theorist.
Exchanges against money are needed for this.

Given that the reduction coefficients are the ratios of micro-MELTs,
then we can also define a reduction coefficient from concrete labour
type i to abstract labour (the MELT):

r_{i,abstract} = v_{i} / MELT  (i=1...k)

Which is related to:

> 1.  Marx assumed that the basic unit of measure of abstract labor, as the
> substance of value", is one hour of simple, unskilled labor, of average
> intensity, and using average conditions of production. (Fred)


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