From: rob@KAMAN.COM (Rob Vienneau)
Newsgroups: sci.econ
Subject: Sraffian Revolution 3/4
Date: 16 Sep 92 00:22:59 GMT

1.0 Introduction

This note presents a classical model of general equilibrium. Since
this is a non-Neoclassical model, assumptions and conclusions at
variance with Neoclassical beliefs are illustrated (I could have
picked a different set of theorems just as well):

  1) The profit-maximizing technique associated with an equilibrium
     position at a low wage may be less labor-intensive than the
     technique associated at a higher wage.

  2) The equilibrium position may be inefficient (defined in physical
     terms)

  3) No economically relevant definition of marginal productivities
     can be calculated prior to the determination of factor prices.

This note demonstrates that none of the Neoclassical beliefs
contradicted by the above theorems follow from maximizing behavior.
It does this by presenting a logically consistent model of a classical
general equilibrium in which none of these beliefs need hold.

I've presented this sort of argument in other forums and obtained blank
incomprehension.  So to forestall that, I present long explanations.  On
the other hand, this note presumes quite a bit of background in
mathematics and economics. 

2.0 Physical Data

This model compares steady state (classical) equilibria.  It is assumed
that each year the economy produces n goods.  The net output of these
goods is Y(1), Y(2), ..., Y(n), where Y(i) is measured in the
appropriate physical units, for example, bushels of wheat, yards of
cloth, or tons of steel.  Assume that all output is consumed (so the net
output of steel is probably zero).  As a consequence all wages and
profits are spent on consumption.  Also assume that all workers and
capitalists desire to consume their income in the proportions indicated
by the Y(i)s.  This last assumption, I believe, is merely a convenience. 
It allows one to separate questions of the distribution of income from
its composition and the level of output.  Different distributions of
income are associated with the same overall final demands.  As a
consequence no assumptions about returns to scale are needed in the
following. 

A technique of production is characterized by a set of n processes, each
process producing a different good.  Assume all processes require a year
to complete.  Firms engaging in a given process must purchase a stock of
each of the n goods and hire each of m types of labor at the beginning
of the year.  Both goods and labor are paid for at the beginning of the
production period.  To be specific, suppose a process produces Q(j)
units of the jth good.  Then the quantity of the ith good input that is
required is a(i, j) Q(j), i = 1, 2, ..., n.  The quantity required of
the ith type of labor is l(i, j) Q(j), i = 1, 2, ..., m. 

The concept of type of labor could use some clarification.  Just because
two jobs require different concrete tasks, the labor employed need not
be of a different type.  For example, busboys at restaurants and
department store clerks perform different tasks.  But, assuming there is
a tendency for clerks and busboys to change jobs in seeking higher
wages, their wages will be equal in a (classical) equilibrium position. 
Hence busboys and clerks perform the same type of (abstract) labor.  On
the other hand, this equalization mechanism does not exist between
engineers and busboys.  So these are two types of labor.  If two sets of
workers take jobs that form "noncompeting groups," then their labors are
of different types. 

2.1 Relationship Between Gross and Net Output

I've now identified a given level and composition of net output (Y(1),
Y(2), ..., Y(n)).  Each of n processes are operated at levels to
produce the gross outputs (Q(1), ..., Q(n)).  Net output is defined as
the surplus for each good over its input requirements, where the net
output is available for paying profit and next year's labor.  Input
requirements are found by summing over the n processes.  Thus, one
obtains the following system of n equations:

   Gross                                                            Net
  Outputs -                   Inputs                            = Outputs

    Q(1) - [ a(1, 1) Q(1) + a(1, 2) Q(2) + ... + a(1, n) Q(n) ] = Y(1)

    Q(2) - [ a(2, 1) Q(1) + a(2, 2) Q(2) + ... + a(2, n) Q(n) ] = Y(2)
                            .
                            .
                            .
    Q(n) - [ a(n, 1) Q(1) + a(n, 2) Q(2) + ... + a(n, n) Q(n) ] = Y(n)

This system can be represent in matrix form:

              [ I - A ] Q = Y.

All vectors in this note are column vectors, except when transposed.
Note each column of A (and of the matrix of labor requirements L)
represents a process.  An alternate technique is represented by
another pair of matices (A, L).  Only matrices in which all quantities
of the vector of net outputs, Y, can be nonnegative for some Q are
considered in this model.  This is a "viability" condition.

Note that the parameters of A, the elements of Q, and the elements of
Y are givens. The above equation only shows a necessary relation
between them.  One can also express Q in terms of A and Y, which is
useful in determining total inputs, (A Q):

                      -1
         Q = [ I - A ]   Y.

The viability condition ensures that the inverse exists.

2.2 Labor Quantities

For my purposes, one needs to be able to determine how many hours of
labor are employed for each technique in stationary conditions.
Letting v(i) denote the hours employed of the ith type of labor
throughout the economy, one has the following system:

     v(1) = l(1, 1) Q(1) + l(1, 2) Q(2) + ... + l(1, n) Q(n)
                         .
                         .
                         .
     v(m) = l(m, 1) Q(1) + l(m, 2) Q(2) + ... + l(m, n) Q(n)

or

                          -1
     v = L Q = L [ I - A ]   Y.

So given the level and composition of net output, and the technique of
production, this model provides a means of calculating the (vertically
integrated) labor that will be employed.

3.0 Equilibrium Prices

In stationary equilibrium, prices will be the same at the beginning and
the end of the year.  Let p(1), p(2), ..., p(n) represent the prices of
the n goods, and w(1), w(2), ..., w(m) denote the money wages of the m
types of labor.  Part of the definition of equilibrium is that each good
will only have one price in all transactions throughout the economy.  (A
generalization of this model permits the introduction of a distinction
between retail and wholesale prices.)

3.1 Numeraire

This model only determines relative prices.  Hence, a degree of freedom
can be fixed by specifying a numeraire commodity.  A common numeraire is
total net output.  That is, assume the value of total output is unity:

    p(1) Y(1) + p(2) Y(2) + ... + p(n) Y(n) = 1.

3.2 Equal Profit Rate Equations

Consider process j. The cost of inputs per unit output is

  In(j) = p(1) a(1, j) + p(2) a(2, j) + ... + p(n) a(n, j)
              + w(1) l(1, j) + w(2) l(2, j) + ... + w(m) l(m, j).

The value of an unit output is p(j).  The net value of an unit output is
merely p(j) - In(j).  Expressing this net value as a percentage of the
value of input, one obtains the rate of profit for process j:

         r = [ p(j) - In(j) ]/In(j).

In effect, a firm operating process j purchase In(j) dollars worth of
inputs at the beginning of the year and, by engaging in production, has
100 r% more dollars at the end of the year. 

If two processes operate with different rates of profit, firms will tend
to discontinue the process with the lower rate and enter the one with
the higher rate.  Classical equilibrium exists when the tendency for
firms to seek the highest profit has leveled the rate of profit to an
equal value in all processes.  This yields the following system of
matrix equations, which defines the "prices of production" obtaining in
an equilibrium system for any given physical list of inputs and outputs:

          T      T               T
       ( p  A + w  L )(1 + r) = p .

With a little manipulation one obtains:

     T                    -1  T
    p  = [ I - (1 + r) A ]   w  L (1 + r).

Multiplying both sides by net output, Y, gives:

                         -1  T
    1 = [ I - (1 + r) A ]   w  L Y (1 + r),

which implicitly defines a function specifying the rate of profit, r, as
a function of the various wage rates. 

This function has some interesting properties.  First, if all wages are
zero, the equal profit rate system defines a maximum rate of profit,
which, usually, will be finite.  Second, if all wages but one are fixed,
increasing that one wage will result in a lower rate of profit.  Third,
if the rate of profit and all wages but two, are fixed, increasing one
of these wages will lower the remaining wage.  Third, if all wages but
one are fixed, that wage will have a maximum value corresponding to a
zero rate of profit.  These relationships can be interpreted as
reflecting the distribution of a physically specified surplus in money
terms.  And they are not as obvious as they may seem, for in a slightly
more general model the rate of profit may increase with an increasing
wage.

4.0 Choice of Technique

The above has shown how, given a physical specification of a technique
by matrices A and L, and given the wages w of all types of labor, one
can determine the rate of profit in the corresponding equilibrium
"prices of production" system.  Given two or more alternative
techniques, (A1, L1), (A2, L2), ..., one can find the rate of profit
corresponding to each system of wages.  Profit maximizing will lead to
the technique with the highest rate of profit becoming the equilibrium
technique corresponding to the given wages and net outputs.  In general,
this technique will vary with wages. 

This process can be illustrated on a graph:

                |
                1
                21
                |21
                |  .
       Rate     |   12
        of      |    1 2
      Profits   |     1  2
       (r)      |       1 2
                |          .
                |           21
                |            2  1
                +--x-------y--2----1--->
                      Wage w(i)

This shows the tradeoff between the wage of one type of labor and the
rate of profits for two technologies for producing the same output.  The
outer envelope curve is the relevant curve for the profit-maximizing
technology.  For wages between x and y, technique 2 will be chosen under
a capitalistic system.  Otherwise, technique 1 wil be chosen. 

Given the chosen technique, one can find the total labor of all kinds
demanded to produce the given output.  Section 2.2 above showed how to
do this.  Thus, one can graph the wage of a given type of labor versus
the labor (of the given type) per unit output (of a given composition). 
Changes in technique will be in indicated by a jump.  Neoclassical
intuition would lead one to expect a step function showing an increasing
labor intensity being associated with a decreasing wage.  In general, no
such relationship will exist.  For example, this graph might look like
so, with little discernable pattern:

          /|\
           |      |
           |        |
     Wage  |     |
     w(i)  |     |
           |              |
           |            |
           +---------------------------->
                Labor of Type i
                per Unit Output

5.0 Conclusions

This note has presented a classical model of equilibrium demonstrating
that the effects on the profitability of alternate techniques associated
with a change in wages provides no basis for beliefs in a well-behaved
demand curve for labor such that lower wages are associated with a
greater demand for labor. 

This model has other destructive consequences for Neoclassical beliefs. 
In particular, examples can be constructed in which two viable
techniques are such that the labor requirements of one technique are
less than the requirements of the other technique for all types of
labor.  Since these two techniques are assumed to produce the same net
output, efficiency demands that the technique with the lower labor
requirements be used.  But, for some combinations of wages, the
technique that maximizes the rate of profit may very well be the other
technique.  In this model of competition, maximizing the rate of profit
may very well lead to the choice of a socially inefficient technique. 

Finally, this model suggests that the marginal productivity theory of
distribution "is all bosh" (Joan Robinson).  No physically specified
unit of measurement of capital can be found such that the marginal
product of capital equals the rate of profits.  Likewise for labor(s).
In fact, any value measurement of marginal products of factors can
only be calculated after determining wages and profits, the very data
that marginal products are supposed to explain.  And the phenomona of
reswitching illustrated above, in which one technique is preferred at
high at low rates of profit, but not at intermediate values, has been
said to overthrow all of Neoclassical capital theory.

So one can perfectly well deny the vision underlying Neoclassical
theory of how prices function, including factor prices, and still
investigate rigorous economic theory.

6.0 References

Luigi Pasinetti, _Lectures on the Theory of Production_, Columbia
University Press, 1977.

Ian Steedman, _Marx after Sraffa_, Verso, 1977, 1978.

                                   Robert Vienneau