From: rob@KAMAN.COM (Rob Vienneau)
Newsgroups: sci.econ
Subject: Sraffian Revolution 4/4
Date: 16 Sep 92 00:23:20 GMT

1.0 Introduction

To clarify my presentation of classical equilibrium, this note
presents a single period, linear programming, Neoclassical/
NeoWalrasian general equilibrium model in which Neoclassical intuition
works.  Once again, the analysis is totally unoriginal with me.  As
usual, I give lots of detail to forestall all the misunderstandings
I've seen whenever I've presented an economic argument in other
forums.  On the other hand, this presentation assumes quite a
background.

2.0 Physical Specifications

Consider an economy in which at the start of the year n goods are
available for use in production, Q(1), Q(2), Q(3), ..., Q(n).  Q(i) is
the quantity of the ith good available, measured in physical units,
for example, hours of unskilled labor, hours of skilled labor of a
particular type, tons of steel, bushels of seed corn, etc.  These
quantities are to be regarded as given constants, not free to be
varied in this model, and they represent factors of production.  The
production processes will totally use up all nonfree quantities of
factors, so no factors from these initial quantities are available at
the end of the year.

Suppose m one-year constant-returns-to-scale processes are available for
use in production.  Each process produces one of k goods, where k < m
(so some goods are produced by more than one process). Each process
requires that certain quantities of factors be purchased at the start of
the year, and produces a certain quantity of its output good at the end
of the year.  Each process is characterized by a production function
relating factors and outputs.  The form of a production function is
assumed to be

  x(j) = Min[ Q(1, j)/a(1, j), Q(2, j)/a(2, j), ..., Q(n, j)/a(n, j) ],

where x(j) is the quantity of the output good produced by the jth
process, j = 1, 2, 3, ..., m; Q(i, j) is the quantity of the ith factor
used as input for the jth process, i = 1, 2, ..., n.  The a(i, j)'s are
given constants characterizing these processes. 

2.1 Production Function Properties

Compare the inputs ( Q(1, j), ..., Q(n, j) ) with ( c Q(1, j), ...,
c Q(n, j) ).  In the second case, all inputs are multiplied by a constant
c.  With the above production function, the output x(j) will also be
multiplied by the constant c.  This is what it means to say that this
production function exhibits constant returns to scale. 

On the other hand, suppose just one factor, say Q(1), is multiplied by a
constant c.  So one compares ( Q(1, j), ..., Q(n, j) ) with ( c Q(1, j),
Q(2, j), ..., Q(n, j) ).  If the first factor is not a binding
constraint or if some other factor is also binding, then the quantity
output will not grow at all.  In other words, these production functions
show an extreme case of decreasing factor returns.

These production functions are referred to in the literature as
"constant coefficient" production functions, but this restriction is not
nearly as special as it seems.  The traditional constant-returns-to-
scale, hyperbolic-isoquant production functions seen in expositions of
calculus versions of neoclassical equilibrium can be approximated
arbitrary closely by linear combinations of these production functions.

2.2 An Efficiency Condition

Suppose the jth process operates to produce x(j) units of output.  This
output can be produced with factor inputs of

  ( Q(1, j), ..., Q(n, j) ) = ( a(1, j) x(j), ..., a(n, j) x(j) ).

If any factor inputs are diminished, then the output is diminished as
well.  On the other hand, if the quantity of a single factor input is
increased, then the quantity output is totally unaffected.  The firm
operating this process, if profit is its goal, would be insane to
purchase more than these inputs where any cost is associated with
these inputs.  So in the remainder, we will assume that when the jth
process is operated at level x(j), the factor inputs are as above.

3.0 Supply Curves for Outputs

The next problem is to determine the quantities of outputs firms will
combine to produce.  One assumes firms are small in comparison to the
market, so they take prices as given.  Form the vector (p(1), ... 
p(m)), where p(j) is the price of the good produced by the jth process. 
(Note that some prices will be repeated, namely for those goods that are
produced by more than one process.) Arbitrage will ensure that the
combined profit-maximizing outputs of all firms will solve the following
Linear Programming problem:

   Choose x(1), ..., x(m) to
   maximize z1 = p(1) x(1) + ... + p x(m)
   subject to

    a(1, 1) x(1) + a(1, 2) x(2) + ... + a(1, m) x(m) < Q(1)
                      .                              -
                      .
                      .
    a(n, 1) x(1) + a(n, 2) x(2) + ... + a(n, m) x(m) < Q(n)
                                                     -
     x(1) > 0, ..., x(m) > 0
          -              -

or, in matrix form,

                    T
     Maximize z1 = p  x
     s.t. A x < Q,  x > 0
              -       -

By solving this system, one can determine the quantities of any output
that will be supplied for any set of prices.  In particular, one can
solve this system for the equilibrium prices (see below).  The solution,
( x(1), x(2), ..., x(m) ) will show what processes are chosen and what
are ignored.  If the solution is unique, only one process will be chosen
to produce each output good that is produced.  Other processes that can
produce those goods will be operated at a level of zero.

4.0 Imputation of Factor Prices (Duality)

Linear Programs have duals. The dual problem to the above is

   chose r(1), r(2), ..., r(n) to
   Minimize z2 = r(1) Q(1) + ... + r(n) Q(n)
   subject to
    a(1, 1) r(1) + a(2, 1) r(2) + ... + a(n, 1) r(n) > p(1)
                      .                              -
                      .
                      .
    a(1, m) r(1) + a(2, m) r(2) + ... + a(n, m) r(n) > p(m)
                                                     -
     r(1) > 0, ..., r(n) > 0
          -              -

or
                   T
    Minimize z2 = r  Q
          T
    s.t. A  r > p,  r > 0.
              -       -

The solution to this problem imputes "rental prices" to the factors of
production.  In particular, given the prices of the output goods, the
solution to this problem reveals the wages of various types of labor.

The primal and dual problems have certain relations.  If a constraint in
the primal problem is met in the solution as a strict inequality, the
corresponding rental price for that factor will be zero.  That is,
factors that are not scarce, will be free.  Or no equilibrium is
possible with unemployment and positive wages.  Similarly, if the
solution of the dual solves a constraint as a strict inequality, the
corresponding process will be operated at a zero level in the solution
to the primal.  In other words, no process will be operated at a loss. 
Finally, the solutions to the primal and the dual are such that z1 = z2.
Economic profit is zero; all revenues are imputed to some factor.

Already this model has certain negative implications.  Taking factors
and their patterns of ownership as given, the equilibrium solution can
very well be that some people are starving.  Likewise, a quantity such
as the surplus of corn harvested over seed corn is of no relevance in
this model.  Equilibrium need not ensure that the conditions of
production are reproduced.  Surplus and viability are not concepts that
belong to Neoclassical economics.  Finally, in this one period model, no
rate of interest exists that can be said to be the price of "capital."
Nor can some subset of factors be identified as capital goods in any
meaningful sense.  Whether a factor was previously produced or is a gift
of nature has no economic relevance. 

5.0 Demand and General Equilibrium

I have stated that the the quantities of factors should be regarded as
given.  So should their endowments, that is who owns what factors.  The
above has then shown how, given prices of outputs, the distribution of
income is determined.  But given individual incomes, tastes, and prices
of outputs, one can calculate the quantity demanded of each output for
each individual.  These individual demands are determined by utility
maximization.  Summing over individuals, one can calculate the total
quantity demanded of each good at the given pattern of prices.

One can then compare quantities demanded with quantities supplied.  If
no excess demand or excess supply exists for any output, then the
system is in (general) equilibrium.

6.0 Some Traditional Theorems

In this model prices can be interpreted as scarcity indices, and one can
calculate the factor intensity of any process.  One can then compare
different equilibrium situations.  In particular, if in position alpha a
particular kind of labor has a high wage (rental price) as compared to
beta, then the production processes chosen in alpha will use that kind
of labor less intensively.  Or, in more obvious terminology, high wages
are associated with a smaller demand for labor.  Another traditional
proposition is the theorem that the equilibrium technique will minimize
the cost of production, as measured by the rental prices.  In this
sense, the equilibrium solution is efficient. 

This model also illustrates the idea that factor prices measure marginal
productivies.  One can compare equilibriums in which the quantities of a
single factor are marginally higher or marginally lower.  Of interest is
the resulting change in the value of the outputs, z1, at given prices.
One finds that, for all i = 1, 2, ... n,

                     +                             -
  [ d( z1 )/dQ( i ) ]  < r(i) < [ d( z1 )/dQ( i ) ]
                       -      -

where r(i) is the imputed rental price obtained by solving the dual. 
This equation shows what it means for factors to be rewarded with their
marginal products in this model.  The rental price for a factor is
bounded above by the change in value of output associated with a
marginal decrease in the factor quantity.  Likewise, it is bounded below
by the increase in value resulting from a marginal increase in the
quantity of that factor. 

7.0 References

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

Vivian Walsh and Harvey Gram, _Classical and Neoclassical Theories
of General Equilibrium_, Oxford University Press, 1980.

                                   Robert Vienneau