Two Dual Linear Programs (LPs) Equivalent To A Linear Complementarity Problem

1.0 Introduction

In a previous post, I mapped a specification of a long run position to a LCP. This specification is in terms of a system of inequalities and equalities, and is in a form appropriate for the application of the direct method to analyze the choice of technique. The LCP supports the application of the Lemke algorithm. Although I have not stepped through the algorithm, I finally understand an aspect of some of Christian Bidard's writings.

This post modifies the LCP so that the matrix M in the LCP has a certain kind of symmetry. With this formulation, the LCP is equivalent to dual LPs.

As far as I know, nobody has written down these dual LPs for analyzing the choice of technique in the special case described by the LCP.

2.0 The Parameters for the Previous LCP

The parameters of a LCP consist of a column vector u and a square matrix M. Where the LCP is equivalent to the specification of a long period position, the column vector is as in Figure 1. The column vector y denotes given final demands for n produced commodities. The row vector a₀ is the direct labor coefficients for each of the m processes comprising the technology.

Figure 1: The Given Vector in the LCP for the Cost-Minimization Problem

The matrix M in the LCP is as in Figure 2. The nxm matrix A is the input matrix. Each column consists of the physical inputs needed to operate a process at unit level. The nxm matrix B is the output matrix. Its columns are the outputs of each process, at a unit level. The scalar r is the given rate of profits.

Figure 2: The Given Matrix in the LCP for the Cost-Minimization Problem

The solution of a LCP consists of two column vectors. In the this case, where the LCP is equivalent to a specification of a long period position, these vectors have a block structure. The components of one solution vector consist of commodity prices and the levels at which each process in the technology is operated in a cost-minimizing solution. A unit of labor is taken as the numeriare.

3.0 A Modification of the LCP

I now consider a variation on the above LCP. Let the matrix M be as in Figure 3. With this modification, the vector y is now total consumption at a point in time along a steady-state growth path. The rate of growth is g.

Figure 3: The Updated Matrix in the LCP for the Cost-Minimization Problem

I make a further assumption that the rate of growth is equal to the rate of profits:

g = r

The matrix M is now skew-symmetric, in which its transpose is equal to its additive inverse. The solution of a LCP in which the matrix parameter has this structure also solves dual LPs.

4.0 Dual LPs

In this case, the primal LP can be written as:

Choose p

To maximize y^T p

Such that:

(1 + r) A^T p + a₀^T ≥ B^T p

p_i ≥ 0, i = 1, 2, ..., n

In other words, prices are set to maximize the value of consumption, while respecting the constraint that the cost of no process, at the given rate of profits, falls below the corresponding revenues.

The dual LP is:

Choose q

To minimize a₀ q

Such that:

[B - (1 + g) A] q ≥ y

q_i ≥ 0, i = 1, 2, ..., m

That is, the levels of operation of the processes comprising the technology are set to minimize total employment, while maintaining the given rate of growth at the point on the steady-state path specified by the consumption basket for the economy. With a unit of labor as numeraire, the objective function of the dual LP can be stated as minimizing total wages.

These dual LPs can be mapped to the LCP with the same mapping as in the previous post, with a couple of modifications. These modifications must be included to include the steady state rate of growth.

5.0 Duality Properties

I want to consider three properties of dual LPs.

If a constraint in the primal LP is met in the solution with an inequality, the corresponding decision variable in the dual LP is zero in the solution. In this context, this duality property is the law of non-operated processes.

If a constraint in the dual LP is met in the solution with an inequality, the corresponding decision variable in the solution to the primal LP is zero. This is the law of free goods.

The values of the objective functions of the dual OPs are equal to one another in the solution. This is Joan Robinson's neo-neoclassical theorem. Given a steady state growth path in which the rate of growth is equal to the rate of profits, the maximum total value of consumption throughout the economy, along that path, is equal to minimum total wages

This trip from the specification of a long period position through an equivalent LCP, the modification of that LCP to have a skew-symmetric matrix, and the consideration of the duality properties of equivalent dual LPs constitutes a novel derivation of Robinson's neo-neoclassical theorem.