Avoiding The Sraffian Combinatorial Explosion With The Lemke Algorithm

1.0 Introduction

I want to return to this example combining extensive and intensive rent. This post specifies parameters in terms of matrices and vectors. All twenty four of the letters of the Greek alphabet are needed to specify techniques.

2.0 Feasibility

In a brute force analysis of the choice of technique, one determines which techniques are feasible, given final demand. For final demand, I consider a ray from the origin in quantity space. Along this ray, the bushels corn in final demand are always numerically equal to the tons iron. Table 1 shows which techniques are feasible, as final demand expands. The bounds are approximate.

Table 1: Feasible Technniques for Final Demands

Starting Final Demand		Ending Final Demand		Feasible Techniques
Tons Iron	Bushels Corn	Tons Iron	Bushels Corn
0	0	6.768	6.768	Alpha, Beta, Gamma, Delta
6.768	6.768	11.985	11.985	Alpha, Beta, Gamma, Eta, Kappa, Phi
11.985	11.985	17.451	17.451	Alpha, Gamma, Epsilon, Eta, Kappa, Mu, Xi, Phi
17.451	17.451	18.753	18.753	Gamma, Epsilon, Eta, Theta, Kappa, Lambda, Mu, Nu, Xi, Phi
18.753	18.753	24.219	24.219	Gamma, Epsilon, Eta, Theta, Lambda, Mu, Nu, Pi, Phi, Chi
24.219	24.219	29.436	29.436	Gamma, Epsilon, Theta, Lambda, Mu, Pi, Sigma, Phi, Chi, Psi
29.436	29.436	36.204	36.204	Gamma, Lambda, Mu, Pi, Sigma, Tau, Upsilon, Phi, Chi, Psi
36.204	36.204	355.392	355.392	Gamma, Lambda, Mu, Tau, Phi, Chi, Psi, Omega
355.392	355.392	367.377	367.377	Zeta, Iota, Lambda, Mu, Tau, Chi, Psi, Omega
367.377	367.377	372.843	372.843	Zeta, Lambda, Omicron, Tau, Psi, Omega
372.843	372.843	384.828	384.828	Omicron, Rho, Tau, Omega

The cost-minimizing technique, at a rate of profits of 14 percent, is in bold. This is the rate of profits I used in my previous analysis. For a rate of profits below approximately 11.16 percent, Omicron is eventually cost-minimizing. Tau is eventually cost-minimizing for a rate of profits below the maximum and greater than approximately 24.43 percent.

3.0 Continuing a Brute Force Approach

For each feasible technique, the system of prices of production can be solved, given the rate of profits.

Is the rent on any type of land negative in any of these solutions? If so, those techniques cannot be cost-minimizing.

For the solutions for the remaining feasible techniques, in which can extra profits be obtained by operating a process not in the technique? Those techniques too cannot be cost-minimizing.

The techniques that remain, if any, are cost-minimizing at the given rate of profits. The cost-minimizing technique is unique in models of circulating capital, of pure fixed capital, and of extensive rent. This uniqueness property may not hold in models of intensive rent and of general joint production. A cost-minimizing technique may not even exist in these latter models; a cycle can arise in which Beta is cost-minimizing at Alpha prices, and Alpha is cost-minimizing at Beta prices.

Table 2: Orders of Fertility and Rentability

Range Index	Range		Cost-Minimizing Technique	Order of Fertility	Order of Rentability
	Starting Rate of Profits (Percent)	Ending Rate of Profits (Percent)
1	0	11.16	Omicron	3, 2, 1	3, 2, 1
2	11.16	13.08	Omega
	13.08	13.24			3, 1, 2
	13.24	13.37			1, 3, 2
	13.37	15.77			1, 2, 3
	15.77	16.05		2, 3, 1
	16.05	18.11		2, 1, 3
	18.11	24.43		1, 2, 3
3	24.43	48.15	Tau
	48.15	50.08		2, 1, 3

Table 2 exhibits details garnered from a brute force analysis. The order of rentability is exactly opposite the order of fertility in one of the subranges in which Omega is cost-minimizing. The intramarginal lands are paid a higher rent than the marginal land.

In the numerical example combining extensive and intensive rent, a cost-minimizing techniquestion exists up to a maximum rate of profits. It is unique.

4.0 Concluding Observations

This post has illustrated complications in the analysis of the choice of technique that are avoided by casting the analysis into the form of a Linear Complementarity Problem (LCP) and applying the Lemke algorithm. For example, you do not need to first define techniques, and determine which are feasible. The algorithm converges to a feasible cost-minimizing technique. The algorithm does not converge if final demand exceeds the upper limit of what is feasible.

I could program the Lemke algorithm to be run at each point on a mesh in the Cartesian product of the quantity space for final demand and the space for the rate of profits. The output of such a program would be something like Table 2.

I do not know how the Lemke algorithm behaves if more than one technique is cost-minimizing. You can see that you are at a switch point because the extra costs in operating a process outside one technique at the switch point are zero.

I am also puzzled about how to set up a LCP for a model with heterogeneous labor.