A Pattern For Non-Uniqueness

Figure 1: The Wage Frontier And Rent

I continue to explore perturbations of an example from Antonio D'Agata. I have found a new type of fluke switch point, in models of intensive rent. In this post, I repeat the data on technology, with a specific parameterization.

Table 1 presents the available technology. Corn is grown on homogeneous land, and three processes are available for producing corn. One hundred acres of land are available, leading to the possibility of two processes being operated side-by-side with positive rent.

Table 1: The Coefficients of Production

Input	Industries and Processes
	Iron	Steel	Corn
	I	II	III	IV	V
Labor	1	1	1	(11/5) e(5/4) - σt	e(1/20) - φt
Land	0	0	1	e(5/4) - σt	e(1/20) - φt
Iron	0	0	1/10	(1/10) e(5/4) - σt	(1/10) e(1/20) - φt
Steel	0	0	2/5	(1/10) e(5/4) - σt	(1/10) e(1/20) - φt
Corn	1/10	3/5	1/10	(3/10) e(5/4) - σt	(2/5) e(1/20) - φt

Table 2 shows the processes operated in each of the six techniques available. (All three corn-producing processes are operated only at a switch point where the Delta, Epsilon, and Zeta techniques are simultaneously cost-minimizing. Iron, steel, and corn are basic commodities in all techniques. Land is never a basic commodity.

Table 2: Techniques

Technique	Process
Alpha	I, II, III
Beta	I, II, IV
Gamma	I, II, V
Delta	I, II, III, IV
Epsilon	I, II, III, V
Zea	I, II, IV, V

Requirements for use are 90 tons iron, 60 tons steel, and 19 bushels corn. Alpha, Delta, and Epsilon can meet requirements for use. That is, one can find levels of operation of the processes comprising these techniques such that the net output of the economy is the previously specified vector and no more than 100 acres of land are farmed. Beta, Gamma, and Zeta are infeasible.

At the specific parameter values illustrated at the top of this post, the switch point between the Alpha and Epsilon techniques occurs at the rate of profits at which the wage curve for the Delta technique intercepts the axis for the rate of profits. This fluke condition arises for a locus in the parameter space in which (φt) is a function of (σt). It reminds me of a fluke case for the order of fertility in models of extensive rent.

At a slightly lower value of (σt) or a higher value of (φt), no range of the rate of profits exists in which both the Alpha and Delta technique are cost-minimizing. A range of the rate of profits does exist in which the Epsilon technique is uniquely cost-minimizing. On the other hand, at a slightly higher value of (σt) or a lower value of (φt), a range of profits exists in which both the Alpha and Delta technique are cost-minimizing, and Epsilon is not uniquely cost-minimizing for any rate of profits. In both cases near this fluke case, a range of profits exists in which Alpha is uniquely cost-minimizing. And a range of the rate of profits exists in which both the Delta and Epsilon techniques are cost-minimizing.

So this fluke case is associated with a variation in the details of of an example in which the cost-minimizing technique is non-unique, and in which no cost-minimizing technique exists even though feasible techniques with positive prices, wages, rate of profits, and rent exist.

References

• D'Agata, Antonio. 1983a. The existence and unicity of cost-minimizing systems in intensive rent theory. Metroeconomica 35: 147-158.
• Kurz, Heinz D. and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge: Cambridge University Press.