Variation Of Prices Of Production With Time In An Example Of Intensive Rent

Figure 1: Variation of the Wage Frontier with Technical Progress

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. Here I explore structural dynamics along a path in which technical change overwhelms the scarcity of land.

In this post, I repeat the data on technology, with a specific parameterization. Table 1 presents the available technology. Iron and steel are produced in processes with inputs of labor and circulating capital. 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

Requirements for use are 90 tons iron, 60 tons steel, and 19 bushels corn.

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

Suppose the coefficients or production in process IV decrease at the rate specified by setting σ to 5/4. And the coefficients of production in process V decrease, with φ set to 1/20.

Figure 1, at the top of the post, illustrates the evolution of the wage frontier with time in this scenario. Table 3 summarizes how the cost-minimizing technique varies with the rate of profits in each region. A discontinuity occurs at the pattern for requirements for use. Alpha, Delta, and Epsilon can satisfy requirements for use in Regions 1, 5, 10, and 11, while Alpha, Beta, Epsilon, and Zeta can satisfy requirements for use in Regions 12, 13, and 4. Finally, Alpha, Beta, and Gamma can satisfy requirements for use in Region 20, which is not shown in Figure 1. Region 20 is an example of a model of circulating capital. Land is in excess surprise, and rent is zero.

Table 3: Regions

Region	Range	Technique	Notes
1	0 ≤ r ≤ Rα	Alpha	No rent.
4	0 ≤ r ≤ Rβ	Beta	No rent.
5	0 ≤ r ≤ r1	Alpha	Rent per acre, when Epsilon is adopted, increases with the rate of profits and decreases with the wage.
	r1 ≤ r ≤ Rε	Epsilon
10	0 ≤ r ≤ Rε	Epsilon	Rent per acre increases with the rate of profits and decreases with the wage.
11	0 ≤ r ≤ r1	Epsilon	A range of the rate of profits exists for which no technique is cost-minimizing. The wage frontier is a non-unique function of the rate of profits. The wage curve for Delta slopes up on the frontier.
	r1 ≤ r ≤ r2	Delta and Epsilon
12	0 ≤ r ≤ r1	Epsilon	Rent per acre is a non- monotonic function of the rate of profits or of the wage. The wage curve for Zeta slopes up.
	r1 ≤ r ≤ r2	Zeta
	r2 ≤ r ≤ Rβ	Beta
13	0 ≤ r ≤ r1	Epsilon	Rent per acre is a non- monotonic function of the rate of profits or of the wage. The wage curve for Zeta slopes down.
	r1 ≤ r ≤ r2	Zeta
	r2 ≤ r ≤ Rβ	Beta
14	0 ≤ r ≤ r1	Zeta	Rent per acre, when Zeta is adopted, decreases with the rate of profits. The wage curve for Zeta slopes down.
	r1 ≤ r ≤ RΒ	Beta
20	0 ≤ r ≤ Rβ	Beta	No rent.

D'Agata's example arises when t is one. As shown in Figure 1, there is a range of the rate of profits in Region 11 in which both Delta and Epsilon are cost-minimizing. Regions 12 and 13 vary in that the wage curve for Zeta slopes up in Region 12 and down in Region 13. The cost-minimizing technique is not a unique function of the wage in Region 12.

Anyways, my approach of partitioning parameter spaces based on fluke cases applies to this example of intensive rent.

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.