Three Patterns Across The Axis For The Rate Of Profits In A Model Of Intensive Rent

Figure 1: Three Patterns Across the r Axis and One Three-Technique Pattern

This post begins a perturbation analysis of an example of intensive rent from D'Agata. I have previously claimed that certain structures in parameter space are universal in some sense.

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. Processes III and IV undergo technical progress through time. Table 2 shows the processes operated in each of the six techniques available.

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: 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. In this parameter range, 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.

Figure 1, at the top of this post, illustrates a part of the parameter space formed by (σ t) and (φ t). Patterns of fluke switch points partition the parameter space into regions in which the wage frontier does not qualitatively vary within each region. (From the numbering, you may correctly guess other patterns of fluke switch points form other partitions off the edges of the graph.)

Table 3: Regions

Region	Range	Technique	Notes
1	0 ≤ r ≤ Rα	Alpha	No rent.
2	0 ≤ r ≤ r1	Alpha	Non-unique cost-minimizing technique. Wage curve for Delta slopes up on frontier.
	r1 ≤ r ≤ r2	Alpha, Delta
5	0 ≤ r ≤ r1	Alpha	Positive rent for some range of the rate of profits.
	r1 ≤ r ≤ Rε	Epsilon
6	0 ≤ r ≤ r1	Alpha	Non-unique cost-minimizing technique. Wage curve for Delta slopes up on frontier.
	r1 ≤ r ≤ r2	Epsilon
	r2 ≤ r ≤ r3	Delta, Epsilon

Figure 2: Wage Frontier and Rent in Region 2

Figure 3: Wage Frontier and Rent in Region 5

Figure 4: Wage Frontier and Rent in Region 6

One can summarize, as in Table 3, which switch points and wage curves appear on the frontier in each region. In region 1, the Alpha technique is cost-minimizing for all rates of profits. Land is in excess supply, and no rent is formed. Technical progress is modeled by a movement to the east, north, or northeast in Figure 1. Technical progress here eventually results in land being scarce, at least for some range of the rate of profits, and landlords receiving a rent. Figures 2, 3, and 4 show the wage frontiers and rent per acre, as a correspondence with the rate of profits, for regions 2, 5, and 6.

Some phenomena arise in regions 2 and 6 that are not possible in models with circulating capital alone. As I understand it, these phenomena are also not possible in pure fixed capital models and in models of extensive rent. I am referring specifically to upward-sloping wage curves on the frontier and a non-unique cost-minimizing technique for some rates of profits.

I like that despite these oddities, the illustrated partition of the selected part of the parameter space is qualitatively similar to partitions for parts of parameter spaces for circulating capital models. Maybe I am exploring something fundamental underlying the analysis of the choice of technique.

Elsewhere

• Many articles from the Thames Papers in Political Economy to 1989 are now available open access.
• The articles in Political Economy: Studies in the Surplus Approach, from 1985 to 1990, are also available open access.
• There is now a Post Keynesian Discord server, whatever that is.
• Here is a Post Keynesian blog, on this newish substack thingy.

A Disconcerting Example of Intensive Rent From D'Agata

Figure 1: The Wage Frontier And Rent

1.0 Introduction

This post is another worked homework example, problem 7.8 in Chapter 10 of Kurz and Salvadori (1995). The example illustrates the possible non-existence of a cost-minimizing technique with intensive rent. I once looked at an example from J. E. Woods of joint production. I claim that that example does not make the desired point, given the possibility of a price of zero for some produced good. I do not think this example of rent can be resolved like that.

Kurz and Salvadori suggest to me how I might apply my perturbation techniques: "...calculate what will happen if either only process (4) or only process (5) were missing."

2.0 Technology, Techniques, and Requirements for Use

Anyways, 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	1
Land	0	0	1	1	1
Iron	0	0	1/10	1/10	1/10
Steel	0	0	2/5	1/10	1/10
Corn	1/10	3/5	1/10	3/10	2/5

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.

3.0 Cost-Minimizing Techniques and the Wage Frontier

When Alpha is used, not all land is farmed. Rent would be zero. But Alpha is never cost-minimizing.

Figure 2: Extra Profits At Delta and Epsilon Prices

To see if a technique is cost-minimizing at a given rate of profits, find prices of production, the wage, and rent for the technique. Then one can calculate extra profits for every process. Costs include the going rate of profits on advances for purchasing capital goods, wages, and rents. Figure 2 plots extra profits for processes for Delta and Epsilon prices.

The left panel illustrates Delta. No extra profits are made or extra costs are incurred in processes I, II, III, and IV. Delta only has a non-negative wage and a non-negative rent between a rate of profits of 1/9 (that is, approximately 11.1 percent) and approximately 52.3 percent. From a rate of profits of approximate 11 percent to 46 percent, extra profits cannot be made in operating process V. Delta is cost-minimizing.

For a higher rate of profits, in a range in which rent is non-negative under Delta prices, process V makes extra profits. Delta is not cost-minimizing. Which technique would be adopted under these conditions? Process V could be be the only corn-producing process, in the Gamma technique. But that technique is not feasible. Suppose process V replaces process III, in the Zeta process. That technique results in more being produced than are needed for requirements for use. Epsilon is the only feasible technique in which land is fully farmed and two corn-producing processes are operated, with a positive rent.

The right panel in Figure 2 illustrates extra profits for all processes under Epsilon prices. Epsilon has a non-negative wage and a positive rent up to a rate of profits of 2/3 (that is, approximately 66.7 percent) In the range of the rate profits from zero to approximately 46 percent, Epsilon is cost-minimizing. For a higher rate of profit, where the wage is still non-negative under Epsilon, process IV makes extra profits. I highlight in this range when Delta is feasible and consistent with a positive wage and positive rent.

The above analysis shows how the wage frontier is constructed in this example. The wage frontier is illustrated in the left panel in Figure 1 at the top of this post. The corresponding rent is shown in the right panel. A range of rate of profits exists in which Delta and Epsilon are both cost-minimizing. The switch point between Delta and Epsilon is at 19/41, (that is, approximately 46.3 percent). Above this rate of profits, no technique is cost-minimizing.

4.0 Conclusion

Between rates of profits of 19/41 and approximately 52.3 percent, Epsilon makes extra profits at Delta prices, and Delta makes extra profits at Epsilon prices. Even though feasible techniques exist that are consistent with positive wages, rates of profits, rent, and prices of production, no cost-minimizing technique need 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.

An Example Of External Intensive Rent From D'Agata

Figure 1: The Wage Frontier And Rent

This post is merely a worked homework example, problem 7.10 in Kurz and Salvadori (1995). I have not considered yet which parameters I want to explore perturbing.

As a matter of history, Anderson, West, Malthus, and Ricardo took extensive rent as the paradigm case, and confined it to land. They imposed no limit on the production of industrial commodities. Ricardo, at least, also discussed the case of intensive rent. The marginalists, on the other hand, took the case of intensive rent as the paradigm case and extended it to all commodities and, sloppily, extended the explanation of rent to payments to capital and labor. I still do not get well-behaved supply and demand relationships in models of intensive rent. Marginalism remains mistaken and lacks a coherent price theory.

Table 1 presents coefficients of production for the technology. One process is known for producing iron, three processes are known for producing steel, and one process is known for producing corn. Iron and steel are industrial commodities, while corn is the single agricultural commodity. One hundred acres of land are available. All processes exhibit constant returns to scale, in corn production up to the limit imposed by the scarcity of land. The scarity of land can result in a combination of two processes being used to produce steel, with land receiving a positive rent. Requirements for use are 10 tons iron, 10 tons steel, and 78 bushels corn. I take requirements for use, that is, net output, as the numeraire.

Table 1: The Coefficients of Production

Input	Industries and Processes
	Iron	Steel			Corn
		I	II	III
Labor	1	1	1/10	11/2	1
Land	0	0	0	0	1
Iron	0	3/10	2/5	1/10	1/10
Steel	1/10	3/10	2/5	1/10	1/10
Corn	0	2/5	3/10	1/5	1/10

Six techniques are available for a sustainable economy. In each technique, the iron-producing and corn-producing processes are operated. The techniques are distinguishable by the steel-producing or combination of steel-producing processes that are operated (Table 2).

Table 2: Techniques

Technique	Steel Process(es)
Alpha	I
Beta	II
Gamma	III
Delta	I, II
Epsilon	I, III
Zea	II, III

The Alpha technique is not feasible; requirements for use cannot be satisified by operating the specified combination of processes while respecting the constraint imposed by land. The Beta, Gamma, Delta, and Epsilon techniques are feasible. When the Beta or Gamma techniques are operated to produce the requirements for use, some land is left farrow and rent is zero. The Delta and Epsilon techniques each require all land to be farmed. Zeta produces more commodities than are required for use. It would only be adopted at a switch point between Beta and Gamma, with a rent of zero.

Prices of production for the Beta and Gamma techniques can be analyzed as in models of circulating capital. For the Beta technique, for example, each of the iron-producing, steel-producing, and corn-producing processes provides a price equation. With the specification of the numeraire, one has four equations in five unknowns: the price of iron, the price of steel, the price of corn, the wage, and the rate of profits. Rent is zero, since land is in excess supply. One can solve for each variable as a function of the rate of profits. As shown in Figure 1 at the top of this post, the wage curves for the Beta and Gamma techniques slope down.

When are the Beta and Gamma techniques cost-minimizing? For a given rate of profits, one can calculate, for each process, the difference between revenues and costs, where costs include a charge for profits on the iron, steel, and corn advanced. The Beta technique, for example, is cost-minimizing only when extra profits cannot be made in operating any process. Figure 2 shows the ranges of the rates of profits at which the Beta and Gamma techniques are cost-minimizing.

Figure 2: Extra Profits At Beta Or Gamma Prices

Prices of production can also be found for the Delta and Epsilon techniques. For the Delta technique, for example, the iron-producing, the first and second steel-producing, and the corn-producing processes provide a price equation. Given the rate of profits and the specification of the numeraire, the wage and the prices of iron, steel, and corn, can be solved for from the iron-producing and steel-producing processes. The corn-producing process then yields the rent per acre of land. I confine my attention to non-negative rents and prices. For the Delta technique, rent is negative at a rate of profits of zero, but not for a certain range of postive rates of profits.

Figure 3: Extra Profits At Delta Or Epsilon Prices

To find when the Delta technique is cost-minimizing, one performs the usual analysis. When can extra profits, at Delta prices, be made by operating a process not in the technique? The left-hand panel in Figure 3 shows that extra profits are available from the third steel-producing processes for start of the range of the rate of profits at which the Delta technique yields positive rents. At the end of this range, the Delta technique is cost-minimizing. One can repeat this analysis for the Epsilon technique. The Epsilon technique is cost-minimizing at towards the end of the range for the rate of profits at which it yields a positive rent, as shown in the right-hand panel in Figure 3.

Even though the choice of technique is not analyzed above by construction of an envelope of wage curves, one can still highlight wage curves for each technique when they are cost-minimizing. The left-hand panel in Figure 1, at the top of this post, shows the resulting wage frontier. The right-hand panel shows rent as a function of the rate of profits.

The wage curve for the Delta technique slopes up, even when it is on the frontier. You can see that there is a certain range of the rate of profits where the Beta, Delta, and Epsilon techniques are each cost-minimizing. The wage could just as well be taken as the independent variable. And there is a range of the wage where the Delta, Epsilon, and Gamma techniques are cost-minimizing. Sraffa was wrong or, at least, misleading in certain comments on intensive rent in his book on intensive rent. Some, but not all, of the analytical tools he built can be used to demonstrate these mistakes.

This post illustrates that in a model of external intensive rent, prices of production, rent, and the wage are not necessarily uniquely determined by the rate of profits. Nor are prices of production, rent, and the rate of profits necessarily uniquely determined by the wage. This non-uniqueness cannot arise in circulating capital models or pure fixed capital systems. It can only arise in a model of extensive rent in a fluke case that I have been calling a pattern for the requirements for use.