Variations In An Analysis Of Intensive Rent With One Type Of Land (Part 1/2)

Figure 1: Variation of the Technique with the Markup in Agriculture

1.0 Introduction

This post is the start of a recreation of a previous post, with a requirement that relative markups lie on a simplex.

These two posts are intended to explain Figure 1, above, which presents a summary of the results of the analysis of the choice of technique, given any level of the relative markup in agriculture, as compared with the relative markups in the non-agricultural industries. My presentation is long enough that I break in down into a couple of posts.

The numerical example illustrates that the interest of landlords are affected by persistent barriers of entry in industry and agriculture, as well as class struggle between workers and capitalists. Other classes care about attempts among capitalists to establish non-competitive market structures, although not in any transparent way.

The example also illustrates the possibility of the existence of multiple cost-minimizing techniques away from switch points and of the non-existence of a cost-minimizing technique. In the first case, a finite number of long-period positions are consistent with a given rate of profits, so to speak. The results have a certain indeterminancy, in this sense. The non-existence of a cost-minimizing technique is compatible with the existence of feasible techniques that yield a positive wage and positive prices of production.

Multiple cost-minimizing techniques and the non-existence of a cost-minimizing technique are possibilities in the theory of joint production. They cannot arise with circulating capital alone, pure fixed capital models, and certain models of extensive rent.

This model is not structured so as to be able to yield variation in the order of rentability with distribution or of the reswitching of the order of rentability.

2.0 Technology, Endowments, and Requirements for Use

I might as well repeat the data. Table 1 shows the coefficients of production. Only one type of land exists, and three processes are known for producing corn on it. Following D'Agata, assume that one hundred acres of land are available and that net output consists of 90 tons iron, 60 tons steel, and 19 bushels corn. The net output is also the numeraire.

Table 1: The Coefficients of Production

Input	Industry
	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

All three commodities must be produced for any composition of net output. Table 2 lists the available techniques. Only Alpha, Delta, and Epsilon are feasible for these requirements for use. Not all land is farmed and only one corn-producing process is operated under Alpha. Two corn-producing processes are operated together under Delta and Epsilon.

Table 2: Techniques

Technique	Processes
Alpha	I, II, III
Beta	I, II, IV
Gamma	I, II, V
Delta	I, II, III, IV
Epsilon	I, II, III, V
Zeta	I, II, IV, V

In the non-competitive case, the relative markups in different industries are taken as given. Let the rates of profits be in proportions of s₁, s₂, and s₃, respectively.

3.0 Prices of Production

Prices of production can be defined for each technique. In what is my usual notation, prices of production must satisfy the following system of equations for Delta:

(p₁ a_1,1 + p₂ a_2,1 + p₃ a_3,1)(1 + s₁ r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2 + p₃ a_3,2)(1 + s₂ r) + w a_0,2 = p₃

(p₁ a_1,3 + p₂ a_2,3 + p₃ a_3,3)(1 + s₃ r) + phi c_1,3 + w a_0,3 = p₃

(p₁ a_1,4 + p₂ a_2,4 + p₃ a_3,4)(1 + s₃ r) + phi c_1,4 + w a_0,4 = p₃

The wage is w. I call r the scale factor for the rate of profits. I assume that the same rate of profits, s₃ r, is obtained in both corn-producing processes operated under Delta. The requirements for use yield an equation for the numeraire:

p₁ d₁ + p₂ d₂ + p₃ d₃ = 1

Lately, I have been imposing the condition that relative markups lie on a simplex:

s₁ + s₂ + s₃ = 1

For the analyses in these posts, I assume that relative markups are the same in both industrial sectors:

s₁ = s₂

The above system of equations are such that the three prices of produced commodities, the wage, and rent per acre can all be expressed as a function of the scale factor for the rate of profits, given the technique. Which technique is cost-minimizing at any given scale factor?

4.0 Competitive Markets

To begin, consider the special case of competitive markets. No barriers to entry or exit exist, or any other mechanism that keeps the rate of profits persistently unequal among industries. In the notation in this example:

s₁ = s₂ = s₃ = 1/3

Figure 2 shows the resulting wage curves for the feasible techniques, and Figure 3 shows the corresponding rent curves. For a small enough scale factor for the rate of profits, the wage for Delta is not non-negative. In this range, Epsilon is cost-minimizing, and land obtains a rent. In the range of the scale factor in which the wage is positive for Delta, up to the switch point between Delta and Epsilon are both cost-minimizing. Landlords would rather have the capitalists adopt Delta, but the model is silent on which cost-minimizing technique will be adopted. For any larger scale factor for the rate of profits, no cost-minimizing technique exists. One could break down this range into three subranges, depending on whether both Delta and Epsilon pay positive rents, only Epsilon pays a positive rent, or neither do. By the way, the competitive case is in region 2 in the figure at the top of this post.

Figure 2: Wage Curves with Competitive Markets

Figure 3: Rent Curves with Competitive Markets

It remains to justify the claims above about which feasible technique is cost-minimizing, given the scale factor for the rate of profits. Suppose prices of production for Alpha prevail. One can calculate, for each of the five processes, the difference between the price of the output and the costs of the inuts (Figure 4). Inputs of iron, steel, and corn are costed up at the going scaled rate of profits in each process. Wages are paid out the surplus at the end of the period. The difference, in each process, is known as supernormal or extra profits.

Figure 4: Extra Profits with Alpha Prices with Competitive Markets

Figure 4 shows that Alpha is never cost-minimizing. Whatever the scale factor, extra profits can always be obtained by growing corn with the fifth process. If the rate of profits is high eneough, extra profits can also be obtained by growing corn with the fourth process. Some capitalists would soon adopt another process to produce corn if Alpha were in operation.

How about if the Delta technique were in operation? Both the third and fourth process would be operated. All land would be farmed, and land would obtain a rent. Figure 5 shows the extra profits in operating the last process under these conditions. (By the way, I always try to draw graphs like these to check the solutions of the price equations.) You can see that for a scale factor for the rate of profits greater than at the switch point with Epsilon, Delta is not cost-minimizing. For a lower rate of profits, Delta is cost-minimizing in the range in which the wage is non-negative.

Figure 5: Extra Profits with Delta Prices with Competitive Markets

Finally, consider prices of production for Epsilon. Figure 6 plots supernormal profits for each process with these prices. Epsilon is cost-minimizing up to the switch point with Delta. Beyond this, price signals indicate that the forth process should be operated. Neither Delta nor Epsilon, much less Alpha, are cost-minimizing in this range.

Figure 6: Extra Profits with Epsilon Prices with Competitive Markets

The above has shown which feasible techniques are cost-minimizing, if any, for the full range of the rate of profits in the competitive case. This analysis essentially replicates D'Agata's example. The next part considers non-competitive markets, including fluke cases in which there are qualitative changes in the analysis of the choice of technique.