Variations In Switch Points With Markups In The 'Corn' Industry

Figure 1: Variation of Switch Points with the Markup in the Corn Industry

1.0 Introduction

I have been re-creating some of my past analyses. The graphs in this post look a bit different because I impose a requirement that the relative markups sum to unity.

2.0 Technology

Consider an economy which produces three commodities, iron, steel, and corn, with the technology specified in Table 1. Two processes are available for producing each commodity. The coefficients of production in a column specify the person-years of labor, tons of iron, tons of steel, and bushels of corn required to produce a unit of output of the given industry.

Table 1: The Technology

Input	Iron Industry		Steel Industry		Corn Industry
	a	b	c	d	e	f
Labor	1/3	1/10	5/2	7/20	1	3/2
Iron	1/6	2/5	1/200	1/100	1	0
Steel	1/200	1/400	1/4	3/10	0	1/4
Corn	1/300	1/300	1/300	0	0	0

Eight techniques (Table 2) are defined for this technology. Each technique is defined by the operation of one process in each of the three industries. All three commodities are Sraffian basics in all techniques. That is, each commodity is a direct or indirect input in the production of all commodities. For example, iron is used directly as an input in the first corn-producing process, and steel is used indirectly in producing corn with this process since steel is an input in either iron-producing process

Table 2: Techniques

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Gamma	a, d, e
Delta	a, d, f
Epsilon	b, c, e
Zeta	b, c, f
Eta	b, d, e
Theta	b, d, f

3.0 Prices of Production

Prices of production are defined here for given ratios of markups among industries. The ratios of rates of profits among industries are assumed stable, but rates of profits are not necessarily uniform. Lack of uniformity in rates of profits can result from variations in evaluations of profits among industries due to idiosyncratic properties of investment; from barriers to entry arising from, for example, secrets in manufacture; and from legal monopolies (D’Agata 2018). Let s₁ r, s₃ r, and s₃ r be the rate of profits in the iron, steel, and corn industries respectively. I call r the scale factor for the rate of profits. The usual system of equations, with labor advanced, must be satisfied for prices of production for a given technique.

As a matter of scaling, suppose the markups lie on a simplex:

s₁ + s₂ + s₃ = 1

Suppose that a bushel of corn is the numeraire. In drawing various graphs, I consider only variations in the markup in the corn industry, with markups in producing iron and steel assumed identical:

s₁ = s₂

The solution to this system, for each technique, has a single degree of freedom, which can be expressed with the wage as a function of the scale factor for the rate of profits

4.0 The Choice of Technique with Competitive Markets

Figure 1 graphs the wage curves for four techniques, given competitive markets. The same relative markups are obtained in all industries. The cost-minimizing technique at a given wage maximizes the scale factor for the rate of profits. The cost-minimizing technique at a given scale factor maximizes the wage. The outer frontier of all wage curves shows the variation of the cost-minimizing technique with distribution. Wage curves are graphed in Figure 1 only for the techniques on the outer frontier. This type of figure, usually for competitive markets, is the most well-known graph in post-Sraffian price theory

Figure 2: Capital-Reversing with Competitive Markets

Around the so-called perverse switch point, the firms in the corn industry switch from the second corn-producing process to the first at a lower wage. That is, they adopt a process that requires less labor to be hired per bushel of corn produced gross. This is known as the reverse substitution of labor (Han and Schefold 2006). For the economy as a whole, the technique adopted at a lower wage requires less labor per unit of net output. This is a consequence of capital-reversing as manifested in a comparison of stationary states (Harris 1973).

5.0 Fluke Cases

Five fluke cases can be found by perturbing the relative markup in the corn industry (Table 3). Figure 3 depicts the wage frontier for the first fluke case. This markup occurs when reswitching is just emerging.

Table 3: Fluke Switch Points

Markup for Corn	Fluke Case
s3 ≈ 0.211996	Reswitching pattern for Gamma vs. Delta.
s3 ≈ 0.249246	Four technique pattern for Gamma, Delta, Eta, and Theta.
s3 ≈ 0.8232415	Alpha vs Beta switch point at wage of zero.
s3 ≈ 0.8696757	Four technique pattern for Alpha, Beta, Gamma, and Delta.
s3 ≈ 0.9307414	Beta vs Delta pattern over r axis

Figure 3: Wage Curves for Gamma and Delta Tangent at Switch Point

6.0 The Choice of Technique with the Full Range of the Markup in the Corn Industry

Figure 1, at the top of this post, is my new type of diagram illustrated for depicting the analysis of the choice of technique. The abscissa is the markup in the corn industry, with given markups of unity in the iron and steel industry. The maximum wage and the wage at switch points along the frontier are plotted. The number and sequence of switch points along the wage frontier are invariant in each numbered region. Fluke switch points partition the numbered regions. Figure 4 enlarges Figure 1 on the right for low wages

Figure 4: Variation of Switch Points with the Markup (Detail)

The qualitative properties of the wage frontier are invariant in each numbered region in Figures 1 and 4. Table 4 describes each numbered region. The cost-minimizing technique along the wage frontier is listed, from a wage of zero to the maximum wage. Some salient properties of switch points and the cost-minimizing technique are summarized in Table 5. Figure 2 depicts the wage frontier for a markup in the corn-industry in region 3, while Figure 3 depicts the wage frontier on the boundary between regions 1 and 2.

Table 4: Variations in the Cost-Minimizing Technique

Region	Range	Technique
1	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ wmax,η	Eta
2	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ w3	Delta
	w3 ≤ w ≤ w4	Gamma
	w4 ≤ w ≤ wmax,η	Eta
3	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ w3	Delta
	w3 ≤ w ≤ w4	Theta
	w4 ≤ w ≤ wmax,η	Eta
4	0 ≤ w ≤ w1	Beta
	w1 ≤ w ≤ w2	Alpha
	w2 ≤ w ≤ w3	Gamma
	w3 ≤ w ≤ w4	Delta
	w4 ≤ w ≤ w5	Theta
	w5 ≤ w ≤ wmax,η	Eta
5	0 ≤ w ≤ w1	Beta
	w1 ≤ w ≤ w2	Delta
	w2 ≤ w ≤ w3	Theta
	w3 ≤ w ≤ wmax,η	Eta
6	0 ≤ w ≤ w1	Delta
	w1 ≤ w ≤ w2	Theta
	w2 ≤ w ≤ wmax,η	Eta

Table 5: Notes on Regions

Region	Summary
1	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.
2	Reswitching of techniques between Gamma and Delta. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta at the lower wage. Process recurrence of the first process in the corn industry.
3	No reswitching. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta. Process recurrence of the first process in the corn industry.
4	No reswitching. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta. Process recurrence of both processes in the corn industry.
5	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.
6	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.

This example allows for a graphical display showing that reswitching arises with an increased markup in corn-production, starting from a markup much less than in other industries. The ‘perverse’ switch point between Gamma and Delta remains on the wage frontier after the other switch point between these techniques falls off the frontier at a higher markup. Eventually, the ‘perverse’ switch point is no longer on the frontier when corn-production has a much higher markup than other industries.

7.0 Conclusion

The properties of the wage frontier might be thought to have some impact on the struggle between capitalists and workers. These properties can be altered both by technical change and by variations in relative market power among capitalists.

Intensive Rent With Two Types Of Land

Figure 1: Wages Curves for Example of Intensive Rent

1.0 Introduction

This post modifies an example from Antonio D'Agata. Two types of land exist, each specialized for producing a specific commodity.

In the example, some wage curves slope upwards, which is not possible in a model with circulating capital alone. The cost-minimizing technique is not found from the outer frontier of the wage curves. For one range of the rate of profits, no cost-minizing technique exists, even though a feasible technique exists in that range with a positive wage and positive prices. If the wage is taken as given, more than one cost-minimizing technique exists in the range of wages where a cost-minimizing technique exists.

This example does not illustrate variation in the order of rentability with the wage or rate of profits. Hence it also does not illustrate the reswitching of the order of rentability.

2.0 Technology, Endowments, Requirements for Use

Table 1 provides the technology for such an example. Each column specifies the quantities of labor, iron, wheat, and rye needed to produce a unit output of the commodity produced by the corresponding industry. The table also specifies the quantity of land that must be rented to operate that process. Constant returns to scale are assumed, with the limitation that the endowments of each type of land are givens.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1 = 1	a0,2 = 1	a0,3 = 2	a0,4 = 1	a0,5 = 1
Type 1 Land	0	c1,2 = 1	c1,3 = 1	0	0
Type 2 Land	0	0	0	c2,4 = 1	c2,5 = 1
Iron	a1,1 = 0	a1,2 = 0	a1,3 = 1/100	a1,4 = 1/10	a1,5 = 1/10
Wheat	a2,1 = 0	a2,2 = 0	a2,3 = 0	a2,4 = 2/5	a2,5 = 1/10
Rye	a3,1 = 1/10	a3,2 = 3/5	a3,3 = 11/20	a3,4 = 1/10	a3,5 = 2/5

I show each type of land as specialized to produce a different kind of agricultural commodity. The givens include the amount of each type of land available. Let t₁ be the acres of type 1 land available and t₂ be the acres of type 2 land available:

• t₁: 97 acres.
• t₂: 100 acres.

The column vector d representing the requirements for use has components:

• d₁: 90 tons iron.
• d₂: 60 quarters wheat.
• d₃: 19 bushels rye.

This vector d of net ouput is also the numeraire.

Table 2 specifies the techniques. All three commodities are Sraffa basics in all techniques. Only the Gamma, Zeta, Eta, and Iota techniques are feasible. For a technique to be feasible, the processes comprising the technique can be operated at a level to produce the required net output.

Table 2: Is Process Operated By Technique?

Input	Industry
	I	II	III	IV	V
Alpha	Yes	Yes	No	Yes	No
Beta	Yes	Yes	No	No	Yes
Gamma	Yes	No	Yes	Yes	No
Delta	Yes	No	Yes	No	Yes
Epsilon	Yes	Yes	No	Yes	Yes
Zeta	Yes	No	Yes	Yes	Yes
Eta	Yes	Yes	Yes	Yes	No
Theta	Yes	Yes	Yes	No	Yes
Iota	Yes	Yes	Yes	Yes	Yes

3.0 Prices of Production

The equations for prices of production vary among the techniques. Both lands are in excess supply and pay no rent for Gamma. Type 2 land is fully farmed under Zeta, and it pays a rent. Type 1 land pays a rent under Eta. Both lands pay a rent under Iota.

In the usual notation, the equations for prices of production for Iota are:

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

(p₁ a_1,2 + p₂ a_2,2 + p₃ a_3,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3 + p₃ a_3,3)(1 + r) + rho₁ c_1,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4 + p₃ a_3,4)(1 + r) + rho₂ c_2,4 + w a_0,4 = p₃

(p₁ a_1,5 + p₂ a_2,5 + p₃ a_3,5)(1 + r) + rho₂ c_2,5 + w a_0,5 = p₃

The price of the numeraire is unity:

p₁ d₁ + p₂ d₂ + p₃ d₃ = 1

These equations can be solved, given either the wage or the rate of profits. Figure 1, at the top of this post, shows the resulting wage curves for the feasible techniques. Figure 2 shows the rent curves.

Figure 2: Rent Curves for the Example

4.0 The Choice of Technique

For a low rate of profits, Iota is the cost-minimizing technique. For a rate of profits greater than that at the switch point for the Iota and Zeta wage curves, the Zeta technique is cost-minimizing, up to the maximum rate for Zeta. No technique is cost-minimizing for a larger rate of profits. Type 2 land has a larger rent per acre than type 1 land in the full range of the rate of profits in which a cost-minimizing technique exists. Type 1 land obtains a rent in the range in which Iota is cost-minimizing. It is free when Zeta is cost-minimizing.

These conclusions are justified by looking at which processes can obtain extra profits when prices of production for a given technique rule. Figure 3 graphs extra profits as a function of the rate of profits with prices of production for the Gamma technique. Either both the third and fifth process can obtain extra profits or the fifth process alone, depending on the level of the rate of profits. Thus, the Gamma technique is never cost-minimizing.

Figure 3: Extra Profits at Gamma Prices

Suppose the rate of profits is given and is in the range from the maximum rate of profits for the Zeta technique to the maximum rate of profits for the Gamma technique. Gamma is feasible, but not cost-minimizing. At prices of production for Gamma, managers of firms would want to operate the fifth process, which produces rye. Firms would adopt either the Delta or the Zeta technique, depending on whether they continue to operate the fourth process at some level. But Delta is infeasible, and prices of production for Zeta are such that the wage is negative in this range. Thus, no cost-minizing technique exists here.

Figure 4 shows extra profits for prices of production for Zeta. At a rate of profits less than the rate at the the switch point for Zeta and Iota, operating the second process obtains extra profits. Iota can be adopted in this range. Zeta is cost-minimizing for a higher rate of profits, up to the maximum rate of profits for Zeta.

Figure 4: Extra Profits at Zeta Prices

Figure 5 graphs extra profits for prices of production for the Eta process. Eta is never cost-minimizing. Operating the fifth process at Eta prices obtains extra profits.

Figure 5: Extra Profits at Eta Prices

Prices of production for Iota are such that extra profits are not obtained in operating any of the five processes in the technology. As long as the wage and rent on both types of land are non-negative, at a given rate of profits, Iota is cost-minimizing.

5.0 Conclusion

Type 2 land obtains a rent for the cost-minizing technique in the full range of the rate of profits where a cost-minimizing technique exists. Type 1 land only obtains a rent for a low range of the rate of profits.

Certain properties of models of circulating capital do not generalize, annoyingly, to the theory of joint production. Are there any such properties that are violated in models of pure joint production that are not also violated in models of intensive rent?

Givens For Two Approaches To The Theory Of Value And Distribution

1.0 Introduction

Broadly speaking, the history of political economy contains two approaches to value and distribution. For purposes of this post, I do not distinguish between classical and Marx's political economy. Institutionalists and those who know about German historical schools, for example, might have a complaint about being ignored.

This post is quite unoriginal. I thought I would just record these properties of two approaches.

2.0 Marginalism

Marginalist economics is about the allocation of given resources among alternatives. In marginalism, the theory of value and distribution is almost co-extensive with economic theory. The givens, for the theory of value and distribution, are:

• Endowments, including distribution of endowments among households.
• Tastes or preferences of each agent.
• Technology.

How to take capital as a given endowment is a difficulty with this approach. It can hardly be taken as a given quantity of value. The theory is supposed to explain prices, including the prices of capital goods. This problem is not just with aggregate theory. It is also a problem with microeconomic theory.

Another approach is to take initial quantities of individual capital goods as given. The neo-walrasian approach abandons the long run and the equalization of the rate of profits among industries. Conceptually, some expectations and plans must have been mistaken before the initial point in time. Yet the theory does not seem to accomodate such mistakes at the given time or into the future. Furthermore, debts and entitlements to future income streams do not seem possible to include among the givens. Disequilibrium processes that change the initial endowments and their distribution do not seem possible to include in the theory either.

3.0 Classical Political Economy

Classical political economics analyzes the conditions needed to ensure the reproduction of society. For the theory of value and distribution, the givens are:

• Technology.
• Requirements for use, which I take as net output.
• Wage or the rate of profits.

The theory of value can be combined with other elements of political economy. The classicals had various theories of wages, combined with demographics. Marx rejected Malthus and developed his theory of the reserve army of labor for similar purposes. The theory is compatible with a rejection of Say's law and enduring unemployment. Many have argued for combining this theory with a long-period interpretation of Keynes' general theory. A theory of growth and the dynamics of technical change can be built upon this theory of value and distribution.

Intensive Rent And The Order Of Rentability

I have thought about what would be the minimal structure of an example that combines extensive and intensive rent. I want to include a commodity produced without land, as well as an agricultural commodity.

This post considers a simpler example. An analysis of extensive rent includes the identification of the order of efficiency and the order of rentability, given the wage or the rate of profits. I take the concept of these orders from Alberto Quadrio Curzio. Can these orders be defined in a model of intensive rent? What would the minimum structure of an example be in which to explore this question? I continue to insist on including an industrial commodity with negligible inputs of land.

I suggest Table 1 provides the technology for such an example. Each column specifies the quantities of labor, iron, wheat, and rye needed to produce a unit output of the commodity produced by the corresponding industry. The table also specifies the quantity of land that must be rented to operate that process. Constant returns to scale are assumed, with the limitation that the endowments of each type of land are givens.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1	a0,2	a0,3	a0,4	a0,5
Type 1 Land	0	c1,2	c1,3	0	0
Type 2 Land	0	0	0	c2,4	c2,5
Iron	a1,1	a1,2	a1,3	a1,4	a1,5
Wheat	a2,1	a2,2	a2,3	a2,4	a2,5
Rye	a3,1	a3,2	a3,3	a3,4	a3,5

I show each type of land as specialized to produce a different kind of agricultural commodity. I am unsure if that specialization is needed for my point. If not, the table defining the techniques below would contain four more techniques. In each, only one of the processes producing the agricultural commodity would be operated.

As noted, the givens include the amount of each type of land available. Let t₁ be the acres of type 1 land available and t₂ be the acres of type 2 land available

The vector d representing the numeraire has components:

• d₁: The quantity of iron in the numeraire.
• d₂: The quantity of wheat in the numeraire.
• d₃: The quantity of rye in the numeraire.

Let net output y consist of a multiple of the numeraire:

y = c d

Net output is among the givens.

Table 2 specifies the techniques. Non-zero coefficients of production in Table 1 should be such that all three commodities are Sraffa basics in all techniques. Not all techniques are feasible for any level of net output.

Table 2: Is Process Operated By Technique?

Input	Industry
	I	II	III	IV	V
Alpha	Yes	Yes	No	Yes	No
Beta	Yes	Yes	No	No	Yes
Gamma	Yes	No	Yes	Yes	No
Delta	Yes	No	Yes	No	Yes
Epsilon	Yes	Yes	No	Yes	Yes
Zeta	Yes	No	Yes	Yes	Yes
Eta	Yes	Yes	Yes	Yes	No
Theta	Yes	Yes	Yes	No	Yes
Iota	Yes	Yes	Yes	Yes	Yes

One can examine which processes are introduced in the cost-minimizing technques as the level of net output expands. That is, is process II adopted before process III or vice versa? Is process IV or process V operated first? Presumably, the answer to these questions depends on the wage or the rate of profits, whichever variable is taken as given in solving the system of equations for prices of production. This model is a model of intensive rent in that, for example, Epsilon or Zeta is cost minimizing, Type 2 land will be fully farmed and obtain a rent. The scarcity of land is shown by having two processes operating side-by-side on a single type of land. Anyways, the analysis outlined here corresponds to determining the order of efficiency in a model of extensive rent.

Suppose the Iota technique is feasible and cost-minimizing. The solution of the equations for prices of production yields the rent per acre for each type of land. Which type of land obtains the larger rent per acre? Does this order vary with the given wage or rate of profits? Is reswitching of this order possible? Does postulating a stable ratio of the rate of profits among industries change the answers, at least in detail? I suggest that this analysis corresponds to determining the order of rentability in a model of extensive rent.

I assume that the order of rentability varies with distribution and that reswitching of this order is indeed possible. As far as I know, nobody has answered these questions or presented a numerical example. I always think that nothing I say would surprise Betram Schefold, Heinz Kurz, or Neri Salvadori, for example. What I try to do is present concrete examples of their more abstract analyses. My identification of fluke cases, of extending the analysis to markup pricing, and presentation of graphs to aid visualization of the results are my own tweaks, I guess.

Elsewhere

• John Cassidy on Donald Harris in the Mew Yorker.
• Dean Dettloff on The Catholic case for communism in America.
• Enrico Bellino and Gabriel Brondino, Circular vs. one-way production processes: two different views on production and income distribution, in Review of Political Economy. I want to recall this for my work on Hayekian triangles.
• Bob Murphy and David Glasner on the Sraffa-Hayek debate. (They spend to much time on monetary policy before getting to the debate.)
• Leland Yeager and Steve Hanke. 2024. Capital, Interest, and Waiting: Controversies, Puzzles, and New Additions to Capital Theory. I probably will not order this.
• Pierangelo Garegnani. 2024. Capital Theory, the Surplus Approach, and Effective Demand. Appears to now be available.

A 1-Dimensional Diagram For Extensive Rent With Markup Pricing

Figure 1: Extensive Rent Example As Relative Market Power Varies Between Industry And Agriculture

This post is an elaboration on this past post.

The analysis of the choice of technique varies with perturbations of relative markups in industry and agriculture. Figure 1 depicts this variation, while Figure 2 is an enlargement of the lower range of the relative markup in industry. The heavy solid lines in Figure 1, other than the horizontal line at the top, are switch points on the inner frontier of the wage curves. They divide the space into areas in which the cost-minimizing technique is labeled in bold. The dashed lines are intersections on the outer frontier of the wage curves. The order of fertility varies across dashed lines. The dotted lines are intersections of rent curves. The order of rentability varies across dotted lines.

Figure 2: Extensive Rent Example As Relative Market Power Varies Between Industry And Agriculture (Enlarged)

Fluke cases partition the graph (Table 1), as shown by the thin vertical lines and the numbering of the resulting regions. To call the intersections of wage curves on the outer frontier 'switch points' is an abuse of language. Quantity flows and the technique do not vary around these intersections. The same types of land are fully farmed, and the type of land that is only partially farmed is farmed to the same extent. The same quantity of iron is produced. Nevertheless, the order of efficiency varies around such intersections. Likewise, to call the intersections of rent curves ‘'switch points' is a similar abuse of language. The order of rentability varies around these intersections.

Table 1: Fluke Switch Points

Regions	Description
Between 1 and 2	Rent curves for Type 1 and 3 lands tangent at switch point.
Between 2 and 3	Switch point for rent curves for Type 1 and 2 lands at wage of zero.
Between 3 and 4	Switch point for wage curves for Beta and Gamma at wage of zero.
Between 4 and 5	Switch point for wage curves for Alpha and Beta at wage of zero.
Between 5 and 6	Three wage curves intersect at single switch point.
Between 6 and 7	Wage curves for Alpha and Beta tangent at switch point.
Between 7 and 8	Switch point for rent curves for Type 1 and 2 lands at wage of zero.

The analysis of the choice of technique is qualitatively invariant in each numbered region. The number and sequence of intersections of wage curves and of rent curves do not vary within a numbered region. Table 2 lists the cost-minimizing technique along the wage frontier, from a wage of zero to the maximum wage, in each numbered region. Variations in the order of fertility and in the order of rentability are also indicated.

Table 2: Variations in the Cost-Minimizing Technique

Region	Range	Technique	Order of Fertility	Order of Rentability
1	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 3, 1, 2
	w1 ≤ w ≤ w2		Type 3, 1, 2
	w2 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
2	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 3, 1, 2
	w1 ≤ w ≤ w2			Type 1, 3, 2
	w2 ≤ w ≤ w3			Type 3, 1, 2
	w3 ≤ w ≤ w4		Type 3, 1, 2
	w4 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
3	0 ≤ w ≤ w1	Beta	Type 1, 3, 2	Type 1, 3, 2
	w1 ≤ w ≤ w2			Type 3, 1, 2
	w2 ≤ w ≤ w3		Type 3, 1, 2
	w3 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
4	0 ≤ w ≤ w1	Gamma	Type 1, 2, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2	Beta	Type 1, 3, 2	Type 1, 3, 2
	w2 ≤ w ≤ w3			Type 3, 1, 2
	w3 ≤ w ≤ w4		Type 3, 1, 2
	w4 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
5	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2		Type 1, 2, 3
	w2 ≤ w ≤ w3	Beta	Type 1, 3, 2	Type 1, 3, 2
	w3 ≤ w ≤ w4			Type 3, 1, 2
	w4 ≤ w ≤ w5		Type 3, 1, 2
	w5 ≤ w ≤ wα, max	Alpha	Type 3, 2, 1	Type 3, 2, 1
6	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w4		Type 1, 2, 3
	w2 ≤ w ≤ w3		Type 2, 1, 3
	w3 ≤ w ≤ w4			Type 2, 1, 3
	w4 ≤ w ≤ w5	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w5 ≤ w ≤ w6			Type 3, 2, 1
	w6 ≤ w ≤ wα, max		Type 3, 2, 1
7	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 1, 2, 3
	w1 ≤ w ≤ w2			Type 2, 1, 3
	w2 ≤ w ≤ w3	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w3 ≤ w ≤ w4			Type 3, 2, 1
	w4 ≤ w ≤ wα, max		Type 3, 2, 1
8	0 ≤ w ≤ w1	Gamma	Type 2, 1, 3	Type 2, 1, 3
	w1 ≤ w ≤ w2	Alpha	Type 2, 3, 1	Type 2, 3, 1
	w2 ≤ w ≤ w3			Type 3, 2, 1
	w3 ≤ w ≤ wα, max		Type 3, 2, 1

In region 1 at the extreme left in Figure 1, agriculture has overwhelming market power, as compared to industry. Owners of Type 2 land do not obtain a rent except when workers make a high wage. Owners of Type 3 land obtain the highest rent per acre, whatever the order of efficiency. In region 8 at the extreme right, owners of Type 3 land do not obtain a rent when workers make a low wage. In this range, industry has much more market power than agriculture. The possibilities are complicated in between these extremes.

Region 5 includes the competitive case, in which s₁ = 1/2. As previously noted, this region exhibits a reswitching of the order of efficiency. Region 2 exhibits a reswitching of the order of rentability. I find worth emphasizing the ranges of the wage in each region in which the orders of efficiency and rentability differ. Such a range exists for each numbered region. Landlords with less fertile land can receive more rent per acre than those with more fertile land. Under capitalism, you are not rewarded, in general, for your contributions or for the contributions of the resources that you own to production. This disconnect applies both to competitive and non-competitive markets.

The analysis of the choice of technique with extensive rent is more complicated than such analysis in a model with only circulating capital:

"The complexity of the outcomes with the potential existence of conflict or concordance among the three major economic categories (earners of wages, profits, and rents) profoundly modifies the traditional analysis of profits and wages." -- Alberto Quadrio Curzio and Fausta Pellizzari (2010).

The quantity and price systems are interconnected. Assumptions on the level of net output are required to determine which techniques are feasible. All three types of land need to be farmed, at least partially, to satisfy the requirements for use in the numeric example.

Introducing relative market power among industries further complicates the analysis of the choice of technique. Extensive rent and which land is marginal, that is, receives no rent, depend on the wage and the relative market power of industry and agriculture. Changes in market power can have similar qualitative effects, such as the introduction of the reswitching of the order of rentability, as structural economic dynamics resulting from technical innovation.

Sraffa's Publications

The following is a list of publications I expect to see in the first volume of Sraffa's collected works:

• Monetary inflation in Italy during and after the war (This is Sraffa's dissertation).
• The current situation of Italian banks. Manchester Guardian. 1922.
• The bank crises in Italy. Economic Journal. 1922.
• On the relations between cost and quantity produced. Annali di economia 1925. (I think an English translation is in the Cambridge Journal of Economics.)
• The laws of returns under competitive conditions. Economic Journal. 1926.
• Increasing returns and the representative firm. Economic Journal. 1930. (This is a symposium with Robertson and Shove.)
• An alleged correction of Ricardo. Quarterly Journal of Economics. (with L. Einaudi).
• Dr. Hayek on money and capital. Economic Journal. 1932.
• Money and Capital: A Rejoinder. Economic Journal. 1932. (This is a rejoinder to Hayek's reply.)
• Malthus on public works. Economic Journal. 1955.
• The Production of Commodities by Means of Commodities. Cambridge University Press. 1960.
• Production of commodities: A comment. Economic Journal. 1962. (This is a response to Harrod's review.)

I expect the three articles in Italian above will be published with Italian and English on facing pages. A couple of letters to editors might be in the first volume. I do not know what I expect for Sraffa's editorial comments in Ricardo's Collected Works or for Garegnani's notes. The second volume should contain Sraffa's lectures in the 1920s, especially those on value.

As I try to recall what was on my crashed hard drive, I'll probably have more lists like this.

Extensive Rent With Markup Pricing: An Example

Figure 1: Choice of Technique with Extensive Rent and Competitive Markets

1.0 Introduction

I am making some slow progress on recreating past posts. For variation, I here take the wage as given.

If I write this up more formally, I intend not to try to relate it to Marx's concept of absolute rent. The point is to illustrate the following comment with long-lasting variations in market power between industry and agriculture:

"The complexity of the outcomes with the potential existence of conflict or concordance among the three major economic categories (earners of wages, profits, and rents) profoundly modifies the traditional analysis of profits and wages." -- Alberto Quadrio Curzio and Fausta Pellizzari. Rent, Resources, Technologies, Springer,2010: 34.

This story illustrates how the owner of Type II land is affected by the distribution between wages and profits, depending on the degree of market power between industry and agriculture. Under competitive markets in the example, this owner receives a rent, no matter what the distribution between wage and profits. With a large enough relative markup in agriculture, as compared with industry, they receive zero rents for some distributions between wages and profits. I also illustrate two fluke cases for relative markups.

2.0 Technology and Endowments

Table 1 presents the technology for the example. The second column shows the inputs of labor, iron, and corn needed to produce a ton of iron. The remaining three columns to the right are the coefficients of production for processes to produce corn. A unit level of operation of a process in agriculture produces a bushel corn and requires an input of one of three types of land, as shown. Constant returns to scale prevail, although the level of operation of the processes producing corn is limited by the available acreage.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV
Labor	1 person-yr.	9/10 person-yrs.	6/10 person-yr.	29/50 person-yr.
Type 1 Land	0	1 acre	0	0
Type 2 Land	0	0	49/50 acre	0
Type 3 Land	0	0	0	2/5 acre
Iron	9/20 ton	1/40 ton	3/2000 ton	29/500 ton
Corn	2 bushels	1/10 bushel	9/20 bushel	13/100 bushel

In the three processes for producing corn, process III requires more labor per acre of land than process II, and process IV requires even more. Output per acre of land also increases across these three processes. Process III requires less seed corn per acre than process II, and process IV requires even less. Given these contrasts, processes II, III, and IV cannot be ranked by physical efficiency alone. Iron inputs per acre do not even vary monotonically among processes II, III, and IV, further illustrating the need for prices to rank lands by efficiency.

I take the numeraire as a commodity basket consisting of 1 ton iron and 1 bushel corn.

The given data includes the land available and the requirements for use. These are such that all three type of land must be at least partially farmed. The given data are in principle observable at a single moment in time. Different types of land are distinguished by how corn is grown on them. No need exists to imagine marginal adjustments

Three techniques, Alpha, Beta, and Gamma, can feasibly satisfy requirements for use. In all three techniques, all four processes are operated. One of the types of land is not fully cultivated in each technique (Table 2). The choice of technique is based on cost-minimization or profit maximization.

Table 2: Techniques of Production

Technique	Land
	Type 1	Type 2	Type 3
Alpha	Partially farmed	Fully farmed	Fully farmed
Beta	Fully farmed	Partially farmed	Fully farmed
Gamma	Fully farmed	Fully farmed	Partially farmed

3.0 Prices of Production

Prices of production are here defined for a given ratio of markups in the industrial and agriculture sectors. The rate of profits in the process producing iron is s₁ r, while the rate of profits in each of the three processes producing corn is s₂ r. I call r the scale factor for the rate of profits. Prices of production satisfy the following system of equations:

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

(p₁ a_1,2 + p₂ a_2,2)(1 + s₂ r) + ρ₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + s₂ r) + ρ₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4)(1 + s₂ r) + ρ₃ c_3,4 + w a_0,4 = p₂

I am assuming that wages and rents are paid out of the surplus at the end of the period of production. The relative market power of industry over agriculture, or vice versa, is expressed by the ratio s₁/s₂. When this ratio is unity, the equations characterize a competitive capitalist economy. Today, I experiment with assuming markups lie on a simplex:

s₁ + s₂ = 1

Each of the processes in the technology contributes an equation to the system of equations defining prices of production. The rate of profits is calculated on the value of the capital goods advanced. Rent and wages are paid out of the surplus. Four equations are defined in terms of seven variables, the prices of iron and corn, the rents per acre on each of the three types of land, the wage, and the scale factor for the rate of profits. The coefficients of production and the markups s₁ and s₂ are taken as given.

Specifying the numeraire removes one degree of freedom:

p₁ + p₂ = 1

All rents must be non-negative and one type of land must pay no rent. This is the type of land not fully cultivated. The following equation specifies that the rent on at least one type of land is zero:

ρ₁ ρ₂ ρ₃ = 0

This constraint removes another degree of freedom. The system of equations for prices of production has one degree of freedom for this model of markup pricing with extensive rent. This degree of freedom can be expressed as a function showing how the wage decreases with the scale factor for the rate of profits, or vice versa. With the wage somehow specified, the rate of profits in industry and agriculture, the price of iron, and rent on the scarce land are determined.

4.0 The Choice of Technique: An Example with Competitive Markets

Figure 1 illustrates wage curves when s₁ and s₂ are 1/2. Markets are competitive. The cost-minimizing technique corresponds to the wage curve on the inner frontier. Figure 2 illustrates the corresponding rent curves.

Figure 2: Order of Rentability with Competitive Markets

Consider a wage of zero or just barely positive. Beta is cost-minimizing. If the requirements for use were small enough that they could be satisfied by only farming Type 2 land, a technique with same wage curve as Beta would be cost-minimizing. No land would pay rent, and the scale fator for the rate of profits would be as shown on the right-most wage curve. With somewhat greater requirements for use, Type 1 land would be taken into cultivation, and the scale factor would be as shown on the wage curve for Alpha. Type 2 would be fully cultivated and pay a rent. Finally, with the originally postulated requirements for use, Type 3 land is cultivated. In the range for the smallest wage, the order of fertility, from most fertile to least fertile land, is Type 2, Type 1, Type 3. The order of fertility is also known as the order of efficiency.

The order of efficiency, for a wage of zero and competitive markets, also orders techniques in decreasing order of the maximum rate of growth. Since land of each type is limited in quantity, the maximum rate of growth must decline, without innovatin.

This is an example of the reswitching of the order of efficiency. When Gamma is cost-minimizing, the order of efficiency varies from Type 2, Type 1, Type 3 lands to Type 1, Type 2, Type 3 lands and back. The inner frontier shows the cost-minimizing technique for this example. The intersections of the wage curves on the outer frontier show variations in the order of efficiency. The wage curve for the Beta technique is never on the inner frontier, and Beta is never cost-minimizing.

The intersections for the rent curves occur at wages different from those for which wage curves intersect. When Gamma is cost-minimizing, the order of rentability varies from Type 1, Type 2, Type 3 to Type 2, Type 1, Type 3. The order of efficiency first deviates from the order of rentability, then matches at a higher wage, deviates at an even higher wage, and matches again at a still higher wage. Whether or not the orders of efficiency and rentability match also varies with the wage in the range where Alpha is cost-minimizing and Type 1 land pays no rent.

5.0 A Three-Technique Fluke Switch Point

Suppose the relative markup in industry is lower. Figures 3 and 4 show the wage and rent curves for a fluke case, in which all three rent curves intersect at a single point. Under Gamma, the order of efficiency, at a wage of zero, starts at Type 2, 1, and 3 lands. The order of rentability, being Type 1, 2, and 3 lands, differs. At a higher wage, the order of efficiency changes, and then matches the order of rentability. When Alpha is cost-minimizing, the orders of efficiency and rentability match. They are both Type 3, 2, and 1 lands.

Figure 3: Choice of Technique for Three-Technique Fluke Case

Figure 4: Order of Rentability for Three-Technique Fluke Case

I suppose the most interesting aspect of this example, other than its fluke property, is that the switch point for the order of efficiency at the low wage w₁ would not exist, in some sense, if the scale factor was taken as given. The scale factor for the rate of profits for this switch point exceeds the maximum scale factor for the Gamma technique.

6.0 A Fluke Switch Point with a Variation in the Order of Efficiency at a Wage of Zero

I next want to consider an even lower markup in industry, for another fluke case. A switch point for the order of efficiency exists in Figure 5 on the axis for the scale factor for the rate of profits. Figure 6 shows the corresponding rent curves.

Figure 5: Choice of Technique for Fluke Case for Order of Efficiency

Figure 6: Order of Rentability for Fluke Case for Order of Efficiency

When Gamma is cost-minimizing at a positive wage, the orders of efficiency and of rentability are Type 1, 2, and 3 lands.

When Beta is cost-minimizing, the orders of efficiency and rentability initially match, with a order of Type 1, 3, and 2 lands. At a higher wage, the order of rentability changes, so they do not match. At an even higher wage, the order of efficiency changes, so the orders once again match, with Beta cost-minimizing.

Alpha is cost-minimizing at the highest range of feasible wages. The orders of efficiency and rentability match, with an order of Type 3, 2, and 1 lands.

7.0 An Example with Higher Market Power in Agriculture

Finally, I want to consider a non-fluke case, with an even lower markup in industry. Figures 7 and 8 show the wage curves and rent curves here. The comments about the orders of efficiency and rentability in the previous section apply, with the exception that the switch point at a wage of zero and disappeared below the axis for the scale factor for the rate of profits.

Figure 7: Choice of Technique for Example with Market Power for Agriculture

Figure 8: Order of Rentability for Example with Market Power for Agriculture

I like that the orders of efficiency and rentability totally reverse, from the lowest to the highest wage.

More qualitative changes in the analyhsis of the choice of technique, including fluke cases, arise for even lower relative markups in industry. But I want to stop here, where the reswitching of the order of efficiency has vanished and Type 2 land obtains no rent for some intermediate range of the wage.

8.0 Conclusion

One can see that the owners of Type 2 land care about competitive forces among capitalists, and the struggle between the workers and the capitalists. If capitalists in agriculture are able to impose lasting barriers to entry, as compared to capitalists in industry, the struggle over the wage can leave them with no rent at all. Furthermore, the specific level of rents per acre, for all types of land, varies at specific wages, depending on the struggle among capitalists. The order of efficiency can vary from the order of lands from the highest rate of growth to the lowest, even at a wage of zero, depending on the struggle among capitalists. I can expand the above write-up to include one of my one-dimension diagrams.

I draw heavily on the work of Alberto Quadrio Curzio in this post. He looked deeply into the theory of rent in post-Sraffian price theory. The book, Resources, Production, and Structural Dynamics (edited by Mauro L. Baranzini, Claudia Rotondi, and Roberto Scazzieri) is dedicated to him. It contains a lot that differs from this kind of formal analysis, considering institutions for fostering innovation to overcome resources scarcities and so on.