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.

The Production of Commodities by Means of Commodity and Money

Money is a medium of exchange (or means of purchase), a unit of account, and a store of wealth. I think Sraffa (1960) implicitly assumes an economy in which money is used. How would one explicitly and formally introduce money into Sraffa's scheme? I think one would want a theory of endogenous money, maybe as in a circuitist theory. How should the references below be extended? Which should I make an effort to read? I am aware that Sinha (2021) has a couple of other chapters about money and that Bellofiore and Passarella (2016) and Giuseppe and Realfonzo (2017) are introductions to special issues of ROKE and Metroeconomica, respectively. Any guidance to the literature, including these pointers, would be useful.

Reference

• Bailly, Jean-Luc, Alvaro Cencini, and Sergio Rossi (eds.) 2017. Quantum Macroeconomics: The legacy of Bernard Schmidt. Routledge.
• Bellofiore, Riccardo and Marco Veronese Passarella. 2016. Introduction: the theoretical legacy of Augusto Graziani, Review of Keynesian Economics 4(3): 243-249.
• Fontana, Giuseppe and Riccardo Realfonzo. 2017. Augusto Graziani and recent advances in the monetary theory of production, Metroeconomica 68(2): 202-204.
• Graziani, Augusto. 2003. The Monetary Theory of Production. Cambridge University Press.
• Moore, Basil. 1988. Horizontalists and Verticalists: The Macroeconomics of Credit Money. Cambridge University Press.
• Panico, Carlo. 1988. Interest and Profit in the Theories of Value and Distribution.
• Pivetti, Massimo (1991). An Essay on Money and Distribution.
• Rochon, Louis-Philippe. 1999. Credit, Money and Production: An Alternative Post-Keynesian Approach.. Edward Elgar.
• Sinha, Ajit (ed.). 2021. A Reflection on Sraffa’s Revolution in Economic Theory. Palgrave-Macmillan.
• Rochon, Louis-Philippe and Mario Seccareccia (eds.). 2013. Monetary Economics of Production: Banking and Financial Circuits and the Role of the State: Essays in Honour of Alain Parguez. Edward Elgar.
• Rogers, Colin. 1989. Money, Interest and Capital: A Study in the Foundations of Monetary Theory. Cambridge University Press.
• Venkatachalam, Ragupathy and Stefano Zambelli (2021). Sraffa, money and distribution. In Sinha (2021).

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.

A Mistake In Kurz And Salvdori (1995)?

On page 299 of Kurz and Salvadori (1995), they write:

System (10.10) is identical with system (8.13).

The above statement is correct only if the steady state rate of growth is zero. The analysis presented around system 8.13 applies to any rate of growth lower than the rate of profits.

Chapter 8 is about joint production in general. Equations 8.13a through 8.13e specify a long-period position for joint production. Equation 8.13c specifies quantity relations and is:

z^T ( B - (1 + g) A) ≥ c^T

Equation 8.13d is a duality condition known as the rule of free goods. It is:

z^T ( B - (1 + g) A) y = c^T y

A full exposition would explain the notation above.

Chapter 10 is about land rent. Equations 10.10a, 10.10b 10.10c, 10.10f, and 10.10g specify a long-period position with land being cultivated. Equation 10.10a specifies quantity relationships and, more or less, is:

x^T ( B - A) ≥ d^T

Equation 10.10b is the rule of free goods for models with rent. It is:

x^T ( B - A) p = d^T p

If the rate of growth were positive in models of rent, a steady state could not be maintained. Eventually, a less efficient technique (at the given rate of profits) must be adopted, and the rate of growth must be lower.

I find I often may explain the dual quantity system for Sraffa's price equation in a confused manner. I often want to consider the trade-off between a steady state rate of growth and consumption per worker, with a given composition of the consumption basket. Given the technique, this trade-off is identical to the wage curve for the technique. On the other hand, one could present the quantity relations for a given level and composition of net output, that is, for given requirements for use. In an exposition, one must choose one of these approached.

Kurz and Salvadori (1995) is comprehensive. Of the mathematics I understand, this is as close as I found to a mathematical mistake. After publication, some argued about what I think are matters of history and judgement in the critique of neoclassical theory in Chapter 14. I think it was demonstrated about half a century ago that most of what most mainstream economists teach in North America is, at best, incorrect. From twitter, I have learned that economics is astrology for white men.

• Heinz D. Kurz and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge University Press.

Geoffrey Harcourt (1931-2021)

This overview of Geoff Harcourt's work is insufficient. He was interested in economics as a means to a better world. Consequently, he offered political advice, sometimes in the form of 'package deals', in the context of Australian politics, which I know nothing about. Also, I do not know Australian rules football, rugby, or cricket. Apparently, he was very good at mentorship and at introducing young scholars to the professional community. Capital theory is a very contentious topic, but Harcourt was on good terms with all sides.

Geoffrey C. Harcourt was born on 27 June 1931. He married Joan Bartrop in July 1955. Their children are Wendy, Robert, Timothy,and Rebecca. I had not known that the economist Claudid Sardoni married Wendy and was his son-in-law. He died on 6 January.

Harcourt attended the University of Melbourne as an undergraduate and came to Cambridge in July 1955. Nicholas Kaldor was his PhD. supervisor for a short while, but he later had Ronald Henderson as supervisor. His dissertation compared depreciation allowances at historical costs with capital consumption at replacement cost. What are the implications of these accounting conventions for the choice of technique and for taxes on profits? In Joan Robinson's golden age, historical cost, replacement cost, and the present value of the revenues expected from the use of capital equipment are all equal. Harcourt (1965) is one paper that emerged from this work.

Harcourt returned to Australia, to a lecturing post in Adelaide, in 1958. The rest of his professional life was shared between Cambridge and Adelaide. He lectured on Kaldor's growth theory and on Robinson's The Accumulation of Capital. I gather that Harcourt quite enjoyed working with some of his Australian colleagues such as W. E. G. Salter and Eric Russell.

From August 1963 to the end of 1966, he was in Cambridge, with a fellowship at Trinity. Although he reviewed Sraffa's book and co-wrote a paper on Sraffa's subsystems, he says he was mostly an observer of the controversies.

Now comes the work that Harcourt is most noted for. Mark Perlman visited Adelaide to convince Harcourt to write a survey article for the relatively new Journal of Economic Literature. I believe another author had backed out of surveying capital theory, and that author would not have focused on the Cambridge controversies. Harcourt wrote the first draft of his 1972 book while visiting Keio University in Japan. Cambridge University Press is in the process of re-issuing this classic, long out of print.

He found Noam Chomsky's 'The responsibility of intellectuals' inspiring and participated in direct action in Australia against the Vietnam war. He helped developed the Adelaide plan in the early 1970s, and was on the National Committee of Inquiry for the Australian Labor Party (ALP) in 1978-1979. Harcourt described his approach to political programs as 'horses for courses'. I think he may have also used this phrase to describe his approach to economic theory. His political programs included an incomes policy and something like the Tobin tax to curb speculation.

He took a year study leave at Clare Hall in 1972-1973, before returning to Adelaide University. Sometime in the 1970s Harcourt edited a conference volume on microfoundations, which, I gather, had quite a different flavor than the work of, say, Robert Lucas. Harcourt and Kenyon (1976) is one of a number of Post Keynesian works of the time relating markup pricing to firms' investment plans.

Harcourt left Adelaide for Cambridge in September 1982 and became a fellow at Jesus College. He retired in September 1998. I suppose I should mention somewhere his interest in intellectual history and his short biographies of economists in the Cambridge school, such as Richard Goodwin and Lorie Tarshis. The book by Harcourt and Kerr (2009) on Joan Robinson is an example. Most of his biographies are articles, though.

On 13 June 1994, Harcourt was awarded an Office in the General Division of the Order of Australia (AO). He was the president of Jesus College in Cambridge for most of 1988 to 1992. I do not know when he was in Cambridge and when he was in Adelaide for the last 20 years.

The list of references I append is very selective. I do not list the many volumes he edited or any articles in which he set out political programs or his views on politics and its relation to economics. Barkley Rosser has an obituary. Lars Syll links to an interview with Harcourt. John Hawkins and Selwyn Cornish describe Harcourt as 'the beating hear of Australian economics.' You can read testimonials here.

References

• Cohen, Avi J. and G. C. Harcourt. 2003. Whatever happened to the Cambridge capital controversies? Journal of Economic Perspectives 17: 199-214.
• Hamouda, O. F. and G. C. Harcourt. 1988. Post-Keynesianism: From criticism to coherence? Bulletin of Economic Research. 40: 1-33.
• Harcourt, G. C. 1965. The accountant in a golden age. Oxford Economic Papers. 17: 66-80.
• Harcourt, G. C. 1969. Some Cambridge controversies in the theory of capital. Journal of Economic Literature. 7: 369-405.
• Harcourt, G. C. 1972. Some Cambridge Controversies in the Theory of Capital. Cambridge University Press.
• Harcourt, G. C. 1982. The Social Science Imperialists: Selected Essays by G. C. Harcourt (ed. by Prue Kerr). Routledge and Kegan.
• Harcourt, G. C. 1986. Controversies in Political Economy: Selected Essays by G. C. Harcourt (ed. by O. F. Hamouda). New York University Press.
• Harcourt, G. C. 1995. Capitalism, Socialism, and Post-Keynesianism: Selected Essays by G. C. Harcourt. Edward Elgar.
• Harcourt, G. C. 2001. Selected Essays on Economic Policy. Palgrave-Macmillan.
• Harcourt, G. C. 2006. The Structure of Post-Keynesian Economics: The Core Contributions of the Pioneers. Cambridge University Press.
• Harcourt, G. C. and Prue Kerr. 2009. Joan Robinson. Palgrave Macmillan.
• Harcourt, G. C. and Vincent G. Massaro. 1964. Mr. Sraffa's Production of Commodities. Economic Record 40: 442-454.
• Harcourt, G. C. and Peter Kenyon. 1976. Pricing and the investment decision. Kyklos. 29: 449-477.

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.

Post-Sraffian Terminology

Terms that include the word 'pattern' are my own creation, as inspired by my research program. The remainder are, as far as I am concerned, standard terminology, some of which you would be introduced to if you were taught price theory properly. (Most of what is in mainstream microeconomic textbooks is, at best, wrong.) The definitions are my own, although obviously inspired by my reading.

• Absolute rent: A price paid for a year's services for land under cultivation due to barriers to entry to agriculture that would be otherwise manifested in persistent higher rates of profits in farming.
• Basic commodity: A commodity that is productively consumed, either directly or indirectly, in the production of each commodity produced in an economy.
• Capital reversing: The association of a higher rate of profits around a switch point with a cost-minimizing technique with a more capital-intensive technique. Also known as a positive real Wicksell effect.
• Circulating capital: Produced commodities that are completely consumed in producing other commodities. Contrast fixed capital.
• Coefficient of production: The amount of a specified commodity that is required as an input to operate a given process at a unit level or the amount of a specified commodity that is produced in operating the given process at a unit level.
• Differential rent of the first kind: See extensive rent.
• Differential rent of the second kind: See intensive rent.
• Extensive rent: A price paid for a year's services for land under cultivation due to the need to cultivate more than one type of land to satisfy requirements for use while prices of production prevail.
• External intensive rent: A price paid for a year's services for land under cultivation due to the need to more than one process, in an industry that uses negligible inputs land, so as to satisfy requirements for use while prices of production prevail. See intensive rent.
• Factor price frontier: See wage frontier.
• Finished good: A produced commodity that is either a consumption good, circulating capital, or a newly produced machine.
• Fixed capital: Produced commodities that are used in producing other commodities and last over more than one production period. A good used as fixed capital is often referred to simply as a 'machine'. Contrast circulating capital.
• Forward substitution of labor: The association of a higher rate of profits, or lower wage, around a switch point with a cost-minimizing technique in which, in one industry, the labor per unit of gross output produced is larger. Contrast with reverse substitution of labor.
• Four-technique pattern of switch points: Occurs when there is a switch point at which four wage curves intersect.
• Intensive rent: A price paid for a year's services for land under cultivation due to the need to operate more than one process on that land to satisfy requirements for use while prices of production prevail.
• Intermediate good: An old machine.
• Joint production: The phenomenon in which some production process produces more than one commodity, such as wool and mutton. Fixed capital, in which a production process produces a finished good and a machine one year older than it was when used as an input is an example. Land, which is both an input to a production process and is an unchanged output, along with a finished good, provides another example.
• Leontief input-output matrix: A matrix of coefficients of production in models of circulating capital, where each coefficient is the amount of a specified commodity needed in the production of a unit amount of another specified commodity. Leontief matrices are often supplemented by vectors of labor coefficients, matrices for land inputs, and so on.
• Market prices: Prices existing in markets at a particular moment in time. Market prices are consistent with inequalities in the quantities supplied and demanded and with momentary variations in the rates of profits among industries. Contrast with prices of production.
• Natural prices: See prices of production.
• Normal prices: See prices of production.
• Order of efficiency: See order of fertility.
• Order of fertility: In models with extensive rent, the order in which lands of different types are taken into cultivation, at a given rate of profits or a given wage, as the quantities in requirements for use expand. Also known as the order of efficiency.
• Order of rentability: In models with extensive rent, the order of lands of different types from high rent per acre to zero rent, at a given rate of profits or a given wage.
• Pattern (of switch points) for the requirements for use: Occurs with an indeterminancy in prices and levels at which processes are operated in the cost-minimizing techniques at a given rate of profits. This indeterminancy arises in models of joint reproduction due to the need to satisfy requirements for use.
• Pattern (of switch points) in the r-order of fertility: Occurs when a switch point associated with a change in the order of fertility of land not on the margin is at the same rate of profits as a switch point on the axis for the rate of profits.
• Pattern (of switch points) in the w-order of rentability: Occurs when a switch point associated with a change in the order of fertility of land not on the margin is at the same wage as a switch point on the axis for the wage.
• Pattern (of switch points) over the axis for the rate of profits: Occurs when there is a switch point at a wage of zero.
• Pattern (of switch points) over the wage axis: Occurs when there is a switch point at a rate of profits of zero.
• Prices of production: Given technology, the rate of profits or the wage, and requirements for use, prices of commodities consistent with the smooth reproduction of a capitalist economy. Contrast with market prices.
• Process: A process of production is specified by the quantities of labor, of a specified type of land, and of specified commodities needed to produce a specified output. Under joint production, the output can consist of more than one commodity. A technique consists of a set of processes.
• Rate of profits: The quotient of the difference between revenue and costs in a process and the costs paid in advances at the start of the production period. The rate of profits is the same for all operated processes when prices of production prevail if there are no barriers to entry or other causes of persistent differences among industries.
• Recurrence of processes: Occurs when a process is in the cost-minimizing techniques, at two disjoint ranges of the rate of profits, while that process is not in the techniques cost-minimizing at the rates of profits between these two ranges. The recurrence of processes always arises when techniques recur, but the recurrence of processes can occur without the recurrence of techniques.
• Recurrence of techniques: Occurs when one technique is cost-minimizing at two disjoint ranges of the rate of profits, while one or more other techniques are cost-minimizing at the rates of profits between these two ranges. The recurrence of techniques always arises when techniques reswitch, but the recurrence of techniques can occur without the reswitching of techniques.
• Requirements for use: The level and composition of net output or of a consumption basket, specified as given in models of production.
• Reswitching of techniques: Occurs when one technique is cost-minimizing at two disjoint ranges of the rate of profits, while another technique is cost-minimizing at the rates of profits between these two ranges.
• Reswitching pattern (of switch points): Occurs when two wage curves are tangent at a switch point.
• Reverse substitution of labor: The association of a higher rate of profits, or lower wage, around a switch point with a cost-minimizing technique in which, in one industry, the labor per unit of gross output produced is smaller. Contrast with forward substitution of labor.
• Scale factor for the rates of profits: When markups among industries hold persistent and stable ratios among themselves, a scale factor that determines the rate of profits from relative markups. See the rate of profits.
• Single production: See circulating capital and contrast with joint production.
• Sraffa effect: The reswitching of techniques, capital reversing, the reverse substitution of labor, the recurrence of techniques, the recurrence of processes, and other effects discovered through the analysis of prices of production that are inconsistent with obsolete marginalist dogmas.
• Sraffa matrix: A Leontief matrix for a viable technique when at least one commodity is basic and the maximum rate of profits for the submatrix of non-basic commodities exceeds the maximum rate of profits for the submatrix for basic commodities. See pp. 123-124 in Kurz and Salvadori (1995).
• Structural economic dynamics: The variation in the relative sizes of industries and in prices of production as the result of technical progress, variation in market structure, variations in the rate of growth, and variation in the relative quantities of commodities in consumption baskets.
• Switch point: A point at which two wage curves intersect. Often defined to apply only to switch points on the wage frontier.
• Technique: A set of processes. In models of circulating capital, a technique contains one process for producing each commodity in the gross output of an economy.
• Three-technique pattern of switch points: Occurs when there is a switch point at which three wage curves intersect.
• Wage curve: For a given technique, the wage as a function of the rate of profits in a system of prices of production. Also known as a wage-rate of profits curve.
• Wage frontier: In models of circulating capital, the outer envelope of wage curves. Also known as the wage-rate of profits frontier or, misleadingly, the factor-price frontier.
• Wicksell effect, price: The variation in the numeraire value of capital goods with the rate of profits for a given technique.
• Wicksell effect, real: The variation in the numeraire value of capital goods with the technique at a given rate of profits. Around a switch point with a negative real Wicksell effect, a higher wage or lower rate of profits is associated with a larger value of capital per person-year employed in a stationary state.

Elsewhere

Why Rationality is Wrong

• Above is a video by "Dr. Skeleman", first in a series.
• Nick Romeo, in The New Yorker, on The CORE textbook.
• Steve Keen's obituary of Janos Kornai.
• J. Barkley Rosser's comments on Kornai's passing. I feel I should have more to say. I recommend autobiography, By Force of Thought: Irregular Memoirs of an Intellectual Journey, although it is somewhat dry.
• J. Barkley Rosser's obituary of Peter Flaschel

Some Kinds Of Rent

Special Cases in the Analysis of Rent

Type	Land	Agricultural Processes	Industrial Processes
Extensive rent	Multiple types of land, each of a given quality	For a given type of land, one process producing corn is available	For a given commodity other than corn, one process for producing it is available
Intensive rent proper	One type of homogeneous land	For the given type of land, multiple processes are available for producing corn	For a given commodity other than corn, one process for producing it is available
External intensive rent	One type of homogeneous land	For the given type of land, one process for producing corn is available	For a given commodity other than corn, multiple processes for producing it are available

Economists have explored several kinds of rent in post-Sraffian price theory (Kurz and Salvadori 1995: 279). Suppose, as a simplifying assumption, that one commodity, 'corn', can be produced on land. Land is a non-produced commodity that emerges from a production process unchanged. Furthermore, assume that no pure joint production occurs otherwise. Let requirements for use be specified as a vector of net outputs.

The table at the head of this post lists three kinds of rent. They are characterized by the appending of three additional assumptions. One assumption deals with whether all land is homogeneous, or whether multiple types of land exist. Another assumption states whether one or more than one process is known for operating on any of the given types of land. A final assumption concerns whether different processes are available to produce commodities that do not require direct inputs of land in their production.

As far as I know, a general model of rent, short of the general theory of joint production, has yet to be developed that considers the relaxation and mixing of these assumptions. Those building on the work of Alberto Quadrio Curzio, I guess, have a ways to go. (I have just started reading the reference below.) Absolute rent may be introduced by postulating persisting, non-uniform ratios of rates of profits across sectors. An obvious generalization would consider the possibility of producing more than one agricultural commodity. In a mixed model of extensive and intensive rent, more than one type of land would exist, and more than one production process would be available for at least some types of land. Furthermore, one might introduce fixed capital, thereby raising the question of the cost-minimizing choice of the economic life of machines.

My impression is that results of the circulating capital model generalize to simple models of extensive rent. The dependence of the price system on requirements for use in models of extensive rent, however, is an important difference in the models. Once one considers other types of rent or any of the above complications, issues that arise in general models of joint production also arise in models of rent. These issues include upward-sloping wage curves on the frontier and the non-uniqueness or the non-existence of a cost-minimizing technique at a given rate of profits.

Reference

• Baranzini, Mauro L., Claudia Rotondo, and Roberto Scazzieri. 2015. Resources, Production and Structural Dynamics. Cambridge: Cambridge University Press.

On David Card's Nobel

The Sveriges Riksbank prize in economic sciences in memory of Alfred Nobel this year goes to David Card, Joshua Angrist, and Guido Imbens. I cannot say much about instrumental variables, Angrist, or Imbens. Since I have been pointing to Card's work with Alan Krueger on minimum wages for decades, I thought I might say somthing about his half of the prize.

I do not have much new to say. I find both natural experiments and meta-analysis intriguing.

Both Card and Krueger's natural experiments with minimum wages and their meta-analysis have been superceded. Maybe 'transcended' or 'replicated' would be better terminology. That is why, in my 2019 paper in Strucutral Change and Economic Dynamics, I reference Andrajit Dube and his colleagues, not Card and Krueger. Also, David Neumark's quibbles with Card are currently uninteresting. (Any reporter talking to Neumark should note he started out with funding from a consortium of fast food joints.)

I object to attempts to explain the lack of impact of minimum wages on employment by the theory of monopsony. Economists have known, for over half a century, that wages and employment cannot, even under ideal conditions, be explained by the interaction of well-behaved supply and demand curves in the labor market. In marginalist theory, the supply of labor is derived from utility-maximizing households trading off leisure and commodities to consume. The demand for labor is supposed to be derived from profit-maximizing firms. But no such valid derivation goes through if firms produce some commodities with the use of previously produced commodities, that is, capital goods. This well-established result is widely ignored, with no pretence at justification.

A Structure in Parameter Space with Three Patterns Across The Wage Axis

Figure 1: Three Patterns Across The Wage Axis And One Three-Technique Pattern

This post continues the approach in this post and in this post. As previously stated, I consider the same two examples. In both examples, three processes are known for producing the numeraire, called "corn". In the example for the left panel, corn is a non-basic commodity, and a different basic commodity is used in each of the three techniques. In the example for the right panel, all three corn-producing processes require inputs of labor power, corn, and iron (in different proportions), and managers firms know of a single process for producing iron. Both commodities are basic in this second example. The examples are also parametrized differently.

In both panels, loci for three patterns of switch points on the frontier and over the wage axis terminate at a point that is also the terminus for a locus for a three-technique pattern of switch points. For the example illustrated by the right panel, a second switch point exists on the wage frontier for a larger rate of profits than the one concerned in the patterns of switch points. This structure in parameter space is also depicted as Figure 2 in my paper, 'Fluke switch points in pure fixed capital systems', for a quite different example.

This generic structure in these parameter spaces is obscured by the dotted line in the panel on the right. Along it, the wage curves for the switch point between the Beta and Gamma techniques intersect at a rate of profits of zero. But this switch point is not on the wage frontier. Below and to the right of the dotted line, the switch point between Beta and Gamma on the frontier exhibits capital-reversing, while above it that switch point has a negative real Wicksell effect. Thus, the dotted line is not associated with a change in the number of sequence of switch points along the wage frontier. But it is associated with the change of the direction of real Wicksell effects around one of these switch points.

So here is another perhaps universal structure, in some sense, in parameter spaces associated with the analysis of the choice of technique in models of prices of production.

A Structure in Parameter Space With Three Patterns Across The Axis For The Rate Of Profits

Figure 1: Three Patterns Across The r Axis And One Three-Technique Pattern

This post continues the approach in this post. I consider the same two examples. In both examples, three processes are known for producing the numeraire, called "corn". In the example for the left panel, corn is a non-basic commodity, and a different basic commodity is used in all three techniques. In the example for the right panel, all three corn-producing processes require inputs of labor power, corn, and labor (in different proportions), and managers firms know of a single process for producing iron. Both commodities are basic in this second example. The examples are also parametrized differently.

These examples are all part of my investigation of how reswitching, capital-reversing, a reverse substitution of labor, process recurrence, and so on can emerge and disappear with perturbations of parameters in post Sraffian models of prices of production.

I call a case when a switch point exists on the wage frontier at a wage of zero a "pattern of switch points over the axis for the rate of profits". When three wage curves intersect on the frontier at a single switch point, I say this is a "three-technique pattern" of switch points.

I claim that the two panels in the figure at the top of this post are the same, at some level of abstaction. Suppose that in the left panel, replace every instance of "Alpha, Gamma" is replaced by S₁. Suppose every instance of "Beta" is replaced with "Gamma" and every remaining instance of "Alpha" with "Beta". Let every instance of S₁ be replaced with "Alpha". Rotate clockwise somewhat and stretch and otherwise distort the regions.

In both panels, one will then have a region labeled with "Alpha", alone. And it will be bounded by two loci, each designating parameters for a pattern of switch points over the axis for the rate of profits. The region diagonally opposite is then bounded by a locus for a third pattern of switch points over the axis for the rate of profits and a locus for a three-technique pattern.

In other words, loci for three patterns of switch points terminate at a point that is also the terminus for a locus for a three-technique pattern of switch points. For the example illustrated by the left panel, a second switch point exists on the wage frontier for a smaller rate of profits than the one concerned in the patterns of switch points.

So here is another generic structure in the parameter spaces relating to the analysis of the choice of technique.

Elsewhere

• Alex Thomas on Krishna Bharadwaj as an ideal economist.
• Her daughter, Sudha Bharadwaj is a political prisoner.
• The first page of this Jeremy Rudd paper is getting noticed. A lot of mainstream economics is "arrant nonsense."
• National Public Radio has a rememberance of Charles Mills.
• Liam Bright has a tribute, too.

A Structure In Parameter Space

Table 1: A Common Structure Example

1.0 Introduction

This post presents partitions of (a part of) parameter space for two examples of models of prices of production with a choice of technique. The examples have a different structure and are parametrized differently. Yet, I want to argue, the partitions are the same, at some level of abstraction.

2.0 Thing 1

The first example is an instance of the Samuelson-Garegnani model. Table 1 presents the coefficients of production for this example. Each coefficient specifies the units of input needed to produce a unit output of the commodity for the given industry. Corn is the numeraire, and is not an input into any industry. It is non-basic in Sraffa's terminology. Three processes are available for producing corn, each distinguished by the capital good used in that process. In a technique, the process that produces that capital good and the given corn-producing process are operated. The techniques are labeled Alpha, Beta, and Gamma, depending on the corn-producing process. The example is fully specified by assigning values to a_2,2 and a_3,3.

Table 1: The Coefficients of Production for First Example

Input	Industry
	Iron	Copper	Uranium	Corn
				Alpha	Beta	Gamma
Labor	1	17328/8281	1	1	361/91	3.63505
Iron	1/2	0	0	3	0	0
Copper	0	a2,2	0	0	1	0
Uranium	0	0	a3,3	0	0	1.95561
Corn	0	0	0	0	0	0

Given the technique (with the parameters assigned numberical values) and, say, a feasible rate of profits, one can solve for the real wage and prices of production. At each feasible rate of profits, the choice of technique is determined by cost. It is the cost-minimizing one. In general, the choice of technique varies with the rate of profits. One could also take the wage as given, and find the rate of profits as a function of the wage.

To go on, I draw upon my publications. I call a switch point in which two wage curves are tangent at that point a reswitching pattern (of switch points). A three-technique pattern (of switch points) is a switch point in which three wage curves intersect. Both of these patterns exist in this example.

The left panel in Figure 1 above partitions a portion of the parameter space for this model. I label each region by the cost-minimizing techniques along the wage frontier, in the order of an increasing rate of profits. The partitions are labeled by the corresponding pattern of switch points. This case illustrates how a perturbation of coefficients of production can lead to the emergence of the reswitching of techniques.

3.0 Thing 2

In the second example, two commodities are produced, and they both are basic commodities in all three techniques. Table 2 presents the coefficients of production for this example. I have introduced technological change in this model The coefficients of production for producing corn with the Beta and Gamma processes decrease exponentially. They decrease at different rates for the two processes, but all coeficients for a process descrease at the same rate.

Table 2: The Coefficients of Production for Second Example

Input	Industry
	Iron	Corn
		Alpha	Beta	Gamma
Labor	1	5191/5770	(305/494) e(3/20) - σt	(19/20) e(3/10) - φt
Iron	9/20	1/40	(3/1976) e(3/20) - σt	(1/40) e(3/10) - φt
Corn	2	1/10	(229/494) e(3/20) - σt	(1/10) e(3/10) - φt

Given the coefficients at a particular time, one can find prices of production and the cost-minimizing technique. Here, too, the analysis of the choice of technique can be performed by constructing the wage frontier. The model is parametrized by (σ t) and (φ t). The right panel in Figure 1 shows a portion of this parameter space for this model.

4.0 Conclusion

I want to say that the two panels in Figure 1 depict the same structure. The labeling of techniques is different, and one panel requires the other be rotated and stretched, as is typical of topological structures. In both cases, the loci for two reswitching patterns intersects. At that intersection, one wage curve on the wage frontier is tangent to two other wage curves, each at a separate switch point. Each locus for the reswitching patterns terminates at a point of tangency for a locus of three-technique patterns. And a three-technique pattern forms an arc connecting those two points of tangency.

Other common structures can be found in the parameter spaces of models of prices of production with the choice of technique.l

John Stuart Mill Illustrates Charles Mills' Racial Contract

Here is John Stuart Mill stating a principle that sounds noble, and then immediately making a strange caveat.

"The object of this Essay is to assert one very simple principle, as entitled to govern absolutely the dealings of society with the individual in the way of compulsion and control, whether the means used be physical force in the form of legal penalties, or the moral coercion of public opinion. That principle is, that the sole end for which mankind are warranted, individually or collectively, in interfering with the liberty of action of any of their number, is self-protection. That the only purpose for which power can be rightfully exercised over any member of a civilized community, against his will, is to prevent harm to others. His own good, either physical or moral, is not a sufficient warrant. He cannot rightfully be compelled to do or forbear because it will be better for him to do so, because it will make him happier, because, in the opinions of others, to do so would be wise, or even right. These are good reasons for remonstrating with him, or reasoning with him, or persuading him or entreating him, but not for compelling him, or visiting him with any evil, in case he do other wise. To justify that, the conduct from which it is desired to deter him must be calculated to produce evil to some one else. The only part of the conduct of any one, for which he is amenable to society, is that which concerns others. In the part which merely concerns himself, his independence is, of right, absolute. Over himself, over his own body and mind, the individual is sovereign.

It is, perhaps, hardly necessary to say that this doctrine is meant to apply only to human beings in the maturity of their faculties. We are not speaking of children, or of young persons below the age which the law may fix as that of manhood or womanhood. Those who are still in a state to require being taken care of by others, must be protected against their own actions as well as against external injury. For the same reason, we may leave out of consideration those backward states of society in which the race itself may be considered as in its nonage. The early difficulties in the way of spontaneous progress are so great, that there is seldom any choice of means for overcoming them; and a ruler full of the spirit of improvement is warranted in the use of any expedients that will attain an end, perhaps otherwise unattainable. Despotism is a legitimate mode of government in dealing with barbarians, provided the end be their improvement, and the means justified by actually effecting that end. Liberty, as a principle, has no application to any state of things anterior to the time when mankind have become capable of being improved by free and equal discussion. Until then, there is nothing for them but implicit obedience to an Akbar or a Charlemagne, if they are so fortunate as to find one. But as soon as mankind have attained the capacity of being guided to their own improvement by conviction or persuasion (a period long since reached in all nations with whom we need here concern ourselves), compulsion, either in the direct form or in that of pains and penalties for non-compliance, is no longer admissible as a means to their own good, and justifiable only for the security of others."

-- J. S. Mill, On Liberty

The second paragraph in that quotation above is no abstract theoretical observation. As I understand it, Mill, following his father, had a day job in the East India Company, eventually becoming Chief Examiner of Correspondence. I gather that that was a fairly prominent position.

Mill, in On Liberty is not writing about a social contract, unlike Hobbes, Locke, and Rousseau, for example. But this book is a classic of liberal political philosophy that, when looked at from a subaltern position, has a dark racial underside. A reader of Charles Mills might be sensitized to see this.

Reference

• John Stuart Mill. 1859. On Liberty
• Charles Mills. 1997. The Racial Contract. Cornell University Press

Summary of Some Conclusions From My Research Program

This blog, over years, presents a welter of fluke cases. I created many of the numerical examples to illustrate the reswitching of techniques, capital reversing, or some such so-called 'perversity'. Fluke cases can be combined. For example, a fluke switch point at a rate of profits of zero can also be a fluke switch point at which three wage curves intersect. Or two switch points on the wage frontier can both be fluke switch points at which four wage curves, not necessarily the same, intersect. Numerical examples remain to be developed for some possibilities.

Selected Fluke Cases

Pattern of switch points over the wage axis

Pattern of switch points for the reverse substitution of labor

Pattern of switch points over the axis for the rate of profits

Reswitching pattern of switch points

Three-techniques pattern of switch points

Four-technique pattern of switch points

Pattern of switch points for the w-order of fertility

Pattern of switch points for the r-order of fertility

Pattern over the wage axis for the order of rentability

Pattern over the axis for the rate of profits for the order of rentability

Pattern for the requirements for use

The analysis and construction of fluke cases yields insights into the analysis of the choice of technique in the system of prices of production. The reswitching of techniques, capital-reversing, process recurrence, the reverse substitution of labor, the extension of the lifetime of a machine at a lower wage, and the divergence between the order of fertility and the order of rentability are not fluke cases. These possibilities can be contrasted with genuine fluke cases, in which the perturbation of parameters destroys characteristics specifying such cases. These fluke cases partition parameter spaces into regions where these so-called 'perverse' phenomena arise.

These phenomena are also more or less independent of one another. Reswitching does not occur without process recurrence, but process recurrence can arise without reswitching. The association of a smaller rate of profits around a switch point with the truncation of the lifetime of a machine may or may not be accompanied by capital-reversing. Capital reversing can arise with or without a reverse substitution of labor and vice versa. Variations in the order of fertility need not accompany variations in the order of rentability. Nor need variations in the order of rentability be accompanied by variations in the order of fertility. The divergence between the order of fertility and the order of rentability can arise in an example of the reswitching of techniques, but reswitching is not necessary for such divergences. These specific examples do not exhaust the possible combinations of Sraffa effects.

The demonstration and visualization of these results is presented in an open and disaggregated model of prices of production. The functional distribution of income between wages, rents, and profits is not specified. The approach illustrated in these blog posts, in some sense, provides an even more open model. Prices of production are consistent with the smooth reproduction of a capitalist economy. In specifying prices of production, technology, relative rates of profits among industries, and requirements for use are frozen. Fluke cases are found, on the other hand, by perturbing parameters that specify these givens for prices of production.

Whatever practical conclusions can be drawn from this widening of the horizon remain on a high level of abstraction. Characteristics of the conflict over the functional distribution of income between wages and profits can depend on struggle within the class of capitalists. Landlords, in as much as they their interests are reflected in the existence and size of the rent of specific types of land, are also affected by the conflicts between workers and capitalists and within the class of capitalists. Variations in technology, in causes of persistent differences in the rate of profits among industries, and in the requirements for can change these characteristics of these conflicts.