Persistent Unemployment: A Flaw In Capitalism

Defenders of capitalism, whether of a liberal reformist or of a more conservative variety, portray it as harmonious, in some sense. One aspect of this portrayal is that under ideal conditions, at least, markets tend to clear. This supposed tendency must apply especially to the labor market. Another aspect, put aside in this post, is the obsolete and explopoded marginal productivity theory of distribution.

Nicholas Wapshott was correct in putting this supposed tendency as a central point of contention between John Maynard Keynes and Friedrich Hayek:

"... the objections of Circus members ... suggested to Keynes what was to become a pivotal element of The General Theory, that overall output was not fixed and could be raised through increased investment to a point where everyone in an economy was employed. It was this first slender thread of thought that led to Keynes's wholesale contradiction of the claim of classical economists like Hayek that an economy, left to its own devices, in the long run inevitably came to rest at a state of equilibrium where there was full employment. Keynes was to argue in The General Theory that in the short and medium terms an economy could reach equilibrium with considerable unemployment and that the full employment equilibrium predicted by classical economists too often proved to be elusive. Keynes believed that the chronic unemployment endured in Britain and America in the 1920s and 1930s was evidence that the full employment equilibrium was a fallacy." -- Nicholas Wapshott. 2011. Keynes Hayek: The Clash That Defined Modern Economics. New York: W. W. Norton and Co.: p. 128.

I can think of three positions to take here:

• No such tendency for markets, including labor markets, to clear in any run.
• Such tendency would exist, but imperfections and rigidities prevent this tendency from manifesting itself.
• Although imperfects and such like exist, they are not empirically important enough to prevent this tendency for markets to clear, if we are patient enough.

Keynes and Piero Sraffa have convincingly argued for the first position, that no tendency exists for markets, including the labor market to clear. Keynes had many arguments. For example, workers and capitalists may have no way of negotiating about the real wage. Suppose unemployment is high. A system-wide drop in the money wage occurs. Prices might drop as a result, leaving the real wage unchanged. If the money supply is exogenous, the real money supply is increased in some sense. You might get money policy implemented by trade unions. But this effect is outside the usual partial equilibrium story about the labor 'market'. Furthermore, capitalists would be inclined not to open new factories in a period of declining wages and prices. They can anticipate that new factories opened tomorrow, instead of today, will have decreased costs and a decreased burden of debt. With declining prices of output, too, capitalists will put off investment.

Sraffa showed that prices, in a long run theory, do not have the properties needed to support just-so stories about supply and demand. Prices of production can support a long run theory in which labor markets do not necessarily clear. Say's law is false.

For the second position, I think of failures of competition, information asymmetries, search costs, incomplete contracts, principal agent problems, and so on. Many misleadingly characterize Keynes as holding this position. I guess those investigating these imperfections often advocate government interventions to remove or mitigate their effects. A radical position is that at least some of these imperfections cannot be removed. Government might then need to simulate how markets would work without these limitations. Is this the position of ordoliberalism?

Those with the third position would be for a more laissez faire approach.

But to return to the first position. Suppose no tendency exists for markets, including the labor market, to clear. Keynes recommended a "somewhat comprehensive socialisation of investment". But he was not clear on what that means.

Fake Switch Points With Fixed Capital And Extensive Rent

Figure 1: An Enlargement Of Wage Curves

1.0 Introduction

This long post is a start at addressing this problem statement. With a bit of improvement on the scholarship in the introduction and an appendix presenting the solutions of the price systems, it would be an article to be submitted to a journal. But I will not do this before exploring other examples. I would like to have an upward-sloping wage frontier, in particular.

Models of the production of commodities with circulating capital have certain nice properties. Models of pure fixed capital and of extensive rent have the same nice properties, for the most part. This article contends that a model that combines pure fixed capital and extensive rent need not have these properties. Many of the issues that arise in models of general joint production also arise in a model combining fixed capital and extensive rent.

The choice of technique can be analyzed in models with circulating capital alone by constructing the outer envelope of the wage curves for each technique. Each wage curve slopes down. The wage, for a technique, is lower the higher the rate of profits. The cost-minimizing technique at a given rate of profits is unique, except at switch points. The "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang 2019). The wage and prices of production are unique functions of the rate of profits. If a feasible technique exists with a defined wage and prices of production at a given rate of profits, then a cost-minimizing technique exists. A market algorithm (Bidard 1990 and Vienneau 2017) converges, without going into a cycle.

None of these properties are necessarily true in a general system of joint production.

Models with fixed capital are special cases of models of joint production. Bidard (2004), Kurz & Salvadori (1995), Pasinetti (1980), Schefold (1989) and Woods (1990) develop the theory of pure fixed capital. Huang (2019) is a survey. In models of pure fixed capital, the choice of technique can be analyzed by constructing the outer frontier of the wage curves for the various techniques.

In the simplest model of extensive rent, a single agricultural commodity is produced, on each type of land, with a single production process. All types of land, except for one, are farmed to the full extent of their endowment. No alternate processes are available on any type of land for producing the agricultural commodity. Prices of production are such that a single rate of profits is made in all operated processes. Rent is obtained by landlords who own the scarce lands. The type of land that is only partially farmed is not scarce and does not pay a rent.

At a given rate of profits, the techniques in a model of extensive rent can be ordered by the wage. This is the order of efficiency, also known as the order of fertility. The cost-minimizing technique is the first technique, in decreasing order of wages, that is feasible and, away from switch points, has positive rents on all lands that are fully farmed. In this sense, the choice of technique can be analyzed by the construction of the wage frontier in models of extensive rent.

The choice of technique cannot generally be analyzed by the construction of the wage frontier in models that combine fixed capital and extensive rent. In particular, an example is given with a fake switch point. The wage curve on the frontier intersects another wage curve. Yet that point of intersection is not a switch point.

2.0 Numeric Example of the Model

This article presents a model combining fixed capital and extensive rent, by a numerical example. The numerical example is specified by the definition of the technology, endowments of land, and requirements for use. An analysis of quantity flows identifies which techniques are feasible at a given level of requirements for use. The analysis of the choice of technique requires the examination of the solutions to the price system for the techniques.

2.1 Technology, Endowments, and Requirements for Use

The parameters for the model specify the technology, endowments of land, and requirements for use. I assume the existence of two types of land. More than one type is required for this model to exhibit extensive rent. With only two kinds of land, contrasting the orders of efficiency and of rentability is uninteresting. The order of efficiency is the order in which different types of land are introduced into cultivation as net output expands. The order of rentability sorts the lands by rent per acre. When both types of land are farmed, one type will be only partially farmed. It has a rent of zero, the other type of land obtains a positive rent. These orders can be in completely reversed order in models with more lands and both extensive and intensive rent.

Fixed capital is another aspect of joint production, in addition to land, in this model. A newly produced machine can be used for two years in production. Machines are assumed not to be consumption goods. In models of pure fixed capital, new machines but not old machines can be consumer goods. This model seems to be of the minimal complexity to investigate a combination of land-like natural resources and fixed capital in a model with the production of multiple commodities that is otherwise of single production alone.

The technology is specified by the coefficients of production for five processes. Each column in Table 1 shows the person-years of labor, acres of either type of land, bushels of corn, and numbers of new and old machines required as inputs to operate a process at unit level. The outputs of corn and machines, new and old, per unit level of each process are shown in Table 2. Machines are an industrial product which needs no land to produce. The laborers produce corn on land from inputs of corn and machines. Old machines are produced jointly with corn from inputs of new machines. Each old machine is of a type customized to the land on which it was produced. Old machines cannot be transferred from one type of land to another. They are assumed to be capable of free disposal. Formally, free disposal of an old machine of, say, type 1 is specified by assuming the existence of another process duplicating the second process, but without an output of an old machine. Each process is assumed to exhibit constant returns to scale (CRS) and to require a year to complete. The coefficients of production for the first three processes, other than those for land, are taken from a reswitching example (Schefold 1980).

Table 1: Inputs for Five Processes Comprise the Technology

Input	Industry
	Machine	Corn
	Process I	Process II	Process III	Process IV	Process V
Labor	a0,1 = 1/10	a0,2 = 43/40	a0,3 = 1	a0,4 = 1	a0,5 = 43/40
Type 1 Land	c1,1 = 0	c1,2 = 1	c1,3 = 1	c1,4 = 0	c1,5 = 0
Type 2 Land	c2,1 = 0	c2,2 = 0	c2,3 = 0	c2,4 = 1	c2,5 = 1
Corn	a1,1 = 1/16	a1,2 = 1/16	a1,3 = 1/4	a1,4 = 1/16	a1,5 = 3/10
New Machines	a2,1 = 0	a2,2 = 1	a2,3 = 0	a2,4 = 1	a2,5 = 0
Type 1 Old Machines	a3,1 = 0	a3,2 = 0	a3,3 = 1	a3,4 = 0	a3,5 = 0
Type 2 Old Machines	a4,1 = 0	a4,2 = 0	a4,3 = 0	a4,4 = 0	a4,5 = 1

Table 2: Outputs for Five Processes Comprise the Technology

Input	Industry
	Machine	Corn
	Process I	Process II	Process III	Process IV	Process V
Corn	b1,1 = 0	b1,2 = 1	b1,3 = 1	b1,4 = 6/5	b1,5 = 4/5
New Machines	b2,1 = 1	b2,2 = 0	b2,3 = 0	b2,4 = 0	b2,5 = 0
Type 1 Old Machines	b3,1 = 0	b3,2 = 1	b3,3 = 0	b3,4 = 0	b3,5 = 0
Type 2 Old Machines	b4,1 = 0	b4,2 = 0	b4,3 = 0	b4,4 = 1	b4,5 = 1

The specification of model parameters is completed with endowments and requirements for use. Assume 100 acres of each type of land exist. The required net output is assumed to be anywhere from 107.5 bushels of corn to 160 bushels. A required net output in this range, but not at the endpoints, is such that all and only the techniques which require both types of land to be farmed are feasible.

2.2 Techniques and Feasibility

A technique is defined by which processes are operated, which type of lands are left unfarmed, which are partially farmed, and which are farmed to the full extent of their endowment. Rents can only be obtained on the last. Twelve techniques (Table 3) are defined for this technology. The capital goods that are used up in operating a technique can be reproduced. A net output remains, consisting, in the example, solely of corn.

Table 3: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2
Alpha	I, II	Partially farmed	Fallow
Beta	I, II, III	Partially farmed	Fallow
Gamma	I, IV	Fallow	Partially farmed
Delta	I, IV, V	Fallow	Partially farmed
Epsilon	I, II, IV	Fully farmed	Partially farmed
Zeta	I, II, III, IV	Fully farmed	Partially farmed
Eta	I, II, IV, V	Fully farmed	Partially farmed
Theta	I, II, III, IV, V	Fully farmed	Partially farmed
Iota	I, II, IV	Parially farmed	Fully farmed
Kappa	I, II, III, IV	Parially farmed	Fully farmed
Lambda	I,II, IV, V	Parially farmed	Fully farmed
Mu	I, II, III, IV, V	Parially farmed	Fully farmed

Only scarce lands obtain a rent, and which lands are scarce varies with the technique. No land is scarce in the Alpha, Beta, Gamma, and Delta techniques. One land is farmed and not to its full extent. Type 1 land is scarce in the Epsilon, Zeta, Eta, and Theta techniques, while type 2 land is scarce in the remaining four techniques. The techniques also vary in the economic life of the machine, one or two years, on each type of land. Under the assumptions, the first four techniques are not feasible. Only Epsilon through Mu are feasible.

2.3 The Price Systems

The modeled economy consists of three classes: workers, landlords, and capitalists. Capitalists buy inputs and hire workers who they direct to produce commodity outputs. In agriculture, farmers pay rent on scarce land. The capitalists choose the processes to operate based on cost. Accordingly, prices must be analyzed.

A system of equations is associated with each technique. As an example of a technique, consider Kappa. The following four equations present its price system:

a_1,1(1 + r) + w_κ(r) a_0,1 = p_1,κ(r)

[a_1,2 + p_1,κ(r)](1 + r) + w_κ(r) a_0,2 = b_1,2 + p_2,κ(r)

[a_1,3 + p_2,κ(r)](1 + r) + w_κ(r) a_0,3 = b_1,3

[a_1,4 + p_1,κ(r)](1 + r) + rho_2,κ(r) c_2,4 + w_κ(r) a_0,4 = b_1,4

Table 4 defines the price variables. These equations show the same rate of accounting profits is obtained on the value of the capital goods advanced at the start of the year. Rent and wages are paid out of the surplus product at the end of the year.

Table 4: Functions for Solutions for Kappa Technique

Function	Defintion
p1,κ(r)	The price of a new machine, in bushels per machine.
p2,κ(r)	The price of an old type 1 machine, in bushels per machine.
p3,κ(r)	The price of an old type 2 machine, in bushels per machine.
rho1,κ(r)	The rent of type 1 land, in bushels per acre.
rho2,κ(r)	The rent of type 2 land, in bushels per acre.
wκ(r)	The wage, in bushels per person-year.

Under Kappa, machines are operated for their full physical life in farming type 1 land. Accordingly, the price of old type 1 machines appears in the price equations, with process II producing old machines jointly with corn. Machines are discarded after one year in farming type 2 land. Consequently, the price of type 2 old machines is zero and their price does not appear in the price equations in Display 1.

Type 2 land is fully farmed under Kappa. It is scarce, and its rent appears in the above equations. Type 1 land, on the other hand, is not scarce. Its rent is zero. The following display expresses that type 2 machines are not used and that that type 1 land is free:

p_3,κ(r) = 0, rho_1,κ(r) = 0

2.4 On the Solutions of the Price Systems

Given the rate of profits, the price system for each technique can be solved. The solution for a technique consists of the functions listed in Table 4. Figure 2 graphs the wage curves for each technique in the example; Figure 1 is an enlargement. The first two intersections of wage curves in Figure 1 cannot be distinguished in Figure 2. The wage frontier is composed of the wage curves for the cost-minimizing techniques. Section 3 demonstrates that the Iota and Kappa techniques are cost-minimizing in the indicated ranges of the rate of profits. In this example, each wage curve is downward-sloping. Wage curves can be upward-sloping off the outer wage frontier in models of fixed capital. In the example with fixed capital and extensive rent, the wage frontier is neither the outer frontier of all wage curves nor the inner frontier. The wage frontier need not be always downward-sloping in other examples.

Figure 2: Wage Curves

The solutions of the price systems also provide rent curves, rent per acre, as a function of the rate of profits. Figure 3 plots the rent curves for the four techniques in which rent is obtained on type 1 land. Only the first quadrant is shown. The rent curves for Epsilon and Zeta lie entirely below the abscissa. Rent is zero for Eta at the intersection of the rent curves for Alpha and Delta. That is, the cost of producing corn is the same with process II or a combination of processes IV and V at this rate of profits. By the same logic, rent is zero for Theta at the intersection of the wage curves for Beta and Delta.

Figure 3: Rent on Type 1 Land

Figure 4 shows the rent curves for the four techniques in which rent is obtained on type 2 land. The zero for the rent curve for Lambda is at the rate of profits at which the wage curves for Alpha and Delta intersect. The zero for the rent curve for Mu is at the fake switch point. These zeros cannot be distinguished by eye in Figure 4. All prices are the same for both techniques at switch points between Iota and Kappa. Thus, the rent curves for type 2 land intersect for these techniques at switch points.

Figure 4: Rent on Type 2 Land

Table 5 summarizes the claims about the variation in the cost-minimizing technique with the rate of profits. Under Kappa, the machine is operated for its full physical life on type 1 land, but only for one year on type 2 land. Type 2 land is scarce. Under Iota, the machine is discarded after one year, no matter which type of land it is operated on. Type 2 land remains scarce. This is a reswitching example. Kappa is adopted at low and high rates of profits. The switch point at a rate of profits of 50 percent is an example of capital-reversing.

Table 5: Cost-Minimizing Techniques

Region	Range	Technique	Old Machines	Rents
1	0 < r < 1/3	Kappa	p2,κ > 0, p3,κ = 0	rho1 = 0, rho2 > 0
2	1/3 < r < 1/2	Iota	p2,ι = 0, p3,ι = 0
3	1/2 < r < 258.8%	Kappa	p2,κ > 0, p3,κ = 0

3.0 Results

The solutions of the price equations can be synthesized to justify claims about the cost-minimizing techniques. In models of extensive rent, the cost-minimizing technique is found, at a given rate of profits, by working downwards through the wage curves until one is found, with non-negative rents, for a feasible technique. That method does not work in this example with fixed capital.

Table 6: Some Properties of Feasible Techniques

Technique	Old Machines	Rent
Epsilon	Not produced	rho1,ε < 0, for all r
Zeta	p2,ζ > 0, for all r	rho1,ζ < 0, for all r
Eta	p3,η < 0, for all r	rho1,η > 0, for r < 24.4%
Theta	p2,θ > 0, p3,θ < 0, for all r	rho1,θ > 0, for r < 24.8%
Iota	Not produced	rho2,ι > 0, for all r
Kappa	p2,κ > 0, for r < 1/3 or r > 1/2	rho2,κ > 0, for all r
Lambda	p3,λ < 0, for all r	rho2,λ > 0, for r > 24.4%
Mu	p2,μ > 0, for r < 1/3 or r > 1/2; p3,μ < 0, for all r	rho2,μ > 0, for r > 24.8%

The wage frontier is composed of the wage curves for the cost-minimizing techniques. The wage frontier for this example of extensive rent is neither the outer frontier nor the inner frontier of the wage curves. A wage curve for a technique contributes to the frontier, at a given rate of profits, only if all old machines produced by the technique have non-negative prices and the rent on land scarce under the technique is non-negative. Only the wage curves for Iota and Kappa satisfy these criteria (Table 6)

Figure 5: Iota Cost-Minimizing at Middling Rates of Profits

Under Iota, process II is operated on type 1 land, and process IV is operated on type 2 land. Type 2 land is fully farmed and obtains a rent. Iota is cost-minimizing if processes III and V do not pay extra profits. Extra profits are the difference between revenues and costs, where advances of capital goods are costed up at the going rate of profits. The following equations define extra profits per unit level of operation for these processes under Iota prices:

Π_{III, ι} = b_1,3 - [a_1,3(1 + r) + w_ι a_0,3]

Π_{V, ι} = b_1,5 - [a_1,5(1 + r) + rho_2,ι c_2,5 + w_ι a_0,5]

As shown in Figure 5, capitalists obtain extra profits by operating process III when Iota prices rule at low and high rates of profits. In other words, capitalists will extend the economic life of the machine when farming type 1 lands.

Figure 6: Extra Profits Not Available at Kappa Prices by Extending Life of Machine

Process V is the only process that is not operated in the Kappa technique. Extra profits in process V, at unit level under Kappa, are defined by the following equation:

Π_{V, κ} = b_1,5 - [a_1,5(1 + r) + rho_2,κ c_2,5 + w_κ a_0,5]

Figure 6 plots these extra profits, as a function of the rate of profits. They are always negative. An analysis of extra profits under Kappa cannot find switch points with Iota, in which the economic life of the machine is truncated on type 1 land. Figure 7 plots the price of type 1 old machines. This price turns negative under Kappa at the switch points with Iota. In models of fixed capital, a negative price of an old machine signals to managers of firms that cost can only be minimized if the economic life of a machine is truncated.

Figure 7: The Price of Type 1 Old Machines

At a switch point, the prices for the two techniques for which the wage curves intersect are the same. At the fake switch point between Kappa and Eta or Theta, the wage and the price of a new machine are invariant among the techniques. The price of a type 1 old machine is equal at a positive price between Kappa and Theta. An old machine is not produced under Eta. Thus, the fake switch point cannot be a switch point between Kappa and Eta. But the price of a type 2 old machine is negative under Theta, not zero. No extra profits are available under Kappa by extending the economic life of the machine in farming type 2 land. So the switch point is a fake. The rents are also inconsistent with the switch point being genuine. Rent on type 1 land is negative under Eta and zero under both Kappa and Theta. But rent on type 2 land is positive under Kappa, not zero, as under Eta and Theta.

In a model combining fixed capital and extensive rent the intersection of a wage curve with the wage frontier for the cost-minimizing technique can be a fake.

Conclusion

A model combining fixed capital and extensive rent can exhibit at least some of the difficulties in models of general joint production that do not arise with fixed capital and extensive rent, considered separately. A fake switch point exists in the reswitching example. A simple examination of wage curves does not reveal which technique is cost-minimizing. The feasible technique with the largest wage is not necessarily cost-minimizing.

References

• D'Agata, A. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory, Metroeconomica, 35: 147-158.
• Baldone, Salvatore. 1974. Il capitale fisso nella schema teorico di Piero Sraffa. Studi Economici XXIX(1): 45-106 (Tr. In Pasinetti 1980).
• Bidard, Christian. 1990. An algorithmic theory of the choice of techniques. Econometrica 58(4): 839-859.
• Bidard, Christian. 1997. Pure joint production. Cambridge Journal of Economics 21(6): 685-701.
• Bidard, Christian. 2004. Prices, Reproduction, Scarcity. Cambridge: Cambridge University Press.
• Bidard, Christian and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28 (1): 88-97.
• Huang, B. 2019. Revisiting fixed capital models in the Sraffa framework. Economia Politica 36: 351-371.
• Kurz, Heinz and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge: Cambridge University Press.
• Pasinetti, Luigi L. (ed.). 1980. Essays on the Theory of Joint Production. New York: Columbia University Press.
• Quadrio Curzio, Alberto. 1980. Rent, income distribution, and orders of efficiency and rentability, (In Pasinetti 1980).
• Quadrio Curzio, Alberto and Fausta Pellizzari. 2010. Rent, Resources, Technologies. Berlin: Springer.
• Salvadori, Neri. 1999. Transferable machines with uniform efficiency paths. Value, Distribution and Capital: Essays in honour of Pierangelo Garegnani (ed. by G. Mongiovi and F. Petri). New York: Routledge.
• Schefold, Bertram. 1980. Fixed capital as a joint product and the analysis of accumulation with different forms of technical progress. (In Pasinetti 1980).
• Schefold, Bertram. 1989. Mr. Sraffa on Joint Production and other Essays. London: Unwin-Hyman.
• Sraffa, Piero. 1960. The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory. Cambridge: Cambridge University Press.
• Varri, Paolo. 1974. Prezzo, saggio del profitto e durata del capitale fisso nello schema teorica di Piero Sraffa. Studi Economici XXIX(1): 5-44 (Tr. In Pasinetti 1980).
• Vienneau, Robert L. 2017. The choice of technique with multiple and complex interest rates. Review of Political Economy 29(3): 450-453.
• Woods, J. E. 1990. The Production of Commodities: An Introduction to Sraffa. Atlantic Highlands, NJ: Humanities Press International.

Catalog of Capital-Theoretic "Paradoxes"

This post lists capital-theory 'paradoxes' or 'perversities'. The label of 'paradox' or 'perverse' merely indicates that they are inconsistent with traditional marginalist theory. I think of the following:

• Reswitching of techniques: Out of a given book of recipes, a technique is cost-minimizing at two ranges of the rate of profits, with some other technique cost-minimizing in-between. A technique is a combination of processes, one for each industry, that characterizes production in the economy as a whole.
• Capital-reversing: A switch point is a rate of profits or wage at which two techniques are cost-minimizing. Around a switch point with capital-reversing, a lower rate of profits is associated with a technique being replaced with one with a lower capital-intensity. Or around such a switch point, a higher wage is associated with a higher labor intensity. You can say that a labor demand curve, for the economy as a whole, slope up.
• Reverse labor substitution: Around a switch point with reverse labor substitution, a higher wage is associated with the adoption of a process in a specified industry with more labor hired per unit gross output in that industry. You can say that a sectorial labor demand function slopes up.
• Process recurrence: In a specified industry, the same process can be in the cost-minimizing techniques at two different rates of profits, with a different process being in the cost-minimizing technique in-between.
• Recurrence of truncation: The cost-minimizing technique, in models of fixed capital, can require that an old machine be discarded before its physical life ends. The recurrence of truncation occurs when the same economic life of a machine is adopted in an industry at two different ranges of the rate of profits, with a different economic life in-between.
• Reswitching of the order of fertility: In models of rent, the order of fertility is the order in which different (types of) land are cultivated, at a given rate of pofits, as output expands. This order can be the same at two ranges of the rate of profits, with a different order in-between.
• Reswitching of the order of rentability: In models of rent, the order of rentability is the order, at a given rate of profits of lands when sorted by rent per acre. This order can be the same at two ranges of the rate of profits, with a different order in-between.
• Association of the lengthing of the economic life of a machine with smaller capital-intensity: The adoption of a longer economic life of a machine can be consistent with a smaller capital-intensity.
• Association of the adoption of a machine requiring a more roundabout process with smaller capital-intensity: The cookbook can have a choice of the use of different types of machines in an industry. The technique that is more roundabout, that is, with a machine that lasts longer, can have a smaller capital-intensity.

The relationships among these different phenomena are tedious to state.. The reswitching of techniques implies that capital-reversing occurs. But capital-reversing can occur without the reswitching of techniques. Reverse labor substitution can occur without either reswitching or capital-reversing. I guess process recurrence cannot occur without reverse labor substitution, but reverse labor substitution can occur without process recurrence. The recurrence of truncation is analogous to process recurrence, but somewhat different. The recurrence of trunctation implies that reverse labor substitution occurs. This recurrence is consistent with an absence of the reswitching of techniques. The reswitching of the order of fertility is consistent with no variations in quantity flows. Which lands are fully farmed and which are only partially farmed need not vary. Thus, the reswitching of the order of fertility does not imply the existence of capital-reversing. I hasten to add that the reswitching of techniques and capital-reversing can arise in models with rent.

The above list is not complete. For example, in models of international trade, a country can specialize as in the theory of comparative advantage, with less output than in autarky. Something like the reswitching of techniques can arise in models of spatial economics. Around a city, two areas at different distances from downtown might be specialized for agriculture, with an area specialized for industry in-between. (I haven't looked into this last case in any detail.)

Many economists may have never seen the economic theory that I draw on. But the above phenomena, which might be called Sraffa effects, follow from the assumptions of mainstream economics. In other words, this post describes some elements of modern economics.

Is The Order Of Efficiency Unique At A Given Rate Of Profits In Models With Multiple Agricultural Commodities?

1.0 Introduction

This post puts forth two research questions that I think I will never get to. And it puts forth one I have started on.

2.0 Is the Order of Efficiency Well-Defined with Mutliple Agricultural Commodities?

I have worked through an example with absolute, extensive, and intensive rent. The orders of efficiency and rentability are uniquely defined in this example. The example has an industrially produced commodity and one commodity, 'corn' produced on land.

Suppose more than one agricultural commodity exists. For example corn and barley can both be produced on land. But assume that no joint production, other than the use of land, exists. Would the order of efficiency still be uniquely defined at a given rate of profits? Would it depend on the composition of corn and barley in net output, as well as the level of net output?

I see even in my example, which techniques are feasible depend on the amount of iron and corn in net output. But would multiple agricultural commodities create more issues for defining the order of efficiency?

3.0 Can Competive Marginal Products Rank Inputs in the Opposite Order of their Productivity?

Frank Hahn argues that marginal productivity, rightly understood, is entirely consistent with Sraffa's model. He concentrated on the case of single production. with a discrete technology, the value of the marginal product of labor, for example, is an interval formed out of the right-hand and left-hand derivatives of production functions.

Hahn considers intertemporal equilibrium paths in which endowments are not given. Rather, they are found as part of the solution for steady state paths. I would rather take them as limit points of non-steady state paths, perhaps with saddle-point stability. But setting these points aside, I cannot see that Hahn is otherwise wrong. Marginal productivity by itself is not a theory of income distribution.

But how does marginal productivity apply to a combination of intensive and extensive rent? In my example, I find a range of the rate of profits in which three types of land are ordered by efficiency in exactly the opposite order as rent per acre. This does not seem consistent with J. B. Clark's idea that, with marginal products, you get rewarded for your contributions (or rather the contributions of the factors you own) to production.

4.0 Which Isues Arising From General Joint Production Arise In Models Combining Extensive Rent and Fixed Capital?

I give an example. In such models, the wage frontier for the cost-minimizing technique is not the outer fronter for wages curves. So I guess, it can be upward-sloping. Can the cost-minimizing technique be non-unique away from switch points? Can I find Bidard's market economy converging to a cycle, with no cost-minimizing technique existing? D'Agata has examples in a model of inetensive rent.

Bidard And Klimovsky's Fake Switch Point

Figure 1: Wage Curves with A Fake Switch Point

1.0 Introduction

I find that, in developing an example on fixed capital and extensive rent, I need to review fake switch points. I am relying on Bidard's book, not on his paper with Klimovsky.

In this post, I just run through their example. They should have been more explicit about which techniques are feasible in their exposition.

2.0 Technology

Bidard and Klimvsky's example is one of pure joint production. The technology consists of the three processes shown in Table 1. Each column shows the inputs of person-years of labor, bushels of corn, and number of hogs needed as inputs for a unit level of operation. The outputs of bushels corn and number of hogs are also shown. Since each process has an output of more than one commodity, this technology is one of joint production. I assume constant returns to scale, and that each process takes a year to complete.

Table 1: Three Processes Comprise the Technology

Input	Processes
	Process 1	Process 2	Process 3
Labor	a0,1 = 1	a0,2 = 1	a0,3 = 1
Corn	a1,1 = 20	a1,2 = 20	a1,3 = 30
Hogs	a2,1 = 20	a2,2 = 20	a2,3 = 30
Outputs
Corn	b1,1 = 21	b1,2 = 23	b1,3 = 36
Hogs	b2,1 = 27	b2,5 = 20	b2,3 = 34

Three techniques (Table 2) can be constructed from these processes, when neither corn or hogs are a free good. Unlike in single production, a single technique cannot produce any composition of net output.

Table 2: Techniques of Production

Technique	Processes
Alpha	1, 2
Beta	1, 3
Gamma	2, 3

Another issue arises in the choice of technique that does not occur with joint production. Suppose the processes in Alpha are being operated. And suppose extra profits can be obtained with the the third process at Alpha prices. Which process in Alpha should be replaced? If the first process is replaced, the Gamma technique is adopted. If the second process is replaced, the Beta process is adopted. In single production, each process is associated with a specific industry, producing a specific commodity. In this example of joint production, considerations of feasibility are sufficient to answer this question.

3.0 Requirements for Use, Prices and the Choice of Technique

I consider two cases. The composition of net output is specified in different proportions in the two cases. Two techniques, out of three, are feasible in each case. But which technique is infeasible varies.

A price system is associated with each technique. Each process in the technique yields an equation. I assume that wages are paid out of the surplus at the end of the year. These equations for prices of production show the same rate of profits being made in both operated processes. I take net output as the numeraire. A third and final equation specifies that the price of the numeraire is unity.

The price equations can be solved, given the rate of profits. The wage, the price of corn, and the price of hogs are each a function of the rate of profits for each technique.

The question posed by the example is whether the outer frontier of the wage curves corresponds to the cost-minimizing technique. It does not. But, in the example when it does not, the wage curve on the outer frontier is not feasible.

3.1 Case 1

Suppose the requirements for use, that is, net output consists of 0.38 bushels corn and 0.62 hogs. Then Alpha is infeasible, and Beta and Gamma are feasible.

Two fluke cases are of interest here. If net output consists of 3/8 bushels corn and 5/8 hogs, all three techniques are feasible. Only the second processes are operated for the Alpha and Gamma techniques. The first and third processes are operated at positive levels for Beta. On the other had, a net output of 3/5 bushels corn and 2/5 hogs is an edge case where Alpha is infeasible, with Beta and Gamma feasible. Only the third process is operated for Beta and Gamma.

For the remainder of this section, I consider the specific intermediate case where Alpha is infeasible. Figure 2 graphs the wage curves. Only one switch point exists, at a rate of profits of 10 percent. Figure 3 shows extra profits for each profits at Beta prices. Beta is cost-minimizing for a non-negative rate of profits up to 10 percent. After that, extra profits can be obtained by operating the second process at Alpha prices.

Figure 2: Wage Curves with Alpha Infeasible

Figure 3: Case 1, Extra Profits at Beta Prices

Figure 4 shows extra profits for each process at Gamma prices. Gamma is cost-minimizing from the switch point at a rate of profits of 10 percent to the maximum rate of profits. For rates of profits smaller than in this rate, extra profits can be obtained by operating process 1.

Figure 4: Case 1, Extra Profits at Gamma Prices

In this case, the wage curve on the outer envelope for a rate of profits between 10 percent and the maximum rate of profits of 18 percent is not the wage curve for the cost-minimizing technique. But it is a wage curve, Gamma, for an infeasible techinque. If only wage curves for feasible techniques were considered, the construction of the outer frontier of wage curves would be a correct analysis of the choice of technique.

3.1 Case 2

Suppose now that net output consists of 0.13 bushels corn and 0.87 hogs. Alpha and Beta are feasible. Gamma is infeasible. (Here a fluke case is a net output of 1/8 bushels corn and 7/8 hogs. Alpha and Beta remain feasible, but only the first process is operated in each case at a positive level. Processes 2 and 3 are operated at a level of 0.)

Figure 1, at the top of this post, shows the wage curves. In this case, the wage curves on the outer frontier intersect at two points, the switch point at 10 percent and the fake switch point at approximately 13 percent.

The price system is identical among techniques at a switch point. Figure 5 shows how the price of corn varies with the rate of profits for the techniques in this case. Figure 6 shows how the price of hogs vary. You can see the switch point on these graphs of prices. No manifestation of the fake switch point is apparent.

Figure 5: Case 2, Price of Corn

Figure 6: Case 2, Price of Hogs

You can graph extra profits for each process, at prices for each technique, here too. Figure 7 graphs extra profits for Alpha. Alpha is cost-minimizing from the switch point at a rate of profits of 10 percent to the maximum at 20 percent. And Figure 8 demonstrates that Beta remains cost-minimizing for rates of profits below 10 percent.

Figure 7: Case 2, Extra Profits at Alpha Prices

Figure 6: Case 2, Extra Profits at Beta Prices

So Bidard and Klimovsky's claim that an intersection of wage curves on the outer frontier can be a fake switch point. In their example, one of the wage curves at the fake switch point is for an infeasible technique. And when the wage curve for the cost-minimizing technique lies below the outer frontier, the curve on the frontier is for that infeasible technique.

4.0 Conclusion

This example illustrates the interaction between requirements for use and the feasibility of techniques of production in models of joint production. The appearance of the wage curves depends on the numeraire. Bidard does not tie the numeraire to the requirements for use, as I do. He considers numeraires (one bushel corn for one case and one hog for the other) which cannot be feasibly produced with any technique. Likewise, Bidard does not consider feasibility. He discusses the angles between certain vectors in quantity space. I do not understand this without more studying, but perhaps it is a matter of feasibility.

Anyways, the example demonstrates the fallacy of analyzing the choice of technique by the construction of the outer wage frontier. In the first case, the cost-minimizing technique is not on the outer frontier for large rates of profits. The technique on the outer frontier is infeasible in this range.

The wage curve for the cost-minimizing technique is not always on the outer frontier in the second case either. Here, too the wage curve on the outer frontier is for a technique infeasible in this range. Furthermore, a fake switch point appears. The wage curves on the outer frontier intersect at a point where the cost-minimizing technique does not change and is unique. The prices of corn and hogs vary with the technique at the fake switch point. This variation is not possible in the single production models.

Which technique is cost-minimizing at a given rate of profits does not depend on the numeraire when the numeraire is disassociated with the net product.

I did not consider techniques in which only one process is operated to satisfy requirements for use. Perhaps there are such techniques in which one good is overproduced and becomes a free good.

References

• Bidard, Christian. 2004. Prices, Reproduction, Scarcity. Cambridge: Cambridge University Press.
• Bidard, Christian and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28 (1): 88-97.

CCC: tl; dr

1.0 Introduction

You can easily find peopole on the Internet asking for a summary of the Cambridge Capital Controversy. An answer is easy.

Supply and demand is balderdash. If you want to understand markets under capitalism, you might as well throw away most microeconomic textbooks and most introductory textbooks.

2.0 Some Expansions

No consistent, valid model in which more than one commodity is produced supports the following two mistaken stories.

Suppose the tastes of those making decisions for households change. They become more future-oriented, more willing to defer consumption. The supply of capital has increased. With an increased supply, the interest rate is driven down. The firms ultimately choose to adopt more capital-intensive techniques. They tend to equip workers with more machinery and to run exisiting machinery longer.

Suppose, again, that the tastes of those making decisions for households change. Workers now prefer consumption over leisure more than they did before. The supply of labor has increased. The real wage is driven down. Firms ultimately choose to adopt more labor-intensive techniques. Equilibrium employment in competitive markets thereby increases.

Numerical examples of capital-reversing and other so-called capital-theoretical paradoxes or perversities are enough to demonstrate that the above stories do not follow from the assumptions of mainstream economics.

3.0 Extension to Technical Discussions

Well-educated economists know that their theory does not support the causal stories in the textbooks. I concentrated above on factor markets. I now go into more technical points. I think that if my powers of exposition and understanding were much greater, this section would still not be clear.

The above refuted stories can be augmented with stories about natural resources and about produced commodities. The assumptions of mainstream economics do not justify explaining equilibrium by the intersection of well-behaved supply and demand curves.

Demonstrations of capital-reversing, for example, are usually presented in open models of competitive, cost-minimizing firms. These models can be closed by assuming households are utility-maximizing. These closures include intertemporal utility-mximizing. Overlapping generations (OLG) models are examples.

Are long-run equilbrium models like this 'neoclassical'? Endowments of capital are not among the givens. The mixture and level of capital goods are found from solving the model. Likewise, the numeraire value of the capital stock is endogeneous, not a parameter.

The claim that, in equilibrium, the rate of interest is equal to the marginal product of capital might be justified as applying to Champernowne's chain index measure of the quantity of capital. This chain index, basically, excludes price Wicksell effects from the measure of the quantity of capital. Both the interest rate and the value of this chain index are found from the solution of the model. They are not part of what needs to be known to find the solution. Furthermore, the chain index is not what is measured in empirical applications of the Solow-Swan model and in measurements of total factor productivity (TFP).

Another argument turns around how capital-theory paradoxes apply to models of intertemporal equilibria, if at all. Can a continuum of equilibria be found in either long-run models or models of intertemporal equilibrium? Do capital-theoretic paradoxes add anything to examination of the stability of intertemporal equilibrium, either for tattonement or for individual paths? Typically, stability cannot be demonstrated, and multiple equilibria exist. No reason exists to expect paths to approach steady states. J. Barkley Rosser Jr. argued that the reswitching of techniques is manifested in intertemporal equilibrium as a cusp catastrophe. Personnally, I think Michael Mandler is more correct than Pierangelo Garegnani about stability.

I finally bring up that you do not have to close long run models with intertemporal utility-maximization. Richard Kahn, Nicholas Kaldor, Luigi Pasinetti, and Joan Robinson had closures related to the Cambridge equation. Or you can take the wage as a matter of social conventions or norms. Or, perhaps, you can have a monetary theory of distribution, in which the monetary authority sets the interest rate. Stephen Marglin had a overdetermined closure that explained stagflation. Questions exist about how some of these closures relate to a generalization of John Maynard Keynes' principle of effective demand to the long run.

I see I have left out debates over the the history of more than two centuries of political economy.

4.0 Conclusion

But the mainstream defenders in these discussion are not defending what is in the textbooks. The simple-minded depiction of well-behaved supply and demand curves determining equilibrium lacks any theoretical or empirical foundation.

Arguments And Debates Among Early Socialists

1. Introduction

Early socialists did not always agree. I provide an account of an idiosyncratic selection of a couple of Ricardian socialists, including a later dispute about priority and a debate with philosophical radicals, that is, liberals. I have been reading John Cassidy's new book. He has a chapter on William Thompson.

2. William Thompson, Karl Marx, Friedrich Engels, and Anton Menger

I start with William Thompson, the author in 1824 of An Inquiry into the Principles of the Distribution of Wealth Most Conducive to Human Happiness. Thompson built upon the ideas of Jeremy Bentham, William Godwin, and David Ricardo. Apparently, Thompson was to first to argue for equality of distribution on the basis of utilitarianism. An extra $100 has a lot more utility to a poor man than a rich one. He coined the term "surplus value", and his political economy had a structure much like Karl Marx's later system, with some differences. Thompson advocated, along with Robert Owen, for mutual co-operation, as in intentional communities.

"it was Ricardo's crude generalisations which gave modern socialism its fancied scientific basis, and provoked, if they did not justify, its revolutionary form… Ricardo ... did more than any intentionally socialist writer to sap the foundations of that form of society which he was trying to explain, and which he believed to be the typical and natural, if not, indeed, the ideal social state...

Thompson was only one of a series of socialist writers, culminating in Marx and Lassalle, who take the Ricardian position as the very basis of their argument. His first section has the familiar Ricardian ring. 'Wealth is produced by labor: no other ingredient but labor makes any object of desire an object of wealth. Labor is the sole universal measure, as well as the characteristic distinction of wealth.' Give the word 'labour' its popular meaning, and it is merely an affair of logic to deduce a large part of modern socialism from this position. Whatever qualifications Ricardo may have made upon it in his own mind, ninety-nine readers out of a hundred took him literally, and the main impression left by his book was that while wealth was almost exclusively due to labour, it was mainly absorbed by rent and other payments to the unproductive classes. This was the text which Thompson and the English socialists proceeded to elaborate." -- H. S. Foxwell

Anton Menger says that Marx basically copied Thompson and other Ricardian socialists:

"Marx is completely under the influence of the earlier English socialists, and more particularly of William Thompson. Leaving out of account the mathematical formulae by which Marx rather obscures than elucidates his argument, the whole theory of surplus value, its conception, its name, and the estimates of its amount are borrowed in all essentials from Thompson's writings. Only Marx, in accordance with the aim of his work, pays special attention to the one form of unearned income (interest on capital), and fails to give either that juridical criticism of private property in instruments of production and useful commodities which is the necessary supplement of the theory of surplus value, or a rigorous exposition of the right to the whole produce of labour. In all these respects Marx is far inferior to Thompson." -- Anton Menger

Engels would have none of it:

"In order, to drag Marx down, his achievements are attributed to other socialists in whom no one is interested, who have vanished from the scene and who have no political or scientific importance any longer. In this way they hope to dispose of the founder of the proletarian world view, and indeed the world view itself. Mr. Menger undertook the task. People are not professors for nothing. They want to make their mark, too...

The present social order gives landowners and capitalists a 'right' to part - the bulk - of the product produced by the worker… So whoever first said that the present right of those who own the soil and the other means of production to part of the proceeds of labour is a wrong is the great man, the founder of 'scientific' socialism. And these men were Godwin, Hall and Thompson. Leaving out all the interminable economic fripperies and getting to the legal residue, Menger finds nothing but the same assertion in Marx. Consequently Marx simply copied these old Englishmen, particularly Thompson, and took care to keep quiet about his source. The proof has been adduced.

We give up any attempt to make this hidebound lawyer understand that nowhere does Marx demand the 'right to the full proceeds of labour', that he makes no legal demands of any kind at all in his theoretical works. Even our lawyer seems to have a faint inkling of this when he reproaches Marx for nowhere giving 'a thorough presentation of the right to the full proceeds of labour'.

In Marx's theoretical studies legal right, which always merely reflects the economic conditions prevalent in a specific society, is only considered as a matter of purely secondary importance; his main concern is the historical justification for certain conditions, modes of appropriation and social classes in specific ages, the investigation of which is of prime importance to anyone who sees in history a coherent, though often disrupted, course of development rather than, as the eighteenth century did, a mere muddle of folly and brutality. Marx views the historical inevitability of, and hence the justification for, the slave-owners of classical times, the feudal lords of the Middle Ages, etc., as the lever of human development for a limited historical period. He thereby also recognises the temporary historical justification for exploitation, for the appropriation of the product of labour by others. Yet at the same time he demonstrates that not only has this historical justification disappeared, but that the continued existence of exploitation in any form, far from furthering social development, is daily impeding it more and more and involving it in increasingly violent collisions. Menger's attempt to force these epoch-making historical investigations into his narrow, legalistic Procrustean bed only goes to show his total inability to understand things that go beyond the narrowest legal horizon." – Friedrich Engels and Karl Kautsky (1887).

Both Menger and Engels and Kautsky have a lot more to say.

3. William Thompson and J. S. Mill

William Thompson and others had a public debate with John Stuart Mill in 1825. Mill was then a teenager, a liberal, a philosophical radical, and an utilitarian.

"There was for some time in existence a society of Owenites. called the Cooperative Society, which met for weekly public discussions in Chancery Lane. In the early part of 1825, accident brought Roebuck in contact with several of its members, and led to his attending one or two of the meetings and taking part in the debate in opposition to Owenism.

Some one of us started the notion of going there in a body and having a general battle: and Charles Austin and some of his friends who did not usually take part in our joint exercises, entered into the project. It was carried out by concert with the principal members of the Society, themselves nothing loth, as they naturally preferred a controversy with opponents to a tame discussion among their own body. The question of population was proposed as the subject of debate: Charles Austin led the case on our side with a brilliant speech. and the fight was kept up by adjournment through five or six weekly meetings before crowded auditories, including along with the members of the Society and their friends, many hearers and some speakers from the Inns of Court.

When this debate was ended, another was commenced on the general merits of Owen's system: and the contest altogether lasted about three months. It was a lutte corps-@agrave;-corps between Owenites and political economists, whom the Owenites regarded as their most inveterate opponents: but it was a perfectly friendly dispute. We who represented political economy had the same objects in view as they had, and took pains to shew it; and the principal champion on their side was a very estimable man, with whom I was well acquainted, Mr. William Thompson. Of Cork, author of a book on the Distribution of Wealth, and of an Appeal in behalf of women against the passage relating to them in my father's Essay on Government.

Ellis, Roebuck, and I, took an active part in the debate, and among those from the Inns of Court who joined in it I remember Charles Villers. The other side obtained also, on the population question, very efficient support from without. The well known Gale Jones, then an elderly man. made one of his florid speeches; but the speaker with whom I was most struck, though I dissented from nearly every word he said, was Thirlwall, the historian, since Bishop of St. David's, then a Chancery barrister, unknown except by a high reputation for eloquence acquired at the Cambridge Union before the era of Austin and Macaulay. His speech was in answer to one of mine. Before he had uttered ten sentences. I set him down as the best speaker I had ever heard, and I have never since heard any one whom I placed above him." – J. S. Mill, Autobiography

In his autobiography, Mill makes clear that those on his side are his friends from university.

3. William Thompson and Thomas Hodgskin

Thompson had a disagreement with Thomas Hodgskin who published in 1825, Labour Defended Against the Claims of Capital. Hodgskin's explanation for profits is a matter of selling commodities over their value, with force and monopoly power.

"But if [Hodgskin] is in favour of competition as the principle by which to determine the division of labour's share between the various ranks of labourers, he is for combination against capital in order to make labour's share as large as possible. By combining, the journeymen 'may reduce or destroy altogether the profit of the idle capitalist ... but they will augment the wages and rewards of industry, and will give to genius and skill their due share of the national produce.'" – H. S. Foxwell This is the sort of explanation and remedy that Thompson rejected. His 1827 book Labour Rewarded is an answer.

"Thompson urges that 'individual competition is incompatible with equal remuneration, as it is also with securing to labor the entire products of its exertions'. 'The author of Labour Defended stands alone, as far as. I know, amongst the advocates of Individual competition, in even wishing that labor should possess the whole of the products of its exertions. All other advocates of individual competition look on the notion as visionary, under the Competitive System'. We know Thompson's solution of the difficulty. Labourers must become capitalists, and unite in communities to regulate their own labour. To ascertain for each the exact product of his own labour is impracticable. If this could be done, then justice would give each individual a property in that product. But moral considerations would force him to share that product with others. The human race could not otherwise be preserved. This voluntary distribution is best carried out under the equitable arrangements of co-operative communities, with their regulated exchanges. 'It is on the regulation of exchanges', he concludes, 'that the industrious classes must depend for realising the general proposition that "the whole produce of labour should belong to the labourer."'" – H. S. Foxwell

We see here the distinction between each individual worker obtaining his product and the working class as a whole obtaining and deciding on the use of their product. Thompson advocates the latter.

4. William Thompson and Robert Owen

Thompson had a later disagreement with Robert Owen. This disagreement, at the 1832 Co-operative Congress in London, arose after Owen's experiences in the USA, at New Harmony, Indiana. Owen thought co-operatives needed much larger initial investments than Thompson.

Selected References

• John Cassidy. 2025. Capitalism and Its Critics: A History from the Industrial Revolution to AI. Farrar, Straus, and Giroux.
• Friedrich Engels and Karl Kautsky. 1887. Lawyer's Socialism. Die Neue Zeit.
• Anton Menger. 1899. The Right to the Whole Produce of Labour. With an introduction by H. S. Foxwell.

Fixed Capital And Extensive Rent

This post is a problem statement.

Models of the production of commodities with circulating capital alone have certain nice properties. I refer to models in which commodities are produced by means of commodities, with a certain circular structure in production. Direct labor inputs are assumed to be necessary to operate each process in the technology.

The choice of technique can be analyzed in models with circulating capital alone by constructing the outer envelope of the wage curves for each technique. Each wage curve slopes down. The wage, for a technique, is lower the higher the rate of profits. The cost-minimizing technique at a given rate of profits is unique, except at switch points. The "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang 2019). The wage and prices of production are unique functions of the rate of profits. If a technique exists with a defined wage and prices of production at a given rate of profits, then a cost-minimizing technique exists. A market algorithm (Bidard 1990) converges, without going into a cycle.

None of these properties necessarily hold in models with joint production. For example, Bidard & Klimovsky (2004) define fake switch points as intersections on the outer wage frontier at which the cost-minimizing technique does not vary.

But they do in a model of pure fixed capital, with the exception that wage curves can slope up when not on the frontier. These models are non-interlocked systems in which old machines cannot be transferred among sectors. Each process produces exactly one finished good, such as a consumption good; a good used as circulating capital; or a new, possibly long-lived machine. Old machines are intermediate goods. Old machines cannot be consumer goods. Every finished good has exactly one primary process for producing it, in which an intermediate good does not enter as an input. In each sector, the secondary processes completely use up the old machine produced by the primary process in that sector, with no other intermediate goods as inputs. They produce the same finished good, possibly jointly with an intermediate good. Joint utilization of machines does not exist in any process. Old machines may be freely disposed of; no cost arises in junking a machine, including before its technical life is complete.

Most of the properties of circulating capital also hold in models of extensive rent. Extensive rent occurs when multiple types of land must be cultivated to satisfy requirements for use. Only one process is operated on each type of land. All but one of the types of cultivated land are fully farmed, except in fluke cases, to the extent of their endowment. With only one price prevailing for corn and only one rate of profits being obtained in the system of prices of production, different amounts of rent per acre must be paid on the different types of land that are fully farmed. The type of land that is only partially farmed is not scarce and does not pay a rent. The cost-minimizing technique can be found from wage curves, but it does not correspond to the technique on the outer frontier. Requirements for use determine which techniques are feasible, and only a feasible technique can be cost-minimizing.

My claim is that a model combining fixed capital and extensive rent lacks more properties characteristic of circulating capital than either does alone. For example, the wage frontier, defined by the wage curves for cost-minizing techniques at each rate of profits, can slope upward. A numerical example can demonstrate this. So I am curious if an elaboration of the model specified in this post works for this purpose.

Tables 1 and 2 specify the technology for a simple model producing multiple commodities and combining fixed capital and extensive rent. A numerical example results by setting each coefficient of production not equal to zero or unity to some positive value.

Table 1: Inputs for Processes Comprising the Technology

Input	Industry
	Machine	Corn
	I	II	III	IV	V
Labor	a0,1	a0,2	a0,3	a0,4	a0,5
Type 1 Land	c1,1 = 0	c1,2	c1,3	c1,4 = 0	c1,5 = 0
Type 2 Land	c2,1 = 0	c2,2 = 0	c2,3 = 0	c2,4	c2,5
Corn	a1,1	a1,2	a1,3	a1,4	a1,5
New Machines	a2,1 = 0	a2,2 = 1	a2,3 = 0	a2,4 = 1	a2,5 = 0
Type 1 Old Machines	a3,1 = 0	a3,2 = 0	a3,3 = 1	a3,4 = 0	a3,5 = 0
Type 2 Old Machines	a4,1 = 0	a4,2 = 0	a4,3 = 0	a4,4 = 0	a4,5 = 1

Table 2: Outputs for Processes Comprising the Technology

Output	Industry
	Machine	Corn
	I	II	III	IV	V
Corn	b1,1 = 0	b1,2	b1,3	b1,4	b1,5
New Machines	b2,1 = 1	b2,2 = 0	b2,3 = 0	b2,4 = 0	b2,5 = 0
Type 1 Old Machines	b3,1 = 0	b3,2 = 1	b3,3 = 0	b3,4 = 0	b3,5 = 0
Type 2 Old Machines	b4,1 = 0	b4,2 = 0	b4,3 = 0	b4,4 = 1	b4,5 = 0

In the tables, each column specfies the inputs and outputs of corn, new machines, and old machines of each type needed to operate that process at a unit level. Inputs of labor and each of the two types of land are also specified. I assume constant returns to scale, up to the limits imposed by endowments of land. Each process requires the same time, a year, to complete. The first process uses inputs of labor and corn to manufacture new machines. The second and third processes produce corn on the first type of land. The remaining two processes also produce corn, but on the other type of land.

Corn is the numeraire and the only consumption good. The total acres of each type of land are also part of the data. Requirements for use are specified by the quantity of corn in the net output of the economy.

Table 3 lists the techniques of production available. In Alpha, Beta, Gamma, and Delta, land is not scarce. Only one type of land is farmed, and the quantity farmed does not exceed its endowment. Beta differs from Alpha in that a machine of the first type is used for its full physical life. Likewise, Delta differs from Gamma in the same way for a machine of the second type. Both types of land are farmed in the remaining eight techniques, and ownership of one type of land obtains a rent. The techniques vary in which land receives a rent and in the economic lifetime of a machine.

Table 3: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2
Alpha	I, II	Partially farmed	Fallow
Beta	I, II, III	Partially farmed	Fallow
Gamma	I, IV	Fallow	Partially farmed
Delta	I, IV, V	Fallow	Partially farmed
Epsilon	I, II, IV	Fully farmed	Partially farmed
Zeta	I, II, III, IV	Fully farmed	Partially farmed
Eta	I, II, IV, V	Fully farmed	Partially farmed
Theta	I, II, III, IV, V	Fully farmed	Partially farmed
Iota	I, II, IV	Parially farmed	Fully farmed
Kappa	I, II, III, IV	Parially farmed	Fully farmed
Lambda	I,II, IV, V	Parially farmed	Fully farmed
Mu	I, II, III, IV, V	Parially farmed	Fully farmed

Which techniques are feasible for a given level of net output? What are the quantity flows? What are the price systems? Which techniques are cost-minimizing? Can I specify numerical values that illustrate interesting phenomena? These questions might be worth answering.

This combination of fixed capital and extensive rent should also demonstrate that approaches to the analysis of the choice of technique customized for each apply to their combination. If the price of some machines or rent on a farmed type of land is negative at a given rate of profits, that technique cannot be cost-minimizing at that rate, for example.