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.