The Centre Of The Solving Subsystem In A Model With Fixed Capital And Scarce Land

1.0 Introduction

This post revisits my example with fixed capital and two types of land. It presents, by means of an example, the concept of the centre of a solving subsystem. Quadrio Curzio & Pellizzari (2010) introduce the solving subsystem in models of rent so as to first solve the price equations without rent. Schefold (1989) introduces the centre of the price system for a pure fixed capital model to, following Sraffa, initially eliminate the prices of old machines from price equations. As far as I know, nobody has combined these concepts before.

The concept of a solving subsystem clarifies how a switch point can lie along a single wage curve. A system of equations for prices is associated with each technique. Each operated process contributes an equation equating revenues and costs. The revenues can include the prices of joint products, and costs include a charge for the rate of profits on advanced capital goods. A last equation specifies the value of the numeraire as unity. In models of extensive rent, a subsystem can be formed from the processes that characterize industrial processes, with no inputs from land, and processes run on land that are not scarce. The resulting subsystem, with the equation for the numeraire concatenated, can be solved, given the rate of profits, for the wage and the prices of produced commodities. In models of intensive rent, the solving subsystem includes the equations for industrial processes and a linear combination of the equations for the processes that operate on one type of land to the limits of its endowment. As Sraffa (1960) explains, a variable for rent is eliminated by this linear combination. In the case of extensive rent, with no joint production otherwise, the solving subsystem also applies to a model of single production. In any case, the solution to the solving subsystem can then be used to find rents. The example in this post, extends the concept of a solving subsystem to a case with extensive rent and fixed capital. I do not know if the concept of a solving subsystem can usefully apply to joint production more generally

The centre of a pure fixed capital system (Schefold 1989) helps solve the price system of a pure fixed capital system. Joint utilization of machines does not exist in any process in a model of pure fixed capital. Old machines are not consumer goods. In the example, a single commodity is a consumption good and acts as numeraire. Old machines may be freely disposed of; no cost arises in junking a machine, including before its technical life is complete. Nice properties of single production systems generalize to such cases of fixed capital. In particular, the "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang, 2019). The cost-minimizing technique can be determined by the construction of the wage frontier. These properties are not retained in the combination of pure fixed capital with scarce land. The centre still helps solve the price system.

2.0 Technology, Endowments, Final Demand

Tables 1 and 2 specify the technology. This technology extends an example of fixed capital from Baldone (1974). Labor uses circulating capital to manufacture a machine in process I. The machine has a physical life of three years. Labor uses circulating capital and the machine to produce corn on type 1 land in processes II, III, and IV. The machine is operated on type 2 land in processes V, VI, and VII. A process that produces corn jointly produces a machine one year older than the machine used as input, up to its physical life. One hundred acres of each type of land are assumed to exist. Final demand is for 87 bushels corn, a level that ensures one or the other type of land is scarce. The numeraire is a bushel of corn.

Table 1: Inputs for Processes Comprising the Technology

Input	Processes
	I	II	III	IV	V	VI	VII
Labor	a0,1 = 0.4	a0,2 = 0.2	a0,3 = 0.6	a0,4 = 0.4	a0,5 = 0.23	a0,6 = 0.59	a0,7 = 0.39
Type 1 Land	0	c1,2 = 1	c1,3 = 1	c1,4 = 1	0	0	0
Type 2 Land	0	0	0	0	c2,5 = 1	c2,6 = 1	c2,7 = 1
Corn	a1,1 = 0.1	a1,2 = 0.4	a1,3 = 0.578	a1,4 = 0.6	a1,5 = 0.39	a1,6 = 0.59	a1,7 = 0.61
New Machines	0	1	0	0	1	0	0
Type 1 1-Yr. Old Machines	0	0	1	0	0	0	0
Type 1 2-Yr. Old Machines	0	0	0	1	0	0	0
Type 2 1-Yr. Old Machines	0	0	0	0	0	1	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	0	1

Table 2: Outputs for Processes Comprising the Technology

Input	Processes
	I	II	III	IV	V	VI	VII
Corn	0	b1,2 = 1	b1,3 = 1	b1,4 = 1	b1,5 = 1	b1,6 = 1	b1,7 = 1
New Machines	1	0	0	0	0	0	0
Type 1 1-Yr. Old Machines	0	1	0	0	0	0	0
Type 1 2-Yr. Old Machines	0	0	1	0	0	0	0
Type 2 1-Yr. Old Machines	0	0	0	0	1	0	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	1	0

3.0 Techniques

Tables 3, 4, and 5 specify the techniques that may be chosen with this technology. Alpha, Beta, and Gamma differ in the economic life of the machine on non-scarce, type 1 land. No processes are operated on type 2 land. Under Delta, Epsilon, and Zeta, on the other hand, type 1 land is not farmed at all, and the economic life of the machine varies among the techniques in the processes operated on type 2 land. The remaining techniques fully cultivate one or the other type of land and require rent to be paid to landlords

Table 3: Techniques of Production with Non-Scarce Land

Technique	Processes	Type 1 Land	Type 2 Land
Alpha	I, II	Partially farmed	Fallow
Beta	I, II, III	Partially farmed	Fallow
Gamma	I, II, III, IV	Partially farmed	Fallow
Delta	I, V	Fallow	Partially farmed
Epsilon	I, V, VI	Fallow	Partially farmed
Zeta	I, V, VI, VII	Fallow	Partially farmed

Table 4: Techniques of Production with Type 1 Land Scarce

Technique	Processes	Type 1 Land	Type 2 Land
Eta	I, II, V	Fully farmed	Partially farmed
Theta	I, II, III, V	Fully farmed	Partially farmed
Iota	I, II, III, IV, V	Fully farmed	Partially farmed
Kappa	I, II, V, VI	Fully farmed	Partially farmed
Lambda	I, II, III, V, VI	Fully farmed	Partially farmed
Mu	I, II, III, IV, V, VI	Fully farmed	Partially farmed
Nu	I, II, V, VI, VII	Fully farmed	Partially farmed
Xi	I, II, III, V, VI, VII	Fully farmed	Partially farmed
Omicron	I, II, III, IV, V, VI, VII	Fully farmed	Partially farmed

Table 5: Techniques of Production with Type 2 Land Scarce

Technique	Processes	Type 1 Land	Type 2 Land
Pi	I, II, V	Partially farmed	Fully farmed
Rho	I, II, III, V	Partially farmed	Fully farmed
Sigma	I, II, III, IV, V	Partially farmed	Fully farmed
Tau	I, II, V, VI	Partially farmed	Fully farmed
Upsilon	I, II, III, V, VI	Partially farmed	Fully farmed
Phi	I, II, III, IV, V, VI	Partially farmed	Fully farmed
Chi	I, II, V, VI, VII	Partially farmed	Fully farmed
Psi	I, II, III, V, VI, VII	Partially farmed	Fully farmed
Omega	I, II, III, IV, V, VI, VII	Partially farmed	Fully farmed

Under techniques Eta through Omicron, type 1 land is fully farmed and pays rent. Under Eta, Theta, and Iota, the machine is operated for only one year on type 2 land and then discarded. The techniques differ on the economic life of the machine on type 1 land. Under Kappa, Lambda, and Mu, the machine is operated for two years on type 2 land, while it is operated for its full physical life of three years under Nu, Xi, and Omicron. Under Pi through Omega, type 2 land is scarce and pays rent. Each technique between Eta and Omicron corresponds to a technique between Pi and Omega in which the same processes are operated. The economic life of the two types of machines are the same in these corresponding techniques. The scale at which the processes are run varies so as to vary which type of land is fully farmed.

4.0 The Price System for Omicron

I consider the price equations for Omicron to illustrate the concepts of the solving subsystem and of the centre. All seven processes are operated under Omicron, and type 1 land is scarce. The following seven displays, in obvious notation, specify the price system for Omicron:

a_1,1(1 + r) + w a_0,1 = p₀

(a_1,2 + p₀)(1 + r) + rho₁ c_1,2 + w a_0,2 = b_1,2 + p_1,1

(a_1,3 + p_1,1)(1 + r) + rho₁ c_1,3 + w a_0,3 = b_1,3 + p_1,2

(a_1,4 + p_1,2)(1 + r) + rho₁ c_1,4 + w a_0,4 = b_1,4

(a_1,5 + p₀)(1 + r) + w a_0,5 = b_1,5 + p_2,1

(a_1,6 + p_2,1)(1 + r) + w a_0,6 = b_1,6 + p_2,2

(a_1,7 + p_2,2)(1 + r) + w a_0,7 = b_1,7

Revenues for operating each process at a unit level are shown on the right-hand side of these equations. Revenues for the first process are obtained by selling new machines. Revenues for the second process result from products of both corn and a type 1 one-year old machine. That type 1 machine, in turn, enters into the advanced costs of the third process, and so on. Type 1 land obtains a rent, and type 2 land is free.

The first equation and the last three of the seven constitute the solving subsystem for Omicron. Given the rate of profits, the solving subsystem specifies the wage, the price of a new machine, and the prices of one-year old and two-year old machines when operated on free type 2 land. The remaining three equations can then be used to find the rent on type 1 land and the prices of one-year old and two-year old machines when operated on type 1 land. The solving subsystem for Omicron is also the solving subsystem for Zeta, Nu, and Xi. In all these techniques, the machine is run for its full physical life of three years on free type 2 land.

The prices of old type 2 machines can be eliminated from the solving subsystem for Omicron. Multiply both sides of the second equation of the solving subsystem by (1 + r)²:

(a_1,5 + p₀)(1 + r)³ + w a_0,5(1 + r)² = b(1 + r)²_1,5 + p_2,1(1 + r)²

Multiply both sides of the third equation of the solving subsystem by (1 + r):

(a_1,6 + p_2,1)(1 + r)² + w a_0,6(1 + r) = b_1,6(1 + r) + p_2,2(1 + r)

Add these two equations and the last equation of the solving subsystem:

where the row vector and matrix in this system of equations is as follows:

The ordered pair consisting of this row vector and matrix is the centre (Schefold 1989) for the solving subsystem for Omicron. Given the rate of profits, this system of matrix equations can be solved for the wage and the price of a new machine. This price system has the form of a price system for a circulating capital model, with the exception of the dependence of the Leontief input matrix and the vector of direct labor coefficients on the rate of profits. Unlike in the model of circulating capital, the wage curve derived from the centre of a pure fixed capital system can slope up for part of its range. The wage frontier of a pure fixed capital system, however, decreases throughout its length (Baldone 1974, Varri 1974).

The prices of old type 1 machines can be similarly eliminated from the full price system for Omicron.

5.0 Conclusion and Questions

The above illustrates the centre of a solving subsystem. In the example, the solving subsystem shows that a system of seven equations for a price system can be decomposed such that a system of four equations is solved first. And the centre of the solving subsystem shows that that system of four equations can be further decomposed so that a system of two equations is solved first.

Perhaps the centre of a solving subsystem can be used to address a theoretical question. Is the wage frontier always decreasing in a model combining fixed capital and rent? Can the wage frontier sometimes slope up?

In a model of extensive rent, the wage frontier is not the outer envelope of the wage curves for the technique. But it is always decreasing. Each wage curve is found from a solving subsystem. And the solving subsystem is from a related circulating capital model. So the wage curves inherit the properties of circulating capital models. The wage frontier is formed from the wage curves of the cost-minimizing techniques and always is decreasing.

In a pure fixed capital model, the wage frontier is the outer envelope of the wage curves for the techniques and is always decreasing. Individual wage curves can be increasing, but the ranges of the rate of profits at which they are increasing is never on the frontier.

I suspect the wage frontier for a model combining extensive rent and fixed capital can be increasing over some range of the rate of profits. This suspicion should be validated by constructing a numerical example. On the other had, if the wage frontier is alwys decreasing in such a model, that should be capable of a proof. And such a proof, if it exists, will probably use the concept of the centre of a solving subsystem.

References

• Baldone, S. (1974), Il capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIV(1): 45-106. Trans. in Pasinetti (1980).
• Huang, B. 2019. Revisiting fixed capital models in the Sraffa framework. Economia Politica 36: 351-371.
• Pasinetti, L.L. 1980. (ed.), 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.
• 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, P. 1974. Prezzi, saggio del profitto e durata del capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIX(1): 5-44. Trans. in Pasinetti (1980).