A Reswitching Example With Extensive And Intensive Rent And Multiple Agricultural Commodities

Figure 1: Wage Curves For Feasible Techniques

1.0 Introduction

This post demonstrates a novel aspect of the reswitching of techniques. The cost-minimizing techniques in the example do not differ in which processes are operated. They differ in which lands are fully cultivated and thus obtain rent. In one technique, two commodities are produced, by distinct processes on the type of land that is fully farmed. In the other, one process, producing one commodity, is operated on the land that pays rent. In other words, the techniques that reswitch pay extensive and intensive rent, respectively.

The reswitching example depends on more than one agricultural commodity being produced. When the technique with extensive rent is cost-minimizing, two processes are operated on type 2 land. Type 2 land is not fully farmed. Two processes producing the same commodity cannot be operated on non-scarce land, away from a switch point, when prices of production prevail.

2.0 Technology, Endowments, Final Demands, and Techniques

Table 1 shows the inputs and outputs for each process known to the managers of firms. Iron is an industrial commodity, produced with no inputs from land. Two types of land are available for producing the agricultural commodities, wheat and rye. Wheat is produced by two processes, each operating on a different type of land. The same is true for rye. Inputs and outputs are specified in physical terms. For example, the inputs for process II, per bushel wheat produced, are 5/2 person-year, the services of one acre of type 1 land, 1/200 ton iron, 1/4 bushels wheat, and 1/300 bushels rye. Each process exhibits constant returns to scale (CRS), up to the limits imposed by the endowments of the lands.

Table 1: Processes Comprising the Technology

Inputs	Industries
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1 = 1/3	a0,2 = 5/2	a0,3 = 7/20	a0,4 = 1	a0,5 = 3/2
Type 1 Land	0	c1,2 = 1	0	c1,4 = 2	0
Type 2 Land	0	0	c2,3 = 5	0	c2,5 = 1
Iron	a1,1 = 1/6	a1,2 = 1/200	a1,3 = 1/100	a1,4 = 1	a1,5 = 0
Wheat	a2,1 = 1/200	a2,2 = 1/4	a2,3 = 3/10	a2,4 = 0	a2,5 = 1/4
Rye	a3,1 = 1/300	a3,2 = 1/300	a3,3 = 0	a3,4 = 0	a3,5 = 0
OUPUTS	1 ton iron	1 bushel wheat	1 bushel wheat	1 bushel rye	1 bushel rye

The specification of the problem is completed by defining the available endowments of land and the level and composition of final demand. Accordingly, assume 100 acres of each type of land are available. Suppose the required net output, also known as final demand, consists of 8 bushels wheat and 60 bushels rye.

Table 2 shows the available techniques for these parameters. Land is free for techniques Alpha, Beta, Gamma, and Delta. Techniques Epsilon, Zeta, Eta, and Theta pay extensive rent. Intensive rent is obtained by landlords for Iota, Kappa, Lambda, Mu, and Nu. Under Nu, intensive rent is obtained on both types of land.

Table 2: Technique

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

Consider the Delta technique, for an example of relationships between these techniques. Under Delta, both wheat and rye are grown on type 2 land, but generally not to the limit imposed by the endowments of land. These same wheat and rye-producing processes are operated in Eta and Theta. In both of these techniques with extensive rent, type 2 land is still not farmed to the extent of its endowment. In Eta, wheat is produced on type 1 land to the extent of the endowment of type 1 land. In Theta, rye is produced on type 1 land to the extent of its endowment. But compare Theta to Mu. In Mu, the processes in Delta are also supplemented by growing rye on type 1 land, as in Theta. Type 1 land is not totally farmed, however, under Mu. Type 2 land is totally farmed to the extent of its endowment, with both wheat and rye contining to be produced in parallel on that type of land.

3.0 Quantity Flows and Feasible Techniques

The space of the final demand vector can be partitioned into regions by where each technique is feasible. Assume the final demand for iron is zero. Figure 2 shows a partition of the two-dimensional space for net output of wheat and rye. The point of final demand is indicated for the reswitching example in this post, with a final demand of 8 bushels wheat and 60 bushels rye.

Figure 2: Final Demands for Feasible Techniques

At the lowest level of final demand, Alpha, Beta, Gamma, and Delta are feasible. Land is in excess supply, and the model reduces to a model of circulating capital. As output expands, towards the specified final demand for the example, Delta becomes infeasible. At the limit of Delta's fesibility, net outputs of wheat and rye can be traded off. Lambda and Mu become feasible, with type 2 land obtaining intensive rent.

As final demand continues to expand, Alpha becomes infeasible. Here, too, at the limit for Alpha, the net outputs of wheat and rye can be traded off. Iota and Kappa are the other techniques that pay intensive rent, and they relate to Alpha in the same way that Lambda and Mu relate to Delta. The feasible techniques are now Beta, Gamma, Iota, Kappa, Lambda, and Mu.

As output continues to expand towards the specified point of final demand, Gamma becomes infeasible, with a constraint imposed by the endowment of type 1 land. Iota becomes infeasible, as well. Epsilon and Theta, which pay extensive rent, become feasible. For the reswitching example, Beta, Theta, Kappa, Lambda, and Mu are feasible.

4.0 Price Equations

A system of equations is defined for prices of production for each technique. I take Mu as an example.. Processes I, III, IV, and V are operated under Mu. Processes III and V combine to bring the entire endowment of type 2 land under cultivation. The prices of production for Mu, in obvious notation, satisfy the following equations

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

(p₁ a_1,3 + p₂ a_2,3 + p₃ a_3,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

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

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

Each equation applies to a process operated under Mu. It is assumed that wages and rents are paid out of the surplus product, at the end of the period of production. The prices of advanced capital goods incur a charge for the rate of prices. A final equation equates the price of the numeraire to unity.

p₁ d₁ + p₂ d₂ + p₃ d₃ = 1

The above equations specify prices of production for Mu. A similar price system characterizes prices of production for each of the other techniques in the example.

The price system for Mu consists of five equations in six variables, r, w, p₁, p₁, p₁, and rho₂. The solution has one degree of freedom. If the rate of profits is taken as given, each of the other five variables can be expressed as a function of the rate of profits. The wage curve is a function of the rate of profits for a given technique. The rent curve is the corresponding function for rent.

The wage curves for the feasible techniques are plotted in Figure 3, at the top of this post. Wage curves are only shown for a technique for the ranges of the rate of profits in which rents are positive. Figure 3 shows graphs of the rent curves for the feasible technique. No rent is paid under Beta, and Beta is never cost-minimizing. Theta is cost-minimizing at high and low rates of profits. Mu is cost-minimizing for intermediate rates of profits.

Figure 3: Rent Curves for Feasible Techniques

5.0 Cost-Minimizing Techniques

Assertions above about the ranges of the rates of profits in which Theta and Mu are cost-minimizing have yet to be demonstrated. The cost-minimizing technique at a given rate of profits is:

• Feasible
• Such that no price of a produced commodity, wage, rate of profits, or rent is negative
• Such that extra profits cannot be obtained, at prices associated with the technique, by operating processes outside the technique

Extra profits exist when the difference between revenues and costs for a process is positive. Costs include a charge on the prices of the advanced capital goods for the given rate of profits, and the difference is taken for a unit level of operation of the process. Prices of production for a technique are such that extra profits are zero for the processes operated by that technique.

Figure 4 illustrates extra profits for Beta. Processes III and IV are not operated by Beta. Extra profits can always be obtained, under Beta, by operating process IV, whatever the rate of profits for which prices are found. Beta is never cost-minimizing.

Figure 4: Beta is Never Cost-Minimizing

Figure 5 shows that Theta is cost-minimizing whenever rent is positive for prices of production for the technique. The graph also makes obvious that the conditions, that rent be non-negative and that extra profits are not available, are independent of one another.

Figure 5: Theta is Cost-Minimizing at High and Low Rates of Profits

Figures 6 and 7 demonstrate that Kappa and Lambda are never cost-minimizing. Figure 8 demonstrates that Mu is cost-minimizing for the range of the rate of profits in which rent is positive for prices of production associated with the technique.

Figure 6: Kappa is Never Cost-Minimizing

Figure 7: Lambda is Never Cost-Minimizing

Figure 8: Mu is Cost-Minimizing at Intermediate Rates of Profits

6.0 The Demand for Labor and for Capital

The analysis of the choice of technique need make no mention of supply and demand functions. But the results allow us to plot the real wage against employment, for the given final demand (Figure 9). The relation can be interpreted as an economy-wide demand curve for labor. The switch points appear as horizontal segments in this graph. The 'perverse' switch point is a step function approximation to an upward-sloping demand curve for labor. The widespread tendency to draw downward-sloping labor demand functions, given ideal assumptions such as competitive markets, no search costs, and so on, lacks a coherent justification.

Figure 9: The Demand for Labor

The rate of profits can also be plotted against the value of advanced capital goods (Figure 10). Here, too, switch points are horizontal segments. The value of capital is the iron, wheat, and rye advanced at the start of the production period and evaluate at prices of production. The variation of the value of capital between switch points is known as a price Wicksell effect. The variation at a switch point across techniques is the real Wicksell effect. What happens at the 'perverse' switch point has also been called reverse capital deepening. At any rate, justification for drawing downward-sloping demand curves for capital, by default, is also lacking.

Figure 10: The Demand for Capital

7.0 Conclusion

This post extends the well-established critique of economic theory to which Sraffa (1960) is a prelude. In this example, the techniques that reswitch do not differ in which processes are operated. They differ in the scale at which these processes are operated, thereby resulting in a variation in which lands are scarce. The usual consequences of 'perverse' switch points appear in which, for example, a higher wage is associated with firms wanting to employ more workers to produce a given final demand.