Innovation Hurting Workers Or Capitalists

Figure 1: The Wage Frontier Is The Inner Envelope Of The Wage Curves For Feasible Techniques

1.0 Introduction

This post presents a solution to the homework problem 7.13 in Kurz & Salvadori (1995), Chapter 10. They assign credit for this problem to Antonio D'Agata. I extend it to include a negligible industrial commodity, as in my outline of a model with rent, multiple lands, and multiple agricultural commodities.

Two agricultural commodities, wheat and rye, are produced by the example economy. If only processes II and V existed, each commodity could only be produced on one type of land. With the given final demand and the given endowments of land, no land would be scarce and no landlords could obtain rent. Suppose innovations introduce new processes, III and IV, so that each commodity could be produced on each type of land. As a result, no long period exists in a range of the rate of profits towards its maximum, and landlords obtain rent at a lower rate of profits.

Nobody else, as far as I know, has considered the orders of efficiency and of rentability in a model with multiple agricultural commodities. Maybe I need to read further in Quadrio Curzio & Pellizzari's book.

2.0 Technology, Endowments, Final Demands, and Techniques

Table 1 shows the inputs and outputs for each process known to the managers of firms. 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 one person-year, the services of one acre of type 1 land, a tiny fraction of a ton iron, 3/10 bushels wheat, and 1/10 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 = 0.0001	a0,2 = 1	a0,3 = 3/2	a0,4 = 1/10	a0,5 = 1/2
Type 1 Land	0	c1,2 = 1	0	c1,4 = 1	0
Type 2 Land	0	0	c2,3 = 5	0	c2,5 = 2
Iron	a1,1 = 0.00001	a1,2 = 0.00001	a1,3 = 0.00001	a1,4 = 0.00001	a1,5 = 0.00001
Wheat	a2,1 = 0.00001	a2,2 = 3/10	a2,3 = 1/10	a2,4 = 1/10	a2,5 = 1/5
Rye	a3,1 = 0.00001	a3,2 = 1/10	a3,3 = 3/10	a3,4 = 1/5	a3,5 = 1/10
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 15 bushels wheat and 35 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 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

Suppose only processes I, II, and V are known by managers of firms. Then only the Beta technique is available. Wheat is grown on type 1 land, and rye on type 2 land. The endowments of land provide an upper limit on the level of final demand that can be feasibly satisfied. The innovations that make processes III and IV available provide a choice of technique, including which grains should be grown on which lands. The limits to feasible final demands are increased.

3.0 Quantity Flows

Beta, Kappa, Lambda, and Nu are feasible for the given final demand. Figure 2 shows which techniques are feasible for final demand consisting of any combination of specified quantities of wheat and rye. One type of land is farmed under both Alpha and Delta, with wheat and rye each produced on that type. The maximum final demand for each of these techniques is a downward-sloping straight line. The outer limits for Beta and Gamma have segments where constraints for each type of land kick in.

Figure 2: Feasible Final Demands

Techniques in which extensive rent is paid extend the two techniques with non-scarce land in which both lands are farmed. Epsilon and Theta become feasible when Gamma is no longer feasible. They differ in which type of land, still non-scarce, becomes used to grow both wheat and rye. The other type of land is cultivated to the extent of its endowment. Eventually the non-scarce land, on which both wheat and rye are produced, becomes scarce Zeta and Eta have the same relationship to Beta.

The limits of the final demand for techniques which pay intensive rent also relate to the boundaries on final demand for the other techniques. The maximum final demand for Alpha is the minimum for Iota and Kappa. Iota and Kappa both grow wheat and rye on type 1 land, as in Alpha, but to the full extent of its endowment. They vary with whether non-scarce type 2 land is used to produce wheat or rye. In the same way, the maximum final demand for Delta is the minimum for Lambda and Mu. When the maximum final demand for Iota is the maximum for Gamma (the minimum for Theta), type 2 land is not a constraint. When the maximum for Iota is the maximum for Epsilon (the minimum for Nu), both types of land are constraints. The maximum final demand for Kappa relates to the maximum for Beta and for Zeta in the same way. Likewise, the maximum for Lambda relates to the maximum for Beta and Eta. Finally, the maximum Mu is either the maximum for Gamma or Theta.

4.0 Price Systems

A system of equations for prices is associated with each technique. The going rate of profits is made in each process operated in the technique. I assume wages are paid out of the surplus product at the end of the period required to operate a technique. Rent is also paid on scarce land at the end of the period. A final equation sets the price of the numeraire to unity.

The variables defined by the price system consist of the rate of profits; the wage; the prices of iron, wheat, and rye; and the rents per acre of the two types of land. They are defined up to a degree of freedom. As usual, I take the dependence of the wage on the rate of profits as expressing that degree of freedom. Figure 1, at the top of this post, plots the wage curves for the four feasible technques. Figures 3 plots the rent curves.

Figure 3: Rent Curves

5.0 The Choice of Technique

For a technique to be cost-minimizing at a given rate of profits, the following must be true:

• It must be feasible.
• No price of a commodity, wage, or rent of a type of land can be negative.
• Extra profits cannot be obtained, at the prices associated with the technique, by operating a process not in the technique.

Extra profits are obtained if the difference between revenues and costs for a process, with the going rate of profits charged on advances for capital goods, is positive. Accordingly, I check whether a technique is cost-minimizing by plotting the difference between revenues and costs, for each process, with the prices of that technique.

Process IV pays extra profits throughout the range of the rate of profits in which the price system for Beta has positive orices and a positive wage (Figure 4). Gamma results from replacing the rye-producing process in Beta with process IV. Zeta and Kappa result from producing rye of both types of land. Gamma would be cost-minimizing at a rate of profits greater than approximately 65 percent if it were feasible. But only Kappa, out of Gamma, Zeta, and Kappa, is feasible. Above a rate of profits of approximately 100 percents, processes III and IV both obtain extra profits under Beta. So Beta is not cost-minimizing at any rate of profits.

Figure 4: Extra Profits at Beta Prices

At Kappa prices, process III obtains extra profits at a rate of profits above approximately 18.17 percent. Kappa is cost-minimizing only for rates a profits below this switch point.

Process IV obtains extra profits at Lambda prices for the entire range at which the rent on type 2 land is non-negative for the Lambda price system. Lambda is never cost-minimizing.

All processes are operated under the Nu technique. So extra profits cannot be obtained. But the rent on type 2 land is positive under Nu only when the rate of profits is greater than 18.7 percent. Thus, Nu is not cost-minimizing for a low rate of profits.

The maximum rate of profits for Nu is approximately 153.8 percent. The maximum for Beta is approximately 167.9 percent. Between these limits, Beta is feasible. The wage and the prices of the three produced commodities are positive in the price system for Beta. Both lands are free. Nevertheless, Beta is not cost-minimizing, and a long period position does not exist.

6.0 The Orders of Efficiency and Rentability

The wage curve for the Alpha technique lies on the outer envelope curve, for small rates of profits. The second wage curve, up to a rate of profits of approximately 65 percent, is Gamma's. After that rate of profits, up to the maximum, the wage curve for Gamma is the outermost. These wage curves are not shown in Figure 1.

Only type 1 land is farmed under Alpha. Both types are farmed in Gamma. As final demand expands for a low rate of profits, first type 1 land is cultivated and then both types are farmed. For a larger rate of profits, both types are initially cultivated together. Thus, the order of efficiency varies from type 1, 2 lands to an order in which they are tied (Table 3).

The order from high rent to low rent lands is type 1, 2, whether Kappa or Nu is cost-minimizing. Under Kappa, type 2 land is not scarce and is free. The orders of efficiency and rentability match for low rates of profits, up to a rate of profits in the range in which Nu is cost-minimizing. They differ for higher rates of profits insofar as the order of rentability is not tied.

Table 3: The Choice of Technique

Rate of Profits (Percent)		Technique	Order of Efficiency	Order of Rentability
Minimum	Maximum
0	18.2	Kappa	Type 1, 2	Type 1, 2
18.2	65.1	Nu
65.1	153.8		Type 1 and Type 2 tied

Suppose you take the order of efficiency as showing which lands, at a given rate of profits, contribute most to production. Since the order of rentability can differ from the order of efficiency, prices in competitive markets do not necessarily reward you for the contributions of the factors of production that you own. This conclusion is aside from the doctrine of Henry George.

This example is extremely restricted, when it comes to examining the orders of efficiency. The posibility of ties in the order of efficiency is a new possibility raised by the existence of multiple agricultural commodities. (I suppose you could look for different techniques having identical wage curves in a model of extensive rent and one agricultural comodity.)

7.0 Conclusion

The example shows how innovations create complications in the analysis of long run positions. In the example, innovations lead to the possibility of a class of landlords to come into existence. I doubt this happened like this anywhere. For a certain range of the rate of profits, a long period position no longer exists.

Kurz and Salvadori (1995) have additional numeric examples with multiple agricultural commodities. I should be able to create more. I am interested in examples with the reswitching of the order of fertility, as well as examples in which techniques with extensive and intensive rent are simultaneously feasible.