An Extensive Rent Example

Figure 1: Wage Curves and Rent for an Example of Extensive Rent

The analysis of the choice of technique in models of extensive rent can be based on the construction of wage curves, even though the outer envelope does not represent the cost-minimizing technique. The orders of fertility and rentability are emphasized here. The order of fertility is the order in which different qualities of land are introduced into production as requirements for use expand. The order of rentability specifies the sequence of different qualities of lands from high rent per acre to low rent per acre. Both orders may vary with the distribution of income. Table 1 presents coefficients of production for an example. Technical progress is assumed to reduce coefficients of production in each of the corn-producing processes, at different rates.

Table 1: The Coefficients of Production

Input	Iron Industry	Corn Industry
	I	II	III	IV
Labor	1	(5191/5770)e1-φ t	(305/494) e1-θ t	(3/2)e1-σ t
Type 1 Land	0	e1-φ t	0	0
Type 2 Land	0	0	e1-θ t	0
Type 3 Land	0	0	0	e1-σ t
Iron	9/20	(1/40)e1-φ t	(3/1976) e1-θ t	(3/20)e1-σ t
Corn	2	(1/10)e1-φ t	(229/494)e1-θ t	(1/3)e1-σ t

The quantity of produced corn is constrained by the available quantities of each type of land. Suppose endowments include one hundred acres of each type of land, a stationary state prevails, and net output is somewhere between 322 and 444 bushels of corn. Then all three types of land must be farmed with the parameters specified in Figure 1. One type will be only partially farmed. The iron-producing process must be operated in each of the three economically viable techniques sustaining such a stationary state. Table 2 describes which type of lands are fully cultivated and which type of land is left partially farrow in each of the Alpha, Beta, and Gamma techniques.

Table 2:

Technique	Land
	Type 1	Type 2	Type 3
Alpha	Fully farmed	Fully farmed	Partially farmed
Beta	Partially farmed	Fully farmed	Fully farmed
Gamma	Fully farmed	Partially farmed	Fully farmed

For a given technique, the rent on a type of land only partially farmed is zero since it is not scarce. The wage curves in the left pane in Figure 1 are constructed from the price equations provided by the iron-producing process and the corn-producing process for the land that pays no rent in each technique. The choice of technique depends on both income distribution and requirements for use (Quadrio-Curzio 1980). Suppose the rate of profits is taken as given. Then the r-order of efficiency or fertility is the order of the wage curves downwards, until requirements for use are satisfied. Since all three types of land must be somewhat cultivated in the example, the wage frontier is the inner envelope of the wage curves. For the illustrated parameters, Alpha is cost-minimizing, and the order of fertility between the switch points is Type 1, Type 2, and Type 3 lands.

One can calculate the cost of capital goods at the given rate of profits and the cost of labor inputs for corn-producing processes on each of Type 1 and Type 2 lands. Since coefficients of production are specified per bushel corn produced, the revenues from each of these processes, at a unit level, are the same as the process operated on Type 1 land. Rent is the difference between revenues and non-land costs on Type 1 and Type 2 lands. Rent per acre is plotted in the right pane for Figure 1. The order of rentability is the order of lands by rent per acres. The order of rentability is the same as the order of fertility between switch points for the given parameters.

The two fluke switch points in Figure 5 do not lie on the wage frontier. The maximum rate of profits for the wage curve for Alpha is the maximum rate of profits for this example. One of the switch point for the wage curves for the Beta and Gamma techniques is at this maximum rate of profits. As seen in Figure 2, this is another edge case, a fluke that arises in models of extensive rent. By the way, at other parameter ranges the curves for rent as a function of the rate of profits in the right pane in Figure 1 can intersect. The order of rentability can vary with distribution, and fluke cases in which these curves intersect at a maximum rate of profits or a rate of profits of zero can arise.

Figure 2: The Parameter Space

Figure 2 illustrates a region of the parameter space around this fluke case. To the northwest, one switch point between the wage curves for Beta and Gamma has vanished over the wage axis, and the other is at a rate of profits exceeding the maximum rate of profits for Alpha. The order of fertility matches the order of rentability. The southwest is of a reswitching example. For such small perturbations, the order of rentability does not change in this example. Thus, the order of fertility matches the order of rentability only at intermediate rates of profits with reswitching here. The northeast and southwest also illustrate parameters for which the order of fertility varies with the rate of profits. The order of fertility varies from the order of rentability at one or the other extreme, as indicated.

The reverse substitution of labor, reswitching, capital reversing, the association of the truncation of the economic life of machine with a more capital-intensive technique, and a divergence between the order of fertility and the order of rentability are not fluke cases. These posts demonstrate this conclusion by contrasting these possibilities with genuine fluke cases.