Fixed Capital And Extensive Rent

This post is a problem statement.

Models of the production of commodities with circulating capital alone have certain nice properties. I refer to models in which commodities are produced by means of commodities, with a certain circular structure in production. Direct labor inputs are assumed to be necessary to operate each process in the technology.

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 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) converges, without going into a cycle.

None of these properties necessarily hold in models with joint production. For example, Bidard & Klimovsky (2004) define fake switch points as intersections on the outer wage frontier at which the cost-minimizing technique does not vary.

But they do in a model of pure fixed capital, with the exception that wage curves can slope up when not on the frontier. These models are non-interlocked systems in which old machines cannot be transferred among sectors. Each process produces exactly one finished good, such as a consumption good; a good used as circulating capital; or a new, possibly long-lived machine. Old machines are intermediate goods. Old machines cannot be consumer goods. Every finished good has exactly one primary process for producing it, in which an intermediate good does not enter as an input. In each sector, the secondary processes completely use up the old machine produced by the primary process in that sector, with no other intermediate goods as inputs. They produce the same finished good, possibly jointly with an intermediate good. Joint utilization of machines does not exist in any process. Old machines may be freely disposed of; no cost arises in junking a machine, including before its technical life is complete.

Most of the properties of circulating capital also hold in models of extensive rent. Extensive rent occurs when multiple types of land must be cultivated to satisfy requirements for use. Only one process is operated on each type of land. All but one of the types of cultivated land are fully farmed, except in fluke cases, to the extent of their endowment. With only one price prevailing for corn and only one rate of profits being obtained in the system of prices of production, different amounts of rent per acre must be paid on the different types of land that are fully farmed. The type of land that is only partially farmed is not scarce and does not pay a rent. The cost-minimizing technique can be found from wage curves, but it does not correspond to the technique on the outer frontier. Requirements for use determine which techniques are feasible, and only a feasible technique can be cost-minimizing.

My claim is that a model combining fixed capital and extensive rent lacks more properties characteristic of circulating capital than either does alone. For example, the wage frontier, defined by the wage curves for cost-minizing techniques at each rate of profits, can slope upward. A numerical example can demonstrate this. So I am curious if an elaboration of the model specified in this post works for this purpose.

Tables 1 and 2 specify the technology for a simple model producing multiple commodities and combining fixed capital and extensive rent. A numerical example results by setting each coefficient of production not equal to zero or unity to some positive value.

Table 1: Inputs for Processes Comprising the Technology

Input	Industry
	Machine	Corn
	I	II	III	IV	V
Labor	a0,1	a0,2	a0,3	a0,4	a0,5
Type 1 Land	c1,1 = 0	c1,2	c1,3	c1,4 = 0	c1,5 = 0
Type 2 Land	c2,1 = 0	c2,2 = 0	c2,3 = 0	c2,4	c2,5
Corn	a1,1	a1,2	a1,3	a1,4	a1,5
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 Processes Comprising the Technology

Output	Industry
	Machine	Corn
	I	II	III	IV	V
Corn	b1,1 = 0	b1,2	b1,3	b1,4	b1,5
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 = 0

In the tables, each column specfies the inputs and outputs of corn, new machines, and old machines of each type needed to operate that process at a unit level. Inputs of labor and each of the two types of land are also specified. I assume constant returns to scale, up to the limits imposed by endowments of land. Each process requires the same time, a year, to complete. The first process uses inputs of labor and corn to manufacture new machines. The second and third processes produce corn on the first type of land. The remaining two processes also produce corn, but on the other type of land.

Corn is the numeraire and the only consumption good. The total acres of each type of land are also part of the data. Requirements for use are specified by the quantity of corn in the net output of the economy.

Table 3 lists the techniques of production available. In Alpha, Beta, Gamma, and Delta, land is not scarce. Only one type of land is farmed, and the quantity farmed does not exceed its endowment. Beta differs from Alpha in that a machine of the first type is used for its full physical life. Likewise, Delta differs from Gamma in the same way for a machine of the second type. Both types of land are farmed in the remaining eight techniques, and ownership of one type of land obtains a rent. The techniques vary in which land receives a rent and in the economic lifetime of a machine.

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

Which techniques are feasible for a given level of net output? What are the quantity flows? What are the price systems? Which techniques are cost-minimizing? Can I specify numerical values that illustrate interesting phenomena? These questions might be worth answering.

This combination of fixed capital and extensive rent should also demonstrate that approaches to the analysis of the choice of technique customized for each apply to their combination. If the price of some machines or rent on a farmed type of land is negative at a given rate of profits, that technique cannot be cost-minimizing at that rate, for example.