I thought I would go through another example in which firms want to hire more workers (per unit output) at a higher wage. Since the last time I presented such an example, I've learned more about how to present mathematics in a blog post.
This example was originally developed by Metcalfe and Steedman. I've renamed the inputs so as to present it as a case of non-competing types of heterogeneous labor. I don't think reswitching need occur for the behavior illustrated in Figure 5 in the third part to occur.
2.0 Data On Technology
Consider a very simple economy that produces a single consumption good, corn, from inputs of two categories of labor, steel, and (seed) corn. All production processes in this example require a year to complete. Two production processes are known for producing steel. These processes require the inputs shown in Table 1 to be available at the start of the year for each ton steel produced and available at the end of the year. Two processes, as shown in Table 2, are also known for producing corn.
Process A | Process B | |
Category 1 Labor | (1/100) Person-Year | (33/50) Person-Year |
Category 2 Labor | 1 Person-Year | (6/5) Person-Year |
Steel | 0 Ton | (3/10) Ton |
Corn | (71/100) Bushel | (1/50) Bushel |
Process C | Process D | |
Category 1 Labor | 1 Person-Year | (7/10) Person-Year |
Category 2 Labor | 1 Person-Year | (4/5) Person-Year |
Steel | (1/10) Ton | (1/5) Ton |
Corn | 0 Bushel | (1/10) Bushel |
Technique | Processes |
Alpha | A, D |
Beta | A, C |
Gamma | B, C |
Delta | B, D |
Appendix A
The production functions that describe the technology in this example exhibit common neoclassical features: Constant Returns to Scale and non-increasing marginal returns to each factor. For purposes of constructing the production function for corn, the quantities of both categories of labor, steel, and corn available as input to corn production are taken as given. Let Q01, Q02, Q1, and Q2 be these quantities of category one labor, category two labor, steel, and corn, respectively. All quantities are measured in physical units (person-years, tons, bushels). The following Linear Program expresses the problem of maximizing the output of corn from these inputs: choose X1 and X2, the quantities of corn produced by processes C and D, respectively, to maximize:
(A-1)such that
(A-2)
(A-3)
(A-4)
(A-5)
(A-6)The production function for corn, f( Q01, Q02, Q1, Q2 ) is the value of the objective function, X* for the solution of the above Linear Program, expressed as a function of the physical inputs into corn production. These parameters define the right-hand-side of the linear constraints in Displays A-2 through A-5. The left-hand-side of these constraints is defined by the parameters in Table 2.
One standard method of visualizing production functions is with isoquants. An isoquant for this production function is a four dimensional surface, which I find difficult to draw. Accordingly, suppose the inputs of category 1 labor and corn are not binding for the level of output for which isoquants are drawn. Then Figure A-1 shows the isoquants for the corn production function in the remaining two dimensions. The dashed rays from the origin correspond to the two processes available for producing corn. If only one process for producing corn was known (a Leontief production function or "fixed coefficients" of production), an isoquant would consist of two rays, one extending horizontally right from the dashed line and the other extending vertically upward from the dashed line for that process. For the two known processes, an isoquant also contains the line segment shown sloping downward to the right. This line segment corresponds to a linear combination of the processes C and D, in which coefficients of production continuously vary.
Figure A-1: Isoquants For Corn Production Function |
Figure A-2: Outputs As Category 2 Labor Varies |
1 comment:
Whereas in the Sraffian theory a single technology is always most efficient (except in the switching point), the optimization problems with scarce resources may yield solutions with a mixed technology. Apparently the maximization of the objective function, for instance with linear programming, does not guarantee the realization of the largest profit rate. The maximization may result in the use of inefficient technology. This is true even when the function is the value of the net product itself. In addition the mix reacts tot scarcity by means of substitution. I wonder what this means or implies.
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