|Figure 1: The Choice of Technique in a Model with One Switch Point|
This post presents an example in which vertically-integrated firms producing a consumption good have a choice between two techniques. The wage-rate of profits curves for the techniques have a single switch point, at which they are tangent. I, and, I dare say, most economists who are aware of the illustrated possibility, consider this a fluke, a possibility that cannot be expected to arise in practice.2.0 Technology
The technology in this example has the structure of Garegnani's generalization of Samuelson's surrogate production function. One commodity, corn, can be produced from inputs of labor and a single capital good. Two processes are known for producing corn, and each process requires a different capital good, called "iron" and "copper". Each capital good is produced, if at all, by a process that requires inputs of labor and that capital good. Each process requires a year to complete, and the services of the capital good fully consume that capital good during the course of the year. No stock of iron or copper remains at the end of the year to carry over into the next year.
Constant Returns to Scale are assumed for each process. Table 1 shows the coefficients of production for the four processes specified by the technology. Each column corresponds to a process. The coefficients of production specify the input of the row commodity that is needed to produce a unit output of the commodity for the column.
Two techniques are available for producing corn (Table 2). The Alpha technique consists of the process for producing iron and the corn-producing process that requires an input of iron. The Beta technique consists of the copper-producing process and the corn-producing process using services provided by copper.
3.0 Prices and the Choice of Technique
The technique, as usual, is chosen by managers of firms to minimize costs. Corn is taken as the numeraire, and wages are paid at the end of the year. Prices of production, in which all extra profits above a common rate have been competed away, are used to calculate costs. For analytical convenience, in this pose I take the rate of profits as given.
Suppose the Alpha technique is chosen. Under the above assumptions, the price of iron and the wage must satisfy the following system of two equations:
(1/2) piron (1 + r) + wα = piron
3 piron (1 + r) + wα = 1
A similar system arises for the Beta technique, but as applied to the price of copper and the coefficients of production for the Beta technique.
For a non-negative rate of profits, up to a certain maximum rate that depends on the technique, one can solve each system of equations for the wage and the price of the relevant capital good. The resulting wage-rate of profits curve for the Alpha technique is:
wα = (1 - r)/(7 + 5 r)
The maximum wage for the Alpha technique, 1/7 bushels per person-year arises for a rate of profits of zero in the above equation. The maximum rate of profits, for the Alpha technique, is 100% and occurs when the wage is zero.
The wage-rate of profits curve for the Beta technique is:
wβ = (43 - 48 r)/361
For what it is worth, the maximum wage for the Beta technique is 43/361 bushels per person-year. The maximum rate of profits is 43/48, approximately 90%. Both the maximum wage and the maximum rate of profits for the Beta technique are dominated by the corresponding values for the Alpha technique.
Figure 1, at the top of this post, graphs the wage-rate of profits curves for both techniques. Since the coefficients of production in copper-production are a constant multiple (48/91) of the coefficients of production in the process for producing corn from copper, the wage-rate of profits curve for the Beta technique is a straight line. The wage-rate of profits curve for the cost-minimizing technique forms the outer envelope in Figure 1. The Alpha technique minimizes costs for all feasible rates of profits and wages.
One switch point arises in this example. It is at 50%, half the maximum rate of profits for the Alpha technique. The wage is 1/19 bushels per person-year at the switch point, and the slope of both wage-rate of profits curves has a value -48/361 at the switch point. One can find the rate of profits for the switch point by equating the functions for wα and wβ. A quadratic equation arises for the rate of profits, and 50% is a repeated root for this polynomial. Both the Alpha and Beta techniques are cost-minimizing at the switch point.4.0 The Market for "Capital"
One can find gross outputs of each process needed to produce a bushel of corn. If the Alpha technique is used, gross outputs consist of two tons iron and 1 bushel corn. For the Beta technique, gross outputs consist of 91/43 tons copper and one bushel corn. The quantity of the capital good, in physical units, needed to produce a net output of one unit of the numeraire good is immediately obvious in this technology. The total quantity of labor, over all processes in a technique, for producing a net output of corn is vector dot product of the labor coefficients, for the technique, and the gross outputs.
The quantity of the capital good must be evaluated with prices so as to graph, say, the amount of capital per person-year for each technique in one space. Since the wage-rate of profits curve for the Alpha technique has some non-zero convexity, the price of iron varies with the given rate of profits:
piron = 2/(7 + 5 r)
The price of copper is a constant 48/91 bushels per ton.
Table 2 brings these calculations together. It shows the ratio of the value of the capital good to labor inputs. The horizontal line shows the real Wicksell effect at the switch point. If one wanted, one could remove the price Wicksell effects with Champernowne's chain index measure of capital.
|Figure 2: Capital per Worker versus Rate of Profits|
For completeness, Figure 3 graphs the wage against the amount of labor hired, across all industries, to produce a net output of corn with cost-minimizing techniques. A linear combination of the techniques at the switch point is shown here, also, by a horizontal line.
|Figure 3: Labor per Unit Output versus Wage|
Generic results show a certain structural stability. Qualitative properties, for generic results, continue to persist for some small variation in model parameters. This is not the case for the example. Small variations will lead to either two switch points (that is, reswitching) or no switch points. In the latter case, the Beta technique would be dominated and never cost-minimizing.
I look to the mathematics of dynamical systems for an analogy. One can look at prices of production as fixed points in some dynamical system. For example, consider a classical view of competition, in which firms and investors are able to shift from the production in one industry to production in another. (Literature on such dynamical processes can be found under the keyword of "cross-dual dynamics".) Neoclassical economists might look at prices of production as a special case of an intertemporal equilibrium, in which initial endowments just happen to be such that relative spot prices do not vary with time. Or one can consider prices of production as partially characterizing a fixed point, in a limiting process, as time grows without bound in neoclassical models of intertemporal or temporary equilibria.
At any rate, hyperbolic points are considered generic in dynamical systems. In discrete time, no eigenvalues of the linearization around a hyperbolic point lie on the unit circle. Continuing in the jargon, no center manifold exists for a hyperbolic point. Non-hyperbolic fixed points are important in that they indicate a bifurcation.