Saturday, July 16, 2022

Fixed Capital And The Emergence Of Reswitching

Figure 1: A Wage Frontier With A Fluke Switch Point

A fluke example with fixed capital illustrates the emergence of the reswitching of techniques. Table 1 presents coefficients of production in a perturbation of an example from Schefold (1980). With the first process, workers, under the direction of mangers of firms, manufacture new machines. The remaining two processes are used to produce corn. The last process requires an input of an old machine, which is jointly produced with corn by the second process. Corn is both a consumption good and a capital good, insofar as it is an input into all three processes. Technology improves in this example, as usual, with an exponential decline in specified coefficients of production.

Table 1: The Coefficients of Production
InputMachine IndustryCorn Industry
One ProcessAnother Process
Labor(1/10)e1-σ t(43/40) e1-φ te1-φ t
Corn(1/16)e1-σ t(1/16) e1-φ t(1/4)e1-φ t
New Machines010
Old Machines001
New Machines100
Old Machines010

The choice of technique corresponds here to the choice of the economic life of the machine. This lifetime is truncated to one year for the Alpha technique, while the machine is operated for its full physical life of two years under the Beta technique. In a pure fixed capital model, the choice of technique can be analyzed by the construction of the wage frontier. The cost-minimizing technique at a given rate of profits has a wage curve on the outer frontier, as illustrated by Figure 1 for a specified parametrization. Managers of firms are willing to operate the machine for two years for any feasible rate of profits. At the maximum wage or a rate of profits of zero, the Alpha technique is also cost-minimizing. The single switch point is a fluke in two ways. First, it lies on the wage axis. Second, the wage curves are tangent at the switch point.

The left pane in Figure 2 depicts a part of the parameter space for this example. Although not apparent to the eye, a thin wedge between two partitions extends to the northeast of the point for the parameters corresponding to Figure 1. At the upper edge of this wedge, the two wage curves for the techniques are tangent at a switch point. The example is of reswitching below this partition and within this wedge. Schefold's example lies to the northeast off the graph, when σ t and φ t are both unity. At the lower edge of this wedge, the switch point with the lower rate of profits is on the wage curve. The pane on the right in Figure 2 shows the vertical difference between these two partitions so as to convince the reader of the existence of this region with reswitching.

Figure 2: A Part of a Parameter Space

Reswitching demonstrates the well-known conclusion that no coherent marginal productivity theory of distribution exists. The economic life of the machine is the full two years here for a low and high rate of profits. Truncation occurs for a range of intermediate rates of profits. The specification of which technique is cost-minimizing can be consistent with vastly different functional distributions of income, with other techniques being cost-minimizing for less extreme distributions. Marginal productivity is, at best, an analysis of the choice of technique within a more general framework.

The switch point at the higher rate of profits in the reswitching region of the parameter space illustrates capital-reversing. Around this switch point, a lower rate of profits is associated with the adoption of a less capital-intensive, cost-minimizing technique. At any rate of profits, inputs into production in a stationary state can be evaluated and these evaluations summed for each technique. The ratio of capital per worker, for example, is an index of the capital intensity of a technique. A more capital-intensive technique produces more output per worker, but its adoption is not necessarily encouraged by a lower rate of profits or interest rate. In other words, A higher wage is associated with the adoption of a technique that requires a greater input of labor per bushel corn produced net throughout the economy. Capital-reversing has been shown to occur in other examples without reswitching on the wage frontier. Harcourt (1972) surveys the controversy in which economists, such as Paul Samuelson and Robert Solow, in Cambridge, Massachusetts, struggled to accept these conclusions drawn by other economists, such as Joan Robinson and Piero Sraffa, at the University of Cambridge.

Consider the region to the southeast in the part of the parameter space illustrated in the left pane of Figure 2. A single switch point exists on the wage frontier. Around this switch point, a lower rate of profits is associated with the adoption of a technique with a greater value of capital per person-year and a greater output per worker. Nevertheless, truncating the operation of the machine for one year is associated with a more capital-intensive technique. The invalidity of Austrian capital theory does not even need the phenomena of reswitching and capital-reversing for its demonstration.

No switch points exist in the northwest of the part of the parameter space graphed in Figure 2, and the machine is operated for its full physical life of two years for any feasible distribution of income.

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 are not fluke cases. These posts demonstrate this conclusion by contrasting these possibilities with genuine fluke cases.

No comments: