Switching of Techniques and Consumption Per Head

Abstract

The analysis of reswitching of techniques enables us to show that if stationary states are compared, low rates of interest do not necessarily go together with a high consumption per head. If a stationary state with a high rate of interest gives way to one with a low rate of interest, the corresponding switch need not necessarily be from a technique with a low consumption per head to one with a high consumption per head. It may also be from a technique with a high to one with a low consumption per head. What has not been sufficiently emphasized in the literature, however, is that whether the former or the latter is the case may depend not only on which technique replaces which but also on what is the pattern of consumption. M. Bruno, E. Burmeister, and E. Sheshinski assert I that this is not so. They argue that the hyperplanes representing the opportunity sets of consumption for the techniques which are optimal at different rates of interest never intersect, and that whatever the pattern of consumption the respective point on one hyperplane must always be above, or always below, that on the other plane. The purpose of the present note is to show by means of a numerical example that this assertion is not correct. Consider a circulating-capital model with a generalized Leontief technology. There is no joint production in it; the same goods may be used either as inputs or as consumer goods; alternative methods of production are available for each of them; and labor is the only primary factor of production. Let wages be paid at the end of the period, and all other inputs at the beginning of it. The technique actually used is then that which minimizes prices subject to the constraint that at the given wage and the given rate of interest no production processes are run at a loss. This means that if a is the optimal technique, the vector of prices may be written as (A) pa(r) =wao[I(l+r)a]'where w is the wage, r is the rate of interest, a is the matrix of input coefficients for technique a, and ao is the corresponding vector of labor coefficients. The consumption hyperplane is then L=wao [I-a] -lCa=Pa(O)Ca