Aggregate Production Functions: Some CES Experiments

Abstract

As a purely theoretical matter, aggregate production functions exist only under conditions too stringent to be believed satisfied by the diverse technological relationships of actual economies. There is a summary discussion and bibliography in Fisher [2]. Yet aggregate production functions estimated from real data do appear to give good results, at least sometimes, and to do so in an apparently non-trivial way. Not only do such estimated relationships give good fits to input and output data, but also the calculated marginal products appear to be related to observed factor payments. Alternatively, production functions with parameters estimated from factor payments turn out to fit input and output data pretty well sometimes. It is not a simple matter to decide why this should be so as a matter of theory. Indeed, the problem is sufficiently complicated that perhaps the most promising mode of attack on it is through the construction and analysis of simulation experiments. By constructing simplified economies in which the conditions for aggregation are known not to be satisfied, we can hope to find out inductively the circumstances under which aggregate production functions appear to give good results in the double sense just discussed. Moreover, such experiments can cast light on other aspects of the estimation of aggregate production functions from underlying non-aggregatable data. The related papers of Houthakker [4], Levhari [6] and Sato [8] do not bear directly on the problems here addressed. Those papers show what aggregate production functions can be expected when the distribution of capital over firms with related technologies is fixed (or changes in very restricted ways). Such fixity of distribution, however, can hardly be expected in the real world and is certainly not true in the world of simulation experiments reported below and in Fisher [3]. Here the issue is that of why aggregate production functions should appear to exist at all, rather than that of what form they will take given that a constant distribution of capital over firms ensures their existence. See Fisher [2, pp. 571-574]. The books by Johansen [5] and Sato [9] give excellent discussions of aggregation from several points of view. This programme of research was begun in Fisher [3]; individual firms (with a single homogeneous output and single homogeneous labour but different capital types) were given different Cobb-Douglas production functions, underlying capital and labour data were generated in various ways, and labour was assigned to firms to maximize output. An aggregate Cobb-Douglas production function was then estimated and its wage predictions examined. A number of subsidiary results were found in these experiments, and we shall comment