The Use of Approximation Analysis to Test for Separability and the Existence of Consistent Aggregates

Abstract

Empirical analysis of production functions has typically been pursued by postulating a substitutable relationship between aggregate indices of heterogeneous capital and labor inputs. More recently, Ernst Berndt and David Wood, and Fuss, among others, have included aggregate indices of similarly heterogeneous energy and materials inputs among the postulated factors of production. The use of aggregate input indices requires the assumption that the production function is separable in these aggregates.' Separability implies that marginal rates of substitution between pairs of factors in the separated group are independent of the levels of factors outside that group. Berndt and Laurits Christensen (1973b) have shown that an alternative definition is that Allen partial elasticities of substitution between a factor in the separable group and some factor outside the group be equal for all factors in the group. The separability specification substantially restricts the structure of technology and therefore the possible functional form of the production function. On the other hand, separability permits the use of aggregate data when disaggregated data are unavailable or of poor quality. Separability is consistent with decentralization in decision making, or equivalently, optimization by stages. It opens up the possibility of multistage estimation of production decisions using consistent aggregates in the latter stages. Even when adequate disaggregated data are available multistage estimation may be the only feasible procedure when large numbers of inputs are involved.2 Thus separability is a pivotal concept in production function estimation. Yet until recently separability and the existence of aggregate inputs were assumed a priori in virtually all production function studies. In two recent papers Berndt and Christensen (1973a, 1974) have provided the first empirical tests of separability and the possible existence of consistent aggregates of labor and capital, using one of the currently available flexible quadratic functional forms, the translog function. In carrying out their tests, Berndt and Christensen implicitly assume that the true underlying production function is translog (i.e., the translog function exactly represents the underlying production process). An alternative, more general interpretation of quadratic functional forms is that they are second-order approximations to some unknown arbitrary production function. This approach has been advocated by Lawrence Lau and Christensen, Dale Jorgenson, and Lau (1973, 1975). The distinction is important since the two approaches lead to different separability tests with different characteristics, even when the analysis is organized around the same functional form. In this paper, we demonstrate first that the tests performed by Berndt and Christensen based on an exact interpretation of the translog function are more restrictive than is readily apparent and therefore cannot be accepted as general tests of the separability *Associate professors of economics, University of Toronto; research associates, Institute for Policy Analysis. We wish to acknowledge helpful comments from Ernst Berndt, Lawrence Lau, and an anonymous referee, but retain responsibility for any remaining errors. This paper is a condensed and somewhat simplified version of our working paper with the same title, to which the reader is referred for technical elaboration of the concepts presented. IWe are implicitly assuming that factor prices do not vary in proportion, so that Hicks' aggregation theorem does not provide a means of justifying the use of aggregates. 2For an example see the two-stage procedure used by Fuss.