Substitution, Complementarity, and the Residual Variation: Some Further Results - Phlips, Louis; Rouzier, Philippe (American Economic Review, 2016)
An earlier paper by Phlips reported the results of a principal component analysis of the residual correlation matrix obtained after estimating a system of dynamic demand equations. The purpose was to verify the postulated additive nature of the utility function and to collect some information on possible substitution and complementarity relationships among the commodity groups. The residuals were obtained using the original Houthakker-Taylor (HT) 1966 estimation procedure. This procedure presents some advantages, but also some deficiencies. On the other hand, the correlations (in particular their signs) were interpreted on the basis of the Hicksian definitions of substitutability and complementarity, in terms of the signs of the substitution effects. The object of this note is to present some further results. First, it is of some interest to determine to what extent the results reported in the abovementioned article resist not unimportant changes in the estimation procedure. Secondly, the Hicksian definitions are rather deceptive: it is intuitively more appealing to work with the old (cardinal) notions of substitutability and complementarity (stated in terms of the signs of the second cross partial derivatives of the utility function). A decomposition of the (total) residual correlations into and correlations, the latter corresponding to preference relations defined in cardinal terms, is presented here. This corresponds to a breakdown of the substitution effect into a general and a specific effect. The analysis of this specific effect allows us to check our previous conclusions as to the grouping of commodities for which the assumption of additive preferences is appropriate.