The implications of a third-order translog demand system and some empirical results

In this paper a homogeneity and symmetry unconstrained variant of the “third-order translog”, representing a third-order approximation to an indirect utility function, is derived. Using Christensen, Jorgenson and Lau's (1975) data, and subsequent observations to 1986, we estimate this third-order translog, and show that it dominates the homogeneity and symmetry unconstrained translog. The restrictions implied by homogeneity and symmetry, conditional on our third-order translog are also rejected. These results are of interest since they demonstrate that standard restrictions implied by the theory of consumer behaviour can be rejected even in the context of a fairly general model which has been estimated using aggregate, time-series data. However, it has been found that the inclusion of dynamic effects sometimes leads to non-rejection of theoretical restrictions (Anderson and Blundell 1983). Unfortunately, the use of aggregate, time-series data to estimate a dynamic, third-order translog would be problematic, since exact aggregation restrictions are imposeda priori and observations are limited. This suggests that it would be more appropriate to estimate the third-order translog using cross-sectional micro-data sets with many observations. This type of data is now available, and can be used to estimate systems of consumer demands. Furthermore, these data sets contain extensive information on demographic characteristics which, having recently been found to be important determinants of demand (Barnes and Gillingham 1984) can be included in general demand systems of the third-order translog type.