A generalization of two leading systems of demand functions

Abstract

For theoretical as well as empirical reasons it is often useful to study special classes of systems of demand functions. Thus, in empirical studies of consumer demand several specific systems of demand functions have been employed to explain, more or less satisfactorily, various aggregate data. There are, however, some serious misunderstandings in the literature about the logical structures and various relationships among these leading empirical systems of demand functions. After a close examination, we are left with two leading systems of demand functions from which we can obtain, as particular cases, the other systems surveyed. One is the system by Basmann and the other is the one by Leser-Houthakker. For the theoretical purposes, it is of interest to find a system of demand functions from which we can obtain all the systems surveyed. After specifying the functional form, we prove two theorems, each of which gives a necessary and sufficient condition that the specified form is a system of demand functions. As a side result, we obtain a criterion which shows the functional form every system of demand functions should satisfy. This result indicates an oversight by K. J. Arrow. This generalized system of demand functions was then used in an attempt to organize a detailed set of individual consumer price, quantity and expenditure data. When we tried to organize such data, we found that even this generalization was in poor agreement with data. After a close examination of the data, it appeared that the level of past consumption might be a factor in determining present consumption. Thus, we propose a simple system of demand functions which is obtainable from the utility function whose shift parameters contain the vector of past consumption. This new system seems to organize the data better than any of the systems of demand functions we surveyed. Also it could explain some apparent contradictions to the traditional Hicks-Slutsky theory of consumer demand. Therefore, we suggest that this approach may be more operationally significant, and may have more predictive power than the usual theory based upon a nonvariable consumer preference.