Additive Utility Functions and Linear Engel Curves.

If an individual's utility function is additive and his demand functions exhibit expenditure proportionality, it is well known that his utility function belongs to the "Bergson family". For some problems (e.g., deriving a consumption function from an intertemporal utility function) expenditure proportionality is a useful simplifying assumption. But for demand analysis the assumption that all income elasticities are unity has little merit even as a first approximation. In this paper I investigate the class of additive utility functions yielding demand functions which are locally linear in income, or, equivalently, yielding income-consumption curves which are linear in some region of the commodity space. The assumption that income-consumption curves for broad aggregates of goods are locally linear is not grossly inconsistent with our knowledge of the world, so this class of utility functions and the corresponding demand functions may be of empirical as well as theoretical interest. In the first section I consider various additive direct utility functions which yield demand functions locally linear in income, and, in the second section, the corresponding indirect utility functions. In the third section I prove that the utility functions considered in Section I are the only additive direct utility functions yielding demand functions locally linear in income. In the fourth section I consider the "counterparts" of the utility functions of Section I (i.e. the additive indirect utility functions obtained by replacing each quantity variable by its dual price variable). In Sections I and IV, I present explicit expressions for the demand functions corresponding to each utility function considered. [ABSTRACT FROM AUTHOR]