A Class of Individual Expenditure Functions

This study examines the regularity properties ensuring that individual expenditure functions are legitimate individual cost functions in the context of collective household models. The structure of collective household models entails a scaling of income through a function that describes how resources are shared within the household. This modified income function defines expenditure functions at the individual level. Our study completes previous work on modifying functions by Barten, Gorman, and Lewbel that was limited to the investigation of the scaling of prices and the translation of income without considering the scaling of incomes. We find that the product of the modifying function and the household expenditure function maintains the regularity properties of expenditure functions if the modifying function is positive, homogeneous of degree zero and at least quasi-concave. We also examine how changes in prices affect the curvature of the modified income function and, in turn, inequality in the distribution of resources within the household. An example shows how our results can be used to test the curvature properties of individual expenditure functions as well as to measure the inequality within the household.