All \lambda-separable Frisch demands and corresponding utility functions

Frisch demands depend on prices and a multiplier \(\lambda\)associated with the consumer's budget constraint. The case in whichdemands or expenditures are separable in $\lambda$ is the case ofgreatest empirical interest, since in this case latent variablemethods can be adopted to control for consumer wealth when estimatingdemands. Subject only to standard, modest, regularity conditions, we provide a completecharacterization of all Frisch demand systems and of the utilityfunctions that rationalize these demand systems when either quantitiesdemanded or consumption expenditures is separable in $\lambda$. Quantities demanded are \(\lambda\)-separable if and only if therationalizing utility function is additively separable in thesequantities. In contrast, expenditures are \(\lambda\)-separable ifand only if marginal utilities for these expenditures belong to one oftwo simple parametric families. With $n$ goods, the first family has$2n$ parameters, and corresponds to Houthakker's "direct addilog"utility function. The second family has $3n$ parameters and is new.It corresponds to a family of utility functions which have Stone-Gearyutility as a limiting case.