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# All \lambda-separable Frisch demands and corresponding utility functions

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• Ligon, Ethan

## Abstract

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.

## Suggested Citation

• Ligon, Ethan, 2016. "All \lambda-separable Frisch demands and corresponding utility functions," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt1w13q2f1, Department of Agricultural & Resource Economics, UC Berkeley.
• Handle: RePEc:cdl:agrebk:qt1w13q2f1
as

File URL: https://www.escholarship.org/uc/item/1w13q2f1.pdf;origin=repeccitec
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## References listed on IDEAS

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### Keywords

Social and Behavioral Sciences; Frisch demands; separability; Pexider equations;
All these keywords.

### NEP fields

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