Flexible Consumer Demand Functions and Linear Estimation: Comment

Abstract

In a recent article in this Journal, Swamy and Binswanger (SB) claim that consumer demand equations in real income and relative prices can be obtained directly by differentiating a cost function in which output is replaced by real income. The advantage of this approach, they maintain, is that we can obtain observable demand functions, linear in parameters, which are integrable into wellbehaved expenditure and indirect utility functions. This approach to deriving demand functions, however, is considerably less general than SB maintain. Their method of derivation assumes that changes in utility can be approximated by changes in real income. Changes in utility are proportional to changes in real income for any given price vector, but this proportionality factor will change when prices change unless indifference curves are everywhere parallel. This latter condition obtains only when the direct utility function is homothetic, implying unitary income elasticities throughout. This means, as a description of the real world, SB's method can have only local validity, not global validity like Roy's identity. This also means that the three functional forms proposed by SB are best regarded as statistical demand functions, integrable into an arbitrary neoclassical utility function only at some base point. SB's method of deriving consumer demand functions is to replace the utility level in the Hicksian demand functions by real income. They maintain this approach is possible because certain index numbers can be estimated which, when used to deflate nominal income, provide estimates of changes in real income that correspond exactly to changes in utility levels (p. 676). The problem with this approach is that, in contrast to output in producer theory, utility is unobservable. Thus, arbitrary changes in utility cannot be calculated from changes in real income unless we know the parameters of the utility functions. (This is true regardless of how accurately real income is measured!) This is equivalent to saying that, in general, the relationship between changes in utility and changes in real income is not independent of prices. To verify this claim, consider the total differential of the Hicksian demand function, Xi(U,P), which can be written (1) d(log X,)