Estimating household welfare from disaggregate expenditures

Understanding how marginal utilities of expenditure evolve over time is key to the specification, the estimation, and the testing of dynamic consumer behavior. But in existing dynamic models researchers often (perhaps implicitly) assume that Engel curves are linear. This allows one to specify marginal utilities as a function of nothing more than real expenditures, but is sharply at odds with strong empirical evidence against linearity, including Engel's Law. Here we show how one can use data on disaggregate expenditures to estimate demand systems that may feature highly non-linear Engel curves; this same estimation procedure yields summary measures of household welfare within the period which we call "neediness". Our neediness measure is of interest in its own right, and can also be used to construct measures of inequality and poverty which match conventional measures given prevailing prices, but which also describe how inequality and poverty would be different were prices different. Beyond this, it is intimately related to the marginal utilities of expenditure that are critical in dynamic models. We illustrate the use of these methods using data from Uganda to estimate an incomplete demand system, and to estimate household neediness in different periods. This offers measures of the distribution of welfare and how this distribution changes over time. Our measure of household neediness can also be regarded as an index of the marginal utility of expenditures, which plays a central role in models involving dynamics and risk. We use our estimates to look for evidence of either borrowing or savings constraints, and find no such evidence; separately, we find strong evidence of heterogeneity in relative risk aversion.