Risk sharing tests and covariate shocks

The hallmark of full risk sharing is that agents' marginal utilities of expenditure (MUEs) have a simple factor structure; a Pareto weight is divided by an aggregate price. Take logarithms and full risk-sharing can be easily tested using panel data with two-way fixed effects. The catch is that we don't directly observe MUEs, and must infer these using data on consumption expenditures. The standard approach to this inference problem is to assume some form of homothetic utility, in which case the MUE is a function of total expenditures and a single price index, and all demands have unit price elasticities. This approach works well when the shocks being tested affect agents' budgets without changing prices; i.e., when the shocks are idiosyncratic. But "covariate" shocks may change relative prices, in which case the standard risk-sharing tests which assume that no demands are inelastic will deliver apparently perverse results. What is the class of utility structures that allow one to test risk-sharing using only panel data on expenditures and two-way fixed effects, and does this class included non-homothetic preferences which are consistent with more realistic demand responses to changes in relative prices? We obtain this class, which happens to be semi-parametric and nests the usual homothetic specification, but which allows for highly flexible Engel curves, with $n$ parameters corresponding to the income elasticities of $n$ goods. We provide a simple algorithm to infer both these parameters and the agents' MUEs. We compute these using panel data from Uganda, and show that risk-sharing tests of covariate shocks using our computed MUEs deliver sensible results while the standard tests do not.