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.

Recent research on household `vulnerability' has led to an increased appreciation of the welfare costs of risk. Measuring the risk borne by a particular household has generally involved the use of panel data, and in particular the use of time series variation in household expenditures to estimate the risk borne by the household in any given period. This has led researchers to focus on static measures of vulnerability, since once used to identify the distribution of consumption expenditures in a single period the time series variation can no longer be used to describe the intertemporal profile of the distribution of consumption expenditures--simultaneous estimation of inequality, risk, and time series variation in household vulnerability requires the additional structure of a dynamic model. Unfortunately, our present understanding of the economic circumstances in which most households are situated seems too limited to permit general agreement on what the right dynamic model is. We show that simple restrictions on households' intertemporal smoothing can be used to simultaneously estimate household risk preferences in a manner which is robust to a variety of different assumptions about the economic environment. Further, these simple restrictions and estimated preferences can then be used to robustly characterize the welfare costs of different sorts of variation in consumption expenditures.

Much recent empirical work on intra-household allocation uses the axiomatic Nash Bargaining model to make predictions about how the distribution of consumption within the household will respond to individuals' income shocks. However, one of the basic axioms underlying this approach is that allocations will be Pareto optimal, so forward-looking, risk adverse household members ought to be expected to smooth away any such response to income shocks-Pareto optimality seems to be too strong in a dynamic setting. In this paper we use explicitly dynamic framework and replace the axiom of Pareto optimality with a weaker notion of efficiency. We give a simple algorithm for computing allocations, and construct an extended example, meant to model the effects of Grameen Bank lending on intra-household allocation in Bangladesh. The model resolves a puzzle in the literature, namely, it predicts that women borrowers will often voluntarily surrender control ("pipeline") their loans to their husbands.