Estimation of unit values in cross sections without quantity information and implications for demand and welfare analysis

Household surveys frequently record only expenditure information. The lack of information about quantities purchased precludes the possibility of deriving household specific unit values. The aggregate price indexes derived from sources exogenous to the household survey are often not sufficient to identify all parameters and to provide plausible estimates. We use a theoretical result developed by Lewbel (1989) to construct "pseudo" unit values by 1) reproducing the variability of crosssectional price variation using the variability of the budget shares, and 2) adding the variability to the aggregate price indexes published by the national statistical institute. The study estimates a complete quadratic demand system using a time series of cross-sections of Italian household budgets including, in turn, aggregate price indexes and "pseudo" unit values. The results show that the matrix of compensated price elasticities is negative semidefinite only if "pseudo" unit values are used. In order to have a counterfactual experiment, we then consider a household survey with actual unit values and compare them with "pseudo" unit values. The experiment shows that in most cases "pseudo" values maintain the relevant characteristics of the distribution of actual unit values. Overall, we conclude that "pseudo" unit values are better than aggregate price indexes for sound demand and welfare analysis.