Measuring Consumer Welfare Using Statistical Demands

The equivalent or compensating variation for a price increase is often calculated using the expenditure function from a statistical (i.e. estimated) demand. If the regression errors are due to unobserved heterogeneity, then the variation from the statistical demand does not generally equal the mean variation for households, resulting in inconsistent estimates. We give conditions ensuring that the compensating variation from the statistical demand i) equals the mean compensating variation; ii) bounds the mean compensating variation; iii) is closer to the mean compensating variation than the change in consumers' surplus from the statistical demand. A necessary condition for ii) is that demands become more dispersed as income rises (for each class of households with the same demographic characteristics and income). This plausible necessary condition is not sufficient for either ii) or iii). If however we can write the indirect utility function for each household in the class as additively separable in income and the preference type, then increasing dispersion is equivalent to ii) and implies iii) if the good is normal. If household preferences are random, then the indirect (von Neumann-Morgenstern) utility function must be additive separability in income and the preference type if the compensating variation from the mean demand is even a first-order approximation to the ex ante compensating variation; consumers' surplus can easily be a better approximation when additive separability fails.