A Unied Approach to Estimating Demand and Welfare⇤

The measurement of price changes, economic welfare, and demand parameters is currently based on three disjoint approaches: macroeconomic models derived from time-invariant utility functions, microeconomic estimation based on time-varying utility (demand) systems, and actual price and real output data constructed using formulas that di�er from either approach. The inconsistencies are so deep that the same assumptions that form the foundation of demand-system estimation can be used to prove that standard price indexes are incorrect, and the assumptions underlying standard exact and superlative price indexes invalidate demand-system estimation. In other words, we show that extant micro and macro welfare estimates are biased and inconsistent with each other as well as the data. We develop a uni�ed approach to demand and price measurement that exactly rationalizes observed micro data on prices and expenditure shares while permitting exact aggregation and meaningful macro comparisons of welfare over time. We show that all standard price indexes are special cases of our approach for particular values of the elasticity of substitution, constant preferences for each good, and a constant set of goods. In contrast to these standard index numbers, our approach allows us to compute changes in the cost of living that take into account both changes in the preferences for individual goods and the entry and exit of goods over time. Using barcode data for the U.S. consumer goods industry, we show that allowing for the entry and exit of products, changing preferences for individual goods, and a value for the elasticity of substitution estimated from the data yields substantially di�erent conclusions for changes in the cost of living from standard index numbers. JEL CLASSIFICATION: D11, D12, E01, E31