Marshallian and Kicksian demands and the CPI

The use of price indices as deflators in demand equations is considered and clarified. Deflating prices and income by the CPI imposes a type of homogeneity restriction, but invalidates the interpretation of coefficients of a double-log model as uncompensated elasticities. However, deflating income by Stone's geometric price index-not the Consumer Price Index-means that real income really is held constant. This permits interpreting price coefficients in a double-log demand equation as compensated elasticities, provided the homogeneity restriction is imposed properly. While this result has been known since Stone (1954), it does not appear to be widely recognized in the literature on applied demand analysis. The compensated version of the double-log model has the same form as a share equation from the widely used linear approximate version of the Almost Ideal demand system, facilitating the construction of a test for choosing between these two alternative models. In an illustrative example using Theil's data for U.S. meat consumption, both the double-log and share specifications are rejected.