An Inverse Demand System for U.S. Composite Foods: Reply

elasticity matrix. The approach he uses for verification, however, is conceptually and empirically inadequate. First, Hicks has systematically treated the inverse demand and defined "q-complements" and "q-substitutes" of the Antonelli matrix in contrast to the "pcomplements" and "p-substitutes" of the well-known Slutsky matrix in ordinary demand. As shown in Deaton and Muellbauer (p. 57), the Antonelli matrix and Slutsky matrix are closely linked; they are generalized inverses of each other. Conceptually, if one intends to verify the dual nature of a demand system, one should focus on the generalized inverse relationships between the Antonelli and Slutsky matrices rather than using Young's intuitive approach by comparing the estimated price flexibilities with the derived demand elasticities. Moreover, Young's demand elasticity matrix is derived from a constructed uncompensated price flexibility matrix that departs from the functional form specified in Huang's inverse demand system. I know of no theoretical basis to verify the inverse demand system by using Young's derived demand elasticity matrix that is obviously outside the framework of the initial model specification for the inverse demand system. Second, in general, the empirical demand elasticity and price flexibility matrices obtained from certain well-known estimation procedures are not the reciprocal of one another in a statistical sense because the two sets of regression lines differ from one another. In an ordinary demand system, the sum of residuals is minimized along the quantity axis; whereas, the sum of residuals is minimized along the price axis in an inverse demand system. The problem was discussed in Houck, who summarized some earlier studies and concluded: (a) the reciprocal of the direct price flexibility is not in general the same as the direct price elasticity, and (b) the reciprocal of the price flexibility is absolutely less than the true elasticity if there are discernible cross effects with other commodities. Thus, Young's demand elasticity matrix, which is computed as the reciprocal of a derived uncompens ted price flexibility matrix, has limited practical use because the elasticity matrix may not represent the "true" demand structure as reflected from the sample observations. Perhaps a statement from Waugh (pp. 29-30) best addresses this point: "I prefer to use the p ice flexibilities themselves rather than their reciprocals. If, for any reason, the elasticity of demand is wanted, I would prefer to use the other regression equations, using quantities as the dependent variables."