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One of the principal tasks of empirical demand analysis is the measurement of income and own-price elasticities. In recent years, the tools of this research have been complete systems of demand equations constructed to be consistent with the theory of utility maximization. General models of this class permit a wide range of income and price response but except for the analysis of a very limited number of commodities, require the estimation of too many parameters to allow efficient or precise estimation. Consequently, much empirical work has been based on specializations of the basic hypothesis which restrict the range of allowable responses and the number of parameters to be estimated. The most common of these special assumptions are direct and indirect additivity, but in Deaton (1974b, 1975a), I have shown that these assumptions imply and enforce invalid linear relationships between price and income elasticities which severely distort the measurement of both. An alternative assumption, which specifies demand in terms of" real" income and" real" price, was first proposed by Fourgeaud and Nataf (1959) and has recently been estimated by Nasse (1973). It is shown in the appendix to this paper that this model, like direct and indirect additivity, also requires an approximate linear relationship between own-price and income elasticities. Since income and price elasticities are normally considered to be largely independent