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In this paper, we introduce three flexible consumer demand systems in which expenditures on goods are quadratic functions of income. We view these alternatives as to the demand systems used heretofore in the empirical modeling of rank-three demands, namely those in which expenditure shares are quadratic functions of the logarithm of income. Curvature conditions required by theory can be imposed locally during the estimation for each, and a semiflexible version can be estimated. For illustrative purposes, we estimate various forms of two of the systems using Canadian data on seven categories of goods for the period 1947 to 1995. © 1999 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology