Flexible and semiflexible consumer demands with quadratic Engel curves

In this paper, we introduce three flexible consumer demand systems in which expenditures on goods are quadratic functions of income. We view these alt ernatives as to the demand systems used heretofore in the empirical modelin g of rank-three demands, namely those in which expenditure shares are quadr atic functions of the logarithm of income. Curvature conditions required by theory can be imposed locally during the estimation for each, and a semifl exible version can be estimated. For illustrative purposes, we estimate var ious forms of two of the systems using Canadian data on seven categories of goods for the period 1947 to 1995.