The Comparative Statics of Quantity Constraints and Conditional Demands: Theory and Applications

This paper presents a decomposition of the total effect of a change in a quantity constraint on the money income constant conditional demand function into an income compensated substitution effect and a pure income effect. The substitution effect is shown to be equivalent to the Hicks substitute-complement relation while the income effect is shown to depend directly on the extent of over- or under-supply of the constrained good. The Tobin-Houthakker results on rationing constraints can be generalized in an intuitive manner to cases of non-optimal values of the constraint when the two goods involved are: (1) Hicks substitutes, the constrained good is over-supplied and the unconstrained good is a normal good; or, (2) Hicks complements, the constrained good is under-supplied and the unconstrained good is a normal good. In the case of substitutes and under-supply or complements and over-supply, the sign of the Hicks substicute-complement relation is not sufficient to determine the sign of the partial derivative of interest. Several applications of the theoretical results are also presented.