Working Paper Series

Consumer Demand with Several Linear Constriants:  A Global Analysis



  • Goldman School of Public Policy Working Paper (October 2004)


Economists sometimes find themselves in the position of having to extend the
neoclassical model of consumer demand to settings where, in addition to the conventional budget
constraint, there are one or more additional linear constraints that restrict the consumer’s utility
maximization problem. Examples include point rationing [Tobin-Houthakker (1950-51), Tobin
(1952)]; models of time allocation where the time constraint cannot be collapsed into the budget
constraint [de Serpa (1971); de Donnea (1972); McConnell (1975); Lyon (1978) Larson and
Shaikh (2001)]; and multi-period portfolio allocation problems [Diamond and Yaari (1972)].
Without exception, the existing literature has focused on differential properties of the resulting
demand functions-i.e. issues such as the effect of rationing on demand elasticities, the Le
Chatelier--Samuelson Theorem, the generalization of the Hick-Slutsky decomposition, and other
comparative static results [see Kusumoto (1976), Chichilnisky and Kalman (1978), Hatta (1980),
Wan (1981) and the references cited above]. By employing some “tricks with utility functions”
in the spirit of Gorman (1976), I am able to obtain a global characterization of these demand
functions. Specifically, I develop an algorithm for deriving the demand functions that apply
when there are M linear constraints from those that apply when these is only a single constraint.
The algorithm permits one to derive all of the existing comparative static results in a simple and
compact manner. It also has some value for empirical demand analysis, because it shows how to compute the demand functions associated with maximization problems involving multiple linear constraint based on direct or indirect utility functions associated with known conventional demand functions.

The paper is organized as follows. Section 2 presents some preliminary results which are
needed for the main analysis, but are also of interest as "tricks” in their own right. Section 3
considers the utility maximization problem with two linear constraints, summarizes the existing
comparative static results, develops the new Global Representation Theorem, and shows how
this can be used to derive and sharpen the existing comparative static results. Section 4 considers
a utility maximization problem with three linear constraints and develops the analogous Global
Representation Theorem for the solution to this problem in terms of known demand functions
associated with a conventional single-constraint problem; the results developed here provide the
basis for extension to problems involving more than three linear constraints. Section 5 offers
some concluding observations.

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