Marshallian Demand Functions Research Articles

Smooth indifference sets

Smoothness of indifference sets is proposed as a weaker notion of smoothness for preference orderings. The usual notion of C r preference orderings is consistent with C r utility representations and C r−1 Marshallian demand functions. Although preferences with indifference sets which are C r manifolds do not possess C r utility representations, in general, many of the usual tools from the analysis of smooth utility functions can still be used, such as marginal rates of substitution. The assumption of smooth indifference sets is consistent with smooth Hicksian demand functions but not smooth Marshallian demand functions.

JEVONS ON UTILITY, EXCHANGE, AND DEMAND THEORY: A REASSESSMENT *

The scientific contribution of W. S. Jevons in the area of general demand theory has been a matter of some debate over the past century. Recently published documents and correspondence, together with Jevons's well-known Theory of Political Economy (1871), sketch a new drawing of his achievements in both partial and general equilibrium theories of demand and exchange. This paper employs these materials, along with additional mathematical derivations, to show that Jevons had the tools to develop a Marshallian demand function linking utility with demand and, further, that Jevons stood with Leon Walras in developing a general equilibrium approach to exchange theory. These inventions lead us to conclude that Jevons deserves more respect as one of the most important of a small and select group of theorists who founded the modern science of economics. Copyright 1989 by Blackwell Publishers Ltd and The Victoria University of Manchester

REVEALED INDEPENDENCE AND QUASTILINEAR CHOICE

Revealed Independence requires that revealed preferences satisfy the independence property from expected utility theory. This condition is essentially the same as the quasi-linearity of the choice function; i.e., it is a linear operator on convex combinations of budget sets. We demonstrate the equivalence of the Strong and Weak Axioms of Revealed Preference for quasi-linear choice functions with convex domains. IN GENERAL terms, revealed preference theory derives utility from demand, complementing subjective preference theory, which derives demand from utility. This reversal of procedure has profound operational significance, because it places neoclassical economics upon a more satisfactory behavioral foundation, where all axioms become testable hypotheses about observable variables. Indeed, the utility function is unobservable to the social scientist, but the typical demand function is at least partially observable, lending itself to statistical estimation. Samuelson [15] originally developed revealed preference theory in the context of commodity markets. He unsuccessfully sought to recover the utility function from his Weak Axiom of Revealed Preference (WARP) by integrating an appropriate system of partial differential equations derived from the Hicks-Slutsky substitution matrix. This integrability problem was eventually solved by Houthakker [7], who introduced the Strong Axiom of Revealed Preference (SARP) to replace WARP. Nevertheless, a social scientist still needs a constructive procedure that verifies whether or not a particular collection of choice data satisfies these behavioral axioms. In this regard, WARP would clearly provide a much simpler foundation for consumer demand theory, and Samuelson demonstrated its sufficiency in a world with only two commodities. Unfortunately, Gale [4] constructed an important counterexample, verifying that WARP is insufficient to recover the utility function from the Marshallian demand function in general. The exact characterization of WARP and SARP in terms of the Hicks-Slutsky substitution matrix thoroughly exposes the logical distinction between WARP and SARP in the context of the neoclassical model of the competitive consumer [9]. Yet this matter is not finished for other types of decision problems. We consider here Uzawa's extension of revealed preference theory to a logical setup that encompasses social choice theory [19]. The essential step is the generalization of the concept of a demand function to a choice function,

Welfare Evaluation and Household Production with Non-Constant Returns to Scale

A recent body of literature examining energy-consuming durables investment [2; 3; 4] has fused the household production and utility maximization models to estimate demand functions for end-use products such as heated water and air. All of these studies have made the seemingly necessary simplifying assumption that the household production functions exhibit constant returns to scale (CRS). This assumption provides for a linear budget constraint, a necessary condition for deriving demand functions. However, since it is unlikely that all the household production functions in fact closely approximate CRS, serious measurement error may occur. It is the purpose of this paper to prove that the CRS assumption is unnecessary. A marginal benefit function for end-use products, which is analogous to a Marshallian demand function, can be derived and estimated under a non-linear budget constraint. These demand functions for end-use products are useful in measuring the welfare effects of government programs, for example, rebates for purchases of high efficiency appliances.' It will be shown that measures of consumer's surplus, both Marshallian and exact, can also be estimated under non-linear budget constraints. In sections II and III, we formally derive a marginal benefit function and explain how it and consumer's surplus can be observed. In section IV, Shephard's lemma and Roy's identity are modified to account for the non-linear budget constraint and a method of deriving exact consumer's surplus is shown. Section V is an illustrative example of the described process and section VI concludes the paper.

Mirrored Pairs of Optimization Problems

Often, a solution to one problem of constrained optimization is also a solution to another, closely related, problem. The binocular vision afforded by these mirrored pairs of optimization problems frequently adds clarity and insight to our understanding of the original problem, as it so markedly has the case of consumer's demand. There it is standard to remark that a solution to the problem of maximizing utility subject to given prices and wealth is, in cases (Varian, 1978, p. 92), also a solution to the problem of minimizing expenditures subject to achieving the appropriate level of utility, and vice versa, a remark that leads directly to important identities that tie together the expenditure function, the indirect utility function, the Marshallian demand function, and the Hicksian demand function (Varian, 1978, p. 92). The question then arises of delineating precisely what is meant by these non-perverse cases, which a solution to an original problem is also a solution to its reflection. The conditions for the consumer's problem offered Varian's textbook (1978, p. 112) are really quite sharp, while even those Debreu's classic treatise (1959, pp. 67-71) are still unnecessarily restrictive. Moreover, these discussions do not bring out the symmetries between the conditions under which a solution to an original maximum problem also solves its reflected minimum problem, and those under which a solution to a minimum problem also solves its reflected maximum problem. This paper presents an abstract formulation of such problems, a formulation that leads naturally to a simple, symmetric and complete answer to the question posed above. Because the sole concern is with the interrelations of solutions to different problems, the theory needs to deal only with a few primitive objects, less numerous and less subtle than would be required for deeper inquiries into such matters as the existence, uniqueness or stability of solutions. Thus, the results obtained here do not lie at all deep the mathematical sense, the simplicity of the proofs alone precluding that. Nevertheless, the high degree of abstraction gives the formal scheme considerable generality and power, capable of application a wide variety of concrete specializations, and so eliminating the need to prove similar theorems ad hoc as one encounters each separate application.

Voluntary purchase of public goods

This paper examines the application of Buchanan's ‘independent adjustment’ model of public good provision to individual donations to voluntary or non-profit organizations. An individual's donation function is a simple transformation of the Marshallian demand function; consequently donation functions ‘reveal,’ in principle, preferences for public goods. The existence of a tax-subsidy system sustaining a Pareto optimal level of provision is demonstrated, and the relationship to the existing subsidy scheme in the U.S. is examined. Finally, two implications of the model suggest that it is not appropriate as a representation of actual donor behavior.

The Marshallian Consumer

Ever since the publication of Principles (Marshall, 1890), the Marshallian assumption that the marginal utility of money is constant has been the source of many debates and confusions. Consequently, the Marshallian demand function has been interpreted in different ways by different authors. Some of the exponents of the Marshallian demand theory (e.g. Samuelson, 1942; Green, 1971) assume a Paretian framework, in which the consumer shops for the day with a predetermined expenditure level. They derive the form of the utility function that is consistent with the assumption that the marginal utility of mone'y (expenditure) is invariant with respect to the prices the consumer has to pay. This is essentially a Paretian exercise, the origin of