Real income growth and revealed preference inconsistency

If a smooth demand function violates the strong axiom of revealed preference, the income and prices can follow a cycle and returm to their starting values even though real income is always rising. We show how real income growth along the "worst" revealed preference cycle depends on the range of price variation and on violations of the Slutsky conditions. We relate this result to proposed reforms of the consumer price index and use it to justify a new index of local demand inconsistency. We also use the Slutsky matrix to determine an upper bound on the number of observations required to detect revealed preference inconsistency

Real income growth and revealed preference inconsistency.

If a smooth demand function violates the strong axiom of revealed preference, the income and prices can follow a cycle and returm to their starting values even though real income is always rising. We show how real income growth along the "worst" revealed preference cycle depends on the range of price variation and on violations of the Slutsky conditions. We relate this result to proposed reforms of the consumer price index and use it to justify a new index of local demand inconsistency. We also use the Slutsky matrix to determine an upper bound on the number of observations required to detect revealed preference inconsistency.

Nonrepresentative representative consumers

Representative consumers can be very Pareto inconsistent. We describe a cornmunity, with equal income distribution, where all consumers require 56 % higher aggregate income than the representative consumer requires in order to be compensated for the doubling of a price. Such large inconsistencies are ruled out if the representative consumer is homothetic, or if the consumers' income shares are fixed and all goods are normal. We show that optimality of the income distribution rule is not necessary for Pareto consistency of the representative consumer, and we give a weaker sufficient condition for Pareto consistency in cornmunities with two goods and two consumers

Nonrepresentative Representative Consumers

Single consumer models are often used to focus attention on economic efficiency, leaving aside equity considerations. In general, these ¡°representative consumer" models do not accurately portray the effects of changes in policies, endowments or technology in the multi-consumer economies they are meant to represent; but even when they do, they are not necessarily adequate for evaluating efficiency. The representative consumer can be Pareto inconsistent, preferring a situation B to A even though all the consumers in the represented economy prefer A to B. It is not clear from the literature how serious a defect this can be. The known examples of Pareto inconsistency are not robust. Small changes in the consumers' preferences remove the Pareto inconsistency. It has been an open question whether large robust Pareto inconsistencies are possible. This paper shows that they are. In one example, the actual consumers require 56% more income than the representative consumer requires in order to be compensated for the doubling of a price. But such large Pareto inconsistencies require that there is a Giffen good for the representative consumer. We argue that the inconsistencies of representative consumers in most macroeconomic applications are likely to be small and we give conditions ruling out inconsistencies entirely.

Demand Dispersion, Metonymy and Ideal Panel Data

In a generic competitive economy with constant returns production and "increasing dispersion," market demand satisfies the weak axiom of revealed preference and equilibrium is unique. Increasing dispersion requires, roughly, that when the households' incomes rise slightly their demand vectors move apart. We show how to test for it using panel data with fixed relative prices under a "structural stability" hypothesis due to Hildenbrand and Kneip (1999). We also show how to test for it using cross section data if the households' demand functions and incomes are independently distributed, or under a much weaker condition called "dispersion metonymy." We show that this weaker condition is untestable---even with ideal panel data that allow a direct test of increasing dispersion. Thus, cross section tests of increasing dispersion rely on an assumption that is not potentially falsifiable.Aggregation, Weak Axiom, Increasing Dispersion, Cross Section, Structural Stability

Nonrepresentative representative consumers.

Representative consumers can be very Pareto inconsistent. We describe a cornmunity, with equal income distribution, where all consumers require 56 % higher aggregate income than the representative consumer requires in order to be compensated for the doubling of a price. Such large inconsistencies are ruled out if the representative consumer is homothetic, or if the consumers' income shares are fixed and all goods are normal. We show that optimality of the income distribution rule is not necessary for Pareto consistency of the representative consumer, and we give a weaker sufficient condition for Pareto consistency in cornmunities with two goods and two consumers.Representative consumer; Optimal income distribution rule; Increasing dispersion; Pareto inconsistency;

Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains

This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2011 ElsevierFor functions from the Sobolev space H^s(\Omega­), 1/2 < s < 3/2 , definitions of non-unique generalized and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain­, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the generalized co-normal derivative inherent non-uniqueness are given. It is shown, that the canonical co-normal derivatives, although de¯ned on a more narrow function class than the generalized ones, are continuous extensions of the classical co-norma derivatives. Some new results about trace operator estimates and Sobolev spaces haracterizations, are also presented

How to recognize convexity of a set from its marginals

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex