Afriat’s Theorem and Samuelson’s ‘Eternal Darkness’

Suppose that we have access to a finite set of expenditure data drawn from an individual consumer, i.e., how much of each good has been purchased and at what prices. Afriat (1967) was the first to establish necessary and sufficient conditions on such a data set for rationalizability by utility maximization. In this note, we provide a new and simple proof of Afriat’s Theorem, the explicit steps of which help to more deeply understand the driving force behind one of the more curious features of the result itself, namely that a concave rationalization is without loss of generality in a classical finite data setting. Our proof stresses the importance of the non-uniqueness of a utility representation along with the finiteness of the data set in ensuring the existence of a concave utility function that rationalizes the data.     

A contribution to duality theory,applied to the measurement of risk aversion

This paper determines the precise connection between the curvature properties of an objective function and the ray-curvature properties of its dual. When the objective function is interpreted as a Bernoulli or cardinal utility function, our results characterize the relationship between an agent’s attitude towards income risks and her attitude towards risks in the underlying consumption space. We obtain these results by developing and applying a number of representation theorems for concave functions.The work of Juan E. Martínez-Legaz has been supported by the Spanish Ministry of Science and Technology and the FEDER, project BEC2002-00642, and by the Departament d’Universitats, Recerca i Societat de la Informació, Direcció General de Recerca de la Generalitat de Catalunya, project 2001SGR-00162. He also thanks the Barcelona Economics Program of CREA for its support. John Quah would like to acknowledge with gratitude the financial support of the ESRC (grant number R000271171). He would also like to thank the Department of Economics at UC Berkeley, whose hospitality he enjoyed while completing this project. Both authors would like to thank Simon Cowan for pointing the way to some important references. They are also very grateful to the referee whose insightful suggestions led to a much improved paper     

On the curvature of the reporting function from objective reality to subjective feelings

I suggest the idea of a reporting function, r(.), from reality to feelings. The ‘happiness’ literature claims we have demonstrated diminishing marginal utility of income. I show not, and that knowing r(.)'s curvature is crucial. A quasi-experiment on heights is studied.     

Necessity of hyperbolic absolute risk aversion for the concavity of consumption functions

Carroll and Kimball (1996) have shown that, in the class of utility functions that are strictly increasing, strictly concave, and have nonnegative third derivatives, hyperbolic absolute risk aversion (HARA) is sufficient for the concavity of consumption functions in general consumption-saving problems. This paper shows that HARA is necessary, implying the concavity of consumption is not a robust prediction outside the HARA class.     

Credibility and determinism in a game of persuasion

This paper studies a game of persuasion. A speaker attempts to persuade a listener to take an action by presenting evidence. Glazer and Rubinstein (2006) showed that when the listener's decision is binary, neither randomization nor commitment have any value for the listener, and commented that the binary nature of the decision was important for the commitment result. In this paper, I show that concavity is the critical assumption for both results: no value to commitment and no value to randomization. Specifically, the key assumption is that the listener's utility function is a concave transformation of the speaker's utility function. This assumption holds vacuously in the binary model. The result that concavity implies credibility allows us to dispense with the assumption that the listener's decision is binary and significantly broadens the scope of the model.