Ambiguity,uncertainty aversion and equilibrium welfare

Summary. In order to analyse the effect of ambiguity and uncertainty aversion on equilibrium welfare, a two period, pure exchange one good economy is considered. Agents are Choquet-expected-utility maximizers with same convex capacity and strictly concave utility index. It is proven that equilibrium is indeterminate whenever several probabilities in the core of the capacity minimize the expected value of aggregate endowment and not all agents have same expected endowments under those probabilities. It is further shown that small changes in aggregate endowment may have drastic welfare implications. A more general model is considered in the case of no aggregate uncertainty: agents have a set of priors and are uncertainty averse as modelled by Gilboa-Schmeidler [1989]. In the case of complete markets, it is shown that assets have a spread of equilibrium prices similar to the spread of no-arbitrage prices compatible with absence of arbitrage in markets with imperfections.Received: 2 June 2000, Revised: 27 March 2003, JEL Classification Numbers: D46, D59,D60, G12.I have benefited from conversations with L. Epstein, F. Magnien and J. M. Tallon.     

Intertemporal equilibria with Knightian uncertainty

We study a dynamic and infinite-dimensional model with incomplete multiple prior preferences. In interior efficient allocations, agents share a common risk-adjusted prior and subjective interest rate. Interior efficient allocations and equilibria coincide with those of economies with subjective expected utility and priors from the agents? multiple prior sets. A specific model with neither risk nor uncertainty at the aggregate level is considered. Risk is always fully insured. For small levels of ambiguity, there exists an equilibrium with inertia where agents also insure fully against Knightian uncertainty. When the level of ambiguity exceeds a critical threshold, full insurance no longer prevails and there exist equilibria with inertia where agents do not insure against uncertainty at all. We also show that equilibria with inertia are indeterminate.     

A REPRESENTATION RESULT FOR CONCAVE SCHUR CONCAVE FUNCTIONS

A representation result is provided for concave Schur concave functions on   L ^∞(Ω)  . In particular, it is proven that any monotone concave Schur concave weakly upper semicontinuous function is the infinimum of a family of nonnegative affine combinations of Choquet integrals with respect to a convex continuous distortion of the underlying probability. The method of proof is based on the concave Fenchel transform and on Hardy and Littlewood's inequality. Under the assumption that the probability space is nonatomic, concave, weakly upper semicontinuous, law-invariant functions are shown to coincide with weakly upper semicontinuous concave Schur concave functions. A representation result is, thus, obtained for weakly upper semicontinuous concave law-invariant functions.     

Existence, uniqueness and determinacy of Arrow–Debreu equilibria in finance models

We characterize the Arrow–Debreu equilibria of a pure exchange one good economy where agents have additively separable utilities. It is then shown that under gross substitution hypotheses on utilities (or if relative risk aversion is smaller than one), the excess utility has gross substitute properties. Uniqueness of equilibria thus follows. It is finally proved that generically equilibria are determinate.     

Pareto efficient insurance contracts when the insurer's cost function is discontinuous

Summary. We consider the problem of efficient insurance contracts when the cost structure includes a fixed cost per claim. We prove existence of efficient insurance contracts and that the indemnity function in such contracts is non-decreasing in the damage. We further show that either there is no insurance, or the indemnity is positive for all losses, or efficient insurance contracts have a unique jump. We study variants of the model and provide a generalization to the case of non expected utilities. Our results are then applied to Townsend's model of deterministic auditing. Received: November 8, 2000; revised version: March 12, 2002 RID="*" ID="*" We are grateful to F. Salanié for pointing out an error in the previous version of the paper and for suggesting Proposition 6 to us. Correspondence to: R.-A. Dana     

Optimal risk sharing with background risk

This paper examines qualitative properties of efficient insurance contracts in the presence of background risk. In order to get results for all strictly risk-averse expected utility maximizers, the concept of “stochastic increasingness” is used. Different assumptions on the stochastic dependence between the insurable and uninsurable risk lead to different qualitative properties of the efficient contracts. The new results obtained under hypotheses of dependent risks are compared to classical results in the absence of background risk or to the case of independent risks. The theory is further generalized to nonexpected utility maximizers.     

Uniqueness of Arrow-Debreu and Arrow-Radner equilibrium when utilities are additively separable

This survey paper has three purposes: We first present in finite dimension, different approaches to the problem of uniqueness of Arrow-Debreu equilibrium when agents have additively separable utilities. We then study how, in the specific framework of a two period contingent good economy the results obtained generalize to infinite dimension. We consider economies where agents' consumption space is and agents' utilities are additively separable. Lastly, we show that in some restricted settings, some results may be used to prove uniqueness of Arrow-Radner equilibria when there are incomplete financial markets.