Rose-Anne Dana

Dynamic complexity in duopoly games

Abstract We discuss the occurrence of periodic and chaotic phenomena in infinite horizon duopoly games where firms maximize their discounted sum of profits and use Markov-perfect equilibrium strategies. In the alternating case their corresponding actions are then constrained orbits of a map similar to a Cournot tâtonnement. We show that any behavior is possible for small discount factors and that one obtains the Cournot tâtonnement as the discount factor goes to zero. In the simultaneous case, we show by mean of an example that many Cournot tâtonnements can be viewed as Markov-perfect equilibria of an infinite horizon game and that one obtains the Nash equilibrium of the static game as the discount parameter goes to zero. We are thus led to study in detail the dynamical properties of a Cournot tâtonnement.

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.

OPTIMAL DEMAND FOR CONTINGENT CLAIMS WHEN AGENTS HAVE LAW INVARIANT UTILITIES

We consider a class of law invariant utilities, which contains the rank-dependent expected utility (RDU) and the cumulative prospect theory (CPT). We show that the computation of demand for a contingent claim when utilities are within that class, although not as simple as in the expected utility (EU) case, is still tractable. Specific attention is given to the RDU and to the CPT cases. Numerous examples are fully solved.

The Dynamics of a Discrete Version of a Growth Cycle Model

Macroeconometric dynamic models are generally estimated and simulated with a discrete time basis although they often rest on continuous time theoretical models.

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

Abstract 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 efficiency for the concave order and multivariate comonotonicity

This paper studies efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson (1994) [27], that efficiency is characterized by a comonotonicity condition. The goal of the paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multidimensional case. The multivariate case is more involved (in particular because there is no immediate extension of the notion of comonotonicity), and it is addressed by using techniques from convex duality and optimal transportation.

Structure of Pareto optima in an infinite-horizon economy where agents have recursive preferences

This article generalizes the one-agent growth theory with discounting to the case of several agents with recursive preferences. In a multi-consumption goods world, we show that, under some regularity conditions, any Pareto optimum can be viewed as a function of a trajectory of a dynamical system. The state space can be chosen to be the product of the space of capitals and the unit simplex. We define and study the properties of generalized value functions.

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 agentsʼ 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.

OVERLAPPING SETS OF PRIORS AND THE EXISTENCE OF EFFICIENT ALLOCATIONS AND EQUILIBRIA FOR RISK MEASURES

The overlapping expectations and the collective absence of arbitrage conditions introduced in the economic literature to insure existence of Pareto optima and equilibria with short-selling when investors have a single belief about future returns, is reconsidered. Investors use measures of risk. The overlapping sets of priors and the Pareto equilibrium conditions introduced by Heath and Ku for coherent risk measures are respectively reinterpreted as a weak no-arbitrage and a weak collective absence of arbitrage conditions and shown to imply existence of Pareto optima and Arrow-Debreu equilibria.

Optimal growth and pareto-optimality

Abstract The purpose of this paper is to show that in a stationary intertemporal economy where agents have recursive utilities every Pareto optimum is a solution of a generalized McKenzie problem. An ‘abstract’ state space is introduced as the space of couples of capital stock and utilities that can be reached by n −1 agents from that capital stock. ‘Generalized technological’ conditions are then defined on that abstract space as well as a recursive criterion on sequences of its elements. The criterion generalizes the additively separable one. As Bellmans and Eulers equations still hold, many dynamical results known for the additively separable one-agent case can be generalized.