arXiv subjects
Publications and source records attributed to Ivar Ekeland.
In this paper we prove a local surjection theorem with continuous right-inverse for maps between Banach spaces, and we apply it to a class of inversion problems with loss of derivatives.
The standard Hotelling model assumes that the stock of an exhaustible resource is known. We expand on the model by Arrow and Chang that introduced stochastic discoveries and for the first time completely solve such a model using impulse control. The model has two state variables: the "proven" reserves as well as a finite unexplored area available for exploration with constant marginal cost, resulting in a Poisson process of new discoveries. We prove that a frontier of critical levels of "proven" reserves exists, above which exploration is stopped, and below which it happens at infinite speed. This frontier is increasing in the explored area, and higher "proven" reserve levels along this critical threshold are indicative of more scarcity, not less. In this stochastic generalization of Hotelling's rule, the expected shadow price of reserves rises at the rate of interest across exploratory episodes. However, the actual trajectories of prices realized prior to exhaustion of the exploratory area may jump up or down upon exploration. Conditional on non-exhaustion, expected price arises at a rate bounded above by the rate of interest, consistent with most empirical tests based on observed price histories.
In this paper we introduce a new algorithm for solving perturbed nonlinear functional equations which admit a right-invertible linearization, but with an inverse that loses derivatives and may blow up when the perturbation parameter $ε$ goes to zero. These equations are of the form $F_ε(u)=v$ with $F_ε(0)=0$, $v$ small and given, $u$ small and unknown. The main difference with the by now classical Nash-Moser algorithm is that, instead of using a regularized Newton scheme, we solve a sequence of Galerkin problems thanks to a topological argument. As a consequence, in our estimates there are no quadratic terms. For problems without perturbation parameter, our results require weaker regularity assumptions on $F$ and $v$ than earlier ones, such as those of Hormander. For singularly perturbed functionals, we allow $v$ to be larger than in previous works. To illustrate this, we apply our method to a nonlinear Schrodinger Cauchy problem with concentrated initial data studied by Texier-Zumbrun, and we show that our result improves significantly on theirs.
A general framework is given to analyze the falsifiability of economic models based on a sample of their observable components. It is shown that, when the restrictions implied by the economic theory are insufficient to identify the unknown quantities of the structure, the duality of optimal transportation with zero-one cost function delivers interpretable and operational formulations of the hypothesis of specification correctness from which tests can be constructed to falsify the model.
We propose a multivariate extension of a well-known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals. This involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions. Moreover, we propose to replace the current law invariance, subadditivity and comonotonicity axioms by an equivalent property we call strong coherence and that we argue has more natural economic interpretation. Finally, we reformulate the computation of regular and coherent risk measures as an optimal transportation problem, for which we provide an algorithm and implementation.
This paper exhibits a duality between the theory of Revealed Preference of Afriat and the housing allocation problem of Shapley and Scarf. In particular, it is shown that Afriat's theorem can be interpreted as a second welfare theorem in the housing problem. Using this duality, the revealed preference problem is connected to an optimal assignment problem, and a geometrical characterization of the rationalizability of experiment data is given. This allows in turn to give new indices of rationalizability of the data, and to define weaker notions of rationalizability, in the spirit of Afriat's efficiency index.
We study the optimal design of electricity contracts among a population of consumers with different needs. This question is tackled within the framework of Principal-Agent problems in presence of adverse selection. The particular features of electricity induce an unusual structure on the production cost, with no decreasing return to scale. We are nevertheless able to provide an explicit solution for the problem at hand. The optimal contracts are either linear or polynomial with respect to the consumption. Whenever the outside options offered by competitors are not uniform among the different type of consumers, we exhibit situations where the electricity provider should contract with consumers with either low or high appetite for electricity.
This paper investigates an infinite-horizon problems in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak. It is shown that optimal solutions a class of problems raising from time inconsistency problems in the framework of the neoclassical one-sector model of economic growth, and contains new results in environment economics. Without exogenous commitment mechanism, a notion of the equilibrium strategies instead of the optimal strategies is introduced. We characterized the equilibrium strategies by an integro-differential equation system. For two special criteria, the bi-exponential criteria and the Chichilnisky criteria, we established the existence of the equilibrium strategies.
We prove an inverse function theorem of Nash-Moser type for maps between Fréchet spaces satisfying tame estimates. In contrast to earlier proofs, we do not use the Newton method, that is, we do not use quadratic convergence to overcome the lack of derivatives. In fact, our theorem holds when the map to be inverted is not C^2
This paper considers the portfolio management problem of optimal investment, consumption and life insurance. We are concerned with time inconsistency of optimal strategies. Natural assumptions, like different discount rates for consumption and life insurance, or a time varying aggregation rate lead to time inconsistency. As a consequence, the optimal strategies are not implementable. We focus on hyperbolic discounting, which has received much attention lately, especially in the area of behavioural finance. Following [10], we consider the resulting problem as a leader-follower game between successive selves, each of whom can commit for an infinitesimally small amount of time. We then define policies as subgame perfect equilibrium strategies. Policies are characterized by an integral equation which is shown to have a solution. Although we work on CRRA preference paradigm, our results can be extended for more general preferences as long as the equations admit solutions. Numerical simulations reveal that for the Merton problem with hyperbolic discounting, the consumption increases up to a certain time, after which it decreases; this pattern does not occur in the case of exponential discounting, and is therefore known in the litterature as the "consumption puzzle". Other numerical experiments explore the effect of time varying aggregation rate on the insurance premium.
I present an inverse function theorem for differentiable maps between Frechet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^2, or even C^1, or even Frechet-differentiable.
This paper characterizes differentiable and subgame Markov perfect equilibria in a continuous time intertemporal decision problem with non-constant discounting. Capturing the idea of non commitment by letting the commitment period being infinitesimally small, we characterize the equilibrium strategies by a value function, which must satisfy a certain equation. The equilibrium equation is reminiscent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term leading to differences in qualitative behavior. As an application, we formulate an overlapping generations Ramsey model where the government maximizes a utilitarian welfare function defined as the discounted sum of successive generations' lifetime utilities. When the social discount rate is different from the private discount rate, the optimal command allocation is time inconsistent and we retain subgame perfection as a principle of intergenerational equity. Existence of multiple subgame perfect equilibria is established. The multiplicity is due to the successive governments' inability to coordinate their beliefs and we single out one of them as (locally) renegotiation-proof. Decentralization can be achieved with both age and time dependent lump sum transfers and, long term distorting capital interest income taxes/subsidy.
We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence-Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.
This paper considers the Merton portfolio management problem. We are concerned with non-exponential discounting of time and this leads to time inconsistencies of the decision maker. Following Ekeland and Pirvu 2006, we introduce the notion of equilibrium policies and we characterize them by an integral equation. The main idea is to come up with the value function in this context. If risk preferences are of CRRA type, the integral equation which characterizes the value function is shown to have a solution which leads to an equilibrium policy. This work is an extension of Ekeland and Pirvu 2006.
We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivator behind this numerical method is estimating solutions to Adverse Selection problems within a Principal-Agent framework. Problems such as product lines design, optimal taxation, structured derivatives design, etc. can be studied through the scope of these models. We develop a method to estimate their optimal pricing schedules.
In this paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting. This gives rise to time-inconsistency of the decision-maker. If the decision-maker at time t=0 can commit his/her successors, he/she can choose the policy that is optimal from his/her point of view, and constrain the others to abide by it, although they do not see it as optimal for them. If there is no commitment mechanism, one must seek a subgame-perfect equilibrium strategy between the successive decision-makers. In the line of the earlier work by Ekeland and Lazrak we give a precise definition of equilibrium strategies in the context of the portfolio management problem, with finite horizon, we characterize it by a system of partial differential equations, and we show existence in the case when the utility is CRRA and the terminal time T is small. We also investigate the infinite-horizon case and we give two different explicit solutions in the case when the utility is CRRA (in contrast with the case of exponential discount, where there is only one). Some of our results are proved under the assumption that the discount function h(t) is a linear combination of two exponentials, or is the product of an exponential by a linear function.
We aim to construct a general framework for portfolio management in continuous time, encompassing both stocks and bonds. In these lecture notes we give an overview of the state of the art of optimal bond portfolios and we re-visit main results and mathematical constructions introduced in our previous publications (Ann. Appl. Probab. \textbf{15}, 1260--1305 (2005) and Fin. Stoch. {\bf9}, 429--452 (2005)). A solution of the optimal bond portfolio problem is given for general utility functions and volatility operator processes, provided that the market price of risk process has certain Malliavin differentiability properties or is finite dimensional. The text is essentially self-contained.
This paper characterizes differentiable subgame perfect equilibria in a continuous time intertemporal decision optimization problem with non-constant discounting. The equilibrium equation takes two different forms, one of which is reminescent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term. We give a local existence result, and several examples in the consumption saving problem. The analysis is then applied to suggest that non constant discount rates generate an indeterminacy of the steady state in the Ramsey growth model. Despite its indeterminacy, the steady state level is robust to small deviations from constant discount rates.