Hopf-Lax formula for variational problems with non-constant discount

We provide a Hop-Lax formula for variational problems with non-constant discount and deduce a dynamic programming equation. We also study some regularity properties of the value function.Comment: 12 pages; corrected typo

Caracterización de equilibrios de Nash perfectos no suaves en una clase de juegos diferenciales

Monográfico: Problemas complejos de decisiónEn este trabajo presentamos una generalización del nuevo método desarrollado en [1] para la caracterización de equilibrios de Nash perfectos en el marco de los juegos diferenciales. Los resultados aquí presentados caracterizan equilibrios de Nash perfectos formados por estrategias feedback continuas, de clase el a trozos, como soluciones, en sentido generalizado, de un sistema de ecuaciones en derivadas parciales cuasi lineales. Estudiamos también los equilibrios simétricos en un juego de explotación, en un horizonte limitado, de un recurso no renovable de propiedad común y determinamos la forma analítica de la solución, así como ciertas de sus propiedades cualitativas. Comparamos el equilibrio de Nash con la solución cooperativa, demostrando que la gestión descoordinada da lugar a una sobreexplotación del recurso.-------------------------------------------------------------This paper is devoted to provide an extension to the results developed in [1]. We characterize continuous and piecewise smooth subgame perfect Nash equilibria in differential games as solutions, in a generalized sense, of a quasilinear system of partial differential equiations. We also study a symmetric nonrenewable resource game of common property on" a bounded horizon, being able to determine its analytic solution as well as sorne qualitative properties. We compare the non cooperative solution with the cooperative one, arriving to the well known tragedy of the commons, that is to say, the resource is overexploited with the noncooperative management.Publicad

Characterization of Markovian equilibria in a class of differential games

We consider an alternative method to the classical one for the determination of Markov perfect Nash equilibria. The approach shown in the paper is based on the study of a quasi-linear system of partial differential equations instead of the Hamilton–Jacobi–Bellman system. The simpler structure of the former allows us to determine existence and uniqueness of Nash equilibria in non-renewable resource games under some assumptions. When closed-form solutions are not available, we give a method to obtain numerical solutions.Financial support from Investigation Projects PB98-0393 of Direcci-on General de Ense˜nanza Superior e Investigaci-on Cient-XKca and VA 108/01 of Junta de Castilla y Le-on are gratefully acknowledged. The author is grateful to two anonymous referees for their comments and corrections. I am also indebted to the Associated Editor Tamer BaHsar for his remarks and for the careful reading of the manuscript. The author is responsible for any remaining error.Publicad

Valor de Shapley de un juego capitalista

La ineficiencia del capitalismo como sistema económico desde el punto de vista del óptimo social, puesta de manifiesto por primera vez por Lancaster en 1973 y derivada de la actuación descoordinada de los agentes que intervienen en el conflicto, ha llevado a numerosos autores a proponer soluciones basadas en la cooperación. En esta línea, en el artículo se demuestra la existencia del valor de Shapley en el juego de Lancaster bajo ciertas hipótesis. Además, se pone de manifiesto que la posición negociadora de ambos jugadores depende del valor del límite superior de la tasa de consumo de los trabajadores.Publicad

Differentiability of the value function and Euler equation in non-concave discrete time stochastic dynamic programming

We consider a stochastic, non-concave dynamic programming problem admitting interior solutions and prove, under mild conditions, that the expected value function is differentiable along optimal paths. This property allows us to obtain rigorously the Euler equation as a necessary condition of optimality for this class of problems.Support from the Ministerio de Economía, Industria y Competitividad (Spain), Grants ECO2017-86261-P, ECO2014-56384-P and MDM 2014-0431, and from the Comunidad de Madrid, Grant MadEco-CM S2015/HUM-3444 is gratefully acknowledged

Markov Perfect Nash Equilibrium in stochastic differential games as solution of a generalized Euler Equations System

This paper gives a new method to characterize Markov Perfect Nash Equilibrium in stochastic differential games by means of a set of Generalized Euler Equations. Necessary and sufficient conditions are given

Certainty equivalence principle in stochastic differential games: An inverse problem approach

Producción CientíficaThis paper aims to characterize a class of stochastic differential games, which satisfy the certainty equivalence principle beyond the cases with quadratic, linear, or logarithmic value functions. We focus on scalar games with linear dynamics in the players' strategies and with separable payoff functionals. Our results are based on the resolution of an inverse problem that determines strictly concave utility functions of the players so that the game satisfies the certainty equivalence principle. Besides establishing necessary and sufficient conditions, the results obtained in this paper are also a tool for discovering new closed-form solutions, as we show in two specific applications: in a generalization of a dynamic advertising model and in a game of noncooperative exploitation of a productive asset.Este trabajo se ha hecho con ayuda de los proyectos del Ministerio de Economía, Industria y Competitividad, Grant/Award Number: ECO2017-86261-P, ECO2014-56384-P, y MDM 2014-0431, de la Consejería de Educación, Juventud y Deporte de la Comunidad de Madrid, Grant/Award Number: MadEco-CM S2015/HUM-3444, y de la Consejería de Educación de la Junta de Castilla y León VA148G18

Equilibrium strategies in a defined benefit pension plan game

Producción CientíficaWe study the optimal management of an aggregated overfunded pension plan of defined benefit type as a two-player noncooperative differential game. The model’s key fact is to consider the fund surplus as a strategic variable that makes the pension plan more attractive both for current and future participants. We let the worker participants to act collectively as a single player that claims a share of the surplus, and let the sponsoring firm act as the player that cares about the investment of the surplus fund assets. The union’s objective is to maximize the expected discounted utility of the extra benefits claimed. We solve this asymmetric game under two different assumptions on the preferences of the firm: in the first scenario, the firm aims to maximize expected discounted utility derived from fund surplus; while in the second scenario, the firm cares about minimizing the probability that the fund surplus reaches very low values.Este trabajo se ha hecho con ayuda de los proyectos del Ministerio de Economía Industria y Competitividad (Spain), ECO2017-86261-P , ECO2014-56384-P y MDM2014-0431, y de la Comunidad de Madrid MadEco-CM S2015/HUM-3444 y Comunidad de Castilla y León VA148G18

New approach to stochastic optimal control and applications to economics

This paper provides new insights into the solution of optimal stochastic control problems by means of a system of partial differential equations, which characterize directly the optimal control. This new system is obtained by the application of the stochastic maximum principle at every initial condition, assuming that the optimal controls are smooth enough. The type of problems considered are those where the diffusion coefficient is independent of the control variables, which are supposed to be interior to the control region. The results obtained are applied to the study of the classical consumption–savings model