Jorge Navas

Consumption and portfolio rules for time-inconsistent investors

This paper extends the classical consumption and portfolio rules model in continuous time [Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: The continuous time case. Review of Economics and Statistics 51, 247-257, Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373-413] to the framework of decision-makers with time-inconsistent preferences. The model is solved for different utility functions for both, naive and sophisticated agents, and the results are compared. In order to solve the problem for sophisticated agents, we derive a modified HJB (Hamilton-Jacobi-Bellman) equation. It is illustrated how for CRRA functions within the family of HARA functions (logarithmic and power utilities) the optimal portfolio rule does not depend on the discount rate, but this is not the case for a general utility function, such as the exponential (CARA) utility function.

Time to Complete and Research Joint Ventures: A Differential Game Approach

In this paper we analyze cooperation in R&D in the form of RJVs.We show that the optimal size of an RJV does not only depend on the degree of spillovers, as literature suggests, but also on the cost function of R&D activities. Moreover, the explicit consideration of the fact that R&D projects take time to complete shows that benefits from cooperation in R&D not only allow RJVs to carry out larger R&D projects, but also to reduce the time to completion for projects with a given size and, consequently, to accelerate the acquisition of the benefits associated with the innovation.

Time-Consistent Equilibria in Common Access Resource Games with Asymmetric Players Under Partial Cooperation

Given a differential game, if agents have different time preference rates, cooperative (Pareto optimum) solutions obtained by applying Pontryagin’s maximum principle become time inconsistent. We derive a set of dynamic programming equations in continuous time whose solutions are time-consistent equilibria for problems in which agents differ in their utility functions and also in their time preference rates. The solution assumes cooperation between agents at every time. Since coalitions at different times have different time preferences, equilibrium policies are calculated by looking for Markov (subgame perfect) equilibria in a (noncooperative) sequential game. The results are applied to the study of a cake-eating problem describing the management of a common property exhaustible natural resource. The extension of the results to a simple common property renewable natural resource model in infinite horizon is also discussed.

Consumption, investment and life insurance strategies with heterogeneous discounting

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent’s preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the time-consistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically.

A consumption–investment problem with heterogeneous discounting

We analyze a stochastic continuous time model in finite horizon in which the agent discounts the instantaneous utility function and the final function at constant but different discount rates of time preference. Within this framework we can model problems in which, when the time t approaches to the final time, the valuation of the final function increases compared with previous valuations. We study a consumption and portfolio rules problem for CRRA and CARA utility functions for time-consistent agents, and we compare the different equilibria with the time-inconsistent solutions. The introduction of random terminal time is also discussed. Differences with both the mathematical treatment and agent’s behavior in the case of hyperbolic discounting are stressed.

Heterogeneous discounting in consumption-investment problems. Time consistent solutions

In this paper we analyze a stochastic continuous time model in finite horizon in which agents discount the instantaneous utility function and the final function at constant but different instantaneous discount rates of time preference. Within this context we can model problems in which, when the time t approaches to the final time, the valuation of the final function increases compared with previous valuations in a way that cannot be explained by using a unique constant or a variable discount rate. We derive a dynamic programming equation whose solutions are time-consistent Markov equilibria. For this class of time preferences, we study the classical consumption and portfolio rules model (Merton, 1971) for CRRA and CARA utility functions for time- consistent agents, and we compare the different equilibria with the time-inconsistent solutions. The introduction of stochastic terminal time is also discussed.

Els grups GEI com un experiència docent orientada a estudiants repetidors: el cas de Matemàtiques I en Economia

En aquest treball estudiem les millores introduides en els Grups d’Estudi d’Intensificacio (grups GEI) durant el curs academic 2012-2013. La introduccio d’aquests grups ha estat una aposta de la Facultat d’Economia i Empresa de la Universitat de Barcelona amb l’objectiu de facilitar a l’estudiant el seguiment de les assignatures que repeteix, tenint en compte el perfil d’aquest alumnat i oferir-li una via atractiva que li faciliti el seguiment i superacio de la materia i la seva compaginacio amb les altres assignatures de primera matricula. Despres d’un primer any d’implantacio en el curs academic 2011-2012, es va redefinir el disseny d’aquests grups amb l’objectiu de corregir les ineficiencies observades. Estudiem aixi, per l’assignatura de Matematiques I del grau d’Economia, com aquests canvis han incidit en el seguiment i rendiment comparant els resultats amb els obtinguts durant el passat curs academic.

Cost of capital for incentives on capacity expansion investments

In this paper, the expression for the cost of capital is derived when capacity expansion investments and replacement investments exhibit differences in their effective prices. It is shown that the cost of capital derived by perturbing the optimal stock path should be constructed under an opportunity cost criterion.