A new class of problems in the calculus of variations

Abstract

IntroductionIn economic theory, and in optimal control, it has been customary to discount future gains at a constant rate δ > 0. If an individual with utility function u (c) has the choice between several streams of consumption c (t), 0 ≤ t, he or she will choose the one which maximises the present value, given by:That future gains should be discounted is well grounded in fact. On the one hand, humans prefer to enjoy goods sooner than later (and to suffer bads later than sooner), as every child-rearing parent knows. On the other hand, it is also a reflection of our own mortality: 10 years from now, I may simply no longer be around to enjoy whatever I have been promised. These are two good reasons why people are willing to pay a little bit extra to hasten the delivery date, or will require compensation for postponement, which is the essence of discounting.On the other hand, there is no reason why the discount rate should be constant, i.e. why the discount factor should be an exponential e −δt . The practice probably arises from the compound interest formula lim ε→0 (1 − εδ) t/ε = e −δt , when a constant interest rate δ is assumed, but even in finance, interest rates vary with the horizon: long-term rates can be widely different from shortterm ones. As for economics, there is by now a huge amount of evidence that individuals use higher discount rates for the near future than for the long-term (see [18] for a review up to 2002). There

A time consistent dynamic bargaining procedure in differential games with hterogeneous discounting

“…Finally, for the derivation of the time-consistent dynamic bargaining solution, we substitute the expression of the threat point W 0 and threat strategy φ 0,nc (see Proposition 3) into (9).…”

Resource conservation across generations in a Ramsey–Chichilnisky model

“…a The paper builds on Ekeland et al (2013) and has been circulated under the title "Generations playing a Chichilnisky game". We are grateful for helpful discussions with Graciela Chichilnisky and Bård Harstad and useful comments received at SURED…”

“…However, this part of the Chichilnisky criterion may lead to more stock conservation than the time-discounted utilitarian optimum with a low-productive initial stock. Based on the notion of von Neumann-Morgenstern abstract stability, we obtain uniqueness by assuming that each generation coordinates on an almost best equilibrium and takes into account that future generations will do as well.Keywords and Phrases: Intertemporal decision making; Time inconsistency; Intergenerational equity JEL Classification Numbers: C70, D63, D91, O41, Q01a The paper builds on Ekeland et al (2013) and has been circulated under the title "Generations playing a Chichilnisky game". We are grateful for helpful discussions with Graciela Chichilnisky and Bård Harstad and useful comments received at SURED…”

Time-Inconsistent Stochastic Linear-quadratic Differential Game

“…An "equilibrium" control is therefore one such that any deviation from it at any time instant will be worse off. The study on time inconsistency by economists can be dated back to Stroz [23] and Phelps ([21, 22]) in models with discrete time (see [17] and [18] for further developments), and adapted by Karp ([15,16]), and by Ekeland and Lazrak ([5,6,7,8,9,10]) to the case of continuous time. In the LQ control problems, Yong [24] studied a time-inconsistent deterministic model and derived equilibrium controls via some integral equations.…”