From Frank Ramsey to René Thom: a classical problem in the calculus of variations leading to an implicit differential equation
type: Item
instance of: scholarly article
title: From Frank Ramsey to René Thom: a classical problem in the calculus of variations leading to an implicit differential equation
zbMATH Open document ID: 1202.91087
author: Ivar Ekeland
published in: Discrete and Continuous Dynamical Systems
publication date: 2010-07-23T00:00:00Z
review text: This paper deals with a classical problem model in mathematical economics, Ramsey's optimal investment problem, in which one needs to choose a time-dependent consumption/investment schedule in order to maximize the time-discounted utility. It is noted that despite the fact that this problem is discussed in many textbooks, it has never received a mathematically rigorous treatment, since the theorems providing a ``transversality condition at infinity'' which is used are not valid in the context of this problem. This paper offers a complete treatment of the problem, which uses the Hamilton-Jacobi equation, which in this case is an ordinary differential equation in implicit form, that is, it is not solved for the derivative. A careful geometric analysis is used to characterize the optimal solution of the Ramsey problem. The Ramsey problem is then extended to the case of a non-constant discount rate, in which case the optimal growth is no longer relevant because of time inconsistency, and a concept of equilibrium strategy is introduced. The question of the existence of an equilibrium strategy for a general discount time-dependent rate remains open, but the special case of a discount rate which is a linear combination of two exponentials is studied in detail, the equilibrium strategy is characterized as a solution of a system of implicit differential equations, and the existence of an equilibrium strategy is proved.
reviewed by: Guy Katriel
Mathematics Subject Classification ID: 91B16
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 91B62
zbMATH DE Number: 5760052
zbMATH Keywords: infinite horizon
zbMATH Keywords: calculus of variations
zbMATH Keywords: Ramsey problem
zbMATH Keywords: time-inconsistency
zbMATH Keywords: Hamilton-Jacobi equation
MaRDI profile type: Publication
full work available at URL: dcds.2010.28.1101
OpenAlex ID: W2011375337
DOI: 10.3934/DCDS.2010.28.1101
description: scientific article; zbMATH DE number 5760052