From Frank Ramsey to René Thom: a classical problem in the calculus of variations leading to an implicit differential equation
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.
scientific article; zbMATH DE number 5760052