A New Class of Problems in the Calculus of Variations

TL;DR

This paper explores equilibrium strategies in infinite-horizon calculus of variations problems with a focus on economic growth and environmental concerns, introducing new solution concepts for time-inconsistent problems.

Contribution

It introduces equilibrium strategies for time-inconsistent variational problems, extending previous work and applying to economic growth and environmental economics.

Findings

Existence of equilibrium strategies for bi-exponential and Chichilnisky criteria.

Characterization of equilibrium strategies via coupled differential equations.

Extension of earlier theoretical frameworks to include new solution concepts.

Abstract

This paper investigates an infinite-horizon problems in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak. It is shown that optimal solutions a class of problems raising from time inconsistency problems in the framework of the neoclassical one-sector model of economic growth, and contains new results in environment economics. Without exogenous commitment mechanism, a notion of the equilibrium strategies instead of the optimal strategies is introduced. We characterized the equilibrium strategies by an integro-differential equation system. For two special criteria, the bi-exponential criteria and the Chichilnisky criteria, we established the existence of the equilibrium strategies.