  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.
