Non-existence of optimal programs for undiscounted growth models in continuous time

•We prove that the minimum dimension for non-existence of optima in continuous time is 2.•This confirms a conjecture advanced in 1976 by Brock and Haurie.•We work in the framework of a two-dimensional optimal growth model à la Bruno (1967).

We report an example of a two-dimensional undiscounted convex optimal growth model in continuous time in which, although there is a unique “golden rule”, no overtaking optimal solutions exists in a full neighborhood of the steady state. The example proves, for optimal growth models, a conjecture advanced in 1976 by Brock and Haurie that the minimum dimension for non-existence of overtaking optimal programs in continuous time is 2.