The stochastic turnpike property without uniformity in convex aggregate growth models

Abstract

The proof of the “late” turnpike property in optimal growth theory requires constructing a bounded value-loss process that records a strictly positive value-loss when paths of capital accumulation from different initial stocks diverge. Uniformity assumptions strengthen this sensitivity by ensuring that value-loss is independent of time and state of environment in which the divergence occurs, and are acknowledged as strong restrictions on the model. This paper argues that uncertainty can obviate the need for uniformity. The multiplicity of states afforded by a stochastic framework permits constructing a value-loss process over an “extended” time-line that is a martingale; if capital stocks diverge, then the martingale registers an upcrossing across a band of uniform width on its extended time-line, thereby giving uniform value-loss. The Martingale Upcrossing theorem and the first Borel–Cantelli lemma then clinch the turnpike property.