The existence of the modiﬁed golden rule and the turnpike property are proved for a multi-sector stochastic growth model. The (exogenous) stochastic environment is represented by a stationary stochastic process that influences preferences, technology and resources. A social planner maximizes the expected sum of discounted utilities. The conditions required in order to obtain these results, are the natural strengthening of the stability conditions of the deterministic case. As in the deterministic case, the discount factor must be close to one in order to guarantee the almost sure (and in the mean) convergence of optimal interior programs. It is proved that all optimal interior programs converge to each other. This fact is used to prove the existence of a unique optimal stationary program (the modiﬁed golden rule). These results imply that all optimal interior programs converge to the stationary program (the turnpike property).
Marimon, Ramon, "Stochastic Equilibrium and Turnpike Property: The Discounted Case" (1983). Cowles Foundation Discussion Papers. 919.