# The Phelps-Koopmans Theorem and Potential Optimality

DRI Working Paper No. 46

By Debraj Ray, NYU

**The Phelps-Koopmans Theorem and Potential Optimality **

Can discounted optimal paths spend an “infinite amount of time above” the golden rule?

This paper seeks to answer that question. I show in Proposition 1 that if an optimal path converges, its limit must lie weakly below the minimal golden rule, the lowest capital stock that globally maximizes net consumption. This result is independent of any curvature assumptions, either on the production function or on the utility function. Thus far, then, the intuition of the convex model carries over: convergent programs that are potentially optimal with respect to some utility function cannot stay above and bounded away from the golden rule . . .