Indeterminacy without Externalities

I will set up a discrete-time multi-sector optimal growth model with fixed labor and show the indeterminacy of an optimal steady state. The key assumptions are constant returns to scale technologies and a linear utility setup. Under those assumptions, I will show that there exists a plane, referred to as the von Neumann-McKenzie facet, in which the optimal steady state is embedded. And any path on the facet turns out to be stable and is optimal. Since the prices are fixed on the facet, there exists no corresponding price dynamics of the optimal path, which is usually obtained as the dual of the original problem. This means that an optimal control solution and a descriptive (market) solution of the optimal problem could be diverged. Introducing the adaptive price-adjustment mechanism adopted by Burmeister and Grahm (1975), if the adjustment were slow, then the optimal steady state would turn out to be totally stable. So this implies local indeterminacy. On the other hand, if the adjustment speed were very fast and infinite, the optimal steady state would be saddle-point stable. So this implies local determinacy.