On Equilibrium Determinacy in Overlapping Generations Models with Money

I started to study models of bubble and money in late 2022 and learned the usefulness of the local stable manifold theorem. During my studies, I noticed that there is often hand-waving in applied works. For instance, Blanchard & Fishcer (1989, p. 268, Endnote 16) state

Care must be taken in using a phase diagram to analyze the dynamics of a difference equation system. […] Thus we must check in this case whether the system is indeed saddle point stable […]. This check is left to the reader.

Furthermore, during the review process of our JPE paper on the Bubble Necessity Theorem, we learned about Scheinkman (1980)’s sufficient condition for local determinacy that involves the curvature of the utility function, which was a bit mysterious. So after resubmitting our paper to JPE at the end of December 2023, we started to work on a complete analysis of the local determinacy of equilibria in Tirole (1985)’s model. It turns out that Scheinkman’s sufficient condition does not generalize to production economies: there are robust examples with arbitrary utility functions in which the nonmonetary steady state is locally determinate or indeterminate. In contrast, the monetary steady state is locally determinate under fairly weak conditions.