# Comparing recursive equilibrium in economies with dynamic complementarities and indeterminacy

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## Abstract

We develop a new multistep monotone map approach to characterize minimal state-space recursive equilibrium for a broad class of infinite horizon dynamic general equilibrium models with positive externalities, dynamic complementarities, public policy, equilibrium indeterminacy, and sunspots. This new approach is *global*, defined in the equilibrium version of the household’s Euler equation, applies to economies for which there are no known existence results, and existing methods are inapplicable. Our methods are able to distinguish different structural properties of recursive equilibria. In stark contrast to the extensive body of existing work on these models, our methods make *no *appeal to the theory of smooth dynamical systems that are commonly applied in the literature. Actually, sufficient smoothness to apply such methods cannot be established relative to the set of recursive equilibria. Our partial ordering methods also provide a qualitative theory of equilibrium comparative statics in the presence of multiple equilibrium. These robust equilibrium comparison results are shown to be computable via successive approximations from subsolutions and supersolutions in sets of candidate equilibrium function spaces. We provide applications to an extensive literature on local indeterminacy of dynamic equilibrium.

## Keywords

Recursive equilibrium Supermodularity Monotone map methods Externality Indeterminacy## JEL Classification

D62 D91 E13## Notes

### Acknowledgements

We are grateful to Francesco Agostinelli, Rabah Amir, Łukasz Balbus, Robert Becker, Stefano Bosi, Dilsat Dalkiran, Jean-Pierre Drugeon, Ed Green, Martin Kaae Jensen, Takashi Kamihigashi, Robert Lucas, Olivier Morand, Ed Prescott, Manuel Santos, Yiannis Vailakis, Alain Venditti, as well as the seminar participants at Johns Hopkins, Paris I-Sorbonne, Warsaw School of Economics, 2013 SAET and DIET conferences in Paris, 2014 ANR Nuvo Tempo Workshop on Recursive Methods in Glasgow, 2015 ANR Nuvo Tempo Tempe Arizona, University of Miami Economic Theory conference, 2016 European General Equilibrium Conference in Glasgow, and two anonymous referees for their helpful comments and suggestions. We especially would like to thank M. Ali Khan for extended conversations on this project that reoriented its focus. All remaining shortcomings are our own.

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