# 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.

## References

- Acemoglu, D., Jensen, M.K.: Robust comparative statics in large dynamic economies. J. Polit. Econ.
**123**(3), 587–640 (2015)CrossRefGoogle Scholar - Amann, H.: Fixed point equations and nonlinear eigenvalue problems in order banach spaces. Siam Rev.
**18**(4), 620–709 (1976)CrossRefGoogle Scholar - Antoci, A., Galeotti, M., Russu, P.: Poverty trap and global indeterminacy in a growth model with open-access natural resources. J. Econ. Theory
**146**(2), 569–591 (2011)CrossRefGoogle Scholar - Balbus, Ł., Reffett, K., Woźny, Ł.: Time consistent Markov policies in dynamic economies with quasi-hyperbolic consumers. Int. J. Game Theory
**44**(1), 83–112 (2015)CrossRefGoogle Scholar - Beaudry, P., Portier, F.: When can changes in expectations cause business cycle fluctuations in neo-classical settings? J. Econ. Theory
**135**(1), 458–477 (2007)CrossRefGoogle Scholar - Benhabib, J., Farmer, R.E.A.: Indeterminacy and increasing returns. J. Econ. Theory
**63**(1), 19–41 (1994)CrossRefGoogle Scholar - Benhabib, J., Perli, R.: Uniqueness and indeterminacy: on the dynamics of endogenous growth. J. Econ. Theory
**63**(1), 113–142 (1994)CrossRefGoogle Scholar - Benhabib, J., Dong, F., Wang, P.: Adverse Selection and Self-fulfilling Business Cycles, Manuscript, New York University (2013)Google Scholar
- Benhabib, J.F.D., Wang, P.: Adverse Selection and Self-fulfilling Business Cycles, Manuscript, New York University (2014)Google Scholar
- Boldrin, M., Rustichini, A.: Growth and indeterminacy in dynamic models with externalities. Econometrica
**62**(2), 323–342 (1994)CrossRefGoogle Scholar - Coleman, I.W.J.: Equilibrium in a production economy with an income tax. Econometrica
**59**(4), 1091–1104 (1991)CrossRefGoogle Scholar - Coleman, I.W.J.: Uniqueness of an equilibrium in infinite-horizon economies subject to taxes and externalities. J. Econ. Theory
**95**, 71–78 (2000)CrossRefGoogle Scholar - Coleman, I.W.J.: Equilibria in distorted infinite-horizon economies with capital and labor. J. Econ. Theory
**72**(2), 446–461 (1997)CrossRefGoogle Scholar - Crettez, B., Morhaim, L.: Existence of competitive equilibrium in a non-optimal one-sector economy without conditions on the distorted marginal product of capital. Math. Soc. Sci.
**63**(3), 197–206 (2012)CrossRefGoogle Scholar - d’Albis, H., Augeraud-Veron, E., Venditti, A.: Business cycle fluctuations and learning-by-doing externalities in a one-sector model. J. Math. Econ.
**48**(5), 295–308 (2012)CrossRefGoogle Scholar - Datta, M.: Existence and uniqueness of equilibrium in a dynamic distorted small open economy. Econ. Lett.
**152**, 19–23 (2017)CrossRefGoogle Scholar - Datta, M., Mirman, L.J., Reffett, K.L.: Existence and uniqueness of equilibrium in distorted dynamic economies with capital and labor. J. Econ. Theory
**103**(2), 377–410 (2002)CrossRefGoogle Scholar - Datta, M., Mirman, L.J., Morand, O.F., Reffett, K.L.: Markovian equilibrium in infinite horizon economies with incomplete markets and public policy. J. Math. Econ.
**41**(4–5), 505–544 (2005)CrossRefGoogle Scholar - Debreu, G.: Economies with a finite set of equilibria. Econometrica
**38**, 387–392 (1970)CrossRefGoogle Scholar - Dugundji, J., Granas, A.: Fixed Point Theory. PWN Polish Scientific Publishers, Warsaw (1982)Google Scholar
- Farmer, R.E.A., Guo, J.T.: Real business cycles and the animal spirits hypothesis. J. Econ. Theory
**63**(1), 42–72 (1994)CrossRefGoogle Scholar - Feng, Z., Hoelle, M.: Indeterminacy in stochastic overlapping generations models: real effects in the long run. Econ. Theor.
**63**(2), 559–585 (2017)CrossRefGoogle Scholar - Feng, Z., Miao, Jianjun, Peralva-Alta, A., Santos, M.: Numerical simulation of nonoptimal dynamic equilibrium models. Int. Econ. Rev.
**55**, 83–110 (2014)CrossRefGoogle Scholar - Greenwood, J., Huffman, G.W.: On the existence of nonoptimal equilibria in dynamic stochastic economies. J. Econ. Theory
**65**(2), 611–623 (1995)CrossRefGoogle Scholar - Guo, J.T., Harrison, S.G.: Balanced-budget rules and macroeconomic (in)stability. J. Econ. Theory
**119**(2), 357–363 (2004)CrossRefGoogle Scholar - Guo, J.T., Harrison, S.G.: Indeterminacy with no-income-effect preferences and sector-specific externalities. J. Econ. Theory
**145**(1), 287–300 (2010)CrossRefGoogle Scholar - Guo, J.T., Lansing, K.J.: Indeterminacy and stabilization policy. J. Econ. Theory
**82**(2), 481–490 (1998)CrossRefGoogle Scholar - Heikkila, S., Reffett, K.: Fixed point theorems and their applications to Nash equilibria. Nonlinear Anal.
**64**, 1415–1436 (2006)CrossRefGoogle Scholar - Hirsch, M., Smale, R., Devaney, S.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, London (2013)Google Scholar
- Huang, K.X., Meng, Q.: Increasing returns and unsynchronized wage adjustment in sunspot models of the business cycle. J. Econ. Theory
**147**(1), 284–309 (2012)CrossRefGoogle Scholar - Jaimovich, N.: Firm dynamics and markup variations: Implications for sunspot equilibria and endogenous economic fluctuations. J. Econ. Theory
**137**(1), 300–325 (2007)CrossRefGoogle Scholar - Jaimovich, N.: Income effects and indeterminacy in a calibrated one-sector growth model. J. Econ. Theory
**143**(1), 610–623 (2008)CrossRefGoogle Scholar - Kehoe, T.J., Levine, D.K., Romer, P.M.: Determinacy of equilibria in dynamic models with finitely many consumers. J. Econ. Theory
**50**(1), 1–21 (1990)CrossRefGoogle Scholar - Kubler, F., Schmedders, K.: Stationary equilibria in asset-pricing models with incomplete markets and collateral. Econometrica
**71**(6), 1767–1793 (2003)CrossRefGoogle Scholar - Liu, Z., Wang, P.: Credit constraints and self-fulfilling business cycles. Am. Econ. J. Macroecon.
**6**(1), 32–69 (2014)CrossRefGoogle Scholar - Lloyd-Braga, T., Modesto, L., Seegmuller, T.: Market distortions and local indeterminacy. A general approach. J. Econ. Theory
**151**, 216–247 (2014)CrossRefGoogle Scholar - Lucas, J., Robert, E., Prescott, E.C.: Investment under uncertainty. Econometrica
**39**(5), 659–681 (1971)CrossRefGoogle Scholar - Markowsky, G.: Chain-Complete Posets and Directed Sets with Applications. Algebr. Univ.
**6**(1), 53–68 (1976)CrossRefGoogle Scholar - MasColell, A.: The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs in Pure Theory 9. Cambridge Press, Cambridge (1986)Google Scholar
- Menuet, M., Minea, A., Villieu, P.: Deficit, monetization, and economic growth: a case for multiplicity and indeterminacy. Econ. Theory (2017). doi: 10.1007/s00199-017-1040-5 CrossRefGoogle Scholar
- Mirman, L.J., Morand, O.F., Reffett, K.L.: A qualitative approach to markovian equilibrium in infinite horizon economies with capital. J. Econ. Theory
**139**(1), 75–98 (2008)CrossRefGoogle Scholar - Morand, O.F., Reffett, K.L.: Existence and uniqueness of equilibrium in nonoptimal unbounded infinite horizon economies. J. Monet. Econ.
**50**(6), 1351–1373 (2003)CrossRefGoogle Scholar - Nishimura, K., Seegmuller, T., Venditti, A.: Fiscal policy, debt constraint and expectations-driven volatility. J. Math. Econ.
**61**(C), 305–316 (2015)CrossRefGoogle Scholar - Nourry, C., Seegmuller, T., Venditti, A.: Aggregate instability under balanced-budget consumption taxes: a re-examination. J. Econ. Theory
**148**(5), 1977–2006 (2013)CrossRefGoogle Scholar - Peralta-Alva, A., Santos, M.: Problems with the numerical simulation of heterogeneous agent models and economic distortions. J. Eur. Econ. Assoc.
**8**, 617–625 (2010)CrossRefGoogle Scholar - Phillips, P.C.B.: Folklore theorems, implicit maps, and indirect inference. Econometrica
**80**(1), 425–454 (2012)CrossRefGoogle Scholar - Pintus, P.: Indeterminacy with almost constant returns to scale: capital-labor substitution matters. Econ. Theor.
**28**(3), 633–649 (2006)CrossRefGoogle Scholar - Prescott, E.C., Mehra, R.: Recursive competitive equilibrium: the case of homogeneous households. Econometrica
**48**(6), 1365–1379 (1980)CrossRefGoogle Scholar - Raines, B.E., Stockman, D.R.: Chaotic sets and Euler equation branching. J. Math. Econ.
**46**(6), 1173–1193 (2010)CrossRefGoogle Scholar - Romer, P.M.: Increasing returns and long-run growth. J. Polit. Econ.
**94**(5), 1002–1037 (1986)CrossRefGoogle Scholar - Samuelson, P.: Stability of equilibrium: comparative statics and dynamics. Econometrica
**9**, 97–120 (1941)CrossRefGoogle Scholar - Santos, M.S.: Smoothness of the policy function in discrete time economic models. Econometrica
**59**(5), 1365–1382 (1991)CrossRefGoogle Scholar - Santos, M.S.: Differentiability and comparative analysis in discrete-time infinite-horizon optimization. J. Econ. Theory
**57**(1), 222–229 (1992)CrossRefGoogle Scholar - Santos, M.S.: On non-existence of Markov equilibria in competitive-market economies. J. Econ. Theory
**105**(1), 73–98 (2002)CrossRefGoogle Scholar - Schmitt-Grohe, S., Uribe, M.: Balanced-budget rules, distortionary taxes, and aggregate instability. J. Polit. Econ.
**105**(5), 976–1000 (1997)CrossRefGoogle Scholar - Shimokawa, T.: Existence and local indeterminacy of periodic equilibrium paths in infinite horizon models with external effects. Econ. Theor.
**16**(1), 199–208 (2000)CrossRefGoogle Scholar - Spear, S.E.: Growth, externalities, and sunspots. J. Econ. Theory
**54**(1), 215–223 (1991)CrossRefGoogle Scholar - Stockman, D.R.: Balanced-budget rules: chaos and deterministic sunspots. J. Econ. Theory
**145**(3), 1060–1085 (2010)CrossRefGoogle Scholar - Stokey, N., Lucas, R., Prescott, E.: Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge (1989)Google Scholar
- Stoltenberg-Hansen, V., Lindstrom, I., Griffor, E.: Mathematical Theory of Domains. Cambridge Tracts in Theoretical Computer Science 22. Cambridge Press, Cambridge (1994)CrossRefGoogle Scholar
- Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math.
**5**(2), 285–309 (1955)CrossRefGoogle Scholar - Topkis, D. M.: Minimizing a submodular function over a lattice. Oper. Res.
**26**, 305–326 (1978)Google Scholar - Topkis, D.M.: Supermodularity and Complementarity. Frontiers of Economic Research. Princeton University Press, Princeton (1998)Google Scholar
- Veinott, A.F. Jr.: Lattice Programming: Qualitative Optimization and Equilibria, Manuscript, Stanford University (1992)Google Scholar
- Wang, P., Wen, Y.: Imperfect competition and indeterminacy of aggregate output. J. Econ. Theory
**143**(1), 519–540 (2008)CrossRefGoogle Scholar