Economic Theory

, Volume 66, Issue 3, pp 627–679 | Cite as

Bubbly Markov equilibria

  • Martin Barbie
  • Marten HillebrandEmail author
Research Article


Bubbly Markov equilibria (BME) are recursive equilibria on the natural state space which admit a non-trivial bubble. The present paper studies the existence and properties of BME in a general class of overlapping generations economies with capital accumulation and stochastic production shocks. Using monotone methods, we develop a general approach to construct Markov equilibria and provide necessary and sufficient conditions for these equilibria to be bubbly. Our main result shows that a BME exists whenever the bubbleless equilibrium is Pareto inefficient due to either overaccumulation of capital or inefficient risk sharing between generations.


Asset bubbles Stochastic OLG Production Markov equilibria Pareto optimality 

JEL Classification

C62 D51 E32 



We would like to thank Tim Deeken, Tomoo Kikuchi, Herakles Polemarchakis, Clemens Puppe, Kevin Reffett, Caren Söhner, John Stachurski, and Klaus Wälde for helpful suggestions and discussions and two anonymous referees for their valuable and very constructive comments. We also thank seminar participants at various conferences including the 2013 SAET Conference in Paris, the 2013 Annual Meeting of the Verein für Socialpolitik in Düsseldorf, the 2014 CEF Conference in Oslo, and the 2017 SAET conference in Faro.


  1. Aiyagari, R., Peled, D.: Dominant root characterization of Pareto optimality and the existence of optimal equilibria in stochastic overlapping generations models. J. Econ. Theory 54, 69–83 (1991)CrossRefGoogle Scholar
  2. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. Springer, Berlin (2007)Google Scholar
  3. Ball, L., Elmendorf, D., Mankiw, N.: The deficit gamble. J. Money Credit Bank. 30, 699–720 (1998)CrossRefGoogle Scholar
  4. Barbie, M., Kaul, A.: The Zilcha criteria for dynamic inefficiency reconsidered. Econ. Theory 40, 339–348 (2009). doi: 10.1007/s00199-008-0367-3 CrossRefGoogle Scholar
  5. Barbie, M., Kaul, A.: Pareto optimality and existence of monetary equilibria in a stochastic OLG model: a recursive approach. Working paper, University of Cologne, Cologne (2015)Google Scholar
  6. Barbie, M., Hagedorn, M., Kaul, A.: On the interaction between risk sharing and capital accumulation in a stochastic OLG model with production. J. Econ. Theory 137, 568–579 (2007)CrossRefGoogle Scholar
  7. Bose, A., Ray, D.: Monetary equilibrium in an overlapping generations model with productive capital. Econ. Theory 3, 697–716 (1993). doi: 10.1007/BF01210266 CrossRefGoogle Scholar
  8. Buchanan, H.E., Hildebrandt, T.H.: Note on the convergence of a sequence of functions of a certain type. Ann. Math. 9(3), 123–126 (1908)CrossRefGoogle Scholar
  9. Cass, D.: On capital overaccumulation in the aggregative neoclassical model of economic growth: a complete characterization. J. Econ. Theory 4, 200–223 (1972)CrossRefGoogle Scholar
  10. Chattopadhyay, S., Gottardi, P.: Stochastic OLG models, market structure and optimality. J. Econ. Theory 89, 21–67 (1999)CrossRefGoogle Scholar
  11. Coleman, W.J.I.: Equilibrium in a production economy with an income tax. Econometrica 59, 1091–1104 (1991)CrossRefGoogle Scholar
  12. Coleman, W.J.I.: Uniqueness of an equilibrium in infinite-horizon economies subject to taxes and externalities. J. Econ. Theory 95, 71–78 (2000)CrossRefGoogle Scholar
  13. Demange, G.: On optimality in intergenerational risk sharing. Econ. Theory 20, 1–27 (2002). doi: 10.1007/s001990100199 CrossRefGoogle Scholar
  14. Demange, G., Laroque, G.: Social security, optimality, and equilibria in a stochastic overlapping generations economy. J. Public Econ. Theory 2(1), 1–23 (2000)CrossRefGoogle Scholar
  15. Diamond, P.: National debt in a neoclassical growth model. Am. Econ. Rev. 55(5), 1126–1150 (1965)Google Scholar
  16. Farhi, E., Tirole, J.: Bubbly liquidity. Rev. Econ. Stud. 79, 678–706 (2012)CrossRefGoogle Scholar
  17. Galor, O., Ryder, H.E.: Existence, uniqueness, and stability of equilibrium in an overlapping-generations model with productive capital. J. Econ. Theory 49, 360–375 (1989)CrossRefGoogle Scholar
  18. Gottardi, P., Kübler, F.: Social security and risk sharing. J. Econ. Theory 146, 1078–1106 (2011)CrossRefGoogle Scholar
  19. Greenwood, J., Huffman, G.: On the existence of nonoptimal equilibria in dynamic stochastic economies. J. Econ. Theory 65, 611–623 (1995)CrossRefGoogle Scholar
  20. Hauenschild, N.: Capital accumulation in a stochastic overlapping generations model with social security. J. Econ. Theory 106, 201–216 (2002)CrossRefGoogle Scholar
  21. Hellwig, C., Lorenzoni, G.: Bubbles and self-enforcing debt. Econometrica 77(4), 1137–1164 (2009)CrossRefGoogle Scholar
  22. Hillebrand, M.: On the role of labor supply for the optimal size of social security. J. Econ. Dyn. Control 35, 1091–1105 (2011)CrossRefGoogle Scholar
  23. Hillebrand, M.: Uniqueness of Markov equilibrium in stochastic OLG models with nonclassical production. Econ. Lett. 123(2), 171–176 (2014)CrossRefGoogle Scholar
  24. Ikeda, D., Phan, T.: Toxic asset bubbles. Econ. Theory 61, 241–271 (2016). doi: 10.1007/s00199-015-0928-1 CrossRefGoogle Scholar
  25. Kamihigashi, T., Stachurski, J.: Stochastic stability in monotone economies. Theor. Econ. 9, 383–407 (2014)CrossRefGoogle Scholar
  26. Konishi, H., Perera-Tallo, F.: Existence of steady-state equilibrium in an overlapping-generations model with production. Econ. Theory 9, 529–537 (1997). doi: 10.1007/BF01213853 CrossRefGoogle Scholar
  27. Kübler, F., Polemarchakis, H.: Stationary Markov equilibria for overlapping generations. Econ. Theory 24(3), 623–643 (2004). doi: 10.1007/s00199-004-0523-3 CrossRefGoogle Scholar
  28. Li, J., Lin, S.: Existence and uniqueness of steady state equilibrium in a generalized overlapping generations model. Macroecon. Dyn. 16, 299–311 (2012)CrossRefGoogle Scholar
  29. Magill, M., Quinzii, M.: Indeterminacy of equilibrium in stochastic OLG models. Econ. Theory 21, 435–454 (2003). doi: 10.1007/s00199-002-0287-6 CrossRefGoogle Scholar
  30. Manuelli, R.: Existence and optimality of currency equilibrium in stochastic overlapping generations models: the pure endowment case. J. Econ. Theory 51, 268–294 (1990)CrossRefGoogle Scholar
  31. Martin, A., Ventura, J.: Economic growth with bubbles. Am. Econ. Rev. 102, 3033–3058 (2012)CrossRefGoogle Scholar
  32. McGovern, J., Morand, O.F., Reffett, K.L.: Computing minimal state space recursive equilibrium in OLG models with stochastic production. Econ. Theory 54, 623–674 (2013). doi: 10.1007/s00199-012-0728-9 CrossRefGoogle Scholar
  33. Miao, J., Wang, P., Xu, L.: Stock market bubbles and unemployment. Econ. Theory 61, 273–307 (2016). doi: 10.1007/s00199-015-0906-7 CrossRefGoogle Scholar
  34. Michel, P., Wigniolle, B.: Temporary bubbles. J. Econ. Theory 112, 173–183 (2003)CrossRefGoogle Scholar
  35. Morand, O.F., Reffett, K.L.: Existence and uniqueness of equilibrium in nonoptimal unbounded infinite horizon economies. J. Monet. Econ. 50, 1351–1373 (2003)CrossRefGoogle Scholar
  36. Morand, O.F., Reffett, K.L.: Stationary Markovian equilibrium in overlapping generations models with stochastic nonclassical production and Markov shocks. J. Math. Econ. 43, 501–522 (2007)CrossRefGoogle Scholar
  37. Rangazas, P., Russell, S.: The Zilcha criterion for dynamic inefficiency. Econ. Theory 26, 701–716 (2005). doi: 10.1007/s00199-004-0547-8 CrossRefGoogle Scholar
  38. Samuelson, P.A.: An exact consumption-loan model of interest with or without the social contrivance of money. J. Polit. Econ. 66(6), 467–482 (1958)CrossRefGoogle Scholar
  39. Tirole, J.: Asset bubbles and overlapping generations. Econometrica 53(6), 1499–1528 (1985)CrossRefGoogle Scholar
  40. Wang, Y.: Stationary equilibria in an overlapping generations economy with stochastic production. J. Econ. Theory 61(2), 423–435 (1993)CrossRefGoogle Scholar
  41. Wang, Y.: Stationary Markov equilibria in an OLG model with correlated production shocks. Int. Econ. Rev. 35(3), 731–744 (1994)CrossRefGoogle Scholar
  42. Weil, P.: Confidence and the real value of money in an overlapping generations economy. Q. J. Econ. 102, 1–22 (1987)CrossRefGoogle Scholar
  43. Zilcha, I.: Dynamic efficiency in overlapping generations models with stochastic production. J. Econ. Theory 52(2), 364–379 (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Center for Macroeconomic ResearchUniversity of CologneCologneGermany
  2. 2.Department of EconomicsGoethe University FrankfurtFrankfurt am MainGermany

Personalised recommendations