Dynamic Games and Applications

, Volume 9, Issue 2, pp 314–365 | Cite as

Strategic Growth with Recursive Preferences: Decreasing Marginal Impatience

  • Luis AlcaláEmail author
  • Fernando Tohmé
  • Carlos Dabús


This paper studies a two-agent strategic model of capital accumulation with heterogeneity in preferences and income shares. Preferences are represented by recursive utility functions that satisfy decreasing marginal impatience. The stationary equilibria of this dynamic game are analyzed under two alternative information structures: one in which agents precommit to future actions, and another one where they use Markovian strategies. In both cases, we develop sufficient conditions to show the existence of these equilibria and characterize their stability properties. Under certain regularity conditions, a precommitment equilibrium shows monotone convergence of aggregate variables, but Markovian equilibria may exhibit nonmonotonic paths, even in the long-run.


Recursive utility Decreasing impatience Dynamic programming Precommitment strategies Markovian strategies 



Luis Alcalá acknowledges financial support from the Universidad Nacional de San Luis, through Grant PROICO 319502, and from the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), through Grant PIP 112-200801-00655. We thank an associate editor and an anonymous referee whose insightful comments and detailed suggestions led to significant improvements of the paper.

Supplementary material


  1. 1.
    Arrow KJ, Debreu G (1954) Existence of an equilibrium for a competitive economy. Econometrica 22(3):265–290MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Başar T, Olsder GJ (1999) Dynamic noncooperative game theory, classics in applied mathematics, vol 23, 2nd edn. SIAM, PhiladelphiazbMATHGoogle Scholar
  3. 3.
    Banks J, Duggan J (2004) Existence of Nash equilibria on convex sets. Working paper, W. Allen Wallis Institute of Political Economy, University of RochesterGoogle Scholar
  4. 4.
    Barbu V, Precupanu T (2012) Convexity and optimization in Banach spaces. Springer Monographs in Mathematics, 4th edn. Springer, DordrechtzbMATHCrossRefGoogle Scholar
  5. 5.
    Beals R, Koopmans T (1969) Maximizing stationary utility in a constant technology. SIAM J Appl Math 17(5):1001–1015MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Becker R (2006) Equilibrium dynamics with many agents. In: Dana RA, Le Van C, Mitra T, Nishimura K (eds) Handbook on optimal growth 1. Discrete time. Springer, Berlin, pp 385–442CrossRefGoogle Scholar
  7. 7.
    Becker R, Foias C (1998) Implicit programming and the invariant manifold for Ramsey equilibria. In: Abramovich Y, Avgerinos E, Yannelis N (eds) Functional analysis and economic theory. Springer, Berlin, pp 119–144zbMATHCrossRefGoogle Scholar
  8. 8.
    Becker R, Foias C (2007) Strategic Ramsey equilibrium dynamics. J Math Econ 43(3):318–346MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Becker R, Dubey R, Mitra T (2014) On Ramsey equilibrium: capital ownership pattern and inefficiency. Econ Theory 55(3):565–600MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Camacho C, Saglam C, Turan A (2013) Strategic interaction and dynamics under endogenous time preference. J Math Econ 49(4):291–301MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Carlson D, Haurie A (1995) A turnpike theory for infinite horizon open-loop differential games with decoupled controls. In: Olsder GJ (ed) New trends in dynamic games and applications. Birkhäuser, Boston, pp 353–376zbMATHCrossRefGoogle Scholar
  12. 12.
    Carlson D, Haurie A (1996) A turnpike theory for infinite-horizon open-loop competitive processes. SIAM J Control Optim 34(4):1405–1419MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Carlson D, Haurie A, Zaccour G (2016) Infinite horizon concave games with coupled constraints. In: Başar T, Zaccour G (eds) Handbook of dynamic game theory. Springer, Basel, pp 1–44Google Scholar
  14. 14.
    Carlson DA, Haurie AB (2000) Infinite horizon dynamic games with coupled state constraints. In: Filar JA, Gaitsgory V, Mizukami K (eds) Advances in dynamic games and applications. Birkhäuser, Boston, pp 195–212zbMATHCrossRefGoogle Scholar
  15. 15.
    Chade H, Swinkels J (2017) The no-upward-crossing condition and the moral hazard problem. Working paper, Department of Economics, Arizona State UniversityGoogle Scholar
  16. 16.
    Coleman WJ (1991) Equilibrium in a production economy with an income tax. Econometrica 59(4):1091–1104MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Coleman WJ (1997) Equilibria in distorted infinite-horizon economies with capital and labor. J Econ Theory 72(2):446–461zbMATHCrossRefGoogle Scholar
  18. 18.
    Coleman WJ (2000) Uniqueness of an equilibrium in infinite-horizon economies subject to taxes and externalities. J Econ Theory 95(1):71–78MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Das M (2003) Optimal growth with decreasing marginal impatience. J Econ Dyn Control 27(10):1881–1898MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Datta M, Mirman LJ, Reffett K (2002) Existence and uniqueness of equilibrium in distorted dynamic economies with capital and labor. J Econ Theory 103(2):377–410MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Dechert WD (1982) Lagrange multipliers in infinite horizon discrete time optimal control models. J Math Econ 9(3):285–302MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Dockner E, Nishimura K (2004) Strategic growth. J Differ Equ Appl 10(5):515–527MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Dockner E, Nishimura K (2005) Capital accumulation games with a non-concave production function. J Econ Behav Organ 57(4):408–420CrossRefGoogle Scholar
  24. 24.
    Drugeon JP, Wigniolle B (2017) On impatience, temptation and Ramsey’s conjecture. Econ Theory 63(1):73–98MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Epstein LG (1987) A simple dynamic general equilibrium model. J Econ Theory 41(1):68–95MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Erol S, Le Van C, Saglam C (2011) Existence, optimality and dynamics of equilibria with endogenous time preference. J Math Econ 47(2):170–179MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Fesselmeyer E, Mirman LJ, Santugini M (2016) Strategic interactions in a one-sector growth model. Dyn Games Appl 6(2):209–224MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Galor O (2007) Discrete dynamical systems. Springer, BerlinzbMATHCrossRefGoogle Scholar
  29. 29.
    Geoffard PY (1996) Discounting and optimizing: capital accumulation problems as variational minmax problems. J Econ Theory 69(1):53–70MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Greenwood J, Huffman G (1995) On the existence of nonoptimal equilibria in dynamic stochastic economies. J Econ Theory 65(2):611–623zbMATHCrossRefGoogle Scholar
  31. 31.
    Horn R, Johnson C (1990) Matrix analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  32. 32.
    Houba H, Sneek K, Vardy F (2000) Can negotiations prevent fish wars? J Econ Dyn Control 24(8):1265–1280zbMATHCrossRefGoogle Scholar
  33. 33.
    Iwai K (1972) Optimal economic growth and stationary ordinal utility: a Fischerian approach. J Econ Theory 5(1):121–151CrossRefGoogle Scholar
  34. 34.
    Le Van C, Saglam C (2004) Optimal growth models and the Lagrange multiplier. J Math Econ 40(3–4):393–410MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Lucas RE, Stokey N (1984) Optimal growth with many consumers. J Econ Theory 32(1):139–171MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Mantel RR (1967) Maximization of utility over time with a variable rate of time preference. Discussion paper, Cowles Foundation for Research in Economics, CF-70525(2), Yale UniversityGoogle Scholar
  37. 37.
    Mantel RR (1993) Grandma’s dress, or what’s new for optimal growth. Rev Anal Econ 8(1):61–81Google Scholar
  38. 38.
    Mantel RR (1995) Why the rich get richer and the poor get poorer. Estud Econ 22(2):177–205Google Scholar
  39. 39.
    Mantel RR (1999) Optimal economic growth with recursive preferences: decreasing rate of time preference. Económica 45(2):331–348Google Scholar
  40. 40.
    McKenzie L (1959) On the existence of general equilibrium for a competitive market. Econometrica 54:54–71MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Mirman LJ, Morand O, Reffett K (2008) A qualitative approach to Markovian equilibrium in infinite horizon economies with capital. J Econ Theory 139(1):75–98MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Pichler P, Sorger G (2009) Wealth distribution and aggregate time-preference. J Econ Dyn Control 33(1):1–14zbMATHCrossRefGoogle Scholar
  43. 43.
    Ponstein J (1981) On the use of purely finitely additive multipliers in mathematical programming. J Optim Theory Appl 33(1):37–55MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Prescott EC, Reffett K (2016) Preface: special issue on dynamic games in macroeconomics. Dyn Games Appl 6(2):157–160MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Rockafellar RT (1970) Convex analysis. Princeton University Press, PrincetonzbMATHCrossRefGoogle Scholar
  46. 46.
    Rockafellar RT, Wets RJB (1976) Stochastic convex programming: relatively complete recourse and induced feasibility. SIAM J Control Optim 14(3):574–589MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Rosen JB (1965) Existence and uniqueness of equilibrium points for concave \(N\)-person games. Econometrica 3(33):520–534MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Shivakumar P, Sivakumar K (2009) A review of infinite matrices and their applications. Linear Algebra Appl 430(4):976–998MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Sorger G (2002) On the long-run distribution of capital in the Ramsey model. J Econ Theory 105(1):226–243MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Sorger G (2006) Recursive Nash bargaining over a productive asset. J Econ Dyn Control 30(12):2637–2659MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Sorger G (2008) Strategic saving decisions in the infinite-horizon model. Econ Theory 36(3):353–377MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Stern M (2006) Endogenous time preference and optimal growth. Econ Theory 29(1):49–70MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Stokey N, Lucas RE, Prescott EC (1989) Recursive methods in economic dynamics. Harvard University Press, CambridgeCrossRefGoogle Scholar
  54. 54.
    Tohmé F (2006) Rolf Mantel and the computability of general equilibria: on the origins of the Sonnenschein–Mantel–Debreu theorem. Hist Polit Econ 38(Suppl. 1):213–227CrossRefGoogle Scholar
  55. 55.
    Uzawa H (1968) Time preference, the consumption function and optimum assets holdings. In: Wolfe J (ed) Value, capital, and growth: papers in honour of Sir John Hicks. Aldine, Chicago, pp 485–504Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto de Matemática Aplicada-San Luis (IMASL)Universidad Nacional de San Luis-CONICETSan LuisArgentina
  2. 2.Departamento de Economía, Instituto de Matemática de Bahía Blanca (INMABB)Universidad Nacional del Sur-CONICETBahía BlancaArgentina
  3. 3.Departamento de Economía, Instituto de Investigaciones Económicas y Sociales del Sur (IIESS)Universidad Nacional del Sur-CONICETBahía BlancaArgentina

Personalised recommendations