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A Mean Field Capital Accumulation Game with HARA Utility

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Abstract

This paper introduces a mean field modeling framework for consumption-accumulation optimization. The production dynamics are generalized from stochastic growth theory by addressing the collective impact of a large population of similar agents on efficiency. This gives rise to a stochastic dynamic game with mean field coupling in the dynamics, where we adopt a hyperbolic absolute risk aversion (HARA) utility functional for the agents. A set of decentralized strategies is obtained by using the Nash certainty equivalence approach. To examine the long-term behavior we introduce a notion called the relaxed stationary mean field solution. The simple strategy computed from this solution is used to investigate the out-of-equilibrium behavior of the mean field system. Interesting nonlinear phenomena can emerge, including stable equilibria, limit cycles and chaos, which are related to the agent’s sensitivity to the mean field.

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References

  1. Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: numerical methods. SIAM J Numer Anal 48(3):1136–1162

    Article  MathSciNet  MATH  Google Scholar 

  2. Adlakha S, Johari R, Weintraub G, Goldsmith A (2008) Oblivious equilibrium for large-scale stochastic games with unbounded costs. In: Proc 47th IEEE CDC, Cancun, Mexico, pp 5531–5538

    Google Scholar 

  3. Amir R (1996) Continuous stochastic games of capital accumulation with convex transitions. Games Econ Behav 15:111–131

    Article  MathSciNet  MATH  Google Scholar 

  4. Andersson D, Djehiche B (2011) A maximum principle for SDEs of mean-field type. Appl Math Optim 63(3):341–356

    Article  MathSciNet  MATH  Google Scholar 

  5. Arthur WB (1999) Complexity and the economy. Science 284:107–109

    Article  Google Scholar 

  6. Askenazy P, Le Van C (1999) A model of optimal growth strategy. J Econ Theory 85:24–51

    Article  MATH  Google Scholar 

  7. Balbus L, Nowak AS (2004) Construction of Nash equilibria in symmetric stochastic games of capital accumulation. Math Methods Oper Res 60:267–277

    Article  MathSciNet  MATH  Google Scholar 

  8. Bardi M (2012) Explicit solutions of some linear-quadratic mean field games. Netw Heterog Media 7(2):243–261

    Article  MathSciNet  MATH  Google Scholar 

  9. Barro RJ, Sala-I-Martin X (1992) Public finance in models of economic growth. Rev Econ Stud 59(4):645–661

    Article  MATH  Google Scholar 

  10. Benhabib J (ed) (1992) Cycles and chaos in economic equilibrium. Princeton University Press, Princeton

    Google Scholar 

  11. Bensoussan A, Frehse J, Yam P (2012) Overview on mean field games and mean field type control theory. Preprint

  12. Bensoussan A, Sung KCJ, Yam SCP, Yung SP (2011) Linear-quadratic mean-field games. Preprint

  13. Brock WA, Mirman LJ (1972) Optimal economic growth and uncertainty: the discounted case. J Econ Theory 4:479–513

    Article  MathSciNet  Google Scholar 

  14. Buckdahn R, Cardaliaguet P, Quincampoix M (2011) Some recent aspects of differential game theory. Dyn Games Appl 1(1):74–114

    Article  MathSciNet  MATH  Google Scholar 

  15. Cardaliaguet P (2012) Notes on mean field games

    Google Scholar 

  16. Carmona R, Delarue F (2012) A probabilistic analysis of mean field games. Preprint

  17. Cass D (1965) Optimum growth in an aggregative model of capital accumulation. Rev Econ Stud 32(3):233–240

    Article  Google Scholar 

  18. Chow YS, Teicher H (1997) Probability theory: independence, interchangeability, martingales, 3rd edn. Springer, New York

    Book  MATH  Google Scholar 

  19. Day RH (1982) Irregular growth cycles. Am Econ Rev 72(3):406–414

    Google Scholar 

  20. Dockner EJ, Nishimura K (2005) Capital accumulation games with a non-concave production function. J Econ Behav Organ 57:408–420

    Article  Google Scholar 

  21. Duffie D, Fleming W, Soner HM, Zariphopoulou T (1997) Hedging in incomplete markets with HARA utility. J Econ Dyn Control 21:753–782

    Article  MathSciNet  MATH  Google Scholar 

  22. Gomes DA, Mohr J, Souza RR (2010) Discrete time, finite state space mean field games. J Math Pures Appl 93:308–328

    Article  MathSciNet  MATH  Google Scholar 

  23. Gast N, Gaujal B, Le Boudec J-Y (2012) Mean field for Markov decision processes: from discrete to continuous optimization. IEEE Trans Autom Control 57(9):2266–2280

    Article  Google Scholar 

  24. Guéant O, Lasry J-M, Lions P-L (2011) Mean field games and applications. In: Carmona AR et al. (eds) Paris-Princeton lectures on mathematical finance 2010. Springer, Berlin, pp 205–266

    Chapter  Google Scholar 

  25. Hart S (1973) Values of mixed games. Int J Game Theory 2:69–86

    Article  MATH  Google Scholar 

  26. Hernández-Lerma O, Lasserre JB (1996) Discrete-time Markov control processes: basic optimality criteria. Springer, New York

    Book  Google Scholar 

  27. Huang M (2010) Large-population LQG games involving a major player: the Nash certainty equivalence principle. SIAM J Control Optim 48(5):3318–3353

    Article  MATH  Google Scholar 

  28. Huang M, Caines PE, Malhamé RP (2003) Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In: Proc 42nd IEEE CDC, Maui, HI, pp 98–103

    Google Scholar 

  29. Huang M, Caines PE, Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ε-Nash equilibria. IEEE Trans Autom Control 52(9):1560–1571

    Article  Google Scholar 

  30. Huang M, Caines PE, Malhamé RP (2012) Social optima in mean field LQG control: centralized and decentralized strategies. IEEE Trans Autom Control 57(7):1736–1751

    Article  Google Scholar 

  31. Huang M, Malhamé RP, Caines PE (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221–251

    MathSciNet  MATH  Google Scholar 

  32. Jovanovic B, Rosenthal RW (1988) Anonymous sequential games. J Math Econ 17:77–87

    Article  MathSciNet  MATH  Google Scholar 

  33. Kizilkale AC, Caines PE (2013) Mean field stochastic adaptive control. IEEE Trans Autom Control 58(4):905–920

    Article  MathSciNet  Google Scholar 

  34. Kolokoltsov VN, Li J, Yang W (2011) Mean field games and nonlinear Markov processes. Preprint

  35. Lambson VE (1984) Self-enforcing collusion in large dynamic markets. J Econ Theory 34:282–291

    Article  MathSciNet  MATH  Google Scholar 

  36. Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229–260

    Article  MathSciNet  MATH  Google Scholar 

  37. Levhari D, Srinivasan TN (1969) Optimal savings under uncertainty. Rev Econ Stud 36(2):153–163

    Article  Google Scholar 

  38. Li T, Zhang J-F (2008) Asymptotically optimal decentralized control for large population stochastic multiagent systems. IEEE Trans Autom Control 53(7):1643–1660

    Article  Google Scholar 

  39. Li TY, Yorke JA (1975) Period three implies chaos. Am Math Mon 82(10):985–992

    Article  MathSciNet  MATH  Google Scholar 

  40. Liu W-F, Turnovsky SJ (2005) Consumption externalities, production externalities, and long-run macroeconomic efficiency. J Public Econ 89:1097–1129

    Article  Google Scholar 

  41. Ma Z, Callaway D, Hiskens I (2013) Decentralized charging control for large populations of plug-in electric vehicles. IEEE Trans Control Syst Technol 21(1):67–78

    Article  Google Scholar 

  42. Mendelssohn R, Sobel MJ (1980) Capital accumulation and the optimization of renewable resource models. J Econ Theory 23:243–260

    Article  MATH  Google Scholar 

  43. Merton R (1969) Lifetime portfolio selection under uncertainty: the continuous time case. Rev Econ Stat 51(3):247–257

    Article  Google Scholar 

  44. Nguyen SL, Huang M (2012) Linear-quadratic-Gaussian mixed games with continuum-parameterized minor players. SIAM J Control Optim 50(5):2907–2937

    Article  MathSciNet  MATH  Google Scholar 

  45. Nourian M, Caines PE (2012) ε-Nash mean field game theory for nonlinear stochastic dynamical systems with mixed agents. In: Proc 51st IEEE CDC, Maui, HI, pp 2090–2095

    Google Scholar 

  46. Nourian M, Caines PE, Malhamé RP, Huang M (2012) Mean field control in leader-follower stochastic multi-agent systems: likelihood ratio based adaptation. IEEE Trans Autom Control 57(11):2801–2816

    Article  Google Scholar 

  47. Olson LJ, Roy S (2006) Theory of stochastic optimal economic growth. In: Dana R-A, Le Van C, Mitra T, Nishimura K (eds) Handbook on optimal growth I. Springer, Berlin, pp 297–335

    Chapter  Google Scholar 

  48. Olson LJ, Roy S (2000) Dynamic efficiency of conservation of renewable resources under uncertainty. J Econ Theory 95:186–214

    Article  MathSciNet  MATH  Google Scholar 

  49. Ramsey FP (1928) A mathematical theory of saving. Econ J 38(152):543–559

    Article  Google Scholar 

  50. Saari D (1995) Mathematical complexity of simple economics. Not Am Math Soc 42(2):222–230

    MathSciNet  MATH  Google Scholar 

  51. Solow RM (1956) A contribution to the theory of economic growth. Q J Econ 70(1):65–94

    Article  Google Scholar 

  52. Stokey NL, Lucas RE Jr., Prescott EC (1989) Recursive methods in economic dynamics. Harvard University Press, Cambridge

    MATH  Google Scholar 

  53. Tembine H, Le Boudec J-Y, El-Azouzi R, Altman E (2009) Mean field asymptotics of Markov decision evolutionary games and teams. In: Proc GameNets, Istanbul, Turkey, pp 140–150

    Google Scholar 

  54. Tembine H, Zhu Q, Basar T (2011) Risk-sensitive mean-field stochastic differential games. In: Proc 18th IFAC world congress, Milan, Italy

    Google Scholar 

  55. Wang B-C, Zhang J-F (2012) Distributed control of multi-agent systems with random parameters and a major agent. Automatica 48(9):2093–2106

    Article  MathSciNet  MATH  Google Scholar 

  56. Weintraub GY, Benkard CL, Van Roy B (2008) Markov perfect industry dynamics with many firms. Econometrica 76(6):1375–1411

    Article  MathSciNet  MATH  Google Scholar 

  57. Yin H, Mehta PG, Meyn SP, Shanbhag UV (2012) Synchronization of coupled oscillators is a game. IEEE Trans Autom Control 57(4):920–935

    Article  MathSciNet  Google Scholar 

  58. Yong J (2011) A linear-quadratic optimal control problem with mean-field stochastic differential equations. Preprint

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Acknowledgements

This work was partially supported by Natural Sciences and Engineering Research Council (NSERC) of Canada.

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  1. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Minyi Huang

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  1. Minyi Huang
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Correspondence to Minyi Huang.

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Huang, M. A Mean Field Capital Accumulation Game with HARA Utility. Dyn Games Appl 3, 446–472 (2013). https://doi.org/10.1007/s13235-013-0092-9

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