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Additive habits with power utility: Estimates, asymptotics and equilibrium

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  • Roman Muraviev

Abstract

We consider a power utility maximization problem with additive habits in a framework of discrete-time markets and random endowments. For certain classes of incomplete markets, we establish estimates for the optimal consumption stream in terms of the aggregate state price density, investigate the asymptotic behaviour of the propensity to consume (ratio of the consumption to the wealth), as the initial endowment tends to infinity, and show that the limit is the corresponding quantity in an artificial market. For complete markets, we concentrate on proving the existence of an Arrow-Debreu equilibrium in an economy inhabited by heterogeneous individuals who differ with respect to their risk-aversion coefficient, impatience rate and endowments stream, but possess the same degree of habit-formation. Finally, in a representative agent equilibrium, we compute explicitly the price of a zero coupon bond and the Lucas tree equity, and study its dependence on the habit-formation parameter.

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  • Roman Muraviev, 2011. "Additive habits with power utility: Estimates, asymptotics and equilibrium," Papers 1108.2889, arXiv.org.
  • Handle: RePEc:arx:papers:1108.2889
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    References listed on IDEAS

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    1. Miklos Rasonyi & Lukasz Stettner, 2005. "On utility maximization in discrete-time financial market models," Papers math/0505243, arXiv.org.
    2. Jerome B. Detemple & Fernando Zapatero, 1992. "Optimal Consumption-Portfolio Policies With Habit Formation," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 251-274.
    3. Abel, Andrew B, 1990. "Asset Prices under Habit Formation and Catching Up with the Joneses," American Economic Review, American Economic Association, vol. 80(2), pages 38-42, May.
    4. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    5. Dana, Rose Anne, 1993. "Existence and Uniqueness of Equilibria When Preferences Are Additively Separable," Econometrica, Econometric Society, vol. 61(4), pages 953-957, July.
    6. John Y. Campbell & John H. Cochrane, 1994. "By force of habit: a consumption-based explanation of aggregate stock market behavior," Working Papers 94-17, Federal Reserve Bank of Philadelphia.
    7. Semyon Malamud, 2008. "Long run forward rates and long yields of bonds and options in heterogeneous equilibria," Finance and Stochastics, Springer, vol. 12(2), pages 245-264, April.
    8. Detemple, Jerome B. & Karatzas, Ioannis, 2003. "Non-addictive habits: optimal consumption-portfolio policies," Journal of Economic Theory, Elsevier, vol. 113(2), pages 265-285, December.
    9. Constantinides, George M, 1990. "Habit Formation: A Resolution of the Equity Premium Puzzle," Journal of Political Economy, University of Chicago Press, vol. 98(3), pages 519-543, June.
    10. Duffie, Darrell & Fleming, Wendell & Soner, H. Mete & Zariphopoulou, Thaleia, 1997. "Hedging in incomplete markets with HARA utility," Journal of Economic Dynamics and Control, Elsevier, vol. 21(4-5), pages 753-782, May.
    11. David A. Chapman, 1998. "Habit Formation and Aggregate Consumption," Econometrica, Econometric Society, vol. 66(5), pages 1223-1230, September.
    12. Dana, Rose-Anne, 1993. "Existence, uniqueness and determinacy of Arrow-Debreu equilibria in finance models," Journal of Mathematical Economics, Elsevier, vol. 22(6), pages 563-579.
    13. Roman Muraviev, 2011. "Additive habit formation: Consumption in incomplete markets with random endowments," Papers 1106.2980, arXiv.org, revised Sep 2011.
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