The Annals of Applied Probability

Optimal consumption choice with intertemporal substitution

Peter Bank and Frank Riedel

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We analyze the intertemporal utility maximization problem under uncertainty for the preferences proposed by Hindy, Huang and Kreps. Existence and uniqueness of optimal consumption plans are established under arbitrary convex portfolio constraints, including both complete and incomplete markets. For the complete market setting, we prove an infinite-dimensional version of the Kuhn –Tucker theorem which implies necessary and sufficient conditions for optimality. Using this characterization, we show that optimal plans prescribe consuming just enough to keep the induced level of satisfaction always above some stochastic lower bound. When uncertainty is generated by a Lévy process and agents exhibit constant relative risk aversion, we derive solutions in closed form. Depending on the structure of the underlying stochastics, optimal consumption occurs at rates, in gulps, or in a singular way.

Article information

Ann. Appl. Probab., Volume 11, Number 3 (2001), 750-788.

First available in Project Euclid: 5 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90A10
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Hindy-Huang-Kreps preferences intertemporal utility intertemporal substitution singular control problem Lévy processes


Bank, Peter; Riedel, Frank. Optimal consumption choice with intertemporal substitution. Ann. Appl. Probab. 11 (2001), no. 3, 750--788. doi:10.1214/aoap/1015345348.

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  • Aumann, R. and Perles, M. (1965). A variational problem arising in economics. J. Math. Anal. Appl. 11 488-503.
  • Bank, P. (2000). Singular control ofoptional random measures-stochastic optimization and representation problems arising in the microeconomic theory ofintertemporal consumption choice. Ph.D. thesis, Humboldt Univ. Berlin.
  • Bank, P. and Riedel, F. (2000). Non-time additive utility optimization-the case ofcertainty. J. Math. Econom. 33 271-290.
  • Benth, F., Karlsen, H. and Reikvam, K. (1999). Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach. Research Report 21, Univ. Aarhus, Denmark.
  • Bertoin, J. (1996). L´evy Processes. Cambridge Univ. Press.
  • Constantinides, G. (1990). Habit formation: a resolution of the equity premium puzzle. J. Political Economy 98 519-543.
  • Cox, J. C. and Huang, C.-F. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Economic Theory 49 33-83.
  • Cox, J. C. and Huang, C.-F. (1991). A variational problem arising in financial economics. J. Math. Econ. 20 465-487.
  • Cuoco, D. (1997). Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J. Econom. Theory 72 33-73.
  • Cvitanic, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2 767-818.
  • Dellacherie, C. and Meyer, P. (1975). Probabilit´es et potentiel, Chapitres I-IV. Hermann, Paris.
  • Duffie, D. and Skiadas, C. (1994). Continuous-time security pricing, a utility gradient approach. J. Math. Econom. 23 107-131.
  • F ¨ollmer, H. and Kabanov, Y. (1998). Optional decomposition and Lagrange multipliers. Finance and Stochastics 2 69-81.
  • F ¨ollmer, H. and Kramkov, D. (1997). Optional decompositions under constraints. Probab. Theory Related Fields 1 1-25.
  • Hindy, A. and Huang, C.-F. (1993). Optimal consumption and portfolio rules with durability and local substitution. Econometrica 61 85-121.
  • Hindy, A. Huang, C.-F. and Kreps, D. (1992). On intertemporal preferences in continuous time: the case ofcertainty. J. Math. Econom. 21 401-440.
  • Jacod, J. (1979). Calcul Stochastique et Probl emes de Martinagles. Lecture Notes in Math. 714. Springer, Berlin.
  • Jin, X. and Deng, S. (1997). Existence and uniqueness ofoptimal consumption and portfolio rules in a continuous-time finance model with habit formation and without short sales. J. Math. Econom. 28 187-205.
  • Kabanov, Y. (1999). Hedging and liquidation under transaction costs in currency markets. Finance and Stochastics 2 237-248.
  • Karatzas, I., Lehoczky, J. P. and Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a "small investor" on a finite horizon. SIAM J. Control Optim. 25 1557-1586.
  • Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.
  • Koml ´os, J. (1967). A generalization ofa problem ofSteinhaus. Acta Math. Acad. Sci. Hung. 18 217-229.
  • Korn, R. (1997). Optimal Portfolios: Stochastic Models for Optimal Investment and Risk Management in Continuous Time. World Scientific, Singapore.
  • Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity ofutility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904-950.
  • Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous case. Rev. Econom. Statist. 51 247-257.
  • Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3 373-413.
  • Merton, R. C. (1990). Continuous-Time Finance. Blackwell, London.
  • Sundaresan, S. (1989). Intertemporally dependent preferences and the volatility of consumption and wealth. Rev. Finan. Econom. 2 73-89.