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Consumption and Asset Prices with Recursive Preferences: Continuous-Time Approximations to Discrete-Time Models

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  • Mark Fisher

    (Federal Reserve Bank of Atlanta)

Abstract

This paper presents tractable and efficient numerical solutions to general equilibrium models of asset prices and consumption when the representative agent has recursive preferences. It provides a discrete-time presentation of the approach of Fisher and Gilles (1998), treating continuous-time representations as approximations to discrete-time "truth." First, exact discrete-time solutions are derived, illustrating the following ideas: (i) The price--dividend ratio (such as the wealth-consumption ratio) is a perpetuity (the canonical infinitely-lived asset), the value of which is the sum of dividend-denominated bond prices, and (ii) the positivity of the dividend-denominated asymptotic forward rate is necessary and sufficient for the nonconvergence of value function iteration for an important class of models. Next, continuous-time approximations are introduced. By assuming the size of the time step is small, first-order approximations in the step size provide the same analytical flexibility to discrete-time modeling as Ito's lemma provides for continuous time. Moreover, it is shown that differential equations provide an efficient platform for value function iteration. Last, continuous-time normalizations are adopted, providing an efficient solution method for recursive preferences. Campbell's approximations are explained, and regions of nonconvergence that have been overlooked are identified.

Suggested Citation

  • Mark Fisher, 1999. "Consumption and Asset Prices with Recursive Preferences: Continuous-Time Approximations to Discrete-Time Models," Computing in Economics and Finance 1999 934, Society for Computational Economics.
    • Handle: RePEc:sce:scecf9:934
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    References listed on IDEAS

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    1. Campbell, John Y. & Koo, Hyeng Keun, 1997. "A comparison of numerical and analytic approximate solutions to an intertemporal consumption choice problem," Journal of Economic Dynamics and Control, Elsevier, vol. 21(2-3), pages 273-295.
    2. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, December.
    3. Campbell, John Y, 1993. "Intertemporal Asset Pricing without Consumption Data," American Economic Review, American Economic Association, vol. 83(3), pages 487-512, June.
    4. Duffie, Darrell & Skiadas, Costis, 1994. "Continuous-time security pricing : A utility gradient approach," Journal of Mathematical Economics, Elsevier, vol. 23(2), pages 107-131, March.
    5. David K. Backus & Stanley E. Zin, 1994. "Reverse Engineering the Yield Curve," NBER Working Papers 4676, National Bureau of Economic Research, Inc.
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