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Consumption and asset prices with recursive preferences: Continuous-time approximations to discrete-time models

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  • Mark Fisher

Abstract

This paper presents tractable and efficient numerical solutions to general equilibrium models of asset prices and consumption where the representative agent has recursive preferences. It provides a discrete-time presentation of the approach of Fisher and Gilles (1999), 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 convergence 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 in 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.

Suggested Citation

  • Mark Fisher, 1999. "Consumption and asset prices with recursive preferences: Continuous-time approximations to discrete-time models," FRB Atlanta Working Paper 99-18, Federal Reserve Bank of Atlanta.
    • Handle: RePEc:fip:fedawp:99-18
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    References listed on IDEAS

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    1. 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.
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    3. 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.
    4. Campbell, John Y, 1993. "Intertemporal Asset Pricing without Consumption Data," American Economic Review, American Economic Association, vol. 83(3), pages 487-512, June.
    5. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, December.
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