# Atlanta Fed Working Papers

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

#### Mark Fisher Federal Reserve Bank of Atlanta Working Paper 99-18 November 1999

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.

JEL classification: G12

Key words: recursive preferences, general equilibrium, optimal consumption, term structure of interest rates, asset pricing, Bellman's equation

The author thanks Christian Gilles for his collaboration on Fisher and Gilles (1999), from which this paper heavily draws, and for additional conversations. He also thanks Dan Waggoner for helpful conversations. The views expressed here are the author's and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the author's responsibility.

Please address questions regarding content to Mark Fisher, Senior Economist, Research Department, Federal Reserve Bank of Atlanta, 104 Marietta Street, NW, Atlanta, Georgia 30303-2713, 404/498-8757, mark.fisher@atl.frb.org.