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

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.