Consumption and asset prices with recursive preferences: Continuous-time approximations to discrete-time models - Mark Fisher (Federal Reserve Bank of Atlanta, 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.
Keywords: 330; Asset pricing; Consumption (Economics); Interest rates; Wealth