Optimal Consumption‐Portfolio Policies With Habit Formation1

Jerome Detemple and Fernando Zapatero

Mathematical Finance, 1992, vol. 2, issue 4, 251-274

Abstract: This is a companion paper to the authors ‘Asset Prices in an Exchange Economy with Habit Formation” in Econometrica which focuses on consumption demand and asset pricing when preferences are habit forming. Here we prove existence of optimal consumption‐portfolio policies for (i) utility functions for which the marginal cost of consumption (MCC) interacts with the habit formation process and satisfies a recursive integral equation with forward functional Lipschitz integrand and (ii) utilities for which the MCC is independent of the standard of living and satisfies a recursive integral equation with locally Lipschitz integrand. Result (i) is demonstrated here for the first time. Result (ii) is novel and enables us to consider Cobb‐Douglas utilities without placing lower bounds on the system of Arrow‐Debreu prices. We also review and extend our earlier results in the linear case; in particular, we provide new insights about the structure of optimal portfolios. Additional new features of the model include the possibility of finite marginal utility of consumption at zero and habit formation mechanisms with stochastic coefficients. an extension to a financial market model with general processes is outlined. A byproduct of the analysis is a set of fixed‐point theorems for recursive integral equations with forward functional Lipschitz or locally Lipschitz integrands.

Date: 1992
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