Additive habits with power utility: Estimates, asymptotics and equilibrium

TL;DR

This paper analyzes a power utility maximization problem with additive habits in discrete-time markets, providing estimates, asymptotic behavior, and equilibrium existence results, including explicit asset pricing in representative agent models.

Contribution

It introduces new estimates for optimal consumption, investigates asymptotic consumption behavior, and proves equilibrium existence in markets with habit formation and heterogeneity.

Findings

Optimal consumption estimates in incomplete markets.

Asymptotic propensity to consume as endowment grows.

Explicit asset prices in a representative agent model.

Abstract

We consider a power utility maximization problem with additive habits in a framework of discrete-time markets and random endowments. For certain classes of incomplete markets, we establish estimates for the optimal consumption stream in terms of the aggregate state price density, investigate the asymptotic behaviour of the propensity to consume (ratio of the consumption to the wealth), as the initial endowment tends to infinity, and show that the limit is the corresponding quantity in an artificial market. For complete markets, we concentrate on proving the existence of an Arrow-Debreu equilibrium in an economy inhabited by heterogeneous individuals who differ with respect to their risk-aversion coefficient, impatience rate and endowments stream, but possess the same degree of habit-formation. Finally, in a representative agent equilibrium, we compute explicitly the price of a zero coupon bond and the Lucas tree equity, and study its dependence on the habit-formation parameter.