Optimal Investment and Consumption under a Habit-Formation Constraint

TL;DR

This paper develops an explicit model for optimal investment and consumption considering habit formation, analyzing how addictive and nonaddictive habits influence financial decisions and market behavior.

Contribution

It introduces a novel habit-formation model with explicit solutions for investment and consumption policies under different habit types.

Findings

Addictive habits require more wealth to sustain the same consumption.

Individuals with addictive habits invest less in risky assets.

The model explains the equity-premium puzzle through habit effects.

Abstract

We formulate an infinite-horizon optimal investment and consumption problem, in which an individual forms a habit based on the exponentially weighted average of her past consumption rate, and in which she invests in a Black-Scholes market. The individual is constrained to consume at a rate higher than a certain proportion $\alpha$ of her consumption habit. Our habit-formation model allows for both addictive ($\alpha=1$) and nonaddictive ($0<\alpha<1$) habits. The optimal investment and consumption policies are derived explicitly in terms of the solution of a system of differential equations with free boundaries, which is analyzed in detail. If the wealth-to-habit ratio is below (resp. above) a critical level $x^*$, the individual consumes at (resp. above) the minimum rate and invests more (resp. less) aggressively in the risky asset. Numerical results show that the addictive habit formation requires significantly more wealth to support the same consumption rate compared to a moderately nonaddictive habit. Furthermore, an individual with a more addictive habit invests less in the risky asset compared to an individual with a less addictive habit but with the same wealth-to-habit ratio and risk aversion, which provides an explanation for the equity-premium puzzle.