Optimal Consumption Under a Habit-Formation Constraint

We propose a new optimal consumption model in which the degree of addictiveness of habit formation is directly controlled through a constraint on admissible consumption. In particular, we assume that the individual is unwilling to consume at a rate below a certain proportion 0<α≤1 of her consumption habit, which is the exponentially-weighted average of past consumption rates. α=1 prohibits the habit process to decrease and corresponds to the completely addictive model. α=0 makes the habit-formation constraint moot and corresponds to the non-addictive model. 0<α<1 leads to partially addictive models, with the level of addictiveness increasing with α. In contrast to the existing habit-formation literature, our constraint cannot be incorporated in the objective function through infinite marginal utility, and alters the analysis of the corresponding control problem. Assuming that the individual invests in a risk-free market, we formulate and solve an infinite-horizon, deterministic control problem to maximize the discounted CRRA utility of the individual's consumption-to-habit process subject to the said habit-formation constraint. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (or, equivalently, those with more addictive habits) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (or, equivalently, those with less addictive habits) consume at the minimum rate if their wealth-to-habit ratio is below a threshold, and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either: (1) larger than a high threshold; or (2) below a low threshold and the agent is more risk seeking (that is, less risk averse). Thus, we provide a simple explanation for the consumption hump observed by various empirical studies