We consider an infinite-horizon optimal consumption problem for an individual who forms a consumption habit based on an exponentially-weighted average of her past rate of consumption. The novelty of our approach is in introducing habit formation through a constraint, rather than through the objective function, as is customary in the existing habit-formation literature. Specifically, we require that the individual  consume at a rate that is greater than a certain proportion \alpha\le1 of her consumption habit . Our habit-formation model allows for both addictive ( \alpha=1 ) and non-addictive ( 0<\alpha<1 ) habits. Assuming that the individual invests in a risk-free market, we formulate and solve a 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 (those with sufficiently large utility discount rates) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (those with small utility discount rates) 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  .
We propose a new optimal consumption model in which the degree of addictiveness of habit formation is directly controlled through a consumption constraint. In particular, we assume that the individual is unwilling to consume at a rate below a certain proportion 0<\alpha\le1 of her consumption habit , which is the exponentially-weighted average of past consumption rates. \alpha=1 prohibits the habit process to decrease and corresponds to the completely addictive model. \alpha=0 makes the habit-formation constraint moot and corresponds to the non-addictive model. 0<\alpha<1 leads to partially addictive models, with the level of addictiveness increasing with \alpha. In contrast to the existing habit-formation literature, our constraint cannot be incorporated in the objective function through infinite marginal utility. 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 consumption-to-habit process subject to the habit-formation constraint. Optimal consumption policies are derived explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals 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 less risk averse. Thus, we provide a simple explanation for the consumption hump observed by various empirical studies .
R
consider an infinite-horizon optimal consumption problem for an individual who forms a consumption habit based on an exponentially-weighted average of her past rate of consumption. The novelty of our approach is in introducing habit formation through a constraint, rather than through the objective function, as is customary in the existing habit-formation literature. Specifically, we require
propose a new optimal consumption model in which the degree of addictiveness of habit formation is directly controlled through a consumption constraint. In particular, we assume
A
is unwilling to
R
that is greater than
below
R
\alpha
0<\alpha
R
. Our habit-formation model allows for both addictive (
, which is the exponentially-weighted average of past consumption rates.
R
) and
prohibits the habit process to decrease and corresponds to the completely addictive model. \alpha=0 makes the habit-formation constraint moot and corresponds to the
R
(
model.
R
) habits.
leads to partially addictive models, with the level of addictiveness increasing with \alpha. In contrast to the existing habit-formation literature, our constraint cannot be incorporated in the objective function through infinite marginal utility.
R
a
an infinite-horizon,
D
individual's
D
said
R
We derive the optimal consumption policies
Optimal consumption policies are derived
D
individuals (those with sufficiently large utility discount rates)
D
(those with small utility discount rates)
A
. 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 less risk averse. Thus, we provide a simple explanation for the consumption hump observed by various empirical studies