  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.
