Optimal Investment and Consumption under a Habit-Formation Constraint

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