This paper analyzes a model of a dynamic monopolist who produces at constant unit cost. Each period a new cohort of consumers enters the market. Each entering cohort is identical. Consumers within a cohort have different tastes for the good. My main results are: If players are sufficiently patient, any positive average profit less than the maximum feasible level can be attained in a subgame-perfect equilibrium; in the subset of subgame-perfect equilibria in which players use stationary strategies, the seller cannot make sales at prices significantly greater than the lowest willingness to pay when period length goes to zero; and the seller attains the maximum profit when commitment is feasible by charging the same (static monopoly) price in every period.