This paper investigates the properties of equilibrium in insurance markets where insurers can obtain specific and private knowledge of the loss experience of their customers. We examine the case where firms obtain information over time from insurance claims and use these data in a Bayesian fashion to adjust individual premiums to experience. 

We first consider the case where firms can change their premiums from one period to the next and customers are free to stay or leave as they see fit. We refer to this case as a single period equilibrium. The resulting premium schedule earns monopoly profits even if entry by new firms into the insurance market is perfectly free. We next investigate a myopic multi- period equilibrium, in which firms maximize the present value of the stream of expected profits over the period in which the individual is insured, but individuals select the firm offering the lowest premiums. With free entry, expected profits are zero but premiums are generally too low or too high relative to actuarial values. We also investigate the properties of a full multi- period equilibrium where insurance firms specify premiums in advance for all future periods as a function of the number of claims that a customer has made within a given time span. In this type of equilibrium long-run profits are zero and insurance firms consciously charge actuarially unfair premiums to some of their customers. 

Although these models are illustrated in an insurance context, they also apply to other situations as well, notably labor markets. The concluding sections briefly explores these extensions and draws out lessons for regulatory policy.