The Reinsurer's Monopoly and the Bowley Solution

The reinsurer has a monopoly in the following sense: He will select a random variable P that determines the reinsurance premiums. The first insurer can purchase a payment of R (a random variable) for a premium of pi = E[PR]. For known P, the first insurer chooses R to maximize his expected utility. Knowing this, i.e., the demand for reinsurance as a function of P, the reinsurer chooses P to maximize his utility. The resulting pair (P, R) is called the Bowley solution. Assuming exponential, quadratic and/or linear utility functions, some explicit results are obtained.