DEDUCTIBLE INSURANCE AND NONLINEAR PRICING

In this paper the problem of pricing deductible insurance is studied in the case where individuals are allowed to self-select their level of deductibility. The model uses a continuous state system allowing for both full and partial losses with a slight modification to include positive probabilities for the events "a full loss occurs" and "no loss occurs." These mass points in the density function are shown to be a determining factor in the choice of a policy by the consumer. In choosing an optimal policy the consumer decides how to allocate his initial wealth between consumption purchases, insurance purchases, and savings. This optimal choice depends directly upon the consumer's degree of risk aversion and his loss probabilities. Other things being equal, consumers who are either more risk averse, more prone to incur a loss or possess less initial wealth buy more insurance. Also the consumer's choice is dependent upon the set of available premiums. If a different set of premiums is made available the consumer will most likely alter his insurance purchases. The optimal contract under various premium schedules is determined. The conditions for choosing an optimal insurance contract yield self-selection properties which will be used by insurers in determining an optimal (expected profit maximizing) price.;Even without perfect information partial discrimination is possible via the consumers' self-selection properties. Here it is assumed that insurers know only the distribution of consumer characteristics. If all individuals possess equal loss probabilities (as might well be the case with insuring against certain types of catastrophic loss for example) then they all share identical actuarial values so that a price based on this actuarial value is possible. In the competitive case this is shown to again lead to actuarially fair pricing. However, if individuals differ in their loss probabilities actuarially fair pricing might not be obtainable. This problem is analogous to the Rothschild-Stiglitz adverse selection problem. Although in general an actuarially fair price cannot be guaranteed, necessary and sufficient conditions are established for the existence of such a price.;The optimal pricing conditions for a monopolistic insurer are determined using variational techniques. The optimal pricing rule obtained is shown to be a variation of Baumol and Bradford's inverse elasticity rule for optimal departures from marginal cost pricing. Some special monopoly cases are considered where the set of admissible premium schedules is restricted to premiums of a certain type such as a fixed percentage mark-up over actuarial value. The case of imperfectly competitive insurance markets is discussed briefly and shown in general to be analogous to the perfectly competitive situation.;Society is viewed as a continuum of risk averse individuals. If an insurance company had perfect information whereby it knew each individual's characteristics it could perfectly discriminate by offering each individual a different set of available premiums. In a competitive insurance market this would lead to actuarially fair pricing in which case all individuals would choose to purchase full coverage contracts. If the insurer is a monopolist it would sell each individual a full coverage contract at a price equal to the sum of his actuarial value and his risk premium.