Insurance Model for High and Low-Risk Consumer - Essay Example

?We consider a model with a single firm and a large number of rational risk-averse consumers with standard preferences. The consumers have an initialwealth W and there is a possibility of an accident that causes damage D. We shall set up the insurance model as one of trade where the firm offers the consumers state contingent claims in return for their initial endowments. Consumers The consumers can be of two types, namely high risk (H) and low risk (L). These two groups of consumers have the same initial endowments e, which essentially are state contingent claims to consumption. These consumers are susceptible to an accidental loss in future. The consumers have standard preferences defined over consumption. If P is the probability of the loss, then the consumers expected utility is: Thus, we can have the following indifference curve: It is simple to show that this leads to negatively sloped convex indifference curves. The slope of the indifference curves are: The high risk and low risk groups differ in their probabilities of incurring the loss. The probability of accident of an individual consumer belonging to the high risk group is PH and that of one belonging to the low risk group is PL, where PH> PL. Figure 1 below shows the indifference curves for a particular utility level for representative agents from the two groups. Observe that since PH> PL the indifference curves for the high risk type will have flatter slopes (less negative). Figure 1: Indifference curves for the high risk and the low risk consumers The monopolist The monopolist’s objective is to maximize its expected profits or alternatively minimize its expected costs by trading with the consumer. The monopolist offers a pair of contingent claims (G,B) which realize in the good (No loss) and bad (loss) states in return for the consumers initial endowment. The expected costs of the monopolist are equal to: We can form the Iso-cost function for the monopolist as follows: Evidently, these are straight lines with a slope of . Observe that since PH> PL the Iso-cost line for the high risk type will have a flatter slope (less negative). Thus, the iso-cost lines for the High risk type and the low risk type can be drawn as follows: Figure 2: The iso-cost lines for the monopolist insurer for high risk and low risk contracts – C(H) represents the iso cost line for the high risk types and C(L) represents the isocost line for the low risk type. The separating equilibrium under asymmetric information Recall that asymmetric information is a situation where one or some of the players of the game have private information. In the present context the asymmetric information is manifested in the form of consumers having private information since they know whether they belong to high risk or low risk groups. The firm does not know any particular agents type. However, the monopolist is perfectly aware of the exact probability distribution of consumer types. A separating equilibrium in the present context would be one where the high risk types choose a contract that is different from the contract chosen by the low risk types. The monopolist firm’s objective is to minimize its costs subject to the participation constraint or the individual rationality constraint and the incentive compatibility constraint of the consumer. The participation constraint requires that the contract offered by the firm provides him at least as much expected utility as the consumer’s initial endowment. This implies that for any consumer to accept the firms offer, the contract has to lie on or above the indifference curve through the initial endowment. The incentive compatibility constraint on the other hand requires that consumers of either type do not find it beneficial to accept the contract devised for the other type. It is essentially the satisfaction of this constraint that leads to the separating equilibrium. In terms of indifference curves, the incentive compatibility constraint requires that the contract for the high type lies on or below the low types indifference curve through the initial endowment and that the contract for the low type lies on or below the high type’s indifference curve through the initial endowment. We show the separating equilibrium in the diagram below (figure 3). Figure 3: The separating equilibrium A is the high risk contract while B is the low risk contract. Observe that the participation constraint is binding for the low risk agent but does not bind for the high risk agent. In other words, while the low risk contract lies on the indifference curve through the initial endowment point E, the high risk contract lies on a higher indifference curve compared to the one through the initial endowment point. The incentive compatibility constraint is binding for both types. Note that contract A lies on a lower indifference curve for the low risk agents compared to the indifference curve through contract B. Contract B and A generate the same level of utility for high risk agents. Thus, neither type can gain any additional utility by accepting the other’s contract. Finally note that the high risk type is perfectly insured while the low risk type is imperfectly insured. This is true since the high type’s contract lies on the 45 degree line while the low type’s lies below the 45 degree line. Full information and separating equilibrium Under perfect information, since the types of the consumers are verifiable the insurer does not have to resort to using incentive compatibility constraints. The monopolist insurer can maximize profits/minimize costs subject to only the participation constraint. The resulting equilibrium is shown in the diagram below (figure 4). Figure 4: Full information contracts Under full information, the monopolist will know which type of consumer is seeking insurance. He will thus pick the lowest iso-cost line accessible subject to the consumer’s indifference curve through his initial endowment point. The tangency of the iso-cost lines with the indifference curves occur on the 45 degree line. Thus, both agents are perfectly insured in this instance. Therefore, in conclusion, while under asymmetric information a monopolist insurer offers contracts which provide full insurance only for the high risk type, under perfect information, both types are fully insured. Additionally, as can be verified from the diagram, in case of the low-risk contract, the monopolist is able to reach a lower iso-cost line under full information implying a loss in profits due to asymmetric information. Read More