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The decision-making interaction between the principal and the agent can be described and modelled in the following way. Let us suppose that the agent is deciding about a decision variable e and that this decision determines the outcome x = x(e, θ), where θ is a random variable with a known probability distribution.
If the principal is fully in formed about the possible alternative decisions and their implications, he can force the agent to make the decision that is optimal from the principal's point of view. In the case of decision-making under uncertainty (as is usual in the real economy), the agent usually has information superiority (so-called hidden information), for example about the available technology. A common feature of principal-agent situations is the endeavour of the principal to prevent the agent from abusing this information superiority to the principal's detriment.
The principal knows that this hidden information θ is known to the agent, but he cannot get it from him. The agent makes his decision with knowledge of the value of θ, while the principal can observe neither e nor θ. The principal can therefore try to convince the agent to reveal true value of θ.
If, for example, an insured party — unlike the insurance company — knows his risks, this can lead to "adverse selection": commercial insurance tends to be used above all by customers who are unprofitable from the point of view of the insurance company. This is because customers — unlike the insurance company — have information about their accident probability. The uninformed insurance company has to choose the halfway (averaged) price, which is attractive for the riskiest customers but deters the least risky ones. The insured are therefore more risky than the average population. This applies not only to the insurance market — many new customers of dating agencies have found that the supply of potential partners offered by these agencies is ceteris paribus worse than that in the average population. Generally, adverse selection can be described as a situation in which "undesirable” customers (from the point of view of the principal) are more likely to participate in voluntary exchange.
For simplicity we assume the same initial income of yi = y for all n agents (insured parties) (i = 1, ..., n). All agents are considered to be risk neutral with an identical utility function vi(y) = y. Moreover, those agents face an identical loss Li = L. The size of the loss L here (unlike in the moral hazard model discussed in the next section) is an exogenous variable which the agent cannot influence. We also assume that among the customers there are:
• (1 - γ) • n high-risk individuals who have a high accident probability πh,
• γ • n low-risk individuals with a low accident probability πl, where πl < πh.
Suppose that the principal (insurance company] is risk neutral and that he knows the shape of the agent's utility function v(y) =y and his initial income y, the size of the potential loss L, the proportion of low-risk customers y (0; 1) and the accident probabilities of both types of customers πh and πl, but he is not able ex ante to recognize the type of customer. This prevents the principal from charging high-risk types a premium of ph= πh • L and low-risk types a premium of pi = πl • L; such insurance would correspond to the expected loss and, under perfect information, would be Pareto optimal for all parties.
Moreover, if the insurance company offered the "halfway” insurance premium given by the weighted average of the optimal premium for high-risk and low-risk individuals (with the weights given by the ratio of high-risk to low-risk agents], i.e. a premium of
it could be sure that only the high-risk agents would accept the offer (i.e. adverse selection would occur] and that it would make a loss. This is true for every "pooling” (same for all agents] contract. There cannot be a competitive equilibrium with a pooling contract — in the competitive equilibrium the insurance company neither generates a loss nor loses customers to its competitors, so the expected profit for a randomly chosen customer has to be equal to zero. Competitive equilibrium therefore has to generate zero profit, which cannot be true for any pooling contract
In this case the insurance company has to offer two different ("separating") insurance contracts in such a way that would "unmask" the agent's choice of contract. These insurance contracts will differ not only in premium amount, but also in the level of coinsurance. A high-risk customer will incline towards the contract with lower coinsurance and thus with higher insurance cover.
Let us characterize these two insurance contracts by binary vectors (pl, ql) (the contract designed for low-risk customers] and (ph, qh) (the contract designed for high-risk customers). The first component of the vector describing the insurance contract represents the premium amount and the second component represents the amount the insured party will receive in the event of an accident We will denote the utility of the agent connected with contract (p, q) by V(p, q).
For the insurance company it is necessary that its customers choose the contracts that are designed for them. To ensure this the following "self-selection" constraints have to be fulfilled:
The second of these constraints can be omitted as inactive. It would be profitable from the insurance company's point of view for low-risk agents to choose the "stricter" contract, but rational agents will choose the more favourable contract (pl, ql). The insurance company therefore has to secure itself only against high-risk customers, so only the first constraint is active:
This constraint should ensure that high-risk agents do not accept the contract which gives an advantage to low-risk agents. The self-selection constraint here — given the agents' assumed risk neutrality, i.e. v(y) = y — is therefore:
On the other hand, competition leads to the insurance company offering the best (from the point of view of the high-risk agent) contract with zero expected profit; otherwise a rival insurance company would take away its customers.
The expected utility of a principal that does not know the risk profile of its customer is:
By maximizing this function subject to the self-selection constraint it can be proved that for the principal the best separating contract for high-risk customers is
which is the same as the contract under perfect information — full cover at a price equal to the expected loss. The optimal separating contract for low-risk agents has to have only partial cover (q1 < L) and its premium is equal to the expected value of the insurance contract:
However the existence of such a competitive equilibrium is not assured. It depends on the proportion of low-risk customers y. If it is higher than some trigger value γ* there will be a pooling equilibrium (where the insurance company offers one contract to all) with zero expected profit and this equilibrium will drive separating contracts from the market. Nevertheless, as discussed above, no pooling contract can be an equilibrium. Thus, no competitive equilibrium exists in this case.
If γ Î (0; γ*] the existence of a competitive equilibrium is assured with suitable separating contracts. However, even here this equilibrium is not Pareto optimal. Compared with the case of perfect information, insurance companies are equally well off (zero profit], high-risk agents are not better off either, but low-risk agents are in a worse situation, so the equilibrium cannot be Pareto optimal. Therefore, either a competitive equilibrium does not exist at all, or it is not Pareto optimal.