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If the economic survival of the agent depends on the survival of the principal then there is an overlap of interests of the principal and the agent — an overly "defrauded" principal could suffer economically and even go bust. The agent thus follows [but does not maximize] the interests of the principal in his own interest.
In this section we assume that survival of the principal is a necessary condition for survival of the agent. This does not hold the opposite way round — the principal survives even if the agent does not.
We assume that the agent maximizes his probability of economic survival, while the principal endeavours to achieve at least zero expected income connected with the contract. This is a condition for competitive equilibrium, which would occur in the case of perfect information.
As in the previous sections, however, we assume information asymmetry. The agent knows his accident probability, but the principal does not.
The principal supplies the agents with accident insurance in return for payment (the price of the contract p). We differentiate between two types of agents: high-risk and low-risk.
Similarly as in the previous section:
yp denotes the principal's initial level of income,
ya denotes the agents initial level of income,
L denotes the size of the potential loss,
γ Î (0; 1) denotes the proportion of low-risk agents with low accident probability π1,
πh denotes the accident probability for high-risk agents, where π1 < πh,
ba denotes the agents extinction zone boundary,
bp denotes the principal's extinction zone boundary.
Let us assume again that the principal is not able recognize ex ante the type of agent he is dealing with; he only knows the proportion of high-risk agents Y Î (0; 1). If the principal offers a pooling contract with price p and with full cover, his expected income from the contract is:
The principal's probability of survival is
We assume that the agent cannot survive alone — if the principal perishes, the agent perishes as well:
An agent maximizing the probability of parallel survival of both entities prefers neither an extreme decrease nor an extreme increase in the price of the contract, because:
An increase in the price of the contract p increases the first and decreases the second multiplicand in (*). The agent therefore prefers price p* for which:
The important point here is that both the high-risk and the low-risk agents evaluate the situation in the same way. Both are fully insured against accidents, so they have the same second multiplicand in (*). Any accident will decrease their probability of economic survival only as a consequence of a decrease in the probability of survival of the principal. This probability will decrease by percentage points from. The low-risk agents do not leave system any more than the high-risk agents.
This means that if the survival of the agent is contingent on the survival of the principal, then the problem of adverse selection does not exist for entities maximizing their probability of economic survival.
There is even in some sense an opposite tendency. In the text below we will show that entities with well-below-average income do not demand insurance services when payment of the insurance premium increases their probability of economic extinction even if no accident occurs. An increase in the premium thus leads to an increase in the "reliability” of the insured over the average of the population (and does not decrease it below the average as in the standard adverse selection model). This is true despite the fact that the insurance is not advantageous even for the extremely well off, because they are very insensitive to changes in the premium.
An increase in the premium increases the expected income of the principal. The principal will therefore be better off if the price of the contract increases. By contrast, the number of surviving agents and demand in the insurance market are decreasing in the range (0; p*), whereas they are "modestly” increasing or constant in the range (p*; +∞). This means that an equilibrium does not necessarily exist This is so when S(p*) < D. If an equilibrium premium does exist, then it is unique. At this price the expected profit of the principal is equal to zero.
The principal will offer full cover at a price that is higher for the low-risk agent than it would be in the case of perfect information. That notwithstanding, the agent accepts this price — from the point of view of his economic survival even the low-risk agent will be better off under these conditions.
Consequently, a competitive Pareto equilibrium may exist — neither the principal, nor the low-risk agents — and still less the high-risk agents — are worse off relative to the perfect information case.
In future research we will relax the strong very assumption (*), for example in the sense that extinction of the principal only cancels the accident cover. Such a change will of course generate significant changes in the conclusions of the analysis. Nevertheless, we do not expect this to rule out the possibility of the existence of a Pareto equilibrium.