Finally find the locus of zero-profit contracts y1 yi for

Question

Besides education as a signal, the other classic application of these ideas is the insurance market. The following simple story is usually told. Imagine an individual who is subject to a random loss. Specifically, the individual may have Yi with which to consume, and she may have only Y_i Y_i . This individual is strictly risk averse, with von Neumann­ Morgenstern utility function U defined on the amounts ofmoney she has
to consume. There is a probability 1fi that the individual will have income Yi only (and a probability 1-1fi that she will have income Y_i), where the subscript istands for either H or L, mnemonics for high and low risk. We have 7fH > 7fL . The individual knows whether she is a high- or low-risk individual.

Several insurance companies offer insurance against the bad outcome for this consumer. An insurance contract is very simple: It specifies amounts y₁ and y₂ that the individual is left with if she would otherwise be con­ suming Y_i or Y_i . That is, you can think of y₁ = Y_i - P, where P is the insurance premium, and Y_i = Y_i - P + B, where B is a benefit paid in case the individual would otherwise be left with only Y_i. The insur­ ance companies are competitive and risk neutral. This implies that con­ tracts that are offered and taken in equilibrium on average break even,
or 1r(Y_z - y₂) + (1 - 1r)(Y_i - y₁) = 0, where 1r is the chance that the con­ sumer would otherwise be left with Y_i . The insurance companies do not know what type the consumer is, but have equilibrium conjectures based on the type of contract that the consumer proposes or takes. Insurance firms begin with the prior assessment that the consumer is high risk with probability 1/2.

Adapt first Spence and then Rothschild and Stiglitz to this setting.

The key is to get the right "picture," and we will give you some assistance here. Think of the "commodity space" (replacing (w, e) space) as the space of pairs (y₁, Y_i) where y₁ is the amount of money the consumer has to use in the contingency where she would otherwise have Yi and Yi is her wealth in the contingency where she would otherwise have Yi. Begin by locating the consumers endowment (Y_i, Y_i). Then draw in indifference curves ·for the consumer in this space. Note carefully, the indifference curves are different if the consumer is high-risk or low-. It is useful to compute the slope of the consumers indifference curve at a point (y₁, Y_i) where y1 = Y_i. Finally, find the locus of zero-profit contracts (y₁, Y_i) for the insurance furn if it knows that it is dealing with a high-risk client, with a low-risk client, and with a client who is high-risk with probability 1/2.