Consider a market for disability insurance. Individuals in the economy become di

Question

Consider a market for disability insurance. Individuals in the economy become disabled
with probability q. When healthy, they earn $100, and when disabled their wage falls to $0.
Individuals can purchase insurance from private rms operating in a perfectly competitive
market. Insurance contracts cost p up-front, and provide $100 in the event of (perfectly-
observable) disability. There are three types of people, characterized by a probability of
getting disabled q and a utility function over consumption c:
Type 1: q1 = 60% and U(c) =c^(1/2)

Type 2: q2 = 20% and U(c) =c^(1/2)

Type 3: q3 = 10% and U(c) = c

There are 50 people of each type.
(a) Explain why only type I and type II would bene t from insurance.

Explanation / Answer

A healthy person earns $100. When a person is disabled, he earns nothing but if his disability perfectly observable, he gets insurance of $100.

Given the probability of disability q = 0.6, the expected wealth for type 1 people is:

E(W1) = 0.6*(100) + 0.4(100) = $100

Given the probability of disability q = 0.2, the expected wealth for type 2 people is:

E(W2) = 0.2*(100)+ 0.8(100) = $100

Given the probability of disability q = 0.1, the expected wealth for type 3 people is:

E(W3) = 0.1*(100) + 0.9(100) = $100

This implies that the expected wealth for every type is same.

Now find the expected utilities:

Given the probability of disability q = 0.6, the expected utility for type 1 people is:

E(U1) = 0.6*(100)1/2 + 0.4(100)1/2 = 10

Given the probability of disability q = 0.2, the expected utility for type 2 people is:

E(U2) = 0.2*(100)1/2+ 0.8(100)1/2 = 10

Given the probability of disability q = 0.1, the expected utility for type 3 people is:

E(U3) = 0.1*(100) + 0.9(100) = 100

The certainty equivalent wealth is the certain wealth that gives a person the same expected utility as the uncertain certain he starts out.

So the CE for type 1 is E(W1) - E(U1) = 100 - 10 = $90

CE for type 2 is E(W2) - E(U2) = 100 - 10 = $90

CE for type 2 is E(W3) - E(U3) = 100 - 100 = $0

Since the CE for type 3 people is zero, they are not benefitting from insurance.

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