There are two groups of individuals of equal size, each with a utility
function given by U(M) = sq. root(M), where M = 100 is the initial wealth
level for every individual. Each member of group 1 faces a loss of 36
with probability 0.5. Each member of group 2 faces the same loss with
probability 0.1.
(a) What is the maximum amount of money a member of each group
would be willing to pay to insure against this loss?
(b) Assume that it is impossible to discover which individuals belong
to which group. Will members of group 2 insure against this loss in
a competitive insurance market, where insurance companies o¤er
the same contract to everybody? Explain your answer.
(c) If insurance companies anticipate the result of part (b), what type
of contract will they o¤er in a competitive insurance market?
(d) Now suppose that the insurance companies have an imperfect test
for identifying which individual belongs to which group. If the test
says that a person belongs to a particular group, the probability
that he/she really does belong to that group is p < 1. How large
must p be in order to alter your answer to part (b)?
(a) To find the maximum amount of money a member of each group would be willing to pay to insure against the loss, we need to calculate the expected utility with and without insurance for each group.
For group 1:
- Without insurance: The expected wealth after the loss is (1 - 0.5) * (100 - 36) = 32.
- With insurance: Assuming they fully insure against the loss, they would pay a premium, let's say x, to receive the initial wealth of 100 back if the loss occurs.
- If the loss occurs (probability 0.5), their wealth will be 100 - x - 36.
- If the loss doesn't occur (probability 0.5), their wealth will be 100 - x.
- The expected wealth with insurance is given by 0.5 * (100 - x - 36) + 0.5 * (100 - x).
We can now calculate the utility for group 1 with and without insurance:
- Utility without insurance: U(32) = sq. root(32)
- Utility with insurance: U(0.5 * (100 - x - 36) + 0.5 * (100 - x))
To find the maximum amount they would be willing to pay, we equate the two utilities:
sq. root(32) = U(0.5 * (100 - x - 36) + 0.5 * (100 - x))
Solving this equation will give us the maximum amount x.
We repeat the same process for group 2, but with the probabilities and loss values in their context.
(b) To determine if members of group 2 would insure against this loss in a competitive insurance market, where insurance companies offer the same contract to everybody, we need to compare their expected utilities with and without insurance.
If it is impossible to discover which individuals belong to which group, members of group 2 will not have any incentive to insure against the loss. This is because their expected loss is lower than that of group 1 (probability 0.1 vs. 0.5) and hence their expected utility without insurance is already higher. They would not want to pay any additional premium for insurance.
(c) If insurance companies anticipate that members of group 2 will not insure against the loss, they will likely offer contracts that are more favorable to group 1. This is because group 1 faces a higher probability of loss and is more willing to pay for insurance. So, in a competitive insurance market, the insurance companies will likely offer contracts with higher premiums and better coverage to group 1.
(d) If the insurance companies have an imperfect test for identifying which individual belongs to which group, and the test says that a person belongs to a particular group with a probability of p < 1, then it would alter the answer to part (b).
In this case, members of group 2 may consider insuring against the loss, as there is a probability p that they may be misclassified as group 1, and hence will face a higher probability of loss. The lower the probability p, the more likely it is for group 2 to insure against the loss in order to protect themselves against a potential misclassification.
To determine the exact threshold value of p that alters the answer, we would need to repeat the calculations done in part (a) and compare the expected utilities with and without insurance for different values of p.