Suppose an individual consumer has preferences over consumption c given by u(c) = c^1/2 . The individual faces uncertainty of the following form: With probability π the individual has wealth of ω which she can spend on consumption; with probability 1−π the individual suffers a negative shock which reduces her wealth to zero. An insurance company sells insurance against this shock at a per unit premium of p > 0. A unit of insurance pays a single unit of consumption in the bad state. Let z represent the amount of insurance the individual purchases.
(a) Write out the individual’s expected utility.
(b) Write out the individual’s consumption in each state as a function of the amount of insurance she purchases z. Write out the individual’s budget constraint in this case.
(c) Graph the individual’s decision problem in the space of consumption in the good state (i.e., cG when wealth is equal to ω) and the bad state (i.e., cB when wealth is equal to zero).
(d) Find the individual’s optimal consumption (i.e., consumption levels in each state) as a function of p, π and ω.
(e) Suppose the insurance company selling insurance is in a competitive industry. Explain what will happen to the price of insurance and what this will represent for the consumer.