Calculate the expected level of consumption

FDIC Insurance Could Have Helped The USA to Avoid the Great recession
Recently, a small bank in Maryland went bankrupt, leaving depositors with only a sign saying the bank was out of business.  Unfortunately, the bank also left the federal government with the requirement of paying off insurance claims through the Federal Deposit Insurance Corporation (FDIC).  Luckily most depositors got their money back.  However, newspaper reporters, policy makers, and depositors were all wondering what went wrong, and why no one noticed that the bank was likely to go out of business.
Consider the following depositor in a world without deposit insurance.  This consumer is looking to deposit $2,500 in a local Maryland Bank.  Without FDIC insurance, the depositor faces two possibilities: the bank closes and the depositor loses all her deposits, or the bank remains open, and the depositor loses nothing.  (Assume no transactions costs, and no interest payments.)
The depositor has the following utility function over bank deposits:
U(deposits) = 100(deposits)½.
a).  What type of risk behavior does this depositor exhibit?
b).  Calculate the expected level of consumption, the expected utility level, and the utility of expected consumption if the probability of the bank going out of business is 0.10.  Calculate mathematically and show graphically.
c).  Now consider the following insurance option.  The FDIC comes to Maryland banks and offers an insurance policy that directly costs the consumer nothing (the consumer supports it through taxes), insuring up to $250,000 of deposits. (This means risk free). Will the consumer deposit at insured or un-insured institutions?  What will the consumer’s level of utility be?
d).  How much would the depositor be willing to pay in a premium for insurance?
e).  Are there any problems associated with FDIC insurance such as moral hazards?  If so what are they?

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