Please see the attached file.

Consider two individuals living in the country of Depressia. Depressian banking system is very unstable. On average, 5% of all banks collapse every year. Each of these individuals is considering depositing all their life savings - $10,000 - in a bank for one year. Banks pay fixed interest, so that by the end of the year all would-be bank clients should have their principals back plus interest. If the bank goes bankrupt all money is lost. The only alternative way of making savings is simply by keeping money (cash) in a piggy bank. (There is zero inflation in Depressia).

One of these individuals is risk-neutral, another is risk-averse with the utility of money (wealth) given by U(I) = I0.5.

a) Will either of these individuals be willing to deposit their savings if banks pay 5% annual interest?
For risk neutral individual
Expected Utility from investment = 10000*1.05*0.95+0.05*0=9975
Utility from keeping money = 10000
Since utility from keeping money is more, he will not invest.

For risk averse individual
Expected Utility from investment = (10000*1.05)^0.5*0.95+0^0.5*0.05=97.35
Utility from keeping money = 10000^0.5=100
Since utility from keeping money is more, he will not invest.

Hence nobody with invest at 5% annual interest rate.

b) Will either of these individuals be willing to deposit their savings if banks pay 8% annual interest?
For risk neutral individual
Expected Utility from investment = 10000*1.08*0.95+0.05*0=10260
Utility from keeping money = 10000
Since utility from investment is high, he will invest.

For risk averse individual
Expected Utility from investment = (10000*1.08)^0.5*0.95+0^0.5*0.05=98.73
Utility from keeping money = 10000^0.5=100
Since utility from keeping money is more, he will not invest.

Hence, only risk neutral individual will invest at 8% annual interest rate.

c) Depressian government introduces tougher banking regulations that result in the probability of bankruptcy decreasing to 2.5%. How will this change your answers to (a) and (b)?
At 5% interest rate
For risk neutral individual
Expected Utility from investment = 10000*1.05*0.975+0.025*0=10237.50
Utility from keeping money = 10000
Since utility from investment is high, he will invest.

For risk averse individual
Expected Utility from investment = (10000*1.05)^0.5*0.975+0^0.5*0.025=99.91
Utility from keeping money = 10000^0.5=100
Since utility from keeping money is more, he will not invest.

Hence, only risk neutral individual will invest at 5% annual interest rate.

At 8% interest rate
For risk neutral individual
Expected Utility from investment = 10000*1.08*0.975+0.025*0=10530
Utility from keeping money = 10000
Since utility from investment is high, he will invest.

For risk averse individual
Expected Utility from investment = (10000*1.08)^0.5*0.975+0^0.5*0.025=101.33
Utility from keeping money = 10000^0.5=100
Since utility from investment is high, he will invest.

Hence, both risk neutral and risk averse individuals will invest at 8% annual interest rate.

d) If the risk-averse individual decides to deposit his savings at 8% interest in these new conditions (i.e. when the risk of bankruptcy is only 2.5% per year), will he be interested in buying an insurance against bankruptcy (deposit insurance)? If yes, how much will he be willing to pay (at most) for such an insurance policy? Will a reasonable private insurance company be willing to offer deposit insurance policies?

Yes, the risk-averse individual would be willing to take the insurance for the deposits. The insured amount would be 10000+ interest (10800). Let X is the amount of insurance premium he is willing to pay, then the maximum value of X should be such that the utility of investment is same as keeping cash.
So we have
Expected Utility from investment = (10000*1.08-X)^0.5*0.975-X^0.5*0.025
Utility from keeping money = 10000^0.5=100
Equating both we get
(10800-X)^0.5*0.975- X^0.5*0.025=100
Solving we get X=383.33
Thus, the risk averse person is willing to pay a maximum of 383.33 for the deposit insurance.

Since insurance companies are risk neutral, the expected value of loss should be less than the risk premium.
Expected value of loss = 0.025*10800=270
Since the expected value of loss is less than the risk premium the risk averse depositors is willing to pay, hence private insurance company would be willing to sell the insurance. The deal can be done between 270 to 383.33.

e) What problems may arise in the market for deposit insurance? Discuss. Why do you think (based on the preceding discussion) deposit insurance is offered in the US by the government and not by private companies?
If the deposit insurance is available, there will be problems of adverse selection. Since, there is large number of banks, the bank offer high interest rate in order to attract depositors. The higher the interest rate offered by the bank, higher is the probability of its failure, as it would invest its money in more risky assets. Thus, investors would chose only those banks which are paying higher interest rate and are not bothered whether these banks have higher risk. Thus, insurance companies will end up with insurance for only those deposits which are highly risky.

The second problem will be moral hazards. As their deposits are insured, they would not bother about checking the credentials of the bank (due diligence) before making a deposit. Thus, the investors would be making poor investments.

Since government is the regulatory body for the banks, it can put regulations for the working of the banks and can check whether the banks are following the regulations. This is not feasible for the private companies. Hence, the possibility of moral hazards and adverse selection is reduced to a large extent.

f) Explain why deposit insurance was impossible before the introduction of tougher banking regulation, i.e. when the probability of a bankruptcy in Depressia was 5%? (Consider the case of 8 percent interest paid by the banks.)

Let X is the amount of insurance premium the risk averse individual is willing to pay, then the maximum value of X should be such that the utility of investment is same as keeping cash.
So we have
Expected Utility from investment = (10000*1.08-X)^0.5*0.95-X^0.5*0.05
Utility from keeping money = 10000^0.5=100
Equating both we get
(10800-X)^0.5*0.95- X^0.5*0.05=100
Solving, we could not get a feasible solution as value of X would be -ve. Since no insurance company can offer insurance at -ve premium, insurance is not possible.