Question 1. (78 pts) To analyze bank runs, consider a two-period economy populated
by one billion individuals. An individual is endowed with $1 before period 1 starts
and chooses whether to deposit it in a bank or keep it. If the individual keeps the money,
it will still be $1. The bank offers a demand deposit contract:
(A1) A depositor can withdraw the deposit from the bank at the end of period 1 if she/he
wants. The net interest rate (deposit rate) is 2% on the deposit. If a depositor
withdraws in period 1, she/he gets $102, provided that the bank has some assets. If
the bank’s asset becomes zero, a depositor gets nothing from the time on.
(A2) At the end of period 2, all depositors who have kept their money in the bank divide
the bank’s assets evenly.
The bank lends to businesses or invests in other portfolios. An investment lasts for two
periods and is referred to as a long-term project. If a long-term investment is carried out
to the end of period 2, the net rate of return will be 6%. If a long-term investment is
terminated or liquidated in period 1, only the principal of the investment can be recovered.
All deposits must be made before period 1 starts. The bank does not take deposits at the
end of period 1 or in period 2.
After depositing in the bank, an individual may experience a shock in period 1. If
an individual is hit by the shock, she/he needs money in period 1 but not in period 2.
If an individual is not hit by the shock, she/he will need money in period 2 but not in
period 1. The shocks are independent among individuals and occur to each individual with
probability 15. An individual hit by the shock is called an impatient individual and an
individual not hit by the shock is called a patient individual.
Assume that the bank’s equity or net worth is zero. Also, at the time of making the
deposit decision, all individuals expect that there will be no bank run.
(1) (4 pts) Show that it is optimal for an individual to deposit in the bank rather
than keep the money.
(2) (10 pts) Calculate the amount of money that the bank keeps as reserves in order
to satisfy the withdrawals by impatient depositors. Calculate the amount that the bank
lends to long-term projects.
(3) (15 pts) Assume that everyone expects that all patient depositors will keep their
money in the bank until the end of period 2. Calculate the net rate of return to a patient
depositor. Show that it is indeed optimal for a patient depositor to keep money in the
bank until the end of period 2.
(4) (6 pts) Suppose that in period 1 all individuals realize that the net rate of return
on long-term projects will be 4% instead of 6%. If all patient depositors are expected to
wait until the end of period 2, is it optimal for a patient individual to wait? Support your
answer by calculating the rate of return on waiting.
Go back to the case where the net rate of return on long-term projects is 6%.
(5) (19 pts) Suppose that 300 million patient depositors withdraw in period 1, in
addition to impatient depositors, but the remaining patient depositors will wait until the
end of period 2. Calculate the amount that will be obtained by a patient depositor who
waits. Show that it is indeed optimal for such a depositor to wait.
(6) (24 pts) Let million be the number of patient depositors who withdraw in
period 1, in addition to impatient depositors. Find the critical level of above which it is
optimal for all depositors to withdraw.
Question 2. (2 pts each) For each statement below, answer whether the statement
is true, false, or uncertain for the U.S. economy. Briefly explain your answer.
(1) Bank runs are always caused by irrational fear or panic.
(2) A crisis can happen even in an economy that is sufficiently healthy in the fundamental
(3) According to RR (Reinhart and Rogoff), most defaults on sovereign debt occurred
when the debt payment exceeded a borrowing country’s ability to pay.
(4) Maturity transformation performed by a bank changes short-term deposits into
(5) One role performed by a bank is risk sharing among depositors.
(6) Suspension of convertibility of demand deposits into cash was a policy used to
(7) Suspension of convertibility was as effective as deposit insurance in preventing bank
(8) The evidence in RR shows that abnormally high capital inflows into a country
increase the probability that the country will have a banking crisis.
(9) The moral hazard problem induced by deposit insurance is that a bank does not
screen the depositors.
(10) The problem of “too big to fail” refers to the situation where a sufficiently large
financial institution has enough funds to meet all depositors’ withdrawals.
(11) Bank-run phenomena can occur only to commercial banks that take deposits.