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Assignment 3 for Econ 452

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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 $102, 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 15. 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

elements.

(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

long-term deposits.

(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

control inflation.

(7) Suspension of convertibility was as effective as deposit insurance in preventing bank

runs.

(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.

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(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.

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