Sharif University of Technology
Graduate School of Management and Economics
44710 - Macroeconomics I
Problem Set 10
* Kurlat: Ch 10, Ch 11
** Romer: Ch 12.3, Ch 12.10
*** Mishkin: Ch 15, Ch 19, Ch 20, Ch 21, Ch 22
Walsh: Ch2
Nili: Ch 2
CSV: Ch 19
Mankiw: Ch 4
DLS: Ch 4, Ch 8
GLS: Ch 20
1 Money-In-Utility Theory
Suppose an economy is made up of one representative household. The total
utility for this representative household is given by:
∞
∑ 𝛽 𝑡 𝑈(𝑐𝑡 , 𝑚𝑡 , 𝑙𝑡 )
𝑡=0
Subject to the period-t Intertemporal budget constraint:
𝑃𝑡 𝑐𝑡 + 𝐵𝑡 + 𝑃𝑡 𝑘𝑡+1 + 𝑀𝑡
= 𝑊𝑡 𝑙𝑡 + 𝑉𝑡 𝐾𝑡 + (1 − 𝛿)𝑃𝑡 𝑘𝑡 + (1 + 𝑖𝑡 )𝐵𝑡−1 + 𝑀𝑡−1 + 𝑃𝑡 𝜏𝑡
Total Production function is
𝑌𝑡 = 𝐴𝑡 𝑘𝑡𝛼 𝑙𝑡1−𝛼
𝑠
Aggregate supply of Bond is zero and money supplies is 𝑀𝑡𝑠 = (1 + 𝜇𝑡 )𝑀𝑡−1
where
𝜇𝑡 is the growth rate of money supply at time t:
a. Write down the budget constraint in real terms and define 𝑏𝑡 =
and 𝑚𝑡 =
𝑀𝑡
𝑃𝑡
.
𝐵𝑡
𝑃𝑡
b. Derive the first order conditions (FOCs) with respect to the decision
variables for the HHs and firms.
c. Write down the equilibrium conditions.
d. Show that 𝑚𝑡 =
1+𝜇𝑡
1+𝜋𝑡
𝑚𝑡−1
Suppose that: 𝑈(𝑐𝑡 , 𝑚𝑡 , 𝑙𝑡 ) = ln(𝑐𝑡 ) + 𝜂 ln(𝑚𝑡 ) + 𝛾ln (1 − 𝑙𝑡 )
e. Suppose𝜇𝑡 = 𝜇. Solve for the steady state equilibrium where
𝑐𝑡 , 𝑙𝑡 , 𝑘𝑡 , 𝑏𝑡 , 𝑚𝑡 are all constant.
f. Consider a more general definition of steady state equilibrium where only
𝑐𝑡 , 𝑙𝑡 , 𝑘𝑡 , 𝑏𝑡 are constant (time-varying 𝑚𝑡 ) Show that the only equilibrium
where inflation is constant is the case where𝜇𝑡 = 𝜇.
g. Suppose there is a permanent productivity shock at time zero.
i. Show the path of capital, consumption and output (just a handplotted pattern of the path to the steady state is needed.).
ii. Show the path of real interest rate. (No code)
2 Bank Seigniorage
Suppose that the demand for M1 money is given by
𝑀
𝑝𝑌
= 𝐴−𝜂𝑖 where 𝑀 is the
quantity of M1 money, 𝑝𝑌 is nominal GDP, 𝑖 is the nominal interest rate and 𝐴 and
𝜂 are parameters. Households want to hold a fraction 𝜒 of their M1 money in the
form of cash and a fraction 1 − 𝜒 in the form of checking accounts, which earn no
interest. Banks earn seigniorage by taking checking deposits and investing in assets
that earn the nominal interest rate.
a. Find an expression for the ratio of total seigniorage earned by banks to
GDP. Call this ratio 𝑠.
b. Compute
𝜕𝑠
𝜕𝑖
. Why does this number depend on 𝜂?
c. Look up data for US on M1 and its components in 2018. What is a
reasonable value for 𝜒? What was the value of
𝑀
𝑝𝑌
? If the average nominal
interest was 2%, how much seigniorage did banks earn as a fraction of
GDP?
d. If 𝜂 = 0.2, how much seigniorage would banks earn as a fraction of GDP if
the nominal interest rate went up to 3%?
3 The Cagan Model and Seigniorage
Suppose that we have a money demand specification as follows:
ln 𝑀𝑡𝑑 − ln 𝑝𝑡 = −𝜂𝐸𝑡 𝜋𝑡+1
The central bank sets the money supply, ln 𝑀𝑡𝑠 , exogenously and market is clear
(ln 𝑀𝑡𝑠 = ln 𝑀𝑡𝑑 ).
a) Provide some verbal intuition for why the demand for real balances ought
to be inversely related to expected inflation.
b) Note that 𝜋𝑡+1 = ln 𝑝𝑡+1 − ln 𝑝𝑡 . Derive an expression for the current log
price level as a discounted value of the current and future money supply,
assuming that lim ln 𝑝𝑡+𝑇 < ∞.
𝑇→∞
c) Use your expression from (c) to show that if
ln 𝑀𝑡 = 𝜇 + ln 𝑀𝑡−1 + 𝑒𝑡 ,
𝐸𝑡 [𝑒𝑡 ] = 0
Then 𝐸𝑡−1 [ln 𝑝𝑡 − ln 𝑝𝑡−1 ] = 𝜇.
d) Seigniorage is defined as the revenue that the government receives from
printing money. It is given by:
𝑀𝑡 − 𝑀𝑡−1
𝑠𝑡 =
𝑃𝑡
Assume that there is a constant growth rate, 𝜇. Use the definition of
seigniorage and Cagan’s money demand function to derive an expression for
seigniorage in terms on 𝜇 and 𝜂. Use the approximation that exp 𝜇 = 1 + 𝜇.
e) Find an expression for the rate of money growth, 𝜇, which maximizes
seigniorage. Provide some verbal intuition for your answer.
4 Baumol-Tobin model
In the course of a year, a household will spend Y. This spending is not all at once:
it's spread evenly over the period. Whenever the household wants to pay for
something, it must use money. Whenever it wants, the household can go to the
bank and swap some of its interest-bearing assets (with interest rate i) for money.
3
Assume that there is a fixed cost F of going to the bank. Let N denote the number
of times that the household goes to the bank. Each time the household goes to the
bank, it brings up its money balance to Y/N.
a) Within Baumol-Tobin framework, find optimal N, total revenue, and money
demand.
b) There are 2 banks, A and B. Suppose 𝑌 = 80, 𝑖𝐴 = 0.16, 𝐹𝐴 = 0.064, 𝑖𝐵 =
0.04, 𝐹𝐵 = 0.004. Calculate values of N, total revenue, and money demand
for each bank.
c) Which bank is prefered by households?
d) Which bank's total revenue will change more by increasing its interest rate?
What is the effect of changing interest rate on money demand?
e) Now suppose that households are allowed to invest their money in both
banks. Households invest all their money in bank A, and go to bank to get
money at the begining of each quarter. In order to gain more interest,
households invest that money in bank B during each quarter. What is the
optimal number of times that households go to bank B in each year?
Compare Total revenue and money demand that can be derieved with part
b.
f) According to your answers, what is the effect of banking network
characteristics on money demand?