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Macroeconomı́a II – Problem Set
TA Session 1
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Soledad Alegria Palma (salegria3@uc.cl)
Some of those exercises will be solved on the ayudantia of August 24, 2018.
Money supply
Exercise 1
[Mankiw] What are the three functions of money? Which of the functions of money do
the following items satisfy? Which do they not satisfy?
(a) A credit card;
(b) A painting by Rembrandt;
(c) A subway token.
Exercise 2
[Mankiw] An economy has a monetary base of 1,000 $1 bills. Calculate the money supply
in scenarios 1-4 and answer part 5.
1. All money is held as currency.
2. All money is held as demand deposits. Banks hold 100 percent of deposits as reserves.
3. All money is held as demand deposits. Banks hold 20 percent of deposits as reserves.
4. People hold equal amounts of currency and demand deposits. Banks hold 20 percent
of deposits as reserves.
5. The central bank decides to increase the money supply by 10 percent. In each of the
above four scenarios, how much should it increase the monetary base?
Macroeconomı́a II, 2018/2 1
Exercise 3
Suppose an economy has two types of deposits: demand deposits (Dv) and savings deposits
(Dp). We define the money aggregate M1 as the sum of demand deposits and currency
(C), that is, M1 ≡ Dv + C. We define the money aggregate M2 as M2 ≡ M1 + Dp.
The monetary base is given by H ≡ C +R, where R denote total reserves in the banking
sector. The total amount of deposits D is given by D ≡ Dv +Dp. Assume agents always
keep a ratio of currency (C) to total deposits (D) equal to 1/4, that is:
C
D
= 14 .
Moreover, the ratio of demand deposits to total deposits and savings deposits to total
deposits is constant and given by
Dv
D
= 34 and
Dp
D
= 14 .
Banks keep a ratio of reserves (R) to total deposits equal to some number θ ∈ (0, 1):
R
D
= θ.
1. Compute the M1 money multiplier (that is M1/H) as a function of θ.
2. Compute the M2 money multiplier (that is M2/H) as a function θ.
3. Find a condition that implies a M1 money multiplier smaller than one.
Money demand
Exercise 4
Consider the simple Baumol-Tobin model where an individual spends uniformly his annual
income Y and makes n withdrawals of equal size to minimize his opportunity cost (iY/2n)
plus the linear cost of withdrawing (Zn), where i > 0 denotes the nominal interest rate
and Z > 0.
1. Write the problem of minimizing the costs and identify the trade-off between the
opportunity cost and the linear cost.
Macroeconomı́a II, 2018/2 2
2. What is the most important conclusion of this model and what are its main assump-
tions? How can you justify the cost of withdrawing?
3. How do you think the demand for money would be affected if the quantity of banks
in which people can make withdrawals increased?
4. Suppose know that people can keep their funds in a savings account (that pays the
interest rate i) and access this funds whenever they need it for transactions using a
debt card (let’s call it electronic money). The money is discounted from the account
only at the moment that the transaction takes place, and thus people do not incur
the opportunity cost i. On the other hand, for every dollar discounted from the
account using the debt card, there is a fee τ ∈ (0, 1) that must be paid. What
happens to the demand for currency in this economy? Under which conditions
there is a positive demand for currency in this economy?
5. Suppose the same environment as in the previous item, except that now not all
sellers accept electronic money. Individuals spend a fraction λ of their income on
sellers that accept only currency and a fraction 1 − λ on sellers that accept both
currency and electronic money, where λ ∈ (0, 1). Find the demand for currency as
a function of τ and λ. What happens to the demand for currency when λ→ 0?
Exercise 5
Answer the questions below according to the Baumol-Tobin presented in class (which is
also presented also Section 15.5.3 of de Gregorio’s book). Are the statements below true
or false? Justify.
1. According to the Baumol-Tobin model, a 10% increase in the interest rate causes a
5% increase in the demand for money.
2. According to the Baumol-Tobin model, the higher the cost of going to the bank, the
higher the elasticity of the money demand with respect to income.
3. Assume the amount of transactions agents realize in a given year is equal to their
income. Then, the Baumol-Tobin model predicts that high-income individuals will
hold a larger fraction of their income in money assets than low-income individuals.
Macroeconomı́a II, 2018/2 3
Proposed exercises
Exercise 6
The graph below shows the M1 money multiplier for the US:
2000 20102002 2004 2006 2008 2012 2014 2016 2018
1.0
2.0
0.6
0.8
1.2
1.4
1.6
1.8
2.2
M1	Money	Multiplier
Ra
tio
Shaded	areas	indicate	U.S.	recessions Source:	Federal	Reserve	Bank	of	St.	Louis myf.red/g/iU0a
1. Provide one possible (theoretical) explanation for the multiplier being lower than
one.
2. Enter the FRED website (https://fred.stlouisfed.org/) and using the time series for
the monetary base and M2, compute the M2 money multiplier for the US.
3. Still on the FRED website, find data on:
(a) Excess reserves (e.g., “Excess Reserves of Depository Institutions”);
(b) Reserves (e.g., “Reserve Balances with Federal Reserve Banks”);
(c) Demand deposits (e.g., “Total Checkable Deposits”).
How does the data above helps you explain the fact that money multipliers have
remained very low after the Great Recession?
4. Read the article available on https://tinyurl.com/hd2zsxp and explain why US banks
excess reserves have increased so much in the last years.
Exercise 7
[Mankiw] To increase tax revenue, the U.S. government in 1932 imposed a 2-cent tax on
checks written on bank account deposits. (In today’s dollars, this tax would amount to
about 34 cents per check.)
Macroeconomı́a II, 2018/2 4
https://fred.stlouisfed.org/
https://tinyurl.com/hd2zsxp
1. How do you think the check tax affected the currency-deposit ratio? Explain.
2. Use the model of the money supply under fractional-reserve banking to discuss how
this tax affected the money supply.
3. Many economists believe that a falling in the money supply was in part responsible
for the severity of the Great Depression of the 1930s. From this perspective, was the
check tax a good policy to implement in the middle of the Great Depression?
Exercise 8
Suppose an economy with 10 agents and 10 banks, indexed by 1 to 10. Every bank keeps
50% of deposits in reserves. The central bank gives $10 of currency to agent 1. Agent 1
deposits the money in the bank 1. Then, bank 1 lends half of it agent 2. Agent 2 deposits
all its loan at bank 2, that lends half of it to agent 3, that deposits in bank 3, and so on.
When the money finally arrives to agent 10, he decides not to deposit it in bank 10 and
keep the amount to himself.
1. Compute the increase in the money supply in this economy.
2. Now assume that agent i only deposits a fraction 1/i of the funds it gets in the bank
(instead of the full amount). To simplify, also assume that agent 4 decides not to
use the bank (instead of agent 10). What is the increase in the money supply?
Exercise 9
Imagine an economy with types of deposits only: demand deposits (Dv) and savings
deposits (Dp). Suppose the that the ratio is of demand deposits to reserves is equal to
0.2, and the ratio of currency to demand deposits is 0.25.
1. What is the money multiplier if we define money as M1?
2. Consider now that ratio of reserves to saving deposits is 0.1 and the ratio of currency
to savings deposits is 0.2. What is the money multiplier if we define money as M2?
Macroeconomı́a II, 2018/2 5
Macroeconomı́a II – Solutions
TA Session 1
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Soledad Alegria Palma (salegria3@uc.cl)
Exercise 1
The three functions of money are: unit of account, medium of exchange and store of value.
A credit card is a medium of exchange, since you can use itto buy goods and services.
It is usually not used as a store of value, and clearly it is not a an unit of account. A
Rembrandt painting is a store of value, but it not usually used as an unit of account or a
medium of exchange. A subway token may be an unusual store of value (if you can resell
or use it later). It is not usually used as a medium of exchange or unit of account.
Exercise 2
Item 1
Let’s start deriving again the money multiplier. Let R/D = θ ∈ (0, 1) denote the reserve
to deposits ratio. Let c = C/D denote the currency to deposits ratio. From the definition
of M :
M = C +D
Let H = C + R denote the monetary base. Dividing M = C + D by H = C + R we get
the multiplier:
M
H
= C +D
C +R =
C/D + 1
C/D +R/D =
c+ 1
c+ θ ≡ m
Hence, M = mH. If all money is held as currency, we have that c→∞. Therefore:
M = lim
c→∞
c+ 1
c+ θ1000 = 1000
(You need to use L’Hopital rule in that limit).
Macroeconomı́a II, 2018/2 1
Item 2
In that case we have c = 0 and θ = 1. Therefore:
M = 0 + 10 + 11000 = 1000
Item 3
In that case we have c = 0 and θ = 0.2. Therefore:
M = 0 + 10 + 0.21000 = 5000
Item 4
In that case c = 1 and θ = 0.2. Therefore:
M = 1 + 11 + 0.21000 = 1666, 66
Item 5
Since M = mH we have that∆M = m∆H, which dividing by M = mH yields
∆M
M
= ∆H
H
Hence, if we want the money supply to increase 10%, we need to increase the monetary
base in 10%. Hence, the central bank should increase the monetary base in a $100 in every
case.
Exercise 3
Item 1
The definition of M1 and H are:
M1 = Dv + C (1)
and
H = C +R. (2)
Macroeconomı́a II, 2018/2 2
Dividing (1) by (2), manipulating and using C
D
= 14 ,
Dv
D
= 34 and
R
D
= θ:
M1
H
= Dv + C
C +R =
Dv/D + C/D
C/D +R/D =
3/4 + 1/4
1/4 + θ =
1
1/4 + θ .
Item 2
The definition of M2 is:
M2 = M1 +Dp
Which using the definitions of M1 and D becomes
M2 = D + C (3)
and
H = C +R. (4)
Dividing (3) by (4), manipulating and using C
D
= 14 and
R
D
= θ:
M2
H
= D + C
C +R =
D/D + C/D
C/D +R/D =
1 + 1/4
1/4 + θ .
Item 3
To get M1
H
< 1 we need:
1
1/4 + θ < 1⇒ θ > 3/4.
Exercise 4
Item 1
The agent chooses the number of withdrawals n ∈ R+ to minimize
C(n) = i
(
Y
2n
)
+ Zn
Note that for simplicity we are assuming that n is chosen from R+ instead of the being
chosen from the set of non-negative integers {0, 1, 2, ...} (as if the agent could do one and
a half withdrawals, for example).
Macroeconomı́a II, 2018/2 3
Notice that the term Y2n represents the agent average money holdings.
1 Hence, when
the agent increases the number of withdrawals it reduces the opportunity cost of holding
money, represented by i
(
Y
2n
)
, since she holds less cash on average, but needs to pay the
cost of withdrawing (represented by Zn).
Item 2
Taking the first order condition we find the n that minimizes the cost:
iY
2n2 = Z
And hence, the optimal n, denoted by n∗, is:
n∗ =
√
iY
2Z
Therefore, the average money holdings (that is, the money demand) is given by
MD = Y
2
√
iY
2Z
= Y2
√
2Z
iY
=
√
Y 2
4
2Z
iY
=
√
Y
2
Z
i
(1)
Taking logs:
lnMD = 12 (ln Y + lnZ − ln 2− ln i) (2)
Therefore, we have two main conclusions: (i) the elasticity of the money demand with
respect to income is equal to 1/2; (ii) the elasticity of the money demand with respect to
the nominal interest rate is equal to −1/2.
1To see that, notice that an agent that goes n times to the bank will withdraw Y/n each time (so
that the total amount withdrawed after n withdrawals is equal to Y ). When the exercise says that the
individual spends his income uniformly, we mean that dMtdt is constant between each withdrawals (where
Mt denotes the agent money holdings at some date t, measured in years). Hence, in interval of time
[0, 1/n] (i.e., the interval between the first and the second withdrawal) the agent money holdings M̃t are
a linear function of time that: (i) is equal to Y/n when t = Y/n (since he withdrawals Y/n at t = 0); (ii)
approaches zero when t = 1/n (the moment of the second withdrawal), since she is about to make a new
withdrawal when t = 1/n, and she only withdrawals when there is no more money holdings. Therefore,
M̃t = Yn − Y × t, for t ∈ [0, 1/n]. Hence, the average money holdings in the interval t ∈ [0, 1/n] is:
MD =
∫ 1/n
0
(
Y
n − Y × t
)
dt
1/n = n
[
Y
n
t− Y t
2
2
∣∣∣∣1/n
0
]
= n
[
Y
n2
− Y 12n2
]
= Y2n
Since the average money holdings between any two consecutive withdrawals is the same, the average
money is Y2n .
Macroeconomı́a II, 2018/2 4
Item 3
The would probably reduce the cost of going to the bank (Z). As equation (1) shows, that
should reduce the money demand.
Item 4
The agent now has two decisions to make. She must choose how much of his total yearly
transactions (Y ) she is going to conduct using currency (denoted by XC) and how much
she is going to conduct using electronic money (denoted by XE). (Note that XE and XC
must be such that XE + XC = Y .) Once she has fixed an amount XC , then she chooses
how many times to go to the bank to achieve that amount of transactions using currency.
First, let’s suppose an agent has chosen a given level of XC . From our solution of the
previous items, we know that the agent will go n∗ =
√
iXC
2Z times to bank and her average
money holding is
√
XC
2
Z
i
. Therefore, the total cost of spending XC using currency is:
CC(XC) = i
√
XC
2
Z
i︸ ︷︷ ︸
average money holdings
+Z
√
iXC
2Z︸ ︷︷ ︸
# of time she goes to the bank
The total cost incurred for a given XC and XE (after optimally choosing how much to go
to the bank to achieve a given XC) is given by:
C(XC , XE) = i
√
XC
2
Z
i
+ Z
√
iXC
2Z︸ ︷︷ ︸
CC(XC)
+ τXE︸ ︷︷ ︸
cost of electronic transactions
Which simplifies to:
C(XC , XE) =
√
2iZXC + τXE
The agent chooses XC and XE to minimize C(XC , XE) above subject to XE +XC = Y ,
XE ≥ 0 and XC ≥ 0. Hence, replacing XE = Y −XC in C(XC , XE) we get:
C̃(XC) ≡ C
(
XC , Y −XC
)
=
√
2iZXC + τ
(
Y −XC
)
(3)
Therefore, we can simplify the problem: the agent chooses XC ∈ [0, Y ] to minimize (3)
(and then XE is given by Y −XC). Notice that the function above is concave. This implies
that we can only have corner solutions: either the agent chooses XC = 0 or XC = Y .
Macroeconomı́a II, 2018/2 5
Therefore, the agent chooses XC = 0 whenever:
τY︸︷︷︸
C̃(0)
>
√
2iZY︸ ︷︷ ︸
C̃(Y )
Which yields:
τ >
√
2iZ
Y
Hence, when τ >
√
2iZ
Y
the agent chooses to make all transactions with currency (i.e., she
chooses XC = Y ). When τ <
√
2iZ
Y
she chooses to only use electronic money (i.e., she
chooses XC = 0). When τ =
√
2iZ
Y
the agent is indifferent between XC = 0 or XC = Y .
Item 5
Now the agent solves the same problem as in item 4, with the additional constraint that
XC ≥ λY (since at fraction λ of the income must be spent with currency). Hence,
instead of choosing XC ∈ [0, Y ] to maximize (3), the agent now chooses XC ∈ [λY, Y ]
to maximize (3). As before, we can only have corner solutions to this problem, since the
objective function is concave. Hence, the agent chooses XC = λY if:
√
2iZλY + τ (Y − λY )︸ ︷︷ ︸
C̃(λY )
<
√
2iZY︸ ︷︷ ︸
C̃(Y )
Which becomes:
τ <
(
1−
√
λ
)
(1− λ)
√
2iZ
Y
Hence, when τ < (1−
√
λ)
(1−λ)
√
2iZ
Y
the agent chooses to make as little transactions as
possible with currency and chooses XC = λY . Similarly, when τ > (1−
√
λ)
(1−λ)
√
2iZ
Y
, the
agents chooses to make all her transactions with currency and then chooses XC = Y .
When τ = (1−
√
λ)
(1−λ)
√
2iZ
Y
the agent is indifferent between XC = λY and XC = Y . When
λ → 0 we have that (1−
√
λ)
(1−λ)
√
2iZ
Y
→
√
2iZ
Y
, which is the same cutoff on τ we found in
item 4. Moreover, limλ→0 λY = 0. Hence, the money demand converges to that of item 4
(which is not surprising, of course).
Exercise 5
Macroeconomı́a II, 2018/2 6
Item 1
True. Using equation (2) from our answer of Exercise 4 one can see that the elasticity of the
money demand with respect to the interest rate is constant and equatl −1/2. Therefore,
when i increases 10%, the money demand falls 5%.
Item 2
False.Using (2) one can see that the elasticity of the money demand with respect to
income is constant and equal to 1/2.
Item 3
False. Dividing (1) by Y we get:
MD
Y
= 1
Y
√
Y
2
Z
i
=
√
1
Y 2
Y
2
Z
i
=
√
Z
2Y i
Hence, the share of the income held as money is decreasing in income.
Macroeconomı́a II, 2018/2 7

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