Workbook

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Workbook - Macroeconomics II
Pontificia Universidad Católica de Chile
Professor: Caio Machado (caio.machado@uc.cl)
Coordinator: Claudia Alvarez (cnalvarez@uc.cl)
Teaching assistants: Vicente Herrera (vgherrera@uc.cl), Gabriel Villalobos (gavillalo-
bos@uc.cl) and Wei Xiong (wxiong@uc.cl)
This document contains all the exercises students are expected to try to solve
in the course. Solutions to all exercises but those market as “proposed” will
be provided during the semester. Moreover, the “proposed” exercises will not
be solved during the TA sessions (if they were solved, you would get solutions,
and those exercises would not be marked as proposed). Students should be
aware that simply understanding the solutions is not enough to succeed in this
course, so that I strong advise you put some effort into each exercise before
checking the solutions and going to the TA sessions. This file may be updated
during this semester. Make sure you have the latest version.
Last updated: August 21, 2019
Contents
1 Money supply 3
2 Money demand 9
3 Monetary equilibrium 23
4 Seignioriage 26
5 Exchange rates 30
6 Goods and money markets 33
7 IS-LM 36
8 Mundell-Fleming 44
9 Phillips curve 49
10 IS-LM-PC 52
11 AS-AD (only if time allows) 57
12 Dynamic inconsistency 59
Macroeconomía II, 2019/2 2
1 Money supply
Exercise 1.1
[Adapated from 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;
(d) Bitcoins.
Exercise 1.2
[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?
Macroeconomía II, 2019/2 3
Exercise 1.3
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
=
1
4
.
Moreover, the ratio of demand deposits to total deposits and savings deposits to
total deposits is constant and given by
Dv
D
=
3
4
and
Dp
D
=
1
4
.
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.
Exercise 1.4
Answer TRUE or FALSE to the following statements:
Macroeconomía II, 2019/2 4
1. If during a financial crisis banks restrict the level of credits they grant to
the private sector, we should expect the money multiplier to increase.
2. If banks keep 100% of their deposits as reserves, the money multiplier is
zero.
Exercise 1.5
[Proposed] 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”).
Macroeconomía II, 2019/2 5
https://fred.stlouisfed.org/
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 1.6
[Proposed] 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?
Exercise 1.7
[Proposed] 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?
Macroeconomía II, 2019/2 6
https://tinyurl.com/hd2zsxp
Exercise 1.8
[Proposed, 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.)
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 1.9
[Proposed] 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
Macroeconomía II, 2019/2 7
4 decides not to use the bank (instead of agent 10). What is the increase
in the money supply?
Macroeconomía II, 2019/2 8
2 Money demand
Exercise 2.1
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 be-
tween the opportunity cost and the linear cost.
2. What is the most important conclusion of this model and what are its
main assumptions? 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 peoplecan 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
Macroeconomía II, 2019/2 9
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 2.2
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 (as usual in the Baumol-Tobin model). Then, the
Baumol-Tobin model predicts that high-income individuals will hold a
larger fraction of their income in monetary assets than low-income individ-
uals.
Exercise 2.3
In the money in the utility function model (without labor) seen in class, we
have shown under the optimal choice of money and consumption, the following
equation holds:
um (ct,mt)
uc (ct,mt)
=
it
1 + it
, (1)
Macroeconomía II, 2019/2 10
where um(ct,mt) and uc(ct,mt) denote the marginal utility of real money balances
(mt) and consumption (ct), respectively, and it denotes the nominal interest rate.
We also know that the household choice satisfies a standard Euler equation:
uc(ct,mt) = β (1 + rt)uc(ct+1,mt+1). (2)
1. Show that the household choice satisfies:
um (ct,mt) =
it
1 + πt+1
βuc(ct+1,mt+1).
Interpret this equation.
2. Assume that u(c,m) = ln c − (m − 5)2. What is the nominal interest rate
and inflation rate that maximize the representative household utility in
the steady state? If the nominal interest rate is chosen to maximize the
household’s utility at the steady state, how much money is demanded at
the steady state? Provide an intuition for the optimal nominal interest rate.
(Tip: remember that money is superneutral in this model.)
Exercise 2.4
[Adapted from Walsh] Consider the basic MIU model in Section 2.2 of Walsh’s
book (also seen in class), with one modification: now the real money balances
that the agent decides to hold at date t yield utility only at date t + 1. More
precisely, the instantaneous utility at date t depends on ct and the real money
balances the agent choose at t− 1 (instead of the real money balances he choose
at t). Define Mt as the amount of money the agent choose to hold at date t− 1
(instead of the amount he chooses to held at date t, as in Section 2.2.). Defining
Macroeconomía II, 2019/2 11
Mt that way, the agent utility can be written as before:
∑∞
t=0 β
tu
(
ct,
Mt
Pt
)
.
1. Write the budget constraint in nominal terms (i.e., with the interest rate,
money holdings and bond holdings expressed in nominal units).
2. Write the budget constraint in real terms (i.e., with the interest rate, money
holdings and bond holdings expressed in real units).
3. Show that in the optimal, the equation below holds for every t:
um (ct+1,mt+1)
uc (ct+1,mt+1)
= it.
Interpret it.
Exercise 2.5
Consider the basic MIU model in Section 2.2 of Walsh’s book (also seen in class),
with some modifications: now agents can work to increase their production,
but working causes them some disutility. More precisely, the production
function is now given by yt = f(kt−1, nt), where nt is the number of hours
worked. Normalizing the total number of hours the worker has available to
1, we have nt = 1 − lt, where lt denotes the number of hours spent on leisure.
The instantaneous utility function of the agent is given by u(ct,mt, lt), where
the usual assumptions on u(·) and f(·) to guarantee an interior solution are
satisfied.
1. Write the budget constraint in nominal terms (i.e., with the interest rate,
money holdings and bond holdings expressed in nominal units).
2. Write the budget constraint in real terms (i.e., with the interest rate, money
holdings and bond holdings expressed in real units).
Macroeconomía II, 2019/2 12
3. Derive the equations that determine the steady state level of consumption,
leisure and capital. Under which conditions we have superneutrality?
Compare the results obtained here to the ones obtained in the model
without labor.
Exercise 2.6
Consider a centralized economy where households produce and consume. The
production function Y (Kt,nt) is homogeneous of degree one in both capital
and labor. Here, household incomes are production Yt, monetary transfers
from the government Tt and money balances Mt. On the other hand, the
outcomes are consumption ct, investment in capital It and money balance held
for the next period Mt+1. Capital evolves according to the following expression:
Kt+1 = Kt(1− δ) + It.
Now let’s think of a decentralized economy, where households own the
resources capital Kt and labor nt, which is then rented every period by the
firms receiving as payment interest rate Ptrt for capital and wages Ptwt for labor
(hence, the real rental rate of capital is rt and the real wage is wt). In this
economy, there are transfers Tt and money Mt+1 that households accumulate
in t to use in t+ 1.
1. Present the budget constraint for both centralized and decentralized
economies in nominal terms.
2. Present the budget constraint for both centralized and decentralized
economies in real terms.
3. Show that we can recover decentralized budget constraint from centralized
one in this setting assuming perfect competition.
Macroeconomía II, 2019/2 13
Exercise 2.7
[Proposed] Consider the basic MIU model in Section 2.2 of Walsh’s book (also
seen in class) with one modification: assume that there is no capital and the
output yt is exogenously given. Assume that:
u(c,m) = v(c) + ω lnm,
where v(·) satisfies the usual assumptions that guarantee an interior solution.
1. Compute the current price as a function of current and future money
supplies.
2. Compute the equilibrium nominal interest rate as function of current and
future money supplies.
Exercise 2.8
[Proposed] Consider an economy with the following state at some date t: β = 0.95,
it = 0.07, πet = 0.02. Expected inflation remains constant while nominal interest
rate increases by 1pp each period. Households preferences are logarithmic
on consumption, having then u(ct) = ln(ct) in the CIA model and u(ct,mt) =
ln(ct) + ln(mt) in the MIU model. (The notation is the standard one and the CIA
and MIU model referred to are the ones in Section 2.2 and Section 3.3.1 in
Walsh’s book).
1. For each of the models (CIA and MIU) present the household problem,
estimate the first order conditions and find the Euler condition by relating
current and future consumption.
2. How do both results relate? Explain each difference you find.
Macroeconomía II, 2019/2 14
Exercise 2.9
Consider an economy in which there is a single good that can be used as capital
or for consumption. Time is discrete and indexed by t ∈ {−1, 0, 1, 2, ...}. Consider
the following problem of the representative household:
max
{ct,kt,Bt,Mt}t≥0
∞∑
t=0
βtu (ct)
s.t.Ptct + Ptkt +Bt +Mt = Ptf(kt−1) + (1 + it−1)Bt−1
+Mt−1 + (1− δ)Ptkt−1 + Tt, ∀t ≥ 0
ψPtct ≤Mt−1 + Tt, ∀t ≥ 0
k−1 = k > 0, B−1 = B > 0, M−1 = M > 0
lim
t→∞
(Bt/Pt) = 0,
kt, ct,Mt ≥ 0, ∀t ≥ 0.
The notation is standard and is the same used in class: u(c) is the instantaneous
utility function; f(k) is the production function; Pt denotes the price of the good
at date t; Bt is the nominal amount of bonds the household chooses to hold
at date t; Mt is the quantity of money he chooses to carry from date t to date
t+ 1; it is the nominal interest rate between dates t and t+ 1; ct is the household
consumption at date t; kt is the amount of capital he carries from date t to
t + 1; Tt are cash transfers received at date t; β ∈ (0, 1) is the discount factor
and δ ∈ (0, 1) is the depreciation rate of capital. Define bt ≡ Bt/Pt, mt ≡ Mt/Pt,
πt ≡ (Pt − Pt−1)/Pt−1,τt ≡ Tt/Pt and let rt denote the real interest rate between
dates t and t+ 1.
The constraint ψPtct ≤Mt−1 +Tt is interpreted as a standard cash-in-advance
constraint, except that now only a fraction ψ ∈ (0, 1) of the household consump-
tion is purchased using cash.
In what follows, assume that u(·) and f(·) satisfy all the usual assumptions
Macroeconomía II, 2019/2 15
that guarantee an unique interior solution (and thus the non-negativity con-
straints kt, ct,Mt ≥ 0 never bind in the optimal choice and you can ignore them).
Moreover, assume the cash-in-advance constraint binds at all dates in the
optimal solution (which is true if it > 0, for every t). There is no uncertainty and
the household takes as given the path of all exogenous variables.
1. Rewrite the budget constraint and the cash-in-advance constraint in real
units (that is, in terms of only the real variables {ct, kt, bt,mt, rt, τt}t≥0 and
the inflation rate {πt}t≥0). (Tip: use the Fisher equation to get rid of nominal
interest rates).
2. Denote by {λt}t≥0 the Lagrange multipliers associated to the budget con-
straint and by {µt}t≥0 the Lagrange multipliers associated to the cash-in-
advance constraint. Write down the Lagrangian using the budget con-
straints in real units and derive the first order conditions for ct, kt, bt and
mt.
3. In the optimal solution we get the following Euler equation:
u′(ct)
u′(ct+1)
= β (1 + rt)h(it−1, it, ψ)
where h(it−1, it, ψ) is a function of it−1, it and ψ. Derive the functional form
of h(it−1, it, ψ).
4. Interpret economically the Euler equation you found in the previous item
when ψ = 1. (Tip: you may find useful to rearrange the equation before
your interpret it.)
Macroeconomía II, 2019/2 16
Exercise 2.10
Consider the CIA model in Section 3.3.1 of Walsh’s book (also seen in class)
with one modification: transfers received at date t can not be directly used to
purchase goods at date t, so that the the cash-in-advance constraint takes the
form Ptct ≤Mt−1. Assume that u(c) = ln c. Also, assume that the CIA constraint
always binds.
1. Write down the household’s decision problem.
2. Write down the household’s first-order conditions.
3. Derive the Euler equation.
4. Show that:
Mt
Mt−1
= β (1 + it−1) .
5. Explain what happens to the nominal interest rate in 3 scenarios:
(a) Moneys grows at a constant rate: Mt/Mt−1 = 1 + µ;
(b) Money grows at an increasing rate: Mt/Mt−1 = 1 + µt;
(c) Money grows at a decreasing rate: Mt/Mt−1 = 1 + 1/ (µt).
Exercise 2.11
When we presented the CIA and MIU models in class, we wrote down the
centralized version of it: a single household decided how much to consume,
how much capital, money and bonds to hold. The household had access to a
production function that would transform this capital into consumption goods.
Now we set up a decentralized version of the CIA model.
Macroeconomía II, 2019/2 17
Consider an economy in which there is a single consumption good that
can be used as capital or for consumption. Time is discrete and indexed by
t ∈ {−1, 0, 1, 2, ...}. There is a representative household and a competitive firm in
this economy. Both the firm and the household are price takers.
Household. The household owns the firm and chooses consumption and how
much capital, bonds and money to hold. At each date t, the household rents
the capital it brought from the previous period (kt−1) to the firm at a rental
rate Pt × Rt (hence, the rental rate in units of the consumption good is Rt).
Every period, the household also receives profits Dt from the firm. Hence, the
household problem is:
max
{ct,kt,Bt,Mt}t≥0
∞∑
t=0
βtu (ct)
s.t. Ptct + Ptkt +Bt +Mt = PtRtkt−1 + (1 + it−1)Bt−1
+Mt−1 + (1− δ)Ptkt−1 + Tt +Dt, ∀t ≥ 0
Ptct ≤Mt−1 + Tt, ∀t ≥ 0
B−1 = B > 0, M−1 = M > 0
lim
t→∞
(Bt/Pt) = 0,
ct,Mt ≥ 0, ∀t ≥ 0.
We are assuming that the profits Dt are only received at the end of the period
(and hence that money is not useful for transactions), while the transfers Tt are
received at the beginning of the period (and hence can be used for transactions).
That is why only Tt shows up in the CIA constraint. The notation is standard
and is the same used in class: u(c) is the instantaneous utility function; Pt
denotes the price of the good at date t; Bt is the nominal amount of bonds the
household chooses to hold at date t; Mt is the quantity of money she chooses to
carry from date t to date t+ 1; it is the nominal interest rate between dates t and
Macroeconomía II, 2019/2 18
t+ 1; ct is the household consumption at date t; Tt are cash transfers received
at date t;β ∈ (0, 1) is the discount factor and δ ∈ (0, 1) is the capital depreciation
rate. Define bt ≡ Bt/Pt, mt ≡Mt/Pt, πt ≡ (Pt − Pt−1)/Pt−1,τt ≡ Tt/Pt, dt ≡ Dt/Pt and
let rt denote the real interest rate between dates t and t + 1. Household solve
their problem taking the sequence of prices, interest rates, transfers and profits
as given.
Firm. The firm has access to a production function f(k̃). At each date t, the
firm chooses how much capital k̃t to rent from the household. Hence, at each
date t the firm problem is:
max
k̃t
Πt ≡ Ptf(k̃t)− PtRtk̃t
subject to k̃t ≥ 0.
Market clearing. The following market clearing must hold:
Bt = 0, ∀t ≥ 0
k̃t = kt−1, ∀t ≥ 0
ct + kt = f(kt−1), ∀t ≥ 0
Moreover, Tt = Mt −Mt−1, since the government can only make transfers by
issuing more money.
Assume that u(·) and f(·) satisfy all the usual assumptions that guarantee
an unique interior solution (and thus the non-negativity constraints never
bind in the optimal choice and you can ignore them). Moreover, assume the
cash-in-advance constraint binds at all dates in the optimal solution (which is
Macroeconomía II, 2019/2 19
true if it > 0, for every t). There is no uncertainty and the household takes as
given the path of prices, interest rates and all exogenous variables.
1. Write down the household problem in real units.
2. Derive the first order conditions of the household problem.
3. Write the first order condition of the firm.
4. Show that in equilibrium Rt = f ′ (kt−1) and rt = Rt+1 − δ .
5. Show that the steady state in the decentralized model is the same as in
the centralized model (that is, the model of section 3.3.1 in Walsh’s book,
which is the one seen in class).
Exercise 2.12
[Proposed] Suppose the representative household chooses consumption of cash
goods ({ct}t≥0), consumption of credit goods ({dt}t≥0), nominal bond holdings
({Bt}t≥0) and hours of work ({nt}t≥0) to maximize
∞∑
t=0
βt [u(ct, dt)− γnt]
subject to the nominal budget constraint
Pt (ct + dt) +Mt +Bt = Wtnt + (1 + it−1)Bt−1 + Tt +Mt−1
and a cash-in-advance constraint of the form
Ptct ≤Mt−1 + Tt.
Macroeconomía II, 2019/2 20
Where γ > 0, nominal wages are denoted by Wt and the remaining notation is
the standard one. Denote the real wage by wt and real bond holdings by bt. The
usual non-negativity constraint on c and l must hold, as well as the solvency
constraint (limt→∞ bt = 0). Moreover, bonds and money at the beginning of period
zero are given. Assume that:
u(ct, dt) = α ln ct + (1− α) ln dt, with α ∈ (0,1).
1. Write the cash-in-advance constraint and budget constraint in real units.
2. Derive the first order conditions with respect to ct, nt, dt and bt.
3. Compute the steady state (assuming a constant inflation rate, real wage,
consumption, hours of work, bond and money holdings). Does this model
exhibit superneutrality? Explain.
4. Maximizing utility at the steady state, what is the optimal inflation in this
economy?
5. Write down an expression for the share of cash in total consumption as a
function of the nominal interest rate. How does the share of cash goods
consumed depend on the interest rate?
Exercise 2.13
[Proposed] Consider the CIA model in Section 3.3.1 of Walsh’s book (also seen
in class) with one modification: to accumulate capital agents need to have cash
in advance. In other words, the CIA constraint now is given by:
Ptct + Pt (kt − (1− δ) kt−1) ≤Mt−1 + Tt
Macroeconomía II, 2019/2 21
That is, now not only consumption must be purchased with cash, but also
capital goods.
1. Present the Lagrangian of this problem, denoting by λt and µt the multipli-
ers for the budget constraint and the CIA constraint. Derive the first order
conditions.
2. Show that:
u′(ct) = βu
′(ct+1)
(
1
1 + it+1
)
f ′(kt) + βu
′(ct+1)(1− δ)
3. Assuming a production function f(k) = kα, with α ∈ (0, 1) solve for the
stationary state capital kss. Is money superneutral in this version of the
model? Give some intuition on your result.
4. Now consider the same economy but with the usual CIA constraint Ptct ≤
Mt−1 +Tt. The steady state capital in that economy, with f(kt) = kαt , is given
by:
kss =
(
α
1
β
− 1 + δ
) 1
1−α
Find the inflation rate that equates steady state capital in both economies
and then explain how this relates to Friedman rule.
Macroeconomía II, 2019/2 22
3 Monetary equilibrium
Exercise 3.1
[Mankiw] In the country of Wiknam, the velocity of money is constant. Real
GDP grows by 3 percent per year, the money stock grows by 8 percent per year,
and the nominal interest rate is 9 percent. What is the growth rate of nominal
GDP, the inflation rate and the real interest rate?
Exercise 3.2
[Mankiw] Suppose a country has a money demand function Md/P = κy, where
κ is a constant parameter and y is real income. The money supply grows by 12
percent per year, and real income grows by 4 percent per year.
1. What is the average inflation rate?
2. How would inflation be different if real income growth were higher? Ex-
plain.
3. How do you interpret the parameter κ? What is its relationship to the
velocity of money?
4. Suppose, instead of a constant money demand function, the velocity of
money in this economy was growing steadily because of financial innova-
tion. How would that affect the inflation rate? Explain.
Macroeconomía II, 2019/2 23
Exercise 3.3
Consider the basic Cagan’s model in which the log of the price level (pt) and the
log of the money supply (mt) satisfy the following difference equation:
mt − pt = −η (pt+1 − pt) (1)
where η is a constant larger than zero. For simplicity, suppose that mt is
constant and equal to m̃, for every t. We know that a solution to (1) is given by
pt =
1
1 + η
∞∑
i=0
(
η
1 + η
)i
mt+i = m̃
The solution above is called the fundamental solution, but we know that there
are other solutions to (1). Answer the questions below:
1. Propose another (non-fundamental) solution and show that it also satisfies
(1).
2. Discuss the economic intuition behind the non-fundamental solutions
in which prices increase over time, even though the money supply is
constant. (Tip: you may want to start with “Suppose everyone expects
prices to increase a lot in the future. Then...”)
Exercise 3.4
[Proposed] Consider the basic Cagan’s model in which the log of the price level
(pt) and the log of the money supply (mt) satisfy the following difference equation:
mt − pt = −η (pt+1 − pt) (1)
Macroeconomía II, 2019/2 24
where η is a constant larger than zero. You may assume that mt is bounded.
1. Write pt as a function of the current and future money supply. Interpret
the results.
2. Are there solutions to (1) that depend on variables other than the money
supply and the parameter η (i.e., non-fundamental solutions, bubbles)? If
your answer is yes, provide one example and show that it satisfies (1). If
your answer is no, show that there is no bubbles.
3. Suppose the money supply is constant and equal to m̃ and everyone is
certain it will remain at that level forever. Using your solution in item 1,
characterize the price level.
4. Suppose now that at date 0 the centrak bank announces that the money
supply will be equal to m̃ from date 0 to date T − 1, and that at date T the
money supply will increase to m̃′ > m̃ (and remain at the new level forever).
In a graph and using the fundamental solution, show the evolution of the
price level as a function of time. Intepret the results.
Macroeconomía II, 2019/2 25
4 Seignioriage
Exercise 4.1
Consider an economy with a single final good. The seigniorage revenue in real
terms in a given period is given by
S =
∆M
P
,
where ∆M is the increase in the money supply and P is the price of the good.
The demand for money is given by M/P = Y e−αi, where i is the nominal interest
rate, Y is output and α > 0. Using the Fisher equation (and assuming expected
inflation is equal to actual inflation), we can write M/P = Y e−α(r+π), where r is
the real interest rate and π is the inflation rate. Suppose that in the long run
the interest rate and output are constant (i.e., ∆i = 0 and ∆Y = 0). Find the
inflation rate that maximizes S in the long run.
Exercise 4.2
Consider the following money demand:
Md
P
= y [1− (r + πe)]
where y is the real output, r is the real interest rate and πe is expected inflation.
1. Derive an expression for the seigniorage revenue assuming y = 250, r = 0.05
and πe = 0.1 (and all those are independent of money growth). Compute
the seigniorage revenue for the following growth rates of the money supply:
25%, 50% and 75%. What is the relationship between seigniorage and
Macroeconomía II, 2019/2 26
money growth?
2. Assume now that expected inflation is equal to actual inflation (π = πe),
while the other variables are the same. Compute the seigniorage revenue
for the following growth rates of the money supply: 25%, 50% and 75%.
Why are your answers different from the previous item? Explain.
3. Assuming π = πe, find the rate of monetary expansion that maximizes the
seigniorage revenue.
Exercise 4.3
[Adapted from de Gregorio] Suppose an economy in which agents keep money
as currency and as deposits. The money multiplier is denoted by µ̃. The demand
for real balances in this economy is given by:
L(i, y) = ay (b− i) ,
where y is real output.
1. Compute the seigniorage if the inflation rate is 10%. What assumptions
do you need to make to be able to compute the seigniorage?
2. Suppose b > r, where r is the real interest rate. Compute the inflation rate
that maximizes the government revenue. What happens with the inflation
rate that you found if the real interest rate increases?
3. Suppose now that in reality the multiplier increases. How does this change
your answer in item 1?
Macroeconomía II, 2019/2 27
Exercise 4.4
[Adapted from de Gregorio] The money demand is given by:
M
P
= α− βi+ γy.
The notation is the usual one.
1. Find the seigniorage (S), assuming π = πe, and discuss how π relates to S.
In order for a revenue maximizing seigniorage to exist, do we need some
restriction on parameters?
2. If it exists, compute the inflation rate that maximizes seigniorage.
Suppose now that in this economies output grows at a rate g.
3. Write the seigniorage as a function of the parameters α, β, γ, the real
output y, the growth rate g, the inflation rate π and the nominal interest
rate i. Use the Fisher equation.
4. Find the inflation rate that maximizes seigniorage. Hows does it compare
to item 2?
Exercise 4.5
[Proposed,adapted from de Gregorio] Consider the following money demand
function:
Mt
Pt
= mt = yte
−απet , (1)
where M is the nominal quantity money, P is the price level, m is the real
quantity of money, y is real output (which is normalized to 1 hereafter), πe is
expected inflation and α is a positive constant.
Macroeconomía II, 2019/2 28
Suppose the government wishes to finance a real deficit d by increasing the
nominal money supply at a rate of σ. The seigniorage is Ṁt/Pt (you can omit
the subscript t).
1. Write the government budget constraint as a function of σ and πe and plot
on the planer (πe, σ). Using equation (1) (take derivatives), determine the
steady state and find the maximum value of d that you can finance in
steady state through seigniorage (denote it by dM ). Suppose d < dM . How
many steady states there is? Show your results in the graph.
2. Suppose expectations are adaptive:
πe = β (π − πe) . (2)
Explain that equation. Differentiate equation (1) and using (2) to replace
replace the inflation rate, show what is the dynamic of expected inflation
in the graph, and find two steady states. Show which steady state is stable
and which one is unstable (assume that βα < 1).
3. Suppose there is an increase in the deficit from d to d′, both being smaller
than dM . Show the dynamics of the adjustment (remember that σ can
jump, but πe adjusts slowly). Finally, suppose that d increases some d′′
above dM and show that we get an hyperinflation.
Macroeconomía II, 2019/2 29
5 Exchange rates
Exercise 5.1
[Adapted from de Gregorio] Suppose an economy with three countries, A, B and
C. The value of imports in countries A and B are given by
qAMA = 250− 2qA + 0.4Y A
qBMB = 260− 2qB + 0.4Y B
where Y j denote output in country j and qj is the real exchange rate between
country j and country C. Exports in each country are given by
XA = 1200 + 3qA
XB = 100 + 2qB
I am implicitly assuming that A and B only export to and import from country
C, not between each other (that is why they only care about the real exchange
related to country C). Assume that Y A = 3000 and Y B = 300 are exogenously
given. Investment in each country is given by
IA = 1000− 2rA
IB = 150− rB
Macroeconomía II, 2019/2 30
Domestic savings in each country is given by
SAD = 350 + r
A + 0.2Y A
SBD = 10 + r
B + 0.2Y B
where rj denotes the real interest rate in each country.
1. Suppose these countries (A, B and C) live in financial autarky (they do not
borrow to or lend from the rest of the world). Find the equilibrium real
exchange rate and real interest rate in country A and B.
2. Suppose the barriers that do not allow country A and B to borrow or lend
between each other are removed (but country C is still in financial autarky).
Compute the equilibrium real exchange rates in each country.
3. Suppose now that there was also in change in tariffs after the financial
openness, so that the value of imports in country B fell 16 units (the
function that determined qBMB changed, changing the equilibrium MB).
What will happen with qB?
Exercise 5.2
1. Show that qt =
1+r∗t
1+rt
qet+1 (the notation is the standard one).
2. Express the Marshall-Lerner condition in terms of elasticities. You may
assume that the country has a zero trade deficit (X = qM ).
Exercise 5.3
[Proposed] Suppose that at date t you can buy or sell USD in future markets at
a price ft+1. In other words, agents agree at t that they will buy/sell dollars at
Macroeconomía II, 2019/2 31
t+ 1 at the agreed price ft+1.
1. Suppose all agents in the economy are risk neutral. Show that 1 + it =
(1 + i∗) ft+1
et
.
2. Do you expect that equation to hold in practice?
Exercise 5.4
[Proposed] Suppose inflation in the US is always 3%, while inflation in Chile
is always 2%. Suppose also that the nominal exchange rate between CLP and
USD is fixed. What do you expect to happen to the nominal interest rate in
Chile?
Exercise 5.5
[Proposed] Explain (using a graph) what you expect to happen to the real
exchange rate in each of the scenarios below.
1. The government increases expenditure without increasing taxes. Assume
the increase in government purchases only increase demand for domestic
goods.
2. The government reduces import tariffs. Suppose first it does not affect
other taxes and then that the government compensates the reduction by
increasing other taxes.
3. The productivity increases because the government found a lot of oil.
Macroeconomía II, 2019/2 32
6 Goods and money markets
Exercise 6.1
Consider the basic model of equilibrium in the goods market, which has the
following assumptions: (i) all firms produce the same good that can be used for
consumption, investment or by the government; (ii) the economy is closed; (iii)
firms are willing to supply any amount of the good at a given price level.
Consumption (C) is given by
C = c0 + c1(Y − T ),
where Y denotes the income, T denotes taxes (minus government transfers)
and c0 > 0 and c1 ∈ (0, 1) are parameters. Investment is exogenous and fixed at
some level I and the same applies to government spending, denoted by G.
1. What is the equilibrium level of output Y ?
2. Only on this item, suppose that there is some regulation that forces the
government to run a balanced budget. That is, G must always be equal to
T . Therefore, any increase in government spending must be compensated
by an equal increase in taxes. Can the government still affect the level of
output using fiscal policy (i.e., changing G and T , but keeping the budget
balanced)? Explain.
3. Now suppose that T depends on Y , according to
T = t0 + t1Y, with t0 ≥ 0, and t1 ∈ (0, 1).
Compute the equilibrium level of output Y . Compare the government
Macroeconomía II, 2019/2 33
spending multiplier you got in Item 1 with the one you got here. Which
multiplier is larger?
Exercise 6.2
Suppose that when the nominal interest rate i is larger than zero the money
demand (Md) is given by
Md = $Y (0.25− i),
where the nominal income $Y is $100. When i = 0, agents will demand at least
0.25 × $Y units of money, but are indifferent between holding any amount of
money equal or larger than 0.25× $Y . Suppose the supply of money is $20.
1. What is the equilibrium interest rate?
2. What is the equilibrium interest rate when the money supply is $30?
3. What happens to the equilibrium nominal interest rate when the central
bank increases the money supply from $30 to $40?
4. Suppose the money supply is $25 and the nominal income $Y increases
from $100 to $200. Will the equilibrium nominal interest rate increase?
Show what happens using a demand and supply graph.
Exercise 6.3
Consider the model of Chapter 3 in Blanchard, where investment and govern-
ment are fixed at I and G, respectively, and consumption is given by
C = c0 + c1(Y − T )
Macroeconomía II, 2019/2 34
where Y denotes real output and T are taxes, and c0 > 0 and c1 ∈ (0, 1). Answer
true or false to the statements below, justifying your answers.
1. An increase of 500 units in government expenditure has the same effect
on equilibrium output as a decrease of 500 units in taxes.
2. An increase of 10% in G always leads to an increase of 1/(1−c1)% in output.
3. The higher the marginal propensity to consume, the higher the increase in
GDP after a marginal decrease in taxes.
4. Suppose taxes are a fraction of total income, i.e., T = τY . The government
spending multiplier is larger than when T is fixed.
5. Suppose the government runs a balanced budget always, i.e.,T = G. An
increase of one unit in G causes an increase of one unit in output.
6. Suppose there are two group of people in a country (of equal size and each
earns half of the total income). One group has a marginal propensity to
consume higher than others. A policy that taxes the group with the low
propensity to consume and transfer the amount to the group with the high
propensity to consume will increase output.
Exercise 6.4
[Proposed] Answer the questionsbelow.
1. Consider a bond that promises to pay $100 in one year. What it is the
interest rate on the bond if its price today is $80, $95 and $110?
2. Suppose there are no costs associated with storing currency. Explain why
the nominal interest rate cannot go below zero.
Macroeconomía II, 2019/2 35
7 IS-LM
Exercise 7.1
Consider the basic IS-LM model where the central bank fixes the money supply.
Suppose country A is at the liquidity trap, as described in the graph below.
0
LM
IS
Output, Y
Interest
rate, i
Equilibrium
Now consider the following idea:
“Imagine that the Fed were to announce that, a year from today, it would
pick a digit from zero to 9 out of a hat. All currency with a serial number ending
in that digit would no longer be legal tender. Suddenly, the expected return to
holding currency would become negative 10 percent.
That move would free the Fed to cut interest rates below zero. People would
be delighted to lend money at negative 3 percent, since losing 3 percent is better
than losing 10.” Gregory Mankiw, NY Times, April 18, 2009
(http://www.nytimes.com/2009/04/19/business/economy/19view.html)
Assume that country A initially has notes with 10 different colors and the
total value of the currency of a given color is the same across colors. Based
Macroeconomía II, 2019/2 36
http://www.nytimes.com/2009/04/19/business/economy/19view.html
on the idea above, the central bank implements the following policy. Every
year the central bank chooses randomly one color (with equal probability for
each) and all notes with that color lose its value. Every time the central bank
picks the currency color that will have its legal tender status withdrawn, it
immediately puts the same amount of money back in the economy in notes
created with a color not yet used (think that money is thrown from a helicopter).
Thus, the money supply is fixed. People in that country only use currency to
make transactions.
1. How would that policy affect the equilibrium output in this economy? You
can use your intuition to answer it, or you can base it on your answer to
item 3 below.
2. Draw the money demand curve of this economy before and after the policy.
3. In the same graph, draw the IS and LM curves before and after the policy.
(Obs: the IS and LM curves before the policy are already drawn in the
graph above, but draw it again to compare with the new equilibrium).
Exercise 7.2
In a graph, represents what happens in the IS-LM after each of the shocks
below. Explain how the transition between the old equilibrium and the new
equilibrium occurs, plotting in a graph the path of the nominal interest rate
i and output Y . To derive the transitions, assume that money markets are
always in equilibrium, while goods markets can be not in equilibrium during the
transition. Assume that the bank central fixes the money supply (assumption
A1 in class).
1. A permanent increase in G.
Macroeconomía II, 2019/2 37
2. A permanent increase in M .
Exercise 7.3
Answer the questions below considering the following modifications of the
standard IS-LM model seen in class in which t the central bank fixes the money
supply (assumption A1).
1. Derive the IS curve assuming that the investment does not depend on the
interest rate. What is the impact of monetary and fiscal policy in this case?
2. Derive the LM curve assuming that the money demand does not depend
on the interest rate. What is the impact of monetary and fiscal policy in
this case?
3. Derive the LM curve assuming that the demand for real balances does not
depend on income. What is the impact of monetary and fiscal policy in
this case?
4. Suppose Md/P = aY , where a > 0. Derive the LM curve when a→∞.
Exercise 7.4
Answer the questions below using the IS-LM model for a closed economy in
which the central bank fixes the money supply.
1. Suppose a country wants to reduce its fiscal deficit, but at the same time,
it does not want to reduce output. Suggest a policy (or a mix of policies)
that could achieve those objectives. Explain it using the IS-LM diagram.
2. Suppose now that a government decides to increase spending (G), but to
keep the fiscal deficit constant it also increases a lump-sum tax (T ) in
Macroeconomía II, 2019/2 38
the same amount. Using the IS-LM diagram, explain the effect of these
policies.
3. Suppose the central bank increases the money supply in a permanent way
at some date τ . Represent the new equilibrium in the IS-LM diagram, as
well as the transition for the new equilibrium. To show the transition, plot
output and the nominal interest rate as a function of time. When analyzing
the transition, assume that money markets are always in equilibrium,
while goods markets adjust slow.
Exercise 7.5
Consider a closed economy in which the central bank fixes the money supply.
Short term activity is characterized by the following equations:
C = c0 + c1(Y − T )
I = I0 − I1r + I2Y
G = Ḡ
T = τY
M s
P
=
M̄
P
Md
P
= l0 − l1i
where C represents the aggregate consumption function, Y denotes income,
I aggregate investment, G is total public expenditure, T represents the level
of taxes (note that we are assuming taxes are proportional to income and not
Macroeconomía II, 2019/2 39
lump-sum as usual), M
s
P
and M
d
P
denote money supply and demand in real
terms, and P represents the price level. Finally c0, I0, I1, I2, Ḡ, M̄ , l0 are positive,
and τ, c1 ∈ (0, 1).
1. Define the IS curve. Determine its functional form at this economy.
2. Obtain the government spending multiplier and explain intuitively it repre-
sents.
3. Define the LM curve and determine its functional form.
4. Find an expression for the equilibrium output and nominal interest rate
in this economy, indicating step by step how you did obtain it.
5. Using derivatives, quantify the total effect on income of an increase in
money supply. Give the intuition of your results.
6. Why do we generally use without distinction i or r in this model?
Macroeconomía II, 2019/2 40
Exercise 7.6
Consider an economy in which the central bank fixes the money supply de-
scribed in the following set of equations.
C = 1 + 0.+ 0.8(1− t)Y
I = 2− 0.4i
G = 1
t = 0.2
M s = 6
Md = 0.75Y − 1.5i
The notation is the standard one. Note that income taxes are proportional to
income, instead of lump sum.
1. Provide Y and i values in equilibrium.
2. Assume that autonomous investment decreases by 10%. What level of
government spending restores previous income level?
3. Assume now that private investment is taxed proportionally to the invest-
ment rate. In particular, assume that:
I = 2− 0.4(1 + τ)i (1)
where τ = 0.25. Compute new equilibrium. Compare your results with
those at item 1.
Macroeconomía II, 2019/2 41
4. Assuming the new investment function, what level of money supply re-
stores income to the level of item 1? Explain your results.
Exercise 7.7
[De Gregorio] Consider a closed economy in which the central bank fixes the
money supply characterized by the equations below:
Y = C + I +G
C = C̄ + c1(Y − T )
I = Ī − αr
Md = kY − θi
i = r + πe
where Y represent income, C consumption, I denotes investment, G denotes
government expenditure, T represents taxes, i is nominal interest rate, r is real
interest rate, πe denotes expected inflation and Md is the real money demand
(note that this implicitly assumes a price level of 1). Finally c1, α, Ī, k, θ are
positive parameters. Denote the money supply my M s.
1. Assume initially that expected inflation equals π0. Obtain equilibrium
equations in money and goods markets. Graph each of them in the plane
(r, Y ).
2. Obtain expressions for equilibrium income Y , and real and nominal inter-
est rates.
Macroeconomía II, 2019/2 42
3. A change in economic expectations triggers a drop of expected inflation to
π1 < π0. Analyze how this change affects the economy.
4. What monetary policy should be implemented to stabilize income?
5. Assuming C = 5, I = 1,T = 2, G = 2, M s = 3, c = 0.8, α = 0.8, k = 0.25, θ =
0.75, π0 = 3, π1 = −5, obtain equilibrium variables before and after expecta-
tions adjust.
6. Given numerical results in the previous answer, which constraint restricts
the central bank when stabilizing income? How can the fiscal authority
help to overcome it?
Exercise 7.8
[Proposed] The government of a country, after years running large fiscal deficits,
is forced to reduce its deficit in a given year. Increasing taxes takes a lot of time
and it is not an available option in the short run. Using the IS-LM model where
the central bank fixes the nominal interest rate, explain how this country could
avoid a fall in output in the short run.
Exercise 7.9
[Proposed] Take two countries that are identical except for the behavior of the
central bank: in country A the central bank fixes the nominal interest rate (and
adjusts the money supply to achieve it), while in country B the central bank
fixes the money supply. Initially, both countries are at an equilibrium in which
the interest rate and output are the same. Suppose both countries experience
the same reduction in goverment spending. Which country will suffer a larger
output loss? Explain your answer using the IS-LM model.
Macroeconomía II, 2019/2 43
8 Mundell-Fleming
Exercise 8.1
Consider an open economy characterized by the following equations:
C = c0 + c1 (Y − T )
I = d0 + d1Y
IM = m1Y
X = x1Y
∗
where, C denotes consumption, Y denotes the domestic income, I denotes
investment, IM and X are the quantity of imports and exports, respectively.
The parameters m1 and x1 are the propensities to import and export. Assume
that the real exchange rate is fixed at a value of 1 and treat foreign income, Y ∗,
as fixed. Also assume that taxes, T , are fixed and that government purchases, G,
are exogenous (i.e., decided by the government). Note that differently from the
standard Mundell-Fleming model seen in class, here we assume that investment
does not depend on the interest rate.
1. Write the equilibrium condition in the market for domestic goods and solve
for Y .
2. What is the government spending multiplier? (Assume that 0 < m1 <
c1 + d1 < 1)
3. What is the change in net exports when government purchases increase
by one unit?
Macroeconomía II, 2019/2 44
4. Suppose the government decides to close the economy, not allowing trade
with the rest of the world (i.e., IM = X = 0). Find the government spending
multiplier. Is it larger or smaller than the multiplier you found in item 2?
Explain, in words, why the two multipliers are different.
Exercise 8.2
Suppose a country operates under a fixed exchange rate regime and has perfect
capital mobility. At date t, the central bank spends 100 units of local currency
buying domestic bonds in an open market operation. Suppose that between
dates t+ 1 and t financial investors had time to fully react to the open market
operation.
1. Is the monetary base at t + 1 larger, equal or smaller than the monetary
base at date t?
2. Did the central bank balance sheet change between dates t+ 1 and t? If
yes, how did it change?
Exercise 8.3
To answer the questions below use the Mundell-Fleming model of an open
economy.
1. Suppose a country has perfect capital mobility and operates under a
flexible exchange rate regime. Suppose the country is in a severe recession,
and policymakers are looking for a policy (or a mix of policies) to increase
output. Moreover, they would like to ensure net exports will not be affected
by those policies. Changing the exchange regime from flexible to fixed
Macroeconomía II, 2019/2 45
is not a posssibility. How could this country increase output without
affecting net exports? Explain using graphs and words.
2. Suppose now a country that has perfect capital mobility and operates
under a fixed exchange rate regime. Suppose there is a sudden exogenous
increase in the demand for exports (say that the income in the rest of the
world increased). What will be the effect of this higher demand for exports
on the amount of foreign currency reserves of the central bank? Explain
using graphs and words.
Exercise 8.4
To answer the questions below use the Mundell-Fleming model of an open
economy. You must always assume that Marshall-Lerner condition holds.
1. Consider a small open economy that has perfect capital mobility and
operates under a flexible exchange rate regime. Suppose the government
increases government spending. How does this shock affects equilibrium
output, nominal exchange rates and net exports? Explain using graphs
and words.
2. Suppose the Federal Reserve of the United States decides to increase the
US nominal interest rate. An economist argues that this could trigger a
recession in Chile. Is this prediction consistent with the Mundell-Fleming
model? Explain using graphs and words. In your answer, consider that
Chile is a small open economy that has perfect capital mobility and oper-
ates under a flexible exchange rate regime.
3. Suppose now Chile operates under a fixed exchange rate regime, with
perfect capital mobility. The domestic currency is the Chilean peso (CLP)
Macroeconomía II, 2019/2 46
and the foreign currency is the US dollar (USD). Moreover:
• The exchange rate is fixed at one (1 CLP costs 1 USD);
• Initially the monetary base in Chile is 100 CLP;
• Initially the Chilean central bank holds 50 USD of (dollars) reserves
(those are foreign exchange reserves held by the Central Bank of
Chile).
Explain why the Central Bank of Chile has no control of monetary policy
under this fixed exchange rate regime. You must do it by explaining what
happens after the Central Bank of Chile decides to buy 10 CLP worth
of domestic bonds in an open market operation (say, in an attempt to
increase the monetary base). In your answer you must indicate what is
the level of the monetary base and the dollar reserves of Central Bank of
Chile after the open market operation and after all adjustments have taken
place (you should provide the exact number of those variables).
4. Consider a small open economy that has perfect capital mobility and
operates under a fixed exchange rate regime. Suppose the government
increases government spending. How does this shock affects equilibrium
output? Explain using graphs and words.
Exercise 8.5
[Proposed] Answer the following questions according to the Mundell-Fleming
model with perfect capital mobility and flexible exchange rates seen in class (de
Gregorio, chapter 20).
1. How does an increase in government spending affects output, interest
rates and the exchange rate? Explain using graphs and words.
Macroeconomía II, 2019/2 47
2. How does an increase in the money supply affects output, interest rates
and the exchange rate? Explain using graphs and words.
3. How does an increase in the international interest rate affects output
interest rates and the exchange rate? Explain using graphs and words.
4. Answer items 1, 2 and 3 again assuming a fixed exchange rate regime
instead of flexible exchange rate regime.
5. Still assuming a fixed exchange rate regime and perfect capital mobility,
what happens when the central bank increases the fixed exchange rate?
Explain using graphs and words.
Macroeconomía II, 2019/2 48
9 Phillips curve
Exercise 9.1
Consider the following modified version of the Lucas model. There is a contin-
uum [0, 1] of firms indexed i. Pi denotes the price level of firm i and ysi is the
quantity produced by firm i. Total output is y =
∫ 1
0
ysi di.
The price level P is drawn from a normal distribution with mean µP and
variance 1/τP . The price of each firm is Pi = P + ri, where ri denotes the relative
price of each firm. ri is drawn from a normal distribution with zero mean and a
variance 1/τr (iid across firms).
Each firm observes two informations before deciding how much to produce:
(i) its own price Pi; (ii) a noisy signal wi = ri + ζi, where ζi ∼ N(0, 1/τw) (iid
acrossfirms). The parameter τw is called the precision of the signal wi and it
represents how good is the information received by firm i about its relative
price. If τζ →∞, firms learn almost perfectly about its relative price. This signal
is meant to capture information about their relative prices firms gather from
various sources.
Once firms have set their expectation of ri conditional on Pi and wi (denoted
by E [ri|wi, Pi]) they produce according to
ysi = y + γE [ri|wi, Pi]
Tip: If an agent has prior N(y, 1/τy) and receives two signals xA = z + ηA and
xB = z+ηB, with ηA ∼ N(0, 1/τA) and ηB ∼ N(0, 1/τB) about some random variable
z, then E
[
z|xA, xB
]
= τAx
A+τBx
B+τyy
τA+τB+τy
. The τ ’s are called precisions (it is the inverse
of the variance).
Macroeconomía II, 2019/2 49
1. Compute E [ri|wi, Pi].
2. Write total output as function of P and E [P ] (which is the parameter µP ).
3. What happens to your answer of the previous item when τw →∞? Interpret
it.
Exercise 9.2
In class we have seen three different types of frictions that can generate a
positive relationship between output and inflation: (i) wage rigidities; (ii) price
rigidities (Calvo’s model); and (iii) information rigidities (Lucas’ model). Briefly
explain in words how they can generate a positive relationship between output
and inflation (the Phillips curve). You do not need to solve the models, you
should only clearly explain in words the economic intuition behind it.
Exercise 9.3
[Proposed] Suppose an economy with three different types of firms.
The first type has flexible prices (they can adjust it freely in each date) and
they fix it according to:
pft − pt = κ (yt − y) , κ > 0
where pft denotes the price set by firms with flexible prices; pt is the price level;
yt is output and y is potential output. All variables are expressed in logarithms.
The second type of firms has fixed prices, and they fix their price prt accord-
ing to their expectations of the price level and output:
prt − pet = σ (yet − y) , σ > 0
Macroeconomía II, 2019/2 50
Let’s assume that expected output is always equal to potential output: yet = y
for every t. Let αr denote the fraction of firms with fixed prices
The third type of firms fix their prices at date t according to the past price
level pt−1, readjusting the price by the inflation rate (indexation). In particular
those firms set their price pit according to:
pit = pt−1 + πt−1
The proportion of those firms is αi. Hence, there is a proportion 1− αr − αi of
firms with flexible prices. Since all variables are expressed in logarithms, you
can use the approximations: pt−pt−1 = πt and pt−pet = pt−pt−1−(pet−pt−1) = πt−πet ,
where the superscripts e denote expectations and πt is the inflation rate.
1. Show that:
yt = y +
1
λ
[αi (πt − πt−1) + αr (πt − πet )]
where λ = (1− αr − αi)κ.
2. Interpret the equation you found in the previous item. Compare it with
the situation in which there are no firms that index their prices (αi = 0).
Macroeconomía II, 2019/2 51
10 IS-LM-PC
Exercise 10.1
Consider the IS-LM-PC model and suppose there is an increase the price oil.
Intepret this increase in the price of oil as a change in potential output. Assume
that before the shock output is equal to potential output. Compare the response
of the economy in two cases: (i) expected inflation always equal lagged inflation;
(ii) inflation expectations are anchored at some level π. In the medium run, the
central bank adjusts interest rates to keep it as close as possible to potential
output. You may assume that initially the interest rates are very high, so
that the zero lower bound will not be binding. Write a graph with time on the
horizontal axis and the path of output and inflation on the vertical axis, for
each of the cases mentioned.
Exercise 10.2
Consider the IS-LM model where the central bank fixes the real interest rate.
Assume that inflation expectations are anchored at some level π. Suppose
the government decides to reduce government spending in a permanent way.
Also, assume interest rates are sufficiently high, so that the central will not be
constrained by the zero lower bound (therefore, you can simply ignore the zero
lower bound).
1. What is the effect on output of this change in the short run? Explain your
answer using the IS-LM model.
2. What is the effect on output of this change in the medium run, assuming
the central bank will try to make inflation equal to π? Explain your answer
Macroeconomía II, 2019/2 52
using the IS-LM-PC model.
3. In the medium run, the IS-LM-PC model predicts that investment will
increase or decrease? Justify.
Exercise 10.3
You should provide your answer to this question using the IS-LM-PC model seen
in class. Suppose that at the initial date, an economy has inflation, expected
inflation and real interest equal to 0. The natural real interest rate of this
economy (i.e., the real interest rate that would make output equal to potential
output) is negative. Expected inflation is always equal to the inflation of the
previous period (i.e., πet = πt−1). The central bank always tries to bring output
as close as possible to potential output, but nominal interest rates cannot go
below zero.
1. According to the IS-LM-PC model, after the intial date the real interest
rate will increase, decrease or remain stable? And output will increase,
decrease or remain stable? Explain (drawing a graph may be very useful
for that).
2. Explain, using the IS-LM-PC model: can fiscal policy make output equal
to potential output? How?
Exercise 10.4
Brazil’s policy real interest rate are very high. Many economists argue that part
of that is caused by the fact that there is a development bank in Brazil (called
BNDES) lending at interest rates way below the the market interest rate (for
some selected firms). They argue that it makes monetary policy less powerful,
Macroeconomía II, 2019/2 53
forcing the central bank to increase the interest rate a lot to reduce inflation.
The following simple extension of the IS-LM-PC model is proposed to check if
that intuition survives a more formal treatment and to try to understand better
the mechanisms behind this kind of argument.
Suppose that a fraction 1− λ of firms in the economy borrow at the policy
rate plus the risk premium (r + x), and a fraction λ borrows at (1 − s) (r + x)
from the BNDES, where s denotes the BNDES subsidy. Each firm has a linear
investment function on Y and on the interest rate they borrow, so that the
aggregate investment is equal to:
I = b0 + (1− λ) [b1Y − b2(r + x)] + λ [b1Y − b2(1− s)(r + x)]
The consumption function is standard: C = c0 + c1(Y − T ), with the usual
assumption that b1 + c1 < 1. Also, inflation expectations are anchored at π.
1. Suppose that initially λ = 0 (no BNDES) and output is equal to potential.
At date t λ becomes positive (the development bank is introduced). How
will that affect the equilibrium in the short and medium run?
2. Now suppose that the government decides to increase λ but at the same
time increase taxes, so that the short run equilibrium remains the same.
What happens in the medium run?
3. Now consider the case with λ = 0 and the case where the government
increases λ and increases taxes as in item 2. Suppose there is an increase
in c0 and compare the medium run equilibrium in both cases. Does it
make sense to say that monetary policy became less powerful because of
the subsidized credit?
Macroeconomía II, 2019/2 54
Exercise 10.5
Consider a country that has had a very low level of investment for decades. Some
economists argue that that investment is low because government spending
is very high. Suppose that the government of this country decides to reduce
government spending (G) in a permanent way.
Using the IS-LM-PC model, answer the questions below. Assume that
inflation expectations are anchored at some level π. Also, assume that interest
rates are sufficiently high, so that the centralbank will not be constrained by
the zero lower bound (therefore, unless otherwise stated, you can simply ignore
the zero lower bound). As usual in the IS-LM-PC model, you must assume that
the central bank will not react to shocks in the short run, but will adjust real
interest rates in order to bring output as close as possible to potential output in
the medium run. Also, assume that before the change in government spending,
the economy was in an equilibrium with output equal to potential output.
1. What is the effect on output of the reduction in government spending (G)
in the short and medium run? Explain your answer using the IS-LM-PC
diagram.
2. In the short run, is investment larger or smaller after the reduction in G?
In the medium run is investment larger or smaller after the reduction in
G? In your answer, you must assume that investment is an increasing
function of output and a decreasing function of the real interest rate (as
usual).
Now we replace some assumptions. First, instead of assuming anchored
expectations, we assume adaptative expectations. Second, we assume that at
the initial date inflation, expected inflation, nominal and real interest rates are
Macroeconomía II, 2019/2 55
all equal to zero (using the notation used in class i = π = πe = r = 0 at the initial
date). Hence, you can no longer ignore the zero lower bound. The economy still
has output equal to potential output at the initial date. Answer the question
below.
3. After a permanent decrease in government spending, what will happen
to the real interest rate in the medium run? Explain using the IS-LM-PC
diagram.
Macroeconomía II, 2019/2 56
11 AS-AD (only if time allows)
Exercise 11.1
Consider the basic 3 equation aggregate demand and aggregate supply model
(the notation is the same used in class):

π = πe + θ (y − y) + ε (OA)
y − y = A− φ (i− π) + µ (IS)
i = r + π + a (π − π) + b (y − y) (Taylor rule)
where r = A/φ and the same condition on parameters and shocks assumed in
class is satisfied.
1. Derive the RPM curve seen in class and represent the equilibrium in a
OA-RPM diagram.
2. Suppose the economy is initially at its long run equilibrium π = π = πe,
y = y, r = r = A/φ and i = i = r + π. Represent and discuss in a OA-RPM
diagram the effect of a temporary and positive inflationary shock ε > 0.
3. Can the OA-RPM diagram say something about what happens with the
interest rate after a inflationary shock?
4. Write the output gap y − y and π − π as a function of parameters and
the exogenous shocks ε and µ. Find a condition on the parameter b that
implies an increase in interest rates after positive inflationary shock.
5. Suppose the central bank wants to minimize the variation of the output
gap. Assume the constant a > 1 is fixed, but the central bank can adjust b
Macroeconomía II, 2019/2 57
to achieve his goal. What should the central bank do? You can assume
that πe is constant. Remember that the shocks ε and µ are assumed to be
independent and iid.
Exercise 11.2
[Proposed, adapted from de Gregorio] Consider the basic aggregate demand and
aggregate supply model (the notation is the same used in class):

π = πe + θ (y − y) + ε (OA)
y − y = A− φ (i− π) + µ (IS)
1. Suppose the central bank follows a rule to keep the nominal interest rate
always constant at some level î. What is the implicit inflation target (π)?
Suppose inflation expectations are always equal to the inflation target.
Plot the Phillips curve and the monetary policy rule on the (y, π) space.
Find the equilibrium output and inflation as a function of paramters and
shocks.
2. Suppose now the central bank follows an usual Taylor rule, with a > 1 and
b > 0. Is the variance of inflation higher or lower than in item 1? Remember
that the shocks ε and µ are assumed to be independent and iid.
Macroeconomía II, 2019/2 58
12 Dynamic inconsistency
Exercise 12.1
Suppose the loss function of the central bank is given by
L = π2 + γ (u− un + κ)2 .
The Phillips curve is given by u = un − η (π − πe). The parameters γ,η and κ are
both strictly larger than zero.
1. Compute the equilibrium inflation under discretion.
2. Compute the equilibrium inflation under full commitment.
3. Is the central bank loss larger under discretion or under full commitment?
Justify doing the algebra.
Exercise 12.2
Time starts at date 0. The loss function of the central bank at date t is given by
Vt = π
2
t + λ (yt − y − κ)
2
and the Philips curve is as usual: yt = y + θ (πt − πet ). At each date τ , the central
bank minimizes the discounted sum of Vt:
Wτ =
∞∑
t=τ
βt−τVt
where β ∈ (0, 1) and λ, θ > 0. We know that under discretion in an one period
model, the central bank would choose π = πq = λθκ, and we also know that
Macroeconomía II, 2019/2 59
under full commitment the inflation would be zero. Now suppose people forms
their expectations according to:
πet =

0 if πτ = 0,∀τ < t
λθκ otherwise
In other words, as long as the central bank keeps choosing zero inflation,
people believe it will do that in the future. If it chooses some inflation different
from zero, people believe it will choose the inflation consistent with a static
equilibrium under discretion, which is π = πq = λθκ. Assume that at date τ the
central has chosen zero inflation for every t < τ .
1. What is the value of Wτ if the central bank chooses zero inflation forever
(that is, if it chooses πt = 0 for every t ≥ τ )?
2. Suppose the central bank decides to deviate from the strategy assumed in
item 1 and chooses an inflation rate different from zero at date τ (but he
does the best deviation possible). In that case, what is the value of Wτ?
3. Write a condition on the parameters that makes the value of Wτ you found
in item 2 smaller than the value of Wτ you found in item 1. Interpret.
Exercise 12.3
Time starts at date 0. The loss function of the central bank at date t is given by
Lt = ut + γπ
2
t
Macroeconomía II, 2019/2 60
and the Philips curve is as usual: ut = un−η (πt − πet ). At each date τ , the central
bank minimizes the discounted sum of Lt:
∞∑
t=τ
βt−τLt
We know that under discretion in an one period model, the central bank would
choose π = πd ≡ η
2γ
, and we also know that the optimal inflation is zero. Now
suppose people forms their expectations according to:
πet =

0 if πτ = 0,∀τ < t or πτ 6= 0 for τ ∈ {t− k, ..., t− 1}
η
2γ
otherwise
In other words, as long as the central bank keeps choosing zero inflation,
people believe it will do that in the future. If he chooses some different inflation
at some date, people believe he will choose the inflation consistent with a static
equilibrium under discretion, πe = η
2γ
, for k periods, where k ≥ 1, and then they
will believe he will choose zero again.
Suppose the central bank is at a date τ such that it has chosen zero inflation
in all previous periods. Is it optimal for the central bank to choose zero inflation
in all periods in the future?
Exercise 12.4
Suppose a central banker loss function (L) depends on inflation (π) and unem-
ployment (u) and is given by:
L = u+ γπ2, γ > 0
Macroeconomía II, 2019/2 61
where γ > 0 and π denotes the inflation rate. The Phillips curve in terms of
unemployment is:
u = un − η (π − πe) , η > 0
where u denotes the unemployment rate, un is the natural unemployment and
πe is the expected inflation.
1. Find the equilibrium π and L under discretion.
2. Find the equilibrium π and L under commitment.
3. Suppose the central banker can deviate from the inflation he promised
under commitment. What is its loss L in that case? Which inflation will it
choose?
4. Why do we say the central banker is time inconsistent? Explain in words.
5. Suppose that instead of playing the game with discretion, the central bank
can delegate the conduction of monetary policy to a third party who has
a loss function L̃ = u+ αγπ2, with α > 1 (who will then play a game under
discretion). Wouldthe central banker be willing to do so? Justify your
answer. (10%)
In what follows consider a dynamic version of the problem presented. Time is
discrete and indexed by t ∈ {0, 1, 2, ...}. The central banker loss at function at
any date τ is:
V =
∞∑
t=τ
β(t−τ)Lt =
∞∑
t=τ
β(t−τ)
[
ut + γπ
2
t
]
The Phillips curve is as before but with time subscripts:
ut = un − η (πt − πet )
Macroeconomía II, 2019/2 62
Denote by Ldisc, Lcom and Ldev the losses you found in item 1, 2 and 3, respec-
tively. Denote by πdisc, πcom and πdev the inflation you found in item 1, 2 and 3,
respectively.
6. Suppose the inflation setting game under discretion is played at every date.
Also, assume that agents form expectations according to:
πet =

πcom if πt−j, for all j > 0
πdisc otherwise
Write a condition on β that guarantees that the central bank will prefer to
cooperate (i.e., choose π = πcom at every future date). Write the conditions
in terms of β, Ldisc, Lcomand Ldev. Interpret the condition.
7. In light of your answer to item 6, do you think that guaranteeing that a
central banker will have a long term in his position alleviates or exacerbates
the time inconsistency problem? Explain in words.
Exercise 12.5
Suppose the central bank loss function (L) depends on inflation (π) and unem-
ployment (u) and is given by:
L = u+ γπ2, γ > 0
The Phillips curve in terms of unemployment is:
u = un − η (π − πe) , η > 0 (PC)
Macroeconomía II, 2019/2 63
where u denotes the unemployment rate, un is the natural unemployment rate
and π and πe are inflation and expected inflation, respectively.
1. Find the equilibrium π and L under discretion.
2. Find the equilibrium π and L under commitment.
In what follows, consider the following modification to the model presented.
Suppose the timing is the following:
Stage 1. The central bank announces some inflation πA;
Stage 2. Agents form their expectations πe about inflation, to minimize the
forecast error (π − πe)2;
Stage 3. The central bank fixes the inflation π, and given the inflation
chosen unemployment is determined according to the Phillips curve (PC).
We modify the loss function of the central bank, which is now given by:
L̂ = u+ γπ2 + ψ
(
π − πA
)2
, γ > 0, ψ > 0
Notice that we added the term ψ
(
π − πA
)2 to the central bank loss function. It
captures, in a reduced form way, the reputation losses of deviating from the
announced inflation πA. Moreover, notice that we are not saying that central
bank necessarily needs to honor the announced inflation πA.
3. Find the equilibrium π, πA and πe.
4. In equilibrium, does the central bank loss increase or decrease with ψ?
Interpret this result.
Macroeconomía II, 2019/2 64
Exercise 12.6
[Proposed] A central bank has decided to adopt inflation targeting and is now
debating whether to target 5 percent inflation or zero inflation. The economy is
described by the following Phillips curve:
u = 5− 0.5 (π − πe)
where u and π are the unemployment rate and inflation rate measured in
percentage points. The social cost of unemployment and inflation is described
by the following loss function:
L = u+ 0.05π2
The central bank would like this loss to be as small as possible.
1. If the central bank commits to targeting 5 percent inflation, what is ex-
pected inflation? If the central bank follows through, what is the unem-
ployment rate? What is the loss from inflation and unemployment?
2. If the central bank commits to targeting zero inflation, what is expected
inflation? If the central bank follows through, what is the unemployment
rate? What is the loss from inflation and unemployment?
3. Based on your answers to items 1 and 2, which inflation target would you
recommend? Why?
4. Suppose the central bank chooses to target zero inflation, and expected
inflation is zero. Suddenly, however, the central bank surprises people with
5 percent inflation. What is unemployment in this period of unexpected
Macroeconomía II, 2019/2 65
inflation? What is the loss from inflation and unemployment?
5. What problem does your answer to part 4 illustrate?
Exercise 12.7
[Proposed] Suppose the loss function of the central bank is given by
V = π2 + λ (y − y − κ)2 .
The Phillips curve is given by y = y + θ (π − πe). The parameters λ, θ and κ are
both strictly larger than zero.
1. Compute the equilibrium inflation under discretion.
2. Compute the equilibrium inflation under full commitment.
3. Is the central bank loss larger under discretion or under full commitment?
Justify doing the algebra.
4. What happens when κ = 0? Explain with equations and words.
Macroeconomía II, 2019/2 66
	Money supply
	Money demand
	Monetary equilibrium
	Seignioriage
	Exchange rates
	Goods and money markets
	IS-LM
	Mundell-Fleming
	Phillips curve
	IS-LM-PC
	AS-AD (only if time allows)
	Dynamic inconsistency