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Exercises before final exam (first part)
Caio Machado
Instituto de Econoḿıa
Pontificia Universidad Católica de Chile
Macroeconomia II, 2017
Exercise 1
a. Using the definition of the real exchange rate and some approximations
show that (the notation is standard)
εt −εt−1
εt−1
= Et −Et−1Et−1
+πt −π∗t
In words, the percentage real appreciation equals the percentage nominal
appreciation plus the difference between domestic and foreign inflation.
The definition of the real exchange rate says that:
εt =
EtPt
P∗t
Taking logs
logεt = logEt + logPt − logP∗t (1)
Writing the same expression above for t + 1:
logεt−1 = logEt−1 + logPt−1− logP∗t−1 (2)
Subtracting (2) from (1)
logεt− logεt−1 = (logEt − logEt−1)+(logPt − logPt−1)−
(
logP∗t − logP∗t−1
)
Since logX − logY ≈ (X −Y )/Y , we get to the desired expression.
b. If domestic inflation is higher than foreign inflation, and the domestic
country has a fixed exchange rate, what happens to the real exchange rate
over time? Assume that the Marshall-Lerner condition holds. What
happens to the trade balance over time? Explain in words.
If the domestic country has a fixed exchange rate, the equation shown in
the previous item becomes:
εt −εt−1
εt−1
= πt −π∗t
Thus, if domestic inflation is higher than foreign inflation, the real exchange
rate increases over time. If the Marshall-Lerner condition holds, a real
appreciation induces a worsening of the trade balance (because the
Marshall-Lerner conditions assumes a negative and stronger effect on
imports than on exports).
c. Suppose the real exchange rate is currently at the level required for net
exports (or the current account) to equal zero. In this case, if domestic
inflation is higher than foreign inflation, what must happen over time to
maintain a trade balance of zero?
There are many different things that could happen. In particualr, we could
have nominal depreciation over time, that is:
Et −Et−1
Et−1
=−(πt −π∗t )
This would guarantee a constant real exchange rate, mantaining the zero
trade balance (everything else constant, of course).
Exercise 2
[Policy coordination and the world economy.] Consider an open economy in
which the real exchange rate is fixed and equal to one. Consumption,
investment, government spending, and taxes are given by
C = 10 + 0.8(Y −T ), I = 10, G = 10, and T = 10. Imports and exports are
given by IM = 0.3Y and X = 0.3Y ∗, where Y ∗ denotes foreign output.
a. Solve for equilibrium output in the domestic economy, given Y ∗. What
is the multiplier in this economy? If we were to close the economy—so
exports and imports were identically equal to zero—what would the
multiplier be? Why would the multiplier be different in a closed economy?
To clearly see the multiplier, it is useful to define the Autonomous Domestic
Demand as ADD = 10−0.8T + I + G = 22 (it is the sum of the
components of domestic demand that do not depend on Y).
Equilibrium output will be given by
Y = ADD + 0.8Y + 0.3Y ∗−0.3Y
(1−0.8 + 0.3)Y = ADD + 0.3Y ∗
Y = 2[ADD + 0.3Y ∗] = 2[22 + 0.3Y ∗]
One can see that the multiplier in this economy is 2.
If we close the economy, the equilibrium would be
Y = ADD + 0.8Y
(1−0.8)Y = ADD
Y = 5×ADD = 110
Thus, the multiplier in this economy is 5, much larger than the multiplier in
the open economy. The reason for that is that in the open economy, part of
the increase in demand that comes from higher income does not become
demand for domestic goods, but instead demand for foreign goods.
b. Assume that the foreign economy is characterized by the same equations
as the domestic economy (with asterisks reversed). Use the two sets of
equations to solve for the equilibrium output of each country. [Hint: Use
the equations for the foreign economy to solve for Y ∗ as a function of Y
and substitute this solution for Y ∗ in part (a).] What is the multiplier for
each country now? Why is it different from the open economy multiplier in
part (a)?
Using our previous equation, the equilibrium output in the foreign country is
Y ∗ = 2[ADD∗+ 0.3Y ]
where ADD∗ = ADD = 22 here too. Plugging the equation for equilibrium
output in the domestic country Y = 2[ADD + 0.3Y ∗] we have
Y ∗ = 2{ADD∗+ 0.3[2(ADD + 0.3Y ∗)]}
Y ∗ = 3.125[ADD∗+ 0.6ADD] = 110
Similarly, we can find that
Y = 3.125[ADD + 0.6ADD∗] = 110
The multiplier now is 3.125, for both countries. This is higher than in part
A because now an increase in domestic demand leads also to an increase in
foreign demand (exports), which causes a further increase in output, which
causes a further increase in both domestic and foreign demand and so on.
c. Assume that the domestic government, G , has a target level of output of
125. Assuming that the foreign government does not change G∗, what is
the increase in G necessary to achieve the target output in the domestic
economy? Solve for net exports and the budget deficit in each country.
Since domestic output is given by Y = 3.125[ADD + 0.6ADD∗]. To achieve
Y = 125 we need to solve
3.125[ADD + 0.6ADD∗] = 125 =⇒ ADD = 40−0.6ADD∗ = 26.8
Thus, the Autonomous Domestic Demand needs to go from 22 to 26.8, and
thus the government needs to increase expenditures in 4.8 units.
The output in the foreign country will be given by
Y ∗ = 3.125[ADD∗+ 0.6ADD] = 3.125[22 + 0.6×26.8] = 119
Which implies net exports of NX = 0.3×119−0.3×125 =−1.8 in the
domestic country and NX ∗ = 0.3×125−0.3×119 = 1.8 in the foreign
country.
The budget deficit is 0 in the foreign country and -4.8 in the domestic
country.
d. Suppose each government has a target level of output of 125 and that
each government increases government spending by the same amount.
What is the common increase in G and G∗ necessary to achieve the target
output in both countries? Solve for net exports and the budget deficit in
each country.
To solve for the necessary increase in government expenditures, we assume
that both countries are going to implement the same increase. Thus, ADD
and ADD∗ will remain equal to each other. We can thus solve:
3.125[ADD∗+ 0.6ADD] = 3.125[1.6ADD] = 125
Which implies
ADD = 25
Thus, both countries need to increase their government expenditures in 3
units.
Net exports in both countries will be equal to zero, that is
NX = 0.3×125−0.3×125 = NX ∗.
The budget deficit is -3 in both countries.
e. Why is fiscal coordination, such as the common increase in G and G∗ in
part (d), difficult to achieve in practice?
It is difficult because an increase in G in the domestic country reduces the
incentives of the foreign country to increase G∗, since it already increases
output a bit in the foreign country. For instance, suppose the foreign
country knows the domestic economy will achieve a output of 125 no
matter what. He can cooperate as in item (d) and also increase G∗, in
which case both end up with a zero trade balance, a output of 125 and
budget deficit. But he may prefer the situation of item (c): only one
country do a fiscal expansion, and the other end up with a slightly smaller
output. On the other hand, the country that did not the fiscal expansion
will have no budget deficit and a trade surplus.
Exercise 3
[Flexible exchange rates and foreign macroeconomic events.] Consider an
open economy with flexible exchange rates. Let UIP stand for the
uncovered interest parity condition.
a. In an IS-LM–UIP diagram, show the effect of an increase in foreign
output, Y ∗, on domestic output (Y ) and the exchange rate (E ), when the
domestic central bank leaves the policy interest rate unchanged. Explain in
words.
Remember we can write the IS relation as:
IS relation: Y = C(Y −T ) + I(Y , i) + G + NX
Y ,Y ∗, 1 + i1 + i∗E e︸ ︷︷ ︸
E

When Y ∗ increase, net exports increase, since exports increase. This shifts
the IS curve.
i i
Y E
IS ′IS
UIP
A A′
∆X > 0
LM
b. In an IS-LM–UIP diagram, show the effect of an increase in the foreign
interest rate, i∗, on domestic output (Y ) and the exchange rate (E ), when
the domestic central bank leaves the policy interestrate unchanged.
Explain in words.
The increase in the foreign exchange rate will shift down the UIP curve,
reducing E . This reduction in E (assuming the Marshall-Lerner condition
holds), induces a real depreciation, increasing NX and shifting IS to the
right. If we consider that the higher i∗ decreased foreign output, then we
would also have a force shifting the IS to the left (it is not clear which
effect would dominate). Thus, in the graph below I assume that only i∗
increased (think of the foreign government using fiscal policy to offset the
impact on output).
i i
Y E
IS ′IS
UIP
A A′
∆E < 0
UIP ′
A′ ALM
Exercise 4
[Flexible exchange rates and the responses to changes in foreign
macroeconomic policy.] Suppose there is an expansionary fiscal policy in
the foreign country that increases Y ∗ and i∗ at the same time.
a. In an IS-LM–UIP diagram, show the effect of the increase in foreign
output, Y ∗, and the increase in the foreign interest rate, i∗, on domestic
output (Y ) and the exchange rate (E ), when the domestic central bank
leaves the policy interest rate unchanged. Explain in words.
When the foreign interest rate increases, UIP shifts to the left, which
reduces the nominal exchange rate (and consequently the real exchange
rate). This increases net exports (assuming Marshall-Lerner), shifting the IS
to the left. The increase in Y ∗ further shifts the IS to the left, by increasing
exports.
i i
Y E
IS ′IS
UIP
A A′
∆NX > 0
UIP ′
A′ ALM
b. In an IS-LM–UIP diagram, show the effect of the increase in foreign
output, Y ∗, and the increase in the foreign interest rate, i∗, on domestic
output (Y ) and the exchange rate (E ), when the domestic central bank
matches the increase in the foreign interest rate with an equal increase in
the domestic interest rate. Explain in words.
Remember that the UIP relation is
E = 1 + i1 + i∗E
e
Thus, if the central bank increases i in the same proportion as the foreign
central bank, the equilibrium E will be the same (even the increase is the
same only in absolute terms, the linear approximation of UIP tells us it
should be almost the same). The IS curve will still shift to the right
because the increase in Y ∗. Depending on the size of the shift, the new
output level can be higher or lower. The graph below shows the case in
which it is higher.
i i
Y E
IS ′IS
UIP
A B
∆Y ∗ > 0
UIP ′
B
ALM
LM ′A
′ A′
c. In an IS-LM–UIP diagram, show the required domestic monetary policy
following the increase in foreign output, Y ∗, and the increase in the foreign
interest rate, i∗, if the goal of domestic monetary policy is to leave
domestic output (Y ) unchanged. Explain in words. When might such a
policy be necessary?
The graph below shows the increase in domestic interest rates needed
(LM’) after the IS shifts. Regarding the equilibrium exchange rate in A’
that could be higher, lower or even equal than the exchange rate at A,
depending on the size of the shift in IS and UIP.
i i
Y E
IS ′IS
UIP
A
UIP ′
ALM
LM ′A
′ A′
Exercise 5 (much more difficult)
[This looks similar to the one I did in class, but look carefully]. Time starts at date 0.
The loss function of the central bank at date t is given by
Lt = ut +γπ2t
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−τ Lt
We know that under discretion in an one period model, the central bank would choose
π = πd ≡ η2γ , and we also know that the optimal inflation is zero [show that again to
practice!]. 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 = η2γ , 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?
Suppose that the central bank does not deviate from the strategy. His
payoff will then be
V Cooperate =
∞∑
t=τ
βt−τ un =
un
1−β
If he deviates, he will choose πτ to minimize Lτ . Plugging the Phillips curve
into the loss function and using πeτ = 0
Lτ = un−ηπτ +γπ2τ
The first order condition is
−η+ 2γπτ = 0 =⇒ πτ =
η
2γ
Which yields a loss function of Lτ = un− η
2
2γ +
η2
4γ = un−
η2
4γ . For the next k
periods then, he will choose the discretion inflation πτ = η2γ , but people will
choose πe = η2γ , yielding a loss un +
η2
4γ . After date τ + k, he gets his loss of
cooperation again.
Thus, the payoff of this deviation is
V Deviate =
(
un−
η2
4γ
)
+
τ+k∑
t=τ+1
βt−τ
(
un +
η2
4γ
)
+βk+1V Cooperate
Note we can rewrite
τ+k∑
t=τ+1
βt−τ
(
un +
η2
4γ
)
=β
(
un +
η2
4γ
)
+β2
(
un +
η2
4γ
)
+· · ·+βk
(
un +
η2
4γ
)
= β1−β
(
un +
η2
4γ
)
− β
k+1
1−β
(
un +
η2
4γ
)
= β−β
k+1
1−β
(
un +
η2
4γ
)
Thus, the payoff of deviating for k periods
V Deviate =
(
un−
η2
4γ
)
+ β−β
k+1
1−β
(
un +
η2
4γ
)
+βk+1

V Cooperate︷ ︸︸ ︷
un
1−β

=
(
un−
η2
4γ
)
+ un
β
1−β +
β−βk+1
1−β
η2
4γ =
un
1−β −
η2
4γ +
β−βk+1
1−β
η2
4γ
Remark: of course, we are considering the best deviation he can do (if he deviates
for k + 1 periods). If it is not optimal to deviate k + 1 periods, it will not be
optimal to deviate for longer periods. (If it is not optimal to deviate at date τ , it is
not optimal to deviate at {τ + k, τ + k + 1, ...} since he will face the exact same
problem).
The deviation is not profitable when V Deviate > V Cooperate
un
1−β −
η2
4γ +
β−βk+1
1−β
η2
4γ >
un
1−β
− η
2
4γ +
β−βk+1
1−β
η2
4γ > 0
2β−βk+1 > 1
Thus, if the condition above is satisfied, there is an equilibrium with
cooperation. Two remarks:
• Note that as k → 0 this conditions becomes 2β > 1, which is
equivalent to β > 1/2 (the same condition we had a few classes ago in
a similar exercise).
• As k becomes larger, this condition is more easily satisfied. Intuitively,
if agents punish the central bank for a lot of periods, it is easier to
have cooperation.