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Macroeconomía II – Problem Set
TA Session 7
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Some of those exercises will be solved on the ayudantia of November 16, 2018.
Phillips curve
Exercise 1
Consider the following modified version of the Lucas model. There is a continuum [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 y
s
i 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 across firms). 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+τBxB+τyy
τA+τB+τy . The τ ’s are called precisions (it is the inverse of the
variance).
1. Compute E [ri|wi, Pi].
2. Write total output as function of P and E [P ] (which is the parameter µP ).
Macroeconomía II, 2018/2 1
3. What happens to your answer of the previous item when τw →∞? Interpret it.
Exercise 2
In class we have seen three different types of frictions that can generate a positive relation-
ship 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.
IS-LM-PC
Exercise 3
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 4
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.
Macroeconomía II, 2018/2 2
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 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 5
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 6
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, 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
Macroeconomía II, 2018/2 3
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?
Exercise 7
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 expec-
tations are anchored at some level π. Also, assume that interest rates are sufficiently high,
so that the central bank 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.
Macroeconomía II, 2018/2 4
2. In the short run, is investment larger or smaller after the reduction in G? In the
medium runis 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 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, 2018/2 5
Macroeconomía II – Solutions
TA Session 7
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Exercise 1
Item 1
Let xi = Pi − µP and η = P − µP ∼ N(0, 1/τP ). Note that observing xi gives you the
same information as observing Pi (and then taking expectation conditional on xi is the
same as taking expectation conditional of Pi). Then
Pi = P + ri
Pi − µP = P − µP + ri
xi = ri + η
Using the tip given:
E [ri|wi, Pi] = E [ri|wi, xi] =
τr · 0 + τwwi + τPxi
τr + τw + τP
= τw
τr + τw + τP
wi +
τP
τr + τw + τP
xi
= τw
τr + τw + τP
wi +
τP
τr + τw + τP
(Pi − µP )
= τw
τr + τw + τP
wi +
τP
τr + τw + τP
(Pi − E [P ])
Item 2
Each firm will produce
ysi = y + γ
[
τw
τr + τw + τP
wi +
τP
τr + τw + τP
(Pi − E [P ])
]
Macroeconomía II, 2018/2 1
Total output is then given by
y =
∫ 1
0
ysi di = y + γ
[
τw
τr + τw + τP
∫ 1
0
widi+
τP
τr + τw + τP
(∫ 1
0
Pidi− E [P ]
)]
= y + γ
[
τw
τr + τw + τP
∫ 1
0
(ri + ζi) di+
τP
τr + τw + τP
(∫ 1
0
Pidi− E [P ]
)]
By the law of large numbers,
∫ 1
0 ridi = E [ri] = 0 and
∫ 1
0 ζidi = E [ζi] = 0. By definition,
we also have
∫ 1
0 Pidi = P . Therefore:
y = y + γ τP
τr + τw + τP
(P − E [P ])
y = y + α (P − E [P ])
where α ≡ γ τP
τr+τw+τP .
Item 3
As τw →∞, α goes to zero. Therefore, if agents have very precise information about their
relative prices, the price level affects very little output, since they can almost perfectly
distinguish increases in their relative prices from increases in the price level.
Exercise 2
• Wage ridigities: suppose firms workers have set their wage at some date. Now sup-
pose right after they set the wage, inflation was much higher than expected. This
will reduce the real wage, making firms use much more labor, reducing unemploy-
ment. Hence, taking as given inflation expectations, higher inflation leads to higher
employment, and therefore to higher output.
• Price rigidities: if firms take time to adjust price and supply as much goods as
consumers demand, a higher inflation reduces the real price goods, making consumers
demand more, leading to higher production.
• Information rigidities: firms care about the relative price to decide how much to
produce. They do not perfectly observe the price level. Hence, when the price level
increases (inflation), they are uncertain if that is an increase in relative prices or an
increase in the price level, leading them to produce more.
Macroeconomía II, 2018/2 2
Exercise 3
Before we start, a remark of why makes sense to interpret an increase in the price as
a fall in potential output Yn. Think of potential output as the output level consistent
with long run equilibrium in labor markets with perfect wage rigidity. In case oil prices
increase, firm’s marginal cost increase, shifting the labor demand to left and reducing the
equilibrium employment.
Since the PC curve is given by π−πe = θ (Y − Yn), a fall in Yn shifts up the PC curve,
as illustrated below for anchored (left panel) and adaptative expectations (right panel).
A′ represents the short run response and A′′ the medium run response, after the central
bank has fully reacted.
r
Y
IS
r
LM ′A
′′
π
Y
π
PC ′
A
A′
PC
LMA and A
′
A′′
r
Y
IS
r
LM ′A
′′
∆π
Y
0
PC ′
A
A′
PC
LMA and A
′
A′′
Inflation increases in both cases after the shock and then, when the central bank adjusts
the interest rate it falls in the left panel and remains constant in the right panel. Output
goes back to potential in both cases.
Macroeconomía II, 2018/2 3
Time
Time
π
Y
π
Yn
Y ′n
Adaptative expectations
Anchored expectations
Same curve for both
Exercise 4
Item 1
As shown in the graph below, after the IS curve shifts to left, output falls.
r
Y
ISIS′
r LMAA
′
Item 2
The figure below shows the medium run equilibrium before the cut in government spending
(point A) and after the cut in government spending (point A”), using the IS-LM-PC model.
As one can see, output returns to potential output, so there is no effect on output in the
medium run.
Macroeconomía II, 2018/2 4
r
Y
π
Y
π
ISIS′
r LM
PC
AA′
A and A′′
A′
LM ′′r′′
π′
A′′
Item 3
As shown in the graph above, the medium run equilibrium with lower government spend-
ing (point A”) features lower real interest than the medium run equilibrium with higher
government spending (point A). Thus, IS-LM-PC predicts that the medium run equilib-
rium with lower government spending will have higher investment, since the real interest
rate is lower.
Exercise 5
Item 1
This the classical case that will generate deflate spirals. The graph below shows the initial
equilibrium of the economy (point A). Since output is initially below potential, inflation is
going down, eventually forcing the central bank to increase the real interest rate because
of the zero lower bound and decreasing output (point A’). As inflation continues to go
down, the lowest possible real interest continues to increase, forcing the real interest rate
up and output down as in illustrated in point A”. This process continues. (To save space,
I did not draw all LM curves in the graph). Note that the central bank cannot make
inflation stop decreasing. That would require setting r = rn < 0, which is never possible
because of the zero lower bound and since inflation starts at zero and only decreases.
Macroeconomía II, 2018/2 5
r
Y
∆π
Y
0
0 LM
PC
B
rn B
Yn
A
IS
A′
A′′
A′
A′′
A
Item 2
Yes. If the government increases spending G or decreases taxes T sufficiently he can makes
the natural real interest rate positive, bringing output to potential output already in the
short run. The graph below illustrates the argument.
r
Y
∆π
Y
0
0 LM
PC
B
rn B
Yn
A
IS ′(for G > G′or T < T ′)
IS(for G and T )
A′
Macroeconomía II, 2018/2 6
Exercise 6
Item 1
First thing we need to understand is how this will affect the IS curve. Equilibrium in the
goods market imply
Y = c0 + c1(Y − T ) + b0 + (1− λ) [b1Y − b2(r + x)] + λ [b1Y − b2(1− s)(r + x)] +G
Y = 11− c1 − b1
[c0 − c1T + b0 − b2 (1− λs) (r + x) +G]
r = 1
b2 (1− λs)
[−(1− c1 − b1)Y + c0 − c1T + b0 +G]− x
Note that
dr
dY
= −1− c1 − b1
b2 (1− λs)
Thus, as λ increases, the IS curve becomes more inclined and also shifts up.
r
Y
IS
IS′
LM
A
π
Y
π
PC
A and A′′
A′
A′′ LM ′
A′
Graphically, the equilibrium goes from point A to A’ in the short run, and to A” in
the medium run. The real policy rate is indeed higher with higher λ.
Item 2
The policy rate does not change:
Macroeconomía II, 2018/2 7
r
Y
ISλ=0ISλ>0
LMA
π
Y
π
PC
A and A′
Item 3
Using r = 1
b2(1−λs) [−(1− c1 − b1)Y + c0 − c1T + b0 +G]−x, one can see that when λ > 0,
the vertical shift in the IS is larger, for a given change in c0. The graph below shows the
change in equilibrium:
r
Y
ISλ=0ISλ>0
LMA
π
Y
π
PC
A,B and C
IS ′λ=0
B
C
IS ′λ>0
Note that interest rates are higher at point C (medium run equilibrium after the shock
when λ > 0), than at point B (the medium run equilibrium after the shock when λ = 0).
Monetary policy needs to react more strongly with λ > 0, so it makes sense to say it
became less powerful.
Macroeconomía II, 2018/2 8Exercise 7
Item 1
In the diagram below point A denote the initial date, point A′ the short run equilibrium
after the change G and point A′′ the medium run equilibrium. In the short run output
falls, but since in the medium run the central bank reacts decreasing interest rates, output
goes back to potential in the medium run.
r
Y
Y
π − π
0
IS
IS ′
LM
LM ′′
PC
AA′
A′′
A′
A and A′′
Yn
Item 2
In the short run, investment is lower, since output is lower and the real interest rate is
the same. In the medium run, investment is higher, since output goes back to its previous
level, but real interest rates are lower.
Item 3
In that case, the reduction in government spending will put the economy in a deflation
trap. Point A illustrates the initial equilibrium, and point A′ is the short run equilibrium
after the decrease in G. Since at A′ we have ∆π < 0 in the PC curve, inflation eventually
decreases once prices adjust (causing deflation), which pushes real interest rates up because
of the zero lower bound: i = r + πe ≥ 0 implies r ≥ −πe. That is, since πe is going down
Macroeconomía II, 2018/2 9
because of adaptative expectations and falling inflation , r goes up in the medium run as
time goes by (which is denoted by points A′′ and A′′′).
r
Y
Y
∆π
0
IS
IS ′
LM
PC
AA′
A′
A
Yn
LM ′′
LM ′′′
0
A′′
A′′′
A′′
A′′′
Macroeconomía II, 2018/2 10