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Exercises before the second mid-term
Caio Machado
Instituto de Econoḿıa
Pontificia Universidad Católica de Chile
Macroeconomia II, 2017
Exercise 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)
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 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that will have its legal tender status withdrawn, it immediately puts the same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 Draw the money demand curve of this economy before and after the policy.
2 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).
Answer item 1
Nominal interest rate, i
Real money balances, M/P
Money demand before the policy
Money demand after the policy
0
Because of the policy, bonds become more attractive and money less attractive.
The money demand shifts down. Moreover, when the interest rate is 0%, a
decrease in interest rates will still cause the demand for money to increase (now,
when the nominal interest rate is 0%, bonds still pay a interest rate higher than
money, and therefore a reduction in i will make people substitute bonds for
money). Nominal interest rates can go negative after the policy, but they certainly
can´t go below -10% (otherwise no one would demand bonds).
Answer item 2
Does the IS shift?
The IS curve does not shift after the policy. Consumption is still a function
of available income, and that did not change. For a given level of Y and i ,
firms will be willing to invest the same as before. G did not change as well.
Thus, the only thing that shifts is the LM curve. Obs: one could always say
that the IS shifts for reasons that are not in the model. Here, you are asked
to analyze this policy through the lens of the basic IS-LM model.
How does the LM shift?
Take some interest rate i > 0. For that interest rate, the level of output Y
consistent with equilibrium in the money market will have to be higher after
the policy to make people demand a given quantity of money (to see that,
go back to item 2 and draw the equilibrium in the money market for
different values of Y before and after the policy, assuming a constant
money supply).
Answer item 2
0
LM (before policy)
IS
Output, Y
Interest
rate, i
Equilibrium (before
policy)
LM (after policy)
Equilibrium (after
policy)
0
LM (before policy)
IS
Output, Y
Interest
rate, i
Equilibrium (before
policy)
LM (after policy)
Equilibrium (after
policy)
In the new equilibrium output will be higher. The economy can leave the
liquidity trap, or enter a new liquidity trap with negative interest rates
(depending on how much the LM curve shifts). The two cases are
represented below, but in both cases, output increases.
Exercise 2
Consider the following IS-LM model where the central bank fixes the
nominal interest rate:
C = c0 + c1(Y −T )
I = b0 + b1Y −b2i
Y = C + I + G
i = i
where Y is the output, C is consumption, T is taxes, i is the nominal
interest rate, I is investment and c0, c1, b0, b1, b2 are parameters larger
than zero. Also, c1 + b1 < 1.
1. Solve for equilibrium output.
Equilibrium output is given by
Y = C + I + G
Using the equations for C , I and G the equation above becomes:
Y = c0 + c1(Y −T ) + b0 + b1Y −b2i + G
Solving for Y
Y = 11− c1−b1
[
c0− c1T + b0−b2i + G
]
2. Suppose G increases. Does the central bank needs to increase or
decrease the money supply to keep the interest rate constant?
When G increases, the IS curve shifts up. That increases equilibrium
output. Higher output shifts the money demand curve to the right,
requiring an increase in the money supply to keep the interest rate
constant, as shown below.
M s M s
′
MD(for Y )
MDMD(for Y ′ > Y )
M/P
i
Exercise 3
Consider two bonds, one issued in euros (e) in Germany, and one issued in
dollars ($) in the United States. Assume that both government securities
are one-year bonds—paying the face value of the bond one year from now.
The exchange rate, E, stands at 0.75 euros per dollar. The face values and
prices on the two bonds are given by
Face Value Price
USA $ 10,000 $ 9,615.38
Germany e 10,000 e 9,433.96
1. Compute the nominal interest rate on each of the bonds.
iUS =
10,000−9,615.38
9,615.38 = 4%
iGERMANY =
10,000−9,433.96
9,433.96 = 6%
2. Compute the expected exchange rate next year consistent with
uncovered interest parity.
Let’s use the approximated linear version of uncovered interest rate parity:
iUS ≈ iGERMANY −
E et+1−Et
Et
Using that equation, we have E
e
t+1−Et
Et ≈ 2%, and thus
E et+1 ≈ 0.75(1 + 0.02) = 0.765. You could also use the non linear version:
(1+ iUS) =
Et
E et+1
(1+ i∗GERMANY )⇒ (1+0.04) =
0.75
E et+1
(1+0.06)⇒
E et+1−Et
Et
= 1.9%
3. If you expect the dollar to depreciate relative to the euro, which bond
should you buy?
The german bonds. To see that, compute how much you get in dollars for
each dollar invested in US bonds:
1 + 0.04
And the return in dollars for each invested in german bonds:
(1 + 0.06) EtE et+1
(Note: E et+1 here is you belief about the exchange rate, not others people
beliefs). If you expect the dollar to depreciate, we have EtE et+1 > 1 and thus
(1 + 0.06) EtE et+1
> 1 + 0.04
4. Assume that you are a U.S. investor and you exchange dollars for euros
and purchase the German bond today. One year from now, it turns out that
the exchange rate, E, is actually 0.72 (.72 euros buys one dollar) What is
your realized rate of return in dollars compared to the realized rate of return
you would have made had you held the U.S. bond?
And the gross return in dollars is:
(1 + 0.06) EtEt+1
= (1 + 0.06)0.750.72 = 1.10
Thus, the realized net return is 10%, compared to 4% return you would
have had if invested in US bonds.
5. Are the differences in rates of return in (d) consistent with the
uncovered interest parity condition? Why or why not?
No! The unconvered interest rate parity does not say that the realized
rate of return of the two bonds (in dollars) must be the same. It says that
the expected rate of return (in dollars) of both bonds must be the same.
Exercise 4
Suppose that the Phillips curve is given by
πt = πet + 0.1−2ut
and expected inflation is given by
πet = (1−θ)π+θπt−1
and suppose that θ is initially equal to 0 and π is given and does not
change. It could be zero or any positive value. Suppose that the rate of
unemployment is initially equal to the natural rate. In year t, the authorities
decide to bring the unemployment rate down to 3% and hold it there
forever.
a. Compute the rate of inflation for years t, t + 1, t + 2, and t + 3.
Initially, outputis equal to potential output, so we have that at date t
πt = π+ 0.1−2 ·0.03
πt = π+ 0.04
Since expectations ar anchored at π, we have that
πt = πt+1 = πt+2 = πt+3. (Note the natural rate of unemployment is 5%).
b. Do you believe the answer given in (a)? Why or why not? (Hint: Think
about how people are more likely to form expectations of inflation.)
I do not, but it is not a so bad approximation for a few periods after the
policy. Once the government announced that it would reduce
unemployment, people should anticipate that inflation would accelerate, or
at least, increase their parameter θ, looking more to past inflation to form
their expectations.
Now suppose that in year t + 6, θ increases from 0 to 1. Suppose that the
government is still determined to keep u at 3% forever.
c. Why might θ increase in this way?
Noticing that inflation is above their expectations for 5 periods, people
could decide to start to look to past inflation to form their expectations.
d. What will the inflation rate be in years t + 6, t + 7, and t + 8?
πt+6 = πt+5︸ ︷︷ ︸
πet+6
+0.1−2 ·0.03 = π+ 0.04 + 0.04 = π+ 0.08
πt+7 = π+ 0.08︸ ︷︷ ︸
πet+7=πt+6
+0.04 = π+ 0.12
πt+8 = π+ 0.12︸ ︷︷ ︸
πet+8=πt+7
+0.04 = π+ 0.16
e. What happens to inflation when θ = 1 and unemployment is kept below
the natural rate of unemployment?
Inflation increases.
f. What happens to inflation when θ = 1 and unemployment is kept at the
natural rate of unemployment?
Inflation stabilizes.
Exercise 5
Suppose the economy is operating at the zero lower bound for the nominal
policy rate; there is a large government deficit and the economy is
operating at potential output in period t. A newly elected government vows
to cut spending and reduces the deficit in period t + 1, period t + 2 and
subsequent periods.
a. Show the effects of the policy on output in period t + 1.
b. Show the effects of the policy on the change in inflation in period t + 1.
c. If expected inflation depends on past inflation, then what happens to the
real policy rate in period t + 2? How will this affect output in period t + 3?
d. How does the zero lower bound on nominal interest rates make a fiscal
consolidation more difficult?
We can use the IS-LM-PC model to provide one possible answer. Implicit
here is that when we go from t to t + 1, we have reached the medium run
(prices adjust).
Before providing an answer, let’s draw the IS-LM-PC graph. Point A will be
where the economy is at the beggining of date t + 1; point A´ is where the
economy will be atthe end of date t + 1 (after the shock); point A′′ is where
the economy is at the end of date t + 2 and so on. We will assume
inflations expectations are equal to the inflation the previous period.
r
Y
∆π
Y
0
ISIS′
YnY ′
r LM
PC
AA′
A
A′
r′ A′′
A′′′
A′′
A′′′
r′′
(Too save space I didn’t write all the LM curves). When the economy goes
from A to A’, inflation goes down. Since nominal interest rates were
initially zero, this lower inflation forces the central bank to increase the real
interest rate, and the economy goes to A”. Inflation keep falling, forcing
further increases in r (point A”’).
a. Show the effects of the policy on output in period t + 1.
Output falls, as we have shown in the previous slide.
b. Show the effects of the policy on the change in inflation in period t + 1.
Inflation falls (∆π < 0).
c. If expected inflation depends on past inflation, then what happens to the
real policy rate in period t + 2? How will this affect output in period t + 3?
The real policy rate increases, which makes output fall.
d. How does the zero lower bound on nominal interest rates make a fiscal
consolidation more difficult?
It does not allow fiscal and monetary policy to coordinate. The central
bank cannot implement the decrease in interest rates that would make
output equal to potential output.
Exercise 6
Consider the following IS-LM model where the central bank fixes the money
supply at some level M:
C = c0 + c1(Y −T )
I = b0 + b1Y −b2i
Y = C + I + G
Md
P = m0 + m1Y −m2i
where Y is the output, C is consumption, T is taxes, i is the nominal
interest rate, I is investment, P is the price level, Md is money demand t
and c0, c1, b0, b1, b2 are parameters larger than zero. Also, c1 + b1 < 1.
1. Derive the LM curve (with i on the vertical axis).
Equilibrium in financial markets implies
M
P = m0 + m1Y −m2i
i = m0 + m1Y −M/Pm2
2. Derive the IS curve.
Equilibrium in the goods markets imply that:
Y = c0 + c1(Y −T ) + b0 + b1Y −b2i + G
Y = 11− c1−b1
[c0− c1T + b0−b2i + G ]
i = 1b2
[c0− c1T + b0 + G− (1− c1−b1)Y ]
3. Derive the AD curve.
Using the equilibrium the IS and LM curves we get
m0 + m1Y −Ms/P
m2
= 1b2
[c0− c1T + b0 + G− (1− c1−b1)Y ]
m0 + m1Y −
m2
b2
[c0− c1T + b0 + G− (1− c1−b1)Y ] =
M
P
P = Mm0 + m1Y − m2b2 [c0− c1T + b0 + G− (1− c1−b1)Y ]
Exercise 7
Suppose agregate consumption is given by
C = f (B) + c1(Y −T )
(the notation is the standard one). f (·) is an increasing function of the
amount of loans (B) that bank supply to households. Let A denote the
total values of houses in the economy. Households borrow from banks by
putting their house as collateral. Thus, we assume that
Thus suppose that B = A. Also, assume the central bank fixes the real
interest in this economy.
1. How a decrease in housing prices affects the economy in the short run?
Consumption is reduced because of lower housing prices, the IS curve shifts
to the left and we end up with less output.
2. Now suppose Tom borrowed $100.000 from the bank to buy a house,
and put that house as collateral. The price of this house fell to $50.000.
Suppose there are many similar houses for sale at the same price in Tom’s
neighborhood. Should the bank expect the loan to be repaid?
No. It is better to default on the $100.000 loan, let the bank take the
house and buy a new one for $50.000.
3. Should we expect firms to be affected by a high default rate of
households? (Think a bit outside the model.)
Yes, as banks start to suffer losses from these loans, they are likely to
increase the risk premium, reducing investment (the IS shifts even more to
the left).