Solutions_Chapter7

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Solutions - Chapter 7
Macroeconomics II
Professor: Caio Machado (caio.machado@uc.cl)
Teaching assistants: Wei Xiong (wxiong@uc.cl) and Diego Cussen (dcussen1@uc.cl)
Last updated: January 15, 2019
Exercise 7.1
Item 1
As shown in item 3 below, output increases.
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Item 2
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).
Item 3
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
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the LM curve. 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). 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.
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 (note where the origin is positioned).
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)
Exercise 7.2
Item 1
Let’s start with the new equilibrium. The shock in G shifts the IS curve to the right, and
the equilibrium goes from point A to point A′ in the graph below.
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LM
IS
IS ′
i
Y
A
A′
∆G > 0
Now we look at the transition. Since we are assuming that money market is always
in equilibrium during the transition, we must never leave the LM curve (remember
that the LM curve gives us all combinations of i and Y such that the demand for money
equals the supply of money). Hence, we must move on the LM curve, as represented
below.
LM
IS
IS ′
i
Y
A
A′
∆G > 0
i
time
Y
time
In words what happens is the following. The increase in government spending
increases demand, which increases firms production and income. The higher income
make people demand more money. Since the money supply fixed, as income increases,
interest rates increase to ensure equilibrium in money markets, until production fully
adjusts and we reach the new equilibrium.
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Item 2
Again we start with new equilibrium. The increase in M shifts the LM curve to the
right and the equilibrium goes from point A to A′.
LM
IS
i
Y
A
LM ′
A′
∆M > 0
Again, we must always be on the LM curve. Hence, the representation in the graph
below. Initially, the economy jumps to point B and then slowly converges to A′. Inter-
est rate fall at point B, which makes output increase gradually, since the much lower
interest rate increases demand for goods. As output increases, the demand for money
increases, which forces interest to go up a bit to reduce the demand for money.
i
time
Y
time
LM
IS
i
Y
A
LM ′
A′
∆M > 0
B
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Exercise 7.3
Item 1
If investment does not depend on the interest rate, the IS curve is vertical. To see that,
notice that equilibrium in the goods markets is not affected by an increase in the interest
rate (the demand curve Z of the model of equilibrium in the goods markets seen in class
does not change when we change i). If the IS curve is vertical, fiscal policy still affects
output and interest rates, but monetary policy only affects interest rates, as illustrated
below.
LM
ISi
Y
A
∆G > 0
IS ′
A′
LM
ISi
Y
A
∆M > 0
A′
LM ′
Item 2
If the money demand does not depend on i (but still depends on Y), the LM curve is
vertical. This is so because, for a given money supply, there is an unique Y that clears
the money markets. But notice that when Ms/P increases, the level of Y that clears the
money market increases (since now people need to demand more money), and hence
the LM curve shifts to the right. Hence, fiscal policy is not effective to increase output,
but it increases interest rates. Expansive monetary policy, on the other hand, increases
output and lowers interest rates.
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LM
IS
i
Y
A
∆G > 0
IS ′
A′
LM
IS
i
Y
A
A′
LM ′
∆M > 0
Item 3
If the demand for real balances does not depend on Y then, the interest rate that clears
money markets is the same for any Y. Hence, we get an horizontal LM curve. When
Ms/P increases, the interest rate that clears money markets decreases, shifting LM
down. Expansive monetary leads to lower i and higher Y, while expansive fiscal policy
increases output without raising i.
LM
IS
i
Y
A
∆G > 0
IS ′
A′
LM
IS
i
Y
A
LM ′
A′
∆M > 0
Item 4
Since the LM curve does not depend we know that the LM is vertical at some Y∗ such
that:
M
P
= aY∗ ⇒ Y∗ = 1
a
M
P
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Hence, as a → ∞, Y∗ → 0, implying the LM curve becomes vertical at the origin.
Intuitively, since the LM curve does not depend on i, the income is entirely determined
by the equilibrium in the money markets. As the demand for money explodes for a
given level output, the only way to reach an equilibrium in money markets is to have
an extremely low income.
Exercise 7.4
Item 1
There are (at least) two ways to do it. First, the country could reduce government
spending (or increase taxes) and then expand the money supply. The IS would shift to
the left and the LM to the right. By choosing the right amount of monetary expansion
one can ensure output will not change. This is represented in the graph below.
IS′
IS
A
i
Y
LM ′
LM
A′
Another way to achieve that is to increase both taxes and government spending.
Remember that equilibrium in the goods markets (the IS curve) requires (I am assuming
a linear consumption function just to simplify, this is not needed for the argument, see
the next item)
Y =
1
1− c1
[c0 − c1T + G + I (i, Y)]
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Hence, if taxes increases in ∆T > 0, spending must increase only increase only
∆G = c1∆T to keep the IS relation identical. Hence, since c1 ∈ (0, 1), ∆T > ∆G and
therefore the fiscal deficit improves, without changing the IS curve.
Item 2
The IS relation is given by:
Y = C(Y− T) + G + I(i, Y)
Therefore an equal and infinitesimal increase in T and G changes the RHS in dRHS =
−C′(Y − T) + 1. Since the marginal propensity to consume C′(Y − T) is smaller than
one, dRHS > 0 and thefefore the IS shifts to the right.
IS
IS′
A′
i
Y
LM
A
Item 3
We start with new equilibrium. The increase in M shifts the LM curve to the right and
the equilibrium goes from point A to A′.
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LM
IS
i
Y
A
LM ′
A′
∆M > 0
Given that money markets are always in equilibrium, we must always be on the LM
curve. Hence, the representation in the graph below. Initially, the economy jumps to
point B and then slowly converges to A′. Interest ratefall at point B, which makes out-
put increase gradually, since the much lower interest rate increases demand for goods.
As output increases, the demand for money increases, which forces interest to go up a
bit to reduce the demand for money.
i
time
Y
time
LM
IS
i
Y
A
LM ′
A′
∆M > 0
B
Exercise 7.5
Item 1
The IS curve represents those pairs of interest rate and output that are consistent with
equilibrium in the goods market. To determine this curve we depart from market equi-
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librium and then we solve for Y or r:
Y = c0 + c1(Y− τY) + I0 − I1r + I2Y + Ḡ
Y(1− c1(1− τ)− I2) = c0 + I0 − I1r + Ḡ
Y =
c0 + I0 − I1r + Ḡ
1− c1(1− τ)− I2
Using the Fisher equation we have that r = i− πe and therefore the IS curve becomes:
Y =
c0 + I0 − I1(i− πe) + Ḡ
1− c1(1− τ)− I2
Item 2
In the model of equilibrium in the goods markets, we took the interest rate as given.
In general, in the IS-LM model where the central bank fixes the money supply (like the
one presented in this exercise), when we increase G we also change the interest rate.
But here, since we assumed the money demand does not depend on output, the LM
curve will be flat (as will be clearer when we solve item 3). Hence, interest rates will
not move after we change G. Therefore, at this stage, I will simply compute the change
in output after a change in G, assuming the interest rate does not move. Hence, we get:
dY
dG
=
1
1− c1(1− τ)− I2
If c1(1− τ) + I2 > 1 then an increase in government spending increases output (again,
assuming the interest rate would not move). The increase government spending ex-
pands aggregate demand, and firms respond by increasing production, which increases
income and further increases demand (which further increases production, income and
so on). (Remark: If c1(1− τ) + I2 < 1 this model makes little sense, because for a real
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interest rate of zero, there would be no positive level of output consistent with equilib-
rium in the goods markets. Hence, we will keep that parametric assumption in the rest
of the solution).
Item 3
The LM curve represents all the pairs of i and Y that are consistent with equilibrium
in the money (financial) markets. To determine it we impose equilibrium in the money
market (Md = Ms) and solve for i (Y) as a function of Y (i):
M
P
= l0 − l1i
Since in this exercise the money demand is independent from income (Y), the LM curve
will be horizontal on the plane (Y, i):
i =
l0 − MP
l1
Item 4
Now we will look for the pair (i, Y) that equilibrates both, money and goods markets
at the same time, by intersecting both curves IS and LM curves:
i∗ =
l0 − MP
l1
Y∗ =
c0 + I0 − I1
(
l0−MP
l1
− πe
)
+ Ḡ
1− c1(1− τ)− I2
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Item 5
To answer this question, we take the derivative of the equilibrium Y∗ we found on the
last item with respect to M:
dY∗
dM
=
I1/(l1P)
1− c1 (1− τ)− I2
Assuming 1− c1(1 + τ)− I2) > 0 this is positive, and therefore an expansionary mon-
etary policy reduces interest rate promoting investment and short term growth.
Item 6
The IS-LM model analyzes the short term: the time horizon in which prices are fixed.
Hence, we can completely abstract from inflation, and for simplicity we usually set
πe = 0, which implies i = r by the Fisher equation.
Exercise 7.6
Item 1
From the equilibrium conditions at goods and money markets we get:
Y = 4 + 0, 64Y− 0, 4i
6 = 0, 75Y− 1, 5i
Hence, Y = 10 e i = 1.
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Item 2
Using Y = 10 in goods market equilibrium we can solve for G:
10 = 1 + 0, 64× 10 + 1, 8− 0, 4× 1 + G
We then get that G should get to 1.2 increasing by 20% from initial level.
Item 3
As in item 1 we have now:
Y = 4 + 0, 64Y− 0, 5i
6 = 0, 75− 1, 5i
And then we get Y ≈ 9, 8 and i ≈ 0, 92. This new tax increases the sensitivity of
investment to interest rate. At the previous interest rate investment and output, which
pushes interest rates down, since there is too little demand for money.
Item 4
The exercise consists of solving the system for Ms imposing equilibrium values from
question 1:
10 = 4 + 0, 64× 10− 0, 5i
M = 0, 75× 10− 1, 5i
We obtain M = 6.3. The money supply expands to decrease interest rates, compensat-
ing the negative effect of the tax on investment.
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Exercise 7.7
Item 1
Equilibrium in the goods markets imply:
Y = C̄ + c1 (Y− T) + Ī − αr + G
Which gives the IS curve below (in term of the real interest rate instead of nominal):
r =
1
α
[C̄ + (1− c1)Y− c1T + Ī + G]
Equilibrium in the money market implies:
kY− θ (r + πe) = Ms
Which implies the following LM curve in term of the real interest rate:
r =
1
θ
(
kY−MS
)
− πe
Hence, we get the usual graph:
LM
IS
r
Y
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Item 2
Equilibrium income is given by:
1
α
[C̄ + (1− c1)Y− c1T + Ī + G] =
1
θ
(
kY−MS
)
− πe
Y∗ =
(
k
θ
− 1− c1
α
)−1 {Ms
θ
+ πe +
1
α
[C̄− c1T + Ī + G]
}
To obtain equilibrium r we simply plug Y∗ that into the LM curve:
r =
1
θ
(
kY∗ −MS
)
− πe
Item 3
This shifts the LM curve to the left (as if there was a negative shock in the demand
for money). The equilibrium goes from A to A′. Output falls and the real interest rate
increases.
LM
IS
r
Y
LM ′
A
A′
Item 4
By increasing the money supply, monetary policy can offset the initial shift to the left
of the LM curve, keeping output constant and the economy at the same equilibrium.
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Item 5
Equilibrium values before the change in expectations are Y = 25.7, r = 1.6, i = 4.6.
After inflation expectations fall, new equilibrium values: Y = 12, r = 5, i = 0.
Item 6
Nominal interest rates are at its minimum possible level (they cannot be negative).
Therefore expansionary monetary policy loses effectiveness, since interest rates can-
not fall. Expansionary fiscal policy is still effective. An increase in G displaces the IS
curve to the right stabilizing the output after a negative shock.
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