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Macroeconomía II – Extra Problem Set
Extra exercises about the IS-LM model
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Exercise 1
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 lump-sum as usual), Ms
P
and Md
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 represents.
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.
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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?
Exercise 2
Consider an economy in which the central bank fixes the money supply described 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 investment 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.
4. Assuming the new investment function, what level of money supply restores income
to the level of item 1? Explain your results.
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Exercise 3
[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 interests rates.
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 expectations
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?
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Macroeconomía II – Solutions
Extra exercises about the IS-LM model
Professor: Caio Machado (caio.machado@uc.cl)
TA assigned: Wei Xiong (wxiong@uc.cl)
Exercise 1
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 equilib-
rium 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 + Ḡ1 − 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− π
e) + 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
= 11 − c1(1 − τ) − I2
If c1(1 − τ) + I2 > 1 then an increase in government spending increases output (again,
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assuming the interest rate would not move). The increase government spending expands
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 interest
rate of zero, there would be no positive level of output consistent with equilibrium 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 = M s) 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
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
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Assuming 1 − c1(1 + τ) − I2) > 0 this is positive, and therefore an expansionary monetary
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 2
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.
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.
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Item 4
The exercise consists of solving the system for M s imposing equilibrium values from ques-
tion 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, compensating
the negative effect of the tax on investment.
Exercise 3
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) = M s
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:
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LM
IS
r
Y
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 {
M s
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.
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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.
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). There-
fore expansionary monetary policy loses effectiveness, since interest rates cannot fall. Ex-
pansionary 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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