EC108 Macroeconomics 1
Review Class - Suggested Answers∗
Jorge F. Chavez
May 22, 2014
Question 1: True or false statements
Read the following statements and say whether they are TRUE, FALSE or UNCERTAIN. Briefly
justify your answer.
(a) Aggregate demand falls when the price level rises, because higher prices cut real incomes.
Answer. False. Aggregate demand does falls when the price level rises, but not for the reason given. The
reason is that a higher price level cuts the real money supply; this raises interest rates and so aggregate
demand through investment and the multiplier.
(b) An increase in the price of some imported goods will show up in the GDP deflator but not in the
CPI.
Answer. False. Imported consumption goods enter directly into the CPI, and so if these goods are the
ones that experience a price increase, then CPI will be affected. In principle, the GDP deflator should
be unaffected unless the increased import prices translates into higher prices for intermediate and capital
goods. This will impact future values of the deflator.
(c) According to the Quantity Theory of Money, the rate of inflation must equal the rate of growth
of the nominal money supply.
Answer. False. This follows from M ×V = P ×Y only if both V and Y are constant. In a growing economy,
Y is not constant and nowhere in the Quantity Theory it is assumed that it would be constant.
(d) The IS curve could be fully vertical.
Answer. True. The slope of the IS curve depends on the income and interest rate elasticities of aggregate
expenditure. If investment is completely independent from the interest rate, then the IS curve will be
vertical.
(e) The basic version of the Solow model (no population growth, no technological progress) implies
an economy where growth is not possible.
Answer. False. The model implies that the economy will not exhibit growth in the long-run, but it will
exhibit transitory growth when moving from one steady-state to a new one after a shock. The proposition
would be true if we add the words “sustained” or “long-term” before growth.
∗
Please note that as can be inferred from the title these are only suggested answers.
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EC108
Review May 2014
Question 2: AD-AS model1
Assume an economy that is initially operating at the natural rate of output Ȳ .
(a) Write down the short-run aggregate supply equation.
Answer. Any of the three theories considered (sticky wages, sticky prices, and incomplete information)
yield the following expression:
Yt = Ȳ + α (Pt − Pte ) + ϵt
where ϵt is a transitory shock.
(b) Suppose that this economy is characterized by price stickiness. Other things equal, what would
happen with the equation you wrote in parte (a) if the proportion of firms that follow a sticky-price
rule increases?
Answer. The aggregate supply equation relates deviations of GDP with respect to its potential level (Yt − Ȳ )
and deviations of the price level with respect to the expected price level (Pt − Pte ). The parameter α is
related to how responsive is output to changes in prices. Therefore, other things being equal, if a greater
proportion of firms follows the sticky-price rule, α decreases (AS curve become less responsive or flatter).
In the extreme case in which prices become fully sticky, the SRAS curve becomes completely horizontal.
(c) Use the model of AD-AS to illustrate graphically the short-run and long-run effects on price and
output of an unexpected expansionary monetary policy change.
Answer. This positive AD shock moves output above its natural rate and P above the level people had
expected. Over time P e adjust upwards, the SRAS shifts up, and output returns to its natural level.
Figure 1: Effects on P and Y of an unexpected expansionary monetary policy change
LRAS
SRAS2
P
SRAS1
C
P 3 = P3e
P2
P2e = P1 = P1e
B
AD2
A
AD1
Y3 = Y1 = Ȳ
1
Y2
Y
Based on Q2, exam 2010-2011
Jorge F. Chávez
2
EC108
Review May 2014
Question 3: Unemployment dynamics2
Consider the simplest model of unemployment dynamics. Let Ut denote the number of unemployed
workers at time t, and let s and f denote the job-separation rate and the job finding rate in this
economy. Assume initially that the total size of labor force is constant (Lt−1 = Lt = Lt+1 = L, all t).
(a) Interpret s and f in terms of probabilities.
Answer. The job separation rate s is the probability that a worker will loose its job. The job finding rate
f is the probability that a worker that is actively seeking for a job finds one.
(b) Let µt denote the unemployment rate at time t. Show that the unemployment rate in steady-state
will be a function of s and f only.
Answer. Start from:
Ut+1 = Ut + sEt − f Ut
Replace Et = L − Ut and divide both sides by L to put everything in terms of unemployment rates.
Rearranging we get:
µt+1 = s + (1 − s − f ) µt
In steady-state µ̄ = µt+1 = µt . Hence, solving for µ̄:
µ̄ =
s
s+f
(c) Suppose s = 0.03 (or 3 percent) and f = 0.7 (or 70 percent). Initially (at time t = 0), the
unemployment rate is equal to µ0 = 0.08. Further assume that at time t = 200 (in the long run)
this economy will experience a permanent shock on f such that from period t = 200 onwards,
f = 0.4. Sketch the evolution of the unemployment rate over time in a simple graph (Hint:
consider how would the path of µt would look in a graph with time in the horizontal axis, where
t goes from 0 to 1000).
Answer. With s = 0.03 and f = 0.7 the unemployment rate in steady-state will be 0.041. Hence, at time
t = 0 we are above the steady-state and from our knowledge of how µt evolves over time we know that we
will progressively tend towards the steady-state from above. This might not take too long, so by the time
we are at t = 199 the economy is already in steady-state.
Then, the permanent shock on f translates into an increase of the steady-state rate of unemployment to
µ = 0.697 ≈ 0.07. This means that right after the shock, the economy is below the steady-state, and
therefore the evolution of µt will be such that we will progressively converge to the new steady-state from
below. See figure 2.
(d) In macroeconomics it is always useful to express a dynamic variable (say xt ) in terms of deviations
from its value in steady state: x̃t ≡ xt − x̄. Show that the deviation of the unemployment rate at
t + 1 with respect to its steady-state value can be expressed as a linear function of the deviation
of the unemployment rate at time t, that is µ̃t+1 = γ µ̃t . Hint: Try to express the γ parameter in
terms of s and f .
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Based on Problem Sets
Jorge F. Chávez
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EC108
Review May 2014
Figure 2: Path of µt
µt
0.08
0.07
0.041
t=0
t = 200
t
Answer. It is just a matter of playing with the expression we got above as follows:
µt+1 − µ̄ = s + (1 − s − f ) (µt − µ̄) + (1 − s − f ) µt − µ̄
Note that besides putting µ̄ on both sides of the equality, I am adding and subtracting (1 − s − f )µ̄ from
the RHS. Rearranging appropriately we get:
µt+1 − µ̄ = (1 − s − f ) (µt − µ̄)
Note that s − (s + f )µ̄ = 0 because of the formula for the natural rate of unemployment. Hence γ ≡
(1 − s − f ) < 1. Making sure that γ < 1 is very important for the stability of the path followed by µt , can
you see why?.
(Optional) Now suppose that the labor force grows at a rate n: Lt+1 = (1 + n)Lt all t.
(e) Show that the unemployment rate in steady-state will be a function of s, f and n only.
Answer. See handout posted here.
(f) (Harder) Show that we can also write µ̃t+1 = ξ µ̃t . What is the relationship between ξ and γ?
Answer. Follow similar steps as for the case of no population growth. At the end you will find that both
parameters are exactly the same.
Jorge F. Chávez
4
EC108
Review May 2014
Question 4: Solow’s Growth Model3
Consider how unemployment would affect the Solow growth model. Suppose that output is produced
according to the following production function:
Yt = Ktα [(1 − u) Lt ]1−α
where Kt is the aggregate stock of capital, Lt is the labor force and u is the natural rate of unemployment. The national saving rate is s, the labor force grows at a rate n and capital depreciates at
a rate δ.
(a) Express output per worker yt = Yt /Lt as a function of capital per worker kt = Kt /Lt and the
natural rate of unemployment u.
Answer. Output per worker is obtained dividing total output by the number of workers:
yt ≡ f (kt ) =
Ktα [(1 − u) Lt ]
Lt
1−α
1−α
= ktα (1 − u)
(b) Describe the steady state of this economy and find the golden-rule level of kt . What is the saving
rate that allows the economy to reach the Golden Rule?
Answer. The textbook states that the law of motion that governs the stock of capital per worker in the
Solow model with population growth is:
∆kt+1 = sf (kt ) − (n + δ) kt
Of course you may use it directly and define the steady-state as follows, however it would be infinitively
better if you are able to arrive to this (or similar) expression from first principles.4 In any case, in steadystate ∆kt+1 = 0, then:
1−α
sk α (1 − u)
3
4
= (n + δ) k
Based on an exercise proposed in Mankiw’s textbook
This implies defining the equation that describes aggregate capital accumulation Kt+1 = Kt + sYt − δKt and the divide
both sides by population to put all variables in per-worker terms. However note that we need to divide by Lt+1 in the
LHS and by Lt in the RHS, so we must do this carefully to preserve the equality. It turns out that the procedure is
simple if we take into account that Lt+1 /Lt = 1 + n as follows:
Kt+1 Lt+1
= sf (kt ) + (1 − δ) kt
Lt+1 Lt
Hence we can rewrite this as:
kt+1 =
sf (kt ) + (1 − δ) kt
(1 + n)
To define the steady-state we can either set kt+1 = kt = k and solve for k or we can defined ∆kt+1 = kt+1 − kt :
∆kt+1 ≡ kt+1 − kt
=
=
sf (kt ) + (1 − δ) kt − (1 + n) kt
(1 + n)
sf (kt ) − (n + δ) kt
(1 + n)
Note that the numerator of the last equality is exactly the same expression stated by the textbook.
Jorge F. Chávez
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EC108
Review May 2014
where k denotes the value of kt in steady-state. Solving for k we get:
(
k = (1 − u)
s
n+δ
)1/(1−α)
(1)
Note that unemployment lower the marginal product of capital and hence reduces the amount of capital
the economy can reproduce in steady-state. Note also that when u = 0 we are back to the textbook version
of the model.
The Golden Rule value of kt is the stock of capital per worker that maximizes consumption in steady-state.
In this case, consumption in steady state is:
c = f (k) − (n + δ)k
where I am using (n+δ)k = sf (k). To find the stock of capital per worker that maximizes this consumption
we can take a look at the FONC :
∂c
= f ′ (k) − (n + δ) = 0
∂k
1−α
For y = ktα (1 − u)
(
)
f ′ k GR
k GR
⇒
(
)
f ′ k GR = n + δ
we can get a closed form solution for k GR :
(
)α−1
1−α
= α k GR
(1 − u)
=n+δ
(
)1/(1−α)
α
= (1 − u)
n+δ
(2)
Comparing (2) with (1) we can see that the only way in which the economy can reach this Golden rule
steady-state is by setting the saving rate equal to the elasticity of output with respect to capital (sGR = α).
The concept of the Golden Rule for the case of the Solow model is illustrated in figure 3. There you can
(
)
see three alternative saving rates, and you can visualize the Golden Rule condition f ′ k GR = n + δ: the
slope of f (·) must be equal to the slope of the (n + δ)k line.
(c) Is there any difference if we analyze output per effective worker instead of output per worker?
Why?
Answer. Even though before we made the analogy between the term (1 − u) and a technological shock,
in reality because u is the natural rate of unemployment it is constant. Therefore setting the model in
per-worker (dividing by L as we did) or in per-effective-worker terms (dividing by (1 − u)L) is irrelevant.
In other words, unlike the model with labor-augmenting technological progress in which Yt /(Et Lt ) will
become constant in the long-run (once the economy reaches a steady-state) (unlike Yt /Lt which will not be
constant but growing), here both Yt /Lt and Yt /[(1 − u)Lt ] will be constant.
(d) Suppose that some change in government policy reduces the natural rate of unemployment. Describe how this change affects output both immediately and over time. Is the steady-state effect
on output larger or smaller than the immediate effect? Explain.
Answer. If we lower the natural rate of unemployment from u1 to u2 , the steady-state capital stock per
worker will be larger as seen in the last equation from part a. The steady-state level of income per worker
will also be higher–both because of the rise in k* and the direct effect on the production function. The
immediate effect will be a rise in output at the initial level of the capital stock, but no effect yet from
a higher capital stock. Thus, the immediate effect on output is smaller than the steady-state effect. See
figure 4.
Jorge F. Chávez
6
EC108
Review May 2014
Figure 3: The Golden Rule
f (k)
(n + δ) k
sf (k, u)
c1
(n + δ)k
cGR
s1 f
(k1∗ )
sGR f k
s1 f (k)
= (n + δ)
GR
c2
sGR f (k GR )
= (n + δ)
s2 f (k)
s2 f (k2∗ ) = (n + δ)
k2
k GR
k1
k
Figure 4: Effect of reducing the natural rate of unemployment
(n + δ) k
(n + δ) k
sf (k, u)
sf (k, u2 )
sf (k1∗ , u2 ) = (n + δ)
sf (k, u1 )
sf (k2∗ , u1 ) = (n + δ)
k1∗
Jorge F. Chávez
k2∗
k
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EC108
Review May 2014
Question 5: Phillips curve
Suppose that an economy has the following Phillips curve:
πt = πte − 0.5(ut − 0.06) + vt
(a) What is the natural rate of unemployment (un )?
Answer. In steady-state, inflation will be equal to expected inflation (πt = πte ) and there will be no shocks
(vt = 0). Hence un = 0.06.
(b) Use the Phillips curve diagram to illustrate graphically how the inflation rate π and unemployment
rate u change in the short run to an unexpected expansionary monetary policy.
Answer. Because the shock is unexpected, expectations do not change and as such the Phillips curve does
not change. The resulting outcome is a change along the Phillips curve.
(c) Use the Phillips curve diagram to illustrate graphically how the inflation rate π and the unemployment rate u change in the short run to an expected expansionary monetary policy.
Answer. Unlike the previous case, here expectations change and therefore the Phillips curve shifts upwards
(because the shock came in the form of an expansionary monetary policy).
Question 6: Dynamic AD-AS Model5
Consider a dynamic AD-AS model is characterized by the following equations:
Yt = Ȳt − α (rt − ρ) + ϵt
The demand for goods and services
rt = it − Et πt+1
The Fisher equation
)
(
πt = Et−1 πt + ϕ Yt − Ȳt + vt
The Phillips curve
Et πt+1 = πt
Adaptative expectations
(
)
it = πt + ρ + θπ (πt − πt∗ ) + θY Yt − Ȳt
The Monetary-Policy rule
(a) Write down the DAD and DAS curves and characterize the long-run equilibrium.6
Answer. To get the DAD begin with the demand for goods and services and replace the real interest rate
using the Fisher equation. Then replace the nominal interest rate using the monetary-policy equation and
the expected inflation. Rearranging terms you’ll get:
]
[
]
[
1
αθπ
(πt − πt∗ ) +
εt
Yt = Ȳ −
1 + αθY
1 + αθY
To get the DAS curve replace expectations in the Phillips curve and you are all set:
(
)
πt = πt−1 + ϕ Yt − Ȳ + vt
5
6
Based on exercises proposed in Mankiw’s textbook
That is, say what happens in the long-run once the economy reaches an equilibrium.
Jorge F. Chávez
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EC108
Review May 2014
(b) Suppose the economy is hit by a transitory supply shock vt > 0. Explain graphically how the
economy returns to its short-run equilibrium.
Answer. This situation is illustrated in figure 5. Start from full equilibrium (point A). The economy is hit
by a transitory shock that will have effect only at time t. This is shown with a shift in the DAS curve from
DASt−1 to DASt (the shift is by exactly the size of the shock). The supply shock causes inflation to rise
and output to fall.
Note that according to the model, part of the effects of this shock is transmitted to the economy through
the reaction of monetary policy. The reason is the increase in inflation triggers the central bank’s response
by raising nominal and real interest rates. The higher interest rate reduces demand of goods and service
in the economy which keeps output below its natural level (point B). Slopes here matter because it can be
seen in the graph that the jump in inflation is lower than the size of the shock. The reason for this is that
the reduction in output dampens the inflationary pressure on the economy to some degree.
The shock fades between t and t + 1 (the shock vt+1 = 0 again) but because expectations exhibit inertia,
the economy is unable to return to its original equilibrium, but to an intermediate point (point C in figure
5).
Figure 5: DAD/DAS - A transitory supply shock
Ȳ
π
DASt
DASt+1
πt
B
DASt−1
C
πt+1
A
πt−1
DADt−1 = DADt = DADt+1
Yt
Yt+1
Yt−1
Y
(c) The sacrifice ratio is the accumulated loss in output that results when the central bank lower its
target for inflation by 1 percentage point. What is the sacrifice ratio implied by the dynamic
AD-AS model?7
Answer. The sacrifice ratio measures the cost in terms of output associated with a one-percentage point
reduction in the target (long-run) inflation rate. Hence we need to come up with a measure of the impact
7
Hint: You have to write down an expression for the sacrifice ratio in terms of model parameters.
Jorge F. Chávez
9
EC108
Review May 2014
of πt∗ on Yt from the equations in the model. We can achieve this by replacing the DAS into the DAD:
[
]
[
]
(
(
)
)
αθπ
1
∗
Yt = Ȳ −
πt−1 + φ Yt − Ȳ + υt − πt +
εt
1 + αθY
1 + αθY
Rearrange the terms so that we can separate Yt from πt∗ :
[
]
[
]
[
]
(
)
αθπ
αθπ
1
∗
Yt = Ȳ −
πt−1 − φȲ + υt − πt −
φYt +
εt
1 + αθY
1 + αθY
1 + αθY
Finally,
[
[
]
[
]
]
[
]
(
)
αθπ
αθπ φ
αθπ
1
πt−1 − φȲ + υt +
1+
Yt = Ȳ −
πt∗ +
εt
1 + αθY
1 + αθY
1 + αθY
1 + αθY
Hence the implied sacrifice ratio is:
αθ
π
∂Y
αθπ
1+αθY
=
=
∗
αθ
φ
π
∂πt
1 + αθY + αθπ φ
1 + 1+αθY
Question 7: Deficit and Public Debt8
The government debt of Mediterranea stands at £12, 000, and the long term interest rate is 5 percent.
The governments budget for this year is based on a proportional income tax rate of 20 percent. From
the revenue the government must meet consumption spending commitments of £1, 000 as well as
interest payments on the debt. The economy’s production possibilities are given by its natural rate of
GDP, which is £6, 000.
(a) What is the level of GDP at which the Mediterranean government will balance its budget this
year (5 marks)?
Answer. Recall that:
∆Dt+1
≡
Dt+1 − Dt
=
Bt = Gt + it Dt − Tt
Hence the level of nominal GDP that allows to balance the budget is:
0 = 1, 000 + 0.05 × 12, 000 − 0.20 × Y
where the last term if Tt = τ Yt . Solving for Y we get, Y = 8000.
(b) What is the structural deficit or surplus of the Mediterranean government (5 marks)?
Answer. The structural deficit is the deficit that the economy will obtain if it is producing at its potential
output level. Here potential output in nominal terms if Y P = 6000. Hence:
Btstructural = 1, 000 + 0.05 × 12, 000 − 0.20 × 6000 = 400
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Exam 2009/10
Jorge F. Chávez
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EC108
Review May 2014
So, the economy is running a structural fiscal deficit (net increase in debt).
(c) Given your answers to (a) and (b), explain what is likely to happen in the Mediterranean economy
over the next years on the assumption that the economy has an IS-LM structure and policies remain
unchanged (10 marks).
Answer. Since the economy has a structural deficit, debt accumulation might accelerate in future periods.
If this happens, two effects will take place. First, because debt service payments are received by domestic
households, higher public debt will move the IS curve upwards because the increase in family incomes. This
is inflationary. Second, the LM curve will tend to shift as well because of the growing liquidity preference
as the stock of debt increases. This is a deflationary effect.
If debt continues to rise, the country might eventually face a huge probability of default. That means that
a crisis might be ad portas.
(d) How might your answer change if it turned out that Mediterranean households adhere to a Ricardian view of government borrowing (that it is equivalent to taxation) (5 marks)?
Answer. If households are Ricardian, then the inflationary effect described before will not take place (the
IS curve will not shift). Households will reduce consumption because they expect future tax increases to
finance the increasing debt.
Jorge F. Chávez
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