The University of Sydney
School of Economics
ECON6002
Practice Final Exam - Solutions
1. Consider the role of nominal rigidities and market imperfections in explaining the e↵ects of
monetary policy shocks.
(a) Explain whether you agree or disagree with the following statement: “Nominal rigidity
can cause monetary policy shocks to have sizeable real e↵ects on output.” (3 points)
The statement is invalid. Given only a nominal rigidities, monopolistically competitive
firms will change their prices in response to changes in aggregate demand thus implying
that money is virtually neutral.
In the Lucas imperfect-information model the aggregate demand curve is: y = m p and the
aggregate supply curve is: y = 11
2z
2z+
2
m
(p E[p]) where 11 > 0 is the elasticity of labour
supply with respect to the real wage, 2z > 0 is the variance of the good-specific taste shock,
2m > 0 is the variance of the aggregate demand shock.
(b) Is there a distinction between the aggregate demand curve and the good-specific demand
curve? Explain how di↵erent factors influence the good-specific demand curve in the
Lucas imperfect-information model. (3 points)
In the Lucas imperfect-information model, the good-specific demand curve is yi = (m
p) ⌘(pi p) + zi. Three factors influence the good-specific demand curve. The first
factor is aggregate demand m p where higher aggregate demand results in higher good-
specific demand. The second factor is the relative price of the specific good, where a
higher relative price lowers demand and the relation is governed by the elasticity ⌘. The
third factor is a good-specific demand shock or a taste shock.
(c) Suppose that the volatility of aggregate demand shocks 2m increases relative to the
volatility of good-specific taste shocks 2z . How would the slope of the aggregate supply
curve change? Explain by providing an economic interpretation. (3 points)
An increase is 2m relative to
2
z would imply a decrease in the slop of the aggregate supply
curve. Intuitively, a higher aggregate demand volatility implies that when agents observe
a change in demand they would less likely attribute it to a change in the demand for their
own good and so would not adjust their production much, implying a lower response of
output to changes in prices and thus a smaller slope for the aggregate supply curve.
(d) Discuss the validity of the following statement within the context of the Lucas imperfect-
information model: “Money growth, whether observed by economic agents or not, raises
the price level which consequently translates into higher output.” (3 points)
The statement is invalid. If the change in money growth is expected by economic agents,
they will adjust their expectations implying that the change in money growth will only
have an e↵ect on the price level of the economy and not on the output level.

2. Consider the IS curve and New Keynesian Phillips curve:
y˜t = Et[y˜t+1] 1
✓
rt
⇡t = Et[⇡t+1] + y˜t + u
⇡
t
where u⇡t = ⇢⇡u
⇡
t1 + e⇡t is a cost-push shock. Assume that there is no serial correlation so
that ⇢⇡ = 0. The solution for the model takes the form y˜t = a⇡u⇡t , ⇡t = b⇡u
⇡
t and rt = c⇡u
⇡
t .
Suppose that monetary policy responds to expected inflation and and expected output gap
such that: rt = ⇡Et[⇡t+1] + yEt[y˜t+1].
(a) Use the method of undetermined coecients to solve for a⇡, b⇡ and c⇡ and explain how
a positive cost-push shock a↵ects the output gap, inflation, and the real interest rate.
(4 points)
With ⇢⇡ = 0, Et[y˜t+1] = Et[⇡t+1] = 0, so we get a⇡ = 0, b⇡ = 1 and c⇡ = 0. So
y˜t = 0, ⇡t = u⇡t and rt = 0. A positive cost-push shock would raise the current inflation
rate one-for-one and will have no impact on the output gap since the interest rate is
unchanged. The interest rate is una↵ected by positive cost-push shock in this case as the
shock has no e↵ect on expectations of future inflation or future output gap.
(b) How would an increase in ⇡ a↵ect the response of the real interest rate and inflation to
an unfavourable cost-push shock? (2 points)
The cost-push shock has no e↵ect on the real interest rate and so a change is ⇡ will not
a↵ect the response of the real interest rate.
Assume instead that monetary policy responds to current inflation and current output gap
such that: rt = ⇡⇡t + yy˜t.
(c) Use the method of undetermined coecients to solve for a⇡, b⇡ and c⇡. What is the
e↵ect of a positive cost-push shock in this case? (4 points)
With ⇢⇡ = 0, Et[y˜t+1] = Et[⇡t+1] = 0. So we have y˜t = 1✓rt, ⇡t = y˜t, rt = ⇡⇡t +
yy˜t. Solving the equations using the method of undetermined coecients yields yt =
⇡✓+⇡+y u⇡t , ⇡t =
✓+y
✓+⇡+y
u⇡t , and rt =
⇡✓
✓+⇡+y
u⇡t . A positive cost-push shock would
raise the current inflation rate. This would trigger an increase in the real interest rate
which negatively a↵ects the output gap through the IS curve.
(d) How would an increase in ⇡ a↵ect the response of the real interest rate and inflation to
an unfavourable cost-push shock? (2 points)
Note that @rt@⇡ > 0. This implies that a higher response of the real interest rate to inflation
would imply that the interest rate would react more to an unfavourable cost-push shock
by raising the interest rate more in order to bring back inflation down.
3. Consider the time-inconsistency model of monetary policy. The central bank has a loss
function that is di↵erent to the social loss function. In particular, the central bank, subject
to an aggregate supply constraint, sets inflation ⇡ in order to minimize:
LCB =
1
2
(y y⇤)2 + 1
2
a0(⇡ ⇡⇤)2
where ⇡⇤ is the central bank’s inflation target, y⇤ is socially optimal output, a0 > 0 reflects
the central bank’s preference for stabilizing inflation. The social loss function is:
Lsociety =
1
2
(y y⇤)2 + 1
2
a(⇡ ⇡⇤)2
where a > 0 reflects society’s relative preference for stabilizing inflation around ⇡⇤.
Suppose that the aggregate supply curve takes the form y = yn+(⇡⇡e) where y is aggregate
output, yn is flexible-price output, and ⇡e is inflation expectations.
(a) What are the equilibrium levels of output, y, and inflation, ⇡, if the central bank has
discretion, i.e., chooses policy taking expectations as given? (3 points)
First, substitute aggregate supply equation into loss function and set ⇡ to minimize loss:
min
⇡
⇢
LCB =
1
2
[yn + (⇡ ⇡e) y⇤]2 + 1
2
a0(⇡ ⇡⇤)2

The central bank will set ⇡ such that:
) ⇡ = 1
1 + a0
[(y⇤ yn) + (⇡e + a0⇡⇤)]
Under rational expectations ⇡e = ⇡, so ⇡e = ⇡⇤ + 1a0 (y
⇤ yn). Now substitute the
expression for ⇡e back into the expression for ⇡ to get ⇡ = ⇡⇤ + 1a0 (y
⇤ yn). Finally,
plug the expressions for ⇡ and ⇡e into the aggregate supply equation to get: y = yn
(b) Using the expressions derived for y and ⇡, compute the social loss function in terms of
yn, y⇤, and the parameters of the model. What value of a0 minimizes social loss? (2
points)
Substituting the expressions above into the social loss function gives:
LCB =
1
2
(yy⇤)2+ 1
2
a(⇡⇡⇤)2 = 1
2
(yny⇤)2+ a
2a02
(y⇤yn)2 = 1
2
✓
1 +
a
a02
◆
(yny⇤)2
Note that the social loss function will be minimized when a0 !1.
Suppose instead that the economy is hit by aggregate supply shocks and that the aggregate
supply curve takes the form y = yn + (⇡ ⇡e) + ". Assume that the aggregate supply shock
is i.i.d. with mean E["] = 0 and variance var(") = 2.
(c) Solve for the equilibrium levels of output, y, and inflation, ⇡, if the central bank has
discretion in this case. (3 points)
First, substitute aggregate supply equation into loss function and set ⇡ to minimize loss:
min
⇡
⇢
LCB =
1
2
[yn + (⇡ ⇡e) + " y⇤]2 + 1
2
a0(⇡ ⇡⇤)2

The central bank will set ⇡ such that:
) ⇡ = 1
1 + a0
[(y⇤ yn) + (⇡e + a0⇡⇤) "]
Under rational expectations ⇡e = ⇡ and note that E(") = 0, so ⇡e = ⇡⇤ + 1a0 (y
⇤ yn).
Now substitute the expression for ⇡e back into the expression for ⇡ to get ⇡ = ⇡⇤ +
1
a0 (y
⇤ yn) "1+a0 . Finally, plug the expressions for ⇡ and ⇡e into the aggregate supply
equation to get y = yn + a
0
1+a0 ".
(d) Using the expressions derived for y and ⇡, compute the social loss function in terms of
yn, y⇤, " and the parameters of the model. (2 points)
Substituting the expressions above into the social loss function gives:
LCB =
1
2
(y y⇤)2 + 1
2
a(⇡ ⇡⇤)2 = 1
2
(yn +
a0
1 + a0
" y⇤)2 + 1
2
✓
1
a0
(y⇤ yn) "
1 + a0
◆2
(e) Explain why a hawkish central bank with a0 >>> a would introduce a trade-o↵ between
credibility and flexibility in the presence of aggregate supply shocks. (2 points)
Having a0 ! 1 will imply that ⇡ = ⇡⇤ and thus imply a more credible central bank.
Meanwhile, the central bank will have the greatest flexibility in stabilizing output when
a0 is small, i.e. when a0 ! a or when the central bank and society equally care about
stabilising inflation. This shows that there is in fact a trade-o↵ between having a credible
central bank and stabilising output.
4. Consider an individual with perfect foresight who lives from 1 to T , and whose lifetime utility
is given by U =
PT
t=1
tu(Ct), where u0(·) > 0, u00(·) < 0. The individuals intertemporal
budget constraint is given by:
PT
t=1
⇣
1
1+r
⌘t
Ct  A0+
PT
t=1
⇣
1
1+r
⌘t
Yt where A0 > 0 is initial
wealth, r is the real interest rate and Yt is the individual’s income. Assume that (1+r) = 1.
(a) Using the first order condition of the individual’s optimization problem with respect to
consumption, show that optimal consumption each period is constant. (3 points)
The Lagrangian function for the household’s maximization problem:
L =
TX
t=1
tu(Ct) +

A0 +
TX
t=1
✓
1
1 + r
◆t
Yt
TX
t=1
✓
1
1 + r
◆t
Ct
!
The first-order condition with respect to consumption yields:
tu0(Ct) =
✓
1
1 + r
◆t
) u
0(Ct+1)
u0(Ct)
=
1
1 + r
With (1 + r) = 1, we get u0(Ct) = u0(Ct+1) and so Ct = Ct+1 = C¯.
Assume now that = 1 and r = 0. Suppose also that the individual’s income is constant in
each period such that Yt = Y¯ for all t.
(b) What is the individual’s utility-maximizing level of consumption and savings in each
period? (3 points)
Since u00(·) < 0 and (1 + r) = 1, the analysis from the Permanent Income hypothesis
implies that utility-maximization requires consumption to be constant. The budget con-
straint then implies that consumption at each point in time is equal to lifetime resources
divided by the length of life:
C¯ =
1
T

A0 +
TX
t=1
Yt
!
=
1
T

A0 + Y¯

=
A0
T
+ Y¯
Savings in each period are given as:
St = A0
T
(c) Are savings positive or negative? Explain why. (3 points)
With an initial wealth of A0 > 0 the savings are negative. The individual consumes come
of the initial wealth in each period of life. The individual lives of T periods in this model
and they find it optimal to consume a fraction 1/T of the initial asset in each period of
life. This allows for perfect smoothing of the consumption path.
(d) Suppose the individual decides to leave a bequest (or gift) B for the future generation
only in the last period of life. Compute the individual’s consumption in each period in
this case. (3 points)
Again here, the budget constraint then implies that consumption at each point in time is
equal to lifetime resources divided by the length of life:
C¯ =
1
T

A0 +
TX
t=1
Yt
!
=
1
T

A0 + Y¯ B

=
A0
T
+ Y¯ B
T
If the individual decides to leave B as bequests, then they will choose to smooth con-
sumption and deduct a fraction 1/T of the bequest B from consumption in each period
of their life.
5. Consider the “Q” model of investment with adjustment costs. Equilibrium suggests that
capital K(t) evolves as K˙(t) = C 0(I(t))1(q(t) 1) (normalizing the number of firms N = 1
and assuming no depreciation), while the marginal value of capital, q(t) evolves as q˙(t) =
rq(t)⇡(K(t)), where r is the real interest rate. Note that the capital adjustment cost func-
tion, C(I(t)) satisfies C(0) = 0, C 0(0) = 0, and C 00(·) > 0 and the real profit function, ⇡(K(t)),
satisfies ⇡0(·) < 0. Assume the transversality condition limt!1ertq(t)(t) = 0, where (t)
is the representative firm’s capital stock. Assume initially that the economy is in steady-state.
(a) Draw the phase diagram for this model, explaining the location of the saddle path. (3
points)
The saddle path location is due to the fact that capital is increasing above the K˙ = 0 line
and decreasing below, while q is increasing above the downward sloping q˙ = 0 line and
decreasing below. Thus the saddle path must be downward sloping and between the two
lines. Otherwise the economy will explode to zero capital or q or to infinite capital or q,
thus violating the transversality constraint.
(b) Explain the economics behind the assumptions that C 00(·) > 0 and ⇡0(·) < 0. (3 points)
C 00(·) > 0 implies that it is costly for a firm to increase or decrease its capital stock, and
that the marginal adjustment cost is increasing in the size of the adjustment. Meanwhile,
⇡0(·) < 0 implies decreasing returns to aggregate stock of capital.
At time t1 aggregate output rises unexpectedly to a new level.
(c) If the sudden rise in aggregate output is permanent, explain both the short-run and
long-run dynamics of q(t) and K(t) and draw the transition path for the economy using
a phase diagram. (3 points)
q and K rise. In particular, rise in output causes prices and profits to rise. This shifts
the q˙ = 0 line up. q rises immediately (market price of capital rises). But as capital
is increased and output rises, the relative price falls, reducing q (and profits) until it
returns to 1.
(d) Assume now that this unexpected rise in aggregate output is temporary instead of per-
manent, i.e. it is expected that at some future time t2 > t1, aggregate output will return
to its original level. Explain both the short-run and long-run dynamics of q(t) and K(t)
and draw the transition path for the economy using a phase diagram. (3 points)
Rise is not as much and q falls and capital rises until economy crosses the K˙ = 0 and
capital starts to fall until economy ends up on original saddle path just as the temporary
shock ends.