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Tutorial to the Monetary Policy Lecture May
24-28, 2004
Dr. Julian von Landesberger HVB Group
Economics julian.vonlandesberger_at_hvb.de julian.vo
n-landesberger_at_gmx.de
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Monetary policy problems
Design The policy design problem is to
characterize how the interest rate should adjust
to the current state of the economy. Instrument T
he instrument problem of monetary policy arises
because of the need to specify how the central
bank will conduct its open market operations.
Intermediate target The intermediate target
problem is the choice of a variable, usually a
readily observable financial quantity (or price)
that the central bank will treat, for purposes of
some interim-run time horizon, as if it were the
target of monetary policy.
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The structure of the monetary policy problem
An important complication of the policy design
problem is that the private sector behavior
depends on the current and expected course of
monetary policy. Therefore credibility is crucial
for monetary policy. A key aspect is that wage
and price setting today may depend upon beliefs
about where prices are headed in the future,
which in turn depends on the future course of
monetary policy. Are there gains from enhancing
credibility either by formal commitment to a
policy rule or by introducing some kind of
institutional arrangement ?
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Discretion

• In a discretionary regime the central bank can
  print more money and create more inflation than
  people expect.
• Why would it do this?
• Unanticipated monetary expansions lead to
  increases in real economic activity.
• The natural rate may be viewed as excessive. This
  can occur through distortions from income
  taxation, unemployment compensation, which reduce
  the privately-chosen level of labor and
  production.
• The policy maker can value inflationary finance
  as a method of raising revenues.

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Discretion - Setup
The policymaker trades off benefits and costs in
each period. The loss function is given
by lt (a/2) pt2 - bt(pt-pte) The
policymaker controls a monetary instrument, which
enables him to select the rate of inflation pt in
each period. At this point he does not know
bt. Similarly people form their expectations pte
of the policymakers choice without knowing the
parameter. The decision has to be taken every
period until infinity.
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Discretion - Setup
The policymaker treats the current inflationary
expectations pte and all future expectations as
given when choosing current inflation! pt is
chosen to minimize the expected costs for the
current period Elt while treating all future
costs as fixed. apt - bt 0 pt
b/a Take expectations... pte
b/a Compute the loss... lt
(1/2)(b)2/a
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Commitment
People understand the policymakers incentives,
therefore the surprises - and the benefits - can
not arise systematically in equilibrium.
Enforced commitment on monetary policy
behavior, as embodied in monetary or price rules
eliminate the potential for ex post surprises. A
commitment to fight inflation in the future can
improve the current output/inflation trade-off
that a central bank faces.
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Commitment - Setup
Suppose the policymaker can commit himself in
advance to a rule determining inflation. The
policymaker conditions the inflation rate on
variables that are known also to the private
agents. In fact, the policymaker chooses pt and
pte together subject to the condition that pt
pte. The inflation surprise term in the loss
function is therefore zero by construction. Given
the cost term (a/2) pt2 the best inflation rate
for the central bank to target is zero. pt
0 lRt 0
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The incentive to cheat
If people expect pt 0, then the policymaker has
an incentive to cheat in order to secure some
benefits from the inflation surprise. It reflects
the distortions that make inflation shocks have a
benefit for the policymaker. What does the
policymaker gain from cheating pt
b/a lCt -(1/2)b2/a The temptation to
cheat is E(lR-lC) (1/2)b2/a
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Alternative mechanisms to enhance credibility
The costs under the commitment are lower than
those under discretion. Without commitment, ptgt 0
without benefits resulting. However, no major
central bank makes any type of binding commitment
over the future course of its monetary
policy. What solutions are found in the
literature? First-best equilibrium - remove the
distortions. Second-best equilibrium - commit to
an optimal rule. Third-best equilibrium -
delegate monetary policy to a
conservative central banker Fourth-best
equilibrium - discretionary policy.
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Expectations augmented Phillips curve
If price setting today depends on beliefs about
the future economic conditions, a monetary
authority that is able to signal a clear
commitment to controlling inflation can improve
the short-run output/inflation trade-off. Clarida
/Gali/Gertler (1999) argue that this improvement
arises even the central bank does not have an
incentive to push output above potential. A
central bank that commits to a rule is able to
credibly signal that it will sustain over time an
aggressive response to a supply shock.
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Expectations augmented Phillips curve
The extra kick in the case of commitment to a
policy rule is due to the impact of the rule on
the expectations of the future course of the
output gap. Since inflation depends on the
future evolution of excess demand, commitment to
the rule leads to a bigger fall in inflation per
unit of output reduction today relative to
discretion.
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Taylor overlapping wage model
Overlapping nominal wage contracts. In period t,
set (log) nominal wage wt for two periods.
Average (log) wage Set wages according to
expected average nominal wages
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Taylor overlapping wage model
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Rotembergs quadratic price-adjustment costs model
optimal unrestricted (log) price,
price of particular firm, pt (log) price level.
First-order condition for
Optimal unrestricted price
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Rotembergs quadratic price-adjustment costs model
All firms are identical, therefore
The Phillips curve can be derived as follows
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Calvos staggered contracts model
optimal unrestricted (log) price,
price of particular firm is adjusted in period t
with prob q, pt (log) price level.
First-order condition for
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Calvos staggered contracts model
19
Calvos staggered contracts model
Aggregate price level (not all firms equal)
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Calvos staggered contracts model
Insert into definition of the price level
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The economy
Say that the economy is described by ut is the
unemployment rate, is the natural rate of
unemployment, pt is the inflation rate and
its expected value et is a supply shock, i.i.d.
with mean 0 and variance s2 Agents have
rational expectations.
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Policymakers objective
The policymakers loss function is given by
is the target unemployment rate which for now
we take as being below the natural rate The
target for inflation is normalized to zero,
without loss of generality.
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Question 1 and 2
1. Given the material covered in the first part
of the course,briefly motivate equation 1. Give
reasons for why you may argue that kgt0. 2.
Assume the policymaker observes et when setting
policy pt at each period, but rational agents
dont. What is the optimal discretionary policy
rule? What are the equilibrium levels of
unemployment and inflation? What is the value of
the ex ante expected loss ELt given this policy?
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Question 1 and 2
Equation (1) is a form of the expectations
augmented Phillips Curve, of Friedman and Phelps.
It can be justified from micro foundations with
rational expectations, via a Lucas islands
story. Reasons for a positive wedge between the
target social optimum and natural rates of
unemployment - Distortions in the labor market
(minimum wage, taxes, subsidies, etc) that push
the equilibrium unemployment rate up. - Taxes in
the economy, that generally reduce the level of
output and employment. - Imperfect competition
(e.g. monopoly) so the private production and
employment levels are too low.
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Question 1 and 2
2. The discretionary Central Bank
solves (1) with F.O.C
yielding the optimal policy rule
(2) This is a simple form of a countercyclical
policy. Replace ut from the Phillips curve into
the expression and take expec-tations to obtain
(3)
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Question 1 and 2
Replace this, together with the Phillips Curve
into equation (2), to obtain
(4) which, after rearranging, gives the
solution for inflation. Plugging this into the
Phillips Curve (together with equation 3) you
obtain unemployment
(5) (6)
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Question 1 and 2
Plug these into the loss function to obtain the
expected loss (7) Take the
expectations taking into account that E(et)0 and
E(e2t)s2 to obtain the ex ante expected loss
under discretion (8)
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Question 3
Assume now the policymaker can commit ex ante to
a linear state contingent rule (3) pt
c bet In ex ante designing the optimal policy
to minimize expected loss ELt, what are the
optimal parameters in this rule. Show this policy
achieves a superior outcome (in terms of expected
loss) to the discretionary one, and explain
intuitively why.
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Answer to 3
Replace the inflation rule into the ex ante
expected loss, and take expectations to
obtain (9) Minimizing
this with respect to b and c yields the optimal
rule c 0 and b l /(1l). Equilibrium
unemployment and inflation are
(10) (11) Clearly,
since this policy leads to the same unemployment
but lower inflation than the discretionary one,
it achieves a superior outcome.
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Question 4
Say the Central Bank has limited commitment. It
can only commit to a non-contingent rule of the
form (4) pt c Solve for the optimal rule
and compare its performance with that of
discretion.
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Question 4
Just set b 0 in (9) Minimize with respect to
c to obtain the optimal policy rule c
0. Equilibrium unemployment and inflation
are pt 0 (12)
(13)
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Question 4
Note immediately that this policy leads to lower
inflation than dis-cretion, but unemployment now
fluctuates more in response to supply shocks than
before (1 gt1/1 l ). We expect to find
therefore a trade-off between lower inflation and
higher variance of unemployment. Plugging the
equilibrium into the loss function, and taking
expectations, you obtain the loss under a
rule (14)
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Question 4
(15) The non-state-contingent
0-inflation rule is therefore preferrable to
discretion if LD gtLR (17)
(18)
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Question 4
This will hold if - the wedge between the
natural rate and the target rate of unemploy-ment
is large (k large) leading to a high inflation
bias. - Supply shocks are not very variable. The
first factor makes discretion very costly in
terms of an increase in inflation, and the second
makes the gains from being able to conduct
countercyclical policy small, since supply shock
dont lead to a very large variability of
unemployment. Discretion therefore becomes
undesirable compared with a 0-inflation rule.
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Question 5
Now assume that the Central Bank has no
commitment ability and so solves every period the
problem in question 2 (this will also be true for
all the questions until the end of the problem
set). Still, the Government has an ability to
commit, and it can appoint a Central Banker from
a pool of possible candidates. The candidates
differ in the weight they give to unemployment
vs. inflation variability l. Find the optimally
appointed Central Bankers l(you do not need to
find a closed form solution). Show that 0 ltllt l.
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Question 5
From question 2, we know the appointed Central
Bank will follow the policy (19)
(20) Plug this into the loss function,
noting crucially that the social loss function
still involves l and not l.
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Question 5
Take expectations to obtain the Governments ex
ante expected social loss function
(21) (22) Minimize this with
respect to l to obtain the F.O.C that implicitly
defines the optimally appointed Central
Banker (23)
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Question 5
To prove the claim in the text, note that G
(0) - ls2 lt0 (24) G (l) lk2 gt 0
(25) Moreover, differentiate G(.) with
respect to its argument to obtain the slope of
the function (26)
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Question 5
Note that in the interval 0, l then G0(.)gt0,
i.e. the function is monotonically increasing.
But, if the function in the interval 0, l is
continuous, starts at a negative value, finishes
at a positive value, and is monotonically
increasing, by an application of Bolzanos
theorem, it must have a unique zero, in the
interior of the interval. Thus there is a
unique optimal lsuch that 0 lt llt l ,as we
wanted to show.
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Question 6

• Assume instead now that the Government cannot
  appoint a Central Banker with a l different than
  the social level, but it can offer the Bank a
  contract. Specifically, it can impose a cost on
  the Bank for higher inflation (by e.g. negatively
  indexing the wage of the Banker to inflation, as
  is the case currently in New Zealand). The
  modified Central Banks Loss function is Lt wpt.
  
• What is the optimal w ?
• Can society achieve the optimal outcome in
  question 3 now?
• Why?

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Question 6
The discretionary Central Bank now minimizes the
loss function (27) Follow
exactly the same steps as in question 1, to
obtain, respectively, the policy rule, the
equilibrium inflation and equilibrium
unemployment (28)
(29) (30)
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Question 6
So immediately note that by setting w lk, we
reach the first-best policy defined in question
3. Intuitively, note that the infla-tion bias
problem is non-state contingent (it is lk
whatever et ), but the gains from discretion come
from the ability to have state contingent policy.
The Barro-Gordon proposal in question 4 for a
fixed rule, removes the bias but also state
contingency from policy. The Rogoff proposal
for appointing a conservative Central Bank, by
distorting the relative values of inflation and
unemployment variability, reduces the inflation
bias but also leads to too little discretionary
policy (l/1l ltl/1l ).
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Question 6
The Walsh proposal for a Central Bank contract,
goes to the heart of the problem the penalty in
inflation is linear in the Central Banks loss
function. Therefore it imposes no extra cost of
variable inflation (it is not squared), and so
does not change the countercyclical
state-contingent optimal policy. But it
decreases the loss from the non-state-contingent,
constant, inflation bias, and if adequately set
can fully eliminate it.
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Question 7
Alternatively, say the Government can give the
Central Bank an ex-plicit inflation target
around which the variance of inflation must be
minimised, together with the variance of
unemployment from the target rate. (This is the
currently the case in many countries and notably
the United Kingdom). Again derive the optimal
and discuss the relation to the previous
question.
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Question 7
This has been defended by Svensson (AER) 1997, in
the context of a model only slightly different
from this. The new loss function the Central
Bank minimizes is (31) But, just
expand the quadratic to see this is
just (32)
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Question 7
Yet, the last term ( ) is not under the
control of the Central Bank and so can be dropped
from the minimisation. Set
and you are just back in Walshs case! So you
can again get to the first-best. Therefore, by
giving the Central Bank an explicit inflation
target that is conservative (below the 0 social
optimum inflation rate implicit in the loss
function for this question), the Government can
gain ensure we obtain the first best.
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Question 8
Finally, say that both the Central Bank and
private agents do not observe the natural rate of
unemployment and the supply shock at t. (Do you
know what any of these is, right now?) They only
observe the actual value of the unemployment
rate. Moreover, the Central Bank targets some
optimally formed expected value of the natural
rate, so that now . a) Derive the
discretionary optimal policy rule and the
equilibrium level of inflation. How do
expectational errors in the forecast of the
natural rate affect inflation?
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Question 8
The Central Bank now minimizes (33)
(34) The FOC is (35)
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Question 8
Taking expectations gives (36) The solution
for inflation is therefore (37) First, see
that underestimating the natural rate
leads to higher inflation.Yet, note that
this is not an inflation bias as before. In the
long-run, because the Central Banks expectations
are rational, inflation should average to 0,
whereas in the discretionary solution in question
2 it averages to lk.
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Question 8
The model predicts high inflation in the 1970s
but low inflation in the the 1990s, which fits
the data. The Barro Gordon model is still
driving the dynamics of inflation, but the
inflation bias is now time-varying, allowing
the model to not only explain the great inflation
of the 1970s but also the low inflation of the
late 1990s.
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Question 8
b) At a given period can this model or the model
in question 2 be distinguished from the behavior
of inflation? What about in the long-run? c) It
has been argued that the 1970s were a period
where the natural rate unexpectedly increased and
the Central Bank took a while to catch on, making
a succession of forecast errors. What does
the model predict would happen to inflation?
Similarly, during the late 1990s, estimates seem
to show the natural rate has fallen but Alan
Greenspan repeatedly claimed he believed the
economy was over-heated, suggesting he did not
believe in such a fall and did not update his
natural rate target. What does the model predict
then? How do these predictions fit the broad
trends in inflation over these periods?
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The Canonical Monetary Policy Problem with
Serially Correlated Shocks
Based on Clarida, Gertler, and Gali (1999).
Consider an economy with both supply and demand
shocks in which the presence of some form of
price rigidity implies the existence of a New
Keynesian Phillips Curve. Assume that the
policymaker is trying to solve the following
problem (1.1) s.t. pt ?xt ßEt pt1
ut xt -jit -Et pt1Etxt1 gt where xt is
the output gap, pt is the inflation rate, ß (0
,1) the discount factor, it the nominal interest
rate, ut a supply shock and gt a demand shock.
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The Canonical Monetary Policy Problem - the shocks
Where , and
i.i.d.(0,su2). Similarly,
, and i.i.d.(0, sg2).
Finally demand and supply shocks are
uncorrelated.
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The canonical monetary policy problem without
commitment
Assuming no possibility of commitment, problem
(1.1) is equivalent to an infinite sequence of
problems defined by (1.2) pt ?xt
ft ft is a given constant from the point of
view of the central bank. Why does the absence
of commitment imply that problem (1.1) can be
written as an infinite sequence of one-period
problems like (1.2)?
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The Canonical Monetary Policy Problem with
Serially Correlated Shocks
Without access to a commitment technology, the
central bank is free to reoptimize every period
taking as given previously formed expectations
(discretionary policy). When the expectations in
program (1.1) are taken as given, the problem
boils down to solve (1.2) for every
period. Substituting the Phillips Curve into the
one-period loss function, problem (1.2) reduces
to
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The first order condition
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Solving by forward substitution
The Phillips curve equation is a stochastic
first-order difference equation in pt.There are
several methods to solve this kind of equations.
A simple one is forward substitution
substituting in for pt1 using (1.6) evaluated at
t 1 and then take the expectations, which depend
on pt2, and then repeat the same
procedure. Eventually you need to impose some
terminal condition to get rid of the last term
after an arbitrarily large number of
substitutions. An alternative to this method is
to use lag (and forward) operators.
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Solving by forward operators
Define the forward operator as L-kxt Etxtk.
Using this definition, (1.6) can be written as
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The canonical monetary policy problem without
commitment
Show that the optimal policy without commitment
implies (1.3)
(1.4) What is the relationship between these
equations and the expressions derived in class?
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Solving by forward operators II
Since the forward operator is linear, this
expression implies that
(1.7) With agt0 and , 0 ltaß/(a
?2) lt1. This condition is equivalent to the
terminal condition that we need to impose on the
problem when we apply forward substitution to
solve the equation and implies that
(1.8)
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Solving by forward operators III
Substituting (1.8) into (1.7) yields Since
, where , and
i.i.d.(0,s2u), we know that
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Unwinding the shock
Hence and, therefore, Since ? lt1
and (1.9)
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Solution for the output gap
Substituting (1.9) into (1.5) yields (1.
10) Note that the presence of supply shocks
implies that inflation and output gap move in
opposite directions. The expressions derived in
class are particular cases of (1.9) and (1.10)
when ? 0, i.e., when there is no persistence in
supply shocks.
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The second-moment trade-off

• Show that, given preferences about the
  inflation-output variability (that is, the
  parameter a), there is a second-moment efficient
  frontier characterized by s (xt)/s(pt)?/a ,where
  s(z) denotes the standard deviation of z.
• Plot this equation on the (s(xt)/s (pt),a)-space.
  
• Why do demand shocks not affect the relative
  variability of inflation and output and supply
  shocks do?
• What is the optimal variance of inflation when a
  0?
• What is the optimal variance of xt when a 0?

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The second-moment trade-off
Given (1.9) and (1.10) and These two
expressions imply that s (xt)/s(pt) ?/a
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The second-moment trade-off
s(x)/s(p)
A
B
a
aR
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The second-moment trade-off
When there is a demand shock, the monetary
authority adjusts the nominal interest rate to
keep xt unchanged (through the IS/Aggregate
Demand Curve) and, without any supply shock,
inflation does not change (because nothing
changes in the Phillips Curve). In contrast,
when there is a supply shock the optimal policy
for the central bank implies that inflation and
output gap are moving in opposite directions.
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The second-moment trade-off
Equation (1.9) implies that when a 0 the
optimal variance of pt is 0. This means that
when the central bank does not care about output,
the best policy is total inflation stability.
Similarly, making a 0 in equation (1.10), the
optimal variance of xt is s2(ut)/?2, which is the
variability induced on output to achieve total
inflation stability using the Phillips Curve.
The above figure makes clear that the cost of
appointing a conserva-tive central banker (one
who has a lower a than the median voter) is
higher output volatility.
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Inflation targeting

• Show that the optimal policy, described by (1.3)
  and (1.4), incorpo-rates inflation targeting in
  the sense that it implies gradual conver-gence of
  inflation to its target, i.e., show that, given
  (1.3), (1.4), and the stochastic process for ut,
• What is the rate of convergence of inflation to
  its target when the supply shock is pure white
  noise?
• What is the rate of convergence of inflation to
  its target when the central bank does not care
  about output variability?

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Inflation targeting
Equation (1.9) implies that We showed in part
(a) that . Hence, Since
? lt1, inflation is expected to return to its
target level gradually at exponential rate ?.
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Inflation targeting
When the shock is pure noise ? 0, i.e., without
persistence, convergence is instantaneous, in the
sense that the central bank expects to hit its
target in any future period. When the central
bank does not care about output variability (when
a 0), convergence is instantaneous as well.
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The optimal interest rate policy
Show that the interest rate policy consistent
with (1.3) and (1.4) is given by Where Why
is the coefficient on Et pt1 greater than one?
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The optimal interest rate policy
Using (1.10), we can write Equation (1.9)
implies (1.11)
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The optimal interest rate policy
The IS/Aggregate Demand Curve can be written
as Substituting (1.11) into the above
expression yields