Transcript Tutorial Slides

Tutorial to the
Monetary Policy Lecture
May 24-28, 2004
Dr. Julian von Landesberger
HVB Group Economics
[email protected]
[email protected]
Julian von Landesberger
13.04.2015
1
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:
The 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.
Julian von Landesberger
13.04.2015
2
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 ?
Julian von Landesberger
13.04.2015
3
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.
Julian von Landesberger
13.04.2015
4
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.
Julian von Landesberger
13.04.2015
5
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
Take expectations...
pt
= b/a
pte
= b/a
lt
= (1/2)(b)2/a
Compute the loss...
Julian von Landesberger
13.04.2015
6
Commitment
People understand the policymaker’s 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.
Julian von Landesberger
13.04.2015
7
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
Julian von Landesberger
13.04.2015
8
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
Julian von Landesberger
13.04.2015
9
Alternative mechanisms to enhance credibility
The costs under the commitment are lower than those under
discretion. Without commitment, pt> 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.
Julian von Landesberger
13.04.2015
10
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.
Julian von Landesberger
13.04.2015
11
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.
Julian von Landesberger
13.04.2015
12
Taylor overlapping wage model
Overlapping nominal wage contracts. In period t, set (log)
nominal wage wt for two periods. Average (log) wage wt
1
pt  wt  wt  wt 1 
2
Set wages according to expected average nominal wages
1
wt  wt 1  wt 1 t  xt
2
1
pt  wt  wt 1 
2
1 1
1
pt   wt 1  wt 1 t  xt  wt 2  wt t 1  xt 1 
2 2
2




Julian von Landesberger
13.04.2015

13


Taylor overlapping wage model


1

pt  wt 1  wt 2   wt 1 t  wt  wt  wt t 1   xt  xt 1 
4
2


 

1

pt  2 pt 1  2 pt 1 t  2 p  pt t 1   xt  xt 1 
4
2
pt 



1

pt 1  pt 1 t  p  pt t 1   xt  xt 1 
2
2

pt  pt 1  pt 1 t  pt    xt  xt 1    t
p t  p t 1 t    xt  xt 1    t
 t   pt  pt t 1    t
Julian von Landesberger
13.04.2015
14
Rotemberg’s quadratic price-adjustment costs model
pt optimal unrestricted (log) price,
~p price of particular firm,
t
pt (log) price level.

2
1   ~
2
*
~
~


min
E

p

p

c

p

p

t
t 
t 
t 
t  1
 ~pt  0 2
 0
First-order condition for ~p
t


~
pt  pt*  c ~
pt  ~
pt 1   c ~
pt 1 t  ~
pt  0
1 ~ ~*
~
~
p t  p t 1 t   pt  pt 
c
Optimal unrestricted price:
pt*  pt  xt   t
Julian von Landesberger
13.04.2015
15

Rotemberg’s quadratic price-adjustment costs model
~
~
All firms are identical, therefore pt  pt and p t  p t
The Phillips curve can be derived as follows:

t
c
c
p t  p t 1 t  xt 
Julian von Landesberger
13.04.2015
16
Calvo’s staggered contracts model
pt optimal unrestricted (log) price,
~p price of particular firm is adjusted in period t with prob q,
t
pt (log) price level.
2
1 
 
*
min  (1  q)  Et ~
pt  pt 
~
pt 2  0
First-order condition for ~pt

 (1  q)   ~pt  pt* t   0

 0


 (1  q)   ~pt   (1  q)   pt* t
 0

Julian von Landesberger
13.04.2015
17

Calvo’s staggered contracts model

 (1  q)   

1
1  (1  q) 

 
~




pt  1  1  q    1  q  p *t  t
 0
*
~
pt  1  1  q   pt 



 
1  1  q  1  q   1  q   p *t  1 t 


 0
~
p  1  1  q   p * 1  q ~
p
t
Julian von Landesberger
13.04.2015
t 1 t
t
18
Calvo’s staggered contracts model
pt*  pt  yt   t
~
pt  1  1  q   pt  yt   t   1  q ~
pt 1 t
Aggregate price level (not all firms equal)
pt  q~
pt  1  q  pt 1
q~
pt  p t  qpt 1
q~
pt 1 t  p t 1 t  qpt
Julian von Landesberger
13.04.2015
19
Calvo’s staggered contracts model
Insert into definition of the price level
p t  qpt 1  q1  1  q   pt  yt   t 
 1  q  p t 1 t  qpt


(1  q )p t  q1  1  q  yt   t 
 1  q p t 1 t
p t  p t 1 t
Julian von Landesberger
13.04.2015
1  (1  q)
yt   t 
q
1 q
20
The economy
Say that the economy is described by:
ut  utN  p t  p te    t
ut is the unemployment rate,
utN is the natural rate of unemployment,
pt is the inflation rate and
p te its expected value
t is a supply shock, i.i.d. with mean 0 and variance s2
Agents have rational expectations.
Julian von Landesberger
13.04.2015
21
Policymaker‘s objective
The policymaker’s loss function is given by:
Lt 
p   ut  u
2
t

T 2
t
2
utT is the target unemployment rate which for now we take as bein
below the natural rate:utT  utN  k
The target for inflation is normalized to zero, without loss of genera
Julian von Landesberger
13.04.2015
22
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 k>0.
2. Assume the policymaker observes t when setting policy pt
at each period, but rational agents don’t. 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?
Julian von Landesberger
13.04.2015
23
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.
Julian von Landesberger
13.04.2015
24
Question 1 and 2:
2. The discretionary Central Bank solves:
2
T 2

p t   ut  ut  
min  

pt
2




with F.O.C yielding the optimal policy rule:
(1)
p t   ut  utT 
(2)
This is a simple form of a countercyclical policy. Replace ut
from the Phillips curve into the expression and take expectations to obtain:
p te  k
(3)
Julian von Landesberger
13.04.2015
25
Question 1 and 2:
Replace this, together with the Phillips Curve into equation (2),
to obtain:
p t    p t  k   t  k 
(4)
which, after rearranging, gives the solution for inflation.
Plugging this into the Phillips Curve (together with equation 3)
you obtain unemployment:
p t  k 

t
1 
1
N
ut  ut 
t
1 
Julian von Landesberger
13.04.2015
26
(5)
(6)
Question 1 and 2:
Plug these into the loss function to obtain the expected loss:
2
2
1 

1


E  k 
 t    
 t  k  
2 
1  
1  
 


(7)
Take the expectations taking into account that E(t)=0 and
E(2t)=s2 to obtain the ex ante expected loss under discretion:
(8)
Julian von Landesberger
13.04.2015

1
2
L  1   k 
s 
1    
2
D
2
27
Question 3:
Assume now the policymaker can commit ex ante to a linear
state contingent rule:
(3)
pt = c + bt
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.
Julian von Landesberger
13.04.2015
28
Answer to 3:
Replace the inflation rule into the ex ante expected loss, and
take expectations to obtain:

1 2
2 2 2
2 2
2
c   b  1  b   s  k
2

(9)
Minimizing this with respect to b and c yields the optimal rule: c
=0 and b =  /(1+). Equilibrium unemployment and inflation
are:

pt 
t
1 
(10)
1
N
ut  ut 
t
1 
(11)
Clearly, since this policy leads to the same unemployment but
lower inflation than the discretionary one, it achieves a
superior outcome.
Julian von Landesberger
13.04.2015
29
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.
Julian von Landesberger
13.04.2015
30
Question 4:
Just set b =0 in (9)
1 2
c  k 2 
2
Minimize with respect to c to obtain the optimal policy rule:
c = 0.
Equilibrium unemployment and inflation are:
pt = 0
ut  utN   t
(13)
Julian von Landesberger
13.04.2015
31
(12)
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 >1/1+  ).
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:
L 
R
Julian von Landesberger
13.04.2015

2
E  t  k 
2
32
(14)
Question 4:
LR 

s
2
 k2
(15)
2
The non-state-contingent 0-inflation rule is therefore
preferrable to discretion if:
LD >LR
1   k 2  s  s 2  k 2 
(17)
1   k 2  s 2
(18)
1 
Julian von Landesberger
13.04.2015
33
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 don’t lead to a very large variability of unemployment.
Discretion therefore becomes undesirable compared with
a 0-inflation rule.
Julian von Landesberger
13.04.2015
34
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 *.
Find the optimally appointed Central Banker’s *(you do not
need to find a closed form solution). Show that 0 <*< .
Julian von Landesberger
13.04.2015
35
Question 5:
From question 2, we know the appointed Central Bank will
follow the policy:
*

p t  *k 

* t
1 
ut  utN 
1

* t
1 
(19)
(20)
Plug this into the loss function, noting crucially that the social
loss function still involves  and not *.
Julian von Landesberger
13.04.2015
36
Question 5:
Take expectations to obtain the Government’s ex ante
expected social loss function:
2
2

*

1  *

1

 

E L   E   k 





k

  (21)
t
t
*
*

2 
1  
 
1  


* 2

2
1 *
    2 
2
(22)
E L        k 
s 
2
*
2
1    

Minimize this with respect to * to obtain the F.O.C that
implicitly defines the optimally appointed Central Banker:


*

 2
G *   *k 2 
s
* 3
1   
Julian von Landesberger
13.04.2015
37
(23)
Question 5:
To prove the claim in the text, note that:
G (0) = - s2 <0
(24)
G () = k2 > 0
(25)
Moreover, differentiate G(.) with respect to its argument to
obtain the slope of the function:
G.  k 2 
Julian von Landesberger
13.04.2015
38
1  2*  3
1   
* 4
s2
(26)
Question 5:
Note that in the interval [0, ] then G’0(.)>0, i.e. the function is
monotonically increasing.
But, if the function in the interval [0, ] is continuous, starts at a
negative value, finishes at a positive value, and is
monotonically increasing, by an application of Bolzano’s
theorem, it must have a unique zero, in the interior of the
interval.
Thus there is a unique optimal *such that: 0 < *<  ,as we
wanted
to show.
Julian von Landesberger
13.04.2015
39
Question 6:
Assume instead now that the Government cannot appoint a
Central Banker with a  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 Bank’s Loss function is Lt +wpt.
•What is the optimal w ?
•Can society achieve the optimal outcome in question 3
now?
• Why?
Julian von Landesberger
13.04.2015
40
Question 6:
The discretionary Central Bank now minimizes the loss
function:
2
T 2
p t   ut  ut 
Lt 
 wp t
2
(27)
Follow exactly the same steps as in question 1, to obtain,
respectively, the policy rule, the equilibrium inflation and
p t   ut  utT   w
equilibrium unemployment:
p t  k 

1 
t  w
1
N
ut  ut 
t
1 
Julian von Landesberger
13.04.2015
41
(28)
(29)
(30)
Question 6:
So immediately note that by setting w =k, we reach the firstbest policy defined in question 3. Intuitively, note that the inflation bias problem is non-state contingent (it is k whatever t ),
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 (*/1+ </1+ ).
Julian von Landesberger
13.04.2015
42
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 statecontingent optimal policy.
But it decreases the loss from the non-state-contingent,
constant, inflation bias, and if adequately set can fully
eliminate it.
Julian von Landesberger
13.04.2015
43
Question 7:
Alternatively, say the Government can give the Central Bank
an ex-plicit inflation target p 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).
p
Again derive the optimal
previous question.
Julian von Landesberger
13.04.2015
and discuss the relation to the
44
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:
Lt 

2
1
p t  p 2   ut  utT 
2

(31)
But, just expand the quadratic to see this is just:


1 2
T 2
2
Lt  p t   ut  ut  2p p t  p 
2

Julian von Landesberger
13.04.2015
45
(32)
Question 7:
Yet, the last term ( p 2 ) is not under the control of the Central
Bank and so can be dropped from the minimisation.
Set p  k  w
and you are just back in Walsh’s 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.
Julian von Landesberger
13.04.2015
46
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
.
utT  EutN
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?
Julian von Landesberger
13.04.2015
47
Question 8:
The Central Bank now minimizes:



1 2
T 2
Lt  p   ut  ut 
2
1 2
e
N
N 2
Lt  p    p t  p t   t  ut  Eut 
2
(33)

(34)
The FOC is:
p t    p t  p te   t  utN  EutN 
Julian von Landesberger
13.04.2015
48
(35)
Question 8:
Taking expectations gives:
p te  0
(36)
The solution for inflation is therefore:
(37)
pt 

1 
u
N
t
 EutN   t 
First, see that underestimating the natural urate
leads
t  Eut
to higher inflation.Yet, note that this is not an inflation bias as
before. In the long-run, because the Central Bank’s
expectations are rational, inflation should average to 0,
whereas in the discretionary solution in question 2 it averages
to k.
N
Julian von Landesberger
13.04.2015
49
N
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.
Julian von Landesberger
13.04.2015
50
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?
Julian von Landesberger
13.04.2015
51
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 



2
2 
max  Et    xt   p t  
(1.1)
2 

 0

s.t.
πt =λxt +βEt πt+1 +ut
xt = -j[it -Et πt+1]+Etxt+1 +gt


where xt is the output gap, πt is the inflation rate, β (0 ,1) the
discount factor, it the nominal interest rate, ut a supply shock
and gt a demand shock.
Julian von Landesberger
13.04.2015
52
The Canonical Monetary Policy Problem - the shocks
Where ut  ut 1  uˆt ,  0,1
and uˆ t ~i.i.d.(0,σu2).
Similarly, g t  g t 1  gˆ t ,  0,1
and gˆ t~i.i.d.(0, σg2).
Finally demand and supply shocks are uncorrelated.
Julian von Landesberger
13.04.2015
53
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
max  xt2  p t2 
2
πt =λxt +ft
(1.2)
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)?
Julian von Landesberger
13.04.2015
54
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:

1 2
max  xt  xt  f t 2
2
Julian von Landesberger
13.04.2015
55

The first order condition
The F.O.C. for this problem is:
-[αxt +λ (λxt + ft )] = 0
or:
λ (λxt + ft )= -αxt
Since πt = λxt + ft, the above condition implies that:

xt   p t

(1.5)
Substituting (1.5) into the original Phillips curve, we obtain:
2
p t   p t  Etp t 1  ut

Julian von Landesberger
13.04.2015
56
(1.6)
Solving by forward substitution
The Phillips curve equation is a stochastic first-order difference
equation in πt.There are several methods to solve this kind of
equations.
A simple one is forward substitution: substituting in for πt+1
using (1.6) evaluated at t +1 and then take the expectations,
which depend on πt+2, 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.
Julian von Landesberger
13.04.2015
57
Solving by forward operators
Define the forward operator as L-kxt =Etxt+k.
Using this definition, (1.6) can be written as:
 2
1 
1    L p t  ut


1   L1 p   u
   2  t   2 t
Julian von Landesberger
13.04.2015
58
The canonical monetary policy problem without commitment
Show that the optimal policy without commitment implies:
xt  

ut
(1.3)

pt  2
ut
   1   
(1.4)
   1   
2
What is the relationship between these equations and the
expressions derived in class?
Julian von Landesberger
13.04.2015
59
Solving by forward operators II
Since the forward operator is linear, this expression implies
that:
1

pt 
u
2 t
1   L1    
   2 
(1.7)
  (0,1)
With α>0 and
, 0 <αβ/(α +λ2) <1. This condition is
equivalent to the terminal condition that we need to impose on
the problem when we apply forward
 to solve the
 substitution
1
  1 
equation and implies that
 
L 
2
1   L1   0   

   2 
(1.8)
Julian von Landesberger
13.04.2015
60
Solving by forward operators III
Substituting (1.8) into (1.7) yields:





  1 
pt 
L  ut
2 
2
    0   


  
pt 
Et ut 
2 
2
    0    
Sinceut  ut 1  uˆt
know that:
  0,1
, where
 1
uˆ t
, and
ut    ut    j uˆt   j

j 0
Julian von Landesberger
13.04.2015
61
~i.i.d.(0,σ2u), we
Unwinding the shock
Et ut     ut
Hence
and, therefore,
pt 
ut
  

  2  0   2 


Since |ρ |<1 and
pt 
Julian von Landesberger
13.04.2015

ut
  2 1   

2
   
62

ut
2   1   
(1.9)
Solution for the output gap
Substituting (1.9) into (1.5) yields:
 ut
xt  2
   1   
(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 ρ
=0, i.e., when there is no persistence in supply shocks.
Julian von Landesberger
13.04.2015
63
The second-moment trade-off
Show that, given preferences about the inflation-output
variability (that is, the parameter α), there is a second-moment
efficient frontier characterized by σ (xt)/σ(πt)=λ/α ,where σ(z)
denotes the standard deviation of z.
•Plot this equation on the (σ(xt)/σ (πt),α)-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 α =0?
•What is the optimal variance of xt when α =0?
Julian von Landesberger
13.04.2015
64
The second-moment trade-off
Given (1.9) and (1.10):
and
s ut 
s  xt   2
   1   
s ut 
s p t   2
   1   
These two expressions imply that:
σ (xt)/σ(πt) = λ/α
Julian von Landesberger
13.04.2015
65
The second-moment trade-off
sx/sp
A
B

R
Julian von Landesberger
13.04.2015
66
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.
Julian von Landesberger
13.04.2015
67
The second-moment trade-off
Equation (1.9) implies that when α = 0 the optimal variance of
πt is 0.
This means that when the central bank does not care about
output, the best policy is total inflation stability.
Similarly, making α =0 in equation (1.10), the optimal variance
of xt is σ2(ut)/λ2, 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 α than the
median
Julian von Landesberger voter) is higher output volatility.
13.04.2015
68
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,

Et p t    2
ut
   1   
•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?
Julian von Landesberger
13.04.2015
69
Inflation targeting
Equation (1.9) implies that
Etp t  
Et ut 
2   1   
Et ut     ut
We showed in part (a) that
. Hence,

Et p t    2
ut
   1   
Since |ρ |<1, inflation is expected to return to its target level
gradually at exponential rate ρ.
Julian von Landesberger
13.04.2015
70
Inflation targeting
When the shock is pure noise ρ =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 α =0), convergence is instantaneous as well.

Et p t    2
ut
   1   
Julian von Landesberger
13.04.2015
71
The optimal interest rate policy
Show that the interest rate policy consistent with (1.3) and
(1.4) is given by
it 
Where
1
j
gt  Etp t 1


 1   
1
  1

j 
Why is the coefficient on Et πt+1 greater than one?
Julian von Landesberger
13.04.2015
72
The optimal interest rate policy
Using (1.10), we can write
 1   ut
xt  Et xt 1   2
   1   
 1   

xt  Et xt 1  
ut
2
    1   
Equation (1.9) implies
(1.11)
Julian von Landesberger
13.04.2015
 1   
xt  Et xt 1  
Etp t 1

73
The optimal interest rate policy
The IS/Aggregate Demand Curve can be written as:
xt  Et xt 1  j it  Etp t 1   g t
Substituting (1.11) into the above expression yields
 1   

Etp t 1  j it  Etp t 1   gt

it  
Julian von Landesberger
13.04.2015
  1   
 1 
Etp t 1

j 
j 
gt
74