Chapter 22. The limits to
stabilization policy:
Credibility and uncertainty
ECON320
Prof Mike Kennedy
Overview
• So far we have assumed that policy is effective at stabilizing
output and inflation fluctuations based on the assumptions
that:
– Policymakers have perfect information on the current state of the
economy
– Policy actions have predictable and known quantitative effects
– All announced policy actions are credible (the public believes that the
central bank will do what it says it will do)
• These are strong assumptions that may not be valid in a real
world setting setting
• In what follows, the model will be adjusted to take account of
these features
Credibility:
The time-inconsistency of optimal monetary policy
• What happens when policymakers can undertake discretionary
policy actions, after agents have formed their expectations?
• Start by assuming that all shocks are zero
• The aggregate supply curve when γ = 1:
p t = p t,e t-1 + yt - y
• Goods market equilibrium simplifies to:
yt - y = -a2 (rt - r )
• The central bank can control the current output gap and thus
e
the rate of inflation for any given p t,t-1
• The loss function is written in terms of y*, the desired level of y
SLt = (yt - y*) 2 + kp t2 , k > 0
• Because of distortions we assume that:
y* = y + w, w > 0
Credibility con’t
• From the above we have
e
e
yt = y + p t - p t,t-1
Þ yt - y* = p t - p t,t-1
-w
• Which means that we can write the loss function as
SLt = (p t - p t,e t-1 - w) 2 + kp t2
• If the inflation target, π*, is zero, then the Taylor rule would be:
rt = r + hp t + b(yt - y)
• Assuming that the central bank sticks to the above Taylor rule,
then equilibrium under rational expectations becomes
p t - p t,e t-1 = p * = 0, yt = y
• The question investigated here is: Does the central bank have
an incentive to cheat?
Determining the incentive to cheat
e
• We can calculate the change in SL at the point where p t = p t,t-1
=0
dSL / dp t = -2w < 0
• This says that the social loss can be reduced if the central bank induces
some surprise inflation which moves y closer to y*
• The outcome is that a central bank that can engage in discretionary
policy will not want to stick to the Taylor rule
• Minimizing SL (2nd equation, slide 4) wrt π:
w
w
, yt = y +
‘Cheating outcome’ with surprise inflation p t =
1+ k
1+ k
• The difference between the social loss with the Taylor rule (SLR) and
that with cheating (SLC) measures the temptation to cheat:
w2
Temptation to cheat º SL R - SLC =
1+ k
Time-consistent monetary policy
• From the final equation we see that the greater is ω, the
greater the temptation to cheat
• The important point emerging from the previous slide is that
the policy maker has no incentive to deliver price stability; that
is to follow a Taylor rule
• The problem here is that rational agents will know this or at
least they will figure it out as time passes
• In this case, the time consistent rational expectations
equilibrium is one where inflation is higher and output is back
at potential
pt = p
e
t, t-1
w
= , yt - y
k
• This is now time consistent in that the central bank delivers the
rationally expected inflation rate, which unfortunately is now
greater than zero
Time-consistent monetary policy and credibility
• The equation on the previous slide illustrates the problem of
building credibility when the central bank has discretion
• The final outcome is one where inflation is permanently
higher but there are no output gains
• The outcome is worse than one under a Taylor rule where
expectations would equal π* = 0 and output would be at
potential
• The social loss when the policy maker has discretion is now
• The first term represents the loss due to inefficiently low
output while the second term is the loss due to higher
inflation – how can the second be eliminated?
Building a reputation
• The above assumed that policymakers were short sighted
• If they care about their reputations then the outcome will be
different
• If the central bank pursues a rules-based policy then
e
p t,e t-1 = p R = 0 if p t-1 = p t-1,
t-2
• If under discretion, if the bank cheats, then the optimal π is:
p
e
t, t-1
w
e
= pD =
if p t-1 ¹ p t-1,
t-2
k
• Knowing that this is the public’s expectation of π, the best
thing the central bank can do is deliver it
• In this way, it gains credibility
Building a reputation con’t
• The temptation to cheat when reputation matters can be
written as:
• The first term is the same as derived in slide 5 (final equation)
and shows the gain from using discretion
• The second shows the cost to the policymaker’s reputation
which occurs in the second period
• The term 1 + ρ shows that the policymakers discounts the
future loss – the higher is the discount rate (ρ) the less
reputation is valued and the greater the incentive to cheat
Building a reputation con’t
• The condition can be evaluated based on previous equations
SLR - SLC
SLD - SLR )
w 2 (rk -1)
(
=
1+ r
k 2 (1+ k )(1+ r )
• The policymaker will stick to the policy rule as long as the shortrun gains are less than the next period costs
• The policymaker will not want to cheat if:
– The discount rate (ρ) is low; that is reputation is valued highly, and
– The value of the inflation aversion parameter (κ) is low
• Note that the value of ω, the measure of market distortions,
does not affect the sign of the expression – a higher value of ω
increases the current period gain but also the next period loss
Delegating monetary policy
• The issue here is the government and whether they would
place sufficient weight on the future outcomes of their actions
• For this reason many economist advocate delegating monetary
policy to an independent central bank
• There are varying degrees of independence as shown in Table
22.1, with Canada, along with Japan and the UK, occupying a
middle ground
• Suppose that the bank considers the loss from instability to be
given by
SLB = (yt - y*) 2 + (k + e)p t2 , e > 0
• The parameter ε measures the degree to which the inflation
aversion of the central bank exceeds that of the government –
a measure of the bank’s conservatism
Delegating monetary policy con’t
• Define 0 ≤ β ≤ 1 as a measure of the degree in independence
that the government has given the central bank
• Optimal monetary policy is determined by minimizing the
modified loss function
S˜L = (1- b)×SL + b ×SLB = (yt - y*) 2 + (k + be)p t2 , 0 £ b £ 1
• The time consistent rational expectations equilibrium with
policy delegated to a conservative central banker is:
pt = p
e
t, t-1
w
=
, yt = y
k + be
• Compared to the last equation in slide 8 (the equilibrium with
cheating) we see that we get lower inflation – it would go to
zero as ε approaches infinity, which would be complete
inflation aversion
Delegating monetary policy con’t
• Recall again the government’s loss function
SLt = (yt - y*)2 + kp t2 Þ w 2 + kp t2
• Subbing in the outcome for inflation from the previous slide
we get
2
kw
SL = w 2 +
2
k
+
be
(
)
• The higher is the term βε, the lower will be the loss
• Two conditions give this result:
– The central bank must have some independence (β > 0)
– The central bank must have a greater aversion to risk than the
government (ε > 0)
• We should expect to see a better inflation outcomes in
countries that have more independent central bank, which we
do observe (Fig 22.2 in the text)
An example of delegating monetary policy:
The case of the Bank of England
7.8
7.7
7.6
7.5
7.4
7.3
7.2
7.1
7
6.9
Date of announcement 7 May
10 year interest rate
An example of delegating monetary policy:
The case of the Bank of England
7.5
7.4
Date of announcement 7 May
5 year interst rate
7.3
7.2
7.1
7
6.9
6.8
Credibility versus flexibility
• There is a trade-off between flexibility and credibility which
arises (not surprisingly) in the case of supply shocks with high
variances
• With a conservative central bank, the reaction to the rise in
inflation will exacerbate the variance in output to such an
extent that social welfare falls
• In the end some flexibility may be desirable
• In such cases, communicating with the public and markets will
be very important
The implication of measurement error
• In real time, when policymakers have to make decisions,
current estimates of the state of the economy are likely not
providing an accurate picture
• A particular dramatic illustration of this is shown in Fig 22. 5 in
the text
• Policymakers significantly overestimated the degree of slack in
the economy
• This reflected both errors in measuring actual output but as
well the level of potential, which we now know was
weakening
• The result was a long period of inflation during which inflation
expectations rose and became difficult to bring down
The implication of measurement error
• To study the implications we first start by defining the following
yˆt º yt - y and pˆ t º p t - p *
• Now suppose that each gap deviates from its true values as
follows
yˆte = yˆt + mt , E[ mt ] = 0, E[ mt2 ] = s m2
pˆ te = pˆ t + e t , E[et ] = 0, E[et2 ] = s e2
• The variables μ and ε are random with zero means and constant
variances and they reflect the degree of uncertainty about the
output and inflation gaps
• The Taylor rule now becomes
rt = r + hpˆ te + byˆte Þ rt = r + hpˆ t + byˆt + het + bmt
The implication of measurement error
• Assume that expected inflation equals the bank’s target π*, then
the SRAS becomes
• Allowing for only demand shocks AD becomes
yˆt = zt - a2 (rt - r ), E[zt ] = 0, E[zt2 ] = s z2
• The Taylor rule along with the SRAS and AD curve are a complete
model, which yields the following output gap expression
zt - a 2 he t - a 2 bm t
yˆt =
1+ a 2 (b+ gh)
• From this expression it follows that the variance of output is
2
2 2 2
2 2 2
s
+
a
h
s
+
a
z
2
e
2b s m
2
2
s y º E[ yˆt ] =
[1+ a 2 (b+ gh)]2
The implication of measurement error
• The final equation on the previous slide, the variance of the
output gap, is reproduced here
2
2 2 2
2 2 2
s
+
a
h
s
+
a
b sm
z
e
2
2
s y º E[ ŷt ] =
[1+ a 2 (b+ g h)]2
2
2
• In the absence of errors, the variance of the output gap would
be
s y2
s e2 =s m2 =0
= E[ ŷt2 ] =
s z2
[1+ a 2 (b+ g h)]2
• Thus the errors contribute to output instability by inducing
policymakers to put in place the wrong interest rates
The implication of measurement error
• From the SRAS equation (first equation, Slide 17), it follows
that, if the variance of the output gap can be minimised, then
so will the variance of the inflation gap
• Deriving the first order conditions for minimising the variance
wrt h and b we get
¶s y2
= 0 Þ ha2s e2 [1+ a 2 (b+ g h)]- g (s z2 + a 22 h2s e2 + a 22 b2s m2 ) = 0
¶h
2
2 2 2
2 2 2
ha 2s e2 s z + a 2 h s e + a 2 b s m
=
g
[1+ a 2 (b+ g h)]
¶s y2
= 0 Þ ba 2s m2 [1+ a 2 (b+ g h)]- (s z2 + a 2 h2s e2 + a 2 b2s m2 ) = 0
¶b
2
2
2 2 2
2 2 2
s
+
a
h
s
+
a
b sm
z
e
2
ba 2s m =
[1+ a 2 (b+ g h)]
2
2
2
The implication of measurement error
• From the above two equations it can be shown that
2
gs
h
m
= 2
b se
• The greater is the uncertainty regarding the output gap
relative to the inflation gap, the larger is the central bank’s
response to the inflation gap
• The above equation and the final equation on the previous
slide yield
s z2
gs z2
and h =
b=
2
a 2s m
a 2s e2
The implication of measurement error
• Suppose now that the measurement errors are expected to
persist in the following way
mt = rmt-1 + dt , 0 < r < 1, E[dt ] = 0, E[dt2 ] = s d2
et = qet-1 + kt , 0 < q <1,
E[kt ] = 0, E[kt2 ] = s k2
• We can expand these back to get
mt = dt + rdt-1 + r 2 dt-2 + r 3dt-3 +...
et = dt + qkt-1 + q 2kt-2 + q 3kt-3 +...
• Then the variance will equal
2
s
s m2 º E[ mt2 ] = s d2 (1+ r + r 2 + r 3 +...) = d
1- r
2
s
s e2 º E[et2 ] = s k2 (1+ q + q 2 + q 3 +...) = k
1- q
The implication of measurement error
• This will modify the final two expressions on slide 20 to
s z2 (1- r )
gs z2 (1- q )
b=
and h =
2
a 2s d
a 2s k2
• The message here is that the more persistent are the shocks
the smaller are the optimal levels of b and h
• In this case, go slow