Lecture 9: Stabilization policy with rational
expecations; Limits to stabilization policy;
closed economy case.
• Ch 21 and ch 22 in IAM
• Comprehensive coverage of issues and premises that are central in modern
discussion of stabilization policies
• Limitation: Closed economy.
Ragnar Nymoen
Department of Economics, University of Oslo
• Focus on the issues that are most analytically tractable.
November 1, 2007
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The case against backward-looking expectations
Backward-looking vs forward-looking expectations (IAM
21.1 and 21.2)
Lecture 8 established the reference case with backward-looking expectations.
IAM refers to that model as static expectations, but as we show see below
inflation expectations are really dynamic since they follow the same dynamic
process as πt itself, but with one period lag.
Start with the subsection called “case against backward-looking expectations”
which point out as a weakness that after a policy change expectations adjust
gradually, even when the policy change is perfectly credible.
This motivates the rational expectations hypothesis, REH.
The short-run AD-AS model of lecture 8 consists of
1
πt = π ∗ − (yt − zt),
(1)
α
α2h
with α =
and
1 + α2b
α + vt + α1gt − α2r̄ + α2bȳ
.
zt = 0
1 + α2b
(where the ȳ- term has been corrected, no consequences for last weeks conclusions), and:
πt = πt−1 + γ(yt − ȳ) + st.
(2)
The short-run aggregate demand (SRAD) function, and the short-run aggregate
supply (SRAS) function. They define the equilibrium solutions for yt and πt as
functions of the exogenous and predetermined variables:gt, vt, st and πt−1.
We established last week that the SRAD was invariant to how inflation expectations are formed.
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But SRAS clearly depends on expectations formation, here we have used so
called static expectations πte = πt−1.
The case against backward-looking expectations rests on the behaviour of inflation expectations in the face of a credible change (reduction) in the inflation
target.
To analyze this, we need the dynamic equation for inflation.
To obtain that, write the SRAD schedule with yt on the left-hand side, and
insert into (2). This gives:
It gives insight to re-write (3) in terms of “deviations from target”, as in IAM
eq (9): subtract π ∗ on both sides of the equation to give
1
γ
1
(4)
(πt−1 − π ∗) +
(zt − ȳ) +
st,
1 + γα
1 + γα
1 + γα
IAM uses the notation
1
β=
.
1 + γα
which is positive but less than one under the assumptions we made in lecture
8.
πt − π ∗ =
The autoregressive coefficient is the same as in the final equation that we
derived for yt last week. Is that a coincidence?
The stable steady-state solution for πt,denoted π̄, is:
1
π̄ = π ∗ + (z̄ − ȳ)
α
where ȳ is full capacity output, and z̄ is the steady-state value of zt. As we
discussed in the last lecture, z̄ is an endogenous variable in the steady-state.
Since
α + α1ḡ − α2r̄ + α2bȳ
z̄ = 0
1 + α2b
the endogenous variable is the equilibrium real-interest rate r̄.
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1
γα
γ
1
πt−1 +
π∗ +
(zt − ȳ) +
st,
1 + γα
1 + γα
1 + γα
1 + γα
for t = 1, 2, 3,
πt =
(3)
which is the final equation for πt since it has only predetermined and exogenous
variables on the right hand side.
Expectation errors
e,mod
πt
=
1
γα
γ
πt−2 +
π∗ +
(z e − ȳ), for t > 1
1 + γα
1 + γα
1 + γα t−1
(7)
We now investigate expectations errors, assuming that t = 0 is the initial
situation, and that expectation for period t = 1, 2, .. is made at the start of
period 1, using information that is available at the end of period 0. Forecast
are made for the (infinite) horizon t = 1, 2, 3, ......
For period t = 2, 3, ..., economic agents use solution for πt−1(form (3)) to
generate their forecasts, but with s2 = s3... = 0, and v2 = v3... = 0 since
there is no way that agents (at the start of period 1) can anticipate the sequence
of future supply and demand shocks. zte denotes ztwith vt = 0 (for all t)
Although we have so far referred to the backward-looking expectations formation as static, we are in fact ”using the model” to generate the agents’ forecast
for period t = 2, 3, ....
However, in order to simplify, we assume that gt = ḡ and vt = st = 0 both in
the economy and in the expectations. This is the same simplification as on p
630 in IAM.
Hence, for the purpose of this lecture
πt
e,mod
πte = π0, for t = 1 but
e,mod
, for t > 1
πte = πt
e,mod
where πt
(5)
(6)
e,mod
denotes the model based forecast. πt
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is defined as:
can be written as:
1
γα
γ
e,mod
πt−1 +
π∗ +
(z̄ − ȳ), for t > 1
1 + γα
1 + γα
1 + γα
where z̄ is the constant obtained by imposing gt = ḡ and vt = st = 0 in the
expression for zt. Hence expectations follow the same dynamic equations as
inflation itself:
e,mod
πt
=
8
The expectations error in period t is:
e,mod
eπ
t = πt − πt
= (πt − π ∗) − (πt−1 − π ∗)
γ
= β(πt−1 − π ∗) +
(z̄ − ȳ) − (πt−1 − π ∗)
1 + γα
γ
= (β − 1)(πt−1 − π ∗) +
(z̄ − ȳ)
1 + γα
γ
γ
= (β − 1)β(πt−2 − π ∗) + (β − 1)
(z̄ − ȳ) +
(z̄ − ȳ)
1 + γα
1 + γα
γ
= (β − 1)β(πt−2 − π ∗) + β
(z̄ − ȳ)
1 + γα
Continue with repeated backward substitution to obtain:
The case against (partly) backward-looking expectations
Assume that π0 = π ∗ so that z̄ = ȳ for consistency. Then (8) shows that
expectations generated by (5) and (6) are always accurate.
If π0 6= π ∗ and z̄ 6= ȳ, the forecasts become gradually closer to the inflation target. The inflation forecast during the phase of adjustment may be as depicted
in Figure 21.1 in IAM, for the case of π0 = 3% and π ∗ = 0.
e,mod
eπ
t = πt − πt
The real case is made the following construction:
= (β − 1)β t−1(π0 − π ∗) + β t−1
γ
(z̄ − ȳ)
1 + γα
(8)
Assume that π0 = π ∗ at the time that the agents form their forecast. Then,
0
for t = 2, 3, ....
which is the counterpart to equation (10) in IAM. Note that this result is
based on agents knowing how the economy operates. Hence expectations are
not backward-looking in the strict sense of setting πte = π0 for all t.
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immediately after, the central bank changes the target to π ∗ = 0.
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From period 1 and onwards the economy and expectations evolve according to
Hence it is difficult to claim that this system has a meaningful solution,apart
from the short run model, for t = 1.
1
πt = − (yt − z̄), t = 1, 2, ...
α
(
π0, for t = 1
e,mod
πt
=
1 π e mod + γα π ∗ + γ (z̄ − ȳ) for t = 2, 3
1+γα t−1
1+γα
1+γα
e,mod
+ γ(yt − ȳ) + st
πt = πt
The first equation is the RSAD curve after the policy change at the start of
period 1. The second equation has the expectations evolving according to the
“old version” of the economy. The third line is the Phillips curve.
Now there is a logical inconsistency in the model:
• The demand side of the model is consistent with πt → 0 as t → ∞.
• The supply side, because expectations are linked to the old regime where
e,mod
πt
→ π ∗ and πt → π ∗.
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This seems to be a much stronger case against (what IAM call ) backwardlooking expectations.
However, a minutes reflection reveals that things may not be so dramatic.
First, note that the short-run model is logically consistent, since in period 1,
i.e. the period of the policy change, we have
1
π1 = − (y1 − z̄),
α
e,mod
= π0
π1
π1 = π0 + γ(y1 − ȳ)
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In period 2, the first period after the shock, there is no reason to expect that
agents stick to their old forecasting rule. Instead they will use
γ
e,mod
e mod +
= βπt−1
πt
(z̄ − ȳ), for t > 1,
1 + γα
since, once agents are able to condition their expectations on the policy change,
they will replace their expectations by the “updated forecasts”.Hence we may
re-specify the model with backward-looking expectation and a policy-change
from π ∗ > 0 to π ∗ = 0 at the start of period 1 as:
1
πt = − (yt − z̄), t = 1, 2, ...
α
(
π0, for t = 1
e,mod
=
πt
e mod + γ (z̄ − ȳ) for t = 2, 3
βπt−1
1+γα
e,mod
πt = πt
e,mod
+ γ(yt − ȳ) + st
and πt
→ new target and πt → new target, just as in the analysis with no
policy change. There are no logical inconsistencies.
The feature with gradual adjustment toward the target remains, though. Can
there be a faster adjustment?
Rational expectations
The rational expectations hypothesis (REH) equates agents’ subjective expece , with the mathematical expectation
tations about a variable, for example πt+1
conditional on an information set It.
In macroeconomics it is assumed that the model coincides with the true relationships of the economy, apart from the error terms that are completely
random. With this extra assumption, REH amount to the same thing as model
consistent expectations. We also use this connotation of rational expectations
in the following.
e
denote the rational expectations for period t inflation conditional on
Let πt|t−1
information available at the end of period t − 1.
The short-run model with rational expectations (RE):
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To solve (9)-(13) under the assumption of RE we first write yt and πt in terms
of their respective expectations, and exogenous variables:
yt − ȳ = α1(gt − ḡ) − α2(rt − r̄) + vt, α1 > 0, α2 > 0
e
rt = it − πt+1|t−1
(10)
e
∗
it = r̄ + πt+1|t−1 + h(πt − π ) + b(yt − ȳ), h > 0, b > 0 (11)
mt − πt − pt−1 = m0 − m1it + m2yt, mi > 0, i = 1, 2
e
πt = πt|t−1
+ γ(yt − ȳ) + st
(12)
(13)
where we have adopted the same notion as in IAM also for the product market,
hence
ȳ = α0 + α1ḡ − α2r̄.
In addition, it is assumed that gt = ḡ, as in IAM.
yt − ȳ = −α2h(πt − π ∗) − α2b(yt − ȳ) + vt,
(9)
Inserting into (13) gives
e
πt = πt|t−1
+ γ [−α2h(πt − π ∗) − α2b(yt − ȳ) + vt] + st.
(15)
We have now two equations in the two endogenous variables yt − ȳ and πt:
(1 + α2b)(yt − ȳ) + α2h(πt − π ∗) = vt
e
γα2b(yt − ȳ) + γα2h(πt − π ∗) = πt|t−1
+ γvt + st
(16)
(17)
The solution is:
(yt − ȳ) =
πt − π ∗ =
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(14)
e
vt − α2hst − α2h(πt|t−1
− π ∗)
1 + α2(b + γh)
e
(1 + α2h)(πt|t−1
− π ∗) + (1 + α2b)st + γvt
1 + α2(b + γh)
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(18)
(19)
There are three important properties to note about this solution
As a second step we solve for the inflation expectations under the assumption
that agents do not know the realization of vt and st, they rationally set vt =
st = 0 in (19). Therefore:
e
πt|t−1
= π∗
(20)
Finally, we solve for the two endogenous variables yt and πt, using (20) and
(18) and (19).
vt − α2hst
yt = ȳ +
1 + α2(b + γh)
(1 + α2b)st + γvt
∗
πt = π +
1 + α2(b + γh)
(21)
(22)
which is (35) and (34) in IAM.
1. The RE solution consists of static models for yt and πt. There is no effects
from initial conditions. There is no propagation of shock. The only nonzero interim multipliers from supply and demand shocks are the impact
multipliers.
2. Inflation expectations are equal to the target π ∗, also when πt 6= π ∗.
3. The RE solution is affected by (systematic) monetary policy, since the
“Taylor coefficients” enter into the solutions (21) and (22).
Regarding 1.
e
Expectations in this models are static in the sense that πt|t−1
= π ∗, and
since the only source of dynamics is expectations, the solution is also static.
Therefore the solution does not contains any persistence or propagation of
shocks.
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Regarding 2
Expectations are always equal to the target, so there is no transition period
during which expectations stay below or above target for a long time.
This does not mean that expectations are always correct. For example RE gives
e
πt|t−1
= π∗
Policy ineffectiveness proposition
RE is often association with the policy ineffectiveness proposition, which states
that systematic monetary (or fiscal) policy, in the form of a Taylor-rule for
example, does not affect the solution.
e
πt+1|t−1
= π∗
To invoke this result within our framework we need to replace the Taylor-rule
with this alternative
also in the case when the target is changes to 0 at the start of period t. But
once the expectations are updated, by conditioning on period t, they are fully
adjusted:
e
e
e
it = r̄ + πt+1|t
+ h(πt|t−1
− π ∗) + b(yt|t−1
− ȳ) ⇐⇒
e
πt+1|t
=0
Regarding 3
We will not discuss optimal monetary policy, but refer to IAM pp 637-640.
The conclusion is that the qualitative conclusions of the analysis with backward
looking expectations still apply: If demand shocks dominate, there is no conflict
between price and output stabilization, but there is with supply-shocks.
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e
e
− π ∗) + b(yt|t−1
− ȳ)
rt = h(πt|t−1
which is (14) in IAM.
In this case the central bank is forecasting period t inflation and output–using
the same information set from period t − 1 as the wage-and price setters, who
e
form expectation about πt|t−1
in the Phillips-curve. In the Taylor-rule above
we assume that the central bank observes yt and πt, or that the central bank
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can nowcast these two variables very accurately. The RE solution obtained on
p 634-5 in IAM is
The Lucas critique
One aspect of the Lucas critique is that models with backward looking expectations give wrong predictions about policy effects.
πt = π ∗ + γvt
yt = ȳ + vt
where the policy coefficients b and h do not enter.
The example with a reduction in π ∗ in period t serves as an illustration:
The intuition for policy ineffectiveness is that from the Phillips curve part of
the model, we have:
The AD-AS model with backward-looking expectations then predicts a (long)
period with falling inflation and unemployment (yt+j < ȳ).
³
´
e
+ γ −1st
(yt − ȳ) = γ −1 πt − πt|t−1
showing that the only way that policy can affect output in period t is by
e
generating an inflation surprise: πt − πt|t−1
6= 0. The possibility of this is
e
e
“assumed away” when the Taylor-rule is in terms of (πt|t−1
−π ∗) and (yt|t−1
−
ȳ).
Today, most economy take the view that there is some room for affecting the
πt and yt after the private sector has committed itself to a inflation forecast
for period t. Nominal rigidities is a common term that captures this stance.
But the model with RE shows that expectations adjust fully in period t + 1, or
already in period t if the change was announced in period t − 1.
Moreover, output is unaffected meaning that there is less welfare loss under
RE.
Today, the validity Lucas critique is simply taken for granted in large parts of
the profession.
But it hinges on the REH being a sufficiently good approximation to real world
expectations.
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The limits to stabilization policy (ch 22)
Rules vs discretion and time consistency
Fine tuning the economy to avoid large and persistent deviation from full employment is a difficult task, also in the case where full employment is the only
steady-state of the economy (as the PCM implies). Ch 22 in IAM identifies
three types of assumptions that have been implicit so far:
1. No information lags. The policy maker knows the state of the economy
(the initial situation). There are no serious measurement problems.
2. There are no adjustments lags in policy decisions–and policy instruments
can be adjusted “both up and down”.
Use again the AD-AS model, but simplify even more than before by assuming
vt = st = 0, gt = ḡ, π ∗ = 0 and γ = 1
We assume RE. From (21) and (22) the equilibrium under a conventional
Taylor-rule (with π ∗ = 0) is
yt = ȳ,
(23)
πt = 0
(24)
e
πt|t−1
= 0.
(25)
and from (20)
3. The announced policy is credible.
IAM discusses all three. Here we concentrate on 3.
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Assume that the central bank would like to minimize the loss-function
SL = (yt − y ∗)2 + κπt2
and therefore SL can be written:
n
where
y ∗ = ȳ + ω, ω > 0.
ω is a parameter that reflects labour market distortions/inefficiencies. ȳ is not
“really” the full employment output.
o2
e
SL = (πt − πt|t−1
) + (ȳ − y ∗)
(26)
+ κπt2
(27)
Assume that the central bank has led the public to believe that it will ensure
e
price stability, so πt|t−1
= 0 as in the rule based RE solution. What is the
optimal πt if the central can deviate from its rule?
The “cheating” inflation rate is:
πtc = min{(πt − ω)2 + κπt2}
The value of SL under the Taylor-rule:
SLR = ω2 > 0
or
πtc − ω + κπtc = 0
Is this the minimum SL? Note that SL is given by
ω
>0
1+κ
e
= 0 into the PCM equation
Insertion of πt|t−1
πtc =
SL = {(yt − ȳ) + (ȳ − y ∗)}2 + κπt2
Since γ = 1, we have the PCM
e
πt − πt|t−1
= yt − ȳ
e
πt − πt|t−1
= (yt − ȳ)
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ω
.
1+κ
These “cheating values” of the rate of inflation and output give zero SL, which
is optimal: The effect from distortions ω on output are reduced.
Therefore loss of credibility in monetary policy can be said to lead to an inflation
bias.
gives
ytc = ȳ +
This is a result which has been very influential in practice.
However, the “cheating policy” is self-defeating since agents will learn to form
expectations from
e
(πt − πt|t−1
− ω) + κπt = 0
which is the 1oc for minimization of (27) with respect to πt. Implying that
ω
e
πt|t−1
= = πt
(28)
κ
insertion into the PCM
yt = ȳ
(29)
For example: One of the main rationale for credibility building, which often
involves delegation of monetary policy and central bank independence.
One interesting point made in IAM is that the case for central bank independence with a strict mandate for inflation stabilization is strongest when demand
shocks dominate. With dominating supply shocks, the results may be excess
variability in output.
(28) and (29) define the time-consistent RE equilibrium.
Inflation is higher than in the Taylor-rule equilibrium.
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