Stabilization policy with rational expectations.
IAM ch 21.
Ragnar Nymoen
Department of Economics, UiO
Revised 20 October 2009
ECON 3410/4410: Lecture 7
Backward-looking expectations (IAM 21.1) I
From the notes to IAM Ch 20, we have the …nal equation for
in‡ation
=
t
1
(1 +
)
t 1
+
(1 +
)
zt +
1
(1 +
)
st +
(1 +
)
(1)
or:
(
t
)=
1
(1 +
)
(
where
=
)+
t 1
2h
(1 +
zt =
2 b)
1
,
=
(1 +
1
1+
)
zt +
, and
(gt g ) + vt
(1 + 2 b)
ECON 3410/4410: Lecture 7
1
(1 +
st
)
(2)
Backward-looking expectations (IAM 21.1) II
We now investigate in‡ation expectations errors, assuming
that t = 0 is the initial situation, and that expectation for
period t = 1; 2; :: are made at the start of period 1, using
information available at the end of period 0. Expectations are
made for the (in…nite) horizon t = 1; 2; 3; :::::.
Although we have so far referred to backward-looking
expectations formation as static, we are in fact "using the
model" to generate the agents’expectations for period
t = 2; 3; :::.
In more detail, the expectations formation is
e
t
e
t
=
=
0 , for t
emod
=
t
= 1 but
t 1;
for t = 2; 3 > 1
(3)
(4)
where emod
denotes the model based forecast and t 1 is the
t
period t 1 solution obtained by the use of the …nal equation.
ECON 3410/4410: Lecture 7
Backward-looking expectations (IAM 21.1) III
Note that this do not entail that the agents has to know the
model (i.e. to sit in this auditorium).
Hence expectations are in fact not backward-looking in the
strict sense of et = 0 for all t.
To analyze the expectations errors, we assume that gt = g
and vt = st = 0 both in the economy and in the expectations
formation process. This is the same simpli…cation as on p 630
in IAM.
The period 1 error (remember that 0 is the initial period):
e1
=
=
=
1
e
1
=(
1
(1 +
)
(1 +
)
(
(
0
)
1
)
(
(
e
1
0
0)
ECON 3410/4410: Lecture 7
)
)
(5)
Backward-looking expectations (IAM 21.1) IV
Period 2 error:
e2
=
e
2
2
=
1
1+
= (
2
=
(
=(
)
2
(
2
(
1
(1 +
)
0
)(
0
)
1)(
0
)
e
2
)
)
(
0
)
And generally:
et =
t 1
(
1)(
0
) for t = 2; 3; ::
(6)
which is equation (10) in IAM p 630.
Assume that
0
=
.
Then (5) and (6) shows that the expectations generated by
(3) and (4) are always accurate.
ECON 3410/4410: Lecture 7
Backward-looking expectations (IAM 21.1) V
If 0 6= , the forecasts are biased but become gradually
closer to the in‡ation target. The in‡ation forecast during the
phase of adjustment may be as depicted in Figure 21.1 in
IAM, for the case of 0 = 3% and
= 0.
The critique against backward-looking expectations is that the
forecast bias (systematic error) only goes away gradually, cf
also the Lucas critique at the end of this set.
ECON 3410/4410: Lecture 7
AD-AS model with rational expectations (IAM 21.2) I
The rational expectations hypothesis (REH) equates agents’
subjective expectations about a variable, for example et+1 ,
with the mathematical expectation conditional on an
information set It .
Rational expectations are without systematic errors, which, in
macroeconomics entails that the true model of the economy is
part of the information set.
With this extra assumption, RE amounts to model consistent
expectations. The agents must know the model before they
can form rational expectations.
e
Let etjt 1 and ytjt
1 denote the rational expectations for
period t conditional on information available at the end of
period t 1.
ECON 3410/4410: Lecture 7
AD-AS model with rational expectations (IAM 21.2) II
The short-run AD-AS model with RE is given by
yt
y
=
1
(gt
g)
2
rt
= it
e
t+1jt 1 ;
it
= r+
e
t+1jt 1
e
+b ytjt
t
=
e
tjt 1
(rt
+ (yt
(7)
(8)
+h
1
r ) + vt ;
e
t1jt 1
(9)
y ;
y ) + st :
(10)
vt and st are assumed to have zero expectations. They have
variances 2v and 2s ; and they are uncorrelated. For simplicity
we set
e
gtjt
1 =g
ECON 3410/4410: Lecture 7
Solving the RE model I
Step 1
Express yt and
yt
t
= y+
=
t
in terms of expectations and exogenous variables
1
(gt
h
+ 2 h
e
tjt 1
+
+
h
2 h
g)
e
tjt 1
1
(gt
e
tjt 1
e
+ b(ytjt
i
y ) + vt
1
g)
e
+ b(ytjt
1
(11)
(12)
i
y ) + vt + st :
ECON 3410/4410: Lecture 7
Solving the RE model II
Step 2
Obtain the functional relationships for the mathematical
e
expectations etjt 1 and ytjt
1 from the expressions in Step 1:
e
ytjt
1
= y+
and
e
tjt 1
1
|
e
gtjt
=
+
0 = h
g +
}
1
{z
=0
e
tjt 1
+
n h
2 h
e
tjt 1
2
h
e
tjt 1
h
e
+ b(ytjt
1
y)
i
(13)
1
e
gtjt
e
tjt 1
1
g +
e
+ b(ytjt
e
+ b(ytjt
1
1
y)
ECON 3410/4410: Lecture 7
y)
io
(14)
Solving the RE model III
Together with (13) this is seen to imply:
e
ytjt
1
e
tjt 1
= y;
+ (14)
=
We now have found the rational expectations for GDP and
in‡ation. Since these are variables in the AD and AS equations,
the last step of the solution is to insert these REs into the
expressions in Step 1.
ECON 3410/4410: Lecture 7
Solving the RE model IV
Step 3
Insertion back into (11) and (12) gives the rational expectations
solution:
yt
t
= y+
=
+
1
(gt
1
g ) + vt
(gt
g ) + v t + st
Which is a static model. There are no persistence or business
cycles in this solution
ECON 3410/4410: Lecture 7
(15)
(16)
Policy ine¤ectiveness I
(gt g ) is a …scal policy “surprise”, which does not appear in
IAM since they set gt = g from the outset. We have used the
e
weaker assumption that gtjt
1 = g . A simple model for gt
which is consistent with this is
gt = g + "gt
where "gt has conditional expectation zero. Using,
gt g = "gt ; the RE solution can be expressed as:
yt
t
= y+
=
1 "gt
+ vt ;
+ vt + st :
(17)
(18)
Since none of the policy variables or parameters ( , h or b)
enter into (17), this is called the policy ine¤ectiveness
proposition.
ECON 3410/4410: Lecture 7
Policy ine¤ectiveness II
Systematic monetary policy does not a¤ect GDP.
This is because the rule based monetary policy cannot
generate in‡ation surprises which drives yt via the AS function
(yt
y) =
1
(
t
e
tjt 1 )
+
1
st
which in this form is known as Lucas’supply function.
To avoid the conclusion about policy ine¤ectiveness the model
needs to be modi…ed: Assume that the central bank acts on
the basis of actual in‡ation and output. Instead of the Taylor
rule (9), we use eq (28) in IAM
rt = r + h (
t
) + b (yt
y)
to represent the situation that central bank can react to t
and yt after in‡ation expectations have been formed in the
private sector.
ECON 3410/4410: Lecture 7
Policy e¤ectiveness under RE
With the modi…ed Taylor rule, IAM p 636 and 637, shows that the
RE solution is (use Step 1 - Step 3), and set gt = g as in the
book):
t
yt
(1 + 2 b)st + vt
,
1 + 2 (b + h)
vt
2 hst
= y+
1 + 2 (b + h)
=
+
(19)
(20)
GPD is now in‡uenced by systematic monetary policy.
For example: Increased weight (b) on GDP in Taylor rule
reduces the impact of supply and demand shocks.
The reason is that the central bank can a¤ect the real interest
rate by changing the nominal rate after in‡ation expectations
have been set.
ECON 3410/4410: Lecture 7
Optimal stabilization policy under RE I
The analysis is in terms of the same social loss function as in
ch 20. What matters is the variability of in‡ation and GDP.
Taking variances on both sides of (19) and (20):
2
y
=
2
=
2
v
+(
[1 +
2 2
v
2 h)
2 2
s
;
+ h)]2
+ (1 + 2 b)2
(21)
2 (b
[1 +
2 (b
2
s
+ h)]2
(22)
We also follow the method of …rst analyzing the case of pure
demand shocks, and then the case of isolated supply shocks.
ECON 3410/4410: Lecture 7
Optimal stabilization policy under RE II
Demand shocks only (
2
s
2
y
=
2
=
= 0):
2
v
1+
[1 +
2 (b +
2 2
v
2 (b
h)
+ h)]2
Both variances are reduced by choosing high positive values
for b and h.
Optimal to choose highest possible values of both b and h,
subject to it 0.
No con‡ict between GDP and in‡ation stabilization.
This is the same qualitative conclusion as with static
expectations.
ECON 3410/4410: Lecture 7
Optimal stabilization policy under RE III
Supply shocks only (
2
v
= 0):
From (21) and (22) we have:
2
y
=
2
=
(
2 2
2 h) s
[1 + 2 (b + h)]2
(1 + 2 b)2 2s
[1 +
2 (b
+ h)]2
It is not possible to see, by direct inspection, how choices of h
and b in‡uence these variances. Therfore we need the
derivatives with respects to b and h.
ECON 3410/4410: Lecture 7
Optimal stabilization policy under RE IV
To calculate, it is practical to …rst take logs and both sides
ln
ln
2
y
2
2 2
2 ln [1 + 2 (b + h)]
2 h) s ]
2 2
ln[(1 + 2 b) s ] 2 ln [1 + 2 (b +
= ln[(
=
and then …nd the derivatives.
@ 2y
@b
@ 2y
@h
@ 2
@b
@ 2
@h
=
=
=
=
2 2
[1 + 2 (b + h)]
2
y
< 0 if b > 0
2(1 + 2 b)
2
h [1 + 2 (b + h)] y
2 22 h
2
(1 + 2 b) [1 + 2 (b + h)]
2 2
2
< 0 if b > 0
[1 + 2 (b + h)]
ECON 3410/4410: Lecture 7
h)]
Optimal stabilization policy under RE V
@
2
2
We see that the “direct derivatives” @by and @@h are negative
subject to b > 0 (countercyclical interest rate setting).
Remember that h > 0 from the Taylor principle
@ 2y
@h
2
But
and @@b are both positive (given that b > 0),
meaning that when supply shocks are dominant, there is a
con‡ict of priorities between GDP stabilization and in‡ation
stabilization.
So in general, there is a trade-o¤, re‡ected by the choice of
parameter in the social loss function.
Again, these conclusions are qualitatively the same as in the
case with static in‡ation expectations.
ECON 3410/4410: Lecture 7
A modi…ed Taylor-rule (gradualism) I
In the RE solution, all endogenous variables adjust
instantaneously to a shock.
This includes the nominal interest rate.
IAM contains an important paragraph (“The optimality of a
modi…ed Taylor rule under forward-looking expectations”)
explaining why a Taylor rule that comes closer to empirical
Taylor-rules may actually be optimal.
In empirical studies we often estimate:
it = r + h(
t
) + h(yt
y ) + cit
1;
c >0
(23)
Which corroborates that many central banks normally prefer
to adjust the interest rate in “small but frequent steps”. ie,
gradualism in interest rate setting.
ECON 3410/4410: Lecture 7
A modi…ed Taylor-rule (gradualism) II
The theoretical point is that, with c rather large, an
adjustment of it will signal a higher interest rate in the future.
The result is that a policy interest rate change is likely to
a¤ect mortgage rates and other long term rates more
e¤ectively than would be the case if c = 0.
The policy instrument gets a stronger e¤ect in the monetary
transmission mechanism.
ECON 3410/4410: Lecture 7
The Lucas critique I
The Lucas critique states that models with backward-looking
expectations give wrong predictions about policy e¤ects.
The example with a reduction in
illustration.
in period t serves as an
Remember that the AD and AS equations with static
expectations are:
(yt
y) =
(
=
t 1
t
t
+ (yt
)
zt ,
y ) + st
0
showing that if
is reduced
in period t, GDP is reduced.
This due to increased real interest rate.
Already, this is di¤erent from the RE solution, compare (17)
and (20), which show no reduction in GDP when
is
reduced.
ECON 3410/4410: Lecture 7
The Lucas critique II
The dynamic e¤ects on GDP in the static model are intuitively
clear: There must be a gradual increase in GDP, back to y .
This is achieved by a gradual reduction in in‡ation, and a
reduction of the real interest rate.
It may be confusing that the …nal equation that we (and IAM)
have derived for (yt y ) does not contain
as a parameter,
so the algebraic solution of the model does not seem to match
the economic interpretation.
The explanation is that in the derivation, we have assumed
that
is a constant parameter, and it therefore drops out
from the solution.
ECON 3410/4410: Lecture 7
The Lucas critique III
To allow for a time varying in‡ation target, we write the
model as
(yt
y) =
(
=
t 1
t
t)
t
zt
+ (yt
y ) + st
and re-do the derivation of the …nal equation, which becomes
(yt
y ) = (yt
1
y ) + (zt
zt
1)
st +
for GDP, and
t
=
t 1
+
t
+
z t + st
for in‡ation.
=
1
1+
, and (1
)=
1+
as before.
ECON 3410/4410: Lecture 7
(
t
t 1)
The Lucas critique IV
We now see that a permanent change in the in‡ation target
has the same e¤ect in the backward-looking model as a
permanent negative demand shock.
The sequence of dynamic multipliers are therefore negative.
but diminishing in magnitude. In‡ation is gradually reduced
down to the new target, the long run-multiplier for in‡ation is
long run
=
1
=1
as we know it should be.
The critique is that this gives a misleading analysis of the
e¤ects of a lower in‡ation target, if expectations are in fact
rational.
As we have seen, in the RE solution, expectations adjust fully
in period t if the change was announced in period t 1:
ECON 3410/4410: Lecture 7
The Lucas critique V
GDP is una¤ected, meaning that there is less welfare loss
under RE.
Today, the validity of Lucas critique is taken for granted in
large parts of the economist profession.
But the relevance of the critique hinges on the REH being a
su¢ ciently good approximation to real world expectations.
ECON 3410/4410: Lecture 7