Stabilization policy. IAM ch 20.
Ragnar Nymoen
Department of Economics, UiO
29 September 2009
ECON 3410/4410: Lecture 7
The loss function I
Since the model economy is dynamically stable, and full
employment GDP is independent of both monetary and …scal
policy, the remaining rationale for doing macroeconomic policy
is to reduce the welfare losses which are due to the temporary
disturbances in demand and supply, and their propagation.
Welfare losses are assumed to be linked to variations in
demand and in in‡ation. Formally
SL =
2
y
+
2
,
> 0,
is a parameter that measures the degree to which the
society values stable in‡ation relative to output stability.
2
y
and 2 are the mathematical variances of output and
in‡ation that are implied by our model of the economy.
ECON 3410/4410: Lecture 7
(1)
The loss function II
Since 2y and 2 are derived from the model, we need to consider
the solution of the AD-AS model which in condensed form is
(yt
y) =
(
=
t 1
t
t
) + zt , and
(2)
y ) + st ;
(3)
+ (yt
In the lectures we have already derived the …nal equation for
(yt y ) which is
(yt
y ) = (yt
where
=
y ) + (zt
1
2h
(1 +
zt =
2 b)
1
,
=
zt
1
1+
1)
, and
(gt g ) + vt
(1 + 2 b)
ECON 3410/4410: Lecture 7
st ,
(4)
Deriving the variance of output I
As stated in IAM p 612, 2y and 2 are interpreted as the
unconditional variances of GDP and in‡ation, also called the
marginal variances, or the asymptotical variances.
These variances should not depend on time, so for GDP:
Var [(yt
y )] = Var [(yt
y )] =
1
2
y.
We then take the variance on each side of “=” in (4). This
gives:
2
y
(1
2
)
2
y
=
2 2
y
+
2
=
+
Var [zt
2
Var [zt
zt
1]
zt
+(
1]
+(
)2 Var [st ]
)2 Var [st ]:
ECON 3410/4410: Lecture 7
(5)
Deriving the variance of output II
We also have that (see separate note):
(1
2
)= 2
+(
)2
2
so that (5) can be written as
2
y
=
Var [zt
zt
2
+ 2 Var [st ]
:
+ ( )2
1]
(6)
Finally, introduce the notation that Var [zt ] = 2z and
Var [st ] = 2s , and obtain the same expression as in equation
(11) on page 612 in IAM:
2
y
=
2
2
2
z
+
+(
2 2
s
)2
ECON 3410/4410: Lecture 7
(7)
Deriving the variance of in‡ation
To derive 2 we use the …nal equation for
insertion of (2) in (3):
t
=
1
(1 +
)
t 1
+
(1 +
)
zt +
t
1
(1 +
which we obtain by
)
st +
(1 +
)
Taking the variance on both sides, and collecting terms gives:
2
+ ( )2
(1 + )2
2
2
=
(1 +
and then
2
=
)
2
z
+
1
(1 +
2 2
z
2
+ 2s
+ ( )2
which is (12) in IAM p 612.
ECON 3410/4410: Lecture 7
2
)
2
s
(8)
Summary of results about variances
Loss-function:
2
y
SL =
2
+
,
> 0,
Variances:
2
y
2
y
=
2
=
2
2
2
z
2
2
+ 2 2s
+ ( )2
2 + 2
z
s
+ ( )2
2
Dependency of
and
on the choice parameters b (weight on
(yt y )) and h (weight on t ).
2h
=
(1 + 2 b)
g ) + vt
1 (gt
Var [zt ] = Var [
]
(1 + 2 b)
=
2
v
(1 +
2
2 b)
ECON 3410/4410: Lecture 7
Response to demand shocks
2
s
Assume that there are no supply shocks, so that
2 >0
v
2
y
=
2
=
2 2z
+(
)2
2
2 h [2(1 +
=
2 (1+
2
= 0, but
2
v
2
2 b)
)2
+(
2
2 b)
+
2h
]
GDP variability is reduced by choosing large values of h and b
in in Taylor rule, cf
@ 2y
@ 2y
< 0 and
< 0 when
@h
@b
Since
2
s
=0
2
2
2
, when 2s = 0
2 y
the central bank faces no trade-o¤ between in‡ation and
output stabilization in this case.
=
ECON 3410/4410: Lecture 7
Demand shock: e¤ect of choice of b
Note that the
size of the
vertical shift is
independent of
b:
π
LRAS
SRAS
π1
π1'
π
AD
*
AD
AD
AD
y1
'
y1
0
1
0
1
@ t
@vt
(b = 0 )
=
AD ;yt =y
(b = 0 )
(b > 0 )
(b > 0 )
y
ECON 3410/4410: Lecture 7
So important to
use vertical shift
to analyze
“e¤ect of
choosing high b”
1
2h
Demand shock: e¤ect of choice of h
Note that the
size of the
horizontal shift
is independent
of h :
π
LRAS
SRAS
π
@yt
@vt
*
AD
AD
AD
AD
0
1
0
1
( h high)
=
( h high)
AD ;
tt =
1
>0
1 + 2b
( h low)
( h low)
y
ECON 3410/4410: Lecture 7
So important to
use horizontal
shift to analyse
“e¤ect of
choosing high h”
Response to demand shocks; multipliers
From lecture 6 we have that:
@yt
1
=
@vt
1+
@zt
1
=
@vt
1+
1
1 + 2b
If the government has a strong preference for output
stabilization, then it chooses b to be su¢ ciently high so that
the demand increase (in zt ) triggered by the shock becomes
very small.
Or, it can neutralize the response of yt to any given increase
in zt by choosing a high h, because is increasing in h.
Since
@ t
@yt
=
@vt
@vt
the same applies for in‡ation— so no trade-o¤ also in this
analysis.
ECON 3410/4410: Lecture 7
Response to supply shocks
If
2
v
= 0:
2
y
=
2
=
2 2
s
2
+(
)2
=
2
s
2
1
+
2
=
2
2
s
(1+ 2 b)
2h
+
2
2
s
2
+(
)2
Unlike the case with “demand shock only”, the choice of high
h and b does not reduce both variances:
b "
=)
b "
=)
h "
=)
h "
=)
# =)
2
y
2
#
" =)
2
y
2
"
# =)
" =)
"
#
So in this case there is a trade-o¤ between in‡ation and
output stabilization.
ECON 3410/4410: Lecture 7
Supply shock: stabilization trade o¤
π
t
SRAS0
SR AS1
π
A: weight on
in‡ation
stabilization
*
A
B
AD (h high, b low)
AD (h low, b high)
yt
ECON 3410/4410: Lecture 7
B: weight on
GDP
stabilization.
Optimal monetary policy
In general, the optimal choice of h and b in the Taylor rule
depends on
preferences, as captured by the parameter in the social loss
function,
and the nature of the dominant shocks that drive the business
cycle
Policy preference
Dominant shock
Demand shock
Supply shock
Stable y
h > 0 and b > 0
h ! 0 and b > 0
Stable
h > 0 and b > 0
h > 0 and b
Pro-cyclical policy b < 0 can be optimal in one case (create a
‡at AD schedule)
ECON 3410/4410: Lecture 7
0
Fiscal policy I
Discretionary policy does not require a separate analysis.
Note that, with, -targeting, a …scal expansion will always be
met by a contraction of monetary policy.
Rule based …scal policy rule can be analyzed by inclusion of a
new equation of the form
gt
g =c (
t
) + cy (yt
y)
in the AD-AS model.
This gives a “…scal policy adjusted” AS curve.
Choose high c for the same reason as h high, and cy positive
for the same reasons as b > 0. This is because both …scal and
monetary policy both operate via aggregate demand.
The results in IAM ch 20.3 shows that …scal policy cannot
stabilize the economy more than monetary policy already does.
ECON 3410/4410: Lecture 7
Fiscal policy II
Note however that the choice of b and h in the Taylor rule is
constrained by to concerns:
1
The nominal interest rate cannot be negative
2
The …nancial sector must be functional in the sense that
market interest rates become strongly conditioned by the
policy rate. This …rst link in the transmission chain of
monetary policy, is weakest in a depression, and when there is
little trust among …nancial institutions.
1. and 2. represent “traps” to monetary policy. At least in
the narrow sense of interest rate setting, because quantitative
easing, can help in both cases.
But both the Great Depression and the current crisis shows
relevance of …scal policy as a separate stabilizer.
ECON 3410/4410: Lecture 7