SEMINAR 7Econ 4325
Problem 1
Assume that output is given by
1:1 : yt = ( t Et 1 t) + ut
where Et 1ut = 0:
Consider two alternative speci...cations of the preferences of the monetary authorities:
1:2 : Lt =
1
2
h
( t )
2
+ (yt y )
2
i
1:3 : Lt =
1
2
( t )
2
yt
1. Given an economic interpretation of the di¤erence between the two speci...cations.
2. Derive the solution for ination and output under a discretionary policy the two loss
functions. Compare and discuss the result.
1. The two loss functions are similar with regard to ination. They di¤er in how they regard output.
The formulation in (1.2) implies that there is an optimal level of output. The quadratic formulation ensures
that the marginal disutility increases in the distance from the target. Hence, output may be too high. The
formulation in (1.3) implies that the government wants output to be as high as possible. The marginal utility
(reduction in marginal disutility) from output is constant and equal to .
2. First we ...nd the preferred monetary policy from the loss function. The government minimize the loss
with respect to ination.
(1.2)
min
1
2
h
( t )
2
+ (yt y )
2
i
s:t: : yt = ( t Et 1 t) + ut
FOC : ( t ) + (yt y ) = 0
Solve this for t
t = + y yt
Insert for yt
t = + y [ ( t Et 1 t) + ut]
Collect terms
t 1 + 2
= + y + 2
Et 1 t ut
We can ...nd expected ination from the FOC
t = + y yt
Et 1 t = Et 1 + Et 1 y Et 1 yt
= + y Et 1 [ ( t Et 1 t) + ut]
= + y
1
Insert above and solve for t
t 1 + 2
= + y + 2
[ + y ] ut
= 1 + 2
+ y + 2 3
ut
t = +
+ 2 3
1 + 2
y
1 + 2
ut
= + y
1 + 2
ut
To ...nd yt we do the following: Start with the expression for yt
yt = ( t Et 1 t) + ut
Insert for t
yt = + y
1 + 2
ut Et 1 t + ut
Insert for expected ination Et 1 t = + y
yt = 1 +
2
1 + 2
ut
Collect terms
yt =
1
1 + 2
ut
(1.3)
min
1
2
( t )
2
yt
s:t: : yt = ( t Et 1 t) + ut
FOC : ( t ) = 0
Solve for t
t = +
Insert into the expression for yt
yt = ( + Et 1 t) + ut
Note that from the FOC, the expected ination is given by +
yt = ut
Comparison and comment
There is an ination bias under both loss functions. Expected ination is the same for y = 1. For (1.2)
ination is volatile. For (1.3) ination is not. Mirroring this we see that output is more volatile under (1.3)
than under (1.2). The reason is that under (1.2) the government wants to balance the e¤ects of the shock
on both ination and output.
2. The expected loss under discretion is given by:
ELt =
1
2
E
h
( t )
2
+ (yt y )
2
i
=
1
2
E
"
+ y
1 + 2
ut
2
+
1
1 + 2
ut y
2
#
=
1
2
E
"
y
1 + 2
ut
2
+
1
1 + 2
ut y
2
#
=
1
2
E
2
6
4
( y )
2
2
2 2
1+ 2 y ut + 1+ 2 ut
2
+
1
1+ 2 ut
2
+ 2 1
1+ 2 uty + (y )
2
3
7
5
2
From cov (u; y ) = 0
ELt =
1
2
2
6
4
( y )
2
+ E 1+ 2 ut
2
+
E 1
1+ 2 ut
2
+ (y )
2
3
7
5
=
1
2
"
( y )
2
+
1 + 2
2
2
+
1
1 + 2
2
2
+ (y )
2
#
The expected loss under strict ination targeting is given by:
ELt =
1
2
E
h
( t )
2
+ (yt y )
2
i
=
1
2
E
h
(ut y )
2
i
=
1
2
h
2
+ (y )
2
i
We can try to ...nd the value of y that makes the discretionary the preferred policy
1
2
"
( y )
2
+
1 + 2
2
2
+
1
1 + 2
2
2
+ (y )
2
#
<
1
2
h
2
+ (y )
2
i
( y )
2
+
1 + 2
2
2
+
1
1 + 2
2
2
< 2
( y )
2
< 2
"
1
1 + 2
2
1
1 + 2
2
#
1
1 + 2
2
1
1 + 2
2
=
1 + 2 2
(1 + 2)
2
2
(1 + 2)
2
1
(1 + 2)
2
=
1 + 2 2
+ 2 4 2
1
(1 + 2)
2
=
2
+ 2 4
(1 + 2)
2
=
2
+ 2 4
(1 + 2) (1 + 2)
=
2
(1 + 2)
( y )
2
< 2
2
(1 + 2)
(y )
2
<
2
(1 + 2)
y <
s
2
(1 + 2)
= p
(1 + 2)
If the output target is higher than this, strict ination targeting is better.
3
Interpretation:
If the output target is very ambitious (high), the ination bias is large. Then strict ination targeting is
better because the cost in terms of too little stabilization of shocks is more than o¤set by the gain in terms
of removing the ination bias. If the output target is less ambitious, the cost of a discretionary policy is
smaller.
Simen Markussen
18.04.2007
4