if society places a relatively high weight on inflation stability, while a price level target
dominates if society places a relatively high weight on output stability.
Existing papers addressing the optimal monetary policy response to potential output
uncertainty typically treat the error process of the central bank’s measurement of
potential output as exogenous to the rest of the model. We have shown that the
(sometimes implicit) assumptions that such models make regarding central bank
estimates of potential output may bias policy implications of such models. If the learning
process of the central bank is treated as exogenous and implies a higher weight on
historical data than would be efficient (and therefore forecast errors that are too
persistent), a policy target that corrects past policy errors (such as a price level target)
becomes more desirable. In contrast, if the exogenous learning process implies a lower
weight on historical data than would be efficient, a policy target that propagates past
policy errors (such as a speed limit policy) becomes more desirable.
18

Appendix 1
Suppose potential output follows a random walk
*
*
1
t
t
t
y
y
−
w
=
+
.
(A1)
Each period, the central bank tries to learn about potential output by observing the
inflation rate. For simplicity, the central bank will set policy in period
t
having only
observed
1
t
π
−
and
1
t
y
−
, and we define
/
.
t
t
e
s
κ
=
From (3),
[
]
*
1
1
1
1
1
*
1
1
1
1
(
)
(
)
1
(
)
,
t
t
t
t
t
t
t
t
t
t
t
t
t
E
y
y
e
u
E
y
y
e
π
β
π
κ
π
β
π
κ
κ
−
−
−
−
−
−
−
−
−
−
=
+
−
+
+
−
⇒
−
−
=
−
−
1
t
w
t
(A2)
where the left hand side represents observables and the right hand side unobservables.
This agents and the central bank observe the value of
*
1
t
t
y
e
w
−
−
−
, which provides a
point estimate of
with a variance of
*
t
y
2
e
2
w
σ
σ
+
. More generally, given
t
i
π
−
, the central
bank observes the value of
1
0
t
j
*
i
t
t
i
j
y
e
−
w
−
−
=
+
−
∑
, which provides a point estimate of
*
t
y
with
a variance of
V
i
2
2
i
e
w
σ
σ
=
+
.
Suppose the central bank constructs an unbiased estimate of
*
t
y
efficiently from
these point estimates, with a weight on the point estimate constructed from
t
i
π
−
of
i
f
.
Then their error in estimating potential output is given by
(A3)
*
*
1
1
1
0
ˆ
(
)
t
t
t
i
i
t
t
j
i
j
y
y
f
e
w
ε
∞
−
−
−
=
=
=
−
=
−
−
∑
∑
,
where unbiasedness implies that
1
1
i
i
f
∞
=
=
∑
.
19