If society places a relatively high weight on

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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