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I have question on central bank loss function.

We know that the central bank loss function is

$$L(\pi, \bar{Y})= (\pi- \pi^e)^2+\beta \bar {Y}^2$$

And we know that fisher equation is $$i=r+\pi^e$$

where $r$ and $i$ Are respectively the real and nominal interest rate.

I want to minimize CB’s loss function by choosing $i$

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I tried to solve it. But it’s meaningless and so wrong:(

When I searched on the Net, I always saw that the CB loss function is minimized by substituting Phillips curve and takes derivative with respect to $Y$.

So, I substituted the fisher equation into the Phillips curve i.e. $\pi=a \bar{Y} +\pi^e= a \bar{Y}+i-r$

And then I substituted this equation into the loss function I.e. $L(\pi, \bar{Y})= (a\bar{Y}+i-r- \pi^e)^2+\beta \bar {Y}^2$

And take its derivative with respect to $i$

I got $2(a\bar{Y}+i-r-\pi^e)=0$

$i^*=\pi^e-a\bar{Y}+r$

But I think my way is wrong. Because this result doesn’t make sense.

Please let me show a way how to solve it. Any help would be appreciated.

Thanks a lot.

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You need to differentiate with respect to Y, as you said! You substitute the phillips curve as you did (because it is a constraint for the central bank's optimization problem), then differentiate w.r.t $\bar{Y}$ (which I assume here stands for gap) and set this to 0 to obtain the optimality condition, which in turn will give you the monetary rule. Recall that once the optimal output-inflation combination is determined using the monetary rule, the central bank sets the interest rate to implement its choice.

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