# Central bank loss function (I did a solution, but it doesn’t totally make sense I guess)

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.

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.