<p>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 <span class="math-container">$\bar{Y}$</span> (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.</p>

CC BY-SA 4.0
2019-10-19T20:49:05.597Z
Hi again, you explained very well! If you mind, could you explain this similar question? Thank you https://economics.stackexchange.com/questions/32331/question-on-central-bank-loss-function-interpretation
studentp
CC BY-SA 4.0
2019-10-18T19:14:46.017Z
2019-10-18T19:14:46.017Z
Ali
<p>I have question on central bank loss function. </p>

<p>We know that the central bank loss function is </p>

<p><span class="math-container">$$L(\pi, \bar{Y})= (\pi- \pi^e)^2+\beta \bar {Y}^2$$</span></p>

<p>And we know that fisher equation is <span class="math-container">$$i=r+\pi^e$$</span></p>

<p>where <span class="math-container">$r$</span> and <span class="math-container">$i$</span> Are respectively the real and nominal interest rate. </p>

<p><strong>I want to minimize CB’s loss function by choosing <span class="math-container">$i$</span></strong></p>

<p>——-</p>

<p>I tried to solve it. But it’s meaningless and so wrong:( </p>

<p>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 <span class="math-container">$Y$</span>. </p>

<p>So, I substituted the fisher equation into the Phillips curve i.e. <span class="math-container">$\pi=a \bar{Y} +\pi^e= a \bar{Y}+i-r$</span></p>

<p>And then I substituted this equation into the loss function I.e. <span class="math-container">$L(\pi, \bar{Y})= (a\bar{Y}+i-r- \pi^e)^2+\beta \bar {Y}^2$</span> </p>

<p>And take its derivative with respect to <span class="math-container">$i$</span></p>

<p>I got <span class="math-container">$2(a\bar{Y}+i-r-\pi^e)=0$</span></p>

<p><span class="math-container">$i^*=\pi^e-a\bar{Y}+r$</span></p>

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

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

<p>Thanks a lot. </p>

<macroeconomics><mathematical-economics><self-study><optimization><central-banking>
Central bank loss function (I did a solution, but it doesn’t totally make sense I guess)
studentp
# Central bank loss function (I did a solution, but it doesn’t totally make sense I guess)

(Asked by: username_0 on 2019-10-18)

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 $i$**

——-

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.

**Tags:** macroeconomics, mathematical-economics, self-study, optimization, central-banking

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## 1 answers

### Answer by username_1 on 2019-10-18. Score: 1 | Accepted answer

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.

> **Comments:**
> **username_0**: Hi again, you explained very well! If you mind, could you explain this similar question? Thank you https://economics.stackexchange.com/questions/32331/question-on-central-bank-loss-function-interpretation  

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