<p>The loss function reflects the central bank’s policy objective (i.e. their preferences for stabilizing inflation and some real variable around target levels, in this case output), and it says that the central bank is concerned about inflation fluctuating from their target level (<span class="math-container">$\pi^{T}$</span>) and their perception of the equilibrium output (here reflected in the output gap). </p>

<p>Now, because the objective of the central bank includes two factors (output and inflation), we can know from the loss function how concerned the central bank is about stabilizing the output gap from the size of coefficient <span class="math-container">$\alpha$</span>. Indeed, at varying levels of <span class="math-container">$\alpha$</span> it is easy to see that the reaction of the central bank to the same output gap will differ. Since <span class="math-container">$\alpha$</span> > 0 we know that they are concerned about the welfare loss associated with the output gap (if = 0 they would simply not be concerned about this, and if &lt; 0 this would reflect that bigger output gaps are best since the function is to be minimized). </p>

<p>I can imagine that when looking at the geometry of the loss function in the Phillips curve diagram this may become clearer to you as someone from maths. </p>

<p>And just to note that once you derive the policy rule - the slope of the MR curve - (as in your <a href="https://economics.stackexchange.com/questions/32308/central-bank-loss-function-i-did-a-solution-but-it-doesn-t-totally-make-sense/32315?noredirect=1#comment48815_32315">other</a> question), you will see that this policy rule not only depends on the central bank’s preferences <span class="math-container">$\alpha$</span> but also on the behavior of the private sector, the price flexibility as given by the parameter <span class="math-container">$\theta$</span>.   </p>

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2019-10-20T10:34:21.527Z
In addition to what you said, I also make some interpretation by using the Phillips curve diagram. Are the following statements correct?
studentp
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2019-10-20T10:35:24.043Z
“ $\alpha =1$ means that each indifference curve is a circle with $(x_e, \pi^T)$ at its centre. The loss declines as the circle gets smaller.
studentp
CC BY-SA 4.0
2019-10-20T10:35:40.780Z
When $\pi =\pi^T$ and $x_e=x$, the circle shrinks to a single point (bliss point) and the loss is at minimum at zero. With $\alpha =1$, the central bank is indifferent between inflation 1% above (or below) $\pi^T$ and output 1% below (or above) $x_e$. They are on the same loss circle.
studentp
CC BY-SA 4.0
2019-10-20T10:35:57.473Z
Only when $\alpha =1$ we have indifference circles. But when $\alpha <1$ then the central bank the central bank is indifferent between (say) inflation1% above (or below) $\pi^T$  and output 2% above (or below) $x_e$. This makes the indifference curves ellipsoid. And finally $\alpha >1$ means that central bank with less aversion to inflation will have ellipsoid indifference curves with a vertical rather than a horizontal orientation.
studentp
CC BY-SA 4.0
2019-10-20T10:36:00.963Z
In that case, the indifference curves are steep indicating that the central bank is only willing to trade off a given fall in inflation for a smaller fall in output than in the other two cases.”
studentp
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2019-10-20T10:36:40.513Z
Is this comment also correct for my example? Thank you.
studentp
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2019-10-20T12:19:29.607Z
Yes, this holds perfectly for your example! I read it wrong and was about to comment something but then realised that you saw it perfectly since when <1 this means that the central bank is inflation averse and when $\alpha > 1$, you can see that the central bank is unemployment-averse. Moreover, as you noted, utility declines with distance from the bliss-point.
Ali
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2019-10-20T19:49:59.483Z
I am sorry, this is my last question to you. Do you have an idea about that? https://economics.stackexchange.com/questions/32349/revenue-maximization-problem
studentp
CC BY-SA 4.0
2019-10-20T08:21:02.627Z
2019-10-20T08:21:02.627Z
Ali
<p>In this case, you can interpret this as the central banks' having two mandates: output gap and inflation target.</p>

<p>Here, the best possible scenario that would minimize the CB's loss function is having zero output gap (<span class="math-container">$\hat x^2 = 0$</span>) and inflation is at the target (<span class="math-container">$\pi = \pi^T$</span>). The CB simply wants to get as close to this as it could, with the "importance" of the output gap component given by <span class="math-container">$\alpha$</span>. (What would <span class="math-container">$\alpha &lt; 1$</span> mean, for example?)</p>

<p>I'm not sure how to help with the geometry part, but note that you can plug <span class="math-container">$\pi(\hat x)$</span> from the Phillips curve into the loss function.</p>

CC BY-SA 4.0
2019-10-20T07:53:58.410Z
No no I think this is not answer that I need. Because what you say is related to another question that I linked. For example what is mean $\alpha$ without any calculation ?
studentp
CC BY-SA 4.0
2019-10-20T04:19:10.943Z
2019-10-20T04:19:10.943Z
Art
<p>How do you interpret this central bank’s loss function:</p>

<p><span class="math-container">$$L(\hat x, \pi)=\alpha \hat{x}^2+(\pi-\pi^T)^2$$</span></p>

<p>where <span class="math-container">$\alpha &gt;0$</span> and <span class="math-container">$\hat x$</span> is output gap. </p>

<p>Phillips curve is <span class="math-container">$\pi = \theta \hat{x}+\pi^T $</span> </p>

<p>Sorry I am not economist, I am studying math, and I need its interpretation. </p>

<p>I guess I need to interpret the value of <span class="math-container">$a$</span>, and I need to look at the geometry of the loss function in the Phillips curve diagram. </p>

<p>What is its interpretation in economic view?</p>

<p>Thank you so much!</p>

<macroeconomics><inflation><economic-growth><self-study><central-banking>
Question on Central bank loss function interpretation
studentp
# Question on Central bank loss function interpretation

(Asked by: username_0 on 2019-10-19)

How do you interpret this central bank’s loss function:

$$L(\hat x, \pi)=\alpha \hat{x}^2+(\pi-\pi^T)^2$$

where $\alpha >0$ and $\hat x$ is output gap.

Phillips curve is $\pi = \theta \hat{x}+\pi^T $

Sorry I am not economist, I am studying math, and I need its interpretation.

I guess I need to interpret the value of $a$, and I need to look at the geometry of the loss function in the Phillips curve diagram.

What is its interpretation in economic view?

Thank you so much!

**Tags:** macroeconomics, inflation, economic-growth, self-study, central-banking


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## 2 answers

### Answer by username_1 on 2019-10-20. Score: 1 | Accepted answer

The loss function reflects the central bank’s policy objective (i.e. their preferences for stabilizing inflation and some real variable around target levels, in this case output), and it says that the central bank is concerned about inflation fluctuating from their target level ($\pi^{T}$) and their perception of the equilibrium output (here reflected in the output gap).

Now, because the objective of the central bank includes two factors (output and inflation), we can know from the loss function how concerned the central bank is about stabilizing the output gap from the size of coefficient $\alpha$. Indeed, at varying levels of $\alpha$ it is easy to see that the reaction of the central bank to the same output gap will differ. Since $\alpha$ > 0 we know that they are concerned about the welfare loss associated with the output gap (if = 0 they would simply not be concerned about this, and if < 0 this would reflect that bigger output gaps are best since the function is to be minimized).

I can imagine that when looking at the geometry of the loss function in the Phillips curve diagram this may become clearer to you as someone from maths.

And just to note that once you derive the policy rule - the slope of the MR curve - (as in your [other](https://economics.stackexchange.com/questions/32308/central-bank-loss-function-i-did-a-solution-but-it-doesn-t-totally-make-sense/32315?noredirect=1#comment48815_32315) question), you will see that this policy rule not only depends on the central bank’s preferences $\alpha$ but also on the behavior of the private sector, the price flexibility as given by the parameter $\theta$.

> **Comments:**
> **username_0**: In addition to what you said, I also make some interpretation by using the Phillips curve diagram. Are the following statements correct?  
> **username_0**: “ $\alpha =1$ means that each indifference curve is a circle with $(x_e, \pi^T)$ at its centre. The loss declines as the circle gets smaller.  
> **username_0**: When $\pi =\pi^T$ and $x_e=x$, the circle shrinks to a single point (bliss point) and the loss is at minimum at zero. With $\alpha =1$, the central bank is indifferent between inflation 1% above (or below) $\pi^T$ and output 1% below (or above) $x_e$. They are on the same loss circle.  
> **username_0**: Only when $\alpha =1$ we have indifference circles. But when $\alpha <1$ then the central bank the central bank is indifferent between (say) inflation1% above (or below) $\pi^T$  and output 2% above (or below) $x_e$. This makes the indifference curves ellipsoid. And finally $\alpha >1$ means that central bank with less aversion to inflation will have ellipsoid indifference curves with a vertical rather than a horizontal orientation.  
> **username_0**: In that case, the indifference curves are steep indicating that the central bank is only willing to trade off a given fall in inflation for a smaller fall in output than in the other two cases.”  
> **username_0**: Is this comment also correct for my example? Thank you.  
> **username_1**: Yes, this holds perfectly for your example! I read it wrong and was about to comment something but then realised that you saw it perfectly since when <1 this means that the central bank is inflation averse and when $\alpha > 1$, you can see that the central bank is unemployment-averse. Moreover, as you noted, utility declines with distance from the bliss-point.  
> **username_0**: I am sorry, this is my last question to you. Do you have an idea about that? https://economics.stackexchange.com/questions/32349/revenue-maximization-problem  

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### Answer by username_2 on 2019-10-20. Score: 0

In this case, you can interpret this as the central banks' having two mandates: output gap and inflation target.

Here, the best possible scenario that would minimize the CB's loss function is having zero output gap ($\hat x^2 = 0$) and inflation is at the target ($\pi = \pi^T$). The CB simply wants to get as close to this as it could, with the "importance" of the output gap component given by $\alpha$. (What would $\alpha < 1$ mean, for example?)

I'm not sure how to help with the geometry part, but note that you can plug $\pi(\hat x)$ from the Phillips curve into the loss function.

> **Comments:**
> **username_0**: No no I think this is not answer that I need. Because what you say is related to another question that I linked. For example what is mean $\alpha$ without any calculation ?  

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