<p>There is no &quot;interpetation&quot; for these coefficients, any more than there is one for the 2 in <span class="math-container">$2(P_x'P_y)^{1/2}$</span>.</p>
<p>Both here and in <a href="https://economics.stackexchange.com/questions/54382/compensating-variation-for-u-xy1-2-and-u-xy">your other question</a>, the equation <span class="math-container">$\Delta M = U_02(P'_xP_y)^{1/2} - U_12(P'_xP_y)^{1/2}$</span> is not particularly useful in itself. The compensanting variation is meant to measure the change in income that would be adequate compensate the consumer for the change in price. It could be considered adequate compensation, because it would enable the consumer to reach their pre price change utility level, i.e. <span class="math-container">$U_0 = U_1$</span>. You should definitely use this piece of information.</p>
<p>Using the above and your formula <span class="math-container">$U_0 = \frac{M}{2(P_xP_y)^{1/2}}$</span>, you can calculate <span class="math-container">$\Delta M$</span> as a function of the other parameters. That's it, that shows you how much additional income would compensate the consumer for the price change, given their utility function and all other parameters.</p>

CC BY-SA 4.0
2023-02-08T14:33:09.090Z
Thank you for your answer @Giskard, and for reading my other post! 1) Are my calculations in the other post correct, particularly for $U = xy$? 2) is there any reason that $∆M$ would be proportional to the $∆U$ by the amount of the coefficient? 

I ask, only because for example for the Income effect $-\frac{∆M}{2P'_x}$ appears to have a meaningful economic interpretation. We divide the income change by 2, because the value of the goods are weighted equally, and then we divide this remaining income by $P'_x$ to see how many  $x's$ we could buy at the new price. It's minus as $P_x$ increases.
CormJack
CC BY-SA 4.0
2023-02-08T14:34:33.487Z
As in that's an example of a seemingly abstract formulae having some meaning. If you think there's nothing to it, I won't insist anymore, thanks!
CormJack
CC BY-SA 4.0
2023-02-08T14:44:19.387Z
I don't think there is anything to it, the pattern becomes unwound/more complicated if you use a non-symmetric Cobb-Douglas function.
Giskard
CC BY-SA 4.0
2023-02-08T14:58:16.347Z
If you do get chance to confirm the calculation on the other question, that'd be great. No worries if not!
CormJack
CC BY-SA 4.0
2023-02-08T07:37:27.877Z
2023-02-08T07:37:27.877Z
Giskard
<p>Assume <span class="math-container">$U(x,y) = x^{1/2}y^{1/2}$</span> s.t. <span class="math-container">$P_xx + P_yy = m$</span>
And a price increase from <span class="math-container">$P_x$</span> to <span class="math-container">$P'_x$</span>:</p>
<ul>
<li><span class="math-container">$U_0 = \frac{M}{2(P_xP_y)^{1/2}}$</span></li>
<li>Compensation variation formulae is: <span class="math-container">$\frac{M + ∆M}{2(P_x'P_y)^{1/2}} = U_0$</span></li>
<li><span class="math-container">$∆M = U_02(P'_xP_y)^{1/2} - U_12(P'_xP_y)^{1/2}$</span></li>
</ul>
<p><strong>We can rewrite this as:</strong> <span class="math-container">$∆M = (U_0 - U_1)2(P'_xP_y)^{1/2} = ∆U[2(P'_xP_y)^{1/2}]$</span></p>
<ol>
<li>What is a meaningful way to inteparate this?</li>
<li>E.g. what is the <span class="math-container">$2(P'_xP_y)^{1/2}$</span> coefficient on our <span class="math-container">$∆U$</span>?</li>
</ol>

<utility><optimization><lagrangian>
Compensating Variation - Interpreting the formulae
CormJack
# Compensating Variation - Interpreting the formulae

(Asked by: username_0 on 2023-02-08)

Assume $U(x,y) = x^{1/2}y^{1/2}$ s.t. $P\_xx + P\_yy = m$
And a price increase from $P\_x$ to $P'\_x$:

* $U\_0 = \frac{M}{2(P\_xP\_y)^{1/2}}$
* Compensation variation formulae is: $\frac{M + ∆M}{2(P\_x'P\_y)^{1/2}} = U\_0$
* $∆M = U\_02(P'\_xP\_y)^{1/2} - U\_12(P'\_xP\_y)^{1/2}$

**We can rewrite this as:** $∆M = (U\_0 - U\_1)2(P'\_xP\_y)^{1/2} = ∆U[2(P'\_xP\_y)^{1/2}]$

1. What is a meaningful way to inteparate this?
2. E.g. what is the $2(P'\_xP\_y)^{1/2}$ coefficient on our $∆U$?

**Tags:** utility, optimization, lagrangian


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## 1 answers

### Answer by username_1 on 2023-02-08. Score: 1 | Accepted answer

There is no "interpetation" for these coefficients, any more than there is one for the 2 in $2(P\_x'P\_y)^{1/2}$.

Both here and in [your other question](https://economics.stackexchange.com/questions/54382/compensating-variation-for-u-xy1-2-and-u-xy), the equation $\Delta M = U\_02(P'\_xP\_y)^{1/2} - U\_12(P'\_xP\_y)^{1/2}$ is not particularly useful in itself. The compensanting variation is meant to measure the change in income that would be adequate compensate the consumer for the change in price. It could be considered adequate compensation, because it would enable the consumer to reach their pre price change utility level, i.e. $U\_0 = U\_1$. You should definitely use this piece of information.

Using the above and your formula $U\_0 = \frac{M}{2(P\_xP\_y)^{1/2}}$, you can calculate $\Delta M$ as a function of the other parameters. That's it, that shows you how much additional income would compensate the consumer for the price change, given their utility function and all other parameters.

> **Comments:**
> **username_0**: Thank you for your answer @username_1, and for reading my other post! 1) Are my calculations in the other post correct, particularly for $U = xy$? 2) is there any reason that $∆M$ would be proportional to the $∆U$ by the amount of the coefficient?   I ask, only because for example for the Income effect $-\frac{∆M}{2P'_x}$ appears to have a meaningful economic interpretation. We divide the income change by 2, because the value of the goods are weighted equally, and then we divide this remaining income by $P'_x$ to see how many  $x's$ we could buy at the new price. It's minus as $P_x$ increases.  
> **username_0**: As in that's an example of a seemingly abstract formulae having some meaning. If you think there's nothing to it, I won't insist anymore, thanks!  
> **username_1**: I don't think there is anything to it, the pattern becomes unwound/more complicated if you use a non-symmetric Cobb-Douglas function.  
> **username_0**: If you do get chance to confirm the calculation on the other question, that'd be great. No worries if not!  

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