Intuition behind the Compensating Variation! In Economics, we can calculate the compensating variation (CV), which (to my understanding) is the amount of money we would need to give back to a consumer to keep them at the same level of Utility after a price increase:
For a standard objective function e.g. $U(x,y) = x^{1/2}y^{1/2}$ and budget line  $P_xx + P_yy = M$
Optimal $U_0 = \frac{M}{2(P_xP_y)^{1/2}}$
The formulae given to me in my book for calculating CV is as follows (I understand if it's compensating variation $∆M$ will actually be negative given how the following is constructed):
$\frac{M - ∆M}{2(P_x'P_y)^{1/2}} = U_0 = \frac{M}{2(P_xP_y)^{1/2}}$ and solve for $∆M$

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*Where $P_x'$ is our increased price for $x$

*$U_1 = \frac{M}{2(P_x'P_y)^{1/2}}$ is the new utility with respect to the price increase.

Question 1: Why is it not $\frac{M}{2(P_x'P_y)^{1/2}} - ∆M = U_0$? This seems to be a more strait forward way of adding/subtracting the additional income to keep us on $U_0$. The original formulation seems to be more about adding/subtracting Utility? I.e. we could define a $U_1' = \frac{∆M}{2(P_x'P_y)^{1/2}}$

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*What's the intuition here?

Question 2: Given the Utility function $U(x,y) = xy$ and the same budget constraint, we get optimal utility: $\frac{M^2}{4(P_x'P_y)}$
What would be the correct equation for compensating variation and why:

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*$\frac{M^2 - ∆M}{4(P_x'P_y)} = U_0$

*$\frac{(M - ∆M)^2}{4(P_x'P_y)} = U_0$

*$\frac{M^2 - ∆M^2}{4(P_x'P_y)} = U_0$

*$\frac{M^2}{4(P_x'P_y)} - ∆M = U_0$
The first option works - I have worked through a numerical example and also compared the $∆M$ you get here with the $∆M$ you get from solving it in terms of the expenditure function.

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*But i would love help on the intuition about why it's the first, and not the others (Assuming they don't work!)

Thank you!
 A: First, if I knew nothing about CV, I would say that that your intuition on computing it, namely
$$\frac{M}{2(P_x'P_y)^{1/2}} - ∆M = U_0,$$
looks wrong just on the basis of units.  The second term on the left-hand side has units of dollars, while first term on the left-hand side has units of
$$ {\text{dollars}\over \sqrt{\text{dollars}^2\over (\text{unit of x})\cdot (\text{unit of y}) }}=\sqrt{(\text{unit of x})\cdot (\text{unit of y})}.$$
A quick dimensional analysis sometimes tell us a lot.

Now, I think both your questions will be cleared up with an understanding of the actual definition of CV.  First start with the indirect utility function $v(P,M)$: this is the value of the maximized utility as a function of the vector of prices $P$ and income $M$.  CV is defined as the amount a government would need to tax you after a price change to keep you just as well off as before the price change:
$$v(P_1,M-CV)=U_0=v(P_0,M).$$
In particular, assuming indirect utility decreases in prices, CV will be positive if prices fall; contrastingly, CV will be negative if prices rise (i.e. government would have to subsidize you).
Your examples are special cases of Cobb-Douglas utilities.  You can see option 2 would be the appropriate equation for your question 2.
