# Calculating the Compensating Variation with $M^2$

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 $$∆M$$

• 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 $$U_1' = \frac{∆M}{2(P_x'P_y)^{1/2}}$$

• 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:

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

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.

• 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!

1. The left hand side expression corresponds to the new utility with the increased price and new budget $$M - \Delta M$$.

The income is now $$M - \Delta M$$ because we replaced the old income $$M$$ with $$M - \Delta M$$.

Therefore,

$$\text{budget change} = \text{new budget} - \text{old budget} = (M - \Delta M) - M = - \Delta M$$.

If the new income $$M - \Delta M$$ was a result of giving the agent money, the money we would give the agent to compensate for the price increase is equal to the $$\text{budget change} = - \Delta M$$.

Since we gave the agent money, that quantity has to be positive. For it to be positive, as you said, the quantity $$\Delta M$$ has to be negative.

I do find it a bit weird that the term is introduced with a negative sign rather than a positive one, which would directly yield the budget change as $$\Delta M$$ without the negative sign.

The other expression you gave

$$\frac{M}{2 (P_{x}^{‘} P_{y})^{\frac{1}{2}}} - \Delta M = U_0$$ actually corresponds to adding/subtracting utility.

This is because the new utility after only the price increase is $$\frac{M}{2 (P_{x}^{‘} P_{y})^{\frac{1}{2}}}$$.

Solving for $$\Delta M$$ in that expression would give

$$\Delta M = \frac{M}{2 (P_{x}^{‘} P_{y})^{2}} - U_0 =$$ $$\text{new utility}$$ $$-$$ $$\text{old utility}$$ $$= \text{change in utility}$$,

rather than a change in budget.

1. The correct one is option #2.

This is because if we replace the old income $$M$$ by what would be the new income $$M - \Delta M$$, the squaring function that was applied to the old income $$M$$, would now have to be applied to the new income $$M - \Delta M$$.

I guess it was a bit of luck with your numerical example that #1 happened to give you the correct answer. You should try a few more numerical examples and you’ll find one where option #1 doesn’t give the correct answer but #2 will always do.

• Thanks for your answer @NicolasTorres I'll digest and hopefully understand soon! Commented Jan 25, 2023 at 13:15
• Just read this again @NicolasTorres, super helpful. The detail in your answer is very thoughtful, thank you! I just read through a couple other answers of yours on this site, also very useful! Commented Feb 7, 2023 at 18:20
• I took your insights here @NicolasTorres, and tried to apply them to check my understanding. I have briefly outlined the CV calculation for the two Utility functions described here. Would you have time to quickly note if these are correct as per your guidance here. No problem if you don't have the time! Thanks again for your help: economics.stackexchange.com/questions/54382/… Commented Feb 10, 2023 at 14:12
• @CormJack I’ll check them when I have the time Commented Feb 10, 2023 at 17:05