Calculating the Compensating Variation with $M^2$

Question

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!

Answer 1

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.

2. 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.