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