I'm currently a TA for a class and recently graded a midterm. I gave the answer key back to the teacher, after going over part of the exam in a study hall. I was going to go over the rest of it tomorrow, but while making my own answer key in office hours, I seem to have come to a different answer than the teacher.
We maximize $$u = 2x_1^{1/2} + 4x_2^{1/2}$$ with a normal budget constraint where $p \cdot x \leq w$. We arrive at the Walrasian demand:
$$x^*(p, w) = \left(\frac{p_2 w}{p_1(4p_1 + p_2)} , \frac{4p_1 w}{p_2(4p_1 + p_2)}\right)$$
Suppose $w = 10, \vec p = (1, 4), \vec p' = (3, 2)$.
Thus, $x(p', w) = (\frac{10}{21}, \frac{30}{7})$, and $x(p, w) = (5, \frac{5}{4})$ for our new and old bundles respectively.
So to find compensating variation we find the original utility:
$2 \cdot 5^{1/2} + 4 \cdot (5/4)^{1/2} = 4 \sqrt 5$
and find $w'$ that would get old utility and new prices:
$4 \sqrt 5 = 2(\frac{w'}{21})^{1/2} + 4(\frac{3w'}{7})^{1/2} = 2(\frac{w'}{21})^{1/2} + 12(\frac{w'}{21})^{1/2} = 14(\frac{w'}{21})^{1/2} \implies \\ 4 \sqrt 5 = 14(\frac{w'}{21})^{1/2} \\ 80 = 14^2 \cdot \frac{w'}{21} \\ \boxed{w' = \frac{60}{7}}$
Thus $\boxed{CV = w - w' = 10 - \frac{60}{7} = \frac{10}{7}}$
To find equivalent variation we find the new utility:
$2 \cdot (10/21)^{1/2} + 4 \cdot (30/7)^{1/2} = 2 \cdot (10/21)^{1/2} + 12 \cdot (10/21)^{1/2} = 14 \sqrt {\frac{10}{21}}$
and find $\hat w$ that would get new utility at old prices:
$14 \sqrt {\frac{10}{21}} = 2(\frac{\hat w}{2})^{1/2} + 4(\frac{\hat w}{8})^{1/2} = 4(\frac{\hat w}{2})^{1/2} \\ 14^2 \cdot \frac{10}{21} = 16 \cdot \frac{\hat w}{2} \\ \boxed{\hat w = \frac{70}{6}}$
Thus $\boxed{EV = \hat w - w = \frac{70}{6} - 10 = \frac{5}{3}}$
The problem is if I recall correctly, the CV and EV are supposed to have the opposite sign so that the change in welfare is ambiguous. Where have I gone wrong, if anywhere? (Worth noting that if you do Slutsky decomposition for this question, you find that good 2 is inferior.)
