Question

In a two-good world, the utility function of a consumer is given by

Hint:

Compensating Variation measures the extra income required after a price increase to keep the consumer at the original utility level.

Answer 1

Correct Answer: 41.4

Step 1: Find the starting bundle.
With $u=x_1x_2$, a Cobb Douglas with equal weights, income splits evenly. At $p_1=p_2=1$ and $M=100$, the consumer spends $50$ on each, so $x_1=x_2=50$ and starting utility is $u_0=2500$.

Step 2: Note the new prices.
Now $p_1=1$ and $p_2=2$. We ask how much income is needed to keep utility at $2500$.

Step 3: Use the expenditure function.
For $u=x_1x_2$ it is
\[ e=2\sqrt{u\,p_1 p_2} \]

Step 4: Plug in.
\[ e=2\sqrt{2500\times1\times2}=2\sqrt{5000}=141.42 \]

Step 5: Take the compensating variation.
This is the extra income needed,
\[ CV=141.42-100=41.42\approx41.4 \]
\[ \boxed{41.4} \]