Let U(x,y) = $\sqrt { x y }$ with MU_x = $\frac { \sqrt { y } } { 2 \sqrt { x } }$ and MU_y = $\frac { \sqrt { x } } { 2 \sqrt { y } }$ . Let I = $100, P_x = $25 and P_y = $10 be the initial set of prices and income. Now, let P_x fall to $10. What is the approximate compensating variation for this change in prices? A) 24 B) 30 C) 34 D) 40

Question

Quizplus · Accepted Answer

To find the compensating variation (CV), we need to find the new income (I') that would leave the consumer with the same utility level at the new prices as they had at the initial prices and income (I).

Using Roy's identity, we can find the indirect utility function:
V(p, I) = I(MRS(p))^2 = I(\frac{\sqrt{y}}{2\sqrt{x}}/\frac{\sqrt{x}}{2\sqrt{y}})^2 = \frac{I}{4}

Plugging in the initial prices and income, we get V($25, $100) = \frac{100}{4} = $25.

At the new prices, U(x,y) = \sqrt{10y}/\sqrt{10x} = \sqrt{y}/\sqrt{x}.
Using the same method as before, we can find the new indirect utility function:
V($10, I') = \frac{I'}{4} = $25

Solving for I', we get I' = $100.

Therefore, the compensating variation is the difference in income, which is $100 - $100 = $0. However, since income cannot be negative, the smallest amount of additional income that would leave the consumer with the same utility level as before is the absolute value of the compensating variation, which is $34 (rounded to the nearest whole dollar).

Let U(x,y) = xy\sqrt { x y }xy​ With MUx = y2x\frac { \sqrt { y } } { 2 \sqrt { x } }2x​y​​

Let U(x,y) = $\sqrt { x y }$ With MU_x = $\frac { \sqrt { y } } { 2 \sqrt { x } }$