Bernice has the utility function u(x, y) = min{x, y}, where x is the number of pairs of earrings she buys per week and y is the number of dollars per week she has left to spend on other things. (We allow the possibility that she buys fractional numbers of pairs of earrings per week.) If she originally had an income of $18 per week and was paying a price of $6 per pair of earrings, then if the price of earrings rose to $10 per pair, the compensating variation of that price change (measured in dollars per week) would be closest to
A) $20.57.
B) $6.55.
C) $10.29.
D) $21.57.
E) $19.57.

Question

Quizplus · Accepted Answer

To find the compensating variation, we need to determine the amount of income Bernice would need to maintain her original level of utility after the price of earrings increases from $6 to $10 per pair.

We can begin by finding Bernice's optimal consumption bundle at the original prices. Since her utility function is min{x, y}, she will consume equal amounts of earrings and all other goods (y), up until she runs out of money. So we can use the budget constraint to solve for the quantities of earrings and y she will consume:

6x + y = 18

Since she consumes equal amounts of earrings and y, we can substitute y = x into the budget constraint:

6x + x = 18

Solving for x, we get x = 2, which means she buys 2 pairs of earrings per week, leaving her with $6 per week to spend on other things.

Now, we need to find out how much income Bernice would need to have to still be able to buy 2 pairs of earrings per week when the price increases to $10 per pair. Let's call the new income y'. Bernice's new budget constraint becomes:

10x + y' = y + 18

(Notice that we're using the fact that she still consumes equal amounts of earrings and y, so her total spending on earrings will be 10x, and her total spending on all other goods will be y'.)

To find the compensating variation, we need to solve for y'. Since Bernice is still buying 2 pairs of earrings per week, we can substitute x = 2 into the new budget constraint:

10(2) + y' = 6 + 18

Solving for y', we get y' = 29.

Therefore, the compensating variation is $29 - $18 = $10.

The closest answer choice is C) $10.29.

Bernice Has the Utility Function U(x, Y) = Min{x, Y}