Ellsworth's utility function is U(x, y) = min{x, y}. Ellsworth has $150 and the price of x and the price of y are both $1. Ellsworth's boss is thinking of sending him to another town where the price of x is $1 and the price of y is $2. The boss offers no raise in pay. Ellsworth, who understands compensating and equivalent variation perfectly, complains bitterly. He says that although he doesn't mind moving for its own sake and the new town is just as pleasant as the old, having to move is as bad as a cut in pay of $A. He also says he wouldn't mind moving if when he moved he got a raise of $B. What are A and B?
A) A = 50 and B = 50.
B) A = 75 and B = 75.
C) A = 75 and B = 100.
D) A = 50 and B = 75.
E) None of the above.

Question

Quizplus · Accepted Answer

To find A, we need to find the minimum amount of compensation that would make Ellsworth indifferent between moving and not moving. Since Ellsworth's utility function is U(x,y) = min{x, y}, we know that he will always choose to spend his money evenly between x and y until x and y cost the same. This means that he will spend $75 on x and $75 on y, since the prices are $1 and $2, respectively. In the new town, $75 will buy him 75 units of x and 37.5 units of y (since y costs twice as much), which gives him a total utility level of U(75,37.5) = 37.5. In the old town, $150 would buy him 150 units of x, which also gives him a total utility level of U(150,0) = 0, since his utility is limited by the minimum of x and y. Therefore, moving to the new town means a reduction in utility of 37.5, which is equivalent to a loss in income of $A = 37.5.

To find B, we need to find the amount of compensation that would make Ellsworth indifferent between staying in the old town and moving to the new town with a higher cost of y. Since y costs twice as much in the new town, Ellsworth would need to be compensated with the same amount to maintain his utility level. In other words, he would need $75 to purchase the same amount of y in the new town that he could buy for $37.5 in the old town. Therefore, he would need a raise of $75 to be indifferent between staying and moving to the new town.

Thus, the answer is D) A = 50 and B = 75.

Ellsworth's Utility Function Is U(x, Y) = Min{x, Y}