Charles's utility function is U(x, y)= xy.Anne's utility function is U(x, y)=1,000xy.Diana's utility function is -xy.Elizabeth's utility function is U(x, y)= -1/(xy + 1).Fergie's utility function is xy -010,000.Margaret's utility function is x/y.Philip's utility function is x(y + 1).(The goods x and y are two very expensive goods.We leave you to speculate about what they are.)Which of these persons have the same preferences as Charles?
A)Everybody except Diana
B)Anne and Fergie
C)Anne, Fergie, and Elizabeth
D)None of them
E)All of them

Question

Quizplus · Accepted Answer

To compare the preferences, we need to compare the marginal utilities. Taking the partial derivative of each utility function with respect to x and y, we get:

For Charles: MUx = y and MUy = x
For Anne: MUx = 1000y and MUy = 1000x
For Diana: MUx = -y and MUy = -x
For Elizabeth: MUx = (1/(xy+1)) - (x/(xy+1))^2 and MUy = (1/(xy+1)) - (y/(xy+1))^2
For Fergie: MUx = y - 10000 and MUy = x - 10000
For Margaret: MUx = 1/y and MUy = -x/y^2
For Philip: MUx = y/(y+1)^2 and MUy = x/(y+1)^2

Comparing the marginal utilities, we can see that Charles has the same preferences as Anne, Fergie, and Elizabeth because their marginal utilities have the same functional form. Specifically, all three have MUx = ay and MUy = ax for some positive constant a, which is the same as Charles's marginal utilities. Therefore, the answer is choice C.

Charles's Utility Function Is U(x, Y)= Xy