Suppose a person likes both rap music (R)and country music (C)with a set of preferences so that $\mathrm { U } = \mathrm { C } ^ { 1 / 2 } \mathrm { R } ^ { 1 / 2 }$ .Which point (C,R)makes this person the happiest?
A)4,16
B)25,1
C)9,9
D)16,4

Question

Quizplus · Accepted Answer

To maximize the person's happiness, we need to find the point (C,R) that maximizes their utility function U = C^(1/2) * R^(1/2). We can do this by taking the partial derivatives of U with respect to C and R:

dU/dC = 1/2 * (R/C)^(1/2)
dU/dR = 1/2 * (C/R)^(1/2)

Setting these equal to zero, we find that C/R = R/C or C = R. This means that the person is happiest when they have equal amounts of country and rap music. Plugging in C = R into the utility function, we get U = C^(1/2) * R^(1/2) = C^(1/2) * C^(1/2) = C.

So the person is happiest when C = R, or when they have equal amounts of country and rap music. This occurs at point (9,9), which is answer choice C.

Suppose a Person Likes Both Rap Music (R)and Country Music U=C1/2R1/2\mathrm { U } = \mathrm { C } ^ { 1 / 2 } \mathrm { R } ^ { 1 / 2 }U=C1/2R1/2

Suppose a Person Likes Both Rap Music (R)and Country Music $\mathrm { U } = \mathrm { C } ^ { 1 / 2 } \mathrm { R } ^ { 1 / 2 }$