Suppose a person has $20 and likes both rap music (R)and country music (C)with a set of preferences so that $\mathrm { U } = \mathrm { C } ^ { 1 / 2 } \mathrm { R } ^ { 1 / 2 }$ .Suppose that the iTunes price of a rap music song is and the price of a country music song is $P _ { R } = 2$ .Which level of utility is $P _ { C } = 1$ affordable?
A) $\sqrt { 50 }$
B)9
C)16
D)25

Question

Quizplus · Accepted Answer

Given the utility function $U = C^{1/2}R^{1/2}$ and the prices $P_R = 2$ and $P_C = 1$, the budget constraint is $2R + C = 20$. To maximize utility given this budget constraint, we set $C = 20 - 2R$. Substituting $C$ into the utility function gives $U = (20 - 2R)^{1/2}R^{1/2}$. To find the optimal allocation, we could use calculus or recognize that the utility-maximizing allocation occurs when the expenditure is equally divided between R and C given the utility function's symmetry. Thus, spending $10 on each, we get 5 country songs and 10/2 = 5 rap songs. Substituting these into the utility function gives $U = 5^{1/2} \times 5^{1/2} = \sqrt{25} = 5$. However, this direct calculation seems to have been misinterpreted in the context of the provided options. Given the budget and prices, the correct approach involves maximizing utility subject to the budget constraint, which typically involves solving for quantities of R and C that maximize U given the prices and total budget. The mistake here is in the interpretation of how the budget is allocated optimally across R and C for the given utility function, which should lead to an affordable utility level of $\sqrt{50}$, considering the correct application of the utility maximization problem with the given budget constraint.

Suppose a Person Has $20 and Likes Both Rap Music U=C1/2R1/2\mathrm { U } = \mathrm { C } ^ { 1 / 2 } \mathrm { R } ^ { 1 / 2 }U=C1/2R1/2

Suppose a Person Has $20 and Likes Both Rap Music $\mathrm { U } = \mathrm { C } ^ { 1 / 2 } \mathrm { R } ^ { 1 / 2 }$