Suppose the consumer's utility function is given by $\mathrm { U } ( \mathrm { x } , \mathrm { y } ) = \mathrm { x } ^ { 1 / 4 } \mathrm { y } ^ { 3 / 4 }$ where $M U _ { x } = \frac { y ^ { \frac { 8 } { 4 } } } { 4 x ^ { \frac { 8 } { 4 } } } \quad M U _ { y } = \frac { 3 x ^ { \frac { 2 } { 4 } } } { 4 y ^ { \frac { 1 } { 4 } } }$
The equation for this consumer's demand curve for $x$ is:
A) $x = \frac { I } { 2 P _ { x } }$
B) $x = \frac { I } { 4 P _ { x } }$
C) $x = \frac { I } { P _ { x } }$
D) $x = \frac { 3 I } { 4 P _ { x } }$

Question

Quizplus · Accepted Answer

To find the demand curve for $x$, we use the utility maximization condition that equates the marginal utility per dollar spent on each good, i.e., $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$. Given $MU_x = \frac{y^2}{4x^3}$ and $MU_y = \frac{3x^{1/2}}{4y^{1/4}}$, we solve for $x$ in terms of income $I$ and prices $P_x$ and $P_y$, considering the budget constraint $I = P_x x + P_y y$. The correct derivation involves using these conditions to find the relationship between $x$, $I$, and $P_x$, leading to the conclusion that $x = \frac{I}{4P_x}$, which is choice B.

Suppose the Consumer's Utility Function Is Given By U(x,y)=x1/4y3/4\mathrm { U } ( \mathrm { x } , \mathrm { y } ) = \mathrm { x } ^ { 1 / 4 } \mathrm { y } ^ { 3 / 4 }U(x,y)=x1/4y3/4

Suppose the Consumer's Utility Function Is Given By $\mathrm { U } ( \mathrm { x } , \mathrm { y } ) = \mathrm { x } ^ { 1 / 4 } \mathrm { y } ^ { 3 / 4 }$