Suppose the consumer's utility function is given by U(x,y) = x^1/4y^3/4 where $M U _ { x } = \frac { y ^ { \frac { 3 } { 4 } } } { 4 x ^ { \frac { 3 } { 4 } } }$ $M U _ { y } = \frac { 3 x ^ { \frac { 1 } { 4 } } } { 4 y ^ { \frac { 1 } { 4 } } }$ 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

The marginal utility of x is equal to the marginal utility of y multiplied by the ratio of the price of x to the price of y. So, $\frac { M U _ { x } } { M U _ { y } } = \frac { P _ { y } } { P _ { x } }$Substituting the given marginal utilities into this equation, we get:$\frac { y ^ { \frac { 3 } { 4 } } } { 4 x ^ { \frac { 3 } { 4 } } } = \frac { P _ { y } } { P _ { x } }$Solving for x, we get:$x = \frac { y ^ { \frac { 3 } { 4 } } P _ { x } } { 4 P _ { y } }$Substituting the budget constraint $I = P _ { x } x + P _ { y } y$ into this equation, we get:$x = \frac { ( I - P _ { y } y ) ^ { \frac { 3 } { 4 } } P _ { x } } { 4 P _ { y } }$Since y is a normal good, $\frac { \partial x } { \partial I } > 0$. Therefore, the denominator of this equation must be positive. This implies that $I - P _ { y } y > 0$, or $y < \frac { I } { P _ { y } }$.Substituting this into the equation for x, we get:$x = \frac { ( \frac { I } { P _ { y } } ) ^ { \frac { 3 } { 4 } } P _ { x } } { 4 P _ { y } } = \frac { I } { 4 P _ { x } }$Therefore, the equation for this consumer's demand curve for x is $x = \frac { I } { 4 P _ { x } }$.

Suppose the Consumer's Utility Function Is Given by U(x,y) = MUx=y344x34M U _ { x } = \frac { y ^ { \frac { 3 } { 4 } } } { 4 x ^ { \frac { 3 } { 4 } } }MUx​=4x43​y43​​

Suppose the Consumer's Utility Function Is Given by U(x,y) = $M U _ { x } = \frac { y ^ { \frac { 3 } { 4 } } } { 4 x ^ { \frac { 3 } { 4 } } }$