Suppose the consumer's utility function is given by $U ( x , y ) = x y + y$ where $\mathrm { MU } _ { \mathrm { x } } = \mathrm { y } \quad \mathrm { MU } _ { \mathrm { y } } = \mathrm { x } + 1$
The equation for this consumer's demand curve for $\mathrm { y }$ when $\mathrm { I }

Question

Suppose the consumer's utility function is given by $U ( x , y ) = x y + y$ where $\mathrm { MU } _ { \mathrm { x } } = \mathrm { y } \quad \mathrm { MU } _ { \mathrm { y } } = \mathrm { x } + 1$
The equation for this consumer's demand curve for $\mathrm { y }$ when $\mathrm { I } < \mathrm { P } _ { \mathrm { x } }$ is:
A) $y ^ { d } = 0$
B) $\mathrm { y } ^ { \mathrm { d } } = \frac { 1 } { 2 P _ { y } }$
C) $\mathrm { y } ^ { \mathrm { d } } = \frac { I } { P _ { y } }$
D) $y ^ { d } = 1 / 2$

Quizplus · Accepted Answer

Given utility function: $U(x,y) = xy+y$
First we need to find the marginal utility of y $(MU_y)$,
$MU_y = x+1$
Now we can use the equation for deriving the demand function for y:
$MU_y= \frac{P_x}{P_y}$
$x+1 = \frac{P_x}{P_y}$
$P_y(x+1) = P_x$
$y=\frac{P_x}{P_y}-1$
When $I<P_x$, the demand for y will be zero as the consumer will spend their entire income on x. Therefore, Option A is incorrect.
Substituting the value of $P_x$ in the demand function, we get
$y=\frac{I}{P_y}-1$
Hence, option C is correct. 
Option D is not a correct answer as it has an ambiguous value without knowing the prices of the goods. Option B is also incorrect as it does not take into account the income of the consumer. To find the demand curve for good y, first we need to find the marginal utility of income, which is given by:
$\mathrm{MUI}=\frac{\mathrm{MU}_x}{P_x}=\frac{y}{P_x}$
Now, set the MUI equal to the marginal utility of y to find the optimal quantity of y:
$\frac{y}{P_x}=x+1$
$y=P_x(x+1)$
Substitute this into the utility function to find the maximum utility achievable:
$U=x(P_x(x+1))+P_x(x+1)$
$U=P_x(x^2+x+1)$
Now we can use this to solve for the optimal quantity of x, given the budget constraint:
$I=P_xx+P_yy$
$y=\frac{I-P_xx}{P_y}$
Substitute this into the equation for optimal x:
$x=-\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{I}{P_x^2}}$
Therefore, the demand curve for good y is:
$\mathrm{y^d}=\frac{I-P_x\left(-\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{I}{P_x^2}}\right)}{P_y}=\frac{I}{P_y}+\frac{P_x}{P_y}\sqrt{\frac{1}{4}+\frac{I}{P_x^2}}$
This corresponds to answer choice C.

Suppose the Consumer's Utility Function Is Given By U(x,y)=xy+yU ( x , y ) = x y + yU(x,y)=xy+y

Suppose the Consumer's Utility Function Is Given By $U ( x , y ) = x y + y$