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

Question

Quizplus · Accepted Answer

To find the demand curve of good x, we need to use the equation: 
MUx/Px = MUy/Py
Substitute in the given marginal utilities:
(sqrt(y)/2sqrt(x)) / Px = (sqrt(x)/2sqrt(y)) / Py
Simplifying, we get:
Py/Px = sqrt(x)/sqrt(y)
Rearranging, we get:
Px = (sqrt(x)/sqrt(y)) * Py
Multiplying both sides by y:
Px * y = sqrt(x) * Py * sqrt(y)
Squaring both sides:
(Px * y)^2 = x * Py^2 * y
Simplifying,
x = (Px * y)^2 / Py^2
x = (Px / Py)^2 * y
Now, substitute the budget constraint: 
I = Px * x + Py * y
Substituting the value of x:
I = Px * ((Px / Py)^2 * y) + Py * y
Solving for y:
y = I / (Py * (1 + (Px/Py)^2))
Substituting back into the equation for x:
x = (Px / Py)^2 * (I / (Py * (1 + (Px/Py)^2)))
Simplifying, we get:
x = I / (2 * Px * (1 + (Py/Px)^2))
Thus, the demand curve for good x is:
x = I / (2 * Px * P)

Suppose the Consumer's Utility Function Is Given By U(x,y)=xyU ( x , y ) = \sqrt { x y }U(x,y)=xy​

Suppose the Consumer's Utility Function Is Given By $U ( x , y ) = \sqrt { x y }$