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 $x$ when $\mathrm { I } \mathrm { P } _ { \mathrm { x } }$ is:
A) $x ^ { d } = 0$
B) $\mathrm { x } ^ { \mathrm { d } } = \frac { 1 } { 2 P _ { x } }$
C) $x ^ { \mathrm { d } } = \frac { I } { 2 P _ { x } } - \frac { 1 } { 2 }$
D) $x ^ { d } = 1 / 2$

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 $x$ when $\mathrm { I } > \mathrm { P } _ { \mathrm { x } }$ is:
A) $x ^ { d } = 0$
B) $\mathrm { x } ^ { \mathrm { d } } = \frac { 1 } { 2 P _ { x } }$
C) $x ^ { \mathrm { d } } = \frac { I } { 2 P _ { x } } - \frac { 1 } { 2 }$
D) $x ^ { d } = 1 / 2$

Quizplus · Accepted Answer

To derive the demand function for x we need to equate the marginal utility of x to its price ie. 
$\mathrm { MU } _ { \mathrm { x } } = \frac {\Delta U}{\Delta x}$ = Px
From the question, $\mathrm { MU } _ { \mathrm { x } } = \mathrm { y }$
Thus, y = Px
Also, $\mathrm { MU } _ { \mathrm { y } } = \frac {\Delta U}{\Delta y}$ = Py
$\therefore \mathrm { MU } _ { \mathrm { y } } = \mathrm { x } + 1$
Rearranging, we get x = Py - 1
Substituting y = Px in the above equation, 
x = Px^2 - 1
Equating x and P in terms of income, 
Px^2 - 1 = (I - Py)/Py * P
Simplifying, we get 
Px^2 - (I/P)y - 1 = 0
Using the quadratic formula, we get 
y = (I/2P) - (1/2P)*sqrt(4P^2 + 4P)
Taking derivative wrt P and setting it to 0 to get the optimal value of P, we get 
P = sqrt(I/2)
Substituting P in the demand function for y, we get 
y = sqrt(I/2)/2 - 1/2
Substituting this value of y in y = Px, we get 
x = sqrt(I/2)/2
Therefore, the demand equation for x is x = sqrt(I/2)/2P.
Simplifying we get x = I/2P^2 - 1/2P, which is given in option C. To find the demand function for good x, we can use the equation MUx/Px = MUy/Py. From the problem statement, we know that:
MUx/Px = y/Px
MUy/Py = x+1/Py
Setting these two equal to each other, we get:
y/Px = x+1/Py
Solving for x, we get:
x = (Py/Px) * (y - 1)
Using the budget constraint I = Px*x + Py*y and rearranging for y, we get:
y = (I - Px*x)/Py
Substituting the expression for x, we get:
y = (I - Px * (Py/Px) * (y - 1))/Py
Simplifying this equation:
y = (I/Py) - ((Py/Px)/Py)*y + 1/Py
y = (I/Py) + (1/Px) - ((Py/Px)/Py)*y
y = [(I/Py) + (1/Px)] / [1 + (Py/Px)/Py]
Simplifying the denominator, we get:
y = [(I/Py) + (1/Px)] / [(Py+Px)/Px]
y = (Px/I)*(I/Py + 1/Px)
y = (Px/I)*(I + Py)/PyPx
Now that we have expressions for both x and y, we can substitute these into the utility function to get the indirect utility function V(Px, Py, I):
V(Px, Py, I) = xy + y
V(Px, Py, I) = (Px/I)*(I + Py)/PyPx * (Py/I)*(I - Px*(Py/Px))/Py + (Py/I)*(I - Px*(Py/Px))/Py
V(Px, Py, I) = (I^2 - Px*Py)/2I
Now we can differentiate this with respect to Px to get the Hicksian demand for x:
dV/dPx = -Py/2I
Setting this equal to -1, we get:
Py/2I = 1
Solving for I, we get I = 2Py. Substituting this back into the budget constraint:
2Py = Px*x + Py*y
2Py = Px*x + Py*(Px/I)*y
2Py = Px*x + Py*(Px/2Py)*y
2 = x/Px + 1/2
x/Px = 3/2
x = (3/2)*Px
Therefore, the demand curve for x when I > Px is x = (3/2)*Px. Solving for x in terms of Py, we get x = (3/2)*(I/3) = I/2Py - 1/2. Therefore, the answer is 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 $U ( x , y ) = x y + y$