Question

3. (14 points) A consumer's utility function is given by U(x,y) = x1/2y1/2 (1) Find the consumer's Marshallian demand functions. (2) Find the consumer's compensated demand functions. (3) Suppose the price of good y is Py = $1 per unit and the consumer's income is 1 = $20. Find the total effects on good x and good y when the price of good x increases from px - $1 per unit to p} = $2 per unit.

Answer 1

1. Utility is maximized when MRS = Px/Py

MRS = MUx/MUy = $\frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial y}} = = \frac{(1/2)x^{(1/2)-1}y^{1/2}}{(1/2)y^{(1/2)-1}x^{1/2}} = \frac{x^{-1/2}y^{1/2}}{x^{1/2}y^{-1/2}} = \frac{y^{(1/2)+(1/2)}}{x^{(1/2)+(1/2)}} = \frac{y}{x}$

y/x = Px/Py
So, y = xPx/Py
Budget constraint: Px*x + Py*y = I
So, xPx + Py(xPx/Py) = I
So, 2xPx = I
So, x = I/2Px

y = xPx/Py = (I/2Px)(Px/Py)
So, y = I/2Py

2. U = x1/2y1/2 = x1/2(xPx/Py)1/2 = x1/2x1/2(Px/Py)1/2
So, (Py/Px)1/2U = x(1/2)+(1/2)
So, x = (Py/Px)1/2U

y = xPx/Py = (Py/Px)1/2U(Px/Py) =
So, y = (Px/Py)1/2U

3. Px1 = 1; Py = 1; I = 20
x = I/2Px1 = 20/2*(1) = 10
y = I/2Py = 20/2(1) = 10

Px2 = 2; Py = 1; I = 20
x' = I/2Px2 = 20/2*(2) = 5
y' = I/2Py = 20/2(1) = 10

Total effect on x = x' - x = 5 - 10 = -5
Total effect on y = y' - y = 10 - 10 = 0