Question

Justin has the utility function U = xy, with the marginal utilities MUx = y and MUy = x. The price of x is $2, the price of y is p_y, and his income is 40. When he maximizes utility subject to his budget constraint, he purchases 5 units of y.

(a)  What must be the price of y and the amount of x consumed?

(b)  Prove that this allocation follows the equi-marginal principle.

(c)  What would be the new bundles of x, y if Px was $3 .

Answer 1

a)

We know that budget line be

$M=P_XX+P_YY\\\therefore M=40=2X+5Y\\\therefore 40=2X+5Y.............................(1)\\$

Also we know that

$\frac{MU_X}{P_X}=\frac{MU_Y}{P_Y}$

$\therefore \frac{Y}{2}=\frac{X}{5}\\5Y=2X\implies Y=\frac{2}{5}X$

From equation (1)

$40=2x+5(\frac{2}{5})X\\=2X+2X=4X\\X=\frac{40}{4}=10\\\implies X=10 \space units$

Hence

$Y=\frac{2}{5}X=\frac{2}{5}(10)=4$

Y= 4 units

b)

For the equi-marginal principle

$MU_X=P_X\\\therefore Y=P_X$

$Y=4 \space and\space P_X=4$

This proves that allocation above follows the equimarginal principle.

c)

Now, $P_X=3 \space given$

$\frac{Y}{3}=\frac{X}{5}\\5Y=3X\\$

$Y=\frac{3}{5}X$

Again from (1)

$40=2X+5(\frac{3}{5})\\\therefore 40=2X+3X=5X\\\therefore X=\frac{40}{5}=8\implies X=8$

Now

$Y=\frac{3}{5}X=\frac{3}{5}(8)\\\therefore Y=\frac{24}{5}$