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

What would be the new bundles of x, y if Px was $3

Answer 1

a) $Utility\ function = U=xy$

$MU_x=y$

$MU_y=x$

$P_x= \$2$

$P_y=py$

Income= $40.

Budget line will be $M=P_xx+P_yy$

$40=2x+5y$

$\frac{MU_x}{P_x}=\frac{MU_y}{P_y}$

$\frac{y}{2}=\frac{x}{5}$

$5y=2x$

$y=\frac{2}{5}x$

$40=2x+5(\frac{2}{5}x)$

$40=4x$

$x=10$

Amount of x consumed will be 10 units.

$y=\frac{2}{5}\times 10$

$y=4$

The price of Y will therefore, be $4.

b) In equi-marginal principle, $MU_x=P_x$

Therefore, $y=P_x$

$y=4$ and $P_x=4$. This proves that the allocation above follows the equi-marginal principle.

c) if $P_x=3$

$\frac{y}{3}=\frac{x}{5}$

$5y=3x$

$y=\frac{3}{5}x$

$40=2x+5(\frac{3}{5}x)$

$40=5x$

$x=8$

$y=\frac{3}{5}\times 8$

$y=\frac{24}{5}$