Question

Suppose a consumer has income of $120 per period, and faces prices, price of X=2 and Price of Z =3. Her goal is to maximize her utility, described by the function U = 10X0.5Z0.5.

i. What is the consumer’s budget constraint? (3)

ii. State the formula for finding marginal utilities for goods X and Z (2)

iii. Calculate the marginal utilities for goods X and Z (4)

iv. State the utility maximizing condition for this consumer (4).

v. Calculate the utility maximizing bundle (X*, Z*) (13)

Answer 1

i. Budget constraint is written as;

Y=P_xX+P_zZ

Where: Y=income

P_x=Price of commodity X

P_y=Price of commodity Y

120=2X+3Z........ budget constraint.

ii. Marginal utility(MU) is the change in utility with respect to a change quantity.

MU_x=$\dfrac{dU}{dX}$

MU_y=$\dfrac{dU}{dY}$

iii. MU_x=5X^-0.5Z^0.5

MU_y=5X^0.5Z^-0.5

iv. Utility is maximized when the slope of the budget line equals the slope of the utility function.

Slope of the budget line =$\dfrac{-P_x}{P_z}$

Slope of utility function =$\dfrac{-MU_x}{MU_z}$

Therefore, utility is maximized at;

$\dfrac{-P_x}{P_z}=\dfrac{-MU_x}{MU_z}$

v. Slope of the budget line =$\dfrac{-P_x}{P_z}$ = $\dfrac{-2}{3}$

Slope of utility function =$\dfrac{-MU_x}{MU_z}=-\dfrac{5X^{-0.5}Z^{0.5}}{5X^{0.5}Z^{-0.5}}\\$

$=-\dfrac{Z^{0.5}Z^{0.5}}{X^{0.5}X^{0.5}}=\dfrac{-Z}{X}\\$

We have;

$\dfrac{-2}{3}=\dfrac{-Z}{X}$

Cross-multiplying, we have;

-2X=-3Z

Z=$\dfrac{2}{3}X$

Inserting Z into the budget constraint, we have;

120=2X+3($\dfrac{2}{3}$X )

120=4X

X=30

Z=$\dfrac{2}{3}$ (30)

Z=20

Therefore, the optimizing bundle is;

(X,Z)=(30.20).