Question

suppose a customer has income of N$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 functional U = 10X0.5Z0.5..What is the consumers budget constraint?.. State the formula for finding marginal utilities for goods X and Z.. Calculate the marginal utilities for goods X and Z.. State utility maximizing condition for the consumer... Calculate the utility maximizing bundle (X*, Z*)

Answer 1

A) The consumer's budget constraint is given as:

$2X+3Z=120$

B) The marginal utility of X, $MU_X$ is given as:

$MU_X=\frac{dU}{dX}$

The marginal utility of Z, $MU_Z$ is given as:

$MU_Z=\frac{dU}{dZ}$

C) Given that

$u=10X^{0.5}Z^{0.5}$

$MU_X=5X^{-0.5}Z^{0.5}$

$MU_Z=5X^{0.5}Z^{-0.5}$

D) The utility maximizing condition is given as:

$\frac{MU_X}{MU_Z}=\frac{P_X}{P_Z}$

E) $u=10X^{0.5}Z^{0.5}$ subject to the constraint

$120-2X-3Z$

$\frac{5X^{-0.5}Z^{0.5}}{5X^{0.5}Z^{-0.5}}=\frac{2}{3}$

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

$X=1.5Z$

Substituting x Into the constraint:.$120-2(1.5Z)-3Z=0$

$120=6Z$

$Z^*=20$

$X^*=1.5(20)=30$