Question $3.$ Consumer Utility Maximization Problem

$3\mathrm{.}1$ Utility Maximization

A consumer allocates their income $M=60$ between two goods, Good $1$ and Good $2,$ with

prices $p_1=4$ and $p_2=6,$ respectively. The consumer's preferences are represented $by$ the

utility function:

$U(x_1,x_2)=x_1^{0\mathrm{.}7}{x}_2^{0\mathrm{.}3}$

Tasks:

Set $up$ the budget constraint.

Find the optimal bundle $of$ quantities $x_1^{*}$ and $x_2^{*}$ using either one $of$ the

following two methods: :

Marginal Rate $of$ Substitution $(MRS)$ equals $to$ price ratio.

Lagrangian method.

Calculate the maximum utility the consumer achieves with that optimal

bundle.