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.