Question $1$: Labor Supply Decision in a Partial Equilibrium Model

Consider a worker who chooses how many hours to work, L$,$ given a fixed wage w and a fixed

amount of non$-$labor income M$.$ The worker$$s utility depends on leisure $($H$$L$)$ and consumption

C$,$ where H is the total available time in a day. The utility function is:

U $($C$,$ L$)=()0\mathrm{.}50\mathrm{.}5$

The worker$$s budget constraint is:

C$=$wL$+$M

where C is consumption, w is the wage, L is hours worked, and M is non$-$labor income.

$1.$ Write down the utility maximization problem for the worker and derive the first$-$order

condition for optimal labor supply.

$2.$ Solve for the labor supply function L$($w$,$M$)$ in terms of the wage rate w$,$ non$-$labor

income M$,$ and the total available time H$.$

$3.$ Analyze how an increase in the wage w affects the worker$$s labor supply decision.