Consider an economic agent with the following utility function:

$U=C^{\frac{1}{2}}+2L^{\frac{1}{2}}$

where $C$ is consumption and $L$ is leisure. Assume that the price of the consumption good is $p=3,$ the wage is $w=6$ and the total endowment of time is $T=24.$

$($a$)$ Write down the budget constraint of the agent assuming she does not have any non$-$labour income. Represent this budget constraint graphically, indicate its slope and vertical intercept, and explain their economic meaning.

$($b$)$ Compute the marginal rate of substitution between consumption and leisure and graphically represent the map of indifference curves.

$($c$)$ Derive the demand of consumption goods, the demand of leisure and the labour supply of the agent. Represent the optimal choice graphically.

$($d$)$ Assume now that the agent also has some non$-$labour income equal to $v=9$ per day. Write down the new budget constraint, represent it graphically and compare it to that of question a$.$

$($e$)$ Derive the new optimal choice of consumption and labour supply under the new budget constraint in question d and compare this with the previous one $($question c$).$

$($f$)$ Compute the minimum level of the daily non$-$labour income that would induce the agent to withdraw from the labour force.