Question $1(4$ Points$)$

Consider the labor supply and consumption of a person with the following utility function:

U$($C$,$ L$)=$ C

$2/3$L

$1/3$

$,$

where L is leisure time and C is consumption of goods. The price of consumption is normalized to one $$ in other words, C is consumption of other goods measured in dollars. The

person has V dollars of non$-$labor income and can work for a wage of w$.$ There are T hours

available for either working or leisure.

$1.$ Write down the budget constraint. Draw a graph representing this constraint, making

sure to label the axes and key points.

$2.$ What are the marginal utilities of consumption and leisure? What is the marginal rate

of substitution between leisure and consumption?

$3.$ Write down a condition involving the marginal rate of substitution that characterizes

the person$$s optimal choice. Represent this condition graphically and interpret it in

words.

$4.$ Solve for the optimal choices of leisure and consumption, L

$$ and C

$$

$,$ in terms of T$,$ V $,$

and w$.$

$5.$ Are L and C normal goods? Explain your answer