Sam's preferences for income (Y) and leisure (TB) can be expressed as:

U(Y,TB)=300+2YTB -67Y-100TB

Out of the 365 days in the year, Sam devotes TH0=60 days to health production and spends TL0=15 days sick. The rest of the days are available to split between work and leisure. Sam earns $100 per day after taxes.

(a) Derive the equation for Sam's budget constraint and graph his budget line labeling as many points as you can.

(b) Derive the equation for Sam's marginal rate of substitution between income (Y) and leisure (TB).Does the property of diminishing marginal utility hold? Why? Does the property of diminishing marginal rate of substitution hold? Why?

(c) Write the optimization problem for the consumer and find the optimal allocation of time to market work (Tw) and Leisure (TB), the optimal level of income, and the utility at the optimal point. Graph your result labeling as many points as you can.

(d) Suppose now that Sam's daily wage increases to $120. Show how his equilibrium level of income and leisure-work choice would change. Graph your result labeling as many points as you can.

(e) Go back to the baseline wage of $100. Suppose Sam increases the days to health production to TH1=70 and is able to lower the sick days to TL1=10 days. Show how his equilibrium level of income and leisure-work choice would change. Graph your result labeling as many points as you can. Is Sam better off or worse off, explain.