Only need to solve c, d, and e, thanks!

1. Sergio derives utility from consumption and leisure U(0,1) = (1/312/3. The price of consumption is Pc and for every hour he works he earns wage w. Sergio's total time available is T, so that his budget constraint is: Pcc= (T-1)w. a. Show that the utility function is homothetic. b. Solve for optimal demand for consumption and leisure. c. How do optimal leisure and consumption change when wage increases? What if inflation doubles both Pe and w, how do the optimal quan- tities change? What if Sergio is no longer taking classes at JHU SO his available time I also doubles? d. Draw the Marshallian demand curve for l for various values of w. Is la normal good? e. What is Sergio's expenditure function? (Hint: For this problem, his expenditure is his total value of his time: wT. First, find the indirect utility function, and then solve for time expenditure wT.) f. Find Sergio's compensated demand curve for consumption and for leisure. What do these functions tell us? g. Draw Sergio's budget constraint, and indifference curve with optimal bundle c* and 1*. Draw a second budget constraint with a lower Pc. Draw what the new optimal bundle might look like. Determine the substitution effect for 1 using the compensated demand function from part f, and label the substitution effect in the plot. Label the income effect as well. h. If T = 12, w = 5, P1 = 1 and P2 = 0.5 what is the increase in Sergio's consumer surplus from the fall in the price of consumption? Show his CS on a plot with his demand function for consumption