Two individuals, Chip and Holly, are stranded on an island. The island has plenty of vegetation. Chip and Holly can spend their day either gathering food, which they then eat themselves as they gather it (private good) or collecting firewood for warmth that evening. Both contribute their share of collected wood to a pile at the end of the day and burn the whole pile, the warmth from which they both enjoy together (public good). Each individual sleeps 8 hours a day thus leaving 16 hours to either collect wood (W) or pick food (F). Both individuals have the same preferences, represented by the following utility function:
UC = 5ln(FC) + 3ln(WC + WH)
A. Write the expression for Chip's time allocation budget constraint (in terms of hours). Include subscripts.
B. Use the given utility function and the budget constraint from part A to solve for Chip's best response function for the number of hours collecting firewood taking the number of hours Holly spends (WH) as given.
C. Solve for the privately optimal number of hours Chip spends collecting firewood. Hint: The problem is symmetric for Holly and Chip.
D. Solve for the publicly optimal number of hours Chip and Holly spend collecting firewood combined.