A race of dwarfs lived near a forest where apple trees grew wild. Any dwarf who wanted to could enter the forest and pick apples for himself and his family. One day a giant came, claimed the forest for himself, and began charging the dwarfs for the right to pick apples.
a. Suppose that dwarfs can pick fewer apples when the forest is more crowded. Draw a graph with "Number of dwarfs in the forest" on the horizontal axis and "Apples per dwarf" on the vertical. Draw a curve representing the number of apples picked per dwarf and a curve representing each dwarf's marginal contribution to the apple harvest. Explain intuitively why the latter curve lies below the former.
b. Suppose that all dwarfs have the same opportunity cost to enter the forest. Show on your graph how many dwarfs enter the forest before the giant arrives and how many enter after the giant arrives. Show the giant's revenue.
c. Now drop the assumption that all dwarfs have the same opportunity cost, and assume that some dwarfs' time is more valuable than others'. On the graph you drew for part (a), add the upward-sloping curve that shows the marginal cost of adding dwarfs to the forest. Show the number of dwarfs that enter. Show the producers' surplus that the dwarfs earn as apple-pickers.
d. Continuing to use your graph from part (c), show the optimal number of dwarfs in the forest. Show the entrance fee that achieves this optimal number. Explain why this is the entrance fee that a competitive giant would set.
e. Can the dwarfs be made better off as a result of the giant's arrival and the entry fee? What about society as a whole (consisting of the dwarfs plus the giant)?
f. Assuming straight-line curves, and assuming that the giant sets a monopoly price to enter the forest show that the monopolized forest is more socially efficient than the common-property forest if and only if the "marginal apple harvest" curve is steeper than the dwarfs' marginal cost curve.