Robinson Crusoe has decided that he will spend exactly 8 hours a day gathering food. He can either spend this time gathering coconuts or catching fish. He can catch 1 fish per hour and he can gather 2 coconuts per hour. On the graph below, show Robinson€™s production possibility frontier between fish and coconuts per day. Write an equation for the line segment that is Robinson€™s production possibility frontier.
F + C/2 = 8.

(a) Robinson€™s utility function is U(F,C) = FC, where F is his daily fish consumption and C is his daily coconut consumption. On the graph above, sketch the indifference curve that gives Robinson a utility of 4, and also sketch the indifference curve that gives him a utility of 8. How many fish will Robinson choose to catch per day? _________ How many coconuts will he collect? _____________
(b) Suppose Robinson is not isolated on an island in the Pacific, but is retired and lives next to a grocery store where he can buy either fish or coconuts. If fish cost $1 per fish, how much would coconuts have to cost in order that he would choose to consume twice as many coconuts as fish? _________ Suppose that a social planner decided that he wanted Robinson to consume 4 fish and 8 coconuts per day. He could do this by setting the price of fish equal to $1, the price of coconuts equal to _____________ and giving Robinson a daily income of $ _______
(c) Back on his island, Robinson has little else to do, so he pretends that he is running a competitive firm that produces fish and coconuts. He wonders, €œWhat would the price have to be to make me do just what I am actually doing? Let€™s assume that fish are the numeraire and have a price of $1. And let€™s pretend that I have access to a competitive labor market where I can hire as much labor as I want at some given wage. There is a constant returns to scale technology. An hour€™s labor produces one fish or 2 coconuts. At wages above $ ________ per hour, I wouldn€™t produce any fish at all, because it would cost me more than $1 to produce a fish. At wages below $ __________ per hour, I would want to produce infinitely many fish since I would make a profit on every one. So the only possible wage rate that would make me choose to produce a positive finite amount of fish is $ 1 per hour. Now what would the price of coconuts have to be to induce me to produce a positive number of coconuts. At the wage rate I just found, the cost of producing a coconut is ______. At this price and only at this price, would I be willing to produce a finite positive number of coconuts.€

Coconuts 16 Utility of 4 12 Production possibility frontier. 4 Utility of 8 12 6 Fish