4. [21] Robinson Crusoe has recently become enlightened. He has learned that he can rest while Friday works and still have Friday work as hard as when he was supervised- All that Robbie needs to do is to pay Friday a wage that depends upon the coconut ha nrest. Suppose that if Friday works hard, then he will get 25 coconuts per day with probability .8 but only 10 coconuts per day with probability .2. If he shirks, then he will get 25 coconuts per day with probability .2 and 10 coconuts per day with probability .8. Friday derives utility from the coconuts. Robbie will pay Friday in coconuts. Friday's utility is the square root ofthe number of coconuts that he consumes. So, if he receives 1 coconut, then his utility is 1, if he receives 4 coconuts, then his utility is 2, and so on. Working hard is equivalent to the utility cost of 1 util. Shirking does not cost him any utility. It Friday does not work for Robbie, then his resulting utility level would be 2. a) Write down the technology for Robbie. This is the relationship between Friday's effort and the outcomes in umber of coconuts harvested}. 2 points. b} Use your answer in a} to write down the two constraints that a contract must satisfy to induce Friday to work hard. 4 points. c) Show that if Robbie pays Friday a flat wage of 9 coconuts, Friday will shirk. What is the optimal contract that Robbie offers Friday in order to induce him to shirk? 3 points. d} Show that the contract that pays Friday a wage of (2)2 = :3 coconuts ifthe harvest is 10 and a wage of(?)2 = g if the harvest 25, is optimal to induce him to work hard. 4 points. e} What are Robbie's expected cost and profit ofthe contract in cl]? Does Robbie induce Friday to work hard orto shirk? 3 points. 1') Suppose thatthe island has had a bountiful coconut season. Instead of getting 25 and 10 coconuts with probabilities .2 and .8 if he shirks, Friday gets those coconut harvests with probabilities .2 +.[15 x and .8 - .05 x, respectively, where x is the last digit of your student ID number (if it is zero. then set x = 10]. Will the contract in distill induce Friday to choose the same level ofeffort? Explain. What will happen to the expected cost for Robbie if he wants to induce Friday to work hard? 5 points