III. (Repeated Quantity Competition) This problem extends the setting of Problem II to analyze the repeated game. Suppose that both Xena and Yanni have a weekly discount factor of 5 = 0.9. (10 points) 1) 2) Consider the stage game in which Xena and Yanni can either collude (C) or defect (D), as in Problem II. Draw a payoff matrix of the stage game. Suppose, as law-abiding citizens. Xena and Yanni compete in the market according to Cournot competition every week. what is the present discounted value of their prots? How nmch more (in present discounted value terms) could Xena and Yanni each make if they colluded every period? Discuss whether collusion can be supported in the repeated game. Suppose that their weekly discount factor is now (I : 0.5. Discuss whether collu- sion can be supported in the repeated game. II. (Quantity Competition and Collusion) At a weekly farmer's market, two farmers, Xena and Yarmi, are all selling tomatoes. Demand for tomatoes is P = 120 Q. The farmers decide how many tomatoes to bring to the market. Neither farmer knows the other's quantity before arriving at the market, but both participate every week. Xena and Yarmi are both very talented farmers. so their marginal cost of producing tomatoes is zero. Moreover, assume that tomatoes are a divisible good. (10 points) 1) Suppose, as law-abiding citizens. Xena and Yanni compete in the market according to Cournot competition. Find the best-response functions for Xena and Yanni. What quantities Xena and Yanni bring to market each week in Cournot compe- tition? What are Xena and Yanni's prots per period in this case. Suppose Xena and Yarmi agree to x the price of tomatoes by colluding on how many tomatoes to bring to market. If they agree to split prots equally and both stick to the agreement. how many tomatoes does each bring to market? What are their total prots? Now suppose they make the agreement, while Yanni sticks to the agreement Xena \"defects\" and instead brings maximizes her own prots for the week. knowing that Yanni is bringing her \"collusive\" amount from Part 4. How many tomatoes Xena brings? What are Xena's profits in this case? Is sticking to the agreement an equilibrium of the stage game? Explain wily or why not