Now we extend the analysis of Exercise S7 to allow for defecting in a collusive triopoly. Exercise S9 of Chapter 5 finds the Nash outcome of a VLCC triopoly of Korea, Japan, and China.
(a) Now find the collusive outcome of the triopoly. That is, what total quantity should be set by the three countries collectively in order to maximize their joint profit?
(b) Assume that under the collusive outcome found in part (a), the three countries produce equal quantities of VLCCs, so that each earns an equal share of the collusive profit. How much profit would each country earn? Compare this profit with the amount each earns in the Nash outcome.
(c) Now suppose the three countries are in a repeated relationship. Once per year, they choose production quantities, and each can observe the amount its rivals produced in the previous year. They wish to cooperate to sustain the collusive profit levels found in part (b). In any one year, one of them can defect from the agreement. If the other two countries are expected to produce their share of the collusive outcome found in parts (a) and (b), what is the best defecting quantity for the third to produce? What is the resulting profit for a defecting country when it produces the optimal defecting quantity while the other two produce their collusive quantities?
(d) Of course, the year after one country defects, both of its rivals will also defect. They will all find themselves back at the Nash outcome (permanently, if they use grim-trigger strategies). How much does the defecting country stand to gain in one year of defecting from the collusive outcome? How much will the defecting country then lose in every subsequent year from earning the Nash profit instead of the collusive profit?
(e) For what interest rates will collusion be sustainable if the three countries are using grim-trigger strategies? Is this set of interest rates larger or smaller than that found in the duopoly case discussed in Exercise S7, part (e)? Why?