Now we extend the analysis of Exercise S7 to allow for defecting in a collusive triopoly. Exercise..

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.

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Now we extend the analysis of Exercise S7 to allow for defecting in a collusive triopoly. Exercise..
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(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?