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Exercise 8.2.1

a) Assumptions in the Cournot model are
· There are two firms.
· They set quantities.
· They do so with no information about what the other firm has chosen.
b) To find all Nash equilibria, we reason like this:
· Take the point of intersection between the two reaction functions, in Figure S.8.1 labeled A. That point must be a Nash equilibrium since:
o Firm 1 cannot get a better outcome by unilaterally changing its
* *

strategy. Given that firm 2 has chosen the quantity q2 , q1
optimal choice according to the reaction function.
o The same is true for firm 2.

is the

· Then, take any other choice of quantity. For instance, q2’ for firm 2. No such
point can be a Nash equilibrium since:
o Firm 1’s optimal response can be read off from the reaction function. In this case at point B, i.e. the quantity q1’. Note that, if firm 1 would have chosen any other quantity it would have been able to improve its choice by moving to the quantity q1’. So none of those points can be part of a Nash equilibrium.

o Since firm 2’s reaction function does not intersect firm 1’s in point B, firm 2’s choice of quantity cannot be optimal given firm 1’s choice. Consequently, firm 2 can improve on its situation by unilaterally changing its strategy. Therefore, point B cannot be a Nash equilibrium.
o The same argument can be used for every point in the graph, except for point A, which consequently is the only Nash equilibrium.

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