. Cournot game with n firms facing asymmetric costs. Consider an industry
of n ≥ 2 firms competing a la Cournot. Firms face an inverse demand
curve p(Q) = a − Q, where Q ≥ 0 denotes aggregate output. Every firm i
has a marginal cost of production ci ≥ 0. Assume c1 ≤ · · · ≤ cn and that
demand is large enough, i.e. a > (n + 1)cn − P
i ci.
(a) Set up firm i’s profit-maximization problem and find its first-order.
(b) Find a Nash equilibrium.
(c) Assume ci = c for all i. Derive a Nash equilibrium using the previous
analysis and check if it matches what we have derived in TB1.
(d) An implicit assumption for questions from (a) to (c) is that the price
implied by the inverse demand curve is allowed to be negative. That
is, for Q > a, we allow p(Q) < 0. Sometimes, the exact same phrase
is interpreted to have assumed that the price will be bound at 0 once
Q > a, or equivalently, max(a − Q, 0). (We will be explicit in our
exams!) Write down the profit-maximization problem and find Nash
equilibrium under otherwise the same set of assumptions.
(e) Consider a setting with n ≥ 2 firms facing inverse demand function
p(Q) = 1 − Q, and symmetric marginal production cost c, where
1 > c ≥ 0. When k firms merge, the merged firm act as one firm and
enjoys a lower marginal cost c−x, while the n−k unmerged firms still
face marginal cost c. Find the aggregate output in equilibrium when
k firms merge, and compare it against aggregate output before the
merger. For which parameter values the merger produces an increase
in aggregate output?