Consider a Cournot oligopoly with three firms = 1,2,3. All firms have the same constant marginal cost c 1. The inverse demand = function of the market is given by P = 9-Q, where P is the market price, and Q = ³ is the aggregate output. i=1 qi a) Solve for the Nash equilibrium of the game including firm out- puts, market price, aggregate output, and firm profits (Hint: the NE is symmetric). (b) Now suppose these three firms play a 2-stage game. In stage 1, they produce capacities 1, 2 and 3, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices P1, P2 and P3. The marginal cost for each firm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual demand for firm i is Di (pi, p-i) = = 9 Pi - 9-Pi 3 9- Pi if pip; for all j‡ i if pip for all ji = Pj if pip; for all ji • (Note, here we assume that the efficient/parallel rationing ap- plies). Prove that it is a Nash equilibrium of the second stage subgame that each firm charges the market clearing price p* 9-91 - 92 - 93. =
Consider a Cournot oligopoly with three firms = 1,2,3. All firms have the same constant marginal cost c 1. The inverse demand = function of the market is given by P = 9-Q, where P is the market price, and Q = ³ is the aggregate output. i=1 qi a) Solve for the Nash equilibrium of the game including firm out- puts, market price, aggregate output, and firm profits (Hint: the NE is symmetric). (b) Now suppose these three firms play a 2-stage game. In stage 1, they produce capacities 1, 2 and 3, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices P1, P2 and P3. The marginal cost for each firm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual demand for firm i is Di (pi, p-i) = = 9 Pi - 9-Pi 3 9- Pi if pip; for all j‡ i if pip for all ji = Pj if pip; for all ji • (Note, here we assume that the efficient/parallel rationing ap- plies). Prove that it is a Nash equilibrium of the second stage subgame that each firm charges the market clearing price p* 9-91 - 92 - 93. =
Chapter15: Imperfect Competition
Section: Chapter Questions
Problem 15.3P
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Transcribed Image Text:Consider a Cournot oligopoly with three firms i = 1, 2, 3. All firms
1. The inverse demand
have the same constant marginal cost c =
function of the market is given by P = 9-Q, where P is the market
price, and Q =E=1 9i is the aggregate output.
13
(a) Solve for the Nash equilibrium of the game including firm out-
puts, market price, aggregate output, and firm profits (Hint: the
NE is symmetric).
(b) Now suppose these three firms play a 2-stage game. In stage
1, they produce capacities q1, q2 and 73, which are equal to the
Nash equilibrium quantities of the Cournot game characterised
by part (a). In stage 2, they simultaneously decide on their
prices p1, p2 and p3. The marginal cost for each firm to sell up
to capacity is 0. It is impossible to sell more than capacity. The
residual demand for firm i is
9 - Pi – Eiti; Ij if p; > p; for all j # i
D; (pi, p-i) =
9-Pi
3
if pi = Pj for all j + i .
if p; < p; for all j + i
9 - Pi
(Note, here we assume that the efficient/parallel rationing ap-
plies). Prove that it is a Nash equilibrium of the second stage
subgame that each firm charges the market clearing price p*
9 — q1 — 92 — 9з.
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