Answer to Question #63174 in Microeconomics for Abdu

Question

Consider the following game of duopoly. Two rms produce the same
product at zero costs. Each of them simultaneously determines their
quantities. The demand function for the commodity is P(q1, q2) =
100 - q1 - q2, where qi, (i = 1, 2) denotes the quantity rm i produces.
Firm i's prots are therefore πi(q1, q2) = P(q1, q2)qi. Suppose that
each firm is allowed to choose from three quantities 50, 30, or 0.
(a) Represent the game in normal form/ matrix form.
(b) Are there any strictly dominated strategies for each player? Why,
or why not ?
(c) Find all the Nash Equilibria of the game

Answer 1

Two firms produce the same product at zero costs. The demand function for the commodity is P(q1, q2) = 100 - q1 - q2, where qi, (i = 1, 2) denotes the quantity firm i produces. Firm i's profits are therefore π i(q1, q2) = P(q1, q2)qi. Suppose that each firm is allowed to choose from three quantities 50, 30, or 0.
(a) Represent the game in normal form/matrix form.
Firm 1\Firm 2 q2 = 0 q2 = 30 q2 = 50
q1 = 0 0\0 0\2100 0\2500
q1 = 30 2100\0 1200\1200 600\1000
q1 = 50 2500\0 1000\600 0\0
(b) There are strictly dominated strategies for each player to produce 30 units both at price of $40 and get profits of 1200 both, because if any firm produce 30 units, it will get positive profits at any level of production of its competitor.
(c) The Nash Equilibria of the game is q1 = 30, q2 = 30, π 1 = 1200, π 2 = 1200.