Question #63174

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 prots are therefore πi(q1, q2) = P(q1, q2)qi. Suppose that

each firm 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

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 prots are therefore πi(q1, q2) = P(q1, q2)qi. Suppose that

each firm 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

Expert's answer

Two firms 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 firm i produces. Firm i's profits are therefore π i(q1, q2) = P(q1, q2)qi. Suppose that each firm 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.

(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.

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