Question

Game theoretic approach toward analyzing output behavior of rivals

Firms X and Y are duopolists facing the same two strategy choices. They can either tacitly collude or they can compete in a Cournot fashion.   The market demand for their product, as well as their respective cost curves are presented below.

P = 100 - Q                                         (Market demand), where Q = q_x + q_y

C(q_x) =C(q_y) = 50q_j                            (Firm X and Y’s total cost curves), where j=x or y.

and MC(q_x) =MC(q_y) = 50                 (Firm X and Y’s marginal cost curves)

a.) Calculate the respective output levels of each firm if they collude to set monopoly prices.

b.) Calculate the respective output levels of each firm if they adhere to the Cournot model.

c.) What are the four possible output combinations available in this game?

d.) Derive the four possible profit outcomes for each firm that arises from producing the four possible output combinations available in this game.

e.) Use these profit outcomes to construct a 2x2 normal representative matrix for this game.

f.) Does either firm have a dominant strategy? If so, what is it?

g.) Is there a Nash equilibrium for this game? If so, what is it?

h.) Is the outcome of this game an example of the prisoners’ dilemma? Explain.

Answer 1

E) matrix

Producing half of monopoly output : Cooperation

Cournot game : cheating

X/Y	monopoly (Cooperate)	cournot (deviate)
Monopoly output (Cooperate)	(312.5, 312.5)	(234.375, 351.5625•)
Cournot output (deviate)	(351.5625*, 234.375)	(277.78*, 277.78•)

F) now

Each firm (both X & Y) deviate from Cooperation payoff

each firm has higher payoff from deviation, for any choice of other firm

so both have dominant stategy to produce Cournot output level

G) NE :

( Both produce Cournot output level )

H) yes, it shows Prisoners dilemma

Bcoz at eqm, both firm are worse off,

Bcoz if both Cooperate, then both earn higher payoff, by joining together & sustaining collusion