Question

In a market with a duopoly, if market demand is $P=300-Q$ find the Cournot Reaction curves and the Cournot quantity solutions then deduce the price in the case where Marginal cost curves for either of the duopoly firms is $Mc_{1}=q_{1}+30$ and $Mc_{2}=q_{2}+30$ . Compare your results to the case where a Monopolist that has a $MC=Q+30$ replaces the duopoly. What are the monopoly quantity and price? Which quantities are bigger, Cournot or Monopoly? What is the consumer Surplus in both cases? Set up the oligopoly model in a game-theoretical prisoner's dilemma framework. Explain briefly the strategies and how you reach the Nash Equilibrium.

Answer 1

Demand: P - 300.9,92 Marginal revenue : MR, = 300 -29,-902 MR,= 300 -9,- 292 Equilibrium: MR, = Ma . or, 300 - 29,-9, = 9,+ 30 W, 270 - 39, -Qiz = 0 ::9, - 270,772 ...19 MR2 = мс ar, 300 -9,- 292 - 92+ 30 250-q ..9, = 250-91 ..(3) solving c) and (2) by symmetry: 9,= 92 ::9,322= 20 = 67.5 :: PE 165 :. = ,= 4556' 25 For a monopolist: P 300-Q MR = 300 - 22 MC: Q + 30 E qui librium : MRMC or, 300 - 2Q = Q + 30 ar, 3a = 270 :.Q = 90 i. p = 210 :. T = 8100

Oligopoly consumer swepuus: “So x ( 30o - 16 ) x 13 5 = 425 Monopol, Consumer sween: com : 5*( 300 - 210)*90 : 4050 In game frame work, there are two Strategies for each player. Either they monopolist or cooperate and act like they defect and be duopolist. In cooperation monopoly profit is equally Shared by two firm. Let one firm defect from cooperation ; that othere will this firm assume will output continue to produce cooperative or which is half of monopoly output 9; = = 45

Then the residual demand for defect firm P = 300- 45 - qi - 255 - q's MR;= 255-298 mci = 30 + qs Equilibrium: MR j = MC j. ar, 255-29; = 30+ aj ar, 393 - 225 :: q; - 75 :: Po 180 :: Aj - 5625 ..tai - 4725 Then written as the game in Firm 2 cooperate Defect 4725,5625 4050, 4050 cooperate tep (monopony) Firmi 5625, 4725 Defect 4556' 25, 4556' 25 Coloopov In this case no firm has dominant strategy. The firm act according to other firm. Then there is two NE : {cooperate, Defect } = {44 25 5625}