Duopolies

Duopolies

The Cournot duopoly a. Description and Assumptionsof the model b. The Cournot-Nash equilibrium c. Graphicalanalysis d. Normal gameRepresentation e. Numerical Application Conclusions

a. Description and Assumptionsof the model • H1 Large number of buyers • P = P(Q) = P (q1 + q2) • H2 the strategic variable is the output • H3 the good isperfectlyhomogeneous • H4 Eachfirmwants to maximizeherprofit by anticipating the behavior of the otherfirm.

a. Description and Assumptionsof the model Maximizationproblem • Firm’sobjective : Maximizing profit by choosing the optimal quantity to supply on the market • The twofirmschoosethe quantity non cooperatively Firm 1 max program is: Firm 2 max program is:

b. The Cournot-Nash equilibrium To maximizeherprofit, eachfirmconsiders the production of the other as given The equilibriumneeds to verify the following conditions:

b. The Cournot-Nash equilibrium • The first order condition of the firmrepresentsheroptimal output according to heranticipation of the output of firm 2 • We call it the reactionfunction of firm 1 • We have one reactionfunction for eachfirm

b. The Cournot-Nash equilibrium • Thuswe have:

b. The Cournot-Nash equilibrium • Nowthatwe have the reactionfunction for eachfirmwewant to determinetheir optimal quantity • To find the equilibriumweneed to replace, in the reactionfunction of firm 1, q2 by the reactionfunction of firm 2 whichdepends on q1 :

b. The Cournot-Nash equilibrium • This givesus an optimal q1* thatwe replace in the reactionfunction of firm 2:

c. Graphicalanalysis The cournot Equilibrium Q1 Firm 2 reactionfunction Cournot Equilibrium Firm 1 reactionfunction Q2 0

d. The Cournot Game Firm2 Firm 1

e. Numerical Application • The Cournot competitionisusuallyfoundin agricultural market, where the land is first exploited and thenyouwillsellyour production. Thusitisdifficult to adjustquantityafterwards. • Twofirms are producing the same good, for instance an apple. • 1st case : Collusion • First suppose that the twofirmsdecide to collude. • Theywillmaximizetheir joint profits

e. Numerical Application • Q = q1 + q2 • D Such as P = 1 – 0,001 Q, C(q) = 0,28 q •  = (1 – 0,001 Q)Q – 2 C (Q/2) où qi = Q/2 • Max Such as  QM = , PM = , and =

e. Numerical Application Nowwe assume that the twofirms have leftthe cartel and decidedto selltheirproductindependently, withoutagreeing on the optimal level of production.

e. Numerical Application • Maximization programs of eachfirm changes • Recall D Such as P = 1 – 0,001 Q, C(q) = 0,28 q • Max program for firm 1 is : • Max program for firm 2 is :

e. Numerical Application Land’s calculate the reaction function for the two firms :

e. Numerical Application • Weneed to solve :

Conclusion • Result Pd <PM, qd>qMand d < M

Conclusions • Cournot duopoly efficacy • Welfare perspective: • The welfareisincreasingwith the number of producers • Cournot oligopolyis best than a monopoly • Firms perspective: • It is best to be in monopoly than in Cournot duopoly • But companditionis lower in a Cournot duopoly than in Bertrand duopoly=> to see

The Bertrand Monopoly a. Description and Assumptionsof the model b. The Bertrand-Nash equilibrium c. Graphicalanalysis d. Numericalexample e. Bertrand Paradox in real life Conclusions

a. Description and Assumptionsof the model • H1 No capacityconstraint for the firms • H2 the strategic variable is the price • H3 The productisperfectlyhomogeneous • H4 Eachfirmwants to maximizeher profit

a. Description and Assumptionsof the model H5 The demanddependsonly on the productprice and the consumer isperfectlyinformed • The firmi fixe herpriceat Pi. Whatis the demand for firmj ?

b. The Bertrand-Nash equilibrium Bertrand theorem Under the Assumptionsfrom 1 to 5, itexistsonly one price P1* = P2* = MC Proof : Assume that the twofirmssell the same good, wecan have the following cases: P1 > P2 > Cm P1 = P2 > Cm P1 > P2 = Cm P1 = P2 = Cm

b. The Bertrand-Nash equilibrium P1 > P2 > MC => Demand for firm 1 is 0 and firm 2 provides the wholeDemand P1 = P2 >MC => D for firm 1 = D for firm 2 P1 > P2 = MC =>if firm 2 decreasesitsprice, Demand for firm 1 becomes 0 and firm 2 provides the wholeDemand P1 = P2 = MC => D for firm 1 = D for firm 2

b. The Bertrand-Nash equilibrium The Bertrand Paradox Even if the firms are onlytwo, theyact as if theywere an infinitenumber. Theyact as if theywere in a competitivemarket. The Bertrand equilibriumisalso a Nash equilibrium

c. Graphicalanalysis p2 p1=R1(p2) p2=R2(p1) 0,28 0,28 p1

d. Numericalexample • Q = q1 + q2 D Suchthat P = 1 – 0,001 Q, C(q) = 0,28 q

e. Bertrand Paradox in real life • Capacity constraints. Sometimes firms do not have enough capacity to satisfy all demand.. • Product differentiation. If products of different firms are differentiated, then consumers may not switch completely to the product with lower price.

e. Bertrand Paradox in real life • Dynamic competition. Repeated interaction or repeated price competition can lead to the price above MC in equilibrium. • Perfect information. The consumers know perfectly the price of the otherproduct.

Conclusion • Bertrand equilibriumis optimal • From the consumer perspective, the Bertrand equilibirumis the best. • But the Bertrand Results relies on very strict assumptions.

Cartel Stability • Game theory • Non-cooperativegame and simultaneous • Collusion canberepresented as a prisonerdilemmagame

The prisonerdilemma • Two men are arrestedaftercommitted a crime • The police know thatthey have committed the crime but have no evidence about it • The police deal: =>The one whoconfesswillwalk free whereas the one whodoes not talk will go to prison.

MarketStrategy • The twoprisoners are interrogated in differentrooms • Theycanconfess or deny • If the twodeny, theybothwalk free (gain of 5) • If one confess and one denies • The one whoconfesswalks free (gain of 10), • The other has to go to prison for a long time (loss of 5) • If theybothconfess: • They are bothcondamned but withsomeclemence (loss of 2)

prisoner 2 Confess Does not confess Confess prisoner 1 Does not confess

NumericalApplication Firm 2 Produces 240 Produces 180 Produces240 Firm 1 Produces 180

Sustain the collusion Solution withrepeatedgame • The temptation to cheat • 2 firms in a cartel • They have the choicebetween 2 strategies: • Sustain the collusion • Cheat • Simultaneousgame in one shot • Complete but imperfect information

Sustain collusion, the temptation to cheat and Avec ; and Firme 2 Firme 1

Nash Equilibrium: (Cheat, Cheat) Firm2 Firm1

Sustain the Collusion • Sustainthe collusion: The folk theorem, the punitive strategy • In a cartel, we have to think in a dynamicway • Folk theorem « The folk theorem says that, in the infinitely repeated version of the game, provided players are sufficiently patient, there is a Nash equilibrium such that both players cooperate on the equilibrium path”

Sustain the Collusion One way to sustaincooperation in the cartel is for the firm to threateneachother of sanction/punitive actions in case of defection. For instance a pricewar. In a context of uncertainty about the time of the game, i.e. the life of the market, firms have to balance between the gain of an immediateunilateraldefection over one period and the punitive outcome for the nextperiod. If the punitive action is able to sustain the cartel, the question is to determineits size.

Sustain collusion • Example: « tit for tat » strategy « I cooperate as long as youcooperate, however if youdefect/cheat once, I willcheattoo but for the end of time » • Is thisstrategy a crediblethreat?

Sustain the Collusion • Comparison of the benefits and the cost of the two alternatives cheat or cooperate. • If one of the twofirmscheat, the punition should by applied . Eachfirmwillthenreceive for eachperiod a profit oft-. • This gain isinferior the one of the cartel c+ • One firmshoulddecidewhaether or not she continues to produce the optimal cartel quantity or to increaseits output • If the firmproduced more, shewouldmake a profit of t+ , with t+ > c+

Sustain the Collusion • net present value of alwayscooperatinggives a profit for the firm of : • If the firmcheats, shereceive one time a super profit a defectiont+, but afterwards, shewillreceiveonly the profit of the punitive action (pricewar):

Sustain the Collusion • When the net present value of the cartel willbesuperior to the net present value of the defection ? • When • Whichmeans

MonopolisticCompandition(E.Chamberlin, 1933) • Assumptions: • H1 A large number of firms • H2 Imperfectlysusbstitutablegoods • H3 Decreasingdemand • H4 Fixedcostssufficientlysmallsothat entry is possible but thateachvariandyisproduced by one firm.

Duopolies

Duopolies

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