# 1 Cournot Oligopoly with n firms

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1 BEE07, Microeconomics, Dieter Balkenborg Cournot Oligopoly with n firms firmi soutput: q i totaloutput: q=q +q + +q n opponent soutput: q i =q q i =Σ j i q i constantmarginalcostsoffirmi: c i inverse demand function: p(q firmi sprofit: Π i (q i,q i =p(q q i c i q i =(p(q i +q i c i q i FOCforprofitmaximumgivenq i : = p q i +p c i =0 q i q i Solutiondefinesreactioncurve q i =r i (q i whichisoftendecreasing inq i. Linearcase: p=a Bq=A B(q i +q i p q FOC: p q i = B Bq i +(A B(q i +q i c i = 0 Bq i = A c i Bq i Reactionfunction q i =r i (q i = A c B q i

2 0 q = r (q 6 q Cournot- Nash q = r (q 0 6 q Cournot-Nash equilibrium:. Everyfirmmaximizesprofitgivenherexpectationofq i.. The expectation is correct. This yields the simultaneous system of equations q i =r i (q i foralli=,...,n. InthelinearcasetheFOCyields,sinceq i +q i =q Summation yields Bq +(A Bq c = 0 Bq +(A Bq c = 0 Bq n +(A Bq c n = 0 Bq+n(A Bq n c=0 where c= c +c + +c n n is the average marginal cost in the market. Thuswecandeducethetotalquantityproducedandthepriceinthemarket (n+bq = n(a c n A c q = n+ B p = A Bq= n+ A+ n n+ c cforn.

3 Each firm produces in the n-firm oligopoly q n i = A Bq c i B = A c i B n A c n+ B = n+ A +n( c c i c i B (n+b. Letusnow,forsimplicity,assumethatfirmshaveidenticalmarginalcostsc i = c=c. Then p = n+ A+ n n+ c casn qi n A c = n+ B 0asn ( Π n i = (p cqi n = n+ A+ n n+ c c n+ nπ n i = n (A c (n+ B 0asn A c B = (A c (n+ B The total profit in the industry decreases with every additional firm entering the market since for all n> (n Π n i >nπ n i n (n > n (n+ (n (n+ >n ( n (n+>n n n+n >n n >n whichistruesincen >nforalln>. In particular, it always pays for the firms to form a cartel and share the monopolist profit since nπ n i <Π i. Stackelberg Equilibrium Twofirmswithmarginalcosts. Differenttiming: Firmmovesfirst, firmobservesthemoveand then adapts. Ifarationalfirmobservesthequantityq itwillchoosethequantity Total output is andthepricewillbe q =r (q = A c B q q +q = A c B + q p=a B(q +q =A A c Anticipating this, firm expects to make the profit ( A+c Bq Π (q,r (q = B q = A+c Bq c q = A c Bq q which is maximized for q = A c B

4 yielding the price and the profit Firm produces p= A+c BA c B Π = (A c B = A c q = A c B q = A c B and makes the profit Π = Π = (A c 6 B NoticethatthiswouldnotbeaNashequilibriumiffirmcouldnotobservethequantitychoicebecause firm reacts optimally while firm should produce q =r (q = A c B q = A c B A c B = A c B Totalquantitywouldbe 5 A c B andthethepricewouldreduceto and yield the profit p=a 5 ( ( A+5c Π = c (A c= A+5c A c = 9 (A c B > (A c B B The leader produces in the Stackelberg equilibrium twice as much than the follower and makes twice the profit. IntheCournotduopolythepayoffΠ i = (A c 9 B whichisinbetweentheprofitoftheleaderand the follower. Bertrand competition with differentiated products The two firms have the demand functions Q = 00 P +P Q = 00 P +P andconstantmarginalcostsc=5. Theprofitfunctionforfirmiis Π i (p,p =(P i cq i =(P i 5(00 P i +P j wherej= i. Thefirstorderconditionforaprofitoptimum(takingtheotherfirm spriceasgivenis =(+ (00 P i +P j +(P i 5 ( =0 P i +P j =0, i=, ThesolutiontothissystemofequationsisP =P = 0 =6 0. Eachfirmproduces =7 units andmakestheprofit7 6 6 ismade. Togethertheymaketheprofit576. Iftheywould form a cartel they could make the profit Π (p,p +Π (p,p. Maximizing joint profit leads to the two first order conditions (Π +Π =0 P i +P j +(P i 5=05 P i +P j =0,i=, whichhavethesolutionp =P =5.5. Ofeachcommodity57.5unitsareproducedandthetotalprofit is ( 7 ( 57 =56.5,whichisobviouslyhigherthanincompetition.

5 Bertrand competition with perfect complements. Two price-setting firms produce with constant marginal costs c = produce goods which are perfect complements. Consumers therefore buy equal amounts from both firms. The total amount they by of each commodity is Q=Q(P,P =5 (P +P Theprofitoffirmi=ori=is The first-order condition for a profit maximum is Π i (P,P =(P i Q=(P i (5 (P +P =5 (P +P (P i = P i P j =0 where j = i. By symmetry, P =P inequilibrium, so P i = or P =P =6. It follows that Q=5 =pairsaresoldattheprice6. Eachfirmmakestheprofit(6 =9andthetotal profitintheindustryis. IfamonopolisttakesoverbothplantsandtakesthepriceP perpairhisprofitis Π(P=(P 6(5 P whichismaximizedforp = 5+6 =0.5where5 0.5=.5pairsaredemanded. Consumersurplus is up in the monopoly because they get more at a lower price. Producer surplus goes up because the monopolist sprofitis.5 =0.5>. 5

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