6OLIGOPOLYTraditional trade theory has been developed in models of per-fect competition in which Þrms lack market po wer and do notact strategically. Models of monopolis tic com petition allowÞrm s to hav e a limited degree of market power but rule outstrategic interaction by assumption — each Þrm perceives itselftoo small to effect the prices of other Þrms. Oligopoly theoryadds a second layer of complicatio n by allowing Þrms to actstrategically. Oligopolistic Þrms recognize that their decisionaffects the decisions of their rivals and they reco gn ize th at th e irrivals recognize this and they recogn ize that their rivals recog-nize that they recognize this and so on (common knowledge).Our goal is to examine trade under these conditions, but Þrstwe develop some basic oligopoly models.Cournot M odel (background)Consider an industry comprised of n Þrms, each of which pro-duces a homogen ous good . Demand function facing the indus-try is given by p (Q) where Q =Piqiis total industry output.Each Þrmschoosesitsownoutputqi, taking the output of allits rivals Q−i(n − 1 v ector) as giv en, to maximize its proÞtsπi= p(Q)qi− c(qi)where c(·) is the cost function for Þrm i. Firms recognize thatthey should account for the output decisions of their rivals,yet when making their own decision, they view their rivals’outputs as Þxed. Each Þrmviewsitselfasamonopolistontheresidual demand curve — the de mand left over after s u b tractingthe output of its rivals. The output vector (q1...qn)isaCournotNash equilibrium iff (given Q−i)πi(qi,Q−i) ≥ πi(q0i,Q−i)OLIGOPOLY 1for all i. In a Cournot-Nash equilibrium, Þrms do not ha ve anincentive to unilaterally deviate b y altering their output levels:the c h os en quantity maximizes the proÞts of eac h Þrm give nthe quan tities chosen by the other Þrms.The Þrst order condition (FOC ) for Þrm i is given by∂πi∂qi= p0(Q)qi+ p(Q) − c0i(qi)=0 (6.1)The Cournot Nash equilibrium is found by simultaneously solv-ing the Þrst order conditions for all n Þrms. Let si≡ qi/Q de-note Þrm i’s share of industry output in equilibrium. Rewritethe FO C asp(Q)"1+dp/pdQ/QqiQ#= c0i(qi)and rewrite again asp(Q)"1 −siε(Q)#= c0i(Q) (6.2)Suppose marginal cost is constant. The above condition al-lo w s som e interesting comparis on s. For a monopolist si=1.For the monopolist, the bracketed term is smaller than for aCournot Þrm , and thus price is hig her and output is lower thanthe Cournot equilibrium total output. For a perfectly compet-itive Þrm ε(Q) is inÞn it e. For a perfectly competitive Þrm, thebracke te d term equals exactly 1, so price equals marginal cost,and price is lo wer and output is higher than in the Cournotequilibrium. Thus, Cournot industry output must fall in be-tween monopoly and perf ectly competitive output. Addin g to-gether the FO C s for all Þrms givesnp(Q)+p0(Q)Q =XiciTh us, Q depends only upon the sum of the marginal costs ofproduction and not upon their distribution. In a symmetricequilibrium, Þrms have equ a l market shares si=1/n so thatp(Q)"1 −1nε(Q)#= cTo keep exposition sim ple, we continue with constan t marginalcost and two Þrms.2 OLIGOPOLYCournotDuopolyThe FOC for Þrm 1 isπ11≡∂π1∂q1= p0(Q)q1+ p(Q) − c =0where Q ≡ q1+ q2.TheaboveFOCdeÞnes Þrm 1’s reactionfunction (R1): the set of best responses for different outputlevels of its rivals. We can trace out R1simply by varying q2.Let q1(q2) denote Þrm 1’s reaction function. According to theFOCπ11(q1(q2),q2) ≡ 0Total differentiation givesπ111dq1dq2+ π112=0Solving gives the slope of Þrm 1’s reaction functiondq1dq2= −π112π111For the second order condition to hold, must haveπ111=2p0(Q)+p00(Q) q1≤ 0Thus, the sign of the slope is the same as the sign of π112.Ifp00≤ 0, th e second order condition is automatically satisÞed.Wh en this cross partial is negative π112≤ 0,wesaythatq1and q2are strategic substitutes — an increase in the outputof Þrm 2 lowers the marginal proÞtability of Þrm 1. Clearly,when the outputs of t wo Þrm s are strategic substitutes , thereaction functions will be do wnw ard sloping. The Cournot as-sumption by it self does not mean that the reaction functionsare do wnward sloping. From the FO Cπ112= p0+ p00q1> 2p0+ p00q1= π111Thus, the assumption of strategic substitutes π112≤ 0 imp liesthe second ord er condition. When π112≥ 0,thenq1and q2arestrategic complements.The in tersection of the two reaction functions gives the equi-librium. The comparative stat ics properties of the Cournotequilibriu m depend upon the sign of a particular condition(the stability condition). Suppose some parameter θ effectsthe proÞt functions of both Þrms.TheFOCcanbewrittenasa function of this parame ter.π11(q1(θ),q2(θ); θ) ≡ 0OLIGOPOLY 3Total differentiation of both FOCs yield sπ111dq1dθ+ π112dq2dθ+ π11θ=0whereπ11θ≡∂2π1∂q1∂θand similarlyπ221dq1dθ+ π222dq2dθ+ π22θ=0The above equations can be written as follo ws"π111π112π221π222#|{z }A"dq1∂θdq2∂θ#|{z }x="−π11θ−π22θ#|{z }bWe can use Cr amer’s rule to solve for the elements of x.1∂q1∂θ=¯¯¯¯¯−π11θπ112−π22θπ222¯¯¯¯¯|A|where |A| = π111π222− π112π221> 0 by assumption (stabilitycondition).2Thus the sign ofdq1dθisthesameasthesignofπ222π11θ− π112π22θ. In the special ca se where θ only directly ef-fects the proÞt function of Þrm 1, π22θ=0so that only the signof π11θwill matter. Thus, in this case, w e can differentiate onlythe FOC of Þrm 1 to determine the sign ofdq1dθ(the en velopetheorem). Recall that π222≤ 0 by the SOC for Þrm 2.The Cournot equilibrium is not P areto optim al from theviewpoin t of the t wo Þrms, as seen fro m the isoproÞtcontoursof the t wo Þrms. IsoproÞt con tours trace out combinations ofq1and q2that yield the same lev e l of proÞts for a Þrm.π1(q1(q2),q2) ≡ πTotally differentiating the above iden tityπ11dq1dq2+ π12=01Cramer’s rule: replace the ith column of A by vector b to form thematric Ai;thesolutiontoAx = b is then xi=|Ai||A|.2This condition ensures that the reaction functions in tersect in the rightw ay — if the equilibrium were slightly perturbed, outputs would convergeback to the Cournot equilibrium.4 OLIGOPOLYdetermines the shape of the isoproÞtcontoursforeachÞrmdq1dq2= −π12π11The denominator is negative since the products are substit utesfor one another π11< 0