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Handout 11: Cournot OligopolyPreviously, we have studied markets perfectly competitive markets with a large numberof rms, and markets with a monopolist. In this handout, we study the Cournot model ofoligopoly, where two rms compete in a market. This will illustrate three broader topics:1. Residual demand2. the rm's strategic reaction curve3. Nash equilibriumThe Cournot ModelIn this model of oligopoly, two rmsAandBproduce the same good. Each rm takesas given overall market demand as well as quantities produced by other rms. It choosesits own production quantity.We will use one numerical example throughout. Market demand for the good is given by:P = 200 − Q, Q = QA+ QBwhereQis the total quantity sold in the market, andQAandQBrespectively are thequantities produced by rmAandB.The marginal cost and xed cost of the two rms areMCA= 100, F CA= 0MCB= 100, F CB= 0Both rms are identical, so we can consider things from rmA's point of view (and rmBwill behave symmetrically).A Quick RefresherSuppse rmAbelieves that rmBwill produce nothing, i.e.QB= 0.FirmA0sproblemis now as if it was the only rm, which we have studied as the case of a monopolist.In this case, whas is rmA0soptimal strategy? FirmA :•faces demand curveP = 200 − Qwith marginal revenueMR = 200 − 2Q.•should setMC = M R.•solving:100 = 200 − 2Q∗A,orQ∗A= 50.1Two Firms ActiveSuppose rmAbelieves that rmBwill produceQB= 50. Now, rmAcares about itsresidual demand curve, the demand left for rmA's product once rmB's productionhas been accounted for. If rmBis producingQB= 50, rmA'sresidual demandisP = 200 − 50 − QA= 150 − QAwhich we use to derive its marginal revenue150 − 2QA.What is rmA's optimal production quantity in this case?The Reaction Curve?However, rmAdoes not know how much rmBplans to produce. Instead, given anyvalue ofQB, we can calculate rmA's optimal production quantity. We derive rmA'soptimal strategy as a function of rmB's strategy this way. This function is rmA'sreaction curve.If rmBproducesQB, then rmA's residual demand isP = (200 − QB) − QAwith marginal revenue(200 − QB) − 2QA.To nd the optimal quantity, we setMR =MC :(200 − QB) − 2Q∗A= 100Solving:Q∗A(QB) =200 − QB2.Q∗A(QB)is rmA's reaction curve: it summarises rmA's best strategy given opponentstrategyQB.Nash EquilibriumWe have analysed rmA's best strategy, taking rmB's choices as given. We can alsodo the same for rmB, deriving its reaction curveQ∗B(QA) =200 − QA2.2Notice rmB's reaction curve takes as given rmA's quantity.The two reaction curves are graphed below in red (for rmA) and blue (for rmB).QBQAQ∗A(QB)Q∗B(QA)Q∗BQ∗A50100100501QB2QB1QA2QAHow do we determine what actually happens when these two rms interact?•what happens if we suppose rmBchooses to produce1QB?then, reading the red line, rmA's best strategy is to pick1QAhowever, given rmAchoosing1QA, rmB's best response is2QB.if we suppose rmBupdates its strategy from1QBto2QB, then rmA'soptimal strategy is2QA.what happens if we follow this line of reasoning?•what happens if we suppose rmAchooses to produce atQ∗A?The Nash equilibrium(Q∗A, Q∗B)is stable in two senses:•givenQA= Q∗A,B's best choice isQ∗B; and givenQB= Q∗B,A's best choice isQ∗A that is, neither rm wants to change strategies taking the other rm's strategyas given•if we consider a point that is not a Nash equilibrium and allow rms to adaptivelyupdate strategies, we draw closer and closer to the Nash

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