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HARVARD ECON 1010A - Cournot Oligopoly

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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 pick1QAhowever, 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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