95 reaction functions in the previous example in a

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9–5Reaction functionsIn the previous example, in a one-off game eachplayer had a dominant strategy, to produce high out-put whatever its rival did. This led to a poor outcomefor both players, because they were not co-operatingdespite being interdependent. When the game isrepeated, commitments and punishment strategieshelp players co-operate to find an outcome that isbetter for both of them.In punishing a rival, a player’s actions change inresponse to bad behaviour by the rival. Dominantstrategies are rare. More usually, each player’s bestaction depends on the actual or expected actions ofother players. How a player reacts depend on what itassumes about its rivals’ behaviour. For simplicity weanalyse duopolyin which there are only two players.Cournot behaviourIn 1838 French economist Augustin Cournotanalysed a simple model of duopoly.Imagine a duopoly in which both firms have the sameconstant marginal costs MC. Figure 9–6 draws thedecision problem for firm A. If firm A assumes thatfirm B produces 0, firm A gets the whole industrydemand curve D0. This shows what output firm A can sell given the prices that it charges. From this,firm A calculates the marginal revenue MR0, and produces Q0to equate its marginal cost and marginalrevenue.If instead firm A assumes that firm B makes 3 units,firm A faces a demand curve D3obtained by shiftingthe market demand D0to the left by 3 units. Firm Bgets 3 units and the residual demand is available forfirm A. For this demand curve D3, firm A computesthe marginal revenue curve MR3, and chooses outputQ3to equate marginal cost and marginal revenue.Similarly, if firm A expects firm B to make 5 units,firm A shifts D0to the left by 5 units to get D5, andproducesQ5in order to equate marginal cost and itsmarginal revenue MR5. The larger the output thatfirm 2 is expected to make and sell, the smaller is theoptimal output of firm A. Q5is smaller than Q3whichis smaller than Q0.In the Cournot model, each firm treats the outputof the other firm as given.By repeating this exercise for every possible belief thatfirm A has about the output of firm B, yields the reac-tion function of firm AIn the Cournot model, a rival’s action is its outputchoice. Figure 9–7 shows the two outputs QAand QB.From Figure 9–6, firm A makes less the more it thinksthat firm B will make. In Figure 9–7 firm A’s optimaloutput choice is the reaction function RA. If firm B isexpected to produce 1 unit less, firm A chooses toraise output by less than 1 unit. This ensures totaloutput falls, raising the price. Because this lets firm Aearn more on its previous output units, it is notworth raising its output by as much as it expects theoutput of B to fall. Equivalently, in Figure 9–6 firmA’s demand curve shifts more than its marginal revenue curve, hence its output rise is smaller thanthe conjectured fall in the output of firm B.

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