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Microeconomics 2 John Hey The remaining lectures • • • • • • • • • We have three weeks left this term: This week chapters 31 and 32. Next week chapters 33 and 34. Final week? I propose looking at the second half of the first specimen paper, and general revision (questions in advance please). Next term we have two slots: 13.15 Tuesday the 30th of April and 13.15 Tuesday the 7th of May. I propose to use them for the second specimen paper... ... and for telling you in what way I have strengthened the examination for this year. OK? Comments? Game Theory • In Chapter 29 we talked about games, in which two players have to choose the value of some decision variables, the values of which affect both players. • We introduced the idea of a Nash Equilibrium in which each is optimising given the decision of the other. • We had our doubts about NE in general but in some cases it seems reasonable. • Do note that the NE depends heavily on the ‘rules of the game’, particularly about sequentiality and repetition. • This week’s tutorial reinforces these results. • Today we will apply game theory in Duopoly. Duopoly • We deliberately keep things simple, but we do not lose anything of interest. • Consider a market in which there are two identical sellers – two firms – Firm 1 and Firm 2, selling an identical good. • Suppose the demand curve in the market is given by: • p = a – b(q1 + q2) The Cournot Model • Each firm chooses independently its output. • The price is determined by the demand curve (recall that their output is identical). • What outputs do the firms choose? • Nash: each firm choosing profitmaximising output given the output of the other firm. Profits • Let us denote the profit of firm 1 by π1. • Suppose that its total cost function is given by: C(q1) = cq1 (Note that this is linear and so constant returns to scale.) • Hence its profits are given by: • π1 = pq1 - cq1 = [a – b(q1 + q2)]q1 - cq1 • Assuming the same cost function for firm 2 its profits are given by • π2 = pq2 – cq2 = [a – b(q1 + q2)]q2 – cq2 Isoprofit Curves • • • • • For firm 1 an isoprofit curve is given by: π1.= constant Hence [a – b(q1 + q2)]q1 - cq1 = constant Note that this is quadratic in q1 for given q2. • • • • • For firm 2 an isoprofit curve is given by: π2.= constant Hence [a – b(q1 + q2)]q2 – cq2 = constant Note that this is quadratic in q2 for given q1. Parameters in Maple file • C(q) = 10q (Note linear so Constant Returns to Scale). • and hence the Marginal Cost is 10. • Demand is p = 110 – q. • So a = 110, b = 1 and c = 10. • Hence Revenue = pq = 110q – q2 • and so Marginal Revenue = 110 – 2q. • Hence the monopoly output (given by MR=MC) is 50, and the monopoly price is 60. • Competitive output (given by Price=MC) is 100, and the competitive price) is10. Reaction Curve (showing how a firm should optimally react to the decision of the other) • If Firm 1 chooses its output to maximise its profits given a level of output of Firm 2, we get: • q1 = (a-c-bq2 )/2b • …the reaction curve of Firm 1. Note that its slope is negative. (We will see this graphically in Maple.) • (To find, differentiate π1 = [a – b(q1 + q2)]q1 cq1 with respect to q1 and put the derivative equal to zero.) • Let’s go to Maple... The Nash Equilibrium with quantity-setting • The NE is given by the intersection of the two reaction curves… • Total output = 2(a-c)/3b • With monopoly (marginal cost – marginal revenue) output = (a-c)/2b • With perfect competition (price equal to marginal cost) output = (a-c)/b • The output with a duopoly is between the monopoly output and the competitive output. • In an oligopoly with n identical firms, the NE has a total output of n(a-c)/(n+1)b. Rises with n. The Bertrand Model • Each firm independently chooses its price. • The demand all goes to the firm with the lowest price (recall that the firms produce an identical product). • What prices will the firms choose? • Nash: each firm choosing optimal price given the price of the other firm. What happens with a duopoly • Is very sensitive to the rules of the game… Summary • With quantity-setting rules in the Nash Equilibrium the total output with a duopoly is between the monopoly output and the competitive output. • A collusive outcome is better for both firms – but is unstable. • For a firm it is better to be the leader. • With price-setting the Nash Equilibrium has price equal to marginal cost (and therefore like competition). • Surplus is lost with quantity-setting but not with price-setting. • After this week’s tutorial you might like to ask what happens if the ‘game’ is repeated many times. • Might some kind of (illicit?) agreement be reached? • Who would enforce it? • Legality of such agreements? • Government regulation of duopolies/oligopolies? Chapter 31 • Goodbye!