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18. Oligopoly Varian, Chapter 27 Two firms, two issues • Concentrate on duopoly – easy notation • Two issues: 1. What are firms’ choices? – Choose a quantity/quality of output; or – Choose a price 2. What is the timing of firms’ actions? – Simultaneous decisions; or – Sequential decisions Four interactions Timing Simultaneous Sequential Strategy Prices Quantities Bertrand Stackelberg-p Cournot Stackelberg-q We’ll do these two Costs and profits • Two firms, 1 and 2 • Single good, outputs y1 and y2 • Cost for firm i is c(yi) Market price depends on total output, but not • Inverse demand function is on which firm makes it p(y1+y2) • If outputs are y1 and y2, profits are p1(y1,y2) = p(y1+y2) y1 - c(y1) p2(y1,y2) = p(y1+y2) y2 - c(y2) Quantity leadership: Stackelberg • Firm 1 goes first; firm 2 follows • The follower’s problem: Given y1, choose y2 to max p2(y1,y2) = [p(y1+y2) y2] - c(y2) Revenue Costs • Firm 2’s output satisfies p(y1+y2) + p’(y1+y2) y2 = c’(y2) Marginal revenue Marginal cost Firm 2’s reaction function • Firm 2’s profit-maximizing output depends on firm 1’s choice • That is, y2 = f2(y1) for some function f2(.) • f2(.) is called firm 2’s reaction function Example: linear demand and zero costs • Suppose the inverse demand function is p(y1+y2) = A – B(y1+y2) • Firm 2’s profit is p2(y1,y2) = (A – B(y1+y2) ) y2 = (A - By1) y2 - B y22 • Firm 2’s best choice of output is y2 = (A – By1)/2B = f2(y1) Graphical treatment of linear case y2 Iso-profit lines for firm 2 Profit increasing Firm 2’s reaction function y2 = f2(y1) = (A – By1)/2B y1 The leader’s problem • Firm 1 anticipates firm 2’s reaction to its output choice • So it chooses y1 to max p1(y1,y2) = [p(y1+y2) y1] - c(y1) or max [p(y1+ f2(y1)) y1] - c(y1) Linear demand, zero costs • We know f2(y1) = (A – By1)/2B • So p1 = (A-B(y1+f2(y1)) y1 = {A-By1 – B [(A – By1)/2B ]} y1 = (A/2) y1 - (B/2) y12 • Best choice of y1: y1 = A/(2B) Stackelberg equilibrium y2 Firm 2’s reaction function y2 = f2(y1) = (A – By1)/2B Stackelberg equilibrium Profit increasing Iso-profit lines for firm 1 y1 Stackelberg outcome • Firm outputs y1 = A/(2B) y2 = f2(y1) = (A – By1)/(2B) = A/(4B) • Total industry output YS = y1 + y2 = (3A)/(4B) • Pareto efficient output YP = A/B Why? Pareto efficiency y2 Is the Stackelberg equilibrium Pareto efficient from the perspective of the two firms? Stackelberg equilibrium 2’s Profit increasing Room for a Pareto improvement 1’s Profit increasing y1 Cournot competition • Now both firms choose output simultaneously • We assume their choices constitute a Nash equilibrium • Whatever 1’s output, y1 , firm 2 must do Firm 2’s reaction the best it can: function y2 = f2(y1) • Whatever 2’s output, y2 , firm 1 must do the best it can: Firm 1’s reaction y1 = f1(y2) function Cournot equilibrium y2 y1 = f1(y2) Cournot equilibrium 2’s Profit increasing y2 = f2(y1) 1’s Profit increasing y1 Linear demand, zero costs • 2’ reaction function is y2 = f2(y1) = (A – By1)/2B • 1’ reaction function is y1 = f1(y2) = (A – By2)/2B • Solve these two equations for y1 and y2 : y1 = y2 = A/3B • Industry output YC = y1+y2 = (2A)/(3B) Pareto efficiency y2 Is the Cournot equilibrium y1 = f1(y2) Pareto efficient from the perspective of the two firms? Still room for a Pareto improvement Cournot equilibrium y2 = f2(y1) y1 Maximizing joint profits • Suppose the firms cooperatively choose outputs, y1 and y2 • When costs are zero, they choose aggregate output Y = y1 + y2 like a single monopolist: YM = A/(2B) • Note that YM < YC < YS < YP A/(2B) A/B (2A)/(3B) (3A)/(4B) Comparing output levels y2 y1 = f1(y2) YP YS YC 45o Pareto efficient from firms’ and consumers’ perspective YM y2 = f2(y1) Pareto efficient from firms’ perspective y1 Externalities in competition • Firms produce too much when they compete • Where does the inefficiency come from? – Each firm ignores the effect on the other’s profit when it expands output – i.e., there is a negative externality – Compared to monopoly, oligopoly pushes result closer to perfectly competitive outcome Sustaining a cartel • Beat-any-price clauses – It sounds very competitive – ….but maybe each firm is using consumers to check that other firms are not “cheating” • VERs – voluntary export restraints in Japan – US negotiated with Japan for Japanese firms to reduce sales in US – Benefited US car makers – …..but not US car consumers