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18. Oligopoly Varian, Chapter 27
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Two firms, two issues Concentrate on duopoly – easy notation
What are firms’ choices? Choose a quantity/quality of output; or Choose a price What is the timing of firms’ actions? Simultaneous decisions; or Sequential decisions
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Four interactions Timing Simultaneous Sequential Strategy Prices
Bertrand Stackelberg-p Quantities Cournot Stackelberg-q We’ll do these two
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Costs and profits Two firms, 1 and 2 Single good, outputs y1 and y2
Cost for firm i is c(yi) Inverse demand function is 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) Market price depends on total output, but not on which firm makes it
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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) Firm 2’s output satisfies p(y1+y2) + p’(y1+y2) y2 = c’(y2) Revenue Costs Marginal revenue Marginal cost
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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
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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)
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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
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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)
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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)
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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
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y2 = f2(y1) = (A – By1)/(2B) = A/(4B)
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?
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Pareto efficiency Is the Stackelberg equilibrium
Pareto efficient from the perspective of the two firms? Stackelberg equilibrium 2’s Profit increasing Room for a Pareto improvement y1 1’s Profit increasing
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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 the best it can: y2 = f2(y1) Whatever 2’s output, y2 , firm 1 must do the best it can: y1 = f1(y2) Firm 2’s reaction function Firm 1’s reaction function
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Cournot equilibrium y2 y1 = f1(y2) y2 = f2(y1) y1 Cournot equilibrium
2’s Profit increasing y2 = f2(y1) y1 1’s Profit increasing
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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)
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Pareto efficiency Is the Cournot equilibrium Pareto efficient from the
perspective of the two firms? y1 = f1(y2) Still room for a Pareto improvement Cournot equilibrium y2 = f2(y1) y1
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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)
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Comparing output levels
y2 y1 = f1(y2) 45o YP YS Pareto efficient from firms’ and consumers’ perspective YC YM y2 = f2(y1) Pareto efficient from firms’ perspective y1
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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
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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
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