# Oligopoly: Cournot/Bertrand/Stackelberg

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1 Outline Alternative Market Models Wirtschaftswissenschaften Humboldt Universität zu Berlin March 5, 2006

2 Outline 1 Introduction Introduction Alternative Market Models 2 Game, Reaction Functions, Solution Outline Alternative Market Models Existence, Uniqueness, Rationalizable Strategies* Some Results 3 The Game Two is a large number Rationalizable Strategies 4 The Game Solution

3 Common Structure Outline Alternative Market Models Common Assumptions 1 Risk neutral firms maximize profits 2 Firms produce a homogeneous product 3 All cost functions are known to all firms 4 Firms meet once and cannot make binding agreements Each firm takes into account that their profits depend on actions of the other firms (strategic interaction)

4 Alternative Market Models Outline Alternative Market Models Cournot (1839): firms simultaneously choose quantities 1 Firms simultaneously select quantities (capacities): q i [0, D(0)], i I 2 Price is chosen (by an auctioneer) to clear the market D(p) = I q i Bertrand (1883): firms simultaneously select prices 1 Firms simultaneously select prices: p i [0, p m ], i I 2 The cheapest supplier l gets the whole demand q l = D(p l ) (The lowest bid gets the contract) There are no capacity constraints Each firm can serve the whole market

5 Alternative Market Models Outline Alternative Market Models Stackelberg (1934): firms sequentially select quantities 1 The leader l selects his quantity (capacity): q l [0, D(0)] 2 The followers observe q l and select their quantities: q i [0, D(0)] 3 Price is chosen (by an auctioneer) to clear the market D(p) = I q i

6 Cournot Game: the interaction Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Describing strategic interaction as a game in normal form: Players: Firms A and B (pure) strategies, strategy-space : {q A, q B } S = [0, q] [0, q]; q = D(0) Pay-off functions: π A (q A, q B ) = q A P (q A + q B ) K A (q A ) π B (q A, q B ) = q B P (q A + q B ) K B (q B )

7 Cournot Game: the solution Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Solution: Cournot Nash equilibrium: A combination of strategies (qa, q B ) S so that both firms select an optimal strategy given the choice of the other firm: q A = arg max q A π A (q A, q B), q B = arg max q B π B (q A, q B )

8 Finding the Solution Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results In principle, the solution can be obtained from the Kuhn-Tucker conditions Let q = q A + q B then (q A, q B ) solve π A (q A, q B ) q A = [P (q)q A + P (q) K A] 0, [ ]q A = 0 π B (q A, q B ) q B 0, [ ]q B = 0

9 Reaction Functions Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Useful tools: reaction functions and inclusive reaction functions : Q A (q B ) := arg max q A Q B (q A ) := arg max q B π A (q A, q B ) =: Q i A(q) π B (q A, q B ) =: Q i B(q) The correspondence ψ ψ(q A, q B ) = {Q A (q B ), Q B (q A )} is a mapping from S into itself (qa, q B ) is a fixed point of this mapping

10 Slope of reaction functions Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results From the first order condition dq A dq B = 2 π A q A q B 2 π A ( q A ) 2 By the second order condition: sign dq A dq B = sign 2 π A q A q B = sign P q A + P Assumed to be negative (recall P < 0)

11 Strategic Substitutes Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results strategic substitutes: the firm responds to an increase of the rival s quantity with a reduction of its own quantity in general: If in increase of another players s choice variable leads to a decrease of the player s optimal level of his choice variable

12 Cournot Equilibrium Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results q B Q A (q B ) q B Q B (q A ) q A q A

13 Existence: mixed strategies Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Existence in mixed strategies Since S is compact (closed and bounded) and payoffs are continuous functions of strategies the existence of an equilibrium (possibly in mixed strategies) is assured (Glicksberg (1952)) If S would be replaced by a discrete grid of quantities existence would be assured by Nash (1952)

14 Existence: pure strategies Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Existence in pure strategies Debreu (1952): there exists a pure-strategy equilibrium if 1 S is convex and compact, 2 all payoff functions are continuous in the strategy profiles s S, 3 all payoff functions are quasiconcave in the players own strategies

15 Uniqueness Introduction Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Uniqueness in pure strategies Kohlstad & Mathiesen (1987) provide necessary and sufficient conditions for uniqueness in pure strategies Essentially, a small change of the firm s output must have a larger impact on the firms own profit than a similar change of output on part of the rival firms

16 Uniqueness Introduction Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Assume that for j = A, B: dq j dq = P (q)q j (q) + P (q) P (q) K j (Q j(q)) < 0 Define the aggregate inclusive reaction function: Q i (q) = Q i A (q) + Qi B (q) 1 Q i is a mapping from [0, q M ] into itself Every fixed point q o of this mapping generates a Nash equilibrium (Q i A (q o), Q i B (q o)) and vice versa 2 Q i j (0) > 0, Qi j (qm ) = 0, j = A, B, hence Q i (0) > 0, Q i (q M ) = 0 3 Q i is continuously differentiable, with dq i /dq < 0 The Cournot game has a unique Nash equilibrium, because Q i cuts the 45 degree line exactly once See Wolfstetter, 1999, Topics in Microeconomics Industrial

17 Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Iterated Elimination of Dominated Strategies Example: Linear demand P (q) = a q, constant marginal cost c With two players: The set of rationalizable strategies is equal to the Nash-equilibrium strategies (q c A, qc B ) Iterated elimination of dominated strategies allows you to lower the upper bound and raise the lower bound of rationalizable strategies until you approach the Nash-equilibrium

18 Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Iterated Elimination of Dominated Strategies With three players: The set of rationalizable strategies is equal to [0, q M ] 3 While no player will ever set q i > q M = Q i (0), Q i (2q M ) = 0 so we cannot increase the lower bound Hence, iterated elimination of dominated strategies is of little use to predict behavior For a detailed presentation see Gibbons 1992, Game Theory for Applied Economists, Princeton University Press pp18ff

19 Mark-Up Introduction Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results We look at general case n 2 firms (q = n i=1 q i) FOC : P (q) K i(q i ) + q i P (q) = 0 Industry cost is not minimized (marginal cost are different for different firms) Rewrite (FOC) as mark-up rule (Lerner index): P K i P = q i q 1 ɛ(p ) Mark-up is proportional to market share and inversely proportional to elasticity of demand

20 Symmetric Case Introduction Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results In the symmetric case we obtain: P K P = 1 n 1 ɛ(p ) As the number of firms n becomes larger, we move from monopoly to perfect competition

21 Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Oligopoly is somewhere between perfect competition and monopoly q m < q < q k ; p m > P (q ) > p k ; n π m > π i (q ) > π k i=1 The larger the number of firms the closer we approach perfect competition Results make sense, but price determination is implausible In most markets firms determine prices themselves

22 Horizontal Merger and Market Power Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Suppose we have n identical firms in the market Marginal cost is c and inverse demand is given by: P (q) = a bq q (n) = a c b(1 + n) and π(n) = (P (nq (n)) c)q (n) = (a c)2 b(1 + n) 2 Two firms are contemplating a merger Will it increase market power? Will it increase profits?

23 Mergers don t pay off Game, Reaction Functions, Solution Existence, Uniqueness, Rationalizable Strategies* Some Results Proposition: For n > 2, a horizontal merger of two firms increases the profits of the industry but is not profitable for the merging firms Proof: 1 Obviously it increases profits of the other firms because: π(n 1) > π(n) 2 Industry profits increase: n 2 n 1 > 0 = (n 1)π(n 1) > nπ(n) 3 Merging firms profit decrease: 1 + 2n n 2 < 0 = π(n 1) < 2π(n)

24 Bertrand (1883): firms select prices The Game Two is a large number Rationalizable Strategies Assumptions: 1 Risk neutral firms maximize profits 2 Firms produce a homogeneous product 3 All cost functions are known to all firms 4 Firms meet once and cannot make binding agreements 5 Firms simultaneously select prices: p i [0, p m ], i I 6 The cheapest supplier l gets the whole demand q l = D(p l ) (The lowest bid gets the contract) There are no capacity constraints Each firm can serve the whole market

25 The Bertrand Game: The Game Two is a large number Rationalizable Strategies Game in normal form: Players: Firms A and B (pure) strategies, strategy-space: p A [0, p m ]; p B [0, p m ] Pay-off function (A): 0 if p A > p B 1 π A (p A, p B ) = 2 D(p A)(p A c) if p A = p B D(p A )(p A c) if p A < p B

26 Two is a large number The Game Two is a large number Rationalizable Strategies Proposition: There is a unique Nash-equilibrium in which both players set prices equal to marginal cost and obtain zero profits (p A, p B) = (c, c) Proof: Equilibrium: By decreasing the price below p a firm will incur losses increasing the price over p does not increase profits (In equilibrium both firms play a weakly dominated strategy) Uniqueness: No equilibrium can entail a p < 0 Suppose there is an equilibrium with p > c for at least on firm Then the rival firm deviates by undercutting (ie by setting p ε where ε > 0 is chosen arbitrarily small) Note: In equilibrium firms are indifferent between their equilibrium strategy and any other strategy

27 Profit function Introduction The Game Two is a large number Rationalizable Strategies The optimization problem is not well behaved π A c p B p A

28 reaction functions The Game Two is a large number Rationalizable Strategies Unless there is a price grid with minimal steps ε, the reaction functions are not defined If there is such a grid, then a Nash-equilibria exist so that (p A, p B ) = (c + ε, c + ε) reaction function : R A (p B ) = { pb ε if p B > c + ε c + ε if p B c + ε Since R A (p 1 ) R A (p 0 ) for p 1 p 0 we have strategic complements

29 Reaction Functions The Game Two is a large number Rationalizable Strategies R A, p A p m c c R B R A p m p B, R B

30 The Game Two is a large number Rationalizable Strategies Solving the Game Through Iterated Elimination of Dominated Strategies Without price grid If the strategy space is given by [0, p m ] 2 the set of rationalizable strategies is p A [c, p m ); p B [c, p m ) With a price grid If the strategy space is given by {0, ε, 2ε, 3εNε = p m } 2 the set of rationalizable strategies is p A {c, c + ε}; p B {c, c + ε}

31 Discussion Introduction The Game Two is a large number Rationalizable Strategies With Bertrand competition two firms are enough to obtain the same results as with perfect competition But how robust is this result? Richer strategy space: price matching : The firm sets a price and offers to match any lower price offered by a competitor This sounds good for customers But any price pair can be an equilibrium including the monopoly price!

32 Example Introduction The Game Two is a large number Rationalizable Strategies In 1992 a subsidiary of German Telecom DeTeMedien started selling a telephone directory on CD-ROM for almost DM 4000 per copy These directories contained: names, telephone numbers and addresses of all the people connected to the net in Germany They were very convenient for businessmen As this data was stored digitally anyway, there were little additional cost of production, distribution etc estimated at DM 10 per CD By October 1995 the price was still higher than DM 1500 Telecom was selling a five-digit number of copies At that time TopWare appeared at the market as the only competitor It digitalized traditional telephone directories books by scanner, and offered the CD at the price of DM 50 selling items, within a few weeks Telecom responded by lowering the price to DM 90 and obtained a prelimenary court ruling preventing TopWare from selling, on the grounds that scanning of telephone directories is an infringement of copyrights

33 Example Introduction The Game Two is a large number Rationalizable Strategies With legal procedures underway, TopWare actualized its version of CD early 1996 by having the whole directory retyped by 600 typists in China It offered the new version again for DM 90 Telecom opened a second case against TopWare in court, but lowered the price for its new version to DM 30 in March 1996 So, the price dropped from DM 1500 to DM 30 during 6 months Hence, within six month the price dropped from DM 1500 to DM 30 which is comparable to the competitive market for music CD Do the assumptions of Bertrand competition apply? What alternative strategies could have been adopted? Why have they not been chosen?

34 Stackelberg (1934) The Game Solution Assumptions: 1 Risk neutral firms maximize profits 2 Firms produce a homogeneous product 3 All cost functions are known to all firms 4 Firms meet once and cannot make binding agreements 5 The leader l selects his quantity (capacity): q l [0, D(0)] 6 The followers observe q l and select their quantities: q i [0, D(0)] 7 Price is chosen (by an auctioneer) to clear the market D(p) = I q i Reinterpretation of quantities as capacities

35 The Game Solution Normal Form of the Stackelberg Game: Players: Firms A (leader) and B (follower) (pure) strategies: q A [0, q]; Q B : [0, q] [0, q]; q = D(0) The strategy of the follower B is rule for every possible initial choice of player A, hence a function Pay-off functions: π A (q A, Q B ) = q A P (q A + Q B (q A )) K A (q A ) π B (q A, Q B ) = Q B (q A ) P (q A + Q B (q A )) K B (Q B (q A ))

36 The Solution Introduction The Game Solution Equilibrium is a pair of strategies (qa, Q B ) which is mutually best response Equilibrium-action: In equilibrium we observe only two actions first: (q A then Q B (q A ))

37 The Game Solution The Stackelberg Follower, a Monopolist? There exists a Nash-equilibrium in which B obtains the monopoly profit, with { q M if q q A = 0, Q B (q A ) = A = 0 D(0) q A otherwise It is easy to check, that no one wants to deviate

38 This solution is not convincing The Game Solution Problem: The equilibrium is not convincing It is based on empty threats in situations which are not reached in equilibrium Suppose A would chose the Cournot quantity qa c (which is out of equilibrium) then it would not be in the interest of B to play D(0) qa c but rather qc B

39 The Game Solution The Subgame Perfect Equilibrium SPB To find the SPB, we start with the subgames after the choice of A (backward induction) Since the only remaining action is B s choice of quantity, subgame-perfectness requires, that it is in B s best interest to follow his strategy in every possible subgame, ie for every possible choice of q A Hence we require: Q B = arg max q B π B (q A, q B ), q A which defines the Cournot reaction function Q B The equilibrium strategy of A is found by maximizing against B s reaction function : qa = arg max π A (q A, Q B(q A )) q A

40 The Value of Commitment The Game Solution To compare Cournot and Stackelberg note that q A solves π A (q A, q B) q A + π A(q A, q B) q B dq B (q A ) dq A = 0 The second term indicates the strategic effect Provided that quantities are strategic substitutes, it is positive Hence q A > qc A 1 Leader produces more than under Cournot competition and obtains higher profits 2 Follower produces less and obtains lower profits

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