Oligopoly Theory 2 Bertrand Market Games

Transcription

1 1/10 Oligopoly Theory 2 Bertrand Market Games May 4, 2014

2 2/10 Outline 1 Bertrand Market Game 2 Bertrand Paradox 3 Asymmetric Firms

3 3/10 Bertrand Duopoly Market Game Discontinuous Payoff Functions (1 p i )(p i c i ) if p i < p j, p i < 1 π i (p i, p j ) = α i (1 p i )(p i c i ) if p i = p j < 1 0 otherwise. where α i [0, 1] and α i + α j = 1 (α i = 0, 5: equal sharing of market in case of tie).

4 4/10 Bertrand Market Game: Discontinuous Payoff Functions π i (p i, p 0 j ) π i(p i, p 0 j ) p 0 j p i p 0 j = c

5 5/10 Outline 1 Bertrand Market Game 2 Bertrand Paradox 3 Asymmetric Firms

6 6/10 Bertrand Paradox Proposition Suppose there are at least two firms with constant average costs. Then, the equilibrium price is equal to the second lowest marginal cost ( two is enough for competition ). In the following we rename firms in such a way that c 1 c 2 c n hence, the two lowest average costs are c 1 c 2.

7 7/10 The case of uniform average cost Suppose there is a smallest monetary unit, ε > 0. Equilibria: p 1 = p 2 =c, p j c, j > 2 (I) p i =c + ε, i {1,..., n} (II) pi = c + 2ε is not also an equilibrium if n > 2 or if demand is price elastic, since a unilateral deviation to p = c + ε pays: Q(c + ε)ε 1 Q(c + 2ε)2ε }{{ n } Gain from unilateral deviation to p = c + ε Q(c + ε)ε 1 Q(c + 2ε)2ε 2 = ε (Q(c + ε) Q(c + 2ε)) > 0.

8 7/10 The case of uniform average cost Suppose there is a smallest monetary unit, ε > 0. Equilibria: p 1 = p 2 =c, p j c, j > 2 (I) p i =c + ε, i {1,..., n} (II) pi = c + 2ε is not also an equilibrium if n > 2 or if demand is price elastic, since a unilateral deviation to p = c + ε pays: Q(c + ε)ε 1 Q(c + 2ε)2ε }{{ n } Gain from unilateral deviation to p = c + ε Q(c + ε)ε 1 Q(c + 2ε)2ε 2 = ε (Q(c + ε) Q(c + 2ε)) > 0.

9 7/10 The case of uniform average cost Suppose there is a smallest monetary unit, ε > 0. Equilibria: p 1 = p 2 =c, p j c, j > 2 (I) p i =c + ε, i {1,..., n} (II) pi = c + 2ε is not also an equilibrium if n > 2 or if demand is price elastic, since a unilateral deviation to p = c + ε pays: Q(c + ε)ε 1 Q(c + 2ε)2ε }{{ n } Gain from unilateral deviation to p = c + ε Q(c + ε)ε 1 Q(c + 2ε)2ε 2 = ε (Q(c + ε) Q(c + 2ε)) > 0.

10 7/10 The case of uniform average cost Suppose there is a smallest monetary unit, ε > 0. Equilibria: p 1 = p 2 =c, p j c, j > 2 (I) p i =c + ε, i {1,..., n} (II) pi = c + 2ε is not also an equilibrium if n > 2 or if demand is price elastic, since a unilateral deviation to p = c + ε pays: Q(c + ε)ε 1 Q(c + 2ε)2ε }{{ n } Gain from unilateral deviation to p = c + ε Q(c + ε)ε 1 Q(c + 2ε)2ε 2 = ε (Q(c + ε) Q(c + 2ε)) > 0.

11 7/10 The case of uniform average cost Suppose there is a smallest monetary unit, ε > 0. Equilibria: p 1 = p 2 =c, p j c, j > 2 (I) p i =c + ε, i {1,..., n} (II) pi = c + 2ε is not also an equilibrium if n > 2 or if demand is price elastic, since a unilateral deviation to p = c + ε pays: Q(c + ε)ε 1 Q(c + 2ε)2ε }{{ n } Gain from unilateral deviation to p = c + ε Q(c + ε)ε 1 Q(c + 2ε)2ε 2 = ε (Q(c + ε) Q(c + 2ε)) > 0.

12 7/10 The case of uniform average cost Suppose there is a smallest monetary unit, ε > 0. Equilibria: p 1 = p 2 =c, p j c, j > 2 (I) p i =c + ε, i {1,..., n} (II) pi = c + 2ε is not also an equilibrium if n > 2 or if demand is price elastic, since a unilateral deviation to p = c + ε pays: Q(c + ε)ε 1 Q(c + 2ε)2ε }{{ n } Gain from unilateral deviation to p = c + ε Q(c + ε)ε 1 Q(c + 2ε)2ε 2 = ε (Q(c + ε) Q(c + 2ε)) > 0.

13 7/10 The case of uniform average cost Suppose there is a smallest monetary unit, ε > 0. Equilibria: p 1 = p 2 =c, p j c, j > 2 (I) p i =c + ε, i {1,..., n} (II) pi = c + 2ε is not also an equilibrium if n > 2 or if demand is price elastic, since a unilateral deviation to p = c + ε pays: Q(c + ε)ε 1 Q(c + 2ε)2ε }{{ n } Gain from unilateral deviation to p = c + ε Q(c + ε)ε 1 Q(c + 2ε)2ε 2 = ε (Q(c + ε) Q(c + 2ε)) > 0.

14 8/10 Outline 1 Bertrand Market Game 2 Bertrand Paradox 3 Asymmetric Firms

15 9/10 Asymmetric firms Suppose c 1 < c 2 ε, and c 2 p1 M, (pm 1 : the monopoly price of firm 1), and let ε > 0 be small. The game has two equilibria p1 = c 2 ε, p2 = c 2, pj c 2, j > 2 (I) p1 = c 2, p2 = c 2 + ε, pj c 2 + ε, j > 2 (II) Hence, the equilibrium price is roughly equal to the second lowest average cost c 2. (Of course, if c 2 > p M 1, one has p 1 = pm 1, p 2 = c 2.)

16 9/10 Asymmetric firms Suppose c 1 < c 2 ε, and c 2 p1 M, (pm 1 : the monopoly price of firm 1), and let ε > 0 be small. The game has two equilibria p1 = c 2 ε, p2 = c 2, pj c 2, j > 2 (I) p1 = c 2, p2 = c 2 + ε, pj c 2 + ε, j > 2 (II) Hence, the equilibrium price is roughly equal to the second lowest average cost c 2. (Of course, if c 2 > p M 1, one has p 1 = pm 1, p 2 = c 2.)

17 10/10... however not robust The two is enough for competition property is only a borderline case: introduction of fixed cost no pure strategy equilibrium convex cost function multiple equilibria, some at price below marginal cost capacity constraints even the monopoly price can be an equilibrium outcome (will be covered in detail) product differentiation price guarantees support high prices.

18 10/10... however not robust The two is enough for competition property is only a borderline case: introduction of fixed cost no pure strategy equilibrium convex cost function multiple equilibria, some at price below marginal cost capacity constraints even the monopoly price can be an equilibrium outcome (will be covered in detail) product differentiation price guarantees support high prices.

19 10/10... however not robust The two is enough for competition property is only a borderline case: introduction of fixed cost no pure strategy equilibrium convex cost function multiple equilibria, some at price below marginal cost capacity constraints even the monopoly price can be an equilibrium outcome (will be covered in detail) product differentiation price guarantees support high prices.

20 10/10... however not robust The two is enough for competition property is only a borderline case: introduction of fixed cost no pure strategy equilibrium convex cost function multiple equilibria, some at price below marginal cost capacity constraints even the monopoly price can be an equilibrium outcome (will be covered in detail) product differentiation price guarantees support high prices.

21 10/10... however not robust The two is enough for competition property is only a borderline case: introduction of fixed cost no pure strategy equilibrium convex cost function multiple equilibria, some at price below marginal cost capacity constraints even the monopoly price can be an equilibrium outcome (will be covered in detail) product differentiation price guarantees support high prices.

22 10/10... however not robust The two is enough for competition property is only a borderline case: introduction of fixed cost no pure strategy equilibrium convex cost function multiple equilibria, some at price below marginal cost capacity constraints even the monopoly price can be an equilibrium outcome (will be covered in detail) product differentiation price guarantees support high prices.