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OT2012 1 Anticompetitive consequence of the nationalization of a public enterprise in a mixed duopoly

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OT2012 2 Nationalization of a private firm yields collusive outcome in a Bertrand duopoly (1) Price Leadership Revisited (JoE 2011, joint work with Daisuke Hirata). (2) On the Uniqueness of Bertrand Equilibrium (Operation Research Letters 2010, with Daisuke Hirata) (3) Welfare Implication of Asymmetric Regulation in Mixed Bertrand Duopoly (Economics Letters 2012) (4) Price vs. Quantity in a Mixed Duopoly (Economics Letters 2012, with Akira Ogawa).

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OT2012 3 Bertrand Competition

4 OT2012 4 rationing rule If P 1

P 2, only firm 2 supplies D(P 2 ). If P 1 =P 2, each firm supplies D(P 1 )/2. D(P) is decreasing in P.

5 OT2012 5 rationing rule P 1

P 2 →D 2 =D( Ｐ 2 ), D 1 =max{D(P 1 )-Y 2, 0} P 1 =P 2 →D 1 =D(P 1 )/2+max{D(P 2 )/2-Y 2, 0} Suppose that firm 1 names a lower price. It can choose its output Y 1, which is not larger than D 1 =D(P 1 ), and then firm 2 can choose its output Y 2, which is not larger than the remaining demand D 2 = D 2 =max{D(P 2 )-Y 1, 0}.

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OT2012 6 Bertrand Paradox Symmetric Duopoly, Homogeneous Product Market, Constant Marginal Costs, Price Competition, Simultaneous-Move Game →Perfect Competition (MC=P) ~ Bertrand Paradox

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OT2012 7 Bertrand Equilibrium with Increasing Marginal Costs P Y D 0 supply curve derived from the marginal cost curves of the two firms MC of firm 1

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OT2012 8 Non-Existence of Pure Strategy Equilibrium under Increasing Marginal Costs Symmetric Duopoly, Homogeneous Product Market, Increasing Marginal Costs, Price Competition, Simultaneous-Move Game →No Pure Strategy Equilibrium ~ Edgeworth Cycle

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OT2012 9 Pure Strategy Symmetric Bertrand Equilibrium Ｐ Ｙ Ｄ 0 supply curve derived from the marginal cost curves of the two firms

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OT2012 10 Bertrand Equilibrium with Increasing Marginal Costs Suppose that P 1 =P 2 =MC 1 =MC 2 at a pure strategy equilibrium. →We derive a contradiction Suppose that firm 1 deviate from the strategy above and raises its price →Firm 2 has no incentive to increase its output since its output before the deviation is the best given P 2. →Given Y 2, firm 1 obtains the residual demand. →Since P 1 =MC 1 >MR 1 before the deviation, a slight increase of P1 must increase the profit of firm 1, a contradiction.

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OT2012 11 Pure Strategy Symmetric Bertrand Equilibrium Ｐ Ｙ Ｄ 0 supply curve derived from the marginal cost curves of the two firms

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OT2012 12 Pure Strategy Symmetric Bertrand Equilibrium Ｐ Ｙ Ｄ 0 supply curve derived from the marginal cost curves of two firms

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OT2012 13 pure strategy symmetric Bertrand Equilibrium Suppose that P 1 =P 2 >MC 1 =MC 2 at a pure strategy equilibrium. →We derive a contradiction Suppose that firm 1 deviates from the strategy above and reduces its price slightly. →Firm 1 can increase its demand (demand elasticity is infinite. Since P 1 >MC 1, the deviation increases the profit of firm 1, a contradiction. ⇒ No symmetric Bertrand equilibrium exists.

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OT2012 14 pure strategy asymmetric Bertrand Equilibrium P Y D 0 supply curve derived from the marginal cost curves of two firms P1P1 P2P2 MC of firm 2

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OT2012 15 The deviation increases the profit of firm 2, a contradiction P Y D 0 supply curve derived from the marginal cost curves of two firms P1P1 P2P2 MC of firm 2 Y2Y2 P2*P2* Y2*Y2*

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OT2012 16 pure strategy asymmetric Bertrand Equilibrium P Y D 0 supply curve derived from the marginal cost curves of two firms P1P1 P2P2 MC of firm 2

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OT2012 17 The deviation of firm 1 increases the profit of firm 1, a contradiction The profit of firm 1 is zero, and it has incentive to name the price slightly lower than the rival's ⇒ Neither symmetric nor asymmetric pure strategy Bertrand equilibrium exists.

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OT2012 18 Edgeworth Cycle Consider the symmetric Bertrand duopoly. Consider the following capacity constraint. Marginal cost of firm i is c if Y i ≦ K and ∞ otherwise. If K is sufficiently large, the equilibrium outcome is same as the Bertrand model with constant marginal cost. If K is sufficiently small, then the equilibrium price is derived from 2K=D(P). Firms just produce the upper limit output K. Otherwise →No pure strategy equilibrium (a similar problem under increasing marginal cost case appears) ~a special case of increasing marginal cost.

19 OT2012 19 rationing rule under supply obligation If P 1

P 2, only firm 2 supplies D(P 2 ). If P 1 =P 2, each firm supplies D(P 1 )/2.

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OT2012 20 Bertrand Equilibrium with Increasing Marginal Costs P Y D 0 supply curve derived from the marginal cost curves of two firms MC of firm 1 PEPE

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OT2012 21 Bertrand Equilibrium with Increasing Marginal Costs In the equilibrium both firms name P ＝ P E and obtain the demand D(P E )/2. Suppose that firm 1 raises its price.→The profit is zero, so it has no incentive for raising its price. Suppose that firm 1 reduces its price. →It obtains the demand D(P 1 ). Since P E =C 1 '(D(PE)/2), the profit is maximized given the price. Since C' is increasing, P E D(P E )/2 - C 1 (D(P E )/2) > P 1 D(P 1 ) - C 1 (D(P 1 )).

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OT2012 22 Continuum Equilibrium Both higher and lower prices than the perfectly competitive price can be equilibrium prices. Define P H by P H D(P H )/2 - C 1 (D(P H )/2) = P H D(P H ) - C 1 (D(P H )). If P 1 > P H, then P 1 D(P 1 )/2 - C 1 (D(P 1 )/2) < P 1 D(P 1 ) - C 1 (D(P 1 )). Define P L by P L D(P L )/2 - C 1 (D(P L )/2) = 0. If P 1 > P L, then P 1 D(P 1 )/2 - C 1 (D(P 1 )/2) < 0. Any price P ∈ (P L, P H ) can be an equilibrium price.

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OT2012 23 Bertrand Equilibrium with Increasing Marginal Costs P Y D 0 supply curve derived from the marginal cost curves of two firms Continuum Equilibrium PHPH PLPL

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OT2012 24 Indeterminacy of Bertrand Equilibria Hirata and Matsumura (2010) Does this result (indeterminacy of equilibria) depend on the assumption of homogeneous product? p 1 =a-q 1 -bq 2 p 2 =a-q 2 -bq 1 b ∈ (-1,1] b>0 supplementary products b=1 homogeneous product b represents the degree of product differentiation. If b =1, a continuum of equilibria exists. If b ∈ (0,1), the equilibrium is unique and it converges to Walrasian as b →1. It is also true under more general demand function.

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OT2012 25 Homogeneous Product Market P2P2 Y2Y2 D2D2 0 P1P1

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OT2012 26 Differentiated Product Market P2P2 Y2Y2 D2D2 0 P1P1

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OT2012 27 Bertrand Equilibrium with Increasing Marginal Costs P Y D 0 S supply curve derived from the marginal cost curves of the two firms

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OT2012 28 Asymmetric Supply Obligation We observe supply obligations in many markets, such as postal service (overnight delivery), electric power distribution, natural gas distribution, telecom, water supply, and so on. However, in most cases, this obligation is imposed to only one firm (usually a dominant firm). In mixed oligopoly, only the public firm has this obligation.

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OT2012 29 Asymmetric Supply Obligation Consider a duopoly private market. Suppose that only one firm (firm1) has this supply obligation. →No Pure Strategy Equilibrium exists.

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OT2012 30 Pure Strategy Symmetric Bertrand Equilibrium? Ｐ Ｙ Ｄ 0 supply curve derived from the marginal cost curves of the two firms Question: Does firm 2 have an incentive for changing its price? PWPW

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OT2012 31 Pure Strategy Symmetric Bertrand Equilibrium? Ｐ Ｙ Ｄ 0 supply curve derived from the marginal cost curves of the two firms Question: Does firm 1 have an incentive for changing its price? PWPW

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OT2012 32 Pure Strategy Symmetric Bertrand Equilibrium? Ｐ Ｙ Ｄ 0 supply curve derived from the marginal cost curves of the two firms Question: Does firm 2 have an incentive for changing its price?

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OT2012 33 Mixed Duopoly Ｐ Ｙ Ｄ 0 supply curve derived from the marginal cost curves of the two firms Question:Suppose that firm 1 (2) is a welfare(profit)- maximizing public (private) firm. Does firm 1 have an incentive for changing its price? PWPW

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OT2012 34 Mixed Duopoly Ｐ Ｙ Ｄ 0 supply curve derived from the marginal cost curves of the two firms Question:Suppose that firm 1 (2) is a welfare(profit)- maximizing public (private) firm. Does firm 2 have an incentive for changing its price? PWPW

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OT2012 35 Without Supply Obligation in Mixed Oligopoly The Asymmetric Obligation + Mixed Oligopoly yield the first best outcome. Are both indispensable? Symmetric obligation yields the first best in both mixed and private duopolies (known results). ~but the equilibrium is not unique. Neither mixed and private duopoly yields the first best under asymmetric obligation on the private firm. Mixed Oligopoly without Supply Obligation It is obvious that the first best is not achieved.

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OT2012 36 Without Supply Obligation in Mixed Oligopoly Mixed Oligopoly without Supply Obligation It is obvious that the first best is not achieved. What is the equilibrium outcome? First I think that (as well as in private duopoly) no pure strategy equilibrium exists. →My conjecture turns out to be wrong. ⇒ Monopoly outcome

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OT2012 37 Model Homogeneous Product Market, Mixed Duopoly, Firm 0~welfare maximizer, Firm 1~profit maximizer, common cost function~ increasing marginal cost, Firms independently choose their prices. The firm naming lower price chooses its output, and then the other firm chooses it output. When firms name the same price, the private chooses its output under the constraint y1 ≦ D(P)/2 and the public chooses its output y0 ≦ D(P)-y1.

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OT2012 38 Behavior of Firm 0 Firm 0 prefers y1= y0 for production efficiency. However, as long as y1 is positive, Y=D(p1). Firm 0 can choose the stand alone best outcome where its price is equal to its marginal cost. (Let y0 * denote the output of the public monopolist). Firm 0 chooses the latter if and only if y1= y0 =D(p1)/2 yields the larger welfare the stand alone best.

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OT2012 39 Limit Pricing by Firm 1 Firm 1 chooses the price either the price which maximizes p1 D(p1)/2 -c1( D(p1)/2 ) ~collusive pricing or chooses the price which yields W(D(p1)/2, D(p1)/2)= W(y0 *, 0) ~Limit Pricing. Either Collusive Pricing or Limit Pricing appears in equilibrium.

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OT2012 40 Model Homogeneous Product Market, Mixed Duopoly, Firm 0~welfare maximizer, Firm 1~profit maximizer, constant marginal cost, cost difference between public and private firms Firms independently choose their prices. The firm naming lower price chooses its output, and then the other firm chooses it output. When firms name the same price, the private chooses its output under the constraint y1 ≦ D(P)/2 and the public chooses its output y0 ≦ D(P)-y1.

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OT2012 41 Behavior of Firm 1 As long as p0>p1>c1, firm 1 chooses y1=D(p1).

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OT2012 42 Behavior of Firm 0 Firm 0 prefers firm 1’s production rather than its own production. Firm 0 can choose y0=D(c0) (public monopoly). It can choose y0=0 and then y1=D(p1).

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OT2012 43 Limit Pricing by Firm 1 Firm 1 chooses the price either the price which maximizes p1 D(p1) -c1D(p1) ~monopoly pricing or chooses the price which yields W(0, D(p1))= W(D(c0) *, 0) ~Limit Pricing. Either Monopoly Pricing or Limit Pricing appears in equilibrium.

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OT2012 44 Implication Supply obligation to the public firm is reasonable. If this obligation is abolished without privatization, it can produces huge welfare loss.

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OT2012 45 important property Under Bertrand competition, the public firm (welfare maximizer) becomes less aggressive because its aggressive behavior reduces the resulting production level of the private rival. Cf Under the Cournot competition, the output level of the private firm is given exogenously when the public firm chooses its output. Thus, its aggressive behavior does not reduces the rival’s output.

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OT2012 46 mixed Bertrand and mixed Cournot Competition is less severe under mixed Bertrand competition than under the mixed Cournot competition, contrasting to the standard results in private oligopoly.~ Under Bertrand competition, the public firm (Ghosh and Mitra, 2010 Letters).

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OT2012 47 Endogenous Choice of Price- Quantity Contract Firms choose whether to adopt price contract or quantity contract, and then choose the prices or quantities. Singh and Vives (1984) showed that choosing the quantity (price) contract is a dominant strategy for each firm if the goods are substitutes (complements). Intuition (substitutable goods case) ： Choosing a price contract increases the demand elasticity of the rival, resulting in a more aggressive action of the rival.

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OT2012 48 Endogenous Choice of Price- Quantity Contract in Mixed Duopoly For the private firm, choosing a price contract increases the demand elasticity of the rival, resulting in a less aggressive action of the rival (substitutable goods case). Thus, the private firm has an incentive to choose the price contract, as opposed to the private duopoly. For the public firm, choosing a price contract increases the demand elasticity of the rival, resulting in a more aggressive action of the rival. Thus, the public firm has an incentive to choose the price contract. →Bertrand competition appears in Mixed Duopoly (Matsumura and Ogawa, 2012)

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