anticompetitive consequence of the nationalization of a public enterprise in a mixed duopoly

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

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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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Bertrand Competition

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rationing rule

If P1<P2, only firm 1 supplies D(P1).

If P1>P2, only firm 2 supplies D(P2).

If P1=P2 , each firm supplies D(P1)/2.

D(P) is decreasing in P.

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rationing ruleP1<P2→D1=D(P1), D2=max{D(P2)-Y1, 0}

P1>P2→D2=D( Ｐ 2 ), D1=max{D(P1)-Y2, 0}

P1=P2→D1=D(P1)/2+max{D(P2 )/2-Y2, 0}

Suppose that firm 1 names a lower price. It can choose its output Y1 , which is not larger than

D1=D(P1), and then firm 2 can choose its output Y2, which is not larger than the remaining demand

D2= D2=max{D(P2)-Y1, 0}.

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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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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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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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Pure Strategy Symmetric Bertrand Equilibrium

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0


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Suppose that P1=P2=MC1=MC2 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 P2.

→Given Y2, firm 1 obtains the residual demand.

→Since P1=MC1>MR1 before the deviation, a slight increase of P1 must increase the profit of firm 1, a contradiction.

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0


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supply curve derived from the marginal cost curves of two firms

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pure strategy symmetric Bertrand Equilibrium

Suppose that P1=P2>MC1=MC2 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 P1>MC1 , the deviation increases the profit of firm 1, a contradiction.

⇒No symmetric Bertrand equilibrium exists.

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pure strategy asymmetric Bertrand Equilibrium

P

Y

D

0


P1

P2

MC of firm 2

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The deviation increases the profit of firm 2, a contradiction

P

Y

D

0


P1

P2

MC of firm 2

Y2

P2*

Y2*

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pure strategy asymmetric Bertrand Equilibrium

P

Y

D

0


P1P2

MC of firm 2

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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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Edgeworth CycleEdgeworth CycleConsider the symmetric Bertrand duopoly. Consider the

following capacity constraint. Marginal cost of firm i is c if Yi 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.

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rationing rule under supply obligation

If P1<P2, only firm 1 supplies D(P1).

If P1>P2, only firm 2 supplies D(P2).

If P1=P2 , each firm supplies D(P1)/2.

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P

Y

D

0


MC of firm 1

PE

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In the equilibrium both firms name P ＝ PE and obtain the demand D(PE)/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(P1). Since PE =C1'(D(PE)/2), the profit is maximized given the price. Since C' is increasing,

PE D(PE)/2 - C1(D(PE)/2) > P1D(P1) - C1(D(P1)) .

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Continuum EquilibriumBoth higher and lower prices than the perfectly

competitive price can be equilibrium prices.Define PH by PHD(PH)/2 - C1(D(PH)/2) = PHD(PH) -

C1(D(PH)).

If P1> PH, then P1D(P1)/2 - C1(D(P1)/2) < P1D(P1) - C1(D(P1)).

Define PL by PLD(PL)/2 - C1(D(PL)/2) = 0.

If P1> PL, then P1D(P1)/2 - C1(D(P1)/2) < 0.

Any price P (P∈ L, PH) can be an equilibrium price.

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P

Y

D

0


Continuum Equilibrium

PH

PL

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Indeterminacy of Bertrand EquilibriaHirata and Matsumura (2010) Does this result (indeterminacy of equilibria) depend on

the assumption of homogeneous product?p1=a-q1-bq2 p2=a-q2-bq1 b (-1,1]∈b>0 supplementary productsb=1 homogeneous productb 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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Homogeneous Product Market

P2

Y2

D2

0

P1

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Differentiated Product Market

P2

Y2

D2

0

P1

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P

Y

D

0

S


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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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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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Pure Strategy Symmetric Bertrand Equilibrium?

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Question: Does firm 2 have an incentive for changing its price?

PW

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PW

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Mixed Duopoly

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Question:Suppose that firm 1 (2) is a welfare(profit)-maximizing public (private) firm. Does firm 1 have an incentive for changing its price?

PW

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Mixed Duopoly

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Question:Suppose that firm 1 (2) is a welfare(profit)-maximizing public (private) firm. Does firm 2 have an incentive for changing its price?

PW

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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 ObligationIt is obvious that the first best is not achieved.

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Without Supply Obligation in Mixed Oligopoly

Mixed Oligopoly without Supply ObligationIt 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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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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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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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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ModelHomogeneous 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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Behavior of Firm 1

As long as p0>p1>c1, firm 1 chooses y1=D(p1).

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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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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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ImplicationSupply obligation to the public firm is reasonable. If this

obligation is abolished without privatization, it can produces huge welfare loss.

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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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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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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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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)

anticompetitive consequence of the nationalization of a public enterprise in a mixed duopoly

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