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Asymmetric welfare Asymmetric welfare implication between a implication between a

small number of leaders small number of leaders and a small number of and a small number of

followers in Stackelberg followers in Stackelberg modelsmodels

Joint work with Hiroaki InoJoint work with Hiroaki Ino

2012/3/4 2

Asymmetric welfare implication between small number of leaders

and followers in Stackelberg models, with Hiroaki Ino

(1) What role should public enterprises play in free-entry markets? (2010, JoE) (2) How many firms should be leaders? Beneficial concentration revisited (forthcoming in IER)

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Plan of the presentationPlan of the presentation

(1) Rough sketch of the model and results(1) Rough sketch of the model and results

(2) Motivation(2) Motivation

(3) Overview of related works (3) Overview of related works

(4) Formal explanation of our model(4) Formal explanation of our model

(5) Results and implications

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Rough sketch of the modelRough sketch of the model

m leaders, n followers, quantity-setting competitionm leaders, n followers, quantity-setting competition

(1) m+n=N is given exogenously(1) m+n=N is given exogenously

(2) m is given exogenously and n is determined by (2) m is given exogenously and n is determined by free entries. free entries.

Our main concerns: Relationship between m and Our main concerns: Relationship between m and welfare (consumer surplus plus profits of all welfare (consumer surplus plus profits of all firms).firms).

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HHI (Herfindahl-Hirschman Index )

HHI =∑HHI =∑i=1i=1nn(firm i's market share)(firm i's market share)2 2

Higher HHI→Higher market concentration Higher HHI→Higher market concentration

(1) An increase in the number of the firms (1) An increase in the number of the firms decreases HHIdecreases HHI

(2) An increase in the asymmetries among (2) An increase in the asymmetries among the firms increases HHIthe firms increases HHI

Is an increase of asymmetries among the Is an increase of asymmetries among the firms harmful or beneficial for welfare?firms harmful or beneficial for welfare?

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Fixed number of the firmsIn the first stage m leaders choose their output

independently

After observing the leaders' output, N-m followers choose their output independently.

m=0 or m=N→ Cournot model

m=1 standard Stackelberg model

m=2,3,...,N-1 multiple leadership ~ a variant of Stackelberg model

Stackelberg models yield higher HHI than the Cournot model

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Results Results （（ fixed number of the firms)fixed number of the firms)

(1) Suppose that marginal cost is constant. W(m) > (1) Suppose that marginal cost is constant. W(m) > W(0) for 0 <m <N W(0) for 0 <m <N (beneficial concentration).(beneficial concentration).

(2) W'(m) <0 if m is sufficiently close to N.(2) W'(m) <0 if m is sufficiently close to N.

(3) W'(m) can be either positive or negative when (3) W'(m) can be either positive or negative when m is close to 0. m is close to 0.

→→It is possible that W(m) <W(0).It is possible that W(m) <W(0).

(4) Suppose that the demand is linear and cost is (4) Suppose that the demand is linear and cost is quadratic. Then, for sufficiently large N, quadratic. Then, for sufficiently large N,

(a) W'(0) <0; (b) integer problem on the number of (a) W'(0) <0; (b) integer problem on the number of firms does not matter. firms does not matter.

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(1) m+n=N is given exogenously(1) m+n=N is given exogenously

W

m Welfare at Cournot Equilibrium

constant marginal cost

N

Stackelberg yields larger welfare

→beneficial concentration

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(1) m+n=N is given exogenously(1) m+n=N is given exogenously

W

m

increasing marginal cost

N

Welfare at Cournot Equilibrium

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Results Results （（ free entries of followers)free entries of followers)

W(m) > W(0) for 0 <m as long as the W(m) > W(0) for 0 <m as long as the number of followers is positive. number of followers is positive.

→→beneficial concentration always takes beneficial concentration always takes place.place.

⇒⇒HHI is a less plausible welfare measure at HHI is a less plausible welfare measure at free entry markets than at the markets free entry markets than at the markets with significant entry barriers. with significant entry barriers.

Leadership is more likely beneficial in the Leadership is more likely beneficial in the long run. long run.

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MotivationMotivation

Matsumura and Kanda (2005, JoE) Matsumura and Kanda (2005, JoE)

Ino and Matsumura (2010, JoE)Ino and Matsumura (2010, JoE)

Ino and Matsumura (forthcoming, IER)Ino and Matsumura (forthcoming, IER)

This paper This paper

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Mixed Oligopoly at Free Entry Mixed Oligopoly at Free Entry Markets (JoE, 2005)Markets (JoE, 2005)

Fixed number of private firmsFixed number of private firms

welfare-maximizing behavior of the public firm is welfare-maximizing behavior of the public firm is never optimal.never optimal.

Free entryFree entry

welfare-maximizing behavior of the public firm is welfare-maximizing behavior of the public firm is always optimal.always optimal.

The public firm with deficits should be abolished The public firm with deficits should be abolished rather than privatized. rather than privatized.

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What role should public enterprises What role should public enterprises play in free-entry markets? play in free-entry markets?

Fixed number of private firmsFixed number of private firms

The public firm should be a Stackelberg leaderThe public firm should be a Stackelberg leader

Free entryFree entry

The public firm should not be a Stackelberg leader The public firm should not be a Stackelberg leader

The same principle must be able to apply to pure The same principle must be able to apply to pure oligopoly (although the result is completely oligopoly (although the result is completely different) → Matsumura and Ino (IER)different) → Matsumura and Ino (IER)

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Benchmark ： Fixed number of the firms

W

m 0

N

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Daughety (1990, AER) Daughety (1990, AER)

W

m 0

N

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Lahiri and Ono (1988, EJ ) Lahiri and Ono (1988, EJ )

Welfare-improving production substitutionWelfare-improving production substitution ・ An increase of the output of the firm with lower marginal cost + A decrease of the output of the firm with higher marginal cost →It economizes total production cost and improves welfare

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Welfare-improving production Welfare-improving production substitutionsubstitution

X2

reaction curve of firm 1

reaction curve of firm 2

0

reaction curve of firm 1

X1

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Welfare-reducing production Welfare-reducing production substitutionsubstitution

X1

reaction curve of firm 1

reaction curve of firm 2

0

reaction curve of firm 2

X2

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NotationsNotations

N: Number of total firms m: Number of leaders xi: Firm i's output, X: Total output Ci(xi) : Firm i's production costP(X): demand function πi: Firm i's profitCS: Consumer surplus, W: social surplussubscript L(F,C):Leader(Follower, Cournot)

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The Model (fixed number of the The Model (fixed number of the firms)firms)

Players: identical m ( [0,N]) leaders, ∈ identical N-m followers. Payoff: Its own profits First, leaders choose their output independently. After observing the leaders' outputs followers choose their outputs. The market opens at the end of the game.

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AssumptionsAssumptions

Assumption 1Assumption 1 P(X) is twice differentiable and P(X) is twice differentiable and P'(X)<0 for all X such that P(X)>0.P'(X)<0 for all X such that P(X)>0.

Assumption 2Assumption 2 C(x) is twice differentiable and C(x) is twice differentiable and C'(x)>0, C''(x) ≧ 0 for all x ≧ 0.~a weaker C'(x)>0, C''(x) ≧ 0 for all x ≧ 0.~a weaker assumption is presented in the paper assumption is presented in the paper presented today.presented today.

Assumption 3 (strategic substitutes) Assumption 3 (strategic substitutes) P''(X)x+P'(X)<0 for all X such that P(X)>0 P''(X)x+P'(X)<0 for all X such that P(X)>0 and x ∈ (0,X).and x ∈ (0,X).

Assumption 4 Assumption 4 The model has the unique The model has the unique equilibrium for all m ∈ [0,N] and N >0. The equilibrium for all m ∈ [0,N] and N >0. The equilibrium is symmetric and all firms equilibrium is symmetric and all firms produce positive outputs.produce positive outputs.

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Proposition 1Proposition 1Suppose that Assumptions 1-4 are satisfied.Suppose that Assumptions 1-4 are satisfied.

If C''(x)=0 for all x 0, then W*(m) > W*(0) for all m ≧If C''(x)=0 for all x 0, then W*(m) > W*(0) for all m ≧ (0,N).∈ (0,N).∈

If marginal cost is constant, beneficial If marginal cost is constant, beneficial concentration always takes place. concentration always takes place.

→→This is because Stackelberg model yields larger This is because Stackelberg model yields larger total output than the Cournot.total output than the Cournot.

(generalization of Daughety (1990) for general (generalization of Daughety (1990) for general demand)demand)

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Proposition 2Proposition 2Suppose that Assumptions 1-4 are satisfied.Suppose that Assumptions 1-4 are satisfied.

Then, (i) W*'(m) at m=0 can be either Then, (i) W*'(m) at m=0 can be either negative or positive and (ii) W*'(m) at m=N negative or positive and (ii) W*'(m) at m=N is always negative.is always negative.

(i) Introducing small number of leaders into (i) Introducing small number of leaders into the Cournot model can be either beneficial the Cournot model can be either beneficial or harmful for welfare.or harmful for welfare.

(ii) Introducing small number of followers (ii) Introducing small number of followers into the Cournot model is always beneficial into the Cournot model is always beneficial for welfarefor welfare..

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Proposition 3Proposition 3

Suppose that P=a-X and C(x) =cx+kxSuppose that P=a-X and C(x) =cx+kx22. If k>0, there . If k>0, there exists N' >0 such that W'(m) at m=0 is negative for exists N' >0 such that W'(m) at m=0 is negative for all N > N'. all N > N'.

For any quadratic cost functions, there are cases For any quadratic cost functions, there are cases where leadership is harmful.where leadership is harmful.

Leadership becomes harmful morel likely when the Leadership becomes harmful morel likely when the number of follower is large. number of follower is large.

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Why can introducing a small Why can introducing a small number of leaders into the Cournot number of leaders into the Cournot

model be harmful ?model be harmful ?

Consider the Stackelberg model with one leader Consider the Stackelberg model with one leader (firm 1). Then firm 1 becomes a followers (firm 1). Then firm 1 becomes a followers (Cournot). (Cournot).

→→Production substitutions from firm 1 to the Production substitutions from firm 1 to the other firms. This production substitution other firms. This production substitution improves production efficiency when improves production efficiency when marginal cost is increasing and can dominate marginal cost is increasing and can dominate the positive effect of increasing CS.the positive effect of increasing CS.

~ This welfare-improving production ~ This welfare-improving production substitution effect is strong when the number substitution effect is strong when the number of followers is large.of followers is large.

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Why is introducing a small Why is introducing a small number of followers into the number of followers into the

Cournot model always beneficial?Cournot model always beneficial? Consider the Stackelberg model with one Consider the Stackelberg model with one

follower (firm n). Then firm 1 becomes a follower (firm n). Then firm 1 becomes a leaders (Cournot). leaders (Cournot).

→→Production substitutions from firm 1, 2..,N-1 to Production substitutions from firm 1, 2..,N-1 to firm N. This production substitution improves firm N. This production substitution improves production efficiency when marginal cost is production efficiency when marginal cost is increasing, but this effect is negligible because increasing, but this effect is negligible because limlimm→Nm→N x xLL= lim= limm→Nm→N x xFF= x= xC C (Cournot output).(Cournot output).

~When the number of followers is small, the ~When the number of followers is small, the difference of output level between each leader difference of output level between each leader and follower is negligible. and follower is negligible.

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Why convex?

W

m 0

N

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Why convex ?Why convex ? Consider the Stackelberg model with m leaders. Consider the Stackelberg model with m leaders.

Then firm m+1 becomes a leader. Then firm m+1 becomes a leader.

→→Production substitutions from firm m+2, firm Production substitutions from firm m+2, firm m+3,... firm N m+3,... firm N (firm 1, firm 2,..., firm m)(firm 1, firm 2,..., firm m) to to firm m+1. This production substitution firm m+1. This production substitution worsens worsens (improves)(improves) production efficiency. production efficiency.

The former (latter) effect is weaker (stronger) The former (latter) effect is weaker (stronger) when m is large. when m is large.

~ An increase of the number of the leaders ~ An increase of the number of the leaders more likely improve welfare when m is large. more likely improve welfare when m is large.

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The Models (endogenous The Models (endogenous number of the followers)number of the followers)

Players: identical m (∈[0,N]) leaders, potential new entrants (followers) Payoff: Its own profits

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Why free entries of followers Why free entries of followers rather than the leaders ?rather than the leaders ?

Leader's profit is larger than follower's. If we consider free entry of leaders, no follower enter the market; resulting in the Cournot model.

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The Model 1 (weakly persistent-The Model 1 (weakly persistent-leadership model)leadership model)

Leaders have already enters the market. First, followers choose whether or not to enter the market. After observing the number of follower, all firms plays the same game described in the previous sections.

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The Model 2 (strongly persistent-The Model 2 (strongly persistent-leadership model)leadership model)

Leaders have already enters the market. First, leaders choose their outputs. After observing the leaders' outputs followers choose whether or not to enter the market. After observing the number of follower, followers choose their outputs. The market opens at the end of the game.

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Propositions 4 and 5Propositions 4 and 5

Suppose that the number of followers entering the Suppose that the number of followers entering the market is positive. Suppose that Assumptions 1-5 market is positive. Suppose that Assumptions 1-5 are satisfied. Then W**(m)>Ware satisfied. Then W**(m)>WCC** for all m>0. ** for all m>0.

(Assumption 5: Increasing marginal cost and (Assumption 5: Increasing marginal cost and positive entry cost) positive entry cost)

The Stackelberg model always yields higher welfare The Stackelberg model always yields higher welfare than the Cournot under free entry. than the Cournot under free entry.

Beneficial concentration always takes place.Beneficial concentration always takes place.

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Production substitution under free Production substitution under free entries (model 1)entries (model 1)

Consider the Cournot model. Then firm 1 becomes a Consider the Cournot model. Then firm 1 becomes a leader. leader.

→→Leadership by firm 1 reduces the number of Leadership by firm 1 reduces the number of entering followers entering followers

→ → Production substitutions from Production substitutions from potential entrantspotential entrants to to firm 1 takes place. firm 1 takes place.

Important point: the output of each follower entering Important point: the output of each follower entering the market does not change. Only the number of the market does not change. Only the number of entering the firms is reduced. entering the firms is reduced.

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Leadership is introducedLeadership is introducedP

Y

Follower's AC

Follower's residual demand

0Follower's output

In the long run~reduction of followersIn the long run~reduction of followers

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Why is the leadership always Why is the leadership always beneficial under free entries?beneficial under free entries?

→→Leadership by firm 1 reduces the number of Leadership by firm 1 reduces the number of entering followers entering followers

→ → Production substitutions from Production substitutions from potential potential entrantsentrants to firm 1 takes place. This saves to firm 1 takes place. This saves follower's production costs at average cost follower's production costs at average cost base and increases the leader's production base and increases the leader's production cost at marginal cost base. Since cost at marginal cost base. Since follower's follower's average cost =P ≧ leader's marginal costaverage cost =P ≧ leader's marginal cost, , this production substitution improves this production substitution improves production efficiency ~ improves welfareproduction efficiency ~ improves welfare