Asymmetric welfare implication
between a small number of leaders
and a small number of followers in
Stackelberg models
Joint work with Hiroaki Ino
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Asymmetric welfare implication
between small number of leaders
and followers in Stackelberg models,
with Hiroaki Ino
(1) What role should public enterprises play in freeentry markets? (2010, JoE)
(2) How many firms should be leaders? Beneficial
concentration revisited (forthcoming in IER)
2012/3/4
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Plan of the presentation
(1) Rough sketch of the model and results
(2) Motivation
(3) Overview of related works
(4) Formal explanation of our model
(5) Results and implications
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Rough sketch of the model
m leaders, n followers, quantity-setting competition
(1) m+n=N is given exogenously
(2) m is given exogenously and n is determined by
free entries.
Our main concerns: Relationship between m and
welfare (consumer surplus plus profits of all
firms).
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HHI (Herfindahl-Hirschman Index )
HHI =∑i=1n(firm i's market share)2
Higher HHI→Higher market concentration
(1) An increase in the number of the firms
decreases HHI
(2) An increase in the asymmetries among the
firms increases HHI
Is an increase of asymmetries among the firms
harmful or beneficial for welfare?
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Fixed number of the firms
In 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
（fixed number of the firms)
(1) Suppose that marginal cost is constant. W(m) >
W(0) for 0 <m <N (beneficial concentration).
(2) W'(m) <0 if m is sufficiently close to N.
(3) W'(m) can be either positive or negative when m is
close to 0.
→It is possible that W(m) <W(0).
(4) Suppose that the demand is linear and cost is
quadratic. Then, for sufficiently large N,
(a) W'(0) <0; (b) integer problem on the number of
firms does not matter.
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(1) m+n=N is given exogenously
constant marginal cost
W
Stackelberg yields larger welfare
→beneficial concentration
Welfare at Cournot Equilibrium
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m
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(1) m+n=N is given exogenously
increasing marginal cost
W
N
m
Welfare at Cournot Equilibrium
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Results
（free entries of followers)
W(m) > W(0) for 0 <m as long as the number
of followers is positive.
→beneficial concentration always takes place.
⇒HHI is a less plausible welfare measure at
free entry markets than at the markets with
significant entry barriers.
Leadership is more likely beneficial in the
long run.
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Motivation
Matsumura and Kanda (2005, JoE)
Ino and Matsumura (2010, JoE)
Ino and Matsumura (forthcoming, IER)
This paper
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Mixed Oligopoly at Free Entry
Markets (JoE, 2005)
Fixed number of private firms
welfare-maximizing behavior of the public firm is never
optimal.
Free entry
welfare-maximizing behavior of the public firm is always
optimal.
The public firm with deficits should be abolished rather
than privatized.
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What role should public enterprises
play in free-entry markets?
Fixed number of private firms
The public firm should be a Stackelberg leader
Free entry
The public firm should not be a Stackelberg leader
The same principle must be able to apply to pure
oligopoly (although the result is completely
different) → Matsumura and Ino (IER)
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Benchmark：Fixed number of the
firms
W
0
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m
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Daughety (1990, AER)
W
0
N
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Lahiri and Ono (1988, EJ )
Welfare-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
substitution
reaction curve
of firm 1
X2
reaction curve
of firm 2
reaction curve
of firm 1
0
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Welfare-reducing production
substitution
reaction curve
of firm 1
X2
reaction curve
of firm 2
reaction curve
of firm 2
0
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Notations
N: Number of total firms
m: Number of leaders
xi: Firm i's output, X: Total output
Ci(xi) : Firm i's production cost
P(X): demand function
πi: Firm i's profit
CS: Consumer surplus, W: social surplus
subscript L(F,C):Leader(Follower, Cournot)
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The Model (fixed number of the
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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Assumptions
Assumption 1 P(X) is twice differentiable and P'(X)<0
for all X such that P(X)>0.
Assumption 2 C(x) is twice differentiable and C'(x)>0,
C''(x) ≧ 0 for all x ≧ 0.~a weaker assumption is
presented in the paper presented today.
Assumption 3 (strategic substitutes) P''(X)x+P'(X)<0
for all X such that P(X)>0 and x ∈ (0,X).
Assumption 4 The model has the unique equilibrium
for all m ∈ [0,N] and N >0. The equilibrium is
symmetric and all firms produce positive outputs.
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Proposition 1
Suppose that Assumptions 1-4 are satisfied.
If C''(x)=0 for all x ≧ 0, then W*(m) > W*(0) for all m
∈ (0,N).
If marginal cost is constant, beneficial
concentration always takes place.
→This is because Stackelberg model yields larger
total output than the Cournot.
(generalization of Daughety (1990) for general
demand)
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Proposition 2
Suppose that Assumptions 1-4 are satisfied.
Then, (i) W*'(m) at m=0 can be either negative or
positive and (ii) W*'(m) at m=N is always
negative.
(i) Introducing small number of leaders into the
Cournot model can be either beneficial or
harmful for welfare.
(ii) Introducing small number of followers into the
Cournot model is always beneficial for welfare.
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Proposition 3
Suppose that P=a-X and C(x) =cx+kx2. If k>0, there
exists N' >0 such that W'(m) at m=0 is negative for
all N > N'.
For any quadratic cost functions, there are cases
where leadership is harmful.
Leadership becomes harmful morel likely when the
number of follower is large.
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Why can introducing a small
number of leaders into the Cournot
model be harmful ?
Consider the Stackelberg model with one leader (firm
1). Then firm 1 becomes a followers (Cournot).
→Production substitutions from firm 1 to the other firms.
This production substitution improves production
efficiency when marginal cost is increasing and can
dominate the positive effect of increasing CS.
~ This welfare-improving production substitution effect
is strong when the number of followers is large.
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Why is introducing a small
number of followers into the
Cournot model always beneficial?
Consider the Stackelberg model with one follower (firm
n). Then firm 1 becomes a leaders (Cournot).
→Production substitutions from firm 1, 2..,N-1 to firm N.
This production substitution improves production
efficiency when marginal cost is increasing, but this
effect is negligible because limm→N xL= limm→N xF= xC
(Cournot output).
~When the number of followers is small, the difference
of output level between each leader and follower is
negligible.
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Why convex?
W
0
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m
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Why convex ?
Consider the Stackelberg model with m leaders. Then
firm m+1 becomes a leader.
→Production substitutions from firm m+2, firm m+3,...
firm N (firm 1, firm 2,..., firm m) to firm m+1. This
production substitution worsens (improves)
production efficiency.
The former (latter) effect is weaker (stronger) when
m is large.
~ An increase of the number of the leaders more likely
improve welfare when m is large.
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The Models (endogenous
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
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 persistentleadership 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 persistentleadership 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 5
Suppose that the number of followers entering the
market is positive. Suppose that Assumptions 1-5
are satisfied. Then W**(m)>WC** for all m>0.
(Assumption 5: Increasing marginal cost and
positive entry cost)
The Stackelberg model always yields higher welfare
than the Cournot under free entry.
Beneficial concentration always takes place.
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Production substitution under free
entries (model 1)
Consider the Cournot model. Then firm 1 becomes a
leader.
→Leadership by firm 1 reduces the number of
entering followers
→ Production substitutions from potential entrants to
firm 1 takes place.
Important point: the output of each follower entering
the market does not change. Only the number of
entering the firms is reduced.
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Leadership is introduced
P
Follower's residual demand
Follower's AC
0
Follower's output
In the long run~reduction of followers
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Why is the leadership always
beneficial under free entries?
→Leadership by firm 1 reduces the number of
entering followers
→ Production substitutions from potential entrants to
firm 1 takes place. This saves follower's production
costs at average cost base and increases the
leader's production cost at marginal cost base.
Since follower's average cost =P ≧ leader's
marginal cost, this production substitution improves
production efficiency ~ improves welfare
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