Oligopoly Theory
8. Irrelevance Results in Mixed
Oligopoly
今日の講義の目的
（１）税・補助金が公企業の行動に与える影響を理解す
る
（２） Irrelevance Results が成立するメカニズムを
理解する
OT:Mixed Oligopoly
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Outline of the 8th Lecture
8-1 Tax Effect in Mixed Oligopoly
8-2 Irrelevance Results in Mixed Oligopoly
8-3 Robustness of Irrelevance Results
8-4 Asymmetry of Order in Private Oligopoly and
Irrelevance Results
8-5 Shadow Cost of Public Funding and Partial
Privatization
8-6 Shadow Cost of Public Funding and Irrelevance
Results
8-7 Irrelevance Results in Free Entry Markets
OT:Mixed Oligopoly
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Tax Effect in Mixed Oligopoly
Mujumdar and Pal (1998)
Introducing unit tax t into the Cournot-type model of
De Fraja and Delbono (1998).
Question: Consider the reaction function of a private
firm. Given the outputs of other firms (one public and
m-1 private firms), the optimal output of the private
firm is (increasing in, decreasing in, independent of)
t (tax rate).
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Tax Effect in Mixed Oligopoly :
Public Firm
Introducing unit tax t into the Cournot-type model of
De Fraja and Delbono (1998).
Question: Consider the reaction function of the public
firm. Given the outputs of other firms (m private
firms), the optimal output of the public firm is
(increasing in, decreasing in, independent of) t (tax
rate).
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Tax Effect in Mixed Oligopoly :
Equilibrium Output of Each Private
Firm
Introducing unit tax t into the Cournot-type model of
De Fraja and Delbono (1998).
Question: Consider the equilibrium outputs. The
equilibrium output of each private firm is (increasing
in, decreasing in, independent of) t.
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Tax Effect in Mixed Oligopoly :
Equilibrium Output of the Public
Firm
Introducing unit tax t into the Cournot-type model of
De Fraja and Delbono (1998).
Question: Consider the equilibrium outputs. The
equilibrium output of the public firm is (increasing in,
decreasing in, independent of) t.
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Tax Effect in Mixed Oligopoly :
Welfare Implication
Introducing unit tax t into the Cournot-type model of
De Fraja and Delbono (1998).
Question: Consider the equilibrium welfare (total
social surplus). For t >0, the equilibrium welfare is
(increasing in, decreasing in, independent of) t.
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Optimal Subsidy in Mixed Oligopoly
White(1996)
Introducing subsidy policy into the Cournot-type
model of De Fraja and Delbono (1989).
The government chooses unit subsidy s so as to
maximizes resulting welfare
Results: Privatization affects neither optimal subsidy
rate nor resulting welfare
→Privatization does not matter under optimal
subsidy policy (Irrelevance Results)
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Optimal Subsidy in Mixed Oligopoly
Introducing subsidy policy into the Cournot-type model
of De Fraja and Delbono (1989).
The government chooses unit subsidy s so as to
maximizes resulting welfare. Consider the duopoly
case. Suppose that s* yields the first best in private
duopoly. Let y* denote the optimal output of each
private firm in private duopoly at the first best outcome.
(Henceforth we call it the base case).
Question: Consider the reaction function of the public
firm (firm 0) in mixed duopoly. Suppose that s=s*.
R0(y*) (>,<,=) y*.
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Optimal Subsidy in Mixed Oligopoly
Introducing subsidy policy into the Cournot-type model
of De Fraja and Delbono (1989).
The government chooses unit subsidy s so as to
maximizes resulting welfare. Consider the duopoly
case. Suppose that s* yields the first best in private
duopoly. Let y* denote the optimal output of each
private firm in private duopoly at the first best outcome.
Answer: Consider the reaction function of the public
firm in mixed duopoly. Suppose that s=s*. R0(y*) = y*.
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Optimal Subsidy in Mixed
Oligopoly: Partial Privatization
Tomaru (2006)
Consider the base case. Suppose that the public firm
is partially privatized and its objective is convex
combination of welfare and its own
Question: Consider the reaction function of the semi
public firm (firm 0) in mixed duopoly. Suppose that
s=s*. R0(y*) (>,<,=) y*. .
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Optimal Subsidy in Mixed Oligopoly:
Non-Profit Maximizing Private Firm
Kato and Tomaru (2007)
Consider the base case. Suppose that the private
firm's objective is convex combination of its own profit
and its revenue , a la Fershtman and Judd (1987).
Question: Consider the reaction function of the public
firm (firm 1). Suppose that s=s* (the optimal subsidy
rate in private duopoly). R0(y*) (>,<,=) y*.
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Optimal Subsidy in Mixed
Oligopoly: Product Differentiation
Hashimzade et al. (2007)
Consider the base case. Suppose that the demand of
firm 0 is p0=a-Y0-βY1 p1=a-Y1-βY2 where a and β are
positive constants and β∈(0,1].
Question: Consider the reaction function of the private
firm (firm 1). Suppose that s=s* (optimal subsidy rate
in private oligopoly). R1(y*) (>,<,=) y*.
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Optimal Subsidy in Mixed
Oligopoly: Public Leadership
Poyago-Theotoky (2001)
Consider the base case. Suppose that the public firm
is Stackelberg Leader.
Question: Suppose that s=s*. Consider the
equilibrium output of the public firm (firm 0). y0E (>,<,=)
y*.
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Optimal Subsidy in Mixed
Oligopoly: Private Leadership
Saito and Tomaru (2009)
Consider the base case. Suppose that the public firm
is Stackelberg Follower.
Question: Suppose that s=s*. Consider the
equilibrium output of the private firm (firm 0). y1E
(>,<,=) y*.
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Optimal Subsidy in Mixed
Oligopoly: Private Leadership
Consider the base case. Suppose that the public firm
is Stackelberg Follower.
Answer : Suppose that s=s*. Consider the equilibrium
output of the private firm (firm 0). y1E > y*, since firm 1
can reduce the rival's output by increasing its output.
Irrelevance result on subsidy rate does not holds
under private leadership.
Question: Does irrelevance result on welfare hold
under private leadership?
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Irrelevance Results
The irrelevance result on subsidy rate does not hold
under private leadership in mixed duopoly, but the
irrelevance result on welfare is quite robust.
Exception
Fjell and Heywood (2004): Privatization is relevant
under asymmetric order of moves among private firms.
Asymmetry after privatization of the public firm yields
the relevance result on welfare.
←Tinbergen Theorem
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Matsumura and Tomaru
(unpublished)
Introducing excess burden of taxation. One dollar of
subsidy costs (1+λ) dollars
(1)The first best outcome is not achieved.
(2)The government has an incentive to economizes
subsidy.
(3)λaffects the behavior of the public firm.
The output of public firm is increasing in λ.
λ=0 →standard marginal cost pricing
λ= ∞→profit maximizing
Similar to partial privatization approach.
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The Model
Players: government, firm 0 (public firm), firm 1
(private firm),
Payoffs: welfare (government, public firm),
Its own profits (private firm)
(1) Government sets s.
(2) Given s, firms faces Cournot competition.
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Notations
s: unit subsidy rate
λ: excess burden of taxation
Ri: Firm i's reaction function at Cournot competition
qi: Firm i's output, Q: Total output
Ci(qi) =0.5k(qi)2 : Firm i's production cost
P(Q)=a-Q: linear demand function
πi: Firm i's profit,
CS: Consumer surplus,
W: social surplus,
Superscript M: Equilibrium value in mixed duopoly
Superscript P: Equilibrium value in private duopoly
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Welfare
Before Privatization
W=CS + profits of firms – total subsidy -λ (subsidy for
the private firm).
After Privatization
W=CS + profits of firms – (1+ λ) total subsidy +
λ( revenue from selling the stocks of the former public
firm)
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Result on optimal subsidy
(1) Either sM>sP or sM<sP is possible ~ Privatization
affects optimal subsidy rate (Relevance result)
When λ is large, sM>sP .
The government has a stronger incentive to reduce s in
private duopoly than in mixed duopoly since it must
pay the subsidy for both firms.
When λ is small (but positive) , sM<sP .
In private oligopoly both firms' productions are too
small when s=0. The government has a stronger
incentive to raise s in private duopoly than in mixed
duopoly since it stimulate production of both firms.
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Result on welfare
(2) WM>WP for any λ>0
~ Privatization affects welfare (Relevance result)
The government has to pay subsidy for both firms in
private duopoly.
In mixed duopoly the public firm produces more than
the private firm even when s is small ~ welfare
improving since the government can economizes
subsidy.
Remark: Privatization can improve welfare if the
privatization reduces firm 0's production cost.
Nevertheless, privatization still affects welfare
(Relevance Result still holds).
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Extension 1:Stackelbergs
Consider two Stackelberg models.
One is Public Leadership (Firm 0 is the Stackelberg
Leader) and the other is Private Leadership.
Let superscript L denote the equilibrium value of the
public leadership and let superscript F denote the
equilibrium value of the public followership (private
leadership).
Result WF=WL=WM=WP if λ=0.
WF>WL>WM>WP if λ>0 (Relevance Result).
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Extension 2:Endogenous Timing
Consider the observable delay game.
There are two possible time periods for output choice .
In the first stage, firm i simultaneously chooses
whether it likes to be the leader (ti=L) or the follower
(t=F). If two players' choices are consistent, i.e., one
chooses to be the leader and the other does to be the
follower, they get the equilibrium payoffs of a agreed
timing Stackelberg. Otherwise, they receive the
equilibrium payoffs in Cournot.
After observing the timing the government chooses
optimal tax rate so as to control the outputs of firms.
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Results in Endogenous Timing
Public Leadership constitutes an equilibrium regardless
of λ, while Private leadership is not always.
~Desirable distribution of roles between public and
private firms may not realized in observable delay
game.
→sharp contrast to Pal (1998) and Matsumura (2003b).
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Summary
Introducing shadow cost of public funding (excess
burden of taxation) changes the results in subsidized
mixed oligopoly.
Privatization matters under shadow cost of pubic
funding.
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Free Entry
Even without excess burden of taxation, privatization
matters if we consider free entry (Cato and
Matsumura ,unpublished).
Introducing subsidy into Matsumura and Kanda (2005).
One public firm compete against private firms.
(1)The government chooses subsidy rate s.
(2)Each private firm chooses whether or not to enter the
market.
(3) Firms face Cournot competition.
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Free Entry
Subsidy affects both the number of entering private
firms and the output of each private firm.
Result 1 Optimal subsidy rate is 0 if the demand is linear,
positive if it is concave, and negative if it is convex, in
both mixed and private oligopoly. →Linear demand
yields irrelevance result on subsidy rate but it crucially
depends on the linearity of the demand.
Result 2 Welfare is higher in Mixed Oligopoly than in
Private Oligopoly (Relevance Result on Welfare).
Irrelevance result again holds if we adopt two part taxsubsidy scheme. ←Tinbergen Theorem
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