Oligopoly Theory 8. Irrelevance Results in Mixed Oligopoly

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Oligopoly Theory 8. Irrelevance Results in
Mixed Oligopoly
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Irrelevance Results ???????????????
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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
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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 gt0, 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 ss. R0(y) (gt,lt,) 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 ss. 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 ss. R0(y) (gt,lt,) 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 ss (the optimal subsidy rate in private
duopoly). R0(y) (gt,lt,) 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 p0a-Y0-ßY1 p1a-Y1-ßY2 where a and
ß are positive constants and ß?(0,1. Question
Consider the reaction function of the private
firm (firm 1). Suppose that ss (optimal subsidy
rate in private oligopoly). R1(y) (gt,lt,) 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 ss. Consider the equilibrium output of the
public firm (firm 0). y0E (gt,lt,) 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 ss. Consider the equilibrium output of the
private firm (firm 0). y1E (gt,lt,) 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 ss. Consider the equilibrium output of the
private firm (firm 0). y1E gt 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?) dollars (1)The
first best outcome is not achieved. (2)The
government has an incentive to economizes
subsidy. (3)?affects the behavior of the public
firm. The output of public firm is increasing in
?. ?0 ?standard marginal cost pricing ?
8?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 ? 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 pi 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 WCS profits of firms
total subsidy -? (subsidy for the private firm).
After Privatization WCS profits of firms
(1 ?) total subsidy ?( revenue from selling
the stocks of the former public firm)
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Result on optimal subsidy
(1) Either sMgtsP or sMltsP is possible
Privatization affects optimal subsidy rate
(Relevance result) When ? is large, sMgtsP . 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 ? is
small (but positive) , sMltsP . In private
oligopoly both firms' productions are too small
when s0. 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) WMgtWP for any ?gt0 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 1Stackelbergs
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 WFWLWMWP if ?0.
WFgtWLgtWMgtWP if ?gt0 (Relevance Result).
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Extension 2Endogenous 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 (tiL) or the
follower (tF). 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 ?, 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 tax-subsidy scheme.
?Tinbergen Theorem