Anticompetitive consequence of the nationalization of a public
enterprise in a mixed duopoly
Toshihiro Matsumura∗
Institute of Social Science, the University of Tokyo
May 25, 2012
Abstract
I investigate a mixed duopoly of a homogeneous product market. A welfare-maximizing public
enterprise competes against a profit-maximizing private one in the Bertrand fashion. I find that
the mixed duopoly yields a collusive outcome or limit pricing by the private firm deterring the
monopoly by the public firm. This result implies that the goodwill (welfare-maximizing objective)
of the nationalized firm may result in the worst outcome (collusive outcome) for social welfare. I
also find that both the two possible outcomes in mixed Bertrand competition yield a smaller total
output and a larger profit for the private firm than mixed Cournot competition.
JEL classification numbers: D43, H44, L32
Key words: monopoly outcome, mixed markets, Bertrand
∗
Correspondence: Institute of Social Science, the University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan. Phone: (81)-3-5841-4932. Fax: (81)-3-5841-4905. E-mail: [email protected]
1
1
Introduction
In the recent financial crisis, many private enterprises facing financial problems were nationalized,
either fully or partially. Studies on mixed oligopolies involving both state-owned public enterprises
and private enterprises have recently attracted more attention and have become increasingly popular.
Before the crisis, many public firms existed and competed against private firms in a wide range of
industries. Such mixed markets have been intensively investigated in the literature.1
Merrill and Schneider (1966) is a pioneering study on mixed oligopoly. This study and many
subsequent works assume that a public firm maximizes social welfare (the sum of consumer surplus
and firms’ profits), while a private firm maximizes its own profits. Most papers in this field highlight
a competition accelerating effect of the public firm. The welfare-maximizing firm produces more than
the private firms and this aggressive behavior by the public firm increases the total output. Although
this aggressive behavior can reduce welfare, it always increases total output and improves consumer
welfare.2
In this paper, I show that this is not true when firms compete in terms of price. I investigate a
mixed duopoly in a homogeneous product market. A welfare-maximizing public firm competes against
a profit-maximizing private firm.3 I find that a mixed duopoly yields either collusive (monopoly)
pricing or limit pricing by the private firm that deters the monopoly of the public firm. In other
1
Recently, studies on mixed markets or mixed oligopoly, involving both private and public enterprises, have become
increasingly popular. See Ishida and Matsushima (2009) and the works cited by them for the recent developments in
this field.
2
3
See De Fraja and Delbono (1989) and Matsumura (1998), among others.
I do not consider the case where the government nationalizes the two firms. As pointed out by Merrill and Schneider
(1966), the most efficient outcome is achieved by the nationalization of all firms in the case where nationalization does
not change the firms’ costs (i.e., there is no X-inefficiency in the public firm). The need for an analysis of mixed
oligopoly lies in the fact that it is impossible or undesirable, for political or economic reasons, to nationalize an entire
sector. For example, without competitors, the public firms may lose the incentive to improve their costs, resulting in a
loss of social welfare. As such, I neglect the possibility of all firms being nationalized.
2
words, the nationalization of one private firm may foster collusion and the output level can fall to
the monopoly level. I present an example indicating that collusive outcome appears in equilibrium
for a wide range of parameter values.
Next, I compare this equilibrium outcome with that in a Cournot model. Because the price competition in a mixed duopoly fosters a collusive outcome, it is naturally expected that the total output
under Bertrand competition would be less than that under Cournot. I show that this conjecture is
true.4
The paper is organized as follows. Section 2 formulates the Bertrand model with strictly convex
costs. Section 3 presents the main result. Section 4 compares the Bertrand and Cournot models.
Section 5 investigates an alternative model (constant marginal cost case). Section 6 concludes the
paper.
2
The model
I formulate a duopoly model. Firm 0 is a welfare-maximizing state-owned public firm and firm 1 is
a profit-maximizing private firm. The firms produce perfectly substitutable commodities for which
the market demand function is given by D(p) : R+ → R+ . Let P (Q) denote the inverse demand
function. I make the following assumptions on the demand function.
Assumption 1 D(p) is twice differentiable and decreasing as long as D(p) > 0.
Assumption 2 P (Q) + P Q is decreasing in Q.
Assumption 2 implies that the marginal revenue for the monopolist is decreasing in the output. These
4
Ghosh and Mitra (2010) have already shown that the mixed Bertrand duopoly yields higher equilibrium prices
than the mixed Cournot duopoly in a differentiated product market. In a private market in which both firms are
profit-maximizers, in contrast to in a mixed market, Cournot competition is less severe than Bertrand competition
under moderate conditions. See Cheng (1985), Okuguchi (1987), Singh and Vives (1984), and Vives (1985). However,
even in a private duopoly, Bertrand competition can yield higher prices. See Dastidar (1997, 2001), Häckner (2000)
and López and Naylor (2004).
3
are standard assumptions in the literature.
Two firms have an identical cost function C : R+ → R+ .
Assumption 3 C(0) = 0, and C(·) is increasing, continuously differentiable, and strictly convex.
Assumption 4 C (0) := c > 0 and D(c) > 0.
If Assumption 4 is not satisfied, no firm produces in equilibrium in both the mixed and private
markets.
In what follows, the inverse of C (·) is denoted by Y (·), the supply function of a firm when the
firm is a price taker. The profit of firm i (i = 0, 1) is given by Πi = pi qi − C(qi ) and welfare is given
by the sum of consumer surplus and the total profits of the two firms as
W (q0 , q1 ) =
0
q0 +q1
P (y)dy − C(q0 ) − C(q1 ).
Note that given q0 and q1 , neither p0 nor p1 affects W (both prices affect the distribution of total
social surplus only). Needless to say, prices can be important for welfare because they affect the
firms’ outputs.
Each firm i(i = 0, 1) names its price pi ∈ [c, ∞) simultaneously. I assume efficient rationing, i.e.,
a consumer with a higher willingness to pay consumes the product in an earlier round. Then the
rationing rule is as follows. If pi < pj (i, j = 0, 1, i = j), firm i chooses its output qi ≤ D(pi ) and
firm j chooses its output qj ≤ max{D(pi ) − qi , 0}. If p0 = p1 (i, j = 0, 1, i = j), firm i chooses its
output qi ≤ max{D(pi )/2, D(pi ) − qj }.
Let pW be the Walrasian equilibrium price. It is derived from D(pW ) = 2Y (pW ). Let pM
0 be
the monopoly price of firm 0. Because firm 0 is a welfare-maximizer, the price is derived from
M
D(pM
0 ) = Y (p0 ). Note that the public monopoly is not the first best because the marginal cost is
increasing and production by two firms reduces total production costs.
Let q C be the collusive output of each firm when the two firms maximize joint profits. Let
4
QC := 2q C and pC = p(QC ). q C is derived from
P (QC ) + 2P (QC )q0C = C (q0C ).
(1)
Let pL
1 be the price satisfying the following:
L
M
M
L
W (D(pL
1 )/2, D(p1 )/2) = W (D(p0 ), 0), W (D(p ), D(p )) < W (D(p0 ), 0) ∀p > p1 .
(2)
In other words, the two firms’ production under the price pL
1 yields the same welfare level as under
L
L
public monopoly, and the welfare level is lower for any price higher than pL
1 . Let Q := D(p1 ) (total
output) and q L = QL /2 (the output of each firm). As explained in the next section, setting p1 = pL
1
is “limit pricing” to deter the monopoly by firm 0. Because the demand function is decreasing and
continuous, and the marginal cost function is increasing and continuous, I can show that under
M
L
L
Assumptions 1,3, and 4, pL
1 is uniquely determined and is strictly larger than p0 . Let D(p1 ) be Q
L
and D(pL
1 )/2 be q .
M
W and pC > pW .
Lemma 1 Suppose that Assumptions 1–4 are satisfied. Then, pL
1 > p0 > p
Proof See the Appendix.
I now present a supplementary result on the equilibrium outcomes.
Lemma 2 Suppose that Assumptions 1–4 are satisfied. If there exists an equilibrium where p0 = pa ,
p1 = pb , q0 = q a , and q1 = q b , then there exists another equilibrium where p0 = pA , p1 = pb , q0 = q a ,
and q1 = q b for any pA ≤ pa .
Proof See the Appendix.
Under the price p0 = pA < pa , firm 0 can choose the same output level when p0 = pa , resulting in
the same welfare level. From Lemma 2, I can assume that p0 = c without loss of generality.
5
3
Equilibrium
3.1
Allocation of outputs
I discuss the allocation of outputs given the prices of the two firms. From Lemma 2, I can assume
that p0 = c.
Firm 0 maximizes W subject to q0 ≤ max{D(p0 ) − q1 , 0}. Because q0 affects q1 and W depends
on q1 , firm 0 must consider how q0 affects q1 .
Firm 1 maximizes p1 q1 −C(q1 ) subject to q1 ≤ max{D(p1 )−q0 , 0}. If D(p1 )−q0 ≤ 0, then q1 = 0.
If D(p1 ) − q0 > 0, then q1 = min{Y (p1 ), D(p1 ) − q0 }. Anticipating the above behavior of firm 1,
firm 0 chooses q0 to maximize welfare. If q0 ≥ D(p1 ), then q1 = 0 regardless of q0 . Given the public
monopoly, welfare is maximized when q0 = Y (pM
0 ). If q0 ≤ D(p1 ), then q1 = min{D(p1 ) − q0 , Y (p1 )}.
As long as D(p1 ) − q0 ≤ Y (p1 ), q0 + q1 = D(p1 ) regardless of q0 . Thus, welfare is maximized when
q0 = q1 = D(p1 )/2. If D(p1 ) − q0 > Y (p1 ), then q1 = Y (p1 ). Given the above output of firm 1,
welfare is maximized when the following equation is satisfied:
P (Y (p1 ) + q0 ) = C (q0 ).
(3)
From the definition of pL
1 , I get that the public monopoly yields a higher welfare if and only if
M
L
W ≤ p ≤ pL , firm 0 chooses
p1 > pL
1
1 . Thus, q0 = q0 and q1 = 0 if and only if p1 > p1 . If p
1
q0 = D(p1 )/2 as discussed above. Note that D(p1 ) − q0 D(p1 )/2 ≤ Y (p1 ) when pW ≤ p1 . If p1 < pW ,
q0 is derived from equation (3). As expected, firm 1 never chooses p1 < pW (it is dominated by
p1 = pW ).
3.2
Equilibrium price and the resulting welfare
I now present the main result on the equilibrium outcome. There exists a pure strategy equilibrium
and the equilibrium output is either QC (collusive output) or QL (limit-pricing output).
Proposition 1 Suppose that Assumptions 1–4 are satisfied. Then, there exist pure strategy equilibria,
6
C
and in every equilibrium, p1 = min{pL
1 , p } and q0 = q1 = D(p1 )/2.
Proof See the Appendix.
In equilibrium, both firms produce the same output, either q C or D(pL
1 )/2. The public firm has
an incentive to induce q0 = q1 to minimize total production cost unless p1 > pL
1 . Anticipating this
C
behavior of the public firm, the private firm chooses p1 = min{pL
1 , p }. As a result, the total output
level can fall to the monopoly level under Bertrand competition in a mixed duopoly.
3.3
Examples
Proposition 1 states that in the equilibrium, firm 1’s pricing is either collusive or limit priced. I now
present an example where both collusive and limit-pricing outcomes can appear in equilibrium for a
wide range of parameter values in the case with linear demand and quadratic cost functions.
Suppose that P (Q) = a − bQ and C = dq12 . Then, I have
(b + 2d) − d(b + 2d)
a
L
, Q =a
.
Q =
b + 2d
(b + d)(b + 2d)
C
3
2
2
3
QC > QL and thus, pC < pL
1 if and only if d + 4bd + 2b d − b > 0. A sufficient but not necessary
condition for pC < (>) pL
1 is d > b/3 (d < b/4).
4
Comparison between Bertrand and Cournot
In this section, we discuss the equilibrium outcome when both firms face Cournot competition and
compare it to the outcome obtained in the previous section. Let superscript “Q” denote the equilibrium outcome of this quantity-setting model. The first-order conditions for firm 0 and firm 1
respectively, are
P = C (q0 )
(4)
P + P q1 = C (q1 ).
(5)
7
Assumption 5 (Strategic substitute for the private firm) P + P q1 < 0.
Under Assumption 5, the reaction curve of firm 1 is downward-sloping (strategic substitute).
This is also a standard assumption in the literature. Assumption 5 ensures that the second-order
condition for firm 1 is satisfied. It also ensures the uniqueness of the equilibrium in the mixed Cournot
duopoly model. Note that the reaction curve of firm 0 is downward-sloping and that the second order
condition for firm 0 is satisfied without Assumption 5.
Let qiQ be the equilibrium output of firm i. Let q0Q + q1Q := QQ . I present a result on the comparison between quantity competition and price competition in a mixed duopoly.
Proposition 2: Suppose that Assumptions 1–5 are satisfied. (i) QQ > max{QC , QL }, i.e., Cournot
competition yields a larger equilibrium total output than Bertrand competition. (ii) Cournot competition yields a smaller profit for firm 1 than Bertrand competition.
Proof See the Appendix.
Bertrand competition yields a lower equilibrium output and smaller private firm profit than
Cournot competition in the mixed duopoly, in contrast to the standard results of private oligopolies.
5
Constant marginal cost case
In the previous sections, I assume that firms’ marginal costs are increasing. If the marginal costs
are constant and there is no cost difference between the public and private firms, the marginal cost
pricing by the public monopoly yields the first-best outcome and this always appears in equilibrium.
Thus, it does not make sense to investigate mixed oligopolies in such circumstances. This is why the
assumption of increasing marginal costs is popular in the literature on mixed oligopolies.
However, there is another popular model formulation of mixed oligopolies. The marginal costs
are constant and the marginal cost of the public firm is higher than that of the private firm.5 In this
5
See, among others, Mujumdar and Pal (1998), Pal (1998), and Matsumura (2003).
8
section, I consider this situation.
Assumption 6 the marginal costs of the two firms are constant. The marginal cost of firm 0 is
strictly higher than the marginal cost of firm 1.
Let c0 be the marginal cost of firm 0 and c1 (< c0 ) be the marginal cost of firm 1. Let q1M 2 be
firm 1’s monopoly output, where superscript “2” means the second model (constant marginal cost
2
M2
model). q1M 2 is derived from P + P q1 = c1 . Let D(pM
1 ) := q1 . Assumptions 1 and 2 ensure that
2
(p1 − c1 )D(p1 ) is increasing in p1 for p1 ∈ [c1 , pM
1 ).
If firm 0 is the monopolist, it sets p0 = c0 because firm 0 is a welfare-maximizer.
Let pL2
1 be the price satisfying the following:
L2
W (0, D(pL2
1 )) = W (D(c0 ), 0), W (0, D(p )) < W (D(c0 ), 0) ∀p > p1 .
(6)
L2
Because c1 < c0 , pL2
1 must be larger than c0 . If firm 1 names p1 = p1 , for firm 0, choosing p0 = c0
and producing D(c0 ) is indifferent from choosing q0 = 0 and letting firm 1 produce D(pL2
1 ).
I now discuss the allocation of outputs given the prices of the two firms. I can show that Lemma
2 holds in this setting, too. Thus, I can assume that p0 = c0 , and principles similar to those in
Section 3.1 can apply. I have the following output allocation, depending on the price of firm 1:
(q0 , q1 ) = (D(c0 ), 0) if p1 > pL2
1 and (q0 , q1 ) = (0, D(p1 )) if otherwise.
The underlying intuition is as follows. In contrast to the increasing marginal cost case, the
monopoly by firm 1 is best from the viewpoint of production efficiency. As long as q1 > 0, Q = D(p1 )
regardless of q0 , and welfare is decreasing in q0 . Thus, q0 = 0 is best for welfare. However, if p1 > pL2
1 ,
q0 = Q = D(c0 ) yields higher welfare than q1 = Q = D(p1 ); as such, firm 0 chooses this output.
In contrast to the case of increasing marginal costs, firm 0 has no incentive to divide production
between the two firms. Thus, only firm 1, a more efficient firm, produces in equilibrium.
M 2 L2
If firm 1 chooses p1 > pL2
1 , its profit is zero. As such, firm 1 chooses p1 = min{p1 , p1 } and the
resulting outcome is (q0 , q1 ) = (0, D(p1 )). This yields the following proposition.
9
Proposition 3 Suppose that Assumptions 1, 2, 4, and 6 are satisfied. In equilibrium, only firm 1
M2
produces and q1 = D(min{pL2
1 , p1 }).
Proof See the Appendix.
The joint-profit maximization is achieved by firm 1’s monopoly because firm 1’s marginal cost is
lower than firm 0’s regardless of the firms’ outputs. Thus, we can say that the essential property of
Proposition 1 holds in this setting, although the resulting allocation of production is different.
I now compare this outcome with that in the mixed Cournot duopoly. Consider the quantity
competition. The first-order condition for firm 0 and firm 1, respectively, are
P = c0 ,
(7)
P + P q1 = c1 .
(8)
The second-order conditions are satisfied. Let qiQ2 be the equilibrium output of firm i in this game
and QQ2 := q0Q2 + q1Q2 .
2
Q2 = D(c ) (this is
If c0 < pM
0
1 , the interior solution exists and is unique. In the equilibrium, Q
Q2 Q2
2
derived from (7)). If c0 ≥ pM
1 , the unique equilibrium outcome is the corner solution and (q0 , q1 ) =
2
(0, D(pM
1 )), i.e., the monopoly by firm 1.
2
Q2 >
Proposition 4: Suppose that Assumptions 1, 2, 4, 5, and 6 are satisfied. If c0 < pM
1 , (i) Q
M2
max{D(pL2
1 ), D(p1 )} i.e., Cournot competition yields a larger equilibrium total output than Bertrand
competition. (ii) Cournot competition yields a smaller equilibrium profit for firm 1 than Bertrand
competition.
Proof See the Appendix.
2
Again, a result similar to Proposition 2 is derived. If c0 ≥ pM
1 , the monopoly by firm 1 appears
regardless of whether the two firms face Bertrand or Cournot competition, or whether firm 0 is a
public or a private firm. Except for this case (i.e., unless the cost difference between the two firms is
10
not too high), mixed Cournot yields a larger total output and smaller profits than mixed Bertrand.
2
Finally, I discuss the effect of the privatization of firm 0. Suppose that c0 < pM
1 . Suppose that
firm 0 is privatized and is a profit-maximizer. Then, the equilibrium price is c0 , which is equal to
the price under mixed Cournot duopoly. Thus, under Bertrand competition, privatization increases
the equilibrium total output and improves welfare.
2
Proposition 5: Suppose that Assumptions 1, 2, 4, and 6 are satisfied. If c0 < pM
1 , the privatization
of firm 0 increases total output and improves welfare.
This result is not true if firms face Cournot competition. Under Cournot competition, the privatization of firm 0 reduces total output, and it is ambiguous whether or not privatization improves
welfare.6
6
Concluding remarks
In this paper, I investigate a mixed duopoly wherein a welfare-maximizing public enterprise competes
against a profit-maximizing private firm. I show that in a mixed Bertrand duopoly, the private firm’s
pricing is either collusive pricing (setting the joint-profit-maximizing price) or limit-pricing (setting
the price that deters public monopoly). In either case, the total output level is lower and the total
profit level is higher than that under Cournot. I present an example indicating that collusive pricing
appears for a wide range of parameter values. These results indicate that the nationalization of a
private firm in a private duopoly can be very harmful for welfare, especially when firms face Bertrand
competition. The goodwill (welfare-maximizing objective) of the nationalized firm can result in the
worst outcome (collusive outcome) in terms of social welfare.
I assume that both firms choose their prices simultaneously.7 However, Proposition 1 does not
6
7
See De Fraja and Delbono (1989) and Matsumura (1998).
In the literature on mixed oligopolies, public and private leaderships and endogenous timing are intensively
investigated. See Pal (1998).
11
depend on this assumption. Both price leadership by the public firm and that by the private firm
yield the same equilibrium outcome presented in Proposition 1. Thus, our result does not depend on
the assumption of simultaneous moves in pricing.
In this paper, I assume that both firms are domestic. Nationality plays an important role in
the literature on mixed oligopoly because the public firm maximizes domestic welfare rather than
global welfare. Considering this problem and investigating the relationship between the privatization
and trade policies in this context remain for the future.8
8
See Corneo and Jeanne (1994) and Fjell and Pal (1996).
12
Appendix
Proof of Lemma 1
M
Because D(·) is decreasing, Y (·) is increasing, D(pW ) = 2Y (pW ), and D(pM
0 ) = Y (p0 ), I have
W = C (D(pW )/2), M R = C (D(pC )/2) and the price is higher than the
pW < pM
0 . Because p
M
M
marginal revenue, I have pW < pC . Because W (D(pM
0 )/2, D(p0 )/2) > W (D(p0 ), 0), from the
L
M
definition of pL
1 , I have p1 > p0 .
Q.E.D.
Proof of Lemma 2
Suppose that p0 = pa , p1 = pb , q0 = q a , and q1 = q b constitute an equilibrium. Suppose that firm 0
deviates from this equilibrium strategy and chooses p0 = pA < pa .
Suppose that pa ≤ pb . Firm 0 can choose q0 = q a , and if firm 0 chooses it, firm 1 chooses
q1 = max{Y (pb ), D(pb ) − q a } = q b . Because the deviation does not affect the resulting outcome,
p0 = pA must also be a best reply given p1 = pb . Note that because the outputs before the deviation
are best for welfare given p1 (otherwise, p0 = pa , p1 = pb , q0 = q a , and q1 = q b are not equilibrium
outcomes), firm 0 has no incentive to change its output. Because the pricing of firm 0 does not affect
q0 and q1 , p1 = pb is still a best reply for firm 1.
Suppose that pa > pb . Before the deviation, firm 1 chooses q1 = Y (pb ) = q b . Consider the
situation after the deviation. If pA > pb , then q1 = Y (pb ) = q b . Firm 0 can choose q0 = q a .
Thus, p0 = pA must also be a best reply given p1 = pb . If pA < pb , firm 0 again can choose
q0 = q a . Then, firm 1 chooses q1 = min{Y (pb ) = q b , D(pb ) − q a }. Because q a ≤ D(pa ) − q b ,
q b ≤ D(pa ) − q a < D(pb ) − q a . Thus, q1 must be q b after the deviation. Under these conditions,
p0 = pA must also be a best reply given p1 = pb . Note that because the outputs before the deviation
are best for welfare given p1 , firm 0 has no incentive to change its output. Because the pricing of
firm 0 does not affect q0 and q1 , p1 = pb is still a best reply for firm 1.
Proof of Proposition 1
13
Q.E.D.
C
First, I show that there exists an equilibrium where p0 = c and p1 = min{pL
1 , p }. In the proof of
Lemma 2, I have shown that firm 0 has no incentive to change the above pricing. As such, I discuss
the pricing of firm 1. If firm 1 deviates from the above pricing and chooses p1 > pL
1 , then q1 = 0,
and therefore, firm 1’s profit becomes zero, a contradiction. Suppose that pC < pL
1 . If firm 1 chooses
C
p1 ∈ [pC , pL
1 ), then q0 = q1 = D(p1 )/2. Because pD(p1 )/2 − c(D(p1 )/2) is maximized when p1 = p ,
the deviation from p1 = pC reduces the profit of firm 1, a contradiction. If firm 1 deviates and
C
chooses p1 ∈ [pW , min{pL
1 , p }), then q0 = q1 = D(p1 )/2. Because pD(p)/2 − c(D(p)/2) is decreasing
in p for p ≤ pC , the deviation reduces the profit of firm 1, a contradiction. If firm 1 deviates from
the above pricing and chooses p1 ≤ pW , then q1 = Y (p1 ). Because pY (p) − c(Y (p)) is decreasing
in p, p1 < pW yields a smaller profit for firm 1 than p1 = pW . Because I have already shown that
C
p1 = pW yields a smaller profit for firm 1 than p1 = min{pL
1 , p }, the deviation reduces the profit of
firm 1, a contradiction.
Next, I show that no other equilibrium exists. In the first part of this proof, I have already
C
shown that p1 = min{pL
1 , p } is the unique best reply for firm 1. From Lemma 2, if there is no
equilibrium where p0 = c, p1 = pb , q0 = q a and q1 = q b , then there is no equilibrium where p0 = pa ,
C
p1 = pb , q0 = q a and q1 = q b . Because p0 = c yields the unique equilibrium where p1 = min{pL
1,p }
C
and q0 = q1 = D(p1 )/2, in every equilibrium, p1 = min{pL
1 , p } and the resulting outputs are
q0 = q1 = D(p1 )/2.
Q.E.D.
Before proving Proposition 2, I present two supplementary lemmas.
Lemma A1 q0Q > q1Q .
From Assumption 1, the left-hand side in (4) is larger than that in (5). From Assumption 3, we have
q0Q > q1Q . Q.E.D.
Lemma A2 If QL := 2q L ≥ QQ := q0Q + q1Q , then W (q L , q L ) > W (q0Q , q1Q ).
14
Proof If QL ≥ QQ , then
W (q0Q , q1Q ) < W
QQ QQ ,
≤ W (q L , q L ),
2
2
where the first inequality is derived from Assumption 3 and the second inequality is derived from
P (QQ /2) ≥ P (QL ) > C (q L ) ≥ C (QQ /2) (and thus, welfare is increasing in the output for Q ∈
[QQ , QL ].) Q.E.D.
Proof of Proposition 2
I first show that QQ := q0Q + q1Q > QC := q0C + q1C = 2q0C , and then show that q0Q + q1Q > QL :=
q0L + q1L = 2q0L .
(a) Proof of QQ > QC
First, I show that q0Q > q0C . Suppose that q0Q < q0C . Then from Lemma A1, I have QQ < QC , and
thus P (QQ ) > P (QC ). From (1), I have that P (0C ) = C (q0C )−2P (QC )q0C > C (q0C ) > C (q0Q ) where
the last inequality is derived from C > 0 and q0Q < q0C . Thus, (4) is never satisfied, a contradiction.
Next, we show that QQ > QC . Suppose otherwise. From Assumptions 1 and 2, we have
P (QQ ) + P (QQ )q1Q > P (QC ) + 2P (QC )q0C . q0Q + q1Q ≤ 2q0C and q1Q < q0Q imply q1Q < q0C . Thus,
Assumption 3 implies C (q1Q ) < C (q0C ). Under these conditions, it is impossible that (5) and (1)
hold simultaneously, a contradiction.
(b) Proof of QQ > QL
From Lemma A2, I have that W (q0Q , q1Q ) > W (q L , q L ) implies QQ > QL . I will show that W (q0Q , q1Q ) >
W (q L , q L ).
L
L L
M
From the definitions of pL
1 and q , I have W (q , q ) = W (q0 , 0). Let R0 (q1 ) denote the reaction
function of firm 0 in the Cournot mixed duopoly. W (R0 (q1 ), q1 ) is increasing in q1 as long as
P (R0 (q1 ) + q1 ) > C (q1 ). Because W (q0Q , q1Q ) = W (R0 (q1Q ), q1Q ) and W (q0M , 0) = W (R0 (0), 0), I get
W (q0Q , q1Q ) > W (q L , q L ) = W (q0M , 0). Note that q1Q > 0. Q.E.D.
Proof of Proposition 3
15
L2
If firm 1 sets p1 > pL2
1 , its profit is zero. If it sets p1 = p1 , its profit is strictly positive because
L2
pL2
1 > c0 > c1 . Thus, firm 1 sets p1 ≤ p1 .
M2
M2
M2
L2
M2
If pL2
1 ≥ p1 , from the definition of the monopoly price p1 , firm 1 sets p1 = p1 . If p1 < p1 ,
M2
firm 1 sets p1 = pL2
1 because (p1 − c1 )D(p1 ) is decreasing in p1 for p1 ≤ p1 . Q.E.D.
Proof of Proposition 4
2 ≤ c . Because pL2 > c , pM 2 ≤ pL2 . In this case, in both the Bertrand and
Suppose that pM
0
0
1
1
1
1
2
Cournot models, (q0 , q1 ) = (0, D(pM
1 )), and thus, the equality in Proposition 4 holds.
2 > c . In the Cournot model, the equilibrium price is c and both firms 0 and
Suppose that pM
0
0
1
1 produce in the equilibrium. In the Bertrand model, only firm 1 produces in the equilibrium and
M2
L2
p1 = min{pL2
1 , p1 }. Because p1 > c0 , I have Proposition 4.
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Q.E.D.
References
Cheng, L. “Comparing Bertrand and Cournot Equilibria: A Geometric Approach for Price and
Quantity Competition in a Differentiated Duopoly.” RAND Journal of Economics, Vol. 16(1)
(1985), pp. 146–152.
Corneo, G. and Jeanne, O. “Oligopole mixte dans un marche commun.” Annales d’Economie et de
Statistique, Vol. 33 (1994), pp. 73–90.
Dastidar, K.G. “Comparing Cournot and Bertrand in a Homogeneous Product Market.” Journal
of Economic Theory, Vol. 75(1) (1997), pp. 205–212.
Dastidar, K.G. “Collusive Outcomes in Price Competition.” Journal of Economics, Vol. 73(1)
(2001), pp. 81–93.
De Fraja, G. and Delbono, F. “Alternative Strategies of a Public Enterprise in Oligopoly.” Oxford
Economic Papers, Vol. 41(2) (1989), pp. 302–311.
Fjell, K. and Pal, D. “A Mixed Oligopoly in the Presence of Foreign Private Firms.” Canadian
Journal of Economics, Vol. 29(3) (1996), pp. 737–743.
Ghosh, A. and Mitra, M. “Comparing Bertrand and Cournot in Mixed Markets.” Economics Letters,
Vol. 109(2) (2010), pp. 72–74.
Häckner, J. “A Note on Price and Quantity Competition in Differentiated Oligopolies.” Journal of
Economic Theory, Vol. 93(2) (2000), pp. 233–239.
Ishida, J. and Matsushima, N. “Should Civil Servants be Restricted in Wage Bargaining? A MixedDuopoly Approach.” Journal of Public Economics, Vol. 93(3–4) (2009), pp. 634–646.
López, M.C. and Naylor, R.A. “The Cournot-Bertrand Profit Differential: A Reversal Result in a
Differentiated Duopoly with Wage Bargaining.” European Economic Review, Vol. 48(3) (2004),
pp. 681–696.
Matsumura, T. “Partial Privatization in Mixed Duopoly.” Journal of Public Economics, Vol. 70(3)
(1998), pp. 473–483.
17
Matsumura, T. “Endogenous Role in Mixed Markets: A Two Production Period Model.” Southern
Economic Journal, Vol. 70(2) (2003), pp. 403–413.
Merrill, W.C. and Schneider, N. “Government Firms in Oligopoly Industries: A Short-Run Analysis.” Quarterly Journal of Economics, Vol. 80(3) (1966), pp. 400–412.
Mujumdar, S. and Pal, D. “Effects of Indirect Taxation in a Mixed Oligopoly”, Economics Letters,
Vol. 58(2) (1998), pp. 199–204.
Okuguchi, K. “Equilibrium Prices in the Bertrand and Cournot Oligopolies.” Journal of Economic
Theory, Vol. 42(1) (1987), pp. 128–139.
Pal, D. “Endogenous Timing in a Mixed Oligopoly.” Economics Letters, Vol. 61(2) (1998), pp.
181–185.
Singh, N. and Vives, X. “Price and Quantity Competition in a Differentiated Duopoly.” RAND
Journal of Economics, Vol.15(4) (1984), pp. 546–554.
Vives, X. “On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation.”
Journal of Economic Theory, Vol. 36(1) (1985), pp. 166–175.
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