Equilibrium location with elastic demand in mixed duopoly
Minoru Kitahara∗
Graduate School of Social Sciences, Tokyo Metropolitan University
and
Toshihiro Matsumura
Institute of Social Science, University of Tokyo
April 10, 2010
Abstract
We examine a location-price model with elastic demand in a mixed duopoly where a public
firm competes against a private firm. Existing works with inelastic total demand showed that the
location pattern is efficient for social welfare in a mixed duopoly. Under elastic demand, the private
firm faces more severe price competition and it seems natural to expect that the firm has a stronger
incentive to differentiate itself from its rival to relax price competition. Counterintuitively, the
private firm chooses to locate itself too close to the public firm (thereby making the resulting
degree of product differentiation too small) for social welfare. Moreover, in this study, we discuss
the welfare implications of public leadership and privatization of the public firm.
JEL classification numbers: H44, L13, R32
Key words: product differentiation, location-price model, elastic demand, mixed market
∗
Corresponding author: Minoru Kitahara, JSPS Research Fellow, Graduate School of Social Sciences, Tokyo
Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo 192-0397, Japan. Email: [email protected]
1
1
Introduction
Since the 1980s, we have witnessed a global wave of privatization. Nevertheless, public firms are
still active in developed, developing, and transitional economies; indeed, many of them continue to
compete with private firms. Studies on mixed oligopoly involving both private and public enterprises
have increasingly become popular.1 In addition, in the wake of the recent financial crisis, many private
enterprises facing financial problems have been nationalized, either fully or partially. Studies on mixed
oligopoly involving both state-owned public enterprises and private enterprises have attracted the
interest of researchers.
In a mixed oligopoly, public firms rarely manufacture products that are exactly identical to those
manufactured by private firms; thus, products in such a market are differentiated. Competition
between public and private firms influences the product positioning of firms, and therefore, affects
consumer welfare as well as social surplus. In their pioneering work, Cremer et al. (1991) have
investigated this problem. They adopted a standard location-price model on the Hotelling line,
and showed that in a mixed duopoly, the equilibrium location (and hence, the degree of product
differentiation) is optimal for social welfare. Matsumura and Matsushima (2004) showed that their
result is robust because the location is optimal under any cost difference between public and private
firms. However, both the papers, as well as many other papers on location-price with mill pricing,
assumed that the total demand is inelastic; i.e., each consumer consumes one unit of the product.2
We drop the assumption of inelastic demand of the standard location-price model and investigated
the welfare implication of equilibrium location.3 We found that the location of a private firm is too
1
The analysis of a mixed oligopoly dates back to Merrill and Schneider (1966). Recently, the literature on this topic
has become richer and more diverse. See Gil-Moltó and Poyago-Theotoky (2008), Ishida and Matsushima (2009), and
the works cited by them for recent developments in this field.
2
For recent studies that adopted mill-pricing models, see Fjell and Heywood (2002), Inoue et al. (2009), Lu (2006),
Nilssen and Sørgard (2002), and Ogawa and Sanjo (2007). The delivered-pricing model is another model that has been
employed extensively in this field. See Heywood and Ye (2009a,b) and Matsushima and Matsumura (2003, 2006).
3
We adopt a constant elasticity demand function discussed by Anderson and de Palma (2000) and Gu and Wenzel
2
close to that of a public firm for social welfare, i.e., the equilibrium degree of product differentiation
is too small. Our result indicates that the private incentive for increasing product differentiation is
smaller than the social incentive when the private firm’s competitor is a public firm.
This is a counterintuitive result. Under elastic demand, the private firm faces more severe price
competition. A lower degree of product differentiation further decreases the prices. Thus, it is natural
to expect that the private firm has a stronger incentive to differentiate itself from the public firm in
order to relax price competition when demand is elastic. Counterintuitively, the private firm chooses
lesser product differentiation than the social planner. This is because the competition-accelerating
effect by lesser product differentiation is weaker when the demand is elastic. We explain this point
in detail in Section 3.
Next, we discuss public leadership. We found that if the public firm is the leader and chooses
its location before the private firm, it chooses a position closer to the center, resulting in further
reduction of product differentiation. Despite the lower degree of product differentiation, it improves
welfare.
Finally, we investigate an effect of the privatization of the public firm. In a private duopoly with
inelastic total demand, firms have strong incentives to be located distantly from the rivals in order to
relax price competition; this results in maximal product differentiation (d’Aspremont et al., 1979).
We show that this finding holds true under elastic total demand. Thus, privatization reduces welfare.
The rest of this paper is organized as follows. Section 2 formulates the model, while Section 3
presents our main results. Sections 4 and 5 discuss the effects of public leadership and privatization
respectively. Finally, Section 6 concludes the paper.
(2009) in a spatial context.
3
2
The Model
We formulate a duopoly model with a linear city à la Hotelling. Firm 1 is a profit-maximizing private
firm, whereas firm 2 is a welfare-maximizing public firm. Consumers are uniformly distributed with
density 1 along the interval [0, 1]. Let xi ∈ [0, 1] be the location of firm i (i = 1, 2). A consumer
living at x ∈ [0, 1] incurs a transport cost of t(xi − x)2 when (s)he purchases the product from firm
i. The consumers have the following indirect utility function:
V (p) = v̄ − v(p) = v̄ −
p1−α
,
1−α
where v̄ and α are positive constants, and v̄ is sufficiently large. To ensure the interior solution at the
second stage, we assume that α is not too large.4 Firms 1 and 2 produce the same physical product.
The common marginal cost is constant and normalized to zero.
The utility of the consumer located at x is given by the following:
{
u(x) =
V (p1 ) − t(x1 − x)2 if bought from firm 1,
V (p1 ) − t(x2 − x)2 if bought from firm 2.
(1)
The functional form of indirect utility implies that the conditional demand of each consumer is
−V ′ (p) = v ′ (p) = p−α ,
where α represents the elasticity of the demand as
−
v ′′ (p)p
= α,
v ′ (p)
and the model collapses to that with inelastic demand when α = 0.
For a consumer living at x̄(p1 , p2 , x1 , x2 ), where
V (p1 ) − t(x1 − x̄)2 = V (p2 ) − t(x2 − x̄)2 ,
4
We require α < 1/2, otherwise the assumption of an interior solution is not satisfied. See Footnote 6.
4
(2)
the utility remains the same regardless of the firm chosen. Henceforth, we assume that x1 < x2 . We
can rewrite (2) as follows:
x̄ =
x1 + x2 v(p1 ) − v(p2 )
x1 + x2
p1−α
− p1−α
1
2
−
=
−
.
2
2t(x2 − x1 )
2
2(1 − α)t(x2 − x1 )
(3)
−α
5
Firm i’s demand Di (i = 1, 2) is D1 = p−α
1 x̄, D2 = p2 (1 − x̄).
The game runs as follows. In the first stage, each firm i (i = 1, 2) independently chooses its
location xi . Then, the firms’ locational choices are made public. In the second stage, each firm i
simultaneously chooses its price pi ≥ 0. Firm 1’s payoff is its profit, while firm 2’s payoff is social
surplus (consumer surplus plus profits of firms). Firm 1’s profit is Π1 = p1 D1 = p1−α
x̄. The social
1
surplus is
∫
W
3
= Π1 + Π2 + V (p1 )x̄ + V (p2 )(1 − x̄) −
∫
x̄
1
t(x1 − x) dx −
t(x2 − x)2 dx
0
x̄
∫ x̄
∫ 1
p1−α
p1−α
1−α
1−α
2
1
2
= p1 x̄ + p2 (1 − x̄) + v̄ −
x̄ −
(1 − x̄) −
t(x1 − x) dx −
t(x2 − x)2(4)
1−α
1−α
0
x̄
2
Analysis
We solve the game by backward induction. First, we consider the price competition given x1 and x2 .
The first-order conditions for firms 1 and 2 are respectively6
)
1
v(p1 )
p1−α
1
= 0 ⇐⇒ x̄ −
=0
(5)
(1 −
x̄ −
1 − α 2t(x2 − x1 )
2t(x2 − x1 )
(
)
p1−α
− p1−α
1
v(p1 ) − v(p2 )
−α
2
1
p2
α(1 − x̄) +
= 0 ⇐⇒ α(1 − x̄) − (1 − α)
= 0. (6)
1−α
2t(x2 − x1 )
2t(x2 − x1 )
α)p−α
1
5
(
−α
We implicitly assume that x̄ ∈ (0, 1). Without this assumption, D1 = p−α
1 max{0, min{x̄, 1}} and D2 = p2 (1 −
max{0, min{x̄, 1}}). We can show that without this implicit assumption our results hold and x̄ ∈ (0, 1) in equilibrium.
This assumption is made only for simplicity.
6
For the latter, more precisely,
)
(
)
(
v(p1 ) − v(p2 )
dW
x1 + x2
= p−α
= p−α
− (1 − 2α)x̄ ,
α(1 − x̄) − (1 − α)
α + (1 − α)
2
2
dp2
2t(x2 − x1 )
2
where we apply (3) for the second equality. By ∂ x̄/∂p2 > 0, thus, unless α < 1/2, the solution minimizes the objective.
5
Let the superscript “SE” denote the equilibrium outcome in the second stage given x1 and x2 . From
(3), (5), and (6), we have
1 − α x1 + x2
α
−
,
1 − 2α
2(
1 − 2α
)
1 − α x1 + x2
α
SE
v(p1 ) = 2t(x2 − x1 )
−
,
1 − 2α
2
1 − 2α
(
)
1 x1 + x2
2α
v(pSE
)
=
2t(x
−
x
)
.
−
2
1
2
1 − 2α
2
1 − 2α
x̄SE =
(7)
(8)
(9)
Next, we consider the choice of location at the first stage. Let the superscript “E” denote the
equilibrium outcome in the full game. The first-order condition for firm 1 is
(
)
∂Π1
v(p2SE ) − v(pSE
1
SE 1−α 1
SE
1 )
= (p1 )
+
− v(p1 )
∂x1
2
2t(x2 − x1 )2
2t(x2 − x1 )
(
)
1
1 − α x2 3x1
α
SE 1−α
= (p1 )
−
+
= 0.
x2 − x1 1 − 2α 2
2
1−α
(10)
From (10), we have the following reaction function of firm 1 at the first stage:
1
2 α
x̂∗1 (x2 ) = x2 +
.
3
31−α
(11)
The first-order condition for firm 2 is
∂W
1
SE
= α(1 − x̄) + (1 − α)(v(pSE
+ t(1 − x2 )2 − t(x̄ − x2 )2 = 0.
2 ) − v(p1 ))
∂x2
2t(x2 − x1 )
(12)
Substituting (11) into (12) and rearranging it by using (7), we have
(
)
∂W
3 1−α
1
2
= 2t(1 − 2α) x̄ −
x̄ +
= 0.
∂x2
2 1 − 2α
2
From (13), we have7
3 1−α
−
x̄E =
4 1 − 2α
√(
Substituting (11) and (14) into (7), we have

√
(
)
3 1 9
1 1 − 2α 2
α
E

(xE
,
x
)
=
−
−
+
,
1
2
8 2 16 2 1 − α
1−α
3 1−α
4 1 − 2α
)2
1
− .
2
√
9 3
−
8 2
(13)
1
9
−
16 2
(14)
(
1 − 2α
1−α

)2
+
α 
. (15)
1−α
7
Note that ∂ x̄/∂x2 > 0 by (7), and the coefficient of x̄2 , 2t(1 − 2α), is positive because α < 1/2, as in Footnote 4.
Thus, the larger solution of the quadratic minimizes the objective. A similar manner is followed for the derivation of
(23).
6
As expected,
E
E
lim (xE
1 , x̄ , x2 ) = (1/4, 1/2, 3/4),
α→0
which is in fact the equilibrium location in the standard model with inelastic demand. From (14)
and (15), we obtain the following proposition:
Proposition 1 For α > 0, (i) x̄E < 1/2 and (ii) x̄E < 2xE
1.
8
E
E
E
Proof From (14) and (15), x̄E and x̄E − 2xE
1 are decreasing in α. Since x̄ = 1/2 and x̄ − 2x1 = 0
when α = 0, thus, we obtain Proposition 1.9 Q.E.D.
Proposition 1(i) indicates that the market share of firm 1 is smaller than that of firm 2. The public
firm (firm 2) takes consumer surplus into account, while the private firm (firm 1) focuses solely on
profits. Thus, it is quite natural that firm 2 obtains a larger market share, owing to the lower price
of its products. Proposition 1(ii) might be counterintuitive. Firm 1’s equilibrium location is larger
than the midpoint of its customers (x̄/2).10 We now explain the intuition. Given x̄, the marginal
cost pricing is the best pricing strategy for firm 2. However, this pricing leads to another distortion.
The low price of firm 2’s products attracts consumers and decreases x̄; this leads to an inefficient
transport of consumers. In order to minimize the total transport cost, it is optimal for firm 2 to set
p1 = p2 , because it yields x̄ = (x1 + x2 )/2. These two effects determine the optimal pricing policy of
firm 2. The former effect (price distortion) is more important when 1 − x̄ and α are large. Since 1 − x̄
8
Note that we assume α < 1/2 (see footnote 4).
9
We present an alternative proof for (ii). (7) and (11) imply
x̄SE (x̂∗1 (x2 ), x2 ) − 2x̂∗1 (x2 ) = 2
α 1 SE ∗
[ x̄ (x̂1 (x2 ), x2 ) − 1],
1−α 2
(16)
which is negative as long as α > 0 and x̄SE (x̂∗1 (x2 ), x2 ) ∈ (0, 1). This proof also suggests that (ii) holds irrespective of
the locational choice of the public firm (as long as x̄SE (x̂∗1 (x2 ), x2 ) ∈ (0, 1)). Further, notice that by (7) and (11),
x̄SE (x̂∗1 (x2 ), x2 ) =
which is decreasing in α as long as
10
α
1−α
α
2 1−α
α
2 x2 − 1−α
,
[x2 −
]=
α
3 1 − 2α
1−α
3 1 − 1−α
(17)
∈ (0, x2 ) and x2 < 1.
Firm 1’s optimal location is always larger than the midpoint of its customers (x̄/2), given firm 2’s location regardless
of whether or not firm 2’s location is the equilibrium location. See footnote 9.
7
is decreasing in x1 , firm 1 strategically moves closer to firm 2’s location and reduces the importance
of the former effect discussed above, resulting in a higher p2 . Thus, we obtain Proposition 1(ii).
We now discuss the welfare implication of equilibrium. First, consider the case where the government can choose locations and prices of firms. It is obvious that the first best is achieved and
x1 = 1/4 and x2 = 3/4. The first best location pattern is equal to the equilibrium one if and only if
α = 0. Second, consider the case where the government can choose locations of firms only. Since firm
2 maximizes welfare, it is obvious that ∂W/∂x2 = 0 at the equilibrium given x1 and second-stage
competition. We now consider the welfare implication of firm 1’s location.
Proposition 2 Suppose that the government does not intervene in the pricing policy. Then,
∂W < 0.
∂x1 (x1 ,x2 )=(xE ,xE )
1
2
Proof
∂W
∂x1
)
(
(
)) (
2
(1 − α)x1 − α
2t(x2 − x1 )α
x1 + x2
1
α 1 − x1 +x
2
= 2tα
+ (1 − α)
1−
−
1 − 2α
1 − 2α
2
2 1 − 2α x2 − x1
(
)2
1 − α x1 + x2
α
+t
−
− x1 − t(0 − x1 )2 .
(18)
1 − 2α
2
1 − 2α
Substituting (x1 , x2 ) = (x̂∗1 (x2 ), x2 ) into (18), we have
∂W
α2
= 2t
∂x1
1−α
(
)
1
x̄ − 1 ,
2
(19)
∗ E
and it is negative because x̄ ≤ 1. Note that xE
1 = x̂1 (x2 ). Q.E.D.
Proposition 2 states that a decrease in x1 improves welfare. This implies that in equilibrium, the
private firm chooses a location that is too close to the public firm for social surplus (thereby making
the resulting degree of product differentiation too small). As stated in the discussion just after
Proposition 1, firm 1 has a stronger incentive to increase x1 for a strategic purpose than the social
planner does. This increases the resulting total transport costs in equilibrium and leads to inefficiency.
This is why xE
1 is too large for social surplus.
8
4
Leadership by the public firm
The public firm and the private firm are often an incumbent and a new entrant respectively. Thus,
the former may be able to take the lead in product positioning (location choice). In this section, we
consider a situation where the public firm chooses its location first, followed by the private firm.
Let the superscript “L” (public leadership) denote the equilibrium outcomes of this game. As
discussed in the previous section, xE
1 is too large for social welfare if the firms choose their locations
simultaneously. Thus, the public firm (firm 2) has an incentive for manipulating x2 to reduce x1 .
Since firm 1’s best reply (optimal location of firm 1 given x2 ) is increasing in x2 , firm 2 has an
incentive to reduce x2 for a strategic purpose. This leads to the following proposition.
E
L
E
E
E
L
L
L
E
Proposition 3 (i) xL
1 < x1 and x2 < x2 . (ii) x2 − x1 > x2 − x1 . (iii) π1 < π1 .
Proof From (7) and (11), we have
xE
1 =
α
1 1 − 2α E
x̄ +
2 1−α
1−α
xE
2 =
3 1 − 2α E
α
x̄ +
.
2 1−α
1−α
(20)
xL
1 =
α
1 1 − 2α L
x̄ +
2 1−α
1−α
xL
2 =
3 1 − 2α L
α
x̄ +
.
2 1−α
1−α
(21)
Similarly, we obtain
The first-order condition for firm 2 is
∂W ∗
∂W (x̂∗1 (x2 ), x2 ) ∂W ∗
=
(x̂ , x2 ) + x̂∗′
(x̂ , x2 )
1
∂x2
∂x2 1
∂x1 1
(
)
(
)
3 1−α
1
1
α2
1
∗
2
∗
∗
=2t(1 − 2α) (x̄(x̂1 , x2 )) −
x̄(x̂1 , x2 ) +
+ 2t
x̄(x̂1 , x2 ) − 1
2 1 − 2α
2
3 1−α 2
=0.
(22)
This yields
x̄L =
3 1−α
1
α2
−
4 1 − 2α 12 (1 − α)(1 − 2α)
√(
)2 (
)
3 1−α
1
α2
1 1
α2
−
−
−
< x̄E .
−
4 1 − 2α 12 (1 − α)(1 − 2α)
2 3 (1 − α)(1 − 2α)
9
(23)
Thus, we have
1 1 − 2α E
(x̄ − x̄L ) > 0,
2 1−α
3 1 − 2α E
L
xE
(x̄ − x̄L ) > 0,
2 − x2 =
2( 1 − α )
3 1 1 − 2α E
E
L
L
(xE
−
(x̄ − x̄L ) > 0.
2 − x1 ) − (x2 − x1 ) =
2 2 1−α
L
xE
1 − x1 =
(24)
(25)
(26)
Finally, because
Π1 = (1 − α)v(p1 )x̄,
(27)
L
E
E
E 2
L
L
L 2
ΠE
1 − Π1 = (1 − α)2t((x2 − x1 )(x̄ ) − (x2 − x1 )(x̄ ) ) > 0.
(28)
from (7),(8), (23), and (26), we have
Q.E.D.
By definition, the leadership of firm 2 improves firm 2’s payoff (welfare). However, it reduces the
degree of product differentiation between firms.
5
The effect of privatization
We discuss the effect of privatization of firm 2. Suppose that both firms are private. Let the
superscript “P” denote the equilibrium outcomes in the private duopoly.
Proposition 4 (i) xP1 = 0 and xP2 = 1. (ii) W P < W E < W L .
Proof By (3) and (27), this case is strategically equivalent to the case with α = 0, whose equilibrium
location is known to be (0, 1). Thus, (xP1 , xP2 ) = (0, 1).
W P < W E follows as
E
SE E
SE E
E
W P = W P SE (0, 1) < W P SE (xE
(x1 , 1 − xE
(x1 , xE
1 , 1 − x1 ) ≤ W
1)≤W
2)=W ,
(29)
E
where W P SE (xE
1 , 1 − x1 ) is the second stage social surplus given the locations of firms in the private
oligopoly.
10
Finally, by Proposition 2, W E < W L follows straightforwardly. Q.E.D.
The privatization leads to maximal differentiation and reduces welfare. However, we must emphasize that the second result may depends on the assumption that the public firm is as efficient as
the private firm. If privatization improves the production efficiency of the public firm, privatization
can improve welfare.
6
Concluding remarks
In this paper, we investigated a location-price model in a mixed duopoly. We introduced an elastic
demand into the standard model and showed that the resulting degree of product differentiation was
too small for social welfare. Moreover, we found that the leadership of the public firm at the location
stage further reduces the degree of product differentiation but improves welfare.
As mentioned in the Introduction, many papers have explored a mixed oligopoly with inelastic
demand and have derived many important policy implications. However, the policy and welfare
implication of the results which hold only under inelastic demand must be limited. To determine
whether or not the above results depend on the assumption of inelastic demand, it is essential to
check their robustness and applicability. This remains an issue for future research.
11
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