Munich Personal RePEc Archive

Market transparency in a mixed
oligopoly
Xu, Lili and Matsumura, Toshihiro

29 August 2023

Online at https://mpra.ub.uni-muenchen.de/118415/
MPRA Paper No. 118415, posted 31 Aug 2023 13:48 UTC

Market transparency in a mixed oligopolyLili Xua+ and Toshihiro Matsumurab
Abstract
This study investigates the relationship between market transparency and economic welfare ina mixed duopoly in which a welfare-maximizing public firm competes with a profitmaximizing private firm. We find that the private firm’s market share, consumer surplus, andwelfare increase with market transparency. Further, the relationship between the private firm’s
profit and market transparency has an inverted U shape. This result suggests that profitmaximizing firms may have incentives to improve market transparency, especially when the
degree of market transparency is low, which is in sharp contrast to the results under a private
duopoly.
Keywords: market transparency, mixed oligopoly, product differentiation, unconstrained
Hotelling model, profit-enhancing market transparency, crowding out
JEL classification: L13; L15; L33; R32

 The first author acknowledges the financial support from the Program of National Social Science Foundation ofChina (grant number 21BJY264) and the Key Program of National Natural Science Foundation of China (grant
number 42030409). The second author acknowledges the financial support from JSPS KAKENHI (grant number
21K01398). We thank Editage for English language editing. The usual disclaimers apply.
a Department of Economics and Trade, Dalian Maritime University, 1 Linghai Road, Dalian, 116026, China;
lilixuchonnam@hotmail.com; ORCID: 0000-0001-9660-3145.
b Institute of Social Science, the University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 1130033, Japan;
matsumur@iss.u-tokyo.ac.jp; ORCID: 0000-0003-0572-6516.
+ Correspondence: lilixuchonnam@hotmail.com; Tel.: +86-411-8472-6698; Fax: +86-411-8472-5667.
1

1. Introduction
We investigate how consumer-side market transparency affects market performance in amixed duopoly. One is a state-owned public firm that maximizes welfare, and the other is a
private firm that maximizes profits. To this end, we use a location-price model with a linear
city (Hotelling line) and endogenize the degree of product differentiation and prices. We
consider the effects of market transparency on firms’ locations and prices, consumer surplus,
welfare, and private firms’ profits.
We find that a decrease in market transparency shifts the public (private) firm’s location
toward the center (edge) of the linear city, which leads to a location choice distortion.
Consequently, the private firm’s market share, consumer surplus, and economic welfare
increase with market transparency. However, an increase in market transparency lowers theprivate firm’s price and increases its market share; thus, the effect of market transparency on
the private firm’s profit is ambiguous. Specifically, we demonstrate that the relationshipbetween a private firm’s profit and market transparency is non-monotonic (inverted U-shaped
relationship) in a mixed duopoly. Notably, this non-monotonic relationship does not appear if
firms are profit maximizers in a private oligopoly (Schultz, 2004), suggesting an important
implication from an antitrust perspective. Our results suggest that profit-maximizing private
firms may have an incentive to improve market transparency in a mixed duopoly, which is not
the case in a private duopoly.1
There are two streams of related research. The first refers to market transparency. Since
the seminal work of Varian (1980), the literature on public economics and industrialorganization has intensively discussed how market transparency affects welfare andcompetition structure. Market transparency has attracted broad attention from the viewpoint ofpublic policy, especially competition and consumer protection (Gu and Hehenkamp, 2014; Gu
and Wenzel, 2011, 2012; Schultz, 2004, 2005, 2009, 2017). The price system is complex in

1

The Japanese electricity and gas market is a mixed oligopoly for which the comparison of fees among firms isdifficult because the fee system is complicated. In this sense, market transparency may be low. Enerchange Ltd. runs
a platform that suggests the best electric or gas company for each consumer, depending on several consumer-specificconditions, improving market transparency. Several electric and gas companies, both public and private firms,
commit to and financially support Enerchange Ltd. This suggests that even private firms have an incentive toimprove market transparency. Conversely, the Japanese gasoline retail market is a private oligopoly. GOGO Labs
Inc. provides a similar service in the gasoline retail market, and there is no significant financial commitment by
gasoline companies. Our results explain these facts adequately.
2many markets. Thus, for consumers, firms’ prices and product positions may not be transparent,
which is why discussing market transparency is significant.
Schultz (2004) incorporates market transparency into a location-price model with theunconstrained Hotelling line formulated by Tabuchi and Thisse (1995) and Lambertini (1997).
Schultz (2004) shows that an improvement in market transparency reduces the degree of
product differentiation and private firms’ profits but improves economic welfare and consumer
surplus. Schultz (2009) further investigates a free-entry market, and finds that an improvement
in market transparency reduces the number of entering firms and improves consumer surplus if
the number of firms is greater than or equal to two. Conversely, Gu and Hehenkamp (2014) andGu and Wenzel (2011, 2012) demonstrate that an increase in market transparency may reduce
welfare under plausible conditions because it distorts firms’ entry decisions. Schultz (2017)
states that increasing (decreasing) transparency on the producer (consumer) side facilitates
collusion. However, all these studies investigate private oligopolies, in which all firms are profit
maximizers. We deviate from these discussions by investigating a mixed duopoly in which oneof the players is a welfare maximizer.
The second research stream refers to mixed oligopolies. Although the privatization ofstate-owned public enterprises has been observed globally, many public enterprises owned by
the public sector remain active, and a considerable number of these public enterprises competewith private enterprises in numerous industries. 2 Public enterprises may affect private
enterprises’ product positioning and pricing strategies. This topic has been extensively
discussed since Cremer et al. (1991).
Specifically, Cremer et al. (1991) investigate a location-price model on the Hotelling line,
and show that the equilibrium location pattern is optimal for welfare in a mixed duopoly.
Matsumura and Matsushima (2004) confirm that this result is robust to any exogenous costdifferences between public and private firms. They endogenize the cost structure by introducing
R&D competition and show that a mixed duopoly distorts R&D competition and yields welfare
loss. Lu (2006) discusses the conditions for the existence of pure strategy equilibrium. Inoue et

2

The literature on mixed oligopolies is rich and diverse. See Cho et al. (2022), Xu and Matsumura (2022), Yang
and Huang (2023), and the papers cited therein.
3al. (2009) find that multiple equilibrium location patterns exist if a public firm is owned by the
local government. Ogawa and Sanjo (2011) extend their analysis to investigate the effect ofmarket integration. Moreover, Li and Zhang (2011) formulate a model in which a public and a
private firm enter a market sequentially over an infinite time horizon. Kitahara and Matsumura(2013) and Matsumura and Tomaru (2015) show that the price-dependent demand and shadow
costs of public funds yield distortions in location choices, respectively. Furthermore, Zhang
and Li (2013) investigate endogenous timing in the presence of demand uncertainty.
Additionally, Ma et al. (2021) introduce a costly product differentiation. However, no studies
have hitherto investigated market transparency in mixed oligopolies.
The remainder of this paper is organized as follows. Section 2 describes the model
formulation, and Section 3 examines the equilibrium outcomes. Subsequently, Section 4
presents the main results. Finally, Section 5 concludes the study.
2. The model
We consider a Hotelling model with a unit-length line in a mixed duopoly. Firm 0 is a
public firm that maximizes welfare and firm 1 is a private firm that maximizes its profits. We
assume that both firms have a common constant marginal cost normalized to zero. Suppose
firm i (𝑖 = 0, 1 ) is located at point 𝑙𝑖 . Without loss of generality, we assume 𝑙0 ≤ 𝑙1 . A
consumer living at 𝑧 ∈ [0, 1] incurs transportation cost 𝑡(𝑙𝑖 − 𝑧)2 when purchasing a
product from firm i. Parameter 𝑡 > 0 can be interpreted as a measure of the intensity of
production differentiation. Each consumer derives a surplus from consumption (gross price andtransportation costs) equal to 𝑠 . We assume that 𝑠 is sufficiently large such that every
consumer consumes one unit of the product.3

Following Schulz (2004), we assume two types of consumers: informed and uninformed.
Informed consumers observe firms’ locations and prices and choose whether to purchaseproducts from firms 0 or 1. Uninformed consumers do not observe firms’ locations and prices
and choose whether to purchase a product based on expectations. Let 𝜑 be the fraction of3

The model formulation is the same as in Schultz (2004), except for firms’ payoff function. Following Schultz(2004), we allow firms to locate outside the linear city. In other words, we adopt the unconstrained Hotelling model
formulated by Tabuchi and Thisse (1995) and Lambertini (1997). We briefly discuss the constrained Hotelling modelin which 𝑙𝑖 ∈ [0, 1] in Proposition 2 at the end of Section 4.
4

informed consumers, while (1 − 𝜑) of them are uninformed. Parameter 𝜑 ∈ (0, 1] denotes thedegree of market transparency. The higher 𝜑 is, the more transparent is the market.

𝑙 +𝑙1
𝑝 −𝑝
+ 1 0 , the total cost of buying from2
2𝑡(𝑙1 −𝑙0 )For an informed consumer living at 𝑥 = 0

𝑙 𝐸 +𝑙 𝐸𝑝𝐸 −𝑝𝐸either firm is the same. For an uninformed consumer living at 𝑦 = 0 2 1 + 2𝑡(𝑙1 𝐸 −𝑙0𝐸 ), the total
10

cost of buying from either firm is the same, with superscript E denoting the expectation.
Following the literature, such as Schulz (2004), we assume that uninformed consumers
rationally expect firms’ choices. Thus, the demand of firm 0, 𝐷0, and that of firm 1, 𝐷1, aregiven, respectively, by:

𝐷0 = 𝜑𝑥 + (1 − 𝜑)𝑦,𝐷1 = 1 − 𝐷0.

Welfare, 𝑊, is given by:
𝑥

(1)

1

𝑦

1

𝑊 = 𝑠 − 𝑡[𝜑(∫0 (𝑧 − 𝑙0 )2 𝑑𝑧 + ∫𝑥 (𝑧 − 𝑙1 )2 𝑑𝑧) + (1 − 𝜑)(∫0 (𝑧 − 𝑙0 )2 𝑑𝑧 + ∫𝑦 (𝑧 −𝑙1 )2 𝑑𝑧)].

The profit of firm i (𝑖 = 0, 1) is:

(2)

(3)

𝜋𝑖 = 𝑝𝑖 𝐷𝑖 .

The game proceeds as follows. In the first stage, each firm i (𝑖 = 0, 1) chooses location 𝑙𝑖

simultaneously. For expositional simplicity, we assume that the two firms choose different
locations and that 𝑙0 ≤ 𝑙1 . In the second stage, each firm i (𝑖 = 0, 1) chooses price 𝑝𝑖 ∈[0, ∞) simultaneously. We use the subgame perfect Nash equilibrium as the equilibrium
concept. Thus, we solve the game using backward induction.
3. Equilibrium analysis
3.1. Price competition
We first discuss the second-stage subgame in which firms’ locations are given exogenously.
Firm 0 (1) chooses 𝑝0 (𝑝1 ) to maximize welfare (profit), given 𝑦, 𝑙0 , 𝑙1 , and 𝑝1 (𝑝0 ). The
first-order conditions are as follows:4
𝜙(𝑝 −𝑝 )𝜕𝑊= 2𝑡(𝑙1 −𝑙 0) ,𝜕𝑝0
1 0
4 All second-order conditions are satisfied in this study.

5∂𝜋1𝑡(𝑙 −𝑙 )(2(1−𝑦)+𝜙(2𝑦−𝑙0 −𝑙1 ))+𝜙(𝑝0 −2𝑝1 )
= 1 0
.
∂𝑝1
2𝑡(𝑙1 −𝑙0 )(4)

Let 𝑅0 (𝑅1 ) be the reaction function of firm 0 (1) in the second stage.
𝑅0 = 𝑝0 (𝑝1 ) = 𝑝1 ,𝑅1 = 𝑝1 (𝑝0 ) =1
𝑡(𝑙1 −𝑙0 )(2(1−𝑦)+𝜙(2𝑦−𝑙0 −𝑙1 ))+ 2 𝑝0 .
2𝜙(5)From (5), we observe that an increase in 𝑝1 (𝑝0 ) increases 𝑝0 (𝑝1 )—the strategies in thesecond stage are strategic complements. Additionally, firm 0’s price is independent of 𝜙 and𝑦, whereas firm 1’s price decreases with 𝜙 and 𝑦. Firm 0 tries to choose the same price asfirm 1 because this pricing strategy minimizes transport costs and, thus, maximizes welfare.
The common equilibrium price in the second stage is:𝑝𝑖 =𝑡(𝑙1 −𝑙0 )(2(1−𝑦)+𝜙(2𝑦−𝑙0 −𝑙1 )), where 𝑖 = 0, 1.
𝜙

𝜋1 =𝑡(𝑙1 −𝑙0 )(2(1−𝑦)+𝜙(2𝑦−𝑙0 −𝑙1 ))2
,
2𝜙

(6)

The resulting profit of firm 1 and welfare are, respectively:

𝑊=𝑠−𝑡(4(1−3𝑦(𝑦−𝑙0 )𝑙0 −3(1−𝑦)(1+𝑦−𝑙1 )𝑙1 )−3𝜙(2𝑦−𝑙0 −𝑙1 )2 (𝑙1 −𝑙0 ))
.
12

(7)

3.2. Location choice

In the first stage, firm 0 (1) chooses 𝑙0 (𝑙1 ) to maximize welfare (profit) in (7). The first-

order conditions are as follows:𝜕𝑊𝑡(4𝑦(𝑦−2𝑙0 )−𝜙(2𝑦−𝑙0 −𝑙1 )(2𝑦−3𝑙0 +𝑙1 ))
=
= 0,
𝜕𝑙0
4

𝑡(2(1−𝑦)+𝜙(2𝑦+𝑙0 −3𝑙1 ))(2(1−𝑦)+𝜙(2𝑦−𝑙0 −𝑙1 ))
∂𝜋1
=
= 0.
2𝜙
∂𝑙1

(8)

We have the following reaction function for firm i in the first stage:𝑙0 (𝑙1 ) =𝑙1 (𝑙0 ) =2√𝑦 2 (4−5𝜙+𝜙2 )+𝜙𝑙1 (2𝑦−2𝑦𝜙+𝜙𝑙1 )−4𝑦(1−𝜙)−𝜙𝑙1,
3𝜙2(1−𝑦+𝑦𝜙)
1+ 3 𝑙0 .
3𝜙(9)From (9), we have the following locations:𝑙0 =3√1+8𝑦 2 (1−𝜙)−(1+8𝑦(1−𝜙))√1+8𝑦 2 (1−𝜙)+(5−8𝑦(1−𝜙))and 𝑙1 =
.
8𝜙
8𝜙

(10)

3.3. Rational expectation

Finally, we derive equilibrium 𝑦. Owing to the rational expectation assumption, 𝑝0 = 𝑝0𝐸 ,6𝑙 𝐸 +𝑙1𝐸𝑝𝐸 −𝑝𝐸+ 1 𝐸 −𝑙0𝐸 ) must hold. Substituting the locations
2
2𝑡(𝑙1 0𝑝1 = 𝑝1𝐸 , 𝑙0 = 𝑙0𝐸 , 𝑙1 = 𝑙1𝐸 , and 𝑦 = 0

𝑙 +𝑙

𝑝 −𝑝

1

in (10) and the resulting prices into 𝑦 = 0 2 1 + 2𝑡(𝑙1 −𝑙0 ), we obtain 𝑦 = 1+𝜙. Subsequently,
1

the two firms’ equilibrium locations are:103𝑙0∗ = 2(1+𝜙) and 𝑙1∗ = 2(1+𝜙),(11)

where superscript ∗ denotes the equilibrium outcomes. From (11), we obtain the following
result.

Lemma 1.
(i). 𝑙0∗ < 12 and 𝑙0∗ decreases with 𝜙;(ii). 𝑙1∗ ≥ 34 and 𝑙1∗ decreases with 𝜙.

Lemma 1(ii) suggests that firm 1 chooses the location closer to the center of the city when

𝜙 is larger. The intuition is as follows. An increase in 𝜙 implies that more consumers observe
firm 1’s location. Therefore, a decrease in 𝑙1 increases firm 1’s market share more effectively

when 𝜙 is larger. Consequently, firm 1 has a stronger incentive to locate closer to the city
center to obtain a larger market share when 𝜙 is larger.

Lemma 1(i) suggests that firm 0 has the opposite incentive (i.e., firm 0 chooses the location

further to the center of the city when 𝜙 is larger). We explain the intuition as follows. A pairof the best locations is (𝑙0∗ , 𝑙1∗ ) = (14, 34). When 𝜙 < 1, 𝑙1∗ > 34. In other words, firm 1 dares tochoose a location closer to the edge than the socially optimal location to mitigate pricecompetition. Responding to this rival’s location, firm 0 dares to choose a location closer to thecenter than the first best location to reduce total transport and improve welfare. As an increasein 𝜙 makes firm 1’s location closer to the first best location, firm 0 chooses the location closer
to the first best location, responding to the change in firm 1’s location.
The equilibrium price of each firm is:
2𝑡𝑝𝑖∗ = (1+𝜙)2, where 𝑖 = 0, 1.

(12)The demands of the firm are, respectively:
1

𝐷0∗ = 1+𝜙 and 𝐷1∗ =

𝜙
.
1+𝜙

(13)

Finally, the resulting profit of firm 1 and welfare are, respectively:7𝜋1∗ =

4. Results

2𝑡𝜙(1+𝜙)3and 𝑊 ∗ = 𝑠 −𝑡(1+3𝜙−6𝜙2 +4𝜙3 ).
12(1+𝜙)3(14)

Here, we discuss the implications of market transparency in a mixed duopoly. As abenchmark, we first present the results for a private duopoly, following Schultz (2004).
Result 1. (Schultz, 2004)
Consider a private duopoly in which both firms maximize their profits, where:(i). welfare and consumer surplus increase with 𝜙;(ii). each firm’s profit decreases with 𝜙;(iii). each firm’s market share is independent of 𝜙.

In a private duopoly, each firm chooses a location closer to the center of the city when 𝜙is larger. The intuition is similar to that behind Lemma 1(ii). This economizes the transport
costs and accelerates competition. Additionally, an increase in 𝜙 directly accelerates pricecompetition in the last stage. These arguments lead to Proposition 1.

We present our main results for a mixed duopolistic market in the following proposition:
Proposition 1.
Consider a mixed duopoly, where:(i). welfare and consumer surplus increase with 𝜙;(ii). the private firm’s profit increases (decreases) with 𝜙 if 𝜙 < (>)𝟏𝟐;(iii). the public firm’s market share decreases with 𝜙.
Proof of Proposition 1.
(i).

3𝑡(1−𝜙)𝜙𝑑𝑊 ∗= 2(1+𝜙)4 > 0 . Additionally, consumer surplus can be obtained by𝑑𝜙𝜋0∗ − 𝜋1∗ = 𝑠 −𝐶𝑆 ∗ = 𝑊 ∗ −𝑡(25+27𝜙−6𝜙2 +4𝜙3 )𝑑𝐶𝑆 ∗𝑡(8+11𝜙−3𝜙2 )
.
Then,
we
obtain=> 0;12(1+𝜙)3𝑑𝜙
2(1+𝜙)4𝑑𝜋∗2𝑡(1−2𝜙) >

𝑑𝐷 ∗

1

<1

(ii). 𝑑𝜙1 = (1+𝜙)4 < 0 when 𝜙 > 2;(iii). 𝑑𝜙0 = − (1+𝜙)2 < 0. Q.E.D.

Proposition 1(i) indicates that increasing market transparency on the consumer side leads8to greater welfare and consumer surplus. A decrease in market transparency moves the private
location toward the right side of the linear city (Lemma 1(ii)); in response to this private firm’s
choice, the public firm also moves toward the right side of the linear city (Lemma 1(i)). Suchrelocations increase transportation costs and reduce welfare. Additionally, a decrease in market
transparency increases prices. Changes in both location and price reduce consumer surplus.
Proposition 1(ii) states that the effect of market transparency on the private firm’s profit
depends on the degree of market transparency. When 𝜙 is lower than 12, a decrease in market

transparency reduces the private firm’s profit. A decrease in market transparency raises theequilibrium price of the private firm, which, in turn, increases its profits. However, a decreasein market transparency affects location and reduces private firms’ market share, which, in turn,
reduces their profits. The latter effect dominates the former when 𝜙 is small. Therefore, anincrease in market transparency may increase the private firm’s profit, which does not occur ina private duopoly. Figure 1 shows the profit of the private firm when 𝑡 = 1. In Figure 1, thevertical axis represents 𝜋1∗ and the horizontal one 𝜙.
1

0.30
0.25
0.20
0.15
0.10
0.05

0.2

0.4

0.6

0.8

1.0

Figure 1. The private firm’s profit (𝑡 = 1)

As previously mentioned, a decrease in market transparency induces the relocation of thefirms, which increases the public firm’s market share. In other words, an improvement inmarket transparency induces production substitution from public to private firms.5
5

If we allow a cost difference between public and private firms such as like in the study by Matsumura andMatsushima (2004), this production substitution has an additional welfare-improving effect. Therefore, Proposition
1 (i) is strengthened if we assume that the public firm’s marginal cost is higher than the private firm’s marginal cost.
The model with the public firm’s cost disadvantage is popular in the literature on mixed oligopolies. See Pal (1998),
Mujumdar and Pal (1998), and Haraguchi and Matsumura (2020).
9

Finally, we discuss the outcomes of adopting a Hotelling model with location restriction
and assume 𝑙𝑖 ∈ [0, 1] . As Schultz (2004) shows, both firms locate outside the linear city,
regardless of 𝜙. Therefore, in this constrained Hotelling model, one firm chooses point 0 andthe other chooses point 1, regardless of 𝜙 . Consequently, welfare is independent of 𝜙 .
However, an increase in 𝜙 reduces equilibrium prices, which improves consumer surplus but
reduces firms’ profits. The following results are derived directly from Schultz (2004).

Result 2. (Schultz, 2004)
Consider the private duopoly with a constrained Hotelling model, where:(i). welfare is independent of 𝜙;(ii). consumer surplus increases with 𝜙;(iii). each firm’s profit decreases with 𝜙;(iv). each firm’s market share is independent if 𝜙.

In the mixed duopoly, (11) yields 0 < 𝑙𝑖∗ < 1 when 𝜙 > 12. Therefore, when 𝜙 > 12, thefirms choose locations within the city. Therefore, constraint 𝑙𝑖 ∈ [0, 1] is not binding, and thesame results hold for both the constrained and unconstrained Hotelling models. Nevertheless,when 𝜙 < 12, constraint 𝑙1 ≤ 1 binds. Therefore, the equilibrium location is (𝑙0∗ , 𝑙1∗ ) = (13, 1)

if 𝜙 < 12. By replacing this equilibrium location with the above equilibrium locations, we
obtain Proposition 2.

Proposition 2.
Consider the mixed duopoly with a constrained Hotelling model, where:(i). welfare is non-decreasing in 𝜙 and is increasing in 𝜙 for 𝜙 > 12;(ii). consumer surplus increases with 𝜙;(iii). the private firm’s profit is non-increasing in 𝜙 and is decreasing in 𝜙 for 𝜙 > 12;(iv). the public firm’s market share is non-increasing in 𝜙 and is decreasing in 𝜙 for 𝜙 > 12.

The most interesting result of Proposition 1—the private firm’s profit may increase with

𝜙 —does not hold in the constrained Hotelling model. However, in the mixed duopoly, we
obtain one important implication of the constrained Hotelling model: welfare is independent of

𝜙 in the private duopoly but may increase with 𝜙 in the mixed duopoly. Therefore, in both
10

the unconstrained and constrained Hotelling models, the mixed duopoly yields implications
different from those for the private duopoly.
5. Concluding remarks
This study investigates a mixed duopoly in which a welfare-maximizing public firmcompetes against a profit-maximizing private firm with consumer-side market transparency.
We discuss how market transparency affects firms’ locations and pricing strategies, as well as
the resulting profits and economic welfare. We find that an increase in market transparency
reduces location distortions and induces production substitution from both the public andprivate firms, which improves welfare. Moreover, the private firm’s profit may increase withmarket transparency in a mixed duopoly, which is never the case in a private duopoly.
However, there remains a need to extensive future research due to the following limitations.
In this study, we do not consider factors such as price-dependent demand, the shadow cost of
public funds, R&D investments that affect quality and/or production costs, and various
externalities. These factors may affect location patterns and, thus, the welfare implications ofmarket transparency. Incorporating of these elements into our analysis remains a topic for future
research. Furthermore, we use a location-price model in the Hotelling line as an endogenous
product differentiation model. Extending our analysis to different endogenous productdifferentiation models is another significant direction for future research.6
References
Cho, S., Kim., D., Lee, S.H., 2022. Free licensing strategy and ex-post privatization policy with
passive ownership. Economic Modelling 106, 105676.
Cremer, H., Marchand, M., Thisse, J.F., 1991. Mixed oligopoly with differentiated products.
International Journal of Industrial Organization 9(1), 43–53.
Gu, Y., Hehenkamp, B., 2014. Too much of a good thing? Welfare consequences of market
transparency. Journal of Institutional and Theoretical Economics 170(2), 225–248.
Gu, Y., Wenzel, T., 2011. Transparency, price-dependent demand and product variety.

6

For example, Matsushima and Matsumura (2003, 2006) investigate a location-quantity model in mixed oligopolies,
while Liu et al. (2020) consider endogenous product differentiation in a mixed duopoly using an alternative nonspatial model.
11Economics Letters 110(3), 216–219.
Gu, Y., Wenzel, T., 2012. Transparency, entry, and productivity. Economics Letters 115(1), 7–
10.
Haraguchi, J., Matsumura, T., 2020. Lack of commitment to future privatization policies maylead to worst welfare outcome. Economic Modelling 88, 181–187.
Inoue, T., Kamijo, Y., Tomaru, Y., 2009. Interregional mixed duopoly. Regional Science and
Urban Economics 39(2), 233–242.
Kitahara, M., Matsumura, T., 2013. Mixed duopoly, product differentiation, and competition.
Manchester School 81(5), 730–744.
Lambertini, L., 1997. Unicity of the equilibrium in the unconstrained Hotelling model.
Regional Science and Urban Economics 27(6), 785–798.
Li, C., Zhang, J., 2011. Equilibrium locations in a mixed duopoly with sequential entry in realtime. Economic Modelling 28(3), 1211–1218.
Liu, L.H., Wang, X.H., Zeng, C.H., 2020. Endogenous horizontal product differentiation in amixed duopoly. Review of Industrial Organization 56(3), 435–462.
Lu, Y., 2006. Hotelling’s location model in mixed duopoly. Economics Bulletin 8(1), 1–10.
Ma, H.K., Wang, X.H., Zeng, C.H., 2021. Location choice and costly product differentiation ina mixed duopoly. Annals of Regional Science 66(1), 137–159.
Matsumura, T., Matsushima, N., 2004. Endogenous cost differentials between public andprivate enterprises: A mixed duopoly approach. Economica 71(284), 671–688.
Matsumura, T., Tomaru, Y., 2015. Mixed duopoly, location choice, and shadow cost of publicfunds. Southern Economic Journal 82(2), 416–429.
Matsushima, N., Matsumura, T., 2003. Mixed oligopoly and spatial agglomeration. Canadian
Journal of Economics 36(1), 62–87.
Matsushima, N., Matsumura, T., 2006. Mixed oligopoly, foreign firms, and location choice.
Regional Science and Urban Economics 36(6), 753–772.
Mujumdar, S., Pal, D., 1998. Effects of indirect taxation in a mixed oligopoly. Economics
Letters 58(2), 199–204.
Ogawa, H., Sanjo, Y., 2011. Market integration and location choice: A mixed oligopoly
approach. Letters in Spatial and Resource Sciences 4(3), 157–166.
12

Pal, D., 1998. Endogenous timing in a mixed oligopoly. Economics Letters 61(2), 181–185.
Schultz, C., 2004. Market transparency and product differentiation. Economics Letters 83(2),173–178.
Schultz, C., 2005. Transparency on the consumer side and tacit collusion. European Economic
Review 49(2), 279–297.
Schultz, C., 2009. Transparency and product variety. Economics Letters 102(3), 165–168.
Schultz, C., 2017. Collusion in markets with imperfect price information on both sides. Review
of Industrial Organization 50(3), 287–301.
Tabuchi, T., Thisse, J.F., 1995. Asymmetric equilibria in spatial competition. International
Journal of Industrial Organization 13(2), 213–227.
Varian, H.R., 1980. A model of sales. American Economic Review 70(4), 651–659.
Xu, L., Matsumura, T., 2022. Welfare-reducing price competition under relative performancedelegation with convex costs. Economic Modelling 112, 105871.
Yang, L., Huang, Z., 2023. Quality-improving licensing of an outside innovator in a mixed
Cournot duopoly. Economic Modelling 126, 106432.
Zhang, J., Li, C., 2013. Endogenous timing in a mixed oligopoly under demand uncertainty.
Journal of Economics 108(3), 273–289.

13