HIAS-E-91

Vertically Diﬀerentiated Cournot Oligopoly: Eﬀects of Market
Expansion and Trade Liberalization on Relative Markup and
Product Quality
Ngo Van Long(a),(b) and Zhuang Miao(c)(a) McGill University(b) Hitotsubashi Institute for Advanced Study, Hitotsubashi University(c) School of International Trade and Economics, Central University of Finance and Economics, 39 South
College Road, Haidian District, Beijing, P.R. China 100081

November, 2019

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Vertically Di§erentiated Cournot Oligopoly: E§ects of Market
Expansion and Trade Liberalization on Relative Markup and
Product Quality
Ngo Van Long!

Zhuang Miaoy

November 19, 2019

Abstract
We model an oligopoly where Örms are allowed to freely enter and exit the market and choose
the quality level of their products by incurring di§erent set-up costs. Using this framework, we
study the mix of Örms in the long-run Cournot-Nash equilibrium under di§erent cost structures
and the e§ects of market size on market outcomes. SpeciÖcally, we consider two alternative
speciÖcations of cost structure. In the Örst speciÖcation, quality upgrading requires a large
increment in the set-up cost or R&D investment. Under this cost structure, we show that in the
Nash equilibrium, each Örm specializes in a single quality level, and an increase in the market
size leads to (i) an increase in the fraction of Örms that specialize in the high quality product,(ii) an increase in the market share of the high quality product, and (iii) a reduction in Örmsí
markups and in markup dispersion. Under the second type of cost structure where quality
upgrading only requires higher marginal cost, we Önd that all Örms will produce both types of
product, and the value share of the high-quality product increases as the market expands, but
in quantity terms, the market share of the high quality product does not change. Finally, weÖnd that trade liberalization has broadly similar e§ects to that of a market expansion, but the
supply of the high-quality product from the smaller economy may decrease.
JEL ClassiÖcation: L1, L2, F15.
Keywords: Multiproduct Örms; Cournot competition; Vertical product di§erentiation; Cost
structure; Market size; Trade liberalization.

!

McGill University and Hitotsubashi Institute for Advanced Study, Hitotsubashi University.
Email:
ngo.long@mcgill.ca.
ySchool of International Trade and Economics, Central University of Finance and Economics, 39 South
College Road, Haidian District, Beijing, P.R. China 100081. Email: zhuang.miao@mail.mcgill.ca. Telephone:(+86)1701079110.

1

1

Introduction
Do increased competitive pressure and/or market expansion induce firms to upgrade the quality of theirproducts, leading to an increase the average quality of products in the market place? Can market expansion
lead to the demise of firms that produce the low-quality products? Does trade liberalization has similare↵ects on product quality to those of a pure market expansion? The answers to these questions may welldepend on the nature of the industry under consideration, in particular its cost structure and possibly on
its mode of competition. While there is an existing theoretical and empirical literature that has shed lighton these questions (see, for example, Fajgelbaum et al. 2011; Berry and Waldfogel 2010; Vives 2008), there
remain some gaps in the analysis of the role of the trade-o↵ between fixed cost and marginal cost in thequality upgrading process and of firms’ decision on whether they should o↵er a wide range of qualities, or,instead, specialize in one end or the other of the quality spectrum. Moreover, the existing studies on therelation between market size and products’ quality are incomplete. We apply our theoretical framework to
the study of this issue and expand the relevant literature by adding the analysis under the Cournot-oligopoly
competition mode and allowing variable markups. 1
In this paper, we develop model of Cournot oligopoly with vertically di↵erentiated products and show
how the e↵ect of market expansion on the average quality depends crucially on the industry’s cost structure.
We demonstrate that the equilibrium features of the model depends on whether the additional investments
required to produce the high-quality product involve sinking large costs (i.e., quality upgrading is intensive inoverhead factors, such as R&D) or incurring higher variable costs per unit of quality (i.e., quality upgrading is
intensive in the intermediate inputs). One of our main findings is that when the upgrading process is intensivein the overhead factors, in equilibrium each firm will choose to produce either the high-quality product, or the
low-quality product, but not both. Another major finding is that, under this cost configuration, an increasein the market size will increase the number of high-quality producers, decrease the number of low-quality
producers, leading to a fall in the relative price and relative markup of the high-quality product and anincrease in the market share of high-quality firms. When, on the contrary, quality upgrading is intensive inintermediate inputs and fixed cost is independent of quality, each firm will produce both the high-quality
and the low-quality products, and an increase in market size will increase the relative price, relative markup,
and the market share of the high-quality product in value terms, while leaving the ratio of high-quality to
low-quality outputs unchanged.
Concerning the e↵ects of partial trade liberalization between two countries, we find that, in broad outlines,
the results are similar to that of an increase in market size. However, when the two countries are of di↵erent
size, we find that as the trade costs decrease, in the small country, the ratio of high-quality firms to lowquality firms may change in a non-monotonic fashion. This is due to the relocation decision of high-quality
firms from the small country to the large country, which depends on trade costs. Indeed, firms choose their
1 For example, Fajgelbaum et al. 2011 did not study how the markups and the relative markup change when the market size
increases. In contrast, this issue is fully analyzed in our paper.

2

location based on the balance of market demand and market competition intensity.
Our paper is an extension of the model of strategic product line choice by Johnson and Myatt (2006).
Di↵erent from Johnson and Myatt (2006), our model assumes more general cost structures. Thus, while
Johnson and Myatt (2006) assume that all firms have the same fixed cost,2 we consider two di↵erent cost
scenarios. In the first scenario, firms that want to produce the high-quality product must incur a higher
quality-adjusted fixed cost, and this enables them to produce the high-quality product at a smaller qualityadjusted marginal cost than that of the lower-quality product. In the second scenario, all firms have the same
fixed cost but marginal costs di↵er: the high-quality product requires much more expensive raw materials,
such that the quality-adjusted marginal cost of the high-quality product is higher than that of the low-quality
product. In this alternative scenario, we find that all firms will produce both products.3
Our results are broadly consistent with the empirical literature. Berry and Waldfogel (2010) examine
descriptive data on the relationship between product quality and market size. They find that in one industry(restaurants), where quality is created largely through variable costs, such as food ingredients and skilledlabor time, an increase in market size will increase the number of varieties, and while there will be morehigh-quality restaurants per capita there is “no evidence that average quality increases as well (p. 25)”.
In contrast, in the newspaper industry, where quality is produced by fixed cost rather than marginal cost,
Berry and Waldfogel (2010) find that larger market sizes are associated with higher average quality in themarket. Our results on decreasing markups and increasing product quality under trade liberalization are also
consistent with empirical studies. Indeed, using a model where intermediate good producers are oligopolists,
Edmond et al. (2015) reported in their empirical study of Taiwanese firms that increased exposure to trade
reduces the markup distortion by one-half. Along the same veins, using highly disaggregated firm-productlevel data from China over the period 2000-2006, Li and Miao (2017) found that market expansion in theform of trade liberalization induced China’s high productivity firms to increase the fraction of high quality
goods they exported.
Our analysis has implications for firms’ strategies. Should a firm that is capable of producing both highand low quality products o↵er a large range of quality to cater for all types of consumers, or should it focus
on the high-quality end of the product spectrum? According to our model, if quality is produced mainly
by fixed cost, and if there are other firms in the industry that choose to incur the low fixed cost (which
disqualify them from producing the high-quality product), then the high-quality firms should specialize inthe high-quality product, because supplying both types of products will create an adverse cannibalizatione↵ect, reducing consumers’ demand for their high-quality good.4 This result seems to be applicable to thenewspaper industry, where, as argued by Berry and Waldfogel (2010) , quality is produced mainly by fixed
cost. We do not observe high-quality newspaper firms that sell low-quality versions of their high-quality
products. For industries that operate under a di↵erent cost configuration, where quality is produced byvariable costs while fixed cost is independent of quality, our model predicts that each firm would sell both
2 This assumption is stated clearly in Johnson and Myatt (2006).
3 This is consistent with Johnson and Myatt (2006)’s proposition 9.
4 Of course, there are other reasons, such as preservation of brand names, but these are outside the scope of our model.

3the high-quality version and the low-quality version. This is consistent with the case of plasma televisions.
Indeed, Sony, Samsung, and Mitsubishi all o↵er product lines with varying degrees of resolution.
The rest of this paper is organized as follows. In Section 2, we review the related literature and relate ourmodel’s predictions to other findings. In Section 3, we specify our general model and analyze it under twoalternative cost configurations. In Section 4, we apply our theoretical framework to investigate the e↵ectsof trade liberalization on the average quality of products, on relative markups, and on the relative demandand supply of the high-quality product in two economies of di↵erent sizes. Section 5 o↵ers some concluding
remarks.

2

Related Literature
There is a large literature on vertically di↵erentiated products. A number of papers postulate that all

firms have the same fixed cost, on the assumption that quality is produced only with variable costs(Johnson
and Myatt, 2006; Mussa and Rosen, 1978; Gal-Or, 1983; Champsaur and Rochet, 1989). This happens
when quality improvement is achieved by using more expensive material inputs or more skilled labor. Other
authors adopt the opposite assumption: high-quality can only be produced by incurring higher fixed cost,such as R&D and advertising (Bonanno, 1986; Ireland, 1987).5 Shaked and Sutton (1987) assume that quality
improvement involves mainly an increase in fixed cost: variable costs may increase only at a “modest rate”(p. 136).6
The e↵ect of increased competitive pressure and/or market expansion on innovation was studied in Vives(2008) and Fajgelbaum et al. (2011). Unlike our model, Vives (2008) did not consider investment in quality
upgrading, nor vertical product di↵erentiation. His focus was on R&D expenditure by oligopolists to reduce
production cost (i.e., process innovation).7 Vives (2008) found that “increasing market size increases cost
reduction expenditure per firm and has ambiguous e↵ects on the number of varieties o↵ered”. He also notedthat Bertrand equilibria tend to be more competitive than Cournot equilibria, and that this conclusion must
be qualified when strategic commitments are allowed. Indeed, in Cournot models, since outputs are strategic
substitutes, it pays a firm to over-invest in order to gain an advantage. Thus, Cournot competition may
results in more cost reduction e↵orts. However, Vives (2008) pointed out that increased competition should
not be modelled as a move between Bertrand and Cournot equilibria, because the mode of competition is
likely to vary with the institutional features of the market. Generally, our setting di↵ers from that of Vives(2008) in a number of respects. First, in our model, consumers are heterogeneous while Vives (2008) assumed
identical consumers. Second, we allow firms to choose their product lines strategies, which may lead to an
5 Motta (1993) considered two separate models, one in which quality improvement comes from incurring a higher fixed cost,and one in which it relies only on variable costs.
6 They find that under certain additional conditions, under price competition, at least one firm continues to retain someparticular level of market share, no matter how large the economy becomes Shaked and Sutton (1987) (p. 132). This result isknown as the “finiteness property”. See also Shaked and Sutton (1983).
7 However, Vives (2008) argued that “the cost reduction model can be interpreted as an investment in quality (in terms of
product enhancement) in the context of the Cournot model.”4

outcome where di↵erent firms specialize in di↵erent quality levels and achieve di↵erent levels of markup andshort run profitability, while Vives (2008) assumed that firms are symmetric and behave in a symmetricway. Third, we address the impact of market expansion on the long-run equilibrium outcome while Vives’
focus was on the short run. Fourth, while in Vives’ model, market expansion is through population or
economic growth, we consider not only that form of market expansion, but also market expansion throughtrade liberalization. Our approach allows us to analyze firms’ quality scope strategies (balancing cost savinge↵ect and cannibalization e↵ect) and the e↵ect of market expansion or increasing competitive pressure on
the relative markups and market shares of di↵erent types of firms.
Fajgelbaum et al. (2011) considered a theoretical model in which products that are di↵erentiated bothvertically and horizontally. Following Gabszewicz and Thisse (1979; 1980) and Shaked and Sutton (1982;
1983), they assumed that when a person consumes more of the outside good her intensity of preferencefor quality of the goods provided by the oligopolists increases. This means that rich consumers appreciate
the high quality product more. However, as made clear in Fajgelbaum et al. (2011), they abstracted fromstrategic interaction by assuming monopolistic competition: each firm thinks that its price does not a↵ect
the average price of the industry. In contrast, our model assumes Cournot oligopoly, implying that strategicinteraction plays a key role. In our theoretical framework, each firm’s markup is variable in di↵erent market
conditions. Moreover, di↵erent from our model, Fajgelbaum et al. (2011) imposed the assumption that eachfirm can produce only a single-product. In the case where the industry o↵ers two quality levels, they wereable to determine the long run equilibrium number of the two types of firms. They proved that rich countries
export the high-quality product and import the low-quality product. This is because of the “home markete↵ects”: due to trade costs, the larger home market bestows a competitive advantage to local firms. In theirpaper, comparative advantage is generated on the demand side.8
Recent empirical works by Baldwin and Harrigan (2011) and Johnson (2012) allowed for vertically differentiated products by modifying the model of Melitz (2003). They found that high-productivity firms
export higher-quality products. They explained this by pointing out that these firms have greater incentive
to undertake quality-improving investments. Both papers assumed monopolistic competition. Specifically,
Johnson (2012) considered an extension of Melitz (2003) monopolistic competition model by allowing firms
to be heterogeneous in two dimensions: quality and cost.9
Di↵erent from the above literature, our analysis of market expansion via trade liberalization (in Section4) tackles the issue of how trade liberalization a↵ects the distribution of product quality in a vertically di↵erentiated international Counot oligopoly. This is a significant departure from the bulk of recent trade models
as these focus on monopolitic competition.10 An exception is Eckel and Neary (2010), who study oligopolistic
8 Other models in which richer countries export the luxury goods focus on the supply side. For example, in Markusen (1986)
and Bergstrand (1990), the richer country exports the luxury good because that good is capital intensive.
9 In his data set, quality and quality-adjusted prices are unobserved and can only be inferred.
10 While the monopolistic competition framework is convenient, the CES utility function assumed in this literature (e.g., Melitz(2003)) produces the counterfactual result that markup is a constant, independent of the market size. This is contrary to theempirical evidence, see, e.g. Edmond et al. (2015). In our model, the markups di↵er between low and high-quality products andvary with market size.

5

competition among multiproduct firms. They consider both supply and demand linkage within the firm. The
supply side linkage is characterized by flexible manufacturing, while the demand linkage is via the “cannibalization e↵ect.” According to Eckel and Neary (2010), multiproduct firms (MPFs) internalize demand linkage
between the varieties they produce. This feature is called the cannibalization e↵ect and is a defining featureof MPFs.” Unlike our model, in Eckel and Neary (2010) products are not vertically di↵erentiated. Theiremphasis is on the concept of core competence. They show that increasing trade openness tends to makeeach firm reduce their product varieties in order to better exploit their core competence. This leads to a
trade gain, in the form of raising each firm’s average productivity (since they are better in producing goods
that are in their in their core competence), but there is also an associated welfare loss, because consumers
will be exposed to fewer varieties.11 In a follow-up paper,Eckel et al. (2015), concentrating on short-run
equilibrium, allow firms to invest to improve the consumers’ perception of the firm’s product qualities. These
investments are possibly outlays on advertising. Applying their model to Mexican data, they find that thedata is consistent with the model’s key prediction, namely, firms in the di↵erentiated product sectors exhibit
quality-based competence (meaning that prices fall with distance from core competence), while firms in the
non-di↵erentiated product sectors exhibit the opposite patterns. Performing comparative statics, given the
fixed number of firms, they showed that an increase in the market size raises the output of all firms if firms
are homogeneous, but if firms are heterogeneous, the outcome displays the so-called “super-star” property:
Firms with above-average total output and outputs per variety tend to grow faster than other firms whenthe market size expands. 12
Our model is built on Johnson and Myatt (2006), where firms can choose the quality levels from a discrete
set {S1 , S2 , ..., Sm }, and they compete in quantities, taking the inverse demand function for each quality typeas given. However, in one of our cost configurations, we replace their assumption that all firms incur thesame set-up cost (regardless of the quality of the product that firms o↵er) with a more plausible one: firmsthat wish to specialize in the lower quality product incur a lower set-up cost than that of firms that produce
the high quality product. In addition, while Johnson and Myatt (2006) are mainly concerned with the short
run equilibrium, where the number of firms are fixed, the focus of our model is the long run equilibrium,
where free entry and exit ensures that profit is zero.

3

The model
We consider an oligopoly with vertically di↵erentiated products. Specifically, we assume there are twoquality levels, denoted by qL and qH , where 0 < qL < qH .13 A firm can produce the low-quality product,or the high-quality product, or both, incurring di↵erent variable costs and fixed costs. We will consider two
11 For an empirical test of the core competence model, see Eckel et al. (2016).
12 Long et al. (2011) examine the long run equilibrium in a model where firms are oligopolists with ex-ante cost heterogeneityand examine the e↵ects of trade liberalization on R&D expenditure and industry productivity. Their paper however assumes
that the all the firms in the industry produce the same homogeneous product.
13 In this paper, we follow the approach of Johnson and Myatt (2006) in that we do not address the issue of how the quality
levels qH and qL are determined. The set of quality levels is discrete and given.

6alternative cost configurations, as specified in subsection 3.2 below.

3.1

Consumers

There is a continuum of heterogeneous consumers. They di↵er from each other in terms of their intensityof preference for quality, which is represented by a parameter ✓, where 0  ✓  ✓. Let G(✓) denote thefraction of consumers whose intensity of preference is smaller than or equal to ✓. We assume that G(0) = 0,G(✓) = 1 and G0 (✓) > 0 for all ✓ 2 (0, ✓).

Each consumer buys at most one unit of the good. She must decide whether to buy one unit of thehigh-quality product, or one unit of the low-quality product, or neither product. A consumer of type ✓ places
a value ✓qH on the consumption of a unit of the high-quality product, and a value ✓qL on the low-quality
product. Let PH (respectively, PL ) denote the market price of the high-quality product (respectively, lowquality product). Her net utility is ✓qH

PH or ✓qL

PL , depending on which product she buys.14 We

assume that PH > PL . Let us define the following ratios:
✓L ⌘

PL
PH
PH
, ✓H ⌘, ✓I ⌘
qL
qH
qH

PL
qL

(1)

In what follows, we assume that equilibrium prices are such that 0 < ✓L < ✓H < ✓I < ✓. It is easy to show
that
PH PL
PH
PL
PH
>
()
<
(2)
qH qL
qH
qL
qH
A consumer whose ✓ equals ✓L is indi↵erent between not buying the good and buying one unit of the lowquality product at the price PL . Similarly, a consumer whose ✓ equals ✓H is indi↵erent between not buying
the good and buying one unit of the high-quality product at the price PH . And a consumer with ✓ = ✓I will
be indi↵erent between the two alternative purchases. The fraction of the population who purchases the highquality product is G(✓)

G(✓I ), the fraction who purchases the low quality product is G(✓I )G(✓L ), andthe fraction who does not buy the good is G(✓L ).

3.2

Producers

We consider a two-stage game. We assume that in stage 1, firms that want to produce the high-quality
product must incur an upfront cost (or set-up cost) FH , which represents the cost of R&D and purchases ofequipment. This fixed cost, once incurred, allow them to produce, in stage 2, both the high-quality productand the low-quality product (though in stage 2, it is possible that in equilibrium they choose to specializein the high-quality product). Any firm that wants to produce the low quality product must incur in stage 1
a fixed cost FL , where 0 < FL  FH . In stage 2, firms choose their output levels and compete as Cournot

rivals. The marginal production costs for high and low-quality products are CH and CL respectively, where
CL < CH . We assume that CL < CH < ✓qH .
14 In Subsection 3.4.2 we consider a more general utility function, as in Johnson and Myatt (2006), namely U = u(✓, q )
i

7

Pi .

Let XH and XL denote the industry outputs of the high-quality and low-quality product. Letting
PH (XH , XL ) and PL (XH , XL ) denote the inverse demand functions for the high-quality and the low-quality
product, the profit function for the two types of firm are
(

3.3

i
⇡H

=

j
⇡L

CH ] xiH + [PL (XH , XL )[PH (XH , XL )=[PL (XH , XL )CL ] xjL

CL ] xiL

FH

(3)

FL

Equilibrium

The equilibrium depends on the specifications of the trade-o↵ between fixed cost and marginal cost. Since
this trade-o↵ depends on the quality level, it turns out to be useful to define, for each of the two quality
levels qL and qH , the fixed cost per unit of quality and the marginal cost per unit of quality for each of thetwo product types:
fH ⌘

FH
,
qH

fL ⌘

FL
,
qL

cH ⌘

CH
,
qH

cL ⌘

CL
qL

While there are several possible configurations of (fH , fL , cH , cL ), we will restrict attention to two costconfigurations, which we refer to as Cost Configuration 1 and Cost Configuration 2. Under Cost Configuration
1, higher quality requires higher fixed cost per unit of quality (fHper unit of quality (cH < cL ), even though CH

fL ), but involves lower marginal cost

CL . Under Cost Configuration 2, we assume that cH

cL ,and FH = FL = F , which implies that fH < fL .
3.3.1

Cost Configuration 1: higher quality requires higher fixed cost per unit of quality andinvolves lower marginal cost per unit of quality

We now consider equilibrium under Cost Configuration 1, in which the following inequalities hold:fH

fL and cH < cL < ✓(4)

We assume that any firm j that has invested only FL is not able to produce the high-quality product.
Its output is denoted by xjL . In contrast, any firm i that has invested FH can produce both quality levels.
Its output levels of high and low-quality products are denoted by xiH and xiL . With the cost structure (4),and any general cumulative distribution of preference G(✓) with G(0) = 0, G(✓) = 1 and G0 (✓)

0 for all✓ 2 0, ✓ , we can prove the following Lemma about the equilibrium markups for the two types of product.
Lemma 1: Assume fH

fL , CH > CL , and cH < cL < ✓. For any general cumulative distribution of0

preference G(✓), with G (✓) > 0, in equilibrium, the markup on the high-quality product is greater than themarkup on the low-quality product, PH /CH > PL /CL , and the price per unit of quality is higher for the highquality product, i.e., PH /qH > PL /qL .
Proof: See the Appendix.
Remark 1: Using U.S. trade data, Baldwin and Harrigan (2011) found that, adjusted for quality, higher
quality products are more expensive (i.e., PH /qH > PL /qL ) and more profitable (i.e., PH /CH > PL /CL ),8this is consistent with our Lemma 1.
Our next Lemma states that if the cumulative distribution function G(✓) is convex (or linear, as in thecase of a uniform density function), then in any Cournot equilibrium where both types of firms exist in themarket, any firm that has invested FH will specialize in the high-quality product even though it has theability to produce both products.
Lemma 2: Assume fH fL and cH < cL < ✓ and that G(✓) has the following properties:(a) G(0) = 0, G(✓) = 1 and G0 (✓)(b) G00 (✓)  0.

0 for all ✓ 2 0, ✓ ,Then in a Cournot equilibrium where both types of firms exist in the market, firms that have invested FH
will specialize in the high-quality product.
Proof: See the Appendix.
Remark 2: Lemma 2 is based on the standard linearity assumption that u(✓, qi ) = ✓qi (as in Mussa andRosen (1978)). We will show in Section 3.4 that when we allow for a non-linear u(✓, qi ), the specialization
result of Lemma 2 will hold if some additional assumptions are made on the cumulative distribution function
G(✓).
In what follows, we assume that the distribution of preferences is uniform. Then, using Lemma 2 andgiven the firm entry in the first stage, i.e. nL and nH , the equilibrium outputs are solved from the equations:8h<✓qH 1h:✓q 1
L(nH xH )N(nH xH )Ni(✓qL nL xL )CHNi(nL xL )CL
N==(✓qH xH )N(5)(✓qL xL )N

The zero profit conditions can then be written as8¯ H2<⇡ ⇤ = ✓q
xi⇤
FH =
H
H
N
¯ L
i⇤ 2
:⇡ ⇤ = ✓q
xL
FL
L
N

0

(6)

0

Using the equations (5) and (6) we can solve for the long-run equilibrium numbers of each type of firms:15
8>< n⇤ =
H>: n⇤ =kpp

1/ +[1 k+(kcL cH )/✓ ](cL cH )L

where fH ⌘ FqHH , fL ⌘ FqLL , cH ⌘ CqHH , cL ⌘ CqLL ,

q

1 k

q

(✓N )/fH

1(7)

N/(✓fL ) 1

1 k⌘ ffHL ,and k ⌘ qqHL < 1. Recalling our specification thatcL > cH > 0 and fH > fL > 0 , we can see from equation (7) that n⇤L > 0 and n⇤H > 0 i↵
p
p
p
p
q
q
fH
fL + (1 k) fL
fH⇤⇤
N /✓ ⌘< N/✓ <cL
(cL cH ) + (1 k) ✓ cL
15 For details, please see the Appendix.

9

p

fL

cH

q
⌘ N ⇤ /✓

(8)

The intuition behind the restriction (8) is as follows. For both types of firm to co-exist, the market size
should not be too large, nor too small. Since the high-quality firms have a higher fixed cost, as long as themarket size is below N ⇤ (i.e., as long as N is not very large), there are not enough high-quality firms aroundto push the low-quality firms out of the market. If the market size is too small (i.e., it is below N ⇤⇤ ), there
are not enough customers to support the high fixed cost.
Proposition 1: Assume cH < cL and fH > fL , and that condition (8) is satisfied. Then there exists
a unique equilibrium point (n⇤L , n⇤H ) > (0, 0), and a marginal increase in N will increase n⇤H and decreasen⇤L , and will lead to an increase in the ratio of output of the high-quality product to that of the low-quality
product, XH /XL , implying that the average quality in the market rises.
Proof: see the Appendix.
Proposition 1 constructs an important property under the Cournot competition model, which is reconciledwith a rich number of empirical evidences, and has been shown in Fajgelbaum et al. (2011) with a pricecompetition model. With some restrictions on the parameters to guarantee the existence of unique andpositive solution, the proportions of high quality products and the firms that produce the high quality
products increase in the market size.
How do prices and markups change when the market size expands? When both types of firms co-exist in
the long-run equilibrium, we find that the long-run equilibrium prices are given by
8
< P

H

: P

L

=

C H + qH=

C L + qL

q

✓fH

q N

✓fL
N

(9)

From equation (9), we see that as the market size expands, the price of each product decreases, and so does
the ratio PH /PL . Concerning mark-ups, let us denote them by ⇢H ⌘ PH /CH and ⇢L ⌘ PL /CL . Then
8
< ⇢

H

: ⇢

L

=

1 + c1H=

1 + c1L

q

✓fH

q N

✓fL
N

(10)

Then we have the result that ⇢L < ⇢H and both markups decrease as the market size expands.
We summarize these results in Proposition 2.
Proposition 2: Assume all conditions listed in Proposition 1 hold, so that there exists a unique interior
equilibrium point (n⇤L , n⇤H ). Then an increase in the market size will (i) decrease the price and markups
of both the high and the low-quality products, (ii) decrease the relative price and the relative markup of thehigh-quality product.
Proof: Omitted.
Our results provide a possible theoretical explanation of the empirical finding by Edmond et al. (2015)
concerning the e↵ect of economic integration on firms’ markups and distribution of markups. They reported
that trade liberalization, which brings about an increase in market competition, reduces both the average

10markup and the dispersion of markups. Our model shows that, under Cost Configuration 1, an increase inmarket size leads to a decrease in all firms’ markup, and narrows the gap between high-quality markup andlow-quality markup. The intuition is clear. As the market size expands, more firms enter the market, andcompetition becomes more fierce. On the other hand, the increase in the number of firms is proportionallyless than the expansion of the market size. This makes firms find it more attractive to invest in the R&D fixed
cost to upgrade the quality they o↵er. Thus, high-quality firms face more intensive competition and theirmarkup declines. In contrast, since the number of low-quality firms decreases, the competitive pressure fromrivals of the same type is alleviated. This explains why the relative price PH /PL and the relative markup⇢H /⇢L decrease.
3.3.2

Cost Configuration 2: higher quality product requires higher marginal cost per unit ofquality but does not require a larger fixed cost

We now turn to the opposite case. In this case, FH = FL = F (an assumption made by Johnson andMyatt (2006)), and cH > cL , that is, the upgrading process is input intensive. We assume, as they did, thatany firm that incurs a single fixed cost F can produce both quality levels; it does not have to incur F twice.
As shown in Proposition 9 of Johnson and Myatt (2006), if there are two possible quality levels, all firms willproduce both high-quality and low-quality products. Applying this result to our model, we can solve for theequilibrium output of each product and determine the e↵ect of an increase in market size. We obtain thefollowing results.
Proposition 3: Assume FL = FH = F , cH > cL , (✓cL )/ ✓

cH > qH /qL , and F is small relative tothe market size. Then in equilibrium, each firm produces both low and high quality products, and an increasein market size will lead to(i) a less than proportionate increase in the number of firms,(ii) a fall in both PL and PH , but an increase in the relative price PH /PL and in the relative markup⇢H /⇢L ,(iii) an increase in the value share of the high quality product, PH XH /(PH XH + PL XL ), but no change
in the ratio of outputs, XH /XL .
Proof: See the Appendix.
The results for Cost Configuration 2 is quite di↵erent from the ones we obtained for Cost Configuration
1 in the preceding subsection. Recall that under Cost Configuration 1, an increase in the market size
leads to an increase in the ratio of high-quality output to low-quality output, XH /XL . In contrast, under
Cost Configuration 2, Proposition 3 shows that this ratio is unchanged. The reasons are as follows. First,
under Cost Configuration 2, the low and high-quality outputs share the same fixed cost, therefore the highquality product does not gain a cost-saving advantage over the low-quality product when the market size
increases. Second, under Cost Configuration 2, each firm produces both types of products, so the head tohead competition between the two types of products is substantially internalized within the firm’s boundary.
Thus it is not surprising that the ratio XH /XL does not change when the market size expands.

11

Even though the output ratio XH /XL does not change, there is an increase in the relative price PH /PL
when the market expands, leading to an increase in the number of firms. This generates a fierce competition
in attracting customers with low preference intensity who previously did not buy the good. Therefore thepercentage fall in PL is bigger.

3.4

Discussion

3.4.1

Characteristics of the equilibrium and firms’ strategies

Figure 1 presents a schematic distinction between Cost Configurations 1 and 2. When the cost structureis characterized by a common fixed cost FH = FL = F (which is shared by both types of product, such thatany firm that has invested F can produce both types of product without paying any additional fixed cost),
and the marginal cost, adjusted for quality, is higher for the high-quality product, cH > cL (i.e., the quality
upgrade process is input intensive), each firm will find it optimal to produce both types of good. An increasein market size will attract more firms, but firms do not adjust their output ratio, xiH /xiL . Under this cost
structure, the average quality in the market, defined as
qb =

✓

XH
XH + XL

◆

qH +

✓

XL
XH + XL

◆

qL

(11)

does not change when the market expands.
In contrast, when the quality upgrade is fixed-cost intensive, any firm currently producing the low-quality
product that wishes to upgrade must incur a higher fixed cost per unit of quality, FH /qH > FL /qL . Under
the variable cost specification that CH > CL , but CH /qH < CL /qL , we find that for a firm that has incurred
FH , its optimal product line strategy is to specialize in the high-quality product, even though it is capable ofproducing both vertically di↵erentiated varieties. Thus, under this cost structure, each firm produces a single
type of product. The reason is that it is not worthwhile for a firm that has invested FH to manufacture thelow-quality product, at a higher marginal cost (adjusted for quality), cL > cH , as it would compete with its
own high-quality product. In other words, the firm wishes to avoid the phenomenon called “cannibalization”:
producing the costly low-quality good will add to the market supply of that good, reducing its price, whichswitches consumers away from its high-quality product. In view of this product-line specialization result, it
is clear that an expansion of the market will lead to an increase in the average quality in the market, definedby (11) above. Indeed, the market expansion will lead to an increase in the number of high quality firms
and a fall in the number of low quality firms. As discussed in the previous section, this result is due tothe cost saving e↵ect, i.e., an increasing market demand brings greater benefits to the product that requires
a high fixed cost. From the consumers’ point of view, there are two main gains from market expansion.
First, the prices fall, and thus a higher fraction of consumers are served. Second, the market share of thehigh-quality product increases, which implies that on average, consumers have a greater access to the higher
quality products.

12

Our theoretical results are consistent with the stylized facts in the real world, especially for the typeof industry in which quality upgrading is capital intensive, in the sense that fH > fL . Examples of thistype of industry, where the quality upgrade is largely dependent on a massive fixed investment, include themedia, cosmetics, fashion goods, and pharmaceutical industries. In these industries, each firm is more likely
to specialize in either the high end or the low end of the quality spectrum. For example, one does not observea magazine or a newspaper that is sold in di↵erent version with di↵erent contents and prices. Similarly, inthe pharmaceutical industry, the firms with famous brand names would not introduce a generic version ofits branded drugs, unless the patent has expired. During the patent protection period, they only sell thebranded drugs with the highest quality. As discussed previously, these market strategies aim at avoiding
the cannibalization e↵ect. Berry and Waldfogel (2010) found empirical support for the prediction that in
an industry with several types of firms with each firm specializing in a single quality level, an expansion of
the market size leads to an increase in the average quality in the market. Our prediction that an increase inmarket size leads to lower markups and reduces the dispersion of markups are in line with the calibration
results of Edmond et al. (2015).

Figure 1. Cost structure, firms’ strategies, and the industrial structure

3.4.2

The general form of the utility function

Our result that a firm that has invested FH will specialize in the high-quality product (if fH > fL andcL > cH ) were obtained using the specification that the net utility that a consumer of type ✓ derives fromconsuming a good at quality level qj and price Pi is ✓qj

Pj . A more general net utility specification would

be, as proposed by Johnson and Myatt (2006),
U (Pj , qj , ✓) = u(✓, qj )

Pj

(12)

where the gross utility u(✓, qj ) is increasing in ✓, and, for any given ✓, it is assumed that u(✓, qH ) > u(✓, qL )for qH > qL . In what follows, we will show that our result of specialization in the high-quality product stillholds under this more general utility function, and under a more general distribution of ✓, provided thatsome additional assumptions are added.
13⇥ ⇤Consider any cumulative distribution function G(✓) over a compact set 0, ✓ such that G0 (✓) > 0 for

all ✓ 2 0, ✓ and G(0) = 0, G(✓) = 1. Given (qH , qL ) and the prices (PH , PL ), where PH > PL , let ✓I be

the consumer who is indi↵erent between buying a unit of the high-quality product and buying a unit of the
low-quality product. Then it holds that u(✓I , qH ) > u(✓I , qL ). Let us add the assumption that the partial
derivative of u(✓, qH ) with respect to ✓, evaluated at ✓I , is greater than the partial derivative of u(✓, qL ) withrespect to ✓, evaluated at ✓I :u1 (✓I , qH ) > u1 (✓I , qL )(13)

where u1 stands for the partial derivative with respect to ✓. This property ensures that the consumer with✓ > ✓I will buy the high quality product in preference to the low quality one. For simplicity, let us normalize
the population size to 1. Then G 1 (1XH ) = ✓I and G 1 (1XH XL ) = ✓L . For a representative firm iithat has invested FH and produces the quantity xH , let us denote by (xiH ) the increase in gross utility if
the marginal consumer ✓I switches from the low-quality product to the high-quality product⇥(xiH ) = u G 1 (1XH i

xiH ), qH⇤⇥
u G 1 (1XH i

xiH ), qL⇤(14)This expression is equal to the di↵erence between the equilibrium prices of the two products, PH

PL . Weii
now make the assumption that by increasing xH by one unit and reducing xL by one unit, the gain in revenue
exceeds the increase in cost,
0

where

(xiH ) ⌘(xiH ) > CH

CL .

(15)

(xiH )xiH .

We can now state the following proposition.
Proposition 4: Assume that the general utility function satisfies condition (13) and that the di↵erencein marginal cost is not too large, so that condition (15) is met at the optimal output level xiH . Then thefirm that has invested in the fixed cost FH (which enables them to produce both type of products) will find it
optimal to specialize in the high-quality product.16
Proof: See the Appendix.
16 Our results are di↵erent from the Proposition 11 of Johnson and Myatt (2006), where they claimed that the firms thatare able to produce both types of products will not specialize in producing only one type of product. However, there is an⇤ > Z ⇤ , where
implicit assumption behind that Proposition. In Proposition 11 of Johnson and Myatt (2006), they assume ZL
HZi⇤ denotes the equilibrium quantity of the products whose quality is higher than or equal to i and i = H, L.This assumption
contains an implicit condition, i.e. the marginal-cost di↵erence between the two types of products is large enough. Otherwise,
the low quality products won’t be o↵ered by the high type firms due to the cannibalization e↵ect. In contrast, Proposition 4of our paper assumes that the di↵erence of the marginal cost between the high and low quality products is low enough. If thehigh type firms provide the low quality products, they won’t save a lot from lowering the marginal cost, but su↵er the so calledcannibalization e↵ect.

14

4

The e↵ects of economic integration
In this section, we use our theoretical framework to explore the e↵ects of economic integration, modeled

as fall in trade costs. Intuitively, lowering the trade barriers between two trading countries is similar to an
expansion in the market size faced by the firms from each country. Therefore, if two countries enter intoa free trade agreement and commit to reduce the trade barriers between them, we can expect that there
will be an increase in the market share of the high-quality product in each country, and a reduction in themarkups and in the dispersion of markups. We restrict attention to Cost Configuration 1, so that firms thathave incurred the higher fixed cost will not want to produce the low-quality product, and firms that haveincurred the low fixed cost are unable to produce the high-quality product. If the two economies are of thesame size, we are able to obtain analytical expressions for long run equilibrium prices and markups of eachproduct type in each market, and sales of each type of firm in the domestic and in the export market. Whencountries are of di↵erent sizes, we must resort to numerical simulations, and report our results with graphicalillustrations.

4.1

Trade liberalization with a vertically di↵erentiated international oligopoly

Suppose there are two countries, called the home country, denoted by h, and the foreign country, denotedby f . In each country, there are two industries. Industry 1 produces two vertically di↵erentiated products:
a high-quality product and a low-quality product. Their quality levels are denoted by qH and qL , where
qH > qL > 0. Firms in industry 1 must incur fixed costs, FH and FL , and we assume that fH > fL wherefH = FH /qH and fL = FL /qL . These firms are Cournot oligopolists. Marginal costs are CH and CL , where
CH > CL , and we assume that the quality adjusted marginal costs, cH = CH /qH and cL = CL /qL aresuch that cH < cL . Industry 2 produces a homogeneous good under constant returns to scale, and operates
under perfect competition. The homogeneous good serves as the numeraire good: its price is unity. Thereis only one factor of production, called labour. Each unit of labour can produce one unit of the numeraire
good. Therefore the wage rate is also unity. The costs FH , FL , CH and CL , and the prices PH and PL are
all expressed in terms of the numeraire good. There are Nh consumers in h and Nf consumers in f . 17
To simplify matters, we assume, as is standard in the recent literature on trade liberalization (see, forexample, Krugman (1995), Feenstra (2015) , Long et al. (2011)) that trade barriers take the form of an“ice-berg transport cost” rather than tari↵s, and that only the di↵erentiated products are subject to tradebarriers. When a domestic producer of the di↵erentiated product sends x units of the good to the foreigncountry, only x units arrive, where 0 << 1. Partial trade liberalization is modeled as an increase in .

Each home firm that produces the high-quality product must decide on the quantity xhh
H that it supplies
17 For simplicity we assume that utility derived from consuming the numeraire good is linear in quantity consumed. Eachconsumer is endowed with M units of labor, which they supply inelastically to the labor market. Since we focus in the longrun equilibrium (i.e., firms earn zero profit), there are no positive profits to be distributed to consumers. Consequently, eachconsumer’s income is M . Each consumer spends her entire income on consumption. Each buys at most one unit of the good
produced by industry 1, and the remaining income is spent on the numeraire good. We assume that M > PH > PL so thatbuying one unit of the di↵erentiated good does not exhaust the consumer’s income.

15

to the domestic market, and the quantity xhf
H that it exports to the foreign market. Similarly, each home
hf
f
fh
f
fh
firm that produces the low-quality product must decide on xhh
L and xL . The symbols nH xH and nL xL

represents foreign supplies to the domestic market. Then total supplies of the two products in the domestic
market are
8<X h =H:X h =
Lffh
nhH xhh
H + nH x H

(16)

f
fh
nhL xhh
L + nL x L

The profit functions for home firms are
8
< ⇡h

H

: ⇡h

L

=
=

h ⇣
✓ qH 1

⇣

h
XH
Nh

✓qL 1

⌘

i
h ⇣
Xh
qL NLh xhh
+
✓
qH 1
⌘ H
⇣
h
h
XH
XL
hh
Nh
Nh xL + ✓qL 1

f
XH
Nf
f
XH
Nf

⌘

i
Xf
hf
qL NLf xhf
CH (xhh
H + xH )
H
f ⌘
XL
hf
xhf
CL (xhhFL
L + xL )
L
Nf

FH

(17)

For the foreign firms, similar formulas apply. In the long run equilibrium, profits are zero, and we obtainthe following expressions for long run equilibrium prices in the home country:
8
< Ph

H

: Ph
L

=
=

q

✓qH FHq (1+ 2 )Nh+ CH✓qL FL
(1+ 2 )Nh + CL

Equation (18) shows that equilibrium prices fall as trade costs fall (i.e., as
as

(18)

increases). With Nh = N/2,

tends to 1 (complete liberalization), this equation coincides with the long run price equation in Section 3.

Concerning long run equilibrium quantities sold by each firm in the domestic market and the export markets,
we obtain the following expressions:
8
>
xhh
>
H
>
>
>
>
hh
>
x
>
< L
hf

Thus, as

xH
>
>
>
>
>
>
>
hf
>
: xL

=
=
=
=

q

N h FH2)
N h FL2r ✓qL (1+ )q ✓qH (1+r

N f FH✓qH (1+ 12 )(19)

N f FL✓qL (1+ 12 )increases (trade becomes more liberalized), each firm’s sales in the domestic market fall (becauseof competition from foreign firms) while its sales in the export market increase.

16

Home firms’ average markups are18
8
>
⇢h
>
>
< H
>
h
>
>
: ⇢L

p

=
CH

r

=
CL

FH (1+ 2 )
r

Nh
+ 2
✓qH

p

r

Nf
✓qH

FL (1+ 2 )rNNh+ 2 ✓qf
✓q
L

+1
(20)
+1

L

Concerning the equilibrium number of firms of each type in each country, we are able to obtain an explicit
expression only in the symmetric case, where Nh = Nf = N/2.
8
>
>
<
>
>
:

nH =
nL =

r

p
p
qL )+CL CH ]+ qL FL
q H FHp2) q F
(1
k)(1+
H
H
r
p
p
(1+ 2 )Nk
(CH CL /k)+k qH FH
qL FL
¯
2✓
p
(1 k)(1+ 2 ) qL FL(1+ 2 )N ¯
[✓(qH
¯
2✓

(21)

where k ⌘ qL /qH . Using the above results, we can obtain the following analytical results concerning thelong-run e↵ect of trade liberalization.

Proposition 5: Consider a two-country, two-quality-product trade model where the market equilibriumis in long run, i.e. the firms are allowed to free entry-exit. A decrease of the trade cost in both country willlead to the following changes:[1] price levels of both products in each country decrease;[2] total consumption on di↵erentiated products increase.
Moreover, if the following additional conditions hold, i.e. both countries are symmetric, i.e. they sharethe same population number,Nh =Nf =N/2, and the relative fixed cost of high quality product is large enough,⇣⌘2¯
i.e. > ✓✓¯ ccHL , then we further get:[3] relative number of high quality firms increase;[4] relative quantity of high quality products supplied in each country increases;[5] average markup of each firm decreases;[6] the relative average markup of the high quality firm decreases.
Proof : See the Appendix.
Trade liberalization thus has similar e↵ects to an expansion of market size. It increases consumers’
accessibility to foreign firms’ products and thus increases the intensity of competition. Proposition 5, which
deals with the e↵ects of trade liberalization, o↵er results that are qualitatively similar to Propositions 1 and
2, which deal with the case of a pure market expansion. In the next subsection, we illustrate the e↵ects ofvarious degrees of trade liberalization on a number of key variables, both for the case of two trading partners
with identical population size and for the case where one country is smaller than the other.
18 Since the prices di↵er across markets, and the marginal cost in the foreign market, inclusive of the iceberg transport cost, ishigher than the marginal cost in the domestic market, the markups reported here are computed using the firm’s average priceand average marginal cost.

17

4.2

Simulation results

This subsection reports simulation results. For both the case of two identical trading partners and thecase of two countries that di↵er in population size, we use the following parameter values: ✓ = 1, qH = 10,
qL = 8, FH = 1, FL = 0.1, CH = 0.1, CL = 0.08.
We consider di↵erent values

from the set {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}. Note that we do notconsider the value = 1, for in that case there is no trade barriers, and thus the total number of firms ofeach type in each country become indeterminate. 19 When the two countries have identical population size,
we set Nh = Nf = 105. When their population sizes di↵er, we set Nh = 100 and Nf = 110.
Figures 2 to 6 report results for the case of two countries with the same population size. (To save space,these Figures refer only to the home country, because what happens to the foreign country is the same.)
The upper panel of Figure 2 shows that asincreases (i.e., as trade becomes more and more liberalized),

the number of domestic firms of each type falls (but of course each domestic consumers are served by more
firms thanks to the trade liberalization). The lower panel depicts the ratio of high quality firms to low quality
firms, nH /nL . Figure 3 shows that with more liberalized trade, the price of each type of product falls, butthat of the high-quality product falls by more. The upper panel of Figure 4 shows the falls in markups, andthe narrowing of the gap between the two markups. The lower panel of Figure 4 shows the fall in the relative
markup of the high-quality product. Figure 5 reports the total supply of each type of products by the firmsthat located in each country. It shows that the supplied quantity of the high-quality product increases whilethat of the low-quality product decreases in . The relative quantity supply of high quality products increaseinas well. Figure 6 reports the total demand of each type of products in each country. We observe asimilar result, i.e. the demand for high quality products increases while the demand for low quality products
decreases in .
Figures 7 to 11 report simulation results for the case of two countries with di↵erent population sizes.
Overall, these simulations show similar results to the case of symmetric countries. The main di↵erence isthat in the asymmetric case, when the trade costs decrease (i.e.,

increases) the small country’s ratio of

high-quality firms to low-quality firms decreases if the initial trade barriers are small, i.e., the initialis large(Figure 7). This is reflected in the ratio of the supply of high-quality product to the supply of low-quality
product by the small country (Figure 10). The result is due to the relocation of high-quality firms from thesmall country to the large country in some range of trade cost, because firms choose their location based on
the balance of market demand and market competition intensity.
The top left-hand panel of Figure 7 shows that the smaller country’s firm number of each type declines
when trade is liberalized. Their ratio changes in a non-monotone way, as shown in the bottom left-hand panelof Figure 7. The right-hand panels report what happens to firm numbers in the larger country. Figure 8
shows the e↵ects of trade liberalization on the prices of each type of product in each country. While all prices
fall with trade liberalization, the price of the of both products are lower in the larger country, and they remain
19 When both countries are fully integrated, it’s equivalent for firms to choose either country to locate in. Thus, we cannot
decide the firm number in each country.

18

so even when trade liberalization has substantially reduced trade cost (i.e., at= 0.9). Figure 9 shows thatmarkups and the relative markup of the high quality product decline as trade liberalization proceeds. Figure
10 reports the e↵ects on the total quantity supplied by each type of firm in each country. Interestingly, thesupply of high quality products increases in size when trade costs decrease ( increases), while the opposite
happens to the small country. As stated before, this is because of the relocation of high-quality firms fromthe small country to the large country. Finally, Figure 11 reports changes in the quantity demanded ofeach type of product in each country. The ratio of consumption of the high-quality product to that of the
low-quality product rises as trade liberalization proceeds. In the long run, as firms earn zero profits, eachcountry’s welfare depends only on the prices PH and PL , and since these prices fall monotonically as trade
costs decrease, welfare obviously increases.

5

Concluding Remarks

This paper studies the Cournot equilibrium of an oligopoly producing vertically di↵erentiated products
under free entry and exit. The model highlights how prices, markups and average quality depend on themarket size and on the cost structure.
Our main focus was on the cost configuration where the set-up cost depends on the quality of the product.
In the first stage of the game, firms decide whether they incur the high fixed cost, which enables them toproduce the high-quality product (and also the low-quality one), or only the low fixed cost, which only
permits them to produce the low-quality product. In the second stage, they decide on their output levels.
We showed that in equilibrium firms that have incurred the high fixed cost will specialize in the high-quality
product if the fixed cost for the low-quality product, adjusted for the quality level, is lower than that for thehigh-quality product. Under that cost configuration, we showed that an increase in market size leads to (i)an increase in the fraction of firms that specialize in the high-quality product, (ii) an increase in the market
share of the high-quality product, (iii) a decrease in the prices and markups of both types of product, (iv) a
decrease in the relative markup of high-quality product.
We also considered the opposite case where the fixed cost is independent of quality level while the qualityadjusted marginal cost of the high-quality product is higher than that of the low-quality product. Under
that cost configuration, we found that all firms will produce both quality levels and an increase in market
size will increase the value share of the high quality product, but leaves its quantity share unchanged.
As an extension of our model, we applied our theoretical framework to analyze the e↵ects of tradeliberalization. We found that a reduction in trade costs has similar e↵ects as a market expansion: the prices
and markups of both types of product decrease, while the share of the high-quality product increases.
Our study constitutes an extension of the existing literature on markets with vertical product di↵erentiation in two directions. First, we o↵er a detailed characterization of the long run equilibrium mix of firms
under alternative specifications of the cost of quality upgrading. Second, we explore the e↵ect of market
size on market outcomes, especially on the market shares and markups of high-quality versus low-quality

19

product, when firms are Cournot rivals within groups as well as across groups. The theoretical predictionsof our model are broadly consistent with empirical studies and stylized facts. For example, the firms thathave made massive investment in quality upgrading are more likely to specialize in the high end of the quality spectrum in order to avoid the cannibalization e↵ect. Trade liberalization induces firms to upgrade theproduct quality and reduces firms’ markup. Consumers gain from trade liberalization, not only from lower
prices, but also from greater access to high quality products.
For simplicity, we have assumed that all firms of each type have the same level of productivity. A naturalextension would be to allow for heterogeneity in productivity.

Appendix A
Proof of Lemma 1⇥ ⇤Since G0 (✓) > 0 over 0, ✓ , we can define the inverse function ⌦(y), where y = G(✓). Then the inverse

demand functions are

(

PH
PL

=

⌦ 1

XH
N

=

(qH

qL ) + ⌦ 1⌦ 1

XH +XLN

XH +XLNqL

XH + XL
N

◆✓
k

(A.1)

qL

From the first order conditions of profit maximization, we obtain
PH
CH

 ✓
PL
qH
=
⌦ 1
CL
CH

XH
N

◆

(1

✓
k) + ⌦ 1

cH
cL

◆

(A.2)

Now, since k = qL /qH < 1 and cH  cL , and since ⌦(.) is monotone increasing, the RHS of equation (A.2)
must be positive. It follows that the LHS must be positive, too. Finally,
PH
CH

PL
CL

✓
qH
⌦ 1
CH

XH + XL
N

◆✓
1

cH
cL

◆

> 0.

(A.3)

Q.E.D.
Proof of Lemma 2
Recall that the inverse demand functions are given by equation (A.1) above. Firm i’s profit function is⇡i = (PHThe F.O.C.s areCH )xiH + (PLCL )xiL

FH

(A.4)

@PH
@PL
+ xiL
 0,
@XH
@XH

(= 0 if xiH > 0)

(A.5)

@PL
@PH
+ xiH
 0,
@XL
@XH

(= 0 if xiL > 0)

(A.6)

(PH

CH ) + xiH(PLCL ) + xiL

20

We now show that if both xiH and xiL were strictly positive, a contradiction would emerge. For then conditions(A.5) and (A.6) would give
PH
qH

CH
qH

✓
1 0
⌦ 1
N

XH
N

PL
qL

CL
qL

◆

(qH

✓
1 0
⌦ 1
N

qL ) xiH

✓
1 0
⌦ 1
N

XH + XLN◆

XH + XLN◆

k(xiH + xiL ) = 0(A.7)(xiH + xiL ) = 0(A.8)

Subtracting (A.7) from (A.8), we obtain
CH
qH

⇢
✓
◆
PH
PL
1 k i 0
XH + XL
+
x ⌦ 1
qH
qL
N L
N
⇢
 ✓
◆
✓
◆
1 k i
XH
XH + XL+x H ⌦0 1
⌦0 1
N
N
N

CL
=
qL

⇢

(A.9)

The first bracketed term on the RHS of (A.9) is positive because
PH
qH

PL
= (1
qL

 ✓
k) ⌦ 1

XH
N

◆

✓
⌦ 1

XH + XLN◆> 0.

(A.10)XHN >
XH +XL,N

The second bracketed term on the RHS of (A.9) is also positive, because ⌦0 > 0. Recalling that 1
1

XH +XL, the third bracketed term on the RHS of (A.9) is positive if ⌦0
N
0

1

XH
N

> ⌦0 1

i.e., if ⌦ (y) is increasing in y, i.e., if ⌦(y) is convex, which holds i↵ G(✓) is concave. On the other hand,by assumption, cH  cL , so that the LHS of (A.9) is negative. Therefore, if both xiH and xiL were strictly
positive, we would have a contradiction. Therefore, either xiH > 0 or xiL > 0, but not both. But if xiH = 0
and xiL > 0, it would not make sense for firm i to invest in FH . It follows that G00  0 implies that the firmthat has invested FH will specialize in the high quality product. Q.E.D.

Proof of Proposition 1
Given Lemma 2, each type of firm specializes in one type of product. In this case, the firms’ profit functionin the second stage can be written as:
8<⇡
Take F.O.C., we get:

H =
:⇡ =
L

(PH

C H ) xH

FH

(PLCL ) xLFL

8<(@P /@x ) x + P
CH
H
H
H
H
:(@P /@x ) x + P
C
L

L

L

L

21

L

=0
=0

(A.11)(A.12)

As the inverse demand function is:
8
<P
:P

H

= 1

L

= 1

XH
N
XH
N

XL
N ✓qL

✓qH
XL
N

(A.13)✓qL

we can re-write equations A.12 as:

8
<

✓qH ⇤
N x H + PH
✓qL ⇤N x L + PL:

Which implies that8<⇡ ⇤ =

CH

=0

CL

=0

✓qH ⇤2N xH✓qL ⇤2N xL

H:⇡ ⇤ =
L(A.14)

FH(A.15)FL

The equationsA.14 imply that all firms of the same type will choose the same quantity strategy. In thiscase, we can get XH = nH xH and XL = nL xL . Next, we need to solve for the firm numbers in the firststage. The firm numbers are solved with the free entry-exit condition, i.e. when the profit equals to zero foreach type of firms.
8<⇡ ⇤ =
H:⇡ ⇤ =
L

0(A.16)

0

Combining equations A.13, A.14, A.15, A.16, and the fact that XH = nH xH and XL = nL xL , we cansolve for the equilibrium firm numbers as:
8>< n⇤ =
H>: n⇤ =
L

where fH ⌘ FqHH ,fL ⌘ FqLL ,

k

p

p

1/ +[1 k+(kcL cH )/✓ ](cL cH )q

1 k
N/(✓fL ) 1

q(✓N )/fH

1(A.17)

1 k⌘ ffHL, cH ⌘ CqHH , and cL ⌘ CqLL . Then, under the conditions demonstrated
in Proposition 1, it is easy to obtain that n⇤H > 0, n⇤L > 0, @n⇤H /N > 0, and @n⇤L /N  0. In this case, thenumber share of the high quality firms increases. Next, we will show that the quantity share of high quality
products increases in market size. The quantity share of the high quality products is XH /XL = nH xH /nL xL .
From equations A.15 and A.16, we know that xH /xL is constant. In this case, the quantity ratio is totally
determined by the firm number ratio. Thus, the quantity share of the high quality products increases in themarket size as well.
Q.E.D.
Proof of Proposition 3

22

Our first step is to show that if x⇤H > 0 and x⇤L must be positive. Consider the first order conditions offirm i with respect to xiH and xiL respectively,
✓
✓
xiH
+ 1
N

◆

kXL
N

kxiL
CH
=
N
✓qH

(A.18)◆xiLCL(with equality holding if xiL > 0)N
✓qL

(A.19)XHN

1

XH + XL
N

xiH
N

Subtracting, and using XL = nxiL , we get
CH
✓qH

CL
1
 (XL + xiL )(1
N
✓qL

k) =

1
(nL + 1)xiLN(A.20)

Since cH > cL , it must hold that xiL > 0, which implies that
✓
xiH
+ 1
N

XH + XL
N

◆

CH

CL

xiL
CL
=
N
✓qL

(A.21)

Using equations (A.18) and (A.21), we obtain
XL⇤
✓qH
✓qL
=
N
1 + n1 (1 k)
and

⇤
XH
n
=
N
1+n

n
(1 + n)(1

Then XH > 0 i↵
k<

CH
✓qH
CL
✓qL

1
1

k)

✓

CH
✓qH

(A.22)

CL
k
✓qL

< 1.

◆

(A.23)(A.24)

From equations (A.22) and (A.23), the ratio XL /XH is independent of the market size. Solving for theequilibrium prices, we get

(

⇤
PH

=

PL⇤

=

nCH +✓qH
n+1
nCL +✓qL
n+1(A.25)⇤
Since cL < cH , we obtain PL⇤ /qL < PH
/qH .

With all firms producing both types of product, the zero profit condition for the industry is simply⇤(PH⇤
C H ) XH
+ (PL⇤

23

CL ) XL⇤ = n⇤ F

(A.26)

Thus we can solve for n⇤
n⇤ =
where
⌘

✓

1

CH
✓qH

◆✓
1

k+k

CL
✓qL

s

qH N ✓(1 k)F

CH
✓qH

◆

1

✓
+k 1

(A.27)
CL
✓qL

◆✓

CH
✓qH

CL
✓qL

◆

> 0.

(A.28)

Q.E.D.
Proof of Proposition 4
To simplify our expressions, we let E ⌘ 1

XH and F ⌘ 1

XL . Let G 1 (·) be the inverse of G(.).

XH

Then for a firm i that has invested in FH has the following profit function⇡i = u(G 1 (E), qH ) u(G 1 (E), qL ) + u(G 1 (F ), qL )⇥⇤+ u(G 1 (F ), qL ) CL xiL

CH xiH(A.29)

Consider the partial derivatives with respect to xiH and xiL@⇡i= u(G 1 (E), qH ) u(G 1 (E), qL ) + u(G 1 (F ), qL ) CH
@xiH
n
0
0
+ xiH
G 1 (E)u1 (G 1 (E), qH ) + G 1 (E)u1 (G 1 (E), qL )0

G 1 (F )u1 (G 1 (F ), qL )0

xiL G 1 (F )u(G 1 (F ), qL )⇥@⇡i
= u(G 1 (F ), qL )
i
@xL

CL

⇤

0

xiL G 1 (F )u(G 1 (F ), qL )o0

xiH G 1 (F )u1 (G 1 (F ), qL )(A.30)(A.31)@⇡iwhere u1 (·) denotes the partial derivative of u1 (·) with respect to the first variable. Then @x
i

H

@⇡i
@xiL

has

the same sign asnu(G 1 (E), qH )⇥u(G 1 (E), qL ) + xiH u1 (G 1 (E), qL )o⇤0
u1 (G 1 (E), qH ) G 1 (E)(CH

CL ) (A.32)

If this expression is strictly positive, then the firm will not produce both types of products, and thus xiH > 0⇥⇤0
and xiL = 0. Note that the term inside {...} is (xiH ) , where (xiH ) ⌘ u(G 1 (E), qH ) u(G 1 (E), qL ) xiH .
Under the condition that0(xiH ) > CH

CL , the expression above is positive.

Q.E.D.
Proof of Proposition 5
From equation (18), we get that @Pij /@ < 0 for all i = H, L, and j = h, f . Thus, the price levels of bothproducts in each country decrease. As the price levels of di↵erentiate good decrease, more consumers that
initially consume the numeraire good will turn to buy the di↵erentiate good. Thus, more consumers will beaccess to the di↵erentiate good. Next, we will assume that the two countries are symmetric in population
size, i.e. Nh = Nf = N/2. From equation (21), we get the relative number of the high quality firms as
24

nH /nL =

r

kFL
FH

p

2 ) N/2(1 +⇥✓¯ (qHhk N/2 (1 +

⇤ p p
p
CH )
✓¯ qH FH
q L FLip pp¯CL /k) + ✓ qH FH
q L FLqL ) + (CL2 ) (C

H(A.33)For convenience, we denote the relative number of high quality firms as Rn ⌘ nH /nL . Under condition⇣⌘2¯> ✓✓¯ ccHL , we have @RH /@ > 0. Thus, the relative number of the firms that produce the high quality
products increases. From equations (19) and (21), we can get the relative quantity of the high quality
products sold in each country as:
Rx =
As

q

qL FH
qH FL

s

q L FHRn
q H FL(A.34)

is constant, the sign of @Rx /@ is the same as @Rn /@ , i.e. @Rn /@ > 0. Thus, in response to alower trade cost, the relative quantity of high quality products supplied in each country increases. With theassumption of symmetric countries, we rewrite the markups for high and quality firms as:
8<markupH =:markup =
L

1
CH
1
CL

q

¯ H FH2✓q2) + 1¯
2✓qL FLN (1+ 2 ) + 1

q N (1+(A.35)

Obviously, @markupi /@ < 0 for all i = H, L. Thus, average markup of each firm decreases, when trade
cost reduces. Last, we will show that the relative markup of the high quality firms decreases. Based on theequation (A.35), the indicator for this relative markup is specified as:
2

1
Rm = 4
CH

s

3 2
3
s
¯ H FH¯ L FL2✓q
1
2✓q+ 15 / 4+ 15
N (1 + 2 )CL N (1 + 2 )(A.36)⇣⌘2¯
Recall that the relative fixed cost of high quality products is high enough, i.e. > ✓✓¯ ccHL
1. (cH  cL ) .
⇣ ⌘2
⇣ ⌘2
CH
H
Then we have > 1 > ccHL
. That is qqHL F. Thus, we have @Rm /@ < 0. The relative markup
FL >CLof the high quality firms decreases. Q.E.D.

25

Appendix B

Figure 2: E↵ects of trade liberalization on firm numbers, the case of symmetric countries

26

Figure 3: E↵ects of trade liberalization on price levels, the case of symmetric countries

Figure 4: E↵ects of trade liberalization on markups, the case of symmetric countries

27

Figure 5: E↵ects of trade liberalization on firms’ production, the case of symmetric countries

28

Figure 6: E↵ects of trade liberalization on market demand, the case of symmetric countries

29Figure 7: E↵ects of trade liberalization on firm numbers, the case of asymmetric countries

Figure 8: E↵ects of trade liberalization on price levels, the case of asymmetric countries

30

Figure 9: E↵ects of trade liberalization on markups, the case of symmetric countries

31

Figure 10: E↵ects of trade liberalization on firms’ production, the case of asymmetric countries

32

Figure 11: E↵ects of trade liberalization on market demand, the case of asymmetric countries

33References
Baldwin, R. and J. Harrigan (2011). Zeros, quality, and space: Trade theory and trade evidence. American
Economic Journal: Microeconomics 3 (2), 60–88.
Bergstrand, J. H. (1990). The heckscher-ohlin-samuelson model, the linder hypothesis and the determinantsof bilateral intra-industry trade. The Economic Journal 100 (403), 1216.
Berry, S. and J. Waldfogel (2010).

Product quality and market size.

The Journal of Industrial Eco-nomics 58 (1), 1–31.
Bonanno, G. (1986). Vertical di↵erentiation with cournot competition. Economic Notes 15, 68–91.
Champsaur, P. and J.-C. Rochet (1989). Multiproduct duopolists. Econometrica 57, 533–557.
Eckel, C., L. Iacovone, B. Javorcik, and J. P. Neary (2015). Multi-product firms at home and away: Costversus quality-based competence. Journal of International Economics 95 (2), 216–232.
Eckel, C., L. Iacovone, B. Javorcik, and J. P. Neary (2016). Testing the core-competency model of multiproduct exporters. Review of International Economics 24 (4), 699–716.
Eckel, C. and J. P. Neary (2010). Multi-product firms and flexible manufacturing in the global economy.
Review of Economic Studies 77 (1), 188–217.
Edmond, C., V. Midrigan, and D. Y. Xu (2015). Competition, markups, and the gains from international
trade. American Economic Review 105 (10), 3183–3221.
Fajgelbaum, P., G. M. Grossman, and E. Helpman (2011). Income distribution, product quality, and international trade. Journal of Political Economy 119 (4), 721–765.
Feenstra, R. C. (2015). Advanced international trade: theory and evidence. Princeton University Press.
Gabszewicz, J. J. and J.-F. Thisse (1979). Price competition, quality and income disparities. Journal of
Economic Theory 20 (3), 340–359.
Gabszewicz, J. J. and J.-F. Thisse (1980). Entry (and exit) in a di↵erentiated industry. Journal of Economic
Theory 22 (2), 327–338.
Gal-Or, E. (1983). Quality and quantity competition. The Bell Journal of Economics 14 (2), 590.
Ireland, N. (1987). Product di↵erentiation and non-price competition. Basil Blackwell.
Johnson, J. P. and D. P. Myatt (2006). Multiproduct cournot oligopoly. The RAND Journal of Economics 37 (3), 583–601.

34

Johnson, R. C. (2012). Trade and prices with heterogeneous firms. Journal of International Economics 86 (1),43–56.
Krugman, P. (1995). Increasing returns, imperfect competition and the positive theory of international trade.
In Handbook of International Economics, pp. 1243–1277. Elsevier.
Li, Y. and Z. Miao (2017). Trade scopes across destinations: Evidence from chinese firm.
Long, N. V., H. Ra↵, and F. Stahler (2011). Innovation and trade with heterogeneous firms. Journal ofInternational Economics 84 (2), 149–159.
Markusen, J. R. (1986). Explaining the volume of trade: an eclectic approach. The American Economic
Review , 1002–1011.
Melitz, M. J. (2003). The impact of trade on intra-industry reallocations and aggregate industry productivity.
Econometrica 71 (6), 1695–1725.
Motta, M. (1993). Endogenous quality choice: Price vs. quantity competition. The Journal of Industrial
Economics 41 (2), 113.
Mussa, M. and S. Rosen (1978). Monopoly and product quality. Journal of Economic Theory 18 (2), 301–317.
Shaked, A. and J. Sutton (1982). Relaxing price competition through product di↵erentiation. The Review
of Economic Studies 49 (1), 3.
Shaked, A. and J. Sutton (1983). Natural oligopolies. Econometrica 51, 1469–1483.
Shaked, A. and J. Sutton (1987). Product di↵erentiation and industrial structure. The Journal of Industrial
Economics 36 (2), 131.
Vives, X. (2008). Innovation and competitive pressure. The Journal of Industrial Economics 56 (3), 419–469.

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