Trade with Horizontally and Vertically Di¤erentiated
Goods
Claudiu Tunea
December 17th ; 2007
Abstract
This paper explains the mechanism through which trade liberalization a¤ects the
composition of intra-industry trade and …rms’choices of investment in quality di¤erentiated goods. Equilibrium of a two-stage game is characterized, in which quality
producers choose investment levels in stage 1, and subsequently, all …rms compete in
quantities in stage 2. A trade model involving two countries with ad valorem tari¤s
yields that quality levels rise and the share of trade in vertically-di¤erentiated goods
also rises when tarri¤ protection is lowered and the relative market size of the two
countries exceeds a certain threshold.
1
1
Introduction.
In the background of signi…cant reductions in tari¤ and non-tari¤ barriers over the past
three or four decades, intra-industry trade has accounted for the largest share in the bilateral
trade between industrialized countries. Although numerous empirical studies document this
increased importance and show that the majority of its growth is due to trade in quality
di¤erentiated goods, until recently the trade theoretical literature has given relatively little
attention to this phenomenon.
This paper investigates the changes in intra-industry trade patterns induced by trade
liberalization and provides an explanation for the observed increases in the value of trade
in quality di¤erentiated goods. This is accomplished using a model of imperfect competition between …rms that belong to the same industry and produce goods di¤erentiated either
horizontally or vertically. Firms compete in quantities and product quality levels are endogenous. The model generates a prediction that the elimination of trade barriers leads to
the production of goods of higher quality and to an increase in the value of bilateral trade
in vertically di¤erentiated goods.
Most existing theoretical trade models assume that …rms produce horizontally di¤erentiated (or close substitutes) goods in analyzing the pattern of trade as well as its sensitivity to
the nature of the competition and tari¤ regime.1 Trade in vertically di¤erentiated products
is outside the scope of such models, and the trade literature has made little use of models of
vertical product di¤erentiation to analyze the rise of trade in quality-di¤erentiated goods.
With few exceptions, the concern among trade theorists has been to explain the emergence
of trade between countries either via intra-industry or inter-industry trade and to study the
changes in trade patterns when either tari¤ (or non-tari¤) barriers, factor endowments or
production technologies change.2 In the context of this debate, the central stage belongs
to horizontal di¤erentiation of products via consumers’taste for product variety as in Dixit
and Stiglitz (1977). Trade between countri es then emerges as a result of di¤erences in factor
endowments and internal economies of scale.
In order to answer questions about the role played by the two types of product di¤erentiation in shaping the trade patterns, however, one needs a model in which consumers
choose among both types of goods, rather than only among a homogenous good and goods
1
2
See Brander and Spencer (1985), Eaton and Grossman (1986) and others.
Herguera, Kujal and Petrakis (2002).
2
that are close substitutes. Further, a …rm’s decision regarding the quality level of the good
it produces must take account of the ex-ante investment that is required of it (i.e. higher
quality goods require higher investment such as R&D expenditures). Firms competing in
the quality-good market do so in two dimensions –quality and quantity–and the equilibrium
quantities and qualities respond to changes in income and, when there is trade, to variations
in the level of trade barriers between countries.
The research reported on in this paper is a step in this direction. A model of consumer
behaviour in the presence of both horizontally di¤erentiated goods and goods having a quality
dimension is developed, and the resulting demand behavior is derived. The model is then
expanded to incorporate two countries that trade freely, and then trade barriers in the form
of ad valorem tari¤s are introduced.
Horizontally and vertically di¤erentiated goods can be thought of as goods which are
produced by two distinct industries. For instance, one can think of the car manufacturers
as forming one sector/industry and the producers of bicycles, scooters, light motorcycles,
ATVs and snowmobiles making up another industry. The latter industry can be viewed as
producing goods that are horizontally di¤erentiated: they are substitutes in consumption
but at the same time they are clearly di¤erentiated goods since the consumer uses them
for di¤erent kinds of outdoors activities. Depending on his/her individual preferences, the
consumer would buy the mix of such goods that is most suitable to the mix of outdoors
activities he/she likes to engage in. (Because changes in the mix of goods consumed within
the horizontal good bundle are not of interest in the present paper, it is of course assumed
that all consumers have identical preferences for these.)
The cars produced by the one industry can be viewed as (imperfect) substitutes for the
bicycles, snowmobiles and the other horizontally di¤erentiated goods. They are used by the
consumer for transportation rather than for outdoors activities involving physical e¤ort or
for activities in national parks or in the countryside. If he can a¤ord it, the consumer (or
most consumers) would typically buy one automobile of some quality level.
On the production side, it is reasonable to assume that innovation and improvements in
performance are more complex and expensive in the automobile industry than is the case
for bicycles and snowmobiles. A car’s safety rating, the engine’s power and fuel economy,
and passengers’ comfort can be improved through R&D and the use of new technological
processes. This implies that car producers have to incur higher …xed costs in order to produce
3
superior quality goods.
The trade model with horizontally and vertically di¤erentiated goods and tari¤s is used to
derive predictions about the e¤ect of lowering tari¤s on the quality of goods produced in the
trading countries and on the composition of trade between them –i.e., on the importance of
trade in vertically di¤erentiated goods versus the trade in horizontally di¤erentiated goods.
The model’s major prediction is that trade liberalization increases the equilibrium quality
levels of the vertically di¤erentiated goods as producers of such goods gain less restricted
access to foreign markets, and therefore …nd it pro…table to invest more in quality.
Changes in the (structure of) intra-industry trade as a result of freer trade between the
two countries depend on the relative size of the two markets, as well as on the location of the
quality producers. When the high-quality producer is located in the larger country, if the
two countries are su¢ ciently di¤erent with respect to the incomes then trade liberalization
increases the share of trade in vertically di¤erentiated goods. In case countries are of similar
(or identical) sizes, lowering tari¤s between the two countries increases the share of trade in
horizontally di¤erentiated goods. When the high-quality producer is located in the smaller of
the two countries, trade liberalization increases the share of trade in vertically di¤erentiated
goods.
The paper proceeds as follows. Section 2 is a brief review of the empirical and theoretical
literature on the relation between product di¤erentiation and trade. Section 3 presents a
simple oligopoly model of product di¤erentiation and the derivation of equilibrium in the
two-stage game. Section 4 solves for the equilibrium in one market, in autarky. Section 5
builds a trade model between two countries with di¤erent average incomes and derives the
free trade equilibrium. Trade barriers are then introduced in the model in the form of ad
valorem tari¤s. The impact of variations in the level of protection is analyzed in Section
6. A short conclusion summarizes the main …ndings of the paper and discusses possible
extensions of the model in order to answer a wider range of questions.
2
Literature review.
This section provides an overview of the empirical and theoretical trade literature concerned
with the changes in bilateral trade patterns and with the specialization of countries along the
lines of product di¤erentiation. The necessary connection with the IO literature on quality
4
product di¤erentiation is made.
Fontagné, Freudenberg and Peridy (1997, 1998) have documented empirically that intraindustry trade accounts for about 60-70% of bilateral trade among the E.U. countries and
about two thirds of that share is attributed to trade in vertically di¤erentiated goods. The
share of intra-industry trade has increased since 1985, and the gains stem from the expansion
of trade in quality di¤erentiated goods. (They use 4-digit level SIC data for intra-E.U. trade.)
Martin-Montaner and Orts (2002) and Chiarlone (2000) have similar …ndings for Spain in
relation with the OECD countries and for Italy and its European trade partners, respectively.
The pattern emerging from these three studies shows that the relative importance of highand low-quality goods in a country’s trade with a foreign partner is determined by the
economic and geographic distances between that country and its trade partners. Similarities
of income and geographic proximity are correlated with a larger share of the intra-industry
trade (IIT) in total bilateral trade.
For an industry (or a category of goods), a country either imports higher quality products
or is an exporter of such goods (while importing goods at the other end of quality spectrum).
In this respect, a country’s status depends on each bilateral relation: a country may be the
exporter of high quality goods (and importer of the lower quality) in some industries while
in others it will produce and export mainly low-quality varieties. For example, MartinMontaner and Orts (2002) …nd that Spain is mostly a lower-quality exporter in relation
with its OECD partners, but becomes an exporter of higher-quality goods to poorer, less
developed countries.3
In the trade theory of di¤erentiated products, the e¤orts have been focused on using
the monopolistic competition models or oligopoly models to explain the emergence of trade
between countries, due either to economies of scale in production of di¤erentiated goods
or particular characteristics of the input markets. For example, Krugman (1980) shows
how trade may be generated between countries, even if they have identical technologies and
factor endowments, when product di¤erentiation is costless and in the presence of increasing
returns to scale.
Venables (1987) analyzes the tari¤ policy and the emerging trade pattern between two
3
Petrucci and Quintieri’s (2002) two-country trade model with di¤erentiated goods and with skilled and
unskilled labour as inputs o¤er and explanation for why this can be the case. Their model predicts that
the country with higher skilled to unskilled labour ratio produces the high quality product, while the other
country with produce the lower quality goods.
5
countries producing both a homogeneous good and a di¤erentiated good. The latter good is
produced by a monopolistically competitive industry under increasing returns to scale, and
both countries’demands for these goods are derived from Dixit-Stiglitz preferences.
Jean (2000) investigates the relationship between the patterns of trade and …rm heterogeneity, with due consideration being given to the trade costs and the e¢ ciency gap between
the …rms located in di¤erent countries. He isolates several cases of parameter ranges that
determine whether the trade is inter- or intra-industry in nature. These …ndings complement
those of Krugman (1980, 1981): the more similar the two countries, the more signi…cant the
intra-industry trade becomes and also the greater the gains from enhanced openness to trade.
In the industrial organization literature, models of vertical di¤erentiation have been extensively used by Gabszewicz and Thisse (1979), Shaked and Sutton (1982), Wauthy (1996)
and others to study the quality choice by …rms in two-stage games, under either quantity or
price competition, or to analyze the market structure and the degree of concentration in an
industry producing a quality di¤erentiated good (Shaked and Sutton,1987).
Motta (1993) employs a simple duopoly model to investigate the quality choice arising
under Cournot and Bertrand competition, where …xed or variable costs are incurred when
producing a good of a particular quality. He …nds that …rms have more incentives to differentiate from their competitor(s) under price competition and that welfare is also higher
than under Cournot due to relatively stronger competition.4
Building on Motta’s (1993) model of quality di¤erentiation, Herguera, Kujal and Petrakis
(2002) study the implications of government’s optimal tari¤ for a vertically di¤erentiated
industry with one domestic and one foreign …rm competing in the domestic market. In the
…rst stage of the game, …rms make an investment allowing them to produce the chosen quality
at zero marginal cost. They subsequently compete in either quantities or prices, and the
government chooses the welfare maximizing tari¤ either before or after …rms’quality choice
but before production takes place –in what they call the ex-ante and the ex-post games.
Tari¤s grant the domestic …rm substantial protection against foreign products and enables
it to always produce the higher quality. Moreover, in this setup tari¤s turn out to be welfare
improving (for the home country) when compared to the free trade. The side e¤ect of tari¤s
in the ex post game is that the domestic …rm invests less in quality than under free trade,
if the Nash equilibrium happens to have placed it in the dominant position. In the reversed
4
In a di¤erent setup, Shaked and Sutton (1982) show that competition is softened in these circumstances.
6
case, the tari¤s make large investments in quality very attractive to the domestic …rm and
it thus becomes the quality leader. The ex ante optimal tari¤s confer the domestic …rm the
monopoly power, driving out the foreign competitor.
Herguera, Kujal and Petrakis’ (2002) –referred to as HKP hereafter– results are interesting in showing the e¤ects of tari¤ protection on the quality choice and welfare under
both quality and price competition, but their model lacks the real dimensionality of a trade
model. The analysis is limited to the impact of import tari¤s on the welfare and the market
structure of a single country. It tells virtually nothing about the consequences of the trade
policy beyond the country’s borders or about the trade pattern since the foreign markets
are ignored altogether. Moreover, the game in tari¤s only has the domestic government as
player, which means that the defensive stand foreign governments may take to protect their
own …rms is muted and therefore the impact of retaliation on the equilibrium outcomes in
the home country is ignored. (In fact, this model analyzes a country’s strategic policy when
the rest of the world is passive.)
The …rst step in answering questions regarding the intra-industry trade is to develop a
framework of demand functions derived from consumer preferences over goods di¤erentiated
either by quality or horizontally, and then to model the …rm behaviour in an oligopoly setting
(of either quantity or price competition).
The present research is an advance from the works of Krugman (1980) and HKP (2002)
since both types of product di¤erentiated are incorporated in the model, and this model can
address the questions regarding the impact changes of trade barriers (and other costs) have
on the shares of trade on quality-di¤erentiated goods and on equilibrium quality level.
In contrast, Krugman’s (1980) model cannot be used for this purpose even if some of the
di¤erentiated goods are re-labeled quality goods since the elasticity of substitution between
any two goods in his model is the same. Therefore, all imported goods are a¤ected in the
same way by a tari¤ (or transportation cost) and the shares of trade for the two types of
goods are constant with respect to the trade costs.
Likewise, HKP (2002) cannot address such questions for two reasons. First, the basic
setup of their model uses one vertically di¤erentiated good (available to the consumers in two
varieties) and a homogenous good, and therefore a two-country model would yield one-way
trade in the homogenous good. Second, in their one-country model, trade with the rest of
the world is unidirectional.
7
3
Consumption of horizontally and vertically di¤erentiated goods.
The empirical trade literature makes a distinction between horizontally di¤erentiated goods
and the ones di¤erentiated by quality, and in doing so it has an approach that is closer
to the production side than to the consumers of those goods. An industry’s products are
deemed to have the same set of (possible) characteristics. If a given product is superior to
another product in one dimension while not being inferior in any of the others (within the
set of characteristics), the former is of higher quality than the latter, and thus one deals
with vertically di¤erentiated goods. If however one of them is superior in some dimensions
while the other product is superior in others, the two goods are said to be horizontally
di¤erentiated.
In the theoretical I.O. literature, the distinction between the two types of product di¤erentiation has to do with the way consumers treat those products in consumption. The horizontal goods are treated by all consumers as close substitutes: the relative intensities associated
with the individual characteristics of the products (as they appear in the “production-side
de…nition”) is only re‡ected in the elasticity of substitution between the (horizontal) goods.
For the vertically di¤erentiated goods the interpretation is more straightforward: there is a
consensus among consumers which of two goods is better. Individual consumers may derive
di¤erent utility from the consumption of the same level of quality –depending on a taste
parameter which is di¤erent among consumers, for instance–but the ranking of the quality
goods is the same for everybody.
For example, the typical consumer may treat sport shoes, suburban style shoes and work
boots as close substitutes in consumption, i.e. as horizontally di¤erentiated goods, since
these di¤erentiated goods can be used in a variety of situations and serve similar purposes.
They have characteristics over which ranking of the goods is not always possible –say, the
sport shoes are more comfortable and softer than work boots but would not o¤er good
protection and would wear out faster in certain work environments. The same consumer
can treat other types of shoes as quality-di¤erentiated goods, for instance upper-leather
shoes, designer shoes and lacquer shoes –goods that embody di¤erent characteristics such
as durability, quality of materials used in the manufacturing process, the degree to which
they are fashionable etc, and that can be clearly ranked in consumers’preferences. Similar
8
examples can be easily found among varieties of wines and alcoholic drinks, products of the
textile, automotive and other industries.
In this section, I …rst model consumer preferences over a range of goods that are the
outputs of the same industry –some goods being di¤erentiated horizontally while others are
vertically di¤erentiated; and then turn to the pro…t-maximizing decisions of …rms. Assume
that there are N horizontally di¤erentiated goods being produced, indexed by j, and assume
that each is available at price pj : In addition, there are two possible varieties of the qualitydi¤erentiated good, having qualities s1 ; s2 –where I assume that s1
s2 ; without loss of
generality. It is further assumed that either zero or one unit of at most one of the quality
goods can be consumed by each consumer.
Therefore, consumers face the price vector !
p = (pj )j=1;:::;N for the horizontal goods and
prices ps1 and ps2 for the two varieties of vertically di¤erentiated good. Each consumer
has a monetary endowment m, where m 2 [0; M ]: This monetary endowment uniquely
identi…es the consumer, since preferences are assumed identical across consumers. I assume
that endowment amounts (and hence consumers) are distributed uniformly over the interval
[0; M ] and the mass of each point in this interval is 1. Using these assumptions, M is also
the country size.
A consumer whose income, and thus, total expenditure on goods, equals m, …rst decides
whether to include 1 unit of the quality good in his consumption or not. The money available
to this consumer after buying 1 unit of quality good s at price ps , i.e. m
ps , will then be
spent on horizontal goods.
It is the case that consumers with incomes lower than the price of the low-quality variety
(which in equilibrium necessarily sells at a lower price than the high-quality variety, so
ps1 < ps2 ) cannot a¤ord the consumption of any quality good and will thus spend their
whole budget on horizontally di¤erentiated goods. Consumers with incomes m
ps1 ; it will
be shown, will buy one unit of quality good s –either variety s1 or s2 –and spend y = m
ps
on horizontal goods.
As income rises above ps1 ; the consumer will buy 1 unit of quality variety s1 and increasing
quantities of horizontal goods. When income rises above the price of the higher quality
variety, m
ps2 ; the consumer will choose between consuming even higher quantities of
horizontal goods and switching to the consumption of (1 unit of) the high quality variety s2 .
Since ps2 > ps1 , less money is left available for the consumption of horizontally di¤erentiated
9
goods and thus lower quantities of such goods will be consumed along with quality good s2 .
It is assumed that the utility function that represents all consumers’identical preferences
is of the form:
U
0
!
h ; s = a(s)
exp @
N
X
hj
j=1
!1
1
b(s)A
(1)
where hj is the quantity of horizontal good j consumed, s is the quality level of the vertically
di¤erentiated good consumed –with s = 0 indicating that no such good is consumed. In (1),
a(s) is an indicator function that takes the value 1 when any quality good is consumed, and
0 otherwise:
a(s) =
Function b(s) is de…ned as follows:
b(s) =
Finally, the parameter
(
0
if s = 0
1
otherwise
(
1
if s = 0
s
otherwise
(2)
(3)
2 (0; 1) determines the elasticity of substitution between the hor-
izontal goods hi , which enter as arguments in a C.E.S. sub-utility function (à la Dixit and
Stiglitz (1977)).
Indicator function a(:) can also be interpreted as the maximum level of utility attainable
in each case. By consuming any variety s > 0 of the quality good (in combination with
horizontal goods), the individual’s utility level is higher than the maximum utility he can
otherwise achieve by consuming only horizontal goods. All consumers have the same ranking
of the vertically di¤erentiated goods (s1 < s2 ) and everybody prefers consuming 1 unit of
such goods to the consumption of horizontal goods alone.
This utility function allows the analysis of consumer decisions to proceed as follows.
Suppose a consumer has decided to spend y
m of its income on the h-goods. Conditional
on this decision, its vector of h -consumption will be determined as
H(!
p ; y) = (h1 (y; !
p ); ::; hN (y; !
p ))
(
)
N
X
H(!
p ; y) = arg max
v(h)j
hj pj = y ;
!
h
(4)
j=1
where v(h) is a CES sub-utility function having quantities of horizontal goods hi ; i = 1::N;
as arguments. H(!
p ; y) is independent of the choice of the quality good s made by the same
consumer, and therefore independent of the prices psi and the total income m.
10
As shown in Dixit and Stiglitz (1977), the assumed form of the function v(h) implies that
the actual maximized value of v(:) that follows from the sub-optimization problem is given
by
v (p; y) = v(H(p; y)) = max fv(h)j
h
where P (!
p)=
PN
j=1
(1
pj
(1
)
j pj hj
y
,
P (!
p)
= yg =
)
.
Let the set of quality-price pairs available in the market be then denoted by
f(0; 0); (s1 ; ps1 ); (s2 ; ps2 )g = f 0 ;
(5)
1 ; 2 g.
=
The situation when no quality good is consumed
is modeled as consumer’s choice to buy quality s = 0 and pay p0 = 0:
For given !
p and m and for any i = (si ; psi ) available in the market, the indirect utility
derivable from choosing
i
can be written as:
V (!
p ; m; i ) = U (H(!
p ; m psi ); si )
= F (v(H(!
p ; m psi ); si )
8
m psi
< 1 exp
s
if either 1 or
P (!
p) i
=
m
: exp
if 0 is chosen
P (!
p)
Note that this incorporates the fact that, if any particular
m
i
2
is chosen
is chosen, the remaining income
psi is spent optimally on horizontal goods.
The question of which of the available
i
2
in the market an individual with income
m will actually choose can be answered by determining which
i
yields the highest value
of V (p; m; i ) among the quality goods a¤ordable to that particular individual. This choice
function can be written as:
(!
p ; ; m) = arg max fV (!
p ; m; i )j
i
i
2
and psi
mg
(6)
In words, if the market o¤ers an individual with income m the market possibilities contained
in , then
expresses the choice of
i
he will make and H(p; m
psi ) tells us the bundle of
horizontal goods he will choose.
Further, the critical values of m at which consumers switch the quality good they choose
can be de…ned in the following manner. For income levels m < ps1 ; no quality goods will be
consumed since they are not a¤ordable. For income levels m
ps1 ; the indirect utility levels
are positive if the quality good is purchased, and negative otherwise. Therefore,
m01 (!
p ; ) = min fm
ps1 j V (m; P;
11
0)
< V (m; P;
1 )g
(7)
is the income threshold at which an individual consumer switches between buying 0 or 1 .
Letting m12 (!
p ; ) the income of the individual indi¤erent between 1 or 2 ; we have that:
m12 (!
p ; ) = min fm
ps2 jV (m; P;
1)
V (m; P;
The indirect utility levels associated with the consumption of each available
i = 0; 1; 2, are given by
V (!
p ; m; 0 ) = exp
V (!
p ; m;
1)
=1
exp
PN
j=1 hj (m
PN
j=1 (m
p0 ; !
p)
ps1 ; !
p)
(8)
2 )g
i
2
;
1
=
m
P (!
p)
exp
1
s1
=1
exp
m ps1
s
P (!
p) 1
1
PN
m ps2
)
)
=
1
exp
s
s2 = 1 exp
2
j=1
P (!
p) 2
Since V (!
p ; m; 0 ) < 0 while V (!
p ; m; 1 ) ; V (!
p ; m; 2 ) > 0, the two critical income levels
p ; m;
V (!
derived from (7) and (8) are
m0;1 = ps1
ps2 s2
m1;2 =
s2
It can be shown that
(9)
ps1 s1
s1
@m0;1
@m1;2
@m1;2
> 0;
< 0 and
> 0;
@ps1
@ps1
@ps2
(10)
This means that a higher price charged for a certain variety will, ceteris paribus, induce
the marginal consumers to switch to consuming other varieties. An increase in the price
of the lower quality good, ps1 ; results in decreased demand for this good since some of the
low-income consumers will cease to buy any quality good, and also because some of the
consumers with incomes above ps2 will switch to buying the superior good. Likewise, an
increase in the price of the superior quality good, ps2 ; will induce some of the consumers
with incomes between ps2 and m1;2 to switch to buying the inferior quality good. It is
therefore the case that the demand for each quality good is negatively a¤ected by own price
increases.
Also, one can show that m12 > ps2 and the working assumption here is that available
bundles
i
2
are such that m12 < M :
Given the assumptions on consumers being distributed uniformly over the interval [0; M ]
with mass 1 for each point in the interval, the aggregate quantities of each quality variety
12
demanded economy-wide are
q1 = m12
q2 = M
m01
m12
After the substitution of (9) one obtains
(ps2 ps1 ) s2
s2 s1
ps2 s2 ps1 s1
= M
s2 s1
q1 =
q2
(11)
The inverse demand functions faced by the vertical goods producers will then be
p s 1 = M q1 q2
(M q2 )s2 q1 s1
p s2 =
s2
(12)
Using this demand structure for the quality goods, the next section will be concerned
with the derivation of equilibrium in the two segments of the market, i.e. the equilibrium in
the market for quality goods and the equilibrium in the market for horizontally di¤erentiated
goods. In the entire industry, the competition is in quantities and therefore I solve a Cournot
problem for each market segment.
4
Autarky Equilibrium.
The structure of the two-stage game describing the competition in this particular industry is
as follows.5 In the …rst stage …rms producing quality goods decide what variety to produce.
6
In the second stage, Cournot competition takes place between the incumbents in the
industry.
A …rm that produces a vertically di¤erentiated good (of any quality level) must make a
…xed investment that has two distinct components. First, like any other …rm in the industry,
it must incur the …xed cost of being in the market, F0 . Second, it incurs a cost related to
5
6
The same structure of the game is used in the two-country trade model.
In principle, a more complex game can be solved where …rms decide on the type of goods they produce,
thus yielding an endogenously determined number of quality-goods producers. Such setup is left for future
research.
13
the development of a quality variety. This cost is commensurate to the quality level s of
its output (R&D costs or product improvement costs) and which are assumed equal to
s2
.
2
Once these …xed costs are incurred, quality di¤erentiated goods are produced at constant
marginal cost c > 0 which is constant.
Firms that produce a horizontal good only incur the …xed cost F0 and produce the good
at constant marginal cost ch > 0, independent of the variety they produce.
Since both the horizontal and vertical goods belong to the same industry, the two marginal
costs could be equal or could be very similar. No special assumption is made with respect
to the relation between cs = c and ch :
4.1
Vertically di¤erentiated goods.
Backward induction is used to solve for the equilibrium in the market for vertically di¤erentiated goods. One begins with solving for the equilibrium in the Cournot competition in
the second stage of the game, and then use the Cournot pro…ts and derive the equilibrium
in the quality competition in the …rst stage.
Letting M denote the di¤erence between the maximum income level and the marginal cost
of producing goods, M = M
c, the (inverse) demand functions faced by the quality-goods
producers net of marginal cost can be conveniently written as
pfs1 = M q1 q2
(M q2 )s2 q1 s1
pfs2 =
s2
(13)
This notation will bring clarity to the algebra in subsequent sections of the paper.
In the second stage of the game, for given quality levels s1 and s2 , …rm i = 1; 2 solves
the following pro…t maximization problem
max
qi
si (qi j
qi0 ; s1 ; s2 )
where
s1 (q1 j
q2 ; s1 ; s2 ) = pfs1 q1
s2 (q1 j
q2 ; s1 ; s2 ) = pfs2 q2
14
s21
2
s2
F0 + 2
2
F0 +
:
which is the same as
where M = M
s1 (q1 j
q2 ; s1 ; s2 ) = (M
s2 (q1 j
q2 ; s1 ; s2 ) =
q1
(M
s21
F0 +
2
q2 )q1
q2 )s2
s2
q1 s 1
q2
(14)
F0 +
s22
2
:
(15)
c.
The F.O.C.-s for the maximization problems (14) and (15) yield the following reaction
functions for the Cournot game
M
q1 (q2 ) =
q2
2
M
q2 (q1 ) =
q1 ss21
2
and the equilibrium quantities in the Cournot game are given by
s2
q1Cournot =
q2Cournot
4s2
2s2
=
4s2
(16)
M
s1
s1
M
s1
and with equilibrium prices received by …rms (net of marginal costs) given by
s2
pfs1 =
pfs2
4s2
2s2
=
4s2
(17)
M
s1
s1
M
s1
(Prices paid by the consumers are given by ps1 =
s2
M
4s2 s1
2
+ 3s
4s2
s1
c
s1
and ps2 =
2s2 s1
M
4s2 s1
+
2s2
c.)
4s2 s1
Equilibrium pro…ts for the …rms producing quality goods are then
s1
s2
where M = M
s21
s22 M 2
=
F0 +
2
(4s2 s1 )2
2
2
(2s2 s1 ) M
s22
=
F
+
0
2
(4s2 s1 )2
(18)
c:
In the …rst stage of the game, …rms in the vertical goods market compete in quality
levels by choosing the appropriate R&D investment. Each …rm chooses the quality level it
produces in order to maximize its pro…t as given by (18).
15
The corresponding F.O.C.-s for the two quality producers are
@ s1
@s1
=
@
@s1
@ s2
@s2
=
@
@s2
2s2
s22 M 2
s21
= (4s 2s )3 M 2 s1
2
(4s2 s1 )2
2
1
s22
(2s2 s1 )2 M 2
4s1 (2s2 s1 )
= (4s s )3 M 2
2
(4s2 s1 )2
2
1
=0
s2 = 0
The two reaction functions are implicitly given by the following equations
2s22
2
(19)
= s1
(4s2 s1 )
4s1 (2s2 s1 ) 2
M = s2
(4s2 s1 )3
De…ne
=
s1
s2
3M
(20)
2 (0; 1): Then equations (19) and (20) become
2
s2 (4
4 (2
s2 (4
3M
2
Side-by-side division of the equations above yields
2
=
, or
4 (2
)
1 = 2 2 (2
There is a unique
(21)
= s1
)
)
2
= s2
3M
)
)
2 (0; 1) that solves the equation above (and its approximate value is
0:59697). Using the value of
determined above, the equilibrium in the …rst stage of the
quality game is derived from equations (21) and is given by
s1 =
s2 =
(4
s
)
4 (2
(4
p
1
8
)
3
)
6
2
+
M
M ,
while (second stage) equilibrium quantities are
q1Cournot =
q2Cournot
1
4
2
=
4
(22)
M
M
and equilibrium prices
pCournot
=
s1
pCournot
s2
1
4
2
=
4
M+
M+
16
3
4
c
2
4
c
(23)
clear the market for quality di¤erentiated goods. Prices net of marginal cost are
1
Cournot
=
pf;
s1
M
4
2
=
4
Cournot
pf;
s2
M
Autarky pro…ts for the two …rms producing quality goods are then given by
M2
F0 +
s 12
2
)2
M2
)2
F0 +
s 22
2
1
s1
=
s2
(2
=
(4
(4
)2
(24)
Therefore, we have actually proved the following
Proposition 1.
There is a unique equilibrium in the two-stage game in which …rms
choose the quality levels in the …rst stage and subsequently compete by non-cooperatively
choosing quantities.
In the analysis of the trade model I will assume that conditions related to the nonnegativity of the …rm pro…ts are satis…ed, both in autarky and in the case of trade between
two countries, and that there are only two …rms that choose to produce vertically di¤erentiated goods in equilibrium. A discussion of cases in which more than two …rms produce quality
goods and earn positive pro…ts and the characterization of parameter ranges that make that
possible are interesting, but nevertheless go beyond the scope of the present research.
4.2
Horizontally di¤erentiated goods.
Turning now to the equilibrium in the market for horizontal goods, consumer m’s demand
functions for the horizontal goods are determined as in Dixit and Stiglitz (1977), and the
assumptions on identical marginal costs and same …xed cost are standard in the literature
spawned by that paper.
An individual who spends y to buy quality s at price ps will demand in equilibrium
hj (y; !
p)=
y
1=(1
pj
17
)
P1
units of horizontal good j, where j = 1; ::; N . This result follows from the derivation of
equilibrium in Dixit and Stiglitz (1977) and from the fact that preferences with regard to
horizontal goods are homothetic.
Once the prices and quantities of the quality goods have been pinned down from the
Cournot game between the quality producers, it is easy to calculate the aggregate demands
for the horizontally di¤erentiated products j = 1; ::; N :
Rm
Rm
RM
p )dm + 0 01 hj (m; !
p )dm =
p )dm + m0112 hj (m ps1 ; !
p ; ) = m12 hj (m ps2 ; !
Hj (!
i
hR
R
R
m
m
M
= 1=(11 ) P 1
(m ps2 )dm + m0112 (m ps1 )dm + 0 01 mdm =
m12
pj
=
=
=
1
1=(1
pj
)
P1
)
P1
1
1=(1
pj
1
1=(1
pj
)
M
2
( m2
h
P1
m2
2
M2
2
ps2 m)
2
m12
+ ( m2
m12
ps1 m)
+
m01
m2
2
m01
=
0
M
ps2 (M
m12
ps2 q2
ps1 q1
m12 )
i
ps1 (m12
m01 ) =
for i = 1; :::; N:
Quantities q1 and q2 are those given by equations (11) and are, implicitly, functions of .
Let
M
2
1 6
2 (4
=
=
denote the total amount of money spent on horizontal goods. Then
2
ps2 q2
2
)2
M
2
ps1 q1 =
2
2+
(4
)2
M
2
2
Mc +
5 4 + 2
M (M
(4
)2
7 3
c2 :
(4
)2
7 3
(4
)2
c)
M
c c=
Using this notation, the aggregate demand for horizontal good j is given by
Hj (!
p; )=
1
1=(1
pj
)
P1
for j = 1; ::; N
;
(25)
Under the assumption that the number of producers N is large enough, each producer
faces a constant price elastic demand curve for the variety it sells.7 Thus it will set the
monopoly price equal to
ch
pj =
PN
(See Dixit and Stiglitz, 1977). The price index P (!
p) =
simply written as
P (!
p)=N
Firm j’s sales are given by Hj (!
p; ) =
7
1
ch
( )
1
1
(1
)
N
j=1
ch
(1
)
ch
1
(1
pj
(1
)
=
1
ch N
)
can now be
and its pro…ts
This is also an assumption in Dixit and Stiglitz (1977) and means that the e¤ect of a change in any one
price on the price index for the horizontal goods is negligible.
18
are
1
1
F0 =
F0 :
ch N
N
The relative quantities, as well as prices, of horizontal goods are independent of the total
income spent on such goods, . Therefore the price index P (!
p ) is also independent of .
j
=(
ch
ch )
The interaction between the two sub-markets consists of one element: prices and quantities
in the market for quality goods a¤ect the quantities in the market for horizontal goods via
an income e¤ect.
5
Two-country trade model.
Assume there are two countries, labeled A and B, and the two countries are characterized by
uniform distributions of consumer incomes over the interval [0; M l ]; l = A; B; respectively.
Without loss of generality, also assume that M A
M B and therefore, MA
MB : Individual
consumers have the same preferences over horizontal and vertical goods, as described above.
Markets are segmented and there are no transportation costs.
It is assumed throughout this section that all …rms produce for both markets and there is
one quality producer located in each country, and that the …rm located in the larger country
produces the higher quality, sA = s2 > s1 = sB . There are NA horizontal producers in
country A, and NB horizontal producers in country B. The free trade equilibrium as well as
the analysis of the e¤ects of tari¤s refer to this case.
5.1
Free trade.
Second period Cournot pro…ts for the quality producers are
A+B
s1
s22 MA2
s22 MB2
=
+
(4s2 s1 )2 (4s2 s1 )2
s2 (MA2 + MB2 )
s21
= 2
F
+
0
2
(4s2 s1 )2
s21
F0 +
2
and
A+B
s2
(2s2 s1 )2 MA2
(2s2 s1 )2 MB2
s22
=
+
F
+
0
2
(4s2 s1 )2
(4s2 s1 )2
2
2
2
2
(2s2 s1 ) (MA + MB )
s
=
F0 + 2 ;
2
2
(4s2 s1 )
19
respectively.
Derivation of the free-trade equilibrium in the market for quality goods is reduced to
p
solving the autarky problem for a value of income M = MA2 + MB2 , in the manner explained
in sub-section ??. Relative quality level
will be the same as in autarky, but absolute quality
levels are larger now, since they are increasing functions of the market size M .
For the horizontal goods, the equilibrium quantities and prices in each market are determined as above, and the maximum number of …rms supported in the free-trade world, N ,
can be derived from the zero-pro…t conditions. Since the determination of the equilibrium
number of horizontal …rms (and how it changes when tari¤s change) is beyond the scope of
the present research, it is assumed here that NA + NB < N and the number of …rms resident
in each country does not change. (When no trade barriers separate the two countries, the
location of either type of …rms would not be relevant.)
5.2
Tari¤s.
This sub-section derives the equilibrium when goods crossing the border between A and
B are subject to (non-prohibitive) tari¤s. For simplicity and increased tractability of the
model, tari¤s are assumed to be the same in each country and they apply to all imports net of
marginal cost, irrespective of the type of product di¤erentiation.8 The working assumption is
therefore that the ad valorem tari¤s are the same for all goods in this industry, are symmetric
across countries,
A
=
B
=
In other words, a share
2 (0; 1); and apply to the markup above the marginal cost.
of foreign …rm’s pro…ts are paid to the domestic government.
This simplifying assumption has the advantage of increasing the algebraic tractability of the
model while not altering its qualitative …ndings. It allows one to concentrate on relative
market shares changes that are induced by the …rms’decisions in stage 1 of the game, when
quality levels are chosen by the quality-goods producers.
8
An alternate assumption is that the two countries have di¤erent tari¤s and in the process of liberalization
they lower tari¤s by the same proportion. This and the working assumption that tari¤s are equal are
algebraically equivalent.
Moreover, a credible scenario is that trade liberalization occurs between countries that have a commitment
to bring tari¤s down to levels that are both lower and more similar. This way, the government of the country
with lower tari¤ can secure support of its own industries for further lowering of their own tari¤ protection –
and gaining improved access to foreign markets.
20
Trade in quality di¤erentiated goods.
In the benchmark case, I will analyze the case in which the …rm located in the larger
country produces the higher quality good –or sA = s2 > s1 = sB . (The opposite case can be
treated in the same manner, and its results are discussed in the last section.) Since the two
countries have segmented markets, second period quantities and prices are derived as shown
in the section on the autarky equilibrium.
For given s1 and s2 ; at the beginning of stage 1 the …rms producing vertically di¤erentiated
goods expect to earn the following pro…ts in stage 2 of the game
A+B
(s1 ; s2 )
s1
s22
=
(1
)MA2 + MB2
F0 +
s21
2
)MB2
F0 +
s22
2
s1 )2
s1 )2
2
2 MA + (1
s1 )
(4s2
(2s2
A+B
(s1 ; s2 ) =
s2
(4s2
;
respectively. In the quality competition in the …rst stage, each …rm chooses the quality level
of its own variety taking as given the competitor’s quality level in order to maximize its
pro…ts:
max
si
A+B
(si ; si0 )
si
, for i; i0 2 f1; 2g; i0 6= i:
The F.O.C.-s corresponding to the two maximization problems are given by
@ s1
@s1
@ s2
@s2
=
=
2s22
((1
(4s2 s1 )3
4s1 (2s2 s1 )
(MA2
(4s2 s1 )3
)MA2 + MB2 )
s1 = 0
)MB2 )
+ (1
s2 = 0
Rearranging,
2s22
(4s2
3
s1 )
(1
)MA2 + MB2 = s1
4s1 (2s2 s1 )
MA2 + (1
3
(4s2 s1 )
)MB2 = s2
(26)
(27)
In order to derive the equilibrium quality levels, one must solve the system consisting of
equations (26) and (27).
As before, de…ning
=
s1
s2
2 (0; 1) and the two equations are re-written as
2
s2 (4
4 (2
s2 (4
3
(1
)MA2 + MB2
= s1
)MB2
= s2
)
)
MA2 + (1
)3
21
(28)
and dividing them side by side one obtains
(1
)MA2 + MB2
) MA2 + (1
)MB2
(1
)MA2 + MB2
MA2 + (1
)MB2
2
4 (2
=
, or
= 2
2
(2
It is straightforward to see that equation (29) has a unique solution
@
( )
@
(29)
)
( ) 2 (0; 1). Moreover,
< 0 since the expression on the left-hand side is constant with respect to
while being
decreasing with respect to , and since the right-hand side is strictly increasing with respect
to
for all values in (0; 1).
Equilibrium quality levels are obtained from equations (28) above and are
s
q
2
( )
s2 ( ) = 2
MA2 + (1
)MB2 and
3
(4
( ))
s1 ( ) =
(4
q
( )) 8
The intuition for the result
@
( )
@
(1
)MA2 + MB2
p
2
)MB2
6 ( ) + ( ( ))2 MA + (1
1
(30)
(31)
< 0 when MA > MB –i.e. for when the producer of
the higher quality is located in the larger country–is that when a …rm competes in foreign
markets that are more protected it will want to charge, ceteris paribus, higher prices in order
to recoup the lost revenue. In order to do so, it must di¤erentiate its product from the
incumbent …rm’s to a larger extent than when tari¤ barriers are low. Then, for a given
quality s2 produced by the local …rm, the foreign …rm will choose a quality level s1 lower
than it would choose otherwise. In case the foreign …rm produces the higher quality to begin
with, it will choose to increase the quality of its product.
It then follows that s1 ( ) decreases when the trade barrier
rises, and increases in the
event of a trade liberalization. This is the case since the …rst ratio in (31) is increasing in
and
@
in
is also to decrease its equilibrium value. In short,
( )
@
< 0, while the second ratio is decreasing in . For s2 ( ); the e¤ect of an increase
@si ( )
< 0, for i = 1; 2
@
@ ( )
< 0
@
These results prove the following:
22
(32)
(33)
Proposition 2. Assume that MA
MB and the producer of the higher quality good is
located in the larger country (i.e. sA = s2 > s1 = sB ). When trade is liberalized between the
two countries, equilibrium quality levels rise and the relative quality
increases.
These properties will be useful in the assessment of the impact of trade liberalization on
the intra-industry trade between the two countries, which forms the object of analysis in the
next section.
Having determined the equilibrium quality levels si , i = 1; 2, second period’s quantities
and prices of vertically di¤erentiated goods in each market are determined as shown above
and pro…ts are given by
A+B
s1
A+B
s2
((1
=
( ))2 (MA2 + (1
(4
( ))2
(2
=
(s ( ))2
F0 + 1
2
)MA2 + MB2 )
(4
( ))2
)MB2 )
!
(34)
(s ( ))2
F0 + 2
2
and the value of trade in vertically di¤erentiated goods as function of
!
is
A
B B
T Q( ) = pA
s1 q1 + p s 2 q2 =
=
1
(4
+
(4
2
2 MA
(35)
+
2
(2
(4
( ))
3
2
2 MB +
( ))
(4
( ))
2
M B c:
( ))2
( )
M Ac +
( ))2
Trade in horizontally di¤erentiated goods.
In what follows, I assume there are NA producers located in country A and NB producers
in country B. Their exports to the other country are subject to the same ad valorem tari¤
(net of marginal cost) as the quality goods. The location choice by …rms is beyond the
scope of this paper, and since the focus here is on comparative statics for marginal changes
in the tari¤ rates, the working assumption is that NA and NB are short-, or medium-,
term equilibrium numbers of …rms resident in each country and they do not change due to
marginal increases or decreases in tari¤s. (In a long-run scenario, the number of …rms would
be determined from zero-pro…t conditions and conditions stating that …rms do not gain from
changing location.)
23
2
Let
l
=
Ml
2
ps2;l q2l be country l’s expenditure on horizontal goods, where l =
ps1;l q1l
A; B. Using the equilibrium quality levels, prices and quantities in the market for vertically
di¤erentiated goods derived in the previous section, the amounts spent on horizontal goods
in the two countries are
( )=
l
16 (
2 (4
2
( ))2
Ml
( ))2
Since the tari¤
2+
( ) ( ( ))2
M lc
(4
( ))2
+
7 3 ( ) 2
c,
(4
( ))2
where l = A; B.
does not distort relative marginal costs between domestic and foreign
producers, prices and the quantities produced by the …rms j = 1; :::; NA ; :::; NA + NB are as
derived in Section ??. Aggregate demand and the price for horizontal variety j in country l
are:
Hjl
=
1
1
1
plj =
pj
ch
"
X
pn
1
n=1;::;NA +NB
#
1
l
( )
for j = 1; :::; NA ; NA + 1; :::; NA + NB .
Therefore, the value of total trade in horizontal goods between the two countries (gross
of tari¤s) will be
A
B B
NB p A
B H B + NA p A H A =
NB
NA
=
A( ) +
NA + NB
NA + NB
(36)
T H( ) =
Ratios
NB
NA +NB
and
NB
NA +NB
B
( )
in equation (36) represent the share of expenditure on horizontal
goods which is spent on imported horizontal goods in country A and in country B, respectively.
Using the values of trade in horizontal and vertical goods determined above, in the next
section I will discuss the impact of trade liberalization on the shares of trade for these types
of goods.
6
E¤ects of trade liberalization.
In this section I discuss the implications of changes in the level of trade barriers –present in
this model in the form of ad valorem tari¤s–for the shares of trade in vertically di¤erentiated
goods.
24
As shown above, the values of trade in quality goods and in horizontal goods are given
by
2
2
( ))2 M B + (3
( ))M A c + 2M B c
T Q( ) =
2
(4
( ))
NA
NB
T H( ) =
A( )+
B( ) ,
NA + N B
NA + NB
M A + (2
where
l
2
( ))2
Ml
( ))2
16 (
2 (4
( )=
2+
( ) ( ( ))2
Ml
(4
( ))2
7 3 ( ) 2
c,
(4
( ))2
c+
(37)
(38)
with l = A; B:
The share of trade in quality di¤erentiated goods in total intra-industry trade is given by
T Q( )
=
T Q( ) + T H( )
( ) =
2
1
2
(4
MA +
( ))
2
1
M A + (2
( ))2
(4
=
(4
(39)
2
( ))2
2 MB
( ))
2
( ))2
M B + (43
( ))2
(2
(4
+
3
(4
( )
M Ac
( ))2
( )
M Ac
( ))2
+
2
( ))2
(4
2
+
( ))2
(4
M Bc
M Bc + T H ( )
A …rst step in assessing the impact of a tari¤ change on the share of trade in quality
di¤erentiated goods is taken by analyzing the case when M A
M B and the producer of the
higher quality level is located in the larger country –the situation analyzed in the previous
section. The reverse situation will be discussed afterwards.
The sign of
@
@
is positive if and only if the elasticity of the ratio’s numerator is larger
than the elasticity of the denominator –i.e.
T Q(
)>
TH(
@
@
): In other words,
is positive if
and only if the elasticity of trade in quality goods (with respect to the tari¤ rate ) is greater
than the elasticity of trade in horizontal goods. Algebraically, this condition is written:
@
@
2
1
( ))2
(4
1
(4
(
MA +
(2
(4
2
(2
(4
MA +
))2
Given that " ( )j
( )
in horizontal goods is
2
( ))2
MB
( ))2
2
( ))2
MB
( ))2
2 0; 4(3
(4
0 > "T H ( ) >
2
+
3
(4
( )
M Ac
( ))2
+
+
3
(4
( )
M Ac
( ))2
+
( ))
( )) 2+(
4(3
(4
( )
( ))2 2
( )
2 ( ))
( )) 2 + (
2
( ))2
(4
2
(4
(
and
@
M Bc
"T H ( )
M Bc
))2
( )
@
< 0, the elasticity of trade
@
2
( ))
2
( )
( )
@
:
Then condition (40) implies the following succession of equivalent inequalities:
@
@
2
1
( ))2
(4
M A+
@
@
( ))2
( ))2
M A+
2
1
(4
M A+
2
1
(4
(4
2
1
(4
(2
( ))
2 M A+
(2
(4
(2
(4
(2
(4
2
( ))
2
3
MB+
( ))2
(4
( )
MA
( ))2
c+
2
(4
( ))2
MB c
2
( ))2
3
( )
2
MB+
M A c+
MB c
( ))2
(4
( ))2
(4
( ))2
2
( ))
2
3
( )
2
MB+
M A c+
MB c
( ))2
(4
( ))2
(4
( ))2
2
( ))2
3
( )
2
MB+
M A c+
MB c
( ))2
(4
( ))2
(4
( ))2
25
@
@
(40)
"T H ( )
" ( ( ))
1
( )
(41)
2
2
((2 )MA +4MB ) c
2
( )
MA 4
MB+
(4
)3
( ))3
(4
( ))3
2
2
(2
( ))2
3
( )
1
2
M A+
MB+
M A c+
MB
( ))2
(4
( ))2
(4
( ))2
(4
( ))2
2
(4
(4
c
" ( ( ))
1
( )
" ( ( ))
1
( )
This last inequality implies
2
2
2
( )
MA 4
MB
( ))3
(4
( ))3
2
( ))2
3
( )
MB+
M A c+
( ))2
(4
( ))2
(4
2
(4
(4
2
(2
1
M
+
A (4
( ))2
2
( ))
2 MB c
which in turn implies
2
(4
(4
2
(2
1
M A+
( ))2
(4
MA
;
MB
Letting z =
2
(4
2
2
MA 4
( ))3
(4
2
( ))2
3
MB+
( ))2
(4
(4
2
(2
1
M
+
A
( ))2
(4
2
( )
MB
( ))3
2
( )
M A+
( ))2
(4
2
1
( )
" ( ( ))
2
MB
( ))2
the left-hand side of the inequality is written
2
2
MA 4
( ))3
(4
2
2
( ))
3
MB+
( ))2
(4
2
( )
MB
( ))3
2
( )
M A+
( ))2
(4
2
2
( ))2
MB
=
2
z2
(4
( ))3
( )
(2
z2 +
( ))2
(4
4
(4
4
2
( )
(4
( ))3
2
( ))
+
( ))2
(4
;
2
( ))2
and is an increasing function of z. Substituting above, one obtains a necessary condition for
@
@
>0:
2
(4
1
(4
(
( ))3
z2 +
))2
z2
(2
(4
4 (42
2
( ))
( ))2
( )
( ))3
+
2
(4
( ))2
4(3
(4
2 ( ))
( )) 2 + (
1
( ))2
2
(42)
( )
The expression on the right-hand side of inequality (42) is positive and constant with
respect to z; while the left-hand side is strictly increasing. For a (unique) value of relative
e the expressions on the two sides of inequality (42) become equal, and
country size, z = Z,
p
for z > Ze the inequality is reversed. (It can be easily shown that Ze > 2 (2
( )) > 1,
and thus it is compatible with the assumption that M A > M B .)
In conclusion, z > Ze is a su¢ cient condition for @@ < 0: In other words, for relative
country sizes exceeding a certain threshold, trade liberalization implies an increasing share
of trade in quality goods in total intra-industry trade.
A brief discussion of the reverse situation –when the larger country is the home market
of the lower quality producer– is now appropriate. Keeping sA = s2 > s1 = sB as in the
benchmark case, this situation translates into M A < M B . The …rst implication of the fact
that the smaller country produces a higher quality than the larger country is that
This is straightforward to show from (29) –whose solution
@
( )
@
> 0.
( ) 2 (0; 1) exists and is unique.
(Which means the uniqueness of equilibrium in the two-stage game is not a¤ected.) Lowering
protection of the markets, by lowering tari¤ , has the consequence of equilibrium quality
levels being farther apart on the quality scale: relative quality
With
@
( )
@
( ) decreases.
> 0,
0<" ( )<
4(3
(4
2 ( ))
( )) 2 + (
26
@
2
( ))
2
( )
( )
@
Should it be the case that
2
(4
1
(4
(
( ))3
z2 +
))2
z2
(2
(4
@
@
0; then the following inequality holds:
4 (42
2
( ))
( ))2
( )
( ))3
+
2
(4
( ))2
4(3
(4
2 ( ))
( )) 2 + (
1
( ))2
2
( )
;
(43)
However, this inequality is never satis…ed for z < 1 since the left-hand side of the inequality
is always negative while the right-hand side is positive.
This implies that in this case
@
@
< 0, which means that trade liberalization increases
the importance of trade in quality goods when the producer of the superior quality good is
located in the smaller country.
One can summarize the results obtained in this section by stating the following
Proposition 3.
Given Ze > 1 de…ned above, trade liberalization between countries A
and B increases the share of trade in quality goods in total intra-industry trade either if the
producer of the superior quality is located in the smaller country, or if this producer is located
e
in the larger market and the relative market size exceeds a certain threshold, M A > Z:
MB
This proposition establishes that trade in vertically di¤erentiated goods grows in importance relative to trade in horizontally di¤erentiated goods in due process of liberalization
in two situations. One situation is when the two economies are di¤erent enough (in terms
of market size) and the larger economy produces a vertical variety of higher quality. The
other situation is when –by historical accident, or otherwise–the superior quality variety is
produced in the smaller country.
In the …rst case, lowering bilateral trade barriers fosters the competition between vertical
goods producers: quality varieties are relatively “closer”and equilibrium quality levels both
rise. Producers in both countries gain from increased market access abroad, but the producer
of the lower quality (located in the smaller country) has more to gain from improved access
to the larger market and thus increases investment in quality more than his competitor, the
result being that his quality variety is relatively closer to that of his competitor. The price
of the low-quality variety increases in equilibrium, and so does the demand for this good,
while for the high-quality variety the Cournot price and quantity decline.
In other words, the low-quality producer is the more aggressive of the two competitors,
and the response of the high-quality producer is to accommodate: the latter remains the
27
dominant player in the market for quality goods but sells fewer units at lower prices, but
these units are of better quality than before. When the di¤erence between the two market
sizes is su¢ ciently large, increased value of low-quality exports to the larger economy o¤sets
declining values of high-quality imports into the small economy as well as increases in the
value of trade in horizontally di¤erentiated goods. The overall result is then a higher share
of trade for vertically di¤erentiated goods in (intra-industry) trade.
In the second case, the producer of high-quality good s2 located in the smaller country
and …nds it optimal to di¤erentiate its product (relative to its competitor’s) even more when
trade barriers are lower. Thus, stronger product di¤erentiation of the two products is the
outcome in the …rst stage of the game, and the low-quality producer tries to appeal to poorer
customers (to which he does by selling at lower prices). Lower sales of good s1 to the small
country are o¤set by an increased value of sales of good s2 to the large country; thus, the
value of total trade in quality goods rises. The total amount spent on horizontal goods
decreases in each country, and therefore, the value of trade in horizontal goods declines.
This, of course, implies a higher share
of the trade of vertically di¤erentiated goods in
(intra-industry) trade.
7
Comparison with other trade models.
In a standard model of trade with non-homothetic preferences, in which consumer preferences
over the range of di¤erentiated goods are given like in Tchamourliyski (2002)
U (x1 ; ::; xn ) =
n
X
j (xj
j)
1
j=1
!
1
;
>1
one can obtain changes in the composition of trade similar to those predicted by the model
in this chapter. However, a particular set of assumptions need to be made and the similarity
with the present model is limited.
With consumers facing the price vector !
p = (pj )nj=1 ; a parameter
as a subsistence level of consumption good k. (Some of
k -s
k
> 0 is interpreted
can be zero, or even negative
–in which case goods are considered luxuries.) De…ning = fk = 1; ::; nj k > 0g, the …rst
P
m
e =
pj j dollars are allocated by the consumer to purchase the subsistence levels k for
j2
all varieties k 2
(or as many such varieties as he can a¤ord to buy). The rest of the income
28
is used to add to the consumption above the subsistence levels and to buy non-subsistence
goods — in the same manner as for the utility maximization when we have homothetic CES
function.
Then the expenditure on good k is given by
pk xk = pk
k
+ P
1
k
pk1
1
j
j=1::n
and the amount purchased is
xk =
k
+ P
1
k
j=1::n
1
j
m
pj1
pk
pj
j
j=1
m
p1j
n
X
n
X
pj
j=1
j
!
!
(44)
(45)
The elasticity of substitution between goods is the same as in the case of homothetic
CES functions. For levels of consumption above the subsistence levels, and for income levels
above m;
e the decision on the optimal mix of goods is the same as for the homothetic CES
(i.e. when all -s are equal to zero). However, because the consumer spends the …rst dollars
on the subsistence goods, even price changes that leave the relative prices unchanged will
alter the mix of goods purchased by the consumer (and thus the market shares for each of
the goods).
In analyzing the simplest trade model, lets make the assumption that goods j = 1; :::; h
are subsistence goods (i.e.
(i.e.
j
> 0) and goods j 0 = h + 1; ::; n are non-subsistence goods
0) . (I will refrain from labeling any of these as horizontal or vertical goods.)
Also assume that in the trade equilibrium with domestic prices given by vector !
p = (p )n
j0
j j=1
both types of goods are traded, and that the home country imports both subsistence and
non-subsistence goods made in the other country. Then a reduction of the ad valorem tari¤
on all goods lowers m;
e and therefore in addition to the substitution e¤ect between goods
there will be an income e¤ect, which e¤ect is di¤erent across goods with di¤erent subsistence
levels
i
6=
j
. Everything else equal, the absolute price elasticity of the demand for good
k will be lower if
k
> 0 than when
k
= 0:
A uniform reduction of ad valorem tari¤s
by 1% for imported goods lowers consumer
prices and increases expenditure on imported varieties. Expenditure on good k increases by
xk (
), — the elasticity of the expression in (45) with respect to : If the elasticity of prices
with respect to
with
is the same across all imports, elasticity
> 0 (subsistence goods) than for good with
29
xk (
) is lower for imported goods
0 (ordinary and luxury goods).
Under the set of assumptions laid out above, this model’s implications regarding the
e¤ect of tari¤ changes on the shares of various goods in trade is that the share of trade in
subsistence goods decreases when tari¤s are lowered while the share of non-subsistence goods
in total trade will rise.
The obvious di¤erence with the model in Section 2.3 has to do with the way in which
the goods are di¤erentiated. In the model developed in Section 2.3, it is the distinction
between quality goods and horizontal goodswhich is signi…cant, whereas in this model it is
the distinction between subsistence goods and others that matters. However, note that the
model presented in Section 2.3 also captures the e¤ect of tari¤ changes on the quality of
those goods which are labelled as quality goods (i.e. higher-quality varieties are produced
and traded when tari¤s are lower), a prediction which is absent from the standard model
with non-homothetic preferences.
The trade model in this chapter is a partial equilibrium one. Re-formulating is as a
general equilibrium model would require that countries have balanced trade, endogenizing
the consumers’income by distributing the …rm pro…ts across all consumers (who can be the
…rms’shareholders) and introducing labor market decisions by the consumers (thus a¤ecting
their income, as well as the production costs of the …rms).
It is the case that imperfect competition models of trade with endogenous income levels
or with a labour market become algebraically intractable even if simple preferences are
used to derive the demand structure for the traded goods. In most such models, it is usually
possible to prove the existence of (unique or multiple) equilibria and then to use mathematical
software packages in order to …nd an equilibrium conditional on a set of parameter values
calibrated to …t a group of trading partners. In models which are built to incorporate
complex decisions by the agents (say, in the form of two-period games) proving equilibrium
uniqueness may not be possible, and therefore, understanding the agents’response to changes
in parameters such as the country size or trade barriers is in fact limited by the fact that
there may be more than one equilibrium in each state of the world.
The easiest way of expanding the model presented here is to allow for a redistribution
of the …rm pro…ts equally among the consumers residing in the country where the …rm
is located, and the use of this income to buy horizontal and vertical goods. The income
level that becomes relevant to the determination of aggregate demand functions are then
Mk + '(
k)
, where
k
is the sum of the pro…ts of all producers (of horizontal and vertical
30
goods) located in country k. Then, maintaining the assumptions on the …rm location made
throughout the model, the pro…ts the two vertical goods producers are then a function
s1
s2
where '(
A)
s22
=
(4s2
(2s2
=
(4s2
=
1
MA
(1
) (MA + '(
2
A )) + (MB + '(
2
B ))
) (MB + '(
2
B ))
s1 )2
s1 )2
(MA + '( A ))2 + (1
s1 )2
!
N
PA A
; '( B ) = M1B
s2 +
j
s1
+
j=1
NAP
+NB
j=NA
assumed zero for all horizontal producers, or not –in which case
B
j
s21
2
s22
2
!
, and
k
j
= (
F0
F0
j -s
sk ;
k
could be
j ).
(
k
j
denotes the pro…ts of the other horizontal producers in country k).
k
j
Making the additional assumption that
=
versions of the equations above
"
s22
s2
(1
) MA +
s1 =
(4s2 s1 )2
MA
"
2
(2s2 s1 )2
s2
+ (1
M
+
=
A
s2
(4s2 s1 )2
MA
(
sk ;
k
j)
= 0 , one obtains simpler
2
2
+ MB +
s1
MB
2
) MB +
s1
MB
#
#
s21
2
F0
s22
2
F0
which implicitly de…ne the pro…ts of the vertical producers as functions of the quality levels
and country sizes. However, since explicit solutions cannot be derived for this system, one
cannot characterize the …rm behaviour in the …rst stage of the game when quality levels are
chosen.
The ability to characterize …rm behaviour with respect to the choice of the quality level
of their products has the further consequence that one cannot characterize the changes in
the patterns of trade when the trade barrier
is lowered. Therefore, a partial equilibrium
model is chosen to allow us to analyze the pattern of trade between the two countries.
8
Conclusion.
This paper is concerned with explaining the mechanism through which trade liberalization
a¤ects intra-industry trade and …rms’choice of investment in quality-di¤erentiated goods.
First an I.O. model of consumer preferences over goods di¤erentiated either horizontally or
vertically is developed and used to derive the demand functions for each good, assuming
31
consumers have identical preferences and di¤er with respect to the total income spent on
the goods produced by a given industry.
Then a model of imperfect competition between producers of vertically- and horizontallydi¤erentiated goods is built and the existence and uniqueness of the equilibrium are proved.
Competition between …rms is modeled as a two-stage game in which producers of verticallydi¤erentiated goods choose investment levels in quality in stage 1, and subsequently, all …rms
compete in quantities in stage 2.
This model of competition between two vertically-di¤erentiated goods producers and N
producers of horizontally-di¤erentiated goods is used to construct a trade model involving
two countries that either trade freely or use ad valorem tari¤s. After characterizing the
trade equilibrium in each case, I derive su¢ cient conditions under which the share of trade
in vertically-di¤erentiated goods rises when tari¤ protection is lowered.
In the benchmark case where the producer of the superior quality variety is located in
the larger of the two countries, the most important implication of trade liberalization is the
increased quality levels which arise endogenously in the two-stage game. Intra-industry trade
share of quality-di¤erentiated goods rises when the two countries are su¢ ciently di¤erent
with respect to the size of their markets. (In the opposite case in which the lower quality
producer is located in the larger market, the share of quality-di¤erentiated goods in intraindustry trade rises for all relative market sizes.)
Future work will involve further developing the model of two-stage competition between
…rms to incorporate the …rms’decision on the type of good they produce, and thus to have
the number of quality producers and the range of quality levels for their products endogenous
to the model. Attempts will be made to also endogenize the location choice by …rms in the
two-country trade model.
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34