DP2011-14

No trade, one-way or two-way trade?
Toshihiro OKUBO
Pierre M. PICARD
J acques-François T HISSE
March 22, 2011

No trade, one-way or two-way trade?
Toshihiro Okubo ∗Pierre M. Picard †Jacques-François Thisse‡

December 22, 2010

Abstract
We study how the level of trade costs and the intensity of competition can explain
the existence of two-way, one-way or no trade within the same industry. As trade costs
decrease from very high to very low values, the economy moves from autarky to a regime of
two-way trade, through a regime of one-way trade from the larger to the smaller country.
Trade is less likely when the economy gets more competitive. Finally once capital ismobile across countries, the market delivers an outcome in which capital is too muchconcentrated in the large country.

Keywords: trade regime; country asymmetry; capital mobility
JEL Classification: F12; H22; H87; R12∗

RIEB, Kobe University (Japan). e-mail: okubo@rieb.kobe-u.ac.jp
CREA, University of Luxembourg (Luxembourg) and CORE, Université catholique de Louvain (Belgium).
‡
CORE, Université catholique de Louvain (Belgium), Université du Luxembourg, and CEPR. e-mail:†jacques.thisse@uclouvain.be1

1

Introduction

Living in a globalizing world conveys the impression that trade is the main ingredient of economic life. However trade takes several forms as it may occur within the same sector or between
sectors. For instance, intra-industry trade has steadily increases ever since World War II andhas served as the main motivation for the development of the new trade theories. By contrast,
inter-industry trade has been the focus of standard trade theories, like the Heckscher-Ohlin andRicardian models, that explain one-way trade by the presence of country-specific factor endowments and technologies. In response to such diﬀerences in predictions, empirical studies haveaimed to measure the intensity of these two types of trade.1 No strong conclusion emerges fromthe empirical studies, thus suggesting that both types of trade play a major role in international
relationships. In addition, there is also evidence that many pairs of countries have no trade inmany industries (Haveman and Hummels, 2004; Baldwin and Harrigan, 2007). According to
Helpman et al. (2008), over the last few decades two-way, one-way and no trade at all respectively account for 30 − 40, 10 − 20 and 50 − 60 percent of country pairs in 158 countries. Thus,
we may safely conclude that world trade is not dominated by two-way trade. The existenceof asymmetric trade patterns and the absence of trade between and within sectors are otherimportant features of the trade landscape, which deserve more attention.
To the best of our knowledge, there is no work accounting for three patterns (two-way trade,one-way trade, and no trade) within a single unified framework highlighting the impact of trade
costs on the nature of trade when capital is free to move between countries.2 The Japanese
textile industry provides a good illustration of the various regimes of trade we want to study
in this paper. The Tokugawa Shogun initiated a seclusion policy in 1639 for more than 200
1

See Hummels and Levinsohn (1995), Evenett and Keller (2002), Debaere and Demiroglu (2003), and Debaere(2005).
2
Behrens (2005) studies the one-way and two-way trade cases in the core-periphery model. However, theobjective of his paper is diﬀerent from ours. Helpman et al. (2008) focus on trade when production factors areimmobile.

2

years. Only the Dutch and Chinese were allowed to trade with Japan in a few specific ports.
During this period, cotton textiles were produced and consumed under autarky. Ever since the1853 US naval appearance in Tokyo, the pressure toward trade opening with foreign countries
increased. Eventually, Japan decided to move from autarky to an open economy. To meetdomestic demand, cotton textiles were imported after trade opening. This may be viewed as aperiod involving one-way trade in the textile sector. By way of contrast, in the 1880s and 1890s,
which corresponds to the First Era of Globalization, cotton textiles started being exported andtheir value drastically increased to exceed that of imports from 1910 onward (Fujino et al.,1979). This corresponds to a regime of two-way trade. Such shifts from autarchy, one-way to
two-way trade have been observed in many other manufacturing sectors such as iron-steel andchemical products in Japan. To sum up, those examples suggest that trade liberalization has
been critical in the evolution of the production and trade patterns.
In this paper, we aim to study the impact of trade costs on the nature of trade, but also
on the long run international allocation of capital, which is another major facet of the processof globalization. To achieve our goal, we use a linear model of monopolistic competition, which
captures the following two basic features: competition gets tougher when the number of firms
increases as well as when the level of trade costs falls (Ottaviano et al., 2002). We then combine
this model with a two-asymmetric country setting to capture the well-documented fact thatmarket size plays a major role in trade and foreign direct investments. Depending on the level oftrade costs, we show that the above-mentioned trade patterns emerge as equilibrium outcomes.
More precisely, as trade costs decrease from very high to very low values, the economy moves
from autarky to a regime of two-way trade, through a regime of one-way trade from the larger tothe smaller country. In the one-way trade regime, the country accommodating the larger numberof firms exports toward the country having the smaller number of firms because competition
is softer therein. In other words, even though firms are homogeneous, their choice to export
depends on the intensity of competition on their foreign markets. This is to be contrasted
with Helpman et al. (2008) where the choice to export is independent from the intensity of3competition, but is determined by firms’ idiosyncratic productivity and fixed costs of serving
foreign countries. We thus oﬀer an alternative explanation for the existence of no trade or
one-way trade.
Our setting also allows us to parametrize the toughness of competition through a single
parameter that encompasses the number and the degree of substitutability of the varieties
available in all markets. Using this parameter, we show that the likelihood of regimes involving
two-way or one-way trade decreases as the economy gets more competitive. This is because thepenetration of foreign products becomes more diﬃcult once the mass of domestic firms is large
enough. Although the emergence of two-way trade within various sectors is often interpreted as
evidence of intensified competition, our analysis thus shows that two-way trade may also be theoutcome of defensive strategies adopted by firms to relax competition. By contrast, a stronger
degree of competitiveness fosters the emergence of one-way trade where only the large countries
export. As a result, implementing a tough competition policy before liberalizing trade deter
the entry of foreign competitors in large countries.
Since the international distribution of firms is a major determinant of the nature and intensity of trade, we find it natural to ask how capital is allocated when it is free to move fromone country to the other. This is an important issue because, by changing their investment locations, capital-owners aﬀect the intensity of competition within each country, thus making thepenetration of foreign products easier or more diﬃcult. Put diﬀerently, we recognize that boththe nature and intensity of trade vary with the mobility of capital. For example, under one-way
trade, we show that the larger country is more competitive than the smaller one, which preventsthe latter country from exporting to the former one. We also show that the home-market eﬀect
prevails in the three regimes of trade provided that the economy is not too competitive. In
this case, the large and prosperous country attracts a disproportionate share of capital. The
market size advantage stressed in economic geography thus provides an explanation to Lucas’(1992) query: Why doesn’t capital flow from rich to poor countries? It is worth stressing thatthe above answer to Lucas’ question is independent from the nature of trade between countries.
4

The home-market eﬀect obtained under one-way trade that characterized many poor countries
also concurs with Collier (2005) who observes that the bottom billion countries export theircapital instead of attracting investments. In contrast, when the economy is very competitive
the home-market eﬀect may be reversed in the autarky and one-way trade regimes. In otherwords, the capital flows can run in the opposite direction when national markets are very competitive. Our analysis thus highlights the role played by the intensity of competition, not only
for the nature of trade, but also for the international allocation of capital.
Last, we determine the eﬃcient allocation of capital and show that the market delivers an
outcome in which capital is too much concentrated in the larger country, whatever the trade
regime. This result diﬀers markedly from those derived in the standard model involving perfectcompetition, constant returns and zero transport costs, where capital mobility always leads tothe socially optimal distribution of activities. On the contrary, once we account for imperfectcompetition, increasing returns and positive transport costs, the liberalization of capital mobility
favors the larger country at the expense of the smaller one. This is to be contrasted with thecase of capital immobility where the smaller country gains more than the larger one from tradeliberalization because the small country’s consumers get a better access to a wider range of
foreign varieties.
Our paper proceeds as follows. The next section introduces the model and characterizes
the demand structure under autarky, one-way and two-way trade. Section 3 explores the
product market equilibrium under these various trade regimes, whereas Section 4 focuses on
the equilibrium allocation of capital. Section 5 is devoted to the welfare analysis, while Section
6 concludes.

2

The model

The economy involves two tradeable goods, two production factors, and two countries  =  with a population of size  and  , respectively; without loss of generality, we assume that5 ≥  with  +  = 1. The global supply of capital to normalized to 1. Individuals
are immobile and endowed with some share of capital; they are free to invest their capitalwherever they want. The spatial allocation of capital is endogenous and will be determined as
an equilibrium outcome. By contrast, each individual inelastically supplies a unit of labor in the
country where she resides. To disentangle the various channels through which the equilibrium
occurs, we will distinguish between a short-run equilibrium in which the capital supply is fixedin each country, and a long-run equilibrium when capital is freely allocated across countries.
Both countries produce a homogeneous good and a range of manufacturing diﬀerentiatedvarieties. Following the literature, we assume that the homogeneous good  is produced underconstant returns to scale and perfect competition by using labor as the only input and using no
capital. Our purpose being to investigate how trade costs aﬀect the spatial distribution of thefirms producing the diﬀerentiated varieties, we isolate this eﬀect by working with a setting inwhich workers’ wage is equalized between countries, which is guaranteed by assuming that trading the homogeneous good is costless. The firms’ unit input requirement being normalized to 1,
the homogeneous good is used as the numéraire. As a result, profit maximization implies thatthe price of the homogeneous good and workers’ wages are equal within each country whereas
factor price equalization implies that those prices and wages are equalized across countries sothat  =  = 1,  =   . This set-up is standard in trade and economic geography, and is
not restrictive for our study of the conditions under which various trade structures emerge inthe manufacturing sector.3
The set of manufacturing diﬀerentiated varieties is modeled as a continuum of horizontallydiﬀerentiated varieties indexed by . Each variety is produced by a single firm under increasing
returns and monopolistic competition. To operate, a firm needs  units of capital and hires a3

Introducing positive trade costs for the homogeneous good makes the analysis more involved (Picard andZeng, 2005). Note also that factor price equalization holds provided that each country has enough labor to
support the production of homogenous goods for any international capital distribution, which we assume fromnow.

6quantity of labor proportional to its output. So, the total mass of varieties produced in theeconomy is equal to  = 1. As the global supply of capital is equal to one, a larger value of means either that more capital is allocated to the manufacturing sector or that firms display
a lower degree of increasing returns. Because demands will be shown to be linear, we may
normalize the marginal requirement of labor to zero. Shipping one unit of the manufactured
good between the two countries requires   0 units of the numéraire.
We now turn to the consumer’s preferences and demand for the two types of goods. A
key-feature of the present model is that consumers may be unwilling to purchase all varieties.
So, we need to distinguish the varieties that are available to consumers from those that are
actually purchased by them. Recall that  is the total mass of varieties that are at consumer’s
disposal in the economy. For any given price profile of varieties  (·) faced by consumers incountry , any consumer in  chooses the quantity profile  (·) and  that maximizes her utility


Z 
0


 ()d −
2

Z 
0


[ ()] d −
2
2

∙Z 

 ()d

0

¸2

+ 

(1)

subject to the budget constraint:Z  () ()d +  =  +  + 00where  = 1 and  the capital return defined below,4 and to the nonnegative consumption
constraint: () ≥ 0
The initial endowment of the homogeneous, 0 , is supposed to be large enough for this good
to be consumed in equilibrium. Note that, without loss of generality, the coeﬃcients   0 and  0 may be normalized to one. Indeed, the first coeﬃcient may be factorized, while the unitof the diﬀerentiated good may be chosen for the second coeﬃcient to be one. In this case,   0
expresses both the substitutability between varieties (the higher , the closer substitutes the
4

In this model, the distribution of capital ownership has no impact on the market outcome because of thequasi linear preferences assumed below.

7varieties) and the desirability of the diﬀerentiated good with respect to the homogeneous good(the higher , the lower the desirability of the diﬀerentiated good).
Consumers do not necessarily purchase all varieties at their disposal at the prevailing prices.
Let  ≤  denote the number of varieties that are actually purchased and consumed in country
. As a consequence,  is equal to  when all varieties are consumed and is lower than when some varieties are not consumed. As will be shown below, trade costs will inflate theprice of imported varieties and could make them too expensive to be consumed in equilibrium.
To determine  , we rank the  varieties by increasing price order:   0 ⇔  () ≤  ( 0 ).
The following proposition is proven in Appendix 1.
Proposition 1 There exist a cut-oﬀ variety ̃ =  ≤  and a corresponding cut-oﬀ price ( ) such that consumers purchase all the varieties that have a price below  ( ). The
demand for each variety  is given by⎧⎨ 1 −  () +  R   ()d if   
1+̃
1+ 0 () =⎩0
otherwise(2)

while the cut-oﬀ variety solves the fixed point equation ( ) (1 +  ) = 1 + Z  ()d(3)0

The cut-oﬀ price is smaller than one.
Quite naturally, consumers purchase only the varieties that are not too expensive. It is
important to note that whereas, too expensive varieties remain at consumers’ disposal, they
do not aﬀect the demand for the less expensive varieties that are actually purchased. Indeed,
the cut-oﬀ variety and cut-oﬀ price are defined as the fixed point of (3), which depends only
on the prices of the varieties that are actually purchased (i.e.  ∈ [0 ̃]). By the same token,
the demand function of each purchased variety depends only upon the prices of varieties thatare purchased by the consumer. Intuitively, the consumer is enticed to consume a wider range8

of varieties if the cheapest varieties become less expensive, but still consumes the same rangeof varieties if the most expensive ones become slightly cheaper.
A variety  is consumed in country  if and only if  ()  0, or equivalently if,
Z 
 ()d () (1 +  )  1 + (4)0

The consumer’s demand are () =1−  () +
1 + 
1 + 

where  is the price index
 ≡

Z 

(5)

(6)

 ()d0

The surplus  that a consumer in country  obtains from purchasing the manufactured goodcan be computed as
1

−
 =
2(1 +  ) 1 + 

Z 
0

1
 ()d +
2

Z 
0


[ ()] d −2(1 +  )
2

∙Z 
0

 ()d

¸2

 (7)

which depends on the price  (·) and number  of varieties that she actually purchases.

3

Short-run trade patterns

In this section, we determine the short-run equilibrium where the international allocation ofcapital, hence of firms, is fixed. More precisely, we show how firms set their prices and discuss
how a given international allocation of capital aﬀects the patterns of trade.

3.1

Firms’ prices and export decisions

For convenience, we focus on the product market in country . We first consider the pricedecision of a domestic firm in market  and then discuss the price and export decision of aforeign firm that exports from country  to country . Let  () denote the consumer price ofa variety  produced in country  and  () the consumer price of another variety  exported
9

from country  to country . The demand for a variety in country  is given by (5) and thecorresponding price index by (6). Because each firm is negligible, it chooses its prices and its
export status taking the price indices as parameters.
A domestic firm producing variety  in country  chooses its price  () to maximize its
domestic operating profits given by   () =   () (), treating the price index  parametrically. When the demand for variety  is positive, it follows from (5) that all firms in country
 set the same profit-maximizing price given by () =  =1 1 + 

2 1 + Because marginal costs are zero, domestic producers make positive profits on domestic sales
and always supply their domestic market.
A foreign firm exporting another variety  to country  chooses its  () to maximize its
operating profit earned abroad,  () =   () ( () − )  under the constraint that it has apositive demand:  ()  0. The firm is able to export ( ()  0) if and only if it can set a low
enough price to satisfy condition (4). Because foreign firms are symmetric, the unconstrained
price and export levels are the same across all foreign firms and given by () =  =  +


2

and
 () =1 1 −  +  − 

2
1 + 

By (4), variety  is exported to country  ( ()  0) if and only if

1 + 

1 + 

(8)

When this condition does not hold, variety  is not exported ( () = 0).
In the next section, we aggregate firms’ decisions and determine the product market equilibrium.

10

3.2

Product market equilibrium

Let  be the given capital share in country . Because each firm produces a single variety, is also the share of varieties produced in country . Let also  ∈ [0  ] be the share ofvarieties imported from country . Using those definitions, the number of varieties purchased¡¢in country  is given by the sum of domestic and imported varieties:  =  +  . The
corresponding price index is¡¢ =   +   

Since  =  + 2, the price index is a function of the domestic price  . The market
equilibrium price is determined by the fixed point condition  ( ()) = , which implies thatthe equilibrium domestic price and price index are as follows:∗ =

1 2 +  
2 2 + 

∗ =

and

 +   (1 +  )2 + 

(9)

while the export price is again∗ = ∗ + 
2Therefore, several export regimes may arise according to whether varieties are exported at theequilibrium ( = 0,  =  or  ∈ (0  )). The export regimes are determined by the valueof trade costs. Let
̄( ) ≡2
2 +  (10)which falls with larger  . We show the following lemma in Appendix 2.
Lemma. All foreign firms, some (indeterminate number of) foreign firms or no foreign
firms export in country  if  S ̄( ). Furthermore, the equilibrium prices are continuous andweakly decreasing with respect to the level of trade costs.
In other words, domestic consumers buy all foreign varieties if trade costs are low and/or theshare of domestic competitors  is suﬃciently small. Conversely, foreign firms are unable to export when trade costs are high and/or the share of domestic firms is large enough. Furthermore
it can be shown that, whether or not varieties are exported, the equilibrium domestic prices
11converge toward marginal cost (here zero) when the manufactured good becomes homogeneous( → ∞). Likewise, a smaller requirement  of capital leads to a larger number of available
varieties  and therefore to lower prices. Finally, the market price prevailing in a country decreases with smaller levels of trade costs as long as some foreign firms export (  0). Stateddiﬀerently, for any given allocation of capital, trade opening makes markets more competitive.
Using the foregoing Lemma, the equilibrium trade pattern may be described as follows.
Proposition 2 Assume that the allocation of capital is fixed. Then, there is two-way trade if  min{̄( ) ̄( )}; there is one-way-trade from country  to country  if ̄( ) ≤   ̄( );finally, there is no trade if  ≥ max{̄( ) ̄( )}.
Figure 1 shows how the trade pattern changes with the level of trade costs and the international allocation of capital. It reveals several interesting and new features. First, two-way
trade prevails when trade costs are very low (  ̄(1)). In this case, all firms export from every
country to the other one regardless of the distribution of firms: geographical separation does
not suﬃce to protect domestic firms from foreign competition. Second, there exist no trade
within the manufacturing sector under very high trade costs (  ̄(0) = 1). In this case, nofirm export and domestic firms are always protected against competition from abroad.

Insert Figure 1 about here
Third, one-way trade may arise when ̄(1)    ̄(0). We distinguish between the following
two configurations. On the one hand, when ̄(1)    ̄(12), one-way trade takes place fromcountry  to country  when few firms are located in country  (small  ). This is because
competition is suﬃciently weak in this country to allow country -firms to export. In contrast,competition in country  is too tough to permit the entry of foreign varieties. As firms movefrom  to  ( increases), the economy involves two-way trade. This is because competition isrelaxed in country , while trade costs are not high enough to prevent trade from  to . By
contrast, when most firms are set up in country  (high  ), the corresponding market is too
12competitive to permit imports from . Competition in country  is then suﬃciently soft for
country -firms to export.
On the other hand, when ̄(12)    ̄(0), one-way trade also occurs for small and large
values of  . However, instead of two-way trade, autarky now prevails when countries becomesless dissimilar in term of capital allocation ( close to 12). In this case, competition becomes
tougher in country  but remains intense enough in country , implying that trade costs are
suﬃciently high to prevent foreign firms to export. This analysis shows that the emergenceof one-way trade may be explained by a suﬃciently uneven distribution of firms that resultsin strong diﬀerences in market competitiveness. Note also that country size asymmetry isnecessary for one-way trade to occur. When the countries are symmetric, the economy shifts
directly from autarchy to two-way trade at  = ̄(12).
Before proceeding, observe that the economy moves gradually from autarky to two-way
trade as  steadily decreases. Product diﬀerentiation is, therefore, one of the main drivers ofinternational trade, as shown by Krugman (1979) in the two-way trade case.
The above trade patterns are obtained when the allocation of capital is given. We now
analyze the long-run equilibrium where capital moves to the country oﬀering the higher return.

4

Long-run capital allocation

In the long-run equilibrium, the international allocation of capital  is endogenous. By allowing
capital to move from one country to the other, we allow the competitiveness of each national
market to change through the number of domestic firms, which in turn aﬀects firms’ incentives
to export. Hence, although firms are homogeneous, the conditions for firms to export changewith the international allocation of capital. This in turn implies that the nature of trade may
change with the relocation of firms. We first define the long run equilibrium, then study thisequilibrium for each trade pattern and finally synthesize the results.
Let  be the return of a unit of capital invested in country . For any allocation of capital13 , each firm earns the profit Π −  where Π is the operating profit defined by⎧⎨   +  if country- firms exports to country Π =⎩
if country- firms do not export.


where  =  (∗ )2 and   =  (∗ − )2 . In a competitive capital market, the capital return is determined by the firms’ bidding process for capital, which comes to an end when no firm
can earn a strictly positive profit. That is, firms’ operating profits are entirely absorbed by thecapital return so that  = Π  .
In the long run or spatial equilibrium, no capital-owner can get a higher return by relocating
her capital unit to another country. Formally, at an interior spatial equilibrium (0  ∗  1),
we have  =  = , where  is such that all profits are equal to zero; we have a corner
equilibrium at ∗ = 1 when  exceeds  for all values of  . For conciseness, we will often use
the notation: ≡   ∈ [1 ∞)with  = 1 if and only if the two countries have the same size. We also set ≡ 
which we call the degree of competitiveness in the economy. The larger the mass of firms
and/or the lower the degree of product diﬀerentiation across varieties, the more competitive
the economy.
Three cases may arise according to the nature of the trade pattern.

4.1

Two-way trade

Suppose that the manufactured good is imported from both countries. Proposition 2 thenimplies that the condition  min {̄( ) ̄( )} 

(11)

holds. Since capital-owners seek the highest rate of return, the relocation incentives are givenby the profit diﬀerential (  +  ) − (   +    ) in which the equilibrium prices are given14

by (9) with  =  , and thus  = , for  =   . This expression has a unique zero given
by
∗ =

1
1
2−
+ ( −  )
≥ 
2

2

(12)

Two types of equilibria may arise.
Partial agglomeration. There is partial agglomeration in country  (12  ∗  1)with two-way trade if ∗  1 and if trade costs are suﬃciently low to satisfy (11). After someeasy algebra, those conditions are equivalent to  0 () ≡2 (2 − ) + 
2 (2 − ) − (13)  1 () ≡2 (4 − 3) −  (2 + )

(14)

and

where 0 () increases and 1 () decreases with . The domain of trade costs and countries’
asymmetry that support partial agglomeration is depicted in Figure 2.

Insert Figure 2 about here
A simple observation of the curves 0 () and 1 () in Figure 2, i.e. (13) and (14), leads tothe following conclusions. First, two-way trade is more likely when countries are more similar(lower ). Second, two-way trade is more likely when trade costs take intermediate values.
Indeed, lower trade costs not only facilitate trade but also entice firms to move away from thesmaller country so that no firms would remain to export from this country. Third, country
 exports less on both the extensive and intensive margins. However, the remaining  -firms
keeps exporting to country , which means that the market integration eﬀect dominates thecompetition eﬀect generated by the larger mass of firms located in country . Last, two-way
trade becomes more likely for smaller trade costs and less likely for larger ones when the degreeof competitiveness increases. Indeed, when  rises, the curve 0 () is shifted upward, whereas
the curve 1 () is shifted downward. In sum, when the economy becomes more competitive, a15

few firms seek protection in the smaller country but stop to export because the larger market
remains too competitive.
Finally, under partial agglomeration, there exists a home-market eﬀect because ≡∗ − 122−4
 1 ⇐⇒  =  (12)
1 = 2

2+
 − 2

which holds under (14). However, the home-market eﬀect is weaker when the economy is morecompetitive: firms relax competition by being more dispersed.
Full agglomeration. When  ≥ 0 (), there is full agglomeration in equilibrium (∗ = 1
and ∗ = 0) and, therefore, one-way trade from the larger to the smaller country. For trade toarise, (11) implies that  ̂ ≡2
2+

must hold. Hence, this regime emerges when the two countries are dissimilar enough and trade
costs are suﬃciently low to sustain trade from the larger to the smaller country (see Figure 2).
To sum up, we have shown the following result.
Proposition 3 If   0 () and   1 (), then there exists a unique spatial equilibrium
involving partial agglomeration in the larger country with two-way trade. Furthermore, when ≥ 0 () and   ̂, there is full agglomeration and one-way trade from  to  .

4.2

No trade

Suppose now that no firms export. Proposition 2 then implies that ≥ max {̄( ) ̄( )} 

(15)

Relocation incentives are given by   −    in which the equilibrium prices are now givenby (9) where  = 0, and thus  =  , for  =   . The unique zero of this expression isequal to√√1 1  −  4 + 1√∗ = + √≥ 
2 2  +  
216

Therefore, the international allocation of capital remains biased toward the larger country .
This bias falls when the economy becomes more competitive (larger ). Since ∗ exceeds ∗ ,market  is more competitive than market  . Therefore, it is always less profitable for firms
located in country  to export than for those located in country . This implies that (15) boils
down to ≥ ̄(∗ ) =2

2 + ∗

(16)This condition becomes less stringent as the mass of firms in country  increases, thus deterring
exports to  because competition therein is tougher.
Partial agglomeration. There is partial agglomeration in country  if ∗  1, or
equivalently, if³
 ´2
  2 ≡ 1 +2which is independent of  since there is no trade. Furthermore, by plugging ∗ = 1 − ∗ into
(16), this condition becomes1 ≤ 3 () ≡ [ − 2 (1 − 2)]2 
4Since  ≥ 1, it must be that  ≥ 4( + 4) for this inequality to hold. Thus, as expected, trade
costs must be suﬃciently large for the no-trade case to arise. Note also that 3 ()  2 when  1.
In this case, there is a home-market eﬀect if and only if
 ≡

³p
p ´2
∗ − 12
4

1
⇐⇒

+
  1 + 1 − 2

Clearly, the LHS of the second inequality decreases from 2 to 1 as  increases from 12 to 1,
whereas the RHS takes values above 2 if and only if   4. Hence, if   4, there is a homemarket eﬀect regardless of countries’ asymmetry. This occurs when competition is suﬃciently
soft, either because there are few firms in the economy or because varieties are good substitutes.
By contrast, if   4, the above inequality holds only for large enough  . In this case, there17

is a home-market eﬀect if and only if
1 1
  +
2 2

s

µ ¶2
4
1−



(17)

Hence, country  must be big enough to overcome the competition eﬀect triggered by a highdegree of competitiveness. In contrast, when the two countries are fairly similar, there is areverse home-market eﬀect. Indeed, the domestic country is here the only firms’ market. Whenthe degree of competitiveness  is high and countries have similar sizes, the larger country is nota suﬃciently big outlet to compensate firms for the higher intensity of competition that would
prevail with a proportionate share of firms. In the same vein, (17) shows that the occurrenceof a reverse home-market eﬀect increases when the degree of competitiveness  rises. These
properties, which standard CES models are silent about, stem from the pro-competitive eﬀects
that our model encompasses.
Full agglomeration. If  ≥ 2 , there is full agglomeration in the bigger country (∗ = 1).
In other words, all firms set up in  when this country is suﬃciently large. However, firms are
not enticed to export when  ≥ 1, as shown by (16) in which ∗ = 0.
To sum up, we have:
Proposition 4 If   2 and  ≤ 3 (), then there exists a unique spatial equilibrium withpartial agglomeration and no trade. Furthermore, there is full agglomeration in the larger
country and no trade when  ≥ 2 and  ≥ 1.

4.3

One-way trade

Suppose, last, that firms export only from country  to country . Proposition 2 then implies
̄( ) ≤   ̄( )

(18)

For this condition to hold, it must be that    , which means that the exporting country
hosts more firms than the other.
18

Under one-way trade from country  to country , the relocation incentives are given by theoperating profit diﬀerential ∆  = ( +   ) −   in which the equilibrium prices are given by(9) where country -firms export ( =  ), whereas country -firms do not ( = 0). Therefore,
the numbers of varieties at consumers’ disposal in country  and  are, respectively,  = and  =  . This yields" µ∆  ∝ 

1
2 + 

¶2

− 


2

µ

2 +   
−
2+
2

¶#

(19)

which is monotonically decreasing in  (∆    0). Thus, when only country- firms
export to country , (19) has a unique zero.
In the presence of trade costs, firms always have an advantage to operate from the larger
market because this one oﬀers a better access to the larger pool of local consumers. Thisadvantage attracts more firms but fosters competition in this market, thus reducing the foreign
firms’ incentives to export into this market. In Appendix 3, we prove the following: the long-run
equilibrium involving one-way trade implies trade only from the larger to the smaller country.
Given this result, we can now focus the conditions for one-way trade from  to  to arise
under partial and full agglomeration.
Partial agglomeration. Under one-way trade from  to  , the unique solution to theequation ∆  = 0 involves partial agglomeration if and only if operating profits in country are lower than those made in  when all firms are located in :  +       at  = 1.
Plugging  = 1 in (19) shows that this condition is equivalent to  4 () ≡(2 + )
[2 (2 − ) + ] 
4(20)

Furthermore, the above solution must fulfill the conditions (18) for  =  and  =  . It isreadily verified that these two conditions are equivalent to∗ ≥ 2 (1 − )

1


and

∗  1 − 2 (1 − )

1



(21)

The former inequality holds if and only if  ≥ 1 (), while the latter holds if and only if   3 ().
This can be shown by plugging each bound of (18) in the equilibrium condition ∆  = 0.
19

As shown in Figure 2, the level of trade costs does not have a monotone impact on therange of countries’ size asymmetries sustaining one-way trade. Consider a situation in which steadily decreases to 1 (), the economy shifts from one-way trade to two-way trade. This isbecause the larger country is no longer competitive enough to deter importing. In contrast,
when  steadily decreases to 3 (), the economy shifts to no-trade because the smaller country
is more competitive and, therefore, more diﬃcult to penetrate. Last, it can be shown thatboth the loci 3 () and 4 () are shifted upward when the degree of competitiveness  increases,
while the gap between 4 () and 3 () widens with .
The expression (19) may be used to uncover the impact of countries’ asymmetry and trade
costs. Indeed, it is readily verified that
∆ 
0


∆ 
 0


Thus, we have:d∗∆  ∆ 
=−

0
d



∆  ∆ 
d∗
=−

 0
d



In words, the larger country hosts a growing share of firms when its relative size rises, whereas
lowering trade costs leads the manufacturing firms to get more agglomerated in the larger
country. Consequently, firms that quit the less competitive and smaller country find it profitable
to export their output to their country of origin. To put it diﬀerently, even though firms arehomogeneous, their locational choices determine their attitudes toward exports. This is to becontrasted with the recent literature that explains export structure with firms’ technologicalheterogeneity (Melitz, 2003; Melitz and Ottaviano, 2008).
Determining the domain of parameters for which the home-market eﬀect holds in a one-way
economy is a very hard task because we do not have the explicit expression for ∗ . Nevertheless,
we can make the following inferences. First, we know that there is always a home-market eﬀectunder two-way trade. By continuity, this still holds under one-way trade when trade costs are
not too high (i.e. above but close to 2 () in Figure 2). Second, we know from Section 4.2 thatthe no-trade regime supports a reverse home-market eﬀect when countries are similar enough,20

the economy suﬃciently competitive, or both. Therefore, by continuity, the same eﬀect holds
when trade costs just sustain one-way trade (i.e. above but close to 3 () in Figure 2). Thus,
when trade liberalization deepens, the economy continuously shifts from a reverse home-market
eﬀect to a home-market eﬀect within the one-way trade economy since price and profits arecontinuous with respect to . However, when the degree of competitiveness is low, there isalways a home-market eﬀect in the no-way trade regime. We may thus expect this eﬀect tohold for all values of trade costs.
Figure 3 illustrates how the value of  varies with  for  = 13. When the economy
is not very competitive ( = 2), the value of  exceeds 1 for all , meaning that there
is always a home-market eﬀect. In contrast, when the economy is very competitive ( = 8),
a reverse home-market eﬀect occurs for  larger than 033, i.e. under autarky and a range of
-values sustaining one-way trade.

Insert Figure 3 about here
Full agglomeration. All firms agglomerate in country  (∗ = 1) if  ≥ 4 (). In thiscase, the two conditions (18) boil down to ̂ ≤   1.
We may summarize our results as follows.
Proposition 5 If  ≥ 1 (),   3 () and   4 (), then there exists a unique spatial equilibrium with partial agglomeration and one-way trade from country  to country  . Furthermore,
full agglomeration with one-way trade prevails when  ≥ 4 () and ̂ ≤   1.

4.4

Synthesis

The above analysis describes the equilibrium conditions for the existence of a unique spatial
equilibrium in the case of two-way trade, one-way trade and no trade. As illustrated by Figure
2, these conditions define several domains that form a partition of the positive quadrant in theplane ( ). We may collate those conditions in the following proposition, which shows how
21

globalization aﬀects the nature of trade and economic geography, where we have set̂ ≡ 1 +2
2+which solves 0 (̂) = 1 (̂) = 4 (̂) = ̂ (see Figure 2).
Proposition 6 As trade costs steadily decrease, the economy goes through the following sequence of stages:(i) For small country size asymmetries (  ̂), the economy involves partial agglomeration ofcapital and firms in the larger country with sequentially no trade, one-way trade and two-way
trade; finally, it involves full agglomeration with one-way trade.
(ii) For intermediate values of country size asymmetries (̂    2 ), the economy involves
partial agglomeration in the larger country with first no trade and then one-way trade; finally,
it involves full agglomeration with one-way trade.
(iii) For large country size asymmetries (2  ), the economy involves full agglomeration in
the larger country with first no trade and then one-way trade.

5

Eﬃcient allocation of capitalIn this section, we determine the socially optimal allocation of capital and investigate whether
and how it diﬀers from the long-run equilibrium. Specifically, we suppose that a fully informed
and benevolent planner decides over the production level and over the allocation of products
and capital. The planner maximizes the global welfare  =   +   + Π, which includes
the consumer surpluses in both countries and the world profit, which here amounts to the returnto capital ( = Π). In the eﬃcient allocation, the planner sets the product prices equal totheir marginal costs, which are equal to zero for the domestic varieties and to  for the foreign
ones. As a result, firms make zero profits and the socially optimal return to capital is nil. In
what follows, we consider the cases of two-way trade, no trade and one-way trade.

22

Under two-way trade, all varieties are consumed everywhere ( = ) so that the consumer
surplus (7) in country  is given by
 =

1

1

−
  +  2 −( )22(1 + ) 1 + 
2
2(1 + )(22)

which is a strictly concave function of  . Maximizing  with respect to  gives the optimal
allocation of capital:
 =

1 1
2−
+ ( −  )2 2
where the superscript  describe eﬃcient outcome. It is easy to check that  is smaller thanthe equilibrium allocation ∗ under two-way trade for any  ≥ 12. As a consequence, there
exists excessive concentration of capital in the larger country. This result has already been
noticed by Ottaviano and van Ypersele (2005) under two-way trade. Given the prevalence ofzeros in the matrix of trade flows between countries, it is worth asking what this result becomesunder other trade regimes (Baldwin and Harrigan, 2007; Helpman et al., 2008).
Under no trade, only the domestic varieties are consumed so that the consumer surplus (7)in country  simplifies to
 =


2(1 +  )(23)

which is again a strictly concave function of  . Maximizing  with respect to  yields√√
1 1  −  2 + √ = + √2 2  +  
which is also smaller than the equilibrium allocation under no trade.
Under one-way trade, individuals consume all varieties in the smaller country ( = )and get the surplus (22), whereas they consume only the domestic varieties in the larger country
and get the surplus (23). The first order condition is given byµ¶2µ¶ 2 + 2
1


−
= 0
− 
2 + 2
2
2 + 2
2
The solution  to this equation is the same as the solution to the equilibrium condition (19)when  is replaced by 2. Therefore, we have  () = ∗ (2). If the equilibrium solution23∗ (2) falls with admissible , we get  () = ∗ (2)  ∗ (), so that the eﬃcient allocationof capital is less dispersed than the equilibrium one. Although we are unable to oﬀer a formal
proof of ∗0 (2)  0 under one-way trade, numerical computations show that this property
holds for all admissible  and .
Proposition 7 Any interior equilibrium implies excessive agglomeration of capital and firms
in the larger country under two-way trade, one-way trade and no trade.
Furthermore, if country size asymmetries increase, firms may agglomerate in equilibrium
although the planner keeps them dispersed. In this case, there also is too much agglomerationfrom the welfare viewpoint. If country size asymmetries increase further, firms agglomerate bothin the equilibrium and in the social optimum. The excessive agglomeration result vanishes onlyin this extreme case.
Thus, once it is recognized that product markets are not competitive, production involves
increasing returns and shipping goods is costly, in all trade regimes liberalizing capital mobility does not yield the socially optimal allocation of capital. This runs against the standard
prediction that the social optimum is always reached at the equilibrium allocation of capital.
Such a prediction is commonly made in perfect competition models with constant returns andimperfect competition models with increasing returns that are based Dixit-Stiglitz iso-elastic
preferences (see Baldwin et al. 2003, p. 257). While taxing capital is undesirable in thesesettings, our approach suggests that taxing capital in the larger country or subsidizing capitalin the smaller one may be welfare-enhancing.

6

Concluding remarks

The main purpose of this paper was to show within a unified framework that trade liberalization
may span a large range of cases, which reflects the diversity displayed by commodity flows inworld trade. In particular, we have seen that the nature and intensity of trade does not depend

24only upon the degree of openness of the economy, measured by the level of trade costs. How
firms are distributed across countries is another critical determinant, the reason being that
their locations aﬀect the intensity of competition within each country. For example, foreign
firms will find it more diﬃcult to export to a country that accommodates a larger number ofdomestic firms because local prices are lower. Likewise, if the manufacturing sector supplies
a poorly diﬀerentiated product, there will be little trade because each market is governed byunleashed competition. Last, when more capital is allocated to the manufacturing sector, theintensity of trade is lowered because a larger mass of firms compete in the economy. In sum,
all parameters and policies that influence the degree of competition within national economies
are likely to have a strong impact on trade.
That said, our analysis sheds light on the diﬀerent roles played, on the one hand, by international market integration across countries and, on the other hand, by competition policy
within each trading partner. As market integration deepens, the economy moves from autarky
to two-way trade through one-way trade. By contrast, when national economies become more
and more competitive, trade vanishes and countries operate under autarky. This is so whenfirms sell a good which is almost homogeneous, produce almost under constant returns, or both.
This result is consistent with a standard result in trade theory according to which perfectlycompetitive agents exchange commodities as long as there are diﬀerences in relative endowments, production techniques, or preferences across locations. Otherwise, no trade will occur(Samuelson, 1939, 1962). In the same vein, lowering trade barriers fosters the concentrationof capital within the bigger country. On the contrary, making the economy more competitive
pushes toward a more dispersed distribution of capital because this allows firms to relax competition. Therefore, we may safely conclude that competition policy and trade policy are notgood substitutes.
Last, we have seen how globalization aﬀects the way capital is distributed across countries.
Specifically, our results highlight the advantage of being big in the race for firms. For instance,
for small country size asymmetries, the international distribution of capital remains the same25

as long as trade costs are suﬃciently high for imports to be prohibitively expensive. Once trade
costs are low enough to permit the larger country to export to the smaller one, further decreases
in trade costs lead to a growing concentration of capital in the larger country. The same holds
when the economy operates under two-way trade. Hence, even though the trading partners have
similar sizes, market integration goes hand in hand with a more unbalanced spatial distributionof the manufacturing sector.

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[14] Ottaviano, G.I.P., T. Tabuchi and J.-F. Thisse (2002) Agglomeration and trade revisited.
International Economic Review 43, 409-436.
[15] Ottaviano, G.I.P. and T. van Ypersele (2005) Market size and tax competition. Journal ofInternational Economics 67, 25-46.
[16] Picard, P. and D.-Z. Zeng (2005) Agricultural sector and industrial agglomeration. Journalof Development Economics 77, 75-106.
[17] Samuelson, P.A. (1939) The gains from international trade. Canadian Journal of Economics 5, 195-205.
[18] Samuelson, P.A. (1962) The gains form international trade once again. Economic Journal
72, 820-829.

27

Appendix 1
Let () be the Kuhn-Tucker multiplier of the nonnegative consumption constraint. The consumer’s program may then be re-written as follows:
¸2
∙Z 
Z 
Z
1 

2
max L =
()d −[()] d −()d
()
2 0
2 0
0
Z 
Z 
−
()()d +()()d + 000The maximization problem has a concave objective function and convex constraints. Thus, theKuhn-Tucker conditions are necessary and suﬃcient condition for a maximum:
L
≤ 0
()
L
≥ 0
()

() ≥ 0and() ≥ 0

and

L
=0
()
L
()
=0
()

()

where
L
= 1 − () − ()L= ()
()

Z 
0

()d − () + ()Since both () and () cannot be positive, three cases may arise. First, suppose that
()  0. Then L() = 0, which implies
() = 1 − () − Z ()d

0This expression is strictly positive if and only if
()  ̄ ≡ 1 − Z ()d

0Second, suppose that ()  0. In this case, () = 0 and L() ≤ 0, implying that
() − ̄ ≥ ()  0
Last, suppose that () = () = 0. Then, we haveL() ≤ 0 ⇐⇒ () − ̄ ≥ 028

Accordingly, the solution to the consumer program is given by⎧⎨ 1 − () −  R  ()d if ()  ̄0
() =⎩
0otherwise.

We are now equipped to construct firms’ demands. Since varieties are ranked by increasing

price order, ̃ is the first variety that is not purchased: ≥ ̃ ⇔ () ≥  (̃) ⇐⇒ () = 0
which implies
follows:R0

()d =R ̃
0

()d. The consumer’s demands may therefore be re-written as

() =⎧⎨ 1 − () −  R ̃ ()d0⎩0if ()  ̄otherwise.

Integrating demands over all varieties belonging to [0 ̃], we get
Z ̃0()d = ̃ −which yields

Z ̃()d − ̃

0

Z ̃

()d0R ̃̃ − 0 ()d()d =1 + ̃
0Z ̃

so thatR ̃
1 +  0 ()d̄ =  (̃) =1 + ̃

and() =⎧⎨⎩

1− () + 1+̃
1+̃

0

R ̃
0

()d

if   ̃otherwise.

Since ()   (̃), the consumer buys the domestic varieties, which implies  (̃)  1. Q.E.D.

Appendix 2
Suppose, first, that all foreign firms export to market  ( =  ). Then, the number of varieties
consumed in country  is  =  so that the equilibrium price set by country- firms is equal29to∗ =

1 2 +  
2 2+

(24)

where  ≡ . By (8), this regime occurs if   ̄( ) where̄( ) ≡2
2 +  is the expression in (10). In other words, consumers buy all foreign varieties if trade costs are
low, the share of domestic competitors is not too large, or both.
Second, suppose that no foreign firm exports ( = 0). Then, we have  =   and∗ =

1
2 +  (25)

where ∗  1. By (8), this regime occurs if   ̄( ), which means that no foreign firms export
when trade costs are high, the share of domestic firms is not too small, or both.
Finally, an indeterminate number of foreign firms export to market  if  = ̄( ). Indeed,plugging ̄( ) in (24) yields
∗ =

1
2 +  

which is independent of  . Therefore, the number of exported varieties  can take any value
in the interval [0  ].
The transition between the export regimes have the following interesting properties. On
the one hand, as the value of trade cost increases, the number of varieties exported to country
 suddenly switches from  to 0 at  = ̄( ). On the other hand, since  is here exogenous,
one can checklim ∗ =→̄( )−lim ∗ =→̄( )+1
2 +  

which implies that ∗ is continuous at ̄( ). As a result, production and profit levels are also
continuous functions.

30

Appendix 3
We here prove that the market outcome involving one-way trade implies trade from the larger
to the smaller country. Suppose indeed that there is one-way trade from the smaller country =  to the larger country  = . It then follows from the trade condition (18) that   12hence   12: competition in the bigger market must be soft enough for the foreign firms
to supply this market, while competition in the smaller market is suﬃciently fierce to prevent
-firms from exporting to the smaller market. We know that, under each trade regime, theprofit diﬀerential ∆  is continuous and decreasing with respect to  . Because ∆  can be
written as a continuous function of all quantities, which are themselves continuous functions of , ∆  is continuous at each transition point. Therefore, the locus ∆  is continuous anddecreasing over the whole interval [0 1] and has, therefore, a unique zero. As shown by Figure
1, when  increases from 0, the economy moves either from one-way trade to no trade, orfrom one-way trade to two-way trade. Under two-way trade, it is readily verified that ∆   0
at  = 12, while ∆   0 also holds  = 12 under no trade. Therefore, for all  ≤ 12,
∆  must be positive under one-way trade. This means that there exists no equilibrium with  12.

31

λi
1

1/2

0

One waytradei→j

Two way
trade

t (1)

t (λ j )No tradeOne
way tradej→it (λi )t ( 12 )t (0)Figure 1: Structure of trade in the short run

t

θ≡

θH
θF

One-way trade
Full agglom.

ρ2ρ2ρ 4 (t )No trade
Partial agglom.

One-way trade
Partial agglom.

θˆρ 0 (t )1

No trade
Full agglom.

Two-way trade tˆ
Partial agglom.

ρ1 (t )t ( 12 )ρ 3 (t )1

Figure 2 : Structure of trade in the long run

t

HME
Full
agglomeration

8

6

Two waytrade

Two waytrade

4

One waytrade

2

No trade

n=2

One way
trade

1

n=8

No trade

0.0

0.2

0.4

0.6

0.8

Figure 3 : Home market effect (HME)

1.0

1.2

t