Country Size, Productivity and Trade Share Convergence: An
Analysis of Heterogenous Firms and Country Size Dependent
Beachhead Costs
Anders Akermanyand Rikard Forslidz

Second Draft, December 2008

Abstract
This paper introduces a market size dependent …rm entry cost in the Helpman, Melitz
and Yeaple (2004) (HMY) version of the Melitz (2003) model. This is a relatively small
generalisation, which preserves the analytical solvability of the model. Nevertheless ourmodel yields several new results that are in line with data. First, the average productivityof …rms located in a market increases in the size of the market. Second, the productivityof exporters is U-shaped in the export market size. Exporting to a small market is di¢ cultbecause the …xed entry cost (e.g. standardization) has to be spread over few units. This
problem decreases as the export market grows. However, when the export market is su¢ ciently large, the market size dependent component of the entry cost (e.g. marketing) starts
to dominate, and this again makes it di¢ ult to enter the market as an exporter. Third,
the productivity premium (the di¤erence in average productivity) between exporters andnon-exporters decreases with the home country size. Fourth, we derive a set of new results
related to trade volume. Contrary to what would be the case in the HMY framework, ourmodel generates the well known fact from the empirical literature that the manufacturingexport share decreases in the size of the exporting country. Moreover, it is shown that whenthe …xed entry cost of exporting declines, for instance as the result of economic integration,
export shares converge. This result is shown to be in line with empirical data. Fifth, we use
a multicountry version of our model to derive a gravity equation. Our speci…cation yields agravity equation à la Anderson and van Wincoop (2003), but where GDP per capita enters
as an additional explanatory variable. Our speci…cation thus gives a theoretical rationale forthe common practise of introducing GDP per capita as an additional explanatory variablein gravity regressions of trade ‡ows.
We are grateful for comments from Pol Antras, Karolina Ekholm, Thierry Mayer, Marc Melitz, Jim Markusen,
Diego Puga, participants at the NOITS conference in Stockholm May 2007, the ERWIT Conference in Appenzell
June 2008, as well as from participants at a seminar at the Department of Economics in Stockholm. Financial
support from Jan Wallander’s and Tom Hedelius’Research Foundation is gratefully acknowledged by Akerman.
yStockholm University, email: anders.akerman@ne.su.se.

z

Stockholm University, CEPR; email: rf@ne.su.se.

1

JEL Classi…cation: D21, F12, F15
Keywords: heterogenous …rms, market size, beachhead costs

1

Introduction

It is empirically well established that there are systematic productivity di¤erences among …rms;
see Tybout (2003) for a survey.1 In particular, exporting …rms tend to be more productive,
larger, and live longer than domestic …rms. There is also evidence of multinational …rms tending
to be more productive than exporters (Helpman, Melitz and Yeaple 2004). The theoretical
literature on trade with heterogenous …rms explains these …ndings by either iceberg trade costs
associated with exports, as in Bernard et al. (2003), or higher …xed costs associated with market
entry into a foreign market, as in Melitz (2003) and Yeaple (2005). Only the most productive…rms will …nd it pro…table to pay the additional cost necessary for exports, and export …rms
will therefore, on average, be more productive than non-exporters.
Naturally, it is also the case that …rm productivity may vary because of country-speci…cfactors. E.g. Bernard et al. (2007) show how comparative advantage may strengthen the
productivity gains associated with trade in a Melitz (2003) type model with heterogenous …rms.
In the present paper, we investigate whether market size dependent entry costs may be one
explanation for observed productivity di¤erences between …rms in di¤erent countries.
This paper modi…es a Melitz (2003) type model allowing the market entry cost that …rms
must pay when entering a new market to increase in country size. We think here e.g. of consumer
brands. It is, for instance, considerably more costly to establish a new brand of toothpaste in
the U.S. than to establish the same brand in a small country such as Sweden. This is because
the cost of television advertising is based on the number of viewers, free sampling based on
number samples etc., which makes it more costly to enter a larger market.2 This notion is also
supported by Eaton et al. (2005) who …nd a strong positive relationship between country sizeand entry cost when calibrating their model to French …rm level data as shown in Figure 1.
An immediate implication of the market entry cost increasing in market size is that …rms
on average should be more productive in larger markets due to …rm selection. This notion isconsistent with empirical evidence showing that workers and …rms on average are more productive in larger markets (Head and Mayer 2004, Redding and Venables 2004, Syverson 2004,
2006, and Amiti and Cameron 2007).3
1

Other studies include Aw et al. (2000), Bernard and Jensen (1995, 1999a, 1999b, 2001), Clerides et al. (1998)

as well as Eaton et al. (2004).
2

It is clear that our argument is less convincing for smaller niche products such as e.g. swedish "surströmming",

which is a particular type of fermented …sh. The probability of …nding a buyer with a taste for this productprobably increases in the number of consumers in a market.
3

An alternative explanation for these productivity di¤erences is externalities associated with agglomeration.

Some preliminary evidence using plant level data for French cities indicate, however, that both …rm seletion andagglomeration may play a role (see Puga et al 2008).

2

We model the market entry cost as having a …xed and a market size dependent component.
The …xed part re‡ects costs such as standardization of the product for a particular market or
to creating a marketing message for the market. The market size dependent component of theentry cost is the marketing cost of introducing a new variety in a market (e.g. product samplingor television ads). As argued above, it is quite natural that this cost depends on the size of themarket, and the fact that part of the entry cost depends on market size is normally taken for
granted in the marketing literature, where the marketing cost over sales ratio is a key variable4 .
This paper makes a relatively small generalization of the Helpman, Melitz and Yeaple (2004)(HMY) version of the Melitz (2003) model, which preserves the analytical solvability of themodel. Nevertheless our analysis yields several new results that are supported by data. First,
the average productivity of …rms located in a market increases in the size of the market. Second,
the productivity of exporters is U-shaped in the export market size. Exporting to a small
market is di¢ cult because the …xed entry cost (e.g. standardization) has to be spread over few
units. This problem decreases as the export market grows. However, when the export market issu¢ ciently large, the market size dependent component of the entry cost (e.g. marketing) starts
to dominate, and this again makes it di¢ cult to enter the market as an exporter. Third, theproductivity premium between exporters and non-exporters decreases with the home country
size. Fourth, we derive a set of new results related to trade volume. Contrary to what would
be the case in the HMY framework, our model generates the well known property of data that4

See e.g. Buzzell et al. (1975).

3the manufacturing export share decreases in the size of the exporting country. Moreover, it isshown that, as the …xed entry cost of exporters declines, for instance as the result of economic
integration, export shares converge. Fifth, we use a multicountry version of our model toderive a gravity equation. Our speci…cation yields a gravity equation à la Anderson and van
Wincoop (2003), but where GDP per capita enters as an additional explanatory variable. Our
speci…cation thus gives a theoretical rationale for the common practise of introducing GDP per
capita as an additional explanatory variable in gravity regressions of trade ‡ows.
We confront our main theoretical results with data in various ways. A limiting factor is that
our model produces cross-country predictions on the …rm level, and such a data set is not yet
available. The …rst result is, as mentioned, in line with existing empirical evidence of …rms being
more productive in larger markets, and we do not perform any independent test of this. The
second result that the productivity of exporters is U-shaped in the export market size, implies
that our model in principle is consistent with exporter productivity decreasing in market size as
in Arkolakis (2006) as well as with the opposite case in Melitz and Ottaviano (2005). Again we
do not perform any independent test of this. For the third result that the exporter productivity
premium is larger in a small market, we present some cross-country evidence on wage premia
in the export sector in the empirical section of this paper. The empirical section explicitly
tests result four, showing that there is evidence of manufacturing export shares converging overtime. The section also con…rms the well known property of data that large markets have smaller
manufacturing export shares. Finally, a large number of empirical papers on the gravity model
supports result …ve that GDP per capita should enter the gravity regression.
Our analysis is related to that of Melitz and Ottaviano (2005) who introduce …rm heterogeneity in the model by Ottaviano et al. (2002) with a linear demand system and where theendogenous mark-ups of monopolistically competitive …rms depend on market size. Melitz andOttaviano (2005) …nd that …rms selling in large markets are larger and more productive, sincehigher competition forces the mark-ups in a large market downwards. The same holds in ourmodel for domestic producers, but the mechanism leading to higher productivity in a large
market is instead that …rms need to be more productive to a¤ord the higher market entry costassociated with a larger market. For exporters, our model instead yields a U-shaped relationshipbetween foreign market size and exporter productivity as discussed above. A further di¤erence
as compared to Melitz and Ottaviano (2005) is that the productivity of …rms in a market also
depends on the size of other markets in our model. E.g. a larger foreign market implies morecompetition from imports, which forces up the productivity of domestic …rms. One consequenceof this dependence of foreign market size is that export shares will vary with the market size.
Finally, our result that trade shares converge as the entry cost into foreign markets falls is naturally not present in Melitz and Ottaviano (2005), since they do not employ any market entry
costs.
Arkolakis (2006) uses a formal model of advertising in a model of heterogenous …rms where
the market penetration cost of each …rm is endogenous. The probability of a marketing message

4reaching at least one consumer increases with the population in this model. Thus, a …rm gets a
relatively larger payo¤ for a small investment in marketing when exporting to a large country
as compared to a small one. The marginal exporter, which is just su¢ ciently productive toexport, will therefore export to a large rather than to a small market. This is consistent withthe feature of the …rm level data e.g. in Eaton et al. (2005) that many small …rms export asmall amount to large markets. Our model generates the same result for not too large export
markets, since the …xed market entry cost (e.g. standardization) can be spread over more units
when the market is large. For large enough export markets the variable entry cost (advertising)
dominates in our model, implying that exporter productivity instead has to increase in theforeign market size. We believe that our set-up complements Arkolakis (2006). Clearly forsome products it is more likely to …nd a buyer in a large market as modeled by Arkolakis(2006). Other more standardized products, such as consumers brands, are certainly more costly
to establish in large markets. Except for the results on exporter productivity our speci…cationdi¤ers by the results on trade share convergence, that are supported by data in the empiricalsection. Finally, because our set-up is simpler than that of Arkolakis (2006), we can solve ourmodel for the general equilibrium with free entry of …rms.
The paper is organized as follows: Section 2 contains the model and section 3 presents
the theoretical results. Section 4 contains empirical tests of our prediction that trade sharesconverge. Finally, section 5 concludes the paper.

2

The ModelThis paper employs a modi…ed Helpman et al. (2004) version of Melitz’ (2003) monopolisticcompetition trade model with heterogeneous …rms.

2.1

BasicsThere are m countries. Each country j has a single primary factor of production labour, Lj ,used in the A-sector and the M-sector. The A-sector is a Walrasian, homogenous-goods sector
with costless trade. The M-sector (manufactures) is characterized by increasing returns, DixitStiglitz monopolistic competition and iceberg trade costs. M-sector …rms face constant marginal
production costs and three types of …xed costs. The …rst …xed cost, FE , is the standard DixitStiglitz cost of developing a new variety. The second and third …xed costs are ‘beachhead’costsre‡ecting the one-time expense of introducing a new variety into a market. These costs are here
assumed to depend on the size of the market.
There is heterogeneity with respect to …rms’marginal costs. Each Dixit-Stiglitz …rm/variety
is associated with a particular labour input coe¢ cient –denoted as ai for …rm i. After sinking
FE units of labour in the product innovation process, the …rm is randomly assigned an ‘ai ’froma probability distribution G(a).
Our analysis exclusively focuses on steady-state equilibria and intertemporal discounting is

5

ignored; the present value of …rms is kept …nite by assuming that …rms face a constant Poissonhazard rateof ‘death’.

Consumers in each nation have two-tier utility functions with the upper tier (Cobb-Douglas)
determining the consumer’s division of expenditure among the sectors and the second tier (CES),
dictating the consumer’s preferences over the various di¤erentiated varieties within the M-sector.
All individuals in country j have the utility function1
Uj = CM j CAj
;

where

(1)

2 (0; 1), and CAj is consumption of the homogenous good. Manufactures enter theutility function through the index CM j ; de…ned by
2
ZNj
6
(
CM j = 4 cij

1)=

0

3 =(

7
di5

1)

;

(2)

Nj being the mass of varieties consumed in country j, cij the amount of variety i consumed incountry j; and> 1 the elasticity of substitution.

Each consumer spends a shareof his income on manufactures, and demand for a variety

i in country j is therefore
xij =

pij

Yj ;

Pj1

(3)

where pij is the consumer price of variety i in country j, Yj is income; and Pj

NRj0

the price index of manufacturing goods in country j.

! 11

p1ij diThe unit factor requirement of the homogeneous good is one unit of labour. This good isfreely traded and since it is chosen as the numerairepA = w = 1;

(4)

w being the nominal wage of workers in all countries.
Shipping the manufactured good involves a frictional trade cost of the “iceberg” form: for
one unit of a good from country j to arrive in country k,

jk > 1 units must be shipped. It is

assumed that trade costs are equal in both directions and thatjj = 1: Pro…t maximization by

a manufacturing …rm i located in country j leads to consumer pricepijk =

1

jk ai

(5)

in country k.
Manufacturing …rms draw their marginal cost, a; from the probability distribution G(a) after
having sunk FE units of labour to develop a new variety. Having learned their productivity,…rms decide on entry in the domestic and foreign market, respectively. Firms will enter a market
as long as the operating pro…t in this market is su¢ ciently large to cover the beachhead (market
6entry) cost associated with the market. Because of the constant mark-up pricing, it is easily
shown that operating pro…ts equal sales divided by . Using this and (3), the critical ’cut-o¤’levels of the marginal costs are given by:where FD (Lj )1
aDj
Bj = FD (Lj );(6)

a1Xjk jk Bk = FX (Lk );(7)

F D (Lj ) ; FX (Lj )

Lj
; and
Pj1

F X (Lj ) ; Bj

jk

1
jk

2 [0; 1]

represents trade freeness. The market entry cost (beachhead cost) is assumed to increase in thesize of the market

dFD (Lj ) dFX (Lj )> 0. We will parametrize how the beachhead cost depends
dLj ;
dLj

on market size below. However, note that it is natural that F depends on L, since the marketingcosts of establishing a new brand in a large market, such as e.g. the US, are typically muchhigher than in a small country.
Finally, free entry ensures that the ex-ante expected pro…t of developing a new variety incountry j equals the investment cost:
aDj
Z

a1

Bj

FD (Lj ) dG(a) +

1
jk a

Bk

FX (Lk ) dG(a) = FE :(8)k;k6=j 0

0

2.2

aXjk
X Z

Solving for the Long-run Equilibrium

In this section, we apply two simplifying assumptions. First, the model is solved with twocountries, j and k (appendix 6.1 indicates how the multicountry case is solved). We refer toj as ’Home’ and k as ’Foreign’. Second, we follow HMY in assuming the probability density
function to be Pareto5 :G(a) = a :(9)

Substituting the cut-o¤ conditions (6) and (7) into the free-entry condition (8) gives Bj ,

Bj =
where

1 > 1, andFE FD 1 (Lj ) (1) (11(Lj ) (Lk )FX (Lj ) 1
FD (Lj )(Lj )(Lk ))

!1

;

(10)

2 [0; 1] is an index of trade freeness.

Using (10) and the cut-o¤ conditions gives the cut-o¤ marginal costs:
aDj =
5

(

1) FE
FD (Lj )(1
1(Lk ))(Lj ) (Lk );This assumption is consistent with the empirical …ndings by e.g. Axtell (2001).

7

(11)

aXjk =(

1) (Lk )FE
FX (Lk )(11(Lj ))(Lj ) (Lk )

:

(12)

From these, it is seen that, contrary to the standard model by Melitz (2003), the market size
(Lk )> FD (Lj ) for all j; k:6 This
will a¤ect the cut-o¤ marginal costs. We will assume that FX(Lk)

assumption implies that aXjk < aDj 8j; k.
The price indices may be written as
Pj1

(1

=

1

1
nj aDj+ nk kj aDk

)

aXkj
aDk

+1

!

;

(13)

and the mass of …rms in each country can be calculated using (10), (11), and (12) together withL

the fact that Bj = P 1 j :jnj =1) Lj (1FD (Lj )(1((Lj )) Lk (Lj ) (1
(Lk )):(Lj ) (Lk )) (1(Lj ))(14)

Welfare may be measured by indirect utility, which is proportional to the real wage

wj:
p1Aj Pju

Since pAj = wj = 1 8j, it su¢ ces to examine Pj . Using (11), (12), (13), and (14), we have
Pj =

Lj FD

1

(Lj )FE (

1)

1
1

(Lk )
(Lj ) (Lk )

1
(

1)

:

(15)

This expression shows that, as in the Melitz (2003) model, welfare always increases (P decreases)
with trade liberalization; that is, with a higher

2.3

FX (Lk )jk or a lower FD (Lj ) .

Parametrisation of the beachhead cost

In the following, we parametrise the beachhead costs as:
FeD (Lj )fD + (Lj ) ;FeX (Lk )fX + (Lk ) ;

> 0:

(16)

The variable component of the beachhead cost increases in market size, while the constant term
picks up costs that are independent of market size. It is quite natural that the beachhead cost
would have one …xed and one variable component. The constant f represents the …xed costof standardizing a product for a particular market or the cost of producing an advertisement
tailored to a particular market with its culture and language. The variable cost term Lj
represents the fact that the cost of spreading an advertising message increases with the numberof consumers in a market. For instance, the number of free product samples or advertisingposters increases with the size of the population. Likewise, the cost of television advertising
increases with the number of viewers. We do not put any restriction on the shape of the variable
cost term except

> 0:7

6

The corresponding condition in Melitz (2003) is that FX > FD :7

A simpler alternative would be a multiplicative formulation: FeX (Lj ) = fX

Lj

: Some of our results could

be derived with this speci…cation but, contrary to our speci…cation, it would e.g. imply that trade shares are

8

3

Results

A large number of comparative static results may be derived. Here, we focus on the more novel
aspects of our model, which are related to the e¤ects of market size. From now on, the simpli…ed
notation FDj

3.1

FD (Lj ), FXj

FX (Lj ); and(Lj ) is adopted:j

Productivity

The …rst set of results concerns the productivity of exporters and non-exporters in the twocountries. From (6) and (7)aDj 1 =Bj;
FDj

aXjk1 =

Bk
:
FXk

(17)

A larger market, measured by Lj ; a¤ects the cuto¤s via two channels: First, it increases thedemand facing each …rm (via Bj and Bk , respectively) and, second, it increases the market size
dependent beachhead costs. However, note that the demand increase experienced by a …rm isdampened by the entry of new …rms.
We …rst turn to the in‡uence of domestic market size. The e¤ect of a larger home market
on non-exporters is@aDj< 0 for
@Lj

< 1;

(18)

as shown in appendix 6.3. The negative signs imply that the higher beachhead cost due to alarger market dominates the e¤ect of higher demand, so that the marginal …rm must be moreproductive in a larger market.
Next, from (17)
@aXjk
< 0;
@Lj

(19)

k
since @B
@Lj < 0. A larger mass of domestic exporters implies stronger competition in the foreign

market and the marginal exporter must consequently be more productive. The e¤ects of domestic market size on the productivity of exporters and non-exporters are summarized in Result
1a.
Result 1a: The average productivity of exporters as well as non-exporters increases in thesize of the domestic market as long as< 1.

The aggregate productivity can be expressed as8 :0'j = @sDj

aDj
Z

a1

dG(a jaDj ) + sXjk

00invariant to country size.
8

aZXjk

See Melitz (2003).

9

a1

1 11

dG(a jaDj )A

;

(20)

where sDj is the share of home producers that sells domestically only and sXjk is the sharethat exports to country k. Using that the ratio of exporters to non-exporters is
1

sDj =
1+

aXjk
aDj

aXjk
aDj

; and sXjk =
1+

'j =

aXjk
aDj

1

1

+1

0

aXjk
aDj

B1 +
@
1+

1 11

2 +1

C
A

aXjk
aDj

:

(21)

From (21), it is seen that average productivity increases in Lj since from (18)
@

(25)

aXjk
aDj

@Lj

> 0, and;, we get:

1

aDj

aXjk
aDj

@aDj
@L

< 0, from+ 1 > 0. It is therefore also the case that aggregate productivity

in manufacturing increases in country size.
Result 1b: Aggregate productivity in manufacturing increases in country size.
Next, the e¤ect of the foreign market size on non-exporters is
@aDj
< 0;
@Lk

(22)

@B

from (17), since @Lkj < 0 by inspection of (10). The intuition is that a larger foreign market
implies a larger mass of foreign …rms competing in the home market, which decreases the market
shares of domestic non-exporters.
The e¤ect of foreign market size on the productivity of the marginal exporter is generally
U-shaped, as shown in appendix 6.4:
@aXjk
@Lk
@aXjk
@Lk
where

k

> 0 forj< 0 for

j

1

(
k

1)

(

(

1) k ) >

2

1

(
k

1)

(

(

1) k ) <

2

(23)
;

FX (Lk )FD (Lk ) > 1 measures the relative market access (the relative beachhead cost) of

foreign versus domestic …rms. The left-hand side of the inequality, determining the sign of thederivative, increases ink as is easily shown.

Note also that

lim

Lk !1

k = 1; since the …xed

cost of the market entry cost becomes irrelevant in the limit. aXjk therefore decreases in theforeign market size for large enough Lk ;, simply because …rms need to be more productive toovercome the higher (variable) entry cost in a larger market. In the opposite case, when theforeign market is quite small, the …xed market entry cost fX becomes important. Increasing theforeign market size implies that this cost can be spread over a larger number of units so that the
cut-o¤ productivity for entering the foreign market deceases in the market size. The two cases
are illustrated in Figure 1, which uses (12) to plot aXjk against Lk for some standard parametervalues ( = 0:8; = 3;

= 1; f x = 2; fD = 1; FE = 1; Lj = 2): The …gure also illustrates how alarge foreign market is particularly valuable to the exporter when the mark-up is high, which
implies that aXjk tends to increase in the foreign market size when10

is low.

σ=1.8

σ=1.5

σ=2.5

Figure 1: Exporter cut-o¤ (inverse productivity) in relation to foreign market size ( = 0:8; =3;= 1; f x = 2; fD = 1; FE = 1; Lj = 2):

Thus, the model encompasses both the case when the marginal exporter needs to be lessproductive in larger markets as in Arkolakis (2006), and the case in Melitz and Ottaviano (2005)
where the opposite holds. Result 2 summarizes the e¤ects of foreign market size.
Result 2: The average productivity of non-exporters increases in the size of the foreign
market. The average productivity of exporters is generally U-shaped in the foreign market size.
Next using (11) and (12), the relative cut-o¤ productivity for non-exporters and exporters
in the home country is

aDj
aXjk

=

FXk
FDj k

1
1

k

> 1;

f or

j

FXj
j

> FDk 8 j; k; andj;k < 1:(24)

There is strong empirical support for exporters being more productive than domestic …rms, and11

as mentioned above, we follow Melitz (2003) by making parameter assumptions for this to hold:
FXj> FDk :

jFinally, market size is also of importance for the relative productivity of exporters as compared to non-exporters:
aDj
aXjk

@

@Lj

< 0 f orj;k < 1;(25)

as shown in appendix 6.6. The larger is the home country, the less productive are exporters ascompared to non-exporters. Essentially, the higher …xed cost associated with the larger homemarket will push up the relative productivity of domestic …rms, which makes exporters lookless productive in comparison.
Result 3: Exporters are more productive than producers for the domestic market. However,this e¤ ect decreases in the size of the home country.

3.2

Trade shares

The next set of results concerns the relationship between country size and manufacturing export
share. A home exporting …rm with the marginal cost a; sells a1

jk Bk in the foreign market.

Using (7), the total export volume from home is

VXjk =aZXjk

a1

dG(a j aDj )

0

FXk
=
a1Xjk

aXjk
aDj

1

FXk nj :(26)

Similarly, the total production volume for the home market is

VDj =

aDj
Z

a1

FDj
=
1
aDj

dG(a j aDj )01

FDj nj :(27)

The export share may now be written as
SXjk =VXjk
k (1=VXjk + VDj
1

j)
j

:

(28)

k

Di¤erentiating with respect to country size gives
@SXjk
k(
=
@Lj
(1
@SXjk
1
=
@Lk
(1

1) @ j< 0;2k ) @Lj

k
j

@

k

2 @L

k

j
j

k)

> 0;

(29)

(30)

which implies that a smaller country has a higher manufacturing export share than a larger
one.
Result 4a: The manufacturing export share of a country decreases in its own size, andincreases in the trade partner’s size.
12

Next, note that for fX = fD and the symmetric iceberg trade cost ( jk =

kj ),

j =

k =

: This means, from (28), that SXjk = SXkj ; i.e., manufacturing export shares converge as
fX approaches fD : Moreover, since a falling fX makes exporting easier, export shares converge
upwards.
Result 4b: Falling relative beachhead costs ( fX converging to fD ) imply (upwards) converging
manufacturing export shares.
The intuition for Result 4a and Result 4b derives from the fact that, when selling theirproduct, …rms have to pay two di¤erent sunk costs: a standardization cost that is independentof market size and a marketing cost which depends on the size of the market re‡ecting thehigher cost of reaching more consumers. Also the standardization for a particular market ismore costly for an exporter than for a domestic producer. Since the cost of standardization is
independent of market size, it becomes relatively less important as compared to the marketing
cost when the market is large. The di¤erence in …xed costs between foreign and domestic …rms
is therefore relatively smaller in a large market.
For example, suppose that Sweden and the United States have similar levels of regulation
but di¤erent tastes in the design of labels, packages and instructions. Then, the cost of standardization is similar for an American …rm targeting the Swedish market and for a Swedish…rm targeting the American market. However, the market size dependent marketing cost ismuch higher for …rms selling in the US as compared to those selling in Sweden. The di¤erencein …xed costs for Swedish exporters and American domestic producers, both serving the same
market, is therefore smaller in relative terms than the di¤erence between American exporters
and Swedish domestic producers. Consequently, the smaller country, i.e. Sweden, has a larger
share of manufacturing exports in its production.
Second, since Swedish exporters are more concerned with the larger marketing costs than thestandardization costs, as compared to American exporters to Sweden, it must be the case thatthe decrease in standardization costs for foreign markets (fX approaching fD ) a¤ects American…rms more than Swedish …rms. This means that American …rms will increase their exports at
a greater pace than Swedish …rms and therefore, they will start catching up with their Swedish
counterparts. In the aggregate, the American export share of manufacturing production willthen approach the (larger) Swedish export share and export shares converge across countries.
In the extreme, when the cost of standardization is the same for the domestic and the foreign
market (fX = fD ), export shares converge completely across countries.
It may be useful to compare our results to the standard models. Here, we use the Melitz(2003) model with a homogenous good and freely traded A-sector à la Helpman et al. (2004).
It is easily shown that the manufacturing export shares are independent of country size in thismodel without our assumption of a market size dependent beachhead cost. However, our result
may also be compared to the standard Dixit-Stiglitz trade model without a homogenous good
A-sector (see e.g. Helpman (1987)). Like our model, trade shares are negatively related tomarket size in that model. However, in contrast to our model, manufacturing trade shares

13

diverge as trade costs fall: trade shares increase from zero in autarky to the share of the foreign
market in total demand at free trade.9 As argued below, we believe our prediction of converging
manufacturing export shares to be supported by empirical evidence.

3.3

The gravity equation

Our modi…cation of the Melitz model has implications for the gravity equation. Consider asetting without the A-sector and with m countries indexed by j and k: The factory gate price1 ai wj ; where wj is the factor price in country j: The export from jof each …rm is now pij =to k isEjk = nj
where Vjk

aXjk
R
0

a1i

1

jk wj

Vjk Yk ;

1 Pk

(31)

dG(a j aDj ): The GDP of country j consists of the sum of sales in all

markets including itself:
Yj =

m
X

Ejk =

k=1

m
X

jk wj

nj

1

Vjk Yk :

1 Pk

k=1

(32)

Following the methodology of Anderson and van Wincoop (2003), a gravity equation of trade1

1 wj

may be derived by solving (32) for nj

and substituting the solution into (31). This

gives:
Yj Yk
Ejk = wY
where sk

1
jk

Pm

Pk1

j=1

Pm

Yk
w
Y w ; and Y

j=1 Yj :

Vjk

;

1
jk
Pk1

(33)

sk VjkSolving the integral for Vjk gives
Vjk =Using this and the fact that a1Xjk =gravity equation

Ejk =

Yj Yk
Yw

aXjk
aDj

+1
FXk Pk1
1
jk Yk

Pk

(34)

from the cut-o¤ conditions gives the following

k=1

1

Yk
FXk

jk

Pm

1
aXjk
:

jk

Pk

sk

1

Yk
FXk

1

1

:

(35)

The expression may be compared to the standard Anderson and van Wincoop gravity formulaY Y

tion: Ejk = Yj wk
9

Pk

Pm

+1

1
jk

k=1

jk
Pk

1

: The di¤erence is essentially the term
sk

Yj
FXj

Naturally, in this model there is no beachhead cost that can be a¤ected by trade liberalization.

14

1; whichmultiplies the frictional trade cost.10 This term measures the importance of the exporters’market entry cost in relation to the size of the export market, that is, trade resistance related
to the market entry cost. Note also that a higher

decreases the negative impact of FXj on

trade, as pointed out by Chaney (2008).
Finally, substituting (16) into (35) gives

Yj Yk
Ejk = wY

Yk
fX +Lk

jk

Pk

Pm

k=1

jk

Pk

sk

1

1

Yk
fX +Lk

1

1

:

(36)

This shows that our speci…cation gives a justi…cation for the common usage of GDP per capita
as explanatory variable in the gravity regression. GDP per capita enters because a higher
purchasing power per head makes it worth more to …rms to pay the entry cost, which increases
by heads. However, since the entry cost increases non-linearly in the size of the population, itkrather than exactly GDP per capita that should enter the gravity regression.
is f Y+L
X

4k

Empirical Section

We have derived a set of new results. We will here explicitly test only a few of them. For
the other results we will discuss to what extent they seem consistent with existing empiricalevidence. One obvious limitation is that several of our results ideally should be tested using across country …rm level data set, and such a data set is not yet available.
Our …rst result that …rms on average are more productive in large markets is consistent
with several empirical studies showing that workers and …rms on average are more productive
in larger markets (Head and Mayer 2004, Redding and Venables 2004, Syverson 2004, 2006, andAmiti and Cameron 2007), and we do not here perform any independent test of this. Likewise,
Result 2, that the productivity of exporters is U-shaped in the size of the export market isconsistent with both Arkolakis (2006) indicating that the productivity of exporters decreases inthe export market size, and with the opposite result in Melitz and Ottaviano (2005). Concerning
Result 3, that country size a¤ects the relative performance of exporters and non-exporters, weturn to stylized evidence in the following section. Results 4a,b are explicitly tested in section
4.2. Finally our result that GDP/capita enters the gravity equation is consistent with a numberof empirical studies.
10

A further di¤erence as compared to the standard gravity equation is that our expression for multilateral
m
+1
X
k
1
jk
k
resistance, namely
sk FYXk; does not equal Pj in our set-up. The reason for this is that the
Pk
k=1

exporters’market entry cost FXk depends on population size and is therefore not symmetric when there is trade
between two countries of di¤erent size.

15

Population and exporter wage premia-4log(wage premium)
-3
-2
-1

0

1

As in Schank, Schnabel & Wagner (wp 2006).

7

8

9
10
log(population)

11

12

Source: Schank, Schnabel & Wagner (wp 2006) and Penn World T ables.

Figure 2: Export premiums decrease in country size.

4.1

Empirical evidence of Result 3

We turn to stylized evidence to evaluate Result 3 that country size a¤ects the relative performance of exporters and non-exporters. A study by Schank et al. (2007) o¤ers a literature overview where they measure the wage premium of exporting …rms as compared to nonexporting …rms. Typically, a regression is run on …rm level data with some measure of wages as
the dependent variable, and a dummy variable indicating whether the …rm is an exporter. The
estimated coe¢ cient for this dummy variable is the exporter wage premium as compared to thatof non-exporters. We interpret this wage premium to indicate productivity di¤erences between
exporters and non-exporters.11 Figure 2 plots the exporter wage premium versus market size
(population) of countries in the studies surveyed in the appendix of Schank et al. (2007).12 We
have also added an observation for Sweden using data provided by Statistics Sweden. Naturally,
it must be acknowledged that all regressions are not done with exactly the same methodology
or fully comparable data. Nevertheless, Figure 2 shows a negative correlation between export
premium and population size. Running a regression on this data gives a slope oft-value of
11

0:58 with a

2:93.

This interpretation is consistent with e.g. learning e¤ects as in Malchow-Møller et al. (2007) or by a non-competitive wage setting à la Shapiro and Stiglitz (1984).
12

We use population to measure market size since it most closely corresponds to our model speci…cation.

However, the pattern of exporter wage premia to market size is the same, if GDP is instead used to measuremarket size.

16

4.2

Empirical evidence of Result 4a,b

In this section, we empirically test predictions of our model related to the e¤ects of market size on
trade shares. First, we check that our dataset has the well known property that manufacturingexport shares are negatively correlated with country size, as predicted by Result 4a. Even
though this is well known theoretical result, it does not typically apply in this type of model.
We employ sector level data within the OECD using the STAN database with yearly observations
from 1980 to 2003, and run the simple regression
sist =
where sist

log

X
Y ist

; lit

0+

1 lit + "ist ,(37)

log Lit : The regression is run at the sectorial level. Table 1 shows

the regression of export shares over GDP on a sectorial level in 2001. The regression includes…xed e¤ects for sectors. The coe¢ cient for population, which can be interpreted as a standard
elasticity, is highly signi…cant and of the expected sign.
Year 2001
X
Y

Dependent variable

(1)
lit

-0.15
(0.024)

Sector dummies

Yes

Observations

595

R squared

0.43

The table reports the estimates of the regression ofexport shares on country size controlling for sectorspeci…c …xed e¤ects. The export share is at thesectoral level and population at the country level.
Note: Standard errors in parentheses.
at 10%,

signi…cant at 5%,

signi…cantsigni…cant at 1%.

Table 1: Export Shares and Country Size
Next Result 4b states that the export share of the manufacturing sector across countries
converges as the …xed component of the exporting beachhead cost, fX , approaches the value forthe …xed component of the domestic beachhead cost, fD . Given that this has been happening17

over time, we should observe converging manufacturing trade shares over time. The assumptionthat the relative access cost to foreign markets, as compared to that of the domestic market,
has been falling over time is very much in line with the often cited e¤ect of globalization making
the world more alike. A concrete example supporting this assumption is the process of productstandardization and removal of non-tari¤ barriers to trade within the European Union during thelast 20-30 years. GATT and WTO negotiations have also aimed at reducing nontari¤ barriers totrade during this period. Finally, the rapid improvement of telecommunications, including theinternet, simpli…es business contacts and information gathering about foreign markets, which
may be interpreted as a fall in fX :We look at the evolution of manufacturing export shares over time, at a sectorial level within
the OECD using the STAN database with yearly observations from 1980 to 2003.13 Accepting
the assumption that the process of falling access costs to foreign markets has occurred gradually
over time during the period investigated, we should observe converging manufacturing export
shares. First, we graphically explore the data on manufacturing aggregated to the country
level. We want a balanced panel so that we only include country and sector pairs that havenonmissing data throughout the period 1980 to 2000, before we aggregate to the country level.
For this period, there are data on most sectors for most countries throughout. The appendix
contains a list of the 18 countries that are included. First, Figure 3 plots the evolution of thecoe¢ cient of variation of the distribution of trade shares (exports divided by output) for the
sample of countries. We use the coe¢ cient of variation since it is neutral to scale. The graphgives an indication that the average dispersion of trade shares in manufacturing across countries
decreases throughout the period. This result is driven by the fact that the mean grows more
rapidly over time than the standard deviation. Figure 4 plots histograms of country level tradeshares for …ve equally spaced years in the period. It can be seen that the mean increases whilea change in the absolute level of dispersion is more di¢ cult to detect.
Next, we proceed to use sectorial data to analyse trade share convergence using a regression
framework from the standard empirical growth literature (see e.g. Barro and Sala-i-Martin(1991), Barro and Sala-i-Martin (1992), Bernard and Jones (1996) and Mankiw et al. (1992)).
We use the initial value of the manufacturing export share for which we have data and regress
the average growth rate in export shares, fsis ; on the initial level of trade shares, where theaverage growth rates are computed as the coe¢ cient on the trend dummy in a regression oflogged values on a constant and linear trend, see e.g. Bernard and Jones (1996):
fsis =

0+

1 sis0 +2 Ds + "is .

The errors are clustered at the country level and sector dummies are included. The model
predicts that

1 should be negative since the higher was the initial level, the lower would be

the average change over time if convergence held. In Table 2, it is shown that the growth rate13

We include all manufacturing sectors except those related to the extraction of raw materials since we do notbelieve these to be a¤ected by the dynamics described in this paper.

18

Coefficient of variation in manufacturing export shares

Coefficient of variation for export/gdp
.42
.44
.46
.48

.5

1980-2000

1980

1985

1990
year

1995

2000

Figure 3: The coe¢ cient of variation for country level export shares. Source: OECD STAN.

Distribution of export shares across time3
Frequency1
2

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
export/gdp

1995

2000

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
export/gdp

Frequency1200Frequency
12

3

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
export/gdp

3

1990

0

Frequency
1
2

3

1985

0

0

Frequency
1
2

3

1980

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
export/gdp

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
export/gdp

Figure 4: The mean country level export share. Source: OECD STAN Industrial Database.

19

of export shares depends negatively on the initial level in 1980, thereby suggesting convergencewithin the OECD at the sectorial level.
Years 1980 to 2002]
sist

Dependent variable

si;1980-0.020(0.002)

Sector dummies

Yes

Observations

406

R squared

0.37

The table reports the estimates from the convergence regression of theaverage change of trade shares at the sectoral level on the initiallevels of trade shares. Sector-speci…c e¤ects are controlled for by
sectoral …xed e¤ets.
Note: Standard errors in parentheses. Errors are clustered at thecountry level.

signi…cant at 10%

signi…cant at 5%

signi…cant at 1%.
Table 2: Convergence (Initial Values)

5

Conclusion

This paper has explicitly modelled a market size dependent market access or beachhead costin the heterogenous …rms and trade model of Melitz (2003). We model this cost as having one
variable component that increases in market size, and one …xed component. The …xed component could e.g. be interpreted as the cost of standardizing a product for a particular market,
while the variable cost term e.g. represents the fact that the advertising cost of introducing a
new product increases with the size of the market (the number of consumers).
In essence we make a relatively small change to the framework of Helpman et al. (2004),
which preserves the analytical solvability of the model. Nevertheless it leads to a number of new
results. The productivity of non-exporting as well as exporting …rms will depend on market size,
as will manufacturing export shares. In particular, we show that …rms based in a large market
are more productive than …rms in a smaller market, due to the higher …xed costs related toestablish a brand in these markets. The relationship between exporter productivity and export
market size is U-shaped, however. This is because a larger market means that the …xed entry
20

cost (e.g. standardization) can be spread over more units, while the variable entry cost (cost of
establishing a new brand) increases in the market size. Therefore, for su¢ ciently small export
markets, the productivity of exporters decreases in the export market size, which is consistent
with the feature of the …rm level data in Eaton et al. (2005) that the number of exporting…rms increases in the size of the export market. For su¢ ciently large exports markets, on thecontrary, exporter productivity increases in the export market size in accordance with Melitz
and Ottaviano (2005). Furthermore we …nd that, as in the standard model, exporters are moreproductive than non-exporters, but this productivity premium decreases in the size of the homecountry, which is consistent with the stylized facts presented in the paper. Fourth, we show thatthe manufacturing export share of a country decreases in its own size, and increases in the sizeof the trade partner. This e¤ect decreases as markets are integrated (in the sense that the …xed
beachhead cost of foreign markets declines). Accepting that market access costs into foreign
markets have been falling over time as a result of globalization, the model predicts converging
manufacturing export shares over time. This result is supported by the empirical section of thepaper, where this hypothesis is tested using sector level OECD data from 1980 to 2003. Finally,
we derive a gravity equation from our set-up. This equation contains a measure of GDP per
capita, which implies that our set-up gives a theoretical rationale for the common practise ofincluding this variable in gravity regressions of trade ‡ows.

21

References
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Puzzle, journal=American Economic Review, volume=93, number=1, pages=170-192,.
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Aw, B., Chung, S. and Roberts, M.: 2000, Productivity and Turnover in the Export Market:
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1992, Convergence., Journal of Political Economy

100(2), 223–251.
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Trade, Working Paper .
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22

Eaton, J., Kortum, S. and Kramarz, F.: 2004, Dissecting Trade: Firms, Industries, and Export
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236

Appendix

6.1

The n-country case

The set of equations, using (6), (7) and (8) for n countries, that produce the solutions for thevector B(B1 ; .....; Bn ) for a model with n countries can be generalized intoX
1
Bi FD (Li )1 +
= (
1) FE
ij Bj FX (Lj )
j6=i

n
X

1
ij Bj FD (Lj )

= (

1) FE

8i 2 f1; .....; ng ;i

Bi FD (Li )1

j=1

where

FX (Lj )ij FD (Lj ) andij =ii = 1.

If we de…ne(38), the equations generalize

to:
n
X

ij j = (1;1

1;2::: :::

2;1

:

1) FE

8i 2 f1; .....; ng ,j=1

or, in a matrix form:
2
6
6
6
6
6
6
6
6
6
4

or

1;n 1

1;n

72
7
7
76
:
76
74
7
:
7
7
n 1;n 5
2;n

:

:

:

:
:

n 1;1
n;1

::: :::

n;2

n;n 1

vector

1

3
7
7

2 5=
3

(

1) FE;(40)n;nn n n 1 = n 1(

where

3

(39)

1) FE

(41)

consists of entries where the element on the ith row and the jth column is
1;

2 ; ..... ;

n 1;

0

n

and

ji ,

is a

is an n dimensional vector of ones. Inverting then nmatrix yields the Bj which, in turn, determines the cut-o¤ productivity levels.
The case with three countries would then look as follows
2
32
3
1
12
13
1
7
6
76
7
6 21 176
23 5 4 2 5 = (431

The determinant ofij

system for
2
6
6
4

ij

2

7
1 6
6
7
2 5=
j j4
3

3

12

23

31 +

13

21

32

(

12 +

13 +

23 ) ;(43)

ji is the product of the bilateral trade costs between i and j.

gives:
3

1

(42)

is

j j=1+
where

1

32

1) FE .

1

23

13

23

31

21

1

21

32

31

12

32

12
13

31

24

32

3

12

23

13

13

21

23 5

1

12

7
7 (

Solving the1) FE ,

(44)

which gives the following solutions for market size per …rm, Bi :
2

3

B1

7
6
6 B2 7 =
5
4
B3

(
1+
2

12

23

1

6
6 1
4
1
aDj
aDk

@
@Lk

6.2

> 0 for

31 +

1) FE

13

21

(

32

12 +

13 +

1)

23 +

13 ( 321) +12 ( 2313 +

23 ( 311) +21 ( 1312 +

32 ( 211) +31 ( 12

j;

(45)

23 )

32

1

FD (L1 )76
61) 7
5 4 FD (L2 )

7
5 .

1 7
1

FD (L3 )

1)

3T

k < 1

Proof:
From (??)
aDj
aDk

=

FDk
FDj

1
1

1

k

k

.

j

Di¤erentiating w.r.t. Lk gives:
@
@Lk

FDk
FDj

1
1

k

=

j

Lk 1
FDj (1
j)

1

(

1)

1

1

k

.

1

k

(46)

The sign of the derivative depends on the sign of the term:
1

1

(

k

1)

k

1

1

k

.

1

(47)

The …rst- and second-order conditions for a minimum of this term w.r.t.
1

@
@ k
@2
@2 k

k

1+(

1) 1

k

1+(

1) 1

@
into (46) gives @L
k

6.3

k

1

=

1

=

1

aDj
aDk

k

1
1

= 1 (sincek= 1 ()

1

1

k

1

1

k

1

The minimum is, thus, given by
j;

1
1

k

(Lk ) are:

1

1

=0
> 0:

= 1). Substituting@= 0. Consequently, it must be the case that @L
k

aDj
aDk

(48)
k

= 1

> 0 fork < 1.
@aDj<0@Lj

From (6.2), we have that
@

aDj
aDk

@Lk

=

@aDj 1@Lk aDk@a

Dk
Since from (11) @LDj
< 0; (49) holds i¤ @a@Lk < 0:k

25

aDj @aDk> 0:(aDk )2 @Lk(49)

@aXjk

6.4@Lk

From (12)aXjk =@aXjk@Lk

The sign of(1) k FE
FXk

(1
1

j)
j

=(

1) FE (1

j)

k

1
1

FXk ( k

j)

:

(50)

is therefore determined by the sign of=1@[FXk (@Lkk@[F F 1@Lk Xk Dk

j )]

(51)
FXk

j ]:

7

0

(52)

Now

@
[F F 1@Lk Xk Dk
FXk
FDk

FXk

,

FDk
FXk

(

aDj
aDk

> 1 i¤ Lk > Lj for

j;

7

1)

FXk
(
FDk

6.5

j]

j

,
7

1)

j

k:

k < 1

Proof:First
Lj = Lk ()j;That Lj = Lk ()

aDj
aDk

aDj
aDk

> 1 for

=
j;

k

FDk
FDj

1
1

k

= 1:

j

@
< 1 now follows from @L
k

aDj
aDk

> 0 fork < 1:

@

6.6

aDj
aXjk

@Lj

<0

f or

j < 1.

Proof:
@
@Lj

aDj
aXjk

= Lj

1 FXk (1
2
FDj
k (1

0
)
k
@(
j)

The sign of (53) will depend on the sign of the term:

26

j

1)

1

FDj
FXk

(1

j)

1

1A .

(53)

j

The F.O.C. when maximising
(

1)

j

(1

1

FDj
FXj

0
@

(

1)
(1

is:

(

2
j

1)

FDj
FXj

2
j)

j = 1. Furthermore,

1

1A :

= 0 () 1j ( j = 0) =

(54)

j

(1

j)

1 and

= 0:

(55)

lim

j = 0:(Lj )!1

j 2 [0; 1):

d
dLj

6.7

1

(1

So the only stationary point is

FDj
FXj

j)

j w.r.t.

j)

Therefore, it follows that for

1

j

aDj
aXjk

< 0.

Countries included in Figure 4.

The following countries are included in Figure 4. This is a subset of the full STAN sample but
it is the only set of countries for which there is data for the full length of 1970 until 2002.
Austria
Belgium
CanadaDenmark
FinlandFranceIceland
Ireland
Italy
Japan
Netherlands
New Zealand
NorwayPortugal
Spain
SwedenUnited Kingdom
United States

27