Global COE Hi-Stat Discussion Paper Series 056

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Firm Heterogeneity andCountry Size Dependent Market Entry Costs

Hi-Stat Discussion Paper

Anders Akerman
Rikard Forslid

April 2009

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Firm Heterogeneity and Country Size Dependent Market Entry
Costs
Anders Akermanyand Rikard Forslidz

February 2009

Abstract
This paper introduces a market size dependent …rm entry cost into 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 productivity ofexporters is U-shaped with reference to export market size. Third, the productivity premium(the di¤erence in average productivity) between exporters and non-exporters decreases in
the home country size. Fourth, we derive a set of new results related to trade volume. It
is shown that when the …xed entry cost of exporting declines, for instance as the result of
economic integration, export shares converge. This prognosis is supported by the empiricalsection of the paper. Fifth, we use a multicountry version of our model to derive 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.

JEL Classi…cation: D21, F12, F15
Keywords: heterogenous …rms, market size, market entry 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,
We are grateful for comments from Pol Antras, Karolina Ekholm, Thierry Mayer, Marc Melitz, Jim Markusen,
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 fromJan Wallander’s and Tom Hedelius’Research Foundation is gratefully acknowledged by Akerman.
yResearch Institute of Industrial Economics, email: anders.akerman@ifn.se.

z

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

1

Other studies include Aw, Chung, and Roberts (2000), Bernard and Jensen (1995, 1999a, 1999b) Clerides,

Lach, and Tybout (1998) as well as Eaton, Kortum, and Kramarz (2004).

1

larger, and endure 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 heterogeneous …rms explains these …ndings by either iceberg
trade costs associated with exports (Bernard, Eaton, Jensen and Kortum, 2003), or higher …xed
costs associated with market entry into a foreign market (Melitz, 2003; Yeaple, 2005). Only themost productive …rms will …nd it pro…table to pay the additional cost necessary for exports,
and export …rms will thus 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, Redding, and Schott (2007) show how comparative advantage may
strengthen the productivity gains associated with trade in a Melitz (2003) type model withheterogenous …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 investigates how an economy with …rm heterogeneity, as in Melitz (2003), isa¤ected when market entry costs, that …rms must pay when entering a new market, increasein country size. We are thinking here of, for example, consumer brands. It is considerably
more costly to establish a new brand of toothpaste in the U.S. than in a small country such as
Sweden. This is because the cost of television advertising is based on the number of viewers,
free sampling etc. which makes it more costly to enter a larger market.2 This basic idea isalso supported by Eaton, Kortum, and Kramarz (2005) who …nd a strong positive relationshipbetween country size and entry cost when calibrating their model to French …rms as shown in
Figure 1.
An immediate implication of the concept of market entry cost increasing in market sizeis that …rms on average should be more productive in larger markets due to …rm selection.
This is consistent with empirical evidence showing that workers and …rms on average are moreproductive in larger markets (Head and Mayer, 2004; Redding and Venables, 2004; Syverson,
2004, 2006; and Amiti and Cameron, 2007).3 We model the market entry cost as having a …xed
and a market size dependent component. The …xed part re‡ects costs such as standardizationof the product for a particular market, or creating a marketing message (e.g. a television ad)
for this market. The market size dependent component of the entry cost is the marketing costof introducing a new variety in a market. The notion that this cost depends on the size of themarket is normally taken for granted in the marketing literature; the marketing cost over sales
ratio is often a key variable.4
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 with the number of consumers in a market.
3

An alternative and more common explanation for these productivity di¤erences is externalities associatedwith agglomeration. Some preliminary evidence using plant level data for French cities indicate, however, thatboth …rm selection and agglomeration may play a role, see Combes, Duranton, Gobillon, Puga, and Roux (2008).
4

See e.g. Buzzell, Bradley, and Sultan (1975).

2Figure 1: As in Eaton, Kortum, and Kramarz (2005).

This paper thus generalises the Helpman, Melitz, and Yeaple (2004) (HMY) version of theMelitz (2003) model, while preserving the analytical solvability of the model. Nevertheless, ouranalysis yields several new results that are supported by data. First, the average productivityof …rms located in a market increases in the size of the market. Second, the productivity ofexporters exhibits a U-shaped relation to export market size. Exporting to a small market isdi¢ cult because the …xed entry cost (e.g. standardization) has to be spread over few units.
This problem initially 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¢ cult to enter the market as an exporter. Third,
the productivity premium between exporters and non-exporters decreases in 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 thatthe manufacturing export share decreases in the size of the exporting country. Moreover, it
is shown that NTB liberalisation, modelled as a decline in the …xed part of the market entry
cost, causes export shares to 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 practice of introducing GDP per
capita as an additional explanatory variable in gravity regressions of trade ‡ows.
3

We confront our main theoretical scheme with data in various ways. A limiting factor isthat our model produces cross-country predictions on the …rm level, and such a data set is not
yet available. The …rst result, as mentioned earlier, is in line with existing empirical evidenceof …rms being more productive in larger markets, and we do not perform any independent test
to verify this point. The second result is that the productivity of exporters is U-shaped withrespect to export market size; this implies that our model in principle is consistent with exporterproductivity decreasing in market size, as in Arkolakis (2007) as well as with the opposite casein Melitz and Ottaviano (2005). Again we do not perform any independent veri…cation. We
document the third result that the exporter productivity premium is larger in a small market
with some cross-country evidence on wage premia in the export sector in the empirical sectionof this paper. The empirical section explicitly tests result four, showing that there is evidence of
manufacturing export shares converging over time. Finally, a large number of empirical papers
on the gravity model support result …ve; that GDP per capita should be included the gravity
equation.
Our analysis relates to that of Melitz and Ottaviano (2005) who introduce …rm heterogeneity
in the model by Ottaviano, Tabuchi, and Thisse (2002) with a linear demand system and where
the endogenous mark-ups of monopolistically competitive …rms depend on market size. Melitz
and Ottaviano (2005) …nd that …rms selling in large markets are larger and more productive,
since higher competition forces the mark-ups in a large market downwards. The same reasoning
holds in our model 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 cost associated with a larger market. For exporters, our model instead yields a U-shaped
relationship between foreign market size and exporter productivity, as discussed above. A
further di¤erence is that in our model the productivity of …rms in a market also depends on
the size of other markets. For example, a larger foreign market implies more competition fromimports, which forces up the productivity of domestic …rms. One consequence of this dependence
on foreign market size is that export shares will vary with the market size. Finally, our resultthat 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.
Our paper is also related to Arkolakis (2007) that introduces a formal model of advertising ina model of heterogeneous …rms where the market penetration cost of each …rm is endogenous.
The probability of a marketing message reaching at least one consumer increases with the
population in this model. Thus, a …rm gets a relatively larger payo¤ for a small investment inmarketing when exporting to larger countries. The marginal exporter that is just su¢ ciently
productive to export will therefore prefer to export to a large rather than to a small market. Thisis consistent with the pattern evident in the …rm level data; e.g. Eaton, Kortum, and Kramarz(2005) show that many small …rms typically export a small amount to large markets. Ourmodel 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

4enough export markets the variable entry cost (advertising) dominates in our model, implyingthat exporter productivity instead has to increase in the foreign market size. We believe that our
set-up complements Arkolakis (2007). Clearly, for some products it is more likely to …nd a buyer
in a large market as modelled by Arkolakis (2007). Other more standardized products, such as
consumer brands, are certainly more costly to establish in large markets. Apart from the results
on exporter productivity, our speci…cation di¤ers by the results on trade share convergence thatare supported by data in the empirical section. Finally, because our set-up is simpler than thatof Arkolakis (2007), we can solve our model 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, Melitz, and Yeaple (2004) version of Melitz’ (2003)
monopolistic competition 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
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
;

5

(1)

where2 (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.

! 1
1

p1ij

di

The 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
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:
1
aDj
Bj = FD (Lj );(6)

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

6

where FD (Lj )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

1

a

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 to j
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 ,1) (1FE FD 1 (Lj ) (1(Lj ) (Lk )

Bj =
where

1 > 1, and

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 =aXjk =((1) FE
FD (Lj )(111) (Lk )FE
FX (Lk )(Lk ))(Lj ) (Lk )
(1

1

;

(Lj ))(Lj ) (Lk )

(11)

:

(12)

From these, it is seen that, contrary to the standard model by Melitz (2003), the market size
(Lk )will a¤ect the cut-o¤ marginal costs. We will assume that FX(L
> FD (Lj ) for all j; k:6 Thisk)assumption implies that aXjk < aDj 8j; k.
The price indices may be written as

5This assumption is consistent with the empirical …ndings by e.g. Axtell (2001) or Luttmer (2007).

6

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

7

Pj1

=

1

1
nj aDj(1 )+ 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 in the number ofconsumers in a market. For instance, the number of free product samples or advertising posters
increases in 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
except3> 0:7

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
7

FD (Lj ), FXj

FX (Lj ); andj(Lj ) is adopted:

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 areinvariant to country size.

8

3.1

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.8
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 as9 :0'j = @sDj

aDj
Z

a1

dG(a jaDj ) + sXjk

0

aZXjk
0

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 =8

As shown explicitly in 6.2.

9

See Melitz (2003).

1+

aXjk
aDj

, we get:

9

aXjk
aDj

;

'j =

1

1

1

aDj

+1

0

aXjk
aDj

B1 +
@
1+

2 +1

1 11
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@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 a countervailing e¤ect comes to play. The …xed market entry costfX becomes important, and increasing the foreign market size implies that this cost can be
spread over a larger number of units. It is therefore possible that the cut-o¤ productivity forentering the foreign market deceases in the market size. This is illustrated in Figure 2, whichuses (12) to plot aXjk against Lk for some standard parameter values (= 0:8;= 4;=

6; f x = 4; fD = 1; FE = 1; Lj = 10): The higher , the stronger is the increase in variable cost
as the market becomes large, and the more pronounced is the hump in Figure 2.
Thus, the model encompasses both the case when the marginal exporter needs to be lessproductive in larger markets as in Arkolakis (2007), and the case in Melitz and Ottaviano (2005)
where the opposite holds. Result 2 summarizes the e¤ects of foreign market size.

10γ=0.7γ=1

γ=1.3

Figure 2: Exporter cut-o¤ productivity in relation to foreign market size ( = 0:8;

= 4;

=

6; fX = 4; fD = 1; FE = 1; Lj = 10)

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

FXj

j

j

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

There is strong empirical support for exporters being more productive than domestic …rms,
and as mentioned above, we follow Melitz (2003) by making parameter assumptions for this to
hold:

FXj
j

> FDk : Moreover, market size is also of importance for the relative productivity ofexporters as compared to non-exporters, in accordance with the stylized evidence presented in
Figure 3 in the introduction:
@

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.
11

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

aXjk
aDj

FXk
=
a1Xjk

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

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

k
j

@SXjk
1
=
@Lk
(1

@

k

2 @L

k

j
j

k)

(29)

> 0;

(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.
Next consider a fall in fX , for instance, as a result of NTB-liberalization. For fX = fD
and 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, sincea 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 (e.g. product standardization) and a marketing cost which depends on the sizeof the market re‡ecting the higher cost of reaching more consumers. Also the standardization
for a particular market is more costly for an exporter than for a domestic producer. Since thecost of standardization is independent of market size, it becomes relatively less important ascompared 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.
12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, Melitz, andYeaple (2004). It is easily shown that the manufacturing export shares are independent ofcountry size in this model 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 to market size in that model. However, in contrast to our model, manufacturing
trade shares diverge as trade costs fall: trade shares increase from zero in autarky to the shareof the foreign market in total demand at free trade.10 As shown below in the empirical section,
we believe our prediction of converging manufacturing export shares to be supported by data.

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 priceof each …rm is now pij =1 ai wj ; where wj is the factor price in country j: The export from j

to k isEjk = nj
10

jk wj

1 Pk

1

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

13

(31)

where Vjk

aXjk
R
0

a1i

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

1

jk wj

nj

Vjk Yk :

1 Pk

k=1

(32)

Following the methodology of Anderson and van Wincoop (2003), a gravity equation of trade
may be derived by solving (32) for nj

1

1 wj

and substituting the solution into (31). This

gives:
Ejk =where sk

Yj Yk
Yw

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

Yj Yk
Ejk = w
Y

aXjk
aDj

+1
FXk Pk1
1
jk Yk

(34)

; from the cut-o¤ conditions, gives the following

Pm

jk

k=1

1

1

Yk
FXk

jk

Pk

1
aXjk
:

sk

Pk

Yk
FXk

1

1

:

(35)

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

tion: Ejk = Yj wk

Pk

Pm

+1

1
jk

k=1

jk
Pk

1

: The di¤erence is essentially the term
sk

Yj
FXj

1

; whichmultiplies the frictional trade cost.11 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 = w
Y

11

Yk
fX +Lk

jk

Pk

Pm

k=1

jk

Pk

sk

1

1

Yk
fX +Lk

1

1

:

(36)

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.

14

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, it isYk
fX +Lkrather than exactly GDP per capita that should enter the gravity regression according

to our model.

4

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 (2007) 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 theoretical result that GDP per capita enters the gravity equation is consistent
with the common use of GDP per capita as control variable in empirical applications of thegravity equation (e.g. Head, Mayer, and Ries (2008)).

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, Schnabel, and Wagner (2007) o¤ers
a literature overview where they measure the wage premium of exporting …rms as compared
to non-exporting …rms. Typically, a regression is run on …rm level data with some measureof wages as the dependent variable, and a dummy variable indicating whether the …rm is anexporter. The estimated coe¢ cient for this dummy variable is the exporter wage premium ascompared to that of non-exporters. We interpret this wage premium to indicate productivity
di¤erences between exporters and non-exporters.12 Figure 3 plots the exporter wage premium
versus market size (population) of countries in the studies surveyed in the appendix of Schank,
12This interpretation is consistent with a non-competitive wage setting à la Shapiro and Stiglitz (1984) or bylearning e¤ects as in Malchow-Møller, Markusen, and Schjerning (2007).

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 Tables.

Figure 3: Export premiums decrease in country size.

Schnabel, and Wagner (2007).13 We have also added an observation for Sweden using data
provided by Statistics Sweden. Naturally, it must be acknowledged that all regressions are notdone with exactly the same methodology or fully comparable data. Nevertheless, Figure 3 showsa negative correlation between the exporter wage premium and population size in accordancewith result 3. Running a regression on this data gives a slope of

4.2

0:58 with a t-value of

2:93.

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.
13

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 measure

16

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 happening
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 product standardization and removal of non-tari¤ barriers to trade (NTB-liberalization) within theEuropean Union during the last 20-30 years. GATT and WTO negotiations have also aimed
at reducing nontari¤ barriers to trade during this period. Finally, the rapid improvement oftelecommunications, including the internet, 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 2003.14 Accepting
market size.
14

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.

17

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

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 4 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 5 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, Romer, andWeil (1992)). We use the initial value of the manufacturing export share for which we havedata and regress the average growth rate in export shares, fsis ; on the average growth rate of
population and the initial level of trade shares, where the average growth rates are computed

as the coe¢ cient on the trend dummy in a regression of logged values on a constant and linear
trend, see e.g. Bernard and Jones (1996):
fsis =

0+

1 sis0 +

18

2

fli +

3 Ds + "is .

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

We allow for arbitrary correlation of the error terms over time at the country level byclustering of errors. Sector dummies are included. The model predicts that1should 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 rate of export shares depends
negatively on the initial level in 1980, thereby suggesting convergence within the OECD at thesectorial level.

5

Conclusion

This paper has explicitly modelled a market size dependent market access or beachhead cost in
the heterogeneous …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 in the size of the market (the number of consumers).
In essence we make a relatively small change to the framework of Helpman, Melitz, andYeaple (2004), which preserves the analytical solvability of the model. Nevertheless it leads to anumber of new results. The productivity of non-exporting as well as exporting …rms will dependon market size, as will manufacturing export shares. In particular, we show that …rms based ina large market are more productive than …rms in a smaller market, due to the higher …xed costs
related to establishing 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 cost (e.g. standardization) can be spread over more units, while the variable entry
cost (advertising) increases in the market size. Therefore, for su¢ ciently small export markets,
19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)
the productivity of exporters decreases in the export market size, which is consistent withthe feature of the …rm level data in Eaton, Kortum, and Kramarz (2005) that the number ofexporting …rms increases in the size of the export market. For su¢ ciently large exports markets,on the contrary, 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
more productive than non-exporters, but this productivity premium decreases in the size of thehome country, which is consistent with the stylized facts presented in the paper. Fourth, we showthat the manufacturing export share of a country decreases in its own size, and increases in thesize of the trade partner. This e¤ect decreases as markets are integrated (in the sense that the…xed beachhead cost of foreign markets declines). If market access costs into foreign markets
have been falling over time–as a consequence of globalisation–the model predicts converging
manufacturing export shares over time. This prognosis is supported in the empirical sectionof the paper, 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 practiceof including this variable in gravity regressions of trade ‡ows.

20

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21Clerides, S. K., S. Lach, and J. R. Tybout (1998): “Is Learning by Exporting Important? Microdynamic Evidence from Colombia, Mexico and Morocco,” Quarterly Journal of
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22

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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

ij =

FX (Lj )ij FD (Lj ) andii = 1.

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

to:
n
X

ij j = (1;1

1;2::: :::

2;1

:

1) FE

j=1

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

:

1;n

:
:

n 1;1
n;2

::: :::

3

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

:

:

n;1

1;n 1

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

n;n 1

n;n

23

1

3
7
7

2 5=
3

(39)

(

1) FE;(40)

orn n n 1 = n 1(

where

1) FE

(41)

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

vector1;

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
32
2
3
1
12
13
76 1 7676
6 21 17
23 5 4 2 5 = (431

The determinant ofij

system for
2
6
6
4

ij

3

12

23

31 +

13

21

(

32

12 +

13 +

2

1

1

7
1 6
7
6
2 5=
j j4
3

23

13

32

23

31

21

1

21

32

31

12

12

3

B
6 1 7
6 B2 7 =
4
5
B3

13
31

(
1+
2

12

1

6
6 1
4
1
@
@Lk

(43)

aDj
aDk

> 0 for

31 +

23

23

13

13

21

23 5

1

32

12

13 (

13 +

23 (

12 +

j;

7
7 (

Solving the1) FE ,

(44)

1) FE

13

23 +

3

12

which gives the following solutions for market size per …rm, Bi :6.2

23 ) ;

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

gives:
3

2

(42)

is

j j=1+
where

1

32

1) FE .

21

32

(

12 +

32

1) +

12 (

23

31

1) +

21 (

13

32 ( 211) +31 ( 12

13 +

1)

(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 (11)
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

24

(

1)

k

1

1

k

1

.

(46)

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

1

(

k

1)

1

1

k

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
j;

6.3

1
1

k

1

=

1

=

1

aDj
aDk

@
@Lk

k

1
1

k

= 1 (since

k

1

1

= 1 ()

1

1

k

1

The minimum is, thus, given by(Lk ) are:

1

1

k

=0
> 0:

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

@
@Lk

aDj
aDk

(48)
k

= 1

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

From (6.2), we have that
@

aDj
aDk

@Lk

=

@aDj 1@Lk aDkaDj @aDk> 0:(aDk )2 @Lk

(49)

@a

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

6.4

@aXjk
@Lk

From (12)aXjk =The sign of

@aXjk
@Lk

(

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 (
@Lk
k
@
1
[F F
@Lk Xk Dk

j )]

(51)
FXk

j ]:

7

0

(52)

Now
@
[F F 1@Lk Xk Dk
FXk
FDk

FDk
FXk

FXk

j]

,
(
FXk
(
FDk
25

1)

7

j

,
1)

7

j

k:

aDj
aDk

6.5

> 1 i¤ Lk > Lj for

j;

k < 1

Proof:
First
aDj
aDk

Lj = Lk ()j;That Lj = Lk ()

aDj
aDk

FDk
FDj

=

> 1 for

j;

1
1

k

= 1:

j
aDj
aDk

@
< 1 now follows from @Lkk> 0 fork < 1:

@

6.6

aDj
aXjk

@Lj

<0

f or

j < 1.

Proof:
@
@Lj

aDj
aXjk

= Lj

0
)
k
@(
j)

1 FXk (1
2
FDj
k (1

1

FDj
FXk

(1

j)

j

1)

1

1A .

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

The F.O.C. when maximising
(

1)

j

(1

1

FDj
FXj

0
@

(

1)
(1

is:

(

2
j

j)

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

FDj
FXj

1

(1

So the only stationary point is
Therefore, it follows that forj)

j w.r.t.

1)

FDj
FXj

1

j

(53)

aDj
aXjk

< 0.

Countries included in Figure 5.

The following countries are included in Figure 5. 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
FinlandFrance26

IcelandIreland
Italy
Japan
Netherlands
New Zealand
NorwayPortugal
Spain
SwedenUnited Kingdom
United States

27