A Trade Model with Institutions and Heterogeneous Firms
*
*
Matteo Fiorini , Mathilde Lebrand
**
**
and Alberto Osnago
* EUI
Bocconi University
September 2013
Abstract
This paper studies how institutions can explain asymmetric eects of trade liberalization
on aggregate productivity across dierent countries and industries. We focus on institutions
aecting the ease of doing business such that the provision of intermediate goods becomes
uncertain for the rm producing the nal good. We propose a framework where countries
dier in terms of institutional quality and rms are heterogeneous in productivity: the
quality of institutions and the ex-ante distribution of productivity determine the endogenous
organization of the rms and, in turn, their production decisions in a multi-industry national
economy. With trade liberalization, dierent institutional qualities across countries trigger
the pattern of comparative advantage. The paper conrms a positive eect of trade in the
countries with good institutions. In contrast with the standard results, our model suggests
a negative eect of trade for the countries with a poor quality of institutions. More precisely
this happens in the form of a decreased aggregate productivity, a reallocation of workers
toward less productive rms and a lower real consumption wage.
1
1
Introduction
The reallocation of resources across industries and countries after trade liberalization is a key
factor for the economic development of a country. It is important to understand the eect of
trade openness on rms, depending on their level of productivity and their incentives for industrial specialization. We propose a novel mechanism to look at this eect and, in particular, to
assess its asymmetries across countries and industries.
It is well established in the trade literature that institutions represent a source of comparative
advantage. Institutions aect rms behavior and in particular their production decisions as well
as their industrial specialization. At the country level, they determine how the economic activity
allocates across industries and therefore, at an international level, dierences in the quality of
institutions trigger the pattern of comparative advantage. Taking this into account and allowing
for rms' heterogeneous productivity, we are able to explain the asymmetric eect of trade on
the average productivity both across industries and countries.
We nd that the more productive rms have the incentives to produce in the sector where one
unit of the nal output requires more intermediate goods and services: this is due to their endogenous organization of production which allows them to face higher complexity with lower
marginal costs. When the economies open to international trade those incentives could remain
in place, only partially corrected by the institutions-driven pattern of comparative advantage.
Consistently with the result in Costinot (2009), countries with better institutions develop a comparative advantage in the production of the complex good. One critical eect of this pattern
of international specialization is that the best rms in the countries with low institutions could
suer from being active in the comparative disadvantaged sector. This is due to a reallocation of
workers towards the comparative advantage sector and to a price eect due to the competition
with rms beneting from a better institutional environment.
A second eect of this mechanism is that trade might give rms that previously were not productive enough to produce a new incentive to produce. This result contradicts most of the trade
theoretical results (Melitz 2003, Bernard et al. 2007) that highlight a positive eect of trade
on average productivity.
The comparative advantage in simple goods benet to the rms on
the left of the producitivty distribution, the least productive ones. Therefore it can give new
incentives to the rm with a producitivty at the frontier that before was not good enough to
enter production. These two eects could dampen or even potentially reverse the positive eect
of trade in the countries with bad institutions.
The rst objective is thus to understand the interaction between productivity and institutional
framework in determining the endogenous organization of the rms. We call rms the producers
of nal goods. In order to produce one unit of any nal good, a xed number of homogeneous
intermediate goods (or services) are required. We call production units (PUs) the providers of
intermediate goods.
More precisely, we model a PU as a company made by a single worker
endowed with a xed amount of labor units: this assumption allows us to identify the PU with
its single worker.
A rm optimally organizes its economic activity dividing the provision of intermediate goods
among PUs: it chooses the optimal degree of horizontal specialization. For every intermediate
2
good, the PU that is going to provide it has to learn specic characteristics of that particular
good; this entails a xed learning costs for the rm. In this framework, higher specialization is
a pattern of organization such that there is a bigger number of PUs working for the rm and
each of them is specialized in a smaller set of intermediate goods.
The key trade-o determining rms' optimal organization is the one between the gains and the
costs of specialization (respectively GS and CS).A higher specialization will allow the rm to
split the learning cost across a bigger number of PUs: each of them will thus be left with a
greater amount of labor units for the actual production of the intermediate goods. CS are given
instead by the institutional barriers to doing business: for every PU in the economy there exists
a positive probability that it fails to provide the set of intermediate goods for which it is responsible toward a rm. This can happen because of corruption in bureaucracies, unpredictability
of policies, security of property rights, et cetera. We assume this probability to be independent
on the set of intermediate goods the PU is assigned to. When the provision of all intermediate
goods is assigned to a single PU, the probability of producing the nal good (i.e. of having all the
intermediate goods provided) coincides with the individual non-failing probability of the single
PU. If instead the provision is assigned to many PUs, each of them specialized in a dierent
set of intermediate goods, the probability of getting the nal good produced coincides with the
product of the individual non-failing probabilities of the PUs. The nal good is produced with
a smaller probability in the case of higher specialization.
How do we model rms productivity and how does it aect horizontal specialization?
Each rm's productivity parameter is drawn from an exogenous distribution.
It determines
how eciently the PUs learn the specic characteristics of the intermediate goods needed for
production (learning eect) and how eciently the PUs transform labor units into units of the
intermediate goods (transformation eect). A productive rm is able to provide its suppliers
with clear instructions and tested protocols about the required intermediate goods. This makes
the PUs more ecient in using their labor units' endowment. Given these assumptions, productivity has dierent eects on the two sides of the trade-o. The model will show formally that
the optimal extent of specialization is a decreasing function of productivity.
How do we model institutions and how do they aect horizontal specialization?
Institutions are represented by the ease of doing business (the probability with which a PU
successfully provides the intermediate goods it is responsible for).
eect on the GS side of the trade-o.
Institutions will have no
They will have instead a straightforward eect on the
CS: better institutions imply a lower CS. Consistently with this, we will prove formally that the
optimal extent of division of labor is increasing in the quality of institutions. Furthermore we
will show that this eect will be bigger for lower productivity rms.
As a second step we will present the solution and the analysis of the autarky equilibrium. One
crucial aspect of our model is the fact that rms can choose in which sector to produce and
therefore the rst thing to be understood is which rm produces what under autarky? We will
show in a general equilibrium framework that only the more productive rms produce the varieties in the complex sector. Less productive rms choose instead to produce simple varieties.
Finally, below a certain productivity threshold, rms will decide not to produce. We show that
the autarky equilibrium exists unique. Another important exercise within the autarky equilibrium analysis is to understand how dierent institutional settings aect production decisions of
the rms. We will show through numerical simulation that better institutions decreases the two
3
productivity thresholds that separate respectively non active rms from active ones and rms
producing the simple varieties from those producing the complex ones.
Finally we will consider trade openness. We will rst specify under which conditions a country
develops a comparative advantage in the production of the complex varieties. Better institutions
determine the comparative advantage in the complex industry. The graphs below compare the
share of exports in simple goods and two commonly-used institutional indexes, the Rule of Law
index and the Ease of Doing Business index. They both support the idea of a correlation between
the quality of institutions and the pattern of trade.
Figure 1: Ease of Doing Business and Specialization
This gure shows that the share of exports of simple goods is lower in countries that score a
higher ranking in the Ease of Doing Business index constructed by the World Bank. Countries
are ranked according to that index on the horizontal axis, going from Singapore, the rst country
1 as the
in the ranking, to Senegal, the 166th. The share of exports in simple goods is constructed
share of total exports of all sectors that are in the lowest quartile according to the complexity
index proposed by Costinot.
0
Share of Exports in Simple Goods
.8
.2
.4
.6
1
Rule of Law and Specialization
-1
0
1
2
Rule of Law
Figure 2: The Rule of law and Specialization
1
Export data come from COMTRADE and it is at the 3-digits SIC level in order to match with the complexity
index constructed by Costinot.
4
This gure plots the relationship between rule of law
2 and the share of simple goods exports on
total exports. Here simple goods are the industries below the average level of the complexity
index.
The relationship is clearly negative suggesting that countries with better institutions
export very little simple goods, whereas countries with poor institutions specialize in such industries. The correlation between the two variables is -0.51 (it is -0.42 if we consider as simple
goods only the ones in the bottom quartile of the complexity index).
Given the institutions-driven pattern of comparative advantage, we will then solve and analyze
the economy after trade liberalization. This will allow us to understand how dierences in the
quality of institutions determine asymmetric impacts of openness both across industries and
countries. In particular our model conrms a positive eect of openness in the countries with
good institutions on the one hand while it introduces several negative eects for the countries
with bad institutions. In particular we show that under certain conditions trade in the country
with poor institutions leads to a decrease of the average productivity. Some workers are reallocated towards the least productive rms. The prots of some rms that produce and export
the simple good but would have produced the complex good in autarky decrease with trade.
The prots of some rms in the complex sector decrease because the complex good is not the
comparative advantage of their country. Finally the real consumption wage might decrease if
the decrease in average productivity due to specializationis due strong.
1.1
A case study in support of our mechanism: Paraguay
In order to support our mechanism we shortly describe the case of one country. Paraguay has an
agricultural and an industrial sectors that are equivalent in size and both contributed to around
20% of the GDP in 2012. Most enterprises are small, micro and individual ones. Only 4% of the
Paraguayan labor force works in companies with more than 50 employees. Reagrding its pattern
of trade, the graph below shows that Paraguay has developed a clear comparative advantage in
agriculture in the last decades.
Figure 3: Evolution of exports in manufacture and agriculture since 1962
One of the result of the paper that drives our conclusions is to show that the very productive
rms always produce the complex goods. This does not depend on the comparative advantage
2
Rule of law is the index available in the Worldwide Governance Indicator Dataset.
5
of the country. In the country with poor institutions, rms with a lower level of productivity
produce the simple good for which the country has a comparative advantage. If rms are always
more productive on average in their comparative advantage sector, our mechanism would be
invalidated.
However Paraguay seems to support our mechanism.
We compare the average
TFP of several sectors with the share of each sector in the total exports of the country.
We
notice that Paraguay has a high TFP in major industrial sectors like machinery or chemicals
and pharmaceuticals but exports very little (less than 6% of total exports)in these sectors. On
the contrary the TFP in the food sector is very low and the share of exports is very high. 75% of
the total exports are in the food sector. Food is considered as simple good whereras industrial
goods like machines are complex goods. Therefore Paraguay has a comparative advantage in
food production, a simple good, but the average productivity in this sector is very low compared
to other sectors.
Figure 4: TFP and share of exports
Finally Paraguay is ranked 103 out of 185 countries in the Doing Business 2013 ranking. The
country suers overall from considerable shortcomings in terms of legal certainty, due to weak institutions and widespread corruption. In our model the key mechanism that creates comparative
advantage is the assymetry between countries in terms of institutional quality. A good illustration of such an assymetric exchange is trade between Paraguay and Germany. Indeed Paraguay
has been a popular emigration destination for Germans: in the past for German Mennonites and
people seeking to escape the economic misery caused by the First and Second World Wars and
nowadays for retired people. Paraguay mainly imports vehicle parts, machinery and chemical
products (116.3 million EUR in 2011) and principally exports raw materials, in particular oil
seeds and oleaginous fruits (174.5 million EUR in 2011). Germany is ranked 20 in the Doing
Business 2013 ranking and trade between Germany and Paraguay provides a good exemple of
specialization in simple and complex goods between two countries with a very dierent institutional environment.
Our model seems to be relevant for the case of Paraguay and provides normative results about
the impact of trade on the average productivity of rms, prots and prices. Unfortunately we
do not have micro data for rms in developing countries so any further empirical work is now
delayed.
6
2
2.1
Related literature and the contributions of the paper
The theoretical literature
Our paper relates to the trade literature by using a Ricardian framework to deal with the sources
of comparative advantages and by modeling heterogeneous rms. After the seminal contribution
by Marc Melitz in 2003, Bernard et al. (2007) extended the heterogeneous rms framework to
account for comparative advantage in a two factors - two industries economy. Our paper builds
upon the same Melitz (2003) framework and, similarly to Bernard et al. (2007), it adds comparative advantage to the picture.
One of the key dierences is the source of the comparative advantage: while Bernard et al. (2007)
make use of the standard Hecksher-Ohlin assumptions (cross country heterogeneity in factor
endowements causing the pattern of comparative advantage) we embed in our paper the novel
lessons of the literature on trade and institutions (Levchenko (2007), Nunn (2007), Acemoglu et
al. (2007) and Costinot (2009)). In particular we have that the pattern of comparative advantage
is driven by dierent qualities of institutions across countries.
Among the papers on trade and institutions Acemoglu et al. (2007) studies the theoretical link
between contract incompleteness and rms' technology adoption under trade openness while
Costinot (2009) the one between contract enforcement and rms' division of labor in an open
economy. Both papers are able to capture the role of institutions in shaping some endogenous
aspect of rm's organization that can be interpreted as a proxy of productivity. We focus on
Costinot (2009) that oers a tractable framework in which contract enforcement is a crucial determinant of comparative advantage. In this model, better institutions, represented by a higher
probability of the enforcement of a contract, allow a country to specialize in the production of
more complex goods. These are sectors that require a higher number of tasks (such as research,
design, assembly) to produce a unit of the good.The present work extends this exercise including an exogenous productivity parameter à la Melitz: this allows us to understand not only
how rms react in terms of their dierent levels of productivity but also how economic institutions can be a source of comparative advantage. Countries with better institutions specialize
in industries for which the existence of stable and reliable institutions is more important. Economic literature started examining more actively this source of comparative advantages recently.
The other main dierence with Bernard et al. (2007) deals with the productivity distribution.
A small change in the initial assumption allows us to get very dierent welfare results. Indeed
Bernard et al. (2007) models 2 sectors in each country and each of the sector has a productivity
distribution for which only the initial entry threshold changes. In our model, there is only one
productivity distribution and rms choose in which sector to specialize depending on their initial
productivity level. This sector choice is the key mechanism and is novel. Bernard et al. (2007)
only nds a positive eect of trade on aggregate productivities in both countries whatever their
comparative advantage. On the contrary we nd that the aggregate productivity of the country
with a comparative advantage in the simplest (the least contract intense) good might decrease
when trading with a country with better institutions. This negative result reminds us of Dornbusch et al. (1977) in which an increase in the size of the population in the foreign country
worseits terms-of-trade and lowers real GDP per capita through a change in production specialization. Our mechanism is therefore very dierent as it relies on reallocations through rms
depending on their productivity. Indeed our negative result is due to labor reallocations from the
7
most productive rms that still produce the complex (the most contract-intense) good towards
less productive rms that capture the comparative advantage. The complex-good production of
the most productive rms is now in competition with imports from rms of the country with
better institutions with the comparative advantage in complex goods.
3
The model
3.1
The economic environment
We consider
2
countries indexed by k
∈ H, F .
Every country is characterized by a continuum of
j ∈ [0, 1] and n ∈ [0, L¯k ]. Lk
rms and workers/consumers, indexed respectively with
is the total
population of workers in country k. There is no mobility of workers across countries. Moreover,
i ∈ {S, A},
the economy in both countries consists of two sectors,
and advanced nal good varieties
ω ∈ Ωi .
pooling respectively simple
The two types of good diern by their complexity.
3.1.1 Demand
We assume Cobb-Douglas utility across sectors (⇒ xed expenditure shares across sectors) and
CES across varieties (⇒ Dixit-Stiglitz demand):
U = Sα A1−α
Where
S
and
A
are the two aggregate quantity indexes dened as follows:
"Z
#
c(ω)
S :=
σ−1
σ d
σ
σ−1
"Z
ω
and
c(ω)
A :=
ω∈ΩS
#
σ−1
σ d
σ
σ−1
ω
with
σ > 1.
ω∈ΩA
Every worker in the economy is endowed with xed human capital
h:
technically we interpret
this parameter as the number of units of labor per worker.
3.1.2 Supply: Firms and Production Units (PUs)
φj is disgk (·) on the support (0, +∞).
G(·) the associated cumulative
Firms are heterogeneous with respect to productivity: the productivity parameter
tributed according to a country-specic probability density function
For now let us assume
gH (·) = gF (·) =: g(·).
We denote with
distribution function.
Each sector
zS
<
i ∈ {S, A}
is characterized by a continuum of intermediate goods,
I ∈ [0, z i ],
with
z A . Intermediate goods are provided by PUs, dened as companies in which only one
representative agent works. They operate through a linear technology with labor units as input:
for any intermediate good
provide
1 unit of it.
I,
it takes
1
labor unit to learn its characteristics and
In order to provide any quantity of
1
labor unit to
I , the PU has to learn its characteristics.
The unique source of input for a PU is the labor units' endowment of its worker. Since we identify
PUs through their single workers/consumers we apply to them the same index
the mass of PUs in country
k
will be equal to the mass of workers
n.
For every sector, all the elements of the sector specic intermediate goods' space
be provided in order to produce one unit of any nal good variety.
8
In equilibrium
Lx .
[0, z i ]
have to
Uij (n, I) the set of units of the nal-good-variety produced by rm j in sector i for which
PU n provides the intermediate good I . The following expression gives the amount of labor
units needed by n to learn the characteristics of I and to provide one unit of I for each unit of
i
the nal good in Uj (n, I):
Call
l(Uij (n, I)) :=
Z
u∈Uij (n,I)
1
1
du +
φj
φj
(1)
Firm's productivity aects the learning process within a PU: given intermediate good
I,
it
is easier for a PU to learn how to produce it when a more productive rm is providing better
instructions and protocols on
I.
Additionally, rm's productivity has the standard role of making
the technologies more ecient: xing the number of nal good units for which which PU
to provide the intermediate good
the production of
I
I,
n
has
the actual amount of labor that the PU has to put into
is lower the higher the productivity in the rm. The productivity of a rm
aects the learning and productive activity of its intermediate goods' suppliers.
In order to get the standard monopolistic competition supply structure we assume a sunk xed
cost
f >0
for the rms. The cost
f
is measured in number of PUs and could be interpreted as
3
those intermediate goods or services which are necessary for any nal good production process .
3.1.3 Firms' Organization and Institutions
The production of a rm
- every rm
j
j
active in sector
i
of country
k
is organized as follows:
partitions the sector-specic intermediate goods' space
product ranges (denote the resulting partition
Rji =
[0, z i ] into Nji
dierent
i
i }Nj ), i.e. sets of intermediate
{Rk,j
k=1
goods whose provision to be assigned to the PUs;
Lij ⊂ [0, L̄k ].
pays wk to the
- the rm selects a subset of the PUs,
We assume that every PU can be selected
by one rm only. The rm then
PU irrespectively of the actual provision
of intermediate goods;
- for every selected PU and for each unit of the nal good, the rm species which range
R
of intermediate goods - if any - have to be provided by that particular PU for the that
particular unit of the nal good. Formally the rm designs the mapping
i
i
Oji (·, ·) : Lij × R+ ⇒ {R1,j
, . . . , RN
i ,j , ∅}
j
Uij (n, I), i.e. the set containing
the units of the nal-good-variety - produced by rm j in sector i - for which PU n provides the
4
intermediate good I .
From the mapping
Oji (·, ·)
(2)
we can derive the above dened set
θx ≥ 0. It aects the
n is actually able to do what it is supposed to do (its activity could
The quality of intitutions is captured through a doing business parameter,
probability with which any PU
be hampered by institutional obstacles such as corrupted bureaucracies, unexpected taxation,
3
f gives the increasing returns, otherwise there are constant returns with respect to multiples of the optimal
number of PUs providing intermediate goods for one unit of the nal good.
4 i
i
i
Uj (n, I) = {u ∈ R+ | ∃ t such that I ∈ Rt,j
∧ Rt,j
∈ Oji (n, u)}
9
violation of property rights et cetera). Given
-
∀k, ∀i, ∀j
- a successful provision indicator
(
i
Sk,j
(n, I, u)
n ∈ Lk
for every
PU
n
θk , the institutional quality in country k , we dene
=
and for every pair
with probability
e
0
with probability
1−e
(u, I)
k
(3)
− θ1
k
i (n, I, u) = 1 means
u ∈ Uij (n, I). Sk,j
I for the production of the uth unit of the
such that
is able to provide the intermediate good
good produced by
− θ1
1
that
nal
j.
For the sake of simplicity we assume that a PU that fails the provision of one intermediate good,
fails also in the provision of all the others intermediate goods it was responsible for.
3.2
Equilibrium under autarky
3.2.1 The consumers' problem
We apply the two stages budget procedure using the aggregate price indexes
and income
R
1
P i = [ ω∈Ωi p(ω)1−σ dω] 1−σ
R.
The rst stage problem faced by consumers is the following:
max Sα A1−α
subject to
S,A
P S S + P AA = R
Solving (C1) we get the xed expenditure shares for the two sectors:
P AA
= (1 − α)R =:
(C1)
P S S = αR =: RS
and
RA .
In order to get rid of any demand side eects in determining the comparative advantage under
5 we assume
free trade
α = 1/2.
In the second stage problem consumers solve the following maximization: for every
Z
max
{c(ω)}ω∈Ωi
c(ω)
σ−1
σ d
Z
ω
p(ω)c(ω)dω = Ri
subject to
ω∈Ωi
i ∈ {S, A}
(C2i)
ω∈Ωi
For every sector we get the consumption and the expenditure across varieties:
(
c(ω) =
(
r(ω) =
−σ
S p(ω)/P S
−σ
A p(ω)/P A
1−σ
RS p(ω)/P S
1−σ
RA p(ω)/P A
if
if
ω ∈ ΩS
ω ∈ ΩA
if
if
ω ∈ ΩS
ω ∈ ΩA
(4)
(5)
5
Krugman (1980) shows how the country with higher internal demand for a sector will develop a comparative
advantage in the production of the sector specic varieties.
10
3.2.2 The rms' problem: optimal organization
We start the analysis of the rms' behavior characterizing their optimal organization.
The
following proposition is based upon the results in Costinot (2009).
Proposition 1 Optimal organization implies that each PU selected by a rm provides one and
only one range of intermediate goods for every nal good's unit it is responsible for. Moreover,
each rm optimally allocates the same number of intermediate goods across ranges. Finally,
expected output for any given price level is given by:
Ni
yji
−θj
=e
x
φj i zi h
−
L
j
zi
Nji φj
(6)
Given this result we derive the optimal number of intermediate good's ranges as the solution of
the following maximization problem of rm
Nji
max pij e− θx Lij
j:
hφ
Nji
j
zi
−
with solution
(Nji )∗
zi
=
1+
2hφj
1 − wk (Lij + f )
Nji
r
4θk hφj
1+
zi
(7)
!
(8)
We identify the degree of horizontal specialization with the endogenous quantity
(Nji )∗ .
It is
in fact the optimal number of PUs whose provision of intermediate goods will determine the
production of every nal good's unit.
Observation 1 (Nji )∗ decreases with φj , increases with θx and the mixed partial derivative of
(Nji )∗
with respect to φj and θx is negative.
3.2.3 The rms' problem: production and sector decision
In this subsection we derive the optimal pricing rule as a function of the productivity parameter
φ.
For the two sectors of the economy, we show the qualitative behavior of the prots as functions
of
φ.
This last step allows us to characterize the sets of productivity values for which a rm will
decide to produce and in which sector to do so.
6 with a productivity level
Let us consider a rm
φ
producing a variety in
Ωi .
Given optimal
horizontal specialization, the maximization problem of the rm can be written as follows:
max pi (φ)y i (φ) − wx Lij y i (φ) + f
y i (φ)
Using the expression of expected output given in Proposition 1 we can derive
of the optimal
8 marginal productivity of labor for a rm in sector i:
6
(9)
7 the expression
We omit here the subscript j to lighten the notational burden.
the computations are in annex
8
this level of productivity diers from the initial distribution of productivity parameters φ and results form
the optimal strategy of the rm to organize the production depending on the complexity of the goods
7
11
i
∗
∂Li (φ) y i (φ)
e(N (φ)) /θx
i := β i (φ)
=h
hφ
1
∂y i (φ)
−
(10)
(N i (φ))∗
zi
Following Dixit and Stiglitz (1977) we posit that the market share of each rm is small enough
in order to be neglected in the pricing decision of the others. This assumption (supported by
the innite number of rms in our set up) together with the constant elasticity of substitution
gives us the following expression for the elasticity of demand faced by rm of productivity
φ
producing in sector i:
i (φ) = =
where
(11)
σ−1
σ .
ρ=
Using (
1
1−ρ
??)
and (11) we get the pricing rule as a function of the idiosyncratic productivity and
the sectorial parameters:
pi (φ) =
wx β i (φ)
ρ
(12)
The price is equal to a constant mark-up over marginal cost. The prot function is given by
P i ρ σ−1
R
− wf
π (φ) =
2σ wβ i (φ)
i
(13)
Let us begin our analysis of the prot function with few observations:
Observation 2
ˆ ∀φ, ∀i π i (φ)
ˆ
Moreover
is continuous and monotonically increasing in
limφ→0 π i (φ) = −wf
and
φ.
limφ→+∞ π i (φ) = +∞.
In order to simplify the rest of the computation to characterize the equilibrium, we show that if
there exists an equilibrium it has to verify the following properties:
Proposition 2 In equilibrium ∃! a value φSA such that π S (φSA ) = π A (φSA ) > 0.
This last result is given by Observation 2 and the fact that in equilibrium there must be production in both sectors to clear demand. The following proposition tells us something more about
the behavior of the prot functions in the two sectors:
Proposition 3 In equilibrium there exist two values φeS and φeA such that π S (φeS ) = π A (φeA ) = 0.
Moreover, φeS < φeA .
Proposition 4 If the equilibrium exists, it has to be characterized by the two thresholds:
ˆ φeS
is called the entry threshold and renamed
ˆ φSA
is unique and called the choice threshold.
12
φe
ˆ
A rm with a productivity
φ
st.
φ < φe
ˆ
A rm with a productivity
φ
st.
φe < φ < φSA
ˆ
A rm with a productivity
φ
st.
φ > φSA
does not produce any good.
produces the simple good S.
produces the complex or advanced good A.
The following graph shows qualitatively the behavior of the prots in the two sectors as functions
of productivity.
! " &!!
!"
!!
"#!
"#"
"##
"
$# $ %
Figure 5: Prots as function of productvity
If a rm draws a productivity parameter below
φeS
it will exit the market immediately and
never start production. If the rm's productivity is instead between
φeS
and
φeA ,
the rm will
eA the rm will produce a
produce a simple variety. With a productivity parameter above φ
variety within the advanced sector.
3.2.4 Aggregation
Each period there is a mass M of producing rms with productivity levels higher than the entry
threshold
φe .
The mass of rms M for a variety of a good only depends on the entry threshold
and threfore is the same in the two sectors.
The previous part shows that there exists two important thresholds in the productivity distribution.
φe
is the entry threshold below which rms choose not to enter the production to avoid
a negative prot.
φSA
is the key choice threshold that determines which rms produce which
13
good. Firms with a productivity level below
φSA
produce the simple good S and produce the
complex good A otherwise.
We dene 3 ex-post distributions:
µS
the production process,
and
µ is the distribution of productivity for the rms who entered
µA the distribution of productivity for the rms given their
choice of sectors S or A.
(
µ(φ) =
(
S
µ (φ) =
g(φ)
1−G(φe )
if
0
otherwise
φ > φe
g(φ)
G(φSA )−G(φe )
if
0
otherwise
and
(
A
µ (φ) =
φe ≤ φ < φSA
g(φ)
1−G(φSA )
if
0
otherwise
φ ≥ φSA
The prices
First we need to dene the equilibrium average marginal cost in the two sectors is determined
φe
by the cuto productivity levels
S
S
e
SA
β̃ = β̃ (φ , φ
and
φSA :
1
)=
G(φSA ) − G(φe )
Z
φSA
1−σ
β (φ)
g(φ)dφ
S
A
β̃ = β̃ (φ
SA
1
)=
1 − G(φSA )
Given this notation, we can dene
PS = MS
"
with
MS =
G(φSA )−G(φe )
1−G(φe )
Pi
1
1−σ
pS β̃ S
and
"
and
∞
1−σ
β (φ)
g(φ)dφ
A
MA =
in one of the two sectors. Among them
MS
(14)
1
1−σ
(15)
φSA
the aggregate price index for the sector
#
M
Z
1
1−σ
φe
and
A
PA = MA
1−G(φSA )
1−G(φe )
1
1−σ
i ∈ {S, A}.
pA β̃ A .
(16)
#
M
with M the mass of rms producing
produce a variety of the simple good S and
MA
a
variety of the complex good A.
The prots and revenues
We can derive the aggregate prot and revenue for the whole economy.
G(φSA ) − G(φe ) ¯S 1 − G(φSA ) ¯A
Π=M
π +
π
1 − G(φe )
1 − G(φe )
14
(17)
with
πS
and
πA
the average prots dened by the following expressions:
R φSA
π¯S =
π¯A =
φe
π S (φ)g(φ)d(φ)
[G(φSA ) − G(φe )]
R∞ A
φSA π (φ)g(φ)d(φ)
[1 − G(φSA )]
The free-entry condition
Similarly to the rms dynamics described in Melitz (2003) the expected prot from drawing a
productivity level has to be equal to the cost
wfe
of having a draw. From this we derive the
rm entry condition:
V = [1 − G(φe )]π̄ = wfe
with V the ex-ante utility of the rm over time,
fe
π̄
(18)
the average ex-post prot in the economy and
the xed cost to pay initially to draw a productivity level.
We use the following relations to derive the average prots in the two sectors (cf. annex):
 S e
S e

 π (φ ) = 0 ⇐⇒ r (φ ) = σwf

 π S (φSA ) = π A (φSA ) ⇐⇒ rA (φSA ) =
h
β S (φe )
β S (φSA )
i1−σ
(19)
σwf
We nally have our free-entry condition that only depends on the two thresholds
V =
φe , φSA :
nh β̃ S (φe , φSA ) i1−σ o
nh β̃ A (φSA ) β S (φSA ) i1−σ oo
wf n
SA
[G(φSA )−G(φe )]
−1
+[1−G(φ
)]
−1
= wfe
δ
β S (φe )
β A (φSA ) β S (φe )
(20)
The dynamics of the entry/exit of rms
Following Melitz (2003) we model a process of rms' dynamics. Every period there is a mass
Me
of potential entrants into a national economy. At this stage the potential entrants are iden-
tical. In order to draw a productivity parameter from the distribution
xed entry cost
fe
thereafter sunk.
g(·)
they have to pay a
Once the rm knows its productivity, it decides whether
to engage in production and in which sector to do so. Those decisions are taken anticipating
9 which in turn embeds optimal organization determined taking prices
optimal pricing behavior,
as given. Every period will be characterized by a mass
the rms active in the two sectors:
M=
MA
+
M
of active rms which is the sum of
MS.
9
As in Dixit and Stiglitz (1977) we assume that the market shares of the rms are small enough not to trigger
the strategic consideration of the opponents' pricing behavior.
15
For every active rm in every period, there is a positive probability
the beginning of the period a proportion
δ
of the incumbent rms
δ
M−1
of exogenous death. At
disappears . A mass
of potential new rms that pay the xed entering cost and receive a productivity level
the potential new rms with a productivity level higher than
φe
φ.
Me
Only
nally enter the production
process. There is no time in this model as there is no accumulation or time related decisions.
The dynamics is given by:
M = (1 − δ)M−1 + (1 − G(φe ))Me
We will focus on the steady states of this dynamic process.
M = M−1 ,
At the stationary equilibrium
we have:
[1 − G(φe )]M e = δM
(21)
The labor market condition
Labor is used to enter the production process as well as to produce. The economy in country
L¯x workers. Lei denotes the total amount of workers used in the entry
p
process in sector i, i∈ S, A and Li denotes the total amount of workers used for production in sece
tor i. Only the productivity level determines the choice of the sector and L is not sector-specic.
x has a population of
The labor market clearing conditions are:
Le + Lp = L̄
with
Lp = LpS + LpA
(22)
The good market condition
RS = αR
and
RA = (1 − α)R
(23)
3.2.5 Equilibrium
Proposition:
The equilibrium under autarky is dened through the vector
{φe , φSA , PS , PA , M, ps (φ), pa (φ)}
that veries the optimal behaviors of the agents, the labor
market and good market conditions.
The equilibrium thresholds solve the following system of equations:

(F E)
[1 − G(φe )]π̄ = wfe







 (def φe )
π S (φe ) = 0


(def φSA )






π S (φe ) = π A (φSA )
labor and good market clearing conditions
Theorem 1 The autarky equilibrium exists and is unique.
16
(24)
Proof:
Once we have the pair
(φe , φSA )
we can derive all the other endogenous equilibrium variables.
The number of rms entering and exiting production is given by the stationary equilibrium
equation and pined down by the labor market condition.
Then in order to nd the equilibrium thresholds, we use the following system:



 (F E)


 (def φSA )
wf
δ
(
h
β̃ S (φe ,φSA )
β S (φe )
i1−σ
[1−G(φSA )]
[G(φSA )−G(φe )]
)
[G(φSA ) − G(φe )] − 1 + G(φe )
= wfe
(25)
n
β̃ S (φe ,φSA )
β̃ A (φSA )
oσ−1
=
n
β S (φSA )
β A (φSA )
oσ−1
In the annex we prove that this system denes a unique pair of thresholds
(φe , φSA ).
Theorem 2 At the equilibrium rms with a productivity level below φe do not produce, the rms
with a productivity level between φe and φSA always produce the simple good S and the rms with
a productivity level higher than φSA always produce the complex good A.
3.2.6 Notes on the autarky equilibrium
The autarky equilibrium has the following characteristics:
ˆ
The rms producing the complex good are always the most productive ones;
ˆ
The average prot in the sector A is higher than the average prot in the sector S.
ˆ
The most productive rms have the highest prot and also employs the highest number of
workers. Size of prot and size in terms of workers are useful to charaterize the equilibrium.
Given that we cannot derive an explicit formula for the relevant productivity thresholds of the
autarky equilibrium we perform some comparative statics using numerical simulation.
PARAMETERS FOR THE SIMULATION:
Firms' productivity drawn from a gamma distribution with shape parameter
Parameters
h
σ
zS
zA
f
fe
L̄
Values
1
2.5
10
1000
5
10
500
6
and scale pa-
Table 1: Parameters
rameter
3.
The above gure shows that better institutions decrease the production productivity threshold
φe
SA . Better institutions create a more inclusive
and the simple/complex productivity threshold φ
framework for production. A better business environment attracts rms that could not produce
before.
17
Figure 6: Entry and choice productivity thresholds as a function of institutional quality
3.3
Equilibrium under free trade
In the rest of the paper, we study the equilibria when two countries can trade the two goods.
Countries might develop a comparative advantage in one of the two sectors.
This pattern of
specialization might aect the welfare in each country. We assume that workers are not mobile
across countries. First we study the free-trade equilibrium then we introduce additional costs.
In a free trade equilibrium there is no trade cost (xed initial cost or iceberg-type cost on imports) so all the rms export. We can notice that two rms with the same productivity level
in dierent country might not have the same behavior -the same choice of sector and of pricesgiven the dierence in institutions between the two countries an the lack of mobility of workers.
The previous expressions are similar under Free-trade and autarky, except an additional revenue
from exports such that:
R−
ri (φ) = ri,d (φ) + ri,x (φ) = 1 +
ri,d (φ)
R
with
ri,d
the domestic revenue,
ri,x
the revenue from exports,
R−
the total revenue in the other
country.
Under free trade, prices are the same in the two countries in the two sectors:
i ∈ {S, A}
1
H 1−σ
Pi = PiH = PiF = [MiH pH
+ MiF τ 1−σ pFi (β̃iF )1−σ ] 1−σ
i (β̃i )
18
PiH = PiF
with
with
k ∈ H, F ,
the average marginal cost in sector i in country k
G(φSA )−G(φe )
k
Mk
MS =
1−G(φe )
sector S in country k
MAk =
1−G(φSA )
1−G(φe )
β̃ik ,
the mass of rms in
and the mass of rms in sector A in country k
M k.
3.3.1 The good market clearing condition
We have:
H
for the country H
F = αRH
+ Rs,x
H
F
H
Ra,d + Ra,x = (1 − α)R
Rs,d
F
for the country F
with
k
Ri,d
H = αRF
+ Rs,x
F
H
F
Ra,d + Ra,x = (1 − α)R
Rs,d
the revenue spent by consumers in the country k in the sector i that is produced
domestically (by rms from country k) and
k
Ri,x
the revenue spent by consumers in the country
k in the sector i that is imported from the other country (produced by rms from the other
country).
3.3.2 The labor market condition-similar to autarky
The clearing labor market condition is:
L̄ =
X
Lpi + Le
L̄ is exogenously given as the total number of workers in the economy.
This equation at the equi-
librium will give us the mass of rms M for a certain couple of productivity thresholds (φ
e , φSA ).
3.3.3 The existence of a comparative advantage
:
Result Existence of a comparative advantage: The country with the best institutions has
a comparative advantage in producing the complex good.
We assume that country H has the best institutions. We can prove that country H therefore has
a comparative advantage at producing the complex good A.
19
By denition of the choice threshold
φSA,k
in country k
∈ H, F ,
we have:
k
πSk (φSA,k ) = πA
(φSA,k )
βSk (φSA,k )
PSk
=
⇒
k (φSA,k )
PAk
βA
From the simulations in the autarky part, we can show that at the autarky equilibrium:
F
Aut,SA,F
)
βSH (φAut,SA,H )
β S (φ
>
H (φAut,SA,H )
F (φAut,SA,F )
βA
βA
Consequently we get at autarky:
F
PSH
S
>
PAH
PAF
This denes a comparative advantage in the free trade equilibrium for the country with the best
institutions, country H, to produce the complex good A.
3.3.4 Equilibrium
Proposition:
The equilibrium under free-trade is dened through the vector for
k ∈ H, F
T
T
T
T
(φ)} that veries the optimal behaviors
(φ), pk,F
, PSF T , PAF T , RF T , M k,F T , pk,F
, φSA,F
{φe,F
a
s
k
k
of the agents, the labor market and good market conditions.
The equilibrium thresholds solve the same system of equations:

(F E)
[1 − G(φe )]π̄ = wfe







 (def φe )
π S (φe ) = 0


(def φSA )






(26)
π S (φSA ) = π A (φSA )
market clearing conditions
The average prots for the two sectors:

(
#1−σ
)
"


S
e
SA
−
β̃ (φ ,φ )

R
S

−1

β S (φe )
 π̄ = wf 1 + R
(
#1−σ
)
"


A
SA
S
SA
−
β̃ (φ ) β (φ )

 π̄ A = wf 1 + R
−1

R
β A (φSA ) β S (φe )

with
R− = R = ω L̄
(27)
given that workers are not mobile.
Proposition Existence:
The equilibrium under free-trade exists and is unique.
Proof:
The proofs for the existence and the unicity is similar to the autarky equilibrium.
endogenous variables can be determined.
All the
This proves that the free-trade equilibrium is well
20
dened. Once we get the equilibrium thresholds through the system of equations (FE+equality
of prots), we can compute the average prots in each sector. Then we use the labor market
conditions to get the mass of rms in each country. Finally we can compute the unique price by
using the masses of rms.
Proposition:
The equilibrium thresholds under free-trade are dierent from the equilibrium
thresholds in autarky. Exports aect the aggregate prices in each country as well as the choice
of sector. All the rms export and the prices are the same in both countries. The masses of
rms (M
k,F T ) in country k dier.
We can notice that this result diers from the results found in Bernard et al. (2007). In this
paper, they nd that the threhsolds are similar under autarky and g=free-tradeWe can now
describe how the entry and choice thresholds move when opening to trade without any trade
costs.
Result FT:
Compared to the autarky choice thresholds
φAU T,SA ,
the free-trade choice thresh-
F T,SA increases in the country with the worst institutions and the comparative advantage
old φ
in the simple good, and decreases in the other country.
Figure 7: Change in thresholds for the country with good institutions
Figure 8: Change in thresholds for the country with poor institutions
Proof:
PSF,A
PAF,A
<
PSH,A
PSF T
<
PAF T
PAH,A
21
We use the equality of prots at the productivity level
that the function
φFk T,SA
for each country k and the fact
is strictly increasing.
π S,k (φSA,k ) = π A,k (φSA,k )
β S,k (φSA,k )
P S,k
=
⇒
P A,k
β A,k (φSA,k )
3.4
The costly trade
3.4.1 Comparative advantage under costly trade
By continuity, we have the following relation:
PSF,A
PAF,A
<
PSF,CT
PAF,CT
<
PSH,CT
PSH,A
PSF T
<
<
PAF T
PAH,CT
PAH,A
This denes a comparative advantage for the country H with the best institution in producing
the complex good. The proof is similar to Bernard et al. (2007).
3.4.2 Equilibrium under costly trade
In this section, exporting rms have to pay a variable and xed cost to export. The xed cost
fx
implies that not all the rms export in each sector and the variable cost is added such that
the thresholds are dierent than in the previous equilibria.
The variable and xed costs are
assumed to be equal in the two sectors and across countries. Most of the previous relations are
similar under Free-trade and costly trade.
Not all the rms export any more so there are two new thresholds
φix
for
country that denes the proportion of rms exporting in each sector.
The two additional thresholds are dened by:
φix
for
i ∈ {S, A}
such that

 (def φSx )
πxS (φSx ) = 0 ⇐⇒ rxS (φSx ) = σwfx
 (def φ )
Ax
πxA (φAx ) = 0 ⇐⇒ rxA (φAx ) = σwfx
Not all the rms export, the ones exporting have an additional revenue.
πi (φ) =
ri,d (φ)
ri,x (φ)
− wf +
− wfx
σ
σ
22
i ∈ {S, A}
for each
k
We dene the export premium ∆i
=
−k
τ 1−σ PP k
σ−1 premium decreases with the variable export cost
τ
Rk
R−k
in country k with
k ∈ {H, F }.
and increases with the price ratio
This
P −k
.
Pk
The revenues of the exporting rms are now given by:
ri (φ) = ri,d (φ) + ri,x (φ) = 1 +
∈ {S, A}
Price aggregates in sector i, i
∆ki
ri,d (φ)
in coutries H and F are given by:
"
PiH
=
#
H 1−σ
MiH pH
i (βi )
+
1
1−σ
F 1−σ F
F 1−σ
Mi,x
τ
pi (βi,x
)
"
#
1
1−σ
H 1−σ H
H 1−σ
PiF = MiF pFi (βiF )1−σ + Mi,x
τ
pi (βi,x
)
with
k
Mi,x
the mass of rms exporting from country k in sector i.
k
MS,x
=
Proposition:
[G(φSA,k ) − G(φkSx )] k
Ṁ
[G(φSA,k ) − G(φe,k ] S
The country with the comparative advantage in sector i has:
∆i > ∆j
Country H has the best institutions and therefore a comparative advantage at producing the
complex good A, we have :
PSF,A
PAF,A
<
PSF,CT
PAF,CT
PSH,A
PSH,CT
PSF T
< F T < H,CT < H,A
PA
PA
PA
This implies:
τ
1−σ
PAH,CT
PAF,CT
σ−1
=
∆H
A
>
23
∆H
S
=τ
1−σ
PSH,CT
PSF,CT
σ−1
3.4.3 The free-entry condition
Given the rms dynamics that we have already described in the previous sections, we derive the
rm entry condition for the costly trade equilibrium:
V = [1 − G(φe )]π̄ = wfe
with V the ex-ante utility of the rm over time,
fe
π̄
the average ex-post prot in the economy and
the xed cost to pay initially to draw a productivity level.
1
SA
Ax ¯A
SA ¯A
SA
e ¯S
¯
S
[G(φ ) − G(φ )]πd + [1 − G(φ )]πd + [G(φ ) − G(φSx )]πx + [1 − G(φ )]πx = wfe
V =
δ
In the annex we give the expressions of each part of this expression that is crucial to determine
the equilibrium thresholds. They dier for the country H and the country F.
3.4.4 The market clearing conditions
The good market condition
We have:
H
for the country H
Rs,d
H
Ra,d
F = αRH
+ Rs,x
F = (1 − α)RH
+ Ra,x
H = αRF
+ Rs,x
F
H
F
Ra,d + Ra,x = (1 − α)R
F
for the country F
with
k
Ri,d
Rs,d
the revenue spent by consumers in the country k in the sector i that is produced
domestically (by rms from country k) and
k
Ri,x
the revenue spent by consumers in the country
k in the sector i that is imported from the other country (produced by rms from the other
country).
The labor market condition
The equilibrium
The equilibrium thresholds the following system of equations:
24

(F E)
[1 − G(φe )]π̄ = wfe







(def φe )
π S (φe ) = 0







 (def φSA )
π S (φe ) = π A (φSA )


(def φSx )








(def φAx )





(28)
πxS (φSx ) = 0
πxA (φAx ) = 0
market clearing conditions
Proposition:
In the country with the best institutions and the comparative advantage in the
complex good, the entry threshold when there is costly trade increases compared to the autarky
equilibrium.
The total aggregate productivity increases due to reallocations of labor towards
the most productive rms (Melitz eect).
Proposition:
In the country with the best institutions and the comparative advantage in
the complex good, the choice threshold decreases compared to autarky such that the aggregate
productivity in the complex sector decreases.
Proposition:
In the country with the worst institutions and the comparative advantage in the
simple good, the entry threshold when there is costly trade can either decrease or increase compared to the autarky equilibrium. The choice threshold decreases. Firms that were producing
the complex good in autarky now produce the simple good and export part of their production.
This unusual eect is due to reallocations of labor from the most productive rms towards the
exporting rms that are less productive (opposite of Melitz eect).
Proof: This can be seen by using the FE condition. All the details are in annex
We can now write the expression of the ex-ante innite utility
V CT
for the rm in the costly
trade equilibrium.
We dene
V CT
as a function but also the value at the equilibrium such that
CT,SA CT
V CT = V CT (φCT,e , φCT
, φAx ) = ωfe
Sx , φ
.
⇒ V CT
I(expS)
= V AU T (φe , φSA )
Z ∞ h A SA i
i
hZ ∞ h
β (φ ) σ−1
β A (φSA )β S (φe ) iσ−1
g(φ)dφ − ωfx
g(φ)dφ]
+ I(expS)[ωf
∆S A
S
SA
β (φ)β (φ )
β A (φ)
φSA
φSA
o
+ [G(φSA ) − G(φSx )]π¯xS + [1 − G(φAx )]π¯xA = ωfe
I(expS) = 1 if the rm
I(expS) = 0 otherwise.
is an indicator dened by
exports in the sector S and by
25
with the productivity threshold
φSA
We can prove (annex) that the additional part due to exports in S of the rm with the productivity threshold
φSA
with
I(expS) = 1
is positive.
This gives us our rst result regarding the thresholds when comparing CT and autarky.
1. If the sector choice threshold
φSA
increases when trading then the entry threshold
φe
can
either increase or decrease.
2. If the sector choice threshold
φSA
decreases when trading then the entry threshold
φe
necessarily increases.
Figure 9: Change in thresholds for the country with good institutions
Figure 10: Change in thresholds for the country with poor institutions
3.5
The welfare analysis
3.5.1 Prots of productive rms
Result:
In the country with the comparative advantage in the simple good, the rms that
produce the complex good but do not export lose with trade compared to the autarky case.
This implies that the most productive rm in the sector of simple goods lose with trade even if
it benets a lot from exporting in the comparative-advantage sector.
Proof:
26
One of the previous results was that the best rms always produce the complex good, even in
the country with the comparative advantage in the simple good.
The prot function is given by:
P i ρ σ−1
R
π (φ) =
− wf
2σ wβ i (φ)
i
Therefore the prot for rm
φ
(29)
depends on the price. The aggregate price only depends on the
equilibrium thresholds. In the country with poor institutions, the choice threshold increase so
the price of the complex good decreases. In addition the exports from the good-institution country also decrease the aggregate price, assuming that the institutional dierence is high enough
between the two countries. If the aggregate price of the complex good decrease, the prot of all
rms whose productivity is between
φSA,CT
and
φCT
Ax
decreases with trade compared to autarky.
From this we can also conclude that the rms on the left of the threhsold
φSA,CT
that produce
and export in the sector of simple goods also lose compared to the autarky case in which they
would have produced the complex good.
The prots of the rms that produce and export the complex good can't be compared in autarky and under costly trade. Indeed, the revenues from exporting might compensate the loss
of prots that aects the domestic producers.
3.5.2 The size of the rms
The average size of the rms is given by the average revenues:
Rk = M k,AU T¯˙rk,AU T
Rk = M k,CT¯˙rk,CT
Result The average revenues and therefore the average size of the rms is higher in the country
with the best institutions and the comparative advantage in the complex good.
3.5.3 The real consumption wages
W k,AU T =
W k,CT =
Result:
ω
(PSk,AU T )α (PAk,AU T )(1−α)
ω
(PSk,CT )α (PAk,CT )(1−α)
The real consumption wage might decrease in the country with the worst institutions.
proof: this result comes from the rise in
PSCT
due to a possible decrease in the entry threshold in
the country with the comparative advantage in the simple good. Previously, it was proved that
this threshold might decrease and therefore increase the price of the simple good. In addition
we proved that the other country only exports complex goods. This eect might increase the
aggregate price
(PSCT )α (PACT )(1−α)
and therefore decrease
27
W CT .
3.6
Numerical applications
In this part, we use numerical applications to nd the conditions under which the previous
negative eects of trade might apply.
We use a Gamma distribution as the initial distribution of productivity for rms.
Figure 11: Distribution of initial productivity
There are two interesting parameters that aect the gains form trade for each country: the difference in the quality of institutions between the two countries and the dierence in complexity
between the simple and the complex goods. We focus on the parameter featuring the complexity of production. z represents the number of intermediate goods that have to be supplied for
the production of the nal good S or A. We show that the higher the number of intermediate
goods of the complex good, the larger the gains for the two countries. Indeed the benets from
specialization are higher when the complex goods is very dicult to produce for the country
with the poor institutions.
3 cases are now presented:
1. high complexity:
gains are for both countries and both average productivities increase
with trade
2. medium complexity: the average productivity decreases in the country with the CA in the
simple good but there are gains for both country
3. low complexity: the average productivity and the real consumption wage decrease in the
country with the CA in the simple good
Common parameters:
Parameters
h
σ
zS
f
fe
fx
τ
L̄
Values
10
2.5
10
5
10
15
1.5
500
Table 2: Common parameters
28
Country
φAut
e
φAut
SA
M Aut
P Aut
φCT
e
H
13
25
33.7
0.34
66
F
17
31
35.5
1.68
18
Country
φAut
e
φAut
SA
M Aut
P Aut
φCT
e
H
11
23
30.2
0.40
60
F
14
27
27.4
0.67
12
Country
φAut
e
φAut
SA
M Aut
P Aut
φCT
e
H
11
23
33.6
0.06
64
F
13
25
34.3
0.09
8
φCT
Sx
φCT
SA
φCT
Ax
M CT
P CT
70
76
22.9
0.33
56
66
25.8
0.27
φCT
SA
φCT
Ax
M CT
P CT
62
70
21.3
0.3
46
75
22.4
0.1
φCT
SA
φCT
Ax
M CT
P CT
66
76
52
70
32
Table 3: high complexity z=1000
φCT
Sx
20
Table 4: medium complexity z=500
φCT
Sx
20
0.6
34.1
0.8
Table 5: low complexity z=100
Observation 3 The interesting conclusion of this numerical exemple is that the entry thresholds
decreases with respect to autarky when the complexity of the complex good decreases. The
average productivity of the country specializing in the simple good decreases. In the last case,
the price increases according to what has been said before and the welfare-the real consumption
wage- of this country decreases.
φe
Observation 4 In the rst two cases the average size of rms increases (M decreases) with
trade whereas this eect is not clear any more in the last case. This shows that the usual positive
eects of trade are counterbalanced by a negative impact of specialization in the sector of the
simple good for rms.
Observation 5 The gain from specialization is high when one good is very complex to produce
and become very expensive to produce for the country with the bad institutions.
Observation 6 When going closer to the technological frontier, the country with poor institu-
tions might loose from specializing in the simple good.
29
References
Acemoglu, Daron, Pol Antras, and Elhanan Helpman, Contracts and Technology Adoption,
The American Economic Review, 2007, 97 (3), 916943.
Bernard, Andrew, Stephen Redding, and Peter Schott,
Heterogeneous Firms,
Comparative Advantage and
Review of Economic Studies, 2007, 74, 3166.
Costinot, Arnaud, On the origins of comparative advantage,
, On the origins of comparative advantage,
September 2007. mimeo.
Journal of International Economics,
2009,
77
(2), 255264.
Dixit, Avinash and Joseph Stiglitz,
Diversity,
Monopolistic Competition and Optimum Product
The American Economic Review, 1977, 67 (3), 297308.
Dornbusch, Rudiger, Stanley Fischer, and Paul Samuelson,
Comparative Advantage,
Trade, and Payments in a Ricardian Model with a Continuum of Goods,
Economic Review, 1977, 67 (5), 823839.
Krugman, Paul,
The American
Scale Economies, Product Dierentiation, and the Pattern of Trade,
American Economic Review, 1980, 70 (5), 950959.
The
Levchenko, Andrei, Institutional Quality and International Trade, The Review of Economic
Studies, 2007, 74 (3), 791819.
Melitz, Marc,
Productivity,
The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry
Econometrica, 2003, 71 (6), 16951725.
Nunn, Nathan,
Relationship-Specicity, Incomplete Contracts, and the Patterns of Trade,
The Quarterly Journal of Economics, 2007, 122 (2), 569600.
30
Appendices
A
Proofs
Proof of Proposition 1
See Costinot (2007).
Proof of Proposition 4
Existence in equilibrium of
φe
and
φe
such that
π S (φe ) = π A (φ∗A ) = 0
is a trivial corollary of
Observation 2.
Assume now by contradiction that
φe > φ∗A .
Then,
∀ φ+ > φSA we have that π S (φ+ ) > π A (φ+ ).
Using equation (29) and after some algebra we get the following
π S (φ+ ) > π A (φ+ ) ⇐⇒
Analogously,
∀ φ− < φSA
we have that
PS
β S (φ+ )
> A +
A
P
β (φ )
π S (φ− ) < π A (φ− ).
π S (φ− ) < π A (φ− ) ⇐⇒
(30)
As before
β S (φ− )
PS
<
PA
β A (φ− )
(31)
Combining the two conditions (30) and (31) we get
β S (φ− )
β S (φ+ )
>
β A (φ− )
β A (φ+ )
Dening the function
B(φ) :=
β S (φ)
we can show that
β A (φ)
(32) and completes the proof.
31
B 0 (φ) > 0.
(32)
This contradicts condition
A.1
The autarky equilibrium
The prices
Given this notation, we can dene
the aggregate price index for the sector
∞
Z
i
Pi
i
P =
p (φ)
1−σ
i i
i ∈ {S, A}.
1
1−σ
∀i ∈ {S, A}
M µ (φ)dφ
0
with
MS =
[G(φSA )−G(φe )]
M and
[1−G(φe )]
of the two sectors. Among them
SA
)]
M A = [1−G(φ
[1−G(φe )] M with M the mass of rms producing in one
M S produce a variety of the simple good S and M A a variety
of the complex good A.
Substituting the prices with the rms' optimal pricing rule and rearranging we get
i
P = M
i
1
1−σ
w
ρ
Z
∞
i
β (φ)
1−σ
i
1
1−σ
∀i ∈ {S, A}
µ (φ)dφ
(33)
0
|
{z
}
:=β̃ i
We recognize in the equation above the pricing rule with the average version
β̃ i
of the marginal
cost. We can thus rewrite
PS = MS
1
1−σ
pS β̃ S
PA = MA
and
1
1−σ
pA β̃ A .
(34)
The equilibrium average marginal cost in the two sectors is determined by the cuto productivity
levels
φe
and
φSA :
S
e
SA
β̃ (φ , φ
1
)=
SA
G(φ ) − G(φe )
Z
φSA
1−σ
β (φ)
g(φ)dφ
S
β̃ (φ
SA
1
)=
1 − G(φSA )
Z
1
1−σ
(35)
φe
and
A
∞
A
β (φ)
1−σ
g(φ)dφ
1
1−σ
(36)
φSA
The prots and revenues
We can derive the aggregate prot and revenue for the whole economy.
Z
Π=
∞
π(φ)M µ(φ)d(φ)
(37)
0
G(φSA ) − G(φe ) ¯S 1 − G(φSA ) ¯A
Π=M
π +
π
1 − G(φe )
1 − G(φe )
32
(38)
with
πS
and
πA
the average prots dened by the following expressions:
R φSA
π¯S =
π¯A =
φe
π S (φ)g(φ)d(φ)
(39)
[G(φSA ) − G(φe )]
R∞ A
φSA π (φ)g(φ)d(φ)
(40)
[1 − G(φSA )]
We can now derive average revenues and prots as functions of the productivity cutos:
"
β̃ S (φe , φSA )
r̄S = rS (β̃ S (φe , φSA )) =
|
{z
}
β S (φe )
#1−σ
rS (β S (φe ))
| {z }
or rS (φe )
or rS (β̃ S )
"
A
A
A
r̄ = r (β̃ (φ
SA
β̃ A (φSA )
)) =
β A (φSA )
#1−σ
rA (β A (φSA ))
and
"
β̃ S (φe , φSA )
π̄ S = π S (β̃ S ) =
β S (φe )
#1−σ
"
rS (φe )
−wf
σ
,
β̃ A (φSA )
π̄ A = π A (β̃ A ) =
β A (φSA )
#1−σ
rA (φSA )
−wf
σ
Using the following relations:
 S e
S e

 π (φ ) = 0 ⇐⇒ r (φ ) = σwf

 π S (φSA ) = π A (φSA ) ⇐⇒ rA (φSA ) =
h
β S (φe )
β S (φSA )
i1−σ
(41)
σwf
Finally we have the average prots for the two sectors:

nh S e SA i1−σ
o
β̃ (φ ,φ )

S = wf

−
1
π̄

S
e
β (φ )


#1−σ
)
("

A (φSA ) β S (φSA )
β̃

A

−1

β A (φSA ) β S (φe )
 π̄ = wf
(42)
The free-entry condition
Given the rms dynamics as described in Melitz (2003) we derive the rm entry condition:
V = [1 − G(φe )]π̄ = wfe
with V the ex-ante utility of the rm over time,
fe
π̄
the average ex-post prot in the economy and
the xed cost to pay initially to draw a productivity level.
33
(43)
V =
1
[G(φSA ) − G(φe )]π¯S + [1 − G(φSA )]π¯A ] = wfe
δ
(44)
By reorganizing this expression and using the previous relations, we have:
V =
nh β̃ S (φe , φSA ) i1−σ o
nh β̃ A (φSA ) β S (φSA ) i1−σ oo
wf n
SA
[G(φSA )−G(φe )]
−1
+[1−G(φ
)]
−1
= wfe
δ
β S (φe )
β A (φSA ) β S (φe )
(45)
The dynamics of the entry/exit of rms
We can now describe the dynamics of entry and exit of rms. A proportion
rms
M−1
disappears at the beginning of each period.
δ
of the incumbent
In addition there is a number
potential new rms that pay the xed entering cost and receive a productivity level
them, only the ones with a productivity level higher than
φe
φ.
Me
of
Among
enter the production process.
Finally there is a number M of rms producing this period in one of the two sectors.
M = (1 − δ)M−1 + (1 − G(φe ))Me
At the stationary equilibrium
M = M−1 ,
we have:
[1 − G(φe )]M e = δM−1
(46)
The labor market condition
Labor is used to enter the production process as well as to produce. The economy has a popula-
Lei denotes the total amount of workers used in the entry process in sector i and
p
Li denotes the total amount of workers used for production in sector i. Only the productivity
e
level determines the choice of the sector such that L is not sector-specic.
tion of workers.
The labor market clearing conditions are:
Le + Lp = L̄
with
Lp = LpS + LpA
Every period, each rm in sector i with a productivity level
duce the quantity
q(φ)
φ
needs
(47)
β i (φ)q̇(φ)
workers to pro-
of goods.
Then, the number of workers required in sector i is:
Lpi
=M
i
Z
β i (φ)
34
ri (φ) i
µ (φ)dφ
p(φ)
(48)
ri (φ)
p(φ)
We use the following relation
= αi R[ wβρi (φ) ]σ P iσ−1 .
The nal clearing labor market condition is:
X
Lpi +Le
ρ
= MR
w
!σ " Z
α
φSA
φe
P Sσ−1
g(φ)d(φ)+(1−α)
β S (φ)σ
Z
∞
φSA
#
P Aσ−1
g(φ)d(φ) +M f˙+Me f˙e = L̄
β A (φ)σ
(49)
L̄
is exogenously given as the total number of workers in the economy.
The good market condition
RS = αR
A.2
and
RA = (1 − α)R
(50)
Proof unicity in autarky
Then in order to nd the equilibrium thresholds, we use the following system:



 (F E)
wf
δ


 (def φSA )
(
h
β̃ S (φe ,φSA )
β S (φe )
i1−σ
[1−G(φSA )]
[G(φSA )−G(φe )]
)
[G(φSA )
−
G(φe )]
−1+
G(φe )
= wfe
(51)
n
β̃ S (φe ,φSA )
β̃ A (φSA )
oσ−1
=
n
β S (φSA )
β A (φSA )
oσ−1
In the annex we prove that this system denes a unique pair of thresholds
(φe , φSA ).
We dierentiate the second equation such that we get:
dφe
β S (φSA )1−σ g(φSA )β S (φe )1−σ
=
Re
SA
dφ
β S (φe )2−2σ g(φe ) + (1 − σ)β̇ S 0 (φe )β S (φe )−σ (˙ φ φSA β S (φ)g(φ)dφ)
⇒
Implicit function theorem
→
function
(52)
χ.
We dene the function k, k1 and k2 such that
R∞
A
1−σ g(φ)dφ
φSA β (φ)
k1(φSA ) = R e φSA S 1−σ
β (φ)
g(φ)dφ
φ
k2(φSA ) =
k(φSA ) = k1(φSA )k̇2(φSA )
We have
n β S (φSA ) oσ−1
(54)
β A (φSA )
such that equation 2
0
k2 (φSA ) < 0.
What about
(53)
0
k1 ?
35
⇔ k(φSA ) = 1
(55)
We note
IntS =
0
k1 (φSA ) =
R e φSA
φ
β S (φ)1−σ g(φ)dφ
and
IntA =
R∞
φSA
β A (φ)1−σ g(φ)dφ,
˙ S (φSA )1−σ g(φSA ) − β A (φe )1−σ g(φSA ) ∂φe )
−β A (φSA )σ−1 g(φSA )˙[IntA] − [IntS](β
∂φSA
[IntS]2
(56)
Using Matlab we can prove that there exists a unique solution for this system. This equilibrium
pair of thresholds gives us the prices, the revenues, and the masses of rms (entering and exiting).
A.3
Free-trade equilibrium
A.3.1 The labor market condition-similar to autarky
The clearing labor market condition is:
Lpi + Le
" Z SA
#
Z ∞ Aσ−1
φ
R− ρ σ
P Sσ−1
P
= MR 1 +
g(φ)d(φ) + (1 − α)
g(φ)d(φ) + M f˙ + Me f˙e
( ) α
A
σ
R w
β S (φ)σ
φSA β (φ)
φe
L̄ =
X
A.4
Costly trade equilibrium
Proposition:
The country with the comparative advantage in the complex industry A does not
export goods from the other sector S.
Proof: We use the previous result
∆A > ∆S
for the country with the comparative advantage in
sector A and we show in annex that there is a contradiction when we assume that this country
exports in the sector S.
What about
∆S ?
First,
rxS (φSx ) = σωfx = ∆S rdS (φSx )
and
rdS (φSx ) =
Therefore
∆S =
h β S (φe ) iσ−1
σωf
β S (φSx )
h β S (φ ) iσ−1 f
x
Sx
S
e
β (φ )
f
We have:
Assuming
φSA
exports in the sector S and by denition of
sectors):
36
φSA
(equalization of prots in both
ΠS (φSA )
=
⇒
with
⇒
ΠA (φSA )
rdA (φSA )
rS (φSA )
− ωf = (1 + ∆S ) d
− ω(f + fx )
σ
σ
rxS (φSA ) = ∆S rdS (φSA )
h β S (φe ) iσ−1
rdA (φSA ) = (1 + ∆S ) S AS
σωf − σωfx
β (φ )
rdA (φSA )
h β S (φe ) iσ−1
β S (φAS )
h β S (φe ) iσ−1
⇒
β S (φAS )
=
h β A (φAS ) iσ−1
rdA (φSA )
β A (φAx )
h β A (φAS )β S (φe ) iσ−1
h β A (φAS ) iσ−1
(1
+
∆
)σωf
−
σωfx
⇒ rdA (φAx ) =
S
β A (φAx )β S (φAS )
β A (φAx )
rdA (φAx ) =
rxA (φAx ) = σωfx = ∆A rdA (φ)
h β S (φe ) iσ−1 f
i
h β A (φAS ) iσ−1 h
(1 + ∆S ) S AS
−1 =1
⇒ ∆A A
β (φAx )
β (φ )
fx
with
f
fx
=
h
β S (φSx )
β S (φe )
iσ−1
1
∆S
h β A (φAS ) iσ−1 h (1 + ∆ ) h β S (φ ) iσ−1
i
S
Sx
−
1
=1
β A (φAx )
∆S
β S (φAS )
(1 + ∆S ) h β S (φSx ) iσ−1
1 h β A (φAx ) iσ−1
−
1
=
∆S
β S (φAS )
∆A β A (φAS )
h β S (φ ) iσ−1
1 h β S (φSx ) iσ−1
1 h β A (φAx ) iσ−1
Sx
+
{
−
1}
=
∆S β S (φAS )
β S (φAS )
∆A β A (φAS )
⇒ ∆A
⇒
⇒
Given that
h
β S (φSx )
β S (φAS )
iσ−1
− 1 > 0,
then:
1 h β S (φSx ) iσ−1
1 h β A (φAx ) iσ−1
<
∆S β S (φAS )
∆A β A (φAS )
h β S (φ ) iσ−1 h β A (φ ) iσ−1
Sx
Ax
⇒
< A AS
β S (φAS )
β (φ )
because
∆A > ∆S
by denition hof the comparative
i
hadvantage
i in A.
This is a contradiction because
β S (φSx )
β S (φAS )
>1
and
37
β A (φAx )
β A (φAS )
< 1.
A.5
FE in costly trade equilibrium
Given the rms dynamics that we have already described in the previous sections, we derive the
rm entry condition for the costly trade equilibrium:
V = [1 − G(φe )]π̄ = wfe
with V the ex-ante utility of the rm over time,
fe
π̄
the average ex-post prot in the economy and
the xed cost to pay initially to draw a productivity level.
1
SA
Ax ¯A
SA ¯A
SA
e ¯S
¯
S
[G(φ ) − G(φ )]πd + [1 − G(φ )]πd + [G(φ ) − G(φSx )]πx + [1 − G(φ )]πx = wfe
V =
δ
We can now give the expression of each part of this expression that is crucial to determine the
equilibrium thresholds.
ˆ
The average domestic revenue in S
[G(φ
ˆ
SA
) − G(φ )]π¯dS = ωf
e
"
h β S (φe ) iσ−1
β S (φ)
φe
#
− 1 g(φ)dφ
The average domestic revenue in A
In country F, the rm with
[1 − G(φSA )]π¯dA
SA
φSA
exports in the sector S
"
#
h β A (φSA )β S (φe ) iσ−1
(1 + ∆S ) A
= ωf
− 1 g(φ)dφ
β (φ)β S (φSA )
φSA
Z ∞ h A SA i
β (φ ) σ−1
−ωfx
g(φ)dφ
β A (φ)
φSA
Z
In country H, the rm with
[1 − G(φ
ˆ
φSA
Z
∞
φSA
)]π¯dA = ωf
does not export in the sector S
Z
∞
"
h β A (φSA )β S (φe ) iσ−1
β A (φ)β S (φSA )
φSA
#
− 1 g(φ)dφ
In country F only, the average export revenue in S
[G(φ
SA
) − G(φSx )]π¯xS = ωfx
Z
φSA
φSx
38
"
#
h β S (φ ) iσ−1
Sx
− 1 g(φ)dφ
β S (φ)
ˆ
In both countries H and F, the average export revenue in A
[1 − G(φAx )]π¯xA = ωfx
∞
Z
"
#
iσ−1
)
Ax
− 1 g(φ)dφ
β A (φ)
h β A (φ
φAx
This gives us the full expression of V in the costly trade equilibrium.
Details of the computation:
2.1) Assuming
φSA
[1 − G(φ
exports in the sector S
SA
∞
h
i
β A (φSA )β S (φe ) iσ−1
({1 + ∆S ) A
−
1}g(φ)dφ
β (φ)β S (φSA )
φSA
Z ∞ h A SA i
β (φ ) σ−1
g(φ)dφ
−ωfx
β A (φ)
φSA
)]π¯dA = ωf
hZ
Proof:
Assuming
φSA
exports in the sector S and by denition of
φSA
(equalization of prots in both
sectors):
ΠS (φSA )
=
⇒
with
ΠA (φSA )
rdA (φSA )
rS (φSA )
− ωf = (1 + ∆S ) d
− ω(f + fx )
σ
σ
rxS (φSA ) = ∆S rdS (φSA )
β A (φSA ) σ−1 A SA
)
rd (φ )
β A (φ)
β S (φe ) σ−1
S
and rd (φ) = (
)
ωf
β S (φSA )
h β A (φSA )β S (φe ) iσ−1
h β A (φSA ) iσ−1
A
⇒ rd (φ) =
(1 + ∆S )σωf −
σωfx
β A (φ)β S (φSA )
β S (φ)
rdA (φ) = (
2.2) Assuming
φSA
does not export in the sector S
[1 − G(φSA )]π¯dA = ωf
∞
i
h β A (φSA )β S (φe ) iσ−1
{ A
−
1}g(φ)dφ
β (φ)β S (φSA )
φSA
hZ
3)
[G(φSA ) − G(φSx )]π¯xS = ωfx
Z
φSA
{
φSx
Proof:
39
h β S (φ ) iσ−1
Sx
− 1}g(φ)dφ
β S (φ)
We have
rxS (φ) = ∆S rdS (φ)
(denition of
∆S
)
h β S (φ ) iσ−1
rdS (φ)
rxS (φ)
Sx
S
⇒
r
(φ)
=
=
σωfx
x
rxS (φSx )
β S (φ)
rdS (φSx )
A.6
Results in equilibrium
Proposition:
In the country with the worst institutions and the comparative advantage in the
simple good, the entry threshold when there is costly trade can either decrease or increase compared to the autarky equilibrium. The choice threshold decreases. Firms that were producing
the complex good in autarky now produce the simple good and export part of their production.
This unusual eect is due to reallocations of labor from the most productive rms towards the
exporting rms that are less productive (opposite of Melitz eect).
Proof: This can be seen by using the FE condition.
From the autarky part, we have the expression of the ex-ante innite utility
V AU T
for the rm
in autarky :
V AU T (φe , φSA ) =
nh β̃ S (φe , φSA ) i1−σ o
nh β̃ A (φSA ) β S (φSA ) i1−σ oo
ωf n
SA
[G(φSA )−G(φe )]
−1
+[1−G(φ
)]
−1
δ
β S (φe )
β A (φSA ) β S (φe )
(57)
At the autarky equilibrium we have
V AU T (φA,e , φA,e ) = ωfe
.
We can now write the expression of the ex-ante innite utility
V CT
for the rm in the costly
trade equilibrium.
We dene
V CT
as a function but also the value at the equilibrium such that
CT,SA CT
V CT = V CT (φCT,e , φCT
, φAx ) = ωfe
Sx , φ
.
V CT
o
nh β̃ A (φSA ) β S (φSA ) i1−σ
o
nh β̃ S (φe , φSA ) i1−σ
ωf n
SA
[G(φSA ) − G(φe )]
−
1
+
[1
−
G(φ
)]
−
1
δ
β S (φe )
β A (φSA ) β S (φe )
Z
Z
h ∞ h
i
∞ h A SA iσ−1
β A (φSA )β S (φe ) iσ−1
β (φ )
+ I(expS)[ωf
∆S A
g(φ)dφ
−
ωf
g(φ)dφ]
x
S (φSA )
β
(φ)β
β A (φ)
SA
SA
φ
φ
o
+ [G(φSA ) − G(φSx )]π¯xS + [1 − G(φAx )]π¯xA = ωfe
=
40
⇒ V CT
I(expS)
= V AU T (φe , φSA )
Z ∞ h A SA i
hZ ∞ h
i
β A (φSA )β S (φe ) iσ−1
β (φ ) σ−1
+ I(expS)[ωf
∆S A
g(φ)dφ
−
ωf
g(φ)dφ]
x
S
SA
β (φ)β (φ )
β A (φ)
φSA
φSA
o
+ [G(φSA ) − G(φSx )]π¯xS + [1 − G(φAx )]π¯xA = ωfe
I(expS) = 1 if the rm
I(expS) = 0 otherwise.
is an indicator dened by
exports in the sector S and by
with the productivity threshold
φSA
We can proove that the additional part due to exports in S of the rm with the productivity
threshold
φSA
with
We have (cf above)
I(expS) = 1
∆S =
∞
h
is positive:
β S (φSx )
β S (φe )
iσ−1
fx
f
Z ∞ h A SA i
i
β A (φSA )β S (φe ) iσ−1
β (φ ) σ−1
A = ωf
∆S A
g(φ)dφ
−
ωf
g(φ)dφ]
x
β (φ)β S (φSA )
β A (φ)
φSA
φSA
Z ∞ hh A SA S
β (φ )β (φSx ) iσ−1 h β A (φSA ) iσ−1 i
g(φ)dφ
−
⇒ = ωfx
β A (φ)β S (φSA )
β A (φ)
φSA
hZ
h
is a decreasing function and
In addition
π¯xS > 0
and
φSA > φSx
so
A > 0.
π¯xA > 0.
From all this, we can conclude that
V CT > V AU T .
This gives us our rst result regarding the thresholds when comparing CT and autarky.
1. If the sector choice threshold
φSA
increases when trading then the entry threshold
φe
can
either increase or decrease.
2. If the sector choice threshold
φSA
decreases when trading then the entry threshold
necessarily increases.
41
φe