Technology and the law of comparative
advantage
Antonio Navas
The University of She¢ eld
PRELIMINARY AND INCOMPLETE.
August 7, 2014
Abstract
This paper explores how trade openness a¤ects both product and
process innovation in a standard model of trade with …rm heterogeneity and factor-endowment-driven comparative advantage. Trade openness
expands the pro…t opportunities of the most productive …rms and expels
the less e¢ cient …rms out of the market, making process innovation more
attractive for the most productive …rms in both industries. Incentives,
however, are larger in the industry in which the country has the comparative advantage. Trade also increases the pro…ts of prospective entrants
what it leads to an increase in product innovation in the comparative
advantage industry. In addition I obtain a non-monotonic relationship
between trade costs and the country’s trade pattern: When the level of
trade costs are reasonably high, a reduction in trade costs leads to an
increase in process innovation and R&D intensities in both industries, being stronger in the comparative advantage industry, however, if the trade
costs are lower enough, the e¤ect is stronger in the comparative disadvantage industry. This …nal result could rationalize recent empirical …ndings
suggesting that in the last half century comparative advantage has become
weaker over time.
Keywords: Innovation, Firm Heterogeneity, Comparative Advantage.
JEL Codes: F12, F43
1
Introduction
The Neoclassical trade theories emphasize the role of di¤erences in technology (Ricardo (1817), Jones (1961), Dornbusch and Fischer, (1977) and a more
recently contribution by Eaton and Kortum (2002)) or di¤erences in factor
endowments (Heckscher-Ohlin (1933), and more recent formulation by Vanek
The author would like to thank Chenglu Li for excellent research assistantship. Remaining
errors are mine.
1
(1968)) to explain trade ‡ows and trade patterns across countries. New trade
theories have incorporated scale economies at an industry and …rm level to
explain several features of the data that the neoclassical theories were …nding di¢ cult to explain like the intra-industry trade phenomenon, or the fact
that only a small proportion of …rms within each industry are able to export
(Melitz (2003), Bernard, Jensen, Eaton and Kortum (2002), Melitz and Ottaviano (2008), Chaney (2008)). While these theories have identi…ed di¤erent
institutional factors that may bust or dampen trade volumes across countries
and may a¤ect a country’s specialisation pattern still, technological and factor
endowment di¤erences across countries are at the heart of the determinants of
trade specialisation patterns. At the empirical level, several studies have tried
to dillucidate how these two channels are able to account for trade patterns and
trade volumes across countries. (Tre‡er 1993a, 1995).
The new new trade theory based on internal scale economies developed by
Melitz (2003) and Melitz and Ottaviano (2008), has outlined a new mechanism through which trade increases welfare in trading countries: The impact of
trade on technology through selection. Trade through tougher competition expels the less e¢ cient …rms out of the market reallocating market shares across
the most productive surviving incumbents. By concentrating the production
in the most e¢ cient industry units, this increases an industry’s average productivity. A recent paper by Bernard, Redding and Schott (2007) outlines the
importance of this mechanism to establish a link between the previously discussed two potential sources of comparative advantage: Di¤erences in factor
endowments through selection generates a Ricardian comparative advantage.
Tougher selection in the comparative advantage industry leads to a relatively
larger increase in the average productivity of that industry after trade openness.
In this paper I want to explore further the link between technology and
factor endowments, by expanding a 2x2x2 standard model of trade, factor proportions and …rm heterogeneity to allow …rms to upgrade their current state
of technology. In our setup there are two …nal good industries producing each
of them a continuum of di¤erentiated goods. The structure of each of the two
industries, is similar to Melitz (2003): Firms pay a …xed cost to create a new
variety which allows them to enter in the market. Associated with this variety
the …rm uses a linear technology to produce that involves the use of an industry speci…c intermediate input. However, the unitary input requirements are
uncertain for the …rm at the moment of entry. The industry-speci…c intermediate inputs are produced according to a Cobb-Douglas production function that
uses both skilled and unskilled labour in di¤erent proportions across both …nal
goods. After entry, the intermediate input unitary requirements are revealed to
the …rm and the …rm chooses whether to stay in the market and paying a per
period …xed cost to produce. Once the …rm has decided to stay, I assume that
it has the option to upgrade its current state of technology by paying a …xed
cost. In this framework I distinguish between process innovation (technology
2
upgrading) and product innovation (the creation of new varieties).
Our results suggest that the selection e¤ect found Melitz (2003) or Melitz
and Ottaviano (2008) leads to a rise in technology upgrading in both industries. Interestingly, technology upgrading is stronger in the industry in which
the country has a comparative advantage. The reason behind this result is
the fact that trade expands the business opportunities for the most productive
…rms in both sectors. However, trade expands it to a greater extent in the
comparative advantage industry since the economy is able to o¤er these goods
relatively cheaper than the foreign counterpart. This rises the expected pro…ts
of prospective entrants and induces a disproportionate entry in the comparative
advantage industry. As a consequence, the relative demand for the abundant
factor rises what it has a positive impact on the relative factor remuneration.
Domestic …rms see their pro…ts reduced and need to exit. The combination
of a stealing business e¤ect in the foreign country and a reallocation of market shares from the less e¢ cient …rms which exit, towards the most productive
ones, induce a larger proportion of …rms to upgrade their technology. Overall
technology upgrading rises in both industries and the comparative advantage
industry enjoys more product and process innovations compared to an autarkic
scenario. Using R&D intensities as a measure of innovation activity, my model
predicts that trade openness increases the relative R&D intensity favouring the
comparative advantage industry one. These last results reinforce the Ricardianled-factor endowment comparative advantage by having a positive e¤ect on the
within plant productivity improvements.
In a further section in the paper, I explore the evolution of the comparative
advantage by considering a reduction of trade costs when both industries are
opened to trade. The results establishes a non-monotonic relationship between
the level of trade costs and the evolution of comparative advantage: When the
trade costs are high, a reduction in trade costs increases technology upgrading
and R&D intensities relatively more in the comparative advantage industry.
This induces TFP divergence across sectors. However, if the trade costs are lower
enough a reduction in trade barriers increases technology upgrading and R&D
intensities in the comparative disadvantage industry leading to TFP convergence
across industries. Overall the average productivity, the proportion of …rms that
upgrade their technology and the R&D intensities increases in both industries
as trade costs fall. However, provided that there is self-selection into exporting
markets, these three dimensions are always larger in the comparative advantage
industry. This suggests that a gradual reduction in trade costs may eventually
strengthened the pattern of comparative advantage at the initial stages while
reducing it when the trade costs become su¢ ciently low.
This paper is related to several literatures. The …rst one is the literature on
the e¤ects of trade openness and trade liberalisation on innovation. A recent
literature based on models with …rm heterogeneity outlines the importance of
selection e¤ects in promoting process innovation (Atkeson and Burstein (2011),
3
Bustos (2011), Impulliti and Licandro (2011), Navas and Sala (2013) and Long
et al. (2011) among others). Unlike those papers we study the role played by
factor endowments in determining the e¤ect that trade has on innovation at the
industry level. My model suggests that di¤erences in factor intensities across
industries and the vector of local factor endowments the home country enjoys
relative to its trade partners, may generate asymmetries in productivity and
innovation across industries that in an autarkic equilibrium exhibit identical
productivity distributions. The second dimension is clearly a determinant of
the dynamic evolution of the industry’s average productivity.
This paper is related to the literature that incorporates di¤erences in factor endowments in models of trade with economies of scale.(Krugman (1981),
Helpman and Krugman (1985) (HK)). Those papers obtain the interesting result that most of the …ndings in the Heckscher-Ohlin model are also present
in an environment in which there are increasing returns to scale at the …rm
level. More recently, Bernard, Redding and Schott (2007) (BRS) incorporates
a factor proportions theory into a standard Melitz (2003) model of trade with
…rm heterogeneity and …nds that the same H-O results are also present when we
allow productivity to vary across …rms. They also …nd that di¤erences in factor
endowments together with trade openness can generate a Ricardian comparative advantage as explained above. My paper goes in line with the above papers
and reinforces the idea that the H-O results are robust to richer environments.
In addition, by including the possibility of …rms to technology upgrade, this
paper …nds that di¤erences in factor intensities across industries and factor endowments across countries generates a di¤erent impact on trade-induced plant
productivity improvements across …rms with the same initial productivity. This
result reinforces the early …ndings by BRS on Ricardian-led-factor endowment
comparative advantage by having an impact on average industry productivity
while it also suggests that this Ricardian advantage may persist along time.
Finally, this paper is related to a recent literature that investigates the evolution of comparative advantage. In a very interesting study, Levchenko and
Zhang (2013) obtain that, contrary to the common knowledge, in the last 50
years on average, productivity has increased by more in the countries’revealed
comparative disadvantage industries. Our paper suggests a non-monotonic relationship between trade costs and the pattern of comparative advantage and
for su¢ ciently low level of trade costs, a reduction in trade barriers may bene…t
the comparative disadvantage industry, narrowing the di¤erences in TFP across
industries within a country. The evidence of Levchenko and Zhang (2013) could
be consistent within this framework with a gradual reduction in trade barriers
across countries provided that the initial level of trade costs were su¢ ciently
low in the 1960s.
Section 2 of the paper describes the main elements of the model in autarky.
On section 3, I discuss the main properties of the model in free trade, obtaining
that as in BRS and HK the main results for the Heckscher-Ohlin model are held
4
in this environment. In section 4 I discuss the most realistic case in which the
economy is open to trade but trade is costly (both in variable and …xed trade
barriers). In section 5 I analyze the e¤ect of trade liberalisation (understood as
a reduction in trade barriers). Section 6 concludes.
2
The model
Consider an economy inhabited by a continuum of consumers. There are two
…nal goods. Let denote with Ci the consumption of good i = 1; 2: Each Ci is a
composite good de…ned over a continuum of varieties belonging to the set i :
Preferences over these goods are given by the following utility function:
U (C1 ; C2 ) = (C1 ) (C2 )
0
C1 = @
!
0
C2 = @
!
Z
Z
(q1 (!))
1
1
1
d! A
1
(q2 (!))
1
1
1
1
d! A
2
Solving the consumer’s problem we arrive to the standard CES aggregate
demand system for each variety of the composite good:
qi (!) =
Ri
Pi
pi (!)
Pi
where Ri denotes consumer’s expenditure dedicated to good i. Under CobbDouglas preferences Ri = i R where R denotes total revenue.
To produce, …rms use an intermediate input (xi ) that is homogeneous to all
products within the same industry but di¤er across industries. This intermediate input is produced competitively combining both capital and labor using the
following Cobb-Douglas technology:
with Ai = ( i )
i
(1
1
1
1
2
x1 = A1 (S1 )
1
(L1 )
x2 = A2 (S2 )
2
(L2 )
i)
1
i
:
We assume without loss of generalization that 1 > 2 that is the industry 1 uses intermediate inputs that are more skilled labour intensive. Perfect
competition in the intermediate input sector implies that:
pmi = (ws )
5
i
(wl )
1
i
The production side in the …nal good sector is identical to Melitz (2003). To
enter in the market, a …rm needs to invest fe units of the intermediate input to
create a new variety. Once the …rm has created this variety the …rm bbtains the
monopoly rights to produce it. Associated with this variety there is a technology
to produce which is linear in the intermediate input. More precisely:
qi ('i ) = 'i xi
However, …rms’productivity 'i is unknown before the creation of this new
variety. More precisely, the …rm knows that the productivity parameter ' follows a random process with support [0; 1) and a cumulative continuous distribution function G('): After the investment in the creation of this variety is
undertaken, the productivity is revealed to the …rm. The creation of new varieties of the same composite good is considered here as product innovation. To
operate the technology the …rm needs to pay a per period …xed investment of
fd units of the intermediate input. At this moment she needs to decide whether
to stay and produce.
Once the …rm decides to stay and produce we are going to consider that
the …rm has the possibility to adopt a new technology which improves their
productivity by a factor of i by investing a …xed amount fI of the intermediate
input of units. The process of a …rm’s technology upgrading will be denoted as
process innovation. In this version of the model we consider that all activities
within an industry (product, process innovation, production and exporting when
applies) uses the same intermediate input. Consequently, all activities within
an industry has the same factor intensity. However, these activities di¤er in
factor intensities across industries.
The …rms’ problem is solved by backward induction. First, I solve for the
…rm’s decision of technology upgrading. Then I solve for the the …rm’s decision
of staying in the market taking into account its future decision of technology
upgrading. Finally, I solve for the …rm’s entry decision, taking into account the
‡ow of expected pro…ts in the industry.
Since the variety produced by each …rm is unique, a …rm charges the standard
monopoly price:
pmi
!i
=
pi (') =
1 '
1 ( i )d '
where d is the indicator function taking the value of 1 if the …rm adopts the
1
i
new technology and ! i = (ws ) i (wl )
. The variable pro…t associated to a
…rm will be given by the following expression:
vi (')
=
Ri
1
(Pi )
d
( i) '
1
1
(! i )
=
rid (')
:
Notice that the ratio between the revenue of two non innovative …rms is
1
'i
id ('i )
given by productivity ratio as in Melitz (2003) ( rrid
):
(' ) = '
j
6
i
A …rm decides to adopt the new technology i¤:
( i)
1
rid (')
1
fiI ! i
(1)
with equality if the …rm is indi¤erent between adopting the technology or
staying with its current technology. Let denote with 'iI the value of the productivity of the indi¤erent …rm, which we call the marginal innovator.
The …rm is indi¤erent between staying or exiting the market when:
rid ('id )
= fiD ! i
(2)
This condition is known in the Melitz (2003) model as the ZP condition.
Dividing (1) and (2) we have that:
'iI
'iD
1
fiI
fiD
=
1
( i)
1
1
Notice that the proportion of surviving …rms undertaking process innovation
is independent of the factor prices and therefore on factor endowments in autarky. This is the consequence of the fact that both activities are using the same
intermediate input and therefore they use the production factors with the same
intensity. Allowing for di¤erences in factor intensities across activities within
an industry breaks this result. We leave this possibility for future research.
Finally, a …rm decides to enter in the industry i¤ E(V )
fei ! i : In this
model we focus on steady state solutions. In steady state a …rms’value function
is given by the following expression:
Vi = max 0;
2.1
vi
(')
;
vi
( ')
fiI ! i
(3)
Equilibrium in a Closed-Economy Model
One of the interesting properties of the Melitz-type models is that the equilibrium of the economy, in our case perfectly characterized by the two productivity thresholds 'iI ; 'iD , can be summarised with two conditions: the Zero Pro…t
Condition (ZP) (Condition (2) in our model) and the Free Entry condition (FE).
In this framework however, we need an extra equation that comes from the Zero
Innovation Pro…ts condition (Condition (1) in our model). The FE condition in
this model becomes:
!
!
1
1
(1 G('iI ))
'
~ iI
fei
'
~ iD
1 fiD +
1 fiI =
:
'iD
(1 G ('iD ))
'iI
(1 G('iD ))
7
The left hand side (lhs) of our FE condition is similar to a standard heterogenous…rm trade model. There is, however, an extra term, the second one, which represents the innovation pro…ts. The possibility of technology upgrading increases
the expected value of pro…ts from entry by increasing the pro…ts of the most
productive …rms. The appendix shows that there exists a unique productivity
threshold level 'iD and consequently a unique value for 'iI : Compared to Melitz
(2003), the possibility of technology upgrading reallocates market shares from
the less productive …rms to the most productive ones, making survival more
di¢ cult in this economy. Consequently, 'iD is larger in this case.
Despite the fact that this model exhibits a larger average productivity due
to the …rm’s possibility of technology upgrading, both sectors share the same
productivity thresholds, 'iD ; 'iI and consequently the same average productivity, provided that the rest of the parameters are identical in both industries.
In autarky, di¤erences in factor endowments across countries are not generating di¤erences in average productivity across industries.1 In the skilled-labour
abundant country, initially …rms have larger expected bene…ts in the comparative advantage industry (Industry 1) because marginal costs of production in
that industry are relatively smaller. Consequently, …rms can charge relatively
lower prices and have relatively larger sales. However, the costs of entry are also
smaller in that industry, and this together with the rise in the expected pro…ts
of the representative …rm increases entry. These two e¤ects o¤set the positive
e¤ect that the comparative advantage mechanism is having on …rms’pro…ts.
However, as discussed above, there is more entry in the industry in which the
economy has the comparative advantage. Thus, the model generates di¤erences
in the mass of surviving …rms in equilibrium. To see this, notice that:
'
~ 1D
'1D
R1 r2
M1
=
=
M2
R2 r1
1
'
~ 2D
'2D
1
f1D (! 1 )
=
1
f2D (! 2 )
1
ws
wL
2
1
(4)
Our country is skilled-labor abundant. We show in the appendix that this
implies that:
wsF
wsH
<
:
H
F
wL
wL
This implies that:
M1H
M2H
>
M1F
M2F
:
This result is already present in standard models of trade with imperfect
competition and increasing returns to scale (Helpman and Krugman, 1985).
Unlike existing work, the innovation resources in this economy are not constant
across industries. The comparative advantage industry invests more resources
in both product and process innovation. R&D expenditures in each sector are
given by:
1 The
same result has been found in Bernard, Redding and Schott (2007).
8
R&D expi = (fei Mei + fiI MiI )
|
{z
}
(! i )
|{z}
Resource cost
am ount of resources
Considering the stationarity condition for each sector and rearranging terms:
R&D expi =
(1
fei
(1
+ fiI
G('iD ))
(1
G('iI ))
G ('iD ))
Mi (! i )
Since 'iD ; 'iI are identical across industries, considering the ratio of R&D expi
across industries we have that:
R&D exp1
M1
=
R&D exp2
M2
ws
wL
1
2
=
1
:
While the relative R&D expenditures just depend on the size of the sector
the amount of resources invested is larger in the industry in which the economy
has a comparative advantage. To see this, consider the simpler case in which =
1
2 : In this case the economy is investing the same amount of income in innovation
in both industries. However, in the industry in which the economy has the
comparative advantage, the cost of resources is cheaper, and consequently this
industry is investing in more resources.
A common measure to explore the presence of innovation activities within an
enditures
industry used in the empirical literature is R&D intensities ( R&D exp
).
sales
This measure is used to avoid the fact that R&D expenditures are larger because
the industry is larger in the economy. The model suggests that R&D intensities
are identical across industries as it can be seen below.
R&D exp1 R2
R&Dint1
=
= 1:
R&Dint2
R&D exp2 R1
3
Costless Trade
In this section I explore the implications of the model for innovation when we
consider a movement from autarky to free trade. Common to the new literature on …rm-heterogeneity and factor endowments, including innovation in this
environment does not alter the main properties of the Heckscher-Ohlin model.
Moreover, R&D intensities are still invariant across industries and innovation
does not have any impact on average productivity. In contrast to the one derived in this section, under costly trade, R&D intensities di¤er according to the
comparative advantage. This has a clear impact on average productivity across
industries.
Consider the possibility that the …rm can serve the foreign market at no
cost. The variable pro…t of a domestic …rm in the domestic market is now given
by:
9
H
rid
(') =
RH
1
(PiH )
1
d
( i) '
1
!H
i
and the variable pro…t of a domestic …rm in the foreign market is given by:
F
rid
(') =
RF
1
P
( iF )
1
d
( i) '
!H
i
1
:
The marginal innovator in the Home country (H) must satisfy the following
condition:
1+
RF
RH
PiF
PiH
1
1
( i)
1
rid ('iI )
H
1
= fiI ! H
i
1
R
( 'iI )
!H
1
i
(PiH )
The marginal survivor, the one indi¤erent between staying on leaving the
market is de…ned by the following condition:
where rid ('iI ) =
1+
RH
RF
PiH
PiF
1
rid ('id )
= fid ! H
i
Then we have that:
'iI
'iD
1
=
fiI
fiD
1
( i)
1
1
!
(5)
which is the same as in autarky. In fact since the variable pro…t for each …rm
is a constant times the variable pro…t in autarky we have that the FE condition
is given by:
!
!
1
1
'
~ iD
(1 G('iI ))
'
~ iI
fei
1 fiD +
1 fiI =
'iD
(1 G ('iD ))
'iI
(1 G('iD ))
(6)
which is identical to the one in autarky. Therefore productivity thresholds
are unchanged after trade openness when trade is costless. This implies that the
productivity distributions remain unchanged after trade openness but costless
trade.
The standard results in the Heckscher-Ohlin model are held in this environment. Factor Price Equalisation holds provided that we do not have factor
intensity reversals (i.e. factor endowments not to be very di¤erent across countries). Unlike previous studies, trade has an impact on innovation. Trade promotes investment in product innovation in those industries in which the country
has a comparative advantage.
In contrast, the relative R&D intensities will be unchanged after trade openness. To see this notice that the R&D intensities ratio is given by:
R&Dint1
M1 R 2
=
R&Dint2
M2 R 1
ws
wL
But Ri = Mi ri ; then substituting we have that:
10
2
1
R&Dint1
R&Dint2
=
R&Dint1
=
R&Dint2
r2
r1
ws
wL
'
~ 2D
'2D
'
~ iD
'iD
1
2
1
f2D +
(1 G('2I ))
(1 G('2D ))
'
~ 2I
'2I
f1D +
(1 G('1I ))
(1 G('1D ))
'
~ 1I
'1I
1
1
f2I
(! 2 )
!1
1
f1I
(!1)
!2
=1
Costless trade does not have any impact on process innovation because it
does not alter the distribution of pro…ts within the industry. When trade is
costless, trade openness widens the pro…t opportunities of all …rms although
this increase is more pronounced in the industry in which the country has the
comparative advantage because the relative cost of factors is cheaper. This
induces entry and an increase in the relative demand for skilled labour. The
increase in entry perfectly o¤sets the increase in pro…t opportunities and leaves
the market share of each …rm in each market unaltered. Since the global size of
the …rm is unchanged under this setting, …rms’incentives to undertake process
innovation activities have not been altered This result is challenged in the next
section.
4
Costly Trade
The recent literature on trade and …rm heterogeneity has suggested that both
…xed and variable trade costs are important in international trade activities
(Roberts and Tyebout, 1998). In this section, I introduce both types of costs
and show that the main implication of it, self-selection into exporting markets,
together with di¤erences in factor endowments generate important consequences
for innovation. Self-selection into exporting creates asymmetries across …rms
within an industry. Di¤erences in factor endowments create asymmetries across
sectors. The interaction between both expand business opportunities of the
most productive …rms to a larger extent in the comparative advantage industry.
This induces di¤erences in innovation outcomes across industries and consequently di¤erences in the average productivity across industries. Unlike BRS
these di¤erences in productivity arise through two channels: an e¤ect on the
survival productivity threshold due to tougher selection which creates a reallocation e¤ect towards most productive plants, and an e¤ect on the within-plant
productivity.
In this economy, to get one unit of the product sold in the foreign market, a
…rm must ship i 1 units of the product incurring into a variable trade cost
of i 1: To serve the foreign market the …rm needs to incur also in a …xed cost
fix units of the intermediate input used in production. As commented before,
we assume that exporting activities uses the same intermediate input as innovation and production activities within the same industry. A great part of this
11
…xed cost of exporting consists on advertisement and complying with regulation
standards. I assume that these costs are proportional to the unitary production
cost. To outline the role played by factor endowments on innovation outcomes,
we assume that sectoral structural parameters other than factor endowments
are identical across countries.
As it is discussed in Navas and Sala (2013) this model exhibits di¤erent
equilibria depending on the parameter con…guration. These are associated with
di¤erent partitions of …rms according to innovation and export status. In the
world I describe here, the variety of equilibria becomes more interesting since different industries could in principle sustain di¤erent type of equilibria depending
on the value of …xed costs of exporting, innovation and trade barriers. Rather
than describing a large variety of cases, in this paper, I focus on a symmetric equilibrium (in which all industries share the same structural parameters)
and an equilibrium in which innovators are a subset of the most productive exporters for both industries and countries, in line with recent evidence found by
Aw, Roberts and Xu (2011). Consequently, both industries are characterized by
a partition of …rms across status given by the following hierarchy: Innovators
and exporters (the most productive ones), exporters and domestic …rms. In a
further section I describe the conditions under which this equilibrium holds,
and through simulations we show that this equilibrium holds provided that the
level of variable trade costs are low enough.2
For further analysis we denote with superscript j = H; F the variables associated with the home country and with superscript k = H; F the variables
associated with the destination country (both of them can be either Home (H)
or Foreign (F)).
In this equilibrium, the marginal innovator is an exporter. Consequently,
the marginal innovator in country j and industry i is de…ned by the following
condition:
0
! 11
!
j
k
k
)
r
('
P
R
id
1
1
i
iI
A ( i)
@1 + i
1
= fiI ! ji
i = 1; 2
Rj Pij
(7)
where we have used the fact that R1j = Rj and R1k = Rk The marginal
exporter in country j is described by the following expression:
0
! 11
!
j
k
k
R
P
r
('
)
id
1
i
ix
A
@ i
= fix ! ji
(8)
Rj Pij
and the marginal surviver is given by the following condition:
rid ('jiI )
= fid ! ji
(9)
2 Robustness checks analyse how the results vary under symmetry with di¤erent hierarchical
structures.
12
Dividing (8) and (9) we …nd that:
'jix
'jid
=
Pij
Pik
i
|
!
'jid
!
1
(10)
}
j
i
Dividing (7) and (9) I obtain:
'jiI
1
Rj fix
Rk fid
{z
1
fiI
=
1
fid ( i )
1
j
i
1+
1
(11)
fix
f id
Notice that as a consequence of trade openness, there is a larger proportion
of …rms undertaking process innovation in both industries, and this result is
independent of factor endowments. This is the consequence of the fact that
innovators have access to a larger market where they can take advantage of
the increasing returns to scale nature of innovation. Taking the ratio across
industries we have that:
'j1I
'j1d
1
'j2I
'j2d
1
1+
j
2
1+
j
1
=
fix
f id
(12)
1
fix
f id
1
'j1I
'j1d
and therefore we can conclude that
1
<
'j2I
'j2d
1
i¤
j
1
<
j
2:
This
implies that those industries exhibiting a larger proportion of …rms innovating
are also those industries in which there is a larger proportion of …rms exporting.
In the appendix I discuss the aggregation properties of the model under
costly trade. Compare to the benchmark case of …rm heterogeneity without
technology upgrading, I observe that the di¤erence in pro…ts between autarky
and trade is larger in this setup due to the e¤ect that trade has on process innovation. Trade openness increases the size of the market for the most productive
…rms and consequently their sales. For a given innovation productivity threshold, the innovators are able to exploit their knowledge advantage across more
production units since they are able to sell more. This increases pro…ts. Substituting the expression for pro…ts in the Free Entry condition and rearranging
terms it can be obtained:
j
id
+ pjix
j
ix
+ pjiI
j
iI
=
fei
1
G('jiD )
(13)
Looking at this condition we can deduct several properties. First, trade
openness improves the average productivity in both industries by increasing the
productivity threshold to survive in the market. Second the inclusion of process
13
innovation increases the e¤ect that trade has on average productivity. This
is due to the fact that innovation opportunities interacting with trade creates
new opportunities to the most productive …rms helping them to increase their
market share in detrimental of the local competitors.
Speci…c to this paper is the asymmetric impact on innovation across industries. More precisely I show in the appendix that:
Proposition 1 Under costly trade:
1. The increase in the survival productivity threshold is larger in the industry
in which the economy has a comparative advantage.
2. In the industry in which the economy has a comparative advantage there is
a relative larger share of incumbent …rms undertaking process innovation.
3. Assuming a Pareto-Distribution for productivity, the R&D intensities are
larger in the sector in which the economy has comparative advantage and
this is due to a joint e¤ ect of more product and process innovation.
Proof. See Appendix.
The intuition behind these results underlies on the fact that when the economy opens to trade, …rms are asymmetrically exposed to di¤erent industry
opportunities. In the home skilled-abundant country, the marginal cost of production in industry 1 is lower than in the Foreign Country. When the economy
opens up to trade, …rms see their opportunities expanded in trade because the
access to a larger market allows them to exploit the increasing returns to scale
associated with both production and innovation. However, these pro…t opportunities are larger in the industry in which the economy has the comparative
advantage since this industry is able to o¤er the good cheaper than its analogous
counterpart in the foreign country (Industry 1). This promotes a disproportionate entry, and consequently more product innovation. The massive entry of …rms
makes pro…ts fall and it becomes more di¢ cult to survive. The less productive
…rms can no longer make positive pro…ts and consequently the productivity
threshold needed to survive in the market increases. The expulsion of the less
e¢ cient …rms generates a reallocation of market shares across the most productive …rms. This increases process innovation due to a combination of larger
opportunities and market share reallocation.
For this equilibrium to hold the following parameter constraints need to be
satis…ed:
1. The marginal innovator must be an exporter. (i.e.
implies:
j
i
1
fid
f ix
+1
<
fiI
fix ( i )
14
'jiI
'jiX
1
1
1
> 1): This
Substituting (10) and rearranging terms, we have that:
1
i
Rij
Rik
!
Pij
Pik
!
1
fiI
<
fix ( i )
1
1
1
2. The productivity threshold of an exporter must be larger than the domestic one which implies
'jiX
'jiD
1
i
1
>1:
Rij
Rik
!
Pij
Pik
!
1
>
fid
fix
Writing both conditions together we have that:
fid
<
fix
1
i
Rij
Rik
!
Pij
Pik
!
1
fiI
<
fix ( i )
1
1
(14)
1
If the following condition is satis…ed:
1
i
Rij
Rik
!
Pij
Pik
!
1
<
fid
fix
then all …rms will be able to export. In that case the economy will be in an
equilibrium with costly trade but no selection into exporting markets.
Condition (14) depends on four endogenous variables and the model does
not exhibit a closed form solution for these variables. The next simulation exercise suggests that this equilibrium holds provided that transportation costs are
not large enough and there are no substantial di¤erences in factor endowments
across countries. In the next section I look at the properties of this equilibrium. As it becomes apparent, while trade has introduced technological divergence across industries, due to a combination of trade barriers and di¤erences
in factor intensity usage, these technological di¤erences across industries could
become smaller as the trade costs are reduced. This suggests that while trade
openness create a ricardian comparative advantage across industries, the width
of the comparative advantage depends clearly on the level of trade costs. If the
initial situation is one of low trade costs, this comparative advantage becomes
weaker over time, provided that trade costs are reduced over time. This could
be consistent with recent empirical evidence found by Levchenko and Zhang
(2013).
15
Paremeter
'1D
'2D
'1I
'2I
'
~1
'
~2
M1
M2
M1e
M2e
Autarky
0.42886
0.42886
1.20211
1.20211
0.79419
0.79419
499.376
481.494
152.93
152.27
Free Trade
0.5171
0.50675
1.13761
1.13963
1.0876
1.06432
271.409
113.380
171.511
66.87
Percentage Variation
20.57
18.16
-5.36
-5.19
36.95
34.01
-45.65
-77.29
12.63
-56.09
Table 1: A movement from autarky towards FreeTrade
5
Simulation Exercises
In this section I undertake several simulation exercises which corroborates the
results discussed in the theoretical part. Firstly, I compare the results in autarky
with the results in Free Trade and second, I discuss the e¤ects of a partial
trade liberalisation experiment (a reduction in trade costs). In this subsection
we depart for the parameter values provided by Bernard, Redding and Schott
(2007). These ones can provide us with a better comparison between the two
models and the role played by innovation in the productivity convergence across
industries.
Table 1 shows the results in autarky and free trade for the home country
(similar ones you can …nd for the foreign country). Notice that a movement
towards free trade increases the survival productivity cuto¤ and reduces the
innovation productivity cuto¤ promoting technology upgrading. However, these
e¤ects are not the same across industries. In the comparative advantage industry
there is more selection due to an increase in the mass of entrants (attracted
by larger expected pro…ts) and a lower innovation productivity cuto¤ (since
trade expand the business opportunities of local …rms more in the comparative
advantage industry). The e¤ects on the average productivity for the benchmark
case are large (taking into account that under the current parametrization only
3.75 and 3.5% of the incumbent …rms undertake process innovation). In the
comparative advantage industry there is an increase in the mass of varieties
created while in the comparative disadvantage industry the mass of each period
new varieties is clearly reduced. This re‡ects the di¤erences in pro…tability
between both industries which reallocates potential entrants for the comparative
disadvantage industry to the comparative advantage one. However, although
the proportion of surviving …rms is clearly large in the comparative advantage
industry, there is a clear drop in surviving in both industries. 3
3 As
commented above, I have used BRS paramter values for the common parameters in the
16
Zero-profit and Export Productivity Cutoffs
0.8
0.75
0.7
productivit y
0.65
0.6
0.55
0.5
0.45
0.4
1
1.05
1.1
1.15
1.2
1.25
1.3 1.35 1.4 1.45 1.5
Variable Trade Costs tau
1.55
1.6
1.65
1.7
1.75
Figure 1: Export and domestic productivity cuto¤s: The continuous line represents the survival productivity cuto¤s for both industries while the discontinuous
line represents the export productivity cuto¤ for both industries. The red line
represents the industry in which the economy has the comparative advantage.
As it can be seen, in the industry in which the economy has the comparative
advantage the survival productivity threshold is de…nitely larger. However, the
export productivity cuto¤ is smaller.
17
Figure 1 shows the export and domestic productivity cuto¤s for both industries in the home country. The continuous line displays the survival productivity
thresholds for both industries while the discontinuous one shows the export productivity cuto¤s. The red lines represent the industry in which the industry has
the comparative advantage (industry 1) while the green one does it for the industry with the comparative disadvantage (industry 2). It is clear that the survival
productivity cuto¤ is larger and the export productivity cuto¤ is smaller in the
industry in which the economy has the comparative advantage. The former
re‡ects tougher competition in that industry due to the larger expanded opportunities for …rms in that industry. Compared to a model without innovation,
the survival productivity cuto¤s in both industries have increased considerably.
In the comparative advantage industry for the case of the trade costs equal
to 20% the productivity cuto¤ is 3.8% larger while in the comparative disadvantage industry it is 3.67% larger. Although small, we can also observe that
when we introduce the possibility of …rms to upgrade their own technologies
in a model with …rm heterogeneity and comparative advantage the di¤erence
between productivity cuto¤s due to comparative advantage mechanisms across
industries exacerbates.
Considering a more general case in which the …xed costs of exporting are
not equal to the …xed costs of production, just con…rm the qualitative results
that we have obtained as the …gure 2 provides.
Figure (3) displays the relative survival productivity cuto¤s of Industry 1
versus Industry 2 for both the home and the foreign country. It becomes apparent that for high level of transportation costs a reduction in transportation
costs increases the survival productivity cuto¤ by more in the comparative advantage industry. However for su¢ ciently low levels of transportation costs, the
opposite happens. This suggests that when the trade costs are high selection
becomes tougher in the industries in which the economy has the comparative
advantage but as the trade costs fall survival is more di¢ cult in the comparative
disadvantage industry.
Figure (4) displays the innovation productivity cuto¤ across both industries
for the home country. The continuous line represents the innovation productivity cuto¤ in the industry 1 while the discontinuous line represents the productivity cuto¤ in industry 2 for di¤erent values of trade costs. It becomes
apparent that a reduction in trade costs, decreases the innovation productivity
cuto¤s in both industries, or in another words increases the mass of …rms that
upgrade their technology. Yet, it can be also observed that the innovation productivity cuto¤ is smaller in the industry in which the home country has the
model. This provides a more accurate interpretation of the results by comparing a model with
and without process innovation. For the value innovation jump we have used 20% ( = 1:2)
and for the innovation cost we have used 25 times the cost of entry. Changes in the parameter
values do not generate qualitative changes in the results, provided that the economy is in the
analysed equlibriu. Robustness checks are available on request.
18
Zero-profit and E xport P roductivity Cutoffs
1.3
1.2
1.1
productivity
1
0.9
0.8
0.7
0.6
0.5
0.4
1
1.05
1.1
1.15
1.2
1.25
1.3 1.35 1.4 1.45 1.5
V ariable Trade Costs tau
1.55
1.6
1.65
1.7
1.75
Figure 2: Export and domestic productivity cuto¤s (Selection into exporting
with = 1): The continuous line represents the survival productivity cuto¤s for
both industries while the discontinuous line represents the export productivity
cuto¤ for both industries. The red line represents the industry in which the
economy has the comparative advantage. As it can be seen, in the industry
in which the economy has the comparative advantage the survival productivity
threshold is de…nitely larger. However, the export productivity cuto¤ is smaller.
Relative productivity cutoffs
1.05
1.04
1.03
P roductivity cutoffs
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3 1.35 1.4 1.45 1.5
V ariable Trade Costs tau
1.55
1.6
1.65
1.7
1.75
Figure 3: Relative productivity cuto¤s. The …gure displays the relative productivity cuto¤s of Industry 1 vs Industry 2 for both the home country (continuous
line) and foreign country (discontinuous line).
19
Innovation P roductivity C utoffs
1.18
1.175
1.17
productivity
1.165
1.16
1.155
1.15
1.145
1.14
1.135
1
1.05
1.1
1.15
1.2
1.25
1.3 1.35 1.4 1.45 1.5
V ariable Trade C osts tau
1.55
1.6
1.65
1.7
1.75
Figure 4: Innovation Productivity Cuto¤s: Industry1 productivity cuto¤ is represented by the continuous line while the discontinuous line represents the industry 2 productivity cuto¤.
comparative advantage and consequently the mass of …rms engaging in process
innovation is larger in the comparative advantage industry. As commented in
the sections above this result is the consequence of a tougher selection process
in this industry.
Figure (5) represents instead the relative innovation productivity cuto¤s (industry 1 versus industry 2) for both the home country (continuous line) and the
foreign country (discontinuous line). The results suggest an interesting …nding.
While the relative innovation cuto¤ is systematically larger in each country’s
comparative disadvantage industry there is a non-monotonic relationship between the trade costs and the relative evolution of both cuto¤s, which is a
measure of the process innovation activity. When the trade costs are high, a
reduction in trade costs decreases by more the innovation cuto¤ in the comparative advantage industry, (decreasing the relative cuto¤ for industry 1 in
the home country and increasing the relative cuto¤ in the foreign country).
However, when the trade costs are low enough we …nd the opposite result: A
reduction in trade costs decreases the innovation cuto¤ more in the comparative disadvantage industry (and consequently the relative cuto¤ increases in the
home country and decreases in the foreign country).
These results suggest that a reduction in trade costs increases process innovation in both industries. When the trade costs are substantially high however, the reduction in trade costs favours the comparative advantage industry
and when the trade costs are low the trade cost reduction favours instead the
comparative disadvantage industry. The implications for the evolution of the
average productivity across industries are straightforward: A process of globalisation induces an increase in TFP in both industries provided that the trade
20
Relative Innovation cutoffs
1.01
1.008
1.006
Innovation cutoffs
1.004
1.002
1
0.998
0.996
0.994
0.992
1
1.05
1.1
1.15
1.2
1.25
1.3 1.35 1.4 1.45 1.5
Variable Trade Costs tau
1.55
1.6
1.65
1.7
1.75
Figure 5: Relative Innovation Cuto¤s. This …gure displays the relative innovation cuto¤s (Industry 1 vs Industry 2) for both the Home (continuous line) and
the Foreign country (discontinuous line) as a function of the trade costs.
costs are not relatively high (we are in the parameter con…guration consistent
with this equilibrium). Yet, globalisation induces TFP divergence across sectors
in countries relatively less open to trade, but it induces TFP convergence across
sectors in countries relatively more open to trade . This can justify the empirical
…ndings of Levchenko and Zhang (2013). Figure 6 illustrates this point.
Figure 7 shows the e¤ects of trade liberalisation on product innovation in
both industries. As it becomes apparent, product innovation is already larger
in the comparative advantage relative to the comparative disadvantage industry
in each of the countries. As trade costs are reduced the di¤erences between
product innovation across industries are enlarged: Product innovation becomes
larger in the comparative advantage industry for the home country (and the
same happens for the foreign country). This is the consequence of the fact that
trade liberalisation expands the opportunities of the most productive …rms in
the comparative advantage sector and the increase in the expected pro…ts in this
sector promotes entry. In the comparative disadvantage sector domestic …rms
face disproprotionate tougher competition, the average expected pro…t falls and
consequently entrants shift away from the comparative disadvantage sector to
the comparative advantage one.
A common measure of innovative activity across industries is to look at
R&D intensities. Figure (8) displays the evolution of the R&D intensities for
both industries in the home country. Notice that R&D intensities increase in
both industries as trade costs fall. Although the di¤erences across industries
are not substantially large there are still di¤erences in R&D intensities when
21
A verage P roductivity
1.15
1.1
P roductivity
1.05
1
0.95
0.9
1
1.05
1.1
1.15
1.2
1.25
1.3 1.35 1.4 1.45 1.5
V ariable Trade Costs tau
1.55
1.6
1.65
1.7
1.75
Figure 6: Evolution of the Average Productivity in Industry 1 (Continuous line)
and Industry 2 (Discontinuous line) in the Home Country.
P roduct Innovation
180
160
Mass of new varieties
140
120
100
80
60
1
1.05
1.1
1.15
1.2
1.25
1.3 1.35 1.4 1.45 1.5
V ariable Trade Costs tau
1.55
1.6
1.65
1.7
1.75
Figure 7: Product Innovation in the Home Country. This …gure displays the
creation of new varieties for di¤erent variable trade costs for both industries,
industry 1 (continuous line) and industry 2 (discontinuous line).
22
R&D Intensities
23.16
23.14
23.12
Percent
23.1
23.08
23.06
23.04
23.02
23
22.98
1
1.05
1.1
1.15
1.2
1.25
1.3 1.35 1.4 1.45 1.5
Variable Trade Costs tau
1.55
1.6
1.65
1.7
1.75
Figure 8: R&D intensities across industries in the home country. The …gure
displays the R&D intensities for industry 1 (continuous line) and industry 2
(discontinuous line) for the home country.
the economy is open to trade favouring the comparative advantage industry in
the home country. This is the consequence of the fact that in the comparative
advantage industry there is more product and process innovation.
If we compare the relative evolution of R&D intensities in the home country
(continuous line) and the foreign country (discontinuous line) we obtain a similar message (Figure 9). The …gure displays a similar functional form to …gure
(3). When the trade costs are large, a reduction in the trade costs increases the
R&D intensity by more in the comparative advantage industry in each country.
However, if the trade costs are su¢ ciently small, the reverse happens and the
R&D intensities increases by more in the comparative disadvantage industry.
When trade costs are high further liberalisation leads to divergence in innovative activity across industries but this result is reversed if the trade costs are
su¢ ciently small.
6
Conclusions
This paper introduces technology upgrading into a standard model of trade,
factor proportions and …rm heterogeneity to explore how factor endowments
shape the impact that trade has on innovation at an industry level.
Our results suggest that factor endowments a¤ect the distribution of innovative activity across industries within a country when the economy opens to
23
R e lativ e R &D inte ns itie s
1 .0 0 2
1 .0 0 15
Re la tive R&D in te n sitie s
1 .0 0 1
1 .0 0 05
1
0 .9 9 95
0 .9 9 9
0 .9 9 85
1
1 .0 5
1 .1
1 .1 5
1 .2
1 .2 5
1 .3 1 .3 5 1 .4 1 .4 5 1 .5
Var ia b le Tr a d e C o s ts ta u
1 .5 5
1 .6
1 .6 5
1 .7
1 .7 5
Figure 9: Relative R&D intensities. This …gure displays Relative R&D intensities (Industry 1 vs Industry 2) for both the home (continuous line) and the
foreign country (discontinuous line).
trade. More precisely, …rms in the industry where the economy has a comparative advantage undertake more product and process innovation. This reinforces
previous results that outline the importance of the relative factor endowments
in generating a ricardian comparative advantage.
In addition I explore how the evolution of technology is a¤ected by a reduction in trade costs under the presence of di¤erences in factor endowments
across countries and factor intensities across industries. The results suggest
that the reduction in variable trade costs promotes technology upgrading and
increases R&D intensities in both industries. However, the results establish
a non-monotonic relationship between the pattern of comparative advantage
and trade costs: When the trade costs are high, a reduction in trade costs
pushes technology upgrading more in the comparative advantage industry leading to TFP divergence across industries. However, when the trade costs are low
enough, a reduction in trade costs pushes technology upgrading in the comparative disadvantage industry leading to a process of TFP convergence across both
industries.
This paper could be extended in several directions. First, the paper has considered a simple process of technology upgrading where the degree of technology
upgrade is …xed across …rms. Generalising the results to a richer environment in
which the …rm can choose how much to upgrade is a promising area for future
research. Second, the paper could allow for a more complete and realistic de24
scription of the intermediate input sector and establishes the complementarities
between importing, exporting and innovation under the presence of di¤erences
in factor endowments. Both dimensions are currently taking part on my research
agenda.
7
Appendix
7.1
Aggregation in a Closed Economy Model
De…ne the following productivity distributions:
(
Pr ('='
Pr ('='
'iD ) =
'iI ) =
iD (')
iI (')
=
=
g(')
1 G('iD ) :
if '
g(')
1 G('iD ) :
if '
0
(
0
'iD
otherwise
'iI
otherwise
)
)
the second one, that is more associated to heterogenous …rm models of
process innovation will be the ex-ante productivity distribution of innovators.
We can show that paralell to the Melitz (2003) model the aggregation property also holds in this model. This allows to write the sectoral aggregate variables of the model as a function of the average productivity of the sector. The
right average productivity of each industry is given by the following expression:
i
(1 G('iI )) h
1
1
1
'
~ = (~
'iD )
+
( i)
1 (~
'iI )
(1 G ('iD ))
where:
(~
'iD )
1
=
Z1
1
'
(') d'
'iD
and
(~
'iI )
1
Z1
= '
1
I
(') d'
'iI
In equilibrium we must have that:
E(Vi ) = fei ! i
This condition is known in the Melitz (2003) framework as the Free Entry
condition (FE). It can be rearranged to:
2
3
innovation pro…ts
z
}|
{
7
(1 G('D )) 6
(1 G('iI ))
r (~
'iI )
6
7
1
E(V i ) =
'iD ) +
( i)
1
fiI 7= f ei ! i
6 (~
(1 G ('iD ))
4
5
|
{z
i
25
}
Compared to a standard model of …rm heterogeneity without technology
upgrading, this new FE condition exhibits a new element in the left hand side
of the equation. More precisely, the second element captures the expected profits from technology upgrading which are given by the probability of innovate
conditional on surviving and the innovation rents.
8
Aggregation in Costly Trade.
De…ne the following productivity distribution:
'jix
Pr '='
=
ix (')
(
=
g(')
1 G('jix )
if '
)
'jix
0
otherwise
This is the conditional productivity distribution of exporters to country j:
De…ne also the following average productivity:
1
'
~ jix
Z1
=
'
1
x
(') d'
'jix
and consider as well the following productivity averages:
'
~ jixi
2
j
6 Mix
1
'
~ jix
=4
1
+ ( i)
1
j
MiI
1
'
~ jiI
3
1
1
7
5
j
Mix
Then the aggregate price index for each sector in each market will remain
as follows:
Pij
1
= Mij p '
~ ji
1
1
k
+ Mix
p '
~ kixi
The expected pro…ts from a potential entrant are given by:
E(Vij )
where
=
(1
2
6
6
6
6
4
G('jiD )) 6
'
~ jiD =
|
'
~ jiD
'jiD
'
~ jiD
+
pjix
'
~ jix
+
pjiI
z
2
innovation pro…ts
4 ( i)
1
1
j
i
1 (1 +
{z
}|
0
~ jiI
fix @ r '
)
fid
i
1
1 fiD =
j
id
and
'
~ jix =
'
~ jix
'jix
1
1 fix =
j
ix :
Compare to autarky this equation has two extra terms. The second term
captures the pro…ts from exporting. The third term captures the innovation
26
3
{
137
7
7
fiI A57
7
5
}
pro…ts in which we can distinguish between the innovation gains in the domestic
market and the ones in the foreign market. Both are positive so this implies
that the expected pro…ts increase after trade openness.
Proposition 2 A threhold productivity level 'jiD
exists and it is unique.
Proof. It is identical to Navas and Sala (2013).
Proposition 3 Under costless trade FPE holds.
Proof. As in HK (1985) or BRS (2007), …rst we are going to characterize the
equilibrium in an hypothetical integrated economy. Then we are going to show
that there is a vector of prices and initial allocations that support this equilibrium
in the non-integrated economies.
The very …rst thing to show is that total revenue equals total expenditure in
each sector. To see this notice that:
wL Lpi + wS Spi + wL Lei + wS Sei + wL LIi + wS SIi = Ri
Notice that in this case the following needs to hold:
wL Lpi + wS Spi + wL LIi + wS SIi = Ri
i
since
i
h
(1 G('iI ))
') fD ! i (1
i = Mi r (~
G('iD )) fI ! i
On the other hand we know that the FE implies that:
Mi fei ! i
i = Mi i = (1 G(' ) = Mei fei ! i = wL Lei + wS Sei :
iD
Then we have that
wL Li + wS Si = Ri
aggregating across industries implies that:
wL L + wS S = R
(15)
Now let us characterize the equilibrium of the integrated economy:
Cobb-Douglas technologies ensure that:
wL Li = (1
wS Si = i Ri
i ) Ri
Labor market clearing condition implies:
(1
)
1)
2 )(1
R + (1 w
R=L
wL
L
)
R + 2 (1
R=S
wS
and dividing both:
1
wS
wL
(1
=
wS
1)
+ (1
+ 2 (1
1
2 ) (1
)
)S
L
(16)
Notice also that manipulating both labour market clearing conditions we …nd
that:
R
L
R
S
wL = (1
) ; wS = 1 + 2 (1
)
1 ) +(1
2 )(1
which leads to:
(1
1)
1
L1 = (1
) +(1
)(1
) ; s1 =
+ (1
)
1
L2 =
(1
2
(1
2)
1 ) +(1
2 )(1
1
);
s2 =
1
2
)
2 (1
+ 2 (1
)
27
Consider w.l.o.g. that wS = 1::Notice that the productivity and innovation
cuto¤ s ('iD ) ; ('iI ) can be easily solved using (6) and (5) and with this we can
obtain p 'ki :Equation (15) can be used together with (16) to obtain R. Using
(6) and the information about ('iD ) ; ('iI ) we can also obtain an expression for
ri : This help us to compute Pi ; '
~ iD ; '
~ iI : Witth this the integrated equilibrium is
characterized.
4 pins down the relative mass of surviving …rms.
Now let us show that this relative wage satis…es the equilibrium conditions
of the open economy but costless trade.
To do so let us consider without loss of generalisation that.wSH = wSF = 1:
Notice that perfect competition in the intermediate input sector implies that:
H
SiH = i RiH ) wL LH
i )Ri
i = (1
SH
L
= 1 i wL :
then LiH = 1 i w
i wS
i
i
The factor market conditions can be expressed as:
!
!
S1j
Sj
S2j
j
j
+
1
=
L1
L1
Lj
Lj1
Lj2
where
j
Li
Lj1 =
=
Sj
wL
Lji
Lj
2
1
2
1
1
S1j =
2
1
1
1
1
j
S1
Lj
and
1
Sj
S1j
!
=
Sij
:Given
Sj
Lj1
j
S1
1
1
2
1
1
1
2
1
j
wL L
2
1
1
Lj
1
1
Lj2
S2j
!
Lj
Sj
=
that we are able to get:
1
; Lj2 =
1
1
+ 1
2
1
j
L1
Sj
wL
2
1
2
; S2j =
;
1
1
2
1
1
2
wL Lj
1
1
2
2
1
2
Sj
2
1
1
2
Aggregate Income in each country equals aggregate revenue and so:
Rj = S j + wL Lj
We know that total indutry payments to unskilled labour are a constant fraction (1
i ) of the total industry revenue and the total industry revenue is a
fraction of the total revenue. Substituting the values of Lk1 in
F
wL LH
S H + S F + wL LH + LF
1)
1 + L1 = (1
we obtain an expression for wL that coincides with the one obtained in the
integrated equlibrium. Notice that this determines p 'ji which determines Pi :
We have shown that there exists a vector of allocations and competitive prices
that replicates the equilibrium of the integrated economy.
8.1
Proof of
H
1
<
H
2
and
F
1
>
F
2
This proof is very similar to a the analogous in BRS (2007). The …rs step is to
obtain the relative aggregate price indexes in autarky. Notice that:
28
P1j
=
P2j
M1j
M2j
!11
p1 (~
'j1 )
p2 (~
'j2 )
Since the innovation and the survival productivity cuto¤s are the same, the
price of the average …rm in each industry is only determined by di¤erences in
the cost of the intermediate input and consequently,
P1j
P2j
M1j
=
M2j
Substituting the expression for
P1j
P2j
=
In Costless trade
In Costly trade:
h
1
M1j (p('
~ j1 ))
P1j
h
j =
1
P2
~ j2 ))
M2j (p('
=
j
wL
j
wL
P1H
P2H
!
P1F
P2F
<
!
( 1
1
since
and consequently,
+pk
~k
ix (p('
2xi ))
2
2)
H
wS
H
wL
P1H
P2H
<
=
F
wS
F
wL
P1F
P2F
and
1
>
2:
:
i
1
1
1
and rearranging terms it can be obtained:
1
+pk
~k
1x (p('
1xi ))
'iX
'iD
where pjix =
F
wS
F
wL
M1k
M2k
wSj
wSj
and consequently in autarky:
H
wS
H
wL
!11
i
: Notice that if
! 1; pix ! 0 and
P1j
P2j
=
1
M1j
M2j
1
j
wS
j
wL
since the productivity and the innovation cuto¤s are the same across industries
PH
PF
(autarkic case). Then P1H < P1F : In the case of ! 1;and fiX ! 0 then pjix ! 1
2
2
and 'jiX = 'jiD 8 i; j: We are consequently in the case of free trade where the
relative prices are equalised across countries. For intermediate cases of …xed
and variable trade costs the relative prices underlie between the two values: the
autarkic and the jfree tradej value.
P
Notice that j1 = 1 P1j provided that the …xed and variable trade costs are
2
2
the same across industries. Consequently in costly trade we have that
F
and F
1 > 2:
8.2
8.2.1
H
1
<
H
2
Proof of Proposition 1
Point 1.
To see the …rst point, notice that we can express condition (13) in terms of the
productivity threshold 'iD : Using equation (10) and the expression for ix I
can express the expected value of pro…ts from exporting as:
1
1
(1 G( j ' ))
'
~ jix
j
1 fix ! ji
pjix jix = 1 G 'i j iD
j
i
'iD
(
( iD ))
29
1
2
This expression is decreasing in ji : Using (11) I obtain the expected pro…ts
from innovation, which are given by:
1
fix
1
fid (( i ) 1 1) 1+( ji )
(1 G('j ))
'
~ jiI
f id
1 fiI
pjiI jiI = 1 G 'jiI
j
fiI
'iI
(
( iD ))
that is also decreasing in ji :
H
F
F
Since before we have shown that H
1 < 2 and 1 > 2 , then we have that
the left hand side of condition (13) is smaller in industry 1 than in industry
2 in the home country. Consequently the productivity threshold is larger in
industry 1 than in industry 2.
Consequently, The second part of the statement (2) is just a consequence of
this result together with the condition (12).
8.2.2
Point 2.
Condition (12). suggests that
H
2
and
8.2.3
F
1
>
F
2,
then
'1I
'1d
H
'j1I
'j1d
1
<
'2I
'2d
<
H
'j2I
'j2d
1
and
'1I
'1d
j
1
i¤
F
>
<
j
2:
'2I
'2d
Since
H
1
<
F
:
Point 3
Let us focus on the home country but the result is similar for the foreign country.
We therefore omit the superscript j for simplicity. The relative R&D intensities
are given by:
R&Dint1
R&Dint2
=
(1
(1
(1
(1
(1
fei
+ fiI
G('2D ))
(1
fei
+
G('1D ))
fiI
G('1I ))
G('1D ))
M1 (! 1 ) R2
M2 (! 2 ) R1
G('2I ))
G('2D ))
Using the fact that Ri = Mi ri and rearranging terms we have that:
R&Dint1
R&Dint2
=
(1
(1
(1
(1
(1
fei
+ fiI
G('2D ))
(1
fei
+
G('1D ))
fiI
G('1I ))
G('1D ))
r2 ! 1
r1 ! 2
G('2I ))
G('2D ))
The …rst term is larger in industry 1 since on the one hand, '1D > '2D and
therefore in principle there’s more entry in industry 1 and on the other hand we
(1 G('1I ))
(1 G('2I ))
have that (1
G('1D )) > (1 G('2D )) , and then the proportion of …rms undertaking process innovation is larger: However, the average revenue in industry 1 is
larger since this is the industry in which the economy has a comparative advantage. To derive the net e¤ect we are going to focus on the particular case of the
Pareto Distribution. Assuming that the productivity cumulative distribution
function takes the following functional form:
Pr('
'0 ) = 1
m
'0
30
>
1
the average revenue is given by the following expression:
ri =
(
1)
[fid + pix fix + piI fiI ] (! i )
(17)
and the average pro…t is given by the following expression:
i
=
1
(
1)
[fid + pix fix + piI fiI ] (! i )
(18)
Using (13), (17) and (18) we have that:
ri =
(
1) (1
fei (! i )
G('id ))
and consequently:
R&Dint1
R&Dint2
=
(1
(1
Clearly,
(1
(1
(1
fei
+fiI
G('2D ))
(1
fei
+fiI
G('1D ))
R&Dint1 H
R&Dint2
G('1I ))
G('1D ))
(1 G('1D ))
(1 G('2D ))
G('2I ))
G('2D ))
H
> 1 i¤ ('1I )
=
( fei + fiI (1 G('1I )))
( fei + fiI (1 G('2I )))
H
< ('2I ) .
Under the Pareto Distribution, the survival productivity cuto¤s are given
by:
'iD =
1
(
1
fid + pix fix + piI fiI
1)
fei
m
and using (11), the innovation cuto¤ is given by:
1
'iI = (piI )
(
1
fid + pix fix + piI fiI
1)
fei
m
Rearranging terms, I got:
1
'iI =
Below
d'iI
d i
(
" fid
piI
1)
+
pix
piI fix
+ fiI
fei
#! 1
m
(19)
> 0: is shown.
For the sake of simplicity, let us make the following transformation. Let call
1
d'iI
d'iI
fix
z = ( i)
fid : Notice that then show d i > 0 is equivalent to show dz < 0:
Without loss of generalization I am going to show it for sector 1. Recall that:
0
p1I = @
f1I
f1d ( 1 )
1
31
1 (1 + z)
1
A
1
and that
0
Notice that
d'iI
dz
p1x
=@
p1I
f
z f1I
1
( 1)
1x
1 (1 + z)
< 0 is equivalent to show that
df
dz
1
1
A
1
< 0;where f = (1 + z)
(f1d )
1
1
Taking derivatives we have that:
1
(1 + z)
1
1
1
(f1d )
1
1
+z
1
(f1x )
1
+
1
z
1
1
1
(f1x )
(1 + z)
1
Rearranging terms we get:
(1 + z)
1
1
(1 + z)
1
1
(f1d )
1
+z
1
1
(f1x )
1
1
+ z
z
1
1
(f1x )
1
Multiplying the entire expression by z and rearranging terms:
(1 + z)
1
z
(f1d )
1+z
1
1
1
+
1
z
1+z
1
1
(f1x )
1
This is negative if the second term is negative, which implies
1
(
z
1)
1
f1x
f1d
1
<1
1
1
This condition therefore requires that:
1
f1x
zf1d
< 1 or expressed in an-
other terms: ( i )
< 1: Since >
1; this condition is satis…ed as to
get the partition between exporters and nonexporters we need that i > 1.
Notice that as ji increases the innovation productivity threshold increases.
H
F
F
Since in this free trade equilibrium H
1 <
2 and
1 >
2 then we can
H
H
F
F
conclude that '1I < '2I and '1I > '2I : Therefore, there is a larger proportion
of …rms undertaking process innovation in the comparative advantage industry.
32
1
1
+z
1
(f1x )
1