Imported Intermediate Goods and Product Innovation
with Heterogeneous Firms
Jose Daniel Rodriguez-Delgado and Murat Seker∗
October 2010
Abstract
In this study, we build a structural model of multi-product firms that illustrates how
access to new foreign intermediate goods contributes to introduction of new product
varieties. We establish a stochastic dynamic model of firm and industry evolution allowing
firms to be heterogeneous in their efficiency levels. Introducing importing decision to this
dynamic framework, we show that importing intermediate goods has two contributions to
firms: i) it increases the revenues per each product created and ii) through the knowledge
spillovers obtained from the imported intermediate goods, firms get more innovative.
Our simulation exercise which compares autarky with free trade equilibrium shows that
average size and product creation rates increase whereas entry rate decreases in trade.
These findings are in close accordance with the recent empirical evidence from Indian
manufacturing dataset provided by Goldberg et al. (2010a, 2010b).
JEL Classification: F12, F13, L11, O31
Keywords: Firm dynamics, heterogeneous firms, innovation, endogenous product scope,
importing intermediate goods, trade liberalization.
∗
Correspondance: J. Daniel Rodriguez-Delgado: Western Hemisphere Department/Caribbean II, IMF,
[email protected]. Murat Seker: Enterprise Analysis Unit, Finace and Private Sector Vice Presidency World Bank, [email protected]. Adress: 1818 H Street NW, MSN F4P-400, Washington DC 20433
1
Introduction
An important role of international trade is the exposure to new goods. The purpose of this
study is to build an analytical framework to illustrate the potential gains in an economy that
international trade can generate as new products are introduced resulting from the access
to new foreign intermediate goods. We illustrate this mechanism through a simulated trade
liberalization episode. We show how aggregate innovation rates as well as individual firm’s
dynamics change after the liberalization period.
The role of trade expanding the set of traded goods has been the subject of recent studies.
Kehoe and Ruhl (2009) find that trade liberalization episodes are characterized by a significant
increase in the traded volume of goods not previously traded. They further document that
such role of the extensive margin of trade seems proper of structural changes in the economy
(e.g. trade liberalization) and absent from events like business cycles. Arkolakis (2009) derives
a similar result within his model of heterogeneous firms and costly access to trade. In our
framework, trade brings access to new intermediate goods previously unavailable to firms,
provided they find it profitable to pay the fixed costs associated with international trade.
At the firm level, Goldberg et al. (2010b) establish a robust relationship between the access
to new foreign intermediate inputs and the introduction of new products by domestic firms.
Using data for India, they find that 31 percent of the expansion of products by firms could be
explained by declining input tariffs. Similar to their empirical specification, at the center of our
theoretical framework is a set of heterogeneous multi-product firms that endogenously decide
their investment in the development of new products. In our model access to new intermediate
inputs increase firms’ revenues of existing products and they also improve firms’ innovation
technologies.
The concept of innovation used throughout our paper is one of horizontal product innovation; that is, an innovation or discovery consists of the knowledge required to manufacture
a new final good that does not displace existing ones. This type of framework is a natural
complement to models of vertical innovation where the “creative destruction” process makes
existing products obsolete. Horizontal innovation models have been extensively used in growth
theory, where Romer (1990) and Grossman and Helpman (1991) represent seminal studies.
Furthermore, it has been applied in trade theory to explain wage inequality, cross-country
1
productivity differences among other topics (see Gancia and Zilibotti (2005)). Among disaggregated firm-level models, Luttmer (2010) and Seker (2009) are relevant references. Both of
these papers follow from Klette and Kortum (2004) which present a stylized model that explains
the regularities in firm and industry evolution. In their model, an establishment is defined as
a collection of products and each product evolves independently. Every product owned by an
establishment can give rise to a new product as a result of a stochastic innovation process or
can be lost to a competitor. This birth and death process of the products is the source of firm
evolution. Through this model of innovation, they explain various stylized facts that relate
R&D, productivity, and growth. The extension introduced by Seker (2009) allows exogenous
heterogeneity in firms’ efficiency levels which provides a better fit to the data on explaining
firm size distribution as well as correlations between size, age, exit and firm growth1 . Our
basic framework, which draws extensively from Seker (2009), is meant to illustrate the role of
international trade in the introduction of new products. It is also relevant to mention that in
contrast to frameworks of learning externalities, ours is one in which firms fully internalize the
dynamic consequences of their innovation efforts.
In our framework international trade affects both the marginal benefit and the marginal
cost of innovation efforts. As in Seker (2009), each individual firm decides how much resource
to invest into innovation, where a higher investment increases the probability of successfully
introducing a new product and consequently enjoying a new stream of revenue. Our model is one
in which a higher variety of intermediate goods benefits a given firm with larger returns to scale
(productivity) both in its existing products as well as in any new product. Within the empirical
literature, Kasahara and Rodrigue (2008) find a robust and significant increase in productivity
among Chilean firms that import intermediate goods; and Halpern et al. (2009) find, among
Hungarian firms, that such gain comes mostly from an increased variety of intermediates and
to a lesser extent from quality improvements.
Our model also assumes that the exposure to foreign intermediate goods reduces, ceteris
paribus, the cost of innovation. As firms learn from the knowledge embodied in such goods, they
could obtain a higher probability of success investing the same amount of resources. Empirical
support for this assumption can be found in Goldberg et al. (2010b). They find that: 1) firms
1
Lentz and Mortensen (2008) present a model that also introduces heterogeneity in firms’ innovation capacities. See Seker (2009) for a comparison of both models.
2
that faced stronger input tariff reductions were found, ceteris paribus, more likely to introduce
new products; 2) such firms were more likely to invest in R&D; and 3) the main channel of these
effects was through new varieties of intermediate goods and not through the trade expansion
of previously traded ones.
A central element of our analysis is the heterogeneity among firms. Besides their exogenous component of productivity level, firms also differ in their portfolio of goods currently in
production. A seminal work in the area of heterogeneous firms and trade is Melitz (2003). In
his framework, more productive firms are self-selected into export markets as only they find
it profitable to pay the fixed costs associated with exporting. In our study, we focus on the
importing side of trade rather than exporting. However the mechanism is quite similar to the
one presented in Melitz (2003). Firms face fixed costs of importing. Only the efficient ones can
compensate these costs and import goods. In line with Melitz (2003) our analysis highlights
that trade liberalization can have a redistribution effect, where some firms could end worse-off
(lower profits) after the trade liberalization episode even if they import intermediate goods. We
further document a similar redistribution of innovation efforts toward most productive firms
away from the least productive.
The remainder of the paper is organized as follows. Section 2 describes the model economy in
two main parts: 1) the firm’s static problem that includes both selecting the mix of intermediate
goods and labor, as well as firm’s decision whether to import intermediate goods from abroad;
2) the firm’s dynamic problem where the optimal level of innovation efforts is selected. Section
3 presents the simulation exercise in which parameters are selected broadly in line with relevant
empirical papers and with the only goal to illustrate the mechanism in terms of its aggregate
consequences and its implications for firm dynamics. Section 4 concludes. We further present
in the appendix, detailed derivations of some of the key equilibrium conditions of the model.
2
Model
Recent empirical evidence presented in a series of studies by Goldberg et al. (2009, 2010a,
2010b) show that imported intermediate goods lead to higher innovation rates of new products
and hence faster growth. In this study, we present a structural model of firm evolution that
can explain this evidence. Following Klette and Kortum (2004) and their extension by Lentz
3
and Mortensen (2008) we introduce a model of firm growth that accounts for the heterogeneity
in producers’ efficiencies as well as entry and exit decisions. In introducing the heterogeneity
in efficiencies, we follow Melitz (2003). There is a continuum of final good producers each of
which produce a different variety. Unlike Melitz (2003), here firms produce multiple products.
The dynamic nature of the model with heterogeneous firms allows us to capture the differences
in the evolution of importers and non-importers and contribution of importing to the aggregate
growth.
The model is presented in a general equilibrium framework. First, we discuss the demand
side of the economy. Then following Kasahara and Lapham (2008), we introduce the static
profit maximization problem of producers. We incorporate this static problem into a dynamic
framework to discuss the growth patterns of importers and non-importers.
2.1
Preferences
Consider an economy with  + 1 identical countries. In a continuous time setup, each country
consists of a representative consumer with an intertemporal utility function given as
 =
Z
∞
−( −) ln   

where  is the discount rate and  is the aggregate consumption of the composite good.
Instantaneous utility obtained from consumption at time  is ln   The consumer is free to
borrow or lend at the interest rate   Aggregate expenditure at time  is  =   where  is
the price of the composite good. The optimization problem of the consumer yields
̇

=  − 
We let total expenditure be the numeraire and set it to a constant for every period.
In each country there is a the mass of available products each of which is indexed by . The
total mass of products is represented by  Consumers have a taste for variety and consume  ()
units of variety  Goods are substitutes with elasticity of substitution   1 The composite
good is determined by the following constant elasticity of substitution production function
 =
µZ
 ()
∈ 
4
−1



¶ −1

(1)
Composite good producer makes zero profit. Solution of this problem yields
 () = 
µ
 ()

¶−

Price of the composite good  can be found as follows
  =
 =
Z
 ()  () 
µZ
1−
 ()
∈ 
2.2

1
¶ 1−

(2)
Production: Static Decision
The production process has a static and a dynamic component. In this section, we solve the
producer’s static optimization problem. The specification for the static problem follows from
Kasahara and Lapham (2008) which extends Melitz (2003) by incorporating importing decision to the firm’s optimization problem. The difference in our model is that here firms produce
multiple products. There are two groups of producers in the economy. The final goods sector
is combined of a continuum of monopolistically competitive firms producing horizontally differentiated goods. The production of each variety requires employment of labor and intermediate
goods which might be either domestically produced or imported. Producers of the final goods
are distinguished from each other by their efficiency levels, indexed by   0 which is randomly
drawn from a continuous cumulative distribution  ()  As in Melitz (2003) higher efficiency
level means producing a symmetric variety at a lower marginal cost. We assume that efficiency
levels exogenously grow at rate  for all firms. Solving the monopolistic competition model
yields revenue  () and profit  () from each product  as follows2
 () =  ()  () = 
 () =
 ()


2
µ
 ()

¶1−
for ∀  ∈ 
Since in this section we only solve for firm’s static optimization problem, we exclude the time subscript 
from the variables for brevity.
5
Each product is produced by a single producer, hence we can write the equilibrium levels of
revenue and profit from each product as a function of firm’s efficiency level 
Firms employ labor and intermediate goods in the production of final goods. We follow
the macro growth and trade literatures and assume that there is increasing returns to variety
in intermediate goods. All intermediate goods enter symmetrically in the production function,
implying that none of them is intrinsically better or worse than any other. Similar approach
has been used in Ethier (1982) and Romer (1987) among many others3 . This specification is
consistent with the empirical findings in Amiti and Konings (2008), Halpern et al. (2009),
and Kasahara and Rodrigue (2005) which associate the use of foreign intermediate goods with
higher productivities.
Importing firms incur a fixed cost  and iceberg transport cost   1 for every period they
import any intermediate good. Let  ∈ {0 1} refer to the import status of the firm. Production
function for final goods is given as
 ( ) = 
∙Z
1
 ()
−1

 + 
0
Z

 ()
−1

0

¸ (1−)
−1
(3)
where  shows the labor,  () shows the domestic and  () shows the imported intermediate
goods employed in production of variety . The output elasticity of labor in the production
is represented by 0    1 and the elasticity of substitution between intermediate inputs is
  1.  shows the number of partner countries that are traded with.
In the intermediate goods sector, firms are perfectly competitive. All firms have identical
linear technologies and have the same productivities which is set to one. Labor is the only input
used in the production. As it is shown in equation 3, there is a unit continuum of domestic
intermediate goods produced within a country. There is also free entry in this sector. Each
intermediate good producer is a price taker hence price of these products equals to marginal
cost which is the wage rate . Interaction among the intermediate good producers happens
through the labor market as they compete for the same resource.
The solution of the final good producer’s static optimization problem is given in the appendix. In the solution we get  () =  and  () =  for ∀ . These goods are related with
3
See Kasahara and Lapham (2008) for a discussion of the benefits of using this specification.
6
the following equation  =  −   With this result the production function simplifies to
1−
¡
¢ 1−
 ( ) =  1 +  1− −1  [ +   ]1− 
Here  (1 +  1− ) −1 could be interpreted as a total factor productivity term4 . Price of a
final good which only uses domestic intermediate goods is  () for a −type producer. This
price can be found as
 () =




 − 1  (1 − )1− 
If imported intermediates are used in production of the good, then the price is  () =
1−
1−
 () (1 +  1− ) −1  Since (1 +  1− ) −1  1  ()   () for ∀  Once the prices are
determined, it is straightforward to find revenues gained by an importing and non-importing
firm. Define  () as the revenue generated by a −type firm that does not import. It is
calculated as
µ

 () =
 ()

¶1−
=
µ



 − 1  (1 − )1− 
¶1−
(4)

Similarly, the revenue of an importing firm  () can be found as
 () =
Ã
1

µ

−1
¶Ã

(1−)
!!1−
 (1 − )1−  (1 +  1− ) −1
µ
¶1−
¡
¢ (1−)(−1)


1−
−1
= 1 + 

 − 1  (1 − )1−  
¢ (1−)(−1)
¡
−1
 ()
= 1 +  1−
=   ()
where  = (1 +  1−
 )
(1−)(−1)
−1

(5)
 1 From equations 4 and 5 we see that revenue is linear
in firm’s efficiency level. This allows us to determine the cutoff value of the efficiency level ∗
for the decision to import. Firms with the cutoff efficiency level will be indifferent between
4
This interpretation of imported intermediate goods increasing productivity of firms is used in Kasahara
and Lapham (2008).
7
importing and not importing which is shown as
  −   = 


−   =


1
µ
¶ −1


 
∗
 =

1−
 ( − 1)
 (1 − )   − 1
2.3
(6)
Production: Dynamic Decision
The dynamic optimization problem of the firm follows from Klette and Kortum (2004) and their
extension in Lentz and Mortensen (2008)5 . The monopolistic competition in the final goods
sector results in each product being produced by a single firm. Firms can produce multiple
varieties. The number of products  determines the portfolio of the producer. This portfolio
increases by innovating new products and it decreases by destruction of the existing products.
This process determines firm evolution. Firm’s innovation rate depends on investment in research and development (R&D)  which is measured in labor units and knowledge capital. The
innovation production function is strictly increasing and strictly concave in 6  It is strictly
increasing in the knowledge capital and homogeneous of degree one in  and knowledge capital.
For non-importing firms knowledge capital is measured by the total knowledge accumulated
through past innovations. We use the number of products innovated  to represent this capital
stock. For the importing firms, it is the product of (1 + )  Here (1 + ) represents the
spillover from the knowledge embodied in the imported intermediate goods. Grossman and
Helpman (1991) discuss several ways in which international knowledge spillovers are possible.
They argue that residents of a country may find occasions to learn technical information from
meeting with foreign counterparts that contributes to their stock of general knowledge. Also the
use of differentiated intermediate goods that are not available in the domestic market can increase the insights that local researchers gain from inspecting and using these goods. Grossman
and Helpman (1991) use the cumulative volume of trade between countries as the international
spillover for innovation. We use a function of the total flow of intermediate goods that are
5
In Lentz and Mortensen (2008) innovations arrive as improvements in quality levels. In our model innovations arrive as new varieties which is similar to the setup in Seker (2009).
6
The properties of the innovation production function follows from Klette and Kortum (2004).
8
imported. We measure the magnitude of the spillover by   0.
Goldberg et at. (2010b) provide empirical evidence that motivates the inclusion of imported
intermediate goods in the innovation function. They show that increase in the availability
of foreign intermediate products have increased the innovation capacities of firms in Indian
manufacturing sector. They find that reduction of input tariffs have led to imports of new
varieties in the economy and this has led to an expansion of within-firm product scope by
almost 8 percent.
As a result of firm’s R&D investment and knowledge capital new products arrive at a Poisson
rate of  where  is the innovation intensity. The innovation function for an importing firm
can be formalized as
´1−
³
 =  (1 + ) 
where 0    1 and   0
 =  1− (1 + )(1−)
! 1
Ã

 =

(1 + )(1−)
If we define  =
1

 1 then
1−

=  − 1  0 We can rewrite R&D investment as
=

(1 + )(−1)

In this equation higher use of imported intermediate goods reduces the cost of R&D and
increases the probability of innovating new varieties. In parameterizing the R&D cost, we use
1+
1

 ()  = 0 (1+)
1 for 0  1  0 The R&D cost for the non-importing firm is same as the
cost for importing firm without the spillover term  ()  = 0 1+1 
Each producer faces a Poisson hazard rate  of losing any of its products. The hazard rate
 is same for all firms and all products. Each firm takes the value of  as given. However its
value is determined by the aggregate creation in the economy which will be discussed below.
Firms exit from the economy if all of their products are destroyed. There is no re-entering once
exit occurs.
The dynamic maximization problem is similar to Lentz and Mortensen (2008). The state
of a producer is determined by its current portfolio of products  Assume that r = {1   }
9
is the vector for the revenue realization of each product. Differences in the revenue realizations
can be caused by the heterogeneity in fixed costs of importing, transportation costs, and the
differences in product specific efficiency realizations of the firm. Let r−1
− represent the same
vector excluding the  element and r+1 = {1    } represent the addition of a new product
with revenue  For a particular −type producer and a constant interest rate , the Bellman
equation is formulated as follows7
 (r ) = max
≥0
⎧
⎪
⎪
⎪
⎪
⎪
⎨

P
 () − ()+
=1
 [ (r+1 ) −  (r )]
∙
¸
⎪
⎪
¢
⎪
P¡
⎪
−1

⎪
 (r− ) −  (r )
⎩ +
=1
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭

(7)
This equation shows that current value of firm is equal to the sum of three terms8 . The first
term on the right hand side shows the current profit net of R&D costs. The other two terms
show the net future value of the firm. The second one is the gain in value caused by the
innovation of a new variety and the last one is the expected loss associated with a loss of a
randomly chosen product9 .
Since the value function is linear in the number of products, it is possible to obtain an analytical solution to optimization problem. Following Lentz and Mortensen (2008), we conjecture
that the value function is given as

 (r ) =

X
 ()
=1
+
+ Θ ()
(8)
where Θ () is the type conditional continuation value of innovation. Then we incorporate this
conjecture into the Bellman equation which simplifies to
½ µ
¶
¾
 ()
+ Θ () −  ()
( + ) Θ () = max 
≥0
+
( ()
)
−

()
+
Θ () = max

 ≥ 0
+−
7
(9)
Constancy of interest rate is obtained from the consumer’s optimization problem.
The only difference in the Bellman equation for an importing and non-importing firm is in the profit and
the cost of innovation. We do not show them explicitly for expositional simplicity.
9
Existence of a solution for this dynamic optimization problem under heterogeneous firms is proven in Lentz
and Mortensen (2005).
8
10
In order to have the conjectured value function solve this problem, all variables have to be
stationary. We have assumed that efficiency rates exogenously grow at rate . In the appendix
we show how the economy grows on a balanced growth path. There we also show that profit
levels and wage rate are constant which makes it possible to have an analytical solution of the
model. Solving the simplified Bellman equation, we get
 ()
+ Θ () = 0 ()
+
We define  () =
()
+
(10)
+ Θ () as the expected value of a product for a −type producer. It
is the sum of the discounted stream of the profits and the innovation option value. Using the
value of Θ () from equation 9 and implementing it into equation 10 we get
0 () =
 () −  ()

+−
(11)
Klette and Kortum (2004) shows that the optimal value of  is an increasing function of profit
level which monotonically increases in  This result shows that firms with high efficiencies are
more likely to innovate. Plugging the value of  () =
0 1+1
(1+)1
for an importing firm into this
equation we get,
0 (1 + 1 ) 1
(1 + )1
1
0 (1 + 1 ) 
1+1
=
0 
 () −  (1+)
1
+−
(1 + )1  () − 0 1+1
=
for   ∗ 
+−
(12)
Similarly for the non-importing firm the solution is
 () − 0 1+1
for  ≤ ∗ 
0 (1 + 1 )  =
+−
1
The difference between the solution for importing and non-importing firms is the spillover term
which is captured by (1 + )1  1 It enters into the equation as a positive multiplier of the
profit level. Hence importing gives extra advantage to firms to innovate.
11
2.4
Entrant’s Problem
From a potential pool of entrants, successful ones enter the economy as a result of an innovation
of a new variety. Firms discover their efficiency types once they enter. Although all entrants
start with a single product, they do not necessarily have the same startup size because of
their varying efficiency levels. In our model initial size increases in efficiency levels which is
not captured in Lentz and Mortensen (2008). Entrants face the same innovation cost function
as incumbent firms and they innovate at rate   Entry rate is determined by the free entry
condition which is given as
0
 ( )
| {z }
=
marginal cost of innovation
Z
|
(13)
 ()  ()  
{z
}
net gain from innovation
Here  (·) is type distribution for entrants and  () is the value of a single product for a −type
producer. Entry rate  is the product of mass of potential entrants  and innovation rate of
entrants    =    From incumbent firm’s optimization problem we had 0 ( ()) =  () 
Using this result in equation 13 we can solve for the innovation rate of entrants as a function
of innovation rates of incumbent firms
0 (1 +
1 ) 1

=
Z
=
"Z
0 ( ())  () 
∙Z
¸11
1
=
 ()  () 
∗
 ()1  ()  +
Z
∗
0
 ()1
∞
(1 + )1
#11
 () 

This result shows that keeping all else constant, higher knowledge spillovers from trade decrease
the innovation rate of entrants, hence reduces the entry rate.
2.5
General Equilibrium
Firms evolve as a result of a birth and death process of products. Define  () as total
P
mass of firms of type  with  products.  () = ∞
=1  () is the total mass of −type
firms and  is the total mass of all firms. We can define  () =  ()  as the steady
12
state type distribution. Mass of products produced by −type firms can be represented as
P
Λ () = ∞
=1  (). To guarantee the existence of a stationary size distribution, we assume
that    () for ∀  If this condition is violated size and age of some establishments diverge
to infinity which precludes having a stationary size distribution10 . In steady state, the rate of
product destruction  should be equal to the sum of the product creation rates of entrants and
incumbent firms
=+
Z
 () Λ ()  () 
(14)
In equilibrium, total mass of products produced by −type firms is found as11
Λ () =
 ()

 −  ()
(15)
In the appendix, we also derive the entry type distribution  (·) as a function of the steady
state type distribution  (·) The final equilibrium condition is labor market clearing condition.
There is a fixed measure of workers denoted as  Labor is allocated across four activities for
every -type incumbent firm: final good production  , intermediate good production  , R&D
investment  (), and for those firms that import the fixed cost of importing  . There is also
a part of the labor force allocated to research and development for potential entrants  ( ) 
The labor market clearing condition can be stated as
=
Z
∞
( () +  () +  ( ()) +  ()  ) Λ ()  ()  +   ( )
(16)
0
where  () = 1 if   ∗  Given these equilibrium conditions, a stationary equilibrium for this
economy consists of aggregate destruction rate , wage rate  and interest rate  such that
for given values of (  )  ) any −type incumbent producer chooses the optimal innovation
rate  () and solves equation 11 to maximize its value, ) potential entrants choose  in
solving equation 13 and break even in expectation, ) representative consumer maximizes
utility subject to budget constraint, ) labor market clears as in equation 16, and ) equations
14 and 15 hold. The equilibrium value of aggregate price index is derived in the appendix.
The condition needed to guarantee    () for ∀  is 0 ()  −()
which follows from equation 11.

Derivation of this equation which can be found in Lentz and Mortensen (2008) is reproduced in the
appendix.
10
11
13
3
Quantitative Model and Simulation
The purpose of this simulation exercise is to show the dynamics of firm evolution and the
contribution of importing to innovation and growth. Hence the parameter values are not chosen
particularly to match any data. In the following versions of this paper, we will test the model’s
prediction power on explaining the innovation and trade performance of Indian manufacturing
firms.
We can compare the profit levels of importing firms in autarky and trade. Define   (),
    as profit, aggregate price, and equilibrium wage in trade equilibrium. Replacing
superscripts to  refers to the same variables in autarky. Aggregate expenditure  is normalized
to a constant. The profit levels   () and   () are calculated as
¸
∙
¢ (1−) −1
  − 1  
1− ¡
1− −1
 (1 − )

 () =
1 +  

 
∙
¸−1
  − 1  
1−

 () =
 (1 − )



 

For an importing firm, the comparison of these profit levels yield
  () =   () −  
¾−1 ½  ¾−1 #
∙
¸−1 "½
(1−)  
¡
¢

−
1


−1
1 +  1−

 (1 − )1− 
  =
−






∙n
(1−)
−1
)
As long as  = (1 +  1−



o−1
−
©   ª−1

¸
 0 there is an efficiency level ̂  ∗
such that all firms with efficiency levels greater than ̂ are going to gain from trade. Unlike
the setup in Melitz (2003) it is not straight forward to prove that   0 analytically. Hence
we simulate the model to compare the autarky and trade equilibriums.
When comparing the autarky with trade equilibrium we have to set the values of the model’s
©
ª
parameters ∆ =  0  1                 . The description of these parameters
and their values are summarized in Table 1. We assume that efficiency types are lognormally
¡
¢
distributed (˜      ). Annual growth rate in the economy  which is exogenous in the
model is set as 10 percent12 . Among the innovation cost parameters 0 is a scale parameter
12
This value is the average annual growth rate of gross sales between 1990 and 2003 that is calculated for
the Indian manufacturing firms in Goldberg et al. (2010a).
14
and it is set as 3000 and 1 is set as 3.5. Lentz and Mortensen (2008) which use a similar
innovation cost structure estimate 1 as 3.7 for the Danish manufacturing firms. In Seker
(2009), estimation of this parameter for five Chilean manufacturing industries yields values
between 3.8 and 5.7. The value for the knowledge spillover parameter  is set as 1. For the
simulation we assume that there are only two symmetric countries in the market and set 
to 1. The choice for elasticity of substitution between varieties  follows from Seker (2009).
The iceberg cost is set as 1 percent. The last two parameters are from Kasahara and Lapham
(2008). In their estimation exercise for three Chilean manufacturing industries, they find that
elasticity of substitution between intermediate goods vary between 4.6 and 5 and the factor
elasticity of labor around 0.26. When comparing the autarky with trade, we normalize mass of
Table 1: Model Parameters
Parameter
0  1


   
  




Description
Innovation cost parameters
Real interest rate
Growth rate of efficiency levels
Efficiency distribution
Fixed and iceberg costs of importing
Number of trading partner countries
Elasticity of substitution between varieties
Elasticity of substitution between intermediate goods
Labor share in production
Value
3000, 3.5, 1
0.05
10%
3, 4
0.001, 1.01
1
2.5
4
0.3
potential entrants  to 1. In the simulations, we randomly draw 10,000 efficiency values and
use the same set of firms in both autarky and trade. The condition required to have a stationary
size distribution    () for all  can be written as an upper bound on the efficiency type
distribution which makes it easy to implement in the simulation. We present this relationship
in the appendix.
Using the parameter values above, we compute average profit level  () per product and
innovation intensities  () for each efficiency level  Figure 1 shows the relationship between
efficiency levels and profit levels. Cutoff efficiency level ∗ that was found in equation 6 is 17
The graph shows that not all importers gain from trade due to higher costs of production. This
finding is in accordance with Melitz (2003). He shows that among the exporting firms only
the most efficient ones gain from trade ( ≥ ̂). In our simulation this threshold value for the
efficiency levels is ̂ ∼
= 28 This also shows that the term  is strictly positive.
15
0
1
Profit
2
3
Figure 1: Firm Profit Level () versus Efficiency Level ()
0
10
17 20
Efficiency Level
Autarky
28 30
40
Trade
Next in figure 2 we look at how innovation intensities of firms are affected by trade. In
trade equilibrium importing firms innovate at higher rates than they would in autarky. This
finding is in accordance with the empirical evidence presented in Goldberg et al. (2010b).
They show that as the result of the tariff liberalization in 1990’s, India experienced a surge in
imported inputs mostly in inputs that were not available before. Higher usage of new input
varieties has lead to introduction of new products in the domestic market. Another result that
can be obtained from the graph is that although some of importers with efficiency levels  such
that ∗    ̂ observe drops in their per-product profits  (), they may still gain higher
aggregate profits  ()  in trade equilibrium because they are likely to innovate more products
in trade than autarky.
The stochastic innovation process and the exogenous heterogeneity in efficiency levels allow
us to compare a rich set moments on innovation and growth. We perform this comparison in
Table 2. Goldberg et al. (2010a, 2010b) present evidence from Indian manufacturing sector
on the relationship between importing and innovation performance of firms which can be explained by our model. In the last column of Table 2 we present some data moments from their
16
0
Innovation Intensity
.02
.04
.06
Figure 2: Innovation Intensities () versus Efficiency Level ()
0
10
17 20
Efficiency Level
Autarky
30
40
Trade
studies. As it was mentioned before, our simulation exercise does not intend to match these
data moments.
Table 2 shows that average sales and average product per firm increases in trade. Among
the incumbent firms, efficient ones have higher innovation rates caused by the increase in their
profits and the knowledge spillover effects of importing on innovation. This gain leads to an
increase in the average and aggregate innovation intensities in the economy. In the mean time
innovation gets costlier due to the increased wage rates and this decreases entry of new firms.
The higher innovation intensity in trade equilibrium leads to doubling of the average product
growth rate from 1 percent to 2 percent. The model also explains what percent of aggregate
growth in the economy is due to product creation (between share of growth) and efficiency
growth (within share of growth). In trade, between share of growth increases due to higher
innovation rates of incumbents. Indian data yields similar growth decomposition. The last
four rows show some moments that the model could be used to match in the Indian data. 90
percent of the firms did not introduce any new products in Indian data and this value is 94
percent for firms with single products. The final variable is the correlation between extensive
17
margin (number of products produced) and the intensive margin (revenue per product). Trade
equilibrium gives a correlation of 0.39.
Table 2: Autarky vs. Trade Comparison
Moments
Ave Sales
Ave # of products
Ave # of products (multi-product firms)
Ave Innovation Intensity
Aggregate Innovation (by Incumbents)
Entry Rate
Average product growth rate
Within share of growth
Between share of growth
% of firms did not change # of products (all)
% of firms did not change # of products (single)
% of firms did not change # of products (multi)
Extensive & intensive margin correlation
4
Autarky
16260
1.89
12
0.03
0.17
0.06
0.01
0.89
0.10
0.93
0.98
0.54
0.44
Trade
20275
2.29
17
0.05
0.32
0.05
0.02
0.84
0.16
0.90
0.96
0.39
0.39
Indian Data
1.97
3
0.04
0.85
0.15
0.90
0.94
0.86
0.43
Conclusion
In this paper we develop a general equilibrium model of multi-product firms to explain the relationship between importing, innovation, and growth. Following the structural model of Klette
and Kortum (2004) and their extension in Seker (2009), we present a stochastic dynamic model
of firm and industry evolution. In the model we introduce heterogeneity in firms’ efficiency
levels as in Melitz (2003). The novel feature of our model is that unlike many of the recent
trade models that follow Melitz (2003), our model is dynamic. Firms invest in R&D which
results in introduction of new varieties to the economy. Each period these products face a
probability of being destructed. The birth and death of products yield the stochastic growth
process of firms. Incorporating the importing decision in this setup allows us to relate trade
with innovation and growth.
Firm’s efficiency is the main driver of its evolution. Only the efficient firms can participate
in import markets as they can compensate the sunk costs of trade. They are also more innovative. With the learning they obtain through knowledge spillovers from the use of foreign
intermediates, their innovation intensities increase even further relative to non-importing firms.
18
With the additional benefits of importing on their revenues and innovation rates, these firms
grow faster and exit less often.
The model’s ability to explain how importing is related to innovation and growth is presented with a trade liberalization exercise comparing autarky with trade equilibrium. This
simulation exercise shows that as in Melitz (2003) in trade resources are reallocated to more
efficient firms and this leads to an increase in average size and number of products produced by
firms. Moreover, average product creation rate increases which leads to faster growth. Higher
innovation rates of importing firms cause a reduction in the entry rate. Although we do not intend to match any dataset in our simulation exercise, we show that our model has the potential
to explain a rich set of dynamics of trade and firm growth by providing supporting evidence
from Indian manufacturing firms.
In the following versions of this study, we will extend our work in two dimensions. First
we will test our model’s capacity to explain the importing decisions and evolution of Indian
manufacturing firms. Second, we will incorporate the export decision into our model to analyze
the complementarity between two aspects of trade in affecting firm evolution.
19
References
[1] Amiti, Mary and Jozef Konings (2007), "Trade Liberalization, Intermediate Inputs, and
Productivity: Evidence from Indonesia," American Economic Review, vol. 97(5), pg 16111638.
[2] Arkolakis, Costas (2009), "Market Penetration Costs and the New Consumers Margin in
International Trade," NBER Working Paper No.14214.
[3] Ethier, Wilfred J. (1982), “National and International Returns to Scale in the Modern
Theory of International Trade," American Economic Review, 72(3), pp.389-405.
[4] Theory of International Trade,” American Economic Review, 72, No. 3: 389-405.
[5] Gancia, Gino and Fabrizio Zilibotti (2005), "Horizontal Innovation in the Theory of
Growth and Development," Handbook of Economic Growth, Chapter 3, Vol(1a), ed. by
Philippe Aghion and Steven N. Durlauf.
[6] Goldberg, Pinelopi, Amit Khandelwal, Nina Pavcnik, and Petia Topalova (2010a), "Multiproduct Firms and Product Turnover in the Developing World: Evidence from India,"
Forthcoming in Review of Economics and Statistics.
[7] –– (2010b), "Imported Intermediate Inputs and Domestic Product Growth: Evidence
from India," Forthcoming in Quarterly Journal of Economics.
[8] –– (2009), "Trade Liberalization and New Imported Inputs," American Economic Review, 99(2), pp.494-500.
[9] Grossman, Gene M. and Elhanan Helpman (1991) Innovation and Growth in the Global
Economy, Cambridge: MIT Press 1991.
[10] Halpern, Laszlo, Miklos Koren, and Adam Szeidl (2009), "Imported Inputs and Productivity," mimeograph, Department of Economics, University of Berkeley.
[11] Kasahara, Hiroyuki and Beverly Lapham (2008), “Productivity and the Decision to Import
and Export: Theory and Evidence,” CESifo Working Paper Series No.2240.
[12] Kasahara, Hiroyuki and Joel Rodrigue (2008), "Does the Use of Imported Intermediates
Increase Productivity? Plant-level Evidence," Journal of Development Economics, 87,
pp.106-118.
[13] Kehoe, Timothy J. and Kim J. Ruhl (2009), "How Important is the New Goods Margin
in International Trade?" Federal Reserve Bank of Minneapolis Staff Report 324.
[14] Klette, Tor J, and Samuel S. Kortum (2004), "Innovating Firms and Aggregate Innovation," Journal of Political Economy, 112(5): 986-1018.
[15] Lentz, Rasmus and Dale T. Mortensen (2005), "Productivity Growth and Worker Reallocation," International Economic Review, 46(3): 731-751.
20
[16] –– (2008), "An Empirical Model of Growth through Product Innovation," Econometrica,
76(6): 1317-73.
[17] Luttmer, Erzo G. J. (2010), "On the Mechanics of Firm Growth," Federal Reserve Bank
of Minneapolis, Research Department Staff Report 440.
[18] Melitz, Marc J. (2003), "The Impact of Trade on Intra-industry Reallocation and Aggregate Industry Productivity," Econometrica, 71(6): 1695-1725.
[19] Romer, Paul (1987) “Growth Based on Increasing Returns Due to Specialization,” American Economic Review, 77(2): 56-62.
[20] ––(1990), "Endogenous Technological Change," Journal of Political Economy, 98(5):
S71-S102.
[21] Seker, Murat (2009), “A Structural Model of Establishment and Industry Evolution: Evidence from Chile,” World Bank Policy Research Working Paper No. 4947.
A
Solving Firm’s Static Problem
In a symmetric equilibrium, the static profit maximization problem of a final good producer
can be formalized as follows
Z 1
Z 
−1
1
      − 
 ()  − 
 ()  − 
0

 = 

∙Z
1
 ()
−1

0
 + 
0
Z

 ()
−1


0
¸ (1−)
−1

The first order conditions of this maximization problem with respect to  (),  () for ∀ 
and  in respective order are given as follows:
(17)
 =
 − 1 −1 −1  (1 − ) 
  


−1
1

 =
∙Z
1
 ()
0
−1

 + 
Z
0

 ()
−1


¸
(1−)
−1
−1
−1
−1
 ()  −1

(18)
∙Z 1
¸ (1−)
Z 
−1
−1
−1
−1
−1
 − 1 −1 −1  (1 − ) 
−1




 ()  −1

 ()
 + 
 ()



−1

0
0
∙Z 1
¸ (1−)
Z 
−1
−1
−1
1  − 1 −1
  −1 −1
 = 
 ()   + 
 ()  
(19)

0
0
1

From equation 17, we get  () =  for all  and from equation 18 we get  () =  for ∀ .
Then, taking ratios of the equations 17 and 18, for all firms that import intermediate products
21
we get
µ ¶− 1

1
=


−
 =   for all 
(20)
Total output of a final good is found as
h −1
i (1−)
¡ − ¢ −1
−1


+   
 ( ) =  
h¡
¢ −1 i (1−)
−1
=  1 +  1−  
¡
¢ (1−)
=  1 +  1− −1 1− 

(21)
This equation could further be simplified. Taking the ratios of equations 17 and 19 and using
the results that  and  are constant, we get
 (1−)
−1
∙ −1
¸ (1−)
−1
−1
−1


 + 
−1
−1
  −1

∙
¸ (1−)
−1
−1
−1

−1   + 
= 1
Simplifying this ratio and using the finding  =  −   we get
− 1
 (1 − ) 
¸ = 1
∙
−1
−1


 
+ 
− 1
 (1 − ) 
h
i = 1
−1
1−

 (1 +  )
¶
µ
−1 1
1−

1−

(1 +  )  = 

µ
¶

= 
(1 +  1− )
1−
22
(22)
¡ ¢− 1
Now combining the result from equation 19, the finding that  () =    and the result
in equation 22, we get an equation that gives the relationship between price and the wage rate
 =
 =
 =
=
∙Z 1
¸ (1−)
Z 
−1
−1
−1
−1
−1
 ()   + 
 ()  

0
0
(1−)
¡
¢
−1
−1 1 +  1− −1 1−



 − 1 −1 (1 +  1− ) (1−)
−1  1−



 − 1  ¡(1 +  1− ) ¡  ¢¢−1 (1 +  1− ) (1−)
−1  1−


1−


=
 − 1  (1 − )1− (1 +  1− )−1 (1 +  1− ) (1−)
−1


 =

 − 1  (1 − )1− (1 +  1− ) 1−
−1
(23)
From the result obtained from equation 23, we see that the final good price for a good that only


uses domestic intermediate goods is  () = −1
 If imported intermediate goods
 (1−)1− 
1−
are used in production, then the price is  () =  () (1 +  1− ) −1 
Next we derive the equilibrium values of  and  From the solution of the profit maximization problem of the final good producer, replacing the equilibrium value of labor  from
equation 22 into equation 21 we get
µ
¶¶
µ
¡
¢ (1−)

1−
 ( ) =  (1 +  )
1 +  1− −1 1−
1−
µ
¶
¡
¢ −

1− −1
 ( ) =  1 + 
 
(24)
1−
Recall that from composite good producer’s maximization problem we get
 () =  ()−

 1−
(25)

Combining equations 24 and 25, we get
−
 ()

 1−
¡
¢ −
=  1 +  1− −1
 =
 ()−

 1−
−
 (1 +  1− ) −1
µ
¡

1−
¢

1−
¶


We can get rid of the price from this equation by plugging in the value of  () from equation
23
23

"
#−
¶− h
¡
¢ − i−1


1− −1
 1 + 
=
 1−
 − 1  (1 − )1− (1 +  1− ) 1−
−1
Ã
!
µ
¶−
¡
¢ (1−)−+
 − 1  (1 − )1−


−1
1−
−1
=


1 +  
 1− 1 − 


µ


1−
Finally labor value can be found by incorporating  into equation 22
Ã
!
µ
¶1−
1− 

(1−)(−1)
¡
¢
(1
−
)

−
1



−1
−1 1 +  1−

 = 1−

1−


B
Balanced Growth in the Economy
In the model the only source of aggregate growth in a balanced growth path is the exogenous growth in efficiency levels. Although firms evolve as a result of innovations, in steady
state this does not create aggregate growth. We assume that efficiency levels grow at rate 
(̇ = ). Since there is no population growth,  is constant. Then from equation 3 we get
̇ ( )  ( ) =  Equation 1 gives

µZ
¶
¶ −1
−1 µZ
−1
−1

−1
̇ ()
−1


̇ =
 ()

 ()

 ()
−1

 ()
∈ 
∈ 
 −1
̇ =

−1 
̇
= 

Since we normalize aggregate expenditure to a constant in the model  =   implies that 
and  () for ∀  decreases at rate .
 =
µZ
1−
 ()
∈ 
1
̇ =
1−
̇

= − =
µZ

1
¶ 1−
1−
 ()
∈ 
̇ ()
for ∀ 
 ()

1
¶ 1−
−1 µZ
−
(1 − )  ()
From the final good producer’s optimization problem we found that
 () =
 

−1
24
̇ ()
 () 
 ()
¶
Since  () decreases at rate  and  grows at rate , wage is constant. Then profit per product
³
´1−
()

 () = 
is constant. This allows us to get the stationary solution in the Bellman

equation. Note that although wage is constant, real wage which is  and real output
grow at rate .
C
()

Steady State Size Distribution of Firms
Lentz and Mortensen (2008) follow closely Klette and Kortum (2004) in deriving the steady
state size distribution. Klette and Kortum (2004) shows that in steady state mass of firms of
any type converges to
µ
¶−1
 ()  ()
 () =

(26)


Using this equation, we can derive total mass of firms of type  as
µ
¶−1
∞
∞
X
 () X 1  ()
 () =
 () =
 =1 

=1
µ
¶
 ()

=
ln

 ()
 −  ()
(27)
Here convergence to a stationary size distribution requires    () for all  Taking the ratio
of equations 26 and 27 gives the steady state size distribution for the number of products
³
´
1 ()


 ()
´
= ³

 ()
ln
−()
This is probability distribution for the logarithmic distribution with parameter  ()  Finally
total mass of products produced by −type firms is found as
µ
¶−1
∞
∞
X
X
 ()  ()
 ()
Λ () =

(28)
 () =
=


 −  ()
=1
=1
D
Deriving Entry Type Distribution in Steady State
From equation 27 we get
 () =
=
 ()  ()
´
³

ln −()
 ()  ()
´ 
³

ln −()
25
R
Taking the integrals of both sides and using the fact that  ()  = 1 we get
Z
Z
 ()  ()
´ 
³
 =
 ()  = 

ln −()
()()

ln( −()
)
 () = R
E
()()


ln( −()
)
(29)

Deriving Aggregate Price Index
We have defined Λ () as total mass of products produced by −type firms. Plugging the value
of  () from equation 23 into aggregate price index from equation 2, we get

1−
=
Z
∗

1−
 ()
Λ ()  +
0
 =
µ



 − 1  (1 − )1−
Z
∞
 ()1− Λ () 
∗
¶ µZ
∗
−1

Λ ()  +
Z
∞
∗
0
1
¶− −1
³
´−1
1− 1−
(1 +  ) −1
Λ () 

In the simulation exercise, aggregate price index can be computed by replacing Λ () by its
value in equation 28 and  () by its value in equation 29. Ignoring the price differences between
importing and non-importing firms, this would yield
Z ∞
1−
=
 ()1− Λ () 

0
=
Z
∞
1−
 ()
0
F
()()

(−()) ln( −()
)
 R
()()


ln( −()
)

Finding Maximum Possible level of Efficiency Level
Maximum profit level  max is the maximum attainable level of profit that would satisfy  ()  
for all 
 max −
0 1+1
(1+)1
+−
 max
=
0 (1 + 1 ) 1
(1 + )1
0 1
=
( (1 + 1 ) + ) 
(1 + )1
26
The efficiency level that would correspond to this profit level can be found as follows
̄ =  max − 
"
1− #−1
  − 1   (1 − )1−  (1 +  1− ) −1
=
− 



1
h


 i −1
max = (̄ +  )
1−

 − 1   (1 − ) (1 +  1− ) 1−
−1
27