WPS4947
Policy Research Working Paper 4947
A Structural Model of Establishment
and Industry Evolution
Evidence from Chile
Murat eker
The World Bank
Global Indicators Analysis Department
Enterprise Analysis Unit
June 2009
Policy Research Working Paper 4947
Abstract
Many recent models have been developed to fit the basic and the hazard rate of exit, which had eluded existing
facts on establishment and industry evolution. While models. In the model, heterogeneity in producer behavior
these models yield a simple interpretation of the basic arises through a combination of exogenous efficiency
features of the data, they are too stylized to confront differences and accumulated innovations resulting from
the micro-level data in a more formal quantitative past endogenous research and development investments.
analysis. In this paper, the author develops a model in Integrating these forces allows the model to perform well
which establishments grow by innovating new products. quantitatively in fitting data on Chilean manufacturers.
By introducing heterogeneity to a stylized industry The counterfactual experiments show how producers
evolution model, the analysis succeeds in explaining respond to research and development subsidies and more
several features of the data, such as the thick right tail of competitive market environments.
the size distribution and the relations between age, size,
This paper--a product of the Enterprise Analysis Unit, Global Indicators Analysis Department--is part of a larger effort
in the department to understand private sector development in developing countries. Policy Research Working Papers are
also posted on the Web at http://econ.worldbank.org. The author may be contacted at mseker@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
A Structural Model of
Establishment and Industry Evolution:
Evidence from Chile
MURAT SEKER
¸
Enterprise Analysis Unit
World Bank
JEL Classification: L11, L6, O31, C15
Keywords: Industry evolution, Establishment dynamics, Innovation, Endogenous product scope, Parameter
estimation.
I would like to thank to Samuel Kortum, Thomas Holmes and Erzo Luttmer for their advice and guidance. I
also would like to thank to Patrick Bajari, Fabrizio Perri, Minjung Park, Daniel Rodriguez Delgado, participants
of 2008 LACEA/LAMES conference and Macroeconomics seminar participants in the Development Research
Group of the World Bank for their suggestions.
Correspondence: 1818 H Street, NW Washington DC, 20433 MSN: F4P-400 mseker@worldbank.org
http://www.enterprisesurveys.org/
1
1 Introduction
Empirical research using longitudinal firm or plant level data has shown strong regularities in
establishment and industry evolution. Recently, researchers have started to build structural
models to explain these regularities. However, being so stylized has made it difficult for these
models to confront the micro-level data in a more formal quantitative analysis. In this paper, I
present a model that explains several salient features of the data that had eluded the existing
models. Furthermore, using a panel of Chilean Manufacturers, I provide an estimation of the
model parameters and show its quantitative success in fitting the data.
This paper builds on the stylized model of establishment and industry evolution presented by
Klette and Kortum (2004). In their model, an establishment is defined as a collection of prod-
ucts 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 com-
petitor. This birth and death process for the number of products is the source of establishment
and industry evolution. Through this parsimonious model of innovation, they explain various
stylized facts that relate R&D, productivity, patenting, and establishment growth. Their model
also generates heterogeneity in establishments' sizes and a skewed size distribution.
However, the Klette and Kortum (2004) model fails to capture several important features
of the data. Through a parsimonious extension of their model, I succeed in explaining these
features which are: the thick right tail of the size distribution, the independent relation between
age, size and the hazard rate of exit, the relation between the variance of growth rate and size,
and the pre-exit behavior of establishments in a birth cohort. At the root of the improvements
over the Klette and Kortum (2004) model is the introduction of heterogeneity in producers'
efficiency levels1 . As a result of this heterogeneity, producers differ from each other in both
their innovation rates and the revenues generated from each product.
The improvements and how they emerge through this extension can be explained as follows.
The first improvement is in fitting the size distribution. In the model, more efficient producers
are more innovative and they generate more revenue per product. This complementarity be-
tween the innovation rate and the revenue magnifies the variation among establishments' sizes
1
Lentz and Mortensen (2008) also incorporates heterogeneity into Klette and Kortum (2004) model in a
different way which I explain through the paper, is not sufficient for the improvements I specify here.
1
and leads to a closer fit to the observed size distribution, especially in the thickness of the right
tail. On the other hand, the size distribution derived in the Klette and Kortum (2004) model
is logarithmic. This distribution is skewed but the tail is much shorter than the one observed
in the data. Through introducing random sized products, Lentz and Mortensen (2008) improve
the size distribution that emerges in Klette and Kortum (2004), but still they cannot capture
the thickness of the right tail. My model succeeds this without postulating any exogenous
variation in the size of the products.
The second improvement caused by this extension is on the relation between the variance of
growth rate and size. This relation has drawn much less attention than the relation between the
growth rate and size, both in theoretical and empirical research. Some recent work by Stanley
et al (1996), Bottazzi (2001), and Sutton (2002, 2007) illustrate that variance of growth rates
declines at a rather slow rate as size of an establishment increases. The mixing of producers
with different efficiency levels at any size allows my model to explain the flatness of this relation.
On the other hand, the models of Klette and Kortum (2004) and Klepper and Thompson (2007)
yield too steep slopes.
The third improvement of the model is on the independent relation between size, age, and
the hazard rate of exit. Evidence shows that as size and age increase, the hazard rate of exit
decreases2 . In my model, exit is determined by the number of products owned by the producer.
Mixing of establishments with different efficiency-types allows age and size to be correlated with
the number of products through different channels. This yields a negative relation between the
hazard rate of exit and size conditional on age and a negative relation between the hazard
rate of exit and age conditional on size. Klette and Kortum (2004) and Luttmer (2007) can
only explain the relation between the hazard rate of exit and size3 and Klepper and Thompson
(2007) explain both relations through introducing random sized products.
The final improvement is on the pre-exit behavior of establishments within a birth cohort.
Evidence on Chilean manufacturers shows that there exists size dispersion among entrants and
on average, establishments with larger startup sizes live longer than the smaller ones. I explain
these observations through type-heterogeneity. All entrants start with a single product. But,
2
This relation has been demonstrated by several studies including Dunne, Roberts, Samuelson (1988) and
Evans (1987a, 1987b).
3
Without conditioning on size, both models can generate a negative relation between the hazard rate of exit
and age.
2
more efficient producers have larger startup sizes. Moreover, they are more innovative and face
lower hazard rate of exit. As a result, conditional on when they will exit, larger producers
survive longer.
Another novel feature of this paper is its quantitative strength. Most of the current models
derive inferences about the heterogeneity in producer behavior by analyzing broadly defined
sectors such as manufacturing, wholesale and retail, or service. Part of this observed hetero-
geneity could be purely due to different industry structures instead of the intrinsic efficiency
differences across producers. To have a better identification of the source of heterogeneity
across producers, I estimate the model's parameters separately on the five biggest 3-digit man-
ufacturing industries in Chile. These industries differ in their size distributions, growth rate
distributions, and entry rates. Estimation results show that model parameters can successfully
explain different industry structures.
This paper also gives insight into the persistent differences in the performances of the
establishments. It incorporates intrinsic exogenous efficiency differences that are determined
before entering the economy and idiosyncratic innovations that endogenously accumulate during
the life of an establishment. Both features have been used extensively to explain industry
dynamics. My model incorporates both features in the producer's optimization problem. At
the early stages of life, efficiency differences are the main contributors of the variation in size. At
older ages, due to the selection of more efficient producers, contribution of the past innovations
exceeds the contribution of the efficiency differences.
After capturing various features of the data, I perform counterfactual experiments. I analyze
two policies that affect the innovation capacities of the producers. The model allows me to
analyze how the policies affect producers at different sizes. In the first experiment, I look at the
effects of an R&D subsidy. In the second experiment, I increase product market competition in
the economy. In the model, innovations are made by incumbent producers which is in contrast
with most previous models of creative destruction. Hence, the results of these experiments
differ from their results. The experiments show that small producers are affected more from
either policy change than the large producers.
The rest of the paper is organized as follows. Next, I summarize the related literature.
Following that, I formulate the model and show its qualitative implications. Then, I estimate
3
the model using simulated method of moments and discuss the results. Following that, I per-
form a variance decomposition analysis of establishments' sizes and perform the counterfactual
experiments. I finish the paper with some concluding remarks.
1.1 Related Literature
Industry evolution has drawn a lot of attention by researchers since late 1950s. At early stages,
Simon and his coauthors (Simon and Bonnini (1958), Ijiri and Simon (1964, 1977)) succeeded
in generating stochastic growth models providing a good approximation to the size distribution
of large U.S. manufacturing firms. However, these models lacked structural foundations. The
availability of detailed longitudinal panels since the 1980s accelerated the development of theo-
retical models based on optimizing agents and the analysis of the regularities in establishment
and industry evolution. Sutton (1997) presents a detailed summary of the findings of this early
literature.
Many recent studies seek to explain these regularities in a structural way. One of these
models is introduced by Klette and Kortum (2004). Their model is based on the quality-ladder
model of Grossman and Helpman (1991). Producers engage in innovation activity which results
in Poisson arrivals of quality improvements over the existing products. The new quality leader of
a product drives the incumbent producer out of the market and becomes the monopoly supplier.
Lentz and Mortensen (2008) introduce heterogeneity into the Klette and Kortum (2004) model
through different quality step choices of firms. This extension enables them to match several
moments of the data and perform structural aggregate productivity decomposition. Another
model that is related to Klette and Kortum (2004) is Luttmer (2008). In his model, innovations
come as new varieties and the number of varieties in the economy grows at the rate of the
population growth. He characterizes a balanced growth path for an economy where firms grow
by developing new blueprints from their goods. High skilled entrepreneurs can also develop
new blueprints from scratch and set up new firms.
This paper follows all three studies mentioned above. In the dynamics of establishment
evolution, it follows Klette and Kortum (2004). In the way the innovations arrive, it follows
Luttmer (2008). It is similar to Lentz and Mortensen (2008) in introducing heterogeneity into
the Klette and Kortum (2004) setup.
4
However the way heterogeneity is introduced here follows from Melitz (2003). As in his
model establishments are born with different efficiency levels. Here, this heterogeneity generates
different innovation intensities across producers. Hence, I extend Melitz (2003) type static
monopolistic competition models into a dynamic framework where establishments' sizes evolve
over time. Moreover, compared to Melitz (2003) model, a smaller amount of dispersion in
efficiency levels can generate a huge amount of size dispersion.
In the model, producers own multiple products. In a recent study, Bernard, Redding, and
Schott (2006a, b) provide empirical evidence on how multi-product producers dominate total
production in the U.S. economy. They also construct a static model of multi-product firms and
analyze their behavior during trade liberalization. They introduce two margins (intensive and
extensive) to expand size, and these margins are positively correlated with each other. However
what causes size differences across producers in their model is different than the one presented
here.
In another study, Klepper and Thompson (2007) construct a model that explains the subtle
relations between size, age, growth, and survival. In their model, there is no heterogeneity
across producers and their simple framework allows them to analytically characterize a wide
range of regularities on industry dynamics. An establishment is a collection of random sized
products and this randomness allows them to capture the independent relation between size,
age, and the hazard rate of exit.
Fitting the observed size distributions has been an important feature of recent industry
evolution models. Luttmer (2007) presents a model of firm and aggregate growth that is con-
sistent with the observed size distribution of U.S. firms. Firms grow as a result of idiosyncratic
productivity shocks, imitation by entrants, and selection. Using different mechanisms, both
Luttmer (2007) and this model successfully capture the thick right tail of the size distribution.
Luttmer (2007) also characterizes the balanced growth path of the economy while the focus
here is on a single industry. Both models differ in their explanations of the relation between
the variance of growth rates and size.
The model complements the work of Stanley et al (1996), Bottazzi (2001) and Sutton (2002,
2007) on explaining the relation between the variance of growth rates and size. All these studies
propose simple statistical explanations to the observed relation in the data. Here, instead, I
5
Table 1: Industry Details
Industry Code Before After Industry Name
311-312 3453 3244 Food Manufacturing and Food Products
321-322 1827 1735 Manufacturing of Textiles, Apparel except Footwear
331 1157 1092 Manufacturing of Wood and Cork Products except Furniture
341-342 710 646 Manufacturing of Paper, Paper Products, Printing, Publishing
381 1305 1212 Manufacturing of Fabricated Metal Products except Machinery
Total number of observations in the original dataset. Number of observations used in the analysis
after the exclusion of observations due to missing variables and industry switches.
present a structural model incorporating optimizing firms.
1.2 Dataset
In this study, I use data from Chilean Manufacturing Census (Encuesta Nacional Industrial
Anual, ENIA) which is provided by Chile's National Statistics Institute (INE). The dataset is
an unbalanced panel of all establishments with 10 or more workers from 1979 to 1997. I use data
on eight biggest 3-digit industries which are described in Table 1. Data is at the establishment
level. In Chile, most of the firms had single establishments; hence the distinction between a firm
and an establishment is not very crucial. Hsieh and Parker (2007) note that in 1984 only 350
establishments were associated with multi-establishment firms4 . In the original dataset there
were 8452 establishments and after excluding observations due to missing variables and industry
switches, I used 7929 establishments in the analysis. Roughly the same 3-digit industries used
here are analyzed in a study by Bergoeing, Hernando, Repetto (2005) who note that these
industries represent around 60% of total value added in the Manufacturing Census. Table 2
shows the number of establishments of different entry cohorts observed during the span of the
study.
4
Moreover, Caves (1998) points out that most of the results on firm growth and turnover which form the
main discussion in this paper, have been insensitive to the distinction between establishment and firm.
6
Table 2: Number of Establishments Observed at Different Years
Cohorts 1980 1985 1990 1995 1997
- 1980 3225 2061 1452 1251 800
1981-1985 501 268 209 111
1986-1990 769 528 277
1991-1995 827 347
2 The Model
Klette and Kortum (2004) propose a stylized model with a simple interpretation of establish-
¸
ment and industry evolution. In Seker (2006), I showed the quantitative strength of their
qualitative results on establishment dynamics on a dataset of Chilean manufacturers. Here, I
extend their work to be able to explain several other features of the data. To achieve this, I
construct a model which combines the static setup of Melitz (2003) with the dynamic setup of
Klette and Kortum (2004).
In the Klette and Kortum (2004) model, there is a fixed number of products and a producer
expands into new markets through quality improvements on the existing products. Here, the
producer grows by innovating new varieties.
This model focuses on solving the partial equilibrium for a single industry in steady state.
The partial equilibrium analysis simplifies the analytical solution and the computation of the
model and it is more appropriate to use since the focus is on a single industry rather than the
whole economy. In this industry, total labor supply is fixed.
2.1 Producer's Problem
The industry consists of a large group of monopolistically competitive producers. Consumption
of the composite good is determined by the CES production function given in equation 1
µZ ¶ -1
-1
= () (1)
The measure of the set represents the mass of available products each of which is indexed by .
As a result of the steady state solution which will be derived below, is fixed. Consumers have
a taste for variety and consume () units of variety Goods are substitutes with elasticity of
substitution 1
7
Producers are distinguished only by their efficiency levels, indexed by 0 with their
production functions given as = . Producers with same efficiency-types will charge
the same price and make the same profit per product. However, the number of products
they produce may vary as the result of the stochastic innovation process. The static profit
maximization problem for any product market for a given wage rate yields a price () =
¯
¯
(-1)
and revenue
µ ¶1-
()
() =
¯ (2)
where = is the aggregate expenditure of the composite good and is the aggregate
price index. For an establishment with products, aggregate profit is
() = ()
¯ ¯ (3)
Here () is the profit per product given as
¯
µ ¶1-
¯ -1
() =
¯ (4)
( - 1)
The efficiency levels of producers grow at an exogenous rate This rate is fixed and
common across all producers. As will be shown below, without this assumption, an average
producer shrinks in size over time and the growth rate distribution cannot be fitted in the
estimation. Efficiency growth is the only source of growth in the aggregate economy. Hence
the aggregate expenditure and the wage rate grow at this rate.
¯
The number of varieties determines the portfolio of the producer. This portfolio increases
by innovation of new products and it decreases by destruction of the current products. To
succeed in innovation, the producer invests in R&D. This investment determines the Poisson
arrival rate of new innovations and it is formulated as () = 0 1+1 for 0 1 0 This
strictly increasing and convex cost function reflects the labor input required for R&D. Klette
and Kortum (2004) provide motivation for incorporating the number of products in the R&D
cost5 . In the mean time, the producer faces a Poisson hazard rate of losing any product.
The hazard rate is fixed and same for all establishments. Exit from the market occurs when
5
Basically, reflects the knowledge capital of the establishment which stands for the know-how and tech-
niques the producer has learned with its previous innovations.
8
all products are destroyed. There is no reentering the market once exit takes place.
2.2 Value Function
The model can be solved using undetermined coefficients method. To have a simple analytical
()
¯ ¯
solution to the producer's value function, I will define () =
and =
. In the
Appendix, I show how this assumption gives a stationary () () and for . I also
show how () and grow at the same rate. Moreover, in the Appendix, I present a simple
¯
formulation of how this industry of interest can be incorporated with the aggregate economy.
The state of the producer is determined by its number of products . The dynamic maxi-
mization problem of a particular efficiency-type producer, for a constant interest rate 6 is
formulated in the following Bellman equation
() - ()+
() = max [ ( + 1) - ()] (5)
0
+ [ ( - 1) - ()]
In this equation, the producer maximizes current profit net of R&D cost and its net future
value.
To derive the solution of the Bellman equation, I conjecture that the value function is given
as
µ ¶
()
() = + () (6)
+
Here, () is the continuation value of innovating. Substituting this value function into
equation 5, I get the following equation, with the details of the derivation given in the Appendix,
½ µ ¶ ¾
()
( + ) () = max + () - () (7)
0 +
Using this equation, I will find the value of () and solve the stationary industry equilibrium.
Before doing that, let's define the problem of the entrants.
6
Constancy of is shown in the Appendix.
9
2.3 New Entrants
Entry requires an innovation. Establishments discover their efficiency types when they enter
the market. All entrants start with a single product but they don't necessarily have the same
startup size because of their differing efficiency levels. More efficient establishments have larger
product sizes which can be seen in equation 2. The potential entrants innovate at rate They
also face the same innovation cost function as the incumbents7 . The free entry condition given
below determines the entry rate in the industry
Z
0
( ) = () (8)
| {z }
marginal cost of innovation | {z }
net gain from innovation
Here is the value of a single product for a efficiency-type producer, and () is the
efficiency-type distribution of entrants. Entry rate is determined by the multiplication of
and the constant measure of potential entrants .
2.4 Stationary Industry Equilibrium
Recall that the interest rate wage and the hazard rate of exit are constant. A stationary
equilibrium for this industry consists of innovation rate () for all efficiency-types and the
entry rate such that for any given and : () any efficiency-type incumbent producer
solves equation 7 to maximize its value () potential entrants solve equation 8 and break even
in expectation.
Lentz and Mortensen (2005) provide a proof for the existence of equilibrium for a model
closely related to mine. To guarantee the existence, I need that () for Otherwise
size and age of some establishments diverge to infinity which precludes having a stationary size
distribution8 . The first order condition for equation 7 is
()
: + () = 0 () (9)
+
The value of () is derived from equation 7 as follows
7
This is a simplifying assumption and the same structure for the innovation cost of entrants is used in Lentz
and Mortensen (2008).
8
The condition needed to guarantee () for is 0 () -()
10
( () )
+
- ()
() = max (10)
0 +-
Implementing the value of () into equation 9 and after some algebra, I get
() - ()
0 () = (11)
+-
The right hand side of equation 11 is equal to the value of a single product Klette and
Kortum (2004) show that increases in In equation 4, is increasing in Hence, more
efficient producers are more innovative. From equation 2, we know that they also generate more
revenue from each product they produce. This complementarity between innovation rates and
product sizes increase the size differences between low and high efficiency-type producers which
consequently stretches out the right tail of the size distribution. It also implies that a smaller
dispersion in efficiency levels (in models like Hopenhayn (1992) or Melitz (2003)) is required
to explain the observed size dispersion. This property of the model is the main reason for
obtaining the qualitative and quantitative success of the model.
2.5 Dynamics of Size Evolution
After solving the model for the optimal innovation rate, the evolution of an individual producer
conditional on its efficiency-type can be characterized. For ease of notation, let denote the
innovation rate () for a particular -type producer At any moment, the number of products
an establishment produces can stay the same, increase by one unit as a result of an innovation,
or decrease by one unit due to the exogenous destruction rate. Denote (; 0 |) as the
probability that an establishment has products at time conditional on having 0 products
at time 0 and being type This probability changes over time at rate (; 0 |) The following
system of equations describe the evolution of an individual -type producer9 ,
(; 0 |) = ( - 1) -1 (; 0 |) + ( + 1) +1 (; 0 |) - ( + ) (; 0 |) for 1
0 (; 0 |) = 1 (; 0 |) (12)
9
A formal solution of this system of equations is given in Appendix C of Klette and Kortum (2004).
11
The solution to this set of coupled difference-differential equations yields a geometric distribu-
tion for establishment size at time conditional on survival
(; 1|)
= (1 - (|)) (|)-1 1 (13)
1 - 0 (; 1|)
¡ ¢
1 - -(-)
where (|) = 0 (; 1|) =
- -(-)
where (|) is the parameter of the size distribution. The solution of this system can be
used to derive the moments of the growth rate of the number of products an establishment
owns. The expected growth rate and the variance of growth rate of the number of products
conditional on initial size 0 are given as
[ () |0 = 0 ] = -(-) - 1 (14)
+ ¡ ¢
[ () |0 = 0 ] = -(-) 1 - -(-) (15)
0 ( - )
The expected growth rate of total size relative to aggregate expenditure can also be determined.
Size of a -type establishment is given as
µ ¶1-
()
() = () (16)
() grows due to the growth of the number of products.
3 Model's Implications
The model has four novel implications, each of which explains a regularity observed in the
Chilean dataset and several other empirical studies on industry evolution. Below, I describe
three of these regularities, present evidence from te Chilean dataset on each one, and show
how the model explains them. The fourth regularity, which is the long right tail of the size
distribution, is explained in the simulation results section.
12
3.1 Effects of Size and Age on the Hazard Rate of Exit
Size and age are two important observable characteristics of establishments that have been
extensively used to analyze their dynamics. Many studies have shown that the hazard rate of
exit decreases as size and age increase10 . Figure 1 shows the relation for the Chilean dataset
including all industries. Each line represents a size cohort where size is measured as total sales.
As both size and age increase, the hazard rate of exit decreases. The establishments in the
smallest size cohort face higher exit rates than the other cohorts and the decline in hazard
rate as age increases is slower in this size cohort. This is probably due to the small number
of observations in the older ages for that size cohort. For the other cohorts, decrease in the
hazard rate of exit is more pronounced.
Figure 1: Hazard Rate of Exit Conditional on Size and Age (Data)
0.29
S<=200
0.24
0.19
Hazard 200~~1000
0.04
1 2 3 4 5 6 7 8 9 10 11 12
Age
The Klette and Kortum (2004) model can explain the relation between the hazard rate of
exit and size. It also generates the negative relation between age and the hazard rate of exit but
only because age is a proxy for size. Conditional on size, age has no effect on the hazard rate
of exit. Introducing random sized products, Klepper and Thompson (2007) can explain both
relations independently. In my model, the relation holds for a different reason. Establishments
with different efficiency levels produce different numbers of products. Given size, there is a
mixture of producers with different number of products and the older ones are more likely to
10
Caves (1998) reviews the empirical literature on these relations.
13
have more products, hence are less likely to exit. Without the efficiency-type heterogeneity, all
producers would have the same number of products at a given size and thus would face the
same hazard rate of exit no matter how old they are.
I can derive an analytical formula that shows these relations. To construct this formula,
at any age , I need to know the hazard rate of exit, the number of products owned by the
producer, and the efficiency-type distribution of producers. Using the solution of the system
of equations for size evolution given in equation 12, having entered at size one, probability of
exiting within one unit of time is given as
¡ ¢
1 - -(-)
0 (1; 1|) = (17)
- -(-)
From equation 13, the probability of having 1 products at age is (1 - (|)) (|)-1
A -type producer at size has () =k 1-
k products. Age-conditional efficiency-type
( )
distribution with density () can be derived using the type distribution at entry (·) and
the probability of surviving more than years 1 - 0 (; 1|) which is given as
() (1 - 0 (; 1|))
() = R for 1 (18)
() (1 - 0 (; 1|))
Using equations 17 and 18, the hazard rate of exit conditional on age and size ( )
can be derived as follows
Z h i
( ) = 0 (1; 1|)() (1 - (|)) (|)()-1 () (19)
I plotted the graph of ( ) for different size and age values in Figure 211 . For comparison,
I also plotted the lines implied by the Klette and Kortum (2004) model which are labeled as
"KK". Each plotted line shows the hazard rate for a different size level. For the Klette and
Kortum (2004) model, I just showed two size levels 200 and 800. In their model, at any
particular size, the hazard rate is independent of age but the hazard rate decreases as size
increases. On the other hand, in my model both relations hold independently. Conditional on
age, the hazard rate of exit decreases in size and respectively conditional on size, it decreases
11
The parameter values used for the graph are from the simulation results of Food industry which will be
explained in the empirical analysis part.
14
in age.
Figure 2: Hazard Rate of Exit Conditional on Size and Age (Model)
0.025 200
400
800
1600
200 (KK)
800 (KK)
0.02
Hazard Rate
0.015
0.01
0.005
0 2 4 6 8 10 12 14 16
Age
The analytical framework of the model allows me to prove the existence of these relations.
In order to do that, first I present two lemmas which are used for the proof of the proposition.
(-) -1 (-) -1
Lemma 1 For all 0, ( - )
and
is strictly increasing in .
Proof. See Appendix.
Lemma 2 For all 0 the parameter of the size distribution () increases in inno-
()
vation intensity (i.e.
0).
Proof. See Appendix.
Proposition 1 Hazard rate of exit ( ) decreases in age conditional on size and
decreases in size conditional on age for all ages 0 and sizes 0.
Proof. See Appendix.
The underlying reason for getting the independent relation between the hazard rate of exit
and age is the heterogeneity in efficiency levels. Although more efficient establishments survive
longer and grow faster, it's possible for a less efficient producer to get lucky and accumulate
many products. This in return decreases the hazard rate of exit relative to a higher efficiency
producer with only few products. Since it takes time to accumulate many products, that
establishment will be older than the more efficient producer with a few products. Hence at
any given size, it is possible to observe establishments with differing number of products and
differing hazard rates of exit.
15
3.2 Effect of Size on the Variance of Growth Rates
In the literature on establishment and industry evolution, there are many studies that have
analyzed the relation between size and the expected growth rate. However, only recently have
there been studies that try to explain the dispersion in growth rates and how it changes with
size. Stanley et al (1996), Bottazzi (2001), Sutton (2002, 2007) show that the variance of growth
rates decreases as size increases12 . The common feature of all these studies is the introduction
of a statistical model to explain the relation observed in the data rather than using a structural
model based on optimizing firm behavior. Three recent studies by Klette and Kortum (2004),
Luttmer (2008) and Klepper and Thompson (2007) analyze this relation in a more structural
setup. Although all three models qualitatively explain the negative relation, the slope of the
relation implied by these models is too steep compared to the data.
Figure 3 shows the relation between the log of the standard deviation of growth rates and
the log of size observed in all data combined and in all five industries individually. Size is
measured as total sales. Slopes of the fitted lines for each industry vary between -0.15 and
-0.32. For the very small and very large size bins, deviations exist from a linear relation which
is probably caused by the small number of observations in these bins. The relation observed for
the specific industries is not different than the relation observed when all industries are merged.
This fact suggests that the nature of this relation is due to some fundamental property of the
economic dynamics and establishment behavior, which makes it appealing to identify.
Since the growth rate of efficiency is constant, the variance of growth rate of size is de-
termined by the variation in the growth rate of the number of products. Emergence of the
negative relation in the model is explained as follows. The evolution of an establishment is
determined by combining the evolution of each of its products. Hence, the aggregate growth of
an establishment is the average of the growth of these independent components. This leads to
an inverse relation between the variance of the growth rates and initial size.
To make the analysis comparable to the empirical studies, I will look at the relation between
the standard deviation of the growth rates and size. If all the producers had the same innovation
12
Sutton (2002) shows that the slope of the fitted line between log of standard deviation of growth rates and
log of size measured in sales varies between -0.15 and -0.21. Stanley et al (1996) performs the same analysis
using employment as size and finds the slope as -0.16.
16
Figure 3: Standard Deviation of % Growth Rates (All Industries)
10
Standard Deviation
1
0.1
100 1000 10000 100000
Size
All Food Textile Wood Paper Metal
rate, as in Klette and Kortum (2004), this inverse relation would give a slope of -0.5. Figure
4 shows the relation for several innovation rate values. As the innovation rate increases larger
establishments can exist in the market and the line shifts to the right. In my model, since
producers differ in their efficiency levels, they have different innovation rates. Hence, the
relation that emerges here is a mixture of the lines in Figure 4. Furthermore, since high-
efficiency producers attain larger product sizes, they get even larger and this causes the line to
extend even further to the right. This property plays an important role in generating a flatter
relation13 in the model.
In the model, as a result of the mixing, a high-efficiency producer with a single product can
be in the same size bin with a low-efficient producer that luckily survived and gained many
small products. The inefficient producer will exhibit lower variance in growth rate since it has
many products. But the existence of high-efficient producer exhibiting high variance due to
its single product will increase the total variance in that size bin. As a result, the mixing of
producers with different efficiency levels generates a flatter relation between the variance of the
growth rates and size.
I derive the formula for the variance of growth rates as follows. Suppose that there are
different efficiency types in the industry denoted as { } for Z+ . For each type, define
=1
³ ´
the variance of growth rates conditional on initial size as ( ) |~ = Total variation
13
Lentz and Mortensen (2008) introduces heterogeneity in producer innovation intensities however the slope
is still too steep.
17
Figure 4: Standard Deviation of % Growth Rates vs Size (Different Innovation Rates)
-1
0.01
0.05
-1.5 0.15
0.2
log(Std Dev of Growth Rate)
-2
-2.5
-3
1 2 3 4 5 6
log(Size)
can be written as
³ ´ X ³
´h ³ ´ ¡ ¢i
|~= = |~ = ( ) | = + [( )] - 2
~ ¯ (20)
=1
³ ´
where |~ = is the probability of being -type conditional on having initial size
¯
[( )] is the expected growth rate for the -type producer, and is the expectation taken
³ ´
~
over all types. Recall that formula for ( ) | = where is the number of products
was given in equation 15. Since each product size is fixed, it is straight forward to show that
³ ´ ³ ´
~ ~
( ) | = is equal to ( ) | = for all efficiency-types. Hence I can simulate
the model's results easily.
In the estimation part that will be discussed below, this slope was not targeted to match in
the data. However the relation implied by the model improves the Klette and Kortum (2004)
result which is shown in Figure 5. The graph on the left shows the data for the food industry
and the fitted line which has a slope of -0.23. The graph on the right shows the simulated data
and the fitted line to it with slope -0.33. The line labeled as "KK Model" shows the Klette and
Kortum (2004) result with slope -0.5. The model clearly improves Klette and Kortum (2004)
results. However the standard deviation of growth rates for small establishments is much larger
in the data than in the simulation.
18
Figure 5: Standard Deviation of % Growth Rates for Food Industry: Data vs Model
Data Model
3.5
3.5 1.5
1.5
Standard Deviation
Standard Deviation
0.5
0.5
0.1
0.1
0.05
0.05
100 1000 10000 100000 100 1000 10000 100000
Size Size
Data Fitted values Simulation Sim Fit KK Model
3.3 Life Cycle of a Birth Cohort
There are various factors that affect the post-entry performance of establishments such as the
amount of sunk costs, as argued by Dixit (1989) and Hopenhayn (1992), and the innovative
environment of the industry, as observed by Geroski (1995). Empirical evidence on these
hypotheses has been provided by Audretsch (1991, 1995) and Baldwin (1995). On the other
hand, Audretsch (1995) summarize other studies which argue that characteristics specific to
establishments also influence their post-entry performances. Since size is the most important
observable characteristic specific to an establishment, its value at the startup could be signaling
important information about the evolution process. To show how important the startup size
is, in Figure 6, I plot the size evolution of establishments grouped with respect to the age at
which they exit. The graph combines all industries and all birth cohorts from 1980 to 1997. A
similar but noisier picture emerges when Figure 6 is drawn for single industries or for specific
birth cohorts due to small number of observations. To get a nicer picture of the relation, I
combined all birth cohorts in all industries.
The graph shows that the establishments that will survive longer are larger in terms of sales
than the exiting establishments within the same birth cohort at all ages including the startup.
It also shows the shadow of death effect; establishments which will exit in the future start to
shrink in size several years before their exit.
The model captures these two features of the data, as shown in Figure 714 . In the model,
14
I estimate my model for the chosen 3-digit industries individually. This graph shows the model's result
19
Figure 6: Pre-Exit Behavior of Establishments (Data)
7000
6000
5000 >11
4000
Sales 9-11
3000
6-9
2000
3-6
0-3
1000
1 2 3 4 5 6 7 8 9 10 11
Age
heterogeneity in size at the startup occurs by the variation in efficiency levels. The model
captures the shadow of death effect especially for the 3-6 and 6-9 year cohorts, but in the data
this effect is more pronounced. In Klette and Kortum (2004), all producers start with one
product; hence, there is no dispersion in size at startup. Lentz and Mortensen (2008) introduce
randomness in size of each product and heterogeneity. However, there is no relation between
the startup size and the post-entry performance. In the proposition below, I show how the
model explains this relation.
Proposition 2 Consider a cohort of establishments all entering at the same time. At
any age 1 within this cohort, establishments that survive longer are larger in size than the
exiting establishments (i.e. for representing the establishment size, representing the time
of the exit, 0 0 [ () | = ] [ () | = + ]).
Proof. See Appendix.
The positive relation between the startup size and the likelihood of survival is not unique
to Chilean Data. Audretsch (1995) performs a logit estimation using US establishments and
concludes that startup size has an impact on the likelihood of survival. Similar results are
found in Dunne, Roberts, and Samuelson (1989). This evidence shows that it's an important
feature that needs to be understood in order to explain causes of size dispersion, heterogeneous
responses of producers to exogenous shocks, and responses to policy changes. The model has
the potential to explain these issues.
for Food industry not the all industries combined because I intend to show how model explains this relation
qualitatively.
20
Figure 7: Pre-Exit Behavior of Establishments (Simulation)
3000
2500
2000 >11
Sales
1500 9-11
1000
6-9
500 3-6
0-3
0
1 2 3 4 5 6 7 8 9 10 11
Age
4 Empirical Analysis of Industry Evolution
In this section, I estimate the model separately for five 3-digit industries in the Chilean Man-
ufacturing sector. I analyze whether the dispersion observed in establishment behavior in the
aggregate manufacturing sector also holds at the 3-digit industries. All the values of sales and
wages are given in thousands of 1986 real Chilean peso. The nominal values are deflated by
the aggregate GDP deflator from World Bank Development Indicators database.
4.1 Industry Comparison
Five industries that will be analyzed, their 3-digit SIC codes, and the total numbers of estab-
lishments observed were given in Table 1. Doing the same estimation exercise five times aims
to capture the flexibility of the model in explaining different industry structures.
In the Appendix, I show the data analysis on the shape of the size distributions, on turnover,
and on growth rates across industries. The size distribution of each industry is estimated non-
parametrically (using Kernel regressions) (see Figure 10). The shapes of these distributions
change very little over time. Food and Paper industries differ from the other three industries in
several ways with the most outstanding difference being on the shape of the size distribution.
The average sales and the variation of sales are larger and the coefficient of skewness of the
log size distributions is higher in these two industries (see Figure 11). The ranking of the
industries is preserved for the variation in size when I analyze the industry data with respect
21
to their means (see Figure 11). This implies that the differences in size distributions are not
purely caused by economies of scale but are due to some intrinsic differences across industries.
Further industry analysis shows that Food and Paper industries have lower turnover rates
(see Figure 11). Capital intensities vary across industries but do not explain the differences
in the shape of the size distribution. Using establishment data for the U.S. economy, Rossi-
Hansberg and Wright (2007) show that the larger the capital intensity in a sector, the thinner
is the right tail of the size distribution. They define sectors at 2-digit SIC level. The industries
analyzed here are more narrowly defined and at this level the relation suggested by their
model does not hold. I looked at the average capital intensities of establishments that were
in the market in 1979 and that entered in 198015 . I found the capital-output ratio of every
establishment and then plotted the average ratio for every industry (see Figure 12). The Paper
industry, which is the most capital intensive, has the thickest right tail. The other industry
with a thick right tail is the Food industry and its capital intensity is among the lowest. Hence
capital intensity does not play a distinguishing role in explaining the differences in the shape
of size distributions for narrowly defined Chilean industries.
4.2 Estimation
The model introduced above is estimated using simulated method of moments. With this
method, I try to find the values of the model parameters that bring the vector of moments
from the simulated data closest to those from the data.
4.2.1 Data Moments
Eight moments are chosen for parameter estimation. They are prominent in explaining the
industry structure and they also reflect the cross-industry differences. As one of the targets of
this paper is fitting the industry size distribution, 10, 25, 50, 75 and 99 percentiles are chosen.
The shape of the size distribution is affected by the exogenous destruction rate of products,
aggregate expenditure, efficiency type distribution, and innovation cost parameters. Another
moment is the entry rate which plays an important role in identification of the destruction
rate. Two other moments, the mean and the variance of establishment growth rates, are
15
Capital data was only available for these establishments in the dataset.
22
closely affected by the innovation cost parameters, destruction rate, and the efficiency type
distribution.
The values of these moments for Food and Food Products industry (SIC 311-312) are given
in Table 3. Following Horowitz (2001), the standard errors of the moments are estimated
by 1000 bootstrap repetitions which are given in parentheses. Since the model incorporates
growth in average establishment size, only year 1979 is used to estimate the moments for the
size distribution. The rest of the moments are constructed by averaging over the 1979 -1997
period16 . The annual values of these moments didn't show a strong trend over time which is
in accordance with the steady state assumption of the model. The growth rate distribution is
found as the annual increase in sales and it includes the exiting establishments (i.e. -1 is placed
in the year that the establishment exits). Moment vectors for the other industries are given in
the Appendix.
© ª
The parameter vector to be identified is = 0 1 which are described
¯
in Table 4. Since there are just eight moments identifying eight parameters, the system is just
¡ ¢
identified. Efficiency types are lognormally distributed (~ ) with representing
¯
the minimum efficiency level.
Other than these industry specific moments, the real interest rate is fixed at 5% In the
dataset, I had information on the average wage rates and the numbers of blue and white collar
workers. Using that information, I found the average annual wage rate for each industry in
thousands of real 1986 Chilean peso. Real wages for the industries are: 3068 for Food, 303 for
Textile, 271 for Wood, 489 for Paper, and 371 for Metal industries. Since aggregate expenditure
growth in an industry is equal to exogenous efficiency growth rate , this parameter is directly
estimated from the data with values of 78% for Food, 55% for Textile, 12% for Wood, 11%
for Paper, and 95% for Metal industries.
4.2.2 Simulation Method and Algorithm
n
o1997
The data is comprised of sales for all producers. Denote this panel by = { }=1
=1979
where refers to the producer, refers to the year, and is the number of observations
16
Chile went through a financial crisis in 1982-1983. I excluded these years from finding the average values
of the moments.
23
Table 3: Data Moments for Food Industry
Moment Definition Value
pctile(10) 10 Percentile 128.80 (4.72)
pctile(25) 25 Percentile 207.43 (4.85)
pctile(50) 50 Percentile 346.15 (10.58)
pctile(75) 75 Percentile 894.56 (72.30)
pctile(99) 99 Percentile 31291.11 (2846.05)
E[g] Average Growth Rate 0.027 (0.0057)
Std[g] Std Dev of Growth Rate 0.75 (0.047)
Entry Rate 0.054 (0.0018)
Values in the parentheses show the standard errors.
Table 4: Model Parameters
0 1 Innovation cost
Destruction rate
Efficiency distribution
¯
Minimum efficiency level
Elasticity of substitution
Aggregate Expenditure
in the data at time . Using the panel, I calculated the vector of data moments denoted as
© ª
^
(). Then, given a vector of parameters I simulated a panel of sales = =1 for
= 10 000 establishments and repeated the simulation for = 10 times. Using the simulated
^
panel, I found the moment values and averaged them over the simulation size giving () as
follows:
1 X^
^
() = ( ())
=1
Finally, the estimator ^ is found as the solution to the following criterion function,
³ ´0 ³ ´
^ = arg min () - () () - ()
^ ^ ^ ^
where is the identity matrix. Since the system is exactly identified, identity matrix is used as
the weighting matrix which gives the equally weighted minimum distance (EWMD) estimator.
Standard errors of the estimator are estimated by bootstrap method. Given the data a
bootstrap sample is drawn with replacement. The draws are made block over time. This
means that if a particular establishment is selected, the entire time series for this establishment
24
is included in the constructed sample. Then for each drawn sample17 , the estimator vector ^
is found by solving
³ ¡ ¢ ¡ ¢´0 ³ ¡ ¢ ¡ ¢´
^ = arg min - -
^ ^ ^ ^
The model is highly nonlinear. Hence, down-hill simplex method (amoeba) is used for
optimization. The steps to compute the industry equilibrium are given as follows:
1. The parameter vector is initialized and the vertices of the simplex are determined.
2. For each parameter vector, an upper bound of the efficiency type distribution satisfying
() for is determined.
3. Given the bounds of the efficiency type distribution and the parameter vector, an aggre-
gate price index is found.
4. For 10,000 establishments, the value function is solved and innovation intensities are
obtained.
5. Using these values, establishment and industry related moment values are determined.
6. The value of the criterion function is checked and using the amoeba routine, the simplex
of parameter vectors is updated.
7. The system is iterated until either the value of the criterion function or the parameter
vector converged.
4.2.3 Simulation Results
The estimated parameter vectors for each industry analyzed are given in Table 5. The standard
errors18 are given in parentheses. Simulation results show that the parameter values for Food
and Paper industries differ from the other industries. These two industries have lower innovation
costs which allow them to have more innovative producers. Combined with the higher efficiency
levels, this contributes to the existence of larger producers. Lower destruction rate explains the
17
For this exercise 50 bootstrap repetition is used.
18
I found the standard errors for only two of the industries (Food and Wood).
25
smaller turnover rates. The stochastic innovation process together with the higher variation
in efficiency levels explains the greater dispersion in size for these industries. It is also seen
that part of the differences in the size distribution is attributed to the differences in aggregate
expenditures. On the other hand, the variation in the elasticity of substitution and the minimum
efficiency level parameters are relatively small across industries.
Table 5: Parameter Estimates for All Industries
Food Paper Textile Wood Metal
2.19 2.37 2.6 2.4 2.3
(0.44) (1.09) (0.46) (0.69) (0.48)
c0 47085.71 44713.4 71896.1 71999.6 74367.9
(1569) (1846.9) (1514.8) (6081) (3214.2)
c1 3.95 3.82 5.45 5.73 5.32
(0.02) (0.4) (0.06) (0.11) (0.25)
0.125 0.13 0.184 0.18 0.176
(0.002) (0.01) (0.002) (0.01) (0.01)
2.03 1.82 0.87 1.55 0.88
(0.17) (0.22) (0.22) (0.33) (0.42)
4.59 4.25 1.77 1.11 1.43
(0.36) (1.05) (0.17) (0.39) (0.27)
¯
1.19 1.21 2 1.43 1.62
(0.04) (0.17) (0.07) (0.17) (0.14)
1293.2 2239.3 931.7 517.6 827
(46) (100.4) (73.1) (47.6) (74)
The values in the parentheses show the standard errors.
Estimation results for each industry are given in the Appendix. Figure 8 shows the observed
and simulated size distributions for Food industry. I also plot the logarithmic size distribution
implied by the Klette and Kortum (2004) model and the Pareto distribution. As the graph
shows, the model captures the fat right tail quite well. Logarithmic distribution cannot generate
enough variation in size and cannot generate very large sized establishments. The Pareto
distribution which was used by Luttmer (2007) to fit the size distribution of the U.S firms,
can generate large sizes19 . But it cannot fit the data at medium-sized classes. Graphs for
other industries with the simulation results are given in the Appendix. Although the model
performs well in capturing the shape of the size distributions, it cannot generate very small
sized producers (i.e. with total sales less than 100).
19
Pareto distribution used here has coefficient 0.7 which corresponds to a straight line with slope -0.7.
26
Figure 8: Size Distribution for Food Industry
1 0.1
Pr(size>S)
0.001 0.0001
1 100 1000 10000 100000
Size
Simulation Data
Pareto(-0.7) Logarithmic (0.95)
In the model, innovation is done by the incumbent producers. When innovation cost is
low, these producers get more innovative and there is less room for entry. Hence larger indus-
tries have lower turnover rates as observed in the data. The model performs relatively well
in capturing the entry rates across industries. However, it underestimates the growth rate
distribution.
5 What Causes Size Dispersion?
In this section, I elaborate a novel feature of the model. The model incorporates two forces
that generate persistent differences in establishments' performances. In his review of models on
establishment evolution, Sutton (1997) lists these forces as: () intrinsic efficiency differences
that are determined before entering the economy () differences that are generated through
idiosyncratic innovations that accumulate through the life of the establishment. Both views
have drawn great attention in the literature20 . Among the first group of models, Lucas (1978)
and Jovanovic (1982) link the differences in efficiencies to the differences in the skills of entre-
preneurs. In the second group of models, performance is driven by producer specific learning,
R&D, and innovation. Some recent models that follow this view are Ericson and Pakes (1995),
Klepper (1996), Klette and Kortum (2004), and Klepper and Thompson (2007).
The model introduced here distinguishes the contributions of exogenous idiosyncratic effi-
ciency differences and accumulated innovations to explain size dispersion. It generates disper-
20
For a review and comparison of both types of models see Klette and Raknerud (2002).
27
sion in the startup sizes due to efficiency differences. As establishments grow old, the innovation
process induces dispersion among producers of the same efficiency-type. As a result, both fac-
tors contribute to the total variation of size.
To see how the contribution of each part evolves over time, I analyze the life-cycle of pro-
ducers within a birth cohort. First, I look at the variation at birth (age=0). All establishments
0
start with a single product. Hence, within type variation, is zero. Total variation in
size only reflects dispersion between efficiency types, which is given as
Z
0
= ( () - )2 () (21)
R
where () is sales per product for a -type producer, = () () is the expected
size at the entry, and () is the probability density at the entry.
As the establishments in the same birth cohort grow older, as a result of the stochastic
innovation process, dispersion will emerge among the establishments of the same efficiency
type. Let be the number of products owned by a -type producer at age which is a
random draw from the geometric distribution with parameter (|) This distribution has
1 (|)
mean 1-(|)
and variance (1-(|))2
Conditional on type, the expected size and the variance
of size at age are given as
1
[ () |] = () (22)
1 - (|)
(|)
[ () |] = ()2 (23)
(1 - (|))2
R
¯
Total size is = [ () |] () Using equations 22 and 23, total variation in size
at age is determined as
Z Z
¡ ¢2
=
[ () |] () +
[ () |] - ¯ () (24)
| {z } | {z }
The total variation in size is plotted in Figure 9. The graph shows that, as the establishments
get older, the share of within type dispersion increases. It increases monotonically and exceeds
the between type dispersion after 23 years. At the end of 50 years of survival, within type
28
variation accounts for 57% of the total variation in size. Since producers with low efficiency
don't live as long, between type heterogeneity decreases at older ages. Hence more of the total
variation is explained by within type dispersion. This analysis shows that both the intrinsic
efficiency differences and the accumulated innovations contribute to explain the variation in
size, but their contributions change with the establishments' ages.
Figure 9: Total Variation in Size over Time
0.9
Within
Between
0.8
0.7
Fraction of Total Variation
0.6
0.5
0.4
0.3
0.2
0.1
0 5 10 15 20 25 30 35 40 45 50
Age
6 Counterfactual Experiments
A distinguishing feature of Klette and Kortum (2004) model from the earlier work in endoge-
nous growth models is the research done by the incumbent establishments. In the models
of Grossman and Helpman (1991) and Aghion and Howitt (1992), innovations are done only
by new establishments and it is hard to reconcile this property with the persistence of large
establishments in industries21 . With this feature of the model, I will demonstrate the effects
of two policy changes on the establishments' innovation capacities. In the first experiment,
I introduce a 25% subsidy on research investment. In the second experiment I decrease the
price-cost margin by 10%. In the model, for each product, establishments charge the same
markup. Hence price-cost margin is equal to -1
and is constant. 10% decrease in the price-
cost margin can be generated with a 15% increase in the elasticity of substitution. The results
of these counterfactual experiments are given in Table 6. Table shows the percentage changes
21
See Klette and Griliches (2000) for further discussion of this difference between the previous literature
and the new studies.
29
Table 6: Counterfactual Experiments
25% Research Subsidy 10% decrease in PCM
(% change) (% change)
Average Growth 4 -3
Std Dev of Growth 0.07 -0.07
Entry Rate -30 17
Exit Rate -10 7
(By Size Quartiles)
1 Quartile
Size 111 -63
Research Int 45 -40
Innov Rate 33 -26
2 Quartile
Size 81 -53
Research Int 40 -36
Innov Rate 27 -21
3 Quartile
Size 47 -37
Research Int 34 -32
Innov Rate 20 -15
4 Quartile
Size 17 -13
Research Int 11 -21
Innov Rate 6 -0.05
Average growth rate is calculated conditional on survival
in the selected variables after the policy change. I divide the establishments into size quartiles
both before and after the change and show how the evolution of the different parts of the size
distribution change. For each quartile, I look at the change in average size, research intensity,
and innovation rate.
In most of the existing models of creative destruction, since the innovation is done by the
outside firms, subsidies encourage them to do more research. The model's implications are in
contrast with these views. Here, subsidies on R&D investment increase the innovative capacity
of the incumbent establishments. With higher innovation intensities, incumbent establishments
grow faster by 4% and survive longer (exit rate decreases by 10%). This leaves less room for
30
potential entrants to enter the market. The effect of the policy change varies for different parts
of the size distribution. Although it improves size, R&D intensity and innovation rate for all
parts of the distribution, the gain is larger for the small producers. Due to the larger increase
in their innovation rates, grow rates of small producers also increase more.
The impact of competition on R&D expenditures and the rate of innovation has been
debated for a long time. Increased elasticity of substitution will cause tougher competition in
the product market which will lower the flow of profits of the incumbent producers. Lower
profits will lead to lower R&D expenditures and less innovation. As a result, average growth
rate of establishments is going to decrease by 3%. Having lower innovation rates, they will
exit more often (7% more) and there will be more room for new entrants to the industry.
Looking at the quartiles of the size distribution, it's seen that tougher competition hurts the
small establishments the most. Average establishment size and the innovation rates in the first
quartile shrink by around five times more than the respective variable in the fourth quartile.
7 Conclusion
This study improves recent models of industry evolution in explaining several regularities ob-
served in the data which have been hard to capture by the existing models. Through a parsimo-
nious extension of a highly stylized model introduced by Klette and Kortum (2004), I construct
a model that succeeds in explaining: () the fat tail in the size distribution, () the independent
relation between size, age, and the hazard rate of exit, () post-entry performance of a birth
cohort, and () the negative relation between the variance of the growth rates and size. The
model is consistent with many empirical regularities. It demonstrates a good framework for
understanding the micro foundations of industry evolution and it is analytically tractable.
In this paper, I also intended to show the quantitative strength of the model in explaining
the data. The model performs well in capturing various moments of the size distribution, entry
rates, and growth rates for five 3-digit industries in Chilean Manufacturing sector. Comparison
of the five industries shows that innovation structure, the destruction rate, and efficiency type
distribution play a role in explaining the differences across industries.
In the model, establishments are defined as legal entities formed of multiple products. Their
evolution is the sum of the evolution of each of their products. In this respect, the model
31
complements several existing models explaining product scope. The way the multi-product
producers are modeled here is closest to the one introduced in Bernard, Redding, and Schott
(2006a). Although the factors driving variation in size are different, both models introduce two
margins that contribute to expand establishment size. However, their model lacks a dynamic
framework and focuses on an analysis of trade liberalization.
The model has several interesting extensions that are worth pursuing. Introducing aggregate
uncertainty to the economy can extend our understanding of the response of the economy to
negative shocks and the recovery from economic slowdowns. Another fruitful area is adding
the financial side to the model. This would bring a more comprehensive understanding of
the establishment dynamics. One promising work in this field is done by Cooley and Quadrini
(2001). They show how the combination of persistent shocks and financial frictions can account
for the simultaneous dependence of the establishment dynamics on size conditional on age and
on age conditional on size. However, their model has some limitations such as not being able
to predict the effect of age on the hazard rate of exit. Finally the model could be carried into
an open economy to understand how the technology imported through intermediate products
affects the establishment and industry evolution.
32
References
[1] Aghion, Philippe and Peter Howitt (1992), "A Model of Growth through Creative De-
struction," Econometrica, 60: 325-351.
[2] Audretsch, David B (1991), "New Firm Survival and the Technological Regime," Review
of Economics and Statistics, 73(4):520-526
[3] Audretsch, David B (1995), Innovation and Industry Evolution, Cambridge, Mass.:MIT
Press.
[4] Baldwin, John R. (1995), The Dynamics of Industrial Competition: A North American
Perspective, Cambridge: Cambridge University Press.
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A Growth rate of ()
Aggregate industry production function was given as
µZ ¶ -1
-1
= ()
Taking the derivative with respect to time, I get
µZ ¶ -1 -1 µZ
¶
-1 -1 -1
-1 ()
= () () ()
-1 ()
Since for every product () = and is constant I get () = and = Since
()
³R ´ 1-1
is also growing at rate and = = ()1- is constant. This implies
¯
that () = (-1) has to be constant. Hence, the wage rate must grow at the same rate
¯
³ ´1-
as the efficiency level . It's also seen that () = )
¯ (
grows at rate and hence
()
¯
() =
is constant.
B Aggregate Economy
A possible setup for the aggregate economy is given below.
35
Consumer's Problem
The economy consists of a unit continuum of consumers. Intertemporal utility of a repre-
sentative consumer is Z
= -( -) ln
where is the discount rate and is the aggregate consumption at time ln measures the
instantaneous utility at time Every consumer maximizes utility subject to an intertemporal
budget constraint Z Z
-[ - ]
-[ - ] +
¯
R
where = 0 is the aggregate interest rate up to time , is the price of the final
consumption good , is the wage rate, and is the value of the household's asset holdings.
¯
Total value of spending at time is = The optimization problem of the consumer
yields
= -
Final Good Producer
The final good sector is perfectly competitive. Cobb-Douglas production function for this
sector is given as
= 1-
where is the consumption of the homogeneous good and is the consumption of the com-
posite good. Let and represent the prices of these goods respectively. Profit maximizing
allocations of these goods are given as
= and = (1 - )
The only factor in production is labor which is perfectly mobile across sectors and across estab-
lishments in the composite good sector. Homogeneous good sector is also perfectly competitive
and one unit of output requires a single unit of labor implying = . ¯
In this setup, the relation between the growth rates of composite good industry and ag-
gregate economy can be easily acquired. Implementing the demand values of and into
aggregate production function, I get
µ ¶ µ ¶1-
= (1 - )
which can be simplified to
¡ ¢
(1 - )1- = 1-
Taking the time derivatives of both sides of this equation, I get
= + (1 - )
Since wage grows at rate and
= 0 I get
= Then using = = (1 - )
36
I find the growth rate of the final good
= = - = - = (1 - )
Following this result, growth rate of the homogenous good industry is
= + (1 - )
1
= ((1 - ) - (1 - ) ) = 0
Finally, the growth rate of = = (1 - ), as defined in the consumer's problem
is equal to This implies that = and it is constant.
C Value Function Solution
Implementing the conjectured value function into equation 5, I get
µ ¶
()
+ () = max { () - ()
+ 0
µ ¶ µ ¶
() ()
+ + () - + () }
+ +
After cancelling in both side of the equation, I get
µ ¶ ½ µ ¶¾
() ()
( + ) + () = max () - () + + ()
+ 0 +
½ µ ¶ ¾
()
( + ) () = max + () - ()
0 +
D Relations between Hazard Rate of Exit, Size and Age
(-) -1 (-)
Lemma 1 For all 0, ( - )
and -1 is strictly increasing in .
(-) (-)
Proof. Using l'Hopital's rule, lim -1 = lim (-) ( - ) Taking the derivative of
1
=
0 0
the term with respect to , I get
µ ¶ ¡ ¢
(-) - 1 (-) ( - ) - (-) - 1
=
2
(-) (( - ) - 1) + 1
=
2
It will be sufficient to show that for 0 (-) (( - ) - 1) + 1 0. Defining =
( - ) 0 I need to show - + 1 0
37
2 3
Taylor series expansion of = 1 + 1! + + for - Implementing this into
2! 3!
the inequality above, I get
µ ¶ µ ¶
2 3 2 3
1+ + + + - 1 + + + + 1
1! 2! 3! 1! 2! 3!
µ ¶ µ ¶ µ ¶
2 1 3 1 1 1 1
= 1- + - + + - +
2! 2! 3! ( - 1)! !
³ ´
1 1
Since (-1)! - ! 0 for 1 the inequality holds. Note that only when
(-) -1 (-)
and as lim
= lim (-)1
= under which the inequality also holds.
Lemma 2 For all 0 the parameter of the size distribution () increases in inno-
vation intensity (i.e. () 0).
Proof. In equation 13, it was shown that the parameter of the size distribution () =
(1--(-) )
--(-)
. Then
¡ ¢¡ ¢ ¡ ¢¡ ¢
() 1 - -(-) - -(-) - -(-) - - -(-) --(-) - -(-)
= 2
( - -(-) )
¡ ¢ ¡ -(-) ¢¡ ¢
- -(-) + - - -(-) - -(-) - + -(-)
= 2
( - -(-) )
¡ ¢ ¡ -(-) ¢
- -(-) + - - -(-) ( - )
= 2
( - -(-) )
After the cancellations the derivative simplifies to
¡ ¢
() 1 - -(-) - -(-) ( - )
= 2
( - -(-) )
¡ ¢2
I need to show that this term is greater than zero. Since - -(-) 0 I only need to
show that
¡ ¢
1 - -(-) -(-) ( - )
¡ ¢
(-) - 1
( - )
(-)
Using the result of Lemma 1, as 0 -1 ( - ) Hence, the inequality holds. Since
³ (-) ´
-1
0 the inequality holds for 0
Proposition 1 Hazard rate of exit ( ) decreases in age conditional on size and
decreases in size conditional on age for all ages and sizes ( 0 and 0).
Proof. The proposition has two parts. First I will prove that conditional on size, hazard rate
of exit decreases in age.
Klette and Kortum (2004) show that 0 () 0 In Lemma 2, I showed that () 0 Also,
38
Klette and Kortum (2004) prove that is uniquely determined for the profit per product
values. In this model, since is uniquely determined by the efficiency level is increasing
in
At any size
= ³ ´-1
(-1)
Among the establishments with size more efficient producers (with higher ) will own fewer
products.
To simplify the proof, I assume that there are two types of establishments with low and
high efficiency levels { }. Denote the age of the low-type producer as and high-type
as For any ( ) ( ) ( ) Since is the parameter of the geometric
distribution22 , this relation implies that size distribution for high type producers stochastically
dominates the size distribution of low types.
( ) ( ) 1 - ( ) 1 - ( )
Note that is a monotonically increasing function in time. Hence low efficiency producers
being more likely to have more products than the high efficiency producers is possible when
In this case, when the difference between and is large enough
( ) ( )
1 - ( ) 1 - ( )
Klette and Kortum (2004) show that the hazard rate of exit at age is (1 - ()) Using the
conclusion that at among the producers with size and ( ) ( ), the hazard
rate of exit is lower for the older producers,
(1 - ( )) (1 - ( ))
The second part of the proposition is more straightforward. At any age , () ()
because is increasing in Hence, more efficient producers are more likely to have more
products and are less likely to exit
(1 - ()) (1 - ())
22
Probability mass function of geometric distribution is Pr( = ) = (1 - ) -1 Cummulative distribution
is Pr( ) = 1 -
39
E Pre-exit Behavior of Establishments
Proposition 2 Consider a cohort of establishments all entering at the same time. At any
age 1 within this cohort, establishments that survive longer are larger in size than the
exiting establishments (i.e. for representing the establishment size, representing the time
of the exit, 0 0 [ () | = ] [ () | = + ]).
Proof. At the startup ( = 0), all establishments start with a single product. From equation
16, size of a -type establishment relative to aggregate expenditure at the startup will be
³ ´ 1-
()
1 () =
. For any , [1 ( )] [1 ( )] Also, since -type producers
are more innovative23 , they're less likely to exit than the -type producers. Hence, more
efficient producers start larger and they are more likely to survive longer. This generates the
dispersion at the startup.
This difference between high and low efficient producers persists through their life spans.
Klette and Kortum (2004) show that expected size of an establishment at any age , conditional
on survival is given as
X
(; 1|) 1
=
=1
1 - 0 (; 1|) 1 - (|)
At any age 0 comparing establishments with efficiency levels [ () | ]
[ () | ] and (1 - ( )) (1 - ( )) Since the revenue per product is also
higher for the more efficient producers, we get [ ( )] [ ( )]
Now consider any two establishment at age of the same efficiency level. For an estab-
lishment that exits at time and 0 instantaneous hazard rate of exit is given as24
0
Pr ( = ) = 1-(;1) = (1 - ()). This probability increases in time. Then the expected
0 (;1)
size of a -type establishment at age conditional on exiting at is
X (; 1)
[ () | = ] =
=1
(1 - ())
For 0
1 1
(1 - ()) (1 - ( + ))
which leads to [ () | = ] [ () | = + ] for 0
23
Klette and Kortum (2004) show that innovation rate increases in profit which increases in efficiency
rate
24
For simplicity I drop the efficiency level from the equations that will follow.
40
F Industry Comparison
Figure 10: Size Distribution of Industries
Size Distribution: Food Industry Size Distribution: Paper Industry
.25
.25
.2
.2
.15
.15
Density
Density
.1
.1
.05
.05
0
0
10 100 1000 10000 100000 10 100 1000 10000 100000
Size Size
79 82 83 86 87 89 79 82 83 86 87 89
Size Distribution: Textile Industry Size Distribution: Wood Industry
.25
.25
.2
.2 .15
.15
Density
Density
.1
.1 .05
.05
0
0
10 100 1000 10000 100000 10 100 1000 10000 100000
Size Size
79 82 83 86 87 89 79 82 83 86 87 89
Size Distribution: Metal Industry
.25
.2 .15
Density
.1 .05
0
10 100 1000 10000 100000
Size
79 82 83 86 87 89
41
Figure 11: Graphs on Industry Comparison
16000 Average Sales Across Industries Standard Deviation of Sales Across Industries
40000
8000
20000
Sales
Stdev
4000
10000
2000
5000
79-82 83-86 87-90 91-94 95-9 8 79-82 83-86 87-90 91-94 95-98
Years Years
Food Textile W ood Paper Metal Food Textile W ood Paper Metal
Skewness of Log(Size) for Industries
1.2
1
Skewness
.6 .8
.4
79-82 83-86 87-90 91-94 95-98
Years
Food Textile
Wood Paper
Metal
Standard Deviation- Industries (wrt Industry Mean) Average Entry and Exit Rates
4.00 3.77 0.120
3.50 entryrate 0.096 0.091
0.100 0.093 0.092
2.98 2.81 2.72
3.00 exitrate 0.082
2.45 0.080 0.075
2.50 0.067
0.060 0.063
2.00 0.060 0.054
1.50
0.040
1.00
0.020
0.50
0.00 0.000
Food Paper Textile Wood Metal Food Paper Textile Wood Metal
42
Figure 12: Capital-Output Ratio across Industries
80 100
6040
Ratio
20
79 80 81 82 83 84 85 86
Years
Food Textile W ood Paper Metal
G Estimation Results for Industries
Table 7: Simulation Results
Food (311) Paper (341)
Moments Data Simulation Data Simulation
pctile(10) 128.80 73.2 107.36 45.5
pctile(25) 207.43 133 186.74 88.1
pctile(50) 346.15 368.2 380.76 295.4
pctile(75) 894.56 1217.2 1026.46 1305.7
pctile(99) 31291.11 31276 44379.6 44370
0.054 0.043 0.063 0.064
E[g] 0.027 0.011 0.02 0.02
Std[g] 0.75 0.39 0.5 0.40
E[Y] 2427.0 2117.1 3192.10 2964.3
Std[Y] 7569.0 9912.3 13549.7 18172
Textile(321) Wood(331) Metal (381)
Moments Data Simulation Data Simulation Data Simulation
pctile(10) 106.94 118 62.50 72.3 86.2 143
pctile(25) 193.09 180.1 119.52 125.5 154.3 213.4
pctile(50) 370.63 357.2 267.54 267.8 343.8 400
pctile(75) 890.32 900.1 673.43 660.2 1021.5 908.7
pctile(99) 20498.3 20498 13421.26 13422 16141.43 16144
0.067 0.058 0.082 0.044 0.092 0.06
E[g] -0.01 -0.02 0.05 0.045 0.017 0.015
Std[g] 0.70 0.47 0.90 0.48 0.83 0.47
E[Y] 1502.6 1530.5 1136.8 1007 1598.1 1361.4
Std[Y] 3716.4 7407.7 3552.5 4069 4052.9 6205
43
Figure 13: Industry Size Distributions: Model Fit
Food Industry Paper Industry
1
1 0 .1
0.1
Pr(size>S)
P r(si ze >S)
0.0 01
0 .001 0 .000 1
0.0 001
1 10 0 1 000 1 0000 100 000 1 100 1 000 1 0000 10 0000
Si ze S ize
Si mu lation Data Simula tion Data
Textile Industry Wood Industry
1
1
0 .1
0.1
Pr(size>S)
P r(si ze >S)
0 .001
0.0 01
0 .000 1
0.0 001
1 100 100 0 10 000 100 000 1 1 00 10 00 100 00 100 000
Si ze S ize
Si mu lation Data Simula tion Data
Metal Industry
1 0.1
Pr(size>S)
0.001 0.0001
1 100 1000 10000 100000
Size
Simulation Data
44
~~