Inter-Firm Knowledge Spillove and Firm Size Heterogeneity
Jie (April) Cai
Feb. 2008
Abstract
Firm level data con…rm that every sector has a distinct and stable Pareto
…rm size distribution. Observing the inter-…rm learning by patent citation data,
I …nd that knowledge spillover is more abundant in sectors where …rm size is more homogeneous: …rms cite 3 to 7% more to other …rms’ patents; the geographic distance between citing and cited …rms are 100-300 kilometers longer than …rms in heterogeneous sectors. I attribute the di¤erence in knowledge spillover e¢ ciency to the distinction in …rms citation network structure.
I build a one-sector model with …rm’s endogenous innovation and imitation to characterize the sector speci…c …rm size dynamics. Firm size distribution will converge to a Pareto distribution driven by this …rm size dynamics. The evidence from patent citation data support the model predictions: when imitation contributes a bigger share in gross growth rate, the …rm size heterogeneity is smaller and surviving …rm’s growth rate drops faster as …rm grows bigger.
Considering cross-sector knowledge spillover, a multi-sector model can mimic the Pareto …rm size distribution of all …rms in the economy.
KEYWORDS: …rm size heterogeneity, knowledge spillover, network structure, innovation, imitation, Pareto distribution, scale dependent growth rate
Axtell (2001) show that …rm size distribution in the entire economy follows a
Pareto distribution with parameter close to one; meanwhile, Helpman, Melitz and Yeaple (2004) points out that every sector has a diferent Pareto …rm size distribution. With …rm level data, I con…rm that sectoral …rm size heterogeneity measured by standard deviation of log scale …rm size does not vary substantially by di¤erent …rm size proxies, over time and across countries. Measuring the inter-…rm learning by patent citation data, I …nd that knowledge spillover is more abundant in sectors where …rm size is homogeneous: …rms cite 3-5% more to other …rms’patents; the geographic distance between citing and cited
…rms are 300-500 miles longer in those sectors. I attribute the di¤erence in knowledge spillover e¢ ciency to the distinct citation network structure. In the homogeneous sectors, …rm’s citation network is less clustered; …rms build more new connections with randomly picked unknown …rms, and less with friends’ friends in the last period. These features help information di¤use evenly across all …rms and reduce sectoral …rm size heterogeneity. The one-sector model in this paper extends Klette and Kortum (2004) by allowing …rms imitate in addition to innovation. Firm size dynamics and eventually …rm size distribution are determined by …rm’s endogenous choice of innovation and imitation inputs, according to their relative costs. Firm size heterogeneity is diminished when
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imitation plays a bigger role in gross growth rate. The model predictions are con…rmed by the evidence from NBER Patent Data. When I include the crosssector knowledge spillover in the multi-sector model, the …rm size distribution of all …rms will converge to a Pareto Distribution too.
Within a narrowly de…ned sector, the …rm size distribution also follows
Pareto distribution, but di¤erent sector has a distinct and stable distribution parameter . By the nature of Pareto distribution, the commonly used …rm size heterogeneity measure, standard deviation of log scale …rm sales, is
1
. In
French and Chilean …rm level data, this sectoral …rm size heterogeneity measure varies little when the …rm size is proxied by number of employees, operational turnover, and value added. Within the same country, sectoral …rm size heterogeneity measure is stable over time in French, U. S., and Chilean …rm level data.
In the same year, sectoral …rm size heterogeneity measures of French and U. S.
manufacturing sectors are highly correlated, with correlation coe¢ cient more than 0.7.
In this paper I want to proof that sectoral …rm size distribution is the equilibrium outcome of a sector speci…c …rm size dynamics. In the literature of …rm size dynamics, imitation and innovation are the two important factors a¤ecting
…rm’s growth dynamics. Speci…cally, I want to analysis how …rm’s endogeneous innovation and imitation a¤ect …rm size dynamics.
My belief is that abandunt inter-…rm knowledge spillover provides …rms more opportunities to learn from each other, which gradually reduces the sectoral …rm size heterogeneity. To illustrate this point, consider two extreme cases. In one case, it costs a …rm in…nity hours of labor to access private knowledge of other
…rms. The …rm size di¤erence originates only from …rm’s risk in innovation by its own private knowledge. Small …rms can not learn from the big ones, therefore the size di¤erence will persist. In the other case, if all knowledge is publicly available within the industry, when any …rm discovers a new technology, all the other …rms will learn about it immediately at zero cost. There will be no size di¤erence between …rms at all. In reality, learning cost is between zero and in…nity, therefore inter-…rm knowledge spillover lies somewhere between these extreme cases.
The learning cost includes searching, absorbing and adopting costs. For example, it is intuitive that higher worker’s mobility reduces …rm’s searching cost for outside information. Figure 3 shows that labor turnover rate varies signi…cantly between sectors. For example, the construction sector has 67% annual job separation rate, while the manufacturing sector has only 32% annual job separation rate. The sectors with high labor turnover rate tend to have homogeneous …rm size. Absorbing and adopting cost depend on whether the knowledge is codi…ed and standarized. In a sector where technology is designed to be …rm speci…c instead of standarized within sector, …rms have di¢ culty applying knowledge from other …rms in their own production.
Other than labor turnover rate, inter-…rm patent citations provide more direct measure of knowledge spillover between …rms. In the empirical section,
I compare the 14 most heterogeneous sectors ( H sectors) with the 14 most homogeneous sectors ( L sectors). I measure …rm size by the number of patents
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granted to that …rm. I …nd that …rms in L sectors are 3-7% more likely to cite other …rm’s patents than …rms in H sectors. I get similar results, when
I conduct the same cross sector comparison with …rms in France and Japan.
Geographically, the great circle distance between citing and cited U. S. inventors are 100-200 kilometers further in L sectors than in H sectors. Regressions with inter-…rm citations from all sectors and controling for citing …rm characteristics also support the above comparasion.
I attribute the di¤erence in knowledge spillover e¢ ciency to the distinction in sectoral …rm network structure. I use inter-…rm patent citations as directed connections between …rms and construct a panel of citation network topology for every sector in every year. Statically, the networks in L sectors are less clustered than those in H sectors, which means …rms in L sectors are more likely to cite
…rms outside the clique, and cite less to clique members. Dynamicly, …rms in
L sectors build more new connections with randomly picked unknown …rms, but less with friends’friends in the last period. As a result, the information is shared not only within the clique, but also di¤uses evenly across cliques within the sector.
A one-sector model is built on the basis of Klette and Kortum (2004), in which …rms grow by endogeneous innovation, but imitation plays no role. I allow …rms to grow by both imitation and innovation. In innovation, …rms use interior knowledge; in imitation, …rms use public knowledge. Firms choose inputs in both types of R&D endogeneously: when it is relatively cheaper to imitate than to innovate, …rms input more in imitation than innovation, and vice versa. In equilibrium, the marginal returns from both growth methods are equalized. The …rm size dynamics and eventually the …rm size distribution are decided by the endogenous innovation and imitation rates. According to
Kesten (1973), when the shocks in imitation and innovation rates are identically independent over time and across …rms, …rm size distribution will converge to a Pareto distribution with parameter , regardless of what initial …rm size distribution is. The commonly used …rm size heterogeneity measure, standard deviation of log scale …rm size
1
, is inversely related to imitation’s share in gross growth rate and positively related to the variance of innovation shock.
These relationships shows that the innovation risk volatility is the origin of …rm size heterogeneity, but imitation is the o¤setting force that moderate the size di¤erences.
The …rm size dynamics in one-sector model also implies a scale dependent growth rate, which means surviving …rm’s growth rate decreases as the …rm grows larger. Growth rate decreases as …rms are bigger because every …rm learns from the same public knowledge pool, but smaller …rms bene…t more than larger
…rms. The …rm growth rate decreases faster in sectors where learning is easier.
The model’s predictions are tested by the …rm’s innovation and imitation rates infered from NBER Patent dataset.
In the multi-sector model, …rms develop products in multiple sectors and knowledge spillover cross sectors.
Firm size becomes the summation of its branches in all sectors. The multi-sector model generates the …rm size distribution of all …rms in the economy, which is also a Pareto distribution.
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This paper is related to the literatures in the Zipf’s law in social science. Zipf’s law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table.
Pareto distribution is the continuous time version of Zipf’s law. Axtell (2001) …nd that
…rm size distribution in the U. S. economy follows a Pareto distribution with parameter close to one. Gabaix (1999) studies the dynamics of city size and give two possible explanations to why city size distribution follows Pareto distribution with parameter one. One route is that when productivity di¤erence is unbounded, cities are constant return to scale in the upper tail, they create new jobs proportional to existing jobs; the second route is when productivity and externality shocks are both unbounded, but these two forces o¤set each other and each city is equally attractive to immigrants. Kesten (1973) gives the conditions for a vector renewal process to converge to Pareto Distribution.
Helpman, Melitz and Yeaple (2004) point out that sectorial heterogeneity measure is di¤erent across sectors; in heterogeneous sectors, more …rms choose to use FDI instead of exporting to serve export market.
The one-sector model is closely related to the literature of heterogeneous
…rms with …rm size dynamics. Klette and Kortum (2004) develops a parsimonious model of innovation to match the …rm level evidence and capture the dynamics of individual heterogeneous …rms. I build my one-sector model on the basis of this well structured paper. I borrowed the Cobb-Douglas knowledge production function from them; and get the same prediction that innovation investment is proportional to …rm size. What di¤erent is that, in their paper
…rms rely solely on interior knowledge to develop new goods, there is no externality from the public knowledge. Therefore …rm size dynamics are exposed to only innovation shock, the …rm size distribution ends up with logarithmic distribution. They can’t explain the cross sector di¤erence in …rm size heterogeneity either.
Luttmer (2006) allows new entrance to imitate from incumbent …rms, but incumbent …rms don’t learn from each other. The …rm size distribution is decided by the imitation e¤ect, but the imitation investment is not …rms’ endogenous decision. His model explains the Zipf’s law by relatively high entry cost and very few imitation chances; It also emphasizes that selection e¤ect is as important a factor as innovation in economic growth.
Rossi-Hansberg and Wright (2005) is closely related to this paper in explaining scale dependent …rm growth rate. They use mean reverse of industry speci…c human capital to model the scale dependent growth rate, exit and …rm size distribution. They use the bigger diminishing return to industry speci…c human capital to explain why physical capital intensive industry’s …rm size distribution has thinner tail or are more heterogeneous. They use a macro view to deal with …rm size dynamics. Households instead of the …rms make the investment decisions on human capital and physical capital. In the model with heterogeneous …rms, it generates the same result as if …rms make investment decision, only when …rm’s productivity shock is identically independent over time and
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across …rms. In reality, …rm size or …rm productivity is quite persistent. More importantly, the productivity shock in their model is exogenous. Firms don’t endogenously invest in R&D to invent new goods or improve their productivity.
The authors assume that human capital has the same share in labor aggregation for any industry. If this assumption is relaxed, standard deviation of log
…rm size is also increasing in human capital’s share 1 or decreasing in pure labor’s share . If the variation of pure labor’s share is bigger than that of capital’s share ; or even worse, and highly correlated, distinction in capital share alone can’t explain the di¤erence in …rm size heterogeneity. In table 2 of their paper, capital share doesn’t explain the variance of heterogeneity within manufacturing sector. In the learning-by-doing section, higher contribution of industrial output in human capital accumulation induces more scale dependent growth rate and smaller …rm size heterogeneity. The author didn’t expend and provide empirical proof for it.
Another stream of related literature is about information di¤usion in network. The empirical section of this paper borrowed and empirically tests the predictions in dynamic network formation literature. Cowan and Jonard (2004)
…nds that nodes have more heterogeneous information sets in a more clustered
"small world" network. Jackson and Rogers (2007) …nds that information diffuses faster in a network with homogeneous degree (number of connections each node has) distribution. I construct a panel of …rm citation network from inter-
…rm citation data.
L sectors have di¤erent network structure from H sectors.
With this network panel, I empirically test the predictions in the literature, L sectors have less clustered network structure (Cowan and Jonard 2004); …rms in L sector makes more new friends with randomly picked …rms (Jackson and
Rogers 2007a); L sector has more homogeneous degree distribution (Jackson and Rogers 2007).
The rest of the paper is organized as follows. In the empirical section, I will show the robustness of the sector speci…c heterogeneity measure; compare the inter-…rm citation pattern, growth scale dependency, and the structure of citation networks between homogeneous and heterogeneous sectors. In the onesector model section, endogenous innovation and imitation inputs generate a
…rm size distribution following Pareto distribution, where …rm size are more homogeneous when the relative cost of imitation to innovation is lower. In the multi-sector section, …rm size is the summation of branches in all sectors. Firm size distribution of all …rms in the economy is a Pareto distribution too. Lastly, it is the conclusion.
This section is organized as follows: …rstly, I will introduce the data sets; secondly, I proof the robustness of sectoral heterogeneity measure by di¤erent size proxies, over time and across countries; thirdly, I compare the the frequency, time lag and distance of patent citations between homogeneous and heterogeneous sectors; fourthly, I compare the structure and dynamic formation process
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of the sectoral citation network; lastly, I will compare the distinct growth pattern between homogeneous and heterogeneous sectors;
In this section, I estimate the …rm size heterogeneity measure for manufacturing sectors in France, Chile and U. S.. These three data sets contain di¤erent statistics about …rm size. I need to estimate …rm size heterogeneity measure by di¤erent methods according to the data availability. There are two methods to estimate : one is to use the reciprocal of standard deviation of log scale
…rm size for all …rms within one industry. The other is to sort …rms in one industry in descending order of size; keep their rank in the sorting; then run an OLS regression: log(rank) = b 1 *log(…rm size)+ b 2 . By the nature of Pareto distribution, the absolute value of b 1 ’ s OLS estimation is the estimation of for that industry.
In French and Chilean data sets I can use both methods. In the Amadeus data set provided by BUREAU van DIJK, there are …rm level data for all …rms in French manufacturing sectors from 1997 to 2005. In the Chilean manufacturing …rms data set provided by Chile Instituto Nacional de Estadistica, there are only …rms with more than 10 employees from 1979 to 1996. If …rm size distribution measured by number of employees follows Pareto distribution, this truncation doesn’t a¤ect the estimation of , because Pareto distribution has a special feature that when the distribution is truncated from the left, the rest of the distribution on the right tail is still a Pareto distribution with the same parameter, except that the new distribution starts with a higher minimum level.
In U. S. data set I can only use the second method, because I don’t have size information for every …rm. U. S. Economic Census in 1997 and 2002 reported
"Industry Statistics by Employment Size". For each 6 digit NAICS manufacturing industry, this report gives the number of …rms in 10 employment size categories: 1 to 4 workers, 5 to 9 workers, 10 to 19 workers, 20 to 49 workers, 50 to 99 workers, 100 to 249 workers, 250 to 499 workers, 500 to 999 workers, 1000 to 2499 workers, and 2500 workers and above. From these numbers, I can …gure out the rank of the …rms with 1, 5, 10, 20, 50, 100, 250, 500, 1000, 2500 number of employees in their 6 digit NAICS industry. Although there are only at most
10 observations for each industry, the estimated Pareto distribution parameters are still quite consistent over time. By the same method, I get the estimation for 4 digit NAICS industries.
In the French and Chilean data sets, I use number of employees, value added, sales and operational turnover as alternatives to check whether this heterogeneity measure is robust by di¤erent …rm size measures. sdlnl is the abbreviation for standard deviation of log(number of employees);sdlny is the abbreviation of standard deviation of log(operational turnover); sdlns is the abbreviation of
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standard deviation of log(sales); sdlnva is the abbreviation of standard deviation of log(value added). In the French data set (…gure 1), these 4 measures for 81 4-digit NAICS sectors are highly correlated; with correlation coe¢ cient higher than 0.9. In Chilean data set (…gure 2), the correlations among sds, sdy and sdva are high as those in French data set, but the correlations between sdl and the other three measures are as low as 0.6 to 0.8. A possible reason is that
…rm data are truncated, because only …rms with more than 10 employees are included in the data set.
The …rm size distribution for a given industry is quite stable over time within one country. Constrained by the data coverage in time series, I compare the starting year with ending year in each data set. In U. S. data set (…gure 8), I compare the estimations for 4 digit NAICS manufacturing industries in 1997 and 2002. In French data set (…gure 9), I compare estimations for 4 digit
NAICS manufacturing industries in 1997 and 2005. The outlier 3122 in graph 2 represents tobacco manufacturing sector. Its heterogeneity measure drops from
3.1 in 1997 to 0.9 in 2005. There are big policy changes in this sector during this period, which might induce the signi…cant change in …rm size heterogeneity. In 2001, Brussels passed a law, soon to take e¤ect, banning mass-media advertising of tobacco and requiring large warning labels on cigarette packages.
To discourage potential new smokers, governments throughout Europe have increased their cigarette taxes in 2003. In Chilean data set (…gure 10), I compare estimations for 4 digit ISIC manufacturing industries in 1979 and 1996. The proxy for …rm size is the number of employees in all data sets. After 20 years, the heterogeneity measure roughly keeps a one-to-one relation with its value in
1979. The outlier typically have less than 100 …rms.
Helpman, Melitz and Yeaple (2004) compares …rm size distribution across industries both in U. S. and France. They found that although U. S. and France have di¤erent economic policy and institutions, …rm size distributions for the same industry are highly correlated across countries, with correlation coe¢ cient more than 0.5. I compare heterogeneity measure between U. S. and France manufacturing industries. There are 81 NAICS 4 digit manufacturing sectors in total. I use standard deviation of log scale number of employees (use sdlnl as abbreviation) as the heterogeneity measure for sectors in both countries. Figure
11 and 12 show that the correlation between sdlnl in the two countries. The correlation coe¢ cients are 0.74 in 1997 and 0.72 in 2002.
Ideally, I would like to compare the heterogeneity measure in U.S., France, and Chile, but U. S. and France data sets both report …rm’s industry classi…cation by NAICS 2002, Chile’s data set uses 4 digit ISIC. Unfortunately, the concordance between NAICS and ISIC classi…cation is not one to one, I can’t compare Chile with the other two countries.
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Inter-…rm patent citations make it possible to track the knowledge ‡ows between
…rms. Ja¤e, Tratjenberg and Fogarty (2000) and Duguet and MacGarvie (2005) justi…es patent citations as an indicator of knowledge spillover, by surveying the patent inventors in U. S. and …rms in France. Their conclusions are that a part of the citations are associated with real knowledge ‡ow, and aggregate patent citation is a noisy measure of knowledge ‡ow. As long as patent citations are equally noisy in all the sectors, I still can infer the di¤erence between sectors.
NBER Patent Data comprises detailed information on almost 3 million patents granted by U. S. Patent O¢ ce between January 1963 and December
1999, all citations made to these patents between 1975 and 2002 (over 16 million). For each patent, there is information about inventor, the organization who owns it, SIC87 industry code, and apply year. From the apply year of citing and cited patents, I know the time lag for citing inventor to use the cited patent. From the latitude and longitude data of inventors city, I calculate the great circle distance between citing and cited patent inventors. I use the percentage of inter-…rm citation and great circle distance between citing and cited inventors for each time lag to measure the knowledge spillover e¢ ciency. Higher percentage of inter-…rm citation means more chances to learn from other …rms; longer distance between citing and cited inventors means the knowledge travels further for a given time.
The standard deviation of log scale patent stock sdlnps is the measure of
…rm size heterogeneity measure
1
. I sort the 42 sectors by their …rm size heterogeneity, then compare the percentage of outside citations and great circle distance between citing and cited patents in the most heterogeneous 14 sectors
(call them H sectors) with those in the lest heterogeneous 14 sectors (call them
L sectors). The H sectors includes Industrial inorganic chemistry, Industrial organic chemistry, Plastics materials and synthetic resins, Soaps, detergents, cleaners, perfumes, cosmetics and toiletries, Paints, varnishes, lacquers, enamels, and allied products, Miscellaneous chemical products, Petroleum and natural gas extraction and re…ning, Engines and turbines, O¢ ce computing and accounting machines, Electrical transmission and distribution equipment, Radio and television receiving equipment except communication types, Ordinance except missiles, and Aircraft and parts. The L sectors includes Primary ferrous products, Farm and garden machinery and equipment, Household appliances,
Refrigeration and service industry machinery, Guided missiles and space vehicles and parts, Ship and boat building and repairing, Railroad equipment,
Motorcycles, bicycles, and parts, Miscellaneous transportation equipment, Miscellaneous machinery, except electrical, Electrical lighting and wiring equipment,
Drugs and medicines, and All other SIC’S.
I group the citations by the year of time lag. For any time lag group, I calculate the percentage of citations made to other …rms. The …rms in the L sectors cite more percentage to others than the …rms in the H sectors, especially when they are citing relatively new patents. In the U. S. comparison (…gure 13),
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the …rms in L sectors cite about 7% to 3% more to others …rms’patents than the …rms in H sectors. I …nd similar patterns when I do the same comparison for Japan (…gure 14), and France (…gure 15).
Another aspect of knowledge spillover is the distance that the knowledge travels for a given time. From U. S. Census Bureau’s Census 2000 U.S. Gazetteer
Files, I match U. S. inventor’s geographic location with the city level latitude and longitude. For each citation, I calculate the great circle distance between citing and cited patent inventors with their latitudes and longitudes. Great circle distance is the shortest direct ‡ight distance between two cities. In …gure 16, on average knowledge travels 100 to 300 kilometers further in L sectors than in
H sectors until the time lag is larger than 23 years.
Figure 4 and 5 give corresponding regression results about the distance between cited and citing patents, and the probability to cite outside patents. The latitude and longitude data from U. S. Census Bureau only cover about 43% inventors’ cities reported in Patent Database. The change and co-work in the regressions are derived from the inventor’s information of NBER Patent Database. On average every inventor applied 4.46 patents. I assume that the inventor works for the patent assignee. If at year t , an inventor’s assignee is di¤erent from the inventor’s last assignee, I mark this inventor as "change" a job at year t . If one inventor has more than one assignees in year t , I mark this inventor as "co-work" at year t . I control for the relative size of industry and citing …rm itself, because a bigger outside knowledge pool attracts more citations to other
…rms.
In …gure 4, distance between cited and citing patents is bigger when sectoral sdlnps is smaller, when the cited patent gets older, and when the inventor of citing patent changes a job or co-works with other organizations. In …gure 5,
…rms are more likely to cite other …rms when sectoral heterogeneity sdlnps is smaller, when the cited patent gets older, and when the citing patent inventor just changes a job or co-works with other organizations.
So far, it is not very clear why there is cross-sector di¤erence in knowledge spillover e¢ ciency. In order to open this black box, I look into the structure and dynamic formation of …rm citation network. The literature of social network
…nd that network structure a¤ects the e¢ ciency of information di¤usion. The citation network is constructed as follows. Every …rm is a node in the network. If
…rm f 1 ’s patent p 1 cites …rm f 2 ’s patent p 2 , then there is a connection pointing from …rm f 1 to f 2 . The n by n matrix M s;t
…rms in sector s at time t .
M s;t
( i; j ) = 1 if …rm describes the network with i cites …rm j , M s;t
( i; j n
) = 0 otherwise.
Statically, L and H sectors have di¤erent network structures in a given year.
In the citation network, …rms cluster into many cliques, they cite more often within cluster and less often outside cluster. The …rst two rows of able 8 show
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that clique members are more correlated in size, and locate closer than nonclique members. In L sectors, …rms are less clustered than …rms in H sectors, in the sense that …rms communicate more often with non-clique members. As a result, information di¤uses more evenly across …rms in the sector. While in H sectors, information is only shared within the clique, but di¤erent cliques have heterogeneous information sets. There are many measures of clustering in the network literature, I use the three clustering coe¢ cients in Jackson and Rogers
(2007). They all measure how often friends’ friends are also friends. The …rst one is
CC 1 ( M s;t
) =
P i ; j
P k = i;j
M s;t i ; j = i ; k = i;j
( i; j
M
) s;t
M
( s;t i; j )
( j; k
M s;t
) M s;t
( j; k )
( k; i )
.
The second clustering coe¢ cient is
CC 2 ( M s;t
) =
P i ; j
P k = i;j s;t i ; j = i ; k = i;j
( i; j ) ^ s;t
( s;t i; j
( j; k
) ^ s;t
) ^ s;t
( j; k )
( k; i )
, where M s;t
( i; j ) = max ( M s;t
( i; j ) ; M s;t
( j; i )) . The third one is
CC 3 ( M s;t
) =
1 n
X i
P j = i
P
= i;j s;t j = i ; k = i;j
( i; j
M
) ^ s;t
( s;t i; j
( j; k
) ^ s;t
) ^ s;t
( j; k )
( k; i )
.
The denominator of CC 1 is the total number of node pair i , k with a common connected node j . The numerator of CC 1 is the total number of node triple i , j , k , among which any two of them are connected. It measures the percentage of node pairs with a common friend are also friends themselves. The …rst and second de…nitions are the same when the network connections are not directed.
The third one gives equal weight to every node; while the …rst two de…nitions give bigger weights to nodes with more connections. Figure 22, 23 and 24 plot the positive relation between the above three clustering coe¢ cients and standard deviation of log scale patent stock ( sd ln ps ) across sectors in 1990.
sd ln ps is the standard deviation of log scale patent stock.
Dynamically, …rms in L and H sectors build new connections by di¤erent dynamic processes. In L sectors, …rms build more new connections with randomly picked …rms ( RF ), and less with friends’ friends or network-based new friends ( F F ) in the previous period. I de…ne the two …rms i , j as RF if M s;t
( i; j ) = 1 , and M s;t 1
( i; j ) = 0 , and
I de…ne the two …rms i , j as F F if M s;t
( i; j
M s;t 1
) = 1
( i; :)
, and
M
M s;t 1 s;t 1
(: ; j ) = 0
( i; j ) = 0 ,
.
and M
M s;t 1 s;t 1
’s i th
( i; :) M row and s;t 1 j th
(: ; j ) > 0 .
M s;t 1
( i; :) M s;t 1
(: ; j ) is the product of column, which is equal to the number of i , j ’s common friends at time t 1 in sector s . This pattern con…rms the prediction in Jackson and Rogers (2007), that higher randomly picked new friends to network-based new friends ratio generates more homogeneous degree distribution and better information di¤usion e¢ eciency.
New connections with randomly picked friends help knowledge spread evenly across …rms, because connections with old friends and friends’ friends tend to
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constrain the domain of knowledge di¤usion. The last three rows in table 8 show that, compared with friends’ friends and old friends, randomly picked cited …rms are uncorrelated with the citing …rms in size; citing and cited …rms locate the furthest apart; the citation lag is the shortest. Cowan and Jonard
(2004)’s information barter model also predicts that two nodes from further away distance have more heterogeneous information sets, therefore have more incentives to exchange information. With homogeneous information sets in L sectors, the smaller …rms can catch up with the bigger …rms, therefore sectoral
…rm size heterogeneity is smaller in those sectors. Figure 25 shows the negative relation between log ( RF=F F ) and sd ln ps across sectors in 1990.
Di¤erent sector have fundamentally di¤erent …rm growth dynamic process, the sector speci…t size distribution is the outcome of the sectoral speci…c growth dynamics process. In this subsection, I compare the distinct growth pattern cross sectors. I …nd that in homogeneous L sectors, surviving …rm’s growth rate drops faster as …rm grows bigger than …rms in heterogeneous H sectors.
Rossi-Hansberg and Wright (2005) also identify and explain the di¤erent growth scale dependency across sectors. They point out that growth rate and exit rate decrease faster with scale in physical capital intensive sectors than in human capital intensive sectors. In my comparison, I …nd that …rm growth rate in homogeneous sectors is more scale dependent than heterogeneous sectors.
Their proxy for scale of …rm size is number of employees. Here I use relative
…rm size, which is absolute …rm size devided by average …rm size in the sector.
Relative …rm size has some advantage, because di¤erent industries have di¤erent number of …rms and total sector size. The absolute size can’t capture each …rm’s relative position in its own industry. For example, in Chemical Manufacturing sector, a …rm with EURO 4,477,000 operational turnover is just a median …rm, but in Computer and Electronic Product Manufacturing sector, a …rm with the same turnover is already a top 20% …rm. If I compare …rms from di¤erent sectors but with the same absolute size, I might be comparing a junior apple with an adult orange.
I choose representatives for the homogeneous, median, and heterogeneous sectors in both the U. S. patent database and French …rm database. In the patent database I use relative patent stock (patent stock of the …rm/average patent stock of the industry) as the proxy of relative …rm size. In …gure 17, I pick the Petroleum and natural gas extraction and re…ning, Paints, varnishes, lacquers, enamels, and allied products, and Drugs and medicines to represent the homogeneous, median and heterogeneous sectors. In each industry, I sort
…rms by their size, and divide …rms into 20 bins; then calculate the average 5 year growth rate for surviving …rms, the average relative …rm size for each bin per year. For example, in 1990 surviving …rms’ 5 year growth rate in Petroleum and natural gas extraction and re…ning (homogenous sector) slows down the fastest; while surviving …rms’ 5 year growth rate in Drugs and medicines
11

(heterogeneous sector) smoothly decreases; Paints, varnishes, lacquers, enamels, and allied products (median sector) lies between the other two sectors.
In the French …rm data set, the relative …rm size is given by …rm’s operational turnover/average turnover in the industry. In …gure 18, I choose fabricated metal product manufacturing, machinery manufacturing, and computer and electronic product to represent homogeneous, median and heterogeneous sectors respectively. sd in the graphs is the abbreviation for standard deviation of log (turnover). Surviving …rm’s growth rate in fabricated metal product manufacturing (homogeneous sector) drops the fastest as the …rm becomes relatively bigger; while …rm growth rate in computer and electronic product manufacturing (heterogeneous sector) drops very slowly as the …rm grows bigger. Machinery manufacturing (median sector) lies between the other two sectors.
In the one sector model section, I will show that …rm growth rate drops faster in homogeneous sectors because learning contributes more to the …rm growth there than in heterogeneous sectors.
The representative consumer maximize her utility by solving the following problem
U = max
C t
; L t
; S f;t
Z
0
1
Z
M
F s:t: L t
W t
+ t dt f;t df = C t
, t = 0 , 1 , ::: , 1
0 log ( C t
)
1
L t
(1)
(2) where is the time preference of the representative consumer; C t sumption of …nal goods; L t is the conis the hours of labor worked, which I assume is a constant over time. The wage rate in the labor market is W t
.
f;t is …rm f’s pro…t The representative consumer owns all the intermediate …rms indexed by f within [0 , M
F
] .
f;t is the pro…t from …rm f at time
…rms. Consumer problem’s …rst order conditions are: t .
M
F is the mass of
C t
= W t
(3)
Y t
=
Z
I t x it di
0
!
1
(4) x it is the input of intermediate goods i at time t . Each product i is sold at price p i;t t .
1
. There are I t
1 number of intermediate goods in the economy at time is the elasticity of substitution between intermediate goods, < 1
12

to ensure that elasticity of substitution is greater than one. Final goods price is normalized to 1 . The pro…t maximizing demand for input x i;t is Y t
( p i;t
) .
There is only one sector in the economy. There are large number of …rms in the economy, so that each …rm is in…nitesimal. In the monopolistic competitive market, each goods have the same market share. Firms grow by inventing new goods. Firm f produces goods in the economy I t
Z
=
R
0
M number of goods at time
F Z f;t df , where M
F t . The total number of is the mass of …rms. Table 3 here borrows the Table 2 in Broda and Weinstein (2007), it shows that big …rms carries larger numbers of products, and develop products in multiple sectors.
Firms do two types of R&D to invent new goods: innovation or imitation.
Innovation uses …rm’s private knowledge capital and N f;t while imitation uses industrial public knowledge capital Z
I;t research hour. I use number of goods Z f;t units of research hour; and M f;t units of to measure the size of …rm f ’s private knowledge capital, and average …rm size Z
I;t in the industry to measure the size of public knowledge pool, which is common to every …rm. Here imitation refers not only to simple reverse engineering and replication; it also includes the improvement and upgrading on other …rms’products, because in reality the patent law doesn’t protect simple replication, a …rm has to upgrade existing products to a certain degree, so that there is enough originality in that new product.
Firms’private knowledge di¤uses to the public knowledge pool through many channels. In Duguet and MacGarvie (2005), there are 12 channels listed: external R&D, cooperative R&D, patents and licenses, analysis of competing, experts, equipment acquisition, hiring employees, communication with suppliers, communication with customers, mergers and acquisitions, joint ventures and alliances, and personnel exchange. Firms may or may not voluntarily reveal their private knowledge to the public, but interactions with other economic agents always generate a steady knowledge ‡ow from each …rm’s private pool to the public pool.
I borrow the new knowledge production function in Klette and Kortum
(2004). The production function of innovative R&D and imitative R&D are
Cobb-Douglas.
E Z
N f;t
= A
N
N f;t
Z
1 f;t
(5)
E Z
M f;t
= A
M
M f;t
Z
1
I;t
(6)
Z N f;t and Z M f;t are the number of new goods invented by innovation and imitation respectively.
A
N is the industry speci…c productivity of innovative
R&D; A
M is the industry speci…c productivity of imitative R&D. By assuming di¤erent e¢ ciency in the two types of R&D, I imply that …rms use private and public knowledge at di¤erent costs. The cost includes the searching cost to …nd the related existing knowledge; the reverse engineer cost to absorb the existing
13

knowledge; and the invention cost to add novelty on top of the existing goods.
Normally, …rm borders block knowledge spillover, therefore it is more productive to use internal knowledge instead of external knowledge, or A
N
> A
M
. On average, …rms tend to use private knowledge more frequently; and they use more up-to-date private knowledge than public knowledge. A few numbers from the patent citation data will illustrate this point: on average, every organization owns 0.17% of the patent stock in the industry, but they cite over proportionally
11.1% to its own old patents. If the cited and citing patents are owned by the same organization, the average citation time lag is 5.86 years; otherwise the average time lag is 9.06 years. Citation lag is the application year of citing patent minus the application year of cited patent, which tells how long it takes the citing …rm to know the cited patent.
The Cobb-Douglas knowledge production functions implies two assumptions: one is that innovation research hour N f;t has decreasing marginal productivity, because 0 < , < 1 . The other assumption is that, with the same amount of research hour, bigger …rms has advantage in innovation, because they have more former experience in R&D Z f;t
. As a result, the research hour that each
…rm spends on innovation is proportional to its private knowledge pool size, because these two forces o¤set each other. For the same reason, every …rm’s input in imitation is proportional to the size of public knowledge pool Z
I;t
.
Cobb-Douglass production function also ensures that there are positive inputs in both types of R&D, because the marginal productivity of input is positive in…nity when the input level is zero.
There are risks in the successful rates of innovation and imitation on the basis of existing knowledge capital.
Z N f;t
Z f;t
=
A
N
N f;t
Z
1 f;t
Z f;t
+ en f;t
(7)
Z
M f;t
Z
I;t
=
A
M
M f;t
Z
1
I;t
Z
I;t
+ em f;t
(8) en f;t and em f;t are zero mean random variables, describing the risks in innovation and imitation respectively. When …rm f ’s manager chooses these two types of research inputs, she knows the distribution of doesn’t know the actual realizations of them.
en f;t +1 and em f;t +1
, but she
In order to produce one unit of x i;t
, …rm f has to hire one unit of production worker. The pro…t maximizing price for each goods is
W t . The pro…t from each product is
Z
I;t
1 Y t
I t
.
Z f;t
Z f;t
I t is the market share of …rm f at time t .
Z
I;t
I t is the relative size of the public knowledge pool to the complete knowledge pool. I will normalize but I still use
I;t of market share is
Z
I;t to 1 in the following part of the model, to clarify the notation. The marginal pro…t from
1
Y t
Z f;t
~ f;t percent
. I assume that …rms get full compensation for their
…rm value when liquidated, therefore the possibility of exiting in the future won’t a¤ect their current innovation and imitation decision. Firm f chooses
14

the optimal investments in innovation N f;t and imitation M f;t to maximize …rm value V ( ~ f;t
) . Firm f discounts future …rm value by
C t
C t +1
. Another assumption is that, labor productivity in R&D grows proportional to the total number of goods in the industry I t
. This assumption keeps the number of R&D workers a constant in the general equilibrium, while number of goods is growing at constant rate. Denote the following problem
N f;t
=
N f;t
I t and M f;t
=
M f;t
I t
. Firm f
’ s manager solves max
N f;t
; M f;t
V (
Z f;t
I t
) =
Y t
Z t
I t
+
C t
C t +1
E [ V (
Z f;t +1
I t +1
)] s. t.
Z f;t +1
= Z f;t
+ Z
N f t
+ Z
M f t
W t
I t
( N f;t
+ M f;t
)
Z N f;t
Z f;t
=
A
N
N f;t
Z
1 f;t
Z f;t
+ en f;t +1
Z M f;t
Z
I;t
=
A
M
M f;t
Z
1
I;t
Z
I;t
+
In terms of market share Z f;t
, the problem is en f;t +1
N max f;t
; M f;t
V ( ~ f;t
) =
Y t
Z f;t
+
C t
C t +1
E [ V ( ~ f;t +1
)] W t
N f;t
M f;t s. t.
Z f;t +1
=
I t
I t +1
Z f;t
+
N f t
+
M f t
N f;t f;t
=
A
N
N f;t
Z
1 f;t
Z f;t
+ en f;t +1
M f;t
~
I;t
=
A
M
M f;t
Z
1
I;t
Z
I;t
+ em f;t +1
The …rst order conditions for intermediate …rm’s problem are:
W t
=
C t
I t
C t +1
I t +1
E [ V
0
( ~ f;t +1
)] A
N
N f;t
1 ~ 1 f;t
W t
=
C t
I t
C t +1
I t +1
E [ V
0
Z f;t +1
)] A
M
M f;t
1 ~ 1
I;t
(9)
(10)
(11)
(12)
15

V
0
( ~ f;t
) =
Y t
+
C t
I t
C t +1
I t +1
E [ V
0
Z f;t +1
)] h
1 + (1 ) A
N
N f;t
Z f;t i
(13)
The …rst (second) equation tells that the marginal cost of innovation (imitation) is equal to the expected marginal return from innovation (imitation). The third equation implies that one percent current market share’s marginal value is equal to marginal pro…t in current period plus the discounted contribution to next period’s market share innovation.
One educated guess for …rm value is a linear function V ( ~ f;t
) = v t
~ f;t
The …rst order conditions and the …rm value function can be rewritten as:
+ u t
.
N f;t
=
C t
I t
C t +1
I t +1
A
N v t +1
W t
1
1
Z f;t
(14) v t
=
Y
M t f;t
=
C t
I t
C t +1
I t +1
A
M
+
C t
I t
C t +1
I t +1 v t +1 h
1 + (1 v t +1
W t
1
1
Z
I;t
) A
N
N f;t f;t i
(15)
(16)
The input in innovation and input in imitation
N f;t
M f;t is proportional to …rm’s private knowledge is proportional to public knowledge expected market share expanded by innovation and imitation are
Z
I;t
Z f;t
.
The
;
E
N f t
= A
N
N f;t
Z
1 f;t
= A 1
N
1 C t
I t
C t +1
I t +1 v t +1
W t
1
Z f;t
(17)
E
M f t
= A
M f;t
Z
1
I;t
= A 1
M
1 C t
I t
C t +1
I t +1 v t +1
W t
1
Firm f ’s expected innovation rate r f;t and imitation rate l f;t are:
Z
I;t
(18) r f;t
E
Z f;t
N f t
= A 1
N
1
C
C t +1
I t
I t t +1 v t +1
W t
1
(19) l f;t
E
Z f;t
M f t
= A 1
M
1
C
C t +1
I t
I t t +1 v t +1
W t
1
I;t
Z f;t
(20)
The expected innovation rate r f;t is a constant, which is independent of …rm size; but the expected imitation rate l f;t is scale dependent. As …rm f grows bigger, its imitation rate drops because the public knowledge pool Z
I;t becomes smaller relative to its own size Z f;t
.
The dynamic process of …rm f ’s market share in (10) can be summarized by f;t +1
= R t +1
~ f;t
+ L f;t +1
(21)
16

where
R
L f;t +1 f;t +1
I t
I t +1
(
1 + A 1
N
1
I t
I t +1
(
A 1
M
1
C t
I t
C t +1
I t +1
C t
I t
C t +1
I t +1 v t +1
W t v t +1
W t
1
1
+ en f;t +1
)
~
I;t
+ em f;t +1
)
The constant part of the …rm value function is given by
(22)
(23) u t
=
C t
C t +1 u t +1
+ (1 )
1
A
M
C t
I t
C t +1
I t +1 v t +1
W t
1
1
Z
I;t
W t
(24) u t is the discounted future pro…t from all the imitated products.
words, u t is the public knowledge pool’s rent, shared by each …rm.
u t
In other must be smaller than or equal to the …xed entry cost F t
. Because potential entrants are
…rms with zero products, their expected return from entering is the rent from imitation u t
. When u t is greater than F t
, new entrants will keep entering and lower the average …rm size in the sector, which is by de…nition the size of public knowledge pool Z
I;t
.
Z
I;t will shrink until u t is equal to F t and no more entry happens.
Z
I t
Z
M
F
L = x it di +
0 0 f;t
N f;t df
C t
= Y t
= W t
(25)
(26)
The general equilibrium v
W
, g , r , l ,
C
1
1 is characterized by (27) to (31).
I
The number of goods I t grows at g = r + l .
r and l are the sectoral average innovation rate and imitation rate. The labor market clearance decides that the production amount of each goods x it is shrinking at speed g as the number of goods grows at the same speed. The production and consumption of …nal goods
Y g t
, C t
, wage rate W t
1
1
; and …rm value parameters v t
. I omit the subscript t and in the GE equations.
u t are growing at rate v
W
=
1
1+(1 ) r
1+ g
(27) g = r + l (28)
17

r E ( r f
) = A 1
N
1 v
1 + g W
1
(29) l E ( l f
) = A 1
M
1 v
1 + g W
1
(30)
F
> u =
(1
1
) l
1 + g v (31)
L =
I
1
C
1
+
1 + g r + l v
W
(32)
From (27), marginal …rm value v is increasing in market size W and decreasing in elasticity of substitution . It is decreasing in both average innovation rate r and average imitation rate l , because new goods squeeze market share of existing goods. From (28), (29), and (30), higher R&D productivity
A
N v t and A
M have two con‡icting e¤ects on growth rate
R&D inputs for given marginal …rm value
…rm value v
W share. Between A
M and A
N more than innovation e¢ ciency A
M v
W
; second, they reduces marginal
, imitation e¢ ciency A
M g : …rst, they rise the for given R&D inputs, because new goods dilute current market hurts marginal …rm value
W t market share Z f;t
, because current private knowledge or also contribute to innovation, but not to imitation in the next period. In (31), the average …rm size I
M
F in the sector is positively related with the entry cost F and negatively related with imitation’s share in gross growth rate l
1+ g and marginal …rm value v . In (32), …rms allocate
1
C
1 workers in
I production and
1+ g r + l I
M
F v
W workers in R&D.
f R f;t
; L f;t g in the market size dynamics (21) are independently and identically distributed over time and cross …rms. In the general equilibrium, E ( R
1+ r
1+ r + l
< 1 . By theorem 5 in Kesten (1973), the productivity distribution f;t
Z f;t follows a Pareto distribution with parameter , such that E ( R f;t
1 special case, when f ln ( R f;t
) g
) follows a normal distribution with variance there is an explicit solution for :
= 1 . In a
2 r
, gross growth rate
= 1 l
1+ r + l
2 ln f E ( R f;t +1
) g
2 r
1 +
2 l
1+ r + l
2 r and increasing with innovation’s risk volatility
(33)
For Pareto Distribution with parameter , 1 is the standard deviation of log scale …rm size, which is commonly used as …rm size heterogeneity measure in the literature. Firm size heterogeneity 1 is decreasing in imitation’s share in
2 r
. They
18

represent two o¤setting forces shaping …rm size distribution: the innovation risk volatility generates …rm size di¤erence, while imitation eliminates the di¤erence.
From (29) and (30), when = , imitation’s share in gross growth rate is increasing in the relative productivity of imitation to innovation
A
M
A
N
, and the absolute size of average imitation rate l .
l
1 + r + l
=
1 l
+ A
N
AM
1
1
1
+ 1
This model shows that the …rm size dynamic process is decided by …rm’s inputs in innovation and imitation according to the relative costs in both types of R&D. Given this size dynamics, …rm size within an industry will converge to a Pareto Distribution in the long run. When imitation is more productive relative to innovation, imitation plays a greater role in the gross growth rate.
Small …rms can easily catch up with big ones, therefore the sectoral …rm size heterogeneity is smaller. In the next subsection, I will test the model predictions using patent citation data.
Patent citation data allow me to tell the source of knowledge input in R&D.
According to the knowledge source, I can attribute the growth rate of patent stock to either innovation or imitation. In other words, I can construct the counterparts of r f;t
, l f;t
, and R f;t
, L f;t in the model, and test the model predictions. One prediction by (33) is that standard deviation of log scale …rm size
1 is decreasing in imitation’s share in gross growth rate prediction by ( ??
) and ( ??
) is that innovation rate r f;t l
1+ r + l
. The other is independent of …rm size; while imitation rate l f;t drops as …rms grow bigger. Additionally, surviving
…rm’s growth rate drops faster in sectors where imitation is more productive.
I use patent stock ps f;t
(the total number of patents granted to …rm f by time t ) as the measure of …rm size Z f;t
; use the growth rate of patent stock as the measure …rm growth rate g f;t
; and then attribute g f;t to innovation rate r f;t or imitation rate l f;t
, according to the percentage of outside citations in total number of citations. For example, if …rm f has 10 patents at the beginning of year t , it invented 5 new patents at year t . In these patent applications, it cites
50 existing patents, among which 30 are given to patents owned by other …rms.
The growth rate is rate ^ f;t g f;t
30
50
^ f;t
=
= 30%
5
10
= 50% . Then I split the growth rate into innovation
, and imitation rate l
^ f;t g f;t
50 30
50
= 20% .
In order to control for the quality of information in each citation, I adjust the pure citation counts by giving bigger weight to the citations with shorter time lag. Because shorter time lag might imply higher quality of the knowledge in‡ow in general. I discount every citation by the time lag between citing and cited patents. For example, if the time lag is n years, this citation is given a weight of (1 ) n
.
is the discount rate of intellectual property. The mean and variance of r f;t and l f;t varies little if I vary the discount rates from 0 to 0.5. I pick = 0 : 1 in the discount.
19

3.8.1
Prediction 1: 1 +
2 l
1+ r + l
2 r
(33)
For every sector, I estimate r and l by the average of ^ f;t by the standard deviation of ln (^ f;t
) ; and estimate
1
^ and l
^ f;t
. Estimate ^
2 r by the standard deviation of log ( ps f;t
) . Figure 19 and …gure 20 show that standard deviation of log ( ps f;t
) decreases as l
1+ r + l
^
2 r increases.
3.8.2
Prediction 2: r f;t
(20).
A 1
N
1 h
C t
C t +1
I
I t t +1 v t +1
W t i
1
(19); l f;t
A 1
M
1 h
C t
C t +1
I
I t t +1
The model predicts that innovation rate is independent of …rm size; while imitation rate is negatively correlated with relative …rm size. Since the imitation and innovation rate are non-negative in the patent citation database, I run the following Tobit regressions for every sector I and report b r
I;t and b l
I;t in .
r f;t
= b r
I;t ln ps f;t ps
I;t
+ const: v t +1
W t i
1 l
^ f;t
= b l
I;t ln ps f;t ps
I;t
+ const: ps
I;t is the number of patents in the industry I at time t . Figure 21 shows the result at year 1997, l
I;t are around -0.03 to -0.2 and signi…cantly di¤erent from zero for all sectors; but statistically signi…cant.
r
I;t are around -0.01, and many of then are not
Z
I;t
~ f;t
There are multiple sectors in the real economy. As is shown in …gure 7, many
…rms also develop products in multiple sectors. In the US Patent data, from
1962 to 2002 on average each organization applied patents in 4.7 categories per year. Bigger organizations tend to apply patents in more sectors, when
…rm’s patent stock increases by 10%, it tends to apply patents in 3.5% more industries. Interestingly, when …rms within one sector is considered separately, the size distribution follows Pareto distribution with di¤erent parameters, with the standard deviation of log scale patent stock
1 ranges from 0.29 to 3. When all the …rms in the economy are pooled together, the size distribution also follows a Pareto distribution with a parameter close to 1.68. This result mimics the evidence in Helpman, Melitz and Yeaple (2004) with …rm size measured by employee number. In their paper, every sector has a special Pareto …rm size distribution, while the aggregate economy also has a Pareto …rm size distribution with 1 . These phenomena encourage me to consider the …rm size dynamics in a macro scope greater than just one sector.
20

In the multi-sector model, a representative …rm f is free to develop products in multiple sectors. Firm f uses its private information to innovate, and uses public knowledge to imitate. Assume there are K sectors in the economy, I use a K dimension real vector Z f t sector at time t ; the k th to represent …rm element of Z f t f ’s number of products for each represents the number of products in the k th industry invented by …rm f . A key di¤erence from the one-sector model is that …rm f can apply its knowledge in sector i to invent new goods in any sector j , i; j 2 f 1 , 2 , ::: , K g . Firm f ’s manager choose innovation and imitation inputs in K sectors, so that the expected return from these 2 K types of R&D are equal to their costs.
The dynamics of …rm size in all K sectors can be summarized by
Z f t +1
= R t +1
Z f t
+ L t +1
(34)
R t +1 is a K K matrix, the ( j; i ) element R ji t +1 measures the productivity when
…rm f use its private knowledge in sector i to innovate new products in sector j .
L t +1
2 R K , the k th element L k t +1 measures the number of imitated products in sector k invented by …rm f . For …rm f ’s branch in sector k , the …rm size dynamics is
Z k f t +1
= R k 1 t +1
Z
1 f t
+ ::: + R kk t +1
Z k f t
+ ::: + R kK t +1
Z
K f t
+ L k t +1
(35)
As long as matrix E ( R ) satis…es the restrictions in Kesten(1973) (4.9), then for any vector x with j x j = 1 , there exist some such that x
0
Z follows Pareto distribution with parameter . When I study the …rm size distribution of the k th sector, I use x = (0 ; ::: 0 ; 1 ; 0 :::: 0) with the k th element equal to one. When
I investigate the size distribution of all …rms in the entire economy, I use x = p
1
K
(1 ; ::: 1 ; 1 ; 1 :::: 1) .
In some cases I solve explicitly for the Pareto distribution parameter . In one case, where matrix E ( R ) is diagonal, these K sectors are independent. The distribution of …rm size in each sector is decided purely by its intra-sector characteristics as it is in the one-sector model. The …rm size distribution parameter in the entire economy is the weighted average of each sector’s distribution parameter k
. In another case, when E ( R ) is positive de…nite, there exists a
0 real invertible matrix J such that J 0 E ( R ) J = and J J = E , where is a positive diagonal matrix and E is the K dimension unit matrix with 1 in the diagonal elements and 0 elsewhere. The above system can be transformed to
B f t +1
= R t +1
B f t
+ L t +1
(36) where
B f t
= J
0
Z f t
R t +1
= J
0
R t +1
J
L t +1
= J
0
L t +1
21

I want to call these K independent sectors in B f t dynamics in real sectors Z f t elements in basic sectors B f t
: basic sectors. The …rm size can be expressed as a linear combination of the
Z f t
= J B f t
This equation tells that each real industry combines knowledge from many basic scienti…c …elds, such as mathematics, physics, chemistry, biology, and so on. But di¤erent industry uses di¤erent portfolio of knowledge from basic scienti…c …elds, which is captured by matrix J . For example, drugs and medicines learns more from chemistry; while transportation equipment uses more physics knowledge.
Again, the …rm size distribution parameter in the entire real economy is the weighted average of basic sector’s distribution parameter by J .
k
, the weigh is given
This model has implications for antitrust law in innovative sectors. If the policy makers have concerns that a big portion of market share is taken by a few big …rms, they should encourage information di¤usion across …rms, especially across previously unconnected …rms. The aim is to reduce the relative cost of imitation to innovation, or reconstruct the …rms’network, so that this network is less clustered. Policies that enhance labor mobility, encourage R&D cooperation with previously unconnected …rms, unify industry standard, etc. would do the job.
In the open economy, policy makers should give the priority of opening to the sector with more abundant knowledge spillover. With the same trade cost reduction, they can expect more new exporters to enter export market and higher growth rate incremental in these sectors. I will write the open economy version of this model in another paper.
Using …rm level data of French, U.S. and Chilean manufacturing sectors, I show that sectoral …rm size distribution is stable with alternative …rm size proxies, over time and between countries. Using inter-…rm patent citation as an indicator of knowledge spillover, I …nd that knowledge di¤uses faster and further between
…rms in sectors with homogeneous …rm size. The di¤erent knowledge spillover e¢ ciency originates from the distinction in dynamic network formation process and citation network structure. The …rm citation network in homogeneous sectors are less clustered, …rms build more new connections with randomly picked
…rms and build less new connections with friends’ friend in the last period.
Therefore, in homogeneous sectors, information di¤uses evenly not only within cliques, but also across cliques within the sector.
Adding imitation to Klette and Kortum (2004), I build a one-sector model to endogenize …rm’s innovation and imitation inputs. Firm size dynamics is
22

characterized by …rm’s optimal innovation and imitation inputs and the risk volatilities in both types of R&D. When the imitation cost is relatively cheaper to the innovation cost, …rms invest more in imitation; imitation rate contributes a bigger share in growth rate, and vice versa. According to Kesten(1973), when innovation and imitation shocks are identically independently over time and across …rm, the …rm size distribution will converge to a Pareto distribution driven by the …rm size dynamics. The sectoral …rm size heterogeneity is inversely related to imitation’s share in gross growth rate, and positively related to the volatility of innovation’s risk. The one-sector model also implies that in sectors where imitation is easier, surviving …rm’s growth rate drops faster as …rms grow bigger. In the multi-sector model, I allow …rms to produce in multiple sectors.
In this case, …rm size is the summation of its branch sizes in every sectors. Firm size distribution in the entire economy also follows a Pareto Distribution.
Axtell, Robert L. 2001. "Zipf Distribution of U.S. Firm Sizes." Science 293:1818—
1820.
Borgatti, S.P., Everett, M.G. and Freeman, L.C. 2002. Ucinet for Windows:
Software for Social Network Analysis. Harvard, MA: Analytic Technologies.
Broda, Christian and Weinstein, David E. 2007, "Product Creation and Destruction: Evidence and Price Implications", NBER Working Paper No. 13041.
Broda, Christian and Weinstein, David E. 2006, "Globalization and the
Gains from Variety," Quarterly Journal of Economics Volume 121, Issue 2 -
May 2006
Cowan, Robin, and Jonard, Nicolas, Network structure and the di¤usion of knowledge, Journal of Economic Dynamics and Control. Volume 28, Issue 8,
June 2004, Pages 1557-1575
Duguet, Emmanuel and MacGarvie, Megan, 2005, "How Well Do Patent
Citations Measure Flows of Technology? Evidence from French Innovation Surveys" Economics of Innovation and New Technology, Volume 14, Number 5 /
July 2005
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94 NO. 1. 300-316
Gabaix, Xavier 1999, "Zipf’s Law for Cities: An Explanation," Quarterly
Journal of Economics, vol. 114, pp. 739-767.
Hall, B. H., A. B. Ja¤e, and M. Tratjenberg (2001). "The NBER Patent Citation Data File: Lessons, Insights and Methodological Tools." NBER Working
Paper 8498.
Ja¤e, Adam B.; Trajtenberg, Manuel and Michael S. Fogarty, "Knowledge
Spillovers and Patent Citations: Evidence from A Survey of Inventors". American Economic Review, Vol. 90, Papers and Proceedings, May 2000, pp. 215-218.
Ja¤e, B. Adam, 1986, "Technology Opportunity and Spillovers of R&D: Evidence from Firms’Patents, Pro…ts, and Market Value." The American Economic
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Review, Vol. 76, No. 5. (Dec. 1986), pp. 984-1001.
Jackson, Matthew and Rogers, Brian W., “Meeting Strangers and Friends of
Friends: How Random are Socially Generated Networks?” American Economic
Review, Vol. 97, No. 3, pp 890-915, June 2007.
Jackson, Matthew and Yariv, Leeat, “Di¤usion of Behavior and Equilibrium Properties in Network Games,” American Economic Review (Papers and
Proceedings), Vol 97, No. 2, pp 92-98, 2007
Kesten, H., “Random Di¤erence Equations and Renewal Theory for Products of Random Matrices,” Acta Mathematica, CXXXI (1973), 207–248.
Klette, Tor J. and Samuel Kortum, 2004, "Innovating Firms and Aggregate
Innovation,.Journal of Political Economy, CXII (2004), 986-1018
Luttmer, Erzo G. J. 2006. “Selection, Growth, and the size distribution of
Firms”, Quarterly Journal of Economics
Melitz, Marc J. 2003. “The Impact of Trade on Aggregate Industry Productivity and Intra-Industry Reallocations.” Econometrica, VOl. 71, No. 6.
1695-1725
Newman, Mark. 2003. “The Structure and Function of Complex Networks.”
SIAM Review, 45(2): 167–256.
Rossi-Hansberg, Esteban and Mark L.J. Wright, 2006, Establishment Size
Dynamics in the Aggregate Economy,. American Economic Review, forthcoming
Watts, D., Strogatz, S., 1998. Collective dynamics of small-world networks.
Letters to Nature 393, 440–442.
J.-P. Onnela, J. Saramaki, J. Hyvonen, G. Szabo, D. Lazer, K. Kaski, J.
Kertesz, and A.-L. Barabasi, 2007, Structure and tie strengths in mobile communication networks, Proceedings of the National Academy of Sciences 104:
7332-7336.
24

Year
1997
1998
1999
2000
2001
2002
2003
2004
2005 corr(sdlnl,sdyln) corr(sdlnl,sdlns) corr(sdlnl,sdlnva) corr(sdlns,sdlny) corr(sdlns,sdlnva) corr(sdlny,sdlnva)
0.964
0.954
0.951
0.924
0.959
0.956
0.924
0.911
0.92
0.961
0.956
0.954
0.926
0.959
0.962
0.922
0.926
0.933
0.97
0.961
0.967
0.926
0.952
0.915
0.945
0.937
0.937
0.998
0.998
0.997
0.998
0.998
0.997
0.997
0.915
0.996
Figure 1 Correlation between Heterogeneity Measures in French Data Set
0.964
0.94
0.918
0.944
0.933
0.911
0.931
0.943
0.92
0.963
0.937
0.912
0.942
0.929
0.906
0.928
0.888
0.907
25

1985
1986
1987
1988
1989
1990
1991
Year
1979
1980
1981
1982
1983
1984
1992
1993
1994
1995
1996 corr(sdlnl,sdlny) corr(sdlnl,sdlns) corr(sdlnl,sdlnva) corr(sdlns,sdlny) corr(sdlns,sdlnva) corr(sdlny,sdlnva)
0.8503
0.8486
0.8268
0.7744
0.8455
0.8139
0.9659
0.9076
0.9422
0.8989
0.9614
0.949
0.8543
0.8259
0.7007
0.7327
0.7712
N. A.
0.7853
0.8057
0.8407
0.8033
0.6233
N. A.
N. A.
0.6678
0.7383
0.8254
0.7933
0.7208
0.6433
0.7351
0.7341
N. A.
0.7249
0.7243
0.7965
0.7532
0.6186
N. A.
N. A.
0.5916
0.6122
0.7687
0.7768
0.7436
0.6938
0.7133
0.7534
N. A.
0.7454
0.7312
0.8173
0.779
0.5515
N. A.
N. A.
0.619
0.3697
0.715
0.9546
0.9353
0.9571
0.9684
0.9628
N. A.
0.9542
0.9293
0.9553
0.9675
0.9602
N. A.
N. A.
0.8591
0.8491
0.9328
Figure 2 Correlation between Heterogeneity Measures in Chilean Data Set
0.8919
0.9519
0.9142
0.9446
0.9418
N. A.
0.8965
0.8865
0.8838
0.8875
0.8958
N. A.
N. A.
0.8579
0.7765
0.8958
0.9131
0.9406
0.9473
0.977
0.9617
N. A.
0.9185
0.9216
0.9338
0.9328
0.9114
N. A.
N. A.
0.9233
0.7468
0.9103
26

Industry Name
Natural resources and mining
Construction
Durable goods manufacturing
Nondurable goods manufacturing
Wholesale trade
Retail trade
Transportation, warehousing and utilities
Information
Finance and insurance
Real estate and rental and leasing
Professional and business services
Educational services
Health care and social assistance
Arts, entertainment, and recreation
Accommodation and food services
Other services
Total
Separation
Rate (%)
36.4
66.4
31.7
33.3
29.9
53.9
36
29.3
29.2
38.4
50.9
22
29.8
74.9
74.6
39.5
Sd.
Dev. of ln(sales)
2.03
1.56
1.94
2.07
1.91
1.33
2.32
2.14
2.36
1.8
1.77
1.23
1.35
1.54
1.19
1.28
Figure 3 Labor Turnover Rate and Firm Size Heterogeneity in U. S. 1990s
27

Dependent Variable: ln(distance between citing and cited patents) ln(cited patent age) change co-work s. d. of log(patent stock)
R²
Number of Observations
Coefficient
(Robust Standard
Error)
0.74 (0.03)
1.91 (0.07)
1.29 (0.13)
-2.27 (0.51)
0.31
1976998
Figure 4 Distance Between Citing and Cited Patents
Dependent Variable: prob(cite an outside patent ) patent stock in the industry/patent stock in the citing firm ln(cited patent age) change co-work s. d. of log(patent stock) in the sector
Wald Chi²(17)
Number of Observations
Coefficient (Robust
Standard Error)
1.19e-06*** (7.47e-
08)
0.033*** (0.004)
0.026*** (0.003)
0.017*** (0.003)
-0.043** (0.020)
5982.07
4804055
Figure 5 Friendship Types in the Citation Network
Marginal e¤ects are reported. Observations are clustered by sector. Year dummies from 1990 to 2002.
28

Figure 6 Friendship Types in the Citation Network
29

Figure 7 Table 2 in Broda and Weinstein (2007)
30

3118
3149
3344
3334
3363
3345
3322
3114
3366
3251
3256
3241
0 1 sdlnl in 2002
Source: U. S. Economic Census 1997, 2002
2 sdlnl in 1997
3
45 degree line
Figure 8 Heterogeneity Measure in U. S. 1997 and 2005
4
31

3361
3211
3117
3161
3219
3379
3371
3399
3114
3312 3336
3311
3363
3364
3122
3391
1 2 sdlnl in 2005 sdlnl in 1997
3
45 degree line
Source: Amadues
Figure 9 Heterogeneity Measure in France 1997 and 2005
4
32

3901
0
3118
3852
3134
3842
3215
3512
3824
3320
3212
3117
3903
3131 3844
3511
3412
3240
3909
3231
3832
3825
3513
3710
3833
3419
3411
3841
3514
3902
3721
3530
1 sdlnl in 1996
Source: Chile's Instituto de Esadistica
2 sdlnl in 1979
3
45 degree line
3696
Figure 10 Heterogeneity Measure in Chile 1979 and 1996
3140
4
33

3361
3115
3335
3369
3113
3371
3219
3251
3252 3336
3231
3323
3211
3372
3279
3346
3272
3366
3326
3345
3391
3122
3364
3352
3311
0 1 sdlnl in France
2 sdlnl in US
3
45 degree line
Figure 11 Heterogeneity Measure in France and U. S. 1997
4
34

3256
3241
3114
3115
3372
3279
3251
3121
3333
3332
3323
3118
3336
3161
3254
3363
3345
3313
3341
3391
3352
0 1 sdlnl in France
2 sdlnl in US
3
45 degree line
Figure 12 Heterogeneity Measure in French and U. S. 2002
4
3361
35

0 10
L sector
20
Citation lag (Year)
H sector
Figure 13 Percentage of Outside Citations in U. S.
30
36

0 10
L sector
20
Citation lag (Year)
H sector
Figure 14 Percentage of Outside Citation in Japan
30
37

0 10
L Sectors
20
Citation Lag (Year)
H Sectors
Figure 15 Percentage of Outside Citation in France
30
38

0 10
L sector
20
Citation lag (Year)
H sector
30
Figure 16 Average Distance from Citing and Cited Inventors in U. S.
39

-4 -2 0 2 4 log(patent stock/industry mean patent stock)
Petroleum and natural gas extraction and refining (sd=1.82)
Paints, varnishes, lacquers, enamels, and allied products (sd=1.69)
Drugs and medicines (sd=1.31)
Source: NBER Patent Database
Figure 17 Surviving Firm’s Growth Rate and Relative Firm Size U. S.
40

-5 0 5 log(turnover/industry average turnover)
Transportation Equipment Manufacturing (sd=2.09)
Computer and Electronic Product (sd=1.69)
Fabricated Metal Product Manufacturing (sd=1.33)
Source: Amadeus French Manufacturing Firms 1999
Figure 18 Surviving Firm’s Growth Rate and Relative Firm Size France
41

7
6
15
8
12
23
53
35
42
54
9
43
27
30
49
1
21
19
25
26
47
11
46
56
20
39
14
24
50
40
31
.02
Source: NBER Patent Data
.04
38
32
.06
.08
.1
imitation's share in gross growth rate/innovation risk
Figure 19 Imitation and Firm Size heterogeneity U. S. 1990
42

15
27 23
6
42 54
12 43
35
11
9
117
30
21
40
49
38
56
39
32
0
Source: NBER Patent Data
.1
52
51
.2
.3
imitation's share in gross growth rate/innovation risk
Figure 20 Imitation and Firm Size Heterogeneity U. S. 2000
43

.5
1 br sdlnps bl
1.5
Figure 21 Scale Dependent Imitation Rate
2
44

51
46
15
1
52 32
1.5
31
49
56
24
38
40
23
25
20
39
19
50
16
11
2
1
36
55
26
54
13
6
42
43
9
12 53
47
2 sdlnps
2.5
8
Figure 22 Clustering Coe¢ cient 1 and Firm Size heterogeneity
3
7
45

15
1
46
11
52
51
40
27
13
35
43
23
32
1.5
31
49
38
25
30
2 1
55
54
6
9
42
20
21
16
39
26
50
36
12
53
47
2 sdlnps
2.5
8
Figure 23 Clustering Coe¢ cient 2 and Firm Size Heterogeneity
3
7
46

15
51
46
11
13
31
49
24
56
38
40
20
17
2
25
39
30
14
19
1
36
55
26
54
23
6
35
27
43
12
53
8
1
52 32
1.5
47
2 sdlnps
2.5
Figure 24 Clustering Coe¢ cient 3 and Firm Size Heterogeneity
3
7
47

49
47
31
54
53
1 1.5
25
40
20
29
30
21
36
14
16
11
2 55
42
13
35
23
27
6
9
43
1
2 sdlnps
2.5
Figure 25 ln(RF/FF) and Heterogeneity Measure
8
15
3
7
48