Spillovers, Absorptive Capacity and Agglomeration

Transcription

1 Spillovers, Absorptive Capacity and Agglomeration Sergey Lychagin Central European University Department of Economics December 18, 2013 Abstract I study knowledge spillovers in an industry where firms are mobile and heterogeneous in their ability to adopt knowledge (absorptive capacity). I develop a model of industry agglomeration in which an attraction force induced by spillovers is counteracted by a force created by local competition. I apply the model to the US software industry and obtain the following results: the data supports localized knowledge spillovers; firms that have higher absorptive capacity are sorted into more agglomerated counties; ignoring firm heterogeneity leads to biased estimates of gains from spillovers; spillovers play a modest role in explaining the geographic distribution of firms. This paper is based upon a chapter of my Penn State University Ph.D. thesis. I am grateful to Kala Krishna and Joris Pinkse for their invaluable advice and support. I also would like to thank Saroj Bhattarai, Stéphane Bonhomme, Edward Coulson, Jonathan Eaton, Miklós Koren, Mark Roberts, Andrés Rodríguez- Clare, Lily Samkharadze, Alexander Tarasov, Neil Wallace and Stephen Yeaple whose comments contributed to this paper. All remaining errors are mine. lychagins@ceu.hu

2 1 Introduction Knowledge spillovers lie at the heart of many economic theories. In models of endogenous growth, cross-firm spillovers are essential in creating increasing returns to scale (Romer (1986)). In theories of urban and regional economics, geographically localized spillovers are used to explain why economic activity tends to be densely concentrated in space (Glaeser (1999)). In Ricardian models of international trade (Eaton and Kortum (2001), to name just one example), the lack of perfect cross-country spillovers is instrumental in generating trade flows. In the development literature (for instance, Feenstra (1996)), localized spillovers are the source of persistent gaps in productivity across countries. In the above theories, it is crucial to know the scope of knowledge spillovers and the magnitude of their economic impact. Quantifying knowledge spillovers is a difficult task as they are almost never directly observed. A usual approach is to correlate each firm s knowledge-generating activity to the performance of its neighbors, assuming that the former causes the latter. However, this approach may be problematic if spillovers affect firms differentially, that is, if firms are heterogeneous in absorptive capacity. An advanced technology firm built in a rural area will not bring any benefits to its geographic neighbors, as they are likely to produce only basic varieties of the final good and not rely on the frontier technology. The same firm located in a megapolis with hundreds of advanced competitors who are eager to find and adopt latest inventions is likely to create a lot of positive externalities. To understand how a spillover from location A affects firms at location B one has to answer two questions: What is the spillover s impact on a firm at B, given this firm s absorptive capacity? What are the absorptive capacities of firms attracted to B? To create a framework for addressing these questions, I construct a structural model of location choice in the presence of R&D spillovers, assuming unobserved heterogeneity of firms in absorptive capacity. I demonstrate that, in equilibrium, firms that are more responsive to spillovers tend to be over-represented in agglomerated locations. I show that this spatial sorting pattern can be used to identify the model. Then, I apply the model to data on production and locations of firms in the US software industry. I find evidence that spillovers within this industry exist and are highly localized in space. I demonstrate that the spatial sorting of firms by absorptive capacity produces substantial differences in the economic impact of spillovers across geographic locations. A researcher who ignores endogeneity of firm locations will tend to overestimate productivity gains from spillovers in 1

3 remote areas and misinterpret the effect of sorting by absorptive capacity as evidence that spillovers decay with distance. I model location choice as a static two-period entry game. There is a fixed mass of firms, which simultaneously choose locations in the first period. In the second period, which can be thought of as consisting of several years, the firms produce varieties of a final good; the firms cannot relocate or exit during this time. The firms have innate differences in absorptive capacity and R&D stocks, which they are endowed with at the time of birth. R&D stocks generate spillovers that decay with distance, which is consistent with a large literature including Jaffe, Trajtenberg and Henderson (1993), Greenstone, Hornbeck and Moretti (2010), Lychagin et al. (2010), to name just a few examples. Locations differ exogenously in the costs of inputs. Firm-specific costs of production are also affected by spillovers and own R&D. In equilibrium, the location choice and hence the spatial distribution of firms is shaped by four forces. First, there is an agglomerative force induced by spillovers. Spillovers between any pair of firms decay with distance; by locating closer to each other, firms enjoy higher productivity gains from spillovers. Since firms vary in absorptive capacity, they respond differentially to this force: a firm who benefits more from spillovers is drawn more strongly to agglomerated regions, even if these regions are expensive and expose the firm to tough competition. Second, there is a counteracting centrifugal force induced by local competition in the final goods market. 1 This force is endogenous; it depends on the equilibrium distribution of firms in space. Third, locations vary in the level of natural advantages (such as a pleasant year-round climate) that attract firm owners irrespective of the equilibrium outcome. Finally, there is a dispersion force caused by idiosyncratic location preferences of firm owners and managers. Since the force induced by spillovers acts differentially on firms, it creates sorting. Identification of the model s main parameters relies on detecting the magnitude of this sorting pattern in the joint distribution of firm locations and absorptive capacities. While absorptive capacities are not directly observed, their distribution can be inferred from a firm-level panel dataset on the production occurring in the second period of the game. The temporal dimension of the data permits the estimation of each firm s absorptive capacity by correlating the variations in the firm s total factor productivity to the variations in the R&D stocks of its 1 One could modify the model in this paper by introducing similar centrifugal forces caused by crowding effects in the labor and capital markets. Thus the centrifugal force based on competition in the final goods market should be viewed as a proof of concept rather than the ultimate explanation why firms might be not willing to settle too close to each other. 2

4 geographic neighbors. Although these firm-level estimates are inaccurate in the short panel setting, they can be used to precisely identify the joint density of firm absorptive capacities and locations, if the number of firms in the sample grows to infinity. This density is then fitted to the predictions of the location choice model in order to identify the model s key parameters. In the empirical application, I focus on the US software industry. Software firms are highly agglomerated around Silicon Valley in California and Boston, Massachusetts. It is commonly believed that knowledge spillovers are partly responsible for this extreme agglomeration. There is a body of anecdotal evidence that spillovers play an important part in software development and that they are highly localized in space (e.g., Saxenian (1996) provides a number of supporting stories). There is also evidence that the software industry features significant firm heterogeneity in absorptive capacity (Matusik and Heely (2005)). The aim of my empirical application is twofold: first, to determine how the impact and the geographic scope of spillovers are affected by the firm heterogeneity, and second, to quantify an overall agglomerative force induced by spillovers. To put it in simple terms, I would like to find (1) how quickly spillovers decline with distance, (2) whether industry productivity is magnified by spillovers more strongly in high-r&d regions, (3) whether the spatial distribution of firms would be very different in the absence of spillovers. The estimation results suggest that spillovers in the software industry are spatially localized: if the receiving firm s absorptive capacity is kept fixed, the spillover s productivity effect declines by half at a distance of 59 kilometers from the spillover s origin. I also find that the pattern of spatial sorting by absorptive capacity is statistically significant. Spatial sorting visibly distorts the scope of spillovers. For instance, simulations of the model show that a spillover from San Jose, the center of Silicon Valley, has an impact in Alameda county almost two times larger than in the less populated Santa Cruz county, although both counties are approximately at the same distance from San Jose. Such difference in the effects of spillovers takes place because Alameda firms have higher absorptive capacity. By modeling knowledge spillovers and location choice in one setting, this paper brings together the literature on knowledge production function, which finds its roots in the work of Zvi Griliches, and the literature on firm location decisions. This paper also contributes to the emerging discussion in urban economics and economic geography of the consequences of the firm- and individual-level heterogeneity on the spatial distribution of economic activity. Finally, my empirical approach is not only applicable in the context of knowledge spillovers. It can be used to identify any other agglomerative force affecting firm locations, for example, 3

5 forces associated with labor market pooling, or input sharing. The literature on the knowledge production function relates each firm s productivity to the firm s own R&D and spillovers from other firms. The effect and the properties of spillovers are inferred by estimating a firm-level production function that includes R&D stocks of geographic neighbors in the firm s total factor productivity term. The most recent examples of papers that employ this technique include Orlando (2004), who studies spillovers within a broad US industry defined by SIC 35, and Lychagin et al. (2010), who estimate how quickly spillovers decline with distance from the origin. Another related paper, Henderson (2003), uses the number of same-industry neighbor firms as a knowledge spillover proxy. All these works find evidence that spillovers exist, have positive impact on productivity, and are geographically localized. They do not explore the issue of firm location endogeneity and assume that within own industry, firms are identical in their absorption of spillovers. My paper contributes to this literature by relaxing these two assumptions and demonstrating that spatial sorting by absorptive capacity seems to have strong impact on gains from spillovers in the US software industry. The literature on firm location decisions focuses on explaining the factors that drive industry agglomeration and dispersion. A common approach is to relate the probability of new entry in a region to the regional characteristics that are expected to attract or discourage entry. Along these lines, Head and Mayer (2004) find that European regions with more same-industry incumbent firms attract affiliates of Japanese multinationals more strongly, even after controlling for conventional factors, such as market potential, wages and taxes. Devereux, Griffith and Simpson (2007) document a similar pattern for new plants opened in the UK between 1986 and My results suggest that the above agglomeration effect is more nuanced: firms locating in industrial clusters and the ones attracted to sparsely populated areas by low taxes or other location incentives differ in absorptive capacity, even if they operate in the same industry. If applied blindly, location incentives may attract firms that do not benefit from knowledge spillovers. Instead of absorbing new ideas and adopting them in the region, the attracted firms will work in isolation from the rest of the world. This paper is also related to a new strand of literature on urban and regional economics that focuses on spatial sorting of heterogeneous firms and individuals across cities, and aims at distinguishing the effect of sorting from the benefits of agglomeration. For example, Baldwin and Okubo (2006) construct a model of reallocation that introduces costly migration into the Melitz framework and shows that the more productive firms are the first to leave their home regions to serve the biggest market. Combes et al. (2012) show that agglomeration 4

6 economies and firm selection have different effects on the distribution of firm productivity within cities, which helps to identify the relative contributions of these factors into the cross-city differences in average productivity. Behrens, Duranton and Robert-Nicoud (2012) study a model of location decisions by heterogeneous workers, which predicts that the more productive workers choose to be in the bigger cities. Arzaghi and Henderson (2008) find that advertising agencies in Manhattan are spatially sorted by size (size is interpreted as a proxy for productivity) at the city block level. The above theories focus on sorting by innate productivity; while I allow for this mechanism to operate in my model, my attention is concentrated on sorting by absorptive capacity. Bayer and Timmins (2007) propose an identification technique to separate the agglomeration effect of spillovers from the force of attraction induced by natural advantages. In the context of R&D spillovers, their estimator allows for firm heterogeneity in absorptive capacity; the estimator is based on the crosssectional discrete choice framework of Berry, Levinsohn and Pakes (1995). In contrast to Bayer and Timmins (2007), my identification technique relies on combining a cross sectional analysis of firm locations with a more direct evidence on absorptive capacity obtained from panel data on production and R&D. Finally, the empirical approach developed in my paper is not restricted to spillovers; it can be applied to infer the strength of any other agglomeration force. To date, the literature that aimed at decomposing the contributions of various forces into the spatial distribution of economic activity relied mostly on reduced form equations where observations represent industries. A usual approach is to relate some measure of geographic concentration in a given industry to a set of proxy variables that indicate the importance of different agglomeration forces. Examples of such work include Audretsch and Feldman (1996), Rosenthal and Strange (2001), Ellison, Glaeser and Kerr (2010). Although cross-industry evidence is valuable in providing a broad, economy-wide assessment of agglomeration forces at work, firm-level studies of narrowly defined industries may be better suited to control for institutional details and industry-specific natural advantages. The identification approach developed in this paper may be viewed as a step towards a general framework for such firm-level analysis. The next section of the paper lays out a model of location choice in the presence of spillovers that can be taken to the data. Section 3 links the model tightly to estimation and shows that its key elements are identified from firm-level data on locations and balance sheet items. Section 4 then proposes an estimation method. This method is applied to data on the US software industry in Section 5. Finally, Section 6 summarizes main findings. 5

7 2 A Model of Location Choice A model developed in this section describes location decisions of firms from a single narrowly defined industry. The narrow focus on a specific industry is needed to ensure that industryspecific agglomeration forces act on all firms in the same way. The only force that is allowed to affect firms differentially is the one caused by spillovers. In order to make the exposition as clear as possible, I also assume that each firm is exogenously endowed with a random R&D stock and an absorptive capacity, which cannot be changed by the firm. There are L locations (counties) indexed by either l or m; 2 l, m {1,..., L}. Counties are populated by immobile consumers with CES preferences defined on all available varieties of a final good. 3 Consumers in county m spend an exogenous amount E m on the final good, which captures the size of local demand. There is a unit mass of infinitesimally small firms indexed by i Ω that are free to choose their locations. Each firm produces one distinct variety of the final good; the varieties can be shipped to any county. When the firm is born, it has a stock of R&D (R i ). To simplify the analysis, R i is assumed to be exogenous. A part of this R&D stock may spill over to other firms, increasing their profitability. Firms differ in their innate ability to adopt and commercialize spillovers; this is captured by a firm specific parameter α i (absorptive capacity). 4 As shown later in this section, different locations provide different advantages to the firm: they may vary in the level of wages, the degree of local competition, or proximity to firms generating high spillovers. Aside from these factors, the firm s location preferences are also affected by an idiosyncratic component ε il. This component captures monetary payoffs (e.g., the firm may be able to secure location incentives from officials at certain places), as well as non-monetary benefits (e.g., a CEO of the firm may prefer the county where he was born) received by the firm at location l. Finally, the firm is characterized by an idiosyncratic productivity shock ϕ i. In total, firms are heterogeneous in L + 3 dimensions: R i, α i, ϕ i, and {ε il } L l=1. 2 The intention is to use l in the context of firm locations and m in the context of final good markets. The reader should be aware, though, that this convention is not being followed strictly. 3 One interpretation of the CES utility function is that it represents a population of heterogeneous consumers who can only buy one variety of the good (Anderson, de Palma and Thisse (1992)). Using CES preferences is a tractable way of modeling markets with differentiated products. 4 An influential strand of literature originating from Cohen and Levinthal (1989) suggests that the firm s R&D effort and absorptive capacity are endogenous and closely interrelated. By investing into R&D, the firm improves its awareness of useful knowledge available for adoption, hence increasing its absorptive capacity. Although in principle this mechanism could be incorporated in the model, it is shut down to keep the identification argument as simple as possible. 6

8 2.1 Timing Location choice and production are modeled as a Nash equilibrium in a static game with the following sequence of events: Period 0 A unit mass of firms is born. At the time of birth, each firm is endowed with a vector of characteristics: [α i, R i, ϕ i, ε i1,..., ε il ]. Period 1 Firms simultaneously choose locations. A firm is allowed to have only one location. Period 2 Given locations, firms set prices in each county, produce output, and ship it to the consumers. 2.2 Demand Each county is inhabited by consumers with CES preferences: ( σ/(σ 1) U m = Q (σ 1)/σ im di), σ > 1 i Ω where index i is used to denote a variety produced by firm i. Given that consumers in county m spend an exogenous amount E m on the final good, one can easily find the demand for each variety: Q im (p im ) = E m P m ( pim P m ) σ (1) The demand depends on P m, a location-specific price index defined in a standard way: 2.3 Supply ( P m = i Ω ) 1/(1 σ) p 1 σ im di There is a unit mass of infinitesimally small single-variety single-location firms. The firms profits are location specific; they are derived below from equilibrium conditions for the final goods market in each county. 7

9 Firms produce output using capital and labor in a constant returns to scale Cobb-Douglas technology with a Hicks-neutral total factor productivity (TFP) term A il : Q i = A il K β k i L β l i Total factor productivity depends on four components: own stock of R&D, R i, spillovers from other firms R&D received at location l, S l, firm- and location-specific productivity shocks, ϕ i and ψ l : A il = R βr i S α i l ϕ iψ l where the parameter α i captures the absorptive capacity of firm i. 5 Spillovers originate from stocks of R&D; their intensity declines with distance from their origin 6 : S l = L e λρ(l,m) R m, R m = m=1 i Ω l i =m R i di (2) where m indexes counties, l i is the location of firm i, ρ(, ) is the geographic distance. The constant returns to scale production technology implies that the marginal cost function is constant and has the following form: mc il = C l R βr i S α i l ϕ = iψ l R βr i c l S α i l ϕ i Apart from the productivity term in the denominator, the marginal cost depends on the price of a cost-minimizing input bundle, C l, which is assumed to be exogenous; c l = C l /ψ l captures overall cost advantages of location l ignoring spillovers. In addition to the cost of production, firms face a cost of transportation 7. The value of a 5 Firms in a single narrowly defined industry can differ in absorptive capacity for a number of reasons. For example, developing some products entails an inherent uncertainty in the type of knowledge needed to complete the development project and to tailor the product to the needs of consumers. Firms designing such products have to rely on the outside knowledge more frequently. Thus, they receive more benefits from being exposed to spillovers. Other determinants of absorptive capacity are closely studied in the management literature. One example studying specifically the US software industry is Matusik and Heely (2005). 6 The decay of spillovers with distance reflects the costs of communication. Although modern technologies make phone calls and electronic correspondence very cheap, some part of knowledge cannot be efficiently transmitted by means other than face-to-face contact. In addition, geographic proximity provides more opportunities for networking: new research ideas can be informally exchanged during an occasional lunch with colleagues from other firms. 7 In the software industry, transportation of the final product is virtually costless. At the same time, software products are often accompanied by support and services that involve interactions between customers 8

10 good produced in county l declines with distance to the county where the good is consumed. This is captured by the iceberg cost τ lm. Producer has to ship τ lm units of the good from the origin county l, in order to have one unit arrive to destination m. 2.4 Payoffs to firms Profit of firm i located in county l earned from sales at market m equals the price-cost markup multiplied by the quantity sold: Π ilm = max p ilm ( p ilm τ lm R βr i c l S α i l ϕ i ) Q im (p ilm ) The firm chooses its profit-maximizing price p ilm, taking the R&D spillovers and the price index as given, where p ilm = σ σ 1 τ lm R βr i c l S α i l ϕ i Substituting the price back to the profit equation yields an expression for the maximized value of profits: Π ilm = E mpm σ 1 σ = 1 σ ( σ 1 σ ( σ 1 σ R βr i S α i l c l R βr i ϕ i S α i l ϕ i ) σ 1 c l τ lm ) σ 1 ( Pm E m τ lm ) σ 1 By summing firm i s profits across all destinations, one obtains the total profit of the firm, given that the firm is located in county l: Π il = 1 σ ( σ 1 σ R βr i S α i l c l ϕ i ) σ 1 m (3) ( ) σ 1 Pm E m (4) τ lm and developers. Spatial proximity facilitates these interactions by reducing the costs of face-to-face contact and thus increasing the total value of the product for consumers. In my sample, 70% of firms are involved in activities other than software development; in particular, 41% of firms are active in computer-related services. The average revenues from these activities constitute respectively 22% and 10% of total revenues. Note, though, that results presented in this paper do not hinge on the assumption of non-zero transportation costs. 9

11 The price index in market m is found by integrating over the price set by firms: P m = σ σ 1 j Ω ( R β r j Sα j l j ϕ j c lj τ lj m ) σ 1 dj 1 σ 1 (5) Note that the R&D stock of firm j affects firm i s profit via two channels: spillovers and prices. First, R j affects firm i s profits positively by increasing spillovers term S l in equation (4). The magnitude of this effect depends on the firm specific absorptive capacity, α i, and thus varies across firms. Second, an increase in R j makes firm j and its neighbors more productive, which drives down the price indices in all counties. Firm j s R&D stock has a direct and an indirect effect on price. If R j is increased, firm j becomes more productive (direct effect). Part of R j spills over to other competitors of firm i making them more productive, too (indirect effect). Both effects put a downward pressure on the price indices in all markets and reduce profits of firm i. The magnitude of the combined effect is the same for all firms at a given location. It declines with distance from the location of firm j, due to increasing transport costs and the decay of spillovers with distance. 2.5 Location choice Before starting production and earning profits, each firm has to settle at some location. The firm s payoff from choosing location l equals a weighted sum of three components: the logarithm of profit, natural advantages of l (e.g., temperate climate or other amenities not directly related to profit) and the idiosyncratic preference shock, drawn at the time of the firm s birth: π il = b log Π il + na l + ε il 10

12 Parameter b captures the importance of profits in the firm s location decision relative to the idiosyncratic shocks. After substituting the expression for profit from (4), one obtains [ ( ) ] [ σ 1 1 σ 1 ( ) ] σ 1 π il = b log σ σ Rβr i ϕ i + b log c 1 σ Pm l E m + na l τ m lm }{{}}{{} f i a l + b(σ 1) α }{{} i log S l + ε il γ To simplify derivations, I assume that location preference shocks ε il have a type I extreme value distribution and are independent across firms and locations. The constant coefficient of the spillover term is denoted as γ. The firm and location-specific parts of the payoff function are collected to form two separate terms denoted f i and a l respectively: π il = f i + a l + γα i log S l + ε il Firm i chooses a location that maximizes the firm s payoff: l i = arg max(f i + a l + γα i log S l + ε il ) (6) l Location choice is driven by three forces. First, there are location-specific factors that are common to all firms, captured by the location fixed effect, a l. These factors include proximity to consumers, prices in the final-good markets, prices of local inputs, and natural advantages. Second, there are knowledge spillovers that the firm expects to receive at location l. Spillovers affect firms differentially: firms with higher values of α i are drawn more strongly to highspillover locations. Parameter γ determines the overall strength of this force. Finally, there is a random shock ε, which reflects the idiosyncratic part of the location preferences. The firm specific term f i does not affect location choice. 2.6 Equilibrium An equilibrium of the game is characterized by 1. Locations of all firms, {l i } i Ω 2. Pricing decisions of every firm in every market m, {p ilm } L l,m=1,i Ω, given an arbitrary own location l (this includes off-equilibrium locations) 11

13 Firms willing to be in city Asymmetric equilibria Symmetric equilibrium Firms in city 1 (a) Finding equilibria Share of firms in city Symmetric equilibrium Asymmetric equilibria Iceberg transportation cost, τ (b) The effect of transportation costs Figure 1: Multiple equilibria in the two-county model Local costs of inputs and consumer expenditures are exogenously given. In principle, markets for inputs could be explicitly modeled here in exactly the same way as the final goods market. However, I choose not to do so to keep the model simple. In equilibrium, each firm chooses the location and the prices that maximize the firm s payoffs, given actions of other firms. This defines a best response mapping from the set of location and pricing decisions of all firms into itself. An equilibrium, by definition, is a fixed point of this best response mapping. The equilibrium is guaranteed to exist; the proof using the Brouwer fixed point theorem can be found in the Appendix. However, it is not always unique. Uniqueness tends to occur if firms are repelled from each other strongly enough. For example, Seim (2006) shows that in a special case of her spatial competition model the equilibrium is unique. However, if the dispersion force induced by competition becomes dominated by the agglomeration force induced by spillovers, multiple equilibria may arise. To illustrate, I set up and solve a simple version of the model with two symmetric counties, assuming that all firms have the same absorptive capacities and R&D stocks (α i = 1, R i = 1). Given τ, the iceberg cost parameter, all equilibria in the game are found as follows. Let f l (1) be a share of firms located in county 1. Since R&D stocks are deterministic and uniform, one can find the aggregate R&D stock and the spillovers for both counties, R l and S l. The spatial distribution of firms, (f l (1), f l (2)), and spillovers, (S 1, S 2 ), determine price indices in equation (5), which are used to obtain location fixed effects (a 1, a 2 ). Given a l and S l, one can solve the location choice problem (6) and find a share of firms that are willing to be located in county 1. This share is depicted in Figure 1a as a solid curve. In equilibrium, this share should be equal to f l (1); that is, the solid curve should intersect the 45 degree line. 12

14 The equilibria are found for a number of values of the iceberg cost parameter. The results are depicted in Figure 1b. When the transportation cost is high, competition in the final goods market is highly localized. Hence, firms try to avoid locating in counties with many competitors, which causes the firm population to spread evenly across the two locations. As the transportation cost falls, it is no longer possible to escape competition in less populated counties. The benefits of receiving more spillovers from geographic neighbors outweigh the negative effect of competition on profits. This gives rise to equilibria with asymmetric distributions of firms across counties. The idiosyncratic shock in the location preferences ensures that every county is always populated by a non-zero share of firms. 2.7 The sorting property All equilibria of the model have one important property that is used later in the empirical exercise to test whether the model s implications are consistent with data. Firms are heterogeneous in the benefits they draw from spillovers. Consequently, firms with high absorptive capacity, α i, are attracted to high-spillover locations more strongly than firms with low α i. This creates spatial sorting of firms by their absorptive capacity. Proposition 1. Consider any equilibrium of the model. Let l 1 and l 2 be two locations such that S l2 > S l1 in this equilibrium. In equilibrium, the firms are spatially sorted by α. In fact, the distribution of firms absorptive capacities at location l 2 stochastically dominates the distribution at l 1 : F α l (α, l 2 ) < F α l (α, l 1 ) Proof. See Appendix. In the empirical exercise, I find evidence that the above sorting pattern does exist in the US software industry: firms with high absorptive capacities tend to be overrepresented in high-spillover counties. 3 Identification This section shows how to identify the key elements of the model that affect location choice and the geographic scope of spillovers. It contains the two main innovations of the paper. First, I show that firm-level data on production can be used to identify the pattern of firm heterogeneity in absorptive capacity and find how this pattern varies across firms in 13

15 geographic space. Unlike the related studies in the agglomeration literature that use proxies, this paper infers absorptive capacity of individual firms directly from fluctuations in firm productivity attributed to spillovers. Second, I demonstrate that the spatial pattern of absorptive capacities found above can be used to identify the location choice problem. Identification does not require the econometrician to observe all factors affecting firm location, such as local wages or consumer demand. 3.1 Production function In order to find how firm productivity is affected by spillovers, I use panel data on output, factor inputs and R&D at the firm level together with cross-sectional data of firm locations. The main objects of interest are the joint distribution of firms absorptive capacities and locations, f α,l (α i, l i ), and the spillovers distance decay rate, λ. Identification relies on the temporal variation in the production function components. To introduce the time dimension, the production period is subdivided into T years, indexed by t = 1,..., T. Available data provide a snapshot of every firm s operations in each of these years. Output of firm i in year t is denoted as Q it. structure as before: Q it = A it K β k it Lβ l it, The production function has the same A it = R βr it Sα i l i t ϕ it Time variation in output comes from a number of sources, working together as follows: 1. The time path of R it is determined exogenously. It is drawn at the time of the firm s birth from some distribution. 2. Variation in spillovers, S li t, is caused by variation in R&D stocks of individual firms. 3. The productivity shock, ϕ it, is a catch-all term that accounts for productivity variation not caused by own R&D or spillovers. It is further subdivided into three parts: ϕ it = U i V t W it, where U i captures firm i s innate efficiency and productivity advantages of location l i common to all firms. V t is a common time-varying component and W it is a residual productivity shock. 4. Capital and labor are adjusted by the firm each year, in response to time variation in factor prices, consumer expenditures and own productivity. 14

16 After taking logarithms, the production function becomes linear: q it = β l l it + β k k it + β r r it + α i s li t + u i + v t + w it (7) Lower-case letters l it, k it, r it and s li t denote the logarithms of labor, capital, own R&D and spillovers. Assumption 1. The error term in the production function equation, w it, and the inputs satisfy the strict exogeneity assumption. This assumption rules out a possibility that w it contains an omitted variable not observed by the econometrician, but observed by the firm at the time when capital and labor are selected. Examples of such variables may include the quality of labor, or intangible assets other than the R&D stock. Equation (7) is used to identify parameters β and λ common to all firms 8, as well as the joint distribution of firm locations and absorptive capacities, f α,l (α i, l i ) Identifying the common parameters To identify common parameters β and λ, I use a standard technique from the literature on random-coefficient panel data models (e.g. Arellano and Bonhomme (2012), Wooldridge (2005)). This technique is equivalent to estimating a linear regression of q it on inputs and year dummies after controlling for firm fixed effects and their interactions with spillovers. In order to derive the identifying condition for β and λ more formally, it is convenient to rewrite equation (7) in the vector form: q i =z i β + x i (λ) [ α i u i ] + w i, where (8) x it (λ) = [s li t(λ), 1] z it = [k it, l it, r it, δ 1t,..., δ T t ] δ 1,..., δ T year dummies I define a transformation matrix M i as M i (λ) = I T x i (x ix i ) 1 x i 8 Recall, that the spillover term is parametrized by λ (see equation (2)) 15

17 Multiplying a vector y by this matrix projects y on a linear space orthogonal to x, or, in other terms, produces an OLS residual from the regression of y it on spillovers s li t for every firm i. It is easy to see from the definition of M i that M i x i = 0. Multiplying both parts of equation (8) by M i removes the term with firm-specific coefficients: M i q i = M i z i β + M i w i (9) The above equation relates the variation in output unexplained by spillovers and firm fixed effects to the unexplained variation in inputs. The error term in this equation picks up the residual productivity variation that is not explained by either inputs, spillovers, year or firm fixed effects. Strict exogeneity of the error term, ensured by Assumption 1, implies a moment condition that I use to identify β and λ: E[M i (λ, l i )(q i z i β) z i, l i ] = 0 (10) Intuitively, this condition implies that firm innate productivity, spillovers and absorptive capacity account for all spatial variation in the residual productivity. After one controls for these variables, as I do in equation 9, firms cannot be systematically different in productivity across locations in any time period. By adjusting λ, one can change the spatial distribution of productivity explained by spillovers and the one left in the residual. The true value of λ is identified by minimizing systematic differences in the residual productivity across locations. If one knows λ, identification of β is trivial: β is the vector of coefficients from the linear regression (9); the error term in this regression is mean-independent of the explanatory variables. In a linear regression, the mean-independence condition identifies the coefficients Identifying the absorptive capacities Knowing common parameters β and λ, one can use equation (8) to obtain information about firm specific absorptive capacities, α i. Identification of α i relies on the dataset s time dimension. In a short panel setting, which is implicitly assumed throughout the paper, it is only feasible to obtain a noisy estimate of α i. To examine this issue more closely, assume that β and λ are known and consider a naive least squares estimator for α i and u i : [ ] α i = (x ix i ) 1 x i(q i z i β), (11) û i 16

18 The estimate of α i is contaminated by a finite error term, ν i, which does not vanish as the number of firms in the sample increases to infinity, while the number of years stays fixed: α i = α i + ν i = α i + T t=1 (s l i t s li )(w it w i ) T t=1 (s l i t s li ) 2 (12) Note, that the magnitude of ν i is inversely proportional to the variance of spillovers received by firm i. If this variance is small, α i is a very noisy approximation of α i. Even though the absorptive capacities cannot be identified separately for each firm, equation (8) still provides enough information to identify the distribution of absorptive capacities. Arellano and Bonhomme (2012) show that under the following independence assumption one should be able to identify the distribution of absorptive capacities for each location, f α l (α i, l i ). Assumption 2. The vector of idiosyncratic productivity shocks, w i, and the absorptive capacity of firm i, α i, are independent given z i, R, and l i. This assumption would fail to hold if, for example, low absorptive capacity is determined by a rigid organizational structure of the firm that at the same time isolates the firm from high productivity shocks. This will create a negative dependence between α i and the covariance matrix of w i, which violates the latter assumption. Intuitively, since w i and α i are conditionally independent, the error term ν i in equation (12) is also conditionally independent of α i. Hence, if the distribution of w i is known or can be somehow identified, one can find the conditional density of ν i and separate it from the conditional density of α i using a non-parametric deconvolution. Arellano and Bonhomme (2012) provide a detailed treatment of this argument. 9 Knowing f α l (α i, l i ), one can obtain the joint density of firm absorptive capacities and locations by taking a product of f α l (α i, l i ) and the share of firms at location l i, f l (l i ). Thus, the joint density f α,l (α i, l i ) is identified. 9 Although this argument fails to identify f α l (α i, l i ), if Assumption 2 is violated, one can still identify correctly the first moment of this distribution under a much milder condition that E[w it l i ] = 0. To see this, note that equation (12) implies E[α i l i ] = E[ α i l i ], irrespective of whether or not α i and w i are independent. Therefore, if one is unwilling to agree with Assumption 2, the prediction of the model that firms are spatially sorted by absorptive capacity can still be tested using the location-specific average of α i as an approximation for E[α i l i ] and comparing these averages for high- and low-spillover counties. 17

19 3.2 Location choice problem The spatial pattern of the firms absorptive capacities found above is used to identify the parameters that shape the geographic distribution of firms in equation (6). These parameters include location fixed effects {a l } L l=1, which capture forces induced by spatial competition, factor prices and natural advantages, and the attraction parameter γ, which reflects the overall influence of spillovers on location choice. proceeds in two steps. First, I show that γ is identified from the pattern of spatial sorting by absorptive capacity. Then, I demonstrate that one can identify {a l } L l=1 given γ Identifying spillover-induced attraction force To find γ, the location choice problem (6) is solved conditional on absorptive capacity, α. Denote as f l α the share of firms choosing location l conditional on their absorptive capacity. Due to the multinomial logit assumption, f l α is found from the location choice problem in a closed form: f l α (l, α) = Let l 1 and l 2 be two locations such that s l2 exp(a l + γαs l ) m exp(a m + γαs m ) (13) s l1 ; let α 1 and α 2 be arbitrary values of absorptive capacity. The latter equation can be used to obtain a relationship between shares of firms with absorptive capacities α 1 and α 2 at locations l 1 and l 2 : This equation can be solved for γ: f l α (l 1, α 1 ) f l α (l 2, α 1 ) = exp [γ(α 1 α 2 )(s l1 s l2 )] f l α(l 1, α 2 ) f l α (l 2, α 2 ) [ ] fl α (l 1, α 1 ) f l α (l 2, α 2 ) γ = log [(α 1 α 2 )(s l1 s l2 )] 1 f l α (l 2, α 1 ) f l α (l 1, α 2 ) Note that the conditional distribution f l α is defined as f l α (l, α) = f α,l (α, l)/f α (α), which implies γ = [ log f α,l(α 1, l 1 ) f α,l (α 2, l 1 ) log f ] α,l(α 1, l 2 ) [(α 1 α 2 )(s l1 s l2 )] 1 (14) f α,l (α 2, l 2 ) Intuitively, the attraction parameter γ relates the variation of firms absorptive capacities across locations to the variation in spillovers these locations offer. The equation is well defined if all four combinations of α and l lie in the support of the joint density f α,l (, ). The right hand side of this equation is identified from the production function for any given 18

20 pair of α 1 and α 2. Hence, γ is also identified Identifying location fixed effects. Given γ, one can identify the location fixed effects from equation (6). Consider the same pair of locations l 1 and l 2 as above. Equation (13) and the definition of conditional probability can be used to obtain the joint density f α,l (, ) at (α, l 1 ): ( ) γα f α,l (α, l 1 ) = e a l 1 a Sl1 l2 f α,l(α, l 2 ) S l2 This equation can then be integrated over α and used to find the fixed effect of county l 2, a l2 : [ ( ) γα Sl1 a l2 = a l1 + log f α,l(α, l 2 )dα] S l2 [ log ] f α,l (α, l 1 )dα This immediately implies that the location fixed effect a l2 location l 2 up to an additive constant. is identified for an arbitrary 4 Estimation This section describes an estimation procedure used in the empirical application. The procedure consists of two main steps: 1. The production function (7) is estimated in a GMM framework using the moment condition given in (10). This yields common production function parameters, β and λ, and naive OLS estimates of firm-specific absorptive capacities, { α i } i Ω. 2. The location choice problem (6) is solved and estimated by a method of maximum likelihood using λ and { α i } i Θ obtained above. Estimated in this step are the attraction parameter, γ, and the location fixed effects, {a l } L l=1. The details on these two steps are outlined in what follows. 4.1 The production function Using GMM to estimate equation (7) is relatively straightforward; the asymptotic properties of this estimator are studied in a detail by Arellano and Bonhomme (2012). 19

21 However, there is one important issue that arises due to data limitations. Instead of using a direct measure of output, this paper has to rely on revenue data. Replacing output with revenue in equation (7) makes the estimation results more difficult to interpret. This issue is generic for a big part of literature studying firm productivity. Firm-level prices or physical output are almost never available in a typical dataset. The consequences of using revenue in place of output are discussed at length in Griliches and Mairesse (1995) and Katayama, Lu and Tybout (2009). In the current setting, replacing output with revenue may bias the estimates of β and α by a factor of (σ 1)/σ. As demonstrated by Griliches and Mairesse (1995), this may result in spuriously observed decreasing returns to scale even if the underlying technology exhibits constant returns, i.e. (β l + β k )(σ 1)/σ < 1, although β l + β k = Location choice problem The production function provides an estimate of the spillover decay parameter, λ, and naive OLS estimates of the firm-specific absorptive capacities, α i. Both λ and α i are used to estimate the location choice problem (6): knowing λ, one can find the spillover proxy s l ( λ) for every location l, while the values of α i serve as noisy measurements of the true absorptive capacities {α i } i. The estimation of parameters specific to location choice is carried out via a maximum likelihood procedure. The likelihood function is obtained from the model s predictions on the spatial distribution of firms stated in (13). Equation (13), though, cannot be used directly; it makes predictions conditional on true absorptive capacities, α i, which are neither observed nor identified. Hence, it has to be transformed to find the model s predictions conditional on α i, the naive OLS estimate of α i. The likelihood function is defined as a joint density of firm locations and α i. Recall that α i is a noisy estimate of α i : α i = α i + ν i. Consequently, the density of α i and l i is found by integrating out ν i from the joint density of α i, l i and ν i and replacing α i with α i ν i : f α,l ( α i, l i ) = f α,l,ν ( α i ν i, l i, ν i )dν i. In order to make use of the model s predictions on firm location decisions, I transform 20

22 the equation above so that the likelihood function is expressed in terms of f l α (l i, α i ): f α,l ( α i, l i ) = f l α (l i, α i ν i )f ν l (ν i, l i )f α ( α i ν i )dν i (15) In this transformation, I use the fact that by Assumption 2, ν i and α i are independent conditional on location. 10 The likelihood function is fully defined once one specifies functional forms for all three densities under the integral sign in equation (15). The first one, f l α, is found from equation (13). Functional forms for the other two densities follow from the next two assumptions. Assumption 3. Idiosyncratic productivity shocks w it mean and an unknown variance σ 2 w. have normal distribution with zero Together with the definition of ν i in (12), this assumption implies that f ν l is a normal density with the zero mean and a location-specific variance. Assumption 4. The marginal distribution of the firms absorptive capacities is normal, with an unknown mean µ α, and an unknown variance σα Taken together, these assumptions ensure that the likelihood function can be computed from equation (15), and then used in a maximum likelihood procedure to estimate parameters specific to location choice, as well as µ α and σα. 2 The variance of the production function error, σw, 2 is estimated separately using residuals from the Arellano-Bonhomme equation. Appendix 7.4 provides finer details on the estimator, its asymptotic distribution and implementation. 5 Data and Results 5.1 Data description The dataset used in the empirical exercise is a subsample of the Standard and Poor s Compustat, focusing on a single 4-digit industry. The sample includes domestic firms whose primary activity is in the development of software products (SIC code 7372, NAICS 4-digit 10 Indeed, according to (12), ν i is a linear function of the components of w i, the error term from the production function, with location-specific coefficients. By Assumption 2, w i and α i are independent conditional on l i. Hence ν i and α i are also independent. 11 In principle, Assumption 3 (together with the ones made earlier) is enough to identify and estimate f α non-parametrically. However, to make the estimation procedure more straightforward, f α is parametrized as well. 21

23 code 5112). The type of primary activity is inferred for each firm from the main Compustat file, the Segments data, and, whenever the first two sources are conflicting or silent, from the companies 10-K reports. Sometimes a firm may switch or slowly drift away from software development to another product or service. In such cases, those time periods are excluded from the firm s data. The choice of industry for the empirical application was dictated by three requirements: high R&D, young age, and large sample size. First, R&D should play an important role in the industry s production process; judging by the ratio of R&D expenditures to sales, software development is one of the most R&D intensive industries in the U.S. economy. As commonly assumed in the literature, R&D to sales ratio reflects the overall importance of spillovers in the industry (Audretsch and Feldman (1996), Rosenthal and Strange (2001)). Second, the software industry is relatively young. In an old industry locations of existing firms may reflect distant history rather than current economic reality. The U.S. software industry as we know it emerged in s. Since these years, the main geographic centers of activity in this industry have not changed much. The forces that attracted firms to Silicon Valley in early years of this industry seem to stay potent over time. Finally, software development is one of the most populous 4-digit industries in the sample of Compustat firms. The requirement that the sample size should be large enough ruled out such candidates as biotechnology and semiconductors. The sample of Compustat firms is by no means representative of the entire industry. Table 1 compares the sample used in this paper to the total universe of software companies in terms of firm size. Compustat firms tend to be an order of magnitude larger both in sales and employment. Table 1 also shows that the total R&D expenditures of the Compustat firms are very close to the total national R&D reported in the NSF Survey. Accounting methods used by the NSF are known to produce conservative figures 12. However, even if the NSF figure is 50% lower than the actual expenditures, the Compustat data still captures a decent share of a research activity in the industry. Summary statistics for all years of the data are presented in Table 2. The stock of R&D is constructed for each firm from its annual R&D expenditures, XR it, 12 For instance, this fact is documented and investigated for the pharmaceutical industry (Congressional Budget Office (2006)). Independent estimates of R&D in this industry are twice as large as the NSF numbers for certain years. 22