The Emergence of Coagglomeration

The Emergence of Coagglomeration Abstract This article uses an agent-based model of intercity firm location to explore the industrial composition of cities. Starting from a random allocation of firms across cities, firms relocate in pursuit of greater profit. There are several key results. First, there is a positive and nonlinear relationship between the strength of inter-industry external economies and coagglomeration, a result that supports using coagglomeration to study the microfoundations of agglomeration economies and to determine the boundaries of industry clusters. Second, the equilibrium level of coagglomeration is less than the efficient level. Third, history matters in the sense that a legacy of homogeneous or heterogeneous cities tilts the economy in favor of the historical pattern. Fourth, an increase in firm size increases coagglomeration. Fifth, an increase in relocation cost increases coagglomeration. 1. Introduction This article uses an agent-based model of intercity firm location to explore the industrial composition of cities. Helsley and Strange (2014) show that the coordination problems inherent in location choice generate multiple equilibria. This article employs agent-based modeling to refine the equilibrium set. Firms experience both intra-industry external economies and inter-industry external economies. Starting from a random allocation of firms across cities, a firm relocates to the city that generates the highest profit, and firms continue to relocate until no single firm can benefit from moving. The result is emergence, the transformation of an initial random pattern into a regular pattern. The computations generate five key results. First, there is a positive and nonlinear relationship between the strength of inter-industry external economies and coagglomeration.Second, the equilibrium level of coagglomeration is generally less than the efficient level. Since inter-industry external economies are weaker than intra-industry external economies, the efficient allocation requires minority status for some firms, and minority firms have an incentive to unilaterally deviate from the efficient allocation by relocating to cities where they are in the majority. Third, history matters: a legacy of heterogeneous or homogeneous cities tilts the economy in favor of the historical pattern. Fourth, an increase in firm size increases coagglomeration because it weakens intra-industry external economies relative to inter-industry external economies. Fifth, an increase in relocation cost increases coagglomeration because the legacy of the initial random allocation of firms across cities persists to a greater extent. These results relate to a number of recent papers that explore the theory of coagglomeration. In Ellison and Glaeser (1997) and Ellison et al. (2010), a discrete-choice model incorporates ‘all-or-nothing’ external economies. A firm that fails to collocate with industries in its partition earns an infinitely negative profit, so the collocation of industries means that there must be spillovers between the industries.1Helsley and Strange (2014) show that this result does not extend to weaker specifications of external economies. Specifically, the equilibrium collocation of industries need not mean that they are ideal partners for each other, or even that they benefit from each other at all. This article’s agent based modeling serves to refine the equilibrium set, establishing that that there is likely to be more coagglomeration of industries that are more strongly linked. This, in turn, provides support for empirical research using observed coagglomeration to learn about the microfoundations of agglomeration economies.2 In this approach, for example, one would see evidence of input sharing in the collocation of industries that are part of the same supply chain. Ellison et al. (2010) use US data to show that proxies for all three of Marshall’s forces are robustly related to the coagglomeration of industries. Faggio et al. (2016) reach similar conclusions about Marshall’s microfoundations using UK data and also present evidence consistent with a positive role of entrepreneurship, innovation and industrial organization in the generation and reception of external economies in production. Coagglomeration is also the focus of a number of other recent papers, including Kolko (2010), Jacobs et al. (2013), Gabe and Abel (2016), Jofre-Monseny et al. (2011), Behrens and Guillain (2016) and the recent review by Behrens (2016). This article supports the key hypothesis behind this line of empirical research, that industries that frequently coagglomerate must be benefitting from each other. The positive relationship between inter-industry external economies and coagglomeration also has implications for the literature on clusters. The key issue is how clusters are defined. Porter (2003, 562) defines clusters as ‘geographic concentrations of linked industries’. The recent paper by Delgado et al. (2016) significantly improves our understanding of the nature of clusters by developing algorithms that create regionally comparable cluster definitions (i.e. definitions that are not specific to particular regions or cities). These algorithms are based on collocation and industry links. This article’s refinement of the equilibrium set provides support for this approach as well. The remainder of the article is organized as follows. Section 2 describes the model and presents some theoretical examples of multiple equilibria. Section 3 outlines the article’s agent-based approach. Section 4 presents the baseline results, focusing on the positive and nonlinear relationship between the strength of inter-industry external economies and coagglomeration. Section 5 presents the other key results. In Section 6, we run the model with different parameter values and functional forms, and show that the results are robust to changes in specification and parameterization. Section 7 concludes. 2. A System of Cities Model 2.1. Fundamentals This section will specify a simple model of firm location in a system of cities. The model will be used to establish the existence of multiple equilibria in city composition in a general setting. It will also be the basis for the article’s agent-based approach to equilibrium selection. The literature on agglomeration and systems of cities begins with Henderson (1974). This literature focuses on how the tradeoffs between agglomerative and dispersive forces interact to determine city size. For the most part, this literature does not address the issue of city composition, except for the polar cases of completely specialized and completely diverse cities. Behrens et al. (2014) consider the vertical composition of cities, where workers are differentiated by their human capital. The self-selection conditions restrict the set of equilibria, leading to a hierarchy of cities according to worker talent. As noted previously, Helsley and Strange (2014) consider horizontal composition, a situation where individual migration leads to multiple equilibrium. They show that coordinated migration through developers or firms can refine the set of equilibria. Our model makes use of an agent-based model to further refine the set of equilibria, generating predictions about industrial composition. Consider a region that contains J cities, indexed j ∈ {1,2,…,J}. N workers inhabit these cities. There are I types of worker, indexed by i ∈ {1,2,…,I}. A firm in industry i hires only type-I workers. The composition of city j in this situation is given by the vector nj = (n1j,n2j,…,nIj), while the population of city j is denoted Nj = ∑inij. The workers are employed by firms. The output of an industry-i firm in city-j is denoted by   qij=θgi(nj)li, (II.1) where θ is a positive constant and li is the number of type-i workers employed by the firm. We take this to be exogenous, and we explore the implications of firm size below. The function gi(-) captures intra-industry (the productivity effect of employment in the same industry, i.e. localization economies) and inter-industry external economies (the productivity effect of employment in related industries). Specifically, we assume that gi(-) is continuously differentiable and strictly concave, and that gi(-) has non-negative first partials. The key terms are ∂gi/∂nk, which capture the link between industry i and industry k. We will impose further structure later. To parallel standard competitive models with single-industry cities, we assume that all industries produce the numeraire commodity. We also suppose that the only cost incurred by the firm is labor. We suppose that labor costs increase with the total population of a city, as captured by the increasing function w(Nj). This can, of course, be given microfoundations through the monocentric model. A Nash equilibrium in locations requires that no firm of a given type be able to improve its profit by moving to another city. The profit of a firm in industry i in city j is   πij(nj)=θgi(nj)li−w(Nj)li. (II.2) We assume that the number of firms, like the number of workers, is fixed. See Helsley and Strange (2014) for a treatment of this issue. Let πij+i (nj) denote the profit earned by a type-i firm in city j when the population of the types is given by the vector nj plus liadditional type-i workers (one firm). In this situation, a Nash equilibrium allocation of employment must satisfy   πij(nj)≥πij’+i(nj)for alliandj′. (II.3) It is worth noting that (II.3) includes both the equal profit and the stability components of traditional systems of cities analysis. We will consider cases where there are relocation costs in the analysis below. If we suppose that a type-i firm has relocation costs equal to ri, then the equilibrium condition (II.3) becomes   πij(nj)≥πij’+i(nj)−rifor alliandj′. (II.4) Relocation costs, thus, enlarge the equilibrium set. The extension to idiosyncratic relocation costs for each firm is straightforward. 2.2. Multiple Equilibrium City Composition: Polar Cases Before turning to the computation, we will present a few analytical results that illustrate the multiple equilibrium problem. We will do this by constructing stylized examples showing the possibility of multiple equilibrium. This is meant to motivate our use of agent-based modeling as a way to select from among the many equilibria that exist. We begin by considering completely specialized cities containing only one type of firm. Let πiS(ni) denote the profit of a type-i firm in a completely specialized city containing only ni type-i workers. If such a specialized city becomes sufficiently large, it will offer no better profit than the profit from forming a new city. We will refer to the latter as the autarky profit, denoted πiA = πiS(li), equal to the profit earned by one isolated firm. Let niA denote the population of a specialized city such that it gives autarky utility: πiS( niA) = πiS(li) = πiA. A system of specialized cities, each with population niA, is a Nash equilibrium since no firm could raise its profit by relocating to a new city or by joining another city. This shows it is possible to construct equilibria where all cities are specialized. It is also possible to construct equilibria where all cities are mixed. Without loss of generality, let I = 2. Suppose that the aggregate populations of the two types are such that it is possible to allocate them to a system of cities such that every city has the same city composition n′ and this results in both types earning their autarky profit levels: π1(n′) = π1A and π2(n′) = π2A. Such a system of mixed cities is an equilibrium by construction. This construction, of course, relies on the strong assumption that it is possible to construct mixed cities such that both types reach the autarky level simultaneously with aggregate population fully allocated. It is also possible to construct an equilibrium in which specialized and mixed cities coexist. Suppose the allocation of firms to mixed cities generates a residual of firms of one type. If the residual firms are allocated to specialized cities that generate the autarky profit level, we again have a Nash equilibrium. In sum, it is possible to construct equilibria where all cities are specialized, where all cities are mixed, and where mixed and specialized cities coexist. This suffices for our purpose of establishing the possibility of a multiple equilibrium problem. See Helsley and Strange (2014) for more on this issue. The remainder of the article will concern the use of agent-based modeling to select from among the many equilibria, a significant departure from Helsley and Strange. As a corollary, it will provide additional examples of multiple equilibrium in city composition. 3. An Agent-based Model of Inter-city Firm Location 3.1. Overview We use an agent-based model to show that the location choices of individual agents transform an initial random location pattern into a regular location pattern. This approach follows Schelling’s (1971) seminal analysis of spatial segregation on a chessboard. Schelling’s analysis establishes the sensitivity of outcomes to changes in initial conditions. In Krugman’s (1993) model of the size and location of cities, an initial advantage in the location of manufacturers is self-reinforcing. Kollman et al. (1997) use an agent-based model to show that when agents choose jurisdictions, the instability of institutions affects the efficiency of public choice. Page (1999) uses an agent-based model to show how agglomeration preferences govern the spatial allocation of activity. Our use of an agent-based model to consider coagglomeration is new to the literature. We are primarily interested in the industrial composition of cities. Let Nidenote total employment in industry i and nij denote the employment in city j and industry i. Let the share of industry i employment found in city j be denoted sij = nij/Ni. Our measure of mixing for two industries i and k is given by   Mik=1−12∑j=1J(sij−skj). (III.1) The second term in equation (III.1) is the familiar index of dissimilarity, which originated in the study of residential segregation (see Massey and Denton, 1988). It has the advantage of being readily interpretable, with a segregation index of s% meaning that s% of workers must to relocate to produce a system of identical cities. Our mixing index is similarly easy to interpret. There are other approaches to characterizing the composition of cities. The Ellison and Glaeser (1997) index of coagglomeration measures the tendency of industries to collocate against a random ‘dartboard’ standard. The exposure index is another alternative. It measures the probability that a firm of one type shares a city with a firm of a second type. We have carried out all of the article’s key analysis using the all three measures, and the results are qualitatively similar. See the online Appendix available as Supplementary material for details. For a discussion of the properties of these alternative indices, see Massey and Denton (1988) and Hutchens (2001, 2004). 3.2 The Computational Model Our computations employ special cases of the theoretical model laid out above. The labor cost (congestion) function is w(N) = Nωlog(N), meaning that the elasticity of the wage with respect to the N is proportional to log(N). The production function shifter is iso-elastic and symmetric. We suppose for simplicity that the own industry agglomeration elasticity λ is the same for all industries. The coagglomeration elasticity will be specific to a pair of industries, i and k. Denote this by νik. For simplicity, we will suppose that all linked industry pairs have the same coagglomeration elasticity, νik = ν, while unlinked industries have νik = 0. Finally, we suppose that the own industry agglomeration elasticity λ is weakly larger than the coagglomeration elasticity ν. In this framework, the productivity of industry i can be written as   gi(nj)=(1+ni)λΠz≠i(1+nz)ν. (III.2) with λ ≥ ν3 In this setup, intra-industry effects are stronger than intra-industry effects in the sense that, ∂gi/∂ni≥ ∂gi/∂nk for k ≠ i and at ni = nk. It is worth noting that we do not assume that ∂gi/∂ni≥ ∂gi/∂nk for any ni and nk, a condition that would make it much more difficult to obtain mixing.4 There are obviously other approaches one might take in developing a numerical model of a regional economy. Our approach economizes on the number of parameters, allowing straightforward sensitivity analysis, such as the computation of alternative allocations with different values of the key parameters. For example, we can vary the congestion forces by changing the value of ω, which determines the elasticity of the wage with respect to city size. Similarly, we can vary the strength of intra-industry localization economies and inter-industry coagglomeration spillovers by changing the values of λ and ν. 3.3 Computational Procedures The computational process of the agent-based model has five steps: Specify values for the relevant parameters, including the elasticities for intra-industry and inter-industry external economies, the wage parameter and the cost of relocation. Set a maximum number of potential city sites and randomly allocate firms across these potential sites. The random allocation employs a uniform distribution. The maximum number is large enough that there are always vacant sites, so availability of sites is not a constraint. The equilibrium number of cities can be less than, greater than or equal to the initial random number of cities, depending on the values of the relevant parameters. For each industry, identify the firm in the region that has the most to gain from a unilateral relocation, and relocate the firm. In each relocation round, one firm in each industry has the opportunity to relocate, and the industry sequence is random. An additional city will develop if for some firm the autarky profit (for a sole occupant of a city) exceeds the maximum profit among existing cities. Naturally, the relocation of profit-maximizing firms can also cause cities to disappear. Repeat step (iii) until there is no incentive for unilateral relocation, and compute the relevant aggregate statistics, for example, a mixing index for each industry pair. Repeat steps (ii)–(iv) for a large number of initial random distributions. This generates a set of equilibrium allocations for a given set of parameter values. We compute equilibria in a model with three industries (I = 3). This is sufficient to explore the tensions between intra-industry and inter-industry external economies. The parameter values for the baseline case are as follows. The common intra-industry external economy elasticity is λ = 0.08. The coagglomeration elasticity (for inter-industry external economies) is positive for industry pair 1 and 2, ν ∈ [0,0.08] and zero for the other two industry pairs. The value of the wage parameter is ω = 0.08. The relocation cost per firm is roughly 1% of firm revenue. Each run of the model starts with a random distribution of a fixed number of firms across cities. There are 5400 firms, and the initial number of cities is between 80 and 100, depending on the random allocation process.5 We run the model for random draws for ν from a uniform distribution on [0,0.08]. As will become clear later, the results are robust to changes in specification and parameterization. 4. Coagglomeration Spillovers and Equilibrium Mixing This is the first of three sections that present the computational results of the agent-based model. This section focuses on the first key result—the positive and nonlinear relationship between the magnitude of inter-industry external economies and coagglomeration. Figure 1 shows two examples of emergence, one for a moderate coagglomeration elasticity, and one for a relatively high coagglomeration elasticity. Figure 1a shows equilibria that arise from a random pattern of initial locations for a coagglomeration elasticity equal to 0.04 (half the agglomeration elasticity). Each city is represented by a horizontal bar, the length of which indicates the number of firms in the city broken down by industry: industry 1 is black; industry 2 is dark gray; industry 3 is light gray. In this particular case, the Nash equilibria involve only specialized cities, giving M12 = 0. It is worth noting that the unlinked industry 3 has relatively small cities. This is a result of path dependence in the emergence process. Type-3 firms are not linked to other types, and early in the transition from the random allocation to the Nash equilibrium, the type-3 firms relocate to establish homogeneous cities. In the meantime, the link between types 1 and 2 generates relatively large heterogeneous cities. Late in the transition process, most of the large heterogeneous cities disappear, but the legacy of relatively large heterogeneous cities generates relatively large homogeneous cities. In contrast, when the coagglomeration elasticity is zero for all industry pairs, the three types of homogeneous cities are roughly the same size. Figure 1 View largeDownload slide (a) Example equilibrium A, (b) Example equilibrium B. Note: This figure shows two examples of equilibrium that arise from random initial locations with the base case parameters and coagglomeration elasticity ν = 0.04 (Figure 1a) and ν = 0.07 (Figure 1b). Figure 1 View largeDownload slide (a) Example equilibrium A, (b) Example equilibrium B. Note: This figure shows two examples of equilibrium that arise from random initial locations with the base case parameters and coagglomeration elasticity ν = 0.04 (Figure 1a) and ν = 0.07 (Figure 1b). Figure 1b shows equilibria for a coagglomeration elasticity of 0.07. In this case, industries 1 and 2 collocate in mixed cities, while the unlinked industry 3 occupies specialized cities. A majority of the mixed cities are roughly balanced with respect to the two linked industries, a common feature of the Nash equilibria that emerge in the computations, regardless of the value of the coagglomeration elasticity. The mixed cities are also larger than the homogeneous cities, another common feature of the equilibria. This pattern arises because the mixed city has larger agglomeration economies in general (from both inter-industry and intra-industry external economies), so the city can support more firms. Table 1 presents results for running the model with several values of the coagglomeration elasticity. For each elasticity value, we run the model 100 times, starting each time with a distinct random initial allocation.6 The results demonstrate the multiplicity of equilibria generated by a given set of parameter values. At ν = 0.03, equilibrium mixing ranges from M12 = 0 to M12 = 0.2850, with a mean value of 0.0995. Moving to ν = 0.05, the range of M12 is 0 to 0.3744, with a mean value of 0.1821. Moving further to ν = 0.07, the range of M12 is 0.6733 to 0.8717, with a mean of 0.7772. The sixth column shows the average number of mixed cities across the 100 runs. As expected, the number increases with the coagglomeration elasticity. The table illustrates two results: (i) for each coagglomeration elasticity, there is a wide range of equilibrium mixing levels; and (ii) there is a positive relationship between the strength of inter-industry external economies and equilibrium coagglomeration. Table 1 Coagglomeration elasticity and equilibrium mixing Coagglomeration Elasticity  M12 Minimum  M12 Maximum  M12 Mean  M12 Standard deviation  Average number of mixed cities  0.01  0  0.0611  0.0049  0.0102  0.33  0.03  0  0.2850  0.0995  0.0552  4.82  0.05  0  0.3744  0.1821  0.0822  7.31  0.07  0.6733  0.8717  0.7772  0.0425  35.45  Coagglomeration Elasticity  M12 Minimum  M12 Maximum  M12 Mean  M12 Standard deviation  Average number of mixed cities  0.01  0  0.0611  0.0049  0.0102  0.33  0.03  0  0.2850  0.0995  0.0552  4.82  0.05  0  0.3744  0.1821  0.0822  7.31  0.07  0.6733  0.8717  0.7772  0.0425  35.45  Note: This table reports equilibrium values for the mixing index M12 and the share of runs with mixed cities for various values of the coagglomeration elasticity. See the text for function forms and parameters. Figure 2 presents the results of running the computational model with 500 random draws of the coagglomeration elasticity in the range 0–0.08. The results reinforce the notion that profit-maximizing location choices generate multiple equilibria. For each coagglomeration elasticity, there are many equilibrium values of the mixing index. The range of the mixing index is largest for intermediate values of the coagglomeration elasticity. Figure 2 View largeDownload slide Equilibrium mixing. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) for the base case parameters as the elasticity of coagglomeration goes from 0 to 0.08. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 2 View largeDownload slide Equilibrium mixing. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) for the base case parameters as the elasticity of coagglomeration goes from 0 to 0.08. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. The figure shows a positive and nonlinear relationship between the coagglomeration elasticity and equilibrium mixing. Starting with unlinked industries (ν = 0), as inter-industry external economies become stronger, mixing increases at an increasing rate. The dashed line in Figure 2 represents the nonlinear relationship. This relationship is obtained using a local linear polynomial computed using the Epanechnikov (1969) kernel and the rule-of-thumb bandwidth (Silverman, 1986), implemented using the lpoly command in Stata. The nature of the nonlinearity is clear: as the coagglomeration elasticity increases from zero, a low marginal effect for industries that are weakly linked becomes a higher marginal effect as the inter-industry external economies increase. As shown in Figure 2 and Table 2, the relationship between the coagglomeration elasticity ν and the value of the mixing index M12 is nonlinear. Table 2 reports OLS estimates of the marginal effects of the coagglomeration elasticity. For low values of the coagglomeration elasticity, the marginal effects are small: the coefficients are 0.96 for the first bin and 3.58 for the second bin. In these cases, a stronger link between the industries has little effect on the amount of mixing observed in an equilibrium. At moderate levels of the coagglomeration elasticity, the relationship becomes much stronger: the coefficient grows to 3.85 in the third bin and then to 23.65 in the fourth bin. Finally, the marginal effect becomes smaller again at higher levels of the coagglomeration elasticity: the coefficient for the top bin is 12.33. All of the marginal effects are statistically significant.7 Table 2 Coagglomeration and equilibrium mixing—marginal effects   (1)  (2)  (3)  (4)  (5)  (6)    Equilibrium mixing  Relocation costs 2x  Relocation costs 3x  Congestion cost 1.1x  Firm size 2x  Firm size 5x  e: 0 to 0.016  0.96  7.29  15.83  1.17  1.59  1.90    (0.21)  (0.68)  (0.97)  (0.20)  (0.24)  (0.40)  N  89  99  93  110  115  105  e: 0.017 to 0.032  3.58  8.67  7.56  8.16  7.06  7.77    (1.07)  (1.01)  (0.88)  (1.06)  (0.79)  (1.60)  N  95  98  107  96  111  100  e: 0.033 to 0.048  3.85  12.16  9.55  3.07  7.74  11.95    (1.58)  (1.26)  (0.83)  (1.45)  (1.49)  (2.01)  N  94  82  114  90  80  86  e: 0.049 to 0.064  23.65  17.10  11.91  13.83  39.74  14.71    (2.24)  (1.04)  (0.84)  (1.43)  (1.92)  (1.73)  N  123  106  94  107  98  86  e: 0.065 to 0.08  12.13  7.84  12.64  17.85  15.76  12.57    (0.88)  (0.84)  (0.73)  (0.80)  (0.89)  (0.58)  N  101  115  92  97  96  123    (1)  (2)  (3)  (4)  (5)  (6)    Equilibrium mixing  Relocation costs 2x  Relocation costs 3x  Congestion cost 1.1x  Firm size 2x  Firm size 5x  e: 0 to 0.016  0.96  7.29  15.83  1.17  1.59  1.90    (0.21)  (0.68)  (0.97)  (0.20)  (0.24)  (0.40)  N  89  99  93  110  115  105  e: 0.017 to 0.032  3.58  8.67  7.56  8.16  7.06  7.77    (1.07)  (1.01)  (0.88)  (1.06)  (0.79)  (1.60)  N  95  98  107  96  111  100  e: 0.033 to 0.048  3.85  12.16  9.55  3.07  7.74  11.95    (1.58)  (1.26)  (0.83)  (1.45)  (1.49)  (2.01)  N  94  82  114  90  80  86  e: 0.049 to 0.064  23.65  17.10  11.91  13.83  39.74  14.71    (2.24)  (1.04)  (0.84)  (1.43)  (1.92)  (1.73)  N  123  106  94  107  98  86  e: 0.065 to 0.08  12.13  7.84  12.64  17.85  15.76  12.57    (0.88)  (0.84)  (0.73)  (0.80)  (0.89)  (0.58)  N  101  115  92  97  96  123  Note: This table reports the estimates for the piecewise linear relationship between coagglomeration elasticity and equilibrium mixing M12 for (i) the base case parameters, (ii) the case with doubled relocation costs, (iii) the case with tripled relocation costs, (iv) the case with 10% higher congestion costs, (v) the case with doubled firm size and (vi) the case with quintupled firm size. Standard errors are in parenthesis. This relationship is the result of two economic forces. The first is the direct effect: a larger coagglomeration elasticity raises the payoff of firms who collocate. The second effect is indirect: a larger coagglomeration elasticity increases mixing, and greater mixing reduces the incentive of a minority firm to move to a city where it is in the majority. For a small coagglomeration elasticity, mixed cities will be nearly homogeneous, with perhaps 10% minority firms. The system is symmetric, so a minority firm has the option of relocating to a city where it would be in a 90% majority. For small values of the coagglomeration elasticity, mixing is difficult to sustain because there is a large gap between the minority profit and the majority profit. As the coagglomeration elasticity rises, the level of mixing rises as well, so instead of a 10-90 split, there is for example a 40-60 split. There is a smaller gap between the minority profit and the majority profit, so there is less incentive for a minority firm to relocate. The indirect effect reinforces the direct effect, generating the curvature seen in Figure 2. A common feature of Schelling-type models of residential location is the sensitivity of location patterns to changes in the strength of the connection between two types of agents. In a Schelling model, a small change in preferences in favor of like neighbors can cause an abrupt change from a city of integrated neighborhoods to a segregated city. In our model of coagglomeration, a small change increase in the coagglomeration elasticity can also cause an abrupt change from a region of highly specialized cities (mixing index close to 0) to a region of highly mixed cities (a mixing index close to 1). The positive relationship between coagglomeration elasticity and equilibrium coagglomeration documented in this section also justifies using equilibrium collocation to define clusters of related industries, as in Porter (2003) and Delgado et al. (2016). As shown by Helsley and Strange (2014), there is no reason to expect that the equilibrium distribution of firms is efficient. This means that it is possible that observed clusters might not include industries that mutually benefit from collocating. The agent-based analysis in this article suggests that even with a large set of potential Nash equilibrium clustering patterns, there is a tendency for industry pairs with stronger coagglomeration spillovers to mix to a greater extent in equilibrium. The nonlinearity of the relationship has implications for both the measurement of agglomeration economies and the definition of clusters. Regarding the former, when industries are linked but only weakly, little mixing will be observed in equilibrium. In fact, for a wide range of coagglomeration elasticities and for a large fraction of observations, there is no mixing at all, as can be seen in Table 1. This means that estimates of the characteristics of industries that collocate will pick up only the strongly linked industries. The picture painted of the microfoundations of agglomeration economies will be incomplete in this way. Similarly, clusters will be defined by the most strongly linked industries. Weakly linked industries will tend to be excluded. 5. Other Key Results 5.1. Efficiency Consider next the efficiency properties of the Nash equilibria. We address this issue in various ways. Starting from a random allocation of firms across cities, the model executes a series of Pareto-improving relocations of individual firms. A Pareto-improving relocation increases the aggregate total profit, equal to the sum of the profits of all firms in all cities. The relocation of a single firm affects the profits of firms in both the origin city and the destination city: all firms in the two cities experience changes in the wage and thus production cost; linked firms experience changes in quantities produced. In a Pareto-efficient allocation, all Pareto-improving moves have been executed. Table 3 reports the features of Pareto-efficient allocations for different values of the coagglomeration elasticity. For each value, we compute efficient allocations for 10 initial random allocations. As shown in the second and third columns, the average equilibrium level of mixing (values transfered from Table 1) is considerably smaller than the average level of mixing in the computed Pareto optima. The differences are largest for intermediate values of the coagglomeration index. For example, for ν = 0.03, the average value for the mixing index is 0.879 in the efficient allocations, compared to 0.0995 for the Nash equilibria. For ν = 0.05, the average value for the mixing index is 0.957 in the efficient allocations, compared to 0.1821 in the Nash equilibria. Table 3 Nash equilibrium versus Pareto efficiency Coagglomeration elasticity  Nash equilibrium: Mean M12  Pareto efficiency: Mean M12  Nash equilibrium: Average number of mixed cities  Pareto efficiency: Average number of mixed cities  Pareto efficiency: Minimum share of industry 1 in mixed cities  0.01  0.0049  0.057  0.33  23.7  0.021  0.03  0.0995  0.879  4.82  20.4  0.377  0.05  0.1821  0.957  7.31  17.8  0.460  0.07  0.7772  0.981  35.45  15.9  0.480  Coagglomeration elasticity  Nash equilibrium: Mean M12  Pareto efficiency: Mean M12  Nash equilibrium: Average number of mixed cities  Pareto efficiency: Average number of mixed cities  Pareto efficiency: Minimum share of industry 1 in mixed cities  0.01  0.0049  0.057  0.33  23.7  0.021  0.03  0.0995  0.879  4.82  20.4  0.377  0.05  0.1821  0.957  7.31  17.8  0.460  0.07  0.7772  0.981  35.45  15.9  0.480  Note: This table reports equilibrium and Pareto efficient values for the mixing index M12 and the share of runs with mixed cities for various values of the coagglomeration elasticity. See the text for function forms and parameters. In the efficient allocations, each mixed city has a minority firm population. As shown in the last column of Table 3, the minimum share of industry 1 in mixed cities increases with the coagglomeration elasticity. The efficient allocations have matched pairs of unbalanced heterogeneous cities. As shown in the last column, for ν = 0.01 (one-eighth the intra-industry agglomeration elasticity), each matched pair has one city with roughly 98% type-1 firms and 2% type-2 firms, and a second city with a mirror image of 2% type-1 firms and 98% type-two firms. Moving to higher values of the coagglomeration elasticity, the city pairs become progressively more balanced, reaching a 48-52 split for ν = 0.07. In contrast, for low values of the coagglomeration elasticity, the equilibria have a relatively small number of roughly balanced mixed cities. Pareto efficiency requires some firms to suffer the relatively low profits of minority status. For ν < λ (inter-industry external economies are weaker than intra-industry external economies), an individual firm’s profit is maximized with majority status, and there is a profit penalty for minority status. As the gap between ν and λ decreases, the profit-maximizing mix moves closer to an equal split (from say 90% majority to 80% majority). In a particular industry, for every minority outcome in one city, there must be a counterbalancing majority outcome in another city. Because the efficient allocation has unbalanced heterogeneous cities, it cannot in general be supported as a Nash equilibrium. Each firm in minority status could earn higher profit in (i) the mirror city where the firm would be in the majority rather than the minority or (ii) a homogeneous city. Starting from a Pareto-efficient outcome, unilateral relocations to escape minority status causes a system of unbalanced heterogeneous cities to be replaced by a mix of (i) a relatively small number of roughly balanced heterogeneous cities and (ii) many homogeneous cities. Unless some firms are willing to be in the minority and thus ‘take one for the team’, efficient allocations are elusive. 5.2. History: Initial Conditions and Coagglomeration So far, our computations have been based on a initial random allocation of firms across cities. To illustrate the possible role of history in coagglomeration, we run the model with three alternative starting points. Suppose that the initial condition is for all cities to be completely diverse, with each containing an equal proportion of the national population of the three types. There is complete mixing in this situation. In this case, the initial allocation by construction gives equal profits to all firms. As long as autarky profit is sufficiently low, this allocation meets the criteria for an equilibrium, although the allocation is likely to be unstable. As an alternative, suppose we start with complete mixing, and then randomly reallocate 10% of firms. As shown in Figure 3b and column (2) of Table 4, the relationship similar to the baseline results, but not identical. In the table, the largest marginal effects occur for ν between 0.033 and 0.048, rather than for higher values, as in the baseline results. This is because starting with more mixing means that a smaller coagglomeration elasticity is required to have mixing in equilibrium. Table 4 Coagglomeration and equilibrium mixing—marginal effects with variation in initial conditions   (1)  (2)  (3)    Nearly homogeneous  Nearly heterogeneous  Diff. initial coagglomeration elasticity  e: 0 to 0.016  0.00  0.57  13.48    (.)  (0.13)  (1.33)  N  92  92  93  e: 0.017 to 0.032  0.00  12.07  72.22    (.)  (2.18)  (3.78)  N  114  92  101  e: 0.033 to 0.048  0.00  43.90  0.20    (.)  (4.32)  (0.07)  N  105  126  108  e: 0.049 to 0.064  2.58  3.08  -0.01    (0.27)  (0.33)  (0.01)  N  101  91  98  e: 0.065 to 0.08  55.08  1.41  0.05    (2.67)  (0.12)  (0.01)  N  88  99  100    (1)  (2)  (3)    Nearly homogeneous  Nearly heterogeneous  Diff. initial coagglomeration elasticity  e: 0 to 0.016  0.00  0.57  13.48    (.)  (0.13)  (1.33)  N  92  92  93  e: 0.017 to 0.032  0.00  12.07  72.22    (.)  (2.18)  (3.78)  N  114  92  101  e: 0.033 to 0.048  0.00  43.90  0.20    (.)  (4.32)  (0.07)  N  105  126  108  e: 0.049 to 0.064  2.58  3.08  -0.01    (0.27)  (0.33)  (0.01)  N  101  91  98  e: 0.065 to 0.08  55.08  1.41  0.05    (2.67)  (0.12)  (0.01)  N  88  99  100  Note: This table reports the estimates for the piecewise linear relationship between coagglomeration elasticity and equilibrium mixing M12 with base case parameters (i) from a nearly homogeneous initial conditions, (ii) from a nearly heterogeneous initial conditions, (iii) with the base case parameters but with an initial coagglomeration elasticity of 0.08. Standard errors are in parenthesis. Figure 3 View largeDownload slide (a) Equilibrium mixing: nearly homogeneous initial cities. (b) Equilibrium mixing: nearly heterogeneous initial cities. (c) Equilibrium mixing with variation in the initial coagglomeration elasticity Note: This figure shows the solutions for equilibrium mixing from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels. Instead of random initial locations, the simulations begin with all cities nearly homogeneous. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 3 View largeDownload slide (a) Equilibrium mixing: nearly homogeneous initial cities. (b) Equilibrium mixing: nearly heterogeneous initial cities. (c) Equilibrium mixing with variation in the initial coagglomeration elasticity Note: This figure shows the solutions for equilibrium mixing from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels. Instead of random initial locations, the simulations begin with all cities nearly homogeneous. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. At the other extreme, suppose that the initial condition is for all cities to be completely specialized, with each containing only one type of firm. There is no mixing at all in this situation. In this case, the initial allocation is an equilibrium, as shown in Section 2. As an alternative, suppose we begin with completely specialized cities, and then randomly reallocate 10% of firms. The results from this exercise are shown in Figure 3a and column (1) of Table 4. For low levels of coagglomeration elasticity, we continue to have M12 = 0. At higher levels of coagglomeration elasticity, there is a positive relationship. In the table, the relationship is highly nonlinear, with a zero marginal effect for low values of coagglomeration elasticity, a moderate marginal effect of 2.58 for ν between 0.049 and 0.064, and a very large coagglomeration elasticity for ν at the top of the range (between 0.065 and 0.080). Beginning with a system of specialized cities, a large coagglomeration elasticity is required to generate mixing. As a third exercise concerning initial conditions, we can illustrate the possibility of path dependence by exploring the effects of an inter-temporal change in the value of the coagglomeration elasticity. Suppose that initially, the coagglomeration elasticity equals the agglomeration elasticity: for ν = λ, the Nash equilibria feature perfectly heterogeneous cities for types 1 and 2, and homogeneous cities for the unlinked type 3. The value of the mixing index is M12 = 1. Suppose that the coagglomeration elasticity for industries 1 and 2 changes, with some new lower value between 0 and 0.08. Figure 3c shows the relationship between the new coagglomeration elasticity and mixing, with the marginal effects reported in column (3) of Table 4. The relationship is positive and nonlinear for coagglomeration elasticities up to roughly 0.03, and then constant for larger values. History matters: for intermediate values of the new coagglomeration elasticity, a history of heterogeneous cities generates higher levels of coagglomeration. Our three exercises with alternative initial conditions generate two key conclusions. First, if we start with an extreme allocation of either perfect homogeneity (M12 = 0) or perfect heterogeneity (M12 = 1), in the new equilibrium there is a weakly positive relationship between the coagglomeration elasticity and the value of the mixing index. Second, history matters in the sense that the legacy of either homogeneity or heterogeneity biases the equilibrium allocation in a predictable direction, with either higher or lower values of the mixing index. The largest legacy effects are for intermediate values of the coagglomeration elasticity. 5.3. Firm Size Consider next the role of firm size in the allocation of firms across cities. So far the decision-makers in the agent-based model are relatively small. There are 5400 firms, so each agent is 1/5400 of the regional economy. To illustrate the effects of firm size, we run the model with larger firm sizes, measured as the number of workers per firm. We increase firm size by a factor of 2 (generating 2700 firms), and then 5 (generating 1080 firms). As shown in Figure 4, an increase in firm size (more workers per firm) increases mixing. Figure 4a shows that doubling firm size increases mixing relative to the base case, while Figure 4b shows that quintupling firm size increases mixing still further. An increase in firm size partly internalizes intra-industry external economies, and thus weakens intra-industry external economies as a firm substitutes internal scale for external scale. As a result, the tug-of-war between intra-industry external economies (pulling a firm toward cities with firms in the same industry) and inter-industry external economies (pulling a firm toward cities with firms in related industries) is more frequently won by inter-industry economies. The result is more mixing (coagglomeration) and less specialization. Marginal effects are reported in the last two columns of Table 2. For low values of ν, we see large coefficients. As usual, an increase in ν has the largest effect for intermediate values. Figure 4 View largeDownload slide (a) Equilibrium mixing with larger firms (2x), (b) Equilibrium mixing with larger firms (5x). Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except firm size, which are twice as large. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 4 View largeDownload slide (a) Equilibrium mixing with larger firms (2x), (b) Equilibrium mixing with larger firms (5x). Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except firm size, which are twice as large. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. We can contrast our firm-size effects with the conventional wisdom about firm size that originates with Chinitz (1961). It is common to view small firms as promoting the concentration of production in cities. The key idea is that small firms engage the local business community to a greater extent because it would be uneconomical for them to internalize all production activities. This external orientation makes the city more attractive to other small firms. See Rosenthal and Strange (2003, 2010), Agrawal et al. (2008), Glaeser and Kerr (2009), Glaeser et al. (2015) and many others for evidence consistent with this virtuous circle of entrepreneurial activity.8 In contrast with the Chinitz framework, small firms in our model promote the clustering of firms in the same industry at the expense of coagglomeration (clustering of firms in related industries). If intra-industry economies are just slightly stronger than inter-industry economies, there is a bias toward specialized cities. The economic forces underlying the bias are similar to the forces in a Schelling model that generate a bias toward residential segregation when integration is efficient (Schelling, 1971, 1978; Zhang, 2004; O’Sullivan, 2009). In our model, an increase in firm size decreases the power of intra-industry external economies and reduces the bias toward specialization, which increases coagglomeration. The result on firm size is related to the Duranton and Puga (2001) analysis of ‘nursery cities’. They show that a new (small) firm can benefit to a greater extent from the diversity of resources available in a large city. As the firm’s production becomes more routinized and scale economies are internalized, the benefit of diversity decreases, prompting the relocation of larger firms to specialized cities. In contrast, in our analysis, large firms can choose a diversified location without giving up the productivity gains associated with localization to as large an extent. The results in Figure 4 can also be related to the role of developers in urban development. As first proposed in Henderson (1974), the idea is that a relatively large (inefficient) city generates profit opportunities for a developer to create smaller, more profitable cities. The strongest version of this result is that only efficient cities can be sustained in a regional equilibrium. This approach assigns to developers an entrepreneurial role by creating startup cities that generate greater value. In our model, an increase in firm size could be interpreted as the formation of coalitions of relocating firms by developers. A coalition partly internalizes intra-industry external economies and thus promotes coagglomeration. The large firms in our analysis play a similar role to the ‘industry parks’ in Rauch’s (1993) paper on why history matters when it matters little. 5.4. Relocation Cost The equilibria computed thus far have been based on relatively low relocation costs, equal to roughly 1% of total revenue. Figure 5 considers the effects of an increase in relocation costs. Figure 5a shows the computed equilibria with a doubling of relocation cost, and Figure 5b shows the equilibria with a tripling of relocation cost. In both figures, it is apparent that an increase in relocation costs results in a more linear relationship between coagglomeration elasticity and equilibrium mixing. With higher relocation costs, a firm does not relocate when differences in profit are modest, and this tends to dampen the forces that generate specialized cities as the legacy of the initial random allocation persists to a greater extent. Figure 5 View largeDownload slide (a) Equilibrium mixing with higher relocation costs (x2), (b) Equilibrium mixing with higher relocation costs (x3). Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except relocation cost, which are twice as large. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 5 View largeDownload slide (a) Equilibrium mixing with higher relocation costs (x2), (b) Equilibrium mixing with higher relocation costs (x3). Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except relocation cost, which are twice as large. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Columns (2) and (3) of Table 2 report marginal effects for the changes in relocation cost. Without relocation costs, column (1) shows that marginal effects differ by a factor of more than 20. In column (2), with doubled relocation costs, the smallest marginal effect is a little less than half of the largest. A similar pattern appears in column (3), where relocation costs are tripled. 6. Extensions In this section, we present computational results based on alternative parameter values and functional forms, and link regimes. As expected, the positive relationship between spillovers and mixing persists. In other words, the results are robust to changes in parameterization and specification. 6.1. Congestion Consider the effects of changes in value of the parameter of the congestion/wage relationship. As shown in Figure 6a 10% increase in the value of the wage parameter increases mixing in general, with the largest effects in the range ν = 0.04 to ν = 0.06. The share of observations with at least one mixed city increases from 0.60 to 0.75. The 10% increase in the value of the wage parameter decreases the average city size by roughly 10%. In general, there is a negative relationship between city size and mixing, which explains the increase in mixing when wages increases more rapidly with city size. Figure 6 View largeDownload slide Equilibrium mixing with 10% higher congestion costs. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except congestion cost, which are 10% larger. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 6 View largeDownload slide Equilibrium mixing with 10% higher congestion costs. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except congestion cost, which are 10% larger. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Consider next the functional form for the congestion/wage relationship. In the baseline computations, the elasticity of the wage with respect to city size proportional to the natural log of city size. An alternative is a specification where the elasticity is proportional to city size. A switch to this alternative does not change our qualitative results: there is still a positive relationship between the coagglomeration elasticity and mixing, and the relationship continues to be convex for low values of the coagglomeration elasticity. The greater sensitivity of wages to city size makes the relationship a bit more convex. 6.2. Production Function The other key functional form is the production function. Our specification has diminishing marginal productivity for both intra-industry and inter-industry external economies. For a sufficiently large number of like firms (same industry), the marginal product of an unlike firm (inter-industry external economy) exceeds the marginal product of a like firm (intra-industry external economy). In other words, the marginal product of the first unlike firm can exceed the marginal product of the last of a large number of like firms. This feature provides an incentive for at least some industry mixing, with the incentive increasing with the value of the coagglomeration elasticity. As an alternative, consider a production function in which the intra-industry external economy is always greater than the inter-industry external economy:   g(nj)=(1+ni+aSz≠jnz)λ (VI.1) where α < 1. Under this specification, there is no incentive for a firm in one industry to locate in the same city as a firm in another industry. Figure 7 shows the relationship between α and mixing for this specification, with α running from 0 to 1. Mixing occurs only for values of α close to 1, where the effective coagglomeration elasticity is close to the agglomeration elasticity. Figure 7 View largeDownload slide Equilibrium mixing with alternative specification of agglomeration economies. Note: This figure shows the solutions for equilibrium mixing from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. The agglomeration economy is as in equation (IV.1), with alpha ranging from 0 to 1 and other parameters at base case. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 7 View largeDownload slide Equilibrium mixing with alternative specification of agglomeration economies. Note: This figure shows the solutions for equilibrium mixing from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. The agglomeration economy is as in equation (IV.1), with alpha ranging from 0 to 1 and other parameters at base case. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. 6.3. Alternative Linkage Regimes So far our results are computed for three industries, two of which are linked in their production technology. Suppose instead that all three industries are linked, with ν denoting the common coagglomeration elasticity for all three industry pairs. In this situation, it is possible to have four kinds of mixed cities, three with two industries collocating and one with all three industries collocating. Figure A3 in the Online Appendix (see the Supplementary material section) shows a positive relationship between the coagglomeration elasticity and the share of cities that are diverse. Again, we see more mixing with stronger links. As an alternative, suppose there are two pairs of linked industries, with links between industries 1 and 2, and links between industries 2 and 3. Unlike the all-linked case discussed above, industries 1 and 3 and not directly linked. As above, the coagglomeration elasticities are common and equal to ν for both of the linked industry pairs. As noted by Behrens (2016), even though there is no direct link between industries 1 and 3, these industries are indirectly linked by the links that industries 1 and 3 have with industry 2. It is possible that an indirectly linked pair will collocate in equilibrium. Our agent-based approach sheds light on this possibility. The simulations in Table A4 in the Online Appendix available as Supplementary material paint a clear picture: there are two types of heterogeneous cities (one with types 1 and 2 and one with types 2 and 3) and three types of homogeneous cities (one for each industry) that appear as Nash equilibria. There are very few cities that include both unlinked industries (1 and 3), a share of roughly 1% of the 500 runs. Starting with an allocation in which some cities host unlinked industries, unilateral relocations occur as firms flee cities where unlinked firms incur a cost (higher wages) without an compensating benefit. At least in the particular environment of our agent-based model, we do not find evidence of indirect links generating collocation. Of course, it is certainly possible for these ‘third industry’ effects to affect equilibrium location patterns in other settings. For instance, if there are all-or-nothing agglomeration effects as in Ellison et al. between industry 1 and 2 and between industries 2 and 3, then industries 1 and 3 would be found together in mixed cities despite the absence of direct links.9 6.4. Four Industries So far our model includes three industries. We have run a number of computational experiments with four industries instead of three. These experiments generate several predictable conclusions. First, adding a fourth industry that is not linked to the other industries does not affect the qualitative results concerning coagglomeration of the two linked industries (1 and 2). Figure 8 shows this relationship. Second, in the case of two pairs of linked industries (1 and 2 linked and 3 and 4 linked), the Nash equilibria have six types of cities: two types of mixed cities (with 1 and 2 and with 3 and 4), and four types of homogeneous cities (one type for each industry). There are no cities that include unlinked firms. Third, in the case of full linkage (all firms are linked), the Nash equilibria have heterogeneous cities (all four types) and homogeneous cities. In the case of links between 3 of 4 industries (1 and 2, 2 and 3, 1and 3), the Nash equilibria have 3-industry heterogeneous cities (1, 2 and 3), 2-industry heterogeneous cities (1 and 2, 2 and 3, 1 and 3), and homogeneous cities for each industry. The unlinked firms (industry 4) locate in homogeneous cities. Fourth, in the case of competing links (1 and 2, 1 and 3, 1 and 4), the Nash equilibria have three types of heterogeneous cities (one for each linked pair) and three types of homogeneous cities (2, 3, 4). Firms in industries 2, 3 and 4 compete for firms in industry 1, so there are no ‘leftover’ industry-1 firms, and thus no homogeneous industry-1 cities. Figure 8 View largeDownload slide Equilibrium mixing with four industries. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) for the base case parameters as the elasticity of coagglomeration goes from 0 to 0.08. The difference in this figure is that we work with four industries instead of three. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 8 View largeDownload slide Equilibrium mixing with four industries. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) for the base case parameters as the elasticity of coagglomeration goes from 0 to 0.08. The difference in this figure is that we work with four industries instead of three. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. 7. Conclusions This article uses an agent-based model to explore the industrial composition of cities. Helsley and Strange (2014) show that the set of equilibrium city compositions is very large. In some situations, industries will collocate when there is no benefit from doing so. In others, industries will fail to collocate when there would be a benefit. Our article employs agent-based modeling to refine the equilibrium set. This shows that there is a positive and nonlinear relationship between inter-industry external economies and coagglomeration. The existence of this relationship supports using observed coagglomeration together with data on links between industries to study the microfoundations of agglomeration economies, as in Ellison et al. (2010) and elsewhere. The relationship also supports using observed patterns of industrial clustering to define industry clusters, as in Delgado et al. (2016). Our computational results generate several other key results. The second result concerns efficiency: the equilibrium level of coagglomeration is less than the efficient level. If inter-industry external economies are weaker than intra-industry external economies, the efficient allocation features unbalanced cities, and a firm in the minority has an incentive to flee to a city where the firm is in the majority. This relocation process causes the efficient outcome to unravel. We show the third result—history matters—by computing equilibria with different initial allocations of firms across cities. The equilibrium coagglomeration is relatively high when we start with relatively heterogeneous cities, and relatively low if we start with relatively homogeneous cities. The fourth result is that an increase in firm size increases coagglomeration: the weakening of intra-industry external economies means that the tug-of-war is more frequently won by the forces pulling firms in related industries together. The final result is that an increase in relocation cost increases coagglomeration: increasing friction in the economy means that the legacy of the initial random allocation of firms persists to a greater extent. Supplementary material Supplementary data for this paper are available at Journal of Economic Geography online. Footnotes 1 Collocation refers to industries that independently choose the same location. 2 See also the surveys by Duranton and Puga (2004), Rosenthal and Strange (2004), Behrens and Robert-Nicoud (2015) and Combes and Gobillon (2015). 3 It is worth noting that the specification in equation (III.3) allows for positive production even when ni = 0 for some industry i as long as one industry has positive employment. This allows for specialized cities to exist. 4 It is also worth noting that a firm's productivity is shifted by gi(-), while gi(-) depends directly on the firm's own employment. Agglomeration effects depend on the scale of activity, which includes the firm's activity. 5 We have considered various alternate migration orders, with no systematic difference in results. 6 The result does not change meaningfully with a greater number of runs. 7 The different numbers of observations in Table 2 are a result of the randomization over elasticity. 8 The main alternative to this view is the idea that large firms are ‘anchors’, in the sense that they generate local externalities. See Agrawal and Cockburn (2003) and Feldman (2003). 9 We thank Kristian Behrens for this observation. Acknowledgements We thank Heski Bar-Isaac, Robert Helsley, Jordan Rappaport, Kristian Behrens and two helpful referees for comments on previous versions of the article. 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For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Economic Geography Oxford University Press

The Emergence of Coagglomeration

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Abstract

Abstract This article uses an agent-based model of intercity firm location to explore the industrial composition of cities. Starting from a random allocation of firms across cities, firms relocate in pursuit of greater profit. There are several key results. First, there is a positive and nonlinear relationship between the strength of inter-industry external economies and coagglomeration, a result that supports using coagglomeration to study the microfoundations of agglomeration economies and to determine the boundaries of industry clusters. Second, the equilibrium level of coagglomeration is less than the efficient level. Third, history matters in the sense that a legacy of homogeneous or heterogeneous cities tilts the economy in favor of the historical pattern. Fourth, an increase in firm size increases coagglomeration. Fifth, an increase in relocation cost increases coagglomeration. 1. Introduction This article uses an agent-based model of intercity firm location to explore the industrial composition of cities. Helsley and Strange (2014) show that the coordination problems inherent in location choice generate multiple equilibria. This article employs agent-based modeling to refine the equilibrium set. Firms experience both intra-industry external economies and inter-industry external economies. Starting from a random allocation of firms across cities, a firm relocates to the city that generates the highest profit, and firms continue to relocate until no single firm can benefit from moving. The result is emergence, the transformation of an initial random pattern into a regular pattern. The computations generate five key results. First, there is a positive and nonlinear relationship between the strength of inter-industry external economies and coagglomeration.Second, the equilibrium level of coagglomeration is generally less than the efficient level. Since inter-industry external economies are weaker than intra-industry external economies, the efficient allocation requires minority status for some firms, and minority firms have an incentive to unilaterally deviate from the efficient allocation by relocating to cities where they are in the majority. Third, history matters: a legacy of heterogeneous or homogeneous cities tilts the economy in favor of the historical pattern. Fourth, an increase in firm size increases coagglomeration because it weakens intra-industry external economies relative to inter-industry external economies. Fifth, an increase in relocation cost increases coagglomeration because the legacy of the initial random allocation of firms across cities persists to a greater extent. These results relate to a number of recent papers that explore the theory of coagglomeration. In Ellison and Glaeser (1997) and Ellison et al. (2010), a discrete-choice model incorporates ‘all-or-nothing’ external economies. A firm that fails to collocate with industries in its partition earns an infinitely negative profit, so the collocation of industries means that there must be spillovers between the industries.1Helsley and Strange (2014) show that this result does not extend to weaker specifications of external economies. Specifically, the equilibrium collocation of industries need not mean that they are ideal partners for each other, or even that they benefit from each other at all. This article’s agent based modeling serves to refine the equilibrium set, establishing that that there is likely to be more coagglomeration of industries that are more strongly linked. This, in turn, provides support for empirical research using observed coagglomeration to learn about the microfoundations of agglomeration economies.2 In this approach, for example, one would see evidence of input sharing in the collocation of industries that are part of the same supply chain. Ellison et al. (2010) use US data to show that proxies for all three of Marshall’s forces are robustly related to the coagglomeration of industries. Faggio et al. (2016) reach similar conclusions about Marshall’s microfoundations using UK data and also present evidence consistent with a positive role of entrepreneurship, innovation and industrial organization in the generation and reception of external economies in production. Coagglomeration is also the focus of a number of other recent papers, including Kolko (2010), Jacobs et al. (2013), Gabe and Abel (2016), Jofre-Monseny et al. (2011), Behrens and Guillain (2016) and the recent review by Behrens (2016). This article supports the key hypothesis behind this line of empirical research, that industries that frequently coagglomerate must be benefitting from each other. The positive relationship between inter-industry external economies and coagglomeration also has implications for the literature on clusters. The key issue is how clusters are defined. Porter (2003, 562) defines clusters as ‘geographic concentrations of linked industries’. The recent paper by Delgado et al. (2016) significantly improves our understanding of the nature of clusters by developing algorithms that create regionally comparable cluster definitions (i.e. definitions that are not specific to particular regions or cities). These algorithms are based on collocation and industry links. This article’s refinement of the equilibrium set provides support for this approach as well. The remainder of the article is organized as follows. Section 2 describes the model and presents some theoretical examples of multiple equilibria. Section 3 outlines the article’s agent-based approach. Section 4 presents the baseline results, focusing on the positive and nonlinear relationship between the strength of inter-industry external economies and coagglomeration. Section 5 presents the other key results. In Section 6, we run the model with different parameter values and functional forms, and show that the results are robust to changes in specification and parameterization. Section 7 concludes. 2. A System of Cities Model 2.1. Fundamentals This section will specify a simple model of firm location in a system of cities. The model will be used to establish the existence of multiple equilibria in city composition in a general setting. It will also be the basis for the article’s agent-based approach to equilibrium selection. The literature on agglomeration and systems of cities begins with Henderson (1974). This literature focuses on how the tradeoffs between agglomerative and dispersive forces interact to determine city size. For the most part, this literature does not address the issue of city composition, except for the polar cases of completely specialized and completely diverse cities. Behrens et al. (2014) consider the vertical composition of cities, where workers are differentiated by their human capital. The self-selection conditions restrict the set of equilibria, leading to a hierarchy of cities according to worker talent. As noted previously, Helsley and Strange (2014) consider horizontal composition, a situation where individual migration leads to multiple equilibrium. They show that coordinated migration through developers or firms can refine the set of equilibria. Our model makes use of an agent-based model to further refine the set of equilibria, generating predictions about industrial composition. Consider a region that contains J cities, indexed j ∈ {1,2,…,J}. N workers inhabit these cities. There are I types of worker, indexed by i ∈ {1,2,…,I}. A firm in industry i hires only type-I workers. The composition of city j in this situation is given by the vector nj = (n1j,n2j,…,nIj), while the population of city j is denoted Nj = ∑inij. The workers are employed by firms. The output of an industry-i firm in city-j is denoted by   qij=θgi(nj)li, (II.1) where θ is a positive constant and li is the number of type-i workers employed by the firm. We take this to be exogenous, and we explore the implications of firm size below. The function gi(-) captures intra-industry (the productivity effect of employment in the same industry, i.e. localization economies) and inter-industry external economies (the productivity effect of employment in related industries). Specifically, we assume that gi(-) is continuously differentiable and strictly concave, and that gi(-) has non-negative first partials. The key terms are ∂gi/∂nk, which capture the link between industry i and industry k. We will impose further structure later. To parallel standard competitive models with single-industry cities, we assume that all industries produce the numeraire commodity. We also suppose that the only cost incurred by the firm is labor. We suppose that labor costs increase with the total population of a city, as captured by the increasing function w(Nj). This can, of course, be given microfoundations through the monocentric model. A Nash equilibrium in locations requires that no firm of a given type be able to improve its profit by moving to another city. The profit of a firm in industry i in city j is   πij(nj)=θgi(nj)li−w(Nj)li. (II.2) We assume that the number of firms, like the number of workers, is fixed. See Helsley and Strange (2014) for a treatment of this issue. Let πij+i (nj) denote the profit earned by a type-i firm in city j when the population of the types is given by the vector nj plus liadditional type-i workers (one firm). In this situation, a Nash equilibrium allocation of employment must satisfy   πij(nj)≥πij’+i(nj)for alliandj′. (II.3) It is worth noting that (II.3) includes both the equal profit and the stability components of traditional systems of cities analysis. We will consider cases where there are relocation costs in the analysis below. If we suppose that a type-i firm has relocation costs equal to ri, then the equilibrium condition (II.3) becomes   πij(nj)≥πij’+i(nj)−rifor alliandj′. (II.4) Relocation costs, thus, enlarge the equilibrium set. The extension to idiosyncratic relocation costs for each firm is straightforward. 2.2. Multiple Equilibrium City Composition: Polar Cases Before turning to the computation, we will present a few analytical results that illustrate the multiple equilibrium problem. We will do this by constructing stylized examples showing the possibility of multiple equilibrium. This is meant to motivate our use of agent-based modeling as a way to select from among the many equilibria that exist. We begin by considering completely specialized cities containing only one type of firm. Let πiS(ni) denote the profit of a type-i firm in a completely specialized city containing only ni type-i workers. If such a specialized city becomes sufficiently large, it will offer no better profit than the profit from forming a new city. We will refer to the latter as the autarky profit, denoted πiA = πiS(li), equal to the profit earned by one isolated firm. Let niA denote the population of a specialized city such that it gives autarky utility: πiS( niA) = πiS(li) = πiA. A system of specialized cities, each with population niA, is a Nash equilibrium since no firm could raise its profit by relocating to a new city or by joining another city. This shows it is possible to construct equilibria where all cities are specialized. It is also possible to construct equilibria where all cities are mixed. Without loss of generality, let I = 2. Suppose that the aggregate populations of the two types are such that it is possible to allocate them to a system of cities such that every city has the same city composition n′ and this results in both types earning their autarky profit levels: π1(n′) = π1A and π2(n′) = π2A. Such a system of mixed cities is an equilibrium by construction. This construction, of course, relies on the strong assumption that it is possible to construct mixed cities such that both types reach the autarky level simultaneously with aggregate population fully allocated. It is also possible to construct an equilibrium in which specialized and mixed cities coexist. Suppose the allocation of firms to mixed cities generates a residual of firms of one type. If the residual firms are allocated to specialized cities that generate the autarky profit level, we again have a Nash equilibrium. In sum, it is possible to construct equilibria where all cities are specialized, where all cities are mixed, and where mixed and specialized cities coexist. This suffices for our purpose of establishing the possibility of a multiple equilibrium problem. See Helsley and Strange (2014) for more on this issue. The remainder of the article will concern the use of agent-based modeling to select from among the many equilibria, a significant departure from Helsley and Strange. As a corollary, it will provide additional examples of multiple equilibrium in city composition. 3. An Agent-based Model of Inter-city Firm Location 3.1. Overview We use an agent-based model to show that the location choices of individual agents transform an initial random location pattern into a regular location pattern. This approach follows Schelling’s (1971) seminal analysis of spatial segregation on a chessboard. Schelling’s analysis establishes the sensitivity of outcomes to changes in initial conditions. In Krugman’s (1993) model of the size and location of cities, an initial advantage in the location of manufacturers is self-reinforcing. Kollman et al. (1997) use an agent-based model to show that when agents choose jurisdictions, the instability of institutions affects the efficiency of public choice. Page (1999) uses an agent-based model to show how agglomeration preferences govern the spatial allocation of activity. Our use of an agent-based model to consider coagglomeration is new to the literature. We are primarily interested in the industrial composition of cities. Let Nidenote total employment in industry i and nij denote the employment in city j and industry i. Let the share of industry i employment found in city j be denoted sij = nij/Ni. Our measure of mixing for two industries i and k is given by   Mik=1−12∑j=1J(sij−skj). (III.1) The second term in equation (III.1) is the familiar index of dissimilarity, which originated in the study of residential segregation (see Massey and Denton, 1988). It has the advantage of being readily interpretable, with a segregation index of s% meaning that s% of workers must to relocate to produce a system of identical cities. Our mixing index is similarly easy to interpret. There are other approaches to characterizing the composition of cities. The Ellison and Glaeser (1997) index of coagglomeration measures the tendency of industries to collocate against a random ‘dartboard’ standard. The exposure index is another alternative. It measures the probability that a firm of one type shares a city with a firm of a second type. We have carried out all of the article’s key analysis using the all three measures, and the results are qualitatively similar. See the online Appendix available as Supplementary material for details. For a discussion of the properties of these alternative indices, see Massey and Denton (1988) and Hutchens (2001, 2004). 3.2 The Computational Model Our computations employ special cases of the theoretical model laid out above. The labor cost (congestion) function is w(N) = Nωlog(N), meaning that the elasticity of the wage with respect to the N is proportional to log(N). The production function shifter is iso-elastic and symmetric. We suppose for simplicity that the own industry agglomeration elasticity λ is the same for all industries. The coagglomeration elasticity will be specific to a pair of industries, i and k. Denote this by νik. For simplicity, we will suppose that all linked industry pairs have the same coagglomeration elasticity, νik = ν, while unlinked industries have νik = 0. Finally, we suppose that the own industry agglomeration elasticity λ is weakly larger than the coagglomeration elasticity ν. In this framework, the productivity of industry i can be written as   gi(nj)=(1+ni)λΠz≠i(1+nz)ν. (III.2) with λ ≥ ν3 In this setup, intra-industry effects are stronger than intra-industry effects in the sense that, ∂gi/∂ni≥ ∂gi/∂nk for k ≠ i and at ni = nk. It is worth noting that we do not assume that ∂gi/∂ni≥ ∂gi/∂nk for any ni and nk, a condition that would make it much more difficult to obtain mixing.4 There are obviously other approaches one might take in developing a numerical model of a regional economy. Our approach economizes on the number of parameters, allowing straightforward sensitivity analysis, such as the computation of alternative allocations with different values of the key parameters. For example, we can vary the congestion forces by changing the value of ω, which determines the elasticity of the wage with respect to city size. Similarly, we can vary the strength of intra-industry localization economies and inter-industry coagglomeration spillovers by changing the values of λ and ν. 3.3 Computational Procedures The computational process of the agent-based model has five steps: Specify values for the relevant parameters, including the elasticities for intra-industry and inter-industry external economies, the wage parameter and the cost of relocation. Set a maximum number of potential city sites and randomly allocate firms across these potential sites. The random allocation employs a uniform distribution. The maximum number is large enough that there are always vacant sites, so availability of sites is not a constraint. The equilibrium number of cities can be less than, greater than or equal to the initial random number of cities, depending on the values of the relevant parameters. For each industry, identify the firm in the region that has the most to gain from a unilateral relocation, and relocate the firm. In each relocation round, one firm in each industry has the opportunity to relocate, and the industry sequence is random. An additional city will develop if for some firm the autarky profit (for a sole occupant of a city) exceeds the maximum profit among existing cities. Naturally, the relocation of profit-maximizing firms can also cause cities to disappear. Repeat step (iii) until there is no incentive for unilateral relocation, and compute the relevant aggregate statistics, for example, a mixing index for each industry pair. Repeat steps (ii)–(iv) for a large number of initial random distributions. This generates a set of equilibrium allocations for a given set of parameter values. We compute equilibria in a model with three industries (I = 3). This is sufficient to explore the tensions between intra-industry and inter-industry external economies. The parameter values for the baseline case are as follows. The common intra-industry external economy elasticity is λ = 0.08. The coagglomeration elasticity (for inter-industry external economies) is positive for industry pair 1 and 2, ν ∈ [0,0.08] and zero for the other two industry pairs. The value of the wage parameter is ω = 0.08. The relocation cost per firm is roughly 1% of firm revenue. Each run of the model starts with a random distribution of a fixed number of firms across cities. There are 5400 firms, and the initial number of cities is between 80 and 100, depending on the random allocation process.5 We run the model for random draws for ν from a uniform distribution on [0,0.08]. As will become clear later, the results are robust to changes in specification and parameterization. 4. Coagglomeration Spillovers and Equilibrium Mixing This is the first of three sections that present the computational results of the agent-based model. This section focuses on the first key result—the positive and nonlinear relationship between the magnitude of inter-industry external economies and coagglomeration. Figure 1 shows two examples of emergence, one for a moderate coagglomeration elasticity, and one for a relatively high coagglomeration elasticity. Figure 1a shows equilibria that arise from a random pattern of initial locations for a coagglomeration elasticity equal to 0.04 (half the agglomeration elasticity). Each city is represented by a horizontal bar, the length of which indicates the number of firms in the city broken down by industry: industry 1 is black; industry 2 is dark gray; industry 3 is light gray. In this particular case, the Nash equilibria involve only specialized cities, giving M12 = 0. It is worth noting that the unlinked industry 3 has relatively small cities. This is a result of path dependence in the emergence process. Type-3 firms are not linked to other types, and early in the transition from the random allocation to the Nash equilibrium, the type-3 firms relocate to establish homogeneous cities. In the meantime, the link between types 1 and 2 generates relatively large heterogeneous cities. Late in the transition process, most of the large heterogeneous cities disappear, but the legacy of relatively large heterogeneous cities generates relatively large homogeneous cities. In contrast, when the coagglomeration elasticity is zero for all industry pairs, the three types of homogeneous cities are roughly the same size. Figure 1 View largeDownload slide (a) Example equilibrium A, (b) Example equilibrium B. Note: This figure shows two examples of equilibrium that arise from random initial locations with the base case parameters and coagglomeration elasticity ν = 0.04 (Figure 1a) and ν = 0.07 (Figure 1b). Figure 1 View largeDownload slide (a) Example equilibrium A, (b) Example equilibrium B. Note: This figure shows two examples of equilibrium that arise from random initial locations with the base case parameters and coagglomeration elasticity ν = 0.04 (Figure 1a) and ν = 0.07 (Figure 1b). Figure 1b shows equilibria for a coagglomeration elasticity of 0.07. In this case, industries 1 and 2 collocate in mixed cities, while the unlinked industry 3 occupies specialized cities. A majority of the mixed cities are roughly balanced with respect to the two linked industries, a common feature of the Nash equilibria that emerge in the computations, regardless of the value of the coagglomeration elasticity. The mixed cities are also larger than the homogeneous cities, another common feature of the equilibria. This pattern arises because the mixed city has larger agglomeration economies in general (from both inter-industry and intra-industry external economies), so the city can support more firms. Table 1 presents results for running the model with several values of the coagglomeration elasticity. For each elasticity value, we run the model 100 times, starting each time with a distinct random initial allocation.6 The results demonstrate the multiplicity of equilibria generated by a given set of parameter values. At ν = 0.03, equilibrium mixing ranges from M12 = 0 to M12 = 0.2850, with a mean value of 0.0995. Moving to ν = 0.05, the range of M12 is 0 to 0.3744, with a mean value of 0.1821. Moving further to ν = 0.07, the range of M12 is 0.6733 to 0.8717, with a mean of 0.7772. The sixth column shows the average number of mixed cities across the 100 runs. As expected, the number increases with the coagglomeration elasticity. The table illustrates two results: (i) for each coagglomeration elasticity, there is a wide range of equilibrium mixing levels; and (ii) there is a positive relationship between the strength of inter-industry external economies and equilibrium coagglomeration. Table 1 Coagglomeration elasticity and equilibrium mixing Coagglomeration Elasticity  M12 Minimum  M12 Maximum  M12 Mean  M12 Standard deviation  Average number of mixed cities  0.01  0  0.0611  0.0049  0.0102  0.33  0.03  0  0.2850  0.0995  0.0552  4.82  0.05  0  0.3744  0.1821  0.0822  7.31  0.07  0.6733  0.8717  0.7772  0.0425  35.45  Coagglomeration Elasticity  M12 Minimum  M12 Maximum  M12 Mean  M12 Standard deviation  Average number of mixed cities  0.01  0  0.0611  0.0049  0.0102  0.33  0.03  0  0.2850  0.0995  0.0552  4.82  0.05  0  0.3744  0.1821  0.0822  7.31  0.07  0.6733  0.8717  0.7772  0.0425  35.45  Note: This table reports equilibrium values for the mixing index M12 and the share of runs with mixed cities for various values of the coagglomeration elasticity. See the text for function forms and parameters. Figure 2 presents the results of running the computational model with 500 random draws of the coagglomeration elasticity in the range 0–0.08. The results reinforce the notion that profit-maximizing location choices generate multiple equilibria. For each coagglomeration elasticity, there are many equilibrium values of the mixing index. The range of the mixing index is largest for intermediate values of the coagglomeration elasticity. Figure 2 View largeDownload slide Equilibrium mixing. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) for the base case parameters as the elasticity of coagglomeration goes from 0 to 0.08. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 2 View largeDownload slide Equilibrium mixing. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) for the base case parameters as the elasticity of coagglomeration goes from 0 to 0.08. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. The figure shows a positive and nonlinear relationship between the coagglomeration elasticity and equilibrium mixing. Starting with unlinked industries (ν = 0), as inter-industry external economies become stronger, mixing increases at an increasing rate. The dashed line in Figure 2 represents the nonlinear relationship. This relationship is obtained using a local linear polynomial computed using the Epanechnikov (1969) kernel and the rule-of-thumb bandwidth (Silverman, 1986), implemented using the lpoly command in Stata. The nature of the nonlinearity is clear: as the coagglomeration elasticity increases from zero, a low marginal effect for industries that are weakly linked becomes a higher marginal effect as the inter-industry external economies increase. As shown in Figure 2 and Table 2, the relationship between the coagglomeration elasticity ν and the value of the mixing index M12 is nonlinear. Table 2 reports OLS estimates of the marginal effects of the coagglomeration elasticity. For low values of the coagglomeration elasticity, the marginal effects are small: the coefficients are 0.96 for the first bin and 3.58 for the second bin. In these cases, a stronger link between the industries has little effect on the amount of mixing observed in an equilibrium. At moderate levels of the coagglomeration elasticity, the relationship becomes much stronger: the coefficient grows to 3.85 in the third bin and then to 23.65 in the fourth bin. Finally, the marginal effect becomes smaller again at higher levels of the coagglomeration elasticity: the coefficient for the top bin is 12.33. All of the marginal effects are statistically significant.7 Table 2 Coagglomeration and equilibrium mixing—marginal effects   (1)  (2)  (3)  (4)  (5)  (6)    Equilibrium mixing  Relocation costs 2x  Relocation costs 3x  Congestion cost 1.1x  Firm size 2x  Firm size 5x  e: 0 to 0.016  0.96  7.29  15.83  1.17  1.59  1.90    (0.21)  (0.68)  (0.97)  (0.20)  (0.24)  (0.40)  N  89  99  93  110  115  105  e: 0.017 to 0.032  3.58  8.67  7.56  8.16  7.06  7.77    (1.07)  (1.01)  (0.88)  (1.06)  (0.79)  (1.60)  N  95  98  107  96  111  100  e: 0.033 to 0.048  3.85  12.16  9.55  3.07  7.74  11.95    (1.58)  (1.26)  (0.83)  (1.45)  (1.49)  (2.01)  N  94  82  114  90  80  86  e: 0.049 to 0.064  23.65  17.10  11.91  13.83  39.74  14.71    (2.24)  (1.04)  (0.84)  (1.43)  (1.92)  (1.73)  N  123  106  94  107  98  86  e: 0.065 to 0.08  12.13  7.84  12.64  17.85  15.76  12.57    (0.88)  (0.84)  (0.73)  (0.80)  (0.89)  (0.58)  N  101  115  92  97  96  123    (1)  (2)  (3)  (4)  (5)  (6)    Equilibrium mixing  Relocation costs 2x  Relocation costs 3x  Congestion cost 1.1x  Firm size 2x  Firm size 5x  e: 0 to 0.016  0.96  7.29  15.83  1.17  1.59  1.90    (0.21)  (0.68)  (0.97)  (0.20)  (0.24)  (0.40)  N  89  99  93  110  115  105  e: 0.017 to 0.032  3.58  8.67  7.56  8.16  7.06  7.77    (1.07)  (1.01)  (0.88)  (1.06)  (0.79)  (1.60)  N  95  98  107  96  111  100  e: 0.033 to 0.048  3.85  12.16  9.55  3.07  7.74  11.95    (1.58)  (1.26)  (0.83)  (1.45)  (1.49)  (2.01)  N  94  82  114  90  80  86  e: 0.049 to 0.064  23.65  17.10  11.91  13.83  39.74  14.71    (2.24)  (1.04)  (0.84)  (1.43)  (1.92)  (1.73)  N  123  106  94  107  98  86  e: 0.065 to 0.08  12.13  7.84  12.64  17.85  15.76  12.57    (0.88)  (0.84)  (0.73)  (0.80)  (0.89)  (0.58)  N  101  115  92  97  96  123  Note: This table reports the estimates for the piecewise linear relationship between coagglomeration elasticity and equilibrium mixing M12 for (i) the base case parameters, (ii) the case with doubled relocation costs, (iii) the case with tripled relocation costs, (iv) the case with 10% higher congestion costs, (v) the case with doubled firm size and (vi) the case with quintupled firm size. Standard errors are in parenthesis. This relationship is the result of two economic forces. The first is the direct effect: a larger coagglomeration elasticity raises the payoff of firms who collocate. The second effect is indirect: a larger coagglomeration elasticity increases mixing, and greater mixing reduces the incentive of a minority firm to move to a city where it is in the majority. For a small coagglomeration elasticity, mixed cities will be nearly homogeneous, with perhaps 10% minority firms. The system is symmetric, so a minority firm has the option of relocating to a city where it would be in a 90% majority. For small values of the coagglomeration elasticity, mixing is difficult to sustain because there is a large gap between the minority profit and the majority profit. As the coagglomeration elasticity rises, the level of mixing rises as well, so instead of a 10-90 split, there is for example a 40-60 split. There is a smaller gap between the minority profit and the majority profit, so there is less incentive for a minority firm to relocate. The indirect effect reinforces the direct effect, generating the curvature seen in Figure 2. A common feature of Schelling-type models of residential location is the sensitivity of location patterns to changes in the strength of the connection between two types of agents. In a Schelling model, a small change in preferences in favor of like neighbors can cause an abrupt change from a city of integrated neighborhoods to a segregated city. In our model of coagglomeration, a small change increase in the coagglomeration elasticity can also cause an abrupt change from a region of highly specialized cities (mixing index close to 0) to a region of highly mixed cities (a mixing index close to 1). The positive relationship between coagglomeration elasticity and equilibrium coagglomeration documented in this section also justifies using equilibrium collocation to define clusters of related industries, as in Porter (2003) and Delgado et al. (2016). As shown by Helsley and Strange (2014), there is no reason to expect that the equilibrium distribution of firms is efficient. This means that it is possible that observed clusters might not include industries that mutually benefit from collocating. The agent-based analysis in this article suggests that even with a large set of potential Nash equilibrium clustering patterns, there is a tendency for industry pairs with stronger coagglomeration spillovers to mix to a greater extent in equilibrium. The nonlinearity of the relationship has implications for both the measurement of agglomeration economies and the definition of clusters. Regarding the former, when industries are linked but only weakly, little mixing will be observed in equilibrium. In fact, for a wide range of coagglomeration elasticities and for a large fraction of observations, there is no mixing at all, as can be seen in Table 1. This means that estimates of the characteristics of industries that collocate will pick up only the strongly linked industries. The picture painted of the microfoundations of agglomeration economies will be incomplete in this way. Similarly, clusters will be defined by the most strongly linked industries. Weakly linked industries will tend to be excluded. 5. Other Key Results 5.1. Efficiency Consider next the efficiency properties of the Nash equilibria. We address this issue in various ways. Starting from a random allocation of firms across cities, the model executes a series of Pareto-improving relocations of individual firms. A Pareto-improving relocation increases the aggregate total profit, equal to the sum of the profits of all firms in all cities. The relocation of a single firm affects the profits of firms in both the origin city and the destination city: all firms in the two cities experience changes in the wage and thus production cost; linked firms experience changes in quantities produced. In a Pareto-efficient allocation, all Pareto-improving moves have been executed. Table 3 reports the features of Pareto-efficient allocations for different values of the coagglomeration elasticity. For each value, we compute efficient allocations for 10 initial random allocations. As shown in the second and third columns, the average equilibrium level of mixing (values transfered from Table 1) is considerably smaller than the average level of mixing in the computed Pareto optima. The differences are largest for intermediate values of the coagglomeration index. For example, for ν = 0.03, the average value for the mixing index is 0.879 in the efficient allocations, compared to 0.0995 for the Nash equilibria. For ν = 0.05, the average value for the mixing index is 0.957 in the efficient allocations, compared to 0.1821 in the Nash equilibria. Table 3 Nash equilibrium versus Pareto efficiency Coagglomeration elasticity  Nash equilibrium: Mean M12  Pareto efficiency: Mean M12  Nash equilibrium: Average number of mixed cities  Pareto efficiency: Average number of mixed cities  Pareto efficiency: Minimum share of industry 1 in mixed cities  0.01  0.0049  0.057  0.33  23.7  0.021  0.03  0.0995  0.879  4.82  20.4  0.377  0.05  0.1821  0.957  7.31  17.8  0.460  0.07  0.7772  0.981  35.45  15.9  0.480  Coagglomeration elasticity  Nash equilibrium: Mean M12  Pareto efficiency: Mean M12  Nash equilibrium: Average number of mixed cities  Pareto efficiency: Average number of mixed cities  Pareto efficiency: Minimum share of industry 1 in mixed cities  0.01  0.0049  0.057  0.33  23.7  0.021  0.03  0.0995  0.879  4.82  20.4  0.377  0.05  0.1821  0.957  7.31  17.8  0.460  0.07  0.7772  0.981  35.45  15.9  0.480  Note: This table reports equilibrium and Pareto efficient values for the mixing index M12 and the share of runs with mixed cities for various values of the coagglomeration elasticity. See the text for function forms and parameters. In the efficient allocations, each mixed city has a minority firm population. As shown in the last column of Table 3, the minimum share of industry 1 in mixed cities increases with the coagglomeration elasticity. The efficient allocations have matched pairs of unbalanced heterogeneous cities. As shown in the last column, for ν = 0.01 (one-eighth the intra-industry agglomeration elasticity), each matched pair has one city with roughly 98% type-1 firms and 2% type-2 firms, and a second city with a mirror image of 2% type-1 firms and 98% type-two firms. Moving to higher values of the coagglomeration elasticity, the city pairs become progressively more balanced, reaching a 48-52 split for ν = 0.07. In contrast, for low values of the coagglomeration elasticity, the equilibria have a relatively small number of roughly balanced mixed cities. Pareto efficiency requires some firms to suffer the relatively low profits of minority status. For ν < λ (inter-industry external economies are weaker than intra-industry external economies), an individual firm’s profit is maximized with majority status, and there is a profit penalty for minority status. As the gap between ν and λ decreases, the profit-maximizing mix moves closer to an equal split (from say 90% majority to 80% majority). In a particular industry, for every minority outcome in one city, there must be a counterbalancing majority outcome in another city. Because the efficient allocation has unbalanced heterogeneous cities, it cannot in general be supported as a Nash equilibrium. Each firm in minority status could earn higher profit in (i) the mirror city where the firm would be in the majority rather than the minority or (ii) a homogeneous city. Starting from a Pareto-efficient outcome, unilateral relocations to escape minority status causes a system of unbalanced heterogeneous cities to be replaced by a mix of (i) a relatively small number of roughly balanced heterogeneous cities and (ii) many homogeneous cities. Unless some firms are willing to be in the minority and thus ‘take one for the team’, efficient allocations are elusive. 5.2. History: Initial Conditions and Coagglomeration So far, our computations have been based on a initial random allocation of firms across cities. To illustrate the possible role of history in coagglomeration, we run the model with three alternative starting points. Suppose that the initial condition is for all cities to be completely diverse, with each containing an equal proportion of the national population of the three types. There is complete mixing in this situation. In this case, the initial allocation by construction gives equal profits to all firms. As long as autarky profit is sufficiently low, this allocation meets the criteria for an equilibrium, although the allocation is likely to be unstable. As an alternative, suppose we start with complete mixing, and then randomly reallocate 10% of firms. As shown in Figure 3b and column (2) of Table 4, the relationship similar to the baseline results, but not identical. In the table, the largest marginal effects occur for ν between 0.033 and 0.048, rather than for higher values, as in the baseline results. This is because starting with more mixing means that a smaller coagglomeration elasticity is required to have mixing in equilibrium. Table 4 Coagglomeration and equilibrium mixing—marginal effects with variation in initial conditions   (1)  (2)  (3)    Nearly homogeneous  Nearly heterogeneous  Diff. initial coagglomeration elasticity  e: 0 to 0.016  0.00  0.57  13.48    (.)  (0.13)  (1.33)  N  92  92  93  e: 0.017 to 0.032  0.00  12.07  72.22    (.)  (2.18)  (3.78)  N  114  92  101  e: 0.033 to 0.048  0.00  43.90  0.20    (.)  (4.32)  (0.07)  N  105  126  108  e: 0.049 to 0.064  2.58  3.08  -0.01    (0.27)  (0.33)  (0.01)  N  101  91  98  e: 0.065 to 0.08  55.08  1.41  0.05    (2.67)  (0.12)  (0.01)  N  88  99  100    (1)  (2)  (3)    Nearly homogeneous  Nearly heterogeneous  Diff. initial coagglomeration elasticity  e: 0 to 0.016  0.00  0.57  13.48    (.)  (0.13)  (1.33)  N  92  92  93  e: 0.017 to 0.032  0.00  12.07  72.22    (.)  (2.18)  (3.78)  N  114  92  101  e: 0.033 to 0.048  0.00  43.90  0.20    (.)  (4.32)  (0.07)  N  105  126  108  e: 0.049 to 0.064  2.58  3.08  -0.01    (0.27)  (0.33)  (0.01)  N  101  91  98  e: 0.065 to 0.08  55.08  1.41  0.05    (2.67)  (0.12)  (0.01)  N  88  99  100  Note: This table reports the estimates for the piecewise linear relationship between coagglomeration elasticity and equilibrium mixing M12 with base case parameters (i) from a nearly homogeneous initial conditions, (ii) from a nearly heterogeneous initial conditions, (iii) with the base case parameters but with an initial coagglomeration elasticity of 0.08. Standard errors are in parenthesis. Figure 3 View largeDownload slide (a) Equilibrium mixing: nearly homogeneous initial cities. (b) Equilibrium mixing: nearly heterogeneous initial cities. (c) Equilibrium mixing with variation in the initial coagglomeration elasticity Note: This figure shows the solutions for equilibrium mixing from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels. Instead of random initial locations, the simulations begin with all cities nearly homogeneous. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 3 View largeDownload slide (a) Equilibrium mixing: nearly homogeneous initial cities. (b) Equilibrium mixing: nearly heterogeneous initial cities. (c) Equilibrium mixing with variation in the initial coagglomeration elasticity Note: This figure shows the solutions for equilibrium mixing from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels. Instead of random initial locations, the simulations begin with all cities nearly homogeneous. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. At the other extreme, suppose that the initial condition is for all cities to be completely specialized, with each containing only one type of firm. There is no mixing at all in this situation. In this case, the initial allocation is an equilibrium, as shown in Section 2. As an alternative, suppose we begin with completely specialized cities, and then randomly reallocate 10% of firms. The results from this exercise are shown in Figure 3a and column (1) of Table 4. For low levels of coagglomeration elasticity, we continue to have M12 = 0. At higher levels of coagglomeration elasticity, there is a positive relationship. In the table, the relationship is highly nonlinear, with a zero marginal effect for low values of coagglomeration elasticity, a moderate marginal effect of 2.58 for ν between 0.049 and 0.064, and a very large coagglomeration elasticity for ν at the top of the range (between 0.065 and 0.080). Beginning with a system of specialized cities, a large coagglomeration elasticity is required to generate mixing. As a third exercise concerning initial conditions, we can illustrate the possibility of path dependence by exploring the effects of an inter-temporal change in the value of the coagglomeration elasticity. Suppose that initially, the coagglomeration elasticity equals the agglomeration elasticity: for ν = λ, the Nash equilibria feature perfectly heterogeneous cities for types 1 and 2, and homogeneous cities for the unlinked type 3. The value of the mixing index is M12 = 1. Suppose that the coagglomeration elasticity for industries 1 and 2 changes, with some new lower value between 0 and 0.08. Figure 3c shows the relationship between the new coagglomeration elasticity and mixing, with the marginal effects reported in column (3) of Table 4. The relationship is positive and nonlinear for coagglomeration elasticities up to roughly 0.03, and then constant for larger values. History matters: for intermediate values of the new coagglomeration elasticity, a history of heterogeneous cities generates higher levels of coagglomeration. Our three exercises with alternative initial conditions generate two key conclusions. First, if we start with an extreme allocation of either perfect homogeneity (M12 = 0) or perfect heterogeneity (M12 = 1), in the new equilibrium there is a weakly positive relationship between the coagglomeration elasticity and the value of the mixing index. Second, history matters in the sense that the legacy of either homogeneity or heterogeneity biases the equilibrium allocation in a predictable direction, with either higher or lower values of the mixing index. The largest legacy effects are for intermediate values of the coagglomeration elasticity. 5.3. Firm Size Consider next the role of firm size in the allocation of firms across cities. So far the decision-makers in the agent-based model are relatively small. There are 5400 firms, so each agent is 1/5400 of the regional economy. To illustrate the effects of firm size, we run the model with larger firm sizes, measured as the number of workers per firm. We increase firm size by a factor of 2 (generating 2700 firms), and then 5 (generating 1080 firms). As shown in Figure 4, an increase in firm size (more workers per firm) increases mixing. Figure 4a shows that doubling firm size increases mixing relative to the base case, while Figure 4b shows that quintupling firm size increases mixing still further. An increase in firm size partly internalizes intra-industry external economies, and thus weakens intra-industry external economies as a firm substitutes internal scale for external scale. As a result, the tug-of-war between intra-industry external economies (pulling a firm toward cities with firms in the same industry) and inter-industry external economies (pulling a firm toward cities with firms in related industries) is more frequently won by inter-industry economies. The result is more mixing (coagglomeration) and less specialization. Marginal effects are reported in the last two columns of Table 2. For low values of ν, we see large coefficients. As usual, an increase in ν has the largest effect for intermediate values. Figure 4 View largeDownload slide (a) Equilibrium mixing with larger firms (2x), (b) Equilibrium mixing with larger firms (5x). Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except firm size, which are twice as large. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 4 View largeDownload slide (a) Equilibrium mixing with larger firms (2x), (b) Equilibrium mixing with larger firms (5x). Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except firm size, which are twice as large. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. We can contrast our firm-size effects with the conventional wisdom about firm size that originates with Chinitz (1961). It is common to view small firms as promoting the concentration of production in cities. The key idea is that small firms engage the local business community to a greater extent because it would be uneconomical for them to internalize all production activities. This external orientation makes the city more attractive to other small firms. See Rosenthal and Strange (2003, 2010), Agrawal et al. (2008), Glaeser and Kerr (2009), Glaeser et al. (2015) and many others for evidence consistent with this virtuous circle of entrepreneurial activity.8 In contrast with the Chinitz framework, small firms in our model promote the clustering of firms in the same industry at the expense of coagglomeration (clustering of firms in related industries). If intra-industry economies are just slightly stronger than inter-industry economies, there is a bias toward specialized cities. The economic forces underlying the bias are similar to the forces in a Schelling model that generate a bias toward residential segregation when integration is efficient (Schelling, 1971, 1978; Zhang, 2004; O’Sullivan, 2009). In our model, an increase in firm size decreases the power of intra-industry external economies and reduces the bias toward specialization, which increases coagglomeration. The result on firm size is related to the Duranton and Puga (2001) analysis of ‘nursery cities’. They show that a new (small) firm can benefit to a greater extent from the diversity of resources available in a large city. As the firm’s production becomes more routinized and scale economies are internalized, the benefit of diversity decreases, prompting the relocation of larger firms to specialized cities. In contrast, in our analysis, large firms can choose a diversified location without giving up the productivity gains associated with localization to as large an extent. The results in Figure 4 can also be related to the role of developers in urban development. As first proposed in Henderson (1974), the idea is that a relatively large (inefficient) city generates profit opportunities for a developer to create smaller, more profitable cities. The strongest version of this result is that only efficient cities can be sustained in a regional equilibrium. This approach assigns to developers an entrepreneurial role by creating startup cities that generate greater value. In our model, an increase in firm size could be interpreted as the formation of coalitions of relocating firms by developers. A coalition partly internalizes intra-industry external economies and thus promotes coagglomeration. The large firms in our analysis play a similar role to the ‘industry parks’ in Rauch’s (1993) paper on why history matters when it matters little. 5.4. Relocation Cost The equilibria computed thus far have been based on relatively low relocation costs, equal to roughly 1% of total revenue. Figure 5 considers the effects of an increase in relocation costs. Figure 5a shows the computed equilibria with a doubling of relocation cost, and Figure 5b shows the equilibria with a tripling of relocation cost. In both figures, it is apparent that an increase in relocation costs results in a more linear relationship between coagglomeration elasticity and equilibrium mixing. With higher relocation costs, a firm does not relocate when differences in profit are modest, and this tends to dampen the forces that generate specialized cities as the legacy of the initial random allocation persists to a greater extent. Figure 5 View largeDownload slide (a) Equilibrium mixing with higher relocation costs (x2), (b) Equilibrium mixing with higher relocation costs (x3). Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except relocation cost, which are twice as large. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 5 View largeDownload slide (a) Equilibrium mixing with higher relocation costs (x2), (b) Equilibrium mixing with higher relocation costs (x3). Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except relocation cost, which are twice as large. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Columns (2) and (3) of Table 2 report marginal effects for the changes in relocation cost. Without relocation costs, column (1) shows that marginal effects differ by a factor of more than 20. In column (2), with doubled relocation costs, the smallest marginal effect is a little less than half of the largest. A similar pattern appears in column (3), where relocation costs are tripled. 6. Extensions In this section, we present computational results based on alternative parameter values and functional forms, and link regimes. As expected, the positive relationship between spillovers and mixing persists. In other words, the results are robust to changes in parameterization and specification. 6.1. Congestion Consider the effects of changes in value of the parameter of the congestion/wage relationship. As shown in Figure 6a 10% increase in the value of the wage parameter increases mixing in general, with the largest effects in the range ν = 0.04 to ν = 0.06. The share of observations with at least one mixed city increases from 0.60 to 0.75. The 10% increase in the value of the wage parameter decreases the average city size by roughly 10%. In general, there is a negative relationship between city size and mixing, which explains the increase in mixing when wages increases more rapidly with city size. Figure 6 View largeDownload slide Equilibrium mixing with 10% higher congestion costs. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except congestion cost, which are 10% larger. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 6 View largeDownload slide Equilibrium mixing with 10% higher congestion costs. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. Parameters are at base case levels, except congestion cost, which are 10% larger. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Consider next the functional form for the congestion/wage relationship. In the baseline computations, the elasticity of the wage with respect to city size proportional to the natural log of city size. An alternative is a specification where the elasticity is proportional to city size. A switch to this alternative does not change our qualitative results: there is still a positive relationship between the coagglomeration elasticity and mixing, and the relationship continues to be convex for low values of the coagglomeration elasticity. The greater sensitivity of wages to city size makes the relationship a bit more convex. 6.2. Production Function The other key functional form is the production function. Our specification has diminishing marginal productivity for both intra-industry and inter-industry external economies. For a sufficiently large number of like firms (same industry), the marginal product of an unlike firm (inter-industry external economy) exceeds the marginal product of a like firm (intra-industry external economy). In other words, the marginal product of the first unlike firm can exceed the marginal product of the last of a large number of like firms. This feature provides an incentive for at least some industry mixing, with the incentive increasing with the value of the coagglomeration elasticity. As an alternative, consider a production function in which the intra-industry external economy is always greater than the inter-industry external economy:   g(nj)=(1+ni+aSz≠jnz)λ (VI.1) where α < 1. Under this specification, there is no incentive for a firm in one industry to locate in the same city as a firm in another industry. Figure 7 shows the relationship between α and mixing for this specification, with α running from 0 to 1. Mixing occurs only for values of α close to 1, where the effective coagglomeration elasticity is close to the agglomeration elasticity. Figure 7 View largeDownload slide Equilibrium mixing with alternative specification of agglomeration economies. Note: This figure shows the solutions for equilibrium mixing from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. The agglomeration economy is as in equation (IV.1), with alpha ranging from 0 to 1 and other parameters at base case. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 7 View largeDownload slide Equilibrium mixing with alternative specification of agglomeration economies. Note: This figure shows the solutions for equilibrium mixing from equation (III.1) as the elasticity of coagglomeration goes from 0 to 0.08. The agglomeration economy is as in equation (IV.1), with alpha ranging from 0 to 1 and other parameters at base case. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. 6.3. Alternative Linkage Regimes So far our results are computed for three industries, two of which are linked in their production technology. Suppose instead that all three industries are linked, with ν denoting the common coagglomeration elasticity for all three industry pairs. In this situation, it is possible to have four kinds of mixed cities, three with two industries collocating and one with all three industries collocating. Figure A3 in the Online Appendix (see the Supplementary material section) shows a positive relationship between the coagglomeration elasticity and the share of cities that are diverse. Again, we see more mixing with stronger links. As an alternative, suppose there are two pairs of linked industries, with links between industries 1 and 2, and links between industries 2 and 3. Unlike the all-linked case discussed above, industries 1 and 3 and not directly linked. As above, the coagglomeration elasticities are common and equal to ν for both of the linked industry pairs. As noted by Behrens (2016), even though there is no direct link between industries 1 and 3, these industries are indirectly linked by the links that industries 1 and 3 have with industry 2. It is possible that an indirectly linked pair will collocate in equilibrium. Our agent-based approach sheds light on this possibility. The simulations in Table A4 in the Online Appendix available as Supplementary material paint a clear picture: there are two types of heterogeneous cities (one with types 1 and 2 and one with types 2 and 3) and three types of homogeneous cities (one for each industry) that appear as Nash equilibria. There are very few cities that include both unlinked industries (1 and 3), a share of roughly 1% of the 500 runs. Starting with an allocation in which some cities host unlinked industries, unilateral relocations occur as firms flee cities where unlinked firms incur a cost (higher wages) without an compensating benefit. At least in the particular environment of our agent-based model, we do not find evidence of indirect links generating collocation. Of course, it is certainly possible for these ‘third industry’ effects to affect equilibrium location patterns in other settings. For instance, if there are all-or-nothing agglomeration effects as in Ellison et al. between industry 1 and 2 and between industries 2 and 3, then industries 1 and 3 would be found together in mixed cities despite the absence of direct links.9 6.4. Four Industries So far our model includes three industries. We have run a number of computational experiments with four industries instead of three. These experiments generate several predictable conclusions. First, adding a fourth industry that is not linked to the other industries does not affect the qualitative results concerning coagglomeration of the two linked industries (1 and 2). Figure 8 shows this relationship. Second, in the case of two pairs of linked industries (1 and 2 linked and 3 and 4 linked), the Nash equilibria have six types of cities: two types of mixed cities (with 1 and 2 and with 3 and 4), and four types of homogeneous cities (one type for each industry). There are no cities that include unlinked firms. Third, in the case of full linkage (all firms are linked), the Nash equilibria have heterogeneous cities (all four types) and homogeneous cities. In the case of links between 3 of 4 industries (1 and 2, 2 and 3, 1and 3), the Nash equilibria have 3-industry heterogeneous cities (1, 2 and 3), 2-industry heterogeneous cities (1 and 2, 2 and 3, 1 and 3), and homogeneous cities for each industry. The unlinked firms (industry 4) locate in homogeneous cities. Fourth, in the case of competing links (1 and 2, 1 and 3, 1 and 4), the Nash equilibria have three types of heterogeneous cities (one for each linked pair) and three types of homogeneous cities (2, 3, 4). Firms in industries 2, 3 and 4 compete for firms in industry 1, so there are no ‘leftover’ industry-1 firms, and thus no homogeneous industry-1 cities. Figure 8 View largeDownload slide Equilibrium mixing with four industries. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) for the base case parameters as the elasticity of coagglomeration goes from 0 to 0.08. The difference in this figure is that we work with four industries instead of three. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. Figure 8 View largeDownload slide Equilibrium mixing with four industries. Note: This figure shows the solutions for equilibrium mixing M12 from equation (III.1) for the base case parameters as the elasticity of coagglomeration goes from 0 to 0.08. The difference in this figure is that we work with four industries instead of three. The solid line depicts a fitted linear relationship while the dashed line depicts a locally fitted linear polynomial. 7. Conclusions This article uses an agent-based model to explore the industrial composition of cities. Helsley and Strange (2014) show that the set of equilibrium city compositions is very large. In some situations, industries will collocate when there is no benefit from doing so. In others, industries will fail to collocate when there would be a benefit. Our article employs agent-based modeling to refine the equilibrium set. This shows that there is a positive and nonlinear relationship between inter-industry external economies and coagglomeration. The existence of this relationship supports using observed coagglomeration together with data on links between industries to study the microfoundations of agglomeration economies, as in Ellison et al. (2010) and elsewhere. The relationship also supports using observed patterns of industrial clustering to define industry clusters, as in Delgado et al. (2016). Our computational results generate several other key results. The second result concerns efficiency: the equilibrium level of coagglomeration is less than the efficient level. If inter-industry external economies are weaker than intra-industry external economies, the efficient allocation features unbalanced cities, and a firm in the minority has an incentive to flee to a city where the firm is in the majority. This relocation process causes the efficient outcome to unravel. We show the third result—history matters—by computing equilibria with different initial allocations of firms across cities. The equilibrium coagglomeration is relatively high when we start with relatively heterogeneous cities, and relatively low if we start with relatively homogeneous cities. The fourth result is that an increase in firm size increases coagglomeration: the weakening of intra-industry external economies means that the tug-of-war is more frequently won by the forces pulling firms in related industries together. The final result is that an increase in relocation cost increases coagglomeration: increasing friction in the economy means that the legacy of the initial random allocation of firms persists to a greater extent. Supplementary material Supplementary data for this paper are available at Journal of Economic Geography online. Footnotes 1 Collocation refers to industries that independently choose the same location. 2 See also the surveys by Duranton and Puga (2004), Rosenthal and Strange (2004), Behrens and Robert-Nicoud (2015) and Combes and Gobillon (2015). 3 It is worth noting that the specification in equation (III.3) allows for positive production even when ni = 0 for some industry i as long as one industry has positive employment. This allows for specialized cities to exist. 4 It is also worth noting that a firm's productivity is shifted by gi(-), while gi(-) depends directly on the firm's own employment. Agglomeration effects depend on the scale of activity, which includes the firm's activity. 5 We have considered various alternate migration orders, with no systematic difference in results. 6 The result does not change meaningfully with a greater number of runs. 7 The different numbers of observations in Table 2 are a result of the randomization over elasticity. 8 The main alternative to this view is the idea that large firms are ‘anchors’, in the sense that they generate local externalities. See Agrawal and Cockburn (2003) and Feldman (2003). 9 We thank Kristian Behrens for this observation. Acknowledgements We thank Heski Bar-Isaac, Robert Helsley, Jordan Rappaport, Kristian Behrens and two helpful referees for comments on previous versions of the article. 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Published: Mar 1, 2018

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