ON SPATIAL DYNAMICS* - Home | Princeton Universityerossi/OSD.pdf · DESMET AND ROSSI-HANSBERG: ON SPATIAL DYNAMICS 45 of cities and the observed distribution of city sizes. He shows

JOURNAL OF REGIONAL SCIENCE, VOL. 50, NO. 1, 2010, pp. 43–63

ON SPATIAL DYNAMICS*

Klaus DesmetUniversidad Carlos III de Madrid, Department of Economics, Calle Madrid 126, 28903Getafe, Spain. E-mail: [email protected]

Esteban Rossi-HansbergPrinceton University, Department of Economics, 309 Fisher Hall, Princeton, NJ 08544.E-mail: [email protected]

ABSTRACT. It has long been recognized that the forces that lead to the agglomerationof economic activity and to aggregate growth are similar. Unfortunately, few formalframeworks have been advanced to explore this link. We critically discuss the literatureand present a simple framework that can circumvent some of the main obstacles weidentify. We discuss the main characteristics of an equilibrium allocation in this dynamicspatial framework, present a numerical example to illustrate the forces at work, andprovide some supporting empirical evidence.

1. INTRODUCTION

Economists have long discussed the relationship between agglomerationand growth. As Lucas (1988) points out, not only are both phenomena related toincreasing (or constant) returns to scale, but in many contexts agglomerationforces are the source of the increasing returns that lead to growth. Krugman(1997), after providing a detailed overview of the different economic forces thatcan explain both phenomena, identifies probably the most important challengeof this literature: the difficulty of developing a common framework that incor-porates both the spatial and the temporal dimensions. In other words, what isneeded is a dynamic spatial theory. In this brief paper, we review the recent lit-erature that has emerged to deal with some of the main links between growthand regional economics, discuss the problems that this literature faces, sketcha framework that we believe can be used to further explore the links betweenthe spatial and temporal dimensions, and provide some empirical evidenceconsistent with the forces present in this framework.

The dynamics of the distribution of economic activity in space have beenstudied using three distinct approaches. A first family of models consists of

*Prepared for the conference “Regional Science: The Next Fifty Years,” on the occa-sion of the 50th anniversary of the Journal of Regional Science. We thank the editor, GillesDuranton, and conference participants for useful comments. Financial support from the Comu-nidad de Madrid (PROCIUDAD-CM), the Spanish Ministry of Science (ECO2008-01300) and theEuropean Commission (EFIGE grant 225343) is gratefully acknowledged.

Received: March 2009; revised: June 2009; accepted: June 2009.

C! 2009, Wiley Periodicals, Inc. DOI: 10.1111/j.1467-9787.2009.00648.x

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44 JOURNAL OF REGIONAL SCIENCE, VOL. 50, NO. 1, 2010

dynamic extensions of New Economic Geography models. These models tendto have a small number of locations, typically two. Agglomeration is drivenby standard Krugman (1991) pecuniary externalities operating through realwages. The models are usually made dynamic by adding innovation in productquality as in Grossman and Helpman (1991a,b). There is a wide variety ofparticular specifications, some of which include capital accumulation or otherforms of innovation. Baldwin and Martin (2004) provide a nice survey of thisliterature. They highlight the possibility of “catastrophic” agglomeration, im-plying that only one region accumulates factors. More generally, agglomerationand innovation reinforce each other, creating growth poles and sinks. The emer-gence of regional imbalances is accompanied by faster aggregate growth andhigher welfare in all regions.1

The contribution of this first strand of the literature is important, as itenhances our understanding of the common forces underlying growth and ag-glomeration. However, the spatial predictions are rather limited. The focus on asmall number of locations does not allow this literature to capture the richnessof the observed distribution of economic activity across space, thus restrictingthe way these models are able to connect with the data. It advances statementsabout how unequal two regions are, but there is no sense in which one can havea hierarchy of agglomerated areas. One could of course try to generalize thesemodels to more than a few regions. The problem is that the analytical tractabil-ity breaks down when one deals with more than two or three regions. Someprogress could be made numerically, using dynamic extensions of continuousspace New Economic Geography frameworks, like the one in Fujita, Krugman,and Venables (2001, Chapter 17), but little has been done so far. Therefore,these models remain mostly useful as analytical tools, rather than as guides todoing empirical work.

A second family of models aims to explain the distribution of city sizes.In general, this literature only models, if at all, space within cities, but notthe location of cities across space. Early contributions include Black andHenderson (1999) and Eaton and Eckstein (1997). Black and Henderson (1999)propose a model of a dynamic economy with cities. Increasing returns in theform of externalities create cities and imply, apart from knife-edge parameterconditions, increasing returns at the aggregate level. Hence, as in the papersabove, agglomeration leads to explosive growth. In contrast to the first strandof the literature, these theories have the advantage of explicitly modeling thecities in each location and allowing for heterogeneity in city characteristics.This comes at the cost of a black box agglomeration effect in the form of aproduction externality.

Within this second strand of the literature, the contribution of Gabaix(1999a) is key in establishing the link between the dynamic growth process

1Readers interested in this strand of the literature should consult Baldwin and Martin(2004), Fujita and Thisse (2002, Chapter 11), and some of the specific papers, such as Baldwin,Martin, and Ottaviano (2001) and Martin and Ottaviano (1999, 2001).

C! 2009, Wiley Periodicals, Inc.

DESMET AND ROSSI-HANSBERG: ON SPATIAL DYNAMICS 45

of cities and the observed distribution of city sizes. He shows that Zipf’sLaw for cities—the size distribution is approximated by a Pareto distributionwith coefficient one—can be explained by models that imply cities exhibit-ing scale-independent growth. For our purposes, the interesting part of thiscontribution is not so much the particular size distribution this growth pro-cess leads to, but rather the link it establishes between the dynamic growthprocess of particular production sites and the invariant distribution of eco-nomic activity in space. It is the growth process that leads to agglomeration(in the form of a size distribution with a fat right tail with many large cities),and not the other way around. Following Gabaix (1999a), many papers havebeen built on this basic insight, which had already been used in other appli-cations in macroeconomics. Eeckhout (2004), for example, proposes a simplemodel in which cities grow by receiving scale-independent shocks, and usesthe Central Limit Theorem to show that the resulting size distribution is lognormal.

Gabaix (1999a,b) and Eeckhout (2004) postulate the growth rate of cities;they do not propose an economic theory of this growth process. The last gen-eration of models in this second strand of the literature addresses this short-coming by successfully establishing a link between economic characteristicsthat determine the growth process and economic agglomeration in cities. Du-ranton (2007) does so by proposing a growth process through the mobility ofindustries across cities as a result of innovations in particular sectors. Rossi-Hansberg and Wright (2007) also produce a particular city growth process asa result of adjustment in optimal city sizes and city entry. Cordoba (2008)discusses general properties that these models need to satisfy in order toyield a growth process consistent with particular characteristics of invari-ant distributions, like Zipf’s Law. Some of these papers also establish a re-verse link between the growth process and agglomeration. In Rossi-Hansbergand Wright (2007), for example, it is the organization of economic activityin cities that leads to the aggregate constant returns to scale necessary togenerate balanced growth. In this sense, agglomeration of economic activ-ity in a particular number and size of cities generates aggregate balancedgrowth.

The main limitation of the dynamic frameworks in this second strandof the literature is the lack of geography. Production happens in particularsites, but these sites are not ordered in space and the trade links betweenthem are either frictionless or uniform. Cities are the units in which pro-duction is organized. The internal structure of cities is sometimes modeledas an area with land as a factor of production and agents facing transportand/or commuting costs. However, geography is only modeled within cities, notacross them. In this sense, these models do not present dynamic spatial the-ories that can be contrasted to the observed distribution of economic activityin space.

The third strand of the dynamic spatial literature incorporates fullyforward-looking agents and factor accumulation into models with a continuum



of geographically ordered locations.2 It also allows for either capital mobility orsome form of spillovers or diffusion between regions (Quah, 2002; Brito, 2004;Brock and Xepapadeas, 2008a,b; Boucekkine, Camacho, and Zou, 2009). Apartfrom these interactions, points in space are still completely isolated from eachother. We review the particular structure of these problems in Section 3 below.For now it suffices to say that progress here has been mostly restricted to for-mulating the necessary and sufficient conditions for efficient allocations and,in some cases, the corresponding conditions characterizing rational expectationequilibria. Few substantive results have been advanced.

The remainder of this paper is organized as follows. In Section 2, we gofurther into the importance of developing spatial frameworks that can be com-pared with the data, some of the difficulties of doing this, and the comparisonwith trade frameworks, like that in Eaton and Kortum (1999). Section 3 dis-cusses some of the setups with continuous space that have been analyzed forthe case of forward-looking agents. Section 4 then proposes a simple endoge-nous growth spatial framework in which innovation decisions are optimally notforward-looking, and it uses a numerical example to shed light on the differ-ent forces present in this framework. Section 5 presents some basic evidencefrom the United States on the forces highlighted in Section 4, and Section 6concludes.

2. THE IMPORTANCE OF SPACE

Incorporating geographically ordered space (or land) is important for twomain reasons. Land at a particular location is a rival and nonreplicable inputof production, and land is geographically ordered in a way that matters foreconomic activity. The latter claim has been documented extensively: patentscite geographically close-by patents (Jaffe, Trajtenberg, and Henderson, 1993),firms co-locate (Ellison and Glaeser, 1997; Duranton and Overman, 2005, 2008),and in general there is ample evidence of substantial trade costs, mobilitycosts, commuting costs, and other costs that increase with distance. The use ofland as a nonreplicable input of production requires, perhaps, some additionalexplanation. Economic activity at a particular location is, of course, endogenous,so the factors operating at a given location can be replicated. Nevertheless, sinceland is an input of production, increasing factors at a given location leads todecreasing returns to scale and therefore dispersion.

It is obviously difficult to incorporate space into dynamic frameworks be-cause it increases the dimensionality of the problem. Another difficulty of incor-porating a continuum of locations in geographic space is that, in the presence of

2We discuss in more detail the importance of using a continuum of locations in the nextsection, but the evidence seems to suggest that the observed patterns are very different when land,and not only cities, is incorporated into the analysis. In particular, Holmes and Lee (2008) showthat the distribution of employment across equal sized squares in space has a significantly lowertail than the one for cities. They also show that for space, and in contrast with cities, growth ratesare not independent of scale (Gibrat’s Law).



mobility frictions like transport or commuting costs, clearing factor and goodsmarkets are not trivial. The reason is that how many goods or factors are lost intransit depends on mobility and trade patterns, which in turn depend on factorprices that are the result of market clearing. Hence, to impose market clearingit is necessary to know the number of goods lost in transit. That is, factor pricesat each location depend on the equilibrium pattern of trade and mobility at alllocations. This yields a problem that in many cases is intractable.

The trade literature has circumvented this difficulty by analyzing the caseof a finite (though potentially large) number of locations in the presence ofrandom realizations of productivity for a continuum of goods (see, e.g, Eatonand Kortum, 2002). In such a framework, the only relevant equilibrium variableis the share of exported and imported goods, which is well determined by theproperties of the distributions of the maximum of the productivity realizations.This has proven to be an effective way to deal with this difficulty. However, itdoes not allow us to talk about trade in particular sectors, since only aggregatetrade flows are determined in equilibrium. This is an important drawbackif we want to study geography models that focus on spatial growth acrossindustries. Since the empirical evidence shows that different sectors exhibitvery different spatial growth patterns, this is a relevant issue (see, e.g., Desmetand Fafchamps, 2006; Desmet and Rossi-Hansberg, 2009a).

Another way of solving this problem is to clear markets sequentially. Sup-pose space is linear and compact. Then we can start at one end of the spaceinterval and accumulate production minus consumption in a given market(properly discounted by transport or commuting costs) until we reach the endof the interval. At the boundary, “excess supply” has to be equal to zero inorder for markets to clear. This method, proposed in Rossi-Hansberg (2005),is fairly easy to apply, but it can only be used in one-dimensional (or two-dimensional and symmetric) compact setups. Extending this formulation tononsymmetric two-dimensional spatial setups (like reality!) is a theoreticalchallenge.

In Section, 4 we sketch a model that uses this form of market clearing.Our view is that it is possible to improve our understanding of dynamic spatialinteractions using fully specified economic dynamic equilibrium models. Incontrast, many geographers rely on so-called agent-based models to capturethe complexities of spatial dynamics. The drawback of these models is thatthey lack economic fundamentals (see Irwin, 2009, in this volume on the use ofagent-based models by economists).

3. SPATIAL MODELS WITH FORWARD-LOOKING AGENTS

The few papers that have studied a fully dynamic setup with a continuum oflocations normally focus on the problem of a planner who allocates resources.We present two examples below. Spatial interactions are introduced in twodifferent ways: a first one by allowing for capital mobility, and a second one byassuming a spatial capital externality. Neither of them introduces land as an



input of production, although given that technology is not necessarily assumedto be constant returns to scale, it could be easily incorporated through absenteelandlords.

The spatial setup is the real line and time is continuous. Let c(!, t) denoteconsumption, L(!, t) population, and k(!, t) capital at location ! and time t. Acentral planner then maximizes the sum of utilities of all agents, all of whomdiscount time at rate !. Production requires only capital, k(!, t), which depreci-ates at rate ". Total factor productivity is given by Z(!, t). The change in capitalat a particular location is therefore equal to production minus depreciationminus consumption plus the capital received from other locations. Boucekkineet al. (2009) show how this last term can be expressed as the second partialderivative of capital across locations: essentially, it is just the difference be-tween the flow of capital from the regions to the left minus the flow of capitalflowing to the regions to the right.3 This law of motion of capital, a parabolicdifferential equation, and in particular the spatial component entering throughthe second order term, introduces space into the problem. In addition, capitalat all locations at time 0 is assumed to be known, and since the real line isinfinite, a transversality condition on capital is required. Hence, the problemsolved by Boucekkine et al. (2009) becomes:

maxc

! "

0

!

RU (c (!, t)) L (!, t) e#!td! dt

subject to

"k (!, t)"t

# "2k (!, t)"!2 = Z (!, t) f (k (!, t)) # "k (!, t) # c (!, t)

k(!, 0) = k0 (!) > 0

lim!$±"

"k (!, t)"!

= 0.

Brock and Xepapadeas (2008b) and Brito (2004) solve similar problems,but with different preferences. In fact, Boucekkine et al. (2009) show that forgeneral preferences this is an “ill-posed” problem in the sense that the initialvalue of the co-state does not determine its whole dynamic path. This is ageneral problem in spatial setups. One can address this issue either by consid-ering particular solutions (like the type of cyclical perturbation analysis foundin many studies) or by putting strong restrictions on preferences. Boucekkineet al. (2009) show that some progress can be made by focusing on the linearcase.

3If a region is an interval in space, the capital received from other regions is the differencein the partial derivatives of capital at the two boundary points. When in the limit a region becomesa point in space, the difference in these partial derivatives equals the second partial derivative.



Brock and Xepapadeas (2008b) study a similar problem in a compact in-terval R, given by

maxc

! "

0

!

RU (k (!, t) , c (!, t) , X (!, t)) L (!, t) e#!td! dt

subject to

"k (!, t)"t

= f (k (!, t) , c (!, t) , X (!, t))

X (!, t) =!

!%R#

"! # !&# k

"!&, t

#d!&

k (!, t) = k0 (!) > 0,

where X(!, t) is an externality that affects production and utility, and f nowrefers to production minus consumption plus an additional term reflecting thedirect effect of the externality on the law of motion of capital. In contrast to theproblem of Boucekkine et al. (2009), there is no capital mobility, which elimi-nates a huge difficulty. Instead, the spatial component is introduced throughthe externality, which is just a kernel of capital at all locations. This is an in-teresting problem, since it incorporates diffusion, although not mobility. As inthe previous case, the authors can derive the Pontryagin necessary conditionsfor an optimum and, under more restrictive assumptions, sufficient conditions.Solving for stable steady states remains, nevertheless, an exercise of findingwhether or not spatially uniform steady states are stable. In other words, theyare unable to fully analyze spatially nonuniform steady states. This is progress,although it does not amount to a complete analysis of the problem.

The lack of a complete solution to the problems above is hardly the faultof the authors working on them. These problems are complicated and, absentmore structure, it is hard to extract general insights. The main problem seemsto be that agents are forward-looking and thus need to understand the wholefuture path to make current decisions. Modeling space implies understandingthe whole distribution of economic activity over space and time for each feasibleaction. One way around this difficulty is to impose enough structure—either onthe diffusion of technology or on the mobility of agents and land ownership—sothat agents do not care to take the future allocation paths into account, giventhat they are out of their control and do not affect the returns from currentdecisions. In the next section, we present an example of such a framework.

4. AN ALTERNATIVE MODEL WITH FACTOR MOBILITYAND DIFFUSION

In Desmet and Rossi-Hansberg (2009b), we introduce a model in whichlocations accumulate technology by investing in innovation in one of two in-dustries and by receiving spillovers from other locations. The key to makingsuch a rich structure computable is that diffusion, together with labor mobilityand diversified land ownership, implies that the decisions of where to locate



and how much to invest in technology do not depend on future variables. Asa result, in spite of being forward-looking, agents solve static problems. Thedynamics generated by the model lead to locations changing occupations andemployment density continuously, but in the aggregate the economy convergeson average to a balanced growth path.

Desmet and Rossi-Hansberg (2009b) study an economy with two sectorsand analyze the sectoral interaction in generating innovation. They use themodel to explain the observed evolution in the spatial distribution of eco-nomic activity in the United States. To give a sense of the forces at workin that model, we here present a simpler version of the setup with onlyone good (and therefore no specialization decision or cross-industry innova-tion effects). In this version of the model, factor mobility is frictionless, andtrade is just the result of agents holding a diversified portfolio of land acrosslocations.

Land is given by the unit interval$0, 1

%, time is discrete, and total popula-

tion is L. We divide space into “counties” (connected intervals in [0, 1]), each ofwhich has a local government. Agents solve

max{c(!,t)}"0

E"&

t=0

!tU(c (!, t))

subject to

w (!, t) + R(t)L

= p (!, t) c (!, t) for all t and !,

where p(!, t) is the price of the consumption good and w(!, t) denotes the wageat location ! and time t. Total land rents per unit of land at time t are denotedby R(t), so that R(t)/L is the dividend from land ownership received by agents,assuming that agents hold a diversified portfolio of land in all locations. Freemobility implies that utilities equalize across regions each period.

The inputs of production are land and labor. Production per unit of land isgiven by

x (L (!, t)) = Z (!, t) L (!, t)$ ,

where $ < 1, Z(!, t) denotes TFP, and L(!, t) is the amount of labor per unit ofland used at location ! and time t. The problem of a firm at location ! is thusgiven by

maxL(!,t)

(1 # % (!, t)) (p (!, t) Z (!, t) L (!, t)$ # w (!, t) L (!, t)) ,

where % (!, t) denotes taxes on profits charged by the county government.The government of a county can decide to buy an opportunity to innovate

by taxing local firms % (!, t). In particular, a county can buy a probability & ' 1of innovating at a cost ' (&) per unit of land. This cost ' (&) is increasing andconvex in &, and proportional to wages. If a county innovates, all firms in thecounty have access to the new technology. A county that obtains the chance



to innovate draws a technology multiplier z(!) from a Pareto distribution withlower bound 1,4 leading to an improved level of TFP, z!Zi(!, t), where

Pr [z < z!] ='

1z

(a

.

The risk-neutral government of county G, with land measure I, will thenmaximize

max&(!,t)

!

G

& (!, t)a # 1

p (!, t) Z (!, t) L (!, t)$ d! # I' (&) .(1)

The benefits of the extra production last only one period. Since a county is byassumption small and innovation diffuses geographically, a county’s innova-tion decision today does not affect its expected level of technology tomorrow.This implies that governments need not be forward-looking when choosing theoptimal level of investment in innovation. Note the scale effect in the previousequation: high employment density locations will optimally innovate more (andso will high-price and high-productivity locations). This is consistent with theevidence presented by Carlino, Chatterjee, and Hunt (2007). They show that adoubling of employment density leads to a 20 percent increase in patents percapita.

The timing of the problem is key. Innovation diffuses spatially betweentime periods.5 So, before the innovation decision, location ! has access to

Zi (!, t + 1) = maxr%[0,1]

e#"|!#r|Z (r, t)

which of course includes its own technology. This means that in a given periodeach location has access to the best spatially discounted technology of theprevious period. Agents then costlessly relocate, ensuring that utility is thesame across all locations. After labor moves, counties invest in innovation.Assuming wages are set before the innovation decision, the fact that agentshold a diversified portfolio of land in all locations implies that they need notbe forward-looking when deciding where to locate. Note also that by holding adiversified portfolio of land, rents are redistributed from high-productivity tolow-productivity locations. As a result, high-productivity locations run tradesurpluses, and low-productivity locations run trade deficits.

In addition to the geographic diffusion of innovations, transport costs areanother source of agglomeration. For simplicity we assume iceberg transportcosts, so if one unit of the good is transported from ! to r, only e#(|!#r| units of thegood arrive in r. Hence, if goods are produced in ! and consumed in r, p(r, t) =e(|!#r| p(!, t). As described in Section 2, goods markets clear sequentially. The

4Using the Pareto distribution simplifies the analytics, but is not essential to the argument.5For early work on the spatial diffusion of technologies, see Griliches (1957) and Hagerstrand

(1967).



stock of excess supply between locations 0 and !, H(!, t), is defined by H(0, t) =0 and by the differential equation

" H (!, t)"!

= ) (!, t) x (!, t) # c (!, t)

)&

i

) (!, t) L (!, t)

*

# ( |H (!, t)| .

At each location the change in the excess supply is the difference between thequantity produced and the quantity consumed, net of the shipping cost in termsof goods lost in transit. Then, the goods market clears if H(1, t) = 0. The labormarket clearing condition is given by

! 1

0L (!, t) d! = L, all t.

Computing an equilibrium of this economy is clearly feasible. Given ini-tial productivity functions, we can solve for production in all locations, for thewages that equalize utility and clear the national labor market, for the pricesthat clear the goods market, and for the resulting average land rents, whichare added to agents’ income. This determines the location of agents and theinvestments in innovation. After productivity is realized, we compute actualproduction, actual distributed land rents, and trade. Overnight there is dif-fusion, which determines the new productivity function. Since decisions arebased on current outcomes only, computing an equilibrium involves solving afunctional fixed point each period, but it does not involve calculating rationalexpectations.

What can we learn from this model? Although the model is extremelysimple, it has two forces that are interesting when thinking about spatialdynamics. On the one hand, although technology is constant returns in land andlabor, it exhibits local decreasing returns to labor, because locally land cannotbe replicated. This is a form of local congestion that spreads employment acrossspace given identical technology levels. On the other hand, agglomeration isthe result of the diffusion of technology. Areas with high levels of employmentinnovate more, since the incentives to innovate are larger there. Since diffusiondecreases with distance, areas close to high-employment clusters become high-productivity areas. This attracts employment and leads to more innovation. Asusual, the balance between the congestion and agglomeration forces determinesthe spatial landscape.

The same forces that lead to particular spatial employment patterns alsoexplain aggregate growth. Dispersion implies more uniform, but smaller, in-centives to innovate. In contrast, concentration implies that fewer locationsinnovate, but each of them innovates more. More diffusion implies that the sec-ond (extensive) effect is less important and that aggregate growth is generallygreater.

Perhaps surprisingly, higher trade costs imply more concentrated produc-tion, which in turn may lead to more growth. Although higher trade costs im-ply static efficiency losses, they also lead to dynamic gains through increased



concentration and innovation, an effect reminiscent of the one in Fujita andThisse (2003). A clear empirical implication emerges from the theory: moreconcentration of employment in surrounding areas leads to higher innovationand growth. This effect is the result of two forces. First, more concentrationas a result of, say, transport costs, leads to more innovation. Second, more in-novation in certain areas leads, through diffusion, to productivity growth inneighboring areas (see, e.g., Rosenthal and Strange, 2008, for evidence on thismechanism).6

The model presented above has only one industry, so by construction itis not suited to study cross-industry effects. In Desmet and Rossi-Hansberg(2009b), we present a version of the model with two industries. In that case,another spatial link between the distribution of economic activity and growthemerges. Locations near clusters of firms in one sector, say, manufacturing,experience high prices of the other good, say, services, since their proximityto manufacturing locations allows them to sell services paying small tradecosts. This channel works through trade: neighboring areas that are specializedin manufacturing will import services, thus pushing up the relative price ofservices. As a result, locations close to manufacturing clusters tend to have highemployment and high prices in services and therefore will tend to innovate inservices. Hence, being near clusters in the other industry is also a source ofgrowth and innovation. However, note that this force operates through imports,whereas the diffusion force operates through employment. In the next section,we present some evidence supporting these predictions.

Figure 1 presents a numerical simulation from the framework with twosectors, manufacturing and services. The model used to compute the figureis identical to the one presented in Desmet and Rossi-Hansberg (2009b).The calibration we use here to illustrate the outcome of the model sets" = 50, ( = 0.005, the elasticity of substitution between manufactures andservices equal to 0.5 and the Pareto parameter a = 35. We use a cost functiongiven by ' (&) = '1 + '2/(1 # &) and set '2 = #'1 = 0.003851w(!, t). Desmetand Rossi-Hansberg (2009b) provides a careful discussion on the effect of mostparameters on equilibrium allocations. The figure shows a contour map of pro-ductivity in time and space. Space is the unit interval (divided in 500 locations),and we run the model for 100 periods. We added a scale that shows how differ-ent levels of productivity are represented. The figure helps to identify whereclusters are located and how they are created and destroyed over time.

We use initial conditions that imply that locations close to the upper boundare good in manufacturing, whereas all locations have an initial productivityin services equal to 1. These initial conditions imply that manufacturing startsinnovating first and only in the upper regions. As we argued, diffusion im-plies that regions that innovate are clustered. As a result, productivity growth

6Duranton and Overman (2005, 2008) present detailed and strong evidence of co-locationin the U.K. Their focus is on regional agglomeration mechanisms within and between industries.Unfortunately, they do not directly address the link between growth and regional agglomeration.



FIGURE 1: An Example.

happens in concentrated areas. This is an expression of the first effect discussedabove. Given that innovation clusters coincide with employment clusters, themodel is able to generate “spikes” of economic activity, which could be inter-preted as cities. The result is then similar to models of systems of cities (see, e.g.,Black and Henderson, 1999), but with the advantage that in our frameworkspace is a continuum.



In period 63 some scattered service areas, which are close to manufacturingclusters, start innovating. This innovation happens in clusters too and, moreimportant, next to manufacturing areas. Relative prices of services are highnext to clusters of manufacturing production as a result of transport costs andtrade. This leads to endogenously higher employment and more innovation inservices. This is an expression of the second effect discussed above.

It is important to understand how productivity growth in the service sectorgets jump-started. Assuming an elasticity of substitution less than one, thesector with the higher relative productivity growth loses employment share.Initially, when only manufacturing is innovating, the share of employment inservices is gradually increasing. Since gains from innovation in a given sectordepend on employment in that sector, at some point the service sector becomeslarge enough, allowing for innovation to take off. This mechanism provides anendogenous stabilization mechanism that tends to increase the productivity ofone of the sectors when the economy experiences fast productivity growth inthe other sector. The result is that by period 100 both sectors are growing at aroughly constant rate of around 3 percent.

In Desmet and Rossi-Hansberg (2009b) we match the model to some ofthe main features of the U.S. economy over the last 25 years. Doing so allowsus to analyze the effect of changes in certain relevant parameter values. Asmentioned before, we show, for example, that higher transportation costs mayyield dynamic welfare gains through increased spatial concentration leadingto more innovation. As argued by Holmes (2009) in this volume, having a fullyspecified theoretical model that can be matched to the data and run on thecomputer has much to offer to the field or regional and urban economics. Mostof the empirical work in the field has taken a reduced-form, rather than astructural, approach.7

5. SOME EMPIRICAL EVIDENCE

The model in Section 4 illustrates two main forces that mediate spatialdynamics. The first one is a “spillover” effect by which locations close to otherlocations in the same sector grow faster because they benefit from innovationinvestments close-by. The second is a “trade” effect by which locations closeto areas that import a particular good experience high prices for that good,thus providing incentives to innovate in that sector. If these effects are thecornerstone of spatial dynamics, as the model above postulates, we should beable to find them in the data.

Using U.S. county data for the period 1980–2000 from the Bureau of Eco-nomic Analysis, we first construct two kernels to measure the importance ofthe “spillover” and the “trade” effect. For each county, the first kernel sumsemployment over all other counties, exponentially discounted by distance. To

7For a recent example of this structural approach in regional economics, see Holmes (2008).



compute the second kernel, we first measure county imports in a particularsector as the difference between the county’s consumption and production inthat sector.8 For each county, the second kernel then sums sectoral imports overall counties, exponentially discounted by distance.9 This constitutes a measureof the excess demand experienced by a county in a particular sector. With thesetwo kernels in hand, we run the following regression:

log Empi!(t + 1) # log Empi

!(t) = * + !1 log Empi!(t) + !2 log

"EKi

!(t)#

+ !3 log"IKi

!(t)#,

where Empi!(t) denotes employment, EKi

!(t) the employment kernel, and IKi!(t)

the imports kernel, for sector i, county !, and period t.10,11

Table 1 presents the results for different discount rates. We fix the discountrate for the employment kernel at 0.1 (implying the effect declines by half every7 km),12 and let the decay parameter for the import kernel vary between 0.07and 0.14 (implying the effect declines by half every 5 to 10 km).13 We presentfour sets of regressions, the first two present the results for the service sectorfor the decades 2000–1990 and 1990–1980, and the last two present the sameregressions for the industrial sector (manufacturing plus construction).

To illustrate our results, focus on the case of a decay parameter in theimport kernel of 0.1 (identical to the one in the employment kernel). In services,we find that for the 1990s a 1 percent increase in the initial employment kernelleads to a 0.006 percent increase in county service employment between 1990and 2000. The coefficient on the employment kernel does not change muchacross different decay parameters and across both sectors. We obtain a differentresult for the 1980–1990 decade, where the coefficients are still positive andsignificant, but the coefficient in industry is substantially larger.14

8A county’s consumption in a given sector is obtained by multiplying the national share ofearnings in that sector by the county’s total earnings. A county’s production in a given sector issimply measured by its earnings in that sector. Note that this calculation does not take into accountinternational trade, most of which is in goods. However, since this changes the level of imports ina similar way in all counties, it should not affect our calculations significantly.

9Note that, according to the theory, the discount rate should be related to transport costs.10Since the import kernel measures a discounted sum of imports in a given sector, this

measure may be positive or negative. We can therefore not simply take the natural logarithm. Inthe regression, we use the natural logarithm of the kernel when the kernel is positive and minusthe natural logarithm of the absolute value of the kernel when it is negative.

11Since we include the log of employment in county ! as a separate regressor, the employmentkernel does not include employment in county !. In contrast, the import kernel does include importsby county !.

12This sharp geographic decline in spillovers is consistent with findings in Rosenthal andStrange (2008), who report that human capital spillovers within a range of 5 miles are four to fivetimes larger than at a distance of 5 to 25 miles.

13Using detailed micro-data, Hillberry and Hummels (2008) document that the value ofshipments within the same 5-digit zip code are three times higher than those outside the zip code.

14Dumais, Ellison, and Glaeser (2002) provide firm-level evidence of a “spillover” effect in themanufacturing industry. For a detailed discussion of the effect of current employment on sectoralgrowth rates, see Desmet and Rossi-Hansberg (2009a).



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We also find a positive and robust “trade” effect. In 1980–1990, the effectseems to be similar in both industries. A 1 percent increase in the importkernel implies roughly a 0.002 percent increase in employment growth over thedecade. In the 1990s, the effect is larger in industry and smaller in services. Inalmost all specifications, the “trade” effect is positive and significant. However,note that the model above leaves out another potential effect, namely, thegrowth effect of easier access to inputs in the same industry. This effect wouldimply, on its own, negative coefficients on the import kernel. The only casein which we obtain such a negative coefficient is when we use a low spatialdiscounting coefficient for the import kernel of services in 1990–2000. Sincein that case the negative coefficient is statistically insignificant, we concludethat the trade effect seems to dominate the growth effects from easier accessto inputs. However, more work is needed to explore these different effects.

Table 2 presents regressions similar to the ones in Table 1, but we nowtake sectoral earnings growth as the dependent variable. The results are sim-ilar, and, if anything, the coefficients are larger than for employment growth.According to the theory this should be the case, since the productivity andemployment effect on innovation are complementary, as are the price and em-ployment effects (see Equation (1)). As before, for virtually all decay parameterswe find positive and significant “spillover” and “trade” effects.

6. CONCLUSION

In this paper, we have discussed the theoretical problems involved in thestudy of spatial dynamics. The literature consists of a set of frameworks thathave only been partially understood and analyzed. To deal with some of themain obstacles in this literature, we have presented a simple framework thatuses two main “tricks”: we make the required assumptions to make decisionsstatic and we clear markets sequentially. This approach allowed us to un-derscore two key links between space and time, for which we have providedempirical support. In particular, we have shown that both the “spillover” andthe “trade” innovation effects seem to be present in U.S. county data.

Undoubtedly, much work is still needed. First, we need to understand thebasic frameworks better. In particular, we need to extract a set of robust insightsfrom a model rich enough to be compared with the data. This requires a modelwith many locations and a distribution of economic activity varied enough tocalculate standard statistics. Having two or three regions without land marketsis not enough. Second, we need better ways of comparing these statistics withthe data. What are the main attributes of the evolution of the distribution ofeconomic activity in space that we should compare with the data? What arethe main statistics across industries that can inform us on spatial-dynamiclinkages? Essentially, we need a tighter connection with the data that goesbeyond reduced-form regressions like the ones in Section 5. These are majorchallenges for the next 50 years of regional science!



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ON SPATIAL DYNAMICS* - Home | Princeton Universityerossi/OSD.pdf · DESMET AND ROSSI-HANSBERG: ON SPATIAL DYNAMICS 45 of cities and the observed distribution of city sizes. He shows