nursery cities: urban diversity, process innovation, and the life

Nursery Cities:Urban Diversity, Process Innovation, and the

Life Cycle of Products

By GILLES DURANTON AND DIEGO PUGA*

This paper develops microfoundations for the role that diversified cities play infostering innovation. A simple model of process innovation is proposed, where firmslearn about their ideal production process by making prototypes. We build aroundthis a dynamic general-equilibrium model, and derive conditions under whichdiversified and specialized cities coexist. New products are developed in diversifiedcities, trying processes borrowed from different activities. On finding their idealprocess, firms switch to mass production and relocate to specialized cities whereproduction costs are lower. We find strong evidence of this pattern in establishmentrelocations across French employment areas 1993–1996.(JEL R30, O31, D83)

A key issue in urban development, as high-lighted by the writings of Jane Jacobs (1969), isthe role that diversified metropolitan areas playin fostering innovation. To date, no one hasspecified the microfoundations for such a role.Most models of urban systems leave no role todiversity. Those that do simply assume thatlocal diversity is essential to the static efficiency

of certain activities.1 In this paper we developmicrofoundations for the role that diversifiedurban environments play in facilitating searchand experimentation in innovation. We combinethis with a role for specialized environments inmass production. As a result, production relo-cates over the product life cycle from diversifiedto specialized cities—a pattern that we find isstrongly supported by data on establishmentrelocations across French employment areas.

Our model builds on two standard static in-gredients. First, the cost of using a given pro-duction process diminishes as more local firmsuse the same type of process because they canshare intermediate suppliers. Second, urbancrowding places a limit on city size. This com-bination of so-called “localization economies”with congestion costs createsstatic advantagesto urban specialization.Most models of urbansystems rely on variations of these two ele-ments, and thus in equilibrium have only fullyspecialized cities (as in J. Vernon Henderson,1987).

The main novelty of our framework is thesimple model of process innovation that wedevelop and combine with those two more tra-

* Duranton: Department of Geography and Environ-ment, London School of Economics, Houghton Street,London WC2A 2AE, United Kingdom (e-mail:[email protected]; website: http://cep.lse.ac.uk/;duranton);Puga: Department of Economics, University of Toronto,150 St. George Street, Toronto, Ontario M5S 3G7, Canada(e-mail: [email protected]; website: http://dpuga.economics.utoronto.ca). We are very grateful to Fre´deric Laineand Pierre-Philippe Combes for kindly providing us with the data on,respectively, plant relocations and sectoral composition ofemployment for French employment areas; also to VernonHenderson for very helpful discussions, and to NancyGallini, Martin Osborne, and Dan Trefler for detailed com-ments on earlier drafts of this paper. We have also benefitedfrom comments and suggestions by Richard Baldwin, JimBrander, Rikard Forslid, Elhanan Helpman, Boyan Jo-vanovic, Maureen Kilkenny, Tomoya Mori, Nate Rosen-berg, Barbara Spencer, Manuel Trajtenberg, as well as otherparticipants at various seminars and three anonymousreferees. Puga gratefully acknowledges financial supportfrom the Canadian Institute for Advanced Research, theSocial Sciences and Humanities Research Council ofCanada, and from the Connaught Fund of the Universityof Toronto.

1 See John M. Quigley (1998), Hesham M. Abdel-Rahman (2000), and Duranton and Puga (2000a) for de-tailed references and discussion.

1454

ditional ingredients.2 We start from the assump-tion that a young firm needs to experiment torealize its full potential—the entrepreneur mayhave a project, but may not know all the detailsof the product to be made, what components touse, or what kind of workers to hire. Of themany possible ways to implement this project,one is better than all others, and this ideal pro-duction process differs across firms. A firm cantry to find its ideal production process by mak-ing a prototype with any one of the types ofprocesses already used locally. If this process isnot the right one, the firm can try differentalternatives. Once a firm identifies its ideal pro-cess, which happens after using this process fora prototype or after exhausting all other possi-bilities, it can begin mass production of itsproduct. The combination of this learning pro-cess that draws from local types of productionprocesses with costly firm relocation createsdynamic advantages to urban diversity.

Firm turnover is introduced by having somefirms randomly close down each period. Opti-mal investment then ensures they are replacedby new firms producing new products. In addi-tion, migration makes workers in all citiesequally well off. Finally, new cities can becreated by competitive developers.

We solve this model, and derive a set ofnecessary and sufficient conditions for a config-uration in which diversified and specialized cit-ies coexist to be a steady state. We then showthat the same conditions guarantee that thissteady state is stable and unique. When diver-sified and specialized cities coexist, it is becauseeach firm finds it in its best interest to locate ina diversified city while searching for its idealprocess, and later to relocate to a specializedcity where all firms are using the same type ofprocess. Location in a diversified city during afirm’s learning stage can be seen as an invest-ment. It is costly because all firms impose con-

gestion costs on each other, but only those usingthe same type of process create cost-reducinglocalization economies. This results in compar-atively higher production costs in diversifiedcities. However, bearing these higher costs canbe worthwhile for firms in search of their idealprocess because they expect to have to try avariety of processes before finding their idealone, and a diversified city allows them to do sowithout costly relocation after each trial. In thissense, diversified cities act as a “nursery” forfirms. Once a firm finds its ideal productionprocess, it no longer benefits from being in adiverse environment. At this stage, if relocationis not too costly, the firm avoids the congestionimposed by the presence of firms using differenttypes of processes by relocating to a city whereall other firms share its specialization.

In contrast with the lack of theoretical work,there is a wealth of empirical work that studiesthe relative advantages of urban diversity andspecialization. Our theoretical framework isconsistent with the established empirical find-ings. For example, detailed micro evidence sup-ports the benefits of diversity for innovationexhibited by our model. The work of BennettHarrison et al. (1996) and Kelley and SusanHelper (1999) is of particular interest. Theystudy the adoption of new production processesby individual establishments in the UnitedStates belonging to three-digit machine-makingindustries (ranging from heating equipment andplumbing fixtures to guided missiles and air-craft). They show that a diversity of local em-ployment contributes significantly towards theadoption of new production processes, whilenarrow specialization hinders it. Similarly,Maryann P. Feldman and David B. Audretsch(1999) find that local diversity has a strongpositive effect, and narrow specialization a neg-ative one, on the development of new productsreported by trade journals in the United States.

While providing microfoundations for thelink between local diversity and innovation, ourmodel also stresses the advantages of an urbansystem in which diversified and specialized cit-ies coexist. This coexistence is a pervasive fact(see Henderson [1988] for evidence for Brazil,India, and the United States; Duncan Black andHenderson [1998] and Duranton and Puga[2000a] for additional U.S. evidence; Fre´dericLaine and Carole Rieu [1999] for evidence for

2 A significant literature addresses firms’ learning abouttheir technology (see, in particular, Boyan Jovanovic [1982]and Jovanovic and Glenn M. MacDonald [1994]). However,previous modeling approaches cannot be easily embeddedin a general-equilibrium model of a system of cities. Fur-thermore, they focus on firms learning in isolation or, in thestrategic learning literature, on interactions based on imita-tions. The focus of this paper is instead on how the urbanenvironment affects learning, and how firms can best choosetheir environment.

1455VOL. 91 NO. 5 DURANTON AND PUGA: NURSERY CITIES

France). The patterns of specialization and di-versity are too marked to be random outcomes(Glenn Ellison and Edward L. Glaeser, 1997),and do not merely reflect comparative advan-tage (Ellison and Glaeser, 1999). Instead, suchpatterns appear to be to a large extent the resultof economic interactions taking place bothwithin and across sectors (Henderson, 1997a).

To assess the relative advantages of diversityand specialization, Glaeser et al. (1992) exam-ine the evolution of urban employment patternsin U.S. cities. They find that diversity fostersurban employment growth. Pursuing this line ofresearch, Henderson et al. (1995) show that,while urban diversity is indeed important forattracting new and innovative activities, a his-tory of similar past specialization appears tomatter more for retaining mature activities.Pierre-Philippe Combes (2000) finds similar re-sults for France. Henderson (1999) takes a dif-ferent approach, and looks at the evolution ofproductivity in manufacturing plants from high-tech and machinery industries in the UnitedStates. He finds that same-sector specializationtends to have a positive effect on productivity.However, when he looks at employmentchanges, he finds once again that diversity isimportant to attract innovative activities. All ofthis is consistent with the predictions of ourmodel, where firms learn about production pro-cesses that will boost their productivity in di-versified cities, but then relocate to specializedcities to exploit such processes.

Cities have been shown to be very stable interms both of their relative sizes (JonathanEaton and Zvi Eckstein, 1997; Black and Hen-derson, 1998), and of their sectoral composition(Sukkoo Kim, 1995; Guy Dumais et al., 1997;Henderson, 1997b, 1999). This stability is incontrast with a high rate of establishment turn-over. Dumais et al. (1997) calculate that nearlythree-fourths of U.S. plants existing in 1972were closed by 1992, and that more than one-half of all U.S. manufacturing employees in1992 worked in plants that did not exist in 1972.They also show that the opening of new plantstends to reduce the degree of agglomeration ofparticular sectors, suggesting that new plantsare created in locations with below-average spe-cialization in the corresponding sector. InFrance, according to the Institut National de laStatistique et des Etudes Economiques (INSEE,

1998) new establishments are also overwhelm-ingly located in more diversified areas. Thesepatterns are entirely consistent with our model.

The model also assigns a central role to es-tablishment relocations, and has a strong pre-diction for these: a tendency for production torelocate over the life cycle from diversified tospecialized cities. A novel data set tracking es-tablishment relocations across France allows usto directly validate this prediction.

This data set on establishment relocationswas extracted from the Syste`me Informatiquepour le Repertoire des ENtreprises et de leursEtablissements (SIRENE) database of theINSEE. It contains the geographical origin anddestination and the sectoral classification of ev-ery single establishment relocation that tookplace in France between 1993 and 1996 (seeLaine, 1998, for a detailed description). Onlycomplete relocations are included in the data(that is to say, episodes in which the completeclosure of an establishment is followed by theopening in a different location of an establish-ment owned by the same firm and performingthe same full range of activities).

The geographical origin and destination ofrelocating establishments is identified at thelevel of employment areas (zones d’emplois).Continental France is fully covered by 341 em-ployment areas, whose boundaries are definedon the basis of daily commuting patterns aroundurban centers. Relocating establishments areclassified by sector according to level 36 of theNomenclature d’Activite´s Franc¸aise (NAF)classification of the INSEE. The 18 sectors westudy cover all of manufacturing and businessservices, with the exception of postal services.To characterize French employment areas interms of diversity and specialization, we usesectoral employment data for each employmentarea from the Enqueˆte sur la Structure des Em-plois (ESE) of the INSEE for December 1993.We measure the specialization of employmentareai in sectorj by the share of the correspond-ing NAF36 sector in local manufacturing andbusiness-service employment,si

j. We measurethe diversity of employment areai by the in-verse of a Herfindahl index of sectoral concen-tration of local employment, 1/¥ j 5 1

85 (sij)2,

calculated in this case at a higher level of sec-toral disaggregation given by the NAF85 clas-sification. In order to identify employment areas

1456 THE AMERICAN ECONOMIC REVIEW DECEMBER 2001

which are particularly specialized in a givensector or particularly diversified, we normalizeboth measures by their median value for allemployment areas.3

Table 1 presents the evidence. Looking firstat the aggregate figures in the bottom row, wesee that complete establishment relocationsacross French employment areas represented4.7 percent of the average stock over this period(29,358 relocations from an average stock of624,772 establishments). Strikingly, 72 percentof these relocations followed the pattern pre-dicted by our model: they were from an areawith above-median diversity to an area with

above-median specialization in the correspond-ing sector.4

Based on our model, more innovative andagglomerated sectors are likely to benefit themost from the advantages that diversity andspecialization offer at different stages of theproduct cycle. As a result, we would expectestablishments in more innovative and agglom-erated activities to have a greater tendency torelocate from particularly diversified to partic-ularly specialized areas. More traditional sec-tors, by contrast, will tend to experience fewer

3 By these measures, Lyon and Nantes are amongFrance’s most diversified areas, Chateaudun has themedian diversity, while Lavelanet is both the leastdiversified and one of the most specialized areas(in textiles, which in 1993 accounted for 84 per-cent of local manufacturing and business-serviceemployment).

4 The relocation of production over a product’s life cycledoes not always require establishment relocations. MasahisaFujita and Ryoichi Ishii (1998) show that the major Japa-nese electronic firms produce prototypes in trial plants thatare located in metropolitan areas which are known to beparticularly diversified. At the same time, their mass-production plants are almost always located in more spe-cialized cities. Thus, our findings on establishment reloca-tion patterns may just reflect the tip of the iceberg in therelocation of production over the life cycle.

TABLE 1—ESTABLISHMENT RELOCATIONS ACROSSFRENCH EMPLOYMENT AREAS 1993–1996

Percentage ofrelocations from

diversified tospecialized areasa

Relocationsas a

percentage ofthe stockb

Geographicconcentrationc

R&D 93.0 8.1 0.023Pharmaceuticals and cosmetics 88.3 6.4 0.020IT and consultancy services 82.1 7.3 0.030Business services 75.8 5.0 0.015Printing and publishing 73.3 5.4 0.026Aerospace, rail and naval equipment 71.6 3.3 0.026Electrical and electronic equipment 69.1 4.2 0.011Motor vehicles 62.5 2.7 0.020Electrical and electronic components 60.9 5.9 0.007Textiles 46.4 2.5 0.024Chemical, rubber and plastic products 38.3 3.9 0.009Metal products and machinery 37.6 3.2 0.005Clothing and leather 36.3 3.4 0.013Food and beverages 34.6 0.8 0.007Furniture and fixtures 32.6 2.7 0.008Wood, lumber, pulp and paper 30.6 1.7 0.009Primary metals 30.0 2.5 0.009Nonmetallic mineral products 27.3 2.0 0.012

Aggregate 72.0 4.7

Source:Authors’ calculations based on data from SIRENE and ESE.a Percentage of all establishments relocating across employment areas that move from an area with above-median diversity

to an area with above-median specialization.b Establishment relocations across employment areas as a percentage of the average number of establishments.c Ellison and Glaeser (1997) geographic concentration index.


relocations and not necessarily with this nurserypattern. That is precisely what comes out ofTable 1 when we split relocations by sector.R&D, pharmaceuticals and cosmetics, IT andconsultancy services, and business services areinnovative sectors where, as reflected in thegeographic concentration indices on the right-most column, firms find it particularly beneficialto co-locate with firms in the same sector. Thesesectors have specially high relocation rates ofbetween 5 and 8.1 percent. And between 75.8and 93 percent of relocations in these sectors arefrom an area with above-median diversity to anarea with above-median specialization. On theother hand, food and beverages, furniture andfixtures, wood, lumber, pulp and paper, primarymetals, and nonmetallic mineral products aremore traditional sectors which are not particu-larly agglomerated. These sectors have reloca-tion rates of only between 0.8 and 2.7 percent,and less than 35 percent of those relocations arefrom an area with above-median diversity to anarea with above-median specialization. Thus, inproviding microfoundations for the role thatdiversified cities play in fostering innovation,our model helps us understand established styl-ized facts about urban systems, as well as pre-viously unexplored features of firms’ locationand relocation patterns.

The remainder of the paper is structured asfollows. We start by setting up the model (Sec-tion I) and deriving a number of basic results(Section II). These serve as the basis to derive aset of necessary and sufficient conditions for aconfiguration in which diversified and special-ized cities coexist to be a steady state (SectionIII). We then show that the same conditionsguarantee that this steady state is stable andunique (Section IV). We finish with some con-cluding remarks (Section V).

I. The Model

There areN cities in the economy, whereN isendogenous,5 and a continuumL of infinitelylived workers, each of which has one ofmpossible discrete aptitudes. There are equal pro-

portions of workers with each aptitude in theeconomy, but their distribution across cities isendogenously determined through migration.Let us index cities by subscripti and workeraptitudes by superscriptj so thatl i

j denotes thesupply of labor with aptitudej in city i . Time isdiscrete and indexed byt (but to make notationless cumbersome, we only index variables bytime when adding over different time periods).

A. Technology

Setting up a firm involves a one-off start-upcost, which enables the firm to start making trialproducts, referred to as prototypes. Perfectlycompetitive and frictionless capital marketsprovide firms with finance for their start-up costand remunerate workers’ savings. A firm mayeventually engage in mass production, withlower production costs, but this involves using acertain “ideal” production process. This idealprocess is firm specific and randomly drawnfrom a set ofm possible discrete processes, withequal probability for each. Each of them pos-sible processes for each firm requires process-specific intermediate inputs from a local sectoremploying workers of a specific aptitude. Thus,through intermediate production, there is a one-to-one mapping between each firm’s possibleproduction processes and workers’ aptitudes.We say that two production processes for dif-ferent firms are of the same type if they requireintermediates produced using workers with thesame aptitude.

A newly created firm does not know its idealproduction process, but it can find this by trying,one at a time, different processes in the produc-tion of prototypes. After producing a prototypewith a certain process, the firm knows whetherthis process is its ideal one or not. Thus, in orderto switch from prototype to mass production afirm needs to have produced a prototype with itsideal process first, or to have tried all of itsmpossible processes except one. Furthermore, weallow for the possibility that a firm decides tostop searching before learning its ideal process.

Firms have an exogenous probabilityd ofclosing down each period (we can think of thisas being due to the death of a shadow entrepre-neur). Firms also lose a period of productionwhenever they relocate from one city to an-other. Thus, the cost of firm relocation increases

5 N is assumed to be a continuous variable, but forsimplicity we shall refer to it loosely as the “number” ofcities. Similarly, we shall talk about the number and not themass of firms even though there is a continuum of them.


with the exogenous probability of closured.This is because a higher value ofd makesfirms discount future profits more relative tothe profits forgone in the period lost inrelocation.

The intermediates specific to each type ofprocess are produced by a monopolisticallycompetitive intermediate sector a` la Wilfred J.Ethier (1982). As in Fujita (1988) and Abdel-Rahman and Fujita (1990), each such interme-diate sector hires workers of aptitudej and sellsprocess-specific nontradable intermediate ser-vices to final-good firms using a process of typej . These differentiated services enter the pro-duction function of final-good producers withthe same constant elasticity of substitution« 1 1

«. Appendix A provides the equations de-

scribing this set up and shows that technology inprototypes can be summarized by the followingcost function:

(1) C?

ij~h! 5 Qi

jx?

ij~h!,

(2) whereQij 5 ~l i

j!2«w ij, « . 0.

We distinguish variables corresponding toprototypes from those corresponding to mass-produced goods by an accent in the form of aquestion mark, ? (firms that can only produceprototypes are still wondering about their idealproduction process). Indexing the differentiatedvarieties of goods byh, we denote output ofprototypeh made with a process of typej in cityi by x

?ij~h!. Qi

j is the unit cost for firms produc-ing prototypes using a process of typej in cityi , andw i

j is the wage per unit of labor for thecorresponding workers. Note thatQi

j decreasesasl i

j increases: there are localization economiesthat reduce unit costs when there is a largersupply of labor with the relevant aptitude in thesame city (which also implies more firms usingthe same type of process in the same city). Thisis because the larger the supply of labor with thesame aptitude in a city, the larger the range ofintermediate varieties produced in equilibrium;this in turn reduces individual firm costs accord-ing to (2).

When a firm finds its ideal production pro-cess, it can engage in mass production at afraction r of the cost of producing a prototype,

where 0, r , 1. Thus the cost function for afirm engaged in mass production is

(3) Cij~h! 5 rQi

jxij~h!,

where xij(h) denotes the output of mass-

produced goodh, made with a process of typej ,in city i .

With respect to the internal structure of cities,there are congestion costs in each city incurredin labor time and parameterized byt (.0).Labor supply,l i

j, and population,Lij, with apti-

tude j in city i are related by the followingexpression:

(4) l ij 5 Li

jS1 2 t Oj 5 1

m

LijD .

This corresponds to a situation in which work-ers live spread along linear cities in land plots ofunit length, work at the city center, have oneunit of labor time, and lose in commuting afraction of their labor time equal to 2t times thedistance traveled. The expected wage income ofa worker with aptitudej in city i is then (12 t¥ j 5 1

m Lij)wi

j, where the higher land rents paid bythose living closer to the city center are offsetby lower commuting costs (see Fujita [1989] fordetails and several generalizations).

B. Preferences

Turning to consumers, we assume that theyhave a zero rate of time preference.6 Each pe-riod consumers allocate a fractionm of theirexpenditure to prototypes and a fraction 12 mto mass-produced goods. We can interpret this

6 Note that there is no form of accumulation in thismodel. All of consumers’ savings are invested in financingfirms’ start-up costs. In order to provide a nonnegativereturn on investment, each firms’ expected profit streammust be sufficient to recover its start-up cost. This limits thenumber of new firms, and hence investment, at every period.Given that when calculating a firm’s expected profit stream,single-period profits are already discounted by the proba-bility that the firm closes down in any period,d, introducingan additional discount rate through intertemporal consumerpreferences would only obscure expressions without chang-ing the nature of our results.


assumption as there being some fraction of pop-ulation that prefers to buy the most novel goods,and another fraction that prefers to consumemore established products. This separation be-tween the prototype and mass-production mar-kets greatly helps us to obtain closed-formsolutions.7 The instantaneous indirect utility ofa consumer in cityi is

(5) Vi 5 P?

2mP2~1 2 m!eij,

whereei denotes individual expenditure,

(6) P?

5 HOj 51

m EE @p?

ij~h!#12s dh diJ 1/~1 2 s!

,

(7) P 5 HOj 51

m EE @p ij~h!#12s dh diJ 1/~1 2 s!

,

are the appropriate price indices of prototypesand mass-produced goods respectively, andp?

ij~h! andp i

j(h) denote the prices of individualvarieties of prototypes and mass-producedgoods respectively. Double integration overhandi and summation overj include in the priceindices all varieties produced with any type of

process in any city. These price indices areequal in all cities because all final goods,whether prototypes or mass produced, are freelytradable across cities. All prototypes enter con-sumer preferences with the same elasticity ofsubstitution s (.2), and so do all mass-produced goods.8

C. Income and Migration

National income,Y, is the sum of expenditureand investment:

(8) Y 5 Oj 5 1

m E Lijei

jdi 1 P?

mP1 2 mFn.

Lij denotes population with aptitudej in city i .

Investment,P?

mP1 2 mFn, comes from the aggre-gation of the start-up costs incurred by newlycreated firms (this start-up cost is incurred onlyonce, when the firm is first created). To come upwith a new product, but not with the ideal wayto produce it, firms must spendF on marketresearch, purchasing the same combination ofgoods bought by the representative consumer(hence the presence of the price indices in thisexpression). Finally,n denotes the total numberof new firms.

To keep matters simple, we make assump-tions about migration to ensure that we have atmost the following two kinds of cities.

Definition 1[Specialized City]: A city is said tobe (fully) specialized if all its workers have thesame aptitude, so that all local firms use thesame type of production process.

Definition 2[Diversified City]: A city is said tobe (fully) diversified if it has the same propor-tion of workers with each of them aptitudes, sothat there are equal proportions of firms usingeach of them types of production process.

To this effect, we assume that workers knowthe population in each city, but they have im-

7 Nevertheless, if we were instead to make prototypesand mass-produced goods indistinguishable from the pointof view of consumers, our results would be substantivelyunaffected. There would still be a tension between diversity,which reduces the cost of searching for the ideal process,and specialization, which reduces production costs. Theonly noteworthy difference is that, with no separate marketfor prototypes, a firm would always switch to mass produc-tion as soon as it found its ideal process. Specifically, interms of our main result in Proposition 1, we would stillneed to impose conditions analogous to 1, 3, 4, and5—which under this alternative specification we could nolonger write down explicitly. We just would not need acondition analogous to 2—which reduces tor , 1, and isthus trivially satisfied. At the same time, it is not difficult tothink of examples where the segmentation between proto-types and mass-produced goods is quite relevant. For in-stance, it is common practice for Japanese electronics firmsto sell prototypes of their goods to consumers before pro-ducing them at mass scale; in doing this they target aspecific group of consumers through distinct distributionchannels. Similarly, some of Microsoft’s customers arewilling to purchase ab-version of its latest operating systemfor US$60, whereas others are more than happy to wait untilthe first service pack is released.

8 The restrictions . 2 is required for a finite equilibriumnumber of firms. Ifs , 2, an increase in the number of

firms reduces the start-up costP?

mP1 2 mF so much as tomake further firm entry ever more profitable.


perfect information about the distribution ofeach city’s workforce across aptitudes. Specif-ically they only know which, if any, is eachcity’s “dominant” specialization, where a city issaid to have a dominant specialization if thelargest group of local workers with the sameaptitude is above some large enough threshold.Further, workers form their expectations aboutincome in each city as if cities with a dominantspecialization were fully specialized, and as ifcities with no dominant specialization werefully diversified. With all workers forming theirexpectations in this way, their expectations turnout to be rational. A city with a dominant spe-cialization attracts only workers with the dom-inant aptitude, and so in steady state is fullyspecialized. A city with no dominant specializa-tion seems equally attractive for workers of allaptitudes, and so in steady state is fully diver-sified. We further assume that each worker canmigrate only every once in a while.9 This re-flects that considerations such as marriage,childbearing, or divorce greatly affect people’sability to migrate at certain periods (Michael J.Greenwood, 1997). It also precludes situationsin which all workers can relocate simulta-neously where there would be nothing by whichto identify a city (so that we remain consistentwith cities being stable in their size and sectoralcomposition). As a result of this migration pos-sibility all workers are equally well off inequilibrium.

D. City Formation

Each potential site for a city is controlled bya different land development company or landdeveloper, not all of which will be active inequilibrium. Developers have the ability to taxlocal land rents and to make transfers to localworkers. When active, each land developercommits to a contract with any potential workerin its city that specifies the size of the city,whether it has a dominant sector and if sowhich, and any transfers.10 It designs its con-

tract so as to maximize its profit, that is totallocal land rent net of any transfers, subject toworkers’ participation constraint. There is freeentry and perfect competition amongst land de-velopers. This mechanism for city creation issupported by the overwhelming evidence re-garding the role of “large” private agents in cityformation in the United States (Joel Garreau,1991; Henderson and Arindam Mitra, 1996;Henderson and Jacques-Franc¸ois Thisse, 2001).At the same time, even in the absence of landdevelopers, municipal governments with tax-raising powers can play an equivalent role.11

Our main results can be derived both with andwithout land developers—we discuss in SectionIV the few changes resulting from eliminatingthem.

E. Equilibrium Definition

Finally, a steady-state equilibrium in thismodel is a configuration such that all of thefollowing are true. Each developer offers a con-tract designed so as to maximize its profits.Each consumer/worker allocates her income be-tween consumption and savings, allocates herexpenditure across goods, and takes her migra-tion decisions so as to maximize expected util-ity. Each firm chooses a location/productionstrategy and prices so as to maximize its ex-pected lifetime profits. All profit opportunitiesare exploited, and the urban structure is constantover time (where by urban structure we meanthe number of new firms,n, the numbers ofprototype producers,n

?ij, the numbers of mass

producers,n ij, and populations,Li

j).

II. Equilibrium City Sizes

We start by deriving a number of static re-sults that will form the basis for the next sec-tion, which deals with the intertemporaldimension of our model. We first derive equi-librium output per worker and use this to obtainoptimal city sizes. We then show that the activ-ities of perfectly competitive land developers

9 This includes migrations within each city, so as toavoid issues related to endogenous neighborhood formationwhich are not the focus of this paper.

10 See Henderson and Randy Becker (2000) for a dis-cussion of various instruments that allow developers andcommunities to control a city’s size and composition.

11 See Fujita (1989) and Becker and Henderson (2000)for a discussion of this issue and for an equivalence resultbetween these two types of institutions.


pin down equilibrium city sizes to the optimallevel.

LEMMA 1 [Output per Worker]: In equilib-rium, output per worker by firms using pro-cesses of type j in city i in a given period is

n?

ijx?

ij 1 rni

jxij

Lij 5 ~Li

j!«S1 2 t Oj 5 1

m

LijD « 1 1

.

PROOF:Total demand for each variety is the sum of

consumer demand, obtained by application ofRoy’s identity to (5) and integration over allconsumers, and demand by newly created firms,obtained by application of Shephard’s Lemmato their one-off start-up cost,P

?mP1 2 mF, and

multiplication by the number of new firms. Theproduct market-clearing conditions for, respec-tively, prototypes and mass-produced goodsmade with a process of typej in city i are

(9) x?

ij 5 m~p

?

ij!2sP

?s 2 1Y,

(10) xij 5 ~1 2 m!~ pi

j!2sPs 2 1Y.

Note that we have dropped indexh, since short-run equilibrium values may vary by city andtype of process/aptitude, but do not vary byvariety. Using (1) and (3), single-period opera-tional profits can be written as

(11) p?

ij 5 ~p

?

ij 2 Qi

j!x?

ij,

(12) p ij 5 ~ p i

j 2 rQij!xi

j.

Maximizing (11) and (12) with respect toprices, and using (9) and (10), gives the profit-maximizing prices for each prototype and foreach mass-produced good firm. They are fixedrelative markups over marginal costs:

(13) p?

ij 5

s

s 2 1Qi

j,

(14) p ij 5 r

s

s 2 1Qi

j.

Substituting (9), (10), (13), and (14) into (11)and (12) yields maximized operational profitsfor prototype and mass-produced good firms:

(15) p?

ij 5 m

1

sFs 2 1

s

P?

QijG s 2 1

Y,

(16) p ij 5 ~1 2 m!

1

s F1

r

s 2 1

s

P

QijG s 2 1

Y.

Demand for labor can be obtained by appli-cation of Shephard’s Lemma to (1)–(3) andintegration over varieties. The labor market-clearing condition for workers with aptitudej incity i is then

(17) l ij 5 n

?

ij

C?

ij

wij 1 ni

jCi

j

wij

5 ~l ij!2«~n

?

ijx?

ij 1 rni

jxij!.

Substituting (4) into (17), rearranging, and di-viding by Li

j yields the result.

We can now use this expression for equilib-rium output per worker to derive optimal citysize.

LEMMA 2 [Optimal City Size]: Optimal citysize is

L* ;«

~2« 1 1!t.

PROOF:For any given shares of local population with

each aptitude, output per worker in each city, asderived in Lemma 1, and hence the utility ofresidents, are maximized forLi 5 L*.

The size of a city affects its efficiency bychanging the balance between the economies of


localization and congestion costs. Optimal citysize increases with localization economies, asmeasured by«, and diminishes with the conges-tion costs parameter,t. Note that optimal sizefor any city is independent of the compositionof its population, and is thus the same for di-versified and for specialized cities.12 Note fur-ther that this optimal size is finite. With freeentry by competitive profit-maximizing devel-opers into a large urban system, if a finite opti-mal city size exists, the Henry George Theoremapplies.

LEMMA 3 [The Henry George Theorem]:Inequilibrium, all cities achieve optimal size, anddevelopers transfer all land rents in their city tolocal workers, filling the gap between the pri-vate and the public marginal product of labor.

This is a classic result in urban economics(Jan Serck-Hanssen, 1969; David A. Starrett,1974; William S. Vickrey, 1977).13 The intu-ition behind it is straightforward (see Beckerand Henderson [2000] for a particularly helpfulderivation, and Chapter 4 in Fujita and Thisse[2002] for a detailed discussion). Free entryforces competitive profit-maximizing develop-ers to operate at optimal city size. To attain thissize they must make transfers that cover the gapopened by localization economies betweenworkers’ social and private marginal products.With zero profits for developers, total land rentsequal total transfers, and thus are just enough tocover that gap.

Before moving on to explore the intertempo-ral dimension of our framework, it is useful toconsider a static benchmark with no learningstage for firms. In terms of the model, this

would require that upon entry every firm knewits ideal process and that there was no marketfor prototypes. In this casen? i

j50, and outputper worker, as given in Lemma 1, as well asdevelopers’ profits, will be maximized when allcities are fully specialized (as in Henderson[1974] and most subsequent models of systemsof cities). In that static framework, diversityonly imposes costs, since adding labor with adifferent aptitude increases congestion withoutfostering localization economies. Learningchanges this, by creating dynamic advantages todiversity: diversity allows learning firms to pro-duce a sequence of prototypes with differentprocesses without costly relocations. This is acrucial innovation of this model. It fundamen-tally affects the equilibrium urban system byproviding a motivation for the coexistence ofdiversified and specialized cities and for thelocation of production to change over the lifecycle.

III. Nursery Cities

Whenever diversified and specialized citiescoexist, diversified cities act as a nursery forfirms by facilitating experimentation. Special-ized cities, on the other hand, provide an envi-ronment where firms can take full advantage oflower production costs due to localization econ-omies. We now consider a configuration wherefirms relocate across these two environmentswhen they find their ideal process. In checkingwhether this is a steady state, we make sure thatit is in firms’ best interest not to relocate at adifferent point in time.

Definition 3 [Nursery Configuration]: Thenursery configuration is characterized as fol-lows. Diversified and specialized cities coexist.The same proportion of cities specializes ineach type of process. Each new firm locates in adiversified city and produces prototypes using adifferent type of production process each pe-riod. As soon as a firm finds its ideal productionprocess, and only then, it relocates to a cityspecialized in that particular type of process andcommences mass production.

The necessary and sufficient conditions forthe nursery configuration to be a steady stateturn out to depend on three elements: the

12 In our model, a larger city size in itself is not helpful,but a larger population with a given aptitude is. This isbecause aggregate production with any one type of processis a homogenous of degree« 1 1 function of net local laborof the corresponding aptitude, as can be seen in (17). Itmight be more realistic (but less tractable) to have theintensity of aggregate increasing returns decreasing with netlocal employment above a given threshold for each type ofprocess. This would yield a larger optimal size for diversi-fied cities than for specialized cities.

13 The best-known version of the Henry George Theo-rem is associated with local public goods (Frank Flatters etal., 1974; Joseph E. Stiglitz, 1977; Richard J. Arnott andStiglitz, 1979).


relative production costs in diversified and spe-cialized cities, the relative number of prototypeto mass producers, and the expected duration ofthe prototype and mass-production stages. Thefollowing three lemmas characterize thesecomponents.

A. Relative Costs, Relative Number of Firms,and Expected Production Periods

To simplify notation, let us replace subindexiwith D for diversified city variables and withSforspecialized city variables. Further, denote byNDthe equilibrium number of diversified cities, andby NS the equilibrium number of cities specializedin each of them types of production processes.

LEMMA 4 [Relative Costs]:Unit productioncosts in diversified cities relative to those inspecialized cities are

QD

QS5 m«.

PROOF:The unit production costs of (2) become

(18) QD 5 FL*

m~1 2 tL* !G2«

wD ,

QS 5 @L* ~1 2 tL* !#2«wS,

when valued in diversified cities and in special-ized cities respectively. Since, by Lemma 2, allcities are of sizeL* and thus workers receivethe same transfers everywhere, migration en-sures that in steady state, wages are equalizedacross cities:wD 5 wS. Taking the ratio of thetwo equations in (18) then yields the result.

By strengthening localization economies, anincrease in the size of each sector present ineach city has a cost-reducing effect. But it alsohas a cost-increasing effect by increasing citysize, worsening congestion, and raising laborcosts. In specialized cities, all firms use thesame type of production process and contributeto both effects. In diversified cities, however,only firms using the same type of process in anyone period (a fraction 1/m of the total) contrib-ute to localization economies, while all firms

impose on each other congestion costs.14 Thus,a diversified city is a more costly place to pro-duce than a specialized city by a factorm«.

LEMMA 5 [Relative Number of Firms]:Theratio of the total number of prototype producersto the total number of mass producers in thenursery configuration is

V ;NDn

?

D

NSnS

5d~m 1 1! 2 1 1 ~1 2 d!m2 1~1 2 2d!

~1 2 d!2@1 2 ~1 2 d!m2 2~1 2 2d!#.

PROOF:The probability that a firm following the

nursery strategy finds its ideal process in periodt, for 1 # t # m 2 2, is equal to the proba-bility of such a firm not having closed down,(1 2 d)t, times the probability that the processit tries in this period is its ideal one, 1/m. If afirm following the nursery strategy gets to pro-ducem 2 1 prototypes and remains in opera-tion [which happens with probability (2/m)(1 2 d)m2 1], it leaves the diversified city atthat point since it has either just found its idealprocess or otherwise knows that the only pro-cess left to try must be its ideal one. The totalnumber of firms relocating from diversified tospecialized cities each period is therefore a frac-tion [¥t 5 1

m2 2 (1/m)(1 2 d)t 1 (2/m)(1 2d)m2 1] of the number of firms starting up eachperiod, n. After simplification, this becomes1 2 d

dm[1 2 (1 2 d)m22(1 2 2d)]n. The num-

ber of firms arriving each period in specializedcities is a fraction (12 d) of those that relocatedfrom diversified cities the previous period, sincea fractiond closes down in the period of idlenessthat makes relocation costly. With a constantnumber of firms in each city, this quantity mustalso equal the number of firms closing down inspecialized cities each period,dmNSnS:

14 For the sake of symmetry, in steady state there must bethe same proportion of firms using each type of process ineach diversified city in any period. Let us suppose that eachfirm chooses the order in which to try different processesrandomly. Since there is a continuum of firms, by the law oflarge numbers, symmetry will be attained.


(19)~1 2 d!2

dm@1 2 ~1 2 d!m2 2~1 2 2d!#n

5 dmNSnS.

In steady state, the number of firms created eachperiod must equal the total number of closures,which is a fractiond of all existing firms:

(20) n 5 dSNDmn?

D 1d

1 2 dmNSnS 1 mNSnSD.

Eliminatingn from (19) and (20) yields the result.

V can be seen as a measure of how unlikelya firm is to find its ideal production process. Sincefirms can engage in mass production only oncethey learn about their ideal process, wheneverfirms are unlikely to find their ideal process, thenumber of prototype producers is large relative tothe number of mass producers.V is a function ofonly two parameters, the number of types of pro-duction process,m, and the probability of a firmclosing down in any period,d. The larger either ofthese two parameters, the less likely that a firmwill see itself through to the mass-production

stageSV

m. 0,

V

d. 0D. Intuitively, if there

are many possibilities for a firm’s ideal produc-tion process or if the closure rate is high, thereis a large chance that a firm will close downbefore it can find its ideal process.

LEMMA 6 [Expected Production Periods]:The number of periods that a firm following thenursery configuration strategy expects to spendproducing prototypes in a diversified city is

D?

5d~m 1 1! 2 1 1 ~1 2 d!m2 1~1 2 2d!

md2 .

The number of periods it expects to engage in massproduction in a city where all workers have theaptitude that corresponds to its ideal process is

D 5~1 2 d!2 2 ~1 2 d!m~1 2 2d!

md2 .

Now consider a firm that instead locates first in aspecialized city, relocates across specialized citiesto try different production processes, and on find-

ing its ideal process fixes its location. The numberof periods that this firm expects to spend produc-ing prototypes in different specialized cities is

D?

OSC51

m~2 2 d!2d2 $~1 1 m!~2 2 d!d 2 1

1 ~1 2 d!2~m2 1!@1 2 2~2 2 d!d#%.

The number of periods it expects to engage inmass production in a city where all workershave the aptitude that corresponds to its idealprocess is

DOSC51

m~2 2 d!d2 $1 2 d

1 ~1 2 d!2~m2 1!@~3 2 d!d 2 1#%.

PROOF:Since the expected duration of a firm is 1/d,

and since in the nursery configuration, firms thatfind their ideal process have a probability(1 2 d) of closing while relocating to a city ofthe relevant specialization, we have

(21) D?

1D

1 2 d5

1

d.

Also,

(22) D?

5 Ot 5 1

m2 2

tF 1

m~1 2 d! t 2 1

1m 2 t

md~1 2 d! t 2 1G

1 ~m 2 1!2

m~1 2 d!m2 2,

~23! D?

OSC5 Ot51

m22

tF1

m~1 2 d!2~t21!

1m2 t

m~1 2 d!2~t21!d~2 2 d!G

1 ~m 2 1!2

m~1 2 d!2~m2 2!,


(24) DOSC5 S Ot 5 1

m2 2 1

m~1 2 d!2t 2 1

11

m~1 2 d!2~m2 1!D1

d.

Expansion and simplification of (22)–(24), to-gether with (21), yields the expressions in thelemma.

To close the model we need to solve for thegeneral-equilibrium level of investment, whichyields the number of new firms created each pe-riod. It is particularly convenient to use JamesTobin’s (1969)q approach. Tobin’sq is the ratioof the value of one unit of capital to its replace-ment cost. In steady state, the general-equilibriumlevel of investment is that for whichq 5 1. Theasset value of a new firm is equal to its expectedstream of operational profits. The cost of its re-placement is the start-up cost. In this context,Tobin’sq 5 1 condition is therefore equivalent toa condition of zero expected net profits for firms:15

(25) q 5D?

p?

D 1 DpS

P?

mP1 2 mF5 1.

We now have everything we need to derivenecessary and sufficient conditions for the nurs-ery configuration to be a steady state.

B. Necessary and Sufficient Conditions forNursery Cities

PROPOSITION 1 [Nursery Steady State]:The nursery configuration is a steady state ifand only if the following five conditions aresatisfied.

Condition 1: Firms relocate to a specialized cityonce they find their ideal process:

m«~s 2 1! $1

1 2 d.

Condition 2: Firms switch to mass productiononce they find their ideal process:

m«~s 2 1! #1 2 m

mV.

Condition 3: Firms stay in diversified citiesuntil they find their ideal process:

m«~s 2 1! #1

1 2 d1

1 2 d

2

1 2 m

mV.

Condition 4: Firms do not give up the search fortheir ideal process:

m«~s 2 1! #md

m 2 1 1 d

3 FD?

1 SD 21 2 d

md D 1 2 m

mVG .

Condition 5: Firms do not search for their idealprocess by relocating across specialized cities:

m«~s 2 1! #D?

D?

OSC

1D 2 DOSC

D?

OSC

1 2 m

mV.

PROOF:To ensure that the nursery configuration is a

steady state, we need to check the unprofitabil-ity of all possible deviations. This requires cal-culating several profitability ratios. From (15),(16), and Lemma 4,

(26)p?

S

p?

D

5pS

pD5 SQD

QSD s 2 1

5 m«~s 2 1!.

In the nursery configuration, there aremn?D

prototype producers and no mass producers ineach of theND diversified cities, and there arenS mass producers and no prototype producersin each of themNS specialized cities. Hence,using (13) and (14), the price indices of (6) and(7) become

15 This property has been used in applications of Tobin’sq approach to endogenous growth models with monopolis-tic competition (Richard E. Baldwin and Rikard Forslid,2000).


(27) P?

5s

s 2 1~ND mn

?

D !1/~1 2 s!QD ,

(28) P 5 rs

s 2 1~mNSnS!1/~1 2 s!QS.

Substituting (27) into (15) and valuing this ina diversified city, and substituting (28) into(16) and valuing this in a specialized city,yields operational profits in the nursery con-figuration for, respectively, prototype andmass producers:

(29) p?

D 5mY

s~ND mn?

D !,

(30) pS 5~1 2 m!Y

s~mNSnS!.

Taking the ratio of (30) and (29), and using thedefinition of V, yields

(31)pS

p?

D

51 2 m

mV.

Let us start ruling out possible deviationsfrom the end of a firm’s life cycle. Consider afirm in a diversified city that knows its idealprocess. It can follow the nursery strategy, re-locate to a city of the relevant specializationand, if it survives the relocation period (whichhappens with probability 12 d), engage inmass production there for an expected 1/d pe-riod. Alternatively, it could engage in mass pro-duction in the diversified city. The latter optionis not a profitable deviation if and only if

(32)pD

d# ~1 2 d!

pS

d.

Substituting (26) into (32) and rearrangingyields Condition 1. Another possible deviationfor a firm that knows its ideal process is tonevertheless keep producing prototypes. Butthis is not more profitable than engaging in massproduction in the same type of city, providedthat

(33)p?

S

pS5

p?

D

pD# 1.

Using (26) and (31), (33) becomes Condition 2.Conditions 1 and 2 jointly guarantee that anyfirm in a diversified city that knows its idealprocess wants to relocate to a city of the rele-vant specialization and engage in mass produc-tion there.

The next issue is whether a firm located in adiversified city stays there until it finds its idealprocess. Alternatively, it could relocate from adiversified to a specialized city after makingm 2 2 prototypes without finding its idealprocess, not yet knowing which of the tworemaining processes is its ideal one. This is notprofitable if and only if

(34) ~1 2 d!p?

S 11

2

~1 2 d!3

dpS

# p?

D 11

2

~1 2 d!2

dpS.

Substituting (26) and (31) into (34) and rear-ranging yields Condition 3.

There are other possible deviations from thenursery strategy for a firm that initially locatesin a diversified city. A firm could relocate froma diversified to a specialized city, not just withtwo processes left to try, but with any number ofuntried processes between 2 andm 2 1. If itdid so and its ideal process did not correspondto this city’s specialization, it could keep tryingto find its ideal process by relocating to otherspecialized cities. At some point it could giveup the search for its ideal process and remain ina specialized city producing prototypes, or itcould eventually return to a diversified city.Appendix B shows that all of these deviationscan be ruled out without imposing any addi-tional parameter constraints on the nurserysteady state beyond those of Conditions 1–5.

The final step is to show that a firm does notfind it profitable to locate initially in a special-ized rather than in a diversified city. AppendixB rules out deviations involving relocation froma specialized to a diversified city, or a firmgiving up the search for its ideal process afterproducing more than one prototype. Given this,


there are only two other possible deviationsfrom the nursery strategy.

A firm could locate initially in a specialized cityand remain there regardless of the outcome of thefirst trial. If it manages to survive the first period ina specialized city (which happens with probability1 2 d), it will be able to engage in mass produc-tion with probability 1/mor else will keep produc-ing prototypes. This is not a profitable deviationfrom the nursery strategy if and only if

(35) p?

S 1 ~1 2 d!F 1

m

pS

d1

m 2 1

m

p?

S

dG

# D?

p?

D 1 DpS.


The remaining alternative is for a firm tolocate initially in a specialized city and to searchfor its ideal process solely in specialized cities,which would mean relocating from one special-ized city to another between prototypes in orderto try different production processes until find-ing the ideal one, and then staying in a city ofthe relevant specialization to engage in massproduction. This is not a profitable deviationfrom the nursery strategy if and only if

(36) D?

OSCp?

S 1 DOSCpS # D?

p?

D 1 DpS.


Therefore, if Conditions 1–5 are satisfied,with all firms following the nursery strategy, nofirm finds it profitable to deviate from thisstrategy.

C. Discussion

Condition 1 says that for a firm to want torelocate when it finds its ideal process, unitproduction costs need to be sufficiently lower inspecialized cities so as to make the relocationcost (d) worth incurring.

Condition 2 ensures that firms switch to massproduction as soon as they can. For it to besatisfied, mass production needs to be suffi-ciently attractive. The attractiveness of mass

production relative to prototype production de-pends on relative costs, relative market sizes@(1 2 m!/m#, and the relative number of compet-itors in each market~V 5 ND mn

?D /NSmnS!.

Condition 3 considers relocation from a di-versified to a specialized city by a firm that hasfailed to find its ideal process and still hasanother two possibilities to try. This firm relo-cates one period earlier than under the nurserystrategy, and so has a lower probability, 12 d,of making its next prototype. On the other hand,if it makes this next prototype, it will do so at alower cost. However, it may turn out (withprobability 1⁄2) that its ideal process is not thisnext one but the one it left to try last. Then it hasto relocate once more than under the nurserystrategy, delaying the mass-production stage.Condition 3 is satisfied when mass production iscomparatively attractive, when the cost advan-tage of specialized cities is not too large, andwhen (if Condition 4 is satisfied) relocationcosts (as measured byd) are not too low.

According to Condition 4, a firm is deterredfrom giving up the search for its ideal processwhen mass production is comparatively attrac-tive. It is also more likely to stick to the nurserystrategy when the additional costs associatedwith diversified cities are not too large, andwhen the expected number of periods producing

prototypes, D?

, and the expected additional

periods engaged in mass production under thenursery strategy,D 2 (1 2 d)/md, are largerelative to the expected number of periods pro-ducing prototypes if it only tries one process,(m 2 1 1 d)/md.

Finally, a firm may consider searching for itsideal process by relocating from one specializedcity to another. Condition 5 says that it must thentake into account a lower probability of finding itsideal process and, conditional on survival, a longerexpected time before it discovers it. Conse-quently, this deviation is deterred by a low costadvantage of specialized cities, and by massproduction being relatively attractive.

On the whole, the nursery strategy can beseen as a risky investment. Whether it is worth-while or not depends on its cost, on the payoffif successful, and on the likelihood of success.The nursery strategy is costly because, in adiversified city, all firms impose congestioncosts on each other, but only those using the


same type of process create cost-reducing lo-calization economies, and this results in com-paratively higher production costs. If the costadvantage of specialized cities is too large, afirm may find it worthwhile to produce inspecialized cities before finding its ideal pro-cess (Conditions 3–5). On the other hand, ifthe cost advantage is too small, a firm maynever want to incur the cost of moving awayfrom a diversified city (Condition 1). Thepayoff to learning is also important, and thisincreases with the size of the market for mass-produced goods relative to the market forprototypes [(12 m)/m]. It also depends onhow crowded each market is, as measured bythe relative number of firms (V, which in turndepends onm and d). Finally, the likelihoodof a firm finding its ideal process depends onthe number of alternatives (m), and thechances of closure in any period (d). Figure1 illustrates the dependence of Conditions1–5 onm andd.16 The area where the nurseryconfiguration is a steady state is shaded ingray.

A larger value of m makes finding theideal production process more difficult forfirms. Consequently, the nursery strategy be-comes more attractive than other strategiesthat involve relocations while producingprototypes.17

Regardingd, a low value of this parametermakes searching for the ideal process acrossspecialized cities a less costly alternativeto the nursery strategy (Condition 3 anddownward-sloping portion of Condition 5).It also implies that, with all firms followingthe nursery strategy, a higher proportion ofthem will get to the mass-production stage.

This makes it more attractive for a firm todeviate and stop looking for its ideal process(downward-sloping portion of Condition 4),or to keep on producing prototypes even if itfinds its ideal process (Condition 2). On theother hand, a high value ofd makes it unlikelythat a firm makes it to the mass-productionstage. This increases the importance of get-ting higher operational profits while produc-ing prototypes, encouraging firms to searchfor their ideal process across specialized cit-ies (upward-sloping portion of Condition 5)or discouraging them from searching alto-gether (upward-sloping portion of Condition4). It is therefore for intermediate values ofdthat the nursery configuration is a steady state.

IV. Stability and Uniqueness

Having derived necessary and sufficient con-ditions for the nursery configuration to be asteady state, we now show that it has a numberof desirable properties. In particular, those sameconditions guarantee the stability and unique-ness of this steady state.

Regarding stability, while mobility ensuresthat workers in all cities are equally well off,there might be configurations where this equal-ity can be broken by a small perturbation in thespatial distribution of workers. Equilibrium sta-

16 For visual clarity, we ignore thatm is an integer.Conditions 1–5 depend only on three parameters other thanm and d: m, «, and s. Figure 1 is plotted form 5 0.2(prototypes represent 20 percent of the market, with mass-produced goods accounting for the remaining 80 percent),« 5 0.07 (a 1-percent increase in the amount of labor witha certain aptitude net of commuting costs increases a city’soutput from that labor by 1.07 percent), ands 5 4 (firmsmark up marginal costs by1⁄3).

17 There are also circumstances (in particular, a verylarge share of demand being allocated to prototypes) underwhich the increased uncertainty associated with a largervalue of m can deter a firm from trying to find its idealprocess (a violation of Condition 4).

FIGURE 1. DEPENDENCE OFCONDITIONS 1–5 ON m AND d


bility with respect to such perturbations isclosely related to optimal city size.

PROPOSITION 2 [Stability]:The steady stateis stable with respect to small perturbations inthe spatial distribution of workers.

PROOF:From Lemmas 1 and 2, local output per

worker, and hence the utility of residents, area concave function of city size, which reachesa global maximum forL 5 L*. Thus, utilityequalization across cities for workers with thesame aptitude implies that cities of the sametype (diversified cities, or cities with the samespecialization) can be, at most, of two differ-ent sizes: one no greater than the optimal size,L*, and another one no smaller than this.Taking the derivative of output per worker, asderived in Lemma 1, with respect to the num-ber of workers of each type in each city,shows that in a city smaller than optimal size,a small positive perturbation makes the citymore attractive to workers [and by (18), alsoto firms] relative to other cities with a similarcomposition. As workers and firms move in,the size of this city increases. A small nega-tive perturbation has the opposite effect ofpushing more and more workers and firmsaway from such a city. By contrast, in a cityno smaller than the optimal size, a smallpositive perturbation makes the city less at-tractive, while a small negative perturbationmakes the city more attractive to workers andfirms. Consequently, stability requires that allcities of the same type are of the same size,and that no city is smaller thanL*. Lemma 3ensures that these two conditions are satisfied.

PROPOSITION 3 [Uniqueness]:Wheneverthe nursery configuration is a steady state, it isthe unique steady state.

PROOF:Suppose that the nursery configuration is a

steady state—Conditions 1–5 are satisfied—andconsider any candidate for a second steady statewhere at least some fraction of firms does notfollow the nursery strategy. Let us use a9 todistinguish variables under this candidatesteady state from those that apply when all firmsfollow the nursery strategy. Denote byl and

1 2 l the fractions of prototype producers lo-cated in diversified and specialized cities re-spectively. Condition 1 ensures that every massproducer fixes its location in a specialized cityregardless of the strategies of other firms. Recallthat the nursery strategy involves no relocationswhile a firm searches for its ideal process. Afirm could also avoid relocating while searchingfor its ideal process by fixing its location in aspecialized city from the beginning. However,Condition 4 ensures that in this case a firmexpects to engage in mass production for fewerperiods than if it follows the nursery strategyregardless of what other firms do. Thus, giventhat the total expected duration of firms is aconstant 1/d, subject to mass production takingplace in specialized cities as guaranteed by Con-dition 1, the fraction of a firm’s total durationdevoted to mass production is maximized byfollowing the nursery strategy. Consequently,with Conditions 1 and 4 satisfied, any steadystate where at least some fraction of firms doesnot follow the nursery strategy must be charac-terized by a higher ratio of prototype to massproducers than the nursery steady state:V9 .V. In this candidate for a second steady state,just like in the nursery steady state, by (15),(16), and Lemma 4,

(37)p?

9S

p?

9D5

p9Sp9D

5 m«~s 2 1!.

However, from (6), (7), and (13)–(16),

(38)p9S

p?

9D5

1 2 m

mV9@l 1 ~1 2 l!m«~s 2 1!#.

By following step by step the proof of Propo-sition 1, we can derive a set of five conditionsfor the nursery strategy to provide the highestexpected profits for any individual firm underthis candidate for a second steady state, and thusfor the nursery strategy to be a profitable devi-ation for any firm not already following it.These conditions are entirely analogous to Con-ditions 1–5 but withV replaced byV9[l 1(1 2 l) m«(s 2 1)], since the only difference isthat the profitability ratio in equation (38) re-


places the one in (31). Given thatV9 . V andthat [l 1 (1 2 l)m«(s 2 1)] . 1, if Conditions1–5 are satisfied, it is optimal for every firm tofollow the nursery strategy regardless of whatother firms do. Hence, the nursery configurationis the unique steady state.

The expected time it takes to find the idealprocess is minimized by sticking to a diver-sified city while trying different processes.Condition 1 ensures that mass production al-ways takes place in specialized cities becauseproduction costs are sufficiently lower. Thus,in steady state, if at least some fraction offirms is not following the nursery strategy,there will be relatively fewer firms engaged inmass production, and getting to be one ofthem will be even more attractive. Hence, ifthe nursery strategy is optimal when all otherfirms are following it as well (as guaranteedby Conditions 1–5), then this strategy makeseven more sense when some fraction offirms is not following it. Consequently thenursery strategy is a profitable deviationfrom any other configuration. This results inuniqueness.

While our main results can be rederived with-out competitive developers, the uniqueness re-sult of Proposition 3 does rely on their presence.This is because, when Conditions 1–5 are sat-isfied, everyone prefers to have both diversifiedand specialized cities. However, in the absenceof land developers, configurations with onlydiversified or only specialized cities could alsobe equilibria, simply because there would be nocoordinating mechanism to create cities of theother type.18 Yet, even in this case, the nursery

configuration would unambiguously provide ahigher level of welfare than configurations withonly diversified or with only specialized cities(see the discussion paper version of this paper,Duranton and Puga [2000b] for a formalproof).19 It would thus be natural to expect theemergence of some mechanism for city creationthat would act as a coordinating device. Such amechanism, like competitive developers, wouldrestore uniqueness.

V. Concluding Remarks

In the empirical literature and economic policydiscussions about which are the best urban eco-nomic structures, the debate has been mostlyframed in terms of diversity versus specialization,as if the answer was one or the other. This papersuggests instead that both diversified and special-ized urban environments are important in a systemof cities. There is a role for each type of localeconomic environment but at different stages of aproduct’s life cycle. Diversified cities are moresuited to the early stages of a product’s life cycle,whereas more specialized places are better to con-duct mass production of fully developed products.A “balanced” urban system may thus not be onewhere all cities are equally specialized or equallydiversified, but one where both diversified andspecialized cities coexist. In such a system, somecities specialize in churning new ideas and newproducts (which requires a diversified base),whereas other cities specialize in more standard-ized production (which, in turn, is better carriedout in a more specialized environment). For man-ufacturing and services, unlike for agriculture,“sowing” and “reaping” can take place in differentlocations.

18 Similarly, equilibrium city sizes could be above opti-mal size because of a coordination failure preventing anincrease in the number of cities. Note that if Conditions 1–5are not satisfied in equilibrium, there may be only diversi-fied or only specialized cities even without a coordinationfailure. The discussion paper version of this paper charac-terizes these alternative equilibria.

19 The discussion paper version only introduces com-petitive developers at the very end, so the interestedreader can find there our main results rederived withoutdevelopers.


APPENDIX A: UNIT COSTS ASINTERMEDIATE PRICES

Here we derive from first principles the reduced-form cost functions presented in the text. Supposethere arem monopolistically competitive intermediate service sectors. Each of these sectors employsworkers with one of them possible aptitudes to produce differentiated varieties. The cost functionof a sectorj intermediate firm producing varietyg in city i is

(A1) CSij~g! 5 @a 1 byi

j~g!#wij,

whereyij( g) denotes the firm’s output. The expression in parenthesis is the unit labor requirement,

which has both a fixed and a variable component. Thus, there are increasing returns to scale in theproduction of each variety of intermediates.

Each of them types of production process for final-good firms (whether prototype or mass producers)corresponds to the use as inputs of services from one of them intermediate sectors. Intermediate servicesare nontradable across cities. All potential varieties (only some of which will be produced in equilibrium)enter symmetrically into the technology of final-good firms with a constant elasticity of substitution(« 1 1)/«. The cost function of a final-good firm producing prototypeh, using a process of typej, in cityi is

(A2) C?

ij~h! 5 Qi

jx?

ij~h!,

and that of a final-good firm mass producing goodh, which uses a process of typej , in city i is

(A3) Cij~h! 5 rQi

jxij~h!,

where

(A4) Qij 5 H E @qi

j~g!#21/« dgJ2«

,

0 , r , 1, « . 0, andqij( g) is the price of varietyg of intermediate sectorj produced in cityi .

Demand for each intermediate variety is the sum of demand by prototype producers and demandby mass producers, obtained by application of Shephard’s Lemma to (A2) and (A3), respectively,and integration over final-good firms. The market-clearing condition for each intermediate variety isthen:

(A5) yij 5 n

?

ij

C?

ij

qij 1 ni

jCi

j

qij 5 S q

?

ij

Q?

ijD2~« 1 1!/«

~n?

ijx?

ij 1 rni

jxij!,

where pricesq? ij still need to be replaced by their profit-maximizing values, and we have dropped

indexg since all variables take identical values for all intermediate firms in the same sector and city.The profit-maximizing price for each intermediate is a fixed relative markup over marginal cost:

(A6) qij 5 ~« 1 1!bw i

j.

Free entry and exit in intermediates drives maximized profits to zero. From the zero-profitcondition, the only level of output in intermediates consistent with zero profits is


(A7) yij 5

a

b«.

Demand for labor can be obtained by application of Shephard’s Lemma to (A1) and integrationover varieties. Using (A7), the labor market-clearing condition becomes

(A8) l ij 5 si

jCSi

j

w ij 5 si

j@a 1 byij# 5 si

ja« 1 1

«,

wheresij is the equilibrium number of sectorj intermediate firms in cityi .

By choice of units of intermediate output, we can setb 5 («/a)«(« 1 1)2(«11). Using (A8) and(A6), the price indices of (A4) simplify into

(A9) Qij 5 @si

j~qij!21/«#2« 5 ~l i

j!2«w ij.

Equations (A2), (A3), and (A9) are the reduced-form cost functions of equations (1)–(3) in the maintext.

APPENDIX B: FURTHER DEVIATIONS FROM THE NURSERY STRATEGY

This Appendix describes further deviations from the nursery strategy besides those directly ruledout by Conditions 1–5, and shows that these do not impose additional parameter constraints on thenursery steady state.

A firm that has been producing prototypes in a diversified city without finding its ideal processcould relocate to a specialized city, not just with two processes left to try (which is ruled out byCondition 3), but with any numbert of untried processes for 2# t # m 2 1. If it did so and itsideal process did not correspond to this city’s specialization, this firm could keep trying to find itsideal process by relocating to other specialized cities, it could at some point give up the search forits ideal process and remain in a specialized city producing prototypes, or it could eventually return

to a diversified city. Denote byD?

~t! the expected remaining periods of prototype production for thisprototype producer witht untried processes if it continues to follow the nursery strategy, which

results from replacingm by t in D?

, as defined by Lemma 6. Similarly, denote byD(t) the expectednumber of periods of mass production if it continues to follow the nursery strategy, which resultsfrom replacingm by t in D. Thus, the expected profits for this firm over its remaining operating life

if it continues to follow the nursery strategy areD?

~t!p?

D 1 D~t!pS. If instead this firm relocatesto a specialized city where all firms are using a type of process it has not yet tried, and stays therewhatever happens, its expected profits over its remaining operating life are

~1 2 d!Hp?

S 1 ~1 1 d!F1

t

pS

d1

t 2 1

t

p?

S

dGJ .

Using (26) and (31), these are no greater than the expected profits of continuing to follow the nurserystrategy if and only if

(B1) m«~s 2 1! #td

~1 2 d!~t 2 1 1 d! FD?

~t! 1 SD~t! 2~1 2 d!2

td D 1 2 m

mVG .

If Conditions 1 and 2 are satisfied, the right-hand side of (B1) is decreasing int: if a firm is to giveup the search for its ideal process, it prefers to do so as early as possible. Hence, we only need to


check that (B1) is satisfied fort 5 m 2 1. But, if Condition 1 is satisfied, locating in a diversifiedcity for just one period and then relocating permanently to a specialized city is dominated bypermanent location in a specialized city, which is ruled out by Condition 4. We also show below thatConditions 1 and 3–5 rule out strategies that involve relocating from a specialized to a diversifiedcity. Thus, the only remaining possibility for a firm that relocates from a diversified to a specializedcity before finding its ideal process is to relocate across specialized cities until it finds its ideal

process, and then engage in mass production. Denote byD?

OSC~t! the expected remaining periods ofprototype production if it does this, conditional on remaining in operation after the first relocation,

which result from replacingm by t in D?

OSC. Similarly, denote byDOSC(t) the expected periods ofmass production, which results from replacingm by t in DOSC. Thus, the expected profits over itsremaining operating life for a firm that relocates from a diversified to a specialized city not yetknowing which of thet processes it has not yet tried is its ideal one, with the intention of finding this

by relocating across cities of different specialization, is~1 2 d!@D?

OSC~t!p?

S 1 DOSC~t!pS#. Thisis not a profitable deviation from the nursery strategy if and only if

(B2) ~1 2 d!@D?

OSC~t!p?

S 1 DOSC~t!pS# # D?

~t!p?

D 1 D~t!pS.

Substituting (26) and (31) into (B2) and rearranging yields

(B3) m«~s 2 1! #D?

~t!

~1 2 d!D?

OSC~t!1

D~t! 2 ~1 2 d!DOSC~t!

~1 2 d!D?

OSC~t!

1 2 m

mV.

Condition 4 impliesd , 0.5. And, withd , 0.5, dependence of the right-hand side of (B3) ont issuch that, if it is satisfied fort 5 2 (which is guaranteed by Condition 4) and fort 5 m 2 1 (whichis guaranteed by Condition 5), then it is satisfied for all 2# t # m 2 1. Thus Conditions 1–5 ruleout any deviations involving relocation from a diversified to a specialized city before a firm finds itsideal process.

Let us turn to deviations involving initial location in a specialized city beyond those directly ruledout by Conditions 4 and 5. A firm locating initially in a specialized city could search for its idealprocess by relocating across specialized cities for up to its firstt prototypes and, if it has not foundits ideal process by then and is still in operation, relocate to a diversified city and follow the nurserystrategy thereafter. The expected lifetime profits of a firm following this deviation are

~B4!t

m S ~1 2 d!@1 2 ~1 2 d!2t#

t~2 2 d!d2 p?

S 1~1 1 t!~2 2 d!d 2 1 1 ~1 2 d!2t@1 2 ~2 2 d!d#

t~2 2 d!2d2 pSD1

m 2 t

m S1 2 ~1 2 d!2t

~2 2 d!dp?

S 1 ~1 2 d!2t~D?

~m 2 t!p?

D 1 D~m 2 t!pS!D .

With Conditions 1, 4, and 5 satisfied, this expression is no greater than the left-hand side of (36),which in turn is no greater than the expected profits of a firm following the nursery strategy. Thisis therefore not a profitable deviation.

This brings us back to the deviation mentioned above but not formally discussed, in which a firmfollows the nursery strategy for some periods, if unsuccessful continues the search for its idealprocess in specialized cities for some periods, and if still unsuccessful comes back to the nurserystrategy. The expected profits for such a firm from the point at which it first deviates from the nurserystrategy are the result of replacingm in (B4) by the number of processes the firms has not yet triedat that point, replacingt by the number of processes it intends to try if necessary in specialized cities,


and multiplying the result by 12 d. But, if Conditions 1 and 3–5 are satisfied, this is no greater thanthe left-hand side of (B2), which in turn we have already shown is no greater than the expectedprofits of continuing to follow the nursery strategy. So Conditions 1 and 3–5 guarantee that thisdeviation is not profitable either.

This also eliminates all other deviations involving location in diversified cities after some periodsof prototype production in specialized cities. Given the strategy of other firms, the profitability of anystrategy only depends on a firm’s current location and, if it does not know its ideal process, on howmany possibilities there are left. Deviations that result from replacing the nursery strategy for someother strategy after moving from a specialized to a diversified city are therefore covered by theconditions derived so far.

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