Agglomeration and Trade Revisited∗

Gianmarco Ottaviano†, Takatoshi Tabuchi‡

Jacques-François Thisse§

First submission 14 December 1999. Final revision 14September 2000

AbstractThe purpose of this paper is twofold. First, we present an alter-

native model of agglomeration and trade that displays the main fea-tures of the recent economic geography literature while allowing forthe derivation of analytical results by means of simple algebra. Sec-ond, we show how this framework can be used to permit (i) a welfareanalysis of the agglomeration process, (ii) a full-fledged forward look-ing analysis of the role of history and expectations in the emergenceof economic clusters, and (iii) a simple analysis of the impact of urbancosts on the spatial distribution of economic activities.

Keywords: agglomeration, trade, monopolistic competition, self-fulfillingexpectations, urban costs.

J.E.L. Classification: F12, L13, R13.∗We should like to thank four referees for their comments as well as R. Baldwin, G.

Duranton, M. Fujita, P. Jehiel, A. Linbeck, K. Middelfart-Knarvik, T. Mori, D. Peeters, P.Picard, D. Pines, K. Stahl, D.-Z. Zeng, participants to seminars at University of Colorado,Kyoto University, IUI in Stockholm, CEPR ERWIT in Rotterdam, SITE in Stanford,ASSET in Bologna, and Kagawa University. The first author is grateful to the EU TMRprogramme, the second author to the Japan Society for the Promotion of Science throughthe Future Program, and the first and third authors to FNRS (Belgium) for funding. Therevision was made when the first author was Jean Monnet Fellow at EUI Florence (Italy).

†Università ‘L.Bocconi’ Milano, CORE, Université catholique de Louvain, and CEPR.‡Faculty of Economics, University of Tokyo.§CORE, Université catholique de Louvain, CERAS, Ecole nationale des ponts et

chaussées, and CEPR.

1

1 Introduction

The agglomeration of activities in a few locations is probably the most dis-tinctive feature of the economic space. Despite some valuable early contribu-tions made by Hirschman, Perroux or Myrdal, this fact remained unexplainedby mainstream economic theory for a long time. It is only recently that econo-mists have become able to provide an analytical framework explaining theemergence of economic agglomerations in an otherwise homogenous space.As argued by Krugman (1995), this is probably because economists lackeda model embracing both increasing returns and imperfect competition, thetwo basic ingredients of the formation of the economic space, as shown bythe pioneering work of Hotelling (1929), Lösch (1940) and Koopmans (1957).However, even though several modeling strategies are available to study

the emergence of economic agglomerations (Fujita and Thisse, 1996), theirpotential has not been really explored as recognized by Krugman (1998,p.164) himself:

To date, the new economic geography has depended heavily on thetricks summarized in Fujita, Krugman and Venables (1999) with theslogan “Dixit-Stiglitz, icebergs, evolution, and the computer”

The slogan of the new economic geography is explained by the follow-ing methodology (see Fujita et al., 1999). First, the main tool used inthe new economic geography is a particular version of the Chamberlinianmodel of monopolistic competition developed by Dixit and Stiglitz (1977) inwhich consumers love variety and firms have fixed requirements for limitedproductive resources (hence, “Dixit-Stiglitz”). Love of variety is capturedby a CES utility function which is symmetric in a bundle of differentiatedproducts. Each firm is assumed to be a negligible actor in that it has noimpact on overall market conditions. Second, transportation is modeled asa costly activity that uses the transported good itself: in other words, acertain fraction of the good melts on the way (hence, “icebergs”). Takentogether, these assumptions yield a demand system in which the own-priceelasticities of demands are constant, identical to the elasticities of substitu-tions and equal to each other across all differentiated products. This entailsequilibrium prices that are independent of the spatial distribution of firmsand consumers. Though convenient from an analytical point of view, sucha result conflicts with research in spatial competition which shows that de-mand elasticity varies with distance while prices change with the level of

2

demand and the intensity of competition. Moreover, the iceberg assumptionalso implies that any increase in the price of the transported good is ac-companied by a proportional increase in its trade cost, which is unrealistic.Third, the stability analysis used to select spatial equilibria rests on myopicadjustment processes in which the location of mobile factors is driven bydifferences in current returns. Despite some analogy with evolutionary gametheory (hence, “evolution”), this approach neglects the role of expectations(Krugman, 1991a; Matsuyama, 1991), which may be crucial for locationaldecisions since they are often made once-and-for-all.1 Last, notwithstandingtheir simplifying assumptions, the models of the new economic geographyare often beyond the reach of analytical resolution so that authors have toappeal to numerical investigations (hence, “the computer”).2

The purpose of this paper is to propose a complementary modeling strat-egy which allows us to go beyond some of the current limits of the neweconomic geography. In particular, we do that by presenting a model ofagglomeration and trade that, while displaying the main features of the core-periphery model by Krugman (1991b), differs under several major respects.First, preferences are not CES in that we adopt an alternative specificationof the preference for variety, namely the quadratic utility model, which isalso popular in industrial organization (Dixit, 1979; Vives, 1990), in inter-national trade (Anderson et al., 1995; Krugman and Venables, 1990) as wellas in demand analysis (Phlips, 1983). Moreover, while firms are still con-sidered as negligible actors, we adopt a broader concept of equilibrium thanthe one in Dixit and Stiglitz (1977). Second, trade costs are assumed toabsorb resources that are different from the transported good itself. Takentogether, our specifications allow us to disentangle the economic meanings ofthe various parameters thus leading to clear-cut comparative static resultsthat are likely to be easier to test than those based on Dixit and Stiglitz(1977).3 They also entail elasticities of demand and substitution that varywith prices, while equilibrium prices now depend on all the fundamentals ofthe market.

1See, however, Ottaviano (1999) for the analysis of a special case.2The most that can be obtained within this framework without resorting to numerical

solutions has been probably achieved by Puga (1999). See also some chapters of Fujita etal. (1999).

3As an additional example, in Krugman (1991b) the same parameter turns out tomeasure not only the elasticities of demand and substitution but also (inversely) the returnsto scale that remain unexploited in equilibrium.

3

Using this framework, we are able to derive analytically the results ob-tained by Krugman (1991b). Going beyond them, our setting allows usto provide a neat welfare analysis of agglomeration. While natural due tothe many market imperfections that are present in new economic geographymodels, such an analysis is seldom touched due to the limits of the stan-dard approach.4 What we show is that the market yields agglomeration forvalues of the trade costs for which it is socially desirable to keep activitiesdispersed. Hence, while they coincide for high and low values of the tradecosts, the equilibrium and the optimum differ for a domain of intermediatevalues. In this case, there is room for regional policy interventions groundedon both efficiency and equity considerations.In addition, our framework allows us to study forward-looking location

decisions and to determine the exact domain in which expectations matterfor agglomeration to arise. Specifically, we show that expectations influencethe agglomeration process in a totally unsuspected way in that they havean influence on the emergence of a particular agglomeration for intermediatevalues of the trade cost only. For such values, and only for them, if (forwhatever reason) workers expect the lagging region to become the leading one,their expectations will reverse the dynamics of the economy provided that thedifference in initial endowments between the two regions is not too large.Finally, our model is sufficiently flexible to establish a bridge between the

new economic geography and urban economics. We show that it can be easilyextended to accommodate urban costs (Fujita, 1989). This trade-off leadsto a set of results richer than the core-periphery model. When the manufac-tured goods’ trade costs decrease, the economy now displays a scheme givenby dispersion, agglomeration, and re-dispersion (Alonso, 1980).5 Such a re-sult confirms and extends preliminary explorations undertaken by Helpman(1998), Tabuchi (1998), and Puga (1999). It also agrees with the observations

4While natural due to the many market imperfections that are present in new economicgeography models, such an analysis is seldom touched due to the limits of the standardapproach. The utilitarian approach is difficult to justify in the CES case because farmersand workers have different incomes, hence a different marginal utility for the numèraire.See, however, Krugman and Venables (1995) and Helpman (1998) for some numericaldevelopments about the welfare implications of agglomeration in related models. Seealso Trionfetti (2000) for a particular example of a more general insight we will developin the present paper, namely, that in some cases market forces may lead to inefficientagglomeration.

5Other reasons leading to a similar scheme are discussed in Ottaviano and Puga (1998)and Fujita et al. (1999).

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according to which some developed economies (especially the UK) would ex-perience re-dispersion (Geyer and Kontuly, 1996).The organization of the paper reflects what we have said in the foregoing.

The model is presented in the next section while the equilibrium prices andwages are determined in Section 3 for any given distribution of firms andworkers. The process of agglomeration is analyzed in Section 4 by using thestandard myopic approach in selecting the stable equilibria. In Section 5,we compare the optimum and market outcomes. In Section 6, we introduceforward-looking behavior and show how our model can be used to comparehistory (in the sense of initial endowments) and expectations in the emergenceof an agglomeration. The impact of urban costs associated with the formationof an agglomeration is investigated in Section 7. Section 8 concludes.

2 The model

The economic space is made of two regions, called H and F . There aretwo factors, called A and L. Factor A is evenly distributed across regionsand is spatially immobile. Factor L is mobile between the two regions andλ ∈ [0, 1] denotes the share of this factor located in regionH. For expositionalpurposes, we refer to sectorA as ‘agriculture’ and sector L as ‘manufacturing’.Accordingly, we call ‘farmers’ the immobile factor A and ‘workers’ the mobilefactor L. We want to stress the fact, however, that the role of factor A isto capture the idea that some inputs (such as land or some services) arenontradeable while some others have a very low spatial mobility (such as low-skilled workers). Hence our model, as Krugman’s one, should not necessarilybe interpreted as an agriculture-oriented model.There are two goods in the economy. The first good is homogenous. Con-

sumers have a positive initial endowment of this good which is also producedusing factor A as the only input under constant returns to scale and perfectcompetition. This good can be traded freely between regions and is chosenas the numéraire. The other good is a horizontally differentiated product;it is supplied by using L as the only input under increasing returns to scaleand imperfect competition.Each firm in the manufacturing sector has a negligible impact on the

market outcome in the sense that it can ignore its influence on, and hencereactions from, other firms. To this end, we assume that there is a continuumN of potential firms, so that all the unknowns are described by density

5

functions. There are no scope economies so that, due to increasing returnsto scale, there is a one-to-one relationship between firms and varieties. Sinceeach firm sells a differentiated variety, it faces a downward sloping demand.Formally, since there is a continuum of firms, each firm is negligible and

the interaction between any two firms is zero. However, aggregate marketconditions of some kind (here average price across firms) affects any singlefirm. This provides a setting in which individual firms are not competitive(in the classic economic sense of having infinite demand elasticity) but, atthe same time, they have no strategic interactions with one another.Each variety can be traded at a positive cost of τ units of the numéraire

for each unit transported from one region to the other, regardless of thevariety, where τ accounts for all the impediments to trade.Preferences are identical across individuals and described by a quasi-linear

utility with a quadratic subutility which is supposed to be symmetric in allvarieties (see the appendix for more details):

U(q0; q(i), i ∈ [0, N ]) = α

Z N

0

q(i)di− β − γ

2

Z N

0

[q(i)]2di (1)

−γ

2

·Z N

0

q(i)di

¸2

+ q0

where q(i) is the quantity of variety i ∈ [0, N ] and q0 the quantity of thenuméraire. The parameters in (1) are such that α > 0 and β > γ > 0. Inthis expression, α expresses the intensity of preferences for the differentiatedproduct, whereas β > γ means that consumers are biased toward a dispersedconsumption of varieties. Suppose, indeed, that an individual consumes atotal mass of Nq of the differentiated product. If consumption is uniform on[0, x] and zero on (x,N ], then the density on [0, x] is Nq/x. Equation (1)evaluated for this consumption pattern is

U = α

Z x

0

Nq

xdi− β − γ

2

Z x

0

µNq

x

¶2

di− γ

2

·Z x

0

µNq

x

¶di

¸2

+ q0 (2)

= αNq − β − γ

2xN2q2 − γ

2N2q2 + q0

which is strictly increasing in x since β > γ. Hence, regardless of the values ofq and N , (2) is maximized at x = N where variety consumption is maximal.We may then conclude that the quadratic utility function exhibits love of

6

variety as long as β > γ. Finally, for a given value of β, the parameterγ expresses the substitutability between varieties: the higher γ, the closersubstitutes the varieties.6

We use a quasilinear utility that abstracts from general equilibrium in-come effects for analytical convenience. Although this modeling strategygives our framework a fairly strong partial equilibrium flavor, it does notremove the interaction between product and labor markets, thus allowing usto develop a full-fledged model of agglomeration formation, independently ofthe relative size of the manufacturing sector.Any individual is endowed with one unit of labor (of type A or L) and

q0 > 0 units of the numéraire. His budget constraint can then be written asfollows: Z N

0

p(i)q(i)di+ q0 = y + q0

where y is the individual’s labor income, p(i) is the price of variety i while theprice of the agricultural good is normalized to one. The initial endowmentq0 is supposed to be sufficiently large for the equilibrium consumption of thenuméraire to be positive for each individual. By this assumption we want tofocus on interior solutions only. This has some costs in terms of generalitybut, as we will see, larger benefits in terms of simpler analysis. It is alsoconsistent with the idea that each individual is interested in consuming bothtypes of goods. Note that this assumption does not imply that the share ofthe manufacturing sector must be small. It only requires that the equilibriumexpenditure share on the differentiated product is smaller than one.Solving the budget constraint for the numéraire consumption, plugging

the corresponding expression into (1) and solving the first order conditionswith respect to q(i) yields

α− (β − γ)q(i)− γ

Z N

0

q(j)dj = p(i) i ∈ [0, N ]

Therefore, the demand for variety i ∈ [0, N ] is:

q(i) = a− bp(i) + cZ N

0

[p(j)− p(i)]dj (3)

6When β = γ, substitutability is perfect. Indeed, (1) degenerates into a utility funtionthat is quadratic in total consumption

RN

0q(i)di, which is exactly what one would expect

with a homogeneous product.

7

where a ≡ α/[(β + (N − 1)γ], b ≡ 1/[β + (N − 1)γ] and c ≡ γ/(β − γ)[β +(N − 1)γ].7 Increasing the degree of product differentiation among a givenset of varieties amounts to decreasing c. However, assuming that all pricesare identical and equal to p, we see that the aggregate demand for the differ-entiated product equals aN − bpN , which is independent of c. Hence (3) hasthe desirable property that the market size in the industry does not changewhen the substitutability parameter c varies. More generally, it is possibleto decrease (increase) c through a decrease (increase) in the parameter γ inthe utility U while keeping the other structural parameters a and b of thedemand system unchanged. The own price effect is stronger (as measuredby b + cN) than each cross price effect (as measured by c) as well as thesum of all cross price effects (cN), thus allowing for different elasticities ofsubstitution between pairs of varieties as well as for different own elasticitiesat different prices.The indirect utility corresponding to the demand system (3) is as follows:

V (y; p(i), i ∈ [0, N ]) =a2N

2b− a

Z N

0

p(i)di+b+ cN

2

Z N

0

[p(i)]2di (4)

− c2

·Z N

0

p(i)di

¸2

+ y + q0

Turning to the supply side, technology in agriculture requires one unit ofA in order to produce one unit of output. With free trade in agriculture, thechoice of this good as the numéraire implies that in equilibrium the wage ofthe farmers is equal to one in both regions, that is, wAH = w

AF = 1. Technology

in manufacturing requires φ units of L in order to produce any amount ofa variety, i.e. the marginal cost of production of a variety is set equal tozero. This simplifying assumption, which is standard in many models ofindustrial organization, entails no loss of generality when firms’ marginalcosts are incurred in the numéraire. Clearly, φ is a measure of the degree ofincreasing returns in the manufacturing sector.Let nH and nF be the mass of firms in regions H and F , respectively.

Labor market clearing implies that

nH = λL/φ (5)7Notice that when γ = β the indirect demand cannot be inverted in terms of each

variety’s quantity q(i) but only in terms of total quantityR N

0q(i)di̇. Once more, this

is what one would expect with homogeneous products and explains why parameter cdegenerates to infinity as γ tends to β.

8

and

nF = (1− λ)L/φ (6)

Consequently, the total mass of firms (varieties) in the economy is fixed andequal toN = L/φ. This means that, in equilibrium, φ can also be interpretedas an inverse measure of the mass of firms. As φ→ 0 (or L→∞), the massof varieties becomes arbitrarily large. In addition, (5) and (6) show that theregion with the larger labor market is also the region accommodating thelarger proportion of firms. In addition, (5) and (6) imply that any change inthe population of workers located in one region must be accompanied by acorresponding change in the mass of firms.As to equilibrium wages, they are determined as follows. Due to free

entry and exit, profits are zero in equilibrium. As in Krugman (1991b), theequilibrium wages corresponding to (5) and (6) are determined by a biddingprocess between firms for workers, which ends when no firm can earn a strictlypositive profit at the equilibrium market prices. In other words, all operatingprofits are absorbed by the wage bills.Since trade costs are positive, firms have the ability to segment markets,

that is, each firm is able to set a price specific to the market in which itsproduct is sold. Indeed, even for very low trade costs, empirical work showsthat firms succeed in price discriminating among spatially separated markets(Head and Mayer, 2000; McCallum, 1995; Wei, 1996).In the sequel, we focus on regionH. Things pertaining to region F can be

derived by symmetry. Using the assumption of symmetry between varietiesand (3), demands faced by a representative firm located in H in region H(qHH) and region F (qHF ) are given respectively by:

qHH = a− (b+ cN)pHH + cPH (7)

andqHF = a− (b+ cN)pHF + cPF (8)

where

PH ≡ nHpHH + nFpFH

PF ≡ nHpHF + nFpFF

9

Clearly, PH/N and PF/N are the average prices prevailing in region H andF so that PH and PF can be interpreted as the corresponding price indicessince N is fixed. Finally, the profits made by a firm in H are defined asfollows:

ΠH = pHHqHH(pHH)(A/2+λL)+(pHF −τ)qHF (pHF )[A/2+(1−λ)L]−φwH(9)

where A/2 stands for the number of farmers in each region and wH for thewage prevailing in region H.

3 Short-run price equilibria

In this section, we study the process of competition between firms for a givenspatial distribution of workers. Prices are obtained by maximizing profitswhile wages are determined as described above by equating the resultingprofits to zero. Since we have a continuum of firms, each one is negligible inthe sense that its action has no impact on the market. Hence, when choosingits prices, a firm in H accurately neglects the impact of its decision overthe two price indices PH and PF . In addition, because firms sell differenti-ated varieties, each one has some monopoly power in that it faces a demandfunction with finite elasticity.When Dixit and Stiglitz use the CES, the same assumption implies that

each firm is able to determine its price independently of the others becausethe price index enters the demand function as a multiplicative term. This nolonger holds in our model because the price index now enters the demandfunction as an additive term (see (7) and (8)). Stated differently, a firmmust account for the distribution of the firms’ prices through some aggregatestatistics, given here by the price index, in order to find its equilibriumprice. As a consequence, our market solution is given by a Nash equilibriumwith a continuum of players in which prices are interdependent: each firmneglects its impact on the market but is aware that the market as a whole hasa non-negligible impact on its behavior. As a result, the equilibrium priceswill depend on key aspects of the market instead of being given by a simplerelative mark-up rule.Since profit functions are concave in own price, solving the first order

conditions for profit maximization with respect to prices yields the equilib-rium prices (denoted by *). In order to illustrate the type of interaction that

10

characterizes our model of monopolistic competition, we describe how theequilibrium prices are determined. First, each firm i in region H maximizesits profit ΠH , assuming accurately that its price choice has no impact on theregional price indices PH and PF . By symmetry, the prices selected by thefirms located within the same region are identical; hence, they are respec-tively given by two linear expressions p∗HH(PH) and p

∗HF (PF ). Second, these

prices must be consistent, that is, they must satisfy the following relations:

nHp∗HH(PH) + nFp

∗FH(PH) = PH

nHp∗HF (PF ) + nFp

∗FF (PF ) = PF

Given (5) and (6), it is then readily verified that

p∗HH =1

2

2a+ τc(1− λ)N

2b+ cN(10)

p∗FF =1

2

2a+ τcλN

2b+ cN(11)

p∗HF = p∗FF +

τ

2(12)

p∗FH = p∗HH +

τ

2(13)

Consequently, the equilibrium prices under monopolistic competition dependon the demand and firm distributions between regions. In particular, boththe prices charged by local and foreign firms fall when the mass of localfirms increases (because price competition is fiercer) but the impact is weakerwhen τ is smaller. In the limit, when τ is negligible, the relocation of firmsin H, say, has almost no impact on market prices. In this case, prices are‘independent’ of the way firms are distributed between the two regions.Equilibrium prices also rise when the relative desirability of the differ-

entiated product with respect to the numéraire, evaluated by a, gets largeror when the degree of product differentiation, inversely measured by c, in-creases provided that trade occurs (see (14) below). All these results are inaccordance with what is known in industrial organization and spatial pricingtheory.

11

Furthermore, there is freight absorption since only a fraction of the tradecost is passed on to the consumers. Indeed we have:

p∗HF − p∗HH = τb+ cλN

2b+ cN< τ which is equal to

τ

2when λ = 1/2

p∗FH − p∗FF = τb+ c(1− λ)N

2b+ cN< τ which is equal to

τ

2when λ = 1/2

It is well known that a monopolist facing a linear demand absorbs exactly one-half of the trade cost. By contrast, we see that monopolistic competition leadsto more (less) freight absorption than monopoly when the foreign market isthe small (large) one: in their attempt to penetrate the distant market,competition leads firms to a price gap that varies with the relative size of thehome and foreign markets.By inspection, it is readily verified that p∗HH (p

∗FF ) is increasing in τ be-

cause the local firms inH (F ) are more protected against foreign competitionwhile p∗HF − τ (p∗FH − τ) is decreasing because it is now more difficult forthese firms to sell on the foreign market. Observe also that arbitrage is neverprofitable since price differentials are always lower than trade costs. Finally,our demand side happens to be consistent with identical demand functionsat different locations but different price levels, as in standard spatial pricingtheory.Deducting the unit trade cost τ from the prices set on the distant markets,

i.e. (12) and (13), we see that firms’ prices net of trade costs are positiveregardless of the workers’ distribution if and only if

τ < τ trade ≡ 2aφ

2bφ+ cL(14)

which depends only upon the primitives (A,L,α,β, γ,φ) once a, b, c, and Nare replaced by their values. The same condition must hold for consumers inF (H) to buy from firms in H (F ), i.e. for the demand (8) evaluated at theprices (10) and (11) to be positive for all λ. From now on, condition (14) isassumed to hold. Consequently, there is intra-industry trade and reciprocaldumping, as in Anderson et al. (1995). However, there must be increasingreturns for trade to occur. Indeed, when φ = 0 all potential varieties areproduced in each region that becomes autarchic. More generally, it is readilyverified that

dτ tradedφ

> 0dτ tradedγ

< 0

12

so that trade is more likely the higher are the intensity of increasing returnsand the degree of product differentiation.It is easy to check that the equilibrium gross profits earned by a firm

established in H on each separated market are as follows:

Π∗HH = (b+ cN)(p∗HH)

2(A/2 + λL) (15)

where Π∗HH denotes the profits earned in H while the profits made fromselling in F are

Π∗HF = (b+ cN)(p∗HF − τ)2[A/2 + (1− λ)L] (16)

Increasing λ has two opposite effects on Π∗HH . First, due to tougher com-petition, the equilibrium price (10) falls as well as the quantity of each varietybought by each consumer living in region H. At the same time, the totalpopulation of consumers residing in this region increases so that the profitsmade by a firm located in H on local sales might rise. What is at work hereis an aggregate local demand effect due to the increase in the local popula-tion that may compensate firms for the adverse price effect as well as for theindividual demand effect generated by a wider array of competing varieties.The individual consumer surplus SH in region H associated with the

equilibrium prices (10) and (13) is then as follows (a symmetric expressionholds in region F ):

SH(λ) =a2L

2bφ− aL

φ[λp∗HH + (1− λ)p∗FH ]

+(bφ+ cL)L

2φ2

£λ(p∗HH)

2 + (1− λ)(p∗FH)2¤

−cL2

2φ2 [λp∗HH + (1− λ)p∗FH ]

2

Differentiating twice this expression with respect to λ shows that SH(λ) isconcave. Furthermore, (14) implies that SH(λ) is always increasing in λ overthe interval [0, 1].The equilibrium wage prevailing in region H may be obtained by evalu-

ating w∗H(λ) = (Π∗HH +Π∗HF )/φ, thus yielding the following expression:

w∗H(λ) =bφ+ cL

4(2bφ+ cL)2φ2

½[2aφ+ τcL(1− λ)]2

µA

2+ λL

¶+ [2aφ− 2τbφ− τcL(1− λ)]2

·A

2+ (1− λ)L

¸¾13

which, after simplifying, turns out to be quadratic in λ. Standard, but cum-bersome, investigations reveal that w∗H(λ) is concave and increasing (convexand decreasing) in λ when φ is large (small) as well as when τ , c, A, and Lare small (large). This implies that both SH(λ) and w∗H(λ) increase with λwhen τ is small, while they go in opposite directions when τ is large.

4 When do we observe agglomeration?

The distribution λ ∈ [0, 1] is a spatial equilibrium when no worker may geta higher utility level by changing location. Given that the indirect utility inregion H is as follows:

VH(λ) = SH(λ) + w∗H(λ) + q0

a spatial equilibrium arises at λ ∈ (0, 1) when∆V (λ) ≡ VH(λ)− VF (λ) = 0

or at λ = 0 when ∆V (0) ≤ 0, or at λ = 1 when ∆V (1) ≥ 0.In order to study the stability of a spatial equilibrium, we assume that

local labor markets adjust instantaneously when some workers move fromone region to the other. More precisely, the number of firms in each regionmust be such that the labor market clearing conditions (5) and (6) remainvalid for the new distribution of workers. Wages are then adjusted in eachregion for each firm to earn zero profits everywhere. For now, we assume amyopic adjustment process, that is, the driving force in the migration processis workers’ current utility differential between H and F :

.

λ ≡ dλ/dt = ∆V (λ) if 0 < λ < 1min{0,∆V (λ)} if λ = 1max{0,∆V (λ)} if λ = 0

(17)

when t is time. Clearly, a spatial equilibrium implies.

λ = 0. If ∆V (λ) ispositive, some workers will move from F to H; if it is negative, some will goin opposite direction. In the sequel, we assume that individual consumptionof the numéraire is positive during the adjustment process. This assumptionis made to capture the idea that individuals need both types of goods.8

8 An analytically convenient way to achieve that is to assume that each consumerreceives the same endowment q0 at each point in time.

14

A spatial equilibrium is stable for (17) if, for any marginal deviation fromthe equilibrium, this equation of motion brings the distribution of workersback to the original one. Therefore, the agglomerated configuration is alwaysstable when it is an equilibrium while the dispersed configuration is stable ifand only if the slope of ∆V (λ) is nonpositive in a neighborhood of this point.The forces at work are similar to those found in the core-periphery model.

First, the immobility of the farmers is a centrifugal force, at least as long asthere is trade between the two regions. The centripetal force finds its originin a demand effect generated by the preference for variety. If a larger numberof firms are located in region H, there are two effects at work. First, less va-rieties are imported. Second, (10) and (13) imply that the equilibrium pricesof all varieties sold in H are lower. Observe that the latter effect does notappear in Krugman’s model. This, in turn, induces some consumers to mi-grate toward this region. The resulting increase in the number of consumerscreates a larger demand for the industrial good in the corresponding region,which therefore increases operating profits (hence, wages) and leads to morefirms to move there. In other words, both backward and forward linkages arepresent in our model.It is readily verified that the indirect utility differential can be written as

follows:

∆V (λ) ≡ VH(λ)− VF (λ) ≡ SH(λ)− SF (λ) + w∗H(λ)− w∗F (λ)= Cτ(τ ∗ − τ) · (λ− 1/2) (18)

where

C ≡ [2bφ(3bφ+ 3cL+ cA) + c2L(A+ L)] L(bφ+ cL)

2φ2(2bφ+ cL)2> 0

and

τ ∗ ≡ 4aφ(3bφ+ 2cL)

2bφ(3bφ+ 3cL+ cA) + c2L(A+ L)> 0

which can also be restated in terms of the primitives of the economy.It follows immediately from (18) that λ = 1/2 is always an equilibrium.

Since C > 0, for λ 6= 1/2 the indirect utility differential has always the samesign as λ− 1/2 if and only if τ < τ ∗, otherwise it has the opposite sign. Inparticular, when there are no increasing returns in the manufacturing sector(φ = 0), the coefficient of (λ − 1/2) is always negative since τ ∗ = 0 so that

15

dispersion is the only (stable) equilibrium. This shows once more the impor-tance of increasing returns for the possible emergence of an agglomeration.It remains to determine when τ ∗ is lower than τ trade. This is so if and

only if

A/L >6b2φ2 + 8bcφL+ 3c2L2

cL(2bφ+ cL)> 3 (19)

where the second inequality holds because b/c = β/γ − 1 ∈ (0,+∞). Thisinequality means that the population of farmers is large relative to the pop-ulation of workers. When (19) does not hold, the coefficient of (λ− 1/2) in(18) is always positive for all τ < τ trade.When τ < τ ∗, the symmetric equilibrium is unstable and workers agglom-

erate in region H (F ) provided that the initial fraction of workers residing inthis region exceeds 1/2. In other words, agglomeration arises when the tradecost is low enough, as in Krugman (1991b) and for similar reasons. In con-trast, for large trade costs, that is, when τ > τ ∗, it is straightforward to seethat the symmetric configuration is the only stable equilibrium. Hence, thethreshold τ ∗ corresponds to both the critical value of τ at which symmetryceases to be stable (the ‘break point’) and the value below which agglom-eration is stable (the ‘sustain point’); this follows from the fact that (18) islinear in λ.

Proposition 1 Assume that τ < τ trade. Two cases may arise:(i) When (19) holds, we have the following: if τ > τ ∗, then the symmetric

configuration is the only stable spatial equilibrium with trade; if τ < τ ∗ thereare two stable spatial equilibria corresponding to the agglomerated configura-tions with trade; if τ = τ ∗ then any configuration is a spatial equilibrium.(ii) When (19) does not hold, any stable spatial equilibrium involves ag-

glomeration.

The reverse of (19) plays a role similar to the ‘black hole’ condition inKrugman and Venables (1995) and Fujita et al. (1999): regardless of thevalue of the trade costs, the region with the larger initial share of the man-ufacturing sector always attracts the whole sector. As in their case, moreproduct differentiation (lower c) and stronger increasing returns (higher φ)make the black hole condition more likely. Although the size of the indus-trial sector is captured here through the relative population size A/L and notthrough its share in consumption, the intuition is similar: the ratio A/Lmustbe sufficiently large for the economy to display different types of equilibria

16

according to the value of τ . Our result does not depend on the expenditureshare on the manufacturing sector because of the absence of general equilib-rium income effects: small or large sectors in terms of expenditure share mayeither be agglomerated when τ is small enough. This does not strike us asbeing implausible.Furthermore, when τ trade > τ ∗, trade occurs regardless of the type of equi-

librium that is stable. However, the nature of trade varies with the type ofconfiguration emerging in equilibrium. In the dispersed configuration, thereis only intra-industry trade in the differentiated product; in the agglomer-ated equilibrium, the region accommodating the manufacturing sector onlyimports the homogenous good from the other region.When increasing returns are stronger, as expressed by higher values of

φ, τ ∗ rises since dτ ∗/dφ > 0. This means that the agglomeration of themanufacturing sector is more likely, the stronger are the increasing returnsat the firm’s level. In addition, τ ∗ increases with product differentiation sincedτ ∗/dγ < 0. In words, more product differentiation fosters agglomeration. Inparticular, γ very small implies that τ trade < τ ∗ so that agglomeration alwaysarises under trade.The best way to convey the economic intuition behind Proposition 1 is

probably to make use of a graphical analysis.9 Figure 1 depicts the aggre-gate inverse demand in region H for a typical local firm after choosing, forsimplicity, the units of L so that b+ cN = 1:

pHH = a+ cPH(nH , τ)− qHHA/2 + φnH

(20)

Since pFH > pHH and the total number of firms is fixed by labor marketclearing, the price index PH is a decreasing function of nH at a rate whichincreases with τ :

∂PH(nH , τ)

∂nH< 0 and

¯̄̄̄∂2PH(nH , τ)

∂nH∂τ

¯̄̄̄> 0 (21)

Insert Figure 1 about here

The horizontal and vertical intercepts of (20) are respectively [a+cPH(nH , τ)]·(A/2+φnH) and [a+cPH(nH , τ)]. The equilibrium values of qHH and pHH are

9For illustrative purposes, we neglect the impact of relocation on the firm’s profit in Fsince this one is typically smaller than the impact on its profit in H.

17

shown as q∗HH and p∗HH . They are found by setting marginal revenue equal

to marginal cost. The operating profits are shown by the shaded rectangleand accrue to the workers while, as usual, the above triangle represents theconsumer surplus enjoyed by both types of workers.Although its depicts a partial equilibrium argument, Figure 1 is a powerful

learning device to understand the forces at work in our model. To see why,start from an initial situation where regions are identical (nH = nF ). Supposethat some firms move from the foreign to the home region so that nH risesand nF falls. For these firms to want to stay in the home region, operatingprofits, thus wages, have to increase. Indeed, were this not the case, the firmswould rather go back to the foreign region.As revealed by Figure 1, an increase in nH has two opposite effects on

operating profits, hence on wages. First, as new firms enter the home region,the price index PH(nH , τ) decreases. Ceteris paribus, this would shift theinverse demand (20) toward the origin of the axes and operating profits,hence wages, would shrink. This effect is due to increased competition in thehome market and stems from the fact that fewer firms now face trade costswhen supplying the home market. But this negative competition effect is notthe only effect. For some firms to move to the home region, some workershave to follow since nH = λL/φ. This means that, as nH increases, λ alsogoes up so that the market of the home region expands. Ceteris paribus,the horizontal intercept of the inverse demand would move away from theorigin and operating profits, thus wages, would expand. This is a positivedemand effect which is induced by the linkage between the locations of firmsand workers’ expenditures.Since the two effects oppose each other, the net result is a priori ambigu-

ous. But we can say more than that. In particular, we can assess which effectprevails depending on parameter values. Start with the competition effectthat goes through [a + cPH(nH , τ)]. This effect is strong if c is large, i.e. ifvarieties are good substitutes. It is also strong if |∂PH(nH , τ)/∂nH | is large.As shown in (21), this happens if τ is large, because, when obstacles to tradeare high, competition from the other region is weak and home firms care a lotabout their competitors being close rather than distant. As to the demandeffect, it will be strong if φ is large because each new firm brings along manyworkers, and if A is small because immigrants have a large impact on thelocal market size.We can therefore conclude that the demand effect dominates the com-

petition effect, when goods are bad substitutes (c small), increasing returns

18

are intense (φ large), the farmers are unimportant (A small) and trade costsare low (τ small). Under such circumstances, the entry of new firms in oneregion would raise the operating profits of all firms, hence wages. Higheroperating profits and wages would attract more firms and workers, thus gen-erating circular causation among locational decisions. Agglomeration wouldthen be sustainable as a spatial equilibrium.This argument establishes a sufficient condition for agglomeration. Since

the impact of firms’ relocation on consumer surplus is always positive, ag-glomeration could still arise even when operating profits, hence wages, de-crease with the size of the local market because the demand effect is dom-inated by the competition effect. Furthermore, it is likely to hold for mostdownward sloping demand functions.

5 Optimality vs. equilibrium

We now wish to determine whether or not such an agglomeration is sociallyoptimal. To this end, we assume that the planner is able (i) to assign anynumber of workers (or, equivalently, of firms) to a specific region and (ii) touse lump sum transfers from all workers to pay for the loss firms may incurwhile pricing at marginal cost. Observe that no distortion arises in the totalnumber of varieties since N is determined by the factor endowment (L) andtechnology (φ) in the manufacturing sector and is, therefore, the same atboth the equilibrium and optimum outcomes. Because our setting assumestransferable utility, the planner chooses λ in order to maximize the sum ofindividual indirect utilities:

W (λ) ≡ A2[SH(λ)+1]+λL[SH(λ)+wH(λ)]+

A

2[SF (λ)+1]+(1−λ)L[SF (λ)+wF (λ)]

(22)in which all prices have been set equal to marginal cost:

poHH = poFF = 0 and poHF = p

oFH = τ

thus implying by (15) and (16) that operating profits are zero, and hencewoH(λ) = w

oF (λ) = 0 for every λ so that firms do not incur any loss. Hence

(22) becomes:

W (λ) = Coτ(τ o − τ)λ(λ− 1)L+ constant (23)

19

whereCo ≡ L

2φ2 [2bφ+ c(A+ L)]

and

τ o ≡ 4aφ

2bφ+ c(A+ L)

The function (23) is strictly concave in λ if τ > τ o and strictly convex ifτ < τ o. Furthermore, since the coefficients of λ2 and of λ are the same (upto their sign), this expression has always an interior extremum at λ = 1/2.As a result, the optimal choice of the planner is determined by the sign ofthe coefficient of λ2, that is, by the value of τ with respect to of τ o.Hence we have:

Proposition 2 If τ > τ o, then the symmetric configuration is the optimum;if τ < τ o any agglomerated configuration is the optimum; if τ = τ o anyconfiguration is an optimum.

In accordance with intuition, it is socially desirable to agglomerate themanufacturing sector into a single region once trade costs are low, increas-ing returns in the manufacturing sector are strong enough (dτ o/dφ > 0)and/or the output of this sector is sufficiently differentiated (dτ o/dγ < 0).In particular, the optimum is always dispersed when increasing returns vanish(φ = 0).A simple calculation shows that τ o < τ ∗. This means that the market

yields an agglomerated configuration for a whole range (τ o < τ < τ ∗) of tradecost values for which it is socially desirable to have a dispersed pattern of ac-tivities. Accordingly, when trade costs are low (τ < τ o) or high (τ > τ∗) noregional policy is required from the efficiency point of view, although equityconsiderations might justify such a policy when agglomeration arises. On thecontrary, for intermediate values of trade costs (τ o < τ < τ ∗), the marketprovides excessive agglomeration, thus justifying the need for an active re-gional policy in order to foster the dispersion of the modern sector on boththe efficiency and equity grounds.This discrepancy may be explained as follows. First, workers do not

internalize the negative external effects they impose on the farmers who stayput in the workers’ region of origin, nor do they account for the impactof their migration decisions on the residents in their region of destination.Hence, even though the workers have individual incentives to move, these

20

incentives do not reflect the social value of their move. This explains whyequilibrium and optimum do not necessarily coincide. Second, the individualdemand elasticity is much lower at the optimum (marginal cost pricing) thanat the equilibrium (Nash equilibrium pricing), so that regional price indicesare less sensitive to a decrease in τ . As a result, the fall in trade costs mustbe sufficiently large to make the agglomeration of workers socially desirable,which tells us why τ o < τ ∗.We will return to the debate spatial equilibrium vs. optimum in Section

7.

6 The impact of workers’ expectations on theagglomeration process

The adjustment process (17) is often used in new economic geography. Yet,the underlying dynamics is myopic because workers care only about theircurrent utility level, thus implying that only history matters. This is a fairlyrestrictive assumption to the extent that migration decisions are typicallymade on the grounds of current and future utility flows. In addition, thisapproach has been criticized because it is not consistent with fully rationalforward-looking behavior (Matsuyama, 1991). In this section, we want tosee how the model presented above can be used to shed more light on theinterplay between history and expectations in the formation of the economicspace when migrants maximize the intertemporal value of their utility flows.In particular we are interested in identifying the conditions under which,when initially regions host different numbers of workers, the common beliefthat workers will eventually agglomerate in the smaller region can reversethe historically inherited advantage of the larger region. Formally, we areinterested in determining the parameter domains for which there exists anequilibrium path consistent with this belief, assuming that workers have per-fect foresight (self-fulfilling prophecy).For concreteness, let us consider the case in which initially region F is

larger than H. The opposite case can be studied in a symmetric way. There-fore, we want to test the consistency of the belief that, starting from t = 0,all workers will end up being concentrated in H at some future date t = T ,that is, there exists T ≥ 0 such that, given λ0 < 1/2,

21

·λ(t) 6= 0 and λ (t) < 1 for t ∈ [0, T ) (24)·λ(t) = 0 and λ (t) = 1 for t ≥ T

Let VH (t) and VF (t) be the instantaneous utility levels of a worker cur-rently in regions H and F respectively at time t ≥ 0.10 Furthermore, let V Cbe the instantaneous utility level in region H at λ = 1. Then, under (24):

VH(t) = VC for all t ≥ T

Since workers have perfect foresight, the easiest way to generate a nonbang-bang migration behavior is to assume that, when moving from oneregion to the other, workers incur a utility loss that depends on the rateof migration (Mussa, 1978). In other words, a migrant imposes a negativeexternality on the others by congesting the migration process. Specifically,we follow Krugman (1991a) and assume that the utility loss for a migrant at

time t is equal to

¯̄̄̄ ·λ(t)

¯̄̄̄/δ, where δ > 0 is the speed of adjustment. Thus,

under (24), the intertemporal utility of a worker who moves from F to H attime t ∈ [0, T ) is given by:

u (t) =

Z t

0

e−ρsVF (s) ds+Z T

t

e−ρsVH (s) ds+ e−ρTV C/ρ− e−ρt·λ (t) /δ (25)

where ρ > 0 is the rate of time preference.We are now ready to characterize the equilibrium migration process. At

the initial time 0, each worker residing in F decides at which date t ≥ 0 tomigrate from F to H. In so doing, she believes that at date T all workers willend up being in H. As argued by Fukao and Bénabou (1993) and Ottaviano(1999), for this belief to be consistent with the equilibrium outcome, a workermust be indifferent between moving at any date t ≥ 0 or at the final expecteddate T . In the former case, she expects to receive (25). In the latter, sheforecasts:10To ease notation, when variables depend on time t through λ(t), we only report their

dependence on time as long as this does not lead to any ambiguity. For example, Vr(t)stands for Vr (λ(t)). Moreover, throughout this section, we keep on assuming that theconsumption of the numéraire is always positive in both regions at any time. Formally,this implies that we assume away any form of intertemporal trade.

22

u (T ) =

Z T

0

e−ρsVF (s) ds+ e−ρTV C/ρ (26)

that is, the limit of (25) as t approaches T where the moving cost disappearssince migration stops. Notice that, since the assumed belief has all workersin H from T onwards, in both cases the worker expects a utility flow VH (t)for all t ≥ T .Subtracting (26) from (25), for each t ≥ 0, we get:

u(t)− u(T ) =

Z T

t

e−ρs[VH(s)− VF (s)]ds− e−ρt.

λ(t)/δ (27)

= e−ρt[vH(t)− vF (t)]− e−ρt.

λ(t)/δ

= e−ρt∆v(t)− e−ρt.

λ(t)/δ

where

vr(t) ≡Z T

t

e−ρ(s−t)Vr(s)ds+ e−ρ(T−t)V C/ρ r = H,F

is what residence in r buys to a worker who stays in H (r = H) or to aworker in F who plans to move to H at T (r = F ) from t onwards, while∆v(t) ≡ vH(t)− vF (t) and ∆v(T ) = 0.Since in equilibrium a worker moving at t must be indifferent between

migrating at that date or at any other date, until the final expected date T ,along an equilibrium path it must be that u (t) = u (T ) for all t ∈ [0, T ),which, using (27), implies

·λ(t) = δ∆v(t) for all t ∈ [0, T ) (28)

with λ(T ) = 1.Furthermore, differentiating ∆v(t) with respect to time yields

·∆v (t) = ρ∆v (t)−∆V (t) (29)

where ∆V (t) ≡ VH(t)−VF (t) stands for the expected instantaneous indirectutility differential flow given by (18). Hence we obtain a system of two lineardifferential equations, instead of the first order differential equation (17),with the terminal conditions λ(T ) = 1, ∆v(T ) = 0.

23

Since λ = 1/2 implies ∆V = 0, the system (28) and (29) has always aninterior steady state at (λ,∆v) = (1/2, 0) which corresponds to the dispersedconfiguration. While for τ > τ ∗ it is the only steady state, for τ < τ ∗ twoother steady states exist at (λ,∆v) = (0, 0) and (λ,∆v) = (1, 0). In the lattercase, as in Fukao and Benabou (1993), since all workers are concentratedwithin the same region, there is no reason to hold that

·∆v(T ) = 0

in (29). In fact, by the terminal conditions, we have

·∆v (T ) = −∆V (T ) = −Cτ(τ ∗ − τ)/2 < 0

which ensures that the spatial equilibrium involves no worker in F . Therefore,the assumed belief (24) can be consistent only in the latter case on which weconcentrate from now on.In order to identify the conditions under which the belief that workers

will eventually agglomerate in the initially smaller region H can reverse thehistorically inherited advantage of the larger region F , we have to study theglobal stability of system (28) and (29). In so doing we exploit the fact that,since the system is linear, local and global stability properties coincide andwe focus on the former. The eigenvalues of the Jacobian matrix of the system(28) and (29) are given by

ρ±pρ2 − 4δCτ(τ ∗ − τ)

2(30)

When τ < τ ∗ two scenarios may arise. In the first one, ρ > 2p

δCτ(τ ∗ − τ)so that the two eigenvalues are still real but both positive. The steady state(1/2, 0) is an unstable node and there are two trajectories that steadily goto the endpoints, (0, 0) or (1, 0), depending on the initial spatial distribu-tion of workers, say λ0. In this case, only history matters: from any initialλ0 6= 1/2, there is a single trajectory that goes towards the closer endpointas in the case where the dynamics is given by (17). This means that belief(24) is inconsistent with the equilibrium path, whence the myopic adjustmentprocess studied in the previous section provides a good approximation of thequalitative evolution of the economy under forward looking behavior.Things turn out to be quite different in the second scenario in which

ρ < 2p

δCτ(τ ∗ − τ). Since Cτ(τ ∗ − τ) = 0 at both τ = 0 and τ = τ ∗, the

24

equation Cτ(τ ∗− τ)− ρ2/4δ = 0 has two positive real roots in τ , denoted τ e1and τ e2, that are smaller than τ ∗:

τ e1 ≡τ ∗ −D2

and τ e2 ≡τ ∗ +D2

whereD ≡

p(τ ∗)2 − ρ2/Cδ

stands for the size of the domain of values of τ for which expectations matter;it shrinks as the discount rate ρ increases or as the speed of adjustment δdecreases. Indeed, for τ ∈ (0, τ e1) as well as for τ ∈ (τ e2, τ ∗), both eigenvaluesare real positive numbers and the steady state (1/2, 0) is an unstable nodeas before. However, for τ ∈ (τ e1, τ e2), they become complex numbers witha positive real part so that the steady state is an unstable focus. The twotrajectories spiral out from (1/2, 0). Therefore, for any λ0 close enough to,but different from, 1/2, there are two alternative trajectories going in op-posite directions. It is in such a case that expectations decide along whichtrajectory the system moves, so that belief (24) is self-fulfilling.. In otherwords, expectations matter for λ close enough to 1/2, while history mattersotherwise. The corresponding domains are now described.The range of values for which expectations matter, called the overlap by

Krugman (1991a), can be obtained as follows. As observed by Fukao andBénabou (1993), the system must be solved backwards in time starting fromthe terminal points (0, 0) and (1, 0). The first time the backward trajectoriesintersect the locus ∆v = 0 allows for the identifications of the endpoints ofthe overlap:

λL ≡ 12(1− Λ) λH ≡ 1

2(1 + Λ)

where

Λ ≡ expÃ− ρπp

4δCτ(τ ∗ − τ)− ρ2

!is the width of the overlap, which is an interval centered around λ = 1/2.The overlap is nonempty as long as τ ∈ (τ e1, τ e2). Thus, the width of the

overlap is increasing in δ, C, and τ ∗, while it decreases with ρ. Moreover, it is∩-shaped with respect to τ , reaching a maximum at τ = τ ∗/2. Since Cτ(τ ∗−τ) > 0 is the slope of ∆V and since this one measures the strength of theforward and backward linkages pushing towards agglomeration, we see that

25

expectations matter more when such linkages are stronger. Consequently, wehave shown the following result:

Proposition 3 Let λ0 be the initial spatial distribution of workers. If τ <τ ∗ and ρ < 2

pδCτ(τ ∗ − τ), there exist τ e1 ∈ (0, τ ∗/2), τ e2 ∈ (τ ∗/2, τ ∗),

λL ∈ (0, 1/2), and λH ∈ (1/2, 1) such that workers’ beliefs about their futureearnings influence the process of agglomeration if and only if τ ∈ (τ e1, τ e2) andλ0 ∈ [λL,λH ].

Hence, history alone matters when τ and λ are large enough or smallenough. In other words, the agglomeration process evolves as if workers wereshort-sighted when obstacles to trade are high or low and when regions areinitially quite different. Instead, as long as obstacles to trade take inter-mediate values and regions are not initially too different, the equilibrium isdetermined by workers’ expectations and not by history.11

The existence of the range (τ e1, τe2) may be explained as follows. Suppose,

indeed, that the economy is such that λ0 < 1/2 and ask what is needed toreverse an ongoing agglomeration process currently leading towards λ = 0.If the evolution of the economy were to change direction, workers wouldexperience falling instantaneous indirect utility flows for some time periodas long as λ < 1/2. The instantaneous indirect utility flows would startgrowing only after λ becomes larger than 1/2. Accordingly, workers wouldfirst experience utility losses followed by utility gains. Since the losses wouldcome before the gains, they would be less discounted. This provides the rootfor the intuition behind Proposition 3. When the forward and backwardlinkages lead to substantial wage rises (that is, for intermediate values of τ),the benefits of agglomerating at λ = 1 can compensate workers for the lossesthey incur during the transition phase, thus making the reversal of migrationpossible. On the contrary, when these linkages get weaker (that is, for low orhigh values of τ), the benefits of agglomerating at λ = 1 do not compensateworkers for the losses. As a consequence, the reversal in the migration processmay occur only for intermediate values of τ .11When τ > τ∗, we know that the equilibrium path is inconsistent with workers’ ex-

pectations under (24). And, indeed, the two eigenvalues are real and have opposite signsso that the steady state is a saddle point. Under the assumption of perfect foresights,regardless of the initial distribution λ0, the system moves along a single stable trajectorytoward the symmetric steady-state. In other words, the system converges towards thedispersed configuration, thus implying that neither expectations nor history matter forthe final outcome.

26

As to the remaining comparative static properties of the overlap, they areexplained by fact that proximity to the endpoint increases the time periodover which workers bear losses, large rate of time preference gives more weightto them, and a slow speed of adjustment extends the time period over whichworkers’ well being is reduced.

7 The impact of urban costs

So far, we have assumed that the agglomeration of workers into a singleregion does not involve any agglomeration costs. Yet, it is reasonable tobelieve that a growing settlement in a given region will often take the formof an urban area, typically a city. In order to deal with such an aspect ofthe process of agglomeration, we extend the core-periphery model by addingthe central variables suggested by urban economics (Fujita, 1989). In orderto keep the analysis short, we go back to the myopic adjustment process ofSection 4.Space is now continuous and one-dimensional. Each region has a spatial

extension and involves a linear city whose center is given but with a variablesize. The city center stands for a central business district (CBD) in which allfirms locate once they have chosen to set up in the corresponding region (seeFujita and Thisse, 1996, for various arguments explaining why firms wantto be agglomerated in a CBD). The two CBDs are two remote points of thelocation space. Interregional trade flows go from one CBD to the other.Housing is a new good in our economy and is described by the amount of

land used by workers. While firms are assumed not to consume land, work-ers, when they live in a certain region, are urban residents who consume landand commute to the regional CBD in which manufacturing firms are located.Hence, unlike Krugman (1991b) but like Alonso (1964), Helpman (1998) andTabuchi (1998), each agglomeration has a spatial extension that imposes com-muting and land costs on the corresponding workers. For simplicity, workersconsume a fixed lot size normalized to unity, while commuting costs are lin-ear in distance, the commuting cost per unit of distance being given by θ > 0units of the numéraire. Without loss of generality, the opportunity cost ofland is normalized to zero.When λL workers live in H, they are equally distributed around the H-

CBD. In equilibrium, since all workers residing in region H earn the samewage, they reach the same utility level. Furthermore, since they all consume

27

one unit of land, the equilibrium land rent at distance x < λL/2 from theH-CBD is given by

R∗(x) = θ(λL/2− x)Hence, a worker located at the average distance λL/4 from theH-CBD bearsa commuting cost equal to θλL/4 and pays the average land rent θλL/4(Fujita, 1989; Papageorgiou and Pines, 1999). When the land rents go toabsentee landlords, individual urban costs, defined by commuting cost plusland rent at each residence x, are given by θλL/2. In order to avoid work-ing with absentee landlords, we assume that all the land rents are collectedand equally redistributed among the H-city workers.12 Consequently, theindividual urban costs after redistribution are equal to θλL/4.Since the urban costs prevailing in regions H and F are not equal, the

incentive to move from one region to the other are no longer given by (18).Indeed, we must account for the difference in urban costs between H and F ,namely

λθL/4− (1− λ)θL/4 = (λ− 1/2)θL/2which must be subtracted from (18) to obtain the actual utility differential:

∆Vu(λ) = [Cτ(τ∗ − τ)− θL/2] · (λ− 1/2)

As in Section 4, agglomeration is a spatial equilibrium when the slope of∆Vu(λ) is positive. This is the case as long as τ falls within the two values

τu1 ≡τ ∗ −E2

and τu2 ≡τ ∗ +E2

where τu1 , τu2 ∈ (0, τ ∗) and

E ≡p(τ ∗)2 − 2θL/C

measures the domain of values of τ for which ∆Vu(λ) > 0. Notice thatsuch domain shrinks as the commuting cost per unit of distance θ increases.Consequently, we have:12Assuming, as in Helpman (1998), that the total land rent across regions is equally

redistributed among workers (hence controlling for the resulting fiscal externality) doesnot affect the utility differential whereas the difference in urban costs becomes twice aslarge. Hence, θ is to be replaced by 2θ in the foregoing developments. This reducesthe value of E and shrinks the interval (τu

1 , τu2 ). As a result, Proposition 4 shows that

agglomeration is less likely to occur.

28

Proposition 4 If 2θL < C(τ ∗)2, there exist τu1 ∈ (0, τ ∗/2), τu2 ∈ (τ ∗/2, τ ∗)such that agglomeration (dispersion) is the only stable equilibrium if andonly if τ ∈ (τu1 , τu2) (τ /∈ (τu1 , τu2)). For τ = τu1 or τ = τu2 , any distributionof workers is a spatial equilibrium. If 2θL > C(τ ∗)2, dispersion is the onlyspatial equilibrium.

Thus, the existence of positive commuting costs within the regional centersis sufficient to yield dispersion when the trade costs are sufficiently low. Thisimplies that, as trade costs fall, the economy involves dispersion, agglomera-tion, and re-dispersion. An increase (decrease) in the commuting costs fostersdispersion (agglomeration) by widening (shrinking) the left range of τ -valuesfor which dispersion is the only spatial equilibrium. Also, sufficiently highcommuting costs always yield dispersion.It is interesting to point out that while dispersion arises for both high and

low trade costs, this happens for very different reasons. In the former case,firms are dispersed as a response to the high trade costs they would incurby supplying farmers from a single agglomeration. In the latter, firms aredispersed as a response to the high urban costs workers would bear within asingle agglomeration.In the present context, we may assume that there is no farmers with-

out annihilating the dispersion force. If A = 0, it is readily verified thatτ trade < τ ∗/2 < τu2 so that dispersion does not arise when trade costs arehigh. Consequently, the economy moves from agglomeration to dispersionwhen trade costs fall, thus confirming the numerical results obtained by Help-man (1998).Finally, it is worth revisiting the debate about the social desirability of the

market outcome once we account for urban costs. Computing the first-ordercondition for the social optimum as in Section 5 in which we now account forthe urban costs θλL/4 in region H and θ(1− λ)L/4 in region F , we obtainthe new critical values:

τuo1 ≡τ o − F2

and τuo2 ≡τ o + F

2

where τuo1 , τuo2 ∈ (0, τ o) and

F ≡p(τ o)2 − 2θL/Co

and we fall back on the condition obtained in Section 5 when θ = 0. As aresult, we have:

29

Proposition 5 Assume (τ o)2 > 2θL/Co. Then, there exist τuo1 ∈ (0, τ o/2)and τuo2 ∈ (τ o/2, τ o) such that the agglomerated configuration is the optimumwhen τ ∈ (τuo1 , τ

uo2 ). When τ = τuo1 , τ

uo2 , any configuration is an optimum.

Otherwise, the symmetric configuration is the optimum.

Observe that the domain for which agglomeration is optimal shrinks asthe commuting cost θ increases.The comparison between the optimum and equilibrium outcomes is less

straightforward than it was in Section 5 with Figure 2 showing the differentpossible patterns. First, as in Section 5, it is readily verified that excessiveagglomeration arises for intermediate values of the trade costs. Second, whenurban costs are positive, the equilibriummay yield either suboptimal agglom-eration or suboptimal dispersion depending on the parameter values of theeconomy. In particular, if we set A = 0, the interval I2 in Figure 2 disappearsso that inefficiency arises only a range of low trade costs for which “marketforces do not generate enough agglomeration” (Helpman, 1998, p. 46).13

Insert Figure 2 about here

8 Concluding remarks

Recent years have seen the proliferation of applications of the ‘Dixit-Stiglitz,iceberg, evolution, and the computer’ framework for studying the impact oftrade costs on the spatial distribution of economic activities. While theseapplications have produced valuable insights, they have often been criticizedbecause they rely on a very particular research strategy.We have proposed a different framework that is able not only to confirm

those insights but also to produce new results that could barely be obtainedwithin the standard one. Specifically, we have used this framework to dealwith the following issues: (i) the welfare properties of the core-peripherymodel, (ii) the impact of expectations in shaping the economic space, and(iii) the effects of urban costs on the interregional distribution of activities.13These results are a novel example of the ambiguous welfare properties of agglom-

erations that have been pointed out especially in urban economics. As argued recentlyby Papageorgiou and Pines (2000), such an ambiguity finds its origin in the simultane-ous working of many potential sources of distortions such as various kinds of external(dis)economies, indivisibilities and nonreplicabilities.

30

This suggests that our framework is versatile enough to accommodate otherextensions.So, we have shown that the main results in the literature do not depend

on the specific modeling choices made, as often argued by their critics. Inparticular, the robustness of the results obtained in the core-periphery modelagainst an alternative formulation of preferences and transportation seemsto point to the existence of a whole class of models for which similar resultswould hold. However, we have also shown that those modeling choices canbe fruitfully reconsidered once the aim is to shed light on different issues.The model used in this paper still displays some undesirable features to

which it should be remedied in future research. First, there is a fixed massof firms regardless of the consumer distribution. Furthermore, by ignoringincome effects, our setting has a strong partial equilibrium flavor.

References

[1] Alonso, W. Location and Land Use (Cambridge (Mass.): Harvard Uni-versity Press, 1964).

[2] Alonso, W. “Five Bell Shapes in Development,” Papers of the RegionalScience Association 45 (1980), 5-16.

[3] Anderson, S.P., N. Schmitt and J.-F. Thisse “Who Benefits from An-tidumping Legislation?” Journal of International Economics 38 (1995),321-337.

[4] Dixit, A.K. “A Model of Duopoly Suggesting a Theory of Entry Barri-ers,” Bell Journal of Economics 10 (1979), 20-32.

[5] Dixit, A.K. and J.E. Stiglitz “Monopolistic Competition and OptimumProduct Diversity,” American Economic Review 67 (1977), 297-308.

[6] Fujita, M. Urban Economic Theory. Land Use and City Size (Cam-bridge: Cambridge University Press, 1989).

[7] Fujita, M., P. Krugman and A. Venables The Spatial Economy. Cities,Regions and International Trade (Cambridge (Mass.): MIT Press,1999).

31

[8] Fujita, M. and J.-F. Thisse “Economics of Agglomeration,” Journal ofthe Japanese and International Economies 10 (1996), 339-378.

[9] Fukao, K. and R. Bénabou “History versus Expectations: A Comment,”Quarterly Journal of Economics 108 (1993), 535-542.

[10] Geyer, H.S. and T.M. Kontuly Differential Urbanization: IntegratingSpatial Models (London: Arnold, 1996).

[11] Head, K. and T. Mayer “Non-Europe. The Magnitude and Causes ofMarket Fragmentation in the EU,” Weltwirtschaftliches Archiv, forth-coming, 2000.

[12] Helpman, E. “The Size of Regions,” in D. Pines, E. Sadka and I.Zilcha, eds., Topics in Public Economics. Theoretical and Applied Analy-sis (Cambridge: Cambridge University Press, 1998), 33-54.

[13] Hotelling, H. “Stability in Competition,” Economic Journal 39 (1929),41-57.

[14] Koopmans, T.C. Three Essays on the State of Economic Science (NewYork: McGraw-Hill, 1957).

[15] Krugman, P. “History versus Expectations,” Quarterly Journal of Eco-nomics 106 (1991a), 651-667.

[16] Krugman, P. “Increasing Returns and Economic Geography,” Journalof Political Economy 99 (1991b), 483-499.

[17] Krugman, P. Development, Geography, and Economic Theory (Cam-bridge (Mass.): MIT Press, 1995).

[18] Krugman, P. “Urban Concentration: The Role of Increasing Returnsand Transport Costs,” International Regional Science Review 19 (1996),5-30.

[19] Krugman, P. “Space: The Final Frontier,” Journal of Economic Per-spectives 12 (1998), 161-174.

[20] Krugman, P. and A.J. Venables “Integration and the Competitivenessof Peripheral Industry,” in C. Bliss and J. Braga de Macedo, eds., Unitywith Diversity in the European Community (Cambridge: CambridgeUniversity Press, 1990), 56-75.

32

[21] Krugman, P. and A.J. Venables “Globalization and the Inequality ofNations”, Quarterly Journal of Economics 110 (1995), 857-880.

[22] Lösch, A. Die Räumliche Ordnung der Wirtschaft (Jena: Gustav FischerVerlag, 1940). English translation: The Economics of Location (NewHaven (Conn.): Yale University Press, 1954).

[23] Matsuyama, K. “Increasing Returns, Industrialization, and Indetermi-nacy of Equilibrium,” Quarterly Journal of Economics 106 (1991), 617-650.

[24] McCallum, J. “National Borders Matter: Canada-US Regional TradePatterns,” American Economic Review 85 (1995), 615-623.

[25] Mussa, M. “Dynamic Adjustment in the Heckscher-Ohlin-SamuelsonModel,” Journal of Political Economy 86 (1978), 775-791.

[26] Ottaviano, G.I.P. “Integration, Geography and the Burden of History,”Regional Science and Urban Economics 29 (1999), 245-256.

[27] Ottaviano, G.I.P. and D. Puga “Agglomeration in the Global Econ-omy: A Survey of the ‘New Economic Geography’,”World Economy 21(1998), 707-731.

[28] Papageorgiou, Y.Y. and D. Pines An Essay in Urban Economic Theory(Dordrecht: Kluwer Academic Publishers, 1999).

[29] Papageorgiou, Y.Y. and D. Pines “Externalities, Indivisibility, Non-replicability and Agglomeration”, Journal of Urban Economics, forth-coming, 2000.

[30] Phlips, L. Applied Consumption Analysis (Amsterdam: North-Holland,1983).

[31] Puga, D. “The Rise and Fall of Regional Inequalities,” European Eco-nomic Review 43 (1999), 303-334.

[32] Tabuchi, T. “Agglomeration and Dispersion: A Synthesis of Alonso andKrugman,” Journal of Urban Economics 44 (1998), 333-351.

[33] Trionfetti, F. “Public Procurement, Market Integration, and IncomeInequalities”, Review of International Economics, forthcoming.

33

[34] Vives, X. “Trade Association Disclosure Rules, Incentives to Share Infor-mation, and Welfare,” Rand Journal of Economics 21 (1990), 409-430.

[35] Wei, S.-J. “Intra-national versus International Trade: How Stubborn areNations in Global Integration,” National Bureau of Economic ResearchWorking paper N◦ 5531, 1996.

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APPENDIX

In the case of two varieties, the symmetric quadratic utility is given by:

U(q1, q2) = α(q1 + q2)− (β/2)(q21 + q

22)− γq1q2 + q0

In the case of n > 2 varieties, this expression is extended as follows:

U(q) = αnXi=1

qi − (β/2)nXi=1

q2i − (γ/2)

nXi=1

nXj 6=iqiqj + q0

= αnXi=1

qi − [(β − γ)/2]nXi=1

q2i − (γ/2)

nXi=1

Ãqi

nXj=1

qj

!+ q0

= αnXi=1

qi − [(β − γ)/2]nXi=1

q2i − (γ/2)

ÃnXi=1

qi

!2

+ q0

When γ → β, this utility boils to a standard quadratic utility for a homoge-nous good. Letting n → ∞ and qi → 0 , we obtain (1) in which N standsfor the mass of varieties.

35

pHH

0

a + cPH

qHH

p∗HH

q∗HH (a + cPH)(A/2 + φnH)

Figure 1: Determination of the equilibrium outcome

0τ

τu2τuo

2τuo1 τu

1

B∗

Bo

I1 I2

0 ττu2τuo

2τuo1τu

1

B∗

Bo

I2 I2

0

τ

τu2τuo

2τuo1 τu

1

B∗

Bo

I1 I2

Figure 2: Divergence between equilibrium and optimum: the three cases