Multiple equilibria, stability, and asymmetries in Krugman's core ...

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80, 425–438 (2001) c RSAI 2001. Multiple equilibria, stability, and asymmetries in Krugman's core-periphery model. *. Luis Fernando Lanaspa, Fernando Sanz.

Papers Reg. Sci. 80, 425–438 (2001)

c RSAI 2001 

Multiple equilibria, stability, and asymmetries in Krugman’s core-periphery model Luis Fernando Lanaspa, Fernando Sanz Faculty of Economics and Business Studies, University of Zaragoza, Doctor Cerrada 3, 50005 Zaragoza, Spain (e-mail: [email protected]) Received: 23 June 1998 / Accepted: 5 October 1999

Abstract. Paul Krugman developed a general equilibrium model with two sectors and two regions in 1991, from which two patterns of industrial localization could be endogenously deduced, dispersion at 50% and total concentration. The introduction of transport costs, which depend on the size of the population, are meant to capture effects produced by the trade-off between congestion costs and advantages derived from the possession of infrastructure, thus generates stable asymmetric multiple equilibria. The outcome of asymmetric stable multiple equilibria demonstrates the fruitfulness of this extension of the original model. JEL classification: F12, R12 Key words: Industrial concentration, infrastructures, stable asymmetries 1 Introduction A study of the real world easily reveals the existence of processes tending towards the concentration of consumers and firms in space. By omission, when these processes do not occur, or when they occur with insufficient intensity, the resultant structure is a more disperse distribution of economic agents. We therefore encounter centrifugal and centripetal forces, the relative importance of which endogenously evokes a specific economic landscape. This landscape, in turn, gives rise to a certain pattern of production and exchange, and consequently to a specific geographical distribution of economic activity. The economic literature, as well as that devoted to other disciplines, attempts to analyze the agglomeration mechanisms that define a given spatial configuration. The pioneer in this field is Von Th¨unen (1826), who considered a single city  Financial support from grant 261-26 of the University of Zaragoza and from DGES grant PB971028 are gratefully acknowledged. We would like to thank two anonymous referees for their comments and suggestions.

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surrounded by concentric rings of agricultural land, so that those closest to the industrial center have the highest incomes and produce goods with the highest transport costs. This idea has been further developed in urban economics in the last 30 years. A second significant contribution is that of Marshall (1920), who emphasized the role of local external economies to explain the concentration of firms. Finally, Christaller (1933) and L¨osch (1940) introduced the central place theory, according to which consumers are uniformly distributed and firms locate at the vertices of a regular lattice, such that a hierarchical system of nested market areas results, provided there is a sufficient number of goods. It is evident that economic geography, considered in its broadest sense, has produced a much larger number of relevant contributions than those cited above (see, for example, Weber 1909; Isard 1956; or the literature on market potential). However, even today its stature as a discipline fully established within standard economic theory is not problem-free (see Krugman 1995, Chapter 2). Two main factors are responsible for this situation. First, the absence of a micro-foundation: usually spatial configurations are taken as an a priori starting point, but this converts the analysis into a mere geometry exercise with limited responsive economic behavior. Second, it is only due to relatively recent contributions in the field of industrial organization that it has been possible to introduce essential aspects, such as economies of scale or non-competitive market structures, into models in such a way that it is both rigorous and tractable. The contribution of Krugman (1991) successfully overcomes these two limitations. While Krugman’s model does not solve all problems, it marks a path towards potential construction of solid explanatory models. Krugman’s ideas are not novel, but his overall approach certainly is. His simplicity is hallmark; he explains with ease core-periphery structures that emerge in an endogenous manner – in accordance with the interaction of certain parameters – reflecting the size of transport costs, the level of the economies of scale, and the weight of the externalities of demand. The basic mechanisms involved are as follows. In the presence of economies of scale, firms have incentives to concentrate production at one site. If, furthermore, transport costs also exist and demand is not uniformly distributed, the optimum location is where ever demand is greatest. However, demand is greatest where firms are already located. Thus, we are faced with a circular causation which can result in an industrialized core. We take the work of Krugman (1991) as a starting point and a reference point, so that the essential model is not subject to change. However, we suggest two aspects that are in need of improvement, as these relate to the results. First, Krugman presents only two possible situations of long-run equilibrium: either total concentration of manufacturers in one region or else there is dispersion, with multiple equilibria not being admitted.We here defend the existence of multiple equilibria, thus introducing the possibility of generating various equilibria in one single scenario, with certain values being given for the relevant parameters. Secondly, when dispersion occurs, it has to be equal; in other words, each region will control 50% of the industrial sector.

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We demonstrate here how to overcome both limitations simply by showing that the Krugman framework is entirely appropriate for aspects related to economic geography, but it is simultaneously sensitive to its starting hypotheses. Small changes in the model can account for the two above-mentioned phenomena, which are not reflected in the original approach. We next outline the most important aspects of the Krugman (1991) model and analyze the situation where transport costs are not constant, as supposed in Krugman, but depend instead on the size of each region. This consideration is justified in detail at a later stage. The underlying idea is summarily that small regions with limited industry do not suffer high congestion costs, but neither do they benefit from sufficient infrastructure. This can be incorporated into the model by way of non-constant, iceberg-type, transport costs which properly value the interaction of both effects. The consideration of this type of transport costs satisfies two objectives. First, it emphasizes the sensitivity of the final result (concentration as against dispersion) to the functional form of the transport costs. Second, it simultaneously accounts for the two situations not considered by Krugman, thus giving rise to the possibility of finding various stable long-run equilibria with asymmetric distributions of the manufacturing sector. Conclusions and extensions to the model are provided last. 2 The Krugman (1991) model We now describe the most salient points in Krugmans’s work, in a highly summarized form. Where this is not possible, we revert to the original work, particularly where it is necessary to justify any part of the model. Let us consider two regions and two sectors: one agricultural with constant returns, and the other manufacturing with increasing returns. All individuals have the same utility function: U = CMµ CA1−µ

(1)

where CA is consumption of the agricultural good and CM is consumption of a manufactured aggregate:  σ  N  (σ−1) (σ−1) CM = ci σ (2) i =1

where ci is consumption of the i th variety. The agricultural technical coefficient is 1 and the peasants are immobile within each region in the magnitude (1 − µ)/2. Manufacturing workers are perfectly mobile between regions. Let the workers of the j -th region, j = 1, 2, be represented by Lj where L1 + L2 = µ. The production of the i th manufacture is governed by: LMi = α + βxi (3) where LMi is the quantity of work employed in producing xi units of the i th manufactured good.

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With respect to transport costs, we suppose that the agricultural good, as the numeraire, is not subject to transportation cost, while manufacturing goods present iceberg transport costs, such that, for each unit sent from one region to the other, only a fraction τ < 1 reaches its destination. There are many firms within the manufacturing sector, each produces a different brand under a market structure of monopolistic competition with free entry. The conditions of maximum profit and of zero profit, respectively lead to: pj = xj =

σ σ−1 βwj α(σ−1) β

j = 1, 2

(4)

j = 1, 2

(5)

where pj and wj are, respectively, the price of the manufactures and the wage rate of workers in j . 2.1 Short-run equilibrium Short-run equilibrium is defined as an equilibrium where the allocation of workers between regions is given. The six equations characterizing short-run equilibrium are: −(σ−1)   w1 τ L1 Z11 = (6) L2 w2 −(σ−1)   w1 L1 Z12 = (7) L2 w2 τ      Z11 Z12 Y1 + Y2 (8) w1 L1 = µ 1 + Z11 1 + Z12      1 1 Y1 + Y2 (9) w2 L2 = µ 1 + Z11 1 + Z12   1−µ +w1 L1 (10) Y1 = 2   1−µ + w2 L2 (11) Y2 = 2 where Z1j , j = 1, 2, is the ratio between the expenditure of region j on manufacturing goods from 1 and its expenditure on manufacturing goods from 2, and where Yj is the income of the j -th region. On the basis of the above system of equations, we determine the endogenous variables Z11 , Z12 , w1 , w2 , Y1 and Y2 . 2.2 Long-run equilibrium The interest of the model lies in deducing the movements of the manufacturing labor force between regions, without supposing, as we do with short-run equilibrium, that the allocation is given. The criterion which guides workers is that of

Multiple equilibria, stability, and asymmetries

429

moving to the region offering higher real wages. If ωj represents the real wage of the j -th region, Pj represents the price index of the manufactures aggregate for consumers of j , and f represents the proportion of the industrial sector possessed by region 1 (f = L1 /µ), we have: w1 = P1µ 

ω1 =

w1 −(σ−1)

µ w −(σ−1) − (σ−1)

f (w1 ) + (1 − f ) τ w2 w2 ω2 = µ =  µ − (σ−1)

P2 −(σ−1) f wτ1 + (1 − f ) (w2 )−(σ−1)

(12)

2

(13)

The system given by Equations (6) to (13) cannot be resolved analytically in an explicit way (except in the case where f is equal to 0.5, where all wages, both nominal and real, are equal to 1). However, it is possible to resolve them numerically for each group of three of the parameters governing the process (τ , µ and σ), postulating different values for f , to observe how ω1 /ω2 evolves. Krugman considers only two cases, both paradigmatic. Thus, with σ = 4, µ = 0.3 and τ = 0.5 or 0.75, he obtains the following results, shown in Fig. 1. 1.06

ω1/ω

2

τ=0.75

1.04 1.02

τ = 0.6

1 0.98 0.96 0.94

τ =0.5

0.92

f 0

0.5

1

Fig. 1. Krugman’s result (σ = 4, µ = 0.3)

Logically, we find equilibria for those distributions of the manufacturing sector for which the workers have no incentive to migrate to the other region, when ω1 = ω2 . For τ = 0.5, the equilibrium in f = 0.5 is stable, given that any worker who migrates to the other region in this situation is motivated – when analyzing the evolution of real wages – to retrace his journey and return to an equal distribution situation (dispersion). By contrast, if τ = 0.75, the equilibrium in f = 0.5 is unstable. The final result is of the core-periphery type, with total concentration of the manufacturing sector in one region. Can ω1 /ω2 (f ) cross the horizontal in 1 more than once, giving rise to multiple equilibria, for a given group of three parameters (τ , µ, σ)? Certainly it could, as Krugman has recognized, where he comments that he bypasses this question. However, he imprecisely states that “there could be several stable equilibria in which both regions have nonzero manufacturing production”. The reason for this imprecision is that Krugman’s original model can generate multiple equilibria with asymmetric distribution. However, these are always unstable, so that, in the final analysis, everything is reduced to two extreme cases: total concentration or

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equal distribution. Purely by way of illustration, for τ = 0.6, and maintaining the value of the other two parameters, we therefore obtain (see Fig. 1) a single stable equilibrium with dispersion at 50% and two unstable equilibria in f = 0.18 and f = 0.82. This gives rise to what we might consider as a minimum size for the industrial sector (in this specific case, 18% of world production) in order to be viable in a region. Below this value, we again have a core-periphery pattern. 3 Asymmetric and stable multiple equilibria: Non-constant transport costs From the model we have just reviewed, we can easily derive a single asymmetric and stable equilibrium. We consider that the allocation of natural resources in both regions is sufficiently heterogeneous so that – with the exogeneous total of the world agricultural population being given – the said population is unequally distributed between the regions. However it is the region with the best land quality that captures a higher percentage of peasants, who remain immobile (Lanaspa and Sanz 1999). We attempt in this section to deduce multiple equilibria that are stable and asymmetric. To accomplish this we modify the parameter τ . In the original Krugman (1991) model, transport costs in the manufacturing sector are, as already seen, constant and independent of all other elements in the model. We consider that these costs depend on the working population of the region in question, formulated in Fig. 2 (we can easily include the total population in this context, given that the agricultural sector is again equally distributed). τ

f 0

0.5

1

Fig. 2. Non-constant transport costs

From the outset, we clarify the types of economic behavior we are seeking to capture with this function. The elements we seek to account for with a function such as τ (f ) exceed the exclusive consideration of transport costs: we also try to reflect the interaction of two other elements that realistically model the economic size of the regions: congestion costs and infrastructure advantages. The first element, congestion costs, has a long tradition in urban and regional economic models, as a centrifugal force to penalize agglomerations and shun undesirable by-products such as pollution or high cost housing. The second element, infrastructure, undoubtedly plays a relevant role when describing the processes of industrial concentration. The empirical works of Garc´ıa-Mil´a and McGuire (1992) and, above all, Wheller and Mody (1992) clarify these processes.

Multiple equilibria, stability, and asymmetries

431

Bearing these ideas in mind, let us return to Fig. 2. We recall that τ denotes the rate per one of each unit sent to a region that is effectively received and can be consumed. Infrastructure is usually identified as public investments (airports, motorways, communications networks), that reduce the cost of transport and which obviously require a certain population in order to be executed. In a recent theoretical work, Martin and Rogers (1995), interpret infrastructure in a broad sense, as anything facilitating the connection between producer and consumer. According to this definition, not only do high-speed trains, ports and harbors qualify as infrastructure, but so do more private entities such as hyper-markets, banks, public notaries or chemists. All of them must be of a minimum size in order to be viable. Thus, in the Figure, τ starts to increase as the size of the region increases because in this way, the region acquires the size necessary to justify major investments while, simultaneously not baring high congestion costs. The transport cost is minimum for f = 0.5, increasing thereafter until f = 1, and it is here that congestion costs are essentially operative. If the region exceeds a certain size, the negative effects of agglomeration surpass the advantages derived from infrastructure, and the ‘lost’ portion of the manufactured good (in somewhat simplistic terms, we can consider excess traffic) upon arrival at large regions increases. However, this sort of relationship may be scale-dependent, while the model is not. By this we mean that in a very small two-region economy there may be no congestion at any distribution; meanwhile in a very large economy there may be adequate infrastructure at any distribution point. Our explanation for justifying the form of τ (f ) might seem interesting and credible, but ultimately, the definitive empirical answer is unknown. However, two aspects in particular regarding its formulation remain arbitrary. First, if we consider that τ (f ) is maximum when the two regions are identical, noteworthy analytical advantages are available to us. We can work with one τ without distinguishing between τ1 and τ2 . In effect, when region 1 is 40% of the manufacturing sector, the other is 60% and, because of the symmetry of τ (f ) around 0.5, both values coincide. Nevertheless, the interaction of congestion costs and infrastructure does not necessarily give rise to this special configuration. In other words, the size of the region that minimizes transport costs can be in f = 0.4 or in f = 0.7. We have indeed found the model’s conclusions to be robust with respect to the different specifications which do not add anything new to the results from a qualitative viewpoint, but they do offer a new analytical and computational complexity. From now on we will therefore limit ourselves exclusively to the case represented in Fig. 2. A second arbitrary aspect is that the influence of the population over the transport cost is a matter of degree; given the same form of inverted U for τ (f ), the difference between the minimum transport cost in f = 0.5 and the maximum in f = 0 or 1 can be either obviously important or not very important. As a matter of prudence, we consider here that the influence of the population exists but is negligible; in all cases we present from now on, τ (0) is never less than

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three quarters of τ (0.5). The conclusions of the model would not be modified if we failed to adopt this hypothesis. Specifically, we postulate the functional form for τ (f ) as: τ (f ) =

A , (f − 0.5)2 + B

0f 1

(14)

with B > A > 0 being a sufficient condition to guarantee that τ is between zero and one. This function is symmetric around f = 0.5; τ (0) = τ (1) = A/(0.25 + B ) and τ  (f ) > 0 if f < 0.5 and τ  (f ) < 0 if f > 0.5, thus τ (f ) is maximum at f = 0.5. We can analytically confirm that an expression such as (14) fits the representation of τ (f ) given in Fig. 2. Of the set of Equations (6) to (13), which govern the behavior of the model, only (6), (7), (12) and (13) are changed, precisely those in which the parameter τ appears, with (14) being substituted for this parameter. The next objective is to analyze how the results change with respect to those obtained by Krugman if we introduce the modification of transport costs that depend on the share of employees. A priori we can anticipate arguments asserting that total concentration is more difficult to achieve; in this case the regions suffer higher transport costs with respect to other, less extreme, configurations. But before considering this question, we have to define the appropriate framework for the comparison between what can be deduced from the original model and what is derived from its modified form. With what constant value of τ , as postulated in the initial formulation, is τ (f ) comparable? Let us define τK as the constant value of τ equivalent to τ (f ) in Krugman’s model, if the following expression holds: 1 τK = τ (f )df . (15) 0

The sense given to the equivalence is such that, considering the possible interval of variation in both cases of f in [0, 1], it ensures that the percentage of goods shipped and effectively arriving is the same. Fig. 3 conveys our idea in a simple visual form, illustrating that, as an average, the loss of the good due to transport costs is neutral with respect to the influence – according to the second approach – exerted by the size of the manufacturing sector f (R + T = S ). The functional form of τ (f ) in (14) being given, the associated expression for τK turns out to be:

 2A (16) τK = 0.5 arctan 0.5B −0.5 B In these conditions we can now turn to a typification of the five different cases in which discrepancies are produced between the use of the function τ (f ) and the constant and equivalent τK . To isolate the variations due solely to the different treatment given to the parameter τ , the values of µ and σ are left unaltered in the comparison between both types of behavior. We stress that the consideration of other values for σ and µ does not modify the results.

Multiple equilibria, stability, and asymmetries

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τ 1 A/B

S

τk

T

R

τ(f)

f 0

0.5

1

Fig. 3. Equivalence between τ (f) and τk

In a significant majority of situations both formulations engender coincidental results in terms of concentration-dispersion. Here we will only reflect the divergencies because they provide evidence on how the incorporation of τ (f ) affects the model. Furthermore, within these divergencies we have searched for a group of three parameters as close as possible to the two cases proposed by Krugman (µ = 0.3, σ = 4 and τ = 0.5 or 0.75). Results obtained for particular combinations of these parameters are presented in Figs. 4 to 8. τK = 0.75, τ (f) =

0.7 , µ = 0.175 (f − 0.5)2 + 0.85566

and

σ = 4.

ω1 /ω 2 1.03 1.02 1.01 1

τk

0.99 0.98

τ (f) f

0.97 0

0.5

1

Fig. 4. Comparison between τ (f ) and τk

The first conclusion, no less important for being trivial, is precisely the existence of a variety of scenarios in which the specification of the functional form of the transport costs influences the results. This conclusion has already been detected by D’Aspremont et al. (1979), Economides (1986) and Gabszcewicz and Thisse (1986) in Hotelling-type models of spatial competition, different from our proposed model. The remainder of our observations refer specifically to the outcome that has been obtained. The computational resolution of this type of model clarifies that the movement from a pattern of total concentration to one of dispersion at 50% , with the values of one of the parameters changing appropriately, usually occurs via multiple equilibria, in which the function ω1 /ω2 (f ) crosses the horizontal three

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τK = 0.75, τ (f) =

0.7 , µ = 0.2 (f − 0.5)2 + 0.85566

σ = 4.

and

ω1 /ω 2 1.02 1.01

τk

1 0.99

τ (f) f

0.98 0

0.5

1

Fig. 5. Comparison between τ (f ) and τk

τK = 0.5, τ (f ) =

0.7 , µ = 0.3 (f − 0.5)2 + 1.3205

and

σ = 3.3.

ω1 /ω 2 1.02 1.01

τk

1 0.99

τ (f) f

0.98 0

0.5

1

Fig. 6. Comparison between τ (f ) and τk

τK = 0.5, τ (f ) =

0.7 , µ = 0.4 (f − 0.5)2 + 1.3205

and

σ = 4.2.

ω1 / ω2 1.02 1.01

τk

1 0.99

τ (f)

0.98

f 0

0.5

Fig. 7. Comparison between τ (f ) and τk

1

Multiple equilibria, stability, and asymmetries

τK = 0.5, τ (f ) =

435

0.7 , µ = 0.4 (f − 0.5)2 + 1.3205

and

ω1 /ω 2 1.03

σ = 4.

τk

1.02 τ (f)

1.01 1 0.99 0.98

f

0.97 0

0.5

1

Fig. 8. Comparison between τ (f ) and τk

times in 1. In the five cases we have detected, at least one of the functions corresponds to this situation. Thus we find asymmetric and stable equilibria with τ (f ) as against the other three possible results in τK ; that is, dispersion (Fig. 4), concentration (Fig. 5) and multiple equilibria (Fig. 6) which are asymmetric but unstable in the case of Krugman. By contrast, the roles are reversed in Figs. 6, 7 and 8; asymmetric and unstable equilibria with τK , as against asymmetric stable equilibria, dispersion and concentration, respectively, with τ (f ). This situation illustrates that, as remarked upon earlier, the introduction of τ (f ) does not represent a strong break with the original model because for divergence to exist between the two approaches, at least one of them must produce a frontier or bridge result not corresponding to archetypal patterns of equal dispersion or total concentration. A less anticipated and less intuitive conclusion can be deduced from changes in the localization pattern which τ (f ) incorporates with respect to the standard model. In principle, we could expect that a formulation as given in (14) would encourage the development of equi-spaced settlements, given that the loss of manufacturing goods is minimized. However, this is applicable to the illustrated case in Fig. 7, and then only in part, since dispersion at 50% coexists with total concentration in the original model; meanwhile only the first result is obtained if we employ τ (f ). In the other situations the predominant effect is the emergence of stable asymmetric equilibria, in that one region corners around 25% of the manufacturing sector, and the other region captures the rest. The reason for deducing these asymmetric distributions – rather than others which minimize transport cost – is explained through Fig. 9. In the Krugman model high transport costs favor, ceteris paribus, dispersion, and low transport costs favor concentration. Thus, if f = 0.5 the transport cost is minimal and tends towards concentration (see arrows a in the Figure); and if f is close to 0 or to 1, the opposite effect happens (see arrows b). Justification is thus given to the relatively high frequency with which stable multiple equilibria with

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L. F. Lanaspa, F. Sanz τ

b 0

a

a 0.5

b

f 1

Fig. 9. Effects of the non-constant transport costs on concentration/dispersion

25–75% distribution appear precisely where the two opposite forces compensate each other. Finally, there are a number of noteworthy observations on the scope and significance of the emergence of stable asymmetric equilibria through the introduction of τ (f ), a phenomenon Krugman’s original model cannot generate. First, even at the theoretical level, the existence of a pure concentrationdispersion dichotomy without intermediate cases, gives rise to inter-regional trade patterns that are hardly credible. Indeed, if we have dispersion at 50%, there is no exchange in the agricultural good and there is only intra-industry trade in manufactures. On the basis of the definition given to variables Z11 and Z12 , and knowing that each region dedicates a portion µ of its income to the purchase of manufacturing goods, the monetary flow, identical in both directions, can be quantified as: µY1 µY2 Z12 = . (17) 1 + Z11 1 + Z12 If we take total concentration as the starting point, there can be no more than inter-industry trade. From among the various possibilities, the simplest way to quantify these exchanges monetarily is to note that the purchases of manufactures made by the solely agricultural region, let us say region 1, necessarily have to come from region 2; they are given by: µY1 = µ

(1 − µ) , 2

(18)

so that (18) constitutes the trade flow, identical in value in both directions, although different with respect to the good being exchanged. In this case if the three parameters are such that concentration takes place, volume of trade only depends on µ and is maximum when µ = 0.5. Stable asymmetric equilibria are indeed capable of generating intra- and interindustry trade simultaneously, in accordance with what actually occurs in the world as well as in accordance with the theoretical model of Helpman and Krugman (1985, Chapters 7 and 8). Unfortunately, it is now not possible to obtain an analytical expression without endogenous variables, and capable of reflecting the magnitude of trade flows. Each case therefore has to be resolved numerically.

Multiple equilibria, stability, and asymmetries

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Second, and finally, we are aware of other works, within the same Krugman framework that also consider the appearance of stable an asymmetric multiple equilibria, namely that of Brakman et al. (1994) and Alonso-Villar (1996). The fundamental supposition of Brakman et al. is to consider that the productivity of workers of one region decreases with the number of firms located there. These negative feedbacks operate starting from the first firm, a view which, we think, does not seem realistic. Our approach of considering a minimum size of the industrial sector for the congestion effects to work is more appropriate. AlonsoVillar incorporate education into the basic model, thus relating it to the human capital literature. Our approach achieves an identical result, although we follow a different route: one we think is properly argued in so far as the key hypothesis is concerned. We achieve this result by making only slight changes to the original Krugman (1991) model, in order to avoid unnecessary complexity. 4 Conclusions and extensions We have set out here to complete and extend the range of possible results that Krugman’s (1991) innovative model of economic geography can generate. Slight changes to the original formulation, taken as a reference point, give rise to appreciable changes regarding resultant patterns of specialization. We introduce non-constant transport costs that contrast with Krugman, and depend on the size of the manufacturing sector of the region in question. Through this means we try to scrutinize the interaction between two elements that exert influence over the localization decisions of agents, and whose relevance and intensity is not neutral to the size of the regions. The two elements are congestion costs which increase with the size of the region) and infrastructure (which require a certain population in the region to be executed). The main result of this approach is that, with certain specific values of the relevant parameters being given, we obtain various asymmetric stable equilibria, another aspect not contemplated in the original model. In this way we find theoretical justification for economic landscapes in which large industrial belts coexist with smaller ones. We realize that our work can be improved upon in many respects. It would be useful to analyze the effects produced by non-constant transport costs, in models with more regions and in models that give greater credence to the spatial component; this task could be properly approached within this framework. We suspect that the qualitative results would remain unchanged. Other relevant extensions, such as the presence of a larger number of goods, or the possibility of deriving the size and the localization of the agricultural sector endogenously, are very interesting but beyond the scope of this article. Infrastructure has been treated as an exogenous variable in our analysis; this simplification should be overcome by an explicit consideration of infrastructure financing. The public sector should be introduced within a model along with the consequent effects the public sector provokes on the geographical distribution of economic activity.

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And lastly, separate analysis of intraregional and interregional transportation costs as a function of the size of the industrial sector would be a useful contribution to research.

References Alonso-Villar O (1996) El papel de la educaci´on en la aglomeraci´on urbana. Working Paper 96-05, Carlos III University, Madrid Brakman S, Garretsen H, Gigengack R, van Marrewijk C, Wagenwoort R (1994) Negative feedbacks in the economy and industrial location. Journal of Regional Science 36: 631–651 Christaller W (1933) Central places in southern Germany. (Translated into English in 1966) Fischer, Prentice-Hall D’Aspremont C, Gabszewicz J, Thisse J F (1979) On Hotelling’s stability in competition. Econometrica 47: 1145–1150 Economides N (1986) Minimal and maximal product differentiation in Hotelling’s duopoly. Economics Letters 21: 67–71 Gabszewicz J, Thisse J F (1986) On the nature of competition with differentiated products. Economic Journal 96: 160–172 Garc´ıa-Mil`a T, McGuire T J (1992) The contribution of publicly provided inputs to states’ economies. Regional Science and Urban Economics 22: 229-241 Helpman E, Krugman P (1985) Market structure and foreign trade. MIT Press, Cambridge, MA Isard W (1956) Location and space economy. MIT Press, Cambridge, MA Krugman P (1991) Increasing returns and economic geography. Journal of Political Economy 99: 483–499 Krugman P (1995) Development, geography, and economic theory. MIT Press, Cambridge MA Lanaspa L F, Sanz F (1999) Krugman’s core-periphery model with heterogeneous quality of land. Urban Studies 36: 499–507 Losch A (1940) The economics of location. (Translated into English in 1954) Fischer, Yale University Press Marshall A (1920) Principles of economics. MacMillan, London Martin P, Rogers C A (1995) Industrial location and public infrastructure. Journal of International Economics 39: 335–351 Von Th¨unen J H (1826) The isolated state. (Translated into English in 1966) Pergamon, Perthes Weber A (1909) The theory of the location of industries. T¨ubingen (Translated into English in 1929) Chicago University Press, Chicago Wheeler D, Mody A (1992) International investment location decisions. Journal of International Economics 33: 57–76

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