Trade openness and city size whit taste heterogeneity and amenities
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TRADE OPENNESS AND CITY SIZE WITH TASTE HETEROGENEITY AND AMENITIES Willy W. Cortez Yactayo and Mauricio Ramírez Grajedaa Universidad de Guadalajara May, 2009 AbstractThis paper incorporates taste heterogeneity and amenities, as dispersion forces, into Fujita‟s et al. (1999,chap. 18) international trade model. By doing so, agglomeration of both firms and workers is the result ofmarket and non-market interactions. We analyze the outcomes of the original model vis-à-vis the outcomesof its extension. In particular, we assess the impact of international trade openness on cities‟ size. Threemain general predictions arise from such a departure: First, the breakpoint with taste heterogeneity is higherthan the breakpoint associated without taste heterogeneity. Second, for low levels of trade openness urbanagglomeration is attenuated with taste heterogeneity. And third the dispersed equilibrium is not feasiblewith both taste heterogeneity and amenities. Finally, based upon particular values of the parameters of themodel, market outcomes converge to the optimal social welfare outcomes as taste heterogeneity increases.Keywords: agglomeration economies, amenities, city size, taste heterogeneity and trade openness.JEL Classification: R12; F15; F121. IntroductionThis paper sheds light on the impact of international trade openness on inter urbanstructure. Our theoretical framework is a departure from Fujita´s et al. (1999, chap. 18)New Economic Geography (NEG) model, where location decisions are not longerexclusively driven by pecuniary considerations. In particular, we numerically solve forFujita et al. (1999) with taste heterogeneity and amenities and compare its outcomes witha model where the only centrifugal force is an exogenous congestion cost parameter.1Furthermore, under particular conditions it is possible to generalize the features observeda Corresponding author. Address: Universidad de Guadalajara, Centro Universitario de Ciencias EconómicoAdministrativas, Departamento de Métodos Cuantitativos. Periférico Norte 799, Módulo M, segundo nivel, C.P. 45100,Zapopan, Jalisco, México. E-mail: ramirez-grajeda.1@osu.edu. Tel. +52 (33) 3770 3300 ext. 5223.1 In the NEG literature, a centripetal force is an agglomeration cost; a centrifugal force is a gain of agglomeration. Bothforces are directly reflected in real wages.
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in our numerical examples. Finally, by increasing taste heterogeneity we find that marketoutcomes may converge to socially optimal outcomes. Within the (henceforth, NEG) literature, taste diversity and amenities involvedealing with an alternative design of migration. Traditionally, migration is driven by alaw of motion, which takes place as long as real wages are different across locations. Itestablishes that a gap in real wages drives a fraction of workers from locations with lowreal wages to those with higher real wages.2 Fujita et al. (1999), for example, define J(1) j j ( ) j 1and, d j(2) ( j () ) j , dtwhere j is the fraction of labor in location j, j() is the real wage function net ofcongestion costs in location j and denotes the speed of convergence. denotes thedistribution of the population over J locations. (1) denotes a weighted sum of real wagesacross J locations. (2) denotes the labor share dynamics over time at location j=1, …, J.By construction, at t, the sum of (2) over J locations is zero. If real wage at t in oneparticular location is, for example, greater than (1), then it will be a net receiver ofworkers. The distribution of the population across locations determines real wages as a2 Seminal NEG papers such as Krugman (1991), (Krugman & Venables, 1995), Venables (1996) or Puga (1999) use asimilar migration mechanism. 2
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result of the interplay of agglomeration economies and, for example, congestion costs orimmobile labor.3 Spatial equilibrium is reached when (2) is zero for every j=1,…,J.4 Several shortcomings are worth mentioning with regard to this particular migrationmechanism. One, labor flows are solely governed by pecuniary considerations: realwages. Consequently, there is no room for migration to low real wage locations. Two,migration is not the result of an individual decision because the law of motion randomlyselects those workers who must move out. Hence migration does not reflect any kind ofmicro foundations as rational expectations or strategic behavior. In addition, the fractionof the population that flees from a particular city positively depends upon the real wagegap. Three, workers focus only on current real wage gaps (Fujita & Thisse, 2009). Four,equilibrium is defined when workers stop moving across locations. And finally, the set ofequilibria is determined by the initial distribution of the population; in other words, thereis path dependence. As a matter of fact this specific mechanism could fall intoevolutionary game theory, see Weibull (1995).5 Why, then, is this “myopic” Ottaviano etal. (2002) mechanism applied? Because, it keeps things simple Fujita et al. (1999). Onthe other hand, however, Baldwin (2001) points out that under the conventional migrationlaw the number of multiple equilibria is drastically reduced. Five lines of research are encountered within the NEG literature to explain anincome gap in equilibrium. First, by incorporating migration decisions based on aforward looking behavior Ottaviano et al. (2002); Ottaviano (2001); Baldwin (1999) andOyama (2009). In this case, initial beliefs determine the long-run spatial distribution of3 Agglomeration economies arise due to increasing returns to scale, trade costs and love for variety.4 In equilibrium, however, a real wage gap is feasible if the equilibrium is a corner solution.5 Brakman et al. (2001) and Fujita et al. (1999) make the same observation. 3
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economic activities, which might mimic the outcomes under the conventional migrationmechanism. Second, by assuming a quasi-linear utility function such that consumersurplus is a component in migration decisions as well Ottaviano et al. (2002).6 Third, bydropping the assumption of costless migration such that the larger the migration flow, thehigher the costs Ottaviano (1999). Fourth, by assuming that regions exhibit differentnatural and cultural amenities (Tabuchi & Thisse, 2002).7 And fifth, by combining tasteheterogeneity and discrete probability models (Tabuchi & Thisse, 2002); Murata (2003)and Mossay (2003). In this paper, we devote our attention to the latter 2 departures that overcome thefirst four shortcomings of the traditional NEG law of motion mentioned above. Tasteheterogeneity, which entails a non-pecuniary component of migration decisions, is adegree of attachment to or perception of a particular location due to a wide variety ofreasons such as place of birth, marital status or risk attitude Greenwood (1985). However,individual´s choices on places could be correlated to income as Rosen (2002) suggests.Focusing on amenities, on the other hand, is not a new idea. Sjaastad (1962) conceivesthat a non-monetary factor also affects migration such as climate, pollution or congestion.For example, (Fujita & Thisse, 2009) explain that there is uneven distribution ofimmobile resources (exogenous amenities) like natural harbors. Jacobs (1969) claims thatsocial factors determine the configuration of cities (endogenous amenities), for example,by attracting creative and talented people. Fortunately, all these factors can be aggregated6 In this case, individuals move according to the value of the indirect utility function, which is equivalent to the nominalwage plus the consumer surpluses plus endowments. See Ottaviano et al., (2002); (Tabuchi & Thisse, 2002); (Picard &Zeng, 2005).7 Cultural amenities could be measured, for example, by the Bohemian Index. 4
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in order to assess their impact on the spatial distribution of both workers and firms. To doso, NEG and probability choice theory can be combined. It is worth recalling that there is an almost limitless set of factors that we mightconjecture would impact on urban attractiveness to potential migrants. For example,(Glaeser & Redlick, 2008) claim that more educated people have higher incentives tomigrate. In particular, they show that education level is an important determinant ofmigration to US urban areas. Our main findings in the long-run are four-fold. First, the breakpoint with tasteheterogeneity is higher than the breakpoint associated with Fujita et al. (1999). Second,for low levels of trade openness urban agglomeration is attenuated with taste diversity.Third, the dispersed equilibrium is not feasible with taste heterogeneity and amenities.And finally, based upon particular values of the parameters of the model, marketoutcomes converge to the optimal social welfare outcomes as taste heterogeneityincreases. The reminder of this paper is divided into the following sections. In section 2, webriefly explain the main ingredients of NEG models and discrete choice models ofmigration, and the way in which both paths are combined. We also discuss theimplications of unifying both paradigms by reviewing the related literature. Section 3 isthe theoretical framework that incorporates taste heterogeneity and amenities. In Section4, we report and analyze the numerical solution of the model. Furthermore, under someparticular assumptions we present some general features of the extended model. Insection 5 there are some final remarks. 5
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2. NEG and discrete choice modelsKrugman‟s (1991) core-periphery general setting is seminal within the NEG literature. Itassumes j locations; two sectors, manufacturing and agricultural. The former ismonopolistically competitive, whose technology exhibits increasing returns of scale andonly employs workers. The latter is perfectly competitive, whose technology exhibitsconstant returns to scale and exclusively employs peasants. Workers can migrate acrosslocations but not across sectors. Peasants can move neither across locations nor acrosssectors. Both workers and peasants have the same preferences over N manufacturingvarieties and a homogenous-good produced in the agricultural sector. Such preferencesare represented by a Cobb-Douglas utility function where the component related tomanufacturing goods is a CES utility function. Trade costs are of the Samuelson (1952)type. There are two types of equilibria: short-run and long-run. The former arises whenboth workers and peasants maximize their utility, firms maximize profits, and the productand labor market clearing conditions are satisfied. The level of real wages in themanufacturing sector associated with the short-term equilibrium in location j areexpressed as j () for j = 1,...,j.8 The latter is defined as the short-run equilibrium and(2), labor migration over time, equal to zero. The immobility of peasants constitutes adispersion or centripetal force of spatial agglomeration. If trade costs are high enough theonly long-run equilibrium is the dispersed one; below a threshold a core-peripheryeconomy suddenly arises: most workers cluster together in a single location.98 In particular, Krugman (1991) does not have a closed-form solution. Therefore, the level of utility associated with theshort-term equilibrium cannot be expressed as a function of . However, its properties can be inferred.9 There is a range where the long-run equilibrium can be either dispersed or concentrated in a single city. 6
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Fujita et al. (1999), among others give a prominent role to the effects ofinternational trade costs on the distribution of population between cities. They assume j-1cities in the home country and 1 in the foreign country. Migration takes place onlybetween cities in the home country. There is only one sector, manufacturing. The mainoutcome is that high levels of such costs foster agglomeration in a single city. By thesame token, Venables (2000) investigates the effects of external trade costs on the shareof manufacturing employment. A single city will have a high amount of employmentwhen the economy of a country is closed to external trade. However, when the economyhas access to imports due to lower trade costs the amount of industrial employment goesdown. The economy develops a duocentric structure if it is open to external trade.Alonso-Villar (2001) suggests that the negative relationship between trade openness andcity size depends upon the relative size of the home country. If it is low with regard to therest of the world, a dispersed equilibrium is not sustainable, given low levels of tradecosts. Mansori (2003) introduces a fixed and a marginal trade cost that may cause thefollowing two outcomes after trade barriers fall. One is that a megalopolis that is alreadyin equilibrium does not shrink in size; Buenos Aires and Bangkok are examples of thisoutcome. The other is that cities in the dispersed equilibrium become a megalopolis likeSeoul.10 Krugman (1991) is a benchmark for many other papers with different assumptionsand, consequently, different outcomes. There are two broad divisions in the NEGliterature: at the international level, migration across sectors is allowed but not across10 Contrary to what Mansori (2003) theoretically claims, Henderson et al. (2001) find that Korea has experienced aprocess of deconcentration of manufacturing due to infrastructure improvements. 7
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countries, see (Krugman & Venables, 1995); (Krugman & Livas, 1996); Puga (1999); atthe regional level, workers can migrate across locations Krugman (1991). On the other hand, according to (Brakman & Garretsen, 2003), the conventionalmigration rule used in the literature discussed above is still not satisfactory and a stepforward would be to incorporate heterogeneity across workers. As mentioned above, tasteheterogeneity is a non-pecuniary component of the intercity location problem thatrepresents a location taste. Hence some workers will stay put even though they may earna higher monetary income in other places. Particularly, once individual monetary incomelevel gets sufficiently high, workers tend to pay more attention to non-pecuniaryattributes of their environment. Low trade costs and more heterogeneous individuals canbe considered as being closely linked to higher levels of economic progress. Taste heterogeneity and amenities in NEG models can be divided into two strands.On the one hand, migration incentives depend upon an overall utility function thatincorporates both pecuniary and non-pecuniary components (Tabuchi & Thisse, 2002);Murata (2003). The theoretical framework of this paper applies this particular approach.On the other hand, stochastic migration models ignores migration driving forces butexplicitly set the distribution of migration movements Mossay (2003). Regarding the first strand, preferences are conceived in several dimensions. On theone hand, they depend on both the level and variety of consumption. Under thisdimension workers are assumed to be homogenous. (Ottaviano & Thisse, 2003) considerthat this is an unappealing and implausible assumption. On the other hand, preferencesare also associated with non-pecuniary factors. Formally, at t and assuming J=2 (2locations), preferences of worker k on location j brakes down as follows: 8
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a) A deterministic pecuniary component, which is represented by j .11 It dependsupon the population distribution across locations and its value is equal among allindividuals located at j, but not necessarily equal among all individuals in other locations. b) The taste component is an idiosyncratic perception or level attachment to aparticular location.12 It is represented by a random variable, jk, i.i.d. according to thedouble exponential distribution with zero mean and variance equal to 22/6. Therealization of this variable is different over time. c) An exogenous level of amenities, aj, associated with a location j such as naturalamenities that do not change over time. In order to isolate the effects generated by thebalance between the centrifugal and centripetal forces, NEG assumes identical regions,but it does not include the impact of differentials in amenities. However, empiricalevidence shows that geographical advantage, such as a coastal location, good climate andgood access to economic centers, may explain the spatial distribution of industry Gallupet al. (1998).13,14 These preferences can be represented by an overall utility for worker k in location jas(3) V jk (t ) j ( ) k a j , 15 j11 It is also referred to as a market or homogenous incentive component.12 It is also referred to as a non-market or heterogeneous incentive component.13 Haurin (1980) explains theoretically that climate partially determines the distribution of population.14 It is possible to endogenize amenities. For example, talented people in a particular location attract talented people,see Florida (2002).15 If J=2 then =, which is the fraction of the population in city 1. 9
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provided that t denotes the fraction of the population in city j at t. Worker k decides tolive in region j at t+1, for instance, if the overall utility in that region is larger than inregion j’,(4) V jk (t ) V jk (t ) . If this is the case, then ykj=1 and ykj’=0, however, such a condition is satisfiedrandomly. By applying qualitative binary response theory, see (Maddala & Flores-Lagunes, 2001), the probability that location j is chosen by the worker k is denoted by (5) P( ykj 1) P(V jk (t ) V jk (t )) f ( z )dz. j ( ) a j j ( ) a j where z=rk-r’k. McFadden (1974) and (Miyao & Shapiro, 1981) show that such aprobability can be expressed as exp(( j ( ) a j ) / )(6) Pj ( ) Pr(V jk ( ) V jk ( )) , exp(( j ( ) a j ) / ) exp(( j ( ) a j ) / )where the parameter is the degree of heterogeneity. If aj - aj’ = 0 and 0, workerstend to be equal among them, then migration decisions exclusively depend uponpecuniary considerations. If aj - aj’ = 0 and , workers tend to be different from eachother and the monetary component weight within the overall utility tends to zero, thenmigration decisions exclusively depend upon individual taste related to both locations. It is possible to know how the distribution of workers evolves over time due to thelaw of large numbers. Therefore, labor changes according to the following new law ofmotion, 10
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d exp(( ( ) (a j a j )) / )(7) (t ) (1 t ) P (t ) t P2 (t ) 1 t , dt exp(( ( ) (a j a j )) / ) 1where (1-t)P1(t) is the population that is leaving city 2 to city 1 and tP2(t) is thepopulation leaving city 1 to city 2, and ()=1()-2(). This setting yields a two-direction gross migration. Equilibrium is defined when (7) is zero. Several shortcomings of the traditional migration rule are overcome by substitutingit with (7). Under this new equation, pecuniary and non-pecuniary considerations are partof migration decisions; migration is the result of individual decisions; and migration inequilibrium can take place but the net result is that cities‟ size remains unchanged. Regarding the second strand of the literature, migration is driven by two rules: apecuniary force or a random force. More precisely, with probability 1-, worker kmigrates according to utility differentials; with probability , workers migrate randomly,where the potential locations to migrate and their associated probabilities are exogenous.The NEG literature with taste heterogeneity and amenities(Tabuchi & Thisse, 2002) model is a similar setting to Krugman (1991), the difference isthat agents preferences are represented by a quasi-linear utility function. Under such anassumption the model has a closed-form solution but there are no income effects.Migration is modeled according to (7). They conclude that if exogenous amenities aredifferent across locations (the asymmetric case) and one location has a larger populationthan the other, then the populous location will always be larger irrespective of trade costs.If there is no an amenity gap, then a bell-shaped curve arises, where for intermediatetrade costs a core-periphery pattern arises, otherwise only the dispersed equilibrium is 11
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feasible. If social amenities are positive and different across locations, and both the lovefor variety and increasing returns are low, it is possible that exist a range in which thesize of the populous city is above the social optimum. Without differential amenities thedispersed equilibrium is socially efficient. Murata (2003) maintains the same form of the utility function used by Krugman(1991), but eliminates the agricultural sector and exogenous amenities. 16 In this case,there is no analytical solution, however, the properties of the overall utility function canbe obtained. For high levels of taste heterogeneity, only the core-periphery pattern isallowed; for intermediate levels of taste heterogeneity a bell-shaped curve arises: finally,for low levels of taste heterogeneity, a dispersed equilibrium is feasible for low levels oftrade costs. For high levels of taste heterogeneity, the social optimal populationdistribution coincides with the market allocation; for low levels of heterogeneity themarket equilibrium never coincides with the social optimum. Tabuchi (1986) uses a system of simultaneous differential equations to explainintercity migration due to differences in utilities, which are expressed as a function of citysize. A deterministic specification of the utility leads to an unstable distribution of citysizes, whereas a stochastic specification does not. In this vein, Mossay (2003) designs astochastic continuous model, where locations are distributed along a circle. In eachlocation the characteristics of preferences, product and labor market of Krugman (1991)also takes place. Workers can make three random movements: left, right or not moving.The main outcome of this paper is that the intuition behind Krugman (1991) can beextended in a more complex world: taste heterogeneity represents a dispersion force. An16 Actually, these modifications lead to Krugman (1980). 12
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extreme case is when migration is exclusively driven by pecuniary considerations, whereagglomeration is expected in few locations.3. TheoryIn this section, we outline Fujita et al.‟s (1999) model assuming taste heterogeneity andendogenous amenities, which makes migration decisions depend upon pecuniary andnon-pecuniary considerations. 17 It focuses on intercity migration within a country whichtrades with the rest of the world. The economy embeds increasing returns to scale, tradecosts and love for variety in a general equilibrium setting. There are j locations and one sector which is monopolistically competitive à la(Dixit & Stiglitz, 1977). Lj denotes labor (consumers/workers) in location j, and λj is thefraction of the population that lives this location. Trade costs are of the Samuelson (1952) type: Tjj´≥0 denotes the amount of anyvariety dispatched in location j per unit received in location j’. 18 If j=j’ then Tjj’=1 andTjj’= Tj’j. It is worth mentioning four implications of assuming this type of trade costs.First, it avoids the introduction of a transportation industry which might complicate themodel to solve for the equilibrium. Second, it is a necessary condition for preserving aconstant elasticity of the aggregate demand. This feature simplifies the conditions ofprofit maximization.19 Third, Tjj´ may represent an explicit ad valorem tariff whoserevenues are redistributed among economic agents but dissipated as a consequence of17 Fujita et al. (1999) heavily draws on (Krugman & Livas, 1996).18 For (Limao & Venables, 2001) the cost of doing business across countries depends on geography, infrastructure,administrative barriers (eg. tariffs) and the structure of shipping industry (eg. carriage, freight and insurance).19 A constant elasticity of aggregate demand means that firms maximize profits by setting a price that is a constantmark-up over marginal cost. A specific level of production satisfies this condition. 13
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rent-seeking.20 And fourth, trade costs are not related to the product variety or distancebetween locations. The representative agent in location j derives her pecuniary utility fromconsumption represented by N 1 1(8) U j cnj , n 1 where σ is the elasticity of substitution between any pair of varieties and cnj is theconsumption of each available variety, n, in location j. Under these preferences, desire forvariety is measured by (σ-1)/σ. Under these preferences, desire for variety is measured by(σ-1)/σ. If it is close to one, for example, varieties are nearly perfect substitutes. At the level of the firm, technology exhibits increasing return to scale.21 Thequantity of labor required to produce q units of variety n in region j is(9) l jn F q jn ,where F and v are fixed and marginal costs, respectively. The firm that produces variety nin region j pays nominal wage, wjn, for one unit of labor. In order to characterize theequilibrium, F = 1/σ and = (σ-1)/σ.22 The number of firms in location j, nj, isendogenous. N = n1+…+nJ is the total number of available varieties. 2320 Agents devote resources (lobbying expenses, lawyer‟s fees and public relations costs) to obtain these tariff revenues.21 Increasing returns to scale are essential in explaining the distribution of economic activities across space. This isknown as the “Folk Theorem of Spatial Economics”.22 To assume a particular value of F means to choose units of production such that solving for the equilibrium is easier.To assume a particular value of v allows us to characterize the equilibrium without loss of generality.23 In equilibrium each firm produce a single variety. 14
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There are two types of prices: mill (or f.o.b) and delivered (or c.i.f.). 24 The formerare charged by firms. The latter, paid by consumers, are defined as(10) p n ´ p nT jj´ , jj jwhere pnj denotes the mill price of a good of variety n produced in location j. pnjj´ is thedelivered price in location j´. By the assumptions on trade costs both prices are equalwhen j=j´. Real wages in location j are defined as(11) w j w j ( )G 1 jwhere Gj is a price index, which is the minimum cost of achieving one unit of utilitygiven N varieties and N prices associated with them.25 We define(12) j ( ) w j j ( , ),j(,) is a congestion deflator function in location j where is an exogenous parameterassociated with congestion cost. Wages deflated by prices and congestion costs arepositively related to utility levels.26The short- run equilibriumThe economy reaches its short-run equilibrium when agents and firms optimizerespectively their pecuniary utility and profit functions, such that the aggregate excessdemands in the labor and product markets are zero.24 f.o.b stands for free on board and c.i.f. for carriage, insurance and freight.25 G is defined in (13). In an economy with two cities, j’ is equivalent to (1-)26 15
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The model does not have a closed-form solution. For J=3 the equilibrium mustsatisfy the following system of 3x2 non-linear equations instead: 1 2 1 1 (13) G j s wsT js s 1 and 1 2 w j Ys T js 1(14) Gs 1 , s 1 where(15) Yi Li wi . (13) represents a price index in location j that measures the minimum cost ofobtaining a unit of utility. (14) is the wage equation, which generates zero profits givenprices, income and trade costs. Real wages across locations might be different. Wechoose w3 as a numeraire.The long-run equilibriumUp to this point there are no movements of workers. When (13) and (14) are satisfiedthere is no interaction between locations. Put another way, it is the (Dixit & Stiglitz,1977) setting for multiple regions. Therefore, (7) (instead of equation 2) is added up toconnect locations by equalizing it to zero in equilibrium in the long-run. Workers decideto move according to (4), where both pecuniary and non-pecuniary are taken intoconsideration. 16
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Trade openness and city sizeIn order to relate urban agglomeration to trade openness, two assumptions areincorporated. First, there are 2 countries termed, foreign and home. Only one city islocated in the foreign country, and 2 cities in the home country. L0 is the population in theforeign country and, L1 and L2 in the home country cities. Trade between cities in thehome country involves the same Samuelson (1952) type trade costs, T. But trade costsbetween a particular city in the home country and the unique city in the foreign country isT0. Second, it is assumed that migration is allowed between cities within the homecountry but not across countries. By using MATLAB we numerically solve both the original and the extended modelfor different levels of international trade openness, T0.27 What happens in the foreign cityis neglected. The value of the parameters are assumed to be L0=2, L1+L2=1, δ=0.1, σ=5,T=1.25, =30. Concerning taste heterogeneity and amenities, the analysis is conducted fordifferent values o f a1, a2 and . We assume that 1(,) = (1-) and 2(,) = ().4. The extended modelTaste heterogeneity without amenitiesIn this first case, there are no amenity differentials between both cities, then a1-a2 = 0. Acongestion cost and taste heterogeneity are the only dispersion forces of location. Forconciseness of exposition we focus on city 1. Figures 1-2 relate the fraction of thepopulation in city 1, , to migration over time, d/dt, for different levels of international27 The MATLAB programs are available upon request. We used some routines provided by (Miranda & Fackler, 2002). 17
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trade costs with and without taste heterogeneity, = 0.005 and = 0, respectively. Thesefigures are consistent with the short-run equilibrium, when there is room for internalmigration. Most NEG papers, the short-run equilibrium is depicted as the relationshipbetween and the equilibrium real wage gap, which is equal across all individuals in oneparticular location. In turn, with taste heterogeneity the individual overall utility V()could be different across all individuals irrespective of their location. FIGURES 1-2 In figure 1, international trade is costless, T0 = 1. If d/dt>0 the net effect ofworkers‟ movements shifts the population in city 1 up; if d/dt<0 implies that thepopulation shrinks. Equal distribution of the population, * = 0.5, is associated with nochanges in the population distribution, d/dt=0 (the long-run equilibrium); furthermore, itis a stable equilibrium because d(d/dt)/d<0 and unstable would be the other wayaround. The only difference between the original model and its extension around theequilibrium point is the rate of convergence to the steady state. The speed of convergencefor = 0 depends upon the parameter . 28 Figures 1-2 describe the transition from aunique long-run equilibrium to multiple equilibria, for a higher value of T0. In figure 2, with low levels of trade openness, T0 = 2, there are 3 equilibria: 1unstable and 2 stable. The difference between the curves associated with = 0.005 and = 0 is that the set of stable equilibria are closer to = 0.5 with taste heterogeneity. Inother words, the extreme outcomes of Fujita et al. (1999) are attenuated.28 We have chosen = 30 for the sake of exposition. A low speed of convergence is not visually adequate. The set oflong-run equilibria does not depend upon the value of such parameter but determines the way in which the distributionof the population evolves over time. In this case, it converges in an oscillatory fashion. 18
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Figures 1-2 can be summarized in figure 3. Hence the analysis can be conductedfrom a different perspective. Figures 3-4, depict the relationship in the long-run betweentrade openness, T0, and the population distribution across cities, LR, instead, which isconsistent with the correspondence c: T0 {(0,1)=LRstable and/or =LRunstable}.The set of long-run equilibria without taste heterogeneity, >0, is depicted in black; withtaste heterogeneity, =0, in green. A cross denotes an unstable long-run equilibrium; astar denotes a stable long-run equilibrium. Without heterogeneity agglomeration in onecity takes place for low levels of trade openness as well. As trade costs decreasesdispersion across cities is the only feasible long-run equilibrium. The breakpoint, T0*,that divides the sets of T0’s where the dispersed equilibrium is stable or unstable is higherwith taste heterogeneity than without it: T0*>0>T0*=0. With heterogeneity dispersionrequires lower levels of trade openness (see proposition 1). In addition to this, for lowlevels of trade openness agglomeration is attenuated with taste heterogeneity:agglomeration is below the levels featured by Fujita et al. (1999) (see proposition 2). FIGURES 3-4 The characteristics of trade in Fujita et al. (1999) associated with the long-runequilibrium are still valid when taste heterogeneity is positive. All varieties are consumedin every location. The trade balance is zero between both cities. The number of varietiesis fixed because it exclusively depends upon the technology and the total population ofthe economy. 19
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In this paper, we assume that the parameters generate a T0*=0>0; andparticularly, >1 and should not be too large. Therefore we can present the followingpropositions.Proposition 1. if (G/())(-1) is increasing in T0, then the breakpoint associated withtaste heterogeneity, T0*>0, is higher than the breakpoint associated without tasteheterogeneity, T0*=0.Proof. The long-run equilibrium satisfies (7) equal to zero. At = 0.5, the stability of themodel depends upon the sign of d( = 0.5)/d. Without taste heterogeneity, the breakpoint, T0*=0, satisfies (see appendix) d ( ) Z (1 )(1 )(16) 0, d ( ) ( Z 1)where Z is defined by 1 1 G Z (1 T 1 ). 29 2 ( ) (17) With taste heterogeneity and following Murata (2003) the break point satisfies d ( 0.5) 1 d ( ) (18) 1 0. d 4 d ( ) 0.5Such a condition implies that (G/*())-1>0>(G/())-1=0, then T0*>0>T0*=0,because d(= 0.5)/d is increasing Z. Note that (16) is increasing in Z.29 (16) and (17) corresponds to (18.11) and (18.12) of Fujita et al. (1999). 20
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Proposition 2. If proposition 1 holds, then LRstable(T0)=0 > LRstable(T0)>0 for>T0>T0*>0.Proof. If proposition 1 holds, then LRstable(T0) =0>0.5 and LRstable(T0) >0>0.5.Suppose that LRstable(T0)=0=LRstable(T0)>0=LRstable . By definition, the long-runequilibrium without heterogeneity satisfies(19) j (LRstable) j (LRstable) 0hold. Therefore, d(20) 0.5 LRstable 0. dtThe long-run equilibrium condition is not satisfied in the presence of taste heterogeneity.Now suppose that LRstable(T0)=0 < LRstable(T0)>0. Again, by definition(21) ( ) 0 j (LRstable 0 ) j (1 LRstable 0 ) 0 ,hold. Therefore,(22) ( ) 0 j (LRstable 0 ) j (LRstable 0 ) 0 ,and d exp( (LRstable 0 ) / )(23) LRstable 0 0. dt exp( (LRstable 0 )) / ) 1The first member of (23) falls between zero and 0.5. Therefore, the long-run equilibriumcondition with taste heterogeneity is not satisfied. 21
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It is worth mentioning that this economy falls into the Murata‟s (2003) typeeconomy with endogenous location of the demand if T0. Furthermore, if , theonly distribution of the population consistent with the long-run equilibrium is thedispersed one.Taste heterogeneity with amenitiesIn this case, another dispersion force is added. More precisely, we assume that city 1 hasamenities, a1=0.01, and city 2 has no amenities, a2=0. Figures 5-6 depict the short-runequilibria for different values of trade openness, T0=1, 2, respectively, and =0.0, 0.005.In figure 5, international trade is costless and the only long-run equilibrium without bothtaste heterogeneity and amenities is the dispersed one, = 0.5. With taste heterogeneityand amenities there is a single long-run equilibrium as well: > 0.5, which is an intuitiveoutcome because city 1 has an advantage over city 2. In contrast with the original model,the agglomeration equilibrium outcomes are not symmetric under the extended model.Put another way, city 1 size is always larger than city 2 size (see proposition 3). Figures5-6 are summarized in figure 7 .In figure 8, multiple long-run equilibria of Fujita et al.(1999) are eliminated when the level of taste heterogeneity is high enough, = 0.02 forhigh levels of trade openness. FIGURES 5-6Proposition 3. If >0 and a1-a2>0, the dispersed stable equilibrium is not feasible.Proof. The condition (7) equal to zero never holds if = 0.5 because 22
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d exp( a1 a2 / ) 1(24) 0. dt exp( a1 a2 / ) 1 2 Recall that real wages are equal across locations when = 0.5. FIGURES 7-8Social welfareFollowing (Small & Rosen, 1981), the social welfare in the home country can be definedas 2 ( ) (a1 a2 ) ~ ~ 1 ( ) (a1 a2 )(25) W ( ) ln exp( ) exp( ) , which is the sum of individual utility functions. Figure 9 compares the populationdistribution in the home country that maximizes (25) with the equilibrium outcomes withtaste heterogeneity. For low levels of trade openness the maximum level of social welfareis associated with the dispersed equilibrium. For high levels of trade openness thesocially optimal distribution involves agglomeration. However, as taste diversity getshigher, the range in which the dispersed equilibrium differs from the social optimumoutcome gets narrower. Figure 10 depicts the long-run equilibria with taste heterogeneity and amenitydifferentials, and the socially optimal outcomes. As international trade costs decline theoutcomes converge to the social optimum; below a threshold both outcomes diverge. Figures 11-12 show that given T0 = 1.035 and a1-a2 = 0, the higher the level of tasteheterogeneity the higher the welfare associated with the long-run equilibrium outcome.For low levels of trade openness the dispersed equilibrium is equal to the social optimum. 23
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In fact, at the dispersed equilibrium, W’(0.5)=0, for any level of trade openness, seeMurata (2003), d ( ) d ( ) d ( )(26) W ( ) G ( ) 1 (1 ) 2 . d d d If the dispersed equilibrium is not optimal, then W(0.5) is a local minimum. If thedispersed equilibrium is optimal, then W(0.5) is obviously a global maximum. FIGURES 9-10 With amenities, (26) equal to zero does not necessarily hold and the social welfareassociated with the market equilibrium increases as heterogeneity gets higher, given T0 =1.035 (see figures 13-14). FIGURES11-145. Final remarksMigration in the NEG has traditionally been modeled by (2). By introducing tasteheterogeneity, dispersion forces different from labor immobility or congestion costs,several unrealistic features of migration are eliminated and the extreme outcomes ofKrugman (1991) or Fujita et al. (1999) moderated. Discrete choice theory is a way to incorporate non-pecuniary factors in migrationdecisions. By doing so, the advantages of agglomeration are either equal or reduced givena particular level of international trade openness. When trade openness is low,agglomeration economies push real wages up, however, Fujita et al. (1999) incorporate acongestion cost. But the net effect of clustering is positive because the demand is highlyconcentrated in the national market. Taste heterogeneity and amenities reduce the 24
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agglomeration economies because the pecuniary component of migration decisionsreduces its weight. The extreme case is when the variance of taste heterogeneity isinfinite. In such a case, real wages have almost no weight in location decisions. Whentrade openness is high the advantages of agglomeration is low, therefore costs incrementsoffset real wage gains of concentration. Such real wages gains of concentration arereduced even more with taste heterogeneity and amenity gap. Thus we conclude that thebreakpoint even is higher and agglomeration moderated when taste heterogeneity isincorporated in migration decisions. Furthermore if amenities differences are alsoincorporated the dispersed equilibrium is not feasible. The rationale behind this result isthat any gap in amenities turns one city more attractive for any level of trade openness. An important finding is the uniqueness of equilibrium when both tasteheterogeneity and amenities are incorporated into Fujita‟s setting. In this case, figure 8describes agglomeration only in city 1 for a specific range of international trade opennessand sufficiently high levels of taste heterogeneity. For low levels of trade opennessextreme agglomeration is present and, for high levels agglomeration is mild. Figure 7suggest that there is room for agglomeration in both cities if there are differentials inamenities but taste heterogeneity, given low levels of trade openness. The main messageis that NEG explains agglomeration, however, its direction operates outside the model.But amenities could explain the attraction of workers to a single location. For low levels of trade openness, trade liberalization moves equilibrium outcomes,in terms of the distribution of the population, closer to outcomes that maximize socialwelfare, = 0.5. However, for high levels of trade openness, agglomeration maximizessocial welfare which diverges with the equilibrium outcomes. Increasing taste 25
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heterogeneity is an alternative way to improve social welfare. In this sense, determinantsof migration are quite different among countries. Some conjectures on such matterswould be differentials in amenities and taste heterogeneity.Appendix. The breakpoint derivation for two local cities and one foreign city.Without both taste heterogeneity and amenities the analytical characterization of thebreakpoint point is obtained by using the equations (A.1)-(A.6) (see Fujita et al. (1999)).(A.1) Y0 L0 ,(A.2) Y1 w1 ,(A.3) Y2 (1 )w2 , 1 1(A.4) G0 [ L0T0 ( w1T0 )1 (w2T0 )1 ]1 , 1 1 1(A.5) G1 [ L0T0 w1 ( w2T )1 ]1 , 1 1 1 1(A.6) G2 [ L0T0 ( w1T )1 w2 ] , 1 1 1 1(A.7) w1 [Y0G0 T0 Y1G1 Y2G2 1T 1 ] and 1 1 1 1 (A.8) w2 [Y0G0 T0 Y1G1 T Y2G2 1 ] . At =0.5, due to a deviation from the dispersed equilibrium changes of theendogenous variables are equal in magnitude but with different sign: dG =dG1= -dG2 and 26
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dY =dY1=-dY2. Then we pay attention to city 1 because it is not necessary to keep track ofvariables associated with city 2. Therefore, by totally differentiating (A.1), (A.5), (A.7)and (12), we obtain the following expressions:(A.9) dY dw dw ,(A.10) (1 )G dG [ (1 )w dw dw1 (1 )(1 )w dwT 1 d (wT )1 ] ,(A.11) w 1dw [dYG 1 Y ( 1)G 2dG dYG 1T 1 Y ( 1)G 2T 1 dG]and dw d dG(A.12) d . w G In order to eliminate dY, (A.9) is plugged into (A.10) and taking = 0.5 and. wesolve for dG/G and dw by arranging (A.10) and (A.11): 2Z 1 Z dG d (A.13) Z G 1 . Z 1 dw 2Z d 1 Using Cramer‟s rule and plugging the solution into (A.12) we obtain (16). 27
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ReferencesAlonso-Villar, O. (2001) Large Metropolises in the Third World: An Explanation, Urban Studies, 38, 1359-1371.Baldwin, R. E. (1999) Agglomeration and Endogenous Capital, European Economic Review, 43, 253-280.Baldwin, R. E. (2001) The Core-Periphery Model with Forward Looking Expectations, Regional Science and Urban Economics, 31, 21-49.Brakman, S., Garretsen, H. & Van-Marrewijk, C. (2001) An Introduction to Geographical Economics. Cambridge: Cambridge University Press.Brakman, S. & Garretsen, H. (2003) Rethinking the „New‟ Geographical Economics, Regional Studies, 37, 637-648.Dixit, A. K. & Stiglitz, J. E. (1977) Monopolistic Competition and Optimum Product Diversity, The American Economic Review, 67, 297-308.Florida, R. (2002) The Economic Geography of Talent, Annals of the Association of American Geographers, 92, 743-755.Fujita, M. & J-F Thisse (2009) New Economic Geography: An Apprisal on the Ocassion of Paul Krugman´s 2008 Nobel Prize in Economics Sciences, Regional Science and Urban Economics, 39, pp. 109-119.Fujita, M., Krugman, P. & Venables, A. J. (1999) The Spatial Economics: Cities, Regions and International Trade. Cambridge, MA: MIT Press. 28
29.
Gallup, J. L., Sachs, J. D. & Mellinger, A. D. (1998) Geography and Economic Development, NBER Working Paper Series 6849, NBER, Cambridge, MA.Glaeser, E. & Redlick C. (2008) Social Capital and Urban Growth. Preliminary Draft.Greenwood, M. (1985) Human Migration: Theory, Models and Empirical Studies, Journal of Regional Science, 25, 521-544.Haurin, D. R. (1980) The Distribution of Population, Migration and Climate, The Quarterly Journal of Economics, 95, 294-380.Henderson, J. V., Shalizi, Z. & Venables, A. J. (2001) Geography and Development, Journal of Economic Geography, 1, 81-105.Jacobs, J. (1969) The Economy of Cities. New York, Random House.Krugman, P. (1980) Scale Economics, Product Differentiation and the Pattern of Trade, American Economic Review, 70, 950-959.Krugman, P. (1991) Increasing Returns and Economic Geography, Journal of Political Economy, 99, 483-499.Krugman, P. & Venables, A. J. (1995) Globalization and the Inequality of Nations, Quarterly Journal of Economics, 110, 857-880.Krugman, P. & Livas-Elizondo, R. (1996) Trade Policy and The Third World Metropolis, Journal of Development Economics, 49, 137-150.Limao, N. & Venables, A. J. (2001) Infrastructure, Geographical Disadvantage, Transport Costs, and Trade, The World Bank Economic Review, 15, 451-479. 29
30.
Maddala, G. & Flores-Lagunes, A. (2001) Qualitative Response Models, in: B. Baltagi (ed.) A Companion to Theoretical Econometrics, chapter 17, Malden and Oxford, Blackwell Publishers.Mansori, K. S. (2003) The Geographic Effects of Trade Liberalization with Increasing Returns in Transportation, Journal of Regional Science, 43, 249-268.McFadden, R. D. (1974) Conditional logit analysis of qualitative choice behavior, in: P. Zarembka (ed.) Frontiers in Econometrics, New York, Academic Press.Miranda, M. J. and Fackler P. L. (2002) Applied Computational Economics and Finance, Cambridge, MA: MIT Press.Miyao, T. & Shapiro, P. (1981) Discrete Choice and Variable Returns to Scale, International Economic Review, 22, 257-273.Mossay, P. (2003) Increasing Returns and Heterogeneity in a Spatial Economy, Regional Science and Urban Economics, 33, 419-444.Murata, Y. (2003) Product Diversity, Taste Heterogeneity, and Geographic Distribution of Economic Activities: Market vs. Non-Market Interactions, Journal of Urban Economics, 53, 126-144.Ottaviano, G. T. (1999) Integration, Geography and the Burden of History, Regional Science and Urban Economics, 29, 245-256.Ottaviano, G. T. (2001) Monopolistic Competition, Trade and Endogenous Spatial Fluctuations, Regional Science and Urban Economics, 31, 51-77.Ottaviano, G. T., Tabuchi, T. & Thisse, J.F. (2002) Agglomeration and Trade Revisted, International Economic Review, 43, 409-435. 30
31.
Ottaviano, G. T. & Thisse, J.F. (2003) Agglomeration and Economic Geography, Unpublished Manuscript.Oyama, D. (2008) History versus Expectations in Economic Geography Reconsidered, Journal of Economic Dynamics & Control, 33, pp. 394-408.Picard, P. M. & Zeng, D.Z. (2005) Agricultural Sector and Industrial Agglomeration, Journal of Development Economics, 77, 75-106.Puga, D. (1999) The Rise and Fall of Economic Inequalities, European Economic Review, 43, 303-334.Rosen, S. (2002) Markets and Diversity. American Economic Review, 92, pp.1-15.Samuelson, P. A. (1952) The Transfer Problem and Transport Costs: The Terms of Trade when Impediments are Absent, Economic Journal, 62, 278-304.Sjaastad, L. (1962) The Costs and Returns of Human Migration, Journal of Political Economy, 70, 80-93.Small, K. A. & Rosen, H. S. (1981) Applied Welfare Economics with Discrete Choice Models, Econometrica, 49, 105-130Tabuchi, T. (1986) Existence and Stability of City Size Distribution in the Gravity and Logit Models, Environment and Planning A., 18, 1375-1389.Tabuchi, T. & Thisse, J.F. (2002) Taste Heterogeneity, Labor Mobility and Economic Geography, Journal of Development Economics, 69, 155-177.Venables, A. J. (1996) Equilibrium Locations of Vertically Linked Industries, International Economic Review, 37, 341-359. 31
32.
Venables, A. J. (2000) Cities and Trade: External Trade and Internal Geography in Developing Economies, Unpublished Manuscript. London, School of Economics.Weibull, J. (1995) Evolutionary Game Theory. Cambridge, MA: MIT Press. 32
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Calculations carried out in MATLABFigures 1-2. Short-run equilibria for different levels of trade openness. 33
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Calculations carried out in MATLABFigures 3-4. Long-run equilibria for different levels of taste heterogeneity. 34
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Calculations carried out in MATLABFigures 5-6. Short-run equilibria with taste heterogeneity and amenities for differentlevels of trade openness. 35
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Calculations carried out in MATLABFigures 7-8. Long-run equilibria with taste heterogeneity and amenities 36
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Calculations carried out in MATLABFigure 9. Maximum social welfare with taste heterogeneity. 37
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Calculations carried out in MATLABFigure 10. Maximum social welfare with taste heterogeneity and amenities. 38
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Calculations carried out in MATLABFigures 11-12. Short-run equilibrium and maximum social welfare. 39
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Calculations carried out in MATLABFigures 13-14. Short-run equilibrium and maximum social welfare with taste heterogeneity and amenities. 40
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