Trade openness and city size whit taste heterogeneity and amenities

TRADE OPENNESS AND CITY SIZE WITH TASTE

HETEROGENEITY AND AMENITIES

Willy W. Cortez Yactayo and Mauricio Ramírez Grajedaa
Universidad de Guadalajara
May, 2009

Abstract
This 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 of
market and non-market interactions. We analyze the outcomes of the original model vis-à-vis the outcomes
of its extension. In particular, we assess the impact of international trade openness on cities‟ size. Three
main general predictions arise from such a departure: First, the breakpoint with taste heterogeneity is higher
than the breakpoint associated without taste heterogeneity. Second, for low levels of trade openness urban
agglomeration is attenuated with taste heterogeneity. And third the dispersed equilibrium is not feasible
with both taste heterogeneity and amenities. Finally, based upon particular values of the parameters of the
model, 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; F12

1. Introduction

This paper sheds light on the impact of international trade openness on inter urban

structure. Our theoretical framework is a departure from Fujita´s et al. (1999, chap. 18)

New Economic Geography (NEG) model, where location decisions are not longer

exclusively driven by pecuniary considerations. In particular, we numerically solve for

Fujita et al. (1999) with taste heterogeneity and amenities and compare its outcomes with

a model where the only centrifugal force is an exogenous congestion cost parameter.1

Furthermore, under particular conditions it is possible to generalize the features observed

a
Corresponding author. Address: Universidad de Guadalajara, Centro Universitario de Ciencias Económico
Administrativas, 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. Both
forces are directly reflected in real wages.

in our numerical examples. Finally, by increasing taste heterogeneity we find that market

outcomes may converge to socially optimal outcomes.

Within the (henceforth, NEG) literature, taste diversity and amenities involve

dealing with an alternative design of migration. Traditionally, migration is driven by a

law of motion, which takes place as long as real wages are different across locations. It

establishes that a gap in real wages drives a fraction of workers from locations with low

real wages to those with higher real wages.2 Fujita et al. (1999), for example, define

J
(1)     j  j ( )
j 1

and,

d j
(2)   ( j ()   ) j ,
dt

where j is the fraction of labor in location j, j() is the real wage function net of

congestion costs in location j and  denotes the speed of convergence.  denotes the

distribution of the population over J locations. (1) denotes a weighted sum of real wages

across 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 one

particular location is, for example, greater than (1), then it will be a net receiver of

workers. The distribution of the population across locations determines real wages as a

2
Seminal NEG papers such as Krugman (1991), (Krugman & Venables, 1995), Venables (1996) or Puga (1999) use a
similar migration mechanism.

2

result of the interplay of agglomeration economies and, for example, congestion costs or

immobile 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 migration

mechanism. One, labor flows are solely governed by pecuniary considerations: real

wages. 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 randomly

selects those workers who must move out. Hence migration does not reflect any kind of

micro foundations as rational expectations or strategic behavior. In addition, the fraction

of the population that flees from a particular city positively depends upon the real wage

gap. 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 of

equilibria is determined by the initial distribution of the population; in other words, there

is path dependence. As a matter of fact this specific mechanism could fall into

evolutionary game theory, see Weibull (1995).5 Why, then, is this “myopic” Ottaviano et

al. (2002) mechanism applied? Because, it keeps things simple Fujita et al. (1999). On

the other hand, however, Baldwin (2001) points out that under the conventional migration

law the number of multiple equilibria is drastically reduced.

Five lines of research are encountered within the NEG literature to explain an

income gap in equilibrium. First, by incorporating migration decisions based on a

forward looking behavior Ottaviano et al. (2002); Ottaviano (2001); Baldwin (1999) and

Oyama (2009). In this case, initial beliefs determine the long-run spatial distribution of

3
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

economic activities, which might mimic the outcomes under the conventional migration

mechanism. Second, by assuming a quasi-linear utility function such that consumer

surplus is a component in migration decisions as well Ottaviano et al. (2002).6 Third, by

dropping the assumption of costless migration such that the larger the migration flow, the

higher the costs Ottaviano (1999). Fourth, by assuming that regions exhibit different

natural and cultural amenities (Tabuchi & Thisse, 2002).7 And fifth, by combining taste

heterogeneity 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 the

first four shortcomings of the traditional NEG law of motion mentioned above. Taste

heterogeneity, which entails a non-pecuniary component of migration decisions, is a

degree of attachment to or perception of a particular location due to a wide variety of

reasons 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) conceives

that a non-monetary factor also affects migration such as climate, pollution or congestion.

For example, (Fujita & Thisse, 2009) explain that there is uneven distribution of

immobile resources (exogenous amenities) like natural harbors. Jacobs (1969) claims that

social factors determine the configuration of cities (endogenous amenities), for example,

by attracting creative and talented people. Fortunately, all these factors can be aggregated

6
In this case, individuals move according to the value of the indirect utility function, which is equivalent to the nominal
wage 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

in order to assess their impact on the spatial distribution of both workers and firms. To do

so, NEG and probability choice theory can be combined.

It is worth recalling that there is an almost limitless set of factors that we might

conjecture would impact on urban attractiveness to potential migrants. For example,

(Glaeser & Redlick, 2008) claim that more educated people have higher incentives to

migrate. In particular, they show that education level is an important determinant of

migration to US urban areas.

Our main findings in the long-run are four-fold. First, the breakpoint with taste

heterogeneity 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, market

outcomes converge to the optimal social welfare outcomes as taste heterogeneity

increases.

The reminder of this paper is divided into the following sections. In section 2, we

briefly explain the main ingredients of NEG models and discrete choice models of

migration, and the way in which both paths are combined. We also discuss the

implications of unifying both paradigms by reviewing the related literature. Section 3 is

the theoretical framework that incorporates taste heterogeneity and amenities. In Section

4, we report and analyze the numerical solution of the model. Furthermore, under some

particular assumptions we present some general features of the extended model. In

section 5 there are some final remarks.

5

2. NEG and discrete choice models

Krugman‟s (1991) core-periphery general setting is seminal within the NEG literature. It

assumes j locations; two sectors, manufacturing and agricultural. The former is

monopolistically competitive, whose technology exhibits increasing returns of scale and

only employs workers. The latter is perfectly competitive, whose technology exhibits

constant returns to scale and exclusively employs peasants. Workers can migrate across

locations but not across sectors. Peasants can move neither across locations nor across

sectors. Both workers and peasants have the same preferences over N manufacturing

varieties and a homogenous-good produced in the agricultural sector. Such preferences

are represented by a Cobb-Douglas utility function where the component related to

manufacturing 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 when

both workers and peasants maximize their utility, firms maximize profits, and the product

and labor market clearing conditions are satisfied. The level of real wages in the

manufacturing sector associated with the short-term equilibrium in location j are

expressed 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 a

dispersion or centripetal force of spatial agglomeration. If trade costs are high enough the

only long-run equilibrium is the dispersed one; below a threshold a core-periphery

economy suddenly arises: most workers cluster together in a single location.9

8
In particular, Krugman (1991) does not have a closed-form solution. Therefore, the level of utility associated with the
short-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

Fujita et al. (1999), among others give a prominent role to the effects of

international trade costs on the distribution of population between cities. They assume j-1

cities in the home country and 1 in the foreign country. Migration takes place only

between cities in the home country. There is only one sector, manufacturing. The main

outcome is that high levels of such costs foster agglomeration in a single city. By the

same token, Venables (2000) investigates the effects of external trade costs on the share

of manufacturing employment. A single city will have a high amount of employment

when the economy of a country is closed to external trade. However, when the economy

has access to imports due to lower trade costs the amount of industrial employment goes

down. The economy develops a duocentric structure if it is open to external trade.

Alonso-Villar (2001) suggests that the negative relationship between trade openness and

city size depends upon the relative size of the home country. If it is low with regard to the

rest of the world, a dispersed equilibrium is not sustainable, given low levels of trade

costs. Mansori (2003) introduces a fixed and a marginal trade cost that may cause the

following two outcomes after trade barriers fall. One is that a megalopolis that is already

in equilibrium does not shrink in size; Buenos Aires and Bangkok are examples of this

outcome. The other is that cities in the dispersed equilibrium become a megalopolis like

Seoul.10

Krugman (1991) is a benchmark for many other papers with different assumptions

and, consequently, different outcomes. There are two broad divisions in the NEG

literature: at the international level, migration across sectors is allowed but not across

10
Contrary to what Mansori (2003) theoretically claims, Henderson et al. (2001) find that Korea has experienced a
process of deconcentration of manufacturing due to infrastructure improvements.

7

countries, see (Krugman & Venables, 1995); (Krugman & Livas, 1996); Puga (1999); at

the regional level, workers can migrate across locations Krugman (1991).

On the other hand, according to (Brakman & Garretsen, 2003), the conventional

migration rule used in the literature discussed above is still not satisfactory and a step

forward would be to incorporate heterogeneity across workers. As mentioned above, taste

heterogeneity is a non-pecuniary component of the intercity location problem that

represents a location taste. Hence some workers will stay put even though they may earn

a higher monetary income in other places. Particularly, once individual monetary income

level gets sufficiently high, workers tend to pay more attention to non-pecuniary

attributes of their environment. Low trade costs and more heterogeneous individuals can

be 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 that

incorporates 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 but

explicitly set the distribution of migration movements Mossay (2003).

Regarding the first strand, preferences are conceived in several dimensions. On the

one hand, they depend on both the level and variety of consumption. Under this

dimension workers are assumed to be homogenous. (Ottaviano & Thisse, 2003) consider

that this is an unappealing and implausible assumption. On the other hand, preferences

are also associated with non-pecuniary factors. Formally, at t and assuming J=2 (2

locations), preferences of worker k on location j brakes down as follows:

8

a) A deterministic pecuniary component, which is represented by  j .11 It depends

upon the population distribution across locations and its value is equal among all

individuals 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 a

particular location.12 It is represented by a random variable, jk, i.i.d. according to the

double exponential distribution with zero mean and variance equal to 22/6. The

realization of this variable is different over time.

c) An exogenous level of amenities, aj, associated with a location j such as natural

amenities that do not change over time. In order to isolate the effects generated by the

balance between the centrifugal and centripetal forces, NEG assumes identical regions,

but it does not include the impact of differentials in amenities. However, empirical

evidence shows that geographical advantage, such as a coastal location, good climate and

good access to economic centers, may explain the spatial distribution of industry Gallup

et al. (1998).13,14

These preferences can be represented by an overall utility for worker k in location j

as

(3) V jk (t )   j ( )   k  a j , 15
j

11
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

provided that t denotes the fraction of the population in city j at t. Worker k decides to

live in region j at t+1, for instance, if the overall utility in that region is larger than in

region j’,

(4) V jk (t )  V jk' (t ) .

If this is the case, then ykj=1 and ykj’=0, however, such a condition is satisfied

randomly. 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 a

probability 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, workers

tend to be equal among them, then migration decisions exclusively depend upon

pecuniary considerations. If aj - aj’ = 0 and , workers tend to be different from each

other and the monetary component weight within the overall utility tends to zero, then

migration 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 the

law of large numbers. Therefore, labor changes according to the following new law of

motion,

10

d exp((   ( )  (a j  a j ' )) /  )
(7)  (t )   (1  t ) P (t )  t P2 (t ) 
1  t ,
dt
exp((   ( )  (a j  a j ' )) /  )  1

where (1-t)P1(t) is the population that is leaving city 2 to city 1 and tP2(t) is the

population 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 substituting

it with (7). Under this new equation, pecuniary and non-pecuniary considerations are part

of migration decisions; migration is the result of individual decisions; and migration in

equilibrium 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: a

pecuniary force or a random force. More precisely, with probability 1-, worker k

migrates 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 is

that agents preferences are represented by a quasi-linear utility function. Under such an

assumption 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 are

different across locations (the asymmetric case) and one location has a larger population

than 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 intermediate

trade costs a core-periphery pattern arises, otherwise only the dispersed equilibrium is

11

feasible. If social amenities are positive and different across locations, and both the love

for variety and increasing returns are low, it is possible that exist a range in which the

size of the populous city is above the social optimum. Without differential amenities the

dispersed 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 can

be obtained. For high levels of taste heterogeneity, only the core-periphery pattern is

allowed; 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 of

trade costs. For high levels of taste heterogeneity, the social optimal population

distribution coincides with the market allocation; for low levels of heterogeneity the

market equilibrium never coincides with the social optimum.

Tabuchi (1986) uses a system of simultaneous differential equations to explain

intercity migration due to differences in utilities, which are expressed as a function of city

size. A deterministic specification of the utility leads to an unstable distribution of city

sizes, whereas a stochastic specification does not. In this vein, Mossay (2003) designs a

stochastic continuous model, where locations are distributed along a circle. In each

location 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 be

extended in a more complex world: taste heterogeneity represents a dispersion force. An

16
Actually, these modifications lead to Krugman (1980).

12

extreme case is when migration is exclusively driven by pecuniary considerations, where

agglomeration is expected in few locations.

3. Theory

In this section, we outline Fujita et al.‟s (1999) model assuming taste heterogeneity and

endogenous amenities, which makes migration decisions depend upon pecuniary and

non-pecuniary considerations. 17 It focuses on intercity migration within a country which

trades with the rest of the world. The economy embeds increasing returns to scale, trade

costs 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 the

fraction of the population that lives this location.

Trade costs are of the Samuelson (1952) type: Tjj´≥0 denotes the amount of any

variety dispatched in location j per unit received in location j’. 18 If j=j’ then Tjj’=1 and

Tjj’= 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 the

model to solve for the equilibrium. Second, it is a necessary condition for preserving a

constant elasticity of the aggregate demand. This feature simplifies the conditions of

profit maximization.19 Third, Tjj´ may represent an explicit ad valorem tariff whose

revenues are redistributed among economic agents but dissipated as a consequence of

17
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 constant
mark-up over marginal cost. A specific level of production satisfies this condition.

13

rent-seeking.20 And fourth, trade costs are not related to the product variety or distance

between locations.

The representative agent in location j derives her pecuniary utility from

consumption 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 the

consumption of each available variety, n, in location j. Under these preferences, desire for

variety 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 The

quantity 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 n

in region j pays nominal wage, wjn, for one unit of labor. In order to characterize the

equilibrium, F = 1/σ and  = (σ-1)/σ.22 The number of firms in location j, nj, is

endogenous. N = n1+…+nJ is the total number of available varieties. 23

20
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 is
known 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

There are two types of prices: mill (or f.o.b) and delivered (or c.i.f.). 24 The former

are charged by firms. The latter, paid by consumers, are defined as

(10) p n ´  p nT jj´ ,
jj j

where pnj denotes the mill price of a good of variety n produced in location j. pnjj´ is the

delivered price in location j´. By the assumptions on trade costs both prices are equal

when j=j´.

Real wages in location j are defined as

(11) w' j  w j ( )G 1
j

where Gj is a price index, which is the minimum cost of achieving one unit of utility

given 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 parameter

associated with congestion cost. Wages deflated by prices and congestion costs are

positively related to utility levels.26

The short- run equilibrium

The economy reaches its short-run equilibrium when agents and firms optimize

respectively their pecuniary utility and profit functions, such that the aggregate excess

demands 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

The model does not have a closed-form solution. For J=3 the equilibrium must

satisfy 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 of

obtaining a unit of utility. (14) is the wage equation, which generates zero profits given

prices, income and trade costs. Real wages across locations might be different. We

choose w3 as a numeraire.

The long-run equilibrium

Up to this point there are no movements of workers. When (13) and (14) are satisfied

there 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 to

connect locations by equalizing it to zero in equilibrium in the long-run. Workers decide

to move according to (4), where both pecuniary and non-pecuniary are taken into

consideration.

16

Trade openness and city size

In order to relate urban agglomeration to trade openness, two assumptions are

incorporated. First, there are 2 countries termed, foreign and home. Only one city is

located in the foreign country, and 2 cities in the home country. L0 is the population in the

foreign country and, L1 and L2 in the home country cities. Trade between cities in the

home country involves the same Samuelson (1952) type trade costs, T. But trade costs

between a particular city in the home country and the unique city in the foreign country is

T0. Second, it is assumed that migration is allowed between cities within the home

country but not across countries.

By using MATLAB we numerically solve both the original and the extended model

for different levels of international trade openness, T0.27 What happens in the foreign city

is 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 for

different values o f a1, a2 and . We assume that 1(,) = (1-) and 2(,) = ().

4. The extended model

Taste heterogeneity without amenities

In this first case, there are no amenity differentials between both cities, then a1-a2 = 0. A

congestion cost and taste heterogeneity are the only dispersion forces of location. For

conciseness of exposition we focus on city 1. Figures 1-2 relate the fraction of the

population in city 1, , to migration over time, d/dt, for different levels of international

27
The MATLAB programs are available upon request. We used some routines provided by (Miranda & Fackler, 2002).

17

trade costs with and without taste heterogeneity,  = 0.005 and  = 0, respectively. These

figures are consistent with the short-run equilibrium, when there is room for internal

migration. Most NEG papers, the short-run equilibrium is depicted as the relationship

between  and the equilibrium real wage gap, which is equal across all individuals in one

particular 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 of

workers‟ movements shifts the population in city 1 up; if d/dt<0 implies that the

population shrinks. Equal distribution of the population, * = 0.5, is associated with no

changes in the population distribution, d/dt=0 (the long-run equilibrium); furthermore, it

is a stable equilibrium because d(d/dt)/d<0 and unstable would be the other way

around. The only difference between the original model and its extension around the

equilibrium point is the rate of convergence to the steady state. The speed of convergence

for  = 0 depends upon the parameter . 28
Figures 1-2 describe the transition from a

unique 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: 1

unstable 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. In

other 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 of
long-run equilibria does not depend upon the value of such parameter but determines the way in which the distribution
of the population evolves over time. In this case, it converges in an oscillatory fashion.

18

Figures 1-2 can be summarized in figure 3. Hence the analysis can be conducted

from a different perspective. Figures 3-4, depict the relationship in the long-run between

trade openness, T0, and the population distribution across cities, LR, instead, which is

consistent 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; with

taste heterogeneity, =0, in green. A cross denotes an unstable long-run equilibrium; a

star denotes a stable long-run equilibrium. Without heterogeneity agglomeration in one

city takes place for low levels of trade openness as well. As trade costs decreases

dispersion 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 higher

with taste heterogeneity than without it: T0*>0>T0*=0. With heterogeneity dispersion

requires lower levels of trade openness (see proposition 1). In addition to this, for low

levels 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-run

equilibrium are still valid when taste heterogeneity is positive. All varieties are consumed

in every location. The trade balance is zero between both cities. The number of varieties

is fixed because it exclusively depends upon the technology and the total population of

the economy.

19

In this paper, we assume that the parameters generate a T0*=0>0; and

particularly, >1 and  should not be too large. Therefore we can present the following

propositions.

Proposition 1. if (G/())(-1) is increasing in T0, then the breakpoint associated with

taste heterogeneity, T0*>0, is higher than the breakpoint associated without taste

heterogeneity, T0*=0.

Proof. The long-run equilibrium satisfies (7) equal to zero. At  = 0.5, the stability of the

model depends upon the sign of d( = 0.5)/d. Without taste heterogeneity, the break

point, 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  ( )  0.5

Such 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

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-run

equilibrium without heterogeneity satisfies

(19)    j (LRstable)   j ' (LRstable)  0

hold. Therefore,

d
(20)  0.5  LRstable  0.
dt

The 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 )) /  )  1

The first member of (23) falls between zero and 0.5. Therefore, the long-run equilibrium

condition with taste heterogeneity is not satisfied.

21

It is worth mentioning that this economy falls into the Murata‟s (2003) type

economy with endogenous location of the demand if T0. Furthermore, if , the

only distribution of the population consistent with the long-run equilibrium is the

dispersed one.

Taste heterogeneity with amenities

In this case, another dispersion force is added. More precisely, we assume that city 1 has

amenities, a1=0.01, and city 2 has no amenities, a2=0. Figures 5-6 depict the short-run

equilibria 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 both

taste heterogeneity and amenities is the dispersed one, = 0.5. With taste heterogeneity

and amenities there is a single long-run equilibrium as well: > 0.5, which is an intuitive

outcome 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). Figures

5-6 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 for

high levels of trade openness.

FIGURES 5-6

Proposition 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

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 7-8

Social welfare

Following (Small & Rosen, 1981), the social welfare in the home country can be defined

as


 2 ( )  (a1  a2 ) 
~ ~
 1 ( )  (a1  a2 )
(25) W ( )   ln exp( )  exp( ) ,
   
 

which is the sum of individual utility functions. Figure 9 compares the population

distribution in the home country that maximizes (25) with the equilibrium outcomes with

taste heterogeneity. For low levels of trade openness the maximum level of social welfare

is associated with the dispersed equilibrium. For high levels of trade openness the

socially optimal distribution involves agglomeration. However, as taste diversity gets

higher, the range in which the dispersed equilibrium differs from the social optimum

outcome gets narrower.

Figure 10 depicts the long-run equilibria with taste heterogeneity and amenity

differentials, and the socially optimal outcomes. As international trade costs decline the

outcomes 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 taste

heterogeneity 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

In fact, at the dispersed equilibrium, W’(0.5)=0, for any level of trade openness, see

Murata (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 the

dispersed 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 welfare

associated with the market equilibrium increases as heterogeneity gets higher, given T0 =

1.035 (see figures 13-14).

FIGURES11-14

5. Final remarks

Migration in the NEG has traditionally been modeled by (2). By introducing taste

heterogeneity, dispersion forces different from labor immobility or congestion costs,

several unrealistic features of migration are eliminated and the extreme outcomes of

Krugman (1991) or Fujita et al. (1999) moderated.

Discrete choice theory is a way to incorporate non-pecuniary factors in migration

decisions. By doing so, the advantages of agglomeration are either equal or reduced given

a particular level of international trade openness. When trade openness is low,

agglomeration economies push real wages up, however, Fujita et al. (1999) incorporate a

congestion cost. But the net effect of clustering is positive because the demand is highly

concentrated in the national market. Taste heterogeneity and amenities reduce the

24

agglomeration economies because the pecuniary component of migration decisions

reduces its weight. The extreme case is when the variance of taste heterogeneity is

infinite. In such a case, real wages have almost no weight in location decisions. When

trade openness is high the advantages of agglomeration is low, therefore costs increments

offset real wage gains of concentration. Such real wages gains of concentration are

reduced even more with taste heterogeneity and amenity gap. Thus we conclude that the

breakpoint even is higher and agglomeration moderated when taste heterogeneity is

incorporated in migration decisions. Furthermore if amenities differences are also

incorporated the dispersed equilibrium is not feasible. The rationale behind this result is

that 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 taste

heterogeneity and amenities are incorporated into Fujita‟s setting. In this case, figure 8

describes agglomeration only in city 1 for a specific range of international trade openness

and sufficiently high levels of taste heterogeneity. For low levels of trade openness

extreme agglomeration is present and, for high levels agglomeration is mild. Figure 7

suggest that there is room for agglomeration in both cities if there are differentials in

amenities but taste heterogeneity, given low levels of trade openness. The main message

is 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 social

welfare, = 0.5. However, for high levels of trade openness, agglomeration maximizes

social welfare which diverges with the equilibrium outcomes. Increasing taste

25

heterogeneity is an alternative way to improve social welfare. In this sense, determinants

of migration are quite different among countries. Some conjectures on such matters

would 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 the

breakpoint 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 the

endogenous variables are equal in magnitude but with different sign: dG =dG1= -dG2 and

26

dY =dY1=-dY2. Then we pay attention to city 1 because it is not necessary to keep track of

variables associated with city 2. Therefore, by totally differentiating (A.1), (A.5), (A.7)

and (12), we obtain the following expressions:

(A.9) dY  dw  dw ,

(A.10) (1   )G  dG  [ (1   )w dw  dw1  (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. we

solve 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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32

Calculations carried out in MATLAB

Figures 1-2. Short-run equilibria for different levels of trade openness.

33


Figures 3-4. Long-run equilibria for different levels of taste heterogeneity.

34


Figures 5-6. Short-run equilibria with taste heterogeneity and amenities for different
levels of trade openness.

35


Figures 7-8. Long-run equilibria with taste heterogeneity and amenities

36


Figure 9. Maximum social welfare with taste heterogeneity.

37


Figure 10. Maximum social welfare with taste heterogeneity and amenities.

38


Figures 11-12. Short-run equilibrium and maximum social welfare.

39


Figures 13-14. Short-run equilibrium and maximum social welfare with taste
heterogeneity and amenities.

40

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