transport technologies and location - uab

Transport Technologies and Location∗

Pedro Pita BarrosUniversidade Nova de LisboaandCEPR

Xavier Martinez-GiraltUniversitat Autonoma de Barcelona

November 1997

Abstract

We present a simple model of spatial competition to analyze the impact of a structural changein transaction (transportation) costs on the location of firms. The distinctive feature of the model isthe existence of two markets which are non-connected. This means that a firm willing to sell in theother market must take into account a fixed fee in addition to the usual quadratic transport costs.Two different formulations for quadratic costs are explored, yielding different results for locationanalysis. In particular, the principle of maximum differentiation may not hold depending on thenature of transport costs.

Keywords: Spatial Competition, Transport technologies, Market Integration.

JEL Classification: F12, F15, L13.

MAILING ADDRESSES:

CODE andDepartament d’Economia Faculdade de EconomiaUniversitat Autonoma de Barcelona Universidade Nova de LisboaEdifici B Travessa Estevao Pinto-Campolide08193 Bellaterra 1070 LisboaSpain. Portugal.Fax: 34-3-581 24 61 351-1-388 60 73e-mail:[email protected] [email protected]

pp. 1 - 25

∗We thank several seminar participants, J. Bouckaert, L. Cabral and K. Stahl for useful comments and suggestions. Theauthors acknowledge partial support from Acc¸ao Integrada Luso-Espanhola E-64/95 (Pedro P. Barros), HP94-011 (Minis-terio de la Presidencia), SRG96-75 (Generalitat de Catalunya) (Xavier Martinez-Giralt), and EC Human Capital Network2/ERBCHRXCT930297. The usual disclaimer applies.

1

Transport Technologies 2

One of the main activities involved in developed economies is the transportation of commodities.

Often, this activity involve the use of different means of transport to reach the market where the com-

modities will finally be sold. Think for instance of a good transported by boat (or plane) to the harbor

and then downloaded to a truck that will carry it to its final destination. Other examples of different

types are goods that change trains (or trucks) at the border between two countries; the levy of a tariff

on imports at the border; tolls in highways. All these examples, although different in nature, have a

common feature. This is the cost involved in changing the means of transport, the toll or the tariff.

A similar situation arises when there appears a structural change in transportation such as the build-

ing of the Channel Tunnel, high speed trains, new highways or mountain tunnels. As a consequence

the overall cost of transportation is lowered, at least in terms of the time it takes for the commodities

to reach the market.

In this paper we intend to present a framework to analyze the impact of these type of cost on the

location of firms. We think this is an important issue because the building of infra-structures in less de-

veloped regions often have the aim of inducing location of economic activities in these regions. Facil-

ities that may become available in one region may give rise to positive externalities if they incentivate

the location of new industries in the neighboring regions. They may also impose negative externalities

if such facilities turn into a relocation of industries from the neighboring regions to the one providing

the new facilities. There may exist, thus, relocation across countries. Whether the latter is always a

reasonable expected effect, or not, is the focus of the paper.

We propose a model where two firms are located at two distant points in a space described by a

non-connected set, where one of the components of this space is larger in terms of potential demand

than the other.

We look into the location choice of firms in a very stylized way. Nonetheless, we hope it will pro-

vide more economic ground to discuss some policies directed to change firms’ location decisions (for

example, the appearance of business parks close to the Tunnel were backed by local authorities).

We choose to cast our analysis in the simplest economic model of location choice – the location-

price game introduced by Hotelling (1929) under quadratic transport costs (d’Aspremont et al., 1979).

The distinctive features of our work are twofold. First, we introduce a set of possible locations that is

non-connected. That is, there are some points in the space that cannot be used as locations for either

consumers or firms. Second, consumers have to pay a (linear) transport cost to cross such space and


quadratic transport costs in each market. Two different specifications of transport costs are investi-

gated, yielding distinct implications to firms’ location decisions.

We can view the linear cost as a forfait illustrating the payment of a toll, tariff, or the cost involved

in downloading the commodity from the boat into the truck. Also, we choose to model transport cost

in the quadratic fashion to avoid the usual lack of quasi-concavity of the payoff functions associated to

the linear transport costs. In fact, considering linear transport cost together with the forfait would only

make the existence problems of the price equilibrium subgame analyzed in Martinez-Giralt, Garella

and Svoronos (1986) more severe.

More in general, this work addresses the question of the impact on firms’ locations when the econ-

omy faces a structural change in transport costs. In other words, we examine the principle of mini-

mum differentiation under non-homogeneity of the space of characteristics. In this sense we can relate

our paper to the literature on the principle of minimum differentiation.1 A particular instance of this

more general body of literature is provided by Bouckaert and Degryse (1995) where conditions under

which banks offering phonebanking are studied, (where phonebanking constitutes a new technology

for transport costs made available).

The structure of the paper is as follows. Next section presents the basic model and two alternative

formulations of the transportation costs. Sections three and four analyze the equilibria in the price

game and the tendencies for firm relocation under the alternative specifications of the transport costs.

Section five contains a discussion of the features of the equilibria in the different model formulation

and draws some conclusions.

1 The model.

1.1 The basic structure.

The model we propose departs from a space composed of two separated markets. One of these mar-

kets is larger in terms of potential demand than the other. We assume that there is a continuum of con-

sumers uniformly distributed on each market. The density of consumers in each market is normalized

to unity. All consumers are identical except for their location (best preferred variety). The (common)

reservation price is high enough to allow all consumers to buy one unit of a (horizontally differentiated)

commodity.

1For a survey of this line of work, see Gabszewicz and Thisse (1986) or Martinez-Giralt (1995).


Transport costs within every market are modeled as a quadratic function of distance.2 Exporting

or importing the commodity implies to add to the transport cost a fixed amount associated with a tariff

(or some other cost associated with the crossig the border3).

We denote by0 andL the extreme points of the small market;T denotes the tariff;L andkL denote

the extremes of the big market (implying thusk > 2). All distances are measured from zero. The

smaller country is termed country 1, the other is country 2. Finally, “firmd” is the firm located in

country 1, or if both firms are located in the same country, it denotes the firm located closest to 0. The

other firm is “firmf ”.4

The starting point for comparison purposes, will be the case of isolated markets (i.e.T = ∞),

where the reservation price is high enough to sustain an equilibrium where each firm is located in the

mid-point of its (domestic) market.5

We consider the (partial) integration of the two markets. The distribution of consumers on the space

remains the same though. Now firms in both markets (countries) have access to the neighboring market

paying a fixed fee ofT monetary units. That is to say, the firm in countryj willing to reach a consumer

in marketi has to pay a transport cost defined as the sum of a fixed fee and the corresponding costs

associated to the distance between consumeri and firmj.

1.2 Transport Technologies

For the analysis of market integration and location of firms, specification of transport technologies

does matter. Previous literature has identified trade-offs, in location decisions, between market size

and transport costs.6 Moreover, as it will be apparent below, different ways of computing transport

costs gives substantially different equilibrium characterizations.

(a) The first possible specification for transport technology is described by quadratic transport costs

(cumulative) in all the path. For example, if a firm is located atd in country 1 and serves a

2We should also specify that the transport cost rate is defined per unit of commodity. Since all consumers buy exactlyone unit, such an assumption would be redundant.

3We ignore, at no cost for our analysis the differences that arise from real barriers and tariffs, as the later generats revenueand the former does not.

4Symmetric cases will be omitted, unless otherwise noticed.5Although such an assumption is arbitrary, the location at the center of the market has the appealing feature of minimizing

the aggregate transport costs.6Some of this literature also considers wage differences across countries and how such differences interact with mar-

ket size - transport costs tradeoffs. See Cordella and Grilo (1995), Horstmann and Markusen (1987, 1992), Krugman andVenables (1991), Motta (1992) and Rowthorn (1992), among others.

0 L kL

(a)

(b)

id

T


consumer located ati in country 2, the associated transport costs are

c(i− d)2 + T

This implies that the transport cost at the margin is the same at both sides of the “border”.

(b) The alternative specification is to have independent quadratic transport costs in each market,

besides the fee. For the same firm and consumer of the previous example, we have

c(i− L)2 + T + c(L− d)2

In this case, at the margin, the transport cost differs on either side of the border. Both cases are

depicted in Figure 1.

Figure 1: Transport technologies.

To illustrate the first technology think of a consumer located ati > L, and of a truck transporting

a commodity from firmd to consumeri. Total transport cost will be given by the quadratic function

of the distance plus the fixed fee at the border. We will refer to this modeling as CAR transport costs

(standing for “Continue After a Rest”).

The second way of specifying the transport costs can be illustrated again by our consumer living in

i > L. Transporting the commodity to his location can be seen as using a boat until the harbor (border)

plus and load it in a truck (at a fixed cost ofT ) that will transport it to the location of the consumer.

We will call this CAB transport cost (standing for “Change At the Border”).

Of course these examples serve only for illustrative purposes, as they correspond to two feasible

ways of transporting goods from firms to final retailing outlets (identified with consumer locations).


Formally, the total CAR transport cost function for a consumeri buying from firmd is

CAR(d, i) =

{c(i− d)2 + T if i ≥ L,c(i− d)2 if i ≤ L

(1)

and

∂CAR(d, i)∂(i− d)

∣∣∣L+

=∂CAR(d, i)∂(i− d)

∣∣∣L−

= 2c(L− d) (2)

whereL+ refers to the partial derivative atL from the right, andL− refers to the partial derivative at

L from the left.

Total CAB transport cost function for a consumeri buying from firmd is given by

CAB(d, i) =

{c(i− L)2 + T + c(L− d)2 if i ≥ L,c(i− d)2 if i ≤ L

(3)

and, at the border, marginal transport costs differ:

∂CAB(d, i)∂(i− d)

∣∣∣L+

= 0 <∂CAB(d, i)∂(i− d)

∣∣∣L−

= 2c(L− d) (4)

Note that the CAR transport costs constitute the more natural generalization of quadratic transport

costs, in the sense thatT = 0 recovers the traditional quadratic transport costs schedule. On the other

hand, the CAB transport costs exhibit the feature that the border matters, even forT = 0. The CAR

transport costs make the assessment of the effect of a non-homogeneous space of locations in the con-

text of the well-known benchmark of d’Aspremont et al. (1979). We do not claim that one specification

is superior (in modeling) to the other. The two cases are chosen to illustrate that industry relocation

patterns can also be sensitive to alternative specifications of transport technologies, as we will show

below.

2 Equilibrium under CAR transport costs.

In this section, the first formulation for transport costs is used. The alternative formulation receives

attention in the next section. To simplify notation, we perform, hereafter, two normalizations:c =

1, L = 1. For expositional purposes the different cases will treated in separated subsections.


2.1 Firm f and firm d in their markets, the indifferent consumer in the big market.

Suppose each firm is located in its domestic market and the indifferent consumer,z, is located in the

big country. The profit function of each firm is, respectively,

Πd = pdz; Πf = pf (k − z)

where the indifferent consumer is defined by

z =f2 − d2 + pf − pd − T

2(f − d)

The solution to the price game yields

pd =(f − d)(f + d+ 2k)− T

3

pf =(f − d)(4k − d− f) + T

3

Next, we will examine tendencies in locations for firmsd andf at those equilibrium prices.7

Proposition 1. Let each firm be located in its domestic market and the indifferent consumer,z, be

located in the big country. Then,

a) firm d tends to locate towards zero;

b1) firm f locates towards1 if α < Γ < β;

b2) firm f locates towardsk otherwise;

where

α ≡ −3f + d+ 4k Γ ≡ T

f − d, and β ≡ f + d+ 2k − 6.

Proof. See the appendix.

7We cannot compute a subgame perfect equilibrium of a location-price game, since demands are not quasi-concave. Tosee it, consider how demand for, say, firmd evolves given a price of firmf . It is easy to compute the minimum price thatprevents firmd from capturing any demand, as it lowers the price its demand increases. At a certain price, firmdwill captureall consumers in the national market. Further increases in its demand, requires to lower the prices enough to compensate forthe tariff. Therefore, there is a range of prices of firmd for which demand is completely inelastic. Such a feature provides akink in the demand function that destroys its concavity.


2.2 Firm f and firm d in their markets, z in the small market.

Consider now the case of the indifferent consumer located in the small country. Suppose that firms

are located in different countries. In a similar fashion as in the section above, the solution to the price

game is

pd =f2 − d2 + 2k(f − d) + T

3

pf =d2 − f2 + 4k(f − d)− T

3

We now turn to the incentives of firms to change location. To this effect, we investigate the value

of change in profits from a marginal change in the location of the firm.

Proposition 2. Let each firm be located in its domestic market and let the indifferent consumer,z, be

located in the small country. Then

(a) firm d tends to locate towards 0;

(b1) firm f locates towardsk if 3 > 2f − k;

(b2) firm f locates towards1 if 3 < 2f − k.


2.3 Both firms in the same market.

The analysis of the case where firms are in the small market mimics the case where firms are located in

the big market. The equilibrium prices emerging from the corresponding profit functions are always

well defined. In terms of location the following result can be stated:

Proposition 3. Let both firms locate in the same country. Then firms tend towards different endpoints

of the market.


3 Equilibrium under CAB transport costs.

In this section we assume that transport costs are independent in each market, besides the linear fee

associated with crossing the border.


3.1 Firm f and firm d in their markets, z in the big market.

Suppose each firm is located in its domestic market and the indifferent consumer,z, is located in the

big country. The indifferent consumer position is defined by the solution to

pd + (1− d)2 + T + (z − 1)2 = pf + (f − z)2

This gives

z =f2 − 1− (1− d)2 − T + pf − pd

2(f − 1)

The profit function of each firm is, respectively, for firmd and firmf

Πd = pdz; Πf = pf (k − z)

The solution to the price game yields

pd =(f − 1)(f + 1 + 2k)− (1− d)2 − T

3

pf =(f − 1)(4k − 1− f) + (1− d)2 + T

3

Next, we will examine tendencies in locations for firmsd andf at those equilibrium prices.

Proposition 4. Let each firm be located in its domestic market and the indifferent consumer,z, be

located in the big country. Then,

a) firm d tends towards 1;

b1) firm f tends towards 1 ifγ < ∆ < φ;

b2) firm f tends towardsk otherwise;

where

γ ≡ −3f + 1 + 4k, ∆ ≡ T + (1− d)2

f − 1, and φ ≡ f + 2k − 5.



3.2 Firm f and firm d in their markets, z in the small market.

We now proceed to the case of firms are located in different countries and the indifferent consumer is

located in the small country. In a parallel way as in the section above, the solution to the price game is

pd =(2k + 1 + d)(1− d) + (f − 1)2 + T

3

pf =(1− d)(4k − 1− d)− (f − 1)2 − T

3

We now look at the incentives of firms to change location.

Proposition 5. Let each firm be located in its domestic market, the indifferent consumer be located

in the small country and transport costs be CAB. Then, whenever the equilibrium of the price game is

defined,

(α) Firm d tends to locate towards 1;

(β) Firm f locates towards 1.

Proof. See appendix.

3.3 Both firms in the same market.

This is like a standard Hotelling model. The type of transport cost does not matter to solve the model,

and accordingly, tendencies in locations will remain the same as in the CAR transport costs.

4 Equilibrium locations: CAB vs. CAR

The incentives for location changes may differ between the two possible specifications for transport

costs. Namely, this is the case if firms are located in different countries. Table 1 summarizes the ten-

dencies for location.8

A first feature of location tendencies is the difference between CAR and CAB transport cost when

firms are located in different countries. In particular, while under CAR transport costs, firmd tends

towards 0, under CAB transport costs, she moves closer to the border. Moreover, this difference in the

location incentives of firmd is not affected by the size of the fee, provided conditions for equilibrium

existence are satisfied.8Conditions for equilibrium existence are not shown. See previous sections and the appendix on this.


Table 1: Location tendenciesCAR transport costs CAB transport costs

Firms in their markets

z in the big market ∂Πd∂d < 0; ∂Πf

∂f

><

0 ∂Πd∂d > 0; ∂Πf

∂f

><

0

z in the small market ∂Πd∂d < 0; ∂Πf

∂f

><

0 ∂Πd∂d > 0; ∂Πf

∂f < 0

both firms in the same market ∂Πd∂d < 0; ∂Πf

∂f > 0 ∂Πd∂d < 0; ∂Πf

∂f > 0T −→ 0Firms in their markets

z in the big market ∂Πd∂d < 0; ∂Πf

∂f > 0 ∂Πd∂d > 0; ∂Πf

∂f > 0

z in the small market ∂Πd∂d < 0; ∂Πf

∂f > 0 ∂Πd∂d > 0; ∂Πf

∂f < 0

Incentives for location of firmf are ambiguous, except for CAB transport and indifferent consumer

in the small economy. Also important is that with CAB transport costs existence of equilibria problems

may appear, as firms tend to move closer to each other. Making the border having negligible transport

costs (T −→ 0) allows to have a clearer identification of location incentives for firmf . In the case of

CAR transport costs, locations tend towards the endpoints (as expected). Under CAB transport costs,

for the indifferent consumer in the big market, firmd gets closer to the border and firmf wants to

move away from the border. In addition, no equilibrium which has the indifferent consumer in the

small country is feasible (both firms tend to move closer to the border and the price equilibrium breaks

down).

Finally, when both firms are in the same market (be it the large or the small country) they tend to

move apart, towards endpoints of the market.

The intuition for these differences in location incentives between CAR and CAB as well as differ-

ences in equilibrium location choices are developed below for the special case ofT = 0.

The previous reasoning has established that we may restrict our attention to corner solutions. In

order to highlight the crucial differences for location equilibrium analysis between the two types of

transport costs, we assume heretofore perfectly integrated markets (T = 0). Of course, the basic in-

sight will hold forT small.

Under CAR transport costs, the analysis ofT = 0 falls into the case of d’Aspremont, Gabszewicz

and Thisse (1979) and firms locate at the endpoints of the market(0, k). For CAB transport costs, the

following proposition results.


Table 2: Location choicesfirm f

0 1 k

0 — (a) (b)firm d 1 (a∗) — (c)

k (b∗) (c∗) —

Table 3: Equilibrium profitsΠd Πf equilibrium conditions

(a) — — no equilibrium

(b) (3k2−2k−1)2

18(k−1)(3k2−4k+1)2

18(k−1)

(c) (k−1)(1+3k)2

18(k−1)(3k−1)2

18 k ≥ 3.3

Proposition 6. The Nash equilibria in pure strategies of the location game, when firms are restricted

to corner locations andT = 0, are defined by:

(a) the location pair (0, k) for k < 3.3;

(b) the location pair (1, k) for k ≥ 3.3.

Proof. According to tendencies of location discussed earlier, forT = 0, three locations are relevant:

0, 1, k. Given our convention on firms, the cases to be investigated are described in Table 2, where *

denotes reversal in the roles of firms.

The cases of location coincidence can be easily dismissed, as Bertrand competition drives firms’

profits to zero. Equilibrium profits are shown in Table 3.

For candidate (a), it is the case that firmd wants to undercut firmf and a Edgeworth cycle re-

sults. No pure strategies equilibrium in prices exists. The possibility of price undercutting must also

be considered in equilibrium candidate (c). Computation of profitable price deviations show that no

pure strategies price equilibrium exists fork < 3.3. Thus, for the location pair (1,k) to be a meaningful

equilibrium, countries must be sufficiently different in size. Thus, equilibrium candidates are (b) and

(c). For small asymmetries across countries, more precisely fork < 3.3, only the location pair (0, k)

is compatible with a pure strategies price equilibrium. Take nowk > 3.3, then it is straightforward to

check thatΠd(c) > Πd(b), implying that for sufficiently asymmetric countries, the equilibrium loca-

tions are (1, k).


Hence, forT = 0, two Nash equilibria may arise. The intuition for location choices is not difficult

to follow. By locating at 1, firmd keeps his domestic market and obtains a higher market share in the

bigger market. On the other hand, price competition is stiffer, which creates an incentive for firms to

relax it. If countries are not too different in size, firmf has an incentive, if the location pair is (1, k), to

undercut firmd and get all the demand (from both markets). In this case, firmdmoves to0 as a way to

relax price competition. The notion of countries of similar size is made precise by the conditions onk

for equilibrium existence. If the larger country is big enough then it does not pay to undercut the other

firm, as the price to consumers in the small country would have to account for the (large) transport cost

of crossing the large country.

In some sense, as the size difference between countries increases, the smaller country counts less

and less in the location decisions of firms. Consider, for example, the case ofk being very large. The

small country is almost irrelevant to firms’ profits and hence to location decisions. According to the

nature of transport costs, the location pair can be sustained as an equilibrium (CAB transport costs) or

not (CAR transport costs).

The main result under CAB transport costs must be explained clearly: making the fixed cost of

moving from one country to the other equal to zero does not recover the standard result of location at

endpoints, that is, firmd choosing to locate at 0 and firmf choosingk (d’Aspremont, Gabszewicz,

Thisse (1979)).

This difference to the standard case is a direct consequence of the particular specification of trans-

port costs used. Instead of full quadratic transport costs, we have quadratic transport costs in each

country, while a fixed fee must be paid to cross the border. Transport costs on the other side of the bor-

der are computed with reference to the border of the country. This means that atT = 0, transport costs

will exhibit a kink at the border between the two countries. At this value, the market is essentially con-

tinuous, with a border between countries, or regions, at 1. This situation is depicted in Figure 2. In the

standard model, transport costs are given by[abc] and[gh] for firm d and firmf respectively. If firm

d deviates to1 (the border), transport costs to firmd (at the right hand side) are given by[be], and the

price equilibrium will be different. Price competition is increased and the deviation is not profitable.

Consider now the specification of CAB transport costs. It is represented by[abe], as by definition

crossing the border leads to recomputation of transport costs. In this case, the indifferent consumer

is aty. Suppose that firmd deviates to location 1. Transport costs do not change and the indifferent

0 k

p1

1

ab

c

d

e

g

h

firm firm fy

0 k

p1

1

ab

c

d

e

g

h

firm firm fy


consumer remains aty. Thus firmf has the same profits as before. However, firmd sells the same

quantity at a higher price. Therefore, the location pair(0, k) is not a Nash equilibrium, while the lo-

cations(1, k) remain a viable candidate for Nash equilibrium.9 Note that this effect is independent of

whether CAB increases transport costs after the border. That is, incentives for firmd do not depend on

the steepness of the transport cost function at the border (see Figure 3).

Figure 2: Nash equilibria with CAB transport costs

Figure 3: Nash equilibria with CAB transport costs (II)

9CAB transport costs produce, in some sense, a convexification of transport costs. Note this is not essential to the result.The driving feature of the equilibrium is the change in the nature of transport costs.


5 Final remarks

We presented a simple two-country model of firms’ location. The distinctive feature of the model is

the existence of two markets which are separated. Generally speaking, the assumption is intended to

represent non-homogeneity of characteristics’ space in linear-city models of horizontal product differ-

entiation.

The implications in terms of modeling are straightforward. For a firm in one market to be able to

sell in the other market it must pay a fixed fee, in addition to the usual quadratic transport costs. Two

different formulations for transport costs are explored, yielding different results for location analysis,

although both formulations share some qualitative features.

The issue addressed is how location decisions are affected by market integration. There is a well-

identified tradeoff between market size and transport costs that induces relocation of firms from one

country to the other. However, this strand of literature puts the location choice in a quite simple way.

The set of possible locations is usually constituted by two points (countries), at most three.10 In this

paper, we allow for a wider set of locations, building directly on the well-known Hotelling model. We

sacrifice, in turn, general equilibrium analysis. This approach is followed in order to assess relocation

of firms across countries and also within the country. This seems to be highly desirable as improve-

ment of communication infrastructures (roads, trains and the like) may (is expected to) induce reloca-

tion within the country. It is also often believed by Governments that improvement in communication

infrastructures will motivate firms to establish new plants in less developed, interior regions (identified

with the small country in our model).

On the expected effect of relocation of firms within countries to locations closer to the border, at

least for the smaller economy, our results point out that such effect should be present if transport costs

are better approximated by the concept of CAB transport costs than by CAR transport costs. Under

the latter, the firm in the small country has always an incentive to move away from the border. This

force must be balanced against the desire to relax price competition, which drives the firm in the small

economy away from the border. If countries are roughly of the same size, the latter force dominates,

while for sufficient asymmetry in size between countries the former effect prevails.

On policy grounds, the results suggest that relocation of firms can be more difficult to predict (and

induce) than usually believed. They also cast some doubts on investment efforts by local authorities

10For a model with three locations see Elizondo and Krugman (1992).


like the ones deployed in Kent’s business parks, on the expectation of U.K. firms moving closer to the

Tunnel.

In terms of infrastructure investment in communications between less developed regions and core

regions, it points out that other features must be present in these regions if they are to attract firms

(these can be lower land rents, cheaper labor, etc. . . ).

On more technical grounds, the analysis has highlighted that the locational tradeoff of lower trans-

port costs versus market size needs to take into account a wider set of locations available and the type

of transport costs.11 The results show that the principle of maximum differentiation of d’Aspremont,

Gabszewicz and Thisse (1979) does not hold under an alternative formulation of transport costs (which

we termed CAB transport costs). In this case, the existence of a border does matter (while it does not

in the standard model in the literature). The superiority of one type of transport costs over the other is

an empirical matter.

Summing-up, market integration through reduction of physical transport costs can originate dis-

tinct location patterns according to relative market size across regions and to the nature of transport

costs. To the best of our knowledge there is no formal evidence on this latter matter. On the top of this

tradeoff, we show that relocation of firms may follow unexpected patterns.

11Although, it may be argued that Hotelling-like models are quite less workable under assumptions of non-homogeneityof the location space. The algebraic complexities of the model easily tend to clout the economic intuition behind models andtheir results.


Appendix

A Equilibrium under CAR transport costs.

A.1 Proof of Proposition 1.

DefineΓ ≡ T

f − d . It is easy to verify thatpd is always positive. Also,pf is positive if

Γ < f + d+ 2k (5)

Evaluatingz(pd, pf ) at the candidate equilibrium price pair, we obtain,

z(pd, pf ) =(f − d)(f + d+ 2k+)− T

6(f − d)

and for the equilibrium to be well definedz(pd, pf ) ∈ [1, f ], which holds if

2k + d− 5f ≤ Γ ≤ f + d− 6 + 2k (6)

It is trivial to check that conditions (6) imply condition (5). Also,k > 2.5 is sufficient to ensure that

the upper bound in conditions (6) is positive. Thus, it will be assumed throughout.

Substituting the equilibrium price pair into profit functions defines the equilibrium profits accruing

from the price subgame

Πd(d, f) =p2d

2(f − d); Πf (d, f) =

p2f

2(f − d)

Consider the derivative of firmd’s profits with respect to its location. It is given by,

∂Πd(d, f)∂d

=pd[(f − d)(f − 3d− 2k)− T ]

6(f − d)2< 0.

where the negative sign follows fromk > f > 1 andd < 1. Thus, firmd tends to go towards zero,

independently of the location of firmf in the larger market.

Take, in turn, firmd’s location as given and consider the derivative of firmf ’s profits with respect

to its location. It is given by,

∂Πf (d, f)∂f

=pf [(f − d)(d− 3f + 4k)− T ]

6(f − d)2

Two cases may arise. Either this derivative is positive or negative.

Γ

Γ if k - 2f + 3 < 0

(6)if k - 2f + 3 > 0

(8)

< 0

(10)

2k - 5f + d 2k + f + d - 6

d + 4k - 3f

∂∏ f∂ f

> 0

∂∏ f∂ f > 0

∂∏ f∂ f


Case I:∂Πf (d, f)

∂f> 0. The derivative is positive if(f −d)(d−3f + 4k)−T > 0. That is to say, if

Γ < d− 3f + 4k (7)

Combining (7) and (6) we obtain that if−2f + k+ 3 > 0 then (6) implies the above derivative

is positive. If, on the contrary−2f + k + 3 < 0, the derivative is positive if

d− 5f + 2k < Γ < d− 3f + 4k (8)

Case II:∂Πf (d, f)

∂f< 0. The derivative is negative if(f −d)(d−3f +4k)−T < 0. This condition

can be expressed as

Γ > d− 3f + 4k. (9)

Combining (9) and (6), we obtain

−3f + d+ 4k < Γ < f + d+ 2k − 6. (10)

This is a well-defined interval if−2f + k + 3 < 0.

The above discussion can be summarized in figure 4.

Figure 4:

As a reference point, note that forT = 0,∂Πf (d, f)

∂f> 0, so that we recover the standard result

of d’Aspremont et al. (1979).



Prices must be positive, which is guaranteed by the following condition:

T

f − d < 4k − (f + d) (11)

The condition requires the border fee to be small or the size difference between countries to be large

(k big).

We must now impose that, at equilibrium prices, the indifferent consumer position is consistent

with initial assumptions:1 ≥ z ≥ d. This amounts to

6− f − d− 2k ≥ T

f − d ≥ 5d− f − 2k (12)

provided country 2 is not too large (the exact condition being2k < 6 − f − d). The interval is well-

defined asd ≤ 1.

We now turn to the incentives of firms to change location. Equilibrium profits are

Πd =p2d

2(f − d); Πf =

p2f

2(f − d)

A small change in the location of firmd implies

∂Πd

∂d=

pd2(f − d)

[pd − 4(f − d)

d+ k

3

]Then, it is straightforward to establish

∂Πd

∂d< 0 if

T

f − d < 2k + 2d− (f − d) (13)

From the conditions for equilibrium to be well-defined (see expression (12)), it results that the above

condition always holds. Thus,∂Πd/∂d < 0, whatever the location of the other firm, relative size of

countries size and tariff, provided equilibrium exists. Since this effect has a constant sign, firmd will

locate away from the border (under the restriction that each firm locates in a different country and the

indifferent consumer is in the small country).

Consider a small change in the location of firmf . Computations reveal that

∂Πf

∂f=

pf2(f − d)2

[−pf + 2(f − d)

−2f + 4k3

]


The sign of the expression hinges upon the sign of

(f + d) + 4k +T

f − d − 4f (14)

Combining this condition with the requirement of a well-defined equilibrium, the following basic char-

acterization results:

∂Πf∂f > 0 if

(a) 2f − k − 3d < 0;

(b) 3 > 2f − k > 3d and Tf−d > 3f − 4k − d

∂Πf∂f < 0 if

(c) 2f − k > 3.

(d) 3 > 2f − k > 3d and Tf−d < 3f − 4k − d

We may now investigate in more detail the stated conditions. Take first∂Πf/∂f > 0. It is straight-

forward to establish that

0 > 2f − k − 3 > 3f − 4k − d

SinceT > 0, the second part in condition (b) is redundant. Consider∂Πf/∂f < 0. Then,

0 > 2f − k − 3 > 3f − 4− d

which means that condition (d) is never satisfied.


The indifferent consumer is defined by

z =f + d

2+

pf − pd2(f − d)


A.3.1 Firms in the big market

Maximizing and solving for the equilibrium prices, we get

pd =(f − d)(d+ f + 2k)

3

pf =(f − d)(4k − f − d)

3

Prices are always positive. It is, nevertheless, necessary thatz ∈ [d, f ], which amounts to

5f − d > 2k > 5d− f. (15)

It implies a very narrow range fork. Equilibrium profits are

Πd =(f − d)(d+ f + 2k)2

18; Πf =

(f − d)(4k − f − d)2

18

The tendencies for changes of location, at the margin, can be obtained after straightforward deriva-

tions:

sign∂Πd

∂d= sign (f − 3d− 2k) < 0

sign∂Πf

∂f= sign (4k − 3f + d) > 0

Therefore, firms tend towards the endpoints of the market.

A.3.2 Firms in the small market.

Solving for the equilibrium prices, we get

pd =(f − d)(d+ f + 2k)

3

pf =(f − d)(4k − f − d)

3.

Prices are always positive. It is, nevertheless, necessary thatz ∈ [d, f ] for the equilibrium to be

well defined, which amounts tok ≤ (5f − d)/2. It implies a very narrow range fork, as at the most

k can be 2.5 (by definition,k has a lower bound at 2). Equilibrium profits are

Πd =(f − d)(d+ f + 2k)2

18; Πf =

(f − d)(4k − f − d)2

18


The tendencies for changes of location, at the margin, can be obtained after straightforward deriva-

tions:

sign∂Πd

∂d= sign (f − 3d− 2k) < 0

sign∂Πf

∂f= sign (4k − 3f + d) > 0

The signs of expressions result fromd, f ≤ 1 andk ≥ 2. Thus, both firms tend towards the endpoints

of the market.

B Equilibrium under CAB transport costs.

B.1 Proof of Proposition 4.

Define∆ ≡ T + (1− d)2

f − 1. Prices are positive if

1 + f − 4k < ∆ < f + 1 + 2k (16)

Evaluatingz(pd, pf ) at the candidate equilibrium price pair, we obtain,

z(pd, pf ) =(f − 1)(2k + 1 + f)− (1− d)2 − T

6(f − 1)

andz(pd, pf ) ∈ [1, f ] if

2k − 5f + 1 < ∆ < 2k + f − 5 (17)

It is trivial to check that conditions (17) imply condition (16). Also, we will assume thatk > 2.5 to

ensure that the upper bound in (17) is positive.

Substituting the equilibrium price pair into profit functions defines the equilibrium profits accruing

from the price subgame

Πd(d, f) =p2d

2(f − 1); Πf (d, f) =

p2f

2(f − 1)

Consider the derivative of firmd’s profits with respect to its location. It is given by,

∂Πd(d, f)∂d

=2pd(1− d)3(f − 1)

> 0.

Thus, whatever the location of firmf , firm d always tends to move towards the border.


Take firmd’s location as given and consider the derivative of firmf ’s profits with respect to its

location. It is given by,

∂Πf (d, f)∂f

=pf [(f − 1)(1 + 4k − 3f)− (1− d)2 − T ]

6(f − 1)2

Two cases may arise. Either this derivative is positive or negative.

Case I:∂Πf (d, f)

∂f> 0. The derivative is positive if(f − 1)(1 + 4k− 3f)− (1− d)2−T > 0. This

condition can be expressed as

∆ < 1 + 4k − 3f. (18)

Combining conditions (18) and (17), we obtain that if2f−k−3 < 0, then condition (17) implies

that the above derivative is positive. If on the contrary2f − k − 3 > 0 then the derivative is

positive if

2k − 5f + 1 < ∆ < 1 + 4k − 3f (19)

Case II:∂Πf (d, f)

∂f< 0. The derivative is negative if(f − 1)(1 + 4k − 3f) − (1 − d)2 − T < 0.

That is to say, if

∆ > 1 + 4k − 3f (20)

The combination of (20) and (17) gives rise to the following condition

1 + 4k − 3f < ∆ < 2k + f − 5 (21)

The above discussion can be summarized in the Figure 5

B.2 Proof of Proposition 5.

DefineΦ ≡ T + (f − 1)2

1− d . Prices must be positive, which is guaranteed by the following condition:

Φ < 4k − 1− d (22)

We must now impose the following conditions so that, at equilibrium prices, the indifferent con-

sumer position is consistent with initial assumptions (1 ≥ z ≥ d):

5− d− 2k > Φ > 5d− 1− 2k (23)

∆

∂∏ f∂ f

> 0

∆ if 2f - k - 3 > 0

∂∏ f∂ f > 0

(18)if 2f - k - 3 < 0

(20)

∂∏ f∂ f < 0

(22)

2k - 5f+ 1 2k + f - 5

1 + 4k - 3f


Figure 5:

It is easy to check the interval is well defined. In addition, it is necessary thatk < (5 − d)/2, as the

tariff is positive.12 It is straightforward to establish that the left-hand side condition onΦ also implies

positive prices.

We now look at the incentives of firms to change location. Equilibrium profits are

Πd =p2d

2(1− d); Πf =

p2f

2(1− d)

A small change in the location of firmd implies

sign∂Πd

∂d= sign[(1− d)(1− 3d+ 2k) + (f − 1)2 + T ] > 0 (24)

Consider now a small change in the location of firmf . Computations reveal

∂Πf

∂f=

pf3(1− d)

∂pf∂f

=2pf

3(1− d)(f − 1) < 0.

Of course, at values ofd close to one and off close to 1 the equilibrium price breaks down.13

12Although it is not sufficient for existence of equilibrium.13For5− d− 2k becomes negative ford = 1. SinceΦ > 0, condition (23) is not satisfied.


References

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