# Location choices under quality uncertainty

**ABSTRACT** We examine a linear city duopoly where firms choose their locations to maximize expected profits, uncertain about how consumers will assess the relative quality of their products. Equilibrium locations depend on the ratio of the expected quality superiority to the strength of horizontal differentiation. When it is small, firms locate at opposite endpoints. As it becomes larger, agglomeration around the centre also emerges as an equilibrium and, eventually, agglomeration becomes the only equilibrium.

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**ABSTRACT:**We set-up a linear city model with duopoly upstream and downstream. Consumers have a transportation cost when buying from a retailer, and retailers have a transportation cost when buying from a wholesaler. We characterize the equilibria in a five-stage game where location and pricing decisions (wholesale and retail) by all four firms are endogenous. The usual demand and price competition effects are modified and an additional strategic effect emerges, since the retailers' marginal costs become endogenous. Firms tend to locate farther away from the market center relative to the vertically integration case. When the wholesalers choose locations before the retailers, each wholesaler locates closer to the market center relative to the retailer locations, and relative to when the wholesalers cannot move first. Each wholesaler does this to strengthen the strategic position of its retailer by credibly pulling him towards the market center. As a result, the intensity of competition is higher and industry profit is lower when upstream locations are chosen before downstream locations. Variations of the model and welfare analysis are provided.C.E.P.R. Discussion Papers, CEPR Discussion Papers. 01/2010; - SourceAvailable from: aueb.gr[Show abstract] [Hide abstract]

**ABSTRACT:**We examine a linear city model with duopoly in the upstream and down- stream level. We set up a …ve-stage game where location and pricing decisions of both upstream and downstream …rms are determined. We show that the unique equilibrium outcome is maximum dierentiation by upstream and downstream …rms. Apart from the standard "demand" and "price competition" eect when …rms change their locations, there is also a third force that aects the whole- sale prices that upstream …rms charge. The interaction of these forces give the equilibrium result. Under price discrimination by the upstream …rms, wholesale and …nal prices reduce and the equilibrium locations move towards the centre of the line. Price discrimination may be not anticompetitive, as it further reduces the social transportation cost for some parameter values. - [Show abstract] [Hide abstract]

**ABSTRACT:**It is known that if exogenous cost heterogeneities between the firms in a spatial duopoly model are large, then the model does not have a pure-strategy equilibrium in location choices. It is also known that when these heterogeneities are stochastically determined after firms choose their locations, spatial agglomeration can appear. To tackle these issues, the current paper modifies the spatial framework by allowing firms to exchange the cost-efficient production technology via royalties. It is shown that technology transfer guarantees the existence of a location equilibrium in pure strategies and that maximum differentiation appears in the market. KeywordsLocation model-Asymmetric firms-Licensing-Royalty-R&D JEL ClassificationD43-D45Journal of Economics 01/2010; 99(3):267-276. · 0.58 Impact Factor

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No. 4323

LOCATION CHOICES UNDER

QUALITY UNCERTAINTY

Charalambos Christou and Nikolaos Vettas

INDUSTRIAL ORGANIZATION

Page 2

ISSN 0265-8003

LOCATION CHOICES UNDER

QUALITY UNCERTAINTY

Charalambos Christou, Athens University of Economics and Business

Nikolaos Vettas, Athens University of Economics and Business and CEPR

Discussion Paper No. 4323

March 2004

Centre for Economic Policy Research

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Copyright: Charalambos Christou and Nikolaos Vettas

Page 3

CEPR Discussion Paper No. 4323

March 2004

ABSTRACT

Location Choices under Quality Uncertainty

We examine a linear city duopoly where firms choose their locations to

maximize expected profits, uncertain about how consumers will assess the

relative quality of their products. Equilibrium locations depend on the ratio of

the expected quality superiority to the strength of horizontal differentiation.

When it is small, firms locate at opposite endpoints. As it becomes larger,

agglomeration around the centre also emerges as an equilibrium and,

eventually, agglomeration becomes the only equilibrium.

JEL Classification: L13 and L15

Keywords: linear city, location, product differentiation and quality uncertainty

Charalambos Christou

Department of Economics

Athens University of Economics

and Business

Patission 76

Athens 10434

GREECE

Email: cachristou@econ.uoa.gr

For further Discussion Papers by this author see:

www.cepr.org/pubs/new-dps/dplist.asp?authorid=159166

Nikolaos Vettas

Department of Economics

Athens University of Economics

and Business

Patission 76

Athens 10434

GREECE

Tel: (30 210) 8203179

Email: nvettas@aueb.gr

For further Discussion Papers by this author see:

www.cepr.org/pubs/new-dps/dplist.asp?authorid=121328

Submitted 25 February 2004

Page 4

1Introduction

Horizontal differentiation in the context of geographical locations or in that of product char-

acteristics has been studied in an extensive and important literature (see e.g. Gabszewicz and

Thisse, 1992 for a review). Hotelling’s (1929) classic model has suggested that firms’ com-

petition for the in-between consumers implies minimal differentiation. Subsequent work has

stressed an opposite incentive, with firms locating away from their rivals, to relax price com-

petition (see e.g. d’Aspremont et al., 1979). While various aspects of the interaction of these

two effects have been studied, including multi-dimensional contexts1, the effect of uncertainty

about a second characteristic on location decisions is not as well understood. In this paper,

we show that locations depend critically on quality uncertainty. For instance, a restaurant’s

incentive to locate close to its rivals may depend critically on how likely it is that consumers

will significantly favor one restaurant to another, for reasons other than their locations.

When quality uncertainty is relatively low, the equilibria in our model are as in the

d’Aspremont et al. (1979) model of pure horizontal differentiation. However, as relative qual-

ity uncertainty increases, agglomeration emerges as an equilibrium outcome, with the set of

equilibrium locations becoming larger for higher uncertainty. By locating at the same location

as its rival, a firm risks obtaining zero profit if its quality proves inferior, but takes full advan-

tage of its superior quality when this event occurs. Importantly, our model offers an economic

rationale and characterization for agglomeration also at points other than the city center.2

The remainder of the paper is as follows. Section 2 sets up the model. Section 3 presents the

equilibrium. Section 4 modifies the game so that firms choose locations sequentially. Section

5 concludes. The calculations underlying the locations equilibria are standard but relatively

1See e.g. Economides (1989), Neven and Thisse (1990), Irmen and Thisse (1998) and Ansari et al. (1998).

2The effect of uncertainty on location choices has been studied in Balvers and Szerb (1996), but in a model

that suppresses price competition and, thus, obtains very different results (risk aversion drives firms away

from agglomeration). Heterogeneity in consumers’ preferences drives firms to central locations in de Palma et

al. (1985), however, the strategic effects in their model are very different as there is no aggregate demand

uncertainty.

2

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tedious (since several cases have to be distinguished); they are contained in the Appendix.

2The model

Consider two firms, A and B, each producing a single product in a (new) market. Firms locate

their products on the [0,1] line; consumers are uniformly distributed on the line segment with

transportation cost quadratic in distance. A consumer located at point x on the line and

purchasing a unit of firm i’s product obtains surplus equal to

u(x,i) = R − t(x − xi)2+ qi− pi,(1)

where xidenotes the location of firm i, and pi, and qiits product’s price and quality, respec-

tively. Thus, products are differentiated both horizontally and vertically.

We assume that the basic reservation value, R, is high enough and that each consumer

purchases one unit of the product, the one that offers the highest net surplus — in other words,

consistent to the main body of the product differentiation literature, the market is “covered”.

In order to capture the effect of relative quality uncertainty on location choices, we assume

that the quality of product i, qi, is a random variable, the realization of which firms ignore

when they choose their locations. The basic game has the following stages:

1. Firms simultaneously choose the locations of their products.

2. The quality difference, qi− qj, is revealed and becomes common knowledge.

3. Firms simultaneously choose their product prices.

4. Having observed the firms’ locations and the product qualities and prices, each consumer

purchases the product that gives him the highest net surplus.

The structure of the model is consistent with the view that location choices have longer

run characteristics than pricing. It also assumes that these locations are costly to change after

certain aspects of the products (quality) become revealed to the consumers.

For reasons of tractability, we further assume that the random variable qi−qjis uniformly

distributed on the interval [−h,h], h > 0. Implicit in this assumption is our treatment of firms

as a priori symmetric. Note that the ratio h/t captures the importance of vertical (quality)

3

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differentiation relative to horizontal (location) differentiation. Since the quality difference is a

random variable, the profits associated with each location pair are also random, at the time

the location decisions are made. We assume that unit costs are zero (so that higher quality

does not have to cost more per unit). Firms are risk neutral.

3 Equilibrium

We proceed backwards, looking for the subgame perfect Nash equilibrium.

3.1Price equilibrium

Since each firm, A or B, can choose any location on the line, it is convenient to denote for

the price competition part of the analysis the firms as 1 and 2, where firm 1 is to the left of

2 (x1≤ x2). When turning to the location choices, we will then allow each firm to choose a

location to the left or the right of its rival (that is, to assume the role of firm 1). Now, given

the firms’ locations their demand functions are as follows. Let z be the demand of firm 1 —

then, firm 2 has demand 1 − z. For z ∈ (0,1), demands are determined by the location of the

consumer indifferent between the two products. From (1), this indifferent consumer is located

at point z, with the property

p1+ t(z − x1)2= −q + p2+ t(z − x2)2,(2)

where q ≡ q2− q1.

The location of the indifferent consumer depends on firms’ locations and prices, as well as on

the transportation cost parameter and the quality difference, that is, z = z(p1,p2,x1,x2,q,t);

for ease of notation we suppress the arguments of this function. For z ∈ (0,1), by solving (2),

we obtain

z =x1+ x2

2

+p2− q − p1

2t(x2− x1).

Taking also into account the possibility that all consumers choose one of the products, the

firms’ profit functions are

π1= p1z and π2= p2(1 − z),(3)

4

Page 7

where

z =

0 if

x1+x2

2

+

p2−q−p1

2t(x2−x1)≤ 0

p2−q−p1

2t(x2−x1)∈ (0,1)

p2−q−p1

2t(x2−x1)≥ 1.

x1+x2

2

+

p2−q−p1

2t(x2−x1)

if

x1+x2

2

+

1if

x1+x2

2

+

(4)

The equilibrium prices, as functions of locations and the realized q, are as follows.3

Proposition 1: For 0 ≤ x1≤ x2≤ 1 the equilibrium prices are

and

p∗

1=

t£(x2− 1)2− (x1− 1)2¤− q

1

3

if q < q−

if q ∈ [q−,q+]

if q > q+,

£t£(x2+ 1)2− (x1+ 1)2¤− q¤

0

(5)

p∗

2=

0 if q < q−

if q ∈ [q−,q+]

if q > q+,

1

3

£t£(x1− 2)2− (x2− 2)2¤+ q¤

q − t(x2

2− x2

1)

(6)

where

q−≡ t£(x2− 2)2− (x1− 2)2¤

and

q+≡ t£(x2+ 1)2− (x1+ 1)2¤.

Proof. See Appendix A1.

(7)

Proposition 1 implies that there a unique price equilibrium for each pair of locations and

each realization of the quality difference. It nests as special cases two extremes. When differ-

entiation is only horizontal (q = 0), we recover the d’Aspremont et al. (1979) characterization

of price equilibria. When differentiation is only vertical (x1= x2), the firm selling the higher

quality product charges a price equal to the quality difference, q, and serves all consumers.

3.2 Equilibrium locations

We now proceed to the first stage of the game, where firms choose locations. Taking as given

equilibrium pricing in the second stage (for any quality realization), each firm’s location should

maximize its expected profit, given the location of its rival.

3This result modifies and extends one in Vettas (1999) to the case of arbitrary quality differences.

5

Page 8

We denote the expected profit function of firm i by Eπi(x1,x2), suppressing in the notation

of the arguments the transportation cost parameter, t, and the quality difference parameter,

h. The expectation is taken over the stochastic quality q. The expected profit function of firm

1 is

Eπ1(x1,x2) =

Zq−

min{−h,q−}

πm

1(x1,x2)dF +

Zmin{q+,h}

max{−h,q−}

πc

1(x1,x2)dF,(8)

where F(x) = (x + h)/(2h) is the cumulative distribution function of the variable q (since it

is uniformly distributed in [−h,h]). Depending on the quality realization, we can have either

an interior solution (where both firms sell their products) or a corner solution (with only the

high quality firm making positive sales). πm

1(x1,x2) denotes firm 1’s profit in the case that

only firm 1 sells its product, and πc

1(x1,x2) denotes the profit of firm 1 when both firms sell

their products. Specifically, by substituting the equilibrium prices from the relevant ranges of

(5) and (6) into the profit functions (3), we obtain

πm

1(x1,x2) = −q + t£(x2− 1)2− (x1− 1)2¤

πc

1(x1,x2) =

18(x2−x1)t

and

1

¡q + t£(x1+ 1)2− (x2+ 1)2¤¢2.

(9)

The bounds of the integrals in (8) are set to be min{−h,q−}, max{−h,q−} and min{q+,h}, to

allow for the possibility that some type of price equilibria do not occur for some values of the

parameters (x1,x2,h,t). If, for example, h is relatively low and the distance between the two

firms’ locations is relatively large, that is, q−is low compared to −h, then even if q equals −h,

both firms sell their products in equilibrium regardless of the realized value of q.

Similarly, the expected profit function of firm 2 is

Eπ2(x1,x2) =

Zmin{q+,h}

max{−h,q−}

πc

2(x1,x2)dF +

Zh

min{q+,h}

πm

2(x1,x2)dF.(10)

Since a firm can choose to locate either at the same location, to the left, or to the right of

its rival, the expected profit functions of firms A and B, over the entire range of locations, are

where Eπ1(·,·) and Eπ2(·,·), are given by (8) and (10).

A key characteristic of the functions Eπ1(x1,x2) and Eπ2(x1,x2) is that they are quasi-

Eπi(xi,xj) =

Eπ1(xi,xj)ifxi≤ xj

xi≥ xj, i,j ∈ {A,B}Eπ2(xj,xi)if

(11)

convex in x1and x2, respectively, a property implying that they are maximized at the extrema

6

Page 9

of their domains.4’5Since Eπ1(x1,x2) is quasi-convex in x1, firm 1’s best response is to locate

either at 0 or x2. Similarly, Eπ2(x1,x2) achieves its maximum value either at point x1or 1. It

follows that given the location of firm B, firm A’s best response belongs to {0,xB,1}, that is,

either it chooses the same location as its rival or one of the endpoints.

Specifically, the best response correspondence of firm i given the location of firm j, i,j =

A,B is as follows:

• For h/t < r1

Ri(xj) =

1 ifxj≤ 1/2

xj≥ 1/2.

0if

(12)

• For h/t ∈ [r1,r2]

Ri(xj) =

1ifxj≤ 1 − x∗

xj∈ [1 − x∗,x∗]

xj≥ x∗,

xj

if

0if

(13)

where

x∗≡ argx∈[1

2,1]

½h2+ 3x2(2 + x)2t2

54xt

=h

4

¾

.(14)

• For h/t > r2

Ri(xj) = {xj} for allxj∈ [0,1].

(15)

The values r1and r2are the solutions with respect to x, to

h2+3x2(2+x)2t2

54xt

=h

4for x = 0.5

and for x = 1, respectively (r1 ≈ 0.786 and r2 ≈ 2.442). The explanation for the equality

yielding the critical thresholds is as follows. The expected profit of a firm if it is located at the

same point as its rival, is Eπ1(x,x) =h

4, whereas Eπ1(0,x) can be easily shown to be equal to

h2+3x2(2+x)2t2

54xt

, and increasing in x. Also, the difference Eπ1(x,x)−Eπ1(0,x) is increasing in h/t,

that is, the greater the maximum quality difference, the greater the incentive for a firm to locate

4In principle, the fact that the profit functions are not quasi-concave poses threats for the existence of

equilibria. However in the present model there always exists at least one equilibrium.

5The complete form of function Eπ1(x1,x2) is derived in Appendix A2 and the details for the proof of

quasi-convexity are in Appendix A3.

7

Page 10

at the same point as its rival. Thus, given the quasi-convexity of Eπi(xi,xj) with respect to xi

and the symmetry of the profit function Eπi(xi,xj) (that is, Eπi(xi,xj) = Eπj(1−xj,1−xi))

a firm’s best response depends on the distance from its rival. Further, given the location of

firm j at a point x ∈ [0.5,1], the best response of firm i will be either at point 0, or at point x;

location at point 1 gives lower profit than location at point 0 (see Appendix A4). For h/t < r1,

Eπ1(x,x) < Eπ1(0,x), for every x, and for h/t > r2, Eπ1(x,x) > Eπ1(0,x), for every x (see

Appendix A5). In the remaining case, when h/t ∈ (r1,r2), if firm j locates at a point x,

close enough to an endpoint firm i would choose the opposite endpoint. For example, if firm

2 locates close enough to endpoint 1, so that x > x∗, then Eπ1(0,x) > Eπ1(x,x) holds and

the best response of firm 1 is to locate at point 0. On the other hand, if x ∈ (1 − x∗,x∗),

then Eπ1(0,x) < Eπ1(x,x), and the best response of firm i is to locate at the same point x

as its rival. Finally, by implicit differentiation of the equality (14) that defines x∗, we observe

that x∗is strictly increasing in h/t. Thus, a higher value of h/t increases the attractiveness of

locations near the center.

The following diagrams illustrate the different cases discussed above. Figure 1, presents the

case h/t ∈ [r1,r2]. Observe that there is a continuum of location pairs at which the two best

response correspondences intersect, that is, there is a continuum of location equilibria.

xB*

RA

RB

0

1

xA

1 - xA*

1 - xB*

xA*

0

1

xA

11

xB

xB

Figure 1: Best response correspondences for h/t ∈ [r1,r2].

As h/t approaches r1, x∗

Band x∗

Aapproach 0.5, and the best response correspondence of

firm A becomes, in the limit, as in Figure 2a. Firm B’s best response correspondence, in this

case, is illustrated in Figure 2b. As h/t approaches r2, x∗

Band 1−x∗

Aapproach 1, and the best

response correspondence of firm A, in the limit, becomes as shown in Figure 2c. For h/t > r2,

the graphs of the best response correspondences of the two firms are identical.

8

Page 11

0.5

1

0

1

RB

RA

1

0

1

RA

1

xA

01

RB

0.5

xB

xA

xA

xB

xB

)(a)(b)(c

1

/rth

<

1

/rth

<

2

/rth

>

Figure 2: Best response correspondences for h/t < r1and h/t > r2.

We summarize the previous analysis:

Proposition 2: The location equilibria are as follows: (i) for h/t < r1 there are only two

equilibrium location pairs, the extreme points (xA,xB) = (0,1) and (xA,xB) = (1,0),

(ii) for h/t ∈ [r1,r2] there is a continuum of equilibria (xA,xB) = (x,x), with x ∈

[x∗,1−x∗] where x∗is given by (14), as well as the two extreme points (xA,xB) = (0,1)

and (xA,xB) = (1,0), and, (iii) for h/t > r2each location pair (xA,xB) = (x,x), with

x ∈ [0,1] is an equilibrium.

Thus, the firms tend to choose minimally differentiated locations if the expected gains of

having a successful (high quality) product are sufficiently high. In other words, it pays to

follow an aggressive product choice strategy if the expected value of the (possible) success is

relatively high. Note that, when there are agglomeration equilibria, there is a continuum of

these, in addition to the extreme locations equilibrium.

In models of horizontal differentiation, there are two forces, one pushing the firms close to

each other and one pushing them in the opposite direction. In our formulation, which force

dominates depends on the level of quality difference. When the expected quality advantage of a

high quality firm is small, the price competition effect dominates and maximum horizontal dif-

ferentiation is preferred. As the quality advantage one firm could gain over the other increases,

a high quality firm tends to prefer locations closer to its rival. This fact is captured by the

quasi-convexity of the profit functions: the expected profit Eπ1(x1,x2) is maximized either at

point 0 or at point x2. This, in turn, is due to the fact that the profit πm

1(x1,x2), of a firm that

has quality higher enough than its rival, is increasing in x1. Moreover, the incentive of each

firm to locate at the same point as its rival depends on the location of the latter. The closer the

9

Page 12

low quality firm is to the center, the greater is the incentive of the high quality firm to choose

the same location. Essentially, the high quality firm prefers to take full advantage of its quality

superiority by locating at the same point as its rival, only if it the horizontal differentiation it

can achieve from the rival is relatively small. Taking expectations over quality realizations, the

ratio of the expected profit under agglomeration to the one when choosing different locations

is increasing in h/t. Further, a higher h/t value supports a greater range of locations at which

there can exist equilibria with minimum horizontal differentiation.

Note that in the simultaneous location game, expected profits are not always maximized;

in particular, agglomeration equilibria are dominated in terms of profits by extreme locations

equilibria if the firms could coordinate their decisions they would like to avoid them. Partly

motivated by this observation, we now turn to sequential location choices.

Figure 3, below, depicts the set of locations at which there exist agglomeration equilibria.

When h/t < r1, there only exist equilibria with maximum horizontal differentiation. When

h/t ∈ [r1,r2], apart from the two equilibria with maximum differentiation, there exist equilibria

with minimum differentiation also. The extent of locations at which this type of equilibrium

exists increases with h/t, and for h/t > r2only equilibria with minimum differentiation exist.

0

1

5 . 0

th/

1r

x

2r

Figure 3: Equilibrium location points.

4Sequential location choices

From the above analysis, we know that a follower’s best response has to be to locate either

at an endpoint or at the same point as the first mover. Unlike before, however, the firm that

chooses its location first can effectively choose the location that maximizes its expected profit,

knowing how its rival will respond (note that this location also maximizes the expected profit

of its rival, since the expected profit functions take the same value in equilibrium). We have:

10

Page 13

Proposition 3: When firms choose locations sequentially, the equilibrium locations are: (i) if

h/t < r2, (xA,xB) = (0,1) and (xA,xB) = (1,0), and, (ii) if h/t ≥ r2, (xA,xB) = (x,x),

for any x ∈ [0,1].

Proof. If either h/t ≥ r2or h/t < r1the equilibrium locations are the same as in Propo-

sition 2. Either firm wishes to locate at the same point as its rival (if h/t ≥ r2), or as far from

each other as possible (if h/t ≤ r1). When h/t ∈ [r1,r2], if the two firms end up located at the

same point, each would have expected profit equal to h/4. If the two firms are located at the

opposite endpoints, the expected profit of each firm would be (t/2 + h2/54t). Straightforward

calculations show that h/4 < (t/2 + h2/54t) ⇐⇒ h/t < r2, implying that the firms prefer

locating at the opposite endpoints.

5 Conclusions

We have studied how quality uncertainty affects the location choices of firms. We identify the

ratio of the expected quality differentiation to horizontal differentiation as the key parameter

driving the results. For low values of this variable, we obtain maximum differentiation. For

higher values, agglomeration equilibria also appear, around the center. Interestingly, agglom-

eration may occur not only at the center but also at a wider set of locations symmetrically

defined around the center. This set expands as the possible quality difference increases and,

after a certain threshold, it covers the entire linear city.

While we cannot claim full generality of our exact results (it is a common feature of the

product differentiation literature that the equilibria may significantly depend on the model

formulation), we believe the main conclusions from our analysis are more general than the

model: in industries where locations (or horizontal product characteristics) have to be chosen

before uncertainty concerning the relative product quality is resolved, we expect firms to tend

to locate closer to their rivals relative to industries where differentiation is essentially only

horizontal, in which case the incentive to relax price competition tends to keep firms away

from each other.6

6Gerlach et al. (2004) also study the effect of quality uncertainty on horizontal differentiation. However,

their focus and results are different, primarily because, in their model, firms with ex ante different qualities do

not coexist in the market.

11

Page 14

References

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entiation, Journal of Regional Science 38, 207-230.

[2] Balvers, R. and L. Szerb, 1996, Location in the Hotelling duopoly model with demand

uncertainty, European Economic Review 40, 1453-1461.

[3] D’ Aspremont, C., J. J. Gabszewicz and J.-F. Thisse, 1979, On Hotelling’s stability in

competition, Econometrica 47, 1145-1150.

[4] De Palma, A., V. Ginsburgh, Y. Papageorgiou and J.-F. Thisse, 1985, The principle of

minimum differentiation holds under sufficient heterogeneity, Econometrica 53, 767-782.

[5] Economides, N., 1989, Quality variations and maximal variety differentiation, Regional

Science and Urban Economics 19, 21-29.

[6] Gabszewicz, J. J. and J.-F. Thisse, 1992, Location, in: R. Aumann and S. Hart, eds.,

Handbook of Game Theory, Vol.1, chapter 9 (North-Holland, Amsterdam).

[7] Gerlach, H., T. Ronde and K. Stahl, 2004, Project choice and risk in R&D, discussion

paper, University of Copenhagen.

[8] Hotelling, H.,1929, Stability in competition, Economic Journal 39, 41-57.

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was almost right, Journal of Economic Theory 78, 76-102.

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Duke University discussion paper.

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Page 15

Appendix

Appendix A1: Proof of Proposition 1

Standard calculations imply that the best response correspondence of firm 1 is given by

R1(p2)=

any a ≥ 0

1

2

£t¡(x2− 1)2− (x1− 1)2¢+ p2− q¤

¢+ p2− q¤

When p2≤ q − t¡x2

quently, every p1≥ 0 is a best response for firm 1: firm 2’s price is so low that doing otherwise

would imply a loss.

When p2> q−t¡x2

£t¡x2

1

2

1

£t¡x2

result follows immediately.

ifp2≤ q − t¡x2

2− x2

2− x2

1

¢

£t¡x2

2− x2

1

¢+ p2− q¤

ifp2∈ [q − t¡x2

p2≥ q − q−,

1

¢,q − q−]

if

where

z ∈ (0,1) and£t¡(x2− 1)2− (x1− 1)2¢+ p2− q¤is the highest p1that implies z = 1.

2− x2

1

2

£t¡x2

2− x2

1

is the price p1 that maximizes π1 (solving dπ1/dp1 = 0) for

1

¢), for every p1≥ 0 we have z = 0 (by expression (4)). Conse-

2− x2

1

¢), firm 1’s best response implies either z = 1 or z ∈ (0,1). By the

¢+ p2− q¤

¢+ p2− q¤>£t¡(x2− 1)2− (x1− 1)2¢+ p2− q¤if and only if p2< q − q−, the

previous arguments, when p1<£t¡(x2− 1)2− (x1− 1)2¢+ p2− q¤firm 1’s profit is π1= p1,

hence, if

2

1

>

£t¡x2

1

2

1

1

2− x2

£t¡(x2− 1)2− (x1− 1)2¢+ p2− q¤, the best response is

2− x2

2− x2

¢+ p2− q¤and£t¡(x2− 1)2− (x1− 1)2¢+ p2− q¤, otherwise. Since

Similarly, the best-response correspondence of firm 2 is

R2(p1) =

any a ≥ 0

1

2

£t¡x2

ifp1≤ −q − t¡(x1− 1)2− (x2− 1)2¢)

£t¡(x1− 1)2− (x2− 1)2¢+ p1+ q¤

1− x2

ifp1∈ [−q − t¡(x1− 1)2− (x2− 1)2¢,−q + q+]

p1≥ −q + q+,

2

¢+ p1+ q¤

if

where1

0) for z ∈ (0,1) and£t¡x2

analogous to that of the previous case, for firm 1.

2

£t¡(x1− 1)2− (x2− 1)2¢+ p1+ q¤is the price p2that maximizes π2(solving dπ2/dp2=

1− x2

2

¢+ p1+ q¤is the highest p2that implies z = 0. The logic is

It remains to show that there exists a unique equilibrium pair (p1,p2), for every vector

(x1,x2,q,t). First, note that the candidate price equilibria involve either z ∈ (0,1), z = 1, or

z = 0. Specifically, if z ∈ (0,1), both firms’ best responses satisfy the first-order conditions. If

13

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