Firms are likely to be uncertain about consumer preferences when launching products. The existing literature models preference uncertainty as an additive shock to the consumer distribution in a characteristic space model. The additive shock only shifts the mean of the consumers' ideal points. We generalize this approach to a state space model in which a vector of parameters can give rise to dierent distributions of consumer tastes in dierent states, allowing other moments of the consumer density to be uncertain. An equilibrium existence result is given. In the case of symmetric distributions, the unique subgame-perfect equilibrium can be described by a simple closed-form solution.Location; Product Dierentiation; Uncertainty; Hotelling

Kieron J. Meagher

Klaus G. Zauner

English

Research Papers in Economics

 
 
 
 
 
 
 
School of Economics 
UNSW, Sydney 2052 
Australia 
 
http://www.economics.unsw.edu.au 
 
ISSN 1323-8949 
ISBN 978 0 7334 2525 7 
 
 
 
 
 
Spatial Equilibrium in a State Space Approach to Demand 
Uncertainty 
 
Kieron J. Meagher and Klaus G. Zauner 
 
School of Economics Discussion Paper: 2007/19 
 
 
 
 
 
 
 
 
The views expressed in this paper are those of the authors and do not necessarily reflect those of the 
School of Economic at UNSW. 
 Spatial Equilibrium in a State Space Approach to
Demand Uncertainty
Kieron J. Meagher
University of New South Wales
Klaus G. Zauner
University of York
This version: May 30, 2007
Abstract
Firms are likely to be uncertain about consumer preferences when
launching products. The existing literature models preference uncertainty
as an additive shock to the consumer distribution in a characteristic space
model. The additive shock only shifts the mean of the consumers' ideal
points. We generalize this approach to a state space model in which a
vector of parameters can give rise to dierent distributions of consumer
tastes in dierent states, allowing other moments of the consumer density
to be uncertain. An equilibrium existence result is given. In the case of
symmetric distributions, the unique subgame-perfect equilibrium can be
described by a simple closed-form solution.
Keywords: Location, Product Dierentiation, Uncertainty, Hotelling.
JEL Classication Numbers: C72, D43, D81, L10, L13, R30, R39.
1 Introduction
The Hotelling model of spatial competition can be thought of as one of the
earliest problems in information economics: rms make strategic choices (lo-
cations/characteristics and prices) without knowing an individual consumer's
type (location). However, in typical formulations, the distribution of consumer
types is common knowledge giving rise to the familiar mill price equilibrium.
Since individual consumer types are not observed in the standard model the
assumption that the distribution of types is observable seems internally incon-
sistent. Furthermore casual empiricism indicates that rms are often less than
perfectly informed about consumer tastes and frequently exert considerable ef-
fort to generate market research data.1 The recent demand location uncertainty
Corresponding author's address: School of Economics, University of New South Wales,
Sydney, NSW 2052, Australia, email: k.meagher@unsw.edu.au, phone: 61-2-93851145, fax:
61-2-93136337.
1Data may be generated externally by market research survey, stated choice experiments
or internally from customer behavior databases or from sta interaction with customers.
1literature recasts the Hotelling model in a more general and realistic setting in
which rms are uncertain not only about the type of an individual but are also
uncertain about the consumer distribution itself.
The existing approaches to preference/demand location uncertainty focus
on an additive shock to consumer types. Early approaches to characteristic
preference uncertainty, such as De Palma et al. (1985), used the law of large
numbers to generate certain demand functions from individual preferences with
idiosyncratic shocks. More recently the demand location uncertainty literature
(Jovanovic (1981), Harter (1996), Casado-Izaga (2000) and Meagher and Zauner
(2004, 2005)) has analyzed situations with perfectly correlated shocks which give
rise to residual aggregate uncertainty.
Thus the existing approaches generate no uncertainty at the distributional
level or uncertainty only about the mean of preferences. The contributions of
this paper are twofold: rst we show by way of an example how even simple situ-
ations fall beyond the scope of the existing `additive-shock' approach. Secondly
we show how a judicious state space formulation gives an existence theorem for
very general forms of uncertainty. For example the distribution of tastes could
change shape across states and need not be from a xed distributional type such
as the uniform or Normal. In the case of symmetric distributions this approach
gives simple closed form equations for equilibrium prices and locations.
By way of a specic example, in Section 3 we consider a Hotelling style linear
city but with the population distributed according to a linear distribution. In
addition to the standard characteristic space interpretation this representation
also has a geographic interpretation as a coastal city such as New York or
Sydney. Two rms, when deciding where to locate, are certain where the coastal
boundary is but are unsure how spread out the city will become by the time
they build their facilities. In this coastal city example both the mean and the
variance of the consumer distribution are uncertain. This issue of uncertainty
beyond the mean has not previously been considered in the literature. We
briey lay out a model below to show how this class of problem can be solved
by extending location theory. We solve the more general case in Section 4
following our specic uncertainty example in Section 3.
2 Common Features of the Model
We present a model of aggregate taste uncertainty in the spirit of Meagher and
Zauner (2004). Both rms and consumers are located at points on the real
line I R. Each consumer demands either one or zero units of the good and has
sucient income to buy one unit of the good. For a consumer with \ideal" point
x, the indirect utility function for consuming rm i's product (located at xi) at
price pi is given by
V (x;xi;pi) = A   pi   (xi   x)2;  > 0: (1)
There are two rms, i = 1;2. The marginal cost of production of each rm
2is constant and normalized to 0. Firms are uncertain about the distribution of
consumers but make decisions on the basis of a common prior, which we describe
below. Firms choose locations xi 2 I R (i = 1;2 and, without loss of generality,
x1  x2) simultaneously, observe the location of their competitor, then choose
prices simultaneously and nally the uncertainty is resolved.
This timing implies that rms are slow in learning about their demand con-
ditions, relative to the product's life cycle, due perhaps to a very short life cycle
or to the diculty or expense of conducting experimentation. Whatever the
reason, the limit case of slow learning is when a rm chooses a price under
uncertainty and is stuck with that price for the life of the product.
Consumers buy from the rm that gives them the highest (net) utility, hence
there exists a unique point , satisfying V (;x1;p1) = V (;x2;p2), where con-
sumers are indierent between buying from rm 1 or rm 2. States of the world
are indexed by S, which might be a vector, with density f(S). In each state of
the world the distribution of consumer locations, x, is given by gS(x).
3 Ice Cream Sellers on an Asymmetric Beach
Consider a variant of the ice cream sellers story so popularly used to motivate
the Hotelling model. Two ice cream sellers must choose locations on a beach of
length 1, represented by the unit interval, and post prices before their customers
arrive for the day. Furthermore assume this is a beach with a car park at the left
end so all consumers enter from the same end and then walk a random distance
to the right, determined jointly by their dislike of walking and crowds.
This scenario could plausibly yield an asymmetric density of consumers on
the interval [0;], 0 <   1. For simplicity we assume consumers are dis-
tributed according to the linear density:
g(x) =
 2
   2x
2 if 0  x < 
0 otherwise
Furthermore, since the ice cream sellers are uncertain who will come to the
beach they are therefore uncertain of which (linear) density of consumers will
occur, which is represented here by a common prior over , denoted f() on
[0;1]. For simplicity and to yield explicit solutions we choose f to be the power
density of order 2, i.e. f() = 32.
Assuming rms are risk neutral the expected prots for rm 1 are
E[1(p1;p2;x1;x2)] =
Z 1
0
Z 
0
p1g(x)f()dxd (2)
=
Z 
0

p1
Z x
0
0d + p1
Z 1
x

2

 
2x
2

32d

dx(3)
= p1

x3   3x2 + 3x

0 (4)
= p1(3   32 + 3) (5)
3and similarly for rm 2
E[2(p2;p2;x1;x2)] = p2

x3   3x2 + 3x
1
 (6)
= p2(1   (3   32 + 3)) (7)
This problem is quite dierent from those considered previously in the lit-
erature in that clearly more than the mean of the consumer distribution varies
with the uncertainty. As the following proposition shows the judicious choice
of functional forms made in specifying this problem does in fact give rise to a
unique equilibrium.
Proposition 1 The ice cream sellers on an asymmetric beach problem has a
unique subgame perfect location-then-price equilibrium.
Proof: See appendix.
The obvious way to establish this proposition is by working directly with
the specied payo functions and their derivatives. Naturally one wonders if
there are more general economic forces at work and if a more general result is
possible. As the next section shows the appropriate formulation of the problem
yields a very general existence result.
4 General State Space Results
Consider the more general setting in which the distribution of consumers over
locations, x, conditional on the value of a vector of parameters M is given by
g(xjM). The marginal density of M, that is, the uncertainty distribution, is
given by f(M) with support S. Without loss of generality assume E[M] = 0.
We analyze the pure-strategy sub-game-perfect Nash equilibria of this game.
Given the above assumptions the distribution of consumers, h(x), is given by:
h(x) =
Z
S
g(xjM)f(M)dM: (8)
For the following proposition it is useful to dene
J(x) 
H(x)(1   H(x))
h(x)
; (9)
where H() is the distribution function associated with the density function h().
Proposition 2 Assume that the distribution of consumers h() is log concave
with mean 0 and support [a;b]. If J(x) is strictly pseudo-concave and limx!a J(x) =
limx!b J(x) then there exists a unique subgame perfect location-then-price equi-
librium.
Proof: The proof establishes the equivalence of our game to another spatial
game with a known solution.
4Prices are state independent thus for a xed M rm i's prot, i, is i =
piQi; i = 1;2, where Qi is the demand for rm i. Since rms are risk neutral
their payos are given by the expectation over M of the state contingent prots.
Locations are also state independent implying  is independent of M. Hence,
we have
EM[Q1(p1;p2;x1;x2;M)] =
Z
S
Z (p1;p2;x1;x2)
 1
g(zjM)f(M)dzdM (10)
=
Z (p1;p2;x1;x2)
 1
Z
S
g(zjM)f(M)dMdz (11)
=
Z (p1;p2;x1;x2)
 1
h(z)dz; (12)
and a similar expression for rm 2. Thus EM[1] = p1H() and EM[2] =
p2(1   H()).
Thus the expected payos and strategies for each rm are the same as a
standard certainty location game with a consumer distribution given by h. Since
the two games are equivalent they will have the same equilibrium which is seen
to be unique by a direct application of Anderson et al. (1997, Proposition 2).
2
Corollary 1 If, in addition to the conditions of Proposition 2, h() is symmetric
then the unique sub-game perfect location-then-price equilibrium is:
 x
1 = x
2 =
3
4h(0)
; (13)
p
1 = p
2 =
3
2h(0)2: (14)
Proof: Proposition 2 and Anderson et al. (1997, Corollary 1). 2
If the density of consumers h(x) is symmetric and neither too concave nor too
convex there is a unique equilibrium in which the prices and the locations depend
only upon the density of the distribution at its mean. The standard spatial
competition model uses a continuum of consumers to aggregate out individual
level uncertainty about consumer locations from rm prot functions. Under
our state space approach rms know even less: rms must still make their
strategic decisions without knowing the locations of individual consumers but
in addition they are uncertain about the distribution of consumers as well.
However, because all decisions are made prior to the resolution of either source
of uncertainty all that matters from the perspective of a rm is the combined
eect of both forms of uncertainty on the density of consumers (at the marginal
location).
5Appendix
Proof of Proposition 1
From Proposition 2 it suces to that h is log concave and that J is strictly
pseudo concave with symmetric limits for its tails. First calculating h:
h(x) =
Z 1
x
p1

2

 
2x
2

32d (15)
= 3x2   6x + 3 (16)
and
H(x) = x3 + 3x2 + 3x (17)
Hence
@2ln(3x2   6x + 3)
@x2 =  
2
(x   1)2: (18)
Which establishes log concavity.
Now for the ice cream sellers problem it can be shown that
J(x) =
H(x)(1   H(x))
h(x)
(19)
=
(x3 + 3x2 + 3x)(1   (x3 + 3x2 + 3x))
3x2   6x + 3
(20)
=  x(x   1)(x2   3x + 3)=3 (21)
Since this function is continuos the limits can be found by simple substitution
J(0) = 0 = J(1): (22)
Finally strict pseudo concavity is easily established since the J function in
this case is in fact concave:
@2( x(x   1)(x2   3x + 3)=3)
@x2 =  4(x   1)2 < 0 (23)
QED
References
Anderson, S. P., Goeree, J. K., and R. Ramer, 1997, Location, location, location,
Journal of Economic Theory 77, 102{127.
Casado-Izaga, F. J., 2000, Location decisions: the role of uncertainty about consumer
tastes, Journal of Economics 71, 31{46.
De Palma, A., Ginsburgh, V., Papageorgiou, Y. Y., and J.-F. Thisse, 1985, The princi-
ple of minimum dierentiation holds under sucient heterogeneity, Econometrica
653, 767{781.
Harter, J., 1996, Hotelling's competition with demand location uncertainty, Interna-
tional Journal of Industrial Organization 15, 327{334.
Jovanovic, B., 1981, Entry with private information, Bell Journal of Economics 12,
649{660.
Meagher, K. J., and K. Zauner, 2004, Product Dierentiation and Location Decisions
Under Demand Uncertainty, Journal of Economic Theory 117(2), pp. 201-216.
Meagher, K. J., and K. Zauner, 2005, Location-then-Price Competition with Uncertain
Consumer Tastes, Economic Theory 25(4), pp. 799-818.
7

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