The Review of Economic Studies Ltd.
Non-existence of Equilibrium for the Two-dimensional Three-firms Location Problem
Author(s): A. Shaked
Source: The Review of Economic Studies, Vol. 42, No. 1 (Jan., 1975), pp. 51-56
Published by: The Review of Economic Studies Ltd.
Stable URL: http://www.jstor.org/stable/2296818
Accessed: 15/09/2008 07:29
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/action/showPublisher?publisherCode=resl.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the
scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that
promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].
The Review of Economic Studies Ltd. is collaborating with JSTOR to digitize, preserve and extend access to
The Review of Economic Studies.
http://www.jstor.org
Non-existenceof
Equilibrium
for
the
Three-rms
Two-dimensional
Location
Problemn
A. SHAKED
Nuffield College, Oxford
Given a distributionof customersand a location of K firmsin the plane, assumethat each
customerbuys one unit fromthe nearestfirmand that the revenueof a firmis proportional
to the numberof customersit has. Underlyingthese assumptionsis the assumptionthat
the customerbears the transportcosts and that these are determinedby a monotonic
functionof the distance.
A locationof K firmsis an equilibriumif no firmcan changeits locationso as to increase
its revenuewhile the other K-1 firmsare kept fixed (Nash equilibrium).
B. C. Eaton and R. G. Lipsey[1] conjecturethatno suchequilibriumexistsfor K > 3.
This note proves that the conjectureis true for K = 3, for a continuousand connected
customerdistribution.
Assumptions
f(x, y) or g(p, 0) will denotethe distributionof customers.
(A.1) f # 0 is continuous all over the plane or in a domain whose boundarydoes not
containany linearsegments.
(A.2) Everytwo points for whichf # 0 can be connectedby a " thick " curvefor whose
pointsf # 0 (connectedness).
1. THE STRANGE NATURE OF EQUILIBRIUM
Consideran equilibriumlocation of the three firms. By the connectednessassumption,
the threefirmscannotbe locatedon a line, nor can two of them be " paired" at one point
becauseit will clearlypay a borderfirmor the non-"paired" one to move in the direction
of the others.
Suppose,therefore,that the threefirmsare located at the points A, B, C. The plane
is, then, divided into three regions by three lines m, n, p intersecting at 0- the centre of the
circleZ circumscribingABC (Figure1).
Note that the angle e = - c, betweenm, n does not change as A moves along Z
betweenB and C.
Keep B, C fixedand allow A to move along Z such that the new separatinglines will
be m'n' (Figure 1). By doing this A loses the revenueM of the area betweenm, m' and
gainsN. Assumingwe startedat an equilibrium:N ? M. Now, let A returnto its original
location and allow C to move along Z so that it loses the revenueN and gains P, and
repeatthe same experimentwith B. Summarizingthe resultswe get:
M?_ P < N < M,
First versionreceivedJune 1973; final versionacceptedFebruary1974 (Eds.).
51
52
REVIEW OF ECONOMIC STUDIES
nA
n
Na
pt
p
p
FIGURE 1
whichimplies:
M = P = N.
That is, A will maintainits maximalrevenuelevel by movingalong Z (keepingB, C fixed
and not allowingA to cross over B or C). The same holds, of course, for B, C. (No
continuityof the distributionfunctionis needed.)
Note that A, B, C can all move a fixed distancein one directionalong Z and thereby
keep theirrevenuesunchangedbut the new locationis not necessarilya Nash equilibrium.
Using the polar representationof the densityfunctiong(p, 0) with 0 as the originand
p as the pole, A's revenueis (takingthe revenueto be the numberof customers):
rA+8
F()=
j
A
'0x
J pg(p, O)dpd0.
...
(1)
o
2
2'
whereA denotesthe anglebetweenm andp as m moves with A.
SinceF(A)is a constantfunctionfor this rangeof A:
00
F'(A) =
00
pg(p, )1+8)dpo
{
pg(p, )Z)dp = 0.
... (2)
Jo
This holds also for a customerdistributionfunction definedand continuouson part of
the plane only.
SHAKED
53
LOCATIONEQUILIBRIUM
m
C
B
(-a,o)
(a,
)
P
FIGURE
2
2. MOVING A OUTSIDE Z
Let BC be an x axis andp a y axis for the coordinatesystemin Figure2.
Let B = (a, 0), C = (-a, 0) and A = (c, d).
The line separatingA and B is:
(a -c)
d2 c2_-a2
2 d+ ( d)x
y(x, c, d) = y(x) = +
2
2d
d
... (3)
and the line separatingA and C is:
y(x, c, d) = y(x) =
A's revenueis:
F(c, d) =
0
d c2- a2
+
2
2d
-
.- oo.(x)
00
C-X
=
-F
dx +
df(x, y(x)) c
... (4)
('0 r0o
(00
f(x, y)dydx +
Jo Jy(x)
+c)(4
x.
d
_(a
50
... (5)
f(x, y)dydx
f(x,
(x))
dx
dX.
.
... (6)
Since, by the argumentof the first section,A maintainsits maximallevel of revenues
for all points of Z betweenB, C: FC 0 for all (c, d) on that part of Z. Transferring
equation (6) to polar coordinateswith origin 0 and a pole parallelto BC (Substituting
x = dp/AB, x -dp/AC
in the two integrals respectively):
cJ'00 (pcf)d
ABo 9( 2 :)p
AB 0 P9P2-d
in
+
C
7
pg
g
+
dp+ pAC
pg
g
+y
dp=O,
,,(7)
54
REVIEWOF ECONOMICSTUDIES
whereAB is the length of the segmentAB and 7z/2+y = (7r/2-f)+s.
(s= =-oe,
is the
angle between m, n).
Denote:
00
pg(p, O)dp
T
.(8)
r(0)=
T
g(p, 0)dp
using that,
AB = 2r sin y
AC = 2r sin ,B
C = Cos
=
r
2
00
(r = radius of Z), and that by equation (2):
pg(p, 0)dp is a periodicfunction of
e
0
with period s, we get:
sinoc
+
sinf
(2
siny(9
2
)
)
To get this we dividedby f pgdp but this cannot be zero as this impliesthat A's portion
of the plane is disconnectedfrom B and C's portionsby two lines with no customerson
them. This violates (A.2) unless A's portion contains no customersat all in which case
this locationcannotbe an equilibriumcontraryto our assumption.
Since equation (9) holds for every position of A betweenB and C, reformulateit,
taking as a parameterthe angle 0 between the right hand separatingline and the axis
through0:
(7 2
sin oc
cosO
r
2
'
_
)
cos (7-oc+0)
r(0)
(10)
r(7r-oc+0)
for
cc-
7r<0<7r
??-
-
2
2
As similarformulascan be writtenfor every pair of separatinglines, we have the
followingthreeequations:
sin oc
cos 0
r
r(0)
sin y
r
cos (7r-oc + 0)
(7r-cO+ 0)
cos (B-oc+ 0) _
cos (7r-oc-y+,+0)
r(7r-oe+0)
0
sin
_
B_
cos(,B-y+0)
r(27r-oc-y+0)
r(27r-oc-y+0)
_
cos
(7r-y
+ 0)
r(27r+0)
...(11)
LOCATIONEQUILIBRIUM
SHAKED
55
Note that the last two equationsdo not have the symmetrypropertiesof the firstbecause
they are modifiedto measure0 startingfrom the axis parallelto BC.
Solvingfor r(0) (for completesolution see appendix):
r(0) = r sin (0+13)
cos y
...(12)
for
a- -
2
?0<
=
- -,2
solvingr(0) for the rays betweenA and C (by movingC and takingan axis paralelto AB):
0)r sin (D+ao)...(3
cos 13
(0 is measuredfrom a line parallelto BC whereas0 is measuredfrom a line parallelto
AB).
These two functionscoincidefor points betweenA and C.
Set:
(2
0oo-+p
)
+
+i3=?fl
0o
\ =y-)+y2,
2
thesewill describea pointhalf-waybetweenA and C. Equalityof the two functionsimplies:
but a +, + y =
it,
sin (y+f)
sin (x+fl+y)
cosy
cosfl
hence:
y+fl=0
or y+f=27.
In eithercase, A, B, C are on a line and hence not an equilibrium. This completesthe
proof for xc = 3.
3. COMMENTS
1. If the firmsbearthe transportcosts andthe revenuefroma givencustomeris a decreasing
function R(d) of his distancefrom the firm, the proof is not valid. I suspect that the
existenceof equilibriumdepends on the relationbetween the customerdistributionand
R(d). If this last function is rapidlydecreasingthere could be an equilibriumwith the
firms well apart: take for examplethe extremecase where a firm has positive revenue
from customerswithina givenradiusonly.
2. Though the proof presentssome interestingpropertiesof a possible equilibrium,
its main failureis that it cannotbe extendedto Kc>3.
REFERENCE
[1] Eaton, B. C. and Lipsey, R. G. " The Principleof MinimumDifferentiationReconsidered: Some New Developmentsin Theory of Spatial Competition", Review
of Economic Studies (this issue).
56
REVIEW OF ECONOMIC STUDIES
APPENDIX
Substitute oc= 7r-o,
1 = 7r-fl,
5 = 7r-y
(a) 1 [sin o cos (oc-,+ 0) cos (-+
r
in equation (13) and eliminate r(O):
0) + sin J cos (c+ 0) cos (0-213)
+ sin
I
cos (a + 0) cos (j-f+
0)]
(10)[cos 0 cos (-fl + 0) cos ( + fl+ 0)-cos (y + 0) cos ( + 0) cos (0-2fl)].
r(O)
All the next equations are obtained by using the formulas for
-
or (cosA+cosB)
(sinA+sinB)
= 27r.
and the fact that a +
Denote the coefficient of l/r(0) by Q:
(b) Q=
Cos 0
[cos (&+ -2f+20)+cos(&-j)]
2
cos (O- 2f) [cos (o + + 20) + cos (oc-7)]
2
or,
(
(c) Q =
24
[cos 0-cos (O-2fl)] +%[cos (E+ -2,f+0)-cos
(E+ 0++2f)],
(d) Q = sin fl sin (0-iJ)[cos ,B-cos (-y)]
sin(0-fl)
-2sinocsin.lsin
Denote the coefficient of llr by R.
=
(e) R =
+
2 [cos (E+3-2fl+20)+cos
2
i2 [cos (oc-2fl+20)+cos
+ 2l [cos (ot+
(o-3)]
(oc+2fl)]
fl + 20)+ cos (oc+ fl-y)
Let R' be the sum of the elements of R free of 0:
(f) R' - '[sin (20c- ) + sin y + sin (2/7- ) + sin + sin 3y -sin y].
(g) R' - '[sin 2y cos f-sin 2y cos (oc-,)]
=
-sin 23 sin o sin ,l.
The rest of R: R"- all the expressions with 0 (to simplify denote ,u = 0- ):
(h) R" = sin (2,t + oc-, ) + sin (Fe+ J-42t)+ sin (2,u+ Fe+ ,) + sin (,B-2,u-oi)
+ sin (2,u+ j) + sin (y-2pu)
Hence,
-sin 23 sin ocsin fl -2 sin ocsin , sin y sin (0-13)
r
r(0)
r sin (0-/TJ)
rrsin (0+13)
(j) r(O)==
cosy
cosy
QED
0.