European Economic Review 11 (1978) 17-35. 0 North-Holland
Publishing Company
DUOPOLY PRICIFKi AND WAITING LINES
David LEVHARI
The Hebrew University, Jerusalem.
Israel
Israel LUSKI
Ben Gurion University, Beer Sheba, Israel
Received September 1977, final version received January 1978
.
The paper deals with a model of duopoly pricing in the context of firms providing services to
consumers. Each of the firms has a waiting line of oustomers arriving randomly. The service
provided by both firms is identical and the service time ‘of both firms is assumed to obey the
same distribution. Different consumers have different tirfje cos*t and have to decide whether or
not to join one of the lines. It is shown that the Court
-Nas ,‘ equilibrium is such that the two
firms charge in general different prices. One of the fir= 1 qkalizes
in servicing individuals with
high cost of time (and the other the rest). Moreover, some examples of nun-existence of
Cournot-Nash equilibrium and permanent oscillations of prices are shown.
1. Introduction
In the following paper we deal with a model of duopoly in the context of
firms providing services to consumers. Each of the firms has a waiting line of
customers arriving randomly. The service provided by both firms is identical,
and it is assumed that in boLh firms the service time is distributed
exponentially with a parameter u. Different consumers have different time
costs and have to decide whether or not to join one of the lines. The
individual will not join either of the lines if it turns out that the net benefit of
waiting in line and obtaining the service is negative. If the net benefit is
positive he will join the line with the higher expected net benefit.
The individual decision on whether to join the line, and which line to join,
is based on an ex-ante net expected benefit. That is, the individual calculates
the expected waiting time in a particular service firm without possessing
knowledge of the waiting line at a particular moment but with a knowledge
of the probability distribution 0; the waiting time. The assumption is that
there is no switching.’ In his long-run behaviour, the individual is choosing a
firm based on his experience about the waiting time.
For simplicity the costs of providing the services are ignored. Each of the
firms assumes that the price of the service of the other firm is given. In this
‘This situation occurs approximately in computer
substantial cost involved in switching and searching.
services and in all cases that there is
18
D. Lethari
and I. Lush.
Duopoly pricirtg
context the question is what is the price equilibrium configuration, i.e., are
the prices charged by the two firms necessarily identical? The equilibrium
under &scussioil is of course a Cournot type equilibrium in the prices or
Nash equi!ibrium
in prices - that is, the equilibrium price is such that it does
not pay each of the firms to alter its price (unless the other firm alters its
price1.
IEach of the firms has a reaction curve giving its maximum profit price as a
function of the other firm price. The Cob-trnot-Nash equilibrium prices are
given by the intersection of the two reaction curves.
The model in its structure has some similarities to the classical model of
Wotelling (1929) in discussing spatial equilibrium. It is shown that the
Cournot---Nash equilibrium is such that the two firms charge, in general,
different prices. One of the lirms specializes in individuals with high cost of
lime. Moreover, in the examples we have, it turns out.that by charging thz
same price the firm obtains a local minimum of its profits. Thus both firms
have incentives of abandoning the equal-prices set-up.’
It is not hard to produce regular examples in which the assumption that
each of the firms seeks maximum global profits using the other firm price as
a parameter would lead to non-existence; of any Cournot-Nash equilibrium.
l%is may seem at the first instance surprising to someone familiar with the
general existence, theorems of Nash equilibrium points. However, one should
notice that in the tiresent case the assumption of continuity of the reaction
curves is not fulfilled (along the line of price equality of both firms, the
reaction functions may be discontinuous). ‘As the two firms are assumed to
be similar in all respects, if it turns out that one of the firms obtains lower
profits. in a proposed price configuration. it may attempt to change the roles
with its competitor. We may thus have ’ permanent oscillation of both firms’
$
prices. On the other hand, we have in all these examples the existence of
local maximum profits for both firms in which it does not pay for any of the
f’s0 firms to change marginally its price (without a shift in the other firm’s
priwj. Thus, for both firms, the situation is either of tacit agreement in the
Coumoz-Nash equilibrium or at csrtain times of “warfare’ where each of the
two firms tries to occupy the position of the one with higher profits.
Besides Luski’s corthcoming paper, the present paper is one of the first to
deal with duopolists in the context of waiting lines of customers.”
‘Thus if we identify the ‘qualit,’ of the service with the length of the waiting line it is seen in
the cxampks brought that in general the duopolists will supply a d#iventiatud product Just in the
se that all arriving customers are being (eventually) served it is true that the duopolists will
an undiflerentiated product.
ur attention has been drawn to the fact that recently Gabszewicz and f’hisse have written a
r Q1977) dealing Hith product quality ar;d product differentiation under duopoly. While
u-ith somewhat similar problems their paper is not in the context of waitkng lines and
he results are not comparable.
D. Levhri
nnd 1. Luski, Duopoly
pricing
19
It is assumed that customers arrive randomly in the firm’s service facility.
The service duration is also assumed to be random. In both, we follow the,
usual assu_nrptions of simple queueing models.
We compare the Cournot-Nash
prices arrived at, with the prices determined by a monopolist operating both service stations. These prices are
also compared with the welfare-maximizing
prices that would ha? *: been
determined by a regulatory agency. As intuitively expected, the monopolist
prices are higher than those determined
in the Cournot-Nash
duopoly
equilibr drn, and the duopo!tsts are still higher than those determmed by
welfare maximization. It should be emphasized that in spite of the fact that
costs are ignored, the welfare-maximizing
prices are positive due to the fact
that joinmg a waiting line has an external effect on all those who later join
that line. These waiting lines have to be taken into account in the
determination of welfare-maximizing prices.
2. The model
Arrivals of potential customers is assunacL”, to follow the usual assumption
of a Poisson distribution with a mean rate af arrivals of A per unit of time.
By an appropriate choice of time units 2 can be assumed to be 1. The length
of service for each customer in horh fi~-ms is assumed to follow an
exponential distribution with a parameter u. Thus the mean service time per
customer m both is 1,/t’.
Let A1 be the rate of arrivals of customers requesting service from the first
firm and & f,-om the second. Hence i, --A, -& = 1 -I1 -JwZ is the rate of
potential customers deciding not to join any of the lines because their time
cost is such that when combined with the service price it does not yield a
positive net benefit for them.
Let W;: be the expected waiting time in the ith firm (w includes both
waiting in line and service time).
Our assumption is the conventional
Queueing Theory model M/M/l.
Using well-known results we obtain [see Saaty (1959, p. 347)]
(1)
iime costs. We assume that the
con su triers may have different
time costs of the coilsumer follow a known distribution function.
If we denote the time cost of a particular individual by C, then f’(C) is the
probability density of individuals with this cost [F(C) is the distribution
function of time cost, i.e., F(C) is the proportion
of consumers poss0sing
time costs t>f C or less].
The benefit from the service in either of the firms is assumed to be R
(measured in dollars) for all customers. Thus, the net benefit of a customer
Different
a 1ternative
D. Lshari
20
and I. Luski, Duopoly pricing
serviced by the ith firm (i = “_,2) is
(2)
Bj=R-CWi-Pi,
where Bi denotes the net benefit and Pi is the service charge of the E’thfirm.
Without loss of generality we assume that PI 29,.
The individual decides on which line to j& according to the higher net
benefit. Thus, if
R-CW,-P,>R-Cw,-P,,
(3)
he prefers the first firm. If also R1 >O, i.e.,
R-CW,
-PI A,
(4)
then he will join the first line. If, on the other hand, (3) holds but B, ~0,
then he would not join any of the lines. If
R-CW,
-PI <R-CW,-
&en he joins the second line.
For a customer to join
,
(5)
the first line we must have rsec Luski
(forthcoming J-J
(6)
For his joining the second line
(7)
For balking (or leaving without service)
K-P,
c>-,--.
I
(W
D. Levhari
und 1. Luski, Duopoly
pricing
21
As he obtains higher benefits from the second firm, then B2 >O. Thus, if hle
joins the second (and longer line), it is not needed to check that his net
benefit is positive.
Using our convention that R= 1, we find from (6) that the proportion of
customers turning to the high-priced, short-line firm is
(9)
The proportion of customers turning to the second firm is [using (7)]
&=F
( >
PI
42
w
2-
w
.
(10)
1
is the proportion of c’tftctomers with a time cost low
enough to decide to stay in any of the Lhes and obtain service. F( (PI
- P2)/( W2- WI)) is the proportion of customers with a time cost low enough
for them to join the firm with long lines. I- F( (R -P, )/WI) is the proportion
of individuals deciding not to join any of the lines due to their high cost of
time.
The expected waiting times in both firms is
FW-WWI)
and
w2=-
1
24~’
(12)
Given a set of prices (al, p2), (9), (lo), (ll), (12) would yield the
parameters I. 11 R2, i-VI, W2. Notice that in the case PI =P2, (PI -P2)/(W2
- W,) is not well defined. In this case the two firms are identical in all
respects,
Ignoring costs, the aim of the ifh~firm is to maxirrke ib per unit of time
expected profit
D. Lcvlzuri and I. Lmki,
22
Duopoly pricing
We assume that each of the firms takes the other firm’s price as a given
parameter and maximizes its p:-ofits with respect to its price. We thus assume
a Coumot-type
behaviour in prices and search for Nash equilibrium points
in which it would pay neither of the firms to change their prices. For this we
use the reaction curves of the first firm for the price of the second firm, and
that of the second firm for the price of the first firm, and we look for ‘stable’
equilibrium points. Moreover, we wish to investigate the profits under the
condition that both urms have to charge the same price.
As eqs. (9), (lo), (11), (12) are hard to solve ; qalytically, to analyse the
equilibrium
we shall use numerical analysis with simple distribution
functions.
3. Numerical analysis of f ‘re reaction functions and equilibrium points
We assume that the time costs of individuals is uniformly distributed over
the range LO, A]. Thus, the density function in this range is l/A. In this
special case eq. (9) assumes the form
1 P,-Pz
-_----1
A W2 - WI A’
R-P,
2, =-w
[assuming (R -PI )/WI <A],” and (10) is
c
2, =
P,--P2
1
wz-w,
2’
(15)
*For given P,, P,, eqs. (1 I), (12), (14) and (15) yield
equations: one for ;iI and one for &. The somtion for AI is
il =
two quadratic
b- (b2 -4ac)g
WI
’
2a
I’,--P2
A
a=
1’
“If (R -PI )/WI > A. then
)_I-I--I
P,--Pz
1
wt--WI
cl-
SEiementary considerations show that for ;.z we have to use the positive root, while for A1 the
tive root.
D. Lechari and 1 Luski, Duopoly pricing
R-P*
-+ A
l
P&P2
_P1--P2
A
A
23
1u2,
and
b’ + (b”- ---._-_
+ 4dd)f
9
211’
i “2 =_----
(17)
where
R-P,
P, --I’?
0’=2+--7--+-_A.
R-P,
A
b’=___-.
c’ =
p*
32
A
l’,
Y12.
Eqs. (16) and (17) give us A1 and IL2as functions of P, and P,. The ith firm
using the other firm price as given wishes to maximize
Max ~i(P1, Pz)=Max
'i
&(P1, P2)*Pi,
i=l,2.
'i
By differentiating the first-order condition for maximization is
For the particular casL:under discussion we get
(191
D. Levhmi
24
and 1. Luski, Duopoly pricing
where
+P,-P2
A2
Similarly, ve obtain for the second firm
Si.2
--__
ZP2
_i$(Sa’/dP,)-i,(i?b’/dP,)2a’& - b’
(&‘/iP,)
9
(20)
where
t?a’ _-.1
---_--_
8P1
A’
36’
_=O,
Zc’ -v2*
--_=-t?P2
A
dP2
Using the particular form of (19) and (20) in (18) we find the reaction
function of each of the firms to the other firm price.
We would like to emphasize that we use the first firm as the one with the
higher price. It is clear that if the first firm elects to charge a lower price
than the second, then their roles are interchanged and we should appropriately change our calculation of the reaction functions. To avoid confusion
we may denote the firms by a and /?. If P, > P,, then P, = P, and P, = P,,
where P, is a solution of the first-order condition,
if Pz c Ps, then Pa = P, where P, is a solution of
Az(P,,
p2 ) + p,
3*2 VI,
_-_-=()o.
;p2
p2 )
It turns out that we obtain two kinds of reaction functions. One of them is
for the case in which none of the consumers decides to leave without service.
In this case the reaction functions always intersect on the 45” line as in fig. 1.
We have a stable equihbrium for both firms in which both charge the same
price. In such a ease, both firms are identical in all respects.
D. Levhari
IL5
\
pricing
25
0’
’
2
I. Luski, Duopoly
0
\
2
and
3
’
’
*
4
’
5
-
pa
6
Fig. 1. The reaction curves of the two firms for v =2.0, R = 30, and a uniform distribution with
A=lO.
e
In the case, which is probably the more relevant and realistic one, where
some potential consumers decide to abandon without being served, it turns
out that the two firms have different reaction curves with discontinuity (on
the 45” line) as shown in fig. 2. It can be seen in this figure that for each of
the firms there is some range of prices in which we obtain two local profitmaximizing prices of firm ocfor a given price of firm 8. One of these prices is
below the 45* line (i.e., P, >P,) and one above the 4S” line (i.e., P, < Pp). It
should be n&iced that there is no ‘stable’ intersection of the two react,ion
lines on the 45” line? The Nash equilibrium point will never be such that a
(‘The proof that the reaction of a to Ps is discontinuous on the 45” line is based on the fact
that as P, = Pz, &(P, P)/W, <&,(I’, P)/dP, [see Luski (forthcoming)]. Thus if for some Pa
the solution of &cl(P I, P)/S, =0 is PI =f@, the solution of dn,(PB, P2)/o”P, =O is such that .P2
<P,. ‘Thus, in fig. 2, the cu’ve MN gives the solution PI (P,) of dA,(P1, P,)/CP, =O. This allso
holds at IV. As &r,(P, P )/JP, < c’n, (P, P)/SP,, we find at N, &r, (P, P)/dP, < 0, and hence INe
have a local maximization for cr to the left of IV.This proves the discontinuity of reaction cur~~-s.
D. Levhari avid I. Lusbci, L\uopoly pricing
?6
Pa (Pp)
OM - Global
LM - Loco1
3
Maximum
/
M
I
I
1
12
I
i3
I
’ Pa
14
-0.6, R=30, ana a uniform distribution
Fig. 2. The reaction curves of the two I’armsfor V-A= 10.
with
and fl charge the same price. One firm specializes in cdnsumers of high time
cost and the other with the rest.’
Let us assume that the firms are just interested in local maximization of
their
profits with the other firm price taken as parameter. Firms take
sequential ‘local’ price changes and find one of two ‘stable’ local equilibria
prices. The two are symmetrical with respect to the 45” line. The two pairs of
symmetrical prices give the prices for both without identifying which firm
~tilJ charge P, and which P,. Thus, one firm serves the consumer with high
time cost and charges high services charges while thr: other served the rest
and charges lower prices.
If we tet each of the firms search their optimal price on the whole range of
ibfe prices so as to get the global maximum the situation in the present
ple might be different. Thus, in the example with uniform distribution of
-Si
r results are obtained in Luski (forthcoming).
D. Levhari and I. Luski, DUO~O~Ypricing
27
time cost and the parameters of fig. 2 (v =0.6, R =30,A = 10)it turns out
that the global profit-maximization reaction curves (the bold lines) have a
discontinuity in general not on the 45O line (see table 2 in the appendix). In
this case the reaction lines (for our particular example) have no intersection
points. Thus, in this case there is no stable Cournot-Nash price equilibrium.
Starting with a particular price for one of the firms we obtain a non-convergent oscillation under and over the 45’. Thus with the parameters of fig. 2
a particular oscillation is given in table 1. In this table it is assumed that
each period just one of the firms react to the price change of the other in the
last period.
Table 1
Period
Yl
P8
1
2
3
4
5
6
7
13.85
12.32
12.32
12.89
12.89
13.85
13.85
12.“U
i 2.w
11 -9.:
13.85
12.32
12.32
12.89
The non-existence of Cournot-Nash equilibrium is not a general result but
is due to the particular shape of time cost distribution with a particular
parameter. With a different type of time cost distribution the reaction curves
for global profit maximization do possess two intersection points (just as the
local in t1g. 2). Thus, for example, numerical analysis of eqs. (9), (lo), (11)
and (12) using a Pareto distribution of time costs,
F(C)=l-C&cBY
with
y=1.8
and
CO=l,
(21)
shows us (fig. 3) that we do get two stable equilibrium points.
The first firm (the firm with the higher price) profits in our particular
example with Pareto distribution of time costs is 7.3 while the second firm
profit is 7.5. The profits are not the same and A:+ may be assumed that each of
the firms would like to occupy the position of the firm with the higher profit.
However, as each takes the other firm price as given, it would not look to
any of them optimal to deviate the equilibrium prices. Which of firms a and
/? would occupy 1 or 2 is not possible to infer. We may sometimes be in a
situation that each of the firms tries to undercut the other firm’s price so as to
occupy position 2. This will be a kind of ‘price war’ as may occur under
duopoly.
D. Lmhari
28
and I. Lush,
Duopoly pricing
3
20
_2+/yy
/
---.*-....
..............*.
/
i
i
/
Pa(Pp)
GM
i
1
19
i
f
Liq.
I
.i
G>/
0
/
ia
.
i
/
GM
/’
I. ’
.r
.-. .
0
/’
,r
/
.
Pa’Pp)
i
//
.’
.. ..
I
Pp(pal
LM......”
. .. .
GM - Global
LM - local
/
1
J
Maximum
I
I
Pa
-
19
20
Fig. 3. The reaction curves for u =OA, R = 30, and a Pareto distribution with y = 1.8 and Co -= 1.
In the examples we find that in equilibrium the two firms charge different
prices for their service. The reaction of rx to a high price quote of j? would
not be to charge a B,= P,.
To one’s surprise, we find that charging P,= P, is a local profit-minimizing
policy. Thus in our numerical anabysis we find that in the region where a has
two Iocal maximums (for a given Pp) P, =P, is a local minimum. With a
given PI 3 r has two local maximums (with P, c P, and P,> Pp)then raising
Pz from rts equilibrium left of the 45’ line profit is declining all the way to
the 45’. Decreasing Pzfrom its equilibrium right of the 45” line, profit is also
declining all the way to the 45’ line. Thus, P,= P, is a local minimum. This
is depicted in fig. 4.
Recapitulating this fact again, with given P, we have PL > P, where ijTt,/‘iTP,
=O, and Pi<Pp where &.JP, =O? From the second-order condition for
maximization, c"nJJP,
> 0 for all Pz such that P, 5 P, < PA, and Jn,/iJP, c 0
for all Pz such that P,”c Pas PD. Thus for P, = P,, (iix~/iiP,) >O and
t&E; P&O, where ‘+’ denotes the right-hand-side derivative and ‘-’ the
M-hand-side derivative.
4. Joint maximization and welfare maximization
One might be interested
in the relationship
QZ-- n,CP,P) in the region Pa > Ps and 71,= zz(P1,
between the Cournot-Nash
P2 ) if P, -c P,.
D. Levhari
and I. Luski, Duopoly
29
pricing
na
,608
.608
\
,804
.598
.596
,594
.592
3.59
,586
.584
I
14
*a
Fig, 4. The revenue of IXfor P, = 13, v=O.6, R = 30, and a uniform distribution with A = 10.
equilibrium points and the prices that would be charged by a monopolist
owning both service establishments, and the price charged by a regulatory
body wishing to maximize social welfare.
The monopolist wishes to maximize the joint profit
NY Pz)=ML
Pz.)+q(P1,
P,).
(22)
Thus, the monopolist is choosing PI and P2 to maximize n(P,, P2). Using
our previous examples with uniform time distribution (u =0.6, R = 30, A
C
D. Levhari and 1. Luski, DuopoQp pricing
30
= 10). we find that for :nax;,-nization of monopolist profit we should set P,
= 18.7 and P2 - 17.5. We did numerical analysis with numerous examples
s.nd consistently (as prob’ably expected) the prices under monopoly are
higher than those determined in a Cournot-Nash equilibrium (in the present
p1 = 13.780 and P2 = 12.846). Also in this case profit-maximizer would
charge different prices in the two service establishments and higher than
under duopoly.
Assume, on the other hand, that the service esta.blishment is controlled by
a regul&ory body wishing to maximize the social welfare. The aim of the
social agency is to maximize
-A* IV,
s
x--~‘.f(‘)
dc
p F-(x)-F(y)
j
@,y
-*2
2
’ --‘.f(‘)
s 0
dC,
(23)
F(Y)
where
x=(R-P,)/W,
As i,, =F(x)-+Q)
and A2=F(J),
-
l
and
W,
y=(&-P2)/(W2-WI).
(23) is reduced to
s;Cf(C)dC-
CV,j; cf(C)dC.
(24)
0 gives the net benefit to all consumers, i.e., the benefit from the service
obtained minus the cost of waiting to all individuals. The firm’s revenue is
just a transfer payment between individuals and the agency providing the
service and is hence ignored. In spite of the fact that cost has been neglected,
it should not be expected that the social optimal prices would be zero as we
have an external effect of joining the line for service which should be taken
into account. Again, in our example. numerical analysis shows that for v
=0.6, R = 30, and A = 10, B, = P1.f: and PZ = 8.6. Again, the prices are
different and lower than those in the Cournot-Nash equilibrium. The
C&mot-Nash equilibrium in numerous examples we have calculated is
between the monopoly and the social optimum prices.’ It is not clear,
however, whether one can prove a general theorem; the question is whether
a monopoly by charging higher prices would more than compensate for the
extemahty. However, in all our examples we do get that the monopoly prices
“Similarresults
were obtained by Naor (1969) for the one-server case.
D. Levhari and I. Luski, Duopoly pricmg
31
are above the social optimal prices and the Cournot-Nash equilibrium is
between them. What can be analytically proven is that the welfare-maximum
prices cannot be the same. That is, ir: the social optimal set-up A, #&.
To prove this, notice that we may assume that the social planner is
choosing x and y (where x > y), such that
4 =Wb-F(y),
(25)
AZ= F(y).
(26)
and
The prices PI and P, can then be determined from x = (R -PI )/WI and y
=(k-~,)l(w,-w,)~
The aim of the social planner is
e(x, y)=R(&
+&)-R,
W,c, -3,,W
cz,
(27)
where ei is the mean time cost of those who are served by the ith service
station. The first-order conditions are
(28)
and
Observe now that
D. Lmhari
32
and I. Luski, Duopoly pricing
Also notice that
s
x
y
‘f(‘)
F(x)-F(y)
&
’
and
Substituting in (29) one finds after cancellations,
If in the optimal solution ii1 =A2, then WI = W2 and 2 W,lAI, = 8 WI,&,
in that case (30) is reduced to
;
"1
ml
aif (y)(C,--x,)=0.
and
(31)
‘1
Fq. (31) implies C, = C, a contradiction as in any non-trivial
Therefore 2, #A2 implying P, #P, (in fact, by construction,
.
24, j.
set-up e, > cr.
P, > P2 and 3,,
D. Levhari and I. Luski, Duopoly pricing
33
Appendix
Table 2
The optimal price of firm a for a given Pi, v =0.6, R = 30, and a
uniform distribution with A = 10.”
11.000
11.300
11.600
11.900
12.000
12.200
12.400
12.6CXl
12.700
12.7-c’O
12.803
12.830
12.830
12.840
12.850
12.900
13.000
13.200
13.4cY)
13.500
13.600
13.800
14.000
13.908
13.905
13.897
13.884
13.878
13.863
13.844
13.820
13.805
13.797
13.788
13.785
13.783
13.781
13.780
13.769
13.747
13.692
13.617
13.567
-
3.392
3.418
3.445
3.474
3.483
3.504
3.525
3.547
3.558
3.564
3.570
3.573
3.574
3.575
3.576
3.582
3.594
3.620
3.647
3.662
-
-
-
11.746
11.800
11.922
12.039
12.156
12.215
12244
12..?73
*. .285
12.291
12.297
12.303
12.332
12.390
12.507
12.624
12.682
12.741
12.857
12.974
3.394
3.411
3.452
3.491
3, $30
3.550
3.560
3.569
3.573
3.575
3.577
3.579
3.589
3.609
3.648
3.687
3.707
3.726
3.766
3.805
aTlje dashes mean that there is no local maximum profit for prices
in that range.
Table 3
The optimal P, for a given P,,, t’=O.6, R = 30, and Pareto distribution with y = 1.8, C,, = 1.”
P, below P,
I’, above I’,{
--
p,
-18.000
18.300
18.600
18.630
18.720
18.750
18.900
19.000
19.660
19.648
19.630
19.620
19.620
19.620
19.612
19.601
7.214
7.258
7.304
7.309
7.323
7.328
7.352
7.367
17.870
18.000
18.138
18.158
18.193
18.217
18.289
18.328
7.134
7.204
8.282
7.290
7.317
7.321
7.361
7.384
34
01. Levhari and I. Luski, Duopoly pricing
l
Table 3 (sontinued)
P, below P,
P, above P,
-
19.200
19.500
19.570
19.620
19.650
19.800
19.830
aNote that
maximum.
-____-_
Cz
%
pa
‘II,
19.590
19.580
19.570
--
7.403
7.455
7.472
--
18.429
18.570
18.610
18.630
18.649
18.719
18.730
7.439
7.517
7.540
7.547
7.555
7.594
7.602
Pm= 19.620
and
P, = 18.630
is
a
stable
Table 4
Firm a’s profit as a function of Pz for a
given P, (Pa =13), u=O.6, R=30, and a
uniform distribution
with A = JO.
12.00
12.10
12.20
12.30
12.39
12.50
12.60
12.20
12.80
12.90
12.95
13.00
13.05
13.10
13.20
13.30
13.40
13.50
13.60
13.70
13.75
13.80
13.90
14.00
“Global maximum.
bLocal minimum.
‘Local maximum.
3.600
3.607
3.607
3.608
3.609”
3.608
3.606
3.603
3.598
3.592
3.588
3.584b
3.585
3.586
3.589
3.591
3.592
3.593
3.594
3.594
3.595”
3.594
3.594
3.593
global
D. Levhari and I. Luski, Duopoly pricing,
35
References
Gabszewicz, J. and J. Thisse, 1977, On product quality and duopoly, CORE Discussion Paper
.7708 (Center for Operations Research and Econometrics, Universitk Catholique, Louvain,
Belgium).
Hotelling, H., 1929, Stability in competition, The Economic Journal 39, 41-57.
Luski, I., forthcoming, On partial equilibrium in a queuerng system with two servers, The
Review of Economic Studies.
Naor, P., 1969, On the regulation of queues size by levying tolls, Econometrica 37, 15-24.
Saaty, T. L., 1959, Mathematical methods of operation research (McGraw-Hill, New York).