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Strategic market coverage in spatial competition
International
Journal
of Industrial
Organization
11 (1993) 299-326.
North-Holland
Strategic market coverage in spatial competition Marcel Boyer Dkpartement de Sciences Economiques, CRDE and CRT, Universitt de Mont&al, C.P.6I28, Montrkal, Quebec H3C 3J7, Canada
Michel Moreaux* GREMAQ, CNRS URA 947, Universitt? de Toulouse I, Place Anatole-France, 31042, Toulouse Cedex, France Final version
received
December
1991
We consider in this paper competition through prices and market coverages in a spatial duopoly. We characterize and compare the Stackelberg equilibria and the Bertrand equilibria. We show that when the cost of informing the consumers is small, duopolists always choose to inform all consumers in the Bertrand equilibrium but the leader in the Stackelberg equilibrium may choose not to inform everyone when the duopolistic competition is intense enough (thereby generating overlapping markets). In equilibrium, prices may be strategic complements, strategic substitutes or be strategically independent. Considering market coverage as chosen by the firms changes the ‘conventional’ relationships between differentiation, profits and markups.
1. Introduction
In the d’Aspremont et al. (1979) reformulation of the canonical spatial competition model of Hotelling (1929),l the firms do not choose their market coverage, that is the percentage of consumers aware of their existence, and the market areas of the firms to not overlap.2 In all these Correspondence to: Marcel Boyer, Dkpartement de Sciences Economiques, CRDE and CRT, Universiti: de Montrkal, C.P.6128, Mont&al, Quebec H3C 357, Canada. *The authors wish to thank Claude d’Aspremont, CORE mathematical economics seminar participants, and two anonymous referees for their comments. We remain of course solely responsible for any errors or shortcomings. ‘See Gabszewicz and Thisse (1986) and Greenhut et al. (1987) for recent surveys of location theory. ‘Overlapping market areas, that is areas in which a given consumer may patronize a firm while the left-hand and the right-hand neighbors patronize another, can be obtained in different ways and for different reasons: (1) if competition is not a price competition but rather a Cournot-like quantity competition, as for example in the network-type model of Weskamp (1985) in which the overlapping property holds in the sense that for some nodes more than one firm have non-zero sales [see also Anderson and Neven (1990) or Hamilton et al. (1989)]; (2) if 0167-7187/93/$06.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
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M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
formulations, either the two sides of the market are perfectly informed or the imperfection of information does not play a crucial role in the determination of the equilibrium. For instance, in de Palma et al. (1985) the firms know the distribution of consumer tastes at each point of the space and it is sufficient to determine the price policies of the firms. More recently Gabszewicz et al. (1989) extending the work of Gabszewicz and Garella (1986) have relaxed the assumption of perfect information of both the consumers and the firms regarding the other side of the market. In their model the consumers know the price quoted by the nearest seller but not the price quoted by the other firms. Hence the partially uninformed consumers must decide whether they will bear the cost of getting additional information or not. They show that at symmetric equilibria the firms quote the highest prices which induce no search by any consumer: the respective markets do not overlap. With asymmetrically (and exogenously) located firms there exist price equilibria where some consumers will search for more information, but even so the markets do not overlap. Schultz and Stahl (1988) develop a model of consumer search over variety and price. Firms choose their location, variety and price and in equilibrium, more than one variety may be offered at one given location and a variety may be offered at more than one location. The consumer preferences are modeled so that more variety means increased demand: a consumer visits only one market place and therefore a visit is more desirable when more variants are offered at that market place. Each consumer is assumed to know, before searching, the location and size (number of varieties offered) of market places; he forms a (true) point forecast of the expected price charged at a given market place and also a forecast of the characteristics of the varieties.3 They consider and compare the perfect Nash equilibria of two market structures: a large number of potential one-product firms, each choosing in a three-stage game to enter or not (in stage 1; an entrant must incur an entry cost) and upon entry, a plant location (stage 2) and both a product variety and a price (stage 3); a one-plant monopolist deciding on varieties and prices. They show that if consumers are choosy enough, then all one-product firms locate at the same location (in any case, there always are two or more firms at each equilibrium location) and may charge prices in excess of that charged by the multi-product monopolist with the same set of
the goods marketed by each firm have some specific characteristics other than the sole location of the firm selling them, as for example in de Palma et al. (1985) [see also Anderson and de Palma (1988) and Anderson et al. (1989a)]; (3) finally if preferences between different location are in part ‘stochastic’, at least as observed by outsiders, then firms may sell to consumers over the whole interval [see Anderson et al. (1989bjl. %ince search costs are a function only of the distance to the marketplace, the overlapping of markets (not explicitly considered by Schultz and Stahl) would likely proceed from the same factors as in de Palma et al. (1985).
’
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
301
variants; they also provide conditions under which the latter offers as many varieties as the former: hence, monopoly may increase welfare.4 In a different context, Boyer et al. (1992a, b) provide an exhaustive analysis, in the space of prior beliefs and cost differentials, of the nature of Perfect Bayesian Intuitive and Divine Equilibria in a two-stage model of sequential location choices (stage 1) and simultaneous choices of delivered pricing schedules (stage 2) when the firms are asymmetrically informed in stage 1 about the production cost of their competitor;5 but again, in the production stage the markets are separated. Casual observation of markets reveals that they more often overlap than not; hence, it seems to us that the usual non-overlapping property of equilibria in most spatial competition models is a weak point of the theory. Moreover, we will allow firms to choose strategically their market coverage. We will assume that consumers are ‘born’ unaware of the prices charged by the firms (and even of their existence) and that they do not search but can become informed of the existence and price charged by a firm if they receive a message from that given firm. In such a framework, each firm chooses not only its price but also the number (proportion) of consumers it will inform. It is this viewpoint of imperfect information that we take in this paper. We suppose that given locations (determined in a preceding stage of the game not modeled here), the firms must inform the consumers about their respective location and price. Hence the firms compete in both prices and ‘market coverage’. We consider only the ‘case of a duopoly but we compare two forms of duopoly markets. In the first case the firms move simultaneously (a la Bertrand) while in the second case they move sequentially (a la Stackelberg) with one of the firms taking the leadership and the other accepting the followership. We assume that the cost of covering the market is quadratic and increasing in a parameter p, which can be let to vary to represent different degrees of technological efficiency in advertising. We show that in Bertrand equilibria, overlapping never occurs when the cost of informing the consumers is small (although it does when the cost of market coverage is high enough), whereas in Stackelberg equilibria the leader (but not the follower) may choose not to inform all the potential buyers of its product and price even if the cost of informing the consumers is zero, a decision which %rtuitively, those results may be understood as an application of the basic result that an mproduct monopolist will sell at lower prices than m one-product oligopolists when the products are complements; in the model of Schultz and Stahl, the products are both substitutes (since a consumer buys only one variety, if any) and complements (since each variety offered at a given location increases the global demand at that market place by making search more profitable, in expected terms). ‘More specifically, the first-mover in stage 1 knows its marginal production cost (equal either to c or c-a) when choosing its location, but the second-mover whose marginal cost is always c can only observe the location chosen by the first-mover firm.
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M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
gives rise to overlapping market areas. We also show that when the cost to the firms of informing the consumers is ‘small’, the Bertrand duopolists always choose a maximal market coverage but the leader in the Stackelberg equilibria will choose under the same cost conditions a partial market coverage when the level of duopolistic competition is ‘intense enough’, that is when the products are good substitutes. Moreover, considering the market coverage as endogenously chosen by the firm with a significant strategic component (made possible here by the Stackelberg structure of the market game) will change in important ways the equilibrium relationships between the level of product differentiation and profits and markups. The two traditional measures of increased product differentiation - ‘an increase in ‘transportation’ costs and a reduction in the reservation price (value) - have different impacts on profits and markups. We believe that the present model not only brings into the economic analysis of spatial (product design) competition an important strategic variable, the decision on market coverage, but also provides a richer analysis of the implications of product differentiation. Grossman and Shapiro (1984) have analyzed the symmetric n-firm Nash equilibria of a model where firms are symmetrically located around a circular city and choose both a price and a level of informative advertising. Their advertising technology is such that it is infinitely costly to inform all the consumers; hence markets always overlap. Although the focus of their paper is different, we can interpret their results in the context of spatial competition with overlapping markets. We will discuss their model and results in relation to our own model and results in the conclusion, The next section is devoted to introducing the model and section 3 presents the simultaneous move (Bertrand) equilibria when all consumers are aware of the existence and prices of both firms; we characterize in section 4 the best-reply function giving one firm’s optimal price as a function of the other firm’s price and partial market coverage level; the stage is then set for the characterization of the simultaneous move (Bertrand) equilibria with partial market coverage and of the leader-follower (Stackelberg) equilibria when firms choose both their price and their market coverage. We conclude by discussing the related literature on informative advertising and ‘rationing’. 2. The model We consider the following context adapted from Butters (1977). Assume a linear city with two firms located at the end points of the city: firm 1 at x=0 and firm 2 at x= 1. The consumers are distributed uniformly on the interval [0,11 and each one demands inelastically one unit of the homogeneous good if the cost is below a reservation price s. In addition to paying the price set by the firm, a consumer located at x1 must bear quadratic transportation
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
303
costs of txf if he buys from firm 1 and t( 1 -X1)’ if he buys from firm 2: the goods are differentiated by their locations only.6 Suppose that at XE(O, 1) the consumer is indifferent between buying form firm 1 or firm 2, that is p,+tXZ=P2+t(l-X)2~;S
or
x=(Pj_Pi+t)/(2t),
and suppose that all consumers are aware of the existence of both firms. Consumers to the left of x will buy from firm 1 and those to the right of x will buy from firm 2. In this particular case, the demand facing firm i may be written as (‘u’ stands for ‘aware of the existence of the other firm’): &I,, pz) =max{O,min (l,(pj--pi+ t)/(2t)}}. More generally, the residual demand function facing firm i given pj and given that all consumers are aware of both firms will depend also on the reservation price s since it may well be the case that for some location the price charged by one firm or both plus the quadratic transport cost exceeds the reservation price s. Moreover, if that is so in equilibrium, then some consumers may even not be served.7 Given pj, the demand to firm i cannot be negative or larger than 1; it cannot be larger than x = [(s - Pi)/t] ‘I2 , the solution to pi+ tx2 = s (the consumer at such a location x is just indifferent between buying the product form firm i and not buying at all), and it cannot be larger than (pi--pi+ t)/(2t) as seen above. Hence dl(p,,p2)=max10,min(P’-~+t,(~)L’i,
I]].
(2.1)
The inverse demand function, when it exists,8 can directly be written as pi(X, pj) =
max (0, min
{S -
tX2, pj + t - 2tX3).
(2.2)
The different cases can be illustrated as in fig. l(a)--(d), where f =((s-pJ/t)"'
6There are two main reasons for using quadratic transportation costs rather than linear ones. First, d’Aspremont et al. (1979) show that maximal product differentiation (the two firms located at the end points of a linear city, that is at x=0 and x= 1, respectively) results from quadratic transportation costs in a model of location choice; hence, although we consider here those endpoint locations as exogenous, they are compatible (in a subgame-perfect sense) with quadratic transportation costs and uniformly distributed consumers. Second, quadratic transportation costs will generate continuous residual demand functions and continuous and concave profit functions contrary to the case of linear transportation costs. ‘The reservation price s will play a role in our model with a leader-follower structure; since we will show that the leader may choose not to inform all consumers while the follower does inform everyone, for p suffkiently small, the profit of the follower (given that the leader is committed to its choice) would be unbounded if s is unbounded. Hence, we cannot simply assume that s is large enough that every consumer, aware of at least one firm, will buy in equilibrium. *No inverse function exists if d;( .) = 1 in (2.1). This case is not illustrated in fig. 2, it would appear for pj > t.
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
304
(b)
I
I
pj
P
-x
h
I
Cc)
Cd)
Pi-Pi+’
di=
2t
-A
: : : : B k;.
h f
I
I”
I
Fig. 1. The market
separation:
r
for ki= kj= 1.
and h = (pj-ppi + t)/2t. The demand itself is represented in fig. 2, where the value of x1 is obtained from pj+t(l -x)‘=s, that is x1 = 1 -((~--p~)/t)l/~ and the value of pi is obtained from j+= s- tx: =pj+ t -2tx1; replacing x1 by its “’ . It is interesting to note that the value, we obtain &=pjt +2t((s-pj)/t) demand function has a kink of Stigler type at (x1, j$). Hence the marginal revenue function MR(x) is discontinuous at x=x1. As pj increases, the kink moves to the right on the E(x) =s - tx2 curve since that curve is independent of pj while the C(x) = pj+ t - 2tx line moves up linearly with pj. Consider again fig. 2 and the demand (2.1). These were derived for a consumer aware of the existence of both firms. A consumer located at x and aware of the existence of firm i only demands one unit of good i if
hf. Boyer and M. Moreaux,
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in spatial competition
pi
/
_----
I I
x
112 x3
\
MR (xl
Fig. 2. The demand function: for t >s and pj < t.
pi+ LX’5s. Hence if all (N = 1) consumers are aware of firm i only, the demand and the inverse demand for good i would be given respectively by (2.2 stands for ‘unaware of the existence of the other firm’): dy(pJ=max
k,min{r+)“‘,
I)],
(2.3)
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M. Boyer and M. Moreaux,
Strategic market coverage
in spatial competition
pi(X)=IllaX{O,.S-tX*}.
(2.4)
If a proportion kjE [0, l] of consumers are aware of the existence of firm j and if those kj consumers are randomly located in the interval [O,l],’ then the residual demand facing firm i will be”’ di(Pi, Pj, kj) =k#?(~i, Pj) +(l -kj)C(pJ,
(2.5)
with d~(pi,pj) given by (2.1) and do given by (2.3). The profit function of firm i, assuming a zero cost of market coverage, can be written as ZiPi, ki; Pj, kj) =(Pt-c)kiddPi,
Pj, kj).
(2.6)
3. The Bertrand equilibria Let us first characterize the full awareness price reaction function, that is the profit-maximizing price pT as a function of pj assuming that ki = kj= 1. Let us assume that c=O. We can refer to fig. 2. Proposition
I.
The profit-maximizing (Pj +
p~(pj;S,t)=’
price pf(pj; s, t) is given by
if OS Pj 5 fij(& t),
t)/2,
~i(Pj;s,t)=pj-_+22t
112
( > ~
,
max{s-_,+s},
if Fj(% t) 5 Pj S pits, t),
if Pj(% t, 2 Pjl
(3.1)l
c
where
-55t+4(t*+tSy’*,
if e&t
0,
otherwise,
FjE
(3.2) l/2
)
1yss3t 0
(3.3)
otherwise. Proof.
All proofs are given in the appendix.
It can easily be checked that (3.1) is continuous. The first portion of (3.1) is linear and increasing with pj, the second portion is decreasing and concave 9We consider only the case where the consumers informed of the existence of a given firm are randomly located over [O,l]; hence the firms cannot aim their advertising effort at a specific consumer group, such as those closer to it; an interesting extension would be to look at information technologies for which the cost of informing a consumer increases with the distance separating the consumer and the firm. “‘There are four groups of consumers: kikj knowing both products, k,(l - kj) knowing only product i, (1- ki)kj knowing only j, and (1- k,)( 1- kj) knowing neither i nor j. The residual demand for product i is defined for ki = 1.
M. Bayer and M. Moreaux,
Fig. 3. The price reaction
Strategic market coverage in spatial competition
function
with full market
coverage
307
(s 5 3t).
in pj, and finally the third portion is constant either at s--t or at 5s. It is important to note that for Pj0; the slope of the linear portion of the reaction function (3.1) is i; therefore, Proposition
equilibrium.
2.
V(s, t),
there
exists
a
unique
symmetric
Bertrand
q
As t increases for a given s, the linear portion of the best-reply function moves up and fij(S, t) and pj(s, t) both increase with t if t
308
M. Boyer and M. Moreaux,
Strategic market coverage
in spatial competition
(b)
Pi *
Pj Pj
Pi
Pj
MiPj
Cc)
Cd)
Fig. 4. The Bertrand-Nash
equilibria.
decrease with t if t>$. We therefore can have four different configurations depending on the relative values of t and s, as represented in fig. 4(a)-(d). Case (a) will prevail if pT(fij)
M. Boyer and M. Moreaux,
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in spatial competition
309
Proposition 3. Assuming full awareness of the consumers, (a) for t &, there is a unique, symmetric and stable Bertrand equilibrium p’= t on the linear portion of the reaction functions; (b) for t E (%s,$s), there is a stable set of unstable equilibria - each element of the set is an unstable equilibrium but the set itself is locally stable (ergodic) - on the concave portion of the reaction functions, indicating that both firms are then at the kink of their respective demand functions; when intersecting, the reaction functions are in fact overlapping; (c) for t Z$s, we have a unique, symmetric and stable equilibrium on the constant portion of the reaction functions; in this case, not all consumers buy the goods and both firms are at their monopolistic equilibrium; (d) the symmetric equilibrium price, as a function of t, is first increasing with t up to a maximum of p’=$s, then decreases with t to a minimum of p’=$s, and stays at that value for larger values oft. 0 4. The characterization
of equilibria in the (price, market coverage) strategy
space Consider now the case where some consumers are unaware (1 - kj) of them. The residual demand facing firm i will be di(Pi, Pj, k, kj) = k[IkjdT(Pi, Pj) + (I -
kj)dY(PJl,
of firm j, say
(4.1)
where dKpi,pj) is given by (2.1) and dy(pi) is given by (2.3). In fig. 5 the demand function of the ‘aware’ is curve DA while the demand function of the ‘unaware’ is curve DU; the associated marginal revenue functions, with respect to price, are respectively MRDA(pi,pj) and ~vIRDU(~,).~’ It is important to recall that although demand curve DU is independent of pj, the position of demand curve DA depends on pj. We have:
if Pi~ElaX(O,Pj-t},
[ min{l,y}, 1
pj+t
~
DA:x=
- sPi3
2t
if max(0,pj-t}5pi~&
: s-p,
l/Z
c-1 t
’
1 0, “Both MRDA(p p,) and MRDU(pJ but the latter is discontinuous.
if fii 5 pi,
if sspi; may be discontinuous; in fig. 5 the former is continuous
M. Bayer and M. Moreaux, Strategic market coverage in spatial competition
310
2 --s 3 r’pj
+t:
Fig. 5. The demand function: for 4s < t x$x
io I
I
l/2
mini,:
DU:x=
,
s--p,
__
(
0,
t
if piSmaX{O,S-t},
112 ) if IIlaX{O,s-t}~pi~Ss, > if ssp,;
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
ifpismaX{O,pj-t},
~pj+t 2t
311
1
if max {O,pj- t> Qisfii,
TPi,
MRDA: x =
( (y)l’2- &P~(?)-“~,
ifjji-Cpi.s,
10,
if S?Zpi.
MRDU:x=<
ifmaX(O,S-t}spisS,
if sSpi.
0, For ki= 1, (4.1) can be rewritten as
/ di, (pi, pj, 1, kj) =
1,
ifp,>t
andp,spj-tt,
kjDA+(l-kj)l,
ifmax(O,pj-t}gpPigs-t,
kjDA + (1 - kj)DU,
ifs-tspig#i, 112
,
if fi;spisS, (4.2)
and for ki-c 1:
di(Pi,Pj, ki, kj) = kiddpi,Pjll, kj)*
(4.3)
It is clear from (4.3) that the best-reply price pi to a given (pj, kj) is independent of ki. Hence we can find the best-reply price for ki= 1. To do so, it will be convenient and useful to distinguish live cases given the nondifferentiability of demand functions DA and DU. Considering (4.2) and
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
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fig. 5,13 we observe that the best-reply axi/api =kjMRDA(pi,pj) +( 1 - kj)MRDU(pi) US define f(p,,pJ as follows:
price pi*(pj, kj) is obtained when =0.14 TO simplify notation, let
which will allow a simpler representation of MRDA(pi,pj) and MRDU(p,). Noting that the profit function ~i( .) is quasi-concave and recalling the definitions of pj> fii and pj from Proposition 1, we can characterize the price best-reply function with the help of the following five cases, each corresponding to a different form of dxi/api=O: Case I: 0 5 pf(pj, kj) < s-t.
This case appears if
CkjS(Pi, Pj)+(l_kj)11,,=,-,<0, that is, if 0 spj<2sby
(4.4)
t((2-t kj)/kj).
In this case, p:(pj, kj) is defined implicitly
[kj(y - ipi)+(I-kj)l]=O;
(4.5)
that is pr(pj, kj)=$(pj+
t) + 2
(4.6)
t. J
Case II: p:(pj, kj) = s - t. This case appears if ’ 5
that
Ckjf(Pi,pj)+(1_kj)lI,i=,-,~o,
(4.7)
Ckjf(Pi, Pj)+(l-kj)g(Pi)]pi=s-t~0,
(4.8)
is,
solving (4.7) and (4.8), if (2~- t[(2+kj)/kJ} In this case, pT(pj, kj) is given by
~pj~
( -3t/kj+
[(l + kj)/kj]s).
13Fig. 5 is drawn for pj=t giving an x-intercept at x=1 but if pj<[>]t. the intercept would be to the left (right) of x= 1. 14Mutatis mutandis at points of discontinuity, or non-differentiability of the demand functions. ISNote that for t
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(4.9)
pZ(pj,kj)=S-t.
Case III: s-t
Ckjf(pi,Pj)+(l-kj)g(pi)l,,=,-,>O, Ckjf(Pi, Pi) +(I
(4.10) (4.11)
-kj)g(pi)lpi=~i(pj)
that is, solving (4.10) and (4.11), if { -3t/kj+[(l +kj)/kj]s)
[kj(y
which after a few manipulations nomial in pi:16
&pi(~)-“‘]]=O;
can be rewritten as a third-degree
(4.12) poly-
(4.13)
B3p?+B2p~+Blpi+Bo=0, where B, =4k;, Bz = Skft-
18kjt +9t-4k;s+4kf-4kfpj,
B,=kj2t2-8kj2st+24kjSt-12st+2k3pjt+4kj2pjs+kj2pf,
Case IV: pF(pj, kj) = ii(
This case appears if
Ckjf(Pi, Pj) + (1 -kj)dpi)lp;=~~(pJ) 160ne can write (4.12) as follows: k,(!!.$ raising
_ +)=
-(l_~j)[(~~‘z_~pi~~)-l’z];
both sides to the power 2, we get k’(q-;p.>‘=(l-kj,‘[+fpi+$(&)].
A few more manipulations
lead to (4.13).
20,
(4.14)
314
M. Boyer and M. Moreaux, Strategic mqrket coverage in spatial competition
’ PI Fig. 6. The price reaction
function
with partial
market
coverage
(s = 2, t = 1).
Ckjf(Pi,Pj)+(l -kj)g(pi)lpi=Bi(pj)=g(pt)lPi=Bi(Pj)~O, that is, solving (4.14) and (4.15), if ~j(kj, S, t) spjs In this case pF(pj, kj) is, from (3.1), given by
[&--t+2(~t/3)“~]
p;(pj,kj)=pj-tt22t
Case V: I7 pi”(pj, kj) =&(pj)
t).
This case appears if
>
(4.17)
‘2
that is if [$s - t + 2(~/3)~“] = pj(s, t) spj. pT(pj,
=jj(s,
(4.16)
=$.
APi) Ipi
(4.15)
In this case, p2(pj,
kj)
is given by
kj) =$s.
(4.18)
Typical price reaction functions with partial market coverage are illustrated in fig. 6. It should be noted that in comparison with the price reaction “Note
that if s 2 3t, then [#s-t
+ 2(st/3)“‘]
> s and therefore
there is no Case V.
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315
function with fulZ market coverage (fig. 3), the boundary value for which prices cease to be strategic complements to become strategic substitutes is now pj rather than fiji, with pjzJj, and that a new interval appears (Case II) where prices are strategically independent. The conditions defining each case may be interpreted as follows: kj is the probability that a consumer knows firm j and f(pi,pj) is the marginal revenue realized by firm i on such a consumer; (1 - kj) is the probability that a consumer is unaware of firm j and the marginal revenue realized by firm i on the latter consumer is 1 if piss-t and is g(p,) otherwise. Therefore, the conditions may be interpreted as stating that pi is chosen so that the ‘expected marginal revenue’ is 0.l’ Assuming that tr$s, we know from Proposition 3 that the symmetric Bertrand full awareness equilibrium is on the linear portion of the price reaction functions as in case (a) of fig. 4. Moreover, since from Case I above we have P*(pj, kj= l)
4.
Observing (pj, kj) chosen by firm j, the best reply of firm i is
if Ospj<2s-t
s-t,
p”(pj, kj) = i root
! 5%
if p-t(y)]spjs[ (B,p?
-;+
yq,
+ B,P,Z
if li,(kj, S,t) 5 Pj 5 Pj(S, t), if
Pj(% t, 5 Pj;
“When kj= 1, we have {2s-t[(2+kj)/kj]}=2s-3t={ -3t/k,+ [(l +kj)/kj]s}, ej(kj,s, t)= pj(s, t) and (4.6), (4.9) and (4.13) collapse as expected to pt(pj) = $pj+ t). Hence, the five cases are then reduced to (3.1). “We assume in this paper the same cost function A(k) for both firms; an interesting extension would be to find out what effects different values of p might generate.
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M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
V(Pj, kj) =min1,f Pidi(Pi,Pj, 1,kj) ; if p is small enough, k:(pj, kj) = 1.
The profit-maximizing price pi and market coverage ki as functions of the other firm’s choice (pj, kj) are then given for Case I above by
pi=f(pj+t)++,
(4.19)
J kjv+(l-kj)
(4.20)
.
Assuming that s is sufficiently large, ” the symmetric equilibrium pi =pj =p” and ki = kj = k’ is obtained [see Tirole (1988)] as for p 2 t/2, for pstt/2,
p’=-
2-k k’
t =(2pt)l”,
kc=
2
2P 1 +(2p/t)“Z’?Cc= (1+(2p/ty’y
1’
[pc=t,kc=1,~c=(t-p)/2].
It is interesting to note that if the market coverage cost parameter p is high enough to make k’< 1, that is if p > t/2, then kc is decreasing with p and since pc decreases with kc, then pc increases with p; both k’ and p” are increasing in t (recall that t is a coefficient of substitutability: the larger is t, the more differentiated the goods are); and finally rccis increasing in p and t. The result that rrc is increasing with p is due to the strategic nature of market coverage in the present model: an increase in p reduces advertising and coverage and therefore increases the ‘endogenous’ product differentiation, thereby releasing the competitive pressure upon firms and resulting in higher prices charged by the firms in equilibrium.‘i On the other hand, if p is small enough to make kc = 1, that is, if p < t/2, then k’ and pc are independent of p and p” increases linearly with t; as for profits, they increase with t but decrease with p: since market coverage is already maximal, an increase in p simply depresses profits. We will use this equilibrium as a benchmark of comparison with the leader-follower equilibrium we characterize next. Rather than pursuing the complete characterization of the Stackelberg “We can make this simplifying assumption to characterize the ‘symmetric’ equilibrium although, as mentioned previously, we cannot in general make it to characterize the ‘asymmetric’ leader-follower equilibrium. ‘IThis effect may explain why some industries or professions encourage increases in advertising costs through restrictions, legal or other, on market coverage by their members.
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
317
Table 1 The numerical evaluation of the Stackelberg equilibrium in prices. A: for variable
t; s= 1 and p = 0
0.113 0.225 0.281 0.309 0.363 0.393 0.400 0.402 0.385 0.344 0.333
0.20 0.40 0.50 0.55 0.65 0.75 0.80 0.85 1.00 1.25 1.33
B: for variable s;
0.30 0.60 0.75 0.82 0.89 0.83 0.80 0.76 0.67 0.67 0.67
0.156 0.312 0.391 0.430 0.456 0.416 0.400 0.384 0.346 0.342 0.333
0.25 0.50 0.62 0.69 0.77 0.79 0.80 0.81 0.82 0.71 0.67
0.38 0.38 0.38 0.38 0.41 0.47 0.50 0.53 0.58 0.52 0.50
0.62 0.62 0.62 0.62 0.59 0.53 0.50 0.47 0.42 0.48 0.50
t = 1 and p = 0
s
“1
n2
PI
P2
D,
D,
0.50 0.75 1.00 1.25 1.30 1.65 +
0.136 0.250 0.385 0.500 0.514 0.563
0.136 0.250 0.348 0.500 0.530 0.781
0.33 0.50 0.67 1.00 1.06 1.50
0.33 0.50 0.82 1.00 1.03 1.25
0.41 0.50 0.57 0.50 0.49 0.38
0.41 0.50 0.43 0.50 0.51 0.62
Table 2 The numerical evaluation of the Stackelberg equilibrium in (P,K) for variable t; s= 1 and p=O.
t
k,
=I
A2
Pl
P2
DI
D,
FAD,
0.20 0.25 0.35 0.40 0.42 0.44 0.45 0.80 1.00
0.34 0.44 0.68 0.82 0.88 0.94 1.00 1.00 1.00
0.199 0.208 0.229 0.240 0.245 0.250 0.255 0.400 0.385
0.542 0.493 0.410 0.369 0.352 0.333 0.352 0.400 0.346
0.62 0.62 0.62 0.62 0.62 0.62 0.67 0.80 0.67
0.80 0.79 0.65 0.60 0.58 0.56 0.57 0.80 0.82
0.32 0.34 0.37 0.38 0.39 0.40 0.38 0.50 0.58
0.68 0.66 0.63 0.62 0.61 0.60 0.62 0.50 0.42
0.05 0.22 0.46 0.54 0.56 0.57 0.62 0.50 0.42
equilibrium for the general setting, a rather tedious and very cumbersome if at all feasible task, we ran some numerical evaluation reported in tables 1 and 2. Table 1 presents the Stackelberg equilibrium in prices with maximal market coverage (exogenous), that is for ki= kj= 1. The values under Di are the quantity demanded of product i. It is interesting to note in table 1 that l As product differentiation increases, that is as t increases, the profit of the first-mover firm n, is increasing up to t =0.85 and then decreasing while
318
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
the profit of the second-mover n2 is increasing up to t =0.65 and then decreasing: there exists for each duopolists a ‘best’ level of product differentiation. This suggests that the duopolists would agree on the desirability of more product differentiation up to t = 0.65 but then disagree for intermediate values of t, that is for t E (0.65,0.85) and then agree again on the desirability of less product differentiation. When product differentiation is too low, competition is harsh and both firms would agree that it is too low. Large product differentiation (t is large) on the other hand means that consumers are very sensitive to product characteristics: if they cannot get a product (location) relatively close to their most preferred one (at their own location), then their net utility (surplus) s-pi-t Ix--xii, i = 1,2, decreases rapidly as they must incur high ‘transportation’ costs; in that sense, product differentiation is too costly for the firms and both would agree that it is too high. For intermediate values, they disagree; the leader would like more differentiation, the follower would want less. l z1z2 (leadership is the preferred position) afterwards:22 prices are strategic complements in the neighborhood of the equilbrium when differentiation is low, t 5%~ = 0.80, since then reaction functions are positively sloped, but strategic substitutes for larger t,23 and moreover low differentiation makes price competition harsher. As a function of the reservation price s, x1 and z2 are both increasing as expected and reach their respective maximum for s= 1.65; leadership is the more advantageous position for small values of s, namely 0.75
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
319
a measure of the spillover effect due to the choice by firm 1 of a partial market coverage. It is interesting to note in table 2 that for t >0.45, the Stackelberg equilibrium in prices and market coverages is the Stackelberg equilibrium in prices (that is, k: = kg = 1) and our previous characterization (table 1) holds; for t co.45 the two Stackelberg equilibria differ quite significantly. Hence, for relatively low product differentiation, that is for t s 0.45: as product differentiation increases, the leader’s market coverage k, is increasing;
although the leader’s profit is increasing and the follower’s profit is decreasing (to a minimum reached for t =0.44), followership is the preferred position; the leader’s price (markup) is constant and the follower’s price (markup) decreases with product differentiation. for t=0.20, the reduction of market coverage k, from full coverage (table 1) to the optimal 34% partial coverage (table 2) brings an increase in the leader’s profits from 0.113 to 0.199; and similarly for ~50.45; when the leader restricts its market coverage below 100x, _ the follower’s profit increases; - both prices increase, with the follower’s price overshooting the leader’s price. Clearly these characteristics of the equilibrium when strategies involve both prices and market coverages would have a significant impact on the subgame-perfect equilibrium choices of location in a two-stage model with locations chosen simultaneously or sequentially in the first stage. 5. The related literature and conclusion
When a firm (such as the first-mover above) chooses a partial market coverage even with a zero cost of market coverage (or information), in the sense that being known to more customers costs nothing, it chooses to restrict the number of its customers. In a sense, this firm is choosing to ration the potential customers by keeping them uninformed of its existence. We showed elsewhere2” that it may be optimal for a firm to behave in such a way in order to reduce the intensity of the competition it faces: by doing 24Boyer and Moreaux (1987, 1988, 1989). The first paper considers the case of a homogeneous good duopoly and the other two, the case of differentiated products. In the second paper we analyze the case where the strategy space of the firms is formed of a price and a production capacity; in the third paper we consider both the strategy space (price, production capacity) and the strategy space (price, serving capacity) and we compare the two types of equilibria (they are always different). When firms choose a price and a serving capacity (a number of customers), rationing appears in equilibrium if product differentiation is not too large; when firms choose a price and a production capacity, all equilibria involve a positive level of rationed customers, and this under very general demand conditions.
320
M. Boyer and M. Moreaux,
Strategic
market coverage in spatial competition
so, it can raise its price above the full market coverage equilibrium level and in spite of the fact that it has fewer customers, it makes more profit (compare table 1 and table 2 for t =0.20 and t =0.40, for example). There are good reasons for believing that partial market coverage, and more generally ‘rationing’, is indeed present in oligopolistic markets. A first reason is that the results are derived in structural models where all variables are under the control of a well-identified agent. This differs from the more usual ‘reducedform’ approach to spatial oligopoly theory, where the firms are modeled as choosing either a price or a quantity (in the present model, the choice of quantity is done through market coverage since each customer consumes either 0 or 1 unit of the good) but not both, supposedly once one variable is chosen, the other is determined ‘by the market’. It is certainly interesting to verify if the equilibria in structural-form models are similar to the equilibria in reduced-form models. Another reason for being interested in the phenomenon of rationing is that it can be disguised under different names. In the present context of spatial competition, this rationing strategy takes the form of partial market coverage; as mentioned above, this is a way to ration potential customers, by keeping them uninformed. But this rationing, without queues of unsatisfied customers, is part of the firm’s strategy to influence the behavior of its competitors. By choosing a partial market coverage, the firstmover is in a sense credibly signaling to the second-mover that he leaves those uninformed customers to the second-mover; this is likely to make this second-mover less aggressive and more accommodating, allowing the firstmover to raise its price above its otherwise equilibrium level. In equilibrium, it happens that both prices in the Stackelberg equilibrium in price and market coverage (table 2) are above (or equal to) their respective levels in the Stackelberg equilibrium in price alone (table 1). There are clearly many other particular forms of such rationing strategies: choosing a non-central location, a lower sales effort, a less favorable set of product characteristics, as compared in each case with the non-strategic level of the same variable. Those results can be related to a more general phenomenon analyzed by Fudenberg and Tirole (1984)25 but with some important differences. In the Fudenberg-Tirole model, an incumbent monopolist in period 1 is facing a potential entrant in period 2; he can advertise in period 1 and create a captive market forever of those customers reached by the advertising. If entry occurs, the incumbent and the entrant compete for the remaining market, but the incumbent cannot price-discriminate between its captive market segment and the duopolistic market. Fudenberg and Tirole show that the incumbent may choose to prevent entry by reducing its advertising level below the level he would choose as a protected monopolist: by so doing, the incumbent 25See also Schmalensee (1983), (1985), Henry (1988) and Fershtman
Gelman and Salop and Muller (1988).
(1983),
d’Aspremont
and
Gabszewicz
M. Bayer and M. Moreaux, Strategic market coverage in spatial competition
321
announces that he will be more aggressive in the duopolistic market and therefore depress prices sufficiently (in a subgame-perfect equilibrium) to make entry unprofitable. By reducing its captive size, the incumbent monopolist avoids being a ‘fat cat’ in period 2. In our case of a single-period spatial competition model with a first-mover and a second-mover, we noted already that the preferred position need not be leadership; whether it is or not depends crucially on the level of product differentiation. When the preferred position is followership (product differentiation is low), then it may be expected that the dominant firm will in fact be the second-mover rather than the leader.26 If that is the case, then the entrant will take the position of the first-mover; by strategically choosing a partial market coverage, it may make the dominant second-mover a ‘fat cat’ willing to accommodate entry: the roles are reversed.27 We mentioned in the introduction the papers which, to our knowledge, endogenize some form of market coverage variable, although none directly addresses this issue. Grossman and Shapiro (1984) characterize the symmetric Nash equilibrium of a circular city model where iz firms, equally spaced around the circle, choose both their price and their informative advertising budget. As in the present context, customers become informed of the existence (location and price) of a firm when they receive a message from it. The advertising technology is such that it is impossible for a firm to make sure that every customer did receive a message from it; the cost of advertising goes to infinity as the market coverage goes to 1; hence, no firm will choose to cover the whole market. Partial market coverage plays no ‘strategic’ - in the sense of modifying the competitors’ behavior - role in Grossman and Shapiro since the firms move simultaneously in a one-shot game. But it is interesting to relate our results to theirs in terms of the relationships between product differentiation (the number n of firms in their mdoel) and market coverage level (informative advertising in their model), prices or markups, and profits. Grossman and Shapiro showed that informative advertising and markups (and therefore profits) increase with product differentiation (a reduction in n). By making the market coverage a strategic variable (by making it endogenous in a Stackelberg market structure), we obtained that the first-mover’s market coverage (the second-mover’s one is always 100x, when it is costless) is non-decreasing with product differentiation (an increase in t) while markups may increase or decrease: when product differentiation is low, the leader’s markup is constant while the follower’s markup decreases with differentiation; then follows a range of product differentiation levels where both markups are increasing; a range where the leader’s markup decreases while the follower’s keeps increasing; and finally, “%ee Boyer and Moreaux (1987). “Somewhat in the spirit of Gelman
and Salop (1983).
M. Boyer and M. Moreaux, Strategic market coverage in spatial
322
competition
the markups decrease to their separate monopoly market levels. As for profits, they increase with product differentiation in Grossman and Shapiro; in our model with market coverage as a strategic variable, as long as the leader’s market coverage is less than 100x, the leader’s profit increases with product differentiation but the follower’s profit decreases with product differentiation, except when the leader’s coverage is very close to 100%. This indicates that the measurement of product differentiation (either through n as in Grossman and Shapiro, or t as in this paper) and the strategic aspect of market coverage or advertising (either through the symmetric Nash equilibrium as in Grossman and Shapiro, or the Stackelberg equilibrium as in this paper) are important determinants of the role of product differentiation in oligopolistic competition.
Appendix
Proof of Proposition 1. In fig. 2, the straight line C(x) of equation pi=pj+t-22tx is the solution to pi+t~‘=pj+t(l-_X)‘; curve E(X) is the solution to pi+ tx2 =s; MRC(xj is the marginal revenue function defined from C(x), that is MRC(xj =Pj+ t-44tx; MRE(xj is the marginal revenue function defined from E(x), that is MRE(xj =s--3tx’. Hence, the marginal revenue function MR(xj is given by MRE(xj for x x1. The profit-maximizing price pT(pj; s, tj will be either at MRC(xj = 0 on the linear portion C(x) of the demand function, or at the kink point where the linear portion C(x) and the concave portion E(x) intersect, or at MRE(xj =0 on the concave portion of the demand function. The value of is 1 +((s-p.j/tjl”. The value of MRC(x,j given by p.+ t hxi[l+((s-p.j/t)l/‘]. therefore MRC(x,j=O if p?+10tp.+(9t2-16tsj=‘O, an expression dhich always has a negative root ani, if sz$t/16, a non-negative one [if s<9t/16, the value of MRC(x,j is negative and hence pi* is already at the kink when pj=O]. When it exists, the non-negative root is given by - 5t + 4(t2 + tsj’j2; hence let -5t+4(t2+tsj1/2,
jjj=
0,
ifsz&t: otherwise,
and SO,for pj & ij, 28 the profit-maximizing pj-t+2t fQ= /
0,
28This value pj is always
price pi” is at the kink value pi:
112
( > ~
(A4
)
ifpj>S--t, otherwise,
less than s since -5t+4(t2+ts)‘~2-s=(3t+s)2>0.
64.2)
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
323
and the demand is given by x1 as long as x1
64.3) For pj~pj, the profit-maximizing p? is constant at s - tx$=$s, unless of course xg > 1, in which case pr=s--t. For a relatively low value of pj, the best-reply price pi is given by MRC(x) = 0 unless s<&t, in which case MRC(x) is already negative at the kink and therefore p: is at the kink value pi; for intermediate values, it is given by the kink point pi (itself a decreasing function of pj); for relatively large pj, that is above pj, it is given by MRE(x) = 0, in which case further increases in pj have no effect on pT which remains constant at +$s.For the case illustrated in fig. 2, pi(x2,pj) =(pj + t)/2, pi(x1,pj)=i)i, P~(X,,P~)=~S. Hence Proposition 1. Q.E.D. Proof of Proposition 2.
Clear from the text.
Proof of Proposition 3.
The equilibrium will be on the linear portion of the
reaction functions if pi
f-jj-t+2t
( ->“* y
which from (3.2) can be written as s+5t-4(t*+t$‘*
(
t
I’*
2’
a condition which is satisfied for t -C$L For larger values of t, the point (fijj,pT(jYj)) is above the 45” line and therefore the intersection with that line will be either in the concave portion of the reaction function or in the flat portion, the former case corresponding to values of t E [$s, $1 and the latter for values tz$s. To show that the concave portion of the two reaction functions when intersecting are in fact overlapping, let us consider pT(pj) for pj in the interval [pji,pj]. From (3.1): 11-2
pi*(pj)=pj-t++t
( > y
.
324
M. Bayer and M. Moreaux, Strategic market coverage in spatial competition
Suppose that pr(pj)E[di,~i]. similarly given by
Then the best reply by firm j to pi=PT(pj) is
S-+(Pj) t
Pj*(Pr(Pj))=PT(Pj)-t+2t C
=pj
l-
C_pY,*= +
I’2 >
if and only if [ s-p~+~-2~o”)“‘, t
Raising both sides to the power 2, we get
which proves that the concave portions when intersecting are in fact overlapping. The borderline between case (b) and case (c) in fig. 4 is obtained when there is a complete overlapping of the concave portions of the reaction functions, that is for ~Js, t) =$v; from 3.2) we have such a case if -St+ 4(t2 + z.Y)‘/~ =$s, that is if t = [( 19 +4 / 10)/27]sxO.978s. And finally case (d) Q.E.D. appears if pj>$s, that is from (3.5) if scat. of Proposition 4. Let i = 2 and j== 1. Suppose the firm 1 chooses first and is committed to its action (pl, k,) which is observed by firm 2 before the latter chooses its own (pz,k2). Given (pl,k,), the optimal price of firm 2 is given by (4.6), (4.9), (4.13), (4.16), (4.18); and from (4.3) its optimal coverage is given by
Proof
k,=min
l,~p,d&,p2,k,)
(A-4)
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
Let us assume that (pl, k,) satisfies p1 52s--t((2+k,)/k,)
325
so that
1-k t P2=3(P1+t)+k, ’ which together with (A.4) characterizes the second-mover’s best reply to (pl, k,). Under the above condition, the best-reply price pZ is less than s--t and therefore the residual demand to firm 2 is the second part of (4.2). Hence (A.4) can be rewritten as k,pl-~~z+t+(l-kl)
.
(‘4.6)
Using (AS) we can rewrite (A.6) as *.
(A.7)
Therefore k2 >O if (k,pf/8t +pl(f-;kJ +(tk,/8 + t/2kl - t/2)) is positive. This expression is a second-degree polynomial in p1 whose roots are always negative for k, E [O,l]; since it is equal to t(k, -2)2/(8k,) >O at p1 =0, we conclude that k,>O for all (pl, k,). Moreover, the polynomial value is non-decreasing with p1 and is therefore minimal at p1 =O; hence, we get ~
1
t(k, -2)2
.
Qk, (‘4.8)
Hence k, = 1 if p 5 t/S, from which we conclude that k, = 1 even in the worst case for firm 2, that is when (pl, k,) =(O, l), in which case we are in Case I as assumed: this means that the second-mover firm always covers the whole market when p is small. Q.E.D.
References Anderson, S.P. and A. de Palma, 1988, Spatial price discrimination with heterogeneous products, Review of Economic Studies 55, 573-592. Anderson, S.P. and D.J. Neven, 1990, Spatial competition a la Cournot: Price discimination by quantity setting duopolists, Journal of Regional Science 30, 1-14. Anderson, S.P., A. de Palma and J.F. Thisse, 1989a, Demand for differentiated products, discrete choice models and the characteristics approach, Review of Economic Studies 56, 21-35. Anderson, S.P., A. de Palma and J.F. Thisse, 1989b, Spatial price policies reconsidered, Journal of Industrial Economics 38, 1-18. Boyer, M. and M. Moreaux, 1987, Being a leader or a follower: Reflections on the distribution of roles in a duopoly, International Journal of Industrial Organization 5, 175-192.
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Boyer, M. and M. Moreaux, 1988, Rational rationing in Stackelberg equilibria, Quarterly Journal of Economics 103, 4099414. Boyer, M. and M. Moreaux, 1989, Endogenous rationing in a differentiated product duopoly, International Economic Review 30, 877-888. Boyer, M., J.J. Laffont, P. Mahenc and M. Moreaux, 1992a, Location distortions under asymmetric information, Regional Science and Urban Economics, forthcoming. Boyer, M., J.J. Laffont, P. Mahenc and M. Moreaux, 1992b, Sequential location equilibria under incomplete information, Mimeo. (Universite de Toulouse (GREMAQ) and Universite de Montreal (CRDE)). Butters, G., 1977, Equilibrium distributions of sales and advertising prices, Review of Economic Studies 44, 465491. d’Aspremont, C. and J.J. Gabszewicz, 198.5, Quasi-monopolies, Economica 52, 141-151. d’aspremont, C., J.J. Gabszewicz and J.F. Thisse, 1979, On Hotelling’s stability in competition, Econometrica 47, 1045-1050. de Palma, A., V. Ginsburgh, Y.Y. Papageorgiu and J.F. Thisse, 1985, The principle of minimum differentiation holds under sufficient heterogeneity, Econometrica 53, 767-781. Fershtman, C. and E. Muller, 1988, The potential disadvantage of becoming too large in duopolistic industry, Discussion paper 794 (Northwestern University). Fudenberg, D. and J. Tirole, 1984, The fat cat effect, the puppy dog ploy and the lean and hungry look, American Economic Review Papers and Proceedings 74, 361-368. Gabszewicz, J.J. and P. Garella, 1986, Subjective price search and price competition, International Journal of Industrial Organization 4, 305-316. Gabszewicz, J.J. and J.F. Thisse, 1986, Spatial competition and the location of firms, in: J.J. Gabszewicz, J.F. Thisse, M. Fujita and U. Schweizer, eds., Location theory (Harwood Academic Publishers) 1-71. Gabszewicz, J.J., P. Garella and C. Nollet, 1989, Spatial price competition with uninformed buyers, Cahier CERPE 14 (Universite du Quebec a Montreal). Gal-Or, E., 1985, First mover and second mover advantages, Inrernational Economic Review 26, 649-653. Gelman, J. and S. Salop, 1983, Judo economics: Capacity limitation under coupon competition, Bell Journal of Economics 14,315-325. Greenhut, M.L., G. Norman and C.S. Hung, 1987, The economics of imperfect competition: A spatial approach (Cambridge University Press). Grossman, G. and C. Shapiro, 1984, Informative advertising with differentiated products, Review of Economic Studies 51, 63-82. Hamilton, J.H., J.F. Thisse and A. Weskamp, 1989, Spatial discrimination, Bertrand vs. Cournot in a model of location choice, Regional Science and Urban Economics 19, 87-102. Henry, C., 1988, Concurrence potentielle et discrimination dans un modele de duopole avec differentiation verticale, in: Melanges Cconomiques: Essais en l’honneur d’Edmond Malinvaud (Economica) 289-3 13. Hotelling, H., 1929, Stability in competition, Economic Journal 39, 41-57. Schmalensee, R., 1983, Advertising and entry deterrance: An exploratory model, Journal of Political Economy 91, 636653. Schultz, N. and K. Stahl, 1988, Consumer search and the spatial organization of markets for differentiated products, Discussion paper (Universitat Mannheim). Tirole, J., 1988, The theory of industrial organization (MIT Press, Cambridge, MA). Weskamp, A., 1985, Existence of spatial Cournot equilibria, Regional Science and Urban Economics 15, 219-228.
Journal
of Industrial
Organization
11 (1993) 299-326.
North-Holland
Strategic market coverage in spatial competition Marcel Boyer Dkpartement de Sciences Economiques, CRDE and CRT, Universitt de Mont&al, C.P.6I28, Montrkal, Quebec H3C 3J7, Canada
Michel Moreaux* GREMAQ, CNRS URA 947, Universitt? de Toulouse I, Place Anatole-France, 31042, Toulouse Cedex, France Final version
received
December
1991
We consider in this paper competition through prices and market coverages in a spatial duopoly. We characterize and compare the Stackelberg equilibria and the Bertrand equilibria. We show that when the cost of informing the consumers is small, duopolists always choose to inform all consumers in the Bertrand equilibrium but the leader in the Stackelberg equilibrium may choose not to inform everyone when the duopolistic competition is intense enough (thereby generating overlapping markets). In equilibrium, prices may be strategic complements, strategic substitutes or be strategically independent. Considering market coverage as chosen by the firms changes the ‘conventional’ relationships between differentiation, profits and markups.
1. Introduction
In the d’Aspremont et al. (1979) reformulation of the canonical spatial competition model of Hotelling (1929),l the firms do not choose their market coverage, that is the percentage of consumers aware of their existence, and the market areas of the firms to not overlap.2 In all these Correspondence to: Marcel Boyer, Dkpartement de Sciences Economiques, CRDE and CRT, Universiti: de Montrkal, C.P.6128, Mont&al, Quebec H3C 357, Canada. *The authors wish to thank Claude d’Aspremont, CORE mathematical economics seminar participants, and two anonymous referees for their comments. We remain of course solely responsible for any errors or shortcomings. ‘See Gabszewicz and Thisse (1986) and Greenhut et al. (1987) for recent surveys of location theory. ‘Overlapping market areas, that is areas in which a given consumer may patronize a firm while the left-hand and the right-hand neighbors patronize another, can be obtained in different ways and for different reasons: (1) if competition is not a price competition but rather a Cournot-like quantity competition, as for example in the network-type model of Weskamp (1985) in which the overlapping property holds in the sense that for some nodes more than one firm have non-zero sales [see also Anderson and Neven (1990) or Hamilton et al. (1989)]; (2) if 0167-7187/93/$06.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
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M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
formulations, either the two sides of the market are perfectly informed or the imperfection of information does not play a crucial role in the determination of the equilibrium. For instance, in de Palma et al. (1985) the firms know the distribution of consumer tastes at each point of the space and it is sufficient to determine the price policies of the firms. More recently Gabszewicz et al. (1989) extending the work of Gabszewicz and Garella (1986) have relaxed the assumption of perfect information of both the consumers and the firms regarding the other side of the market. In their model the consumers know the price quoted by the nearest seller but not the price quoted by the other firms. Hence the partially uninformed consumers must decide whether they will bear the cost of getting additional information or not. They show that at symmetric equilibria the firms quote the highest prices which induce no search by any consumer: the respective markets do not overlap. With asymmetrically (and exogenously) located firms there exist price equilibria where some consumers will search for more information, but even so the markets do not overlap. Schultz and Stahl (1988) develop a model of consumer search over variety and price. Firms choose their location, variety and price and in equilibrium, more than one variety may be offered at one given location and a variety may be offered at more than one location. The consumer preferences are modeled so that more variety means increased demand: a consumer visits only one market place and therefore a visit is more desirable when more variants are offered at that market place. Each consumer is assumed to know, before searching, the location and size (number of varieties offered) of market places; he forms a (true) point forecast of the expected price charged at a given market place and also a forecast of the characteristics of the varieties.3 They consider and compare the perfect Nash equilibria of two market structures: a large number of potential one-product firms, each choosing in a three-stage game to enter or not (in stage 1; an entrant must incur an entry cost) and upon entry, a plant location (stage 2) and both a product variety and a price (stage 3); a one-plant monopolist deciding on varieties and prices. They show that if consumers are choosy enough, then all one-product firms locate at the same location (in any case, there always are two or more firms at each equilibrium location) and may charge prices in excess of that charged by the multi-product monopolist with the same set of
the goods marketed by each firm have some specific characteristics other than the sole location of the firm selling them, as for example in de Palma et al. (1985) [see also Anderson and de Palma (1988) and Anderson et al. (1989a)]; (3) finally if preferences between different location are in part ‘stochastic’, at least as observed by outsiders, then firms may sell to consumers over the whole interval [see Anderson et al. (1989bjl. %ince search costs are a function only of the distance to the marketplace, the overlapping of markets (not explicitly considered by Schultz and Stahl) would likely proceed from the same factors as in de Palma et al. (1985).
’
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
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variants; they also provide conditions under which the latter offers as many varieties as the former: hence, monopoly may increase welfare.4 In a different context, Boyer et al. (1992a, b) provide an exhaustive analysis, in the space of prior beliefs and cost differentials, of the nature of Perfect Bayesian Intuitive and Divine Equilibria in a two-stage model of sequential location choices (stage 1) and simultaneous choices of delivered pricing schedules (stage 2) when the firms are asymmetrically informed in stage 1 about the production cost of their competitor;5 but again, in the production stage the markets are separated. Casual observation of markets reveals that they more often overlap than not; hence, it seems to us that the usual non-overlapping property of equilibria in most spatial competition models is a weak point of the theory. Moreover, we will allow firms to choose strategically their market coverage. We will assume that consumers are ‘born’ unaware of the prices charged by the firms (and even of their existence) and that they do not search but can become informed of the existence and price charged by a firm if they receive a message from that given firm. In such a framework, each firm chooses not only its price but also the number (proportion) of consumers it will inform. It is this viewpoint of imperfect information that we take in this paper. We suppose that given locations (determined in a preceding stage of the game not modeled here), the firms must inform the consumers about their respective location and price. Hence the firms compete in both prices and ‘market coverage’. We consider only the ‘case of a duopoly but we compare two forms of duopoly markets. In the first case the firms move simultaneously (a la Bertrand) while in the second case they move sequentially (a la Stackelberg) with one of the firms taking the leadership and the other accepting the followership. We assume that the cost of covering the market is quadratic and increasing in a parameter p, which can be let to vary to represent different degrees of technological efficiency in advertising. We show that in Bertrand equilibria, overlapping never occurs when the cost of informing the consumers is small (although it does when the cost of market coverage is high enough), whereas in Stackelberg equilibria the leader (but not the follower) may choose not to inform all the potential buyers of its product and price even if the cost of informing the consumers is zero, a decision which %rtuitively, those results may be understood as an application of the basic result that an mproduct monopolist will sell at lower prices than m one-product oligopolists when the products are complements; in the model of Schultz and Stahl, the products are both substitutes (since a consumer buys only one variety, if any) and complements (since each variety offered at a given location increases the global demand at that market place by making search more profitable, in expected terms). ‘More specifically, the first-mover in stage 1 knows its marginal production cost (equal either to c or c-a) when choosing its location, but the second-mover whose marginal cost is always c can only observe the location chosen by the first-mover firm.
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gives rise to overlapping market areas. We also show that when the cost to the firms of informing the consumers is ‘small’, the Bertrand duopolists always choose a maximal market coverage but the leader in the Stackelberg equilibria will choose under the same cost conditions a partial market coverage when the level of duopolistic competition is ‘intense enough’, that is when the products are good substitutes. Moreover, considering the market coverage as endogenously chosen by the firm with a significant strategic component (made possible here by the Stackelberg structure of the market game) will change in important ways the equilibrium relationships between the level of product differentiation and profits and markups. The two traditional measures of increased product differentiation - ‘an increase in ‘transportation’ costs and a reduction in the reservation price (value) - have different impacts on profits and markups. We believe that the present model not only brings into the economic analysis of spatial (product design) competition an important strategic variable, the decision on market coverage, but also provides a richer analysis of the implications of product differentiation. Grossman and Shapiro (1984) have analyzed the symmetric n-firm Nash equilibria of a model where firms are symmetrically located around a circular city and choose both a price and a level of informative advertising. Their advertising technology is such that it is infinitely costly to inform all the consumers; hence markets always overlap. Although the focus of their paper is different, we can interpret their results in the context of spatial competition with overlapping markets. We will discuss their model and results in relation to our own model and results in the conclusion, The next section is devoted to introducing the model and section 3 presents the simultaneous move (Bertrand) equilibria when all consumers are aware of the existence and prices of both firms; we characterize in section 4 the best-reply function giving one firm’s optimal price as a function of the other firm’s price and partial market coverage level; the stage is then set for the characterization of the simultaneous move (Bertrand) equilibria with partial market coverage and of the leader-follower (Stackelberg) equilibria when firms choose both their price and their market coverage. We conclude by discussing the related literature on informative advertising and ‘rationing’. 2. The model We consider the following context adapted from Butters (1977). Assume a linear city with two firms located at the end points of the city: firm 1 at x=0 and firm 2 at x= 1. The consumers are distributed uniformly on the interval [0,11 and each one demands inelastically one unit of the homogeneous good if the cost is below a reservation price s. In addition to paying the price set by the firm, a consumer located at x1 must bear quadratic transportation
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
303
costs of txf if he buys from firm 1 and t( 1 -X1)’ if he buys from firm 2: the goods are differentiated by their locations only.6 Suppose that at XE(O, 1) the consumer is indifferent between buying form firm 1 or firm 2, that is p,+tXZ=P2+t(l-X)2~;S
or
x=(Pj_Pi+t)/(2t),
and suppose that all consumers are aware of the existence of both firms. Consumers to the left of x will buy from firm 1 and those to the right of x will buy from firm 2. In this particular case, the demand facing firm i may be written as (‘u’ stands for ‘aware of the existence of the other firm’): &I,, pz) =max{O,min (l,(pj--pi+ t)/(2t)}}. More generally, the residual demand function facing firm i given pj and given that all consumers are aware of both firms will depend also on the reservation price s since it may well be the case that for some location the price charged by one firm or both plus the quadratic transport cost exceeds the reservation price s. Moreover, if that is so in equilibrium, then some consumers may even not be served.7 Given pj, the demand to firm i cannot be negative or larger than 1; it cannot be larger than x = [(s - Pi)/t] ‘I2 , the solution to pi+ tx2 = s (the consumer at such a location x is just indifferent between buying the product form firm i and not buying at all), and it cannot be larger than (pi--pi+ t)/(2t) as seen above. Hence dl(p,,p2)=max10,min(P’-~+t,(~)L’i,
I]].
(2.1)
The inverse demand function, when it exists,8 can directly be written as pi(X, pj) =
max (0, min
{S -
tX2, pj + t - 2tX3).
(2.2)
The different cases can be illustrated as in fig. l(a)--(d), where f =((s-pJ/t)"'
6There are two main reasons for using quadratic transportation costs rather than linear ones. First, d’Aspremont et al. (1979) show that maximal product differentiation (the two firms located at the end points of a linear city, that is at x=0 and x= 1, respectively) results from quadratic transportation costs in a model of location choice; hence, although we consider here those endpoint locations as exogenous, they are compatible (in a subgame-perfect sense) with quadratic transportation costs and uniformly distributed consumers. Second, quadratic transportation costs will generate continuous residual demand functions and continuous and concave profit functions contrary to the case of linear transportation costs. ‘The reservation price s will play a role in our model with a leader-follower structure; since we will show that the leader may choose not to inform all consumers while the follower does inform everyone, for p suffkiently small, the profit of the follower (given that the leader is committed to its choice) would be unbounded if s is unbounded. Hence, we cannot simply assume that s is large enough that every consumer, aware of at least one firm, will buy in equilibrium. *No inverse function exists if d;( .) = 1 in (2.1). This case is not illustrated in fig. 2, it would appear for pj > t.
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
304
(b)
I
I
pj
P
-x
h
I
Cc)
Cd)
Pi-Pi+’
di=
2t
-A
: : : : B k;.
h f
I
I”
I
Fig. 1. The market
separation:
r
for ki= kj= 1.
and h = (pj-ppi + t)/2t. The demand itself is represented in fig. 2, where the value of x1 is obtained from pj+t(l -x)‘=s, that is x1 = 1 -((~--p~)/t)l/~ and the value of pi is obtained from j+= s- tx: =pj+ t -2tx1; replacing x1 by its “’ . It is interesting to note that the value, we obtain &=pjt +2t((s-pj)/t) demand function has a kink of Stigler type at (x1, j$). Hence the marginal revenue function MR(x) is discontinuous at x=x1. As pj increases, the kink moves to the right on the E(x) =s - tx2 curve since that curve is independent of pj while the C(x) = pj+ t - 2tx line moves up linearly with pj. Consider again fig. 2 and the demand (2.1). These were derived for a consumer aware of the existence of both firms. A consumer located at x and aware of the existence of firm i only demands one unit of good i if
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in spatial competition
pi
/
_----
I I
x
112 x3
\
MR (xl
Fig. 2. The demand function: for t >s and pj < t.
pi+ LX’5s. Hence if all (N = 1) consumers are aware of firm i only, the demand and the inverse demand for good i would be given respectively by (2.2 stands for ‘unaware of the existence of the other firm’): dy(pJ=max
k,min{r+)“‘,
I)],
(2.3)
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Strategic market coverage
in spatial competition
pi(X)=IllaX{O,.S-tX*}.
(2.4)
If a proportion kjE [0, l] of consumers are aware of the existence of firm j and if those kj consumers are randomly located in the interval [O,l],’ then the residual demand facing firm i will be”’ di(Pi, Pj, kj) =k#?(~i, Pj) +(l -kj)C(pJ,
(2.5)
with d~(pi,pj) given by (2.1) and do given by (2.3). The profit function of firm i, assuming a zero cost of market coverage, can be written as ZiPi, ki; Pj, kj) =(Pt-c)kiddPi,
Pj, kj).
(2.6)
3. The Bertrand equilibria Let us first characterize the full awareness price reaction function, that is the profit-maximizing price pT as a function of pj assuming that ki = kj= 1. Let us assume that c=O. We can refer to fig. 2. Proposition
I.
The profit-maximizing (Pj +
p~(pj;S,t)=’
price pf(pj; s, t) is given by
if OS Pj 5 fij(& t),
t)/2,
~i(Pj;s,t)=pj-_+22t
112
( > ~
,
max{s-_,+s},
if Fj(% t) 5 Pj S pits, t),
if Pj(% t, 2 Pjl
(3.1)l
c
where
-55t+4(t*+tSy’*,
if e&t
0,
otherwise,
FjE
(3.2) l/2
)
1yss3t 0
(3.3)
otherwise. Proof.
All proofs are given in the appendix.
It can easily be checked that (3.1) is continuous. The first portion of (3.1) is linear and increasing with pj, the second portion is decreasing and concave 9We consider only the case where the consumers informed of the existence of a given firm are randomly located over [O,l]; hence the firms cannot aim their advertising effort at a specific consumer group, such as those closer to it; an interesting extension would be to look at information technologies for which the cost of informing a consumer increases with the distance separating the consumer and the firm. “‘There are four groups of consumers: kikj knowing both products, k,(l - kj) knowing only product i, (1- ki)kj knowing only j, and (1- k,)( 1- kj) knowing neither i nor j. The residual demand for product i is defined for ki = 1.
M. Bayer and M. Moreaux,
Fig. 3. The price reaction
Strategic market coverage in spatial competition
function
with full market
coverage
307
(s 5 3t).
in pj, and finally the third portion is constant either at s--t or at 5s. It is important to note that for Pj
equilibrium.
2.
V(s, t),
there
exists
a
unique
symmetric
Bertrand
q
As t increases for a given s, the linear portion of the best-reply function moves up and fij(S, t) and pj(s, t) both increase with t if t
308
M. Boyer and M. Moreaux,
Strategic market coverage
in spatial competition
(b)
Pi *
Pj Pj
Pi
Pj
MiPj
Cc)
Cd)
Fig. 4. The Bertrand-Nash
equilibria.
decrease with t if t>$. We therefore can have four different configurations depending on the relative values of t and s, as represented in fig. 4(a)-(d). Case (a) will prevail if pT(fij)
M. Boyer and M. Moreaux,
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309
Proposition 3. Assuming full awareness of the consumers, (a) for t &, there is a unique, symmetric and stable Bertrand equilibrium p’= t on the linear portion of the reaction functions; (b) for t E (%s,$s), there is a stable set of unstable equilibria - each element of the set is an unstable equilibrium but the set itself is locally stable (ergodic) - on the concave portion of the reaction functions, indicating that both firms are then at the kink of their respective demand functions; when intersecting, the reaction functions are in fact overlapping; (c) for t Z$s, we have a unique, symmetric and stable equilibrium on the constant portion of the reaction functions; in this case, not all consumers buy the goods and both firms are at their monopolistic equilibrium; (d) the symmetric equilibrium price, as a function of t, is first increasing with t up to a maximum of p’=$s, then decreases with t to a minimum of p’=$s, and stays at that value for larger values oft. 0 4. The characterization
of equilibria in the (price, market coverage) strategy
space Consider now the case where some consumers are unaware (1 - kj) of them. The residual demand facing firm i will be di(Pi, Pj, k, kj) = k[IkjdT(Pi, Pj) + (I -
kj)dY(PJl,
of firm j, say
(4.1)
where dKpi,pj) is given by (2.1) and dy(pi) is given by (2.3). In fig. 5 the demand function of the ‘aware’ is curve DA while the demand function of the ‘unaware’ is curve DU; the associated marginal revenue functions, with respect to price, are respectively MRDA(pi,pj) and ~vIRDU(~,).~’ It is important to recall that although demand curve DU is independent of pj, the position of demand curve DA depends on pj. We have:
if Pi~ElaX(O,Pj-t},
[ min{l,y}, 1
pj+t
~
DA:x=
- sPi3
2t
if max(0,pj-t}5pi~&
: s-p,
l/Z
c-1 t
’
1 0, “Both MRDA(p p,) and MRDU(pJ but the latter is discontinuous.
if fii 5 pi,
if sspi; may be discontinuous; in fig. 5 the former is continuous
M. Bayer and M. Moreaux, Strategic market coverage in spatial competition
310
2 --s 3 r’pj
+t:
Fig. 5. The demand function: for 4s < t x$x
io I
I
l/2
mini,:
DU:x=
,
s--p,
__
(
0,
t
if piSmaX{O,S-t},
112 ) if IIlaX{O,s-t}~pi~Ss, > if ssp,;
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
ifpismaX{O,pj-t},
~pj+t 2t
311
1
if max {O,pj- t> Qisfii,
TPi,
MRDA: x =
( (y)l’2- &P~(?)-“~,
ifjji-Cpi.s,
10,
if S?Zpi.
MRDU:x=<
ifmaX(O,S-t}spisS,
if sSpi.
0, For ki= 1, (4.1) can be rewritten as
/ di, (pi, pj, 1, kj) =
1,
ifp,>t
andp,spj-tt,
kjDA+(l-kj)l,
ifmax(O,pj-t}gpPigs-t,
kjDA + (1 - kj)DU,
ifs-tspig#i, 112
,
if fi;spisS, (4.2)
and for ki-c 1:
di(Pi,Pj, ki, kj) = kiddpi,Pjll, kj)*
(4.3)
It is clear from (4.3) that the best-reply price pi to a given (pj, kj) is independent of ki. Hence we can find the best-reply price for ki= 1. To do so, it will be convenient and useful to distinguish live cases given the nondifferentiability of demand functions DA and DU. Considering (4.2) and
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
312
fig. 5,13 we observe that the best-reply axi/api =kjMRDA(pi,pj) +( 1 - kj)MRDU(pi) US define f(p,,pJ as follows:
price pi*(pj, kj) is obtained when =0.14 TO simplify notation, let
which will allow a simpler representation of MRDA(pi,pj) and MRDU(p,). Noting that the profit function ~i( .) is quasi-concave and recalling the definitions of pj> fii and pj from Proposition 1, we can characterize the price best-reply function with the help of the following five cases, each corresponding to a different form of dxi/api=O: Case I: 0 5 pf(pj, kj) < s-t.
This case appears if
CkjS(Pi, Pj)+(l_kj)11,,=,-,<0, that is, if 0 spj<2sby
(4.4)
t((2-t kj)/kj).
In this case, p:(pj, kj) is defined implicitly
[kj(y - ipi)+(I-kj)l]=O;
(4.5)
that is pr(pj, kj)=$(pj+
t) + 2
(4.6)
t. J
Case II: p:(pj, kj) = s - t. This case appears if ’ 5
that
Ckjf(Pi,pj)+(1_kj)lI,i=,-,~o,
(4.7)
Ckjf(Pi, Pj)+(l-kj)g(Pi)]pi=s-t~0,
(4.8)
is,
solving (4.7) and (4.8), if (2~- t[(2+kj)/kJ} In this case, pT(pj, kj) is given by
~pj~
( -3t/kj+
[(l + kj)/kj]s).
13Fig. 5 is drawn for pj=t giving an x-intercept at x=1 but if pj<[>]t. the intercept would be to the left (right) of x= 1. 14Mutatis mutandis at points of discontinuity, or non-differentiability of the demand functions. ISNote that for t
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313
(4.9)
pZ(pj,kj)=S-t.
Case III: s-t
Ckjf(pi,Pj)+(l-kj)g(pi)l,,=,-,>O, Ckjf(Pi, Pi) +(I
(4.10) (4.11)
-kj)g(pi)lpi=~i(pj)
that is, solving (4.10) and (4.11), if { -3t/kj+[(l +kj)/kj]s)
[kj(y
which after a few manipulations nomial in pi:16
&pi(~)-“‘]]=O;
can be rewritten as a third-degree
(4.12) poly-
(4.13)
B3p?+B2p~+Blpi+Bo=0, where B, =4k;, Bz = Skft-
18kjt +9t-4k;s+4kf-4kfpj,
B,=kj2t2-8kj2st+24kjSt-12st+2k3pjt+4kj2pjs+kj2pf,
Case IV: pF(pj, kj) = ii(
This case appears if
Ckjf(Pi, Pj) + (1 -kj)dpi)lp;=~~(pJ) 160ne can write (4.12) as follows: k,(!!.$ raising
_ +)=
-(l_~j)[(~~‘z_~pi~~)-l’z];
both sides to the power 2, we get k’(q-;p.>‘=(l-kj,‘[+fpi+$(&)].
A few more manipulations
lead to (4.13).
20,
(4.14)
314
M. Boyer and M. Moreaux, Strategic mqrket coverage in spatial competition
’ PI Fig. 6. The price reaction
function
with partial
market
coverage
(s = 2, t = 1).
Ckjf(Pi,Pj)+(l -kj)g(pi)lpi=Bi(pj)=g(pt)lPi=Bi(Pj)~O, that is, solving (4.14) and (4.15), if ~j(kj, S, t) spjs In this case pF(pj, kj) is, from (3.1), given by
[&--t+2(~t/3)“~]
p;(pj,kj)=pj-tt22t
Case V: I7 pi”(pj, kj) =&(pj)
t).
This case appears if
>
(4.17)
‘2
that is if [$s - t + 2(~/3)~“] = pj(s, t) spj. pT(pj,
=jj(s,
(4.16)
=$.
APi) Ipi
(4.15)
In this case, p2(pj,
kj)
is given by
kj) =$s.
(4.18)
Typical price reaction functions with partial market coverage are illustrated in fig. 6. It should be noted that in comparison with the price reaction “Note
that if s 2 3t, then [#s-t
+ 2(st/3)“‘]
> s and therefore
there is no Case V.
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Strategic market coverage
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315
function with fulZ market coverage (fig. 3), the boundary value for which prices cease to be strategic complements to become strategic substitutes is now pj rather than fiji, with pjzJj, and that a new interval appears (Case II) where prices are strategically independent. The conditions defining each case may be interpreted as follows: kj is the probability that a consumer knows firm j and f(pi,pj) is the marginal revenue realized by firm i on such a consumer; (1 - kj) is the probability that a consumer is unaware of firm j and the marginal revenue realized by firm i on the latter consumer is 1 if piss-t and is g(p,) otherwise. Therefore, the conditions may be interpreted as stating that pi is chosen so that the ‘expected marginal revenue’ is 0.l’ Assuming that tr$s, we know from Proposition 3 that the symmetric Bertrand full awareness equilibrium is on the linear portion of the price reaction functions as in case (a) of fig. 4. Moreover, since from Case I above we have P*(pj, kj= l)
4.
Observing (pj, kj) chosen by firm j, the best reply of firm i is
if Ospj<2s-t
s-t,
p”(pj, kj) = i root
! 5%
if p-t(y)]spjs[ (B,p?
-;+
yq,
+ B,P,Z
if li,(kj, S,t) 5 Pj 5 Pj(S, t), if
Pj(% t, 5 Pj;
“When kj= 1, we have {2s-t[(2+kj)/kj]}=2s-3t={ -3t/k,+ [(l +kj)/kj]s}, ej(kj,s, t)= pj(s, t) and (4.6), (4.9) and (4.13) collapse as expected to pt(pj) = $pj+ t). Hence, the five cases are then reduced to (3.1). “We assume in this paper the same cost function A(k) for both firms; an interesting extension would be to find out what effects different values of p might generate.
316
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
V(Pj, kj) =min1,f Pidi(Pi,Pj, 1,kj) ; if p is small enough, k:(pj, kj) = 1.
The profit-maximizing price pi and market coverage ki as functions of the other firm’s choice (pj, kj) are then given for Case I above by
pi=f(pj+t)++,
(4.19)
J kjv+(l-kj)
(4.20)
.
Assuming that s is sufficiently large, ” the symmetric equilibrium pi =pj =p” and ki = kj = k’ is obtained [see Tirole (1988)] as for p 2 t/2, for pstt/2,
p’=-
2-k k’
t =(2pt)l”,
kc=
2
2P 1 +(2p/t)“Z’?Cc= (1+(2p/ty’y
1’
[pc=t,kc=1,~c=(t-p)/2].
It is interesting to note that if the market coverage cost parameter p is high enough to make k’< 1, that is if p > t/2, then kc is decreasing with p and since pc decreases with kc, then pc increases with p; both k’ and p” are increasing in t (recall that t is a coefficient of substitutability: the larger is t, the more differentiated the goods are); and finally rccis increasing in p and t. The result that rrc is increasing with p is due to the strategic nature of market coverage in the present model: an increase in p reduces advertising and coverage and therefore increases the ‘endogenous’ product differentiation, thereby releasing the competitive pressure upon firms and resulting in higher prices charged by the firms in equilibrium.‘i On the other hand, if p is small enough to make kc = 1, that is, if p < t/2, then k’ and pc are independent of p and p” increases linearly with t; as for profits, they increase with t but decrease with p: since market coverage is already maximal, an increase in p simply depresses profits. We will use this equilibrium as a benchmark of comparison with the leader-follower equilibrium we characterize next. Rather than pursuing the complete characterization of the Stackelberg “We can make this simplifying assumption to characterize the ‘symmetric’ equilibrium although, as mentioned previously, we cannot in general make it to characterize the ‘asymmetric’ leader-follower equilibrium. ‘IThis effect may explain why some industries or professions encourage increases in advertising costs through restrictions, legal or other, on market coverage by their members.
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
317
Table 1 The numerical evaluation of the Stackelberg equilibrium in prices. A: for variable
t; s= 1 and p = 0
0.113 0.225 0.281 0.309 0.363 0.393 0.400 0.402 0.385 0.344 0.333
0.20 0.40 0.50 0.55 0.65 0.75 0.80 0.85 1.00 1.25 1.33
B: for variable s;
0.30 0.60 0.75 0.82 0.89 0.83 0.80 0.76 0.67 0.67 0.67
0.156 0.312 0.391 0.430 0.456 0.416 0.400 0.384 0.346 0.342 0.333
0.25 0.50 0.62 0.69 0.77 0.79 0.80 0.81 0.82 0.71 0.67
0.38 0.38 0.38 0.38 0.41 0.47 0.50 0.53 0.58 0.52 0.50
0.62 0.62 0.62 0.62 0.59 0.53 0.50 0.47 0.42 0.48 0.50
t = 1 and p = 0
s
“1
n2
PI
P2
D,
D,
0.50 0.75 1.00 1.25 1.30 1.65 +
0.136 0.250 0.385 0.500 0.514 0.563
0.136 0.250 0.348 0.500 0.530 0.781
0.33 0.50 0.67 1.00 1.06 1.50
0.33 0.50 0.82 1.00 1.03 1.25
0.41 0.50 0.57 0.50 0.49 0.38
0.41 0.50 0.43 0.50 0.51 0.62
Table 2 The numerical evaluation of the Stackelberg equilibrium in (P,K) for variable t; s= 1 and p=O.
t
k,
=I
A2
Pl
P2
DI
D,
FAD,
0.20 0.25 0.35 0.40 0.42 0.44 0.45 0.80 1.00
0.34 0.44 0.68 0.82 0.88 0.94 1.00 1.00 1.00
0.199 0.208 0.229 0.240 0.245 0.250 0.255 0.400 0.385
0.542 0.493 0.410 0.369 0.352 0.333 0.352 0.400 0.346
0.62 0.62 0.62 0.62 0.62 0.62 0.67 0.80 0.67
0.80 0.79 0.65 0.60 0.58 0.56 0.57 0.80 0.82
0.32 0.34 0.37 0.38 0.39 0.40 0.38 0.50 0.58
0.68 0.66 0.63 0.62 0.61 0.60 0.62 0.50 0.42
0.05 0.22 0.46 0.54 0.56 0.57 0.62 0.50 0.42
equilibrium for the general setting, a rather tedious and very cumbersome if at all feasible task, we ran some numerical evaluation reported in tables 1 and 2. Table 1 presents the Stackelberg equilibrium in prices with maximal market coverage (exogenous), that is for ki= kj= 1. The values under Di are the quantity demanded of product i. It is interesting to note in table 1 that l As product differentiation increases, that is as t increases, the profit of the first-mover firm n, is increasing up to t =0.85 and then decreasing while
318
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
the profit of the second-mover n2 is increasing up to t =0.65 and then decreasing: there exists for each duopolists a ‘best’ level of product differentiation. This suggests that the duopolists would agree on the desirability of more product differentiation up to t = 0.65 but then disagree for intermediate values of t, that is for t E (0.65,0.85) and then agree again on the desirability of less product differentiation. When product differentiation is too low, competition is harsh and both firms would agree that it is too low. Large product differentiation (t is large) on the other hand means that consumers are very sensitive to product characteristics: if they cannot get a product (location) relatively close to their most preferred one (at their own location), then their net utility (surplus) s-pi-t Ix--xii, i = 1,2, decreases rapidly as they must incur high ‘transportation’ costs; in that sense, product differentiation is too costly for the firms and both would agree that it is too high. For intermediate values, they disagree; the leader would like more differentiation, the follower would want less. l z1
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
319
a measure of the spillover effect due to the choice by firm 1 of a partial market coverage. It is interesting to note in table 2 that for t >0.45, the Stackelberg equilibrium in prices and market coverages is the Stackelberg equilibrium in prices (that is, k: = kg = 1) and our previous characterization (table 1) holds; for t co.45 the two Stackelberg equilibria differ quite significantly. Hence, for relatively low product differentiation, that is for t s 0.45: as product differentiation increases, the leader’s market coverage k, is increasing;
although the leader’s profit is increasing and the follower’s profit is decreasing (to a minimum reached for t =0.44), followership is the preferred position; the leader’s price (markup) is constant and the follower’s price (markup) decreases with product differentiation. for t=0.20, the reduction of market coverage k, from full coverage (table 1) to the optimal 34% partial coverage (table 2) brings an increase in the leader’s profits from 0.113 to 0.199; and similarly for ~50.45; when the leader restricts its market coverage below 100x, _ the follower’s profit increases; - both prices increase, with the follower’s price overshooting the leader’s price. Clearly these characteristics of the equilibrium when strategies involve both prices and market coverages would have a significant impact on the subgame-perfect equilibrium choices of location in a two-stage model with locations chosen simultaneously or sequentially in the first stage. 5. The related literature and conclusion
When a firm (such as the first-mover above) chooses a partial market coverage even with a zero cost of market coverage (or information), in the sense that being known to more customers costs nothing, it chooses to restrict the number of its customers. In a sense, this firm is choosing to ration the potential customers by keeping them uninformed of its existence. We showed elsewhere2” that it may be optimal for a firm to behave in such a way in order to reduce the intensity of the competition it faces: by doing 24Boyer and Moreaux (1987, 1988, 1989). The first paper considers the case of a homogeneous good duopoly and the other two, the case of differentiated products. In the second paper we analyze the case where the strategy space of the firms is formed of a price and a production capacity; in the third paper we consider both the strategy space (price, production capacity) and the strategy space (price, serving capacity) and we compare the two types of equilibria (they are always different). When firms choose a price and a serving capacity (a number of customers), rationing appears in equilibrium if product differentiation is not too large; when firms choose a price and a production capacity, all equilibria involve a positive level of rationed customers, and this under very general demand conditions.
320
M. Boyer and M. Moreaux,
Strategic
market coverage in spatial competition
so, it can raise its price above the full market coverage equilibrium level and in spite of the fact that it has fewer customers, it makes more profit (compare table 1 and table 2 for t =0.20 and t =0.40, for example). There are good reasons for believing that partial market coverage, and more generally ‘rationing’, is indeed present in oligopolistic markets. A first reason is that the results are derived in structural models where all variables are under the control of a well-identified agent. This differs from the more usual ‘reducedform’ approach to spatial oligopoly theory, where the firms are modeled as choosing either a price or a quantity (in the present model, the choice of quantity is done through market coverage since each customer consumes either 0 or 1 unit of the good) but not both, supposedly once one variable is chosen, the other is determined ‘by the market’. It is certainly interesting to verify if the equilibria in structural-form models are similar to the equilibria in reduced-form models. Another reason for being interested in the phenomenon of rationing is that it can be disguised under different names. In the present context of spatial competition, this rationing strategy takes the form of partial market coverage; as mentioned above, this is a way to ration potential customers, by keeping them uninformed. But this rationing, without queues of unsatisfied customers, is part of the firm’s strategy to influence the behavior of its competitors. By choosing a partial market coverage, the firstmover is in a sense credibly signaling to the second-mover that he leaves those uninformed customers to the second-mover; this is likely to make this second-mover less aggressive and more accommodating, allowing the firstmover to raise its price above its otherwise equilibrium level. In equilibrium, it happens that both prices in the Stackelberg equilibrium in price and market coverage (table 2) are above (or equal to) their respective levels in the Stackelberg equilibrium in price alone (table 1). There are clearly many other particular forms of such rationing strategies: choosing a non-central location, a lower sales effort, a less favorable set of product characteristics, as compared in each case with the non-strategic level of the same variable. Those results can be related to a more general phenomenon analyzed by Fudenberg and Tirole (1984)25 but with some important differences. In the Fudenberg-Tirole model, an incumbent monopolist in period 1 is facing a potential entrant in period 2; he can advertise in period 1 and create a captive market forever of those customers reached by the advertising. If entry occurs, the incumbent and the entrant compete for the remaining market, but the incumbent cannot price-discriminate between its captive market segment and the duopolistic market. Fudenberg and Tirole show that the incumbent may choose to prevent entry by reducing its advertising level below the level he would choose as a protected monopolist: by so doing, the incumbent 25See also Schmalensee (1983), (1985), Henry (1988) and Fershtman
Gelman and Salop and Muller (1988).
(1983),
d’Aspremont
and
Gabszewicz
M. Bayer and M. Moreaux, Strategic market coverage in spatial competition
321
announces that he will be more aggressive in the duopolistic market and therefore depress prices sufficiently (in a subgame-perfect equilibrium) to make entry unprofitable. By reducing its captive size, the incumbent monopolist avoids being a ‘fat cat’ in period 2. In our case of a single-period spatial competition model with a first-mover and a second-mover, we noted already that the preferred position need not be leadership; whether it is or not depends crucially on the level of product differentiation. When the preferred position is followership (product differentiation is low), then it may be expected that the dominant firm will in fact be the second-mover rather than the leader.26 If that is the case, then the entrant will take the position of the first-mover; by strategically choosing a partial market coverage, it may make the dominant second-mover a ‘fat cat’ willing to accommodate entry: the roles are reversed.27 We mentioned in the introduction the papers which, to our knowledge, endogenize some form of market coverage variable, although none directly addresses this issue. Grossman and Shapiro (1984) characterize the symmetric Nash equilibrium of a circular city model where iz firms, equally spaced around the circle, choose both their price and their informative advertising budget. As in the present context, customers become informed of the existence (location and price) of a firm when they receive a message from it. The advertising technology is such that it is impossible for a firm to make sure that every customer did receive a message from it; the cost of advertising goes to infinity as the market coverage goes to 1; hence, no firm will choose to cover the whole market. Partial market coverage plays no ‘strategic’ - in the sense of modifying the competitors’ behavior - role in Grossman and Shapiro since the firms move simultaneously in a one-shot game. But it is interesting to relate our results to theirs in terms of the relationships between product differentiation (the number n of firms in their mdoel) and market coverage level (informative advertising in their model), prices or markups, and profits. Grossman and Shapiro showed that informative advertising and markups (and therefore profits) increase with product differentiation (a reduction in n). By making the market coverage a strategic variable (by making it endogenous in a Stackelberg market structure), we obtained that the first-mover’s market coverage (the second-mover’s one is always 100x, when it is costless) is non-decreasing with product differentiation (an increase in t) while markups may increase or decrease: when product differentiation is low, the leader’s markup is constant while the follower’s markup decreases with differentiation; then follows a range of product differentiation levels where both markups are increasing; a range where the leader’s markup decreases while the follower’s keeps increasing; and finally, “%ee Boyer and Moreaux (1987). “Somewhat in the spirit of Gelman
and Salop (1983).
M. Boyer and M. Moreaux, Strategic market coverage in spatial
322
competition
the markups decrease to their separate monopoly market levels. As for profits, they increase with product differentiation in Grossman and Shapiro; in our model with market coverage as a strategic variable, as long as the leader’s market coverage is less than 100x, the leader’s profit increases with product differentiation but the follower’s profit decreases with product differentiation, except when the leader’s coverage is very close to 100%. This indicates that the measurement of product differentiation (either through n as in Grossman and Shapiro, or t as in this paper) and the strategic aspect of market coverage or advertising (either through the symmetric Nash equilibrium as in Grossman and Shapiro, or the Stackelberg equilibrium as in this paper) are important determinants of the role of product differentiation in oligopolistic competition.
Appendix
Proof of Proposition 1. In fig. 2, the straight line C(x) of equation pi=pj+t-22tx is the solution to pi+t~‘=pj+t(l-_X)‘; curve E(X) is the solution to pi+ tx2 =s; MRC(xj is the marginal revenue function defined from C(x), that is MRC(xj =Pj+ t-44tx; MRE(xj is the marginal revenue function defined from E(x), that is MRE(xj =s--3tx’. Hence, the marginal revenue function MR(xj is given by MRE(xj for x
jjj=
0,
ifsz&t: otherwise,
and SO,for pj & ij, 28 the profit-maximizing pj-t+2t fQ= /
0,
28This value pj is always
price pi” is at the kink value pi:
112
( > ~
(A4
)
ifpj>S--t, otherwise,
less than s since -5t+4(t2+ts)‘~2-s=(3t+s)2>0.
64.2)
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
323
and the demand is given by x1 as long as x1
64.3) For pj~pj, the profit-maximizing p? is constant at s - tx$=$s, unless of course xg > 1, in which case pr=s--t. For a relatively low value of pj, the best-reply price pi is given by MRC(x) = 0 unless s<&t, in which case MRC(x) is already negative at the kink and therefore p: is at the kink value pi; for intermediate values, it is given by the kink point pi (itself a decreasing function of pj); for relatively large pj, that is above pj, it is given by MRE(x) = 0, in which case further increases in pj have no effect on pT which remains constant at +$s.For the case illustrated in fig. 2, pi(x2,pj) =(pj + t)/2, pi(x1,pj)=i)i, P~(X,,P~)=~S. Hence Proposition 1. Q.E.D. Proof of Proposition 2.
Clear from the text.
Proof of Proposition 3.
The equilibrium will be on the linear portion of the
reaction functions if pi
f-jj-t+2t
( ->“* y
which from (3.2) can be written as s+5t-4(t*+t$‘*
(
t
I’*
2’
a condition which is satisfied for t -C$L For larger values of t, the point (fijj,pT(jYj)) is above the 45” line and therefore the intersection with that line will be either in the concave portion of the reaction function or in the flat portion, the former case corresponding to values of t E [$s, $1 and the latter for values tz$s. To show that the concave portion of the two reaction functions when intersecting are in fact overlapping, let us consider pT(pj) for pj in the interval [pji,pj]. From (3.1): 11-2
pi*(pj)=pj-t++t
( > y
.
324
M. Bayer and M. Moreaux, Strategic market coverage in spatial competition
Suppose that pr(pj)E[di,~i]. similarly given by
Then the best reply by firm j to pi=PT(pj) is
S-+(Pj) t
Pj*(Pr(Pj))=PT(Pj)-t+2t C
=pj
l-
C_pY,*= +
I’2 >
if and only if [ s-p~+~-2~o”)“‘, t
Raising both sides to the power 2, we get
which proves that the concave portions when intersecting are in fact overlapping. The borderline between case (b) and case (c) in fig. 4 is obtained when there is a complete overlapping of the concave portions of the reaction functions, that is for ~Js, t) =$v; from 3.2) we have such a case if -St+ 4(t2 + z.Y)‘/~ =$s, that is if t = [( 19 +4 / 10)/27]sxO.978s. And finally case (d) Q.E.D. appears if pj>$s, that is from (3.5) if scat. of Proposition 4. Let i = 2 and j== 1. Suppose the firm 1 chooses first and is committed to its action (pl, k,) which is observed by firm 2 before the latter chooses its own (pz,k2). Given (pl,k,), the optimal price of firm 2 is given by (4.6), (4.9), (4.13), (4.16), (4.18); and from (4.3) its optimal coverage is given by
Proof
k,=min
l,~p,d&,p2,k,)
(A-4)
M. Boyer and M. Moreaux, Strategic market coverage in spatial competition
Let us assume that (pl, k,) satisfies p1 52s--t((2+k,)/k,)
325
so that
1-k t P2=3(P1+t)+k, ’ which together with (A.4) characterizes the second-mover’s best reply to (pl, k,). Under the above condition, the best-reply price pZ is less than s--t and therefore the residual demand to firm 2 is the second part of (4.2). Hence (A.4) can be rewritten as k,pl-~~z+t+(l-kl)
.
(‘4.6)
Using (AS) we can rewrite (A.6) as *.
(A.7)
Therefore k2 >O if (k,pf/8t +pl(f-;kJ +(tk,/8 + t/2kl - t/2)) is positive. This expression is a second-degree polynomial in p1 whose roots are always negative for k, E [O,l]; since it is equal to t(k, -2)2/(8k,) >O at p1 =0, we conclude that k,>O for all (pl, k,). Moreover, the polynomial value is non-decreasing with p1 and is therefore minimal at p1 =O; hence, we get ~
1
t(k, -2)2
.
Qk, (‘4.8)
Hence k, = 1 if p 5 t/S, from which we conclude that k, = 1 even in the worst case for firm 2, that is when (pl, k,) =(O, l), in which case we are in Case I as assumed: this means that the second-mover firm always covers the whole market when p is small. Q.E.D.
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