Indirect Competition with Spatial Product Differentiation
Author(s): Thomas E. Cooper
Source: The Journal of Industrial Economics, Vol. 37, No. 3 (Mar., 1989), pp. 241-257
Published by: Blackwell Publishing
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THE JOURNAL OF INDUSTRIAL ECONOMICS
March 1989
Volume XXXVII
0022-1821 $2.00
No. 3
INDIRECT COMPETITION WITH SPATIAL
PRODUCT DIFFERENTIATION
THOMAS E. COOPER*
Although two markets may appear to be separate, sometimes one firm
participates in both of them. That firm provides a link between the two
markets. Such a straddling firm transmits indirect competition from each
market to the other since its actions reflect competitive conditions in both
markets.In a model of spatial product differentiation,we show that indirect
competition may make a market perform significantly better than the
number of firms would indicate. Moreover, total consumer surplus
increases if two previously distinct markets are linked by a straddler.
Finally, we consider implications of these results for antitrust analysis.
I. INTRODUCTION
ATTEMPTS
to define a market for antitrust purposes are fraught with problems
because product characteristics vary almost continuously across goods. One
cannot neatly define a market as a group of products each of which is a good
substitute for all others in the group and a bad substitute for those outside.
Instead, the defined market typically includes some products which are fairly
poor substitutes for the good being studied but reasonably close substitutes for
other products already included in the market. Products near the market
boundary often lie in the intersection of two markets, for one could include them
in other markets as well. Differentiation in any product characteristic can create
this situation. For example, videocassette recorders (VCRs) could fall in the
market with stereos and compact disc players (the home entertainment market)
or with theatres (the movie market). The home entertainment and movie
markets would usually seem distinct, but VCRs fall in the gray area where the
markets overlap.
Firms that straddle two markets, as VCR-makers do, must consider both
markets when making price and production decisions. Because their reactions
reflect competitive conditions in both, these straddling firms effectively link the
markets together. Stereo-makers do not compete directly with theatres, but they
do compete indirectly through their impact on VCR-makers. This competitive
pressure from one market which is felt in another is what we call indirect
competition. The purpose of this paper is to study this indirect competition and
demonstrate its importance for market performance.
A spatial model of product differentiation is used to analyze indirect
* This paper was written while I was a member of the faculty at the University of Florida.
Comments from Sanford Berg,Roger Blair,Jonathan Hamilton, Richard Romano, Steven Slutsky,
and three anonymous refereeshave improved this paper. Any errors are, of course, my own.
241
242
THOMASE. COOPER
competition. There are two markets representedby a pair of tangent circles. A
straddlingfirmis located at the point of tangency, and there are two other firms,
each of which sells in only one market. Prices and profits of all firms depend on
conditions in both markets so the intensity of competition in one market does
indirectlyinfluence the price of the firmthat sells only in the other market. In the
example, stereo-makers limit movie prices by restraining VCR prices. More
interesting is the magnitude of this competitive influence. We show that a
market with two firms with extreme indirect competition provides as much
consumer surplus as an isolated market with five firms would.
Indirectcompetition may be beneficialin one market,but its overall impact is
not obvious because two markets are involved. The straddler transmits
competition from each market into the other. Just as severe competition spills
over into one market, so must a lack of competition flow to the other market,
thereby placing a drag on performance there. An overall assessment of indirect
competition must include both of these effects so a suitable reference case is
needed. For this purpose, a similar model with two completely separate
duopolies and no straddling firmsis used for the benchmark. Compared to this
reference case, the linked markets with indirect competition generate greater
aggregate consumer surplus whenever direct competition in the two markets is
different. Therefore, indirect competition increases what might be called
"averagecompetition" as it leads to increased consumer surplus.
These results can be related to the price discrimination literature by noting
that the separate duopoly referencecase is virtually identical to our model if the
straddlercan price discriminate. Comparing our model to the referencecase is
almost the same as examining the impact of a prohibition on price
discrimination by the straddler, as De Graba [1987] does in a similar model.'
For this reason, our results about the effects of indirect competition are similar
to the results of Schmalensee [1981] and Neven and Phlips [1985] concerning
when price discrimination lowers welfare.
The findings about indirect competition provide useful insight for studies
attempting to define markets or to analyze the structure-performance relation.
Some of the methods employed require that firms be classified as completely in
or completely outside a market.For example, Elzinga and Hogarty [1973] focus
on actual shipments of goods. If there is relatively little shipment of a product
into or out of a region, the area is considered to be a market. Firms shipping small
quantities into the area are completely excluded even though they probably
have some influence on competition. Research refiningor extending analysis of
the structure-performancerelation (Salinger[1984], Domowitz, Hubbard,and
1 De Graba's [1987] model has a structure similar to ours, but he uses linear markets instead of
circles.He focuses on the effectof pricediscrimination on the level of pricesand the location decisions
of firms. Our paper takes location as given and examines the effect of indirect competition on
consumer surplus.
Taking location as given, one could derive our results in either model, though the circle model is
analytically simpler. If one wanted to let location vary, he would need to use linear markets.
INDIRECT COMPETITION WITH SPATIAL PRODUCT
DIFFERENTIATION
243
Petersen [1986a, 1986b], and Harris [1986]) still relies on concentration ratios
definedfor particularmarkets. Since concentration ratios depend only on sales
of firms in the market, they cannot capture the discipline imposed by firms
outside the market. Leitzingerand Tamor [1983] take a step toward correcting
this weakness by using world productive capacity to adjust US concentration
ratios. Their approach appears to be an improvement, but merely including all
firmswhich might be in a market also presents problems.Treating all such firms
as full-fledged competitors is as inappropriate as excluding them entirely.
Dichotomous methods of market definition cannot deal well with the fuzzy
market boundaries arising from indirect competition.
Recently some researchers have relied on pricing patterns to determine
market boundaries.2 Since these price-based tests measure the influence on
pricingthat a particularfirmor region has, they can handle the fuzzy-boundary
problem. Some of the results clearly illustrate that boundaries are not neatly
defined. When Spiller and Huang [1986] calculate probabilities that two cities
are in the same market, they find only gradual changes in the probabilities
instead of dramatic drops at a boundary. Similarly, Horowitz [1981] relies on
confidence levels to define markets and shows that small changes in confidence
levels may significantly alter the defined market. In addition to illustrating the
hazy nature of market boundaries, this literatureyields some results consistent
with our theoretical model. Since prices of indirect competitors often move in
the same direction, markets defined by the similarity of price movements can
easily include indirect competitors.3 Indirect competition is probably quite
strong between relatively distant cities in the northeast region of the United
States. As a result, Spiller and Huang were unlikely to find sufficient pricing
differences to establish unequivocally that two cities there are in different
markets. In other cases, pricing tests may imply that indirect competitors are
loosely linked because the ties are weaker.This possibility could explain Slade's
[1986] finding that two moderately distant regions are loosely connected
markets. One final piece of supportive evidence is Uri and Rifkin's [1985]
finding that seemingly separate regions are in the same market. If the links
between indirect competitors are very strong, one could easily find that their
prices are quite closely linked despite wide geographic dispersion. Thus, these
price-based tests appear capable of identifying and measuring the varying
degrees of pricing discipline associated with indirect competition. Economists,
2 Horowitz [1981] and Stigler and Sherwin [1985] focus directly on similar price changes when
defining markets. Uri and Rifkin [1985] and Slade [1986], on the other hand, base their market
definitions on causality. They determine whether past observations of one price help explain the
current value of another price.
3While Uri and Rifkin [1985] and Slade [1986] do not explicitly check for similar price
movements, their method may include firmsin the same market on the basis of price movements. If
pricesadjustslowly, similarpricemovements could generatesufficientexplanatory power to include
both products in one market. Thus, at least partly, similar price movements are a source of market
definition here, also.
244
THOMASE. COOPER
however,may still want to devise ways to incorporate this varyingdisciplineinto
their analysis.
Policy-makers must also consider this effect when examining potential
antitrustviolations. For example, we show that size of a firmin-a market may be
a reflection of the competition it faces in another market. In particular, the
straddler's size is directly related to its competitive impact on a market.
Attempts to identify firms that are large but positive forces in the market could
look for indirect competition as supporting evidence. In addition, we find that
the effects of mergers depend on who the firms are. If the straddling firm is
involved, a mergeris likely to lessen the benefits of indirect competition. On the
other hand, mergers between indirect competitors (and no straddling firms)
seem to enhance performance in the absence of price discrimination.
The paper is organized as follows: Section II presents the spatial model and
shows that all equilibrium prices depend on parameters in both markets.
Section III contains the results showing the impact of indirect competition.
Using an isolated market for comparison, we determine how the benefits of
indirect competition vary with the size or competitiveness of the two markets.
Then there is policy-oriented analysis, looking at the meaning of market share
and the effects of mergers in the presence of indirect competition. Finally, the
concluding discussion summarizesthe results and raisesissues for furtherstudy.
II. THE MODEL
To examine indirect competition and its effects, we employ a spatial model of
product differentiation adapted from Salop [1979]. Two circles represent the
two markets (A and B) in which firms may sell differentiated products. Both
circles have circumferenceof one, and the circles are tangent. The firm which
straddles the markets is located at the point of tangency, and each market
contains one other firmlocated opposite the straddler.The locations of all firms
are fixed throughout this paper.4Figure 1 provides a graphic representation of
this situation. Firms A and B are located in markets A and B respectively,while
Firm S is at the intersection of the markets, where it can easily sell in both
markets.
Demand arisesfrom the utility-maximizingchoices of many consumers. Each
consumer has a preferredproduct representedby a point on one of the circles.
Assigning consumers to the circles by the position of their most preferred
product generates a density of consumers on the circles. We assume that the
resulting density is a constant dA in Market A and a constant dB in Market B,
where dA and dB may differ.If a consumer purchases a product from a seller at a
differentlocation, he must pay the transportation cost to move the product to his
spot on the circlein addition to the purchaseprice.This transportation cost may
represent an actual delivery cost in a geographic model or a utility loss in a
4As noted infootnote 1,De Graba's [1987] model is very similarto this, but he uses linearmarkets
instead of circles.
INDIRECT COMPETITION WITH SPATIAL PRODUCT DIFFERENTIATION
Firm A 4 t
lp
Market A
Market B
245
Firm E
Figure 1
Graphic Representation of Model
product characteristic model. We assume linear transportation costs so a
consumer must pay t'x for delivery of a product located x units of distance from
him in Market i (= A, B). Consumers on one circle who buy a product from the
other circle pay the delivery cost at the rate tA for transportation in Market A
and tBfor transportation in Market B.5 Each consumer has an inelastic demand
for one unit of the product available at the lowest deliveredcost p + tx, wherep is
the seller'sprice and tx is the transportation cost, provided the deliveredprice is
less than a reservation price R. The reservation price is great enough that each
consumer buys one of the three products. This assumption eliminates the
possibility that one firm is insulated from direct competition with the others.
Since competition with another firm is necessary for indirect competition, this
assumption simply focuses our attention on the case of interest here.6
Each firm has a territory of consumers to whom it makes sales. Since
consumers seek the lowest delivered price, the boundaries of these territories
depend on the prices.Let psbe the straddlingfirm'sprice and pibe the price of the
other firm in Market i (= A, B). Then in this model the resulting demand
functions are
DSL= _(dA
+dB)+pd
2
_ps(d
+-)
~~tA
+ pB
tB
for Firm S and
5An alternative approach is to prohibit purchases from travelling across circles. The approaches
give similar results, but this model is somewhat more tractable. Although the possibility of crossmarket sales exists, we shall usually suppress it in the exposition ofthe model and basic results. When
necessary, we shall check that such sales do not occur.
6 There is another issue of whether demand should be elastic. One could model elastic demand
without insulation of a firm from the others, but such a model is more complicated than this one. We
adopt the assumption of inelastic demand to simplify the analysis, though it makes welfare analysis
more difficult.
246
THOMAS E. COOPER
D' = ldi + (ps - pi) -
for Firm i (= A, B).
ti
We assume that all firms have a constant average cost of production, c.
Combining this with the demand functions gives the profit functions
[IS =
pS
-
c) L(d +d )+Pt
p(A
[IL
= (pi c)
+B)]
di+(pSpi)t]
+
tB
for Firm S and
for Firm i (= A, B).
The firms have Nash conjectures about the prices of other firms. Each firm
selects a price to maximize its own profit, taking the other prices as given.
Because of the structureof this model, Firm S must select a best response to the
prices of both other firms,while firms operating in a single market incorporate
only the price of Firm S into their pricing decisions. In the pure-strategy
equilibrium,7the Nash equilibrium prices are
(dA + dB) tAt
(dA +B d tA
d At +dBt
for Firm S and
for Firm i (= A, B).
These prices indicate that the model with tangent circle markets need not
generate unusual behavior. If the markets have identical transportation costs,
the resultant prices are the standard prices for a duopoly on a single circle,
p = C+ ~2t.Prices differfrom those values when the degree of substitutability or
competition in the two markets,as measuredby the transportationcosts, differs.
In the case with tA : tB, all three prices are different.As one would expect, for
ti > t1the ordering is pi > ps > pJ. The firm selling only in the market in which
goods are relatively poor substitutes has the highest price, the straddler has an
intermediate price, and the firm selling only in the market in which goods are
relativelyclose substitutes has the lowest price.Although piis the highest price,it
is lower than it would be in an isolated duopoly because of the indirect
competition from firm j.
7There is a possibility that either Firm A or Firm B would set a price low enough to drive Firm S
out of the market by stealing all its customers. This possibility arises only in the more competitive
market, the one with a lower transportation cost, t'. Provided the density in the more competitive
market is great enough relative to the density in the other market, such a plan is not profitable. If
market i is more competitive, then for any ratio of relative transportation costs h( = tI/ti), there is a
number k such that mill-price undercutting is no threat for d'I/di> k. Moreover, for all h > 0,
equilibrium exists if d' > 2di. Since there is a broad range of parameter values for which a purestrategy equilibrium exists, we assume it exists in order to simplify our analysis.
INDIRECT COMPETITION WITH SPATIAL PRODUCT
DIFFERENTIATION
247
In equilibrium, the prices of all firms depend on parametersin both markets
even though two firms try to sell only to customers in their separate markets.
Firm A must consider the sizes (dA and dB)and transportation costs (tA and tB) for
both markets when setting its price.The dependence of pA on featuresin Market
B arises indirectly. Firm A's reaction function, pA = 4tA + 2C + 2 pS, shows that
Firm A is directly concerned only with characteristicsof its market and the price
set by Firm S. But the straddler (Firm S) sells in both markets, so its price
depends on circumstances in both markets. In fact, if we let p[S denote the
straddler's single market best response to p', then we can re-write pS as
PS = caps + (1- oa)ps, where oc= (dAtB/(dAtB +dBtA)). That is, ps is a weighted
average of best responses to the prices with the weights depending on size and
transportation cost in each market. Because the straddling firm adjusts its price
to conditions in both markets, Firm A's response to ps indirectly involves a
reaction to circumstances in Market B. Consequently, the equilibrium value of
pA depends on conditions in Market B although those factors are not included in
Firm A's reaction function. Firm A must consider factors in Market B when
setting its price because its direct rival (Firm S) bases decisions on both markets.
The straddlingfirm serves as a medium to transmit effectsfrom one market into
another. Indirect competition is the influence of these factors which affect the
equilibrium value of price despite being absent from the reaction function.
In our model, a straddling firm which cannot price discriminate serves as the
medium to transmit indirect competition. One could model this effect
differently.Another approach is to permit firms in narrow markets to compete
indirectly by offering their products to the same consumers. A simple model of
this type is the standard Cournot model. In the equilibrium of that model, the
Lerner Index for firm i is (p - MC)/p = s,/g, where MC[ is the firm's marginal
cost, si is its market share, and e is the market elasticity of demand. This model
permits a distinction between direct and indirect competition. Increasing direct
competition by adding firmsto the market would lead to a lower value of si while
increasing indirect competition by having outside firms lower their prices to
these customers would alter the value of s. For example, if consumers purchase
two goods and have a CES utility function,s = a - t where a is the elasticity of
substitution between the goods and l is the cross-price elasticity of demand. As
increased competition in the outside market drives prices there down, it may
alter il in a way to increaseefor the product being studied.This alternativemodel
of indirect competition would be useful in situations in which there is no
straddlingfirm.If thereis a straddler,however, such a symmetricview of indirect
competition misses an important point. The impact of the indirect competitors
is much greater on straddling firms than on firmscompletely inside the market.
In our model, an attempt to raise all prices in a market by equal percentages
would cost the straddler more dearly than it would cost the other firm.
Consequently, even in markets with a low elasticity, firms may be incapable of
raising prices because the straddler, who must bear the burden of the plan, is
unwilling to go along with an increase. Some price increases that are profitable
248
THOMAS E. COOPER
for the group will not occur because the impact of indirect competition is not
equally felt.
III. IMPACT OF INDIRECT COMPETITION
A usefulway to measurecompetitiveness is to rely on the usual relation between
the number of firmsand the degree of competition. In our model, as the number
of firms increases (ignoring how fixed costs affect desired entry), the market
performanceapproaches the competitive level. Prices fall closer to the constant
marginal cost of production and, hence, consumer surplus rises. Since indirect
competition can generate these same improvements in performance without
changing the number of firms,we measure its effectsin terms of how many firms
would be necessary to yield the same performance. Specifically, we base our
evaluation of indirect competition on what we call the competitive-equivalent
number of firms.For any market situation, the competitive-equivalent number
of firms is the number that would yield the same level of consumer surplus in a
symmetric equilibrium in an isolated market. In the benchmark case of an
isolated symmetric equilibrium, the consumer surplus from n firms is
CS(n) = d(R-c-4
)
where R is the reservation price, d is the density of consumers and t is the
transportation cost, all in the market being studied. Then if a market has
consumer surplusequal to CS*, the competitive-equivalent number of firmsis8
n
5dt
4[d(R-c)-CS*]
This measure provides a neat measure of how competitive a market is. Higher
values of n indicate the marketgeneratesconsumer surpluswhich is closer to the
competitive level.
The competitive-equivalent number of firms is a better measure than
consumer surplus for our purposes.9 First, it is more useful for comparative
static exercises. Since n varies directly with CS*, one may think the measures are
equivalent. If one permits parameters to vary, however, n becomes a superior
measure of competitiveness. CS* would change with the parameters, but one
must compare the new value with maximum attainable consumer surplus to
8 It is quite likely that n will not be an integer. Let us simply treat n as a continuous variable to
simplify the analysis.
9 Our model makes consumer surplus a better measure than total surplus. The assumption of
inelastic demand throughout the relevant price range reduces the total welfare issue to one of
minimizing transportation costs. The model precludesany deadweight loss frommonopoly pricing.
Since total surplus would not depend on the level of prices, this welfaremeasure would not permit
ready generalization of the results. Consequently, we focus on consumer surplus as a welfare
measure reflectingcompetitiveness in a market, which is at least suggestive of total welfare.
INDIRECT COMPETITION WITH SPATIAL PRODUCT
DIFFERENTIATION
249
TABLEI
COMPARATIVE
STATICRESULTS
An
n2 (dA+dB)tBdB
At
20 (dAtB +dBtA)2
[
An n2 (dA+dB)dBtA
At
An
20 (dAtB + dBtA)2
n2 dBtB(tA 7tB)
t2L
20(dAtB+dBt)
An
ad`
n2
7
(dA+dB)tB
dAtB +B
d 1t<+d0t
(dA+d)t
B1
+d
J
dAt
1
dAtB(tB-tA)
20(d AtB
1
(dA+dB)tB
>0
B+
d >t0+dBt
d7-
+dBtA)2
[7(dA+d)t
L
dAtB +d BtAJ
<0
(t
s
s(tA
_ tA)
tB)
Note: "-" means "has the same sign as".
evaluate the change in performance. Since n automatically adjusts for parameter
shifts, it is a simpler measure to use. Second, n is a convenient measure for the
policy analysis of indirect competition. One can look at n to determine how
significant this influence is. If n is larger than the actual number of firms, then
indirect competition has increased the effective number of competitors. One
would expect behavior to be more closely linked to the competitive-equivalent
number of firms than to the actual number of firms. Finally, we can use n to
determine the limits of indirect competition. By taking limiting cases of
parameter changes, we can see how widely the effective number of firms may
vary in duopoly.
For convenience, consider the comparative static results when Firm B
provides indirect competition for Market A. Then the competitive-equivalent
number offirms is that number n which yields the same level of consumer surplus
in an isolated market like Market A as the level of consumer surplus actually
observed in Market A
~~~~~~~~~~~~
C
L
32l d-6
+B)tAtB
(A +Bd)2tB2tA1
dAtB +dBtA + 32 (dAtB+d BtA)2
Equating CSA and CS(n) implicitly defines n as a function of the parameters of
the model. Implicitly differentiating this expression yields the results presented
in Table I. Provided each firm has positive sales, the signs are as shown in the
table. '
The first two derivatives, an/@tA
and an/DtB, confirm our intuition about
indirect competition. Since the straddler faces "outside" competition in Market
B, behavior in Market A depends an circumstances in Market B. As one would
expect, linking Market A to a relatively more competitive outside market results
10 The bracketed term [7- ((dA + d B) tB)/(d AtB + dBtA)] is critical for determining the signs of these
expressions. It is easy to show that the assumption that all sell implies the expression is positive. If
Firm S is located at point 0 on A's circle, the point at which delivered prices of Firm S and Firm A are
+ dBtA). For this to be an interior value allowing Firm S to sell any
equals is 8-8((dA
+dB)tB)/(dAtB
units, the bracketed expression must be positive.
250
THOMASE. COOPER
in more indirectcompetition and, hence, more competitive pricingin Market A.
This notion is confirmed by the results, an/atA > 0 and On/OtB < 0. Since the
explanations are similar,let us discuss an/atA As tA rises,the products in Market
A become poorer substitutes. The natural response to this reduced
substitutability is to raise prices in that market.The straddler,however, cannot
freelyraise price because competitive conditions in Market B have not changed.
Knowing that Firm B will have no immediate desire to raise pB, the straddling
firmmust restrainitself,limiting the increasein pS. This restraintalso reducesthe
increase in pA because Firm A reacts to the limited increase in pS. If the market
were completely isolated and distinct from others, the price increases would be
greater. Indirect competition from Market B holds down the price increase,
thereby raising the competitive-equivalent number of firms.
Thus, Firm B effectivelydisciplines all firmsin Market A although it directly
competes only with Firm S and it has no desire to sell in Market A. An additional
reason for this result arises from the fact that, as tA rises, Market B becomes
relatively more competitive. Products in Market A become poorer substitutes
so the straddler places greater weight on pB when choosing its price. This
adjustment is natural since the penalty for deviating from the single-market
best-response price is greater in the market with closer substitutes. The effect of
the shift in weight becomes clear when one examines an/atA + an/t5B, which has
the sign of tB_ tA. If tB > tA, increasing transportation costs equally raises the
relative transportation cost in Market A, tA/tB. Firm S responds by shifting
weight in its pricing decision to the relatively more competitive market and,
thus, slows the priceincreasesin Market A. Therefore,n risesas both tA and tB do.
Obviously the logic extends to the case of tA > tB, the situation in which the
adjustmentby the straddlerhastens the price increases.In that instance, indirect
competition declines because Firm S becomes less concerned with Firm B;
therefore,the degree of competitiveness in Market A diminishes.
The effect of market size on indirect competition is also quite reasonable. A
relativelysmall outside market provides little discipline on a market.Again, the
resultdepends upon the importance the straddlerassigns to the two markets.As
the density of consumers in a market rises, having the "correct"price for that
market matters more to the straddling firm. Therefore, one would expect to
move closer to the single-market best-response price for the market with
growing density. Suppose Market B is relatively more competitive (tB < tA).
Then shiftingemphasis to Market A as dA rises results in higher values of p5and
pA and, hence, a smaller competitive-equivalent number of firms. Raising dA
relative to dBmakes indirect competition less important. If d5 rises instead, then
the stronger indirect competition improves performance in Market A. By
similarlogic, one can see that if Market B is relatively less competitive (tB > tA),
changes that make Market B smallerin comparison to Market A yield improved
performance.In this case the outside market hinders competition by pulling up
ps. Since indirect competition of this type harms performance, factors that
weaken this force improve performance.
.
pS
INDIRECT COMPETITION WITH SPATIAL PRODUCT
DIFFERENTIATION
251
Having seen that indirect competition affects behavior, one may wonder
whether this influence is significant.To evaluate the potential import of indirect
competition, let us determine limits for the competitive-equivalent number of
firms. Substituting the value of CSA for CS* in the expression defining n gives
14 (dA+d B)tB
F=40 7
L=0[7+14dA tB +dBtA
(dA+dB)tB
\(dAtB+dBt) JA
221-1
j
From the comparative static results, we know that indirect competition
becomes more intense as tB falls. Letting tB go to zero," we find
lim n= 55
tB
o
That is, as the "outside"market nears perfectcompetition, Market A generates
more consumer surplusthan it would with fivefirmssymmetricallylocated. This
duopoly could behave quite competitively because of indirect competition. A
firm in another market provides pricing discipline in the market.
On the other hand, Market B could also drag down performancein Market A.
Asymptotically, we find
lim n =
1503
tB _o
assuming d = 2dB.12 The assumption of an interior solution and the possibility
of selling across circles keep n from falling farther,for Firm S must select a price
that is low enough to retain some sales in Market A. 3 Despite these restrictions
in this model, the result demonstrates that weak indirect competition can drag
down performance in a market. Together these limits reveal that indirect
competition can have significant effects. Depending on its strength, indirect
competition can force a duopoly to performbetter than a market with five firms
or nearly as poorly as a monopoly. As a consequence, the actual number of firms
may be a very weak indicator of performance if indirect competition varies
substantially across markets. The effect does seem to weaken as the number of
firmsin each market rises.If we add to each market another firmwhich sells only
in that market, then the qualitative effects of indirect competition remain.
l As we drive tB to zero, there is a possibility that our proposed interior solution ceases to be an
equilibrium. In the more competitive market, the firm that sells only in that market may wish to
underpricethe straddlerin order to gain access to the less competitive market. Clearly our limiting
results are valid only if undercutting the straddleris not profitable. As discussed in footnote 6, there
are relative densities which ensure this potential problem does not arise. In particular,for all tB > 0,
the interior solution is an equilibriumfor dB > 2dA.This condition ensures that Firm S is sufficiently
aggressive in Market B to make mill-price undercutting unprofitable for Firm B.
12 The assumption on densities ensures that there is an interior solution. Clearly, as dArises, the
value of n also rises.
13 Without these restrictions, the limit would have to be very near to one. As competition in
Market B diminishes, Firm S would willingly price itself out of Market A in order to reap great
returns in Market B, leaving a virtual monopoly in Market A.
252
THOMAS E. COOPER
755L
n for
Market A
2
1103
I
Figure 2
Effect of Indirect Competition on Competitive-Equivalent
Market A
tA/tB
Number of Firms in
However, the upperbound on n rises only to 6A, so the maximum increasein the
effective number of firms declines quickly.
Figure 2 summarizesthe impact of indirectcompetition on n,the competitiveequivalent numberof firmsin Market A. The ratio of transportationcosts (tA/tB)
serves as a measure of indirect competition. Since 1/t1measures substitutability
of products in Market i, tA/tB measures substitutability in Market B relative to
Market A. Direct competition is stronger in the market with lower
transportation costs so higher values of tA/tB represent relatively more intense
direct competition in Market B. This relatively strong competition spills over
and improves performancein Market A. Using the results in Table I, one can
show that higher values of n result from any changes in tA and tB that increase
tA/tB, as indicated by the upward-sloping curve in Figure 2. To interpret the
graph,one must note that an isolated duopoly, one with no indirectcompetition,
has a competitive-equivalent number of firms equal to two. If n > 2, then
indirect competition has improved performance in Market A. Conversely, if
n < 2, then the net effect of influence from Market B is poorer performancein
Market A. From the graph one can see that indirect competition improves
performanceif the linked, outside market is internally more competitive (has a
lower transportation cost) and it lowers performanceif the outside market is less
competitive. One can easily determinewhetherindirectcompetition is a positive
or negative force in the market by comparing price-cost margins. Indirect
competition improves performance in Market A if the price-cost margin for
Firm B is lower than that of Firm A, and it worsens performanceif Firm A has the
INDIRECT COMPETITION WITH SPATIAL PRODUCT
253
DIFFERENTIATION
lower margin. Because the situation is symmetric, indirect competition
deteriorates performance in one market whenever it improves performance in
the other.
Since indirect competition can increase or decrease the level of consumer
surplus, its net effect is not obvious. The ambiguity arises from the fact that
indirect competition flows both ways between markets. If a relatively
competitive situation in Market B improves performancein Market A, then the
less competitive situation in Market A must be dragging down consumer
surplus in Market B. To determine which effect dominates, let us compare
consumer surplus in the two linked markets with its value in two separate
duopolies with similar market characteristics. Based on the earlier expression
for consumer surplus in Market A, we have total consumer surplus of
A
CSA + CSB=
CSA+CSB
(d +d B )[R-C-32
13(dA+dB)tAtl
dAtB+dBtA
J
32(dt+dt). d
7_d___
t)
In a spatial duopoly on a circle, consumer surplus is CS = d(R - c - 8t).
Therefore, total consumer surplus if our two markets were separate duopolies
with no straddling firm would be
CSA+
Whenever
CSB
the
CSA + CSB > CS
= (dA+d B)(R-C)_5(dAtA
transportation
A
+ CS
B.
costs
That is, indirect
+dBtB)
differ
in
competition
the
two
markets,
lowers price averaged
over both markets, resulting in greater total consumer surplus. This result is
quite reasonable if one recalls how the straddling firm chooses its price. The
straddler sets ps equal to a weighted average of the single-market best-responses
to its rivals' prices, with the weights reflecting competitiveness in the markets.
Since the weight is greater and the rival's price lower where competition is
stronger, the lower rival price dominates the straddler's pricing decision. This
domination leads to lower average prices and higher total consumer surplus
than in completely separate markets.
The results can also be explained in terms of price discrimination since we
compare the case in which the straddler cannot price discriminate with the case
in which he can. Our result resembles Schmalensee's [1981] result that price
discrimination by a monopolist lowers welfare if output does not increase.'4
The intuition that price discrimination pushes apart marginal valuations of
consumers applies to this model, though it appears as a shift in the boundaries of
14Schmalensee's results do not apply perfectly here for two reasons. First, he considers a
monopolist while our straddler faces rivals in two markets. The presence of competition affects the
firm's pricing strategy, but the firm still changes prices in the same directions as a monopolist would if
it faced similar markets with corresponding elasticities of demand. Consequently, the intuition
based on directions of price changes seems appropriate, for the responses rivals make serve to
dampen, not eliminate, the allocative effects. Second, we use consumer surplus as a welfare measure
because total surplus depends on relative rather than absolute prices. Since profit increases under
discrimination in his model, his result that welfare falls necessarily implies consumer surplus falls.
254
THOMASE. COOPER
firms' territories.15Linking the markets produces less divergence in marginal
valuations relativeto the valuation of alternativesavailable, therebyresultingin
greater total surplus. Since Firm S earns less than it would by discriminating,
consumer surplus must also rise from tying the markets together, provided
profits of the other firms do not rise substantially. Alternatively, one could
compare our finding to the results of Neven and Phlips [1985]. They find that
allowing duopolists to price discriminate in two markets generates less total
surplus than prohibiting price discrimination. Our case in which markets are
completely separate corresponds to their price discrimination case, but our
model with indirect competition is the same as allowing only one of the
duopolists to discriminate.Since this case falls between theirs,one would expect
it also to be better than complete price discrimination (in terms of surplus).
The notion of indirect competition also has important implications for
antitrustpolicy. We have already found that the number of firmsis a poor proxy
for competition because indirect competition can vary the competitiveequivalent number of firms significantly. Another implication is that the
antitrust assessment of a situation depends on whether a firm is a straddler or
not. Consider the case of a large market share as an indicator of monopoly
power. In a market the point at which delivered prices are equal is the point of
market division. The share of Firm S in Market A is
M
1 (d A+ dB) tB 16
=
4
dAtB+
dBtA
This measuremoves in the same direction as n in response to changes in tA, tB, dA,
or dB. For example, as tA rises (or tB falls),indirect competition from Market B
becomes more intense. By inducing Firm S to lower its prices, the more
competitive situation in Market B raises the competitive-equivalent number of
firms in Market A. Not surprisingly, the lower price by Firm S also raises its
share in Market A, so the straddler's market share moves with n. Similarly,
changes in dA or dB that raise the competitive-equivalent number of firms
achieve this effectby making Firm S more aggressive,which increasesits market
share. A large share for a straddler indicates that firm is quite competitive and
performanceis good, but the influencemust be reversedfor a firmwhich is not a
straddler.Firm A's market share is inversely related to performancemeasured
by n because it varies inversely with Firm S's share.To inferperformancefrom a
firm'smarket share, one must consider whether that firm is a straddler or not.
Recognizing that a firm straddles two markets is also important for merger
5 As Schmalensee notes, this intuition is properly attributed to Joan Robinson [1938].
16 This market share assumes a geographical interpretation of the model since it includes only
sales in Market A. If the differentiationexists only in product characteristic space, one could not
actually distinguish sales in one market from those in the other. Then the market share of Firm S
would be its totalsales as a ratio of the sales of FirmA and Firm S. This approach yields similarresults
except for the effectof changes in dAand dBon market share of Firm S. The differenceoccurs because
the share includes sales in both markets while the exogenous change affects only one market.
INDIRECT COMPETITION WITH SPATIAL PRODUCT
DIFFERENTIATION
255
policy. The effectsof a mergerdepend on whether one of the firmsis a straddler.
Suppose an apparent conglomerate merger actually involves two indirect
competitors. After Firm A and Firm B merge, assume they cannot price
discriminate (so they will change prices). The new situation is completely
symmetric with each firm setting its price at
,(dA + dB) tAtB
+ 2 dAtB +dBtA
PC
the pre-merger value of
Since each firm sells to the nearest half of the
pS.
consumers, consumer surplus in market i is CS' = di(R - p - (t'/8)). Consumer
surplus rises in the less competitive market and falls in the other market, for the
new price is between the pre-mergervalues pA and pB. Overall, the mergerraises
consumer surplus, as CSA + CSB > CSA + CSB. Since consumers benefit,
society must benefit from a merger of indirect competitors. The straddler's
profits do not change, and the other firms must benefit in order to merge. Their
gain must come from economies not included in our model because they
sacrifice pricing flexibility.'7 Unless they enjoy economies in production or
promotion, indirect competitors have no incentive to merge. Thus, a merger
involving indirect competitors is socially desirable.
Now consider merging Firm S and Firm B, another merger which appears
harmless for Market A. Obviously consumer surplus falls in Market B, but one
may wonder what the effect is on Market A. The post-merger prices in this
market are
pA =
tA
C +_+
3
A
dBtA
adt
nd
C +_+2
p
dBtA
The resulting consumer surplus is
= R-C-tA
A~ARc~tAI
_1
dBtA
dB2tA
dA +8dA2
Consumer surplus in Market A falls with the merger (CSAC>CSA) provided
Market B is not almost five times as large (dA > ]7dB).18 Not only do these
consumers lose, but those in Market B also face higher prices if tA exceeds tB.
Consequently, a merger of a straddler and an indirect competitor may hurt
consumers in one or both markets. This result differs from the case in which
indirect competitors merge, for then overall consumer surplus rises. Whether a
17 If we allow economies not specifiedin the model, then this mergeris also socially desirableif the
firmscould price discriminate.They would keep prices unchanged so there must be cost reductions
that provide the impetus for integration.
18
CSA > CSA iff(dAtB)2 (dA
+dBtA2(
For all dA >
j7dB,
3dA2 +dAdB_
7dB)
dB2)
this inequality holds.
+dAtAtB(
0
dAdB +dA2_
2dB2)
256
THOMAS E. COOPER
merging firmis a straddleris an important factor for determiningthe effects of a
merger. Of course, these antitrust results apply only if competition (direct and
indirect)actually has the essential featuresof our model. The one criticalfeature
seems to be that increased competition from Market B induces the straddlerto
behave more competitively toward Firm A.
IV. CONCLUDING DISCUSSION
Whenever one defines a market, the boundaries of the market are somewhat
fuzzy. If any of the products included faces competition from an excluded
product, then all the products in the market face indirect competition from the
outside firm. We have shown that indirect competition may impose significant
discipline on the pricing policies of firmsin a market. This additional constraint
on pricing freedom may improve performance as much as would injecting
several more firms. On the other hand, if there is very little competition from
outside firms, then market performance will be worse than one would expect.
Incorporating this influence could improve antitrust policy. When trying to
determine if a market is workably competitive, policy-makers can look for
evidence of strong indirect competition. If, for example, outside competition
causes a straddling firm to have a large market share, then the market is
performing better than the number of firms would indicate. In addition,
appropriate merger policy often depends on whether a straddler is involved
because the merger may lessen indirect competition. These results hold if
competition is well represented by our model. For cases in which our model is
inappropriate,other models like the one discussed at the end of Section 2 may be
able to give equally valuable (though possibly different) results. Indirect
competition may be less important for antitrust analysis in other models, but
our results indicate it may be worthwhile to investigate the topic further.
Some important issues remain to be solved before indirect competition
becomes very useful for policy. First, to apply this theory to specific cases, one
needs a method for identifying sources of significant indirect competition. One
must be able to recognize an outside market which provides indirect
competition and assess the strength of the resulting indirect competition.
Already we know that it is stronger if the outside market is larger or more
competitive, but one needs a simple, operational rule for deciding whether this
effect is important. Second, one needs a quantitative measure of indirect
competition. Such a measure would permit economists to include indirect
competition in empirical studies of markets and help them to determine how
much pricing discipline it provides. One candidate, the industry elasticity of
demand, is only a rough measure of indirect competition because it ignores the
asymmetric impact of indirect competition. One would like to have a more
complete measure of the discipline provided by indirect competitors. While the
model in this paperneatly demonstrates the significanceof indirectcompetition,
it is too specific for generating these broad measures. A fruitful direction for
INDIRECT COMPETITION WITH SPATIAL PRODUCT
DIFFERENTIATION
257
future researchis to consider indirect competition in a general model suited to
deriving these desirable measures and drawing conclusions with broader
applicability.
ACCEPTED JUNE 1988
THOMAS E. COOPER,
Business Program,
ArkansasCollege,
Batesville,
Arkansas72501, USA
REFERENCES
on Competition Between
National and Local Firms', Rand Journal of Economics, 18, pp. 333-347.
DoMowITZ, I., HUBBARD,R. G. and PETERSEN,B. C., 1986a, 'Business Cycles and the
Relationship Between Concentration and Price-Cost Margins', Rand Journal of
DEGRABA, P. J., 1987, 'The Effects of Price Restrictions
17, pp. 1- 17.
Economics,
B. C., 1986b, 'The Intertemporal Stability
DoMowITZ, I., HUBBARD,R. G. and PETERSEN,
of the Concentration-Margins Relationship', Journal of Industrial Economics, 35, pp.
13-34.
ELZINGA,K. G. and HOGARTY,T. F., 1973, 'The Problem of Geographic Market
Delineation in Antimerger Suits', Antitrust Bulletin, 18, pp. 45-81.
HARRIS,F. H. DE B., 1986, 'Market Structure and Price-Cost Performance Under
Profit Risk', Journal
Endogenous
of Industrial
HOROWITZ,I., 1981, 'Market Definition
Approach',
Southern
Economic
Journal,
Economics,
35, pp. 35-59.
in Antitrust Analysis: A Regression-Based
48, pp. 1-16.
J. J. and TAMOR,K. L., 1983, 'Foreign Competition
LEITZINGER,
Journal
of Law and Economics,
in Antitrust Law',
26, pp. 87-102.
NEVEN,D. and PHLIPS,L., 1985, 'Discriminating Oligopolists and Common Markets',
Journal of Industrial Economics,
ROBINSON, J., 1938, The Economics
34, pp. 133-149.
of Imperfect Competition
London).
SALINGER,M. A., 1984, 'Tobin's q, Unionisation
Relationship',
Rand Journal
of Economics,
(Macmillan and Co., Ltd.,
and the Concentration-Profits
15, pp. 159-170.
SALOP,S. C., 1979, 'Monopolistic Competition with Outside Goods', Bell Journal of
Economics, 10, pp. 141-156.
SCHMALENSEE,
R., 1981, 'Output and Welfare Implications of Monopolistic ThirdDegree Price Discrimination', American Economic Review, 71, pp. 242-247.
SLADE,M. E., 1986, 'Exogeneity Tests of Market Boundaries Applied to Petroleum
Products',
Journal
of Industrial
Economics,
34, pp. 291-303.
SPILLER,P. T. and HUANG,C. J., 1986, 'On the Extent of the Market: Wholesale Gasoline
in the Northeastern United States', Journal of Industrial Economics, 35, pp. 131-145.
STIGLER,G. J. and SHERWIN,R. A., 1985, 'The Extent of the Market', Journal of Law and
Economics,
28, pp. 555-585.
URI, N. D. and RIFKIN, E. J., 1985, 'Geographic
Deregulation',
Review of Economics
and Statistics,
Markets, Causality and Railroad
67, pp. 422-428.