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A model of duopoly and meeting or beating competition
International Journal of Industrial Organization 5 (1987) 399-417. North-Holland
A MODEL
OF DUOPOLY AND MEETING COMPETITION
OR BEATING
T e r r e n c e M. B E L T O N * Federal Reserve System, Washington, DC 20551, USA
Final version received November 1986 This paper analyzes the use of 'meeting and beating competition' clauses in duopoly markets. A differentiated-products, price-setting model is considered, in which firms have the option of precommitting themselves to contracts which bind their own price to be some prespecified function of their competitor's price. The analysis establishes conditions under which these contracts facilitate oligopoly coordination, under which they result in price discrimination, and under which they can actually be procompetitive.
1. Introduction It has become a fairly c o m m o n practice in a n u m b e r of industries for firms to write explicit or implicit contracts that relate their o w n price to that of a competitor. Advertising guarantees that one firm will m a t c h o r beat by some specified a m o u n t the price of a c o m p e t i t o r are the most c o m m o n form of these contracts. Explicit contracts given at the time of p u r c h a s e that guarantee a rebate to the buyer in the event it finds a lower-priced c o m p e t i t o r in the near future are a second c o m m o n form. These implicit and explicit contracts - generally referred to as meeting or beating c o m p e t i t i o n clauses - are seen m o s t frequently in local retail markets and have also appeared at various times in the airline, natural gas, b a n k i n g and other industries, While the use of these clauses appears fairly widespread, there exists, surprisingly enough, little research on their effects, F i r m s that c o m m i t themselves to meeting or beating c o m p e t i t i o n clauses in o l i g o p o l y markets are c o m m i t t i n g their o p p o n e n t s (over s o m e subset of the price space) to a price leadership role. As one m i g h t expect, this choice of roles has i m p o r t a n t implications for b o t h o l i g o p o l y o u t c o m e s and c o n s u m e r welfare. *I would like to thank Bob Avery, Mark Bagnoli, Allen Berger, Larry Benveniste, Jeff Daskin, Mike Goldberg, Ed Green, Meg Guerin-Calvert, Gerald Hanweck, Steve Salop, Hal Varian, and two referees of this journal for helpful comments. The views expressed in this paper are those of the author and do not necessarily reflect those of the Board of Governors or its staff. 0167-7187/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
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Some research on the effect of meeting competition clauses (MCC) can be found in a recent paper by Salop (1982). ~ That analysis, which investigates a number of different pricing contracts commonly used by firms, suggests that meeting competition clauses may facilitate collusive behavior in oligopoly markets. Such behavior can result from two distinct effects of MCCs. First, they may operate as information exchange devices between potential colluders by rewarding consumers for reporting secret price discounts of a competitor. This information exchange reduces incentives for secret price discounts and thereby enhances the stability of tacit agreements. Second, MCCs can operate as incentive management devices by altering the payoff matrices of firms in ways that make non-cooperative equilibrium yield cooperative outcomes. Salop's analysis demonstrates that when duopolists' choice sets and payoffs follow a simple Prisoners' Dilemma structure, an MCC transforms the joint profit maximum into a Nash equilibrium. As a result, MCCs can result in non-cooperative firms achieving collusive outcomes, In this paper, we attempt to generalize and extend the existing literature on the use of meeting and beating competition clauses in three different directions. First, we consider under a differentiated-products duopoly model, the conditions under which MCCs result in cooperative outcomes. The analysis, which focuses primarily on the incentive management effects of MCCs, establishes conditions under which MCCs raise price above noncooperative levels, under which they result in collusive prices, and under which they might actually be procompetitive, Second, the analysis considers the optimal design of a more general class of contracts which we term competitor-based price clauses (CPC). These clauses, which include both meeting and beating competition clauses as special cases, are defined to be any binding commitment made by one firm to set its own price equal to some prespecified function of another firm's price. When designed optimally by a firm, CPCs can be shown, under certain conditions on the information structure and demand and cost functions of the duopolists, to have the following properties. (i) They are linear, i.e., of the form p~ =a+bp2, where Pl denotes the contracting firm's own price and Pz the price of its competitor. (ii)They result in prices for both firms which exceed prices under a Nash-Bertrand outcome and which, from the producers' point of view, are Pareto optimal. 2 (iii) The outcome of the optimal price clause (under certain conditions), weakly dominates all other duopoly outcomes the contracting firm could expect. A final result established in the analysis is to demonstrate that competitor1Related research on most-favored customer clauses is found i n Cooper (1986). 2Throughout the paper, we use the term Pareto optimal to characterize outcomes which are efficient from the producers' point of view. An outcome is Pareto optimal if there is no other feasible outcome which increases both firms' profits,
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401
based price clauses can be used to price discriminate between informed and uninformed consumers. When a firm commits itself to both a price clause and a posted price, consumers who are uninformed about the competitor's price will, in some cases, be charged a higher price than informed consumers. Thus, under certain circumstances, competitor-based price clauses allow the contracting firm to segment the market and capture full monopoly profits from the uninformed consumers and Pareto optimal duopoly profits from the informed consumers. The attention given to the potential anticompetitive effects of competitorbased price clauses in this paper should not be interpreted as precluding other possible reasons for their use. A central assumption in the current analysis is that consumer demand is unaffected by the mere presence of CPCs. In reality, there are at least two cases where this assumption may not hold. First, it may often be the case that meeting or beating competition clauses are viewed by uninformed consumers (correctly or incorrectly) as implying that the contracting firm's posted price is in fact the lowest price in the market. By signalling to consumers some information about relative prices, MCCs could induce uninformed consumers not to incur the search costs involved in pricing a competitor's product and instead purchase the contracting firm's product. Second, when CPCs are retroactive, relative demand for the contracting firm's product can be increased by eliminating the risk associated with future price decreases of a competitor. By guaranteeing that a competitor's future price decreases will result in a rebate to the buyer, a retroactive CPC can increase the incentives of consumers to buy now (from the contracting firm) rather than later. Since these increases in demand come at the expense of one's competitor, it is clear that CPCs need not be purely anticompetitive in their effects. The potential benefits resulting from these effects of CPCs should be weighed against any anticompetitive impacts analyzed here.
2. Duopoly coordination and meeting competition clauses The commitment to match the price of a competitor is probably the most commonly observed competitor-based price clause. This section analyzes the potential effects of these clauses on duopoly outcomes. For the moment, we focus only on the case where all consumers are informed of the announced price policies of each firm, and are able to costlessly enforce any price contract the firm might choose. Consider an industry composed of two firms producing differentiated products 3 where firm 1 is the incumbent firm in the industry and firm 2 is a 3While in practice, MCCs seem most common when products are physically identical, the products are still virtually always differentiated due either to Iocational differences or quality of service differences between sellers,
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402
new entrant. Let P=(Pl, P2) denote the vector of firm prices and qi(P) the demand function for the ith firm's product, In section 3 of the paper, we disaggregate q~(p) into the demand of two types of consumers and present a more formal set of assumptions on the structure of demand. F o r the purposes of this section, however, we need only to assume that each firm's output is decreasing in its own price and increasing in its competitor's price so that the goods are substitutes. Specifically,
(A1) - - < 0 ,
>0
for
q~(p)>O,
i=1,2,
i#j.
OP~ Denote C~(q~) as the cost function of each firm (assumed to be twice differentiable), and rci(p)=piq~-Ci(q~(p)) as firm i's profits. Letting subscripts on 7z denote partial derivatives with respect to the prices, the following three conditions on C i and n are assumed to hold throughout the analysis:
> O, dZCi(qi)>o dq~
(A2)
for
qi>O,
i=1,2,
(A3) 7zii+TzljO,
i=1,2,
i#j,
(A4)
i=1,2,
i#j.
rclj>0
forq~(p) andq2(p)>O,
Assumption (A2) implies increasing marginal cost whenever output is positive. Assumptions (A3) and (A4) together indicate that r~i is concave in Pi and that equal price increases by each firm are less profitable for firm i, the higher is pi. ( A 3 ) a n d (A4) are sufficient conditions for a unique N a s h equilibrium to exist and for the firms' Bertrand type price reaction functions to be upward sloping. 4 These conditions facilitate comparison of the competitor-based price clause equilibrium with the non-cooperative outcome that might occur if MCCs were not used. To examine the impact that MCCs have on equilibrium prices, we define the strategy spaces of the two firms as follows. In the period prior to the entry of firm 2, the incumbent firm (firm I) chooses whether or not to a d o p t an MCC. 5 Firm 2 then makes its entry decision and also chooses whether or not to adopt an MCC. F o r each firm, the M C C is defined to be a binding advertisement which contains (1) a posted price pi, and (2) a guarantee that at the time of purchase if ~>pj, i@j, then the actual price charged to '*The Nash equilibrium concept referred to in this paper is the Bertrand equilibrium extended to differentiatedproducts. ~Section 2 expands the allowable choice of contracts to include both meeting and beating competition clauses,
T.M. Belton, Meeting or beating competition in duopoly
403
consumers by firm i will equal pj. Following the decision on whether to adopt an MCC, prices are set in the following way. If both firms advertise an MCC in the pre-sales period, then the actual price charged to consumers by both firms will equal min{pl,/52}. If firm j does not adopt an MCC, then it is free to choose any pj>O in period 2. Hence, if firm i chooses to advertise but firm j does not, then in period 2, firm j's best reply will solve maxnJ(p) p~
subject to
pi=pj
if Pj
=/~
otherwise.
(1)
By advertising an M C C in the pre-sales period therefore, the firm is assumed to commit itself both to a posted price,/5~, and a p~=pj reaction function for all pi
Proposition 1. Let the costs of each firm be identical and let demand be symmetric in the sense that (i)
qi<(=)qj
if
pi>(=)pj,
(ii)
OQ(pJ)/Opl =8Q(pS)/Op2,
and
where Q = q l +q2 denotes industry demand and pS is the price vector which maximizes joint profits. Then (1) it is optimal for either firm to unilaterally commit itself to an M C C with posted price ~i=p s, and (2) the M C C results in the collusive outcome p = pJ. Proof
See the appendix.
Proposition 1 provides a set of sufficient conditions under which each firm has an incentive to unilaterally commit to an MCC that results in the collusive outcome. The class of demand functions considered are those for
404
T.M. Belton, Meeting or beating competition in duopoly
which (1) the two firms share the market equally when their prices are equal, and (2) when their prices are not equal, the high price firm's market share is smaller. The importance of symmetric demand functions in Proposition 1 lies largely in the fact that when conditions (i) and (ii) of the proposition hold, an MCC constrains both firms to split the market evenly. In this sense, Proposition 1 is consistent with the findings of Osborne (1976) and Spence (1978) who establish that strategy functions chosen by non-cooperative firms which attempt simply to maintain market share can result in collusive outcomes. When the conditions of Proposition 1 hold, an MCC is in fact a strategy which maintains equal market shares and accordingly facilitates oligopoly coordination in much the same way as the Osborne and Spence reaction function equilibria do. In practice, MCCs are frequently observed in industries where products are differentiated due to location or advertising and where the conditions in Proposition I are not met. To examine the impact of MCCs in the more general case when demand is not symmetric, let pn denote the noncooperative Nash-Bertrand equilibrium for a single period model in which firms are price setters. In Proposition 2, we compare pn to the equilibrium that results when each firm's strategy space is expanded to include MCCs. Proposition 2. (a) There is never an equilibrium in which neither firm adopts an MCC. (b) Let p "< i = p j". Then if an M C C is chosen only by firm i, it results in equilibrium prices Pl > P~ and Pz > P~. (e) There do exist equilibria in which p~ > p~ and an M C C is optimal for firm 1 that results in Pl
The proofs of parts (a) and (b) of the proposition are in the appendix. Part (c) is shown below by an example. Several properties of equilibrium are examined in the proposition. First, the proposition states that it is always rational for at least one firm to adopt an MCC. This is because an MCC always allows at least one firm to unilaterally raise its profits above ni(pn). Second, if the firm adopting the clause is the firm with the lowest price in a Nash-Bertrand equilibrium, prices of both firms will always rise as a result of the MCC. The firm with the lower price at p" is the firm with lower marginal costs or lower demand at equal prices. Hence, part (b) of Proposition 2 suggests that when an MCC is adopted only by the firm with lower marginal costs (or lower demand), prices of both firms will rise above non-cooperative Nash-Bertrand levels. Finally, part (c) of the proposition indicates that prices do not always rise as a result of MCCs. A necessary condition for pi
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T.M. Belton, Meeting or beating competition in duopoly
Fig. 1 illustrates the case where an M C C leads to a price increase by both firms. In this example, firm l's profits are maximized by adopting an M C C with posted price /51 =p'i. 6 Given an M C C by firm I moreover, there are no incentives for firm 2 to also adopt one. 7 Instead, 2's best reply is to play the role of Stacketberg leader and select p2=p'2 with equilibrium outcome p'. Because 2's N a s h - B e r t r a n d reaction function is upward sloping, this results in higher prices and profits for both firms. It should be noted that there are two different reasons in this model why MCCs can have the effect of raising prices. First, like other binding commitments studied in oligopoly theory, these contracts can facilitate oligopoly coordination by binding firms to actions which, ex-post, might not be optimal. In this model, when a firm (e.g., firm 1) chooses an MCC, it is committing itself to a particular posted price for all p2>/51. When this commitment precludes firm l from making a competitive response to firm 2's high price choice, it will increase firm 2's incentive to choose the cooperative price. This incentive is enforced both by the commitment to the posted price and by the commitment to match one's competitor whenever P2
/
P,~P,
t
I p.
I p'
1
P1
Fig. 1 6Firm 1 is indifferent between adopting and not adopting an MCC. If 1 does not adopt an MCC, firm 2 will adopt with posted price iO2=p~. This would yield the same equilibrium outcome p'. 7Firm 2 will be indifferent between adopting an MCC with posted price /72=p~ and not adopting an MCC and simply selecting P2 = pt, in period 2.
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T.M. Belton, Meeting or beating competition in duopoly
A second reason why these contracts may result in higher prices is the fact that when only one firm commits itself to a n MCC, the rules of the oligopoly game are changed to a leader-follower arrangement. It is not difficult to show that in a price-setting model like the one considered here, prices under a Stackelberg leader-follower game are always higher than prices under a N a s h - B e r t r a n d o u t c o m e ) Hence, even if p l = p ~ were already a credible reaction function for each firm, the unilateral commitment to an M C C by a first mover can change the oligopoly outcome. This occurs because given an MCC by one firm, the best reply by a competitor is to solve the Stackelberg leader problem. Can an MCC which does not meet the conditions of Propositions 1 or 2b ever be procompetitive? Fig. 2 illustrates one possible case, The Nash equilibrium p" satisfies p~>p~ with the line R i denoting firm i's N a s h Bertrand reaction function. In this example, firm I will find it optimal to adopt an MCC prior to the entry of firm 2 with #1 = P ] resulting in p2=p'z [-assuming n2(p')>0]. To see that this is optimal, note that if 1 does not adopt an MCC, firm 2 will adopt and will choose a posted price where n 2 is
~=
P2
(P')
r
rt
I
I I , , P°P"
~,
P
Fig, 2 SThis followsdirectly from the fact that Nash-Bertrand reaction functions are upward sloping in the model.
T.M. Belton, Meeting or beating competition in duopoly
407
maximized along the R 1 constraint. 9 Since (1)rcl(p")
3. The optimal design of competitor-based price clauses The previous section considered the simplest case where firms could only commit themselves to meet the price of a lower priced competitor. In this section, the model is extended to investigate the effects of a more general class of contracts which we term competitor-based price clauses (CPCs). CPCs include the frequently observed practice where firms commit themselves to contracts that promise to beat the price of a competitor. In practice, these contracts are often written in a form which relates one firm's own price linearly to the price of a competitor. While one would expect that this restriction to linear forms is done primarily for simplicity, it will be seen that, under certain conditions, the commitment to an appropriately designed linear price contract weakly dominates any non-linear CPC the firm might choose. Three effects of CPCs are demonstrated in this section. First, they may result in entry deterrence. Second, entry may occur but prices will exceed prices under a Nash-Bertrand equilibrium. Finally, we show that CPCs may allow the contracting firm to price discriminate between consumers having different information sets. To illustrate these effects, the model of section 2 is extended as follows. In the period immediately preceding that in which sales occur, firms may choose an advertisement which contains (i) a posted price, Pi, and (ii) a price clause of the form p i = a + b p ? The C P C (`6i,a,b) represents a binding commitment by firm i that any consumer with knowledge of pj may purchase good i at a price (restricted to be non-negative), equal to rain[`61, a+ bpj]. As in section 2, the incumbent firm (firm 1) is able to choose its contract prior to the entry of firm 2. We also generalize the demand environment of the previous section by 9profits to firm 2 along R 1 exceed profits along the Pl =P2 line. Hence 2's choice for /~2 will solve max rt2(p) subject to R 1.
T.M. Belton, Meeting or beating competition in duopoly
408
disaggregating q~(p) into the demand of two types of consumers. Let q~(Pi) denote the demand of type me (for monopoly) consumers, qT(p) denote the demand of type d (for duopoly) consumers where qi(P) =-qT'(Pi)+ qT(P). Type m~ consumers are assumed to not incur the search costs necessary to learn pj (either because they dislike good j or because the search costs are too large) and accordingly, their demand for good i is only a function of p~. By contrast, the demand of type d consumers is a function of both p~ and P2. These consumers are assumed to view goods 1 and 2 as substitutes and to be informed at the time of purchase of both p~ and P2. At the time it selects its price, firm i is assumed to be unable to separately identify type m~ from type d consumers. The formal set of assumptions made on consumer demand may be described as follows. Let q~ and qd be non-increasing in p~ for all pj and denote /~(pj), i=1,2, i# j, as the minimum Pi such that q~(p)=0. The function ~(pj) characterizes the price for which firm i just drops out of the duopoly market. Similarly, let ~ denote the lowest price at which q~(pi) equals zero. Letting the notation Ap~<(>)0 refer to the left (right) hand derivative, the formal set of assumptions on qi(P) are as follows? ° da m
(A5) -~' < 0 dp~
=0
if
(a) p i < ~
or
(b) p t = ~ a n d A p i < 0 ,
otherwise,
d
(A6)
°q?<0
Op~
=0 (A7)
t~q~>0
dpj
if
(a) pi<~(pj)or(b)pi=ff~(pj)
Ap~
otherwise, if (a) (b)
=0
and
Pi < P~(Pj) and pj < P~(Pi),
i # j,
pi=p~(p2), Pj 0 ,
or
i-~ j,
otherwise.
These assumptions imply that at positive output levels, the demand for firm i's product is strictly decreasing in its own price and strictly increasing in its competitor's price. When q~ = q~ = 0, however, then increases in firL i's lOq: is assumed to be twice differentiable everywhere except at p=(:],:~). If pl=p'~ and p ~ # ~ , then continuous second partial derivatives exist with respect to p~. With respect to p~, only the left hand derivative is assumed to exist.
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409
own price have no effect on q~. Similarly, when qa =0, decreases in firm j's price have no effect on qt. Fig. 3 illustrates a typical set of isoprofit curves for firm 2 under assumptions (A2)-(A7). For all P2 greater than or equal to the line labeled /~(Pl), demand for firm 2's product by type d consumers equals zero. Over this range, price decreases of firm 1 have no effect on firm 2's demand or profits, (since qT is only a function of P2), and accordingly, 2's isoprofits are represented as straight lines. By contrast, when firm 2 has positive demand from type d consumers, (P2 <~(Pl)), its demand and profits are increasing in pl (for q~ > 0) and its isoprofit curves are drawn to be convex. The price denoted p~' in fig. 3 is the price which maximizes firm 2's profits when it is out of the duopoly market [i.e., p~ solves maxp2p2q"2(p2 ) P2
~2
Fig. 3
Pl
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410
-C2(q"d(p:))], and the profit level rU denotes the c o r r e s p o n d i n g level of profits. ~z can be interpreted as an individual rationality constraint o n firm 2's profit level. Since nzt'mw2,ew= " ~> ~z for all Pl, and p~' is always a price choice which firm 2 can unilaterally make, it is clear that the m i n i m u m profits of firm 2 in any duopoly game (including one in which firm 1 signs a CPC) is ~z. In this model, therefore, the extent to which C P C s can be used as a predatory weapon is limited by the existence of uninformed consumers. We would like to characterize the non-cooperative equilibrium which results when each firm's strategy set equals (t5, a, b). To this end, note that once firm 1 has committed itself to a CPC, with b 4=0, firm 2's best reply can be seen to be a choice of b = 0 a n d a choice for /52 which simply solves the Stackelberg leader problem maxn2(pl,pz) s.t. p l = m i n [ p ~ , a + b p z ] . 11 To solve for firm l's best reply, note that without loss of generality, we can ignore all strategy choices which have the property a + b p z > f i l , 12 This implies that the price paid by firm l's informed consumers can be written simply as a+bp2, and that given a C P C by firm 1, na=n2(a+bpz, p2). The non-cooperative equilibrium in the model can then be characterized as the solution to the following problem. max i01q~(/51)+ Plq~(Pl, P2(a, b)) - C(q'~ + q~),
(2)
a, b, 1J1
subject to (i)
Px =a+bp2(a,b),
(ii)
7C2(a + bp2, P2) >=~2,
(iii) pa(a, b)
= argmax
7~2(a -4- bp'z, P'z).
Pl There are three possible solutions to (2).
Case 1: Entry deterrence. If ~ 2 < 0 , then firm l can prevent firm 2 from entering by posting its m o n o p o l y price and choosing any (a,b) such that nZ(a +bpz,pz)<~ 2 for all P2. By c o m m i t t i n g itself to u n d e r c u t firm 2's price even if undercutting were irrational ex post - firm 1 alters its competitor's payoff matrix in such a way that n 2 < 0 for all P2. This will deter a profit maximizing firm 2 from producing good 2 and will allow firm 1 to reap m o n o p o l y profits, 11By the definition of a maximum, this choice of P2 must be a best reply. Of course, it need not be unique. In general, given a CPC by firm i, there exist choices of b~0 for firm 2 which achieve the same equilibrium price outcomes as the b = 0 equilibrium analyzed here. t2If a+bp2>O~, both informed and uninformed consumers of firm 1 will pay/~t. In this case, if firm 2's reply is in fact best, it will solve max~2(pt,p2). Since the same outcome will occur for any (a,b) such that a+ bpz =/~, we can restrict attention to cases where a+bpz
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411
Case 2: Cooperative, non-discriminatory outcome. If ~2 >0, it is not possible for firm 1 to use a C P C to cause firm 2 to shut down production. In this case however, the use of a linear C P C does allow firm 1 to obtain its more preferred single-price outcome subject to the constraint that firm 2's profits are at least as large as 7~2. Since by assumption, n ~ is convex in p, we know there exists a separating hyperplane, Pl = a + bp2, which goes through firm l's preferred price pair, p+, with the property that n2(p+)>n2(a+bp2,p2) for all P2. Fig. 4 illustrates such an outcome. As can be seen from the figure, if firm 1 chooses the CPC, p l = a + b p 2 , which separates the tangent isoprofits at (p~,p]), and posts a price i61=p~, then it will induce a profit maximizing
P2
Pl * a ÷ bp 2 ! I p+
I
~2
!
I Pl Fig. 4
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412
firm 2 to choose pz=p~. Hence, the CPC a + b P z with posted price p[ allows firm 1 to obtain its best possible duopoly outcome subject to the individual rationality constraint ~z. This outcome will be a producer efficient point which satisfies b=-n~(p+)/n~(p+)=-n~(p+)/n~(p +) and
a=p'~-bp~.
(3)
Profits to firm 1 and prices for both firms under the CPC will be strictly greater than those obtained under the Nash-Bertrand outcome. 13 Case 3: Cooperative, discriminatory outcome. It is always possible for firm 1 to design its CPC and posted price combination so that uninformed, type ml consumers pay a higher price for good 1 than informed, type d consumers. Since informed consumers pay min[p~,a+bp2], and uninformed consumers pay 151, price discrimination will result whenever ~ > a+ bpz. Whether this is a profitable strategy for firm 1, depends, loosely speaking, on whether or not type ml consumers have a greater willingness to pay for good 1 than do type d consumers. Specifically, define (/ST,/~,/3z) to be firm l's preferred discriminatory outcome subject to the individual rationality constraint n2; i.e.,
(t~T,~ , h) solves m m m +plql(pl,pa)_Cl(qT+q~), max Plql(Pl) d d a subject to nZ(P'1, P2) = r?2. I f / ~ >/5 T, firm l's preferred price to informed type d consumers exceeds its preferred price to uninformed type ml consumers. Hence, in this case, firm 1 would never choose a posted price which exceeds the equilibrium price paid by informed consumers. For ~2>0, this implies the optimal CPC is of the cooperative non-discriminatory type outlined above. If/3~>p".
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P2
PI " a + bp2
p÷
2- 21
? ~r2
e
pl(a,b>~ d
!
~l~l
Pt
Fig. 5
maximizing firm 2 will choose p2=/~2, with informed consumers of good 1 paying mini/51, a + bp2] =/sd. Uninformed consumers, who by assumption don't know P2, will pay /51 = / ~ and accordingly firm 1 will achieve the outcome (/ST,/~, P2). In this case, therefore, a CPC allows the contracting firm to segment the market and capture Jits preferred monopoly price from its uninformed consumers and preferred duopoly price from informed consumers. This yields profits to firm 1 which are strictly greater than both the Bertrand-Nash outcome and the single-price cooperative outcome discussed above. Informed
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414
consumers are better off relative to the non-discriminatory outcome (/~d p~).
4. Concluding comments It has been recognized for some time now, in the economics literature, that the use of binding contracts by firms can alter the incentives of oligopolists in ways which make cooperative outcome non-cooperative equilibrium. In this paper, we have provided some conditions under which the use of competitor-based price clauses in a duopoly market facilitates cooperative outcomes and under which they can be used to price discriminate between informed and uninformed consumers. An important assumption of the analysis is that both firms operate under complete information on their opponent's payoff function. In related work [see Belton (1986)], it is shown that when the contracting firm chooses a CPC under incomplete information on its competitor's technology, the results of section 3 must be modified somewhat. First, although the commitment to a CPC remains an optimal strategy for the contracting firm, the CPC will generally not be linear. Second, the optimal CPC need not result in a set of prices which are Pareto efficient. Under certain conditions however, prices do rise as a result of the CPC. Although MCCs and CPCs can be seen to facilitate oligopoly coordination, this analysis also establishes that conditions exist under which MCCs can actually be procompetitive. This is in contrast to the existing literature on these contracts which has tended to focus exclusively on their anticompetitive effects. The fact that MCCs can actually lower prices suggests that public policy needs to proceed slowly with careful attention being given to the cost structure of the firms which choose these contracts.
Appendix Lemma 1. Let the costs of each firm be identical, demand be symmetric as defined in Proposition 1 and define P~(Pl) to solve maxn2(pl,p2). If p~(/~l)>pl, then Pl
__
n
and
P*(/~I)-'PP,
for
/51:~p~.
(A.1)
n~(pJ) < 0 implies * J P2(Pl)
(A.2)
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415
(A.1), (A.2), continuity of p*(p~) and p~ >p~ imply p~(/5~)
if /5~>p~.
(A.3)
Since p~(p~) is upward sloping [from assumptions (A3) and (A4)] (A.1) implies p~(/~,)
if /~,
(A.4)
(A.3) and (A.4) imply that if p~(/Sa)>/~1, then/~t < P~ and p~(i61)< p~. [] We first demonstrate that if firm i adopts an MCC, with /~i=p{, equilibrium prices equal pJ. Second, we demonstrate that zi(pJ) (weakly) dominates any other feasible payoff for firm i so that it is optimal for firm i to choose an MCC with pi=p{. Without loss of generality, we prove this for the case i= 1. The collusive outcome pJ solves Proof of Proposition 1.
max nl(p) + nZ(p).
(A.5)
PI,P2
Differentiating, combining first order conditions and utilizing condition (ii) of the proposition which implies aql/Opl-Oq~/ap2 = Oqz/ap z -Oq2/Op I yields q~ -- q2 + (dC(q2)/dq2 - dC(q ~)/dq~ + p, - P2)(Oq~/OPt - c3q~/aP2) = O.
(A.6)
aql/OPl-Oql/~P2<0, condition (i) of the proposition and the assumption of increasing marginal cost imply that the solution to (A.6) has Pl =P2, ql =q2. This fact implies that (A,5) is equivalent to maxnl(p)+~Z(p)
subject to
pl=p2
(A.7)
P2 = m a x P2q2 + P2q2 -- C ( q 2 ) - C ( q 2 ) P2
=max2gZ(p2,P2),
or equivalently,
P2
max ~2(pz, Pz).
(A.8)
P2
Given an MCC by firm 1 however (with /~a =P~), (A.8) characterizes firm 2's best reply. Hence the equilibrium resulting from the MCC is the same as the collusive outcome pS.
J,I,O.-- C
416
T.M. Belton,Meeting or beatingcompetitionin duopoly
To complete the proof, we show that nt(p)<=nt(p J) for p equal to any other feasible price pair. If firm 1 chooses an MCC with /5i >p~, (A.8) characterizes from 2's best reply resulting in nl=nl(pJ). Hence there are no gains to firm 1 from choosing /~1 >pS. If firm 1 chooses an M C C with /51,/~: P*>/~I implies firm 2's best reply solves maxnZ(~l,p2). From Lemma 1,/5~
Proof of Proposition 2. (a) Let pipj" with ni(~i,p*)>ni(p"). Hence, if firm j does not adopt an MCC, firm i can always do better than n~(p~) if an M C C is chosen. (b) Without loss of generality, let firm 1 be the firm choosing the MCC and pT<=p~. If the MCC adopted by firm 1 is a best reply, then from the proof of (a) nl(p')>nl(p ") where p' denotes equilibrium prices under the MCC. If /~lp7, then the conditions of the proposition are met only if firm 2's best reply satisfies p~
References
Anderson, R., 1985, Quick-response equilibrium, Working paper (The University of California, Berkely, CA). Belton, T., 1986, A model of duopoly and meeting or beating competition, Research Papers in Banking and Financial Economics no. 87 (Financial Studies Section, Board of Governors of the Federal ReserveSystem, Washington, DC), Cooper, T.E, 1986, Most-favored-customer pricing and tacit collusion, The Rand Journal of Economics 17, Autumn, 377-388.
T.M. Belton, Meeting or beating competition in duopoly
417
Friedman, J.W, 1977, Oligopoly and the theory of games (North-Holland, Amsterdam). Harris, M. and R.M, Townsend, 1981, Resource allocation under asymmetric information, Econometrica 49, Jan., 33-64. Kalai, E. and M.A. Satterthwaite, 1986, The kinked demand curve, facilitating practices, and oligopolistic competition, Discussion paper no. 677 (Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, IL). Osborne, D.K., 1976, Cartel problems, American Economic Review 66, Dec., 835-844. Salop, S.C., 1982, Practices that credibly facilitate oligopoly coordination, Paper presented at the IEA symposium, Ottawa. Sappington, D., 1983, Limited liability contracts between principal and agent, Journal of Economic Theory 29, Feb., 1-21. Scherer, F.M., i980, Industrial market structure and economic performance, 2nd ed. (Rand McNally, Chicago, IL). Spence, A.M., 1978, Efficient collusion and reaction functions, Canadian Journal of Economics 11, Aug., 527-533. Weitzman, M.L., 1980, Efficient incentive contracts, Quarterly Journal of Economics 94, June, 719-730.
A MODEL
OF DUOPOLY AND MEETING COMPETITION
OR BEATING
T e r r e n c e M. B E L T O N * Federal Reserve System, Washington, DC 20551, USA
Final version received November 1986 This paper analyzes the use of 'meeting and beating competition' clauses in duopoly markets. A differentiated-products, price-setting model is considered, in which firms have the option of precommitting themselves to contracts which bind their own price to be some prespecified function of their competitor's price. The analysis establishes conditions under which these contracts facilitate oligopoly coordination, under which they result in price discrimination, and under which they can actually be procompetitive.
1. Introduction It has become a fairly c o m m o n practice in a n u m b e r of industries for firms to write explicit or implicit contracts that relate their o w n price to that of a competitor. Advertising guarantees that one firm will m a t c h o r beat by some specified a m o u n t the price of a c o m p e t i t o r are the most c o m m o n form of these contracts. Explicit contracts given at the time of p u r c h a s e that guarantee a rebate to the buyer in the event it finds a lower-priced c o m p e t i t o r in the near future are a second c o m m o n form. These implicit and explicit contracts - generally referred to as meeting or beating c o m p e t i t i o n clauses - are seen m o s t frequently in local retail markets and have also appeared at various times in the airline, natural gas, b a n k i n g and other industries, While the use of these clauses appears fairly widespread, there exists, surprisingly enough, little research on their effects, F i r m s that c o m m i t themselves to meeting or beating c o m p e t i t i o n clauses in o l i g o p o l y markets are c o m m i t t i n g their o p p o n e n t s (over s o m e subset of the price space) to a price leadership role. As one m i g h t expect, this choice of roles has i m p o r t a n t implications for b o t h o l i g o p o l y o u t c o m e s and c o n s u m e r welfare. *I would like to thank Bob Avery, Mark Bagnoli, Allen Berger, Larry Benveniste, Jeff Daskin, Mike Goldberg, Ed Green, Meg Guerin-Calvert, Gerald Hanweck, Steve Salop, Hal Varian, and two referees of this journal for helpful comments. The views expressed in this paper are those of the author and do not necessarily reflect those of the Board of Governors or its staff. 0167-7187/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
400
T.M, Belton, Meeting or beating competition in duopoly
Some research on the effect of meeting competition clauses (MCC) can be found in a recent paper by Salop (1982). ~ That analysis, which investigates a number of different pricing contracts commonly used by firms, suggests that meeting competition clauses may facilitate collusive behavior in oligopoly markets. Such behavior can result from two distinct effects of MCCs. First, they may operate as information exchange devices between potential colluders by rewarding consumers for reporting secret price discounts of a competitor. This information exchange reduces incentives for secret price discounts and thereby enhances the stability of tacit agreements. Second, MCCs can operate as incentive management devices by altering the payoff matrices of firms in ways that make non-cooperative equilibrium yield cooperative outcomes. Salop's analysis demonstrates that when duopolists' choice sets and payoffs follow a simple Prisoners' Dilemma structure, an MCC transforms the joint profit maximum into a Nash equilibrium. As a result, MCCs can result in non-cooperative firms achieving collusive outcomes, In this paper, we attempt to generalize and extend the existing literature on the use of meeting and beating competition clauses in three different directions. First, we consider under a differentiated-products duopoly model, the conditions under which MCCs result in cooperative outcomes. The analysis, which focuses primarily on the incentive management effects of MCCs, establishes conditions under which MCCs raise price above noncooperative levels, under which they result in collusive prices, and under which they might actually be procompetitive, Second, the analysis considers the optimal design of a more general class of contracts which we term competitor-based price clauses (CPC). These clauses, which include both meeting and beating competition clauses as special cases, are defined to be any binding commitment made by one firm to set its own price equal to some prespecified function of another firm's price. When designed optimally by a firm, CPCs can be shown, under certain conditions on the information structure and demand and cost functions of the duopolists, to have the following properties. (i) They are linear, i.e., of the form p~ =a+bp2, where Pl denotes the contracting firm's own price and Pz the price of its competitor. (ii)They result in prices for both firms which exceed prices under a Nash-Bertrand outcome and which, from the producers' point of view, are Pareto optimal. 2 (iii) The outcome of the optimal price clause (under certain conditions), weakly dominates all other duopoly outcomes the contracting firm could expect. A final result established in the analysis is to demonstrate that competitor1Related research on most-favored customer clauses is found i n Cooper (1986). 2Throughout the paper, we use the term Pareto optimal to characterize outcomes which are efficient from the producers' point of view. An outcome is Pareto optimal if there is no other feasible outcome which increases both firms' profits,
T.M. Belton, Meeting or beating competition in duopoly
401
based price clauses can be used to price discriminate between informed and uninformed consumers. When a firm commits itself to both a price clause and a posted price, consumers who are uninformed about the competitor's price will, in some cases, be charged a higher price than informed consumers. Thus, under certain circumstances, competitor-based price clauses allow the contracting firm to segment the market and capture full monopoly profits from the uninformed consumers and Pareto optimal duopoly profits from the informed consumers. The attention given to the potential anticompetitive effects of competitorbased price clauses in this paper should not be interpreted as precluding other possible reasons for their use. A central assumption in the current analysis is that consumer demand is unaffected by the mere presence of CPCs. In reality, there are at least two cases where this assumption may not hold. First, it may often be the case that meeting or beating competition clauses are viewed by uninformed consumers (correctly or incorrectly) as implying that the contracting firm's posted price is in fact the lowest price in the market. By signalling to consumers some information about relative prices, MCCs could induce uninformed consumers not to incur the search costs involved in pricing a competitor's product and instead purchase the contracting firm's product. Second, when CPCs are retroactive, relative demand for the contracting firm's product can be increased by eliminating the risk associated with future price decreases of a competitor. By guaranteeing that a competitor's future price decreases will result in a rebate to the buyer, a retroactive CPC can increase the incentives of consumers to buy now (from the contracting firm) rather than later. Since these increases in demand come at the expense of one's competitor, it is clear that CPCs need not be purely anticompetitive in their effects. The potential benefits resulting from these effects of CPCs should be weighed against any anticompetitive impacts analyzed here.
2. Duopoly coordination and meeting competition clauses The commitment to match the price of a competitor is probably the most commonly observed competitor-based price clause. This section analyzes the potential effects of these clauses on duopoly outcomes. For the moment, we focus only on the case where all consumers are informed of the announced price policies of each firm, and are able to costlessly enforce any price contract the firm might choose. Consider an industry composed of two firms producing differentiated products 3 where firm 1 is the incumbent firm in the industry and firm 2 is a 3While in practice, MCCs seem most common when products are physically identical, the products are still virtually always differentiated due either to Iocational differences or quality of service differences between sellers,
T.M, Belton,Meeting or beatingcompetitionin duopoly
402
new entrant. Let P=(Pl, P2) denote the vector of firm prices and qi(P) the demand function for the ith firm's product, In section 3 of the paper, we disaggregate q~(p) into the demand of two types of consumers and present a more formal set of assumptions on the structure of demand. F o r the purposes of this section, however, we need only to assume that each firm's output is decreasing in its own price and increasing in its competitor's price so that the goods are substitutes. Specifically,
(A1) - - < 0 ,
>0
for
q~(p)>O,
i=1,2,
i#j.
OP~ Denote C~(q~) as the cost function of each firm (assumed to be twice differentiable), and rci(p)=piq~-Ci(q~(p)) as firm i's profits. Letting subscripts on 7z denote partial derivatives with respect to the prices, the following three conditions on C i and n are assumed to hold throughout the analysis:
> O, dZCi(qi)>o dq~
(A2)
for
qi>O,
i=1,2,
(A3) 7zii+Tzlj
i=1,2,
i#j,
(A4)
i=1,2,
i#j.
rclj>0
forq~(p) andq2(p)>O,
Assumption (A2) implies increasing marginal cost whenever output is positive. Assumptions (A3) and (A4) together indicate that r~i is concave in Pi and that equal price increases by each firm are less profitable for firm i, the higher is pi. ( A 3 ) a n d (A4) are sufficient conditions for a unique N a s h equilibrium to exist and for the firms' Bertrand type price reaction functions to be upward sloping. 4 These conditions facilitate comparison of the competitor-based price clause equilibrium with the non-cooperative outcome that might occur if MCCs were not used. To examine the impact that MCCs have on equilibrium prices, we define the strategy spaces of the two firms as follows. In the period prior to the entry of firm 2, the incumbent firm (firm I) chooses whether or not to a d o p t an MCC. 5 Firm 2 then makes its entry decision and also chooses whether or not to adopt an MCC. F o r each firm, the M C C is defined to be a binding advertisement which contains (1) a posted price pi, and (2) a guarantee that at the time of purchase if ~>pj, i@j, then the actual price charged to '*The Nash equilibrium concept referred to in this paper is the Bertrand equilibrium extended to differentiatedproducts. ~Section 2 expands the allowable choice of contracts to include both meeting and beating competition clauses,
T.M. Belton, Meeting or beating competition in duopoly
403
consumers by firm i will equal pj. Following the decision on whether to adopt an MCC, prices are set in the following way. If both firms advertise an MCC in the pre-sales period, then the actual price charged to consumers by both firms will equal min{pl,/52}. If firm j does not adopt an MCC, then it is free to choose any pj>O in period 2. Hence, if firm i chooses to advertise but firm j does not, then in period 2, firm j's best reply will solve maxnJ(p) p~
subject to
pi=pj
if Pj
=/~
otherwise.
(1)
By advertising an M C C in the pre-sales period therefore, the firm is assumed to commit itself both to a posted price,/5~, and a p~=pj reaction function for all pi
Proposition 1. Let the costs of each firm be identical and let demand be symmetric in the sense that (i)
qi<(=)qj
if
pi>(=)pj,
(ii)
OQ(pJ)/Opl =8Q(pS)/Op2,
and
where Q = q l +q2 denotes industry demand and pS is the price vector which maximizes joint profits. Then (1) it is optimal for either firm to unilaterally commit itself to an M C C with posted price ~i=p s, and (2) the M C C results in the collusive outcome p = pJ. Proof
See the appendix.
Proposition 1 provides a set of sufficient conditions under which each firm has an incentive to unilaterally commit to an MCC that results in the collusive outcome. The class of demand functions considered are those for
404
T.M. Belton, Meeting or beating competition in duopoly
which (1) the two firms share the market equally when their prices are equal, and (2) when their prices are not equal, the high price firm's market share is smaller. The importance of symmetric demand functions in Proposition 1 lies largely in the fact that when conditions (i) and (ii) of the proposition hold, an MCC constrains both firms to split the market evenly. In this sense, Proposition 1 is consistent with the findings of Osborne (1976) and Spence (1978) who establish that strategy functions chosen by non-cooperative firms which attempt simply to maintain market share can result in collusive outcomes. When the conditions of Proposition 1 hold, an MCC is in fact a strategy which maintains equal market shares and accordingly facilitates oligopoly coordination in much the same way as the Osborne and Spence reaction function equilibria do. In practice, MCCs are frequently observed in industries where products are differentiated due to location or advertising and where the conditions in Proposition I are not met. To examine the impact of MCCs in the more general case when demand is not symmetric, let pn denote the noncooperative Nash-Bertrand equilibrium for a single period model in which firms are price setters. In Proposition 2, we compare pn to the equilibrium that results when each firm's strategy space is expanded to include MCCs. Proposition 2. (a) There is never an equilibrium in which neither firm adopts an MCC. (b) Let p "< i = p j". Then if an M C C is chosen only by firm i, it results in equilibrium prices Pl > P~ and Pz > P~. (e) There do exist equilibria in which p~ > p~ and an M C C is optimal for firm 1 that results in Pl
The proofs of parts (a) and (b) of the proposition are in the appendix. Part (c) is shown below by an example. Several properties of equilibrium are examined in the proposition. First, the proposition states that it is always rational for at least one firm to adopt an MCC. This is because an MCC always allows at least one firm to unilaterally raise its profits above ni(pn). Second, if the firm adopting the clause is the firm with the lowest price in a Nash-Bertrand equilibrium, prices of both firms will always rise as a result of the MCC. The firm with the lower price at p" is the firm with lower marginal costs or lower demand at equal prices. Hence, part (b) of Proposition 2 suggests that when an MCC is adopted only by the firm with lower marginal costs (or lower demand), prices of both firms will rise above non-cooperative Nash-Bertrand levels. Finally, part (c) of the proposition indicates that prices do not always rise as a result of MCCs. A necessary condition for pi
405
T.M. Belton, Meeting or beating competition in duopoly
Fig. 1 illustrates the case where an M C C leads to a price increase by both firms. In this example, firm l's profits are maximized by adopting an M C C with posted price /51 =p'i. 6 Given an M C C by firm I moreover, there are no incentives for firm 2 to also adopt one. 7 Instead, 2's best reply is to play the role of Stacketberg leader and select p2=p'2 with equilibrium outcome p'. Because 2's N a s h - B e r t r a n d reaction function is upward sloping, this results in higher prices and profits for both firms. It should be noted that there are two different reasons in this model why MCCs can have the effect of raising prices. First, like other binding commitments studied in oligopoly theory, these contracts can facilitate oligopoly coordination by binding firms to actions which, ex-post, might not be optimal. In this model, when a firm (e.g., firm 1) chooses an MCC, it is committing itself to a particular posted price for all p2>/51. When this commitment precludes firm l from making a competitive response to firm 2's high price choice, it will increase firm 2's incentive to choose the cooperative price. This incentive is enforced both by the commitment to the posted price and by the commitment to match one's competitor whenever P2
/
P,~P,
t
I p.
I p'
1
P1
Fig. 1 6Firm 1 is indifferent between adopting and not adopting an MCC. If 1 does not adopt an MCC, firm 2 will adopt with posted price iO2=p~. This would yield the same equilibrium outcome p'. 7Firm 2 will be indifferent between adopting an MCC with posted price /72=p~ and not adopting an MCC and simply selecting P2 = pt, in period 2.
406
T.M. Belton, Meeting or beating competition in duopoly
A second reason why these contracts may result in higher prices is the fact that when only one firm commits itself to a n MCC, the rules of the oligopoly game are changed to a leader-follower arrangement. It is not difficult to show that in a price-setting model like the one considered here, prices under a Stackelberg leader-follower game are always higher than prices under a N a s h - B e r t r a n d o u t c o m e ) Hence, even if p l = p ~ were already a credible reaction function for each firm, the unilateral commitment to an M C C by a first mover can change the oligopoly outcome. This occurs because given an MCC by one firm, the best reply by a competitor is to solve the Stackelberg leader problem. Can an MCC which does not meet the conditions of Propositions 1 or 2b ever be procompetitive? Fig. 2 illustrates one possible case, The Nash equilibrium p" satisfies p~>p~ with the line R i denoting firm i's N a s h Bertrand reaction function. In this example, firm I will find it optimal to adopt an MCC prior to the entry of firm 2 with #1 = P ] resulting in p2=p'z [-assuming n2(p')>0]. To see that this is optimal, note that if 1 does not adopt an MCC, firm 2 will adopt and will choose a posted price where n 2 is
~=
P2
(P')
r
rt
I
I I , , P°P"
~,
P
Fig, 2 SThis followsdirectly from the fact that Nash-Bertrand reaction functions are upward sloping in the model.
T.M. Belton, Meeting or beating competition in duopoly
407
maximized along the R 1 constraint. 9 Since (1)rcl(p")
3. The optimal design of competitor-based price clauses The previous section considered the simplest case where firms could only commit themselves to meet the price of a lower priced competitor. In this section, the model is extended to investigate the effects of a more general class of contracts which we term competitor-based price clauses (CPCs). CPCs include the frequently observed practice where firms commit themselves to contracts that promise to beat the price of a competitor. In practice, these contracts are often written in a form which relates one firm's own price linearly to the price of a competitor. While one would expect that this restriction to linear forms is done primarily for simplicity, it will be seen that, under certain conditions, the commitment to an appropriately designed linear price contract weakly dominates any non-linear CPC the firm might choose. Three effects of CPCs are demonstrated in this section. First, they may result in entry deterrence. Second, entry may occur but prices will exceed prices under a Nash-Bertrand equilibrium. Finally, we show that CPCs may allow the contracting firm to price discriminate between consumers having different information sets. To illustrate these effects, the model of section 2 is extended as follows. In the period immediately preceding that in which sales occur, firms may choose an advertisement which contains (i) a posted price, Pi, and (ii) a price clause of the form p i = a + b p ? The C P C (`6i,a,b) represents a binding commitment by firm i that any consumer with knowledge of pj may purchase good i at a price (restricted to be non-negative), equal to rain[`61, a+ bpj]. As in section 2, the incumbent firm (firm 1) is able to choose its contract prior to the entry of firm 2. We also generalize the demand environment of the previous section by 9profits to firm 2 along R 1 exceed profits along the Pl =P2 line. Hence 2's choice for /~2 will solve max rt2(p) subject to R 1.
T.M. Belton, Meeting or beating competition in duopoly
408
disaggregating q~(p) into the demand of two types of consumers. Let q~(Pi) denote the demand of type me (for monopoly) consumers, qT(p) denote the demand of type d (for duopoly) consumers where qi(P) =-qT'(Pi)+ qT(P). Type m~ consumers are assumed to not incur the search costs necessary to learn pj (either because they dislike good j or because the search costs are too large) and accordingly, their demand for good i is only a function of p~. By contrast, the demand of type d consumers is a function of both p~ and P2. These consumers are assumed to view goods 1 and 2 as substitutes and to be informed at the time of purchase of both p~ and P2. At the time it selects its price, firm i is assumed to be unable to separately identify type m~ from type d consumers. The formal set of assumptions made on consumer demand may be described as follows. Let q~ and qd be non-increasing in p~ for all pj and denote /~(pj), i=1,2, i# j, as the minimum Pi such that q~(p)=0. The function ~(pj) characterizes the price for which firm i just drops out of the duopoly market. Similarly, let ~ denote the lowest price at which q~(pi) equals zero. Letting the notation Ap~<(>)0 refer to the left (right) hand derivative, the formal set of assumptions on qi(P) are as follows? ° da m
(A5) -~' < 0 dp~
=0
if
(a) p i < ~
or
(b) p t = ~ a n d A p i < 0 ,
otherwise,
d
(A6)
°q?<0
Op~
=0 (A7)
t~q~>0
dpj
if
(a) pi<~(pj)or(b)pi=ff~(pj)
Ap~
otherwise, if (a) (b)
=0
and
Pi < P~(Pj) and pj < P~(Pi),
i # j,
pi=p~(p2), Pj
or
i-~ j,
otherwise.
These assumptions imply that at positive output levels, the demand for firm i's product is strictly decreasing in its own price and strictly increasing in its competitor's price. When q~ = q~ = 0, however, then increases in firL i's lOq: is assumed to be twice differentiable everywhere except at p=(:],:~). If pl=p'~ and p ~ # ~ , then continuous second partial derivatives exist with respect to p~. With respect to p~, only the left hand derivative is assumed to exist.
T.M. Belton, Meeting or beating competition in duopoly
409
own price have no effect on q~. Similarly, when qa =0, decreases in firm j's price have no effect on qt. Fig. 3 illustrates a typical set of isoprofit curves for firm 2 under assumptions (A2)-(A7). For all P2 greater than or equal to the line labeled /~(Pl), demand for firm 2's product by type d consumers equals zero. Over this range, price decreases of firm 1 have no effect on firm 2's demand or profits, (since qT is only a function of P2), and accordingly, 2's isoprofits are represented as straight lines. By contrast, when firm 2 has positive demand from type d consumers, (P2 <~(Pl)), its demand and profits are increasing in pl (for q~ > 0) and its isoprofit curves are drawn to be convex. The price denoted p~' in fig. 3 is the price which maximizes firm 2's profits when it is out of the duopoly market [i.e., p~ solves maxp2p2q"2(p2 ) P2
~2
Fig. 3
Pl
T.M, Belton, Meeting or beating competition in duopoly
410
-C2(q"d(p:))], and the profit level rU denotes the c o r r e s p o n d i n g level of profits. ~z can be interpreted as an individual rationality constraint o n firm 2's profit level. Since nzt'mw2,ew= " ~> ~z for all Pl, and p~' is always a price choice which firm 2 can unilaterally make, it is clear that the m i n i m u m profits of firm 2 in any duopoly game (including one in which firm 1 signs a CPC) is ~z. In this model, therefore, the extent to which C P C s can be used as a predatory weapon is limited by the existence of uninformed consumers. We would like to characterize the non-cooperative equilibrium which results when each firm's strategy set equals (t5, a, b). To this end, note that once firm 1 has committed itself to a CPC, with b 4=0, firm 2's best reply can be seen to be a choice of b = 0 a n d a choice for /52 which simply solves the Stackelberg leader problem maxn2(pl,pz) s.t. p l = m i n [ p ~ , a + b p z ] . 11 To solve for firm l's best reply, note that without loss of generality, we can ignore all strategy choices which have the property a + b p z > f i l , 12 This implies that the price paid by firm l's informed consumers can be written simply as a+bp2, and that given a C P C by firm 1, na=n2(a+bpz, p2). The non-cooperative equilibrium in the model can then be characterized as the solution to the following problem. max i01q~(/51)+ Plq~(Pl, P2(a, b)) - C(q'~ + q~),
(2)
a, b, 1J1
subject to (i)
Px =a+bp2(a,b),
(ii)
7C2(a + bp2, P2) >=~2,
(iii) pa(a, b)
= argmax
7~2(a -4- bp'z, P'z).
Pl There are three possible solutions to (2).
Case 1: Entry deterrence. If ~ 2 < 0 , then firm l can prevent firm 2 from entering by posting its m o n o p o l y price and choosing any (a,b) such that nZ(a +bpz,pz)<~ 2 for all P2. By c o m m i t t i n g itself to u n d e r c u t firm 2's price even if undercutting were irrational ex post - firm 1 alters its competitor's payoff matrix in such a way that n 2 < 0 for all P2. This will deter a profit maximizing firm 2 from producing good 2 and will allow firm 1 to reap m o n o p o l y profits, 11By the definition of a maximum, this choice of P2 must be a best reply. Of course, it need not be unique. In general, given a CPC by firm i, there exist choices of b~0 for firm 2 which achieve the same equilibrium price outcomes as the b = 0 equilibrium analyzed here. t2If a+bp2>O~, both informed and uninformed consumers of firm 1 will pay/~t. In this case, if firm 2's reply is in fact best, it will solve max~2(pt,p2). Since the same outcome will occur for any (a,b) such that a+ bpz =/~, we can restrict attention to cases where a+bpz
T.M. Belton, Meeting or beating competition in duopoly
411
Case 2: Cooperative, non-discriminatory outcome. If ~2 >0, it is not possible for firm 1 to use a C P C to cause firm 2 to shut down production. In this case however, the use of a linear C P C does allow firm 1 to obtain its more preferred single-price outcome subject to the constraint that firm 2's profits are at least as large as 7~2. Since by assumption, n ~ is convex in p, we know there exists a separating hyperplane, Pl = a + bp2, which goes through firm l's preferred price pair, p+, with the property that n2(p+)>n2(a+bp2,p2) for all P2. Fig. 4 illustrates such an outcome. As can be seen from the figure, if firm 1 chooses the CPC, p l = a + b p 2 , which separates the tangent isoprofits at (p~,p]), and posts a price i61=p~, then it will induce a profit maximizing
P2
Pl * a ÷ bp 2 ! I p+
I
~2
!
I Pl Fig. 4
T.M, Belton, Meeting or beating competition in duopoly
412
firm 2 to choose pz=p~. Hence, the CPC a + b P z with posted price p[ allows firm 1 to obtain its best possible duopoly outcome subject to the individual rationality constraint ~z. This outcome will be a producer efficient point which satisfies b=-n~(p+)/n~(p+)=-n~(p+)/n~(p +) and
a=p'~-bp~.
(3)
Profits to firm 1 and prices for both firms under the CPC will be strictly greater than those obtained under the Nash-Bertrand outcome. 13 Case 3: Cooperative, discriminatory outcome. It is always possible for firm 1 to design its CPC and posted price combination so that uninformed, type ml consumers pay a higher price for good 1 than informed, type d consumers. Since informed consumers pay min[p~,a+bp2], and uninformed consumers pay 151, price discrimination will result whenever ~ > a+ bpz. Whether this is a profitable strategy for firm 1, depends, loosely speaking, on whether or not type ml consumers have a greater willingness to pay for good 1 than do type d consumers. Specifically, define (/ST,/~,/3z) to be firm l's preferred discriminatory outcome subject to the individual rationality constraint n2; i.e.,
(t~T,~ , h) solves m m m +plql(pl,pa)_Cl(qT+q~), max Plql(Pl) d d a subject to nZ(P'1, P2) = r?2. I f / ~ >/5 T, firm l's preferred price to informed type d consumers exceeds its preferred price to uninformed type ml consumers. Hence, in this case, firm 1 would never choose a posted price which exceeds the equilibrium price paid by informed consumers. For ~2>0, this implies the optimal CPC is of the cooperative non-discriminatory type outlined above. If/3~
T.M. Belton, Meeting or beating competition in duopoly
413
P2
PI " a + bp2
p÷
2- 21
? ~r2
e
pl(a,b>~ d
!
~l~l
Pt
Fig. 5
maximizing firm 2 will choose p2=/~2, with informed consumers of good 1 paying mini/51, a + bp2] =/sd. Uninformed consumers, who by assumption don't know P2, will pay /51 = / ~ and accordingly firm 1 will achieve the outcome (/ST,/~, P2). In this case, therefore, a CPC allows the contracting firm to segment the market and capture Jits preferred monopoly price from its uninformed consumers and preferred duopoly price from informed consumers. This yields profits to firm 1 which are strictly greater than both the Bertrand-Nash outcome and the single-price cooperative outcome discussed above. Informed
T,M. Belton, Meeting or beating competition in duopoly
414
consumers are better off relative to the non-discriminatory outcome (/~d
4. Concluding comments It has been recognized for some time now, in the economics literature, that the use of binding contracts by firms can alter the incentives of oligopolists in ways which make cooperative outcome non-cooperative equilibrium. In this paper, we have provided some conditions under which the use of competitor-based price clauses in a duopoly market facilitates cooperative outcomes and under which they can be used to price discriminate between informed and uninformed consumers. An important assumption of the analysis is that both firms operate under complete information on their opponent's payoff function. In related work [see Belton (1986)], it is shown that when the contracting firm chooses a CPC under incomplete information on its competitor's technology, the results of section 3 must be modified somewhat. First, although the commitment to a CPC remains an optimal strategy for the contracting firm, the CPC will generally not be linear. Second, the optimal CPC need not result in a set of prices which are Pareto efficient. Under certain conditions however, prices do rise as a result of the CPC. Although MCCs and CPCs can be seen to facilitate oligopoly coordination, this analysis also establishes that conditions exist under which MCCs can actually be procompetitive. This is in contrast to the existing literature on these contracts which has tended to focus exclusively on their anticompetitive effects. The fact that MCCs can actually lower prices suggests that public policy needs to proceed slowly with careful attention being given to the cost structure of the firms which choose these contracts.
Appendix Lemma 1. Let the costs of each firm be identical, demand be symmetric as defined in Proposition 1 and define P~(Pl) to solve maxn2(pl,p2). If p~(/~l)>pl, then Pl
__
n
and
P*(/~I)-'PP,
for
/51:~p~.
(A.1)
n~(pJ) < 0 implies * J P2(Pl)
(A.2)
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415
(A.1), (A.2), continuity of p*(p~) and p~ >p~ imply p~(/5~)
if /5~>p~.
(A.3)
Since p~(p~) is upward sloping [from assumptions (A3) and (A4)] (A.1) implies p~(/~,)
if /~,
(A.4)
(A.3) and (A.4) imply that if p~(/Sa)>/~1, then/~t < P~ and p~(i61)< p~. [] We first demonstrate that if firm i adopts an MCC, with /~i=p{, equilibrium prices equal pJ. Second, we demonstrate that zi(pJ) (weakly) dominates any other feasible payoff for firm i so that it is optimal for firm i to choose an MCC with pi=p{. Without loss of generality, we prove this for the case i= 1. The collusive outcome pJ solves Proof of Proposition 1.
max nl(p) + nZ(p).
(A.5)
PI,P2
Differentiating, combining first order conditions and utilizing condition (ii) of the proposition which implies aql/Opl-Oq~/ap2 = Oqz/ap z -Oq2/Op I yields q~ -- q2 + (dC(q2)/dq2 - dC(q ~)/dq~ + p, - P2)(Oq~/OPt - c3q~/aP2) = O.
(A.6)
aql/OPl-Oql/~P2<0, condition (i) of the proposition and the assumption of increasing marginal cost imply that the solution to (A.6) has Pl =P2, ql =q2. This fact implies that (A,5) is equivalent to maxnl(p)+~Z(p)
subject to
pl=p2
(A.7)
P2 = m a x P2q2 + P2q2 -- C ( q 2 ) - C ( q 2 ) P2
=max2gZ(p2,P2),
or equivalently,
P2
max ~2(pz, Pz).
(A.8)
P2
Given an MCC by firm 1 however (with /~a =P~), (A.8) characterizes firm 2's best reply. Hence the equilibrium resulting from the MCC is the same as the collusive outcome pS.
J,I,O.-- C
416
T.M. Belton,Meeting or beatingcompetitionin duopoly
To complete the proof, we show that nt(p)<=nt(p J) for p equal to any other feasible price pair. If firm 1 chooses an MCC with /5i >p~, (A.8) characterizes from 2's best reply resulting in nl=nl(pJ). Hence there are no gains to firm 1 from choosing /~1 >pS. If firm 1 chooses an M C C with /51,
Proof of Proposition 2. (a) Let pi
References
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Friedman, J.W, 1977, Oligopoly and the theory of games (North-Holland, Amsterdam). Harris, M. and R.M, Townsend, 1981, Resource allocation under asymmetric information, Econometrica 49, Jan., 33-64. Kalai, E. and M.A. Satterthwaite, 1986, The kinked demand curve, facilitating practices, and oligopolistic competition, Discussion paper no. 677 (Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, IL). Osborne, D.K., 1976, Cartel problems, American Economic Review 66, Dec., 835-844. Salop, S.C., 1982, Practices that credibly facilitate oligopoly coordination, Paper presented at the IEA symposium, Ottawa. Sappington, D., 1983, Limited liability contracts between principal and agent, Journal of Economic Theory 29, Feb., 1-21. Scherer, F.M., i980, Industrial market structure and economic performance, 2nd ed. (Rand McNally, Chicago, IL). Spence, A.M., 1978, Efficient collusion and reaction functions, Canadian Journal of Economics 11, Aug., 527-533. Weitzman, M.L., 1980, Efficient incentive contracts, Quarterly Journal of Economics 94, June, 719-730.