The most-favored-nation pricing policy and negotiated prices

International Journal of Industrial Organization 9 (1991) 209-223. North-Holland

The most-favored-nation pricing policy and negotiated prices Thomas E. Cooper* Arkansas College, Batesville, AR 72501, USA

Timothy L. Fries* Bellcore

Final version received February 1990

Previous work has viewed the most-favored-nation (MFN) contract as a practice capable of facilitating collusion among sellers, but this paper shows that even a monopoly seller may gain by including the MFN provision in sales contracts. We consider a case in which the seller negotiates price separately with each of two buyers. By including a MFN clause in her contract with the first buyer, the seller raises her cost of granting price concessions to the second buyer. This increases the seller's bargaining strength with respect to the second buyer, thereby helping her negotiate a higher price.

1. Introduction I n t h e i r sales c o n t r a c t s , firms h a v e i n c l u d e d a v a r i e t y of p r i c i n g clauses or g u a r a n t e e s . ~ A l t h o u g h these different price policies m a y r e p r e s e n t active c o m p e t i t i o n for buyers, t h e y are often viewed suspiciously in tight oligopolies, w h e r e tacit c o l l u s i o n is a real possibility. O n e policy t h a t m a y facilitate tacit c o l l u s i o n is the m o s t - f a v o r e d - n a t i o n ( M F N ) p r i c i n g policy. T h e M F N p r o v i s i o n g u a r a n t e e s a c u s t o m e r t h a t h e receives t h e lowest price c u r r e n t l y offered b y t h a t seller t o a n y c u s t o m e r . If a seller offers t h e M F N policy to all *We wish to thank Robert Masson and an anonymous referee for suggestions which have greatly improved the exposition of this paper. We are also grateful to Clarence Adams, Chip Chappell, and Roger Blair for helpful comments on an earlier version of this paper. Any remaining errors are, of course, our own. 1Several of these practices have already been studied. These include meet-or-release clauses (promises to match lower price offers or release the buyer from the contract) in Salop (1986) and Holt and Scheffman (1987), most-favored-customer clauses (promises to lower prices to past customers through a rebate if price falls) in Hay (1982), Cooper (1986), Salop.(1986), and Butz (forthcoming), and 'we-won't-be-undersold" policies (offers to match any lower price currently available) in Golding and Slutsky (1986), and Png and Hirschleifer (1987). Hay (1982), Salop (1986), Holt and Scheffman (1987), and DeGraba (1987) also discuss most-favored-nation policies, the subject of this paper. 0167-7187/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)

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buyers, then he is unable to offer selective discounts in an effort to steal customers from a rival. By eliminating this aggressive form of competition, the M F N policy may facilitate tacit collusion.: This possibility could explain the use of M F N pricing, but a n o t h e r explanation is possible if the oligopolists face only a few potential buyers. In that case there is an element of bilateral m o n o p o l y present. Each buyer-seller pair m a y bargain to reach an acceptable transaction price, setting the price by negotiation. It is in such a bargaining setting that we consider the effect of the M F N policy on negotiated prices and examine the value of the policy to buyers and sellers. The purpose of this paper is to show that the M F N policy could result from negotiation between buyers and sellers even if it does not facilitate collusion. 3 We employ a simple m o d e l of bargaining to assess the impact of the M F N policy on negotiated prices. There is a single seller w h o bargains sequentially with two buyers before m a k i n g any actual shipments (or sales).'* The buyers may have different valuations of the good. Each buyer-seller pair negotiates a price which depends only on the additional surplus resulting from their contract, We assume the participants recognize the N a s h b a r g a i n i n g solution as the only mutually acceptable outcome. W i t h o u t the M F N policy, the negotiators consider the trade-offs associated with price changes when searching for an acceptable price. As they lower the price from some initial high value, the seller's surplus falls while the buyer's surplus rises. The N a s h price is the one where any further reduction costs the seller too m u c h relative to the extra benefit going to the buyer. Offering the M F N policy to the first buyer affects the negotiation of the second contract. For prices below the first c o n t r a c t e d price, lowering price forces the seller to suffer d o u b l y because of the M F N policy. N o t only will profit from the second c o n t r a c t fall, but her payoff from the first also falls since she must match the price reduction for the first buyer. F o r this reason, ZWhether the practice facilitates coordination is unclear because it also lessens the willingness of a firm to match selective discounts offered to its customers. A firm which has the MFN provision is more vulnerable to attempts to steal a few customers. DeGraba (1987) focuses on this aspect of the policy in a model in which a national firm competes with two different local firms. Responding to the national firm's vulnerability, the local firms undertake such intense non-price competition that prices are lower with the MFN policy. In a different model, Holt and Scheffman (1987) consider how this force affects tacit collusion. They show that at least one firm has an incentive to offer selective discounts at any initial supra-competitive prices, if both use the MFN policy, Despite this, the industry may use the policy to support a more profitable equilibrium in which limited discounting occurs [Salop (1986)]. 3Other papers show similar pricing policies could arise for non-collusive reasons. Butz (forthcoming) shows that a durable good monopolist may adopt a most-favored-customer clause to avoid having consumers wait for price redactions. In another paper, Butz (1989) considers most favored treatment clauses offered by a buyer to his suppliers. Again a non-collusive explanation exists as the clause can induce suppliers to invest in firm-specific capital. 4By considering the case of a single seller, we rule out any potential gains from facilitating tacit collusion among sellers.

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her bargaining power increases in the second round of negotiation. M a k i n g price cuts so costly enables the seller to obtain a higher price in the second contract (compared to the case without the MFN). In the first contract, the negotiators anticipate the impact of their decisions on the second contract. Raising the first price helps to pull the second price up, but the seller who earns more from the contract is in a weaker position to seek concessions from the buyer. As a consequence, the first price with the M F N could be higher in some cases and lower in others. The first buyer clearly benefits from the M F N if the price is lower, while the seller benefits if the buyers are sufficiently similar. We derive sufficient conditions for both to benefit from the M F N policy, a case in which the policy would surely be included in their contract. The bargaining framework seems to be a natural setting in which to assess the M F N policy. One reason is that bargaining is one method of resolving disagreements about price under bilateral monopoly. Since the M F N policy causes concern primarily in tight oligopolies, one can focus the analysis on cases with few sellers. Some of those cases will also involve few buyers, making the pricing problems of bilateral m o n o p o l y relevant. Another reason is that one may suspect that buyers suggest or demand the policy in negotiations. The M F N clause provides insurance that the buyer will be able to compete on equal terms with other buyers. It may also lead to a lower price. F o r these reasons, buyers may request it. Finally, there is some evidence from the E t h y l case that sellers offer the policy to help them deny subsequent requests for discounts. 5 Requests for discounts and their acceptance or rejection are components of negotiation, not simple pricesetting by firms. F o r these reasons, we believe it is important to examine this practice in the context of bargaining. The remainder of this paper is organized as follows. In section 2 we present the basic model and the Nash bargaining solution. Section 3 considers the impact of the M F N policy. First, we show how including the policy in the first contract affects the second price. Then we incorporate that effect in a model of sequential negotiation and assess the impact of the policy on the first contract. We go on to consider whether the negotiators would include the policy in the first contract. Section 4 contains a brief numerical example based on two successive monopolies which illustrates the potential for both parties in the first contract to benefit from the M F N . Section 5 is a concluding discussion of the robustness of our results. 2. The model In our model there are three participants, one seller and two potential 5The facts, issues, and opinions concerning the Ethyl case are in "FTC Opinions and Order in re Ethyl Corp.,' (no author), Antitrust & Trade Regulation Report, April 7, 1983, pp. 740-780.

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buyers. There is a pre-market period in which prices are negotiated before any production or exchange occurs. The seller negotiates a price with buyer 1 first. After settling on a price in the first contract, she begins bargaining with the second buyer. Thus, the seller participates in two consecutive cooperative games in which the parties involved must select a price to divide the surplus from the game. To solve this bargaining problem, one must make explicit assumptions about the bargaining process. We assume that the bargaining within each game satisfies the Nash axioms of bargaining and the players accept the Nash outcome. 6 Specifically, suppose the seller receives a payoff rc~(p~) from negotiating price p~ with the ith buyer (where i = 1 or 2). Buyer i's surplus from agreeing on price p~ is U~(p~), where U ~ is assumed to be independent of the other buyer's contracted price, Then the Nash bargaining solution for contract i is the one which maximizes ~zi(p~)U~(p~), the product of the incremental surpluses produced by the contract. To permit us to study these bargaining problems, we assume AI: A2: A3: A4:

Ui(pi) is convex and decreasing in Pd ;~i(p~)is concave in p~ and increasing at Pl = 0; ni(p~)Ui(p 0 is concave in p~,"and, n~(p)UJ(p) is concave in p for i4:j.

The first assumption says buyers prefer low prices while the second implies the seller prefers high prices up to the profit-maximizing price. The last two assumptions ensure that the bargaining problems we consider have unique solutions. We will call prices resulting from these bargains the simple Nash prices, p* and p~. Without the M F N policy, these are the prices set in the pre-market period. Let us now examine the effect of the M F N policy in this model.

3. Impact of the M F N policy The M F N policy links the price paid by the first buyer to the price negotiated by the second buyer. If p2
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o u t c o m e of the second c o n t r a c t when negotiating the first contract. F o r this reason we solve the m o d e l backwards, beginning with the second c o n t r a c t first, Suppose the first c o n t r a c t set price Pl and included the M F N provision. These facts do n o t affect the buyer's payoff function, but they do alter the seller's payoff function. She can receive ~ ( p l ) even if she fails to reach an agreement with the second buyer. H e r incremental surplus f r o m the second contract is therefore [/l:2(P2 ) nt-Tr1 [ m i n ( p ~ , p 2 ) ] - ~ z l ( p l ) ] • C o n s e q u e n t l y , the N a s h price is the one which maximizes [7c2(p2) + ~1 [rain (Pr, P2)] - nt(Px)] x [UZ(P2)]. O n e can e x a m i n e the solution by considering the solutions to the two related constrained m a x i m i z a t i o n problems: 7 I. II.

m a x ~2(p2)U2(pz) subject to P2 > P l , and m a x [~z2(pz) q-hi(p2) -Tr~(pl)]U2(p2) subject to P2 ~ P l .

If we let ~ be the L a g r a n g e multiplier for P r o b l e m I and let fl be the multiplier for P r o b l e m II, we can characterize the solution in terms of a and ft. The solution to the b a r g a i n i n g p r o b l e m is the solution to P r o b l e m I if = 0 and fl > 0 at the solution. Otherwise, the bargaining s o l u t i o n is the price which solves P r o b l e m II. s U s i n g this information, we can characterize the impact of the M F N policy on P2.

Lemma 1. If the seller has signed a first contract with the M F N provision and price Pl, then the price resulting from the second contract is P"(Pt), where (i) p'(pl)=p* / f c ~ = 0 and f l > 0 ; (ii) pm(pl)> p* and p"(pl)=pl / f ~ > 0 and f l > 0 ; (iii) P'(Pl)¢P~ and p"(pl)<_pl / f c t > 0 and f l = 0 . 9 The p r o o f of this l e m m a is in the appendix. This l e m m a describes three types of o u t c o m e s with the M F N policy. In case (i), the policy has no effect on the second contract price. The policy is a binding constraint in the o t h e r two cases. The p r o v i s i o n raises P2 up to Pl in case (ii), and it requires the seller to set Pl equal to P2 in case (iii). Since c t > 0 if pl>p*, the M F N policy affects Pz w h e n e v e r the first c o n t r a c t price exceeds the simple N a s h price p~'. T o 7We are grateful to a referee for suggesting this approach. 8It is possible to have ct= 0 and fl =0 for some payoff functions, so both problems would have interior solutions. In that case there are two candidates for P2. We can eliminate one of these candidates, however, because this situation arises only if Pl exceeds the price which maximizes ~z~. One can see that the solution with p~>p~ would never be observed. The seller would not want so high a value of p~ because it lowers her profit from the first contract without raising her payoff from the second deal (because p2>pt). As a result, one would observe a value of Pl which leads to solutions with ct=fl=0 only if Problem II is the relevant one for pricing. Therefore, we assume the price solves Problem II (p2~_pl) if ct=3=0. 9We ignore the possibility that ct=fl=0 because we show in Lemma 3 in the appendix that the parties to the first contract will not include the MFN provision if that case would occur.

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describe p" further, we need restrictions on values of Pl which may be chosen, We now turn to the first contract to derive these restrictions. For the first contract, the bargainers anticipate how their choices affect the second contract price. The seller could earn surplus of nZ(p~) from the second contract if she fails to find an acceptable first price. For this reason, her incremental surplus from agreeing on price Pl and including the M F N policy is ~zl[min(pl,p'(pl))]+Tr2(p'(pl))--Tz2(p~). The buyer has no other options so his surplus is U l[min(pl,p"(pl))]. The Nash bargaining solution is the value of Pl that maximizes (r~a [min (Pl, pm(pl))] + r~2(P"(Pl))--~2(P*))X (U2[min(pl,p"(pO)]). The solution to this problem is a first-bargain price with the M F N clause, P'I', One question is whether the negotiators would agree to set price p]" and include the M F N clause in the contract or instead choose to set price p~' without the price guarantee. The answer depends on how the M F N policy enters the bargaining. Most logically, one would expect it to enter as the price does, subject to Nash bargaining. In this case, which is most consistent with our model, the participants include the M F N policy in their contract if doing so increases the product of their surpluses. One party may lose from including the provision as long as the other gains enough to justify it. An alternative is to have the policy enter the negotiations subject to a veto by either party. This approach is more restrictive because it requires the policy to increase the surplus product without lowering either party's payoff. We assume the M F N clause is also subject to Nash bargaining, but we will show that it is sometimes possible for the policy to pass the stricter 'no-veto' test for inclusion. We have established that the first negotiators make their decisions with the objective of maximizing the product of their payoffs. This objective restricts the set of pl which they might offer in combination with the M F N clause. Now we can more completely characterize the results of the M F N on the second contract price.

Proposition 1. If the first contract sets price pa with the M F N provision, then (i) p"(pl)>p'~; (ii) dp~'/dpl >O if ~ a is increasing at pl; and (iii) 7zx is increasing at pm(pl).l° (Proof" See appendix.) These results describe some interesting features of the second contract. First note that the second buyer must lose if the first negotiators use the MFN clause. They raise the second price to p" which exceeds p~, thereby 1°For this proposition, we assume the first contract includes the MFN clause only if it increases the bargainers' payoff product. If it would leave that product unchanged, we assume they do not include it.

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reducing that buyer's surplus. The M F N policy enables the first contractors to raise their payoff product by capturing some of the second buyer's payoff. The proposition also reveals how the policy helps to raise the price. If the first contract includes the M F N policy, both buyers will pay the price set in the second contract. This fact helps the seller resist price concessions, for now a price reduction is doubly costly. A lower second contract price yields less profit from both buyers, not just the second. Since price reductions reduce the seller's payoff in two ways, lower prices are less attractive. Thus, the M F N policy alters the impact of price changes in a way which leads to a higher second price. In addition to this marginal change, the M F N policy contributes to a higher price by reducing the seller's second payoff by r~X(pl)-rcl(p2) if nl(pl)>nl(p2). A Nash bargainer can insist on more favorable terms if his payoff is smaller, so this effect also contributes to a higher P2. In these ways, the M F N policy increases the seller's bargaining power, enabling her to obtain a more favorable price in the second contract. Finally, it is clear that the M F N links the prices in the two contracts. Raising Pl (up to the value that maximizes rc1) leads to higher values of p:.11 In fact, this effect occurs even if pm
Proposition 2. If p~ >=p~, the first negotiators will include the M F N policy in their contract if dpm/dpt > 0 at p,~=pe and 7r2(p*)~>~z2(p~). In that case, the first buyer benefits from the policy if 7~2 is decreasing at p~. Both the seller and first buyer benefit if, additionally, rc2 is increasing at p~ and (oua/Qp)+U 1 decreases in p. The proof of this proposition is in the appendix. By assuming -* -* FI > ~-~.F2,

we

11The restriction on Pl to be less than the value which maximizes rcI is unimportant for the main point, the fact that p" varies with p~. The only difference is that, for higher values of p~, p" decreases as Pz increases. Still the first negotiators choose the price for both buyers.

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have restricted our attention to the case where the policy is most likely to matter. The M F N policy precludes price discrimination if the second price would be lower than the first. This condition is satisfied immediately when p*>p~, when the policy would affect P2 even if there were no change in Pl. Very small price increases in the first contract make the policy effective if Pl -P2, so again it is likely to increase the payoff product. In the alternative case (p*Tr2(p~) and there exist values of pm which increase the product of their payoffs. For :zZ(p*) > 7c2(p*), the payoff product from the first contract is greater at p"~=p* because the M F N policy lets the seller extract more surplus from the second buyer. When rc2(p*)=~Z(p*), the product of first contract payoffs is unchanged at p'=p~, but the payoff product is either increasing or decreasing in pro. Therefore, again there are values of pm near p* which raise the product of first contract payoffs and, hence, justify using the M F N policy. The only remaining question is whether the negotiators can raise pm high enough by setting Pa appropriately and offering the policy. The second condition (dpm/dpl > 0 at p"-=pe) e n s u r e s they have sufficient control to move the price to a level beneficial to them. The M F N policy is then an equilibrium component of the first Nash bargain. Although some similarity is necessary to ensure the M F N policy is used, excessive similarity means the first buyer will obtain less surplus under the policy. Consider the effect of the M F N in the case of identical buyers, where p~=p*. Raising the price above p* increases the seller's payoff from both buyers. Since p~' is the price where U Irr 1 is maximized, a price increase will increase Ui[nX(p)+n2(p)-n2(p~)] by raising n2(p)-n2(p~). The Nash bargain price is higher because increasing price yields greater payoffs to the seller from both contracts. For this reason, the seller has greater bargaining power and the price is higher. Obviously the buyer would then lose because of the policy. If, however, the buyers are sufficiently different, price increases above p* lower n 2. In this case, the seller has less cause to request a high price with the M F N provision than she has without it, so the resulting value of p" is below p~' and the buyer benefits from the policy.

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The seller can benefit, too, if the remaining conditions are met. In that case both parties in the first contract prefer the MFN policy. Not only would the policy be included if it is subject to Nash negotiation, but it would also be part of the first contract under a 'no-veto' rule. As our example later illustrates, this is a real possibility. Nevertheless, one should remember that there are many situations in which the MFN policy should be excluded from the first contract. Our proposition applies if p*~>=p*,but we have not determined whether the order of bargaining satisfies that condition. Will the buyer with the higher simple Nash price actually be the first to negotiate? The answer depends on how the participants make that decision. Suppose first that either bargaining sequence produces the MFN clause in the first contract. Then it would appear that each buyer would try to be first because the second buyer loses if the MFN policy matters. However, with an effective MFN clause, each buyer ultimately pays the same price as the other. They would both prefer the order in which their common price is lower, but the seller would prefer the order with the higher price. The order chosen by a majority vote would differ from the order chosen under dictatorship by the seller. Now consider the other interesting case, in which the MFN policy is a binding constraint for only one sequence of negotiation. In the order with the M F N clause, the buyer who is to bargain second would clearly prefer to be first, for that would let him avoid the loss due to the policy. The first contract contains the MFN, however, because it benefits someone in the first round of negotiation. Therefore, at least one participant in the first contract prefers the order that produces the MFN clause. Again there will not be a unanimous preference for one sequence of bargaining. As a result, different methods of selecting the order can produce very different outcomes, possibly even deciding whether the MFN policy is part of the first contract. Having considered the MFN policy in a general model, let us now illustrate our results in a specific example.

4. Example In this illustration the seller deals sequentially with two buyers who are monopoly sellers of a final good, The profits they derive from providing the finished good constitute the buyers' payoffs. The seller negotiates with each buyer over the price of an intermediate good which she produces costlessly. One unit of intermediate good is required to produce a unit of the final good. We assume the cost of the intermediate good is the sole cost the buyers incur. Buyer l's price from selling q units of output is A - B q and his marginal cost is Pl, the price negotiated in his contract. Buyer 1 will earn his greatest

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payoff for any value of Pl ( < A ) by producing (A--pl)/2B units of output. His payoff is UI(pO=(A-pl)Z/4B. The seller's payoff from buyer 1 is pl(A-pO/2B, and the simple Nash price is p* =A/4. Similarly, we assume buyer 2's price is a - b q (where we use lower case parameters here to distinguish between buyers) a n d his m a r g i n a l cost is Pz. Buyer 2 earns the payoff UZ(p2)=(a-p;)2/4b by selling (a-pz)/2b units of output. The seller's payoff zrZ(pz)=pz(a-pz)/2b and the simple N a s h price is

p* =a/4. Using this model, we wilt restrict parameters in accordance with Proposition 2 and show the effects of the policy on payoffs when both the seller a n d the first buyer benefit. First we need p * >=P2, * which requires A>a. F o r wlJ= w2J, we need 3a>A. We have dpm/dpl>O at pe if 2 b > B . 12 The seller's payoff from the second bargain, n 2, is decreasing in price at p* = A/4 if 2a < A. This payoff is increasing at pe as long as pe < a/2, because a/2 is the price which maximizes n 2. This is true if nl(p *) + n 2(P2) , =< n~(a/2) + n2(a/2), which is satisfied for 3A:B<8Aab-4a2b+a2B. F o r the special case with b=B, n 2 is increasing at pe for A<2.2a. Finally, ~U1/~p=-2(A-p)4B, so (DU1/~p)/UI= -2/(A--p) which decreases in p for all p < A. In s u m m a r y , this model meets all the requirements of P r o p o s i t i o n 2 if b = B a n d 2a < A < 2.2a. Broader sets of parameters also satisfy our assumptions, b u t these permit us to consider a numerical example. Suppose A = 5, a = 2.4, a n d b = B = 1. In that case, prices would be p* = 1.25 and p ~ = 0 . 6 without the M F N policy. If the first contract includes the M F N policy, then both contracts yield p"~--1.2017. O b v i o u s l y this lower price benefits the first buyer, as his payoff rises from 3.516 to 3.607, a gain of 2.5~. The seller sees her total payoff n ~+Tr 2 rise from 2.884 to 3.002, an increase of 4.1~. Obviously these gains are c o m i n g at someone's expense, a n d it is the second buyer who is losing. He sees his payoff fall from 0.810 to 0.718, a drop of 11.4~. In this example, the total of the payoffs of the three participants is greater with the M F N policy. Since Nash b a r g a i n i n g has a n objective other than m a x i m i z i n g the sum of the payoffs, this result should not be terribly surprising. Nevertheless, it is interesting to see that the M F N policy could raise total surplus of the contractors when it appears in this type of bargaining situation instead of being used as a facilitating practice.13 12This assumption is satisfied if it is possible to raise p" above pe through the choice of Pt. Since p'3B, using the restrictions that

2a
t3One should not be excessivelyexcited about this result because there are also buyers of the final good to consider. Still, this finding would also result if the U functions were indirect utility functions. Therefore, we have shown it to be possible for the MFN policy to increase total surplus if one merely adapts our model slightly.

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5. Concluding discussion In this paper we have shown that contracts may contain the M F N policy even if there is no possibility of tacit collusion. We have examined the M F N pricing policy under the assumption that prices are set through Nash bargaining. The policy can be very valuable in a bargaining context. By guaranteeing the first buyer that he will pay the lower of the two negotiated prices, the clause establishes a relation between the second price and the seller's payoff from the first contract. This link across contracts increases the seller's bargaining power, thereby helping her to obtain a higher price in the second contract. Incorporating this effect into the negotiation of the first contract, we found sufficient conditions for the M F N policy to be employed and for both the seller and first buyer to benefit from it. Finally, we used a numerical example to illustrate these effects and to demonstrate that our sufficient conditions could easily be satisfied. In addition, our example shows that including the M F N policy in the first contract can increase total surplus. One may wonder how robust our results are. One logical variation on our model would be to consider buyers who compete with each other. In such a model, each buyer's payoff depends on both prices. Then a buyer may value the M F N provision as insurance against paying a higher input price than his rival. This issue requires a complicated model, but we can describe some of the likely effects. In the second contract, assuming the first bargain had a M F N provision, the buyer loses more slowly from price increases because his rival's input price is also going up. The seller gains from both buyers as price rises. Thus, the second contract still seems likely to yield a higher price because of the policy. Provided the buyers are similar enough, the first contract again seems likely to include the M F N clause since the ability to raise P2 is still present. Another possible variation concerns the method of bargaining. Nash bargaining satisfies some desirable axioms, but negotiators could choose other methods for dividing the surplus they create. Changing bargaining models can alter the effects of the policy, Suppose, for example, bargainers seek to maximize the sum of their incremental surplus. If the seller has a • horizontal average cost, the result without the policy would be prices equal to marginal cost and the seller would receive zero profit, Adding the M F N policy would enable the seller to obtain higher prices, so the buyers would both always lose. This result differs from our finding that the first buyer could benefit from the policy. (This bargaining model produces a situation similar to ours with P *l --P 2 *, where the buyers lose with the policy.) Although the results are sensitive to the choice of bargaining model, our results still show that in some cases the M F N policy could appear not as a facilitating practice but as an outcome of bargaining between buyer and seller.

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Appendix Proof of Lemma 1: Consider the first order conditions for the two maximization problems. Problem I yields n2 cqU2/cqp2 + Ua 6~zc2/6~p2 + c~= 0

(1)

for P2 >0, where ~ is the Lagrange multiplier for the constraint (P2 >P~). For P2 > 0, Problem II's solution satisfies

U2(Orc2/@+ c?n~/OP)+ [n2(P2) + nl (P2) - n~(Pl)]OU2/c3P-fl = 0,

(2)

where fl is the Lagrange multiplier for the constraint (Pz 0, then (1) gives p* and (2) is satisfied at pz=p~. Notice p2=pl also falls within the constraint set for Problem I. Since ~ = 0 implies P~>Pl and since rc2U 2 is concave, rcZU2 is non-decreasing at pz=p~, where Problem II is solved. The payoffs in the two problems are the same at p2=p~. Therefore, the overall maximum of n2U 2 comes at p* which is at least as great as p~. (ii) If c~>0 and fl>0, each problem achieves its maximum at p2=p~. The same price maximizes both problems so it is the maximizing value. (iii) If c~>0 and fl=0, then Problem I achieves its maximum at pz=p~. Problem II, however, yields a payoff product at least as great as the one at pz=p~. The solution satisfies (2) which yields p:
Proof of Proposition 1: This proof consists of proving a series of lemmas, all based on the assumptions of the proposition.

Lemma 2. p"(pl)>p~. Proof. In case (i) there is no reason to offer the M F N policy so we assume they do not include it then; therefore, we will not observe the M F N policy if Pl falls in case i. In case (ii), P'(Pl)=Pl and Pl >P~' because c~>0 (p~' solves (1) if c~=0). In case (iii) suppose pm(pO
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bargainer. We know rcZ(p~)>rc2(pm(pl)) because ~zu2 is concave in Pz, pr~(p~)p* in this case. Q.E.D.

Lemma 3.

It cannot be true that c~=0 a n d / 3 = 0 .

Proof.

Suppose e = 0 and/3 = 0. Since (2) is satisfied at a price below the one solving (1) and since U2n 2 is concave, eq. (2) can be satisfied only if U s Or~l/Op+ [ r c l ( p z ) - nl(pl)]OU2/O p < 0, where derivatives are with respect to the price argument of the function, p~ for ~1 and Pz for U 2. This condition can hold only if Onl/Oprcl(pl). Consider whether the first contractors would want the solution to (1) or to (2), given e = 0 =/3. First, consider (1). The M F N policy has no effect on P2 but rc1 must be decreasing at pl. Clearly lowering p~ would benefit both parties to the first bargain, so this cannot be an optimizing outcome. N o w consider the solution to (2). This yields a value of p~(pt) which is less than p~ (which is the solution to (1)). By Lemma 2, this cannot be a strategy which maximizes the product of their payoffs, so there is no solution with e = 0 = / 3 which is consistent with Nash bargaining. This proves the result. Q.E.D.

Lemma 4.

dp'~/dp 1>0 provided •l is increasing at Pl.

Proof First note that the M F N policy does not appear in case (i). In case (ii), pm(pi)=Pi SO dp'/dpi= 1 >0. In case (iii), pm is the solution to (2) with c~=0. By comparative statics, dpm/dpl=(1/A)(&ci/api'OUZ/Op), where A is the derivative of (2) with respect to Pz. We know A < 0 because rcl(p)U2(p) and r~2(p)UZ(p) and - U 2 are all concave and the sum of concave functions is concave. We know au2/ap 0 at Pt, we have dpm/dpl > 0 . Q.E.D. Lemrna 5

nl is increasing at pro.

Proof Suppose Oni/Op P : =P". Since p">p~, U2OnZ/ap+n2OU2/Op
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T.E. Cooper and T.L. Fries, Pricing policy

pz=p"(pl) and buyer 1 also pays p"(p~). The parties effectively choose p" (by their choice of P0, Since rc2(p*) >=rrZ(p*), p¢ exists and lies between p* and p*. Since dpm/dpl > 0 at pc, the first bargainers can set p~ at p~ or above. Consider Ul(p)[rcl(p)+rc2(p)-rcZ(p'~)] at p=p¢, where p is the price ultimately paid, If p¢
OU~/Op[r?(p)+ nZ(p) -

rtZ(p~)] q-- U 1 [6~n 1/ap +

aTr2/Op],

(3)

which is positive at p* because (3U1/Op)nl+Ul(Onl/Op)=O at p*, zcZ(p)= n2(p~,) at Pe--P z *, and 07~Z/Op>O at p*. Therefore, raising price above p~' increases the payoff product above its level at p*, which gives the same product as without the policy. Hence, they select the policy. Now we show the first buyer benefits if rc2 is decreasing at p*. The first buyer would lose with the policy if pm>p~,, which can only occur if (3) is positive at p~'. But, when we substitute p=p*, (3) becomes c3U1/Op[~ 1(p~) h- rcZ(p~") - ~2(p~)] + U 1[ 87~1/0p + O~2/Op]. Since ( 3U1/Op)zcl + U~(O~Z/Op)=O at p~, ~z2(p*)>n2(p*), and n z is decreasing at p*, it is true that (3) is negative at p*. The payoff product is decreasing there, so the price chosen must be lower, p"
Ùu' /ap[~'(pt)] + u'[an~/ap + o~2/ap],

(4)

and we know (4) exceeds the expression 0 U l(pe)/6~p[7~l (p~')] -/'- U i(p~) [0~: l(p~)/Op],

(5)

because ~rz is increasing at pC, ~1 is concave, and n~ is increasing at p*. Factoring out Ul(pe), we find (4) and (3) must be positive if

Ul(p) {O~zl(p~)/Qp+ [(OUI(p)/Op)/U'(p)] [~cl(p~)]}

(6)

is non-negative at p = p". The expression (6) is zero at p = p~', so it is positive at pe if I(OU1/Op)/UII is smaller at pe than at p~'. Since the ratio is negative and assumed to decrease in p, then (6) is positive so (3) must be positive. Price exceeds p~ and we have already shown p"rc2(p*) and the seller's payoffs are concave, the seller earns a greater surplus at this new price, where pe
T.E. Cooper and T.L. Fries, Pricing policy

223

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