Rent Shifting and E ciency in Sequential Contracting

Transcription

1 Rent Shifting and E ciency in Sequential Contracting Leslie M. Marx and Greg Sha er University of Rochester September 2001 Abstract In this paper, we analyze the use of contracts between vertically related rms to shift rents. We focus on a three-party sequential contracting environment in which two sellers negotiate with a common buyer. We nd that overall joint payo is maximized in this environment under general conditions, even though, in equilibrium, contracts are used for rent shifting. The division of surplus depends on the contracting environment, including: the set of feasible contracts, the relationship among the sellers products, and the bargaining power of the rms. We use the model to gain a new perspective on business practices such as slotting allowances, exclusive dealing, quantity discounts, market-share discounts, and predatory pricing, and we consider policy implications. JEL Classi cation Codes: D43, L13, L14, L42 We thank Ray Deneckere, Phil Reny, Sergei Severinov, Ivo Welch, seminar participants at Duke University, University of Iowa, Ohio State University, University of Pittsburgh, University of Rochester, University of Toronto, Washington University, University of Wisconsin, and the participants of the 1998 North American Summer Meeting of the Econometric Society for helpful comments on previous drafts of this paper. We thank the National Science Foundation (Grant SES ) for nancial support. The authors can be reached at the William E. Simon Graduate School of Business, University of Rochester, Rochester, NY 14627, or by at marx@simon.rochester.edu or sha er@simon.rochester.edu.

2 1 Introduction In traditional price theory, the buyer is a price taker. He or she chooses whether and how much to purchase, taking the price as given. In many settings, this is a reasonable approximation. For example, we are all familiar with buying groceries at the local supermarket. In other settings, however, the approximation is not a good one (supermarkets often negotiate their terms of trade when they purchase from manufacturers). Buyer negotiating power has become increasingly evident in recent years with the increase in concentration among supermarket chains, the growth of single-category retailers such as Toys R Us, and the emergence of superstores such as Walmart. Settings in which the terms of trade are negotiated often occur in intermediate goods markets, where buyers and sellers jointly participate in creating value for the end user. This creates a tension that does not arise in traditional price theory: the buyer and seller are partners in creating value, but adversaries in determining how the surplus is apportioned. Thus, when a single buyer negotiates a contract with a single seller, the terms of trade play dual roles, determining how much overall value is created and how that value is divided. Furthermore, if, as in most intermediate goods settings, a buyer negotiates contracts with multiple sellers whose payo s are, or can be made, interrelated, then the terms of trade also play another role: they a ect the buyer s future negotiations with all other sellers. In this paper, we study the economics of rent shifting, where in maximizing their joint payo, the buyer and one seller choose the terms of their contract to extract surplus from a second seller. 1 For example, quantity discounts, which are pervasive in intermediate goods markets, result in rent shifting (regardless of whether rent shifting is the primary intent). Whereas traditional price theory might emphasize how such discounts encourage the buyer to purchase the e cient quantity of a seller s product, these discounts also give the buyer 1 The use of contracts by sellers who are rst-movers in negotiating with a buyer to extract surplus from sellers who are second-movers was rst studied by Aghion and Bolton (1987). In a model in which the buyer purchases at most one unit from one seller, they show that, with complete information, a buyer and high-cost seller jointly can extract all the surplus from a low-cost seller by agreeing to an exclusive-dealing contract in which the buyer must pay a lump-sum penalty to the high-cost seller if it buys from the low-cost seller. 1

3 more leverage in its negotiations with other sellers, allowing it to negotiate better terms of trade with these sellers. For example, when PepsiCo o ers discounts on the purchase of Pepsi products that are increasing in quantity, it reduces the buyer s demand for Coca Cola products, allowing the buyer and PepsiCo to extract additional surplus. 2 As long as PepsiCo does not sell its products to the buyer at below-cost prices, these discounts are legal (otherwise, the discounts may violate antitrust laws against predatory pricing). 3 Examples of rent shifting are not limited to quantity discounts. In fact, any contractual provision between a buyer and seller that increases the buyer s opportunity cost of purchasing from other sellers can jointly increase the buyer and rst seller s payo. For example, PepsiCo s recent acquisition of the Quaker Oats Co., in which it acquired Gatorade, a noncarbonated sports drink, increases the possibilities for rent shifting. 4 In addition to o ering quantity discounts on its Pepsi products, PepsiCo can now o er brand discounts that are conditional on the buyer s purchasing minimum quantities of both its Pepsi products and Gatorade, aggregate rebates that depend on the total volume of purchases accumulated over a period of time of all PepsiCo products, discounts based on the percentage of the refrigerated shelf space devoted to its product line, and market-share discounts based on the percentage it receives of the buyer s total purchase in a given product category. 5 All these strategies have the potential to shift rents from one seller, Coca Cola, to another seller, PepsiCo. The buyer bene ts from rent shifting in two ways. First, it bene ts to the extent that it has bargaining power with the rst seller and can participate in the sharing of the surplus 2 Firms often claim to be harmed by their rivals quantity discounts. For example, Anti-Monopoly, Inc. argued in Anti-Monopoly, Inc. v. Hasbro that, when selling to Toys R Us, it was disadvantaged by Hasbro s practice of o ering volume discounts because its sales would cut into TRU s sales of Hasbro products, which will reduce the percentage of TRU s volume discount, 958 F. Supp. 895 at 901 (S.D.N.Y.), The court ruled against Anti-Monopoly (see previous footnote) stating, absent predatory pricing, an antitrust plainti cannot complain that its competitor s prices are too low. Id. at The Federal Trade Commission declined to investigate the proposed acquisition, nding that carbonated soft drinks and non-carbonated sports drinks are separate markets. See the Statement of Commissioners Swindle and Leary, PepsiCo, Inc./The Quaker Oats Company, File No , August 1, Discounts that are conditional on the buyer s purchase of minimum quantities of more than one product in a seller s product line are a form of bundling and tying. They have been analyzed previously as mechanisms for price discrimination (Stigler, 1968; Adams and Yellen, 1976; McAfee, McMillan, and Whinston, 1989; Armstrong, 1996), or as mechanisms to foreclose rivals (Whinston, 1990). These discounts, and discounts that are contingent on the quantity purchased of a rival s product, e.g., exclusive dealing contracts, are subject to a rule of reason (see Bruckmann, 2000; and Tom, Balto, and Averitt, 2000). 2

4 that is extracted from the other sellers. Second, and perhaps more importantly, it can capture the value to a seller of being the rst to commit to a contract by having the sellers bid for this right. Thus, one can view the widespread use of slotting allowances, which are lump-sum payments by a manufacturer to a retailer ostensibly to obtain shelf space or remain on the shelf as another form of rent shifting, this time from the sellers to the buyer. 6 We study the simplest multiple player, sequential contracting environment that captures the key ingredients of rent shifting: there are three players (a buyer and two sellers), two bilateral negotiations, and interdependencies among the sellers payo s. To study marketshare discounts, we allow for contracts that can depend in a general way on both sellers quantities. To study the e ectiveness of quantity discounts only, we also consider contracts that are restricted to depend only on a seller s own quantity. To study slotting allowances, we endogenize the order of negotiations by having the sellers bid for the right to negotiate rst. We allow for continuous quantities, general cost functions, trade with one or both sellers, any interactions (any manner of substitution or complementarity) among the units sold by the sellers, and any distribution of bargaining power among the contracting parties. 7 We have two main results, with several implications following from each. Our rst main result is that the ability of the buyer and rst seller to shift rents from the second seller depends on the contracting environment. 8 For example, the extent of rent shifting depends on the distribution of bargaining power between the buyer and rst seller, with surplus extraction from the second seller increasing in the rst seller s bargaining power. The extent of rent shifting also depends on the relationship among the sellers products, with greater 6 There is no single accepted de nition of slotting allowances. Some use the term only in connection with new products and analyze their role as a signaling device. We adopt a broader de nition here, using the term to include such things as upfront payments, facing allowances, shelf payments, pay-to-stay fees, and market development funds. These payments are the subject of ongoing investigation by antitrust authorities. 7 Segal (1999) considers a general contracting environment in which the common player makes simultaneous take-it-or-leave-it o ers to multiple agents. The main di erence between our paper s setting and his lies in the sequential nature of our contracting, which introduces a new pecuniary externality in that the second seller s bargaining payo depends on the prior contract between the buyer and rst seller. 8 Thus, the buyer and rst seller do not always extract all the surplus from the second seller as in Aghion and Bolton s (1987) model with complete information. Several factors account for this di erence. We allow the buyer to have bargaining power and allow the sellers products to be imperfect substitutes or complements, and we do not restrict attention to the case in which the buyer purchases at most one unit of one seller s product. 3

5 surplus extraction when the products are substitutes than when they are complements. Finally, the extent of rent shifting is greater when the buyer and rst seller can negotiate market-share discounts than when they are restricted to quantity discounts only, and greater still if the rst seller can o er a menu that includes the o er of large quantities at below-cost prices (useful as a threat to extract rents, but not purchased in equilibrium). Our second main result is that overall joint payo is maximized in all non-pareto dominated equilibria for all contracting environments. Thus, the rent-shifting methods that arise in our model do not distort the buyer s equilibrium quantity choices; the buyer chooses the same quantities that a fully integrated monopolist would choose. 9 This result holds for any distribution of bargaining power, any relationship among the sellers products, and whether or not below-cost pricing, market-share discounts, and slotting allowances are feasible. We can draw policy implications from these two main results because they allow us to consider the e ects of di erent environmental factors (and the interaction among them) on the distribution of payo s among the players and on consumer surplus. For example, our results lend support to Coca Cola s claim that it would be harmed by PepsiCo s acquisition of Gatorade. If one believes that PepsiCo s acquisition will give it more bargaining power with retailers than it had before the acquisition, then our theory of rent shifting suggests that Coca Cola has reason to worry: assuming PepsiCo can outbid Coca Cola for the right to negotiate rst, PepsiCo will be better o as a result of the merger and Coca Cola will be worse o. However, our result that overall joint payo is una ected by rent shifting implies that there will be no e ect on the prices consumers pay, so while Coca Cola may lose, the merger will cause no harm to competition. Our results also suggest that slotting allowances need not lead to higher prices for consumers, as some have argued, and that exclusive dealing contracts and below-cost pricing need not raise rivals costs or drive a rival out of business. To the contrary, we show that for some environments enforcement of antitrust laws against such practices may actually induce market foreclosure instead of preventing it. 9 In contrast, it has been shown in an environment with one seller and two buyers that the seller is often unable to commit not to engage in rent shifting, reducing the overall joint payo and thus preventing overall joint payo from being maximized (O Brien and Sha er, 1992; McAfee and Schwartz, 1994; Segal, 1999). 4

6 The rest of the paper is organized as follows. In Section 2, we describe the model. In Section 3, we illustrate the role of slotting allowances, market-share contracts, and belowcost prices in facilitating rent shifting. In Section 4, we give conditions under which surplus extraction is complete, and we show that overall joint payo is maximized in all non-pareto dominated equilibria. In Section 5, we consider the e ect of a ban on market-share contracts. In Section 6, we endogenize the order of negotiations, allowing the buyer to choose the seller with whom it will negotiate rst. In Section 7, we discuss the implications of our results for slotting allowances and laws against predatory pricing, and in Section 8 we conclude. 2 Model We consider a sequential contracting environment with complete information in which there are two sellers, X and Y; and a single buyer. Seller X sells product X and seller Y sells product Y. If the buyer purchases quantity x of seller X s product and quantity y of seller Y s product, then sellers X and Y incur costs c x (x) and c y (y); respectively. We assume that c x ( ) and c y ( ) are strictly increasing, continuous, and unbounded, with c x (0) = c y (0) = The model consists of three stages. In stage one, the buyer and seller X negotiate a contract T x for the purchase of product X. In stage two, the buyer and seller Y negotiate a contract T y for the purchase of product Y. In stage three, the buyer makes its quantity choices and pays the sellers according to T x and T y. We assume the legal environment is such that sellers are prohibited from selling their product at below-cost prices. 11 consider quantity choices of the buyer such that each seller earns non-negative payo. Thus, we We let denote the set of contracts, where 2f M ; I g: We use = M to denote the case in which contracts can depend on multiple sellers quantities. In this case, T x speci es a payment from the buyer to seller X as a function of the quantities (x; y) purchased by the 10 Our results hold equally if sellers X and Y each sell multiple products and x and y are vectors, or if sellers X and Y each sell a single product and x and y are scalars. 11 This assumption follows from antitrust laws against predatory pricing. Our results also apply in environments with borrowing constraints or limited liability protection. 5

7 buyer. We allow this payment to be an element of R[f1g. 12 Thus, M ft x : R + R +! R [f1gg: In Section 5, we consider how equilibrium outcomes are a ected if contracts are restricted to depend only on the quantity of an individual seller, i.e., T x 2 I where I Tx 2 M j T x (x; y) =T x (x; y 0 ) 8y; y 0 0 ª : We make similar assumptions for T y ; however, because T y is negotiated second, equilibrium outcomes are una ected by whether T y depends on both quantities or only on the quantity of y chosen by the buyer (see Proposition 1 below). Let R(x; y) denote the buyer s gross payo if it purchases the quantities (x; y). 13 Then the buyer s net payo is R(x; y) T x (x; y) T y (x; y). If negotiations with seller Y fail, the buyer s net payo is R(x; 0) T x (x; 0). IfnegotiationswithsellerX fail, the buyer s net payo is R(0;y) T y (0;y). If negotiations with both sellers fail, the buyer s net payo is zero. We assume that R( ; ) is continuous and bounded, with R(0; 0) = 0. Let (x; y) R(x; y) c x (x) c y (y) denote overall joint payo, xy max x;y 0 (x; y) denote its maximized value, and Q xy arg max x;y 0 (x; y) denote the set of maximizing quantity pairs. Similarly, denote the monopoly value and quantities for the buyer and seller X by x max x 0 (x; 0) and Q x arg max x 0 (x; 0), and the monopoly value and quantities for the buyer and seller Y by y max y 0 (0;y) and Q y arg max y 0 (0;y). Given our assumptions on R( ; ) and c i ( ), these values and quantities are well de ned. In the negotiation between the buyer and seller i, we assume that the two players choose T i to maximize their joint payo, and that the division of surplus is such that each player receives its disagreement payo plus a share of the incremental gains from trade (the joint payo of the buyer and seller i if they trade minus their joint payo if negotiations fail), with proportion i 2 [0; 1] going to seller i. 14 Our assumption of a xed division of the gains from trade admits several interpretations. For example, if seller i makes a take-it-or-leave-it 12 Given later boundedness assumptions, a range that includes large nite values su ces. 13 We can accommodate the case in which the buyer purchases more than it consumes (or resells) by letting R(x; y) =max x ~ 0 ;y 0 R(x 0 ;y 0 ) d(x x 0 ;y y 0 ),where0 x 0 x and 0 y 0 y and R(x ~ 0 ;y 0 ) denotes the buyer s utility (revenue) if it consumes (resells) (x 0 ;y 0 ) and incurs disposal cost d(x x 0 ;y y 0 ): 14 These assumptions are consistent with most commonly used bargaining solutions, which typically require players to maximize their bilateral joint payo s and divide the incremental gains from trade. For example, the bargaining solutions proposed in Nash (1953) and Kalai-Smorodinsky (1975) satisfy these conditions. However, the bargaining solution proposed in Binmore, Shaked, and Sutton (1989) does not since the additional surplus above the two players disagreement payo s is not always divided in xed proportion. 6

8 o er to the buyer, then i = 1. If the buyer makes a take-it-or-leave-it o er to seller i, then i =0. And if the buyer and seller i split the gains from trade equally, then i = 1. 2 We solve for the equilibrium strategies (we restrict attention to pure strategies) of the buyer and sellers X and Y by working backwards, taking our assumptions about the outcome of negotiations as given. The equilibrium we identify corresponds to the subgame-perfect equilibrium of the related three-stage game in which the assumed bargaining solution is embedded in the players payo functions. For subgame-perfection, we must restrict attention to contracts such that optimal quantity choices for the buyer in stage three exist. 3 Preliminary Results To gain some intuition, we start by considering the simplest case, a multiple-units extension of Aghion and Bolton s (1987) model with complete information in which both sellers can make take-it-or-leave-it o ers and in which overall joint payo is maximized when only product Y is sold ( xy = y > x ). In this case, the ine cient seller (seller X) and buyer can extract all the surplus by agreeing to a contract that penalizes the buyer if it purchases from seller Y. To see this, suppose seller X o ers, and the buyer accepts, the contract: 8 < c x (x)+ y ; if y>0 T x (x; y) = : c x (x)+ x ; if y =0: Then all gains from trade between the buyer and seller Y are extracted by seller X: Given T x, it is optimal for seller Y to o er contract T y (x; y) =c y (y) in stage two (the buyer rejects Y s o er rather than pay more). Given T x and T y, the buyer purchases from seller Y,the buyer and seller Y s payo is zero, and seller X s payo is y. Seller X has no incentive to o er any other contract, since the contract in (1) extracts all the surplus, and the buyer has no incentive to reject seller X s o er since then seller Y would o er T y (x; y) =c y (y)+ y and seller Y would extract all the surplus. Thus, the contract in (1) is an equilibrium contract. We now consider the role of slotting allowances, market-share contracts, and below-cost prices in facilitating rent shifting under various perturbations of this simple case. 7 (1)

9 Role of Slotting Allowances in Facilitating Rent Shifting Since the buyer plays a critical role in facilitating the transfer of rent, one might think that the buyer should be able to capture some of the surplus for itself. In practice, the buyer can do this by engaging the sellers in a bidding game, where each seller o ers to pay the buyer for the right to negotiate rst. In this bidding game, suppose there is a stage zero in which the o ers are made simultaneously and the buyer chooses with whom to negotiate rst. In stage one, the buyer negotiates with its chosen seller. In stage two, the buyer negotiates with the other seller, and in stage three, the buyer makes its quantity choices. In this case, equilibrium requires that each seller o er y in stage zero (one can show that the ine cient seller s payo is zero and the e cient seller s payo is y if the ine cient seller negotiates second). The buyer accepts one of the o ers and earns payo y. Each seller earns zero. The simplicity of this example belies the importance of its policy implications. Lumpsum payments to buyers, sometimes referred to as slotting allowances, are common in intermediate-goods markets and have received the attention of Congress and antitrust policymakers. In this example, slotting allowances transfer rents from seller X to the buyer and have important distributional consequences, but they have no short-run e ect on consumers. Their feasibility does not alter the buyer s product choice (the buyer always buys the e cient seller s product), nor does it a ect the prices consumers pay. Thus, a policy of banning slotting allowances, while clearly in the interest of seller X, only hurts the buyer. Role of Market-Share Contracts in Facilitating Rent Shifting Contracts that depend on both sellers quantities are sometimes referred to as market-share contracts; the buyer s payment to seller X depends not only on how much the buyer purchases from seller X but also on how much the buyer purchases from seller Y. An example is the exclusive dealing contract de ned in (1), where the buyer must pay a penalty of y x to seller X if it purchases from seller Y. Other examples include contracts in which sellers o er discounts to buyers based on the share they receive of the buyer s total purchases. Laws that curtail the use of market-share contracts have important rent-shifting impli- 8

10 cations, and their reach extends even to situations in which such contracts do not arise. For example, whether or not market-share contracts are feasible in the bidding game above, there exist equilibria in which the buyer negotiates rst with seller Y, seller Y o ers contract T y (x; y) =c y (y) + y in stage one, seller X o ers contract T x (x; y) =c x (x) in stage two, and the buyer accepts both o ers. Although neither contract depends on the other seller s quantity, a law that bans market-share contracts will nevertheless have distributional consequences in such cases because it a ects the equilibrium level of slotting allowances. In particular, slotting allowances are higher when market-share contracts are feasible than when they are not because the gain to seller X from negotiating rst, and thus the loss to seller Y from negotiating second, is less when these contracts are banned. 15 This implies that seller Y will gain and the buyer will lose from a ban on market-share contracts. Role of Below-Cost Pricing in Facilitating Rent Shifting The ability of rms to engage in rent shifting is also a ected by antitrust laws against predatory pricing. To see this, suppose that market-share contracts are feasible, and that the buyer can make a take-it-or-leave-it o er to seller X in stage one. Then if the buyer is to capture all the surplus from seller Y, seller X must accept a contract o er such as 8 < c x (x) if y>0 T x (x; y) = : c x (x)+ x y ; if y =0: The contract in (2), if accepted, would eliminate the buyer s gains from trade with seller Y, allowing the buyer to capture all the surplus. But notice that because y > x,therentshifting mechanism requires the buyer to purchase seller X s product at below-cost prices in the event the buyer does not purchase from seller Y. Such contracts are not credible because 15 One might conjecture that seller Y s surplus can be fully extracted with the non market-share contract ½ cx (x)+ T x (x; y) = y ; if x =0 c x (x)+ x ; if x>0; because, given T y, the most the buyer can earn from purchasing solely from seller X or seller Y is zero. However, if the buyer purchases from both sellers, the buyer s payo is max x;y 0 R(x; y) c x (x) T y (x; y) x, which is strictly positive for some T y (x; y) >c y (y). This implies that seller Y earns positive surplus. (2) 9

11 if the buyer were to purchase seller X s product at below-cost prices, the purchase would be illegal. It is illegal for seller X to sell its product at below-cost prices, and it is illegal for the buyer knowingly to induce seller X to sell its product at below-cost prices. Not only can laws against predatory pricing a ect the ability of rms to shift rents, they can also a ect the incidence of market-share contracts and the incidence and magnitude of slotting allowances. The best the buyer can do in this example is to o er seller X the nonmarket-share contract T x (x; y) =c x (x) (see Section 4.2), earning for itself a payo of x. Seller X earns zero, and seller Y earns y x. If the buyer were to negotiate with seller Y rst, the equilibrium payo s would be unchanged. Thus, in this example, the buyer s payo does not depend on the order of negotiations and neither does the payo of either seller. Slotting allowances would not be observed in this instance even if they were feasible. The common theme of the examples in this section is that rent shifting can take many forms and that the feasibility of certain kinds of contracts can have important e ects on the distribution of surplus in intermediate goods markets, regardless of the incidence of such contracts in practice. In the next section, we extend the model by solving for any relationship among the units sold by the sellers (substitutes, complements, or independent in demand), and we allow for any distribution of bargaining power between the buyer and each seller. 4 Solving the Model Stage three Buyer s quantity choices If the buyer has contracts with both sellers, then we denote the buyer s quantity choices by (x (T x ;T y );y (T x ;T y )): If the buyer only has a contract with seller X; then we denote the buyer s quantity choice by x (T x ); and if the buyer only has a contract with seller Y; then we denote the buyer s quantity choice by y (T y ): For now, we assume that x ;y ;x ; and y are well de ned. Later we verify this for the equilibrium contracts. Consider the case in which at stage three the buyer has contracts with both sellers. Then 10

12 the buyer chooses quantities (x (T x ;T y );y (T x ;T y )), where (x (T x ;T y );y (T x ;T y )) 2 arg max x;y 0 R(x; y) T x (x; y) T y (x; y) subject to T x (x; y) c x (x) 0 and T y (x; y) c y (y) 0: (3) Two stars denote the buyer s quantity choices when contracts are in place with both sellers. If the buyer only has a contract with seller X; it chooses x (T x ); where x (T x ) 2 arg max R(x; 0) T x(x; 0) x 0 subject to T x (x; 0) c x (x) 0: (4) One star denotes the buyer s quantity choice when a contract is in place with only one seller. We refer to R(x ; 0) T x (x ; 0) as the buyer s disagreement payo with seller Y. If the buyer only has a contract with seller Y; it chooses y (T y ); de ned analogously to x (T x ).WerefertoR(0;y ) T y (0;y ) as the buyer s disagreement payo with seller X. The de nition of x, y, x,andy requires that sellers earn non-negative payo. This follows from our assumption that sellers are prohibited from selling their products to the buyer at below-cost prices. The assumption is innocuous when the buyer has contracts with both sellers and there is complete information, because neither seller would sign a contract thatwouldgiveitnegativepayo ontheequilibriumpath. However,aswewillsee,the same cannot be said about a seller s incentive o the equilibrium path. In negotiating rst with the buyer, seller X may have an incentive to sign a contract that would give it negative payo if negotiations with seller Y were to fail because this might allow it and the buyer to extract more surplus from seller Y: Our assumption rules out such contracts because they rely on actions that if carried out would violate antitrust laws against predatory pricing Predatory pricing is a violation of Section 2 of the Sherman Act and, if it occurs in an intermediate goods market, of Section 2(a) of the Robinson-Patman Act. To establish a violation, seller Y must show that its rival s prices are below an appropriate measure of its rival s costs. Although there is some disagreement in the lower courts about whether sunk costs should be included in the de nition of below-cost pricing, all courts would nd below-cost pricing if T x (x ; 0) <c x (x ),giventhatc x (x ) is an incremental cost. 11

13 Stage two Negotiations with seller Y Given the buyer s equilibrium behavior in stage three, and assuming the buyer and seller X negotiate contract T x in stage one, the buyer and seller Y choose T y instagetwotosolve subject to seller Y s earning a payo of ¼ y,where max T y R(x ;y ) T x (x ;y ) c y (y ); (5) ¼ y = y (R(x ;y ) T x (x ;y ) c y (y ) (R(x ; 0) T x (x ; 0))) : (6) Note that ¼ y depends on (x ;y ;x ;T x ;T y ); we suppress the arguments except in some of the proofs. Seller Y earns y times the di erence between the joint payo of the buyer and seller Y when contract T y is in place and the buyer s disagreement payo with seller Y. Proposition 1 Given any T x such that x (T x ) and (x (T x ;c y );y (T x ;c y )) are well de ned, if T y solves (5) subject to (6), then Proof. See the Appendix. (x (T x ;T y );y (T x ;T y )) 2 arg max x;y 0 R(x; y) T x (x; y) c y (y) subject to T x (x; y) c x (x) 0: Given contract T x, Proposition 1 implies that, in equilibrium, the buyer and seller Y will choose a contract in stage two that induces the buyer to choose their joint payo -maximizing quantities in stage three. 17 Because the buyer s negotiations with seller Y occur after its negotiations with seller X, the buyer and seller Y have no incentive to choose a contract that would subsequently distort the buyer s quantity decisions for products X and Y. If there is no rst-stage contract between the buyer and seller X; i.e., if negotiations with seller X fail, the buyer and seller Y negotiate T y to solve max Ty (0;y ); subject to seller Y s earning payo ¼ y = y (0;y ). It is straightforward to show that for any optimal T y in this case, the buyer chooses y 2 Q y in stage three. The buyer s payo is then (1 y) y. 17 For example, the buyer and seller Y might agree on a contract in which seller Y o ers to sell its units to the buyer at cost plus a xed fee, where the xed fee is chosen to satisfy (6). 12

14 Stage one Negotiations with seller X In stage one, the buyer and seller X negotiate T x to maximize their joint payo, subject to dividing the surplus so that each player receives its disagreement payo plus a share of the gains from trade, with proportion x going to seller X. Thus, given the buyer and seller Y s equilibrium strategies in stages two and three, the buyer and seller X choose T x to solve max T (x ;y ) ¼ y ; (7) x2 subject to the buyer s preferring to negotiate with seller Y in stage two, 18 R(x ;y ) T x (x ;y ) c y (y ) R(x ; 0) T x (x ; 0); (8) seller X s earning a payo of T x (x ;y ) c x (x )= x ( (x ;y ) ¼ y (1 y) y ) ; (9) and, from Proposition 1, (x ;y ) 2 arg max x;y 0 R(x; y) T x(x; y) c y (y): (10) Note that, using Proposition 1, x ;y ; and ¼ y can be viewed as functions only of T x. Seller X earns x times the di erence between the joint payo of the buyer and seller X when contract T x is in place with seller X and the buyer s disagreement payo with seller X. In this stage, the buyer and seller X choose their contract T x taking into account its e ect on the buyer s subsequent negotiations with seller Y. We now consider whether the contract will induce (x ;y ) 2 Q xy, and whether surplus extraction will be complete. 4.1 When is surplus extraction complete? One might think that the buyer and seller X can choose contract T x to induce (x ;y ) 2 Q xy in stage three (this maximizes (x ;y )) and extract all of seller Y s surplus (this minimizes 18 Given our assumptions, it is never optimal for the buyer and seller X to negotiate a contract in stage one that precludes negotiations with seller Y in stage two. 13

15 ¼ y ). For example, suppose the buyer and seller X negotiate the contract 8 < c x (x)+f 1 ; if y>0 T x (x; y) = : c x (x)+f 0 ; if y =0; (11) where F 0 = F 1 + x xy. In this case, given an equilibrium contract T y, if the buyer purchases from both sellers, the joint payo of the buyer and seller Y is xy F 1. And if the buyer purchases only from seller X, the joint payo of the buyer and seller Y is also x F 0 = xy F 1. This implies that the buyer gains nothing from trading with seller Y, so any equilibrium involving the contract in (11) must result in zero payo for seller Y. If the buyer can make a take-it-or-leave-it o er to seller X, it would like to o er F 1 =0 and F 0 = x xy.butif xy > x, this would involve seller X selling at below-cost prices if negotiations with seller Y were to fail, violating antitrust laws. At the other extreme, if seller X can make a take-it-or-leave-it o er to the buyer, seller X would like to o er F 1 = xy (1 y) y and F 0 = x (1 y) y. 19 But if x < (1 y) y, then seller X s contract o er would also involve below-cost pricing if negotiations with seller Y were to fail. More generally, it must be that F 0 = F 1 + x xy and F 1 = x( xy (1 y) y ) if full extraction is to occur in equilibrium. However, in many environments this involves below-cost pricing. Since the buyer s disagreement payo with seller Y would then not be credible, it is not possible for the buyer and seller X always to achieve full extraction. Proposition 2 Suppose = M. Then full extraction is achieved if and only if seller Y has no bargaining power or y 0; where y xy x x ( xy (1 y) y ) : Proof. See the Appendix. If y = 0, then seller Y earns zero payo in any equilibrium and full extraction is trivially achieved. In other environments, however, full extraction is not assured, and the reason is that the emphasis of the optimal rent-shifting contract changes when the buyer s 19 If seller X attempted to extract more surplus by asking for a payment of more than F 1 = xy (1 y) y, the buyer would reject seller X s o er and earn its disagreement payo of (1 y) y. 14

16 bargaining power with each seller changes. When the sellers have most of the bargaining power, the emphasis of surplus extraction is on the penalty component of contracts in which the buyer is penalized for purchasing from seller Y. In contrast, when the buyer has most of the bargaining power, the emphasis of surplus extraction is on the below-cost pricing component of contracts in which seller X o ers the buyer low prices on out-of-equilibrium quantities. Both methods are e ective at extracting surplus from seller Y, butatsomepoint the latter method runs up against the legal constraint on below-cost pricing and additional surplus extraction is not possible. For xy > x, this happens when the buyer s bargaining power with each seller is su ciently large (when x and y are su ciently small). To gain further insight, we can write the condition y 0 as xy x ( xy (1 y) y ) x ; (12) which implies that, if seller Y has bargaining power, full extraction is achieved in equilibrium if and only if the di erence between xy and seller X s equilibrium payo is less than or equal to the monopoly value of seller X s product. In other words, if the joint payo of the buyer and seller Y is greater than the monopoly value of seller X s product, then the buyer strictly gains from trading with seller Y and so seller Y necessarily earns positive surplus. We now present two corollaries to highlight when full extraction can be achieved and when not. We begin by considering the implications of Proposition 2 for the degree of substitutability or complementarity between products. We say that the sellers products are perfect substitutes if xy = x = y and perfect complements if xy > 0, x =0; y =0. Corollary 1 If the products are perfect substitutes, full extraction is achieved. If the products are perfect complements, full extraction is achieved if and only if y =0or x = 1. If the sellers products are perfect complements, the inequality in (12) simpli es to xy x xy, which yields the result in Corollary 1. In this case, the buyer must purchase both products to realize positive surplus, which means that its disagreement payo with seller Y is zero. Unless seller X can capture all the surplus for itself in stage one, there will necessarily 15

17 be surplus left over for seller Y. In contrast, if the sellers products are perfect substitutes, then full extraction is always achieved. More generally, since y is increasing in xy,it follows that full extraction is more likely, for a given x and y, the more substitutable are the sellers products (substitutes imply xy < x + y,complementsimply xy > x + y ). In light of the section on preliminary results, it is also useful to consider the case in which overall joint payo is maximized when only one product is sold, i.e., xy =maxf x ; y g. Corollary 2 If y > 0, and overall joint payo is maximized when only one product is sold, then full extraction is achieved if and only if x (1 x y) y. In this case, full extraction is achieved if the monopoly value of seller X s product is weakly greater than the monopoly value of seller Y s product, or if both sellers can make take-it-or-leave-it o ers. Otherwise, full extraction is achieved for a given x, y if and only if the di erence in the monopoly values of the two sellers products is su ciently small. 4.2 Incomplete surplus extraction and e ciency In this section, we consider whether the buyer and seller X will want to maximize overall joint payo when full extraction cannot be achieved, or whether they will want to distort the buyer s equilibrium quantity choices in order to extract more surplus from seller Y. To reduce the dimensionality of the problem, we begin by proving the following lemma. Lemma 1 Suppose = M. Then contract T x is an equilibrium contract if and only if (x (T x ), y (T x ), x (T x ), T x (x ;y ), T x (x ; 0)) solves max (x 2 ;y 2 ) ~¼ y (13) x 2 0;y 2 0;x 1 0;t 2 ;t 1 subject to R(x 2 ;y 2 ) t 2 c y (y 2 ) R(x 1 ; 0) t 1 ; (14) t 1 c x (x 1 )and t 2 c x (x 2 ); (15) ~¼ y = y (R(x 2 ;y 2 ) t 2 c y (y 2 ) (R(x 1 ; 0) t 1 )) ; (16) t 2 c x (x 2 )= x ( (x 2 ;y 2 ) ~¼ y (1 y) y ) : (17) 16

18 Proof. See the Appendix. Lemma 1 simpli es the task of choosing contract T x to the easier task of choosing x, y, x, T x (x ;y ),andt x (x ; 0) to maximize the buyer and seller X s joint payo in (13) subject to the buyer and seller Y having non-negative gains from trade, (14), seller X s earning non-negative payo on and o the equilibrium path, (15), and each seller s earning its bargaining share of the buyer s gains from trade with it, (16) and (17). The buyer and seller X cannot always achieve full extraction because they cannot always choose (x, T x (x ; 0)) to eliminate the buyer s gains from trade with seller Y.From(15)and (16), we see that the buyer and seller X extract surplus from seller Y by decreasing T x (x ; 0) as long as T x (x ; 0) c x (x ) is satis ed. If this constraint binds before surplus extraction is complete, then the best the buyer and seller X candoistochoose(x, T x (x ; 0)) sothat the buyer earns x if negotiations with seller Y fail. Thus, in any equilibrium in which full extraction is not achieved, the buyer s disagreement payo with seller Y is xed. Since the buyer s disagreement payo with seller X, (1 y) y, is also xed, it follows that the buyer and seller X have no incentive to choose their contract T x to distort overall joint payo. Proposition 3 Suppose = M. If seller Y has all the bargaining power, then equilibria exist and overall joint payo is maximized in all non-pareto dominated equilibria. For all y < 1, equilibria exist and overall joint payo is maximized in all equilibria. Proof. See the Appendix. Proposition 3 yields one of the main results of the paper. It implies that in the design of the optimal rent-shifting contract, joint-pro t maximization considerations can be separated from surplus extraction considerations, even if full extraction is not achieved. If y < 1, the buyer and seller X s payo is increasing in (x ;y ), whether or not full extraction is achieved, and thus the buyer and seller X have an incentive to choose their contract T x to induce (x ;y ) 2 Q xy (choosing (x, T x (x ;y )) to distort the buyer s quantity choices 17

19 in equilibrium would lower overall joint payo with no o setting gain to either player). 20 Using Propositions 2 and 3, the payo expressions in (6) and (9), and noting that the buyer s disagreement payo with seller Y is x when full extraction is not achieved, it is straightforward to determine for any non-pareto dominated equilibrium each player s payo when contracts can depend on the quantities the buyer purchases from both sellers. Proposition 4 Letting ¼ M b, ¼M x,and¼ M y denote respectively the buyer s payo, seller X s payo, and seller Y s payo in any non-pareto dominated equilibrium with = M,then ¼ M b = xy ¼ M x ¼ M y ¼ M x = x xy ¼ M y (1 y) y ½ ¾ ¼ M y y = max 0; y : 1 x y The buyer earns the di erence between the joint payo maximum and the sum of the sellers payo s. Seller X earns its share of the buyer s gains from trade with it, which in equilibrium are given by the joint payo maximum minus the sum of seller Y s payo and the buyer s disagreement payo. Seller Y earns the maximum of zero and the amount y (1 x y) y. Thus, we can clearly see the distributional consequences of a law prohibiting sellers from charging below-cost prices. If y 0, then a law against predatory pricing has no distributional e ect and the contract de ned in (11) can be used by the buyer and seller X to extract all of seller Y s surplus. Otherwise, seller Y gains and, if x 2 (0; 1), both seller X and the buyer lose. Surprisingly, a law against predatory pricing has no short-run e ect on overall joint payo, so the policy neither helps nor harms consumers in the short run. 5 Individual Seller Contracts Contracts that depend on both sellers quantities may be infeasible in the sense that a seller (or the courts) may not be able to observe whether or how much the buyer purchases from its 20 If y =1, seller Y captures any gains from inducing the buyer to choose (x ;y ) 2 Q xy in stage three, and so the buyer and seller X are indi erent to choosing T x in stage one to induce (x ;y ) 2 Q xy or not. 18

20 rival. In addition, there may be legal constraints that prohibit making contracts contingent on such information. Thus, it is important to extend the analysis to consider rent shifting in an environment in which contracts can only depend on a seller s own quantity, = I. We refer to contracts in I as individual seller contracts. Because I is a subset of M ; we know that Proposition 1 continues to hold and, because T y is negotiated second, similar restrictions on the domain of T y do not a ect equilibrium outcomes. However, unlike when = M, with individual seller contracts, there is no direct way to penalize the buyer for choosing y>0. As we have shown in the section on preliminary results, this sometimes further limits the ability of the buyer and seller X to extract surplus. As in the previous section, we begin by simplifying the buyer and seller X s task of choosing contract T x to the easier task of choosing x, y, x, T x (x ;y ),andt x (x ; 0). Lemma 2 Suppose = I. Then contract T x is an equilibrium contract if and only if (x (T x ), y (T x ), x (T x ), T x (x ;y ), T x (x ; 0)) solves (13) subject to (14) (17) and Proof. See the Appendix. y 2 2 arg max y 0 R(x 2;y) c y (y): (18) R(x 1 ; 0) t 1 R(x 2 ; 0) t 2 : (19) R(x 2 ;y 2 ) t 2 c y (y 2 ) max y 0 R(x 1;y) t 1 c y (y) (20) With individual seller contracts, there are three additional constraints that must be satis ed: y must maximize R(x ;y) c y (y), the buyer must choose (x ; 0) over (x ; 0) when it only has a contract with seller X; and the buyer must choose (x ;y ) over (x ;y) for any y when it has contracts with both sellers. The rst requirement has no e ect on surplus extraction, and the second requirement does not bind in equilibrium since the incentives of the buyer and seller X are to decrease payments for x. But the requirement that (x ;y ) be chosen over (x ;y) for y 0, while never binding for = M, 21 may be binding with 21 Contracts in M can penalize the buyer for choosing x together with any positive y. 19

21 individual seller contracts. Thus, for = I ; the following constraint can bind: R(x ;y ) T x (x ;y ) c y (y ) max y 0 R(x ;y) T x (x ; 0) c y (y): (21) We refer to the constraint in (21) as the individual-seller-ic constraint. If this constraint binds, then the de nition of ¼ y in (6) implies that seller Y s payo satis es ¼ y = ¼ IC y (x ); where µ ¼ IC y (x ) y max y 0 (x ;y) (x ; 0) : If the individual-seller-ic constraint binds, then the joint payo of the buyer and seller X is (x ;y ) ¼ IC y (x ), which is maximized by choosing T x such that (x ;y ) 2 Q xy and x 2 arg min x 0 ¼ IC y (x): This is because for a given x, inducing the buyer to choose (x ;y ) 2 Q xy in stage three both increases the overall joint payo and relaxes the constraint in (21), allowing the buyer and seller X to extract more surplus from seller Y by decreasing T x (x ; 0). Thus, even with individual seller contracts, the buyer and seller X can separate the maximization of overall joint payo from the minimization of seller Y s payo. This leads to a result for individual seller contracts that is analogous to Proposition 3. Proposition 5 Suppose = I. If seller Y has all the bargaining power, then equilibria exist and overall joint payo is maximized in all non-pareto dominated equilibria. For all y < 1, equilibria exist and overall joint payo is maximized in all equilibria. Proof. See the Appendix. Previous results in the literature suggest that the buyer s quantity choices will be distorted when rms engage in rent shifting and contracts are restricted to the class of two-part tari s (McAfee and Schwartz, 1994; Marx and Sha er, 1999). Marx and Sha er, for example, nd that in equilibrium seller X will charge a wholesale price that is below its marginal cost, accepting a distortion of the joint-pro t maximum in order to increase the buyer s disagreement payo with seller Y. This occurs because seller X has only two instruments (a wholesale price and xed fee) to control three objectives (maximizing overall joint payo, 20

22 dividing the surplus between seller X and the buyer, and extracting rent from seller Y ). Proposition 5 implies that the class individual seller contracts we consider, although more restrictive than the class of contracts that can depend on both sellers quantities, is su ciently less restrictive than the class of two-part tari contracts that it is still possible to separate the maximization of overall joint payo from how much surplus is extracted and how it is divided. What is surprising is that this is so even if full extraction is not achieved (either because the constraint on below-cost pricing binds, or because the constraint that the buyer must choose x over x when it has contracts in place with both sellers binds). If y = 1 or the individual seller IC constraint does not bind, then the stage-one problem of choosing T x to maximize the joint payo of the buyer and seller X is the same for individual seller contracts as for contracts that can depend on both rms quantities, so the equilibrium payo s are the same under individual seller contracts as under the more general contracts. (This is apparent from the proof of Proposition 5.) However, if y < 1 and the individual seller IC constraint binds, then as we have seen seller Y s payo is min x 0 ¼ IC y (x). These observations lead to the following proposition, which is the counterpart to Proposition 4. Proposition 6 Letting ¼ I b, ¼I x, and ¼I y denote respectively the buyer s payo, seller X s payo, and seller Y s payo in any non-pareto dominated equilibrium with = I,then ¼ I b = xy ¼ I x ¼ I y ¼ I x = x xy ¼ I y (1 y) y ½ ¾ ¼ I y = max ¼ M y ; min (x) : x 0 ¼IC y Since ¼ M y y =maxf0; 1 x y y g, a policy prohibiting the use of market-share contracts has distributional consequences if and only if min x 0 ¼ IC y y (x) > maxf0; 1 x y y g.thereare many environments in which this inequality is satis ed. For example, if the sellers products n o are independent ( xy = x + y ) then min x 0 ¼ IC y y (x) > max 0; 1 y x y as long as x; y; x ; and y are positive. 22 In this case, a restriction to individual seller contracts reduces the 22 With independent products, ¼ IC y (x) = y y and y 1 y x y = y y y x 1 y x x : 21