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EAG Discussion Papers are the primary vehicle used to disseminate research from economists in the Economic Analysis Group (EAG) of the Antitrust Division. These papers are intended to inform interested individuals and institutions of EAG's research program and to stimulate comment and criticism on economic issues related to antitrust policy and regulation. The analysis and conclusions expressed herein are solely those of the authors and do not represent the views of the United States Department of Justice. Information on the EAG research program and discussion paper series may be obtained from Russell Pittman, Director of Economic Research, Economic Analysis Group, Antitrust Division, U.S. Department of Justice, BICN 10-000, Washington, DC 20530, or by e-mail at russell.pittman@usdoj.gov. Comments on specific papers may be addressed directly to the authors at the same mailing address or at their e-mail address. Recent EAG Discussion Paper titles are listed at the end of the paper. To obtain a complete list of titles or to request single copies of individual papers, please write to Janet Ficco at the above mailing address or at janet.ficco@usdoj.gov or call (202) 307-3779. Beginning with papers issued in 1999, copies of individual papers are also available from the Social Science Research Network at www.ssrn.com. Abstract When buyers choose the order in which they bargain with suppliers of known characteristics, prices are determined jointly by bargaining power and competitive intensity (the outside option to bargain with rival suppliers). Bargaining power becomes less important to the outcome as competition intensifies; prices fall to marginal cost in the limit. With positive visit costs and weak competition, some buyer power is necessary for trade. Incomplete buyer power may lead to inefficient choice of bargaining order. The robustness of ordered bargaining to the possibility of price posting and auctions, and welfare properties of these alternative pricing institutions are also explored.
Analyses of competition have consumer choice at their heart. Competition authorities recognize that consumers often have well-specified preferences over While consumer choice is central, consumers are commonly treated as passive price-takers. Suppliers are typically portrayed as choosing which prices to post or (in case a buyer organizes an auction) which prices to bid. The key role played by bargaining in determining prices in many markets has gained broader recognition recently, and the development of models of buyer power is on the rise. This paper develops a model of The ordered bargaining model developed here is a static model with a finite number of suppliers who face no effective capacity constraints. This economic setting is common in studies of industrial organization. The competitive bargaining literature, however, focuses on large economies with a continuum of atomistic buyers and sellers. This formulation has helped to make tractable the study of steady states in dynamic settings with continuous inflows and outflows of traders. But the focus on atomistic agents has tended to obscure the potential relevance of bargaining to the classic issues of concentration and market structure in industrial organization. The ordered bargaining model offers a number of intuitively appealing features. It exploits the idea of "competition as outside option." The option of bargaining with rival suppliers is key to determining a buyer's expected price in the model. While the importance of outside options to bargaining outcomes has long been recognized, outside options have typically been treated as exogenously given. According to the well-known "outside option principle," The broader the set of suppliers with whom a buyer could bargain, and the greater the ease with which a buyer could switch bargaining partners (e.g., less differentiation among suppliers; lower costs of visiting them), the stronger is the buyer's outside option. Stronger outside-option competition lowers the expected price the buyer faces in subgame perfect equilibrium of the ordered bargaining game. Outside-option competition and buyer bargaining power both contribute to lowering expected prices. Bargaining power wanes in significance, however, as outside-option competition intensifies. In the limit as the number of suppliers grows without bound and as trading frictions decline to zero, expected prices fall to marginal cost. This result contrasts with the typical finding in the competitive bargaining literature, that equilibria diverge from the perfectly competitive outcome even in the limit as frictions vanish. The ordered bargaining model offers a framework for studying the competitive effects of mergers and business practices in markets where prices are determined by bilateral bargaining. The model illustrates that, as in other market settings, a loss of (outside-option) competition tends to raise prices in ordered bargaining markets. The model offers insights into how the costs of initiating bargaining ("visit costs") and heterogeneity among suppliers affects competitive intensity, as measured by the value of the buyer's outside option to bargain with rival suppliers. The remainder of the paper is organized as follows. Section 2 lays out the economic setting. Section 3 derives the unique subgame perfect equilibrium of the general game and explores the efficiency of equilibrium. Section 4 treats two special cases, one with identical players and one with a particular structure of symmetric differentiation, to illustrate how competitive intensity affects bargaining outcomes. Section 5 augments the ordered bargaining game by introducing the possibility that suppliers can post prices and buyers can organize auctions for their business. Circumstances under which posted prices or auctions emerge in the new equilibrium are then explored, as are welfare implications of the alternative pricing institutions. Section 6 discusses some implications of the preceding results and concludes.
All players have full information and the structure of the game is common knowledge. There is some number (possibly a continuum) of buyers and suppliers indexed by Each buyer In period 1, buyer
The subgame perfect equilibrium of the game with respect to any buyer
If , there clearly is no scope for trade, in which case buyer Now consider buyer
. Note that switching the places of suppliers Proposition 1 states that buyer Buyer
Proposition 2 describes equilibrium with respect to a given buyer
by Proposition 1. By Proposition 2, buyer
by Proposition 1, and so . Thus buyer
by Proposition 1. Buyer Inefficiency can occur when because buyer Define the expected (net) price facing buyer
A key question of interest is how this price depends on the buyer's bargaining power and on "competitive intensity," as measured by the strength of the buyer's outside option. This is explored in the next section for two special cases.
This section treats two special cases of the ordered bargaining game: (1) identical players and (2) a particular structure of symmetrically differentiated goods.
Suppose all buyers are identical, as are all suppliers. Every buyer values every good at Note that if then by recursion p = 1, and no buyer visits any supplier in equilibrium. This is a generalization of Diamond's (1971) paradoxical result: When prices are (always) chosen by sellers and are unobservable to buyers prior to a costly visit, the unique equilibrium is for sellers to set the monopoly price and for no buyer to visit any seller._{1}^{(7)} Diamond's (1971) result holds for any strictly positive visit cost, however small (given ). The result holds more generally whenever buyers choose prices with positive probability but search costs are sufficiently high, . Note that this result does not depend on the number of suppliers, n. If , on the other hand, buyers face a below-monopoly expected price (p u_{1}_{1} > 0) from playing the game. In this case, competition in the sense of n does affect the expected market price. For , equation (3) simplifies to
Note from equation (8) that the smaller is Evaluated at the outset of the game, the buyer's outside option of visiting supplier n. This is because a power buyer is already capturing a large share of the joint surplus, apart from any outside-option competition. For , the series on the right-hand side of equation (8) converges to as . Thus, for ,
By equations (7) and (9),
The limit price in equation (10) falls to zero (marginal cost) as The result also diverges from the classic Bertrand-Nash model of price setting with identical suppliers and zero visit costs, in which the equilibrium price equals marginal cost for any number of suppliers . This has potential implications for the competitive analysis of horizontal merger in industries characterized by homogeneous suppliers. The posted-price Bertrand-Nash model implies that (absent efficiencies) a three-to-two merger would have no effect on price. In contrast, the ordered bargaining model suggests the effect might be substantial. For example if , the expected price would double, from to , with a change from
To highlight the role of product differentiation, assume now that the cost of visiting any supplier is zero, suppliers have common marginal cost of zero, and buyers have common bargaining power . Suppose that any given buyer values suppliers' goods according to a power function of the supplier's ranking. In particular (dropping superscript n! buyer types, each distinguished by a unique permutation in ranking the n suppliers. Differentiation is symmetric in the sense that every supplier has ranking j, j = 1, 2, ..., n, for the proportion 1/n of the buyer population. Maintaining the full information setting, suppliers know their rankings with respect to every buyer. In this example, the analogues to equations (1) and (2) are
and
The buyer would offer a price of zero if chosen as proposer, supplier
The higher is (closer to one), the less differentiated are suppliers' goods from any given buyer's perspective. Note that a buyer's share in joint surplus increases with , given that the buyer's outside option is strengthened thereby. If , a buyer's share in joint surplus also grows with As the number of suppliers grows without bound, the buyer's expected share in joint surplus in equation (13) converges to
and thus the expected price converges to
Perfect competition and one-shot bargaining are polar extreme cases. For , the limit price in equation (15) falls to zero (marginal cost). The outside option is very strong in this case: perfect homogeneity of goods allows buyers in the limit to capture the entire unit joint surplus from trade with supplier 1, by equation (14). At the opposite extreme of , the outside option becomes worthless. In this case, supplier 1 has the position of local monopolist, albeit one that must bargain to make the sale. The buyer then captures only of the joint surplus in the one-shot bargaining game.
This section contributes to the growing literature endogenizing pricing institutions. As shown presently, in some circumstances posted pricing or auctions can supplant the ordered bargaining equilibrium. Issues of key interest are the factors determining which pricing institution emerges in equilibrium, and the welfare consequences of this outcome. Consider first the traditional conception of price posting in which suppliers publicly announce prices at which they stand ready to serve all comers. Suppose that buyers are identical, as are suppliers. If suppliers can post prices at zero posting cost, clearly the ordered bargaining equilibrium cannot be sustained. In the new equilibrium, all suppliers post the common marginal cost and buyers capture the entire surplus without bargaining. With costly posting, there are two possibilities. If posting costs are prohibitively high, the ordered bargaining equilibrium is preserved. If posting costs are small but strictly positive, typically no equilibrium exists. In this case, if multiple suppliers were to post prices, none could recover posting costs given the ensuing fierce competition.
Suppose that suppliers can send advertising leaflets to individual buyers at the common cost per leaflet. A buyer that has not received any leaflet embarks on ordered bargaining as before. A buyer that has received a leaflet chooses whether to redeem it or to engage in ordered bargaining. A leaflet is a posted price targeted to a particular buyer. For notational convenience, assume for the moment that buyers have common bargaining power , although they may differ in other respects. If buyer
Supplier
where is the expected surplus from ordered bargaining (given in equation (3)), and
In equilibrium with full information, at most one supplier sends a leaflet to buyer The lowest price that any supplier
Susbtituting equation (18) into equation (16) yields
as the highest expected surplus that supplier
Recall from Proposition 3 that the equilibrium of the ordered bargaining game may be inefficient. A given buyer may forgo trade altogether, despite the existence of a known supplier with whom joint surplus would be positive. A buyer may also knowingly choose a trade that yields less than maximal joint surplus, in seeking to maximize private gains from trade. Proposition 4(i) implies that if the equilibrium of the ordered bargaining game is inefficient, targeted price posting may improve the efficiency of trade. Such a posting allows the efficient supplier to propose a split in joint surplus to buyer Conversely, Proposition 4(ii) states that if the equilibrium of the original ordered bargaining game was already efficient, introducing targeted price posting weakly reduces efficiency. In this case, buyer The next two subsections each explore a special case of the augmented model, with an eye to factors determining whether targeted price posting emerges in equilibrium.
Consider again the case of identical players discussed in Section 4.1, only augmented now with an initial stage in which suppliers can engage in targeted price posting. Recall from condition (17.1) above that a buyer is willing to redeem a leaflet only if doing so yields expected surplus at least as high as the buyer can obtain from ordered bargaining:
The highest expected surplus from redeeming a leaflet that suppliers could profitably promise is the joint surplus from trade minus visit cost Recall that the expected surplus u increases with the number of suppliers, _{1}n, which improves the outside option. Second, a reduction in visit costs, s, also improves the outside option to visit other suppliers. All else equal, condition (20) is harder to satisfy the greater the intensity of outside-option competition in ordered bargaining. This result is fairly general, as can be seen by inspecting equation (3). In the limit as , recall (equation (9)) that buyers capture the entire joint surplus from trade less the portion . In this limiting case,
discarding the normalization (
Thus the critical posting cost decreases as buyer power grows. The welfare consequences of targeted price posting are subtle. Suppose that
The middle inequality in (23) implies If , on the other hand, . In this case, trade does take place in the original ordered bargaining game. Moreover, there is no question of allocative inefficiency given identical players. Targeted price posting then represents a pure waste, if it occurs in equilibrium of the augmented game. Dissipation of total surplus would be greatest if posting costs were just below the critical value .
Recall from Proposition 3(iii) that in equilibrium of the ordered bargaining game, a buyer may choose a trade that fails to maximize joint surplus, in order to economize on visit costs. The reason is that a buyer bears the full cost of visiting a (distant or inconvenient) supplier, but may capture only a fraction of the joint surplus in ex post bargaining. In this case, by Proposition 4(i), there is scope for targeted price posting to improve allocative efficiency in equilibrium of the augmented game. To simplify notation, let denote the joint surplus from trade between buyer
Inequality (24) can be written as
According to inequality (25), in order for the joint surplus from trade with supplier Note that is a high upper bound on posting costs consistent with supplier A distinct point is that, as in the identical player setting, targeted price posting is less likely to emerge the more intense is outside-option competition in the ordered bargaining game. The reason is that more intense competition (whether from a larger number of potential bargaining partners or from lower visit costs) yields the buyer more of the joint surplus
Auctions, like targeted price posting, guarantee that the efficient trading partner wins the buyer's business. To see this, consider again the symmetric differentiation case of Section 4.2. Recall that in this case suppliers have zero costs and a buyer values the good of the Other qualitative results for targeted price posting derived above generally apply to auctions as well: A buyer is less likely to undertake an auction the higher the cost of organizing the auction and the greater the expected surplus obtained from ordered bargaining. To focus on auctions as an alternative means of extracting surplus from suppliers, assume that auctions can be organized costlessly. In the limit as , ordered bargaining yields the buyer more surplus than does an auction in case
The right-hand side of inequality (26) is the price at which supplier 1 would win the auction, whereas the left-hand side is the expected price the buyer would pay to supplier 1 in ordered bargaining (from equation (15)). Rearranging terms in (26) yields
or equivalently
Thus for given buyer power , ordered bargaining yields a lower price than does an auction if differentiation is sufficiently great (i.e., small; condition (27.1)). Intuitively, the more precipitously the value of the good drops in moving from supplier 1 to supplier 2, the more poorly the auction performs, given that bid competition from supplier 2 (alone) constrains the price paid to supplier 1. Alternatively, for given differentiation, ordered bargaining yields a lower price than does an auction if buyer power is sufficiently great (condition (27.2)). Note that implies ordered bargaining outperforms an auction for any level of differentiation in the limit as . Holding fixed, outside-option competition grows weaker the smaller is
For example if , (28) implies .
A sometime expressed belief is that competition and bargaining are distinct and irreconcilable forces acting upon prices. On this view, prices in a market are determined either by competition, or through bargaining--and if through bargaining, then "any outcome is possible in bilateral monopoly." The dichotomy is a false one, however, as this paper has striven to show. Competition, in the guise of the outside option to bargain with rival suppliers, can work in tandem with buyer power to shape market outcomes. In the general model of of ordered bargaining presented in Sections 2 and 3, the unique subgame perfect equilibrium (with regard to any given buyer's trade) can be computed for an arbitrary specification of valuations, production costs, visit costs and bargaining powers. Section 4 developed some intuition into the workings of the model by exploring two special cases: identical players and a particular structure of symmetric differentiation. The equilibrium expected price is determined in ordered bargaining by both the buyer's power in bargaining with the current supplier as well as the attractiveness of the buyer's outside option to bargain with rival suppliers. The value of the option rises with the number of suppliers, but declines with "frictions" to exercising the option, such as visit costs or product heterogeneity. Large enough frictions render the outside option worthless, resulting in the polar extreme of bilateral monopoly bargaining. At the other extreme, frictionless exercise of the outside option leads to the perfectly competitive outcome in the limit as the number of suppliers grows without bound. The model offers insights to competition authorities for analyses of mergers or business practices in markets where bilateral bargaining is an important determinant of prices. First and foremost, the model reveals that negotiated prices tend to rise with a reduction in (outside-option) competition, as is the case in other, more familiar contexts. The model also illustrates, however, that the link between competition and pricing can differ subtly in ordered bargaining as compared with posted price or auction markets. For example, the standard Bertrand-Nash model of posted pricing in homogeneous goods markets predicts that (absent efficiencies) a three-to-two merger of suppliers will have no effect on price. The ordered bargaining model suggests, on the contrary, that the effect of such a merger might be substantial. Settings in which buyers interact repeatedly with suppliers on the road to securing a deal are frequently analyzed as auction markets. Often, however, the nature of the interaction may better fit the ordered bargaining paradigm. In an auction, the buyer's price is determined (exclusively) by bidding competition between the buyer's top two choices. Thus a merger of suppliers tends to raise the auction price only in case the merging parties are (with some positive probability) the buyer's number one and number two options. Given the model's full-information setting, bargaining delay and breakdown are not at issue. Nevertheless, the equilibrium of the ordered bargaining game may be inefficient. This is because buyers fully bear the cost of visiting a supplier, but may capture only a fraction of the joint surplus in ex post bargaining. This is a generalization of Diamond's (1971) fundamental insight. Not only may valuable trade fail to occur in the ordered bargaining equilibrium, but buyers may choose trade that fails to maximize joint surplus. Section 5 discussed how posted pricing and auctions can eliminate these distortions if they emerge in equilibrium of the augmented game. However, if the ordered bargaining equilibrium is efficient to begin with, the emergence of alternative pricing institutions may well introduce new distortions. The model suggests several areas for future research. It would be interesting to endogenize visit costs, broadly conceived as any cost buyers must sink prior to bargaining with a supplier. A market organized by suppliers alone may not have the proper incentives to reduce costs, however. A reduction in the cost of visiting one supplier imposes a negative pecuniary externality on rivals, by sharpening outside-option competition. Thus collectively as well as individually, suppliers may have inadequate incentives to lower individual visit costs. Another possible area for future research is endogenizing bargaining power. Investments by individual suppliers to improve their bargaining power could be viewed as a form of wasteful rent-seeking. Acemoglu, D., Shimer, R., 1999. Holdups and efficiency with search frictions. 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The role of firm size in bilateral bargaining: a study of the cable television industry. Review of Economics and Statistics 81, 326-340. Corbae, D., Temzelides, T., Wright, R., 2003. Directed matching and monetary exchange. Econometrica 71, 731-756. Davis, D.D., Holt, C.A., 1996. Consumer search costs and market performance. Economic Inquiry 34, 133-151. De Fraja, G., Sakovics, J., 2001. Walras retrouve: Decentralized trading mechanisms and the competititve price. Journal of Political Economy 109, 842-863. Diamond, P.A., 1971. A model of price adjustment. Journal of Economic Theory 3, 156-168. Dobson, P.W., Waterson, M., 1997. Countervailing power and consumer prices. Economic Journal 107, 418-430. Engle-Warnick, J., Ruffle, B.J., 2005. Buyer Concentration as a Source of Countervailing Power: Evidence from Experimental Posted-Offer Markets. Mimeo. Ferris, J.S., 1990. Time, space, and shopping: the regulation of shopping hours. Journal of Law, Economics & Organization 6, 171-187. 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* Economist, Antitrust Division, U.S. Department of Justice. The views expressed are solely my own and not those of the U.S. Department of Justice. I thank Patrick Greenlee and Glen Weyl for helpful discussions and comments. 1. See for example the Horizontal Merger Guidelines, U.S. Department of Justice and the Federal Trade Commission, April 2, 1992 (revised: April 8, 1997). 2. Papers on directed or ordered search are exceptions to this (see e.g. Perry and Wigderson, 1986; Stahl, 1996; Acemoglu and Shimer, 1999; Corbae et al, 2003; Arbatskaya, 2005). 3. A very partial listing: Horn and Wolinsky (1988), Dobson and Waterson (1997), Chipty and Snyder (1999), Inderst and Wey (2003) and Raskovich (2003). See also Normann et al (2005) and Engle-Warnick and Ruffle (2005) for experimental tests of buyer power. 4. Among others, Jackson and Palfrey (1998) use this term to describe the literature. In business circles, however, the term has the colloquial meaning of "hard" bargaining. Papers in the competitive bargaining literature are too numerous to list exhaustively. Among the early papers in this vein are Rubinstein and Wolinsky (1985), Gale (1986), Bester (1988) and Binmore and Herrerro (1988); see Osborne and Rubinstein (1990) for a survey. More recent papers include Jackson and Palfrey (1998) and De Fraja and Sakovics (2001). 5. See the seminal paper by Binmore et al (1989). 6. Gross of any earlier visit costs that have been sunk. In subgame perfect equilibrium of the full information game, no buyer visits more than one supplier. 7. See Davis and Holt (1996) for an experimental test of the Diamond (1971) paradox. 8. Here the merger is assumed to simply reduce the number of bargaining opportunities by one. 9. Papers in this vein include Bester (1993), Wang (1993, 1995), Bulow and Klemperer (1996), Arnold and Lippmann (1998), Neeman and Vulkan (2003) and Raskovich (2006). 10. This is similar to the advertising technology in Butters (1977), except that here the leaflet can be directed to a particular buyer with certainty. 11. This assumes , of course. Otherwise supplier 1 would do at least as well by playing the ordered bargaining game with buyer 12. Given identical suppliers, a buyer's ordering of suppliers is of course arbitrary. The subscript 1 in condition (20) simply indicates the buyer's perspective prior to making the first visit. 13. This result also holds for finite 14. See Waehrer and Perry (2003) for an analysis of mergers in an auction setting. 15. Suppliers also typically must sink some costs prior to bargaining with buyers. Raskovich (2003) shows that a supplier has an incentive to sink as little cost as possible prior to bargaining with a pivotal buyer, suggesting this as an explanation for profit-sharing in the motion picture industry as well as fragmented participation in loan syndicates for project finance. 16. Suppliers may also lobby for restrictive hours-of-service regulation. For analyses of such rules and their deregulation, see Ferris (1990), Kosfeld (2002) and Inderst and Irmen (2005). 17. The analysis is of course symmetric for buyers. 18. However, Raskovich (2006) shows that suppliers may have a collective interest in weakening individual power to commit to a posted price. |