Selling methods by a monopolist

Selling methods by a monopolist Lluís Bru y Ramon Faulí-Oller z and Joel Sandonís x February 18, 005 Abstract In an industry with an upstream monopolist and a continuum of heterogeneous downstream rms, we analyze their incentives to be supplied by two di erent selling methods: a posted price market versus bilateral aggreements. We show that whenever both selling methods imply the same transaction costs, the posted price market unravels. However, if a bilateral aggreement is more costly, both selling methods coexist in equilibrium. J.E.L codes: D4, L11, L14 Keywords: Selling methods, posted price market, bilateral agreements. Financial support from 9/UPV 00035.31-13560/001 and SEJ 004-017/ECON (Sandonís), BEC001-0535, GV01-391 and SEJ 004-017/ECON (Faulí-Oller), SEJ004-07530-C04-04/ECON (Bru) and the IVIE is gratefully acknowledged. y Universitat de les Illes Balears. Departament d Economia de l Empresa. Campus de la carretera de Valldemossa km. 7.5, 071 Palma de Mallorca. Spain. Phone: 34971171330. Fax: 34 971 17 13 89. e-mail: lluis.bru@uib.es z Corresponding author:universitat d Alacant. Departament de Fonaments de l Anàlisi Econòmica. Campus de Sant Vicent. 03071 Alacant. Spain. Phone: 34 965 90 34 00 (ext. 36). Fax: 34 965 90 38 98. e-mail: fauli@merlin.fae.ua.es x Universitat d Alacant. Departament de Fonaments de l Anàlisi Econòmica. Campus de Sant Vicent. 03071 Alacant. Spain. Phone: 34 965 90 34 00 (ext. 33). Fax: 34 965 90 38 98. e-mail: sandonis@merlin.fae.ua.es 1

1. Introduction The great development of the Internet in the last few years has opened di erent new ways of connecting people. Firms have taken notice of these new opportunities to reshape their ties with customers and suppliers. For example, they can now purchase commodities in the Web via either a posted price market or by directly negotiating with suppliers. A challenge for economists is to understand how rms choose between the two di erent selling methods. We have recently seen, for example, that although rms like e-steel and MetalSite have attempted to create anonymous posted price markets, most of the buyers still prefer to rely on the traditional bilateral agreements. In other markets however, both selling methods coexist. This is the case of some agricultural markets and the stock market (Pirrong, 000). When we talk about a posted price market we refer to a selling method such that all the buyers pay the same price for the good. By direct negotiations we refer to a selling method such that the price is adjusted to the characteristics of individual traders. In this paper, we consider an upstream monopolist selling an input to a continuum of downstream rms producing an homogeneous good. We assume that downstream rms are heterogeneous in their production cost. They can obtain the input either in a posted price market or by bilateral negotiations. This last option requires some speci c investment in the form of a link that allows the upstream rm to adjust the supply contract to the individual characteristics of the rms. In other words, the creation of links allows the upstream rm to price discriminate among its linked customers. Price discrimination is pro table because rms di erences in costs translate into di erent elasticity of input demands. In particular, it is the case that the higher the cost of a downstream rm the higher its input demand elasticity. Therefore, the upstream rm would like to adjust upwards the wholesale price for low cost rms and downwards for high cost rm, knowing that the personalized contract will only be accepted if it o ers a discount with respect to the price in the posted price market. In the posted price market, the upstream rm can not discriminate and he has to set the same price for rms with di erent costs. Then, the optimal price will be a decreasing function of the average cost of those rms and, when a rm decides to create a link, it a ects the average cost of nonlinked rms and thus the price in the posted price market. As we

will see, high cost rm are the ones that create links. In this way, they generate a negative externality on the rms that remain in the anonymous market, through an increase in the posted price market they face. Regarding the creation of links, we will analyze two possible cases: on the one hand, the links are created by the upstream rm in a centralized way; on the other hand, each downstream rm decides whether to establish a link with the upstream rm. In the rst case, the upstream rm prefers to connect high cost rms, because low cost rm would reject the personalized contract whenever the posted price market exists. In the limit case, when the cost of a link vanishes, the upstream rm would connect all downstream rms and the posted price market would unravel. In the second case, we have that the gains of creating a link for downstream rms are increasing in their costs, because the higher the cost the higher the discount they will receive in the personalized contract. This explains that, again, market is segmented such that high cost rms create links and are treated personally and low cost rms attend the posted price market. As in the rst case, in the limit case where the cost of the link vanishes, all rms create a link and the posted price market unravels. In this second case, rms that take advantage of the personalized treatment exert a negative externality on those that decide to attend the posted price market. As this negative externality is not internalized by downstream rms when they decide whether to attend the organized market, we nd there is overinvestment in the technology that allows to receive a personalized treatment. In other words, the number of links is higher than the one that would maximize joint downstream pro ts. Regarding welfare implications of our paper we obtain very sharp conclusions: in order to maximize social welfare links should be forbidden. This result is reminiscent to the well known result by Schmalensee (1981) that compares, in terms of our model, the case with only bilateral agreements (third degree price discrimination to all downstream rms) with the case where there is only a posted price market (uniform price). He shows that in a linear environment price discrimination is welfare reducing. We show that this result generalizes to the case of partial price discrimination induced by the market segmentation obtained in 3

equilibrium in our model. Our work is related to three di erent strands of the economic literature: the literature on market microstructure, the research on the formation of networks between rms and the literature on price-discrimination. Regarding the rst strand, there have been many papers trying to understand the microstructure of markets and trading institutions. Among them, the papers more closely related to ours are the ones that compare di erent institutions to determine their structural advantages and allow traders to endogenously choose the institution through which they prefer to conduct their trades. In this line, Wang (1993,1995) compares the pro ts a seller obtains using di erent selling methods. In Wang (1993), he shows that an auction is more pro table than a posted price when buyers are heterogeneous. Wang (1995) compares bilateral bargaining with a posted price market, leading to similar results. He does not consider, however, the possibility that the seller uses more than one selling method simultaneously, which is the main focus of our paper. Two papers that pursue this goal are Rust and Hall (003) and Neeman and Vulkan (001). The former considers several buyers and sellers and models the coexistence of di erent kinds of intermediaries (market makers that post prices and middlemen that bargain with customers). The authors provide evidence of this coexistence in several markets. The latter follows the same approach but reaches a more extreme conclusion: the existence of a posted price market precludes the existence of trade through bilateral negotiations. Notice that our results suggest that the opposite result may emerge, namely, that the posted price is the one that may unravel. The di erence in the models that can explain these striking di erences in results are the following: on the one hand, the pro ts for a buyer in a bilateral agreement only depend on its type, whereas in Neeman and Vulkan (001) they also depend on the pool of participants in bilateral negotiations. On the other hand, in our model it is the posted price that endogenously depends on the pool of participants in the posted price market, whereas Neeman and Vulkan consider an exogenous posted price. Hence, in our models downstream rms form expectations about the level of participation in the posted price market and, consequently, they choose whether to join that market. In 4

Neeman and Vulkan (001), we have that expectations are made about the personalized wholesale price that will be o ered. As a consequence, the dynamics of the externalities among downstream rms work di erently, leading to opposite results. The literature on network formation assume that in order to be supplied some speci c inputs, rms must invest in establishing a relationship ( a link ) with the supplier. Kranton and Minehart (001) analyze rst whether, once links are created, the e cient amount of input is supplied. Second, they analyze whether the optimal network of links is created. In our setup, a link instead of serving to obtain speci c inputs, serves to obtain a speci c contractual treatment. Finally, our paper, as we have mentioned, is related with the literature on price discrimination by a monopolist (Schmalensee (1981)). Observe that in our setting, when a buyer invest in a link, he is giving the seller the possibility to exert third degree price discrimination. The rest of the paper is organized as follows. In the following section, we present the general model solving it for the case where the upstream chooses the links and the case where downstream rms choose the links. Finally, the last section discusses the results and opens new avenues for future research.. The Model We assume we have a market with demand P (Q) = A Q. We assume that there is an upstream monopolist producing an intermediate good at no cost. There also exists a continuum of downstream rms that transform this input on a one-for-one basis into a nal homogeneous good. The cost of this transformation for downstream rm i is given by: C i (q i ) = c i q i + q i Downstream rms are heterogeneous in parameter c i, which is assumed to be uniformly distributed among the rms in the interval [0,1]. The timing of the game is as follows: In the rst stage, the links are created. We will consider two di erent possibilities, 5

namely, either they are chosen in a centralized way by the upstream rm or they are decided individually by downstream rms. In any case, the cost of creating a link is f. In the second stage, the upstream rm decides a uniform wholesale price w to supply the good to the rms attending the posted price market and an individual wholesale price of the form w i = b ac i to be o ered to each of the downstream rms with a link. In the third stage, downstream rms decide how many units to buy from the upstream rm and how many units to sell to nal consumers. We allow linked rms to attend the posted price market and, therefore, they will make use of the personalized contracts only when w b ac i. We look for the Subgame Perfect Nash Equilibrium of the game solving it by backward induction. Solving explicitly this game is complex. Moreover, as we show below, in order to obtain the equilibrium distribution of links, which is our main interest, it is enough to solve a simpli ed version of the game (called Game I) where linked rms do not have the option to be supplied in the posted price market. Very nicely, we will show that there is a strong relationship between the equilibria of Game I and the equilibria of our original game. In order to be able to study the equilibrium of Game I, it is convenient to introduce the following de nitions. We de ne the sets N = fc i [0; 1]= rm i is not linkedg and L = fc i [0; 1]= rm i is linkedg. De ne also z as the mass of set N (hence the mass of set L amounts to 1 z), x = R c N idc i and u = R c L idc i. Finally, de ne r = R L c i dc i. In the third stage, as we have a continuum of rms, every rm behaves as a price taking rm. Linked rm i chooses its output q i to maximize its pro ts: i = P q i c i q i q i w i q i This leads to the following individual supply function for rm i: S i (P ) = P c i w i 6

The individual supply of a non-linked rm j is similarly obtained and amounts to: S j (P ) = P c j w The market clearing condition is given by: Z N P ci w Z dc i + L P ci w i dc i = A P and leads to the following equilibrium price: P (a; b; w) = A + u(1 a) + x + b(1 z) + wz 3 In the second stage, and given the set of linked rms, the monopolist chooses the personalized wholesale and the price to be charged in the posted price market to maximize: Z I (L; a; b; w) = where superscript I denotes Game I: The optimal contracts are: N P Z (a; b; w) c i w P (a; b; w) c i w i w dc i + w i dc i L a = 1, b = A and w (L) = A The equilibrium pro ts of the upstream rm amounts to: x z (.1) I (L; a ; b ; w (L)) = 3x + (A(A 1) + 3r 1 4 )z 4z (.) In the rst stage, we have to derive the equilibrium distribution of links depending on whether the links are created by the upstream rm or by the downstream rms. We analyze each case in turn. 7

.1. The upstream rm chooses the links. We proceed to characterize the rst stage equilibrium. The decisions of the upstream rm are (1) to choose the number of unconnected rms (that amounts to decide the mass z of the set N) and () to decide the distribution of connected and disconnected rms (that amounts to the choice of terms x and r in the pro t expression). We nd in the following lemma which is the distribution of connected rms for a given measure 1 z of linked rms that maximizes (.). Observe that this amounts to maximize x + rz. Lemma.1. For a given measure 1 z of linked rms, the optimal distribution of linked rms in Game I is any set B = [0; s] [ [z + s; 1] for any s [0; 1 z]. Proof. For the distributions not considered in the lemma, there exist numbers 0 a < b < d 1 and 0 < c < d b such that rms in [a; b] [ [b + c; d] are not connected and the ones in [b; b + c] are connected. If z is the measure of unconnected rms and H is the expected cost of unnconected rms other than the ones in [a; b] [ [b + c; d], the payo of the upstream rm can be written as: f(b) = H + Z b a c i dc i + Z d b+c c i dc i + z Z b+c b c i dc i + constant We show that this distribution is not optimal because, the payo can be increased changing b given that the payo is strictly convex in b. f 0 (b) = H + Z b f 0 (b) = c H + c i dc i + a Z b f"(b) = c + cz > 0 a Z d c i dc i + b+c Z d c i dc i (b (b + c)) + z (b + c) b b+c c i dc i + (b + c)cz This implies that this distribution does not maximize the pro ts of the upstream rm. 8

The payo of the upstream rm with the distributions in the Proposition are: h(s) = h 0 (s) = Z z+s s Z z+s s Z s c i dc i + c i dc i + Z 1 0 z+s c i dc i z c i dc i (z + s s) + s (z + s) z h 0 (s) = (z + s) s z + s (z + s) z = 0 All the distributions in the Proposition yield the same payo. The upstream rm is indi erent among all the sets in Lemma 1, because all them allow him to get exactly the same payo. Let us consider again the original game. The following lemma relates the equilibrium payo of the upstream rm in the second stage in both games. Lemma.. If w (L) b a c i for all c i L, then (.1) and (.) are the equilibrium contracts and payo of the upstream rm in the second stage of the original game. Otherwise, the equilibrium payo in the second stage of the original game is strictly lower than (.). Proof. In Game I, let us de ne the maximal value of I (L; a; b; w) as I (L): In the original game, let us de ne the payo of the upstream rm given the contracts (a; b; w) as (L; a; b; w): Assume this function is maximized in (a 0 ; b 0 ; w 0 ). Z = fc i L = c i b0 w 0 g. It is possible to see that: a 0 (L; a 0 ; b 0 ; w 0 ) = I (Z; a 0 ; b 0 ; w 0 ) I (Z) I (L) Let us de ne The rst equality is driven by the fact that in both cases we have the same set of linked rms and the contracts are also the same. In this case, we will have the same equilibrium market price and the same payo s. If Z = L and (a 0 ; b 0 ; w 0 ) = (a ; b ; w (L)), the two inequalities transform into equalities. If Z = L, and (a 0 ; b 0 ; w 0 ) 6= (a ; b ; w (L)), the rst inequality is strict. If Z L the second inequality is strict and comes from the fact that the equilibrium payo s in Game I increase by adding links. To prove that assume that rms in the interval 9

[a; a + t] are connected, rms in [a + t; c] are unconnected, H is the expected costs of unconnected rms other that the ones in [a + t; c] and z its measure. Following (.), then the equilibrium payo s are given by: f(t) = 1 8 0 B H + R c c a+t idc i @ z + c a t where K is a constant + Z a+t a 1 c C i dc i + KA It is immediate to see that f 0 (t) > 0, what proves the second inequality. If the upstream rm connects the downstream rms in J = [z; 1]; he will set in Game I the contracts in (.1), where w (J) = A z A x. But as we have that < A z, these will 4 4 also be the equilibrium contracts in the original game. Therefore he will obtain the payo s given in (.). We are going to show, with the help of Lemmas 1 and, that this is the optimal distribution of links given that the upstream rm connects a mass of 1 z of rms. Observe that any set B in Lemma 1 (other than J = [z; 1]), has an interval [0; s], with s > 0 that includes the rm with the lowest cost. For this rm (and for some neighborhood with strictly positive mass around 0), the intended personalized price w i (c i = 0) = A is strictly higher than w = A x ; hence, these rms would refuse the personalized price z and would attend instead the posted price market. Given Lemma, the upstream rm will obtain, with these distribution of links, less pro ts than in Game I and, therefore, less pro ts than if he connects the rms in [z; 1]. With distributions of links di erent than the ones in Lemma, the pro ts of the upstream rm are bounded above by the pro ts he obtained in Game I and, therefore, they are lower than the ones obtained connecting rms in [z; 1]. Transforming (.) taking into account that only rms in [z; 1] are connected, we can obtain the payo of the upstream rm in the rst stage as a function of z: 1 4 z3 + (A(A 1) + 3) 4 4 (1 z)f (.3) Proposition.3. The upstream rms connects the rms in the set [z; 1] where z = Minf1; 4 p fg. 10

Observe that the size of demand (A) does not a ect the set of linked rms. Of course, it a ects the gains and losses of connecting downstream rms, but in the aggregate these e ects cancel out. Notice that creating a link has a direct positive e ect of allowing the upstream to personalize the contract, which is indeed increasing in A. On the other hand, there is an indirect e ect because the upstream rm will adjust upwards the price in the posted price market. It turns out that this indirect e ect is decreasing in A and it exactly cancels out with the direct e ect... Downstream rms choose the links. We are going to study equilibria such that only rms in [x; 1] pay the cost of the link. For this distribution of links, the optimal wholesale price in Game I is A x and we have that 4 A x 4 > A x A c i ; for all c i [x; 1]: Then, by using Lemma, we know that our original game has the same equilibrium contracts as Game I. With this information, we can compute, for a given x, the payo obtained by a rm with cost c i in the posted market p (c i ; x) and with a personalized contract b (c i ). Observe that as rms are in nitesimal the connection does not a ect the contracts to be set in the second stage. p (c i ; x) = ( A 4c i + x + 4P (a ; b ; w )) ; 64 b (c i ) = (A + c i P (a ; b ; w )) : 16 Observe that the latter pro ts do not depend on the distribution of links, because equilibrium market price (given the optimal contracts) does not depend on x either. The equilibrium is determined by x such that p (c i ; x ) b (c i ) f if c i x ; p (c i ; x ) b (c i ) f if c i x : (.4) 11

When f 8A 13 ; 19 it is easy to see that x is the unique solution in [0,1] of p (x; x) D b (x) f = 0; given that p (c i ; x) D b (c i) c is strictly decreasing in c i. In particular, we obtain: x = 4A + 1 p (4A + 1) 880f : (.5) 15 We have that the number of links created is decreasing in f and increasing in A When f > we have that x = 1 and no rm creates a link. 8A 13 ; (.6) 19 Comparing (.5) and Proposition 1, we can conclude that more links are created when downstream rms create the links than when the upstream creates the link...1. Subsidizing the link In order to set a benchmark to compare the equilibrium results of this section, we analyze the case in which an agent can a ect the decision of downstream rms to create links by subsidizing a proportion s of the cost of each link. If this is the case, we would have the same equilibrium as in this section with the cost of the link reduced to (1 s)f. We consider three possibilities: rst, a welfare-maximizing agent designs the subsidy scheme; second, the upstream rm decides the subsidy and third, the subsidy is set to maximize joint downstream pro ts. Given s, rms in [x S ; 1] will be connected, where x S = minf 4A + 1 p (4A + 1) 880(1 s)f ; 1g: 15 1

We know that the third stage equilibrium price does not depend on x S. Therefore, in order to maximize social welfare it is enough to minimize total costs (production and linking costs). Production costs will be minimized when marginal costs, wholesale price excluded (c i + q i ), are the same in equilibrium for all downstream rms. This can be the case only when all downstream rms receive the same wholesale price i.e. when all rms attend the posted price market. Then he would set a prohibitively large tax so that no rm gets connected. When the upstream chooses the subsidy, we know that he bene ts from the links because this allows him to price discriminate (see.3). However, we have to take into account that downstream rms create more links than the ones he would create himself. Therefore, his incentive to increase the number of links with a subsidy is ambiguous. It turns out that the upstream rm nds optimal to subsidize the links whenever f is high and not many rms create links. When the subsidy is chosen to maximize downstream rms pro ts, we have to take into account the following. Cost minimization, equivalent to industry pro ts maximization, is obtained by reducing the links. We have seen that upstream pro ts are increasing in the number of links. Thus, it follows that downstream rms pro ts should be decreasing in the number of links which it will be achieved with a large enough tax. 3. Discussion and conclusion. It is well-known that market transactions are organized in very di erent ways. In the stock market, for example, one can buy stock in the stock exchange or through intermediaries. In the rst case, the price will be common to all buyers; in the second, the price will depend on the speci c characteristics of the buyer. In other markets however one observes that one of the systems prevail. For example, in the steel industry, although rms like e-steel and MetalSite have attempted to create anonymous posted price markets, most of the buyers still prefer to rely on the traditional bilateral agreements. A challenging task for economist is to explain these phenomena. In this paper, we en- 13

dogenize the choice of selling methods used by buyers and sellers in the markets. With this purpose in mind, we study a simple model where there exists a vertical structure characterized by monopolization upstream and perfect competition downstream. The intermediate good can be obtained either in a centralized posted price market or through individual agreements. All rms attending the posted price market pay the same price for the input, whereas individual agreements allow for price discrimination. We assume that for the personal relationship (link) to be established a xed cost must be incurred (this xed cost represents the di erence between the intermediation costs in a bilateral negotiation and in an organized market). We consider two di erent cases. On the one hand, the upstream rm decides the links to be created and, on the other hand, each individual downstream rm decides whether to establish a link with the upstream rm. It is very intuitive that the upstream rm will bene t from the possibility of price discrimination. It is also the case that some downstream rms may obtain a better deal with a personalized agreement. These rms will be the ones with a higher elasticity of demand for the input i.e. the less e cient rms. Interestingly enough, those rms are also the ones that the upstream rm is interested in treating personally. As the structure of links is the same in both cases, namely that high cost rms are linked, it is possible to draw general conclusions from both models. First, the two selling methods coexist in equilibrium, except in the extreme cases in which the cost of the link is either zero or very large. In the rst case, we obtain that the posted price market unravels. In the second case, no bilateral negotiation takes place. Second, more links are created when downstream rms choose the links. The main di erence between both cases, is that the number of links does not depend on demand conditions when they are chosen by the upstream rm. In the other case, the number of links increases with the size of demand. For simplicity we have analyzed a simple case where the upstream rm perfectly distinguishes the cost of every individual rm; that is we assume a complete information framework. It would be interesting to analyze the polar where the cost is private information of 14

each rm. This will limit the ability of the upstream rm to price discriminate and would change dramatically the structure of the model. This is indeed part of our research agenda but it should be studied in a separate paper. 15

4. References Kranton, R.E. and D.F. Minehart, 001, A theory of buyer-seller networks American Economic Review 91, 485-508. Neeman, Z. and N. Vulkan, 001, Markets versus negotiations: the emergence of centralized markets, working paper, Boston University. Pirrong, C, 000, Third markets and the second best. Manuscript St Louis:Washington University. Rust, J. and G. Hall, 003, Middlemen versus market makers: a theory of competitive exchange, Journal of Political Economy, 111, 353-403. Schmalensee, R. 1981, Output and welfare implications of monopolistic third degree price discrimination, American Economic Review, 71, 4-47. Wang, R., 1993, Auctions versus posted-price selling American Economic Review 83, 838-851. Wang, R., 1995, Bargaining versus posted price selling European Economic Review 39, 1747-1764. 16