Competition in Two-sided Markets With Common Network Externalities

Competition in two-sided markets with common network externalities Abstract We study competition in two sided markets with common network externalities rather than with the standard inter-group e¤ects. This type of externality occurs when both groups bene…t, possibly with di¤erent intensities, from an increase in the size of one group and from a decrease in the size of the other. We explain why common externalities are relevant for the health and education sectors. We focus on the symmetric equilibrium and show that when the externality itself satis…es an homogeneity condition then platforms’ pro…ts and price structure have some speci…c properties. Our results reveal how the rents coming from network externalities are redistributed by platforms from one side to other, according to the homogeneity degree. In the speci…c but realistic case where network externalities are homogeneous of degree 0, platform’s pro…t do not depend on the intensity of the (common) network externalities. This is in sharp contrast to conventional results stating that the presence of network externalities in a two-sided market structure increases the intensity of competition when the externality is positive (and decreases it when the externality is negative). Prices are a¤ected but in such a way that platforms only transfer rents from consumers’group to providers’one. 1 Introduction The theory of two-sided markets has been developed in recent years to investigate market structures in which two groups of agents interact via platforms; see for instance Rochet and Tirole (2006). The central theme of this literature is the notion of (network) externality which re‡ects the property that the bene…t from joining a platform for individual of a given group depends on the size of membership (and/or usage) from the other group (Armstrong, 2006). Prominent examples of sectors in which such intergroup externalities occur range from credit cards and software to dating clubs. We consider a two-sided market with externalities of a di¤erent nature. We shall refer to them as a “common network externalities”. This type of externality occurs when both groups bene…t, possibly with di¤erent intensities, from an increase in the size of one group and from a decrease in the size of the other. Such externalities are relevant in a number of two-sided markets. For instance, in the health care sector, hospitals compete for patients on one side and for providers on the other side (see Pezzino and Pignatoro, 2008). It is a conventional assumption that the quality of health care depends on the providers’“workload”. This is documented, for instance, by Tarnow-Mordi et al. (2000) who use UK data to show that variations in mortality can be explained in part by excess workload in the intensive care unit. Accordingly, health care quality is frequently related to the provider/patient ratio; see Mc Gillis Hall (2004). In other words, it increases when the number of health care professionals increases (for a given number of patients) but decreases when the number of patients increases (for a given number of providers). Both sides bene…t from a higher quality albeit for di¤erent reasons and possibly with di¤erent intensity. This is quite obvious on the patients’ side, where one can expect a higher quality to translate into a improvement in patients’ health state (or at the very least into a reduction in waiting lines for appointments, etc...). Physicians bene…t from a higher quality through a reduction in their workload1 or indirectly, through their altruism. Similar issue may arise in the education sector. Colleges or universities compete 1 See for instance Fergusson-Paré (2004) for the nursing workload. Gri¢ n and Swan (2006) also …nd a strong relationship between nurses’workload and quality of health care. 1 for students on one side and for professors on the other side. The quality of education depends on the pupil/teacher ratio and one can expect both sides to bene…t from a higher quality (Mueller, Chase and Walden, 1988). On one hand, lower pupil/teacher ratios are associated with higher test scores for the children (see for instance Angrist and Lavy, 1999). In addition, it has been observed a smaller class size tend to increase average future earnings (Card and Krueger, 1992). On the other hand, teachers enjoy an improved job satisfaction.2 In this paper, we revisit the Armstrong’s framework but with common network externalities rather than with the standard inter-group e¤ects. Two platforms compete in prices on two distinct Hotelling’s lines. The common externality enters the preferences of both groups as a quality parameter. Each group values the common externality with (possibly) di¤erent intensities but the underlying notion of quality that matters (the functional form that speci…es quality) is the same for both groups. We focus on the symmetric equilibrium and show that when externalities are speci…ed by an homogeneous function, price structure and platforms’ pro…t present some special features. First, network externalities enter in a cumulative way in the price structure. Second, platforms operate a redistribution of this common network externality from one side to other. We identify that the direction of this redistribution depends on the sign of the homogeneity degree of the common network externality. Third, the competition intensity is also a¤ected by this homogeneity degree. Roughly speaking, according to the homogeneity degree, platforms may or may not be able to modify its price structure such that a higher price charged on one side is not outweighed by a lower price on the other side. Finally, our results shed lights on equilibrium properties in education and health sectors where quality is known to mainly depend on consumer/provider ratio i.e. homogeneity degree is 0. In this case, platforms’pro…t do not depend on the intensity of the (common) network externalities. This property is in sharp contrast to the results obtained so far in the two-sided literature. One of the major results …ndings which has been reiterated in many settings is that the presence of network externalities in a 2 Buckingham (2003) …nds that reduction of class size allows to increase slightly achievment but also increase teachers work conditions by lightening their workload and easing classroom managment. 2 two-sided market structure increases the intensity of competition when the externality is positive (and decreases it when the externality is negative). We show that in a context of common network externalities of degree 0, this is not the case. Under this assumption, prices are a¤ected by the externality but in such a way that platforms only transfer rents from one group to the other. Roughly speaking, to come back to the sectors previously mentioned, some rents due to common network externalities are extracted from the “consumers’side” and transferred to “providers”. Before proceeding, let us have a closer look at the relationship of our paper to the existing literature. As pointed out by Rochet and Tirole (2003), the two-sided literature is at the intersection between multi-product pricing and network theories. The main focus of this paper lies on the second aspect. Several types of network externalities have been analyzed in the two-sided literature. The standard one is the inter-group network externalities which we have mentioned above. It has also been pointed out in the literature that negative intra-group network “externalities” can occur in equilibrium. This may be the case when members of a given group compete with each other. An additional member on one side then not only creates a positive inter-group externality but at the same time, it can adversely a¤ect welfare of the other members of the considered group.3 For instance, in Bardey and Rochet (2009), health plans compete for policy holders on one side and for physicians on the other side. When a health plan enlists more physicians, this directly increases welfare of its policy holders. However, at the same time, it may tend to attract riskier policy holders who place a higher value on the diversity of physicians. The induced adverse selection problem can be seen as a negative intra-group network “externality” that occurs, in equilibrium, on the policy holders’side. These intra group e¤ects are of course strictly speaking not externalities as they operate through the price system. However, some recent papers have also considered proper negative intra-group network externalities. Belle‡amme and Toulemonde (2007) develop a model where agents value positively the presence of agents of the other group but may value negatively agents of their own group. For instance, both advertisers 3 Most of time, this e¤ect occurs because it increases the number of competitors. 3 and consumers bene…t from a large representation of the other group (positive interbrand externality) but advertisers are in competition for eyeballs (negative intra-brand externality). Belle‡amme and Toulemonde show that entry of a new platform might be impossible as long as intra-group negative externalities are too strong in comparison with inter-group ones. Kurucu (2008) analyses a matching problem in which an agent on one side prefers more agents on the other side but less on its own side. Such a con…guration of externalities can occur for matrimonial or job matching agencies. Our paper is inspired by Belle‡amme and Toulemonde (2007) and Kurucu (2008) from whom we borrow the presence of negative intra- and positive inter-groups network externalities. However, we combine the same ingredients in a di¤erent way. In our framework, an additional consumer generates a negative intra-group and a positive intergroup network externality. Roughly speaking, the utility of a consumer is increasing in the number of providers and is decreasing in the number of the other consumers a¢ liated with the same platform. On the providers’ side, network externalities work on the opposite direction. In other words, the utility of a provider is increasing in the number of providers a¢ liated to the same platform (positive intra-group network externality), while it is decreasing in the number of consumers present on the other side (negative inter-group network externalities). The combination of these two characteristics leads to our concept of common externality: both groups bene…t, possibly with di¤erent intensities, from an increase in the size of one group and from a decrease in the size of the other group. The paper is organized as follows. The model is presented in Section 2. In Section 3, we determine the equilibrium and study its properties. Some illustrations are provided in Section 4. 2 Model Consider two platforms j = f1; 2g located at both endpoints of the Hotelling’s segment. They compete for two groups of agents i = fA; Bg of mass 1 (group A) and m (group B) respectively. Agents of each group are uniformly distributed over an interval of length 1. The utilities of both groups exhibit quadratic transportation costs with parameters 4 tA and tB respectively. For the sake of simplicity, we shall refer to members of group A as “customers” while group B individuals are considered as “providers”. We shall return to this interpretation later. The utility of a group A individual (a customer), located at z, who patronizes platform j (consumes one unit of its product) is given by V = V + qj Pj where Pj denotes platform j’s price, while tA (z xj )2 ; measures the preference intensity for a quality qj . An individual of group B (a provider), located at y, who works (a given number of hours) for platform j has utility4 U = U + qj + wj tB (y xj )2 ; where wj denotes the wage paid by platform j, while is the preference for quality qj . Without loss of generality, reservation utilities are equal to zero. Consequently, the constants V and U denote the gross utility on sides A and B; they are assumed to be su¢ ciently large to ensure full coverage on both sides of the market. Platforms maximize pro…ts and simultaneously set their price/wage vectors (Pj ; wj ), j = 1; 2: Let nij denote the share of type i = A; B individuals a¢ liated with platform j = 1; 2, while Nji denotes the number of a¢ liates. With our normalizations we have NjA = nA j and NjB = mnB j : The quality o¤ered by platform j depends on its number of a¢ liates in both groups and is determined by B qj = f (NjA ; NjB ) = f nA j ; mnj : This function speci…es what we refer to as a “common network externality” and which is de…ned as follows. De…nition 1 A common network externality, described by the function qj = f (NjA ; NjB ); occurs when both sides value, possibly with di¤ erent intensities, the same network externality. 4 Note that for purpose of simplicity, we do not enter in the details of the providers’time constraint. 5 An important feature of this de…nition is that the functional form f is the same on both sides.5 In other words, customers and providers agree on the ranking of quality levels. However, the taste for quality (measured by and ) can di¤er between customers and providers. Prominent examples of such common externalities can be found in the health care and education markets. In the hospital sector for instance, on can think of nA j as representing the number of patients while mnB j represents the number of physicians. B Alternatively, nA j can be interpreted as the number of students while mnj stands for the number of teachers. In both of these cases one would expect quality to increase with A mnB j and to decrease with nj . A formulation often used in the literature on education A and health is given by qj = cmnB j =nJ . With this speci…cation the quality o¤ered by a hospital or a university depends upon provider/patient or teacher/student ratio, and the function f is homogenous of degree 0.6 More generally, one can assume that the function specifying the quality is homogenous of degree k, which may or may not be positive. For instance when quality is speci…ed by qj = NjB f is homogenous of degree = NJA ; (1) . We do not impose this assumption when determining the equilibrium in the next section. However, it will turn out that the equilibrium has speci…c properties when the common externality is homogenous of degree k. We shall focus more particularly on the realistic case k = 0 which has some strong implications in terms of competition policy. Using subscripts to denote the derivatives of f with respect to its …rst and second arguments (NjA and NjB respectively) and applying Euler’s law yields the following property. 5 Concerning the nature of network externalities, the di¤erence between our framework and Kurucu (2008) and Belle‡amme and Toulemonde (2007) can be understood as follows. In our setting, the two utility functions on both sides i.e. V A (NA ; NB ) and U B (NA ; NB ) are both decreasing in NA and increasing in NB . In their framework (but with our notations), they consider the case where V A (NA ; NB ) is increasing in NB but decreasing in NA while U B (NA ; NB ) is increasing in NA but decreasing in NB . 6 Krueger (2003) provides a cost-bene…t analysis of class size reduction. He shows that the internal rate of return of a class size reduction from 22 to 15 students is around 6%. 6 Property 1 When a network externality is homogenous of degree k then NjA fA (NjA ; NjB )+ NjB fB (NjA ; NjB ) = kf (NjA ; NjB ). 3 Equilibrium First, we characterize the demand functions (market shares) on both sides. Then, we determine the price equilibrium and study the properties of the corresponding allocation. 3.1 Demand functions On group A’s side, the marginal consumer indi¤erent between two platforms is determined by z~ = 1 1 + [ (q1 2 2tA q2 ) (P1 P2 )] ; while in group B, the marginal provider is given by y~ = 1 1 + [ (q1 2 2tB q2 ) + (w1 w2 )] : As both sides are fully covered, demand levels are equivalent to market shares. On side A, we have nA ~ and nA 1 =z 2 =1 z~ while on side B, nB ~ and nB 1 =y 2 = (1 y~). De…ning the quality di¤erential between platforms as B A B g nA 1 ; mn1 = f n1 ; mn1 f 1 nA 1 ; m(1 nB 1 ) = q1 q2 ; the demand functions are determined by the following system of implicit equations 1 1 + 2 2tA 1 1 nB + 1 = 2 2tB nA 1 = Let B g nA 1 ; mn1 (P1 P2 ) ; (2) B g nA 1 ; mn1 + (w1 w2 ) : (3) = ( ; ; tA ; tB ; m) denotes the vector of exogenous parameters. Equations (2)–(3) B de…ne the demand levels of platform 1, nA 1 (P1 ; P2 ; w1 ; w2 ; ) and n1 (P1 ; P2 ; w1 ; w2 ; ) ; as functions of both platforms price/wage vectors and of the exogenous variables7 . With 7 Note that we assume throughout the paper that demands are well de…ned and unique for any price levels. When qj = mnB nA j j , it is straightforward that demands are uniquely de…ned. Appendix A A describes that it is also the case when qj = mnB j =nJ . 7 full market coverage on both sides, demand levels of platform 2 are then also fully determined and given by nA 2 =1 B nA 1 and n2 = 1 nB 1. Totally di¤erentiating (2)–(3), solving and de…ning gA = @g=@N1A , gB = @g=@(N1B ) with = 1 2tB mgB 1 2tA gA mgB ; 4tA tB gA yields the following properties of the demand functions: B g nA @nA 1 ; mn1 1 = @ 2tA B B g nA @n1 1 ; mn1 = @ 4tA tB 1 gA 2tB mgB ; ; 1 2tB mgB @nA 1 = ; @P1 2tA @nB gA 1 ; = @P1 4tA tB @nA mgB 1 = ; @w1 4tA tB 1 2tA gA @nB 1 = : @w1 2tB 3.2 (4) (5) (6) (7) (8) (9) Equilibrium prices and allocation Platform 1 maximizes its pro…t with respect to P1 and w1 and solves max P1 ;w1 1 = P1 n A 1 (P1 ; P2 ; w1 ; w2 ; ) mw1 nB 1 (P1 ; P2 ; w1 ; w2 ; ) : The …rst-order conditions are given by @ 1 @nA @nB 1 1 = nA + P mw = 0; 1 1 1 @P1 @P1 @P1 @ 1 @nA @nB = P1 1 mw1 1 mnB 1 = 0: @w1 @w1 @w1 (10) (11) The …rst two terms of equations (10) and (11) represent the traditional marginal income tradeo¤, while the third terms capture the two-sided market feature. Speci…cally, an increase in the price charged on one side of the market also a¤ects the demand on the other side. Equations (10) and (11) determine platform 1’s best-reply functions: P1 = Pe1 (P2 ; w2 ; ) and w1 = w e1 (P2 ; w2 ; ). Platform 2’s best-reply functions 8 P2 = Pe2 (P1 ; w1 ; ) and w2 = w e2 (P1 ; w1 ; ) can be determined in a similar way by the maximization of 2. Solving these best-reply functions yields the Nash equilibrium (P1 ; w1 ); (P2 ; w2 ). In the remainder of the paper, we concentrate on symmetric equilibria in which both platforms charge the same prices, pay the same wages and equally split the market A B B on both sides (nA 1 = n2 = 1=2 and n1 = n2 = 1=2) so that quality levels are also identical (g = 0). To determine the symmetric equilibrium we solve (10) and (11). B The derivatives of nA 1 and n1 that appear in these expressions are given by equations B (6)–(9); with nA 1 = 1=2 and n1 = 1=2, they are all well determined and the problem reduces to the solution of a system of linear equations.8 Using the Cramer’s rule, we obtain h B i @nB 1 @n1 1 + m 2 @w1 @P1 P1 = ; D w1 = 1 2 h @nA m @P11 + mD @nA 1 @w1 i ; (12) where B B @nA @nA 1 @n1 1 @n1 + : @P1 @w1 @w1 @P1 D= (13) Substituting from (6)–(9) and rearranging yields @nB 1 2tA gA mgA 1 @nB 1 +m 1 = ; 2 @w1 @P1 2 4tA tB @nA @nA 1 2tB + mgB + gB 1 1 m 1 + = ; 2 @P1 @w1 2 4tA tB 1 D= ; 4tA tB (14) (15) (16) where gA and gB are evaluated at (1=2; m=2): Substituting (14)–(16) into (12), simplifying and de…ning gA = gA (1=2; m=2) and gB = gB (1=2; m=2) establishes the following proposition: Proposition 1 Symmetric equilibrium prices are given by 1 ( + m ) gA ; 2 1 tB + ( + m ) gB ; 8j = 1; 2 2 Pj = t A (17) wj = (18) 8 B It is worth noticing that the derivates depend on nA 1 and n1 (which are by de…nition set at 1=2) but not directly on P1 and w1 . The underlying reason for this simpli…cation is that for the determination of demands only di¤ erences in prices and wages matter; see expressions (2)–(3). 9 As usual, on both sides, platforms take advantage of transportation costs to increase their markup. Externalities, on the other hand, a¤ect prices in a more interesting way. Because of their speci…c nature i.e. common externalities, their impact is “cumulative”, as is re‡ected by the factor ( + m ) in the second term. In other words, network externalities a¤ect prices on the side they occur ( + m ) gi and they are partially or entirely shifted to the other side ( + m ) gh , with h 6= i. Using (17) and (18) we can now express equilibrium pro…ts as j 1 P mwj 2 j 1 = (tA + mtB ) 2 = 1 ( + m ) (mgB + gA ) ; 4 (19) so that @ @ j =m @ @ j = 1 (mgB + gA ) : 4 This last expression establishes the following proposition: Proposition 2 The impact of individual valuations of quality and on (symmetric) equilibrium pro…ts is described by m @ @ j = @ @ j S0 if and only if gA S mgB , 8j = 1; 2: Proposition 2 shows that the impact of the externality (or, more precisely of the relevant preference parameters) on pro…ts depend on the relative strength of the externalities created by the membership on the two sides. To interpret this proposition, we shall concentrate on the case where the common externality is homogenous of degree k. According to Property 1 we then have N1B fA + N1A fA = kf . Moreover, at a symmetric equilibrium, we have: gA = 2fA ; gB = 2fB ; so that mgB + gA = 4kf 10 1 m ; 2 2 0: Proposition 3 mgB + gA = 4kf When f homogenous of degree k the symmetric equilibrium implies 1 m 2; 2 sign @ @ , so that j = sign @ @ j = sign (k) ; j = 1; 2 (20) Proposition 1 has shown that common network externalities have a cumulative way on prices. Proposition 3 shows how pro…ts and competition intensity are a¤ected. When the degree of homogeneity of the common network externalities is positive, platforms’ pro…t decrease in the externalities parameters. In other words, common network externalities increase the competition intensity between platforms. This outcome occurs because in this case, platforms have to pay a higher relative price wj /Pj on providers’ side. Returning to the health care (or schooling) example, we have a positive externality created by one side and a negative externality generated by the other side. We can think about the case of k > 0 as that where the global impact of the externality is positive. In other words, if we increase membership on both sides in the same proportion, f (and thus quality) increases. From that perspective we can think of our …nding as a generalization of the conventional result in the literature (relating pro…ts and intensity of competition to the sign of the externality). When k < 0; on the other hand, we have a negative global externality which brings about extra pro…ts for the platforms. The wage paid to providers continues to increase in the network externality parameters. However, this increased cost is now more than fully shifted to the consumers. This is because quality is more sensitive to the number of consumers–and recall that quality decreases with the number of consumers. Consequently, the network externality tends to reduce the intensity of competition on the consumers’side. The …rms are then able to extract more extra rents from the consumers than they have to concede to the providers. Finally, let us consider the special case in which homogeneity degree is equal to 0 i.e. k = 0. Corollary 1 For a symmetric equilibrium, 8j = 1; 2 with f homogenous of degree 0 we have mgB + gA = 0, so that @ @ j = @ @ 11 j = 0: In that case, the intensity of preferences for quality (and thus the intensity of the externality) has no impact on equilibrium pro…ts. Speci…cally, pro…t levels are the same when the externality does not matter at all (in which case = = 0) as when one or both of these parameters are positive. In other words, a common network externality that is homogenous of degree 0, has no impact on the intensity of competition which is in stark contrast with conventional results obtained in the two-sided market literature (for alternative forms of externalities). The expressions for the prices (17) and (18) make it clear why this result emerges. Assuming gB > 0 (providers produce a positive externality) the externality in itself (or an increase in its valuation on either side) increase “rents” on the providers’ side: wages are increased. However, this increase in wages has no impact on pro…ts because it is entirely shifted to consumers: the price increase exactly matches the increase in wages. 4 Examples and illustrations To illustrate the results and provide some additional intuition, we shall now provide the full analytical solution for three special cases. First, we consider the case there the externality simply depends on the ratio between membership on both sides (so that f is homogenous of degree zero). Then we consider a setting with di¤erent degrees of homogeneity. Finally, we provide an example for the non homogenous case. A When qj = mnB j =nj , the common network externality is homogeneous of degree 0. Proposition 1 implies that equilibrium prices are given by Pj = tA + 2 ( + m ) (m) ; wj = tB + 2 ( + m ) (m) 1 : With this price structure, it is clear that platforms only transfer rents from “consumers” to “providers” and have equilibrium pro…ts independent of the network externalities. To con…rm this, note that with this speci…cation 19 reduces to j = 1 (tA + mtB ) , 2 12 j = 1; 2; which does not depend on or . It is worth noticing that (as discussed in the in- troduction) this functional form (with quality depending on the ratio), has interesting applications for education and heath care sectors. We now turn to the more general case where f is speci…ed by (1) so that the externality is homogenous of degree k = . The nice feature about this speci…cation is that it shows that negative levels of k do not have to be ruled out. Equilibrium prices and pro…t levels are given by (j = 1; 2) wj = j = 1 1 2 Pj = t A tB + m ( + m ); (21) 1 (22) 1 1 2 m 1 (tA + mtB ) 2 1 2 ( + m ); m ( + m )( ): (23) Not surprisingly, network externalities a¤ect pro…ts according to the sign of k = . When > , quality o¤ered by the platforms are relatively more sensitive to the number of consumers than to the number of providers. Therefore, platforms can charge a higher relative price on the consumers’side for the quality provided without transferring all the network externalities rents get to the providers. Finally, let us consider a case where the homogeneity property does not hold at all. For instance, think about a case where quality depends positively on the ratio A mnB j =nj but also depends positively on the “volume” of consumers treated by the provider9 . In such a case, we have qj = A + cnA with c small enough to mnB j =nj j ensure that we continue to have a negative intra-group externality on the consumers’ side. Equilibrium prices, wages and pro…ts are now given by c ; 2 tB + 2 ( + m ) (m) ; Pj = t A + 2 ( + m ) m wj = j = 1 (tA + mtB ) 2 ( + m ) c: It is worth noticing that because of the parameter c (which reduces the intensity of 9 This assumption makes sense for a hospital. The quality of care can depend on the volume of patients treated. 13 negative intra-group externality on the consumers’side) the common network externality does not satisfy the homogeneity property. Then, platforms’ pro…t are reduced in equilibrium because they charge a lower price on the consumers’side. This lower price is not outweighed by a lower wage paid on the providers’side. 5 Conclusion Our concept of common network externalities applied to a two-sided market structure allows to point out several issues. One of them concerns the rent redistribution that may occur in a two-sided framework. Our model reveals the rent transfer mechanism, operated by platforms, from consumers to providers according to the homogeneity degree of common network externalities. In particular, we have considered the case, popular in the health and education literature, where the externality is homogeneous of degree 0. Under this assumption, platforms merely transfer rents, generated by network externalities, between the two sides of the market. Roughly speaking, we show that the providers’side gains while the consumers’one pays for the network externalities. If the common externality is not homogenous of degree zero, the platform can extract some rents from the side with the largest marginal impact on the of the common externality. Our analysis could be extended at least in two ways. First, It would be interesting to investigate how the results change when one (or two) of the market is not fully covered. This would for example illustrate the case where the severity of illness of patients is not high enough for them to buy one unit of medical care in both hospitals. On the providers’side, it would capture the case where some of the physicians would prefer to remain independent workers. Second, another issue concerns the type of hospitals that compete in the market for patients. It would be interesting to study the outcome of competition between not for pro…t or physician’s owned hospitals in a setting of mixed oligopolies. This is on our research agenda. 14 References [1] Angrist J.D and V. Lavy, 1999, “Using Maimonides Rule to Estimate the E¤ect of Class Size on Stochastic Achievement”, The Quarterly Journal of Economics, vol 114, n 2, 533-575. [2] Armstrong M, 2006, “Competition in Two-Sided Markets”, Rand Journal of Economics, 37(3), 668–691. 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Appendix A Unicity of the demand A In the ratio case we have q1 = mnB 1 =n1 : From(3), one obtains B g nA 1 ; mn1 = where w = w1 2tB nB 1 1 2 1 2 w 2tB w 2tB w2 . Substituting this expression in (2) yields: nA 1 = where P = P1 1 tB B + n1 2 tA P 2tA (24) A P2 . It is worth noticing that if nB 1 is unique, then n1 is also unique. From (3), one has: nB 1 1 m 2tB 1 1 + A n1 1 nA 1 = 1 1 + 2 2tB Multiplying both sides of this equality by nA 1 1 A nB 1 n1 1 nA 1 m 1 A = n 1 2tB 2 1 16 m + 1 n1A w nA 1 yields: nA 1 1+ w tB tB nA 1 : Substituting (24) gives nB 1 as a solution to a three degree polynomial. Fastidious computations show that this polynomial admit three solutions but only one belongs to the space of real numbers.10 B Second order local conditions We analyze here the second order local conditions for the class of functions f being homogemous of degree k. For the symmetric equilibrium to be a local maximum, one needs that the matrix H de…ned by 2 H=4 be semi de…nite negative i.e @ 2 @2 1 @P12 @2 1 @P1 @w1 2 1 =@P1 3 @2 1 @P1 @w1 @2 1 @w12 < 0; @ 2 2 1 =@w1 5 < 0 and det H > 0. We thus assume that g is homogenous of degree k i.e NjA gA (NjA ; NjB ) + NjB gB (NjA ; NjB ) = kg(NjA ; NjB ): We also know, that gA and gB are homogenous functions of degree k all 1 so that for : gA ( NjA ; NjB ) = k 1 gA ; gB ( NjA ; NjB ) = k 1 gA : Di¤erentiating with respect to and letting = 1, leads respectively to: gA (k 1) = N1A gAA + N1B gAB gB (k 1) = N1A gAB + N1B gBB Thus at the symmetric equilibrium where gA = mgB , one has: N1A gAA + N1B gAB + mN1A gAB + mN1B gBB = 0 10 Note that the market share belongs to [0; 1]. If the solution of the polynomial is negative or superior to 1, it means that we are in presence of a corner solution at the demands level. 17 so that 1 m2 gAA + g + mgAB = 0; 2 2 BB 2 (k 1) gA = gAA + mgAB ; 2 (k (25) (26) 1) gB = gAB + mgBB : (27) The second order conditions are: @2 1 @nA @ 2 nA 1 1 = 2 + P 1 @P1 @P12 @P12 @2 1 = @w12 2m w1 m @nB @ 2 nA 1 1 + P1 @w1 @w12 @2 1 @nA 1 = @P1 @w1 @w1 m @ 2 nB 1 ; @P12 w1 m @ 2 nB 1 ; @w12 @nB @ 2 nA 1 1 + P1 @P1 @P1 @w1 w1 m @ 2 nB 1 : @P1 @w1 Di¤erentiating (6) with respect to P1 yields: @ 2 nA 1 1 = 2 2tA @P1 + 2 m 2 1 2 1 1 2tA 1 + 2tA gBA 2tB 2tB 2tB @nA @nB 1 + mgBB 1 @P1 @P1 mgB 2tB mgB 1 2tB mgB 2tA gA @nA @nB 1 + mgBB 1 @P1 @P1 A @nB @n ; gAA 1 + mgAB 1 @P1 @P1 m gBA 2tA or @ 2 nA 1 1 = 2 2tA @P1 2 2 4 4tA tB gA @nA 1 @P1 2tA According to (26) and (27), we obtain 2 @ 2 nA 1 1 = 2 2tA @P1 Using the fact that @nB (mgBA + gAA ) + m @P11 (mgBB + gAB ) 2 6 6 4 @nA gAA @P11 2 4tA tB 2gA (k + 2tA @nA 1 @P1 + @nB mgAB @P11 1) @nA 1 @P1 @nB 1 @P1 gAA 1+ 2t gA +mgAB 2t gA B B 2tA @nB 1 = @P1 1 2tA it yields @ 2 nA 1 1 = 2 2 @P1 4tA 3 4 g 2 (k 4tA tB A 18 1) + 2tA gAA 3 ! 7 7 5 3 5 Using the same method, one …nds 4 @ 2 nB 1 = @P12 4 @ 2 nA 1 = @w12 4 @ 2 nB 1 = @w12 2 2tB 2 (k gA 4t2A 2 2tA 2 (k gA 2 (k 2gA @ 2 nB 1 = @P1 @w1 1) 2tB (29) 3 1) + 4t2B 1) 2tA 2 2tA m gBB 1) + 2 (k gA 2 (k 2gA @ 2 nA 1 = @P1 @w1 (28) 3 4t2B 4tA tB 2tB gAA 1) + 2 2tB m gBB (30) 3 2tA + 4tB tA 2tA + 4tB tA 2tA mgAB 2tB 3 (31) 2tB mgAB 2tB 3 (32) As a result, using (17) and (18) together with (??) and (28), one ends up with: h i gAA 2 (k 1) + g ( + m) 2 A 2 B tB A 2 @ n1 @ n1 P1 w1 m = 2 2 2 2 @P1 @P1 4tA i h m2 gBB 2 (k 1) + g ( + m) 2 A 2 B tA A 2 @ n1 @ n1 P1 w1 m = 2 2 2 2 @w1 @w1 4tB h i mgAB 2 (k 1) ( + m) + g 2 B 2 A A tA tB 2 @ n1 @ n1 w1 m = P1 2 @P1 @w1 @P1 @w1 4tB tA So 0 det H = @ 0 @ 2 1+ 2tB gA 2tA 2m 1 2tA gA 2tB 0 @ gA ( m 4tA tB ) ( + m) + ( + m) + ( + m) + h 19 h h tA 2 tB gA (k 4t2A 1) + 2 2 tA gA (k 1) + 4t2B + tB gAA 2 4tB tA 2 A m2 gBB 2 2 2 (k gA i1 1) i1 A mgAB 2 i 12 A