Reverse Pricing and Revenue Sharing in a Vertical Market

Transcription

1 Reverse Pricing and Revenue Sharing in a Vertical Market Qihong Liu Jie Shuai January 18, 2014 Abstract Advancing in information technology has empowered firms with unprecedented flexibility when interacting with each other We compare welfare results in a vertical market eg, manufacturers and retailers for several types of pricing strategies depending upon 1 which side retailers or manufacturers chooses retail prices and 2 whether there is revenue sharing or linear pricing between the two sides Our results are as follows Under revenue sharing, retail prices and thus industry profits are higher if and only if they are chosen by the side featuring less competition Under linear pricing, however, retail prices are higher if they are chosen by the side featuring more competition for linear demand functions Relative to linear pricing, revenue sharing always leads to lower retail prices, higher consumer surplus and social surplus However, the comparison on industry profits depends on the demand elasticity ratios Revenue sharing raises industry profits when the elasticity ratios are small, but the results are reversed when the elasticity ratios are large Keywords: Reverse pricing; Revenue sharing; Vertical relationship JEL Classification Codes: D43, L13 We would like to thank the Editor Antony Dnes, two anonymous referees, and Suman Basuroy for very helpful comments and suggestions Department of Economics, University of Oklahoma, 308 Cate Center Drive, Norman, OK qliu@ouedu Phone: Nankai Institute of Economics, Nankai University, #94 Weijin Rd, Tianjin , PR China jshuai@nankaieducn 1

2 1 Introduction The advancing of information technology has dramatically changed how various businesses in vertical markets interact with each other Take the bookselling industry for example See also Jerath and Zhang 2010 for more examples such as store-within-a-store 1 It used to rely on wholesale pricing model which has two features: 1 publishers choose wholesale prices linear pricing and then 2 booksellers choose retail prices retailer pricing In 2010, Apple introduced its ipad and proposed an agency pricing model which differs from the wholesale pricing model on both features First, there is revenue sharing rather than linear pricing Apple takes 30% of the sales Second, retail prices are set by publishers themselves manufacturer pricing or reverse pricing 2 Information technology has made pricing strategies such as reverse pricing and revenue sharing ever easier to implement, especially when the retail transaction is done online Correspondingly, we would expect such pricing strategies to become more relevant and their welfare implications more important over time The aim of this paper is to analyze and compare these pricing strategies in a vertical market structure where there is competition and product differentiation in both the upstream manufacturers and downstream markets retailers We ask the following questions First, how do the welfare results compare when retail prices are chosen by retailers retailer pricing vs when they are chosen by manufacturers manufacturer pricing or reverse pricing? Second, how do the results under linear pricing compare with those under revenue sharing? We first analyze the issue of which side manufacturers or retailers chooses retail prices Under revenue sharing, we find that retail prices are higher if and only if they are chosen by the side featuring less competition This is because, with revenue sharing, there is competition in only one side of the market whose competition intensity then determines the level of retail prices Ranking of retail prices is always opposite to the ranking of consumer surplus and social surplus, but always the same as the ranking of industry profits Under linear pricing, we find that each side prefers to move first first-mover advantage Due to the sequential game nature, we cannot compare retail prices and industry profits under general demand functions Under linear demand, we find that retail prices are higher if and only if they are chosen by the side featuring more competition, opposite to what we find under revenue sharing The sequential game nature introduces an incomplete pass through effect which leads to higher retail prices if and only if the intermediate prices are higher, ie, the side featuring less competition moves first and retail prices are chosen by the other side featuring more competition Similar to that 1 This example is based on the article What is agency pricing? Wall Street Journal, April 11, This is part of the appeal of the agency pricing model to the publishers, since it allows them to avoid what happened with Amazon who offered many titles below cost to boost its sales of Kindle e-readers at the cost of the publishers sales through other channels 2

3 under revenue sharing, higher retail prices also lead to lower consumer and social surplus However, under linear pricing there is both competition and double marginalization Correspondingly, the competitive prices may be above or below the monopoly price, and higher retail prices may lead to higher or lower industry profits We then analyze the issue of linear pricing vs revenue sharing, assuming that the same side chooses retail prices under either case We find that revenue sharing always leads to lower retail prices relative to linear pricing The intuition is simple There is double marginalization which raises retail prices under linear pricing but not under revenue sharing Lower prices always lead to higher consumer surplus and social surplus However, the comparison for industry profits depends on two demand elasticity ratios: one for competition between manufacturers and the other for competition between retailers see Section 32 for more details Linear pricing lowers industry profit relative to revenue sharing when the two elasticity ratios are sufficiently small, but the result is reversed when the two elasticity ratios are sufficiently large In the special case of linear demand functions, we find that an increase in either elasticity ratio always favors linear pricing over revenue sharing in terms of industry profits 11 Literature review Our paper is related to the extensive literature on vertical relationships 3 The traditional market foreclosure theory focused on the anti-competitive implications of vertical mergers, which was later criticized by authors associated with the Chicago School eg Posner 1976 The post-chicago approach eg, Ordover, Saloner and Salop 1990 and Chen 2001 uses modern economic theory to analyze vertical mergers and formally illustrate the welfare implications of vertical mergers With both procompetitive and anti-competitive impacts, the overall effect of vertical mergers on consumer and social welfare can be ambiguous see, for example, Salinger 1988 Within this literature, our paper is mostly related to studies analyzing the potential for vertical mergers to enhance efficiency, in particular, by eliminating double marginalization eg Economides 1999 Our setup allows upstream and downstream firms to rely on revenue sharing so double marginalization can be avoided However, our paper differs from the vertical mergers literature in several perspectives 4 First, revenue sharing is non-exclusive and does not necessarily introduce asymmetric incentives as vertical mergers do For example, a manufacturer would share revenue with all downstream retailers and have no incentive to favor one retailer against another Second, existing studies usually consider either monopoly or homogeneous products in one of the two markets upstream or downstream In contrast, we allow competition and product differentiation in both the upstream and downstream markets 3 See Perry 1989 and Riordan 2005 for two surveys of this literature 4 To some extent, such similarity as well as differences, also exist if one compares this paper with studies on complementary mergers and integration eg, Economides and Salop 1992 and Economides

4 In contrast to the abundance of literature on vertical relationships, little attention has been paid to the issue of which side manufacturers or retailers chooses retail prices An exception is Jerath and Zhang 2010 They consider a setup where a monopoly retailer decides whether to lease space to either of the two competing manufacturers If the retailer leases retail space to a manufacturer store-within-a-store, the manufacturer will make its own retail price decision They find that manufacturer pricing can be supported as part of a subgame perfect Nash equilibrium under certain conditions Similar to their paper, we also consider the possibility that retail prices are chosen by manufacturers However, our setup features competition in the retail sector as well and we also consider general demand function For tractability, we focus on the comparative statics manufacturer pricing vs retailer pricing rather than check which one will emerge endogenously in equilibrium 5 We also focus on different issues They are mostly interested in whether the monopoly retailer, with all the bargaining power, has an incentive to lease store space to manufacturers In contrast, we are interested in how the welfare results compare depending on which side chooses retail prices as well as the ranking of competition intensities in the two sides More recent research focuses on contract terms between the manufacturers and the retailers One issue involves vertical restraints such as resale price maintenance RPM eg, Dobson and Waterson 2007 and Asker and Bar-Isaac 2011 Other issues include the comparisons of linear pricing vs two-part tariff eg, Marx and Shaffer 2007, uniform pricing vs price discrimination Inderst 2010 etc Competition in both upstream and downstream markets, together with general demand functions, prevents us from considering two-part pricing due to the need to calculate the welfare outcomes when the contract offer is declined Since we assume symmetry between the firms in each side of the market, there is no need for price discrimination Correspondingly, we only consider uniform pricing under linear pricing and in the case of revenue sharing, we assume that the contract terms percentage of revenue does not vary across manufacturers or across retailers The rest of the paper is organized as follows We present our model in Section 2 In Section 3, we compare manufacturer pricing with retailer pricing and then compare revenue sharing with linear pricing We conclude in Section 4 Proofs of Lemmas and Propositions are relegated to the Appendix 2 The model We consider a vertical market structure where there is competition in both the upstream and downstream markets In particular, there are two firms in the upstream market called manufactures denoted by j = A, B and two firms in the downstream market called retailers denoted by 5 There are also other differences For example, we rule out in-store service which exists and affects the equilibrium outcome in their setup On the other hand, we allow revenue sharing between manufacturers and retailers which is not considered in their paper 4

5 by i = 1, 2 For simplicity, we assume that manufacturers and retailers both have constant marginal cost which we normalize to zero We allow differentiation both across manufacturers and across retailers Combined, there are 4 products offered to consumers: ij {1A, 2A, 1B, 2B} refers to manufacturer j s product sold at retailer i 6 Let p ij denote the price of product ij Demand for product ij is given by D ij D ij p ij, p kj, p il, p kl, i k, j l We assume that demand functions are symmetric and twice differentiable with D ij p ij < 0; > 0, kl ij We further assume that regularity conditions are satisfied so that there exists a pure strategy equilibrium which is also unique in every game we analyze 7 Due to symmetry across retailers and across manufacturers, we focus on symmetric equilibrium throughout the paper Our model features both vertical relationship across markets and horizontal competition within each market There are multiple scenarios in terms of how prices are chosen and revenues are distributed, which we describe next Manufacturers may choose to charge wholesale prices w j followed by retailers choices of retail prices p ij We call this the wholesale pricing scenario W The sequence of move can also be reversed retailers charge reverse wholesale prices rw i to be paid by manufacturers who then choose retail prices p ij We call this the reverse wholesale pricing scenario RW Different from scenarios W or RW which feature linear pricing, manufacturers and retailers may also share revenue with exogenously fixed percentages of revenue going to manufacturers and retailers respectively Under revenue sharing, there are no intermediate prices so only retail prices will be chosen, by either retailers or manufacturers We call them revenue sharing with retailer pricing RSR and revenue sharing with manufacturer pricing RSM respectively These four scenarios are summarized in Table 1 D ij p kl W RW RSR RSM Retail prices chosen by Retailers Manufacturers Retailers Manufacturers Linear pricing Yes Yes No No Table 1: The four scenarios 6 For illustration purpose we use the term manufacturers and retailers It is equivalent to view our setup as 1 a standard upstream-downstream market where one unit of output from the upstream market is required to produce one unit of output in the downstream market, or 2 a complementary product case where the upstream and downstream products are always consumed in one-to-one ratio 7 More details are provided in the Analysis section after firms profit maximization problem is laid out more clearly A specific example where all regularity conditions are satisfied is when demand is linear in prices In that case, we obtain a unique equilibrium which is also symmetric in each scenario 5

6 3 Analysis Recall that the four scenarios differ on potentially two dimensions: 1 which side chooses retail prices; 2 whether there is linear pricing or revenue sharing between the two sides In this section, we will compare these scenarios, two at a time in a pair which differ on one dimension but not the other 31 Which side chooses retail prices? We start by comparing scenarios which differ only on whether retail prices are chosen by manufacturers or retailers Within each pair for comparison, either both scenarios feature revenue sharing or both feature linear pricing We will start with the former 311 Under revenue sharing We first compare revenue sharing with retailer pricing RSR and revenue sharing with manufacturer pricing RSM In both scenarios, manufacturers receive fixed fractions of the total revenue received 8 Let D ij p kj and D ij p il, i k = 1, 2, j l = A, B, evaluated at equal retail prices, denote the competition intensity across retailers and that across manufacturers respectively The comparative statics results turn out to depend on the comparison of these two competition intensities, as summarized in the next Proposition Proposition 1 Revenue sharing Comparing revenue sharing with retailer pricing and revenue sharing with manufacturer pricing, i Retail prices are higher if and only if they are chosen by the side featuring less competition ii Ranking of consumer surplus and social surplus is opposite to the ranking of retail prices iii Ranking of industry profits is always the same as the ranking of retail prices Proof See the Appendix The intuition for the above results is the following Under revenue sharing, there is competition in only one side of the market the side which chooses retail prices To see this, consider RSR for example Retailers choose retail prices to compete with each other However, a retailer, say retailer 1, only cares about the total profit from selling both manufacturers products, not how it is split between them Correspondingly, there is no competition across manufacturers With competition 8 Our focus is on how the scenarios compare with each other, rather than how the industry profit will be split between manufacturers and retailers, or more generally, what scenarios can be supported as part of a subgame perfect Nash equilibrium Nevertheless, our results can provide useful insights for such analysis 6

7 only in the side which chooses retail prices, retail prices are higher if and only if they are chosen by the side featuring less competition With revenue sharing, there is no double marginalization so competition pushes retail prices below the monopoly level which maximizes industry profit Consequently, industry profit goes up if and only if retail prices go up Results for consumer welfare and social welfare are just the opposite We do not consider how manufacturers profits or retailers profits compare across the scenarios With revenue sharing, an increase in industry profit can be divided in a way to make both retailers and manufacturers better off Our results critically depend on the assumption of firm symmetry in each side manufacturers or retailers 9 In particular, rankings of prices and profits depend on the ranking of competition intensities which are defined at the side level and at symmetric prices, the latter naturally stemming from firm symmetry If firms are asymmetric on either of the manufacturer and retailer side, then new measures of competition intensity need to be introduced 312 Under linear pricing Next, we consider two scenarios where firms rely on linear pricing: wholesale pricing W and reverse wholesale pricing RW 10 Under wholesale pricing, manufacturer j = A, B chooses wholesale price w j to maximize its profit, πj W = w j D 1j + D 2j After observing wholesale prices, retailer i = 1, 2 s problem is max π p ia,p i W = p ia w A D ia + p ib w B D ib ib The order of move is the opposite for reverse wholesale pricing RW First, retailer i = 1, 2 chooses reverse wholesale prices rw i to maximizes its profit π RW i = rw i D ia + D ib After observing reverse wholesale prices, manufacturer j = A, B s problem is max p 1j,p 2j π RW j = p 1j rw 1 D 1j + p 2j rw 2 D 2j 9 Except when the degree of firm asymmetry is sufficiently small in which case by continuity our main results under firm symmetry likely would remain valid 10 We assume that firms rely on linear pricing in the absence of revenue sharing, rather than two-part pricing fee plus price With competition in both the upstream and downstream market, it is intractable to derive the optimal fee for the first-movers under general demand functions In the meantime, we conjecture that the outcomes are likely equivalent to those under revenue sharing, so long as retail prices are chosen by the same side under either case To see this, the optimal intermediate prices would be 0 under two-part pricing so retail prices would be the same under two-part pricing as under revenue sharing 7

8 Next, we introduce regularity conditions which govern how the second movers will respond when first-movers change their intermediate prices Regularity conditions: Let i, k {1, 2} denote retailers and let j, l {A, B} denote manufacturers We impose the following assumptions on profit functions 2 π W i p 2 ij < 0; 2 π W i p ij w j + 2 π W i p ij w l < 2 π RW j p ij p kl > kl 2 π RW j p 2 ij < 0; 2 π W i p ij p kl > 0, kl ij; 2 π W i p ij p kl ; kl 2 π RW j p ij rw i + 2 πi W p 2 ij 2 π W i p ij p il + 2 π RW j p ij p kl > 0, kl ij; 2 π RW j p ij rw k ; 2 πj RW p 2 ij kl 2 π W i p ij p kl < 0, kl; πi W p ij p kj > 2 πi W p ij p kl kl 2 π W i p ij w j, k i, l j 2 2 πi W p il w j 2 π RW j p ij p kl < 0, kl πj RW p ij p il 2 πj RW p ij p kj + 2 π RW j p ij p kl > 2 π RW j p ij rw i, k i, l j 4 2 πj RW p kj rw i Equations 1 and 2 are for wholesale pricing while equations 3 and 4 are for reverse wholesale pricing Let us focus on wholesale pricing Conditions in 1 are standard Profit is concave in own price but convex in cross prices An equal increase in all prices is less profitable for firm i, the higher p ij is Conditions in 2 require more explanation Let us see what they do Retailer 1 s first-order conditions are π 1 p 1j = 0, j = A, B Next, suppose that w A changes by dw A while w B remains fixed dw B = 0 Taking total differentials to the first-order conditions and then imposing symmetry dp ij = dp kj, we can obtain 2 π 1 2 π 1 dp 1A + dp 1B = 2 π 1 dw A, 5 p 2B p 2 1A 2 π 1 dp 1A + p 2 1B dp 1B = 2 π 1 dw A 6 p 2B The coefficients in parentheses are c 1 and c 2 in 5, and c 2 and c 1 in 6 evaluated at equal prices where c 1 = 2 π 1 p 2, c 2 = 2 π 1 1A p 2B 8

9 Solving these two equations for p 1j, we can obtain = c 1 D1A c 2 c 2 1, c2 2 = c 2 D1A + c 1 c 2 1 c2 2 Conditions in 2, together with those in 1 will allow us to pin down the range of p 1j shown in the next Lemma as Lemma 1 Under the regularity conditions 1-4, 1 > p ij hold under wholesale pricing and 1 > wholesale pricing p ij rw i > p kj rw i w j > p il w j > 0 l j = A, B must > 0 k i = 1, 2 must hold under reverse Proof See the Appendix It is intuitive that an increase in w A raises p 1A, ie, > 0 How about? It turns out that dw A has two opposite effects on dp 1B First, substituting dp 1A = 0 into equation 5, we can obtain the direct effect 11 = direct 2 π 1 p 2B < 0 That is, the direct effect says that an increase in w A leads to a reduction in p 1B The indirect effect is that a change in w A affects p ia which in turn affects p 1B It takes the form of = > 0 indirect Substituting dw A = 0 into equation 5, we can obtain = p 2 p 1A 1A p 2B > 0 holds if and only if the indifferent effect dominates the direct effect Summing up the direct and indirect effects, we recover the earlier result that = c 2 D1A + c 1 c 2 1 c One can use either equation 5 or 6 Later on both equations are used when we substitute the expression 9

10 In the case of linear demand functions, it can be shown that p 2 1A > p 2B > p 2B, which always holds since > 0 and p 2B < 0 What ensures that < 1? The relevant condition by substituting i = 1, j = A into the first condition in equation 2 is 2 π W π W 1 w B < 2 π W 1 p kl The left-hand side measures how much a unit increase in w A and w B will impact π 1 12 The right-hand side measures how much a unit increase in all p ij will impact π 1 Note that these two impacts must exactly offset each other since π 1 = 0 holds both before and after the changes in w A and p ij Correspondingly, their relative sizes determine how dp 1A compares with dw A kl One can also substitute the π W 1 expression and transform the above inequality into the following: ij p ij < p 1A w A ij 2 D 1A p ij p 1B w B ij 2 D 1B p ij In the case of linear demand functions, the righthand side is zero and the condition becomes ij p ij < 0 That is, demand goes down when all prices go up by the same amount Comparison for each side Next, we analyze how the two sides would rank the two scenarios In particular, we check whether there is a first mover advantage The answer is yes as shown in the next Proposition Proposition 2 Under linear pricing, each side manufacturers or retailers prefers to be the first mover Proof See the Appendix Recall that the first movers intermediate prices are the second-movers marginal costs Based on Lemma 1, when the second movers face higher costs, they will raise retail prices but not by as much as the cost increases This makes the demand less elastic for the first movers who would then 12 Here we allow both w A and w B to change Let dw A = dw B = dw, we show that dp 1A < 1 See the proof for more details dw < 1 which then implies 10

11 have an incentive to raise their intermediate prices and force the second mover to absorb part of the price increases Doing so benefits the first movers at the cost of the second movers Consequently, each side would prefer to be the first mover Comparison on retail prices and industry profits Next, we compare the final retail prices and overall industry profits across the two scenarios: W and RW Because of the sequential game nature, under general demand functions we cannot obtain explicit solutions for the equilibrium which are required for us to compare retail prices and industry profits Consequently, we employ specific demand functions which we assume to be linear in retail prices, D ij = 1 p ij + a R p kj + a M p il + a RM p kl, k i = 1, 2, l j = A, B 7 Note that a R > 0 captures the competition intensity across the same product sold at different retailers On the other hand, a M > 0 captures the competition intensity across different products sold at the same retailer a RM < min{a R, a M } captures the competition intensity across different products sold at different retailers For simplicity we assume that a RM = a R a M Regularity conditions in 2 require that a R + a M + a RM < 1, ie, an equal increase in all retail prices leads to lower demand for all products The comparative statics on retail prices and industry profits are summarized in the next Proposition Proposition 3 Linear demand Comparing wholesale pricing W and reverse wholesale pricing RW, i Retail prices are higher if and only if they are chosen by the side featuring more competition ii Ranking of consumer welfare and social welfare is always opposite to the ranking of retail prices iii Ranking of industry profits is independent of the ranking of retail prices Proof See the Appendix The intuition for the results in Proposition 3-i is the following In both wholesale pricing and reverse wholesale pricing, final retail prices are determined by: 1 competition intensity across manufacturers; 2 competition intensity across retailers and 3 the ability of the second movers to absorb high prices chosen by the first movers 1 and 2 are the same across wholesale pricing and reverse wholesale pricing 3 exists because the first movers do not sell directly to consumers but to second movers instead The second movers will pass only a fraction of the intermediate price increases down to consumers and absorb the rest, incomplete pass through Second movers ability 11

12 for such absorption decreases with their costs ie, the intermediate prices In particular, when intermediate prices are high, the second movers have little room to absorb increase in intermediate prices and will pass most of the increase down to consumers Therefore, high intermediate prices raise final retail prices through two channels: by raising second movers costs and by reducing second movers ability to absorb higher wholesale prices Combined, retail prices are higher if the side featuring less competition moves first higher intermediate prices Note that the ranking of retail prices under linear pricing in Proposition 3-i is opposite to the ranking under revenue sharing in Proposition 1-i In particular, under revenue sharing, retail prices are higher if chosen by the side featuring less competition On the other hand, under linear pricing, retail prices are lower if chosen by the side featuring less competition The ranking of industry profit across W and RW is independent of the ranking of retail prices In particular, an increase in retail prices does not necessarily imply that industry profit will go up From the perspective that retail prices are competitive prices, they are too low compared to monopoly price But there is also double marginalization so retail prices can be too high Combined, how retail prices compare with monopoly price is ambiguous and would depend on how competition effect and double marginalization effect compare with each other 32 Revenue sharing vs linear pricing In the previous subsection, we have compared the welfare results depending on which side chooses retail prices Next, we have the same side choosing retail prices and focus on the comparison between revenue sharing and linear pricing We will compare wholesale pricing W with revenue sharing retailer pricing RSR, and compare reverse wholesale pricing RW with revenue sharing manufacturer pricing RSM Our goal is to analyze how the presence or absence of double marginalization affects consumer surplus, industry profits and social surplus We will start with the comparison of retail prices Let p s denote the equilibrium retail price under scenario s {W, RW, RSR, RSM} The results are presented in the next Proposition Proposition 4 When retail prices are chosen by the same side, revenue sharing always leads to lower retailer prices, relative to linear pricing That is p W > p RSR, p RW > p RSM Results in Proposition 4 are straightforward applications of Lemma 1 Note that p W is retailers best response to wholesale price of w > 0, while p RSR is their best response to hypothetical wholesale price of 0 Based on Lemma 1, we immediately get p W > p RSR The proof for p RW > p RSM is similar 12

13 While linear pricing always leads to higher retail prices relative to revenue sharing, it is no guarantee that it entails higher industry profits as well Consider the scenarios W vs RSR Let p M denote the monopoly retail price which maximizes industry profit While p RSR < p M always holds, the comparison of p W and p M is ambiguous Ex-ante it is unclear whether p M is closer to p W or p RSR Next, we define two demand elasticity ratios: D 1B / across manufacturers and D 2A / across retailers respectively, both evaluated at equal retail prices pij = p 13 These two elasticity ratios will help determine the comparison of profits across linear pricing and revenue sharing, as summarized in the next Proposition Proposition 5 When elasticity ratios are sufficiently small, linear pricing entails lower industry profit relative to revenue sharing, that is, Π W < Π RSR, Π RW < Π RSM The rankings are reversed when elasticity ratios are sufficiently large Proof See the Appendix Let us focus on the scenarios where retail prices are chosen by retailers W for linear pricing and RSR for revenue sharing to illustrate the ideas Previously we have explained that p RSR < p M always holds while the comparison of p W and p M is ambiguous Let us start with the extreme case where the elasticity ratios are zero, ie, all cross partial derivatives of demand with respect to prices are zero In this case, there is no competition effect and we have p RSR = p M < p W This implies Π RSR = Π M > Π W By continuity the same result Π RSR > Π W would hold when the elasticity ratios are sufficiently small On the other hand, when the elasticity ratios are large enough, the competition effect dominates the double marginalization effect We would have p RSR < p W < p M, implying that Π RSR < Π W < Π M An example of linear demand With general demand functions, we can compare profits only for the extreme cases: when elasticity ratios are either sufficiently small or sufficiently large Next, we consider specific demand functions and explore the whole ranges of elasticity ratios We revisit the linear demand function as given in equation 7, D ij = 1 p ij + a R p kj + a M p il + a RM p kl, k i, l j 13 It is straightforward to verify that evaluated at equal prices, these partial derivative ratios are equivalent to the elasticity ratios For example, D 1B / = D 1B / D 1B p 1A / D 1A p 1A since p 1A = p 1B and D 1A = D 1B 13

14 Since D ij p ij = 1, the two elasticity ratios become ar and a M respectively Without loss of generality, we compare the scenarios W and RSR 14 The results are in the same spirit as those in Proposition 5 In particular, we find that Π RSR Π W decreases with a M and a R This implies that there is unique level curve of a R, a M combinations such that Π RSR = Π W See Figure 1 Industry profit is higher under revenue sharing below the curve, but is higher under linear pricing above the curve W RSR Π >Π RSR W Π >Π Figure 1: Industry profit comparison: Revenue sharing vs Linear pricing 4 Conclusion This paper investigates vertical relationship where there is competition and product differentiation in both the upstream and downstream market We compare the welfare results under several types of pricing strategies depending upon which side manufacturers or retailers chooses retail prices and whether there is linear pricing or revenue sharing between the two sides Under revenue sharing, retail prices are higher if they are chosen by the side featuring less competition Under linear pricing, each side prefers to be the first mover but different from revenue sharing, retail prices are higher if they are chosen by the side featuring more competition under linear demand Higher retails prices always translate into higher industry profits under revenue sharing, but may lead to higher or lower industry profits under linear pricing Comparing linear pricing and revenue sharing, 14 If one switches a M and a R, then scenarios W and RSR under the new parameters become scenarios RW and RSM under the initial parameters 14

15 we find that revenue sharing always generates lower retail prices, but the comparison of industry profits depends on the two demand elasticity ratios in the upstream and downstream markets Revenue sharing leads to larger industry profits when the elasticity ratios are small but the result is opposite when elasticity ratios are large Appendix Proof of Proposition 1 i Let us start with RSR Retailer i = 1, 2 chooses p ia and p ib to maximize its share of profit, which is equivalent to maximizing the overall profit Retailer 1 s problem is max p 1A D 1A + p 1B D 1B p 1A,p 1B FOC implies that D 1A + p 1A + p 1B D 1B = 0 8 Note that competition intensity between manufacturers D ib p ia, i = 1, 2 enters into the FOC and is internalized On the other hand, competition intensity between retailers D 1j p 2j, j = A, B does not and it will determine the equilibrium level of retail prices Imposing symmetry all retail prices are equal and denoted as p RSR, we have p RSR = D1A D 1A + D 1B Note that < 0, increases with D 1B D 1B > 0 and + D 1B < 0 It can be easily verified that p RSR is Next, we consider RSM where manufacturers choose retail prices Manufacturer A s problem max p 1A D 1A + p 2A D 2A p 1A,p 2A FOC implies that D 1A + p 1A + p 2A D 2A = 0 9 Imposing symmetry, we have p RSM = D1A D 1A + D 2A 15

16 Note that with D 2A < 0, D 2A > 0 and + D 2A < 0 It can be verified that p RSM increases Next, we compare p RSR and p RSM If D 1B = D 2A, we must have p RSR = p RSM Moreover, p RSR increases with D 1B and p RSM increases with D 2A D 2A > D 1B to have D 2A Combined, p RSM > p RSR if and only if holds at either p ij = p RSR i, j or p ij = p RSM i, j A sufficient condition is > D 1B at all equal retail prices ii With zero cost of production, rankings for consumer welfare and social welfare are always opposite to the ranking of retail prices iii It can be easily verified that the monopoly price which maximizes industry profit is p M = + D 1B D 1A + D 2A + D 2B, It is easy to see that max{p RSR, p RSM } < p M, which further implies that higher retail prices would lead to higher industry profits Proof of Lemma 1 Wholesale pricing Recall that = c 1 D1A c 2 c 2 1, c2 2 = c 2 D1A + c 1 c 2 1, c2 2 where Note that c 1 = 2 π 1 p 2, c 2 = 2 π 1 1A p 2B 2 π 1 = < 0, 2 π 1 = > 0 Next, we show that 1 > i p 1j > 0, j = A, B > > 0 16

17 Since c 1 < 0, c 2 > 0, c 1 + c 2 < 0, From the regularity conditions, we have p 2 1A < 0, > p 2B > 0, = > 0 must hold The numerator in the expression must be positive, so > 0 ii > This is because = c 1 + c 2 D1A c 2 1 c2 2 > 0 iii < 1 We have shown p ij > 0 Similar conditions will ensure p ij w B > 0 Next, restrict dw A = dw B = dw which leads to dp 1A = dp 1B = dp 2A = dp 2B 15 Taking total differentials to π 1 π1 dp ij + p ij ij ij p ij = 0, we can obtain π1 dw A + π1 w B dw B = 0 dp 1A = w B dw dp 1A dw < 1 if and only if w B < p ij, 10 ij which is established in our regularity conditions Combined with dp 1A dw >, < 1 must hold Reverse wholesale pricing The idea of this proof is very similar to that for wholesale pricing We present it here for the sake of completeness 15 Note that the corresponding dp 1A must be larger than the dp 1A when only w A changes This is because dp 1A + w B > 17 dw =

18 We solve the game backwards In the second stage, manufacturer A maximizes its profit: π A = p 1A rw 1 D 1A + p 2A rw 2 D 2A First-order conditions imply π A = 0, π A = 0 Due to symmetry, we will impose drw 2 = 0 and look at to the first-order conditions, we can obtain πa dp ij + p ij ij ij πa p ij dp ij + πa dp ij drw 1 rw 1 drw 1 = 0, πa rw 1 drw 1 = 0 only Taking total differentials We further impose symmetry in the form of dp ij = dp il, i = 1, 2 The previous two equations then become 2 π A p 2 1A + 2 π A dp1a 2 drw 1 + π A + 2 π A p 2B dp2a 2 π A drw 1 =, 11 rw 1 2 π A + 2 π A dp1a 2 drw 1 + π A p π A dp2a 2A p 2B drw 1 = 2 π A 12 rw 1 Note that 2 π A rw 1 = < 0, 2 π A rw 1 = > 0 Step 1: Derive the rw 1 and rw 1 expressions Written in matrix form, equations 11 and 12 become d 1 d 2 d 2 d 1 p1a rw 1 rw 1 = D1A, where d 1 = 2 π A p 2 1A + 2 π A, d 2 = 2 π A + 2 π A p 2B 18

19 From the regularity conditions, equation 1, we have d 1 < 0, d 2 > 0, d 1 + d 2 < 0 They then imply d 2 1 d2 2 > 0 We can then obtain p1a rw 1 rw 1 = 1 d 2 1 d2 2 d 1 d 2 d 2 d 1 D1A That is, rw 1 = d 1 D1A d 2 d 2 1, d2 2 rw 1 = d 2 D1A + d 1 d 2 1, d2 2 Step 2: Show that 1 > rw 1 > rw 1 > 0 i rw 1 > 0 Since d 1 < 0, d 2 > 0, d 1 + d 2 < 0, From the regularity conditions, we have 2 π A p 2 1A 2 π A + < 0 and > 0, + 2 π A > 2 π A p 2B rw 1 > 0 must hold The numerator in the ii iii rw 1 rw 1 > dp 2A drw 1 This is because rw 1 expression must be positive, so p rw 1 = d 1 + d 2 D1A d 2 1 d2 2 rw 1 < 1: Taking total differentials to π A ij πa p ij dp ij + πa rw 1 drw 1 + 2A rw 1 > 0 > 0 = 0, we can obtain πa rw 2 drw 2 = 0 Next, we restrict drw 1 = drw 2 = drw 16 It must be dp 1A = dp 2A Then we have the following: 2 π A + 2 π A 2 π A dp 1A π A 2 π A dp 2A = + 2 π A drw, p 2B rw 1 rw 2 p 2 1A 16 As we have shown that dp 1j drw 1 > 0, the corresponding dp1a must be larger than the dp1a when only rw1 changes 19

20 2 π A p π A + 2 π A + 2 π A 2 π A dp 1A = + 2 π A drw 1A p 2B rw 1 rw 2 dp 1A < drw if and only if 2 π A + 2 π A < 2 π A rw 1 rw 2 p ij, which is established in our regularity conditions Under linear demand functions, this condition is equivalent to a R + a M + a RM < 1 ij Proof of Proposition 2 We will introduce a hypothetical scenario where all manufacturers and retailers choose their component prices simultaneously We consider a symmetric equilibrium Let p M and p R denote the manufacturer and retailer component prices and let p p M + p R the final retail price Let and π M and π R denote the corresponding manufacturer profit and retailer profit respectively Similarly let p s, πm s and πs R denote the equilibrium retail price, manufacturer profit and retailer profit under scenario s = W, RW Let w denote the optimal wholesale price and rw the optimal reverse wholesale price respectively Step 1: Show that w > p M > p RW rw, rw > p R > p W w We first show that w > p M Due to symmetry, consider manufacturer A only Under the wholesale pricing scenario, its profit is πa W = w A D 1A + D 2A First-order condition implies πa W D1A = D 1A + D 2A + w A + D 2A = 0 We only consider symmetric equilibrium D 2A = D 1A and D 2A = Then, D 1A + w A = 0 w = w A = D 1A 13 Under the hypothetical scenario, manufacturer A s profit is π A = p MA D 1A + D 2A 20

21 First-order condition implies π A p MA = D 1A + D 2A + p MA Note that p MA enters into p 1A and p 2A, so Imposing symmetry, we have, p MA = + D 1A + p MA p MA = 0 D1A + D 2A = 0 p MA p MA p M = p MA = Comparing equations 13 and 14, w > p M if and only if D1A > + D 1A It holds because D 1A = p 2B D1A = + D 1A p 1A D1A + + D 1A p 2B > D1A + > +, the second term is positive + 14 p 2B since + < 0 and 0, 1 Next, we prove that p M > p RW rw We have shown that under wholesale pricing, first movers raise their component prices beyond the level under simultaneous move The same holds for reverse wholesale pricing so rw > p R Going from the hypothetical scenario to the reverse wholesale pricing scenario, retailer component price goes up from p R to rw Based on Lemma 1, final retail price should go up from p to p RW but by not as much, That is, p RW p < rw p R p RW rw < p p R p RW rw < p M Combined, we have w > p MA > p RW rw The proof of rw > p R > p W w is similar and skipped 21

22 Step 2: Show that π W M > π M > π RW M and πw R < π R < π RW R First, we show that π M > πm RW Relative to the hypothetical scenario, under RW manufacturers choose higher retail prices p RW > p so lower sales In the mean time, manufacturers price-cost margin goes down since p RW rw < p p R Combined, manufacturers must be worse off under reverse wholesale pricing relative to the hypothetical scenario π M > π RW M Next, we show that π W M > π M Let π A w A, w B denote manufacturer A s profit when manufacturers wholesale prices are w A and w B under wholesale pricing Then π W M = π A w, w π A p M, w w A = w is best-response to w B = w > π A p M, p M w > p M = π M p R is the best-response to p M So far we have established that πm W > π M > πm RW Following similar steps, it can be verified that πr W < π R < πr RW Therefore, each side always prefers to be the first-mover Proof of Proposition 3 i Prove that retail prices are higher if and only if they are chosen by the side featuring more competition Recall that demand for manufacturer j s product at retailer i is given by D ij = 1 p ij + a R p kj + a M p il + a RM p kl, i k = 1, 2, j l = A, B, where min{a R, a M, a RM } > 0, a RM = a R a M and a R + a M + a RM < 1 We will start with wholesale pricing and solve the game backwards In Stage 2, retailer s profits are given by Solving the first-order conditions π i p ij π 1 = p 1A w A D 1A + p 1B w B D 1B, π 2 = p 2A w A D 2A + p 2B w B D 2B = 0, we can obtain p 1A = 2 2a M a R a M 2w A + a R + a R w A 2a R a M w B + 2a 2 M w A + a 2 M a Rw A a 2 R + a2 R a2 M + 4a Ra 2 M + 4a R + 4a 2 M 4, p 1B = a Ra M 2 + a R + a R a 2 M w B + a R w B 2a R a M w A + 2a 2 M w B 2a M 2w B a 2 R + a2 R a2 M + 4a Ra 2 M + 4a R + 4a 2 M 4, 22

23 p 2A = 2 2a M a R a M 2w A + a R + a R w A 2a R a M w B + 2a 2 M w A + a 2 M a Rw A a 2 R + a2 R a2 M + 4a Ra 2 M + 4a R + 4a 2 M 4, p 2B = a Ra M 2 + a R + a R a 2 M w B + a R w B 2a R a M w A + 2a 2 M w B 2a M 2w B a 2 R + a2 R a2 M + 4a Ra 2 M + 4a R + 4a 2 M 4 In stage 1, manufacturers profits are π A = w A D 1A + D 2A, π B = w B D 1B + D 2B Substituting p ij into π j and solving for manufacturers first-order conditions, we can obtain the following wholesale prices w A = w B 2 2a R a M + a R a 2 M = + 2a2 M + a R a 3 M a2 R 2a2 R a2 M a2 R a M + 2a 2 R + 3a3 M a R 2a R a 2 M + a Ra M 6a R + 2a 3 M 4a2 M 2a M + 4 Substituting the optimal w j into p ij, we can obtain the equilibrium p ij = p W as the following p W 1 = a R a M + 2a M 2 + a R 4a3 M + 4a Ra 3 M + a3 M a2 R 2a2 R a2 M 5a Ra 2 M 6a2 M + 4a Ra M 4a M a 2 R a M + 2a 2 R 7a R + 6 a 3 M a2 R + 3a Ra 3 M + 2a3 M 4a2 M 2a Ra 2 M 2a2 R a2 M a2 R a M + a R a M 2a M + 4 6a R + 2a 2 R For reverse wholesale pricing, following similar steps we can obtain the following equilibrium reverse wholesale prices rw 1 = rw a M + 2a 2 R = + a2 R a M 2a R a M a 3 R a2 M + 3a3 R a M + 2a 3 R 4a2 R 2a2 R a M 2a 2 R a2 M a Ra 2 M + a Ra M 2a R + 4 6a M + 2a 2, M which lead to the following equilibrium retail price: p RW 1 = 2a R + a R a M 2 + a M ar 3 a 2 M + 4a 3 R a M + 4a 3 R 2a2 R a2 M 5a2 R a M 6a 2 R + 4a Ra M 4a R a R a 2 M + 2a2 M 7a M + 6 a 3 R a2 M + 3a3 R a M + 2a 3 R 2a2 R a M 4a 2 R 2a2 R a2 M a Ra 2 M + a Ra M 2a R + 4 6a M + 2a 2 M We then calculate p W p RW and find that it is a multiple of and has the same sign as a R a M With = a R and = a M, it must be that p W > p RW > 23

24 ii It is obvious that an increase in retail prices lowers both consumer surplus and social surplus Correspondingly, the ranking of consumer welfare and social welfare is always opposite to the ranking of retail prices iii An increase in retail price may raise or lower industry profit, as illustrated in the following examples: 1 a R = 035, a M = 03: p W > p RW and Π W > Π RW 2 a R = 035, a M = 015: p W > p RW but Π W < Π RW Proof of Proposition 5 Elasticity ratios are small Suppose that the elasticity ratios are zero, ie, all cross partial derivatives of demand with respect to prices are zero In this case, there is no competition so RSR would lead to the same equilibrium retail price as the monopoly price However, double marginalization remains if firms rely on linear pricing Combined, we have p RSR = p M < p W, which then leads to Π RSR = Π M > Π W That is, when there is no competition across retailers or manufacturers, RSR leads to higher industry profit relative to W By continuity, the results would hold when there is sufficiently weak competition Elasticity ratios are large Relative to monopoly price p M, p W captures two effects: double marginalization effect which pushes p W beyond p M, and competition effect which lowers p W below p M Next, suppose that competition intensity is very strong across manufacturers and across retailers Correspondingly, under wholesale pricing, both manufacturers and retailers can charge prices only slightly above their marginal cost This implies that competition effect dominates the double marginalization effect and we have p W < p M Recall that p RSR captures competition effect but not double marginalization effect It must be p RSR < p W < p M, which further implies Π RSR < Π W < Π M 24

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