Platform Price Parity Clauses with Direct Sales

Transcription

1 Platform Price Parity Clauses with Direct Sales BjØrn Olav Johansen Thibaud Vergé November 2015 [VERY PRELIMINARY DRAFT - PLEASE DO NOT CIRCULATE] 1 Introduction Recent cases: Hotel booking cases (UK, Germany, France-Italy-Sweden,... ). Private Motor Investigation (UK CMA, 2015) Other retail MFN cases? (OFT s Tobacco case for instance). Literature: Most closely related: Price Parity in standard vertical relationships model. Boik and Corts (2015), Johnson (2014), Rey and Vergé (2015),.... Vertical relationships but issue being more switch from wholesale to agency model (Johnson (2013), Foros, Kind and Shaffer (2014), other papers linked to the Apple / E.Books case,... ). Platform models: Edelman and Wright (2015), Wand and Wright (2015),.... See Hviid (2015) for a more exhaustive review. This paper / Overview of the results The authors are grateful to the Norwegian Competition Authority for financial support (SNF grant number XXXX). Thibaud Vergé was part of CRA s team advising Expedia in the recent European cases related to hotel booking platforms. The views expressed in these papers are those of the authors only. University of Bergen and BECCLE. ENSAE ParisTech (UMR EXCESS/CREST) and Norwegian School of Economics (NHH/BECCLE). 1

2 2 Model We consider a setting where two symmetrically differentiated suppliers, 1 and 2, compete for customers. The suppliers production costs are assumed to be linear (i.e., constant marginal cost) and we normalize the marginal production cost to zero. Suppliers can reach final consumers by selling through two symmetrically differentiated online platforms (intermediated sales), A and B, or through their own website (direct sales), D. The marginal distribution costs are normalized to zero for all distribution channels i.e., we assume that selling directly is as efficient as selling through a platform. Therefore, some consumers may be attracted to the platforms simply because they offer a different service from the suppliers direct sales channels. 1 We assume that (inverse) demand functions are linear and that demand for supplier j (j k {1, 2}) on platform i (i h l {A, B, D}) is given by: p ij = 1 (q ij + αq ik + β (q hj + q lj ) + αβ (q hk + q lk )), where α ]0, 1[ measures the degree of inter-brand competition (i.e., between suppliers) and β ]0, 1[ measures the degree of intra-brand competition (i.e., between platforms). Note that this demand specification implies that the platforms and the direct sales channel are symmetrically differentiated. When all six quantities are strictly positive, we obtain the following direct demand for supplier j on platform i when inverting the system of inverse demands. 2 q ij = q (p ij, p ik, p hj, p hk, p lj, p lk ) = (1 α) (1 β) (1 + β) p ij + α (1 + β) p ik + β (p hj + p lj ) αβ (p hk + p lk ) (1 α 2 ) (1 β) (1 + 2β) We assume that the contract between supplier j {1, 2} and platform i {A, B} is purely vertical and boils down to a constant per-unit commission w ij paid by the supplier to the platform for each sale that takes place on platform i. This restriction to linear commissions is not totally innocuous. Our results would not be robust to the introduction of non-linear commissions (for instance two-part commissions specifying a constant fee for the service plus 1 We discuss in section?? how the results would be affected when the marginal cost of selling directly to consumer is strictly positive. 2 This specification is similar to that used in Ziss (1995). Although this demand function has some unusual features (for instance qij p hk < 0), it keeps exposition simple and prevents from adding restrictions on parameter values for α and β. Alternatively we could simply extend to six products the linear demand specification used by Rey and Vergé (2010) and obtain qualitatively similar results. 2

3 a constant per-unit commission). However, we believe it to be a better approximation of actual contracts. Platforms often impose revenue-sharing rules to suppliers where the share of the revenues paid as commission is constant. We could thus have focused on constant revenue-sharing rules (as in Johnson (2014) for instance) but the analysis quickly becomes rather non-tractable. We believe that most of our qualitative results would nevertheless apply to both types of commissions and thus prefer to focus on the simpler per-unit commission (as in Boik and Corts (2015) for instance). Building on earlier literature following Hart and Tirole (1990), we focus on secret contracting, and assume that the commission offered by platform i to supplier j, as well as the supplier s decision to accept that offer, is private information to the two parties. Modelling secret contracting in multilateral relationships is tricky, even in the absence of upstream competition. 3 To keep the analysis as tractable as possible and avoid some technicalities, we adopt the contract equilibrium approach developed by Crémer and Riordan (1987) and Horn and Wolinsky (1988), and which was also used in a context of downstream price competition by O Brien and Shaffer (1992). The timing of negotiations between suppliers and platforms and retail pricing decisions is as follows: 1. Each platform i simultaneously offer a per-unit commission, w ij, to each supplier j. Suppliers then decide which offers to accepts (i.e., on which platform to list). Offers are secret and listing decisions are not observed by the rival supplier. 2. Suppliers simultaneously set retail prices on all platforms on which they are active. We look for the contract equilibria of this two-stage game, defined as follows. Definition 1 (Contract equilibrium) A contract equilibrium (with unobservable offers and acceptance decisions) is a vector of commissions (wa1, w A2, w B1, w B2 ) and an associated vector of retail prices (p A1, p A2, p B1, p B2, p D1, p D2 ) (with the implicit notation that p ij = + if supplier j decides not to list on platform j) such that: In the second stage, for any pair of commission (w Aj, w Bj ) that it has been offered, supplier j s pricing strategy Pj R (w ij, w hj ) is given by: P R j (w ij, w hj ) = arg max (pij,p hj,p Dj) [(p ij w ij ) q (p ij, p ik, p hj, p hk, p Dj, p Dk ) + (p hj w hj ) q (p hj, p hk, p ij, p ik, p Dj, p Dk ) +p Dj q (p Dj, p Dk, p ij, p ik, p hj, p hk )] 3 See McAfee and Schwartz (1995) and Rey and Vergé (2004). 3

4 In the first stage, the commission w ij offered by platform i to supplier j maximizes the platform s profit given the other three equilibrium commissions, the supplier s pricing ( ) Pj R wij, whj, and the rival supplier s equilibrium prices P k, that is: w ij = arg max wij [ wij q ( p R ij + w ik q ( p ik, pr ij ( ) wij, whj, p ik, p R hj ( ) wij, whj, p hk, p R hj ( ) wij, whj, p hk, p R Dj ( ) wij, whj, p Dk, p R Dj ( wij, w hj), p Dk ) ( wij, w hj))] In what follows, we restrict attention to symmetric contract equilibria (i.e., w ij = w ) for which both suppliers are active on all three channels (i.e., on both platforms as well as on the direct sales channels). In this linear demand setting, it is not difficult to prove that there is a unique contract equilibrium for which all suppliers are active and that this equilibrium is symmetric, but focusing directly on a symmetric equilibrium greatly simplifies the exposition. There could however exist other (asymmetric) contract equilibria with fewer than six products offered in equilibrium (i.e., at least one of the suppliers could be inactive on at least one channel). 3 Unrestricted pricing We start by assuming can freely set the prices in all three sales channels. Consider first the pricing decisions made by supplier j when it faces commissions w A1 and w B1. When setting its prices, the supplier anticipates that its rival will set the equilibrium prices, which taking, into account symmetry, are simply p P on the two platforms and p D j thus maximizes: for direct sales. Supplier π j = (p ij w ij ) q (p ij, p P, p hj, p P, p Dj, p D ) + (p hj w hj ) q (p hj, p P, p ij, p P, p Dj, p D ) +p Dj q (p Dj, p D, p ij, p P, p hj, p P ) Given our demand specification, it thus chooses prices given by: p R ij (w ij, w hj ) = 1 α + αp P + w ij 2 for i {A, B} and p R Dj (w ij, w hj ) = 1 α + αp D. 2 The equilibrium retail prices must satisfy p P = pr ij (w, w ) and p D = pr Dj (w, w ), thus: p P = 1 α + w 2 α and p D = 1 α 2 α. The symmetric equilibrium commission maximises the platform s profit, taking as given the other three commissions, the supplier s pricing reaction and the rival supplier s equilibrium 4

5 ( ) prices. Given that p R ij only depends on w ij (and p R Dj wij, whj = p D ), this profit is simply: π i (w) = wq ( p R ij(w), p P, p P, p P, p D, p D) + w q ( p P, p P, pr ij(w), p P, p D, p D ) 2(1 α)(1 β) (1 + β)(2 α)w + (α(1 β) + 2β) w = w 2(2 α) (1 α 2 ) (1 β)(1 + 2β) + w 2(1 α)(1 β) + α(1 + β)(2 α)w (2 2αβ + α2 (1 + β)) w 2(2 α) (1 α 2 ) (1 β)(1 + 2β) The platform still needs to ensure that the supplier is willing to accept this commission and sell through the platform. However, given that the suppliers can freely set prices through all channels, a supplier will never drop on channel unless the price on that channel is excessive. It is easy to check that this will never be the case in equilibrium. The equilibrium commission is thus simply given by: π i (w ) = 0 w = 2 (1 β) 2 (2 + β) α (1 + β). Proposition 1 (Unrestricted pricing equilibrium) When suppliers can freely set prices on all platforms, there exists a unique contract equilibrium for which both suppliers are active on all three channels. In this equilibrium, platforms charge the same commission w, w = 2 (1 β) 2 (2 + β) α (1 + β) and the suppliers set prices p P on the platforms and prices p D p D = 1 α 2 α and p P = p D + w 2 α when selling directly: Unsurprisingly, the equilibrium commission decreases as the two platforms become closer substitutes (i.e., it is a decreasing function of β) and tends to 0 (and thus all prices become equal) as β tends to 1. The equilibrium commission increases as suppliers become closer substitutes (i.e., an increasing function of α). Since, price competition will keep prices low - i.e., closer to the commission on the platforms and closer to zero for direct sales - the platforms can now afford to charge higher commissions as this will have less of an impact on their sales. Platforms then set high commissions in order to compensate for the increase in intra-brand competition. This is similar to the strategic delegation effect à la Bonanno and Vickers (1985) or Rey and Stiglitz (1988, 1995). 5

6 4 Price parity clauses In this section, we assume that both platforms impose price parity clauses to the suppliers. We take these clauses as exogenous and thus do not look at asymmetric situations in which only of the two platforms requires price parity. We consider successively two types of price parity clauses: Wide price parity: under a wide price parity clause, platform i does not allow a supplier to sell at a lower price anywhere else. It must thus be the case that, p ij min {p hj, p Dj }. When both platforms impose wide price parity clauses, a supplier has to charge the same price on both platforms and this common price has to be lower than the price charged for direct sales, i.e., p ij = p hj p Dj. Given that it is cheaper to sell directly, this last condition is binding, thus, under wide price parity, a supplier sets a unique price p j which is then used on all channels (on which the supplier decides to be active). Narrow price parity: under a narrow price parity clause, platform i allows the supplier to freely set its price on platform h and only requires that the price charged on its platform does not exceed the direct sales price, i.e., p ij p Dj. When both platforms impose narrow price parity clauses, the supplier can freely choose the prices charged on both platforms but has to charge a higher price for direct sales than on the most expensive platform, i.e. p Dj max {p ij, p hj }. Again, because it is cheaper to sell directly, this last condition is binding, i.e., p Dj = max {p ij, p hj }. 4.1 Wide price parity clauses We start by looking at wide price parity clauses and focus again on a (symmetric) contract equilibrium for which both suppliers are active on all three channels. If both suppliers are active on all three channels, the demand for supplier j s product in each channel is now simply: ˆq (p j, p k ) = 1 α p j + αp k (1 α 2 ) (1 + 2β) Deriving the equilibrium commission and retail price When it sets its retail price p j, supplier j maximizes its profit, taking as given the commissions it faces and assuming that its rival sets the (symmetric) equilibrium price p W. Supplier j s price is thus given by: ( p R (w ij, w hj ) = arg max p 3 p w ) ij + w hj ˆq ( p, p ) W = α + αp W + w ij + w hj 3 2.

7 Given each supplier sets the same price on all three channels and that these channels are symmetrically differentiated, everything occurs on the suppliers side as if there were only one channel and the supplier s cost was the average commission (with the convention that the commission is equal to 0 for direct sales), that is, supplier j faces a marginal cost w j = w ij+w hj 3. The equilibrium retail price must satisfy: p W = p R ( w W, w W ) p W = 1 α 2 α + 2wW 3(2 α). Consider now the commission w ij = w set by platform i to supplier j, taking as given the other three commissions (set at the equilibrium level w W ). If this offer is accepted, the platform s profit is: π i (w) = wˆq ( p ( R w, w ) W, p ) W + w W ˆq ( p W, p ( R w, w )) W 3(1 α) (2 α)w + αww = w 3(2 α) (1 α 2 ) (1 + 2β) + w W 3(1 α) + α(2 α)w + (2 α2 ) w W 3(2 α) (1 α 2 ) (1 + 2β) Assuming that for exogenous reasons suppliers can only sell through all three channels or not at all (implying that offers are always accepted as long as supplier s profits remain positive), the (unconstrained) equilibrium commission would be given by π i (w) = 0, that is w = 3. Moreover, because 4 α w 2, it would appear as if wide price parity clauses allow 4 α platforms to substantially increase their commissions, in line with the theory of harm that has been developed by competition authorities in some recent cases. However, we have not yet checked that suppliers would accept such high commissions. In particular, consider the case of supplier j having just received the offer w W from platform i and anticipating that all other offers (identical to that one) will be accepted, its rival then charging price p W in all three channels. If it accepts the offer and sells on platform i (as well as through the other two channels), its profit is: π j ( w W ) = (1 α) ( 3 2w W ) 2 3(2 α) 2 (1 + α)(1 + 2β). Alternatively, supplier j could decide not to accept and sell only through the rival platform and the direct sales channels. 4 Given that this deviation would not be observed by its rival, supplier j would continue to anticipate that supplier k sets price p W in all three channels. Its 4 Given that we consider contract equilibria, it is assumed that supplier j takes its (acceptance) decision vis-à-vis the the other platform h as given. 7

8 maximal profit following this deviation is therefore: ( π j = max 2 p ww p 2 ) q ( p, p W,, p W, p, p W ) = ( 12(1 α) (6 7α)w W ) 2 72(2 α) 2 (1 α 2 ) (1 + β). Comparing the two profits when the commission is equal to 3 4 α we find that: ( π 3 ) ( j 4 α > 3 ) (5 4α) 2 πj 4 α 8(4 α) 2 (1 α 2 ) (1 + β) > 3(1 α) (4 α) 2 (1 + α)(1 + 2β) (5 4α) 2 24(1 α) 2 > 1 + β 1 + 2β. The left-hand term of the last inequality is minimized for α = 0, and it is then strictly larger then 1, whereas the right-hand term of the same inequality is strictly less than 1 over the whole support ]0, 1[. This implies that, at the unconstrained commission level, the supplier would always want to deviate and stop selling on platform i. Therefore, the true equilibrium commission is such that the supplier s participation constraint is binding, that is, w W is given by: (1 α) ( 3 2w W ) 2 3(2 α) 2 (1 + α)(1 + 2β) = ( 12(1 α) (6 7α)w W ) 2 72(2 α) 2 (1 α 2 ) (1 + β) 4(1 α) ( 3 2w ) W 12(1 α) (6 7α)w = 2(1 + 2β) W 3(1 + β) Proposition 2 (Wide Price Parity Clauses) Given that both platforms use wide price parity clauses, there exists a unique contract equilibrium for which both suppliers are active on all three channels. In this equilibrium, platforms charge the same commission w W, w W = 12(1 α) (1 σ(β)), where σ(β) = 2(1 α) (4 3σ(β)) + ασ(β) [ ] 2(1 + 2β) 2 3(1 + β) 3, 1, and the suppliers set the same price p W on the platforms and when selling directly: p W = 1 α 2 α + 2wW 3(2 α) When wide price parity clauses are introduced, the supplier s participation constraint is always binding and this constraint thus determines the equilibrium commission level. This is 8

9 not surprising when suppliers are very close substitutes (i.e., α close to 1). Suppose that both platforms offer similar strictly positive commissions. If both suppliers sell on all channels, their retail prices will be set very close to their average marginal cost, that is, 2w. If supplier 3 j decides to stop selling on platform i, it faces a substantially lower average marginal cost (now equal to w ). This allows supplier j to (marginally) undercut its rival on the second 2 platform as well as on the direct sales channel. Although it stops selling through one of the channels, it now almost doubled its sales on the other two channels as well as substantially increasing its margin. Therefore, in order to convince supplier j to sell on its platform as well, platform i needs to drastically cut its commission. As a consequence, in a symmetric equilibrium, the commission is close to 0 when suppliers are very close substitutes. If the result is quite intuitive when suppliers are very close substitutes, it also applies when they are totally differentiated (α = 0). Competition with the rival supplier is no longer a constraint. However, it is still the case that the supplier has the choice between selling on all channels or on two only. If it sells on all three it sets its monopoly price facing an average marginal cost equal to 2w. The alternative to sell only to two-thirds of the market 3 but facing now an average marginal cost of w. If w is too high, the positive effect of leaving 2 the platform on its average margin will dominate the negative effect of reduced sales. Thus, the commission cannot increase too much. As a consequence, the equilibrium commission decreases as the suppliers become closer substitutes and it tends to 0 as α tends to 1. Since the effect of leaving on platform is smaller when platforms are close substitutes, it is also the case that like in the unrestricted case the equilibrium commissions decreases as platforms become closer substitutes, and tends to 0 as β tends to 1. Note that the possibility to sell directly to consumers is crucial for the result. Suppose that suppliers are very close substitutes and can only sell through platforms A and B and look at a supplier s incentives when platform A offers the same commission as platform B. When it stops selling on one platform, the supplier continues to face the same average commission. Therefore, by leaving platform A, the supplier cannot expect to undercut its rival on platform B and increase substantially its sales. Therefore, in any symmetric equilibrium, the participation is not binding and the equilibrium commission is thus equal to the unconstrained commission. In this case, equilibrium commission will always increase when platforms impose price price clauses. 9

10 4.1.2 Effects of the wide price parity clauses Effect on commissions The analysis so far illustrate that the effects of the platforms price parity clauses are not as straightforward as the prevailing theories of harm want to believe. First of all, it is not always true that the equilibrium commission is higher with wide price parity clauses than without. Indeed, when prices are fully flexible (no price parity clauses), the equilibrium commission is always strictly positive (unless platforms are perfectly substitutable) and increases as suppliers become closer substitutes. With wide price parity clauses, the equilibrium commission decreases as suppliers become closer substitutes and tends to 0 as α tends to 1. Therefore, for any level degree of substitution between platforms (β), there exists a threshold α w (β) < 1 such that the commission increases when wide price parity clauses are introduced if and only if α < α w (β). Effect on final prices and consumer surplus Consider now the effects on prices. In the absence of price parity clauses, supplier s prices on the platform and for direct sales (for a given symmetric commission level w are: p P (w) = 1 α 2 α + w 2 α and p D = 1 α 2 α. Given that prices are higher on the platforms than for direct sales, more than a third of the sales will be done directly and less than two thirds on the platforms. The (weighted) average price is thus strictly lower than p(w), where: p(w) = 2 3 p P (w) p D = 1 α 2 α + 2w 3(2 α). When wide price parity clauses are introduced, the symmetric price charged on all three platforms (again a given commission level) is exactly equal to p(w). Keeping the commissions constant, introducing wide price parity clauses reduces the prices charged on the platforms but to increase the prices charged for direct sales but the average price increases. When the equilibrium commission increases (i.e., α α w (β)), an additional effect leading to increases in all prices occurs, implying that the equilibrium price with wide price parity is necessarily above the average price without price parity. When the equilibrium commission decreases (i.e., α > α w (β)), the additional effect lowers the price on all channels. In addition, given that, in the absence of price parity, the equilibrium commission w only depends on the degree of substitution between platforms, the average price is always strictly larger than p(0) = p D. With price parity, the equilibrium commission tends to 0 and therefore the final prices tend to p(0) when suppliers become perfect substitutes. Therefore, there exists a threshold α p (β), 10

11 with α w (β) < α p (β) < 1, such that the average price is lower without price parity if and only if α α p (β). Our linear demand specification results from a standard quadratic utility model with a representative consumer. Given the symmetry in this utility function with imperfectly substitutable products, consumers will usually prefer to consume equal quantities of the different products. This is exactly what occurs with wide price parity clauses since market shares are then identical for all products, whereas more sales occurs through the direct channel than on the platforms in the absence of such price parity clauses. Therefore, consumers may benefit from price parity clauses even if the average price increases. The analysis shows that there exists a threshold α S (β), with α w (β) < α S (β) < α p (β) such that consumers benefit from wide parity clauses whenever α > α S (β). Effect on the platforms profits When wide price parity clauses lead to higher commissions, platforms always benefit from such clauses: in this case, their per-unit margin increases as well as their market share. Because the average price may increase, there is also a negative demand effect (consumers being price sensitive) but this effect tends to be relatively small and is always dominated by the margin and market share effects. Moreover, the fact that the equilibrium commission decreases when using wide price parity clauses, does not necessarily imply that the platforms do not benefit. Platforms may indeed be willing to accept lower commissions in return to higher market shares when using price parity clauses. This will obviously be the case when the commission does not decrease much. However, as suppliers become closer substitutes, the commission and therefore each platform s profit tends to 0 in which case platforms cannot benefit from price parity clauses. Profit comparison shows that exists a threshold α P (β), with α S (β) < α P (β) < 1 such that platforms benefit from wide parity clauses whenever α < α P (β). Effect on the suppliers profits Consider finally the effect of price parity clauses on suppliers profits. If these clauses lead to higher commissions, suppliers are necessarily harmed: they face higher costs for each sale made through a platform and moreover, because of price parity, the share of sales made through the platforms increases. In addition, since the average price also increases, total sales decrease. All effects are thus negative on the suppliers profits. However, when suppliers are very close substitutes, introducing price parity clauses lead to a substantial drop in commissions rates (these tend to 0 as α tend to 1) where as this is also when commissions are highest without these clauses. Although more sales now take place on the platforms, the cost of selling on these platforms is substantially lower and thus suppliers profits increase. Profit comparisons show 11

12 that suppliers benefit from price parity clauses whenever they are sufficiently close substitutes. Moreover, the threshold α S is the same as for consumers, i.e., suppliers benefit from price parity clauses if and only if consumers also benefit. These results are summarized in the following corollary and illustrated by Figure 1. Corollary 1 (Effects of wide price parity clauses) There exists thresholds 0 < α w (β) < α S (β) < α P (β) < 1 and α p > α S, such that wide price parity clauses lead to: Higher equilibrium commissions if and only if α < α w (β). Higher final prices on average if and only if α < α p (β). Lower profits for suppliers and lower consumer surplus if and only if α < α S (β). Higher profits for platforms if and only if α < α P (β). In particular, platforms, suppliers and consumers all benefit from wide price parity clauses whenever α S (β) α α P (β). 4.2 Narrow price parity clauses We now consider narrow price parity clauses by which a platform does not constrain the suppliers pricing decisions on the platforms but simply requires that the price charged on its platform does not exceed that charged for direct sales. Under a narrow price parity clause, platform i thus simply requires that p ij p Dj. As in the previous section, we assume that for exogenous reasons both platforms impose narrow price parity clauses. This implies that supplier j s direct price has to be set above the highest price charged on the two platform, that is, p Dj max {p ij, p hj }. We again focus on a symmetric contract equilibria for which both suppliers are active on all three channels. By symmetric equilibria, we mean that equilibrium commissions are identical, that is, wij N = w N for any i {A, B} and any j {1, 2}. Retail prices will then all be identical as well, that is, p N ij = p N for any i {A, B} and any j {1, 2} and price parity constraints are both binding for each supplier as long as w N > 0. Suppose supplier i has received the offer of w ij = w from platform i, while it has received the (equilibrium) offer w hj = w N from platform h. When setting its prices, supplier j anticipates that its rival has received and accepted the equilibrium commissions, w ik = w hk = w N, and charges the equilibrium prices p ik = p hk = p Dk = p N. Given the narrow price parity constraints, supplier j has two alternative pricing strategies: 12

13 β 1 α w β α S β α P β Platforms, suppliers and consumers all benefit from wide price parity clauses. ½ ½ α p β 1 α Figure 1: Effects of wide price parity clauses 1. Set a lower price on platform i than on platform j, implying: p hj = p Dj = p and p ij = p d (with d 0). 2. Set a lower price on platform j than on platform i, implying: p ij = p Dj = p and p hj = p d (with d 0). Intuitively, charging a lower price on platform i than on platform h should be attractive when w w N. Consider first this strategy. Supplier j then chooses its base price p and the 13

14 level of discount d 0 offered on platform i so as to maximize its profit π j (p, d) where: π j (p, d) = [ (p d w) q ( p d, p N, p, p N, p, p N) + ( p w N) q ( p, p N, p d, p N, p, p N) + pq ( p, p N, p d, p N, p, p N)] Maximizing this profit with respect to p, taking the discount d as given, yields: p R (d) = 1 α + αp N + w + wn d 3. get: When we substitute p R (d) into the profit function and maximize with respect to d, we dπ j ( p R (d), d ) dd = w N 2w 4d 3 (1 β) (1 α) (1 + α). Therefore, supplier j optimally sets a strictly lower price on platform i (i.e., d > 0) if and only if the commission offered by platform i is sufficiently low, namely, w < wn 2. Otherwise, supplier j sets the same prices on all channels (d = 0). Therefore, it would seem intuitive to expect the supplier to offer a strictly positive discount on the cheaper platform, this is not the case if the difference between the two commissions is too low. However, the intuition behind this seemingly un-natural result is relatively simple: because of the narrow price parity constraint imposed by platform h, the supplier has to set the same price on the direct sales channel and on platform h. Given the symmetry of the demand function, the supplier sees the direct sales channel and platform h as a single platform on which its marginal cost is wn 2. When setting its base price and the discount offered on platform i, the supplier thus faces a trade-off between driving sales towards platform i on which its cost is w and the second platform on which the marginal cost is wn. Therefore, it is not tempted to offer a discount if 2 w is larger than wn (but lower than w N ). Since it cannot impose a surcharge either because 2 of the (narrow) price parity clause imposed by platform i, it sets the same prices on all three channels. A similar analysis show that the supplier is not willing to offer a discount on platform h unless the commission becomes extremely high on platform i, i.e., unless w > 2w N. Under narrow price parity clauses, supplier [ j s ] pricing strategy is identical than under wide price w parity clauses whenever w N2, 2wN. It can then easily be shown that the unique symmetric contract equilibrium is the same as with wide price parity clauses (platforms profit functions are not affected for small deviations). 14

15 Proposition 3 (Narrow Price Parity Clauses) When both platforms impose narrow price parity clauses, the unique contract equilibrium for which both suppliers are active on all three channels is the same than when they impose wide price parity clauses, i.e., w N = w W and p N = p W. 4.3 Endogenous adoption? [TO BE WRITTEN...] 5 Extensions / Discussion Some important points for the discussion and potential extensions: Non-linear commissions. Our results are not robust to the use of non-linear commissions. Rey and Vergé (2015) consider a similar setting but without direct sales (the main focus is on interlocking relationships and resale price maintenance) and show that, if platforms and suppliers negotiate over two-part commissions (a fixed component and a per-unit commission), price parity clauses have no impact on final prices. Our model is undoubtedly more restrictive since we only allow constant per-unit commissions. However, we believe that linear commissions are more in line with the motivating examples (hotel booking platforms). Perhaps more important, assuming linear commissions seem also consistent with the prevailing theory of harm put forward by the antitrust authorities and other commentators. Observable decisions to accept/reject. So far, we assume that decisions to list on a platform are not observed by rivals (when final prices are set). One could alternatively assume that these decisions are observed (although the commissions remain unobservable) and that suppliers can thus optimally (re-)adjust their final prices if one of them decides to stop selling on one of the platforms. However, which decision is more long term is not necessarily obvious: it may be as easy (quick) to stop selling on a platform than it is adjust prices. Moreover, although it would make the model less tractable, observable listing decisions should actually strengthen our main mechanism, since it makes it even more tempting for a supplier to reject the platform s offer under price parity, all else equal. Note that, when listing decisions are observable, when supplier j rejects platform i s offer, the rival supplier responds by setting a higher price (under price parity) which in turn increases supplier s j outside option. Assuming that listing 15

16 decisions are unobservable thus seem to make it less likely that the participation constraint binds and thus limits the need for platforms to be aggressive when price parity clauses are imposed. Cost of selling directly. So far we have assume that the cost of selling directly is the same as the distribution cost for platforms (we normalize that distribution cost to 0). Suppose now that it is slightly more costly to sell directly, and that this cost is c > 0. As long as the cost is not too large and that platforms are sufficiently differentiated (β small enough), it is still the case that in the absence of price parity final prices are higher on platforms than for direct sales. We thus expect that results won t be qualitatively affected. When platforms are close substitutes, (β tends to 1), it may be the case that the equilibrium commission is lower than the cost of selling directly (w (c) < c). In this case, price parity clauses only matter across platforms since the suppliers continue to be able to charge higher prices on their own direct sales channel. Narrow price parity clauses are thus useless (they do not bind) and wide price parity clauses should always lead to higher commissions, higher (average) prices and thus harm consumers and suppliers. More than two suppliers and/or platforms. This should not qualitatively affect our results. What if the degree of substitution between platforms is not equal to the degree of substitution between a platform and a direct sales channel? This should not qualitatively affect the analysis of wide price parity clauses. However, it obviously matters when considering narrow price parity clauses. The interesting case is then when the direct sales channel is a closer substitute to a platform than the rival platform in which case the narrow price parity clauses should not be equivalent to wide price parity clauses. More general demand functions. 16

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