Should platforms be allowed to charge ad valorem fees?

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1 Should platforms be allowed to harge ad valorem fees? Zhu Wang and Julian Wright November 27 Abstrat Many platforms that failitate transations between buyers and sellers harge ad valorem fees in whih fees depend on the transation prie set by sellers. Given these platforms do not inur signifiant osts that vary with transation pries, their use of ad valorem fees has raised ontroversies about the effiieny of this pratie. In this paper, using a model that onnets platforms use of ad valorem fees to third-degree prie disrimination, we evaluate the welfare onsequenes of banning suh fees. We find the use of ad valorem fees generally inreases welfare, inluding for alibrated versions of the model based on data from Amazon s marketplae and Visa s signature debit ards. JEL lassifiation: D4, H2, L5 Keywords: platforms, taxation, third-degree prie disrimination Researh Department, Federal Reserve Bank of Rihmond. zhu.wang@rih.frb.org. The views expressed are solely those of the authors and do not neessarily reflet the views of the Federal Reserve Bank of Rihmond or the Federal Reserve System. Department of Eonomis, National University of Singapore. jwright@nus.edu.sg.

2 Introdution Ad valorem platform fees whih depend positively on transation pries are widely used in pratie. Well-known examples inlude online marketplaes suh as Amazon and ebay), payment ard platforms suh as Visa, MasterCard and Amerian Express), and hotel booking platforms suh as Booking.om and Expedia). In these ases platforms typially harge sellers perentage fees, as well as sometimes small fixed per-transation fees. Platform osts, whih are largely fixed or dependent on the number rather than the value) of transations annot explain the levels of ad valorem fees set by these platforms. This has led to ritiisms of the ad valorem fee struture, given it is not ost refletive. In this paper, we explore whether suh ad valorem fees harm welfare, and so whether there may be a ase for banning them. Conerns over the use of ad valorem fees have been raised in the ontext of payment ard platforms. Merhants and poliymakers point out that debit and prepaid ard transations do not provide redit or float and bear very small fraud risk; therefore, they do not warrant a perentage-based fee struture. For instane, Summers 22) ritiizes that Payment shemes owners and infrastruture operators also have monopoly power that an be used to set pries far above their prodution ost. There is abundant evidene of learing and settlement priing that is based not on prodution ost but on methods designed to extrat very high returns for use of the infrastruture. Perhaps the most prominent example is ad valorem priing for payment methods that essentially involve giving bank aount holders diret aess to their deposits and that do not entail bank redit, as in the ase of debit ards. The Canadian Senate Committee on Banking, Trade and Commere June 29) made the following ruling: The Committee believes that there is little rationale for perentage-based interhange, merhant disount and swith fees on debit ards, sine this payment method involves a relatively simple and nearly instantaneous transfer of funds. There is no obvious redit risk and no interest-free period to fund in these transfers...the Committee believes that debit ard transations are inherently less risky and ostly than redit ard transations; onsequently, they do not warrant a perentage-based fee struture, whether at the level of interhange fees or swith fees. To address this issue we make use of the model we developed in Wang and Wright forthoming) in whih a profit maximizing platform designs its fee struture to take into aount heterogeneity in demand aross the many produts sold over its platform. The key idea aptured by the model is that when a market involves many different goods that vary widely in their osts and values that may not be diretly observable, then ad valorem fees and taxes represent an effiient form of prie disrimination beause the value of a transation 2

3 is plausibly proportional to the ost of the good traded. The model implies that the profit maximizing fee struture is affine onsisting of a perentage fee plus a fixed per-transation omponent) if and only if the demand faed by sellers belongs to the generalized Pareto lass that features onstant urvature of inverse demand whih inludes linear demand, onstant-elastiity, and exponential demand as speial ases). This mathes the fee struture used by many platforms, as shown in Table below. Aording to the model, the fixed per-transation omponent is present only beause the platform inurs a marginal ost for proessing eah transation; otherwise a simple perentage fee would be profit maximizing. In Wang and Wright forthoming) we used the model to show that ad valorem fees should also be used by an authority that wants to regulate a platform s fees while allowing for the reovery of a ertain amount of revenue e.g., to over the platform s fixed osts). In ontrast, in this paper we investigate a different issue: What would happen if a poliymaker banned a platform s use of any fee that depended on the value of transations i.e., ad valorem fees) but left the level of the platform s fees unregulated. For poliymakers who want platform fees to be determined on the basis of osts but are onerned about diretly regulating fee levels, this seems to be a natural approah to onsider. However, we show that the welfare results turn out not at all obvious, and are related to the long-standing debate on the welfare effets of third-degree prie disrimination. Table. Platform fee shedules Amazon Visa DVD 5% + $.35 Gas Station.8% + $.5 Book 5% + $.35 Retail Store.8% + $.5 Video Game 5% + $.35 Restaurant.9% + $. Game Console 8% + $.35 Small Tiket.5% + $.4 To address this question, we first show that the onditions for welfare to inrease as a result of banning the use of ad valorem fees turn out to be the same as those whih determine whether banning a monopolist from using third-degree prie disrimination improves welfare, in whih eah good traded is treated as a separate observable market over whih the monopolist an prie. This allows us to draw on the substantial literature on monopolisti third-degree prie disrimination e.g., see Aguirre et al., 2, for a reent analysis). In the setting usually adopted for prie disrimination studies, in whih there are only two markets or a ontinuum of markets, we find welfare is generally higher when platforms are Table reports fees that Amazon and Visa harge to sellers for eah transation on the platform. Note that Visa fees shown in Table are signature debit ard interhange fees for the U.S. market. These fees, set by Visa, are paid by merhants to ard issuers through merhant aquirers. 3

4 allowed to use ad valorem fees. Speifially, we are able to show in the speial ase in whih the platform trades only two goods, welfare is higher whenever demand is exponential or log-onvex within the generalized Pareto lass, and also when demand is log-onave within this lass provided the two goods are suffiiently dispersed. We then extend these results to show similar qualitative findings hold allowing for a ontinuum of uniformly distributed goods. The intuition for the above results relates to the standard intuition for why third-degree prie disrimination is desirable when uniform priing shuts down the low-demand market or in our setting, trade of the low-demand good). For log-onave demand, there is a hoke prie at whih demand beomes zero. As a result, provided the demand aross the two goods is suffiiently different, a monopoly platform that an only set a uniform fee will want to set a fee that leads to the low-demand good not being traded beause setting the fee low enough so that both goods are traded will sarifie too muh of the platform s profits. In this ase, allowing the platform to prie disriminate i.e., by using ad valorem fees) will not only inrease its profit but also onsumer surplus and soial welfare sine the low-demand good is then offered for sale. We show this logi extends to the ase with a ontinuum of uniformly distributed goods provided they are suffiiently dispersed. A similar logi also applies to exponential and log-onvex demands in whih there is no hoke prie, provided the two goods are suffiiently different. This is beause a platform that an only set a single fee will set a fee very lose to the monopoly level for the high-demand good, thereby almost ruling out all sales of the low-demand good. On the other hand, when the two goods are not very different, the existing results of Aguirre et al. 2) an be applied to our exponential and log-onvex demands to explain why welfare will be higher with prie disrimination. More generally, the welfare effets of allowing prie disrimination when there are many goods an be sensitive to the distribution of goods. In pratie, the distribution of pries and sales of goods traded are highly skewed. We therefore use information on the atual distribution of pries and sales measures for two different platforms Amazon s and Visa s), as well as their fee strutures, to alibrate our model. We find, in most ases, welfare would be harmed if ad valorem fees were banned. Our results therefore imply poliymakers should be autious about banning the use of ad valorem fees. Shy and Wang 2) also look at the welfare effets of a platform shifting from a fixed per-transation fee to an ad valorem fee. They onsider a setting with a monopoly platform and imperfetly ompetitive sellers that sell a homogenous good, and demand takes a onstant-elastiity form. They find welfare is higher when the platform an use an ad valorem fee, but their result relies on the property that ad valorem fees help mitigate the double marginalization problem. Thus, their work omplements ours, whih assumes away 4

5 any double marginalization problem by fousing on the ase in whih sellers are idential prie ompetitors. In a similar vein, several studies e.g., Foros et al., 24; Gaudin and White, 24a; and Johnson, forthoming) have explored the advantages of so-alled ageny model used by mass retailers suh as Amazon, where the retailer lets suppliers i.e., sellers) set final pries and reeive a share of the revenue, whih is equivalent to using a perentage fee. Like Shy and Wang 2), they also show that the revenue-sharing used in the ageny model has the advantage of mitigating double marginalization, but they differ by fousing on how the ageny model affets retail pries ompared with the traditional wholesale priing model. Our theory an also be used to justify the adoption of ad valorem taxes rather than unit or speifi) taxes in settings in whih governments seek to maximize tax revenue the so-alled Leviathan hypothesis; see Brennan and Buhanan, 977, 978). In this ase, the tax-revenue maximizing government is idential to a revenue-maximizing platform. Our results imply that an ad valorem tax regime welfare dominates a unit tax regime. In this regard, our finding omplements that of Gaudin and White 24b). They also show ad valorem taxes welfare dominate when governments maximize tax revenue, although they obtain their results in a very different setting, in whih double marginalization rather than prie disrimination is the key driving fore. Finally, our paper also ontributes to the extensive literature on the welfare effets of monopolisti third-degree prie disrimination e.g., Shmalensee, 98; Varian, 985; Shwartz, 99; Layson, 994; Aguirre, 26, 28; Cowan, 27, 26; and Aguirre et al., 2). Aguirre et al. 2) is partiularly relevant. They fous on the ase there are two markets whih are always served and onsider a general demand speifiation. Among the ases for whih they obtain stronger results is the ase in whih demand in eah market has onstant urvature of inverse demand. We study a speial ase of their speifiation in whih all markets have the same onstant urvature of inverse demand and in whih inverse demand only varies by a multipliative term aross eah market. At the same time, we generalize their setting by allowing for a ontinuum of markets and that not all markets are served. Our results also extend the findings of Malueg and Shwartz 994), who onsider a ontinuum of markets eah having a linear demand that varies with a multipliative term, whih turns out to be a speial ase of ours. Moreover, we go beyond purely theoretial disussions by alibrating our model to the data from Amazon s marketplae and Visa s signature debit ards. This allows us to provide quantitative evaluations of the welfare effets of prie disrimination at those platforms based on the atual, highly skewed, distributions of goods traded on these platforms. The rest of the paper proeeds as follows. Setion 2 sets out the model. Setion 3 provides 5

6 some analytial results while Setion 4 provides results based on a alibrated model of the Visa platform and the Amazon platform. Finally, Setion 5 provides some brief onluding remarks. 2 The model We onsider an environment where multiple goods are traded over a platform. For eah good traded, a unit mass of buyers want to purhase one unit of the good. There are multiple idential sellers of eah good who engage in Bertrand ompetition. Different goods sold on the platform are indexed by, whih an be thought of as a sale parameter, so that different goods an be thought of as having similar demands exept that they ome in different sales. In partiular, the per-unit ost of good to sellers whih is known to all buyers and sellers of the good) is normalized to and the value of the good to a buyer drawing the benefit parameter b is + b), so the sale parameter inreases the ost and the buyer s valuation proportionally. We denote the lowest and highest values of as L and H respetively, with H > L >. We assume + b is distributed aording to some smooth i.e., twie ontinuously differentiable) and stritly inreasing distribution funtion F on [, + b ], where b >. We do not require that b is finite.) Only buyers know their own b, while F is publi information. This setup aptures the idea that for a given market that an be identified by the platform, the main differene aross the goods traded is their sale i.e., some goods are worth a little and some a lot). In omparison to the wide range of sales of goods traded, potential differenes in the shapes of demand funtions aross the different goods traded are not likely to be of first-order importane. The assumption that buyers values for a good an be saled by is onsistent with a key empirial finding of Einav et al. 25) who study quasi-experimental observations from a large number of autions of different goods on ebay. In Wang and Wright forthoming), we show that this demand speifiation an also be justified on alternative grounds. Instead of diretly assuming positive orrelation between osts and onsumer valuations of the goods traded, onsider a platform that redues trading fritions, and assume the loss to buyers of using the less effiient trading environment i.e., trading without using the platform) is proportional to the ost or prie of the goods traded. This would apply whenever the alternative trading environment exposes the buyer to some risk or inonveniene that is proportional to the amount she pays for the good. This alternative demand speifiation delivers exatly the same results and helps further larify why the differene in demand aross the goods traded on the platform is mainly determined by their sale. Wang and Wolman 26) provide empirial evidene onsistent with this 6

7 interpretation. Analyzing payment patterns for two billion retail transations, they find that the value of transation is the key to explain onsumers hoie between using payment ards and ash. The number of transations Q for a good is the measure of buyers who obtain nonnegative surplus from buying the good, Pr + b) p ). Therefore, the demand funtion for good is Q p ) Q p ) F p ). ) The orresponding inverse demand funtion for good is p Q ) = F Q ), whih note is proportional to. This form of demand funtion, hinged on a saled prie, is reminisent of the one used in Weyl and Tirole 22), whih they refer to as the streth parametrization of a general demand funtion. Given this setup, in Wang and Wright forthoming) we show that the profit-maximizing platform fee is affine if and only if F takes on the generalized Pareto distribution F x) = + λ σ ) x )) σ. 2) Here λ > is the sale parameter and σ < 2 is the shape parameter. Note that the generalized Pareto distribution implies the orresponding demand funtions for sellers on the platform are defined by the lass of demands with onstant urvature of inverse demand Q p ) = F p ) = + λ σ ) p ) ) σ, 3) where p is the prie of good on the platform and Q p ) is the measure of units of good sold by sellers on the platform at that prie. 2 The onstant σ is the urvature of inverse demand, defined as the elastiity of the slope of the inverse demand with respet to quantity. When σ <, the support of F is [, + /λ σ)] and it has inreasing hazard. Aordingly, the implied demand funtions Q p ) are log-onave and inlude the linear demand funtion σ = ) as a speial ase. Alternatively, when < σ < 2, the support of F is [, ) and it has dereasing hazard. The implied demand funtions are log-onvex and inlude the onstant elastiity demand funtion σ = + /λ) as a speial ase. When σ =, F aptures the left-trunated exponential distribution F x) = e λx ) on the support [, ), with a onstant hazard rate λ. This implies the exponential or log-linear) demand Q p ) = e λp ). Taking as given that demand belongs to the generalized Pareto lass, we allow to take 2 This lass of demands has been onsidered by Bulow and Pfleiderer 983), Aguirre et al. 2), Bulow and Klemperer 22), and Weyl and Fabinger 23), among others. 7

8 on potentially many different values in [ L, H ], with the set of all suh values being denoted C. The distribution of on C is denoted G. We allow for the possibility takes only a finite number of values in C, or that it is ontinuously distributed. We let g apture the probability or density in ase G is ontinuous) orresponding to the realization. The platform inurs a ost of d per transation. 3 If it harges sellers the fee shedule T p ), the platform s profit is Π = T p ) d) Q p ) for good. Note given Bertrand ompetition between sellers, the prie p for good solves p = + T p ). 4) The platform s problem is to hoose T p ) to maximize Π = C g Π. 5) In Wang and Wright forthoming), we show that under these assumptions, the optimal fee shedule is affine, given by T p ) = λd + λ 2 σ) + p + λ 2 σ), 6) whih maximizes 5). Note the platform s optimal fee shedule has a fixed per-transation omponent only if there is a positive ost to the platform of handling eah transation i.e., d > ). Given λ > and σ < 2, the fee shedule is inreasing higher pries imply higher fees are paid) but with a slope less than unity this implies 4) has a unique solution for any given > ). The result in 6) also implies the platform an maximize its profit without knowing the distribution G of goods that are traded on its platform. Finally, note in Wang and Wright forthoming) we show the solution in 6) is equivalent to harging a different fixed per-transation fee T = λd + λ 2 σ) for eah different good, whih would be possible if the platform ould identify eah good and set a different fee aordingly. 3 Note that if the platform is a tax authority, then based on the onventional approah of ignoring olletion osts, we an set d =. 7) 8

9 3 Banning ad valorem fees: analytial results In this setion, we onsider whether banning the use of ad valorem fees i.e., any fees that depend on transation pries) would harm welfare in an otherwise unregulated environment. Without any restritions on its priing, the platform hooses a fee shedule to maximize 5), whih results in the affine fee shedule 6). If instead the platform annot ondition on transation pries in any way, it must hoose a single fixed per-transation fee T aross all goods. In this ase, it will hoose T to maximize Π = g T d) Q p ), 8) C where from 3) and 4), Q p ) = F p ) ) = F + T. Our problem of interest is thus what happens to total welfare in going from the platform s optimal fee shedule 6), whih maximizes 5), to the single fee T, whih maximizes 8). In other words, is banning ad valorem fees desirable? The solution to this problem an be found by solving a dual problem, whih amounts to the welfare effets of banning third-degree prie disrimination. The dual problem involves onsidering a standard monopolist firm that sells in distint and identifiable markets, eah indexed by. It sets T for eah market to maximize profit Π = T d) Q T ). 9) In our ontext, Q T ) an be interpreted as the demand funtion that the platform faes, T is the relevant prie in eah market, and is a parameter whih shifts demand aross different markets. The expression of Q T ) = F ) + T is given by Q T ) = + λ σ ) T ) σ, ) whih also belongs to the generalized Pareto lass. Note that while the platform deals with different goods, one an equivalently think of the platform providing a homogenous servie to different markets with demand in eah market varying by, and the platform s output in terms of transation numbers is addable aross these markets. If third-degree prie disrimination is banned, the monopolist will instead hoose a uniform prie T aross all markets to maximize 8). resulting T denoted T ) is between T L Given our demand speifiation, the and T H, the lowest and highest optimal disrimina- 9

10 tory pries. 4 The following duality result follows from the equivalene between solving for the optimal T for eah observed good or market) and solving for the optimal fee shedule in ase the latter is an inreasing affine funtion of the transation prie with slope less than one. Proposition Duality): The welfare effet of banning a platform from using an ad valorem fee is idential to the welfare effet of banning third-degree prie disrimination in the dual problem in whih a monopolist an observe the demand of eah different market as given by ) and harge different optimal) pries aordingly. This duality result allows us to draw on the existing literature on the welfare effets of monopolisti third-degree prie disrimination. Consider then a monopolist firm i.e., the platform) faing demand in market given by ) where T is the prie set by the monopolist in market and is a demand shift parameter. When prie disrimination is allowed, the monopolist hooses pries T for eah market to maximize its profit g T d)q T ). C If prie disrimination is banned, it must set the same T in eah market. In this setion, to derive analytial results, we first onsider a setting with just two goods L and H being sold on the platform, with H > L, and g = for eah good. We then extend this analysis to the ase with a ontinuum of goods drawn from the uniform distribution on [ L, H ] so that the platform s profit beomes ) H [T d)q T )]d. H L L In eah ase, the results we obtain on prie disrimination diretly apply for the welfare effets of allowing a platform to use ad valorem fees given the duality result above. 3. Two goods Aguirre et al. 2) fous on the ase in whih the monopolist sells in two distint markets, and will ontinue to sell in both markets even with uniform priing. They provide general 4 Given σ < 2, it an be shown that dπ /dt < if T > T and dπ /dt > if T < T, so Π is single-peaked. In this ase, as shown in Nahata, Ostaszewski and Sahoo 99), the optimal uniform prie is bounded above and below by the highest and lowest optimal disriminatory pries. The intuition is straightforward: Sine T is inreasing in, T would never be below T L above T H ) sine a higher T lower T ) an inrease profit in eah market in whih there is trade on the platform.

11 onditions to sign the output and welfare effets of prie disrimination under non-linear demand. When the urvature of inverse demand funtion σ is ommon aross markets, as it is in our ase, a suffiient ondition for total output to inrease is that σ is positive and onstant for eah good. This implies that in our setting with generalized Pareto demand, prie disrimination will always expand output i.e., the number of transations on the platform) provided σ > so demand is more onvex than linear demand). Aguirre et al. 2) also show that when σ >, prie disrimination raises redues) welfare when demand is log-onvex log-onave) if the disriminatory pries are not far apart. This applies to our setting, but given platforms often deal with transations of widely different values, we also need to onsider ases where disriminatory pries ould be far apart. Define k H / L > as the measure of dispersion of the two demand levels. We first establish a new result on the welfare effets of third-degree prie disrimination. The proof, whih is lengthy, is given in Appendix A.) Proposition 2 Welfare effets with two markets): Assume that demand is given by ) and the monopolist has zero marginal osts i.e., d = ). If there are two markets with demand haraterized by L and H, then banning prie disrimination aross the two markets lowers welfare if demand is exponential σ = ) or log-onvex < σ < 2), and will also lower welfare if demand is log-onave σ < ) provided k H / L is suffiiently large. From the duality result in Proposition, Proposition 2 implies that for exponential and log-onvex demand from the generalized Pareto lass given by ) and for log-onave demand given by ) but with high enough k, banning a monopolist platform from using ad valorem fees will lower welfare. Proposition 2 overs two types of ases. In the first ase, demand is log-onave, so σ <. As an example, onsider the ase σ = so that the monopolist faes linear demand for good given by Q T ) = Then there is a hoke prie at whih demand beomes zero. λt ). ) As a result, provided k is suffiiently high, a monopolist that an only set a single prie will want to stop selling the good L i.e., the low-demand good) by setting its single profit maximizing prie at the monopoly level for good H i.e., the high-demand good). This is beause ontinuing to sell the low-demand good will sarifie too muh of the monopolist s profit from the high-demand good. Allowing it to prie disriminate will not only inrease the monopolist s profit but also onsumer surplus and so welfare sine sales of low-demand goods are enabled. The ondition for this to arise in the linear demand example given in ) is k > 3 when d =, so that the

12 dispersion of demand aross goods does not have to be very high for the result to hold. The proposition shows the same logi holds for any log-onave demand. Figure illustrates this for the linear demand ase. 5 our welfare finding ontinues to hold for d >. prie T It also shows as d inreases, the ritial value of k delines, so Figure : Monopoly pries with linear demand two-good ase) uniform prie prie for good H prie for good L d= welfare gain: WPD)-WU)..5 d= prie T d=.5 welfare gain: WPD)-WU)..5 d= prie T d=. welfare gain: WPD)-WU)..5 d= For the exponential and log-onvex ase, the logi is atually similar even though there is no hoke prie at whih demand beomes zero. When demand for the two goods is suffiiently dispersed, a monopolist that an only set a single prie will set a prie very lose to the monopoly prie for the high-demand good, thereby almost ruling out all sales of the low-demand good, and implying a welfare gain of allowing prie disrimination. Figure 2 illustrates this property, showing the monopolist s optimal pries with and without prie disrimination as k varies in the partiular ase of exponential demand. In the proof of Proposition 2 we note this property formally by showing that as k, the uniform prie in the absene of prie disrimination onverges to the prie that the monopolist would set for good H under prie disrimination, both when demand is exponential and when it is 5 To plot the figure, we normalize λ = 4.5 and L =. Note that the values of λ and L just sale the results, but do not affet welfare findings in any of our exerises in this setion. 2

13 log-onvex. This explains the result when the two goods are suffiiently dispersed. On the other hand, when the two goods are not dispersed very muh, we know already from Aguirre et al. 2) that prie disrimination also raises welfare, whih is onsistent with our finding in Proposition 2. prie T Figure 2: Monopoly pries with exponential demand two-good ase) uniform prie prie for good H prie for good L d= welfare gain: WPD)-WU) d= d=.5. d=.5 prie T welfare gain: WPD)-WU) d=.. d=. prie T welfare gain: WPD)-WU) The welfare effets of allowing prie disrimination i.e., an ad valorem fee shedule) for the different ases aptured by Proposition 2 are summarized in Figure 3. The figure onsiders three different values of d and for a range of values of k and σ. The dark blue area in the figure indiates a welfare loss due to prie disrimination, while the light orange area indiates a welfare gain. In the log-onave demand ase σ < ), there is a disrete jump between these two areas when k beomes suffiiently large, refleting that the low-demand good gets shut down if prie disrimination is not allowed. In the log-onave ase with d =, the ritial level of k for whih welfare is higher under prie disrimination than under uniform priing is k > 3.5, so quantitatively we do not require unreasonably high dispersion in the demand for goods to get the welfare result. 6 In the exponential or log-onvex ase, 6 While not shown in the figure, the ritial level of k for whih welfare beomes higher under prie disrimination delines as σ dereases below, so the suffiient ondition k > 3.5 ontinues to hold. 3

14 Figure 3: Welfare omparison for two goods Figure 3 shows that welfare is always higher under prie disrimination regardless of the level of dispersion k. For both ases, Figure 3 shows that the welfare finding extends to d >. 3.2 Continuum of uniformly distributed goods The qualitative onlusions on the welfare-gains of prie disrimination or equivalently, allowing ad valorem fees) an hold when there are many markets or equivalently, goods) rather than just two. In this setion we will assume that is uniformly distributed between L > and H = k L, with k H / L and k >. We first derive the welfare results for the speial ase in whih σ = so demand is linear) and d =. We then establish that welfare is always higher under prie disrimination whenever demand is log-onave provided there is enough dispersion in when d =. In partiular, we show there is a utoff level of equal to x L where < x < k) suh that all markets below the utoff will be shut down by the monopolist, provided that the dispersion in is large enough i.e., k > k ). We show 4

15 that the threshold k depends only on σ, and the utoff value x is a onstant fration of k provided k > k. Finally, we explore graphially the welfare effets of prie disrimination for the full range of σ, allowing d > Linear demand We first onsider the speial ase with d = and σ =, so demand is linear. Then the inverse demand faed by the monopolist for good is T Q ) = Q ). λ Then the problem is stated in exatly the same form as the third-degree prie disrimination problem analyzed by Malueg and Shwartz 994), exept that we allow inverse demand to be multiplied by a onstant positive parameter and we allow that the uniform distribution on does not have to be entered at unity. 7 It turns out what matters for Malueg and Shwartz s results is the ratio of the highest to lowest value of in the support of the distribution, i.e. k. Therefore reinterpreting the relevant part of their Proposition to our setting, it implies that for large enough dispersion k > k, some markets are shut down under uniform priing; in this range, the ratio of welfare under prie disrimination to welfare under uniform priing inreases monotonially with dispersion and exeeds when dispersion is suffiiently large. To alulate these points preisely, define k > whih solves + 2 ln k = k, so k Then the point at whih dispersion is suffiiently large for welfare to inrease under prie disrimination arises when 8 k > 3k 3k 4 k ) k 4 + 3k 4 k ) Thus, provided there is suffiient dispersion in, welfare is unambiguously higher with prie disrimination or equivalently, with ad valorem fees). The result is illustrated in Figure 4, whih repliates Figure for this ontinuum ase Log-onave demand We an generalize Malueg and Shwartz s result on the positive welfare effets of prie disrimination when goods are suffiiently dispersed to log-onave generalized Pareto demands. 7 Their speifiation an be obtained by setting λ =, = a, L = x and H = + x. 8 There is a typo in Malueg and Shwartz s stated formula for this threshold in their footnote 7) whih does not generate the approximate numerial value they state in the footnote. However, their stated numerial value orresponds to ours, whih we derived diretly with our speifiation. I.e. if their threshold is denoted x e and ours is denoted k e, then it an be heked that k e = + x e ) / x e ). 5

16 prie T Figure 4: Monopoly pries with linear demand ontinuum ase) uniform prie prie for good H prie for good L d= welfare gain: WPD)-WU) d= prie T d=.5 welfare gain: WPD)-WU) d= prie T d=. welfare gain: WPD)-WU) d= The demand for eah good is given by ) and inverse demand is T Q ) = Q σ λ σ) ), where σ < given demand is log-onave. With this speifiation, we obtain the following result on the welfare effets of prie disrimination. The proof is given in Appendix B). Proposition 3 Welfare effets with a ontinuum of markets): Assume demand is given by ) and the monopolist has zero marginal osts i.e., d = ). If there are a ontinuum of markets, uniformly distributed between L and H, then banning prie disrimination aross the markets lowers welfare if demand is log-onave σ < ) provided k H / L is suffiiently large. In the proof of the proposition we show that the monopolist will set the prie suh that goods with lower than some utoff level x L will be dropped, provided that the dispersion of demand aross goods is large enough i.e., k > k ). The threshold k depends only on σ, and the utoff value x is a onstant fration of the dispersion k. 6

17 3.2.3 Exponential and log-onvex demand For ases with log-onvex demand, we present the results graphially. The ase orresponding to Figure 2, with exponential demand, is given in Figure 5. Figure 5: Monopoly pries with exponential demand ontinuum ase) prie T uniform prie prie for good H prie for good L d= welfare gain: WPD)-WU) d= d=.5.4 d=.5 prie T welfare gain: WPD)-WU) d=..4 d=. prie T welfare gain: WPD)-WU) More generally, Figure 6 shows that provided k is large enough when demand is logonave, and for all k when demand is exponential or log-onvex, welfare is higher under prie disrimination and so when a platform an use ad valorem fees). In the log-onave ase with d =, the ritial level of k for whih welfare is higher under prie disrimination than under uniform priing is k > 5, and the ritial value of k delines as d inreases, so the dispersion of demand aross goods does not have to be very high for prie disrimination to generate higher welfare than uniform priing. In the exponential or log-onvex ase, Figure 6 suggests welfare is always higher under prie disrimination regardless of the level of k and d. 7

18 Figure 6: Welfare omparison for a ontinuum of goods 4 Banning ad valorem fees: alibrated model In pratie, platforms deal with thousands of different goods, and will not be uniformly distributed aross these goods. For example, there are typially many more transations using Visa debit ards in the $-$5 range than in the $6-$ range. The same is true for goods traded on Amazon s marketplae, where the sales distribution is highly skewed. The welfare effets of prie disrimination in suh settings are largely unexplored in the literature. In this setion, we wish to work out the welfare effets of banning ad valorem fees i.e., banning prie disrimination in the equivalent setting where different aptures distint markets with observably different demand) using realisti sales distributions. The data we use are from Visa signature debit ard transations and DVD listings on Amazon s marketplae. In both ases, the platforms have adopted affine fee shedules, as shown in Table. 8

19 4. Methodology First, we illustrate how our theoretial model an be alibrated to real world data. Rather than assume there is a unit mass of buyers for eah good, we allow for the possibility that for eah good there an be a different number of potential buyers so that even goods with the same an sell different amounts. The number of transations for a distint good i with ost is Q i, = n i, Q and the platform makes a profit Π i, = n i, Π, where Q and Π are the quantity and profit expressions from Setion 2 based on a unit mass of potential buyers, and n i, is the number of potential buyers for good i with ost. We denote the number of distint goods with ost as n. A platform s total profit is therefore Π = C n i= n i, Π. 2) Given 2), all our previous analysis holds exept that we need to hange the mass g to n i= n i,. This follows beause, as shown in Wang and Wright forthoming), the optimal platform fee does not depend on g. Aordingly, an affine fee shedule suh as those in Table ) an be rationalized by the platform faing generalized Pareto demands, and the profit maximizing platform fee is still given by 6). Given an observed platform fee shedule T p ) = a + a p, 6) implies that we an uniquely identify the values of λ and d for a given value of σ. Our welfare omparisons will then onsider all the possible values of σ. 9 Note that λd + λ 2 σ) = a, + λ 2 σ) = a, so λ = /a )/2 σ) and d = a /λ + a 2 σ). Given the value of λ, and the observed prie p and quantity Q i, for eah good traded on the platform, we an then identify the number of potential buyers n i, for eah good. Substituting T p ) = a + a p into Q i, = n i, + λ σ ) T ) σ, 9 Alternatively, we ould pin down d and σ for a given value of λ, and then ondut welfare omparisons based on different values of λ. However, given that σ determines the shape of demand funtions, deriving welfare results based on different values of σ is more informative. It allows us to ompare our results with those in the literature that typially assume speifi demand funtions e.g., linear demand). 9

20 we derive n i, = Q i, [ ] + λσ )a +a p ) σ a )p a. 3) With n i, determined, we an use the weight n i= n i, in plae of g when alulating profit and welfare. Our theory also allows us to identify the lower bound of σ from the data. Reall when σ <, the generalized Pareto demand has finite support on [, + ]. This means λ σ) that if we observe any good with positive sales, its prie has to satisfy p) < + λ σ). Sine p) = + T p ), this requires that T p ) < λ σ). Substituting in that T p ) = a + a p, = a )p a and the expression for λ from above, the equivalent inequality an be written as σ > + a a a )p a. Thus, the minimum prie we observe in the data pins down the minimum value of σ that our model permits. 4.2 Visa debit ards We use data from the Diary of Consumer Payment Choie DCPC), onduted in Otober 22 by the Boston, Rihmond, and San Franiso Federal Reserve Banks to alibrate the model. The DCPC ollets onsumer payments data on the dollar value of purhases, the payment instrument used, and the ategory of expense. A national representative sample of 2,468 U.S. respondents were seleted, who eah reorded all their payments over a three-day period. Sine respondents were spread over the entire month of Otober 22, this sampling methodology provides reasonable probability estimates of all onsumers. For transations made with payment ards, respondents were asked to report the dollar amount, the exat ard type and the ard network s brand name. Based on the DCPC data, we identify,48 Visa signature debit ard transations in four distint market ategories, namely retail, restaurant, gas station, and small tiket, to form our empirial transation distributions. For eah market ategory, we use the interhange fee shedule published by Visa shown in Table ) to infer its platform priing. Given merhant aquirers are highly ompetitive in the U.S. market, the interhange fee shedules posted by Visa mirror very losely the atual fee shedules passed onto sellers. Figure 7 plots the raw density distribution of transation pries in eah market ategory. The distributions are quite skewed. Based on the raw transation distributions and the fee shedules, we then numerially alulate perentage welfare gains under the observed affine fee shedule and the ounterfatual optimal uniform per-transation fee for eah possible value of σ assuming the underlying demand takes the generalized Pareto form. The results are presented in Figure 8. Figure 8 shows that in three out of the four markets, welfare is onsistently higher when 2

21 density Retail 2 3 transation prie density Restaurant transation prie density Gas Station transation prie density Small Tiket transation prie Figure 7: Visa Signature Debit Card Transation Distributions ad valorem fees are allowed for any possible value of σ. The only exeption is for the Restaurant market for lower values of σ. However, this result is driven by a single outlier whih has an unusually large transation prie, as an be seen from Figure 7. If that outlier is removed from the sample, welfare would also be onsistently higher in the Restaurant market when ad valorem fees are allowed. Note that the perentage welfare gain or loss) from allowing ad valorem fees is alulated by assuming that the platform only inurs a marginal ost d but not an overall fixed ost K. The perentage hange in welfare would be even higher one a positive level of K is taken into aount. Moreover, the absolute level of welfare hange is likely to be substantial given the size of the payment ard industry. In 2, debit ards were used in 49 billion transations for a total value of $.8 trillion in the U.S. market, in whih 6 perent were signature debit ard transations, with Visa s share of these being about 75 perent. Note that the implied value of d from our alibrated model varies from zero in the limit as σ tends to its highest allowed value i.e., 2) up to 6 ents as σ tends to its lowest allowed value as determined by the lowest observed prie. This is onsistent with the Federal Reserve s study based on omprehensive ost surveys of debit ard issuers, as mandated by the Durbin Amendment to the Dodd-Frank At. Aording to the study, most issuers inur a marginal ost no more than 2 ents per transation See Federal Reserve Board, 2). It is easy to see for any W > W 2, W K)/W 2 K) is an inreasing funtion of K. 2

22 6 Retail 4 Restaurant 5 3 welfare gain %) welfare gain %) Gas Station 3 Small Tiket welfare gain %) 2.5 welfare gain %) Figure 8: Visa Signature Debit Cards: Welfare Gain from Ad valorem Fees 4.3 Amazon marketplae We also alibrate our model using data from Amazon s marketplae. We fous on DVDs given that it is a well defined market ategory for whih we an be sure all the goods identified are subjet to the same fee shedule, and also sine we an ollet onsistent sales ranks for this ategory. Using a web robot, we olleted data on every DVD that was listed under Movies & TV on Amazon s marketplae in January 24. We seleted New under Condition and de-seleted the Out of Stok option, and ended up with a total of 295,7 distint items. The data olleted inlude the title, unique ASIN number identifying the DVD, the prie, and sales rank of eah DVD. 2 Given shipping fees are often not inluded in the listed prie, we also separately olleted data on only those items where the listed prie inluded free shipping, resulting in a sample with 9,28 distint items. Sine some DVDs are listed with extreme pries, we restrit our sample to DVDs selling for under $,, whih inludes around 99% of the items olleted. For robustness, we also tried alternative prie limits, inluding $5 and $2,, and the results are very similar. 3 2 The prie is taken as the prie posted at Amazon s marketplae for the DVD. It is the prie a buyer will fae when they add the item to their art and go to the hekout i.e., the buy-box prie. 3 A onern with extreme DVD pries is that the pries listed are unlikely to reflet the pries at whih transations atually take plae. For instane, some sellers post extreme pries as plaeholders to avoid a temporary delisting when they are out of stok or away for vaation. Others may be errors in the seller s entry of its pries. 22

23 Given we do not diretly observe the sales of eah DVD, we use a power law to infer it from the sales rank, so Q i, = ar φ i,, where Q i, is the estimated sales of an item and R i, is the orresponding sales rank. 4 The parameter a does not affet our results, so we normalize it by setting a =. We try different values for the parameter φ, inluding φ = where sales rank is assumed to be irrelevant), φ = Zipf s law) and φ =.7 whih is the number suggested by Smith and Telang 29) in an experimental study on DVD sales on Amazon, although it implies very little weight is plaed on items with sales ranks below the top ten). density All DVDs transation prie density DVDs with Free Shipping transation prie density All DVDs DVDs with Free Shipping Figure 9: DVD Sales Distribution transation prie density Figure 9 plots the density of items listed at eah prie whih orresponds to the sales distribution under the assumption that φ =. The distributions are highly skewed with a majority of items listed at pries below $5. With φ = or φ =.7, the distributions beome even more skewed. Based on eah of the sales distributions and the fee shedule from Table, we numerially transation prie alulate perentage welfare gains under the observed affine fee shedule and the ounterfatual optimal uniform per-transation fee for eah possible value of σ assuming the underlying demand takes the generalized Pareto form. The results are presented in Figure, whih shows that one sales ranks are taken into aount, welfare is onsistently higher under an affine fee shedule than under a uniform per-transation fee. 5 4 Power law distributions are widely used to desribe rank data, with the well-known Zipf s law being a speial ase. See Chevalier and Goolsbee 23) for detailed disussions as well as an appliation to online sales data. 5 In the Amazon ase, the implied value of d from our alibrated model varies from zero up to $.65 as σ varies from its highest possible value to its lowest. 23

24 All DVDs =) 3 Free Shipping DVDs =) welfare gain %) 5 welfare gain %) All DVDs =) Free Shipping DVDs =) welfare gain %) welfare gain %) All DVDs =.7) Free Shipping DVDs =.7).5 welfare gain %) welfare gain %) Figure : DVDs: Welfare Gain from Ad valorem Fees 5 Conluding remarks Many platforms that failitate transations between buyers and sellers harge ad valorem fees in whih fees depend on the transation prie set by sellers. Given these platforms do not inur signifiant osts that vary with transation pries, their use of ad valorem fees has raised ontroversies about the effiieny of this pratie. For poliymakers who would want to align platform fees with osts but are onerned about diretly regulating fee levels, it seems natural to onsider regulating fee strutures, suh as banning platforms from using ad valorem fees. However, we have shown that suh regulation tends to have negative welfare outomes, inluding when we alibrate our model to data on sales of DVDs on Amazon s marketplae and data for Visa signature debit ard transations. Therefore, aution should be taken when poliymakers onsider this option. A similar result would also apply to a government that wanted to maximize tax revenue welfare would be higher when it does so using an ad valorem tax. The key feature that drives these results is that when a market involves many different goods that vary widely in their osts and values, ad valorem fees and taxes represent an effiient form of prie disrimination. In omparison, uniform fees or taxes ould adversely affet low-ost low-value goods so that the total welfare is redued. There are several avenues for future researh. First, as Shy and Wang 2) showed, another reason why banning ad valorem fees ould lower welfare is that ad valorem fees help 24

25 to mitigate double marginalization. This suggests that allowing for imperfet ompetition between sellers would add to the welfare loss of banning ad valorem fees. Using our demand assumptions, one ould analyze a ban on ad valorem fees in their environment to evaluate the overall effets. However, in this ase, affine fee shedules will not neessarily maximize platform profits, although they may still do so approximately. Moreover, one would no longer be able to rely on the duality result whih allowed us to draw on the existing literature on third-degree prie disrimination. Thus, ombining these two mehanisms in a single model would be a hallenging exerise for future researh. Seond, one ould onsider demand funtions outside the generalized Pareto lass. The reason that we foused on the generalized Pareto demand is beause it overs a broad family of ommonly used demand funtions that rationalize platforms use of ad valorem fees. In reality, however, ad valorem fees may be used as an approximation to more ompliated optimal fee shedules. Therefore, it might be useful to onsider demand speifiations outside the generalized Pareto lass and ondut robustness heks for our results. Finally, it might be interesting to onsider alternative regulations on platform fees, suh as allowing for ad valorem fees but with a ap for highvalue transations. Suh a regulation may ahieve better welfare outomes, even though there would be the additional ompliation of hoosing the appropriate level of the ap. Referenes [] Aguirre, Iñaki 26). Monopolisti Prie Disrimination and Output Effet under Conditions of Constant Elastiity Demand. Eonomis Bulletin, 423), 6. [2] Aguirre, Iñaki 28). Output and Misalloation Effets in Monopolisti Third-Degree Prie Disrimination. Eonomis Bulletin, 4),. [3] Aguirre, Iñaki, Simon Cowan and John Vikers 2). Monopoly Prie Disrimination and Demand Curvature. Amerian Eonomi Review,, [4] Brennan, G., and J. M. Buhanan 977). Towards a Tax Constitution for Leviathan. Journal of Publi Eonomis, 83), [5] Brennan, G., and J. M. Buhanan 978). Tax Instruments as Constraints on the Disposition of Publi Revenues. Journal of Publi Eonomis, 93), [6] Bulow, Jeremy and Paul Pfleiderer 983). A Note on the Effet of Cost Changes on Pries. Journal of Politial Eonomy, 9,

Economics 2202 (Section 05) Macroeconomic Theory Practice Problem Set 3 Suggested Solutions Professor Sanjay Chugh Fall 2014

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