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Working Paper Series Ad-valorem platform fees, indirect taxes and efficient price discrimination WP 12-08R Zhu Wang Federal Reserve Bank of Richmond Julian Wright National University of Singapore This paper can be downloaded without charge from: http://www.richmondfed.org/publications/ Ad-valorem platform fees, indirect taxes and efficient price discrimination∗ Zhu Wang† and Julian Wright‡ November 2014 Working Paper No. 12-08R Abstract This paper explains why platforms such as Amazon and Visa rely predominantly on ad-valorem fees, fees which increase proportionally with transaction prices. It also provides a new explanation for why ad-valorem sales taxes are more desirable than specific taxes. The theory rests on the ability of ad-valorem fees (or taxes) to achieve efficient price discrimination in the face of transactions that differ widely in costs and values. We show how welfare can be higher due to the use of ad-valorem fees, including when we calibrate our model to data from Amazon and Visa. JEL classification: D4, H2, L5 Keywords: Platforms, Taxation, Ramsey pricing ∗ We thank Boyan Jovanovic, Grace Bin Li, llya Segal, Thomas Tregouet, Glen Weyl and participants at various conferences and seminars for helpful comments, and Venkata Kiran Yedugundla and Ye Ziqiu for their excellent research assistance. † Research Department, Federal Reserve Bank of Richmond. Email: zhu.wang@rich.frb.org. The views expressed are solely those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Richmond or the Federal Reserve System. ‡ Department of Economics, National University of Singapore. Email: jwright@nus.edu.sg. 1 1 Introduction Ad-valorem fees and taxes which depend positively on transaction prices are widely used in practice. Platforms that facilitate transactions between buyers and sellers typically charge sellers ad-valorem fees, with fees being a percentage of the transaction price. Well-known examples include online marketplaces (such as Amazon and eBay) and payment card networks (such as Visa and MasterCard). Given these platforms do not incur significant costs that vary with transaction prices, their use of ad-valorem fees has raised controversies about the efficiency of this practice. Ad-valorem sales taxes are also widely used, with their desirability vis-a-vis per-unit (or specific) taxes the subject of a substantial literature. In this paper, we provide a new rationale for the use of ad-valorem fees and taxes. The theory explains why profit-maximizing platforms favor ad-valorem fees even if they do not incur any costs of enabling trades. The same theory also implies that a regulatory authority that maximizes social welfare subject to covering a fixed cost should similarly make use of ad-valorem fees when regulating fees for such platforms, and that a tax authority should prefer ad-valorem sales taxes rather than per-unit sales taxes. The key idea behind our theory is that when a market involves many different goods being traded that differ widely in their unobserved costs and values (which we call their “scale”), then taxing them based on the transaction price represents an efficient form of price discrimination. The situation captures the usual scenario facing platforms, regulators or tax authorities. For example, Amazon and eBay have goods traded that are worth a few dollars and others that are worth hundreds of dollars, within any specific category (e.g., DVDs). In such a setting, per-unit fees and taxes have the problem that they distort the price elasticity of demand across products, since they add proportionally more to the price of a low-cost low-value item compared to a high-cost high-value item, thus reducing the efficiency of revenue extraction. Advalorem fees and taxes that are proportional to the transaction price do not lead to such a distortion and so can ensure the optimal Ramsey pricing. In fact, we show charging ad-valorem fees and taxes allows the platform, regulator or tax authority to achieve the same level of profit or welfare that could be obtained under third-degree price discrimination as if the relevant authority could perfectly observe the cost and valuation for each good traded and set a different optimal fee for each. 2 We then extend the theory to accommodate the fact that many platforms charge sellers a small fixed fee for each transaction in addition to the main ad-valorem component (Table 1 in Section 4 gives some specific examples). To do so, we allow for the fact that platforms typically incur a small marginal cost per transaction. We show that an affine fee schedule (a fixed fee per transaction plus a fee proportional to the transaction price) is optimal if and only if demand for the sellers belongs to the class with constant curvature of inverse demand (which includes linear demand, constant-elasticity, and exponential demand). We use this result to explore policy questions surrounding the use of ad-valorem fees by platforms. In recent years, a sizeable body of literature has developed to evaluate whether fee setting in the payment card industry involves some market failure (see, for instance, Bedre-Defolie and Calvano 2013, Rochet and Tirole 2002, 2011, Schmalensee 2002, Wang 2010, Wright 2003, 2012). Meanwhile, questions have also been raised about ad-valorem fees: Why do payment card networks set interchange fees (in the case of Visa and MasterCard)1 or merchant fees (in the case of American Express) that depend on transaction prices rather than being a fixed amount per transaction? Costs cannot explain the levels of ad-valorem fees set by brand name credit, debit or prepaid cards. Particularly, merchants and policymakers point out that debit and prepaid card transactions do not provide credit or float and bear very small fraud risk; therefore, they do not warrant a percentage-based fee structure. The issue is of concern to policymakers in many countries who have recently tried to align payment card fees with cost measures, including Australia, Canada and the United States.2 A major issue they face when regulating these platforms is whether to allow an ad-valorem fee in the regulated fee structure. Moreover, regulators are interested in whether restricting platforms to only setting fixed per-transaction fees (i.e., banning the advalorem component) could lead to a welfare improvement. We address these policy questions using our model. We do so first by considering a regulated setting in which the regulator seeks to maximize social welfare but allows the platform to recover its costs, including fixed costs, and has to decide whether to use ad-valorem fees to do so. In contrast to the policymakers’ concerns regarding 1 Interchange fees are paid by merchants to card issuers through merchant acquirers whenever merchants accept Visa or MasterCard card payments. 2 E.g., see “Transparency, Balance and Choice: Canada’s Credit Card and Debit Card Systems,” Chapter 5, the Standing Senate Committee on Banking, Trade, and Commerce, Canada, June 2009. 3 the use of ad-valorem fees, we show that for the class of demand functions that rationalizes a platform’s use of affine fee schedules, welfare in this constrained case is always higher when a proportional fee is used to recover costs, in addition to a fixed per-transaction fee. We then consider the question in an unregulated setting, in which the regulator only considers whether to ban the use of ad-valorem fees but otherwise does not restrict the level of fees chosen. The conditions for welfare to increase as a result of banning the use of ad-valorem fees turn out to be the same as those which determine whether banning price discrimination improves welfare within the class of demands considered. This allows us to draw on the substantial literature on monopolistic third-degree price discrimination (e.g., see Aguirre et al., 2010, for a recent analysis). In the setting usually adopted for price discrimination studies, in which there are only two goods or a continuum of uniformly distributed goods, and restricting to the class of demand functions that rationalizes a platform’s use of affine fee schedules, we find welfare tends to be higher when platforms are allowed to use ad-valorem fees provided the prices of goods are sufficiently dispersed. In practice, however, the distribution of prices and sales of goods traded is likely to be highly skewed. We therefore use information on the actual distribution of prices and sales measures for two different platforms (Amazon’s and Visa’s) to calibrate our model. We find, in most cases, welfare would be harmed if ad-valorem fees were banned. Policymakers should therefore be cautious about introducing a ban on ad-valorem fees. Our paper is distinct from some recent studies that consider the use of proportional fees by platforms. Shy and Wang (2011) focus on the market power of the platform and sellers, and argue that proportional fees are preferred over fixed per-transaction fees since they help mitigate double marginalization. Several papers (e.g., Foros et. al. 2014, Gaudin and White 2014 and Johnson 2014) have explored the optimality of a retailer (in our context, a platform) that lets suppliers (in our context, sellers) set final prices and pay proportional fees on sales (known as the agency model). Again, they assume suppliers (i.e., sellers) are imperfect competitors but compare the agency model to a different alternative—the traditional wholesale model of retailing. Loertscher and Niedermayer (2012) consider a mechanism design approach in an independent private values setup with privately informed buyers and sellers, in which an intermediary’s optimal fees converge to linear fees as markets become increasingly 4 thin. Muthers and Wismer (2012) show that if a platform can commit to proportional fees, this can reduce a hold-up problem that arises from the platform wanting to compete with sellers after they have incurred costs to enter the platform. All these studies propose different rationales for why platforms may choose proportional fees, but they all assume the platform only deals with one type of good (not goods that come in different scales, with different unobserved costs and valuations). Therefore, none of them consider the price discrimination perspective, which we believe is a primary reason for the widespread use of ad-valorem fees. Moreover, our theory provides a natural explanation for why platforms often use affine fee schedules that include both a proportional and a fixed component, which is distinct from these studies that focus exclusively on proportional fees. Our paper also offers a new perspective on the use of ad-valorem sales taxes. One of the oldest literatures in the formal study of public finance is the choice between a perunit (or specific) tax and an ad-valorem tax (a tax proportional to transaction prices). The main focus has been on the role of different forms of imperfect competition among sellers (see Keen, 1998 for a survey).3 In contrast, our paper shuts down such channels by assuming sellers are identical price competitors. Rather, we focus on a previously overlooked aspect that ad-valorem taxes allow the tax authority to efficiently price discriminate across goods with different costs and valuations based on Ramsey pricing principles.4 Thus, our paper contributes to the voluminous literature on indirect taxation, providing a new theory for why ad-valorem sales taxes should be preferred. The remainder of this paper is structured as follows. Section 2 introduces the model. Section 3 provides general results for the optimality of ad-valorem taxes and fees in the absence of per-transaction costs. Section 4 extends the model for platforms that incur a positive marginal cost per transaction. It characterizes the class of demand functions for which affine fee schedules maximize platform profits. Given this class of demands, it considers the Ramsey problem of regulating platform fees to recover costs (including a fixed cost), and whether the use of ad-valorem fees is 3 Important contributions include Suits and Musgrave (1953), Delipalla and Keen (1992), Skeath and Trandel (1994), Anderson et al. (2001) and Hamilton (2009). 4 In reality, we often observe per-unit taxes on alcohol, cigarettes and gasoline. One explanation is that ad-valorem taxes on these goods may promote the consumption of lower quality products that could cause worse side effects. Our paper abstracts from this issue. 5 part of the optimal solution. Section 5 investigates the welfare implications of a ban on ad-valorem fees in an otherwise unregulated environment and calibrates our model to data from Amazon and Visa. Finally, Section 6 offers some concluding remarks. 2 The model We consider an environment where multiple goods are traded in a market. For each good traded, a unit mass of buyers want to purchase one unit of the good. We consider a regulator of the market (it could be a tax authority or a platform operator that facilitates the trades) that wishes to collect some revenue from traders in the market by levying taxes (in the case of a platform, we will refer to these as fees). The market could consist of a narrow category of goods (e.g., DVDs) or a wide variety of different goods that the regulator cannot easily distinguish (e.g., electronics). Within such a market, there are multiple identical sellers of each good who engage in Bertrand competition.5 Different goods within the market are indexed by c, which can be thought of as a scale parameter, so that different goods can be thought of as having similar demands except that they come in different scales. In particular, the per-unit cost of good c to sellers (which is known to all buyers and sellers of the good) is normalized to c and the value of the good to a buyer drawing the benefit parameter b ≥ 0 is c (1 + b), so the scale parameter increases the cost and the buyer’s valuation proportionally.6 We assume the lowest and highest values of c are denoted cL and cH respectively, with cH > cL > 0. We also assume 1 + b is distributed according to some smooth (i.e., twice continuously differentiable) and strictly increasing distribution function F on 1, 1 + b , where b > 0. (We do not require that b is finite.) Only buyers know their own b, while F is public information. Let f = F 0 be the density and h = f / (1 − F ) the hazard rate of F . This setup captures the idea that for a given market that can be identified by the regulator, the main difference across the goods traded is their scale (i.e., some goods are worth a little and some a lot). In comparison to the wide range of scales of goods 5 As shown by Shy and Wang (2011), imperfect competition among sellers provides an additional rationale for ad-valorem fees, to mitigate double marginalization. We shut down this channel so we can isolate the role ad-valorem fees play as a price discrimination device. 6 The assumption b ≥ 0 is an innocuous normalization. In fact, we do not have to consider buyers with b < 0 given their valuation for a product is less than its cost. 6 traded, potential differences in the shapes of demand functions across the different goods traded are not likely to be of first-order importance. The assumption that buyers’ values for a good can be scaled by c is broadly consistent with a key empirical finding of Einav et al. (2013). They examined quasi-experimental observations from a large number of auctions of different goods on eBay and found that the distribution of buyer valuations for a good is proportional to the normal transaction price of the good (taken as the average price across posted-price transactions of the particular good). Since we will show that the transaction price predicted by our model is proportional to c, this finding implies an inverse demand function that is proportional to c, which is the essential property of our model as shown below.7 We assume the regulator cannot directly observe which good is being traded within a given market (i.e., the scale of the good sold), but only the transaction price of each trade. As a result, taxes can only condition on the price of a transaction (i.e., not on the actual cost of sellers or valuation of buyers). Since sellers of a particular good are identical and compete in price, competition between sellers for a good c will result in an equilibrium in which sellers charge buyers a common price pc . In general, the tax schedule can be written as T (pc ) for a given price of the transaction pc , which can be decomposed into a buyer tax schedule T b (pc ) and seller tax schedule T s (pc ). Each tax schedule can consist of two components, a fixed per-transaction component (i.e., a specific tax) which is independent of pc and a variable component which varies with pc (i.e., the ad-valorem component). When the ad-valorem component is linear in pc , we call it a proportional tax.8 Given that identical sellers compete for buyers, any tax charged to sellers will be passed through to buyers. The final price faced by buyers will reflect any taxes, and the buyer treats these the same whether he faces them directly or through sellers. Due to this standard result on the irrelevance of the incidence of taxes across the two sides, we can assume without loss of generality that only the seller side is charged. With this normalization, we write the sellers’ tax schedule as simply T (pc ). 7 In Appendix A we provide an alternative setting which generates the same demand specification in the context that platforms reduce trading frictions. 8 We assume all buyers and sellers have access to the market, so in the case of a platform, we abstract from joining fees. We could allow for entry to a platform by assuming buyers know their draw of b before deciding whether to join the platform. Given that each buyer wants to purchase exactly one unit of a good and assuming no costs to the platform of handling additional buyers or sellers, platform joining fees would not play any role. 7 The number of transactions Qc for a good c is the measure of buyers who obtain non-negative surplus from buying the good, Pr (c (1 + b) − pc ≥ 0). Therefore, we can write the demand function for good c as Qc (pc ) = Q p c c ≡1−F p c c . (1) The corresponding inverse demand function for good c is pc (Qc ) = cF −1 (1 − Qc ), which note is proportional to c. Finally, assume c takes on a finite number of distinct values in [cL , cH ], with the set of all such values being denoted C. The distribution of c on C is denoted G and the associated probability (or mass) for each distinct value of c is denoted gc , with P c∈C gc = 1. 3 The optimality of proportional taxes and fees In this section, we rationalize a simple form of ad-valorem taxes and fees, which are proportional. We show they can be used to maximize revenue and welfare (subject to a minimum revenue constraint) when there is no variable cost associated with the regulator. 3.1 Revenue maximization Consider first the problem facing a regulator that wants to maximize its revenue from taxing the trades in the market. In case the regulator is a profit-maximizing platform operator, this captures the situation in which the variable cost of handling transactions on the platform is zero. If the regulator sets the tax schedule T (pc ) to sellers, its revenue is Rc = T (pc ) Qc (pc ) for good c. The total revenue across all P goods in the market is R = c∈C gc Rc . The regulator’s problem is to choose T (pc ) to maximize R. Consider a hypothetical full-information setting from the perspective of the regulator. This is the case that the regulator can observe c for each good and set a different tax for each good. Denote the tax charged for good c as Tc . Due to Bertrand competition, the price for good c will be pc = c + Tc . Therefore the regulator will set Tc to 8 maximize revenue from good c, Tc Rc (Tc ) = Tc Q 1 + . c This is a monopoly pricing problem. We assume Rc (Tc ) is strictly concave in Tc and there exists a sufficiently high Tc < cb such that Rc0 (Tc ) < 0. This ensures the standard first-order condition characterizes the monopoly price. Denote eTc as the demand elasticity with respect to Tc . Then, the optimal level of Tc , denoted Tc∗ , solves 0 eTc∗ = − Tc∗ c Tc∗ Q 1 + = 1. c Q 1 + Tc∗ (2) c Note (2) will be true for all c ∈ C if Tc∗ = ωc for some constant ω > 0 which solves −ω Q0 (1 + ω) = 1. Q (1 + ω) (3) Sellers of good c have cost c and pay the fee Tc , so Bertrand competition implies pc = Tc + c for each good. Combining this with Tc = ωc, the regulator’s full-information optimal solution can be achieved using the linear ad-valorem (or proportional) tax schedule ω pc , (4) T (pc ) = 1+ω with ω solving (3). Implementation is possible given the slope of the tax schedule is strictly between 0 and 1, so it never pays for a seller to set a lower price in order to pay a lower tax. The findings are formalized in the following proposition. Appendix B contains the remainder of the proof, together with the proofs of the other propositions in the paper. Proposition 1 (Revenue maximization): The proportional tax schedule given by (4) maximizes the revenue that a regulator can extract from a market, where the value of ω solves (3). The result rationalizes why platforms set proportional fee schedules. This allows 9 platforms to set the optimal fee for each different good traded, given it is proportional to the scale of the good. In the absence of any variable costs, this ensures the demand elasticity with regard to the platform fee is equated to unity for each good traded, as the classic Ramsey pricing principle implies, so that revenue is maximized. In the case of a tax authority, this would maximize the tax revenue collected. To better understand why proportional taxes are efficient, consider what would happen under the alternative in which all goods in the market face a fixed pertransaction tax. This would imply the effective rate of taxation (i.e., Tc /c in our setting) would be higher for low-cost goods than for high-cost goods. Given the usual condition that the price elasticity of demand for a given good increases in the price of the good (known as Marshall’s second law of demand), this would translate to higher price elasticities for low-cost goods than for high-cost goods. Compared to this situation, by shifting taxes from low-cost low-value goods to high-cost high-value goods, more revenue would be raised from goods with low price elasticity of demand and less from goods with high price elasticity of demand, thereby improving the efficiency of revenue extraction. This is how proportional tax achieves this result, which is in line with classic Ramsey pricing principles. This logic can also be illustrated by noting that a fixed per-transaction tax would likely result in some very low-value goods not being traded at all due to the high burden a fixed per-transaction tax would imply for such goods. A proportional tax schedule can instead prevent this. All transactions can contribute some tax revenue, with high value transactions contributing proportionally more. The same rationale applies to setting a regulated fee schedule for a platform that has to cover some fixed fees to break even, and likewise to a tax authority that has to raise a fixed amount of tax revenue, which are the cases considered next. To derive the optimality of proportional taxes and fees, a key assumption we made is that the willingness to pay of all types of buyers is assumed to grow proportionally with the cost of the good c. While we do not think this property holds exactly in reality, it serves as a reasonable approximation for the following reasons. First, it captures that consumers’ willingness to pay increases with c. It is intuitive to think that a one dollar increase in taxes or fees will be very important for a $5 item but not very important for a $500 item, and this will be true regardless of any particular demand specification we assume. In fact, Einav et. al (2013) provide some direct 10 evidence supporting that this relationship is indeed approximately linear, as noted above. Second, we only require the assumption holds over some particular market segment, not across all market segments. We typically observe platform fees that vary across different market segments and this is consistent with our analysis applying within each of these market segments but not across different market segments which may have inverse demands with very different shapes. Finally, we show in Appendix A that our results are robust to random demand heterogeneity, in which case we allow each product in the same market segment to have demand that is a bit different (either due to an additive and/or multiplicative term) but not in a systematic way. 3.2 Welfare maximization The Ramsey problem in which a social planner chooses a tax schedule to maximize welfare subject to raising a minimum level K of revenue can be solved in a similar way. This covers the case of a tax authority trying to optimally raise a certain amount of tax revenue from the market. It also covers the case of a regulatory authority that regulates the fee schedule charged by a platform so as to maximize welfare subject to the platform’s break-even condition in case the platform faces a fixed cost of operation of K. Consider again the full-information setting in which the social planner can observe c for each good and set a tax accordingly. The welfare generated by the trade of good c is Z ∞ Tc t dt + Tc Q 1 + . Wc = Q 1+ c c Tc Total welfare is maximized with a set of taxes Tc , one for each good, chosen to P maximize W = c∈C gc Wc − K, subject to the constraint that R ≥ K. Note if we denote the maximum possible revenue as RM , we need to assume RM > K so it is possible to raise at least K in revenue. The classic Ramsey result implies that the first-order condition is the same as for the revenue maximization problem in the previous section except that the tax is set so that the demand elasticity with regard to Tc will be lower in magnitude, reflecting that less than 100% of the weight in the social planner’s objective is put on revenue 11 recovery. In other words, (2) can be replaced by eTcr r Q0 1 + Tcc η = =− , r c Q 1 + Tc 1+η c Tcr (5) where η > 0 is the Lagrangian multiplier on the revenue constraint and Tcr is the social planner’s optimal tax for good c. Note (5) will be true for all c ∈ C if Tcr = ωc, for some constant ω > 0, in which case (5) can be written −ω Q0 (1 + ω) η = . Q (1 + ω) 1+η (6) As with revenue maximization, the proportional tax schedule (4) can implement the full-information solution for the social planner, with the only difference being that the value of ω is now lower to reflect that the right-hand-side of (6) is less than unity. We formalize these results with the following proposition. Proposition 2 (Welfare maximization): The proportional tax schedule given by (4) maximizes welfare subject to raising the revenue K, where the value of ω solves (6). Accordingly, the tax rate ω/(1 + ω) chosen to maximize (constrained) welfare is lower than that chosen to maximize revenue. The proposition provides a new argument in the long-standing debate between specific versus ad-valorem sales taxes. We show that ad-valorem taxes enable the tax authority to charge different specific taxes to different goods based on the value of transactions, which minimizes distortions in our setting. 4 The optimality of affine fee schedules Propositions 1 and 2 above help explain the prevalence of the use of proportional taxes and fees by tax authorities, regulatory authorities and platforms alike. However, in the case of platforms, we often see the use of a fixed per-transaction fee in addition to the proportional component, so that the fee schedule is actually an affine function.9 9 We follow the maths literature in distinguishing affine functions, which have a non-zero intercept term, from linear functions which do not. 12 For example, Table 1 shows some fee schedules that Amazon and Visa currently charge to sellers for using their platform services.10 Table 1. Platform Fee Schedules Amazon Visa DVD 15% +$1.35 Gas Station 0.75% +$0.17 Book 15% +$1.35 Retail Store 0.95% +$0.20 Video Game 15% +$1.35 Restaurant 1.19% +$0.10 Game Console 8% +$1.35 Small Ticket 1.60% +$0.05 In this section we will show how such fee schedules are optimal for a platform once we allow the platform to incur some positive marginal costs of dealing with transactions. We will show the optimality only holds for a restricted class of demand specifications, albeit a reasonably general one. This finding will be particularly useful for our analysis on the welfare effects of regulating or banning the use of ad-valorem fees (which we do in Section 4.2 and Section 5), as they pin down the functional form of the distributions of valuations under the assumption that the platform is employing the optimal fee schedule. 4.1 A profit maximizing platform and affine fees Suppose the platform incurs a cost d > 0 for handling each trade. Such costs likely arise in the examples provided in Table 1 given there is a probability any given transaction will require an intervention by the platform, such as in cases where disputes arise. In the case of Visa, card issuers also incur incremental costs for card processing, clearance and settlement. According to the recent Federal Reserve estimates, the average incremental cost can be as high as 21 cents for a typical debit card transaction 10 Visa fees listed in Table 1 are the interchange fees that Visa sets for its debit card services in the U.S. market for transactions authorized by a cardholder’s signature (i.e., for signature debit cards). Given the U.S. merchant acquirer market is highly competitive, these interchange fees are typically directly passed through to merchants who accept Visa debit cards as payment means. Another card network, MasterCard, has an almost identical fee schedule as Visa. Since October 1, 2011, the Federal Reserve has regulated the U.S. debit card market by setting an interchange fee cap at $0.21 plus 5 basis points for each transaction. However, the regulation only covers card issuers with combined assets greater than $10 billion. In the meantime, Visa and MasterCard continue to use their original affine fee schedules for debit card transactions involving non-covered issuers. 13 in the U.S. market.11 We assume that b > d/cL , so it is efficient for some buyers to trade with sellers, even for the good with the lowest scale. The starting point, as before, is to characterize when the platform’s optimal fees are an increasing affine function of c in the hypothetical case in which the platform can observe c for each good and set a separate fee accordingly. We will then show the same conditions characterize when the platform’s optimal fee schedule is an increasing affine function of the transaction price when the platform cannot directly observe c for each good traded. Given c takes on only a finite number of values, there may be multiple fee schedules that can implement the optimal outcome for the platform. If this is the case, we assume that the platform will choose the fee schedule from among these that remains optimal for as many other values of c between cL and cH (i.e., which are not in the support of G). Obviously the platform can only do better doing so, in case c actually takes on such values. The platform makes a profit which is equal to Πc = (T (pc ) − d) Qc (pc ) for good c. The platform chooses T (pc ) to maximize Π= X gc Πc . (7) c∈C We want to know under what conditions it will choose T (pc ) to be an affine function. Consider again the full-information setting in which the platform can observe c for each good and set a fee accordingly. The first-order condition for optimality can be written as Tc∗ 0 Tc∗ T∗ Q 1 + c = , (8) eTc∗ = − c c Q 1 + Tc∗ (Tc∗ − d) c which requires that (Tc∗ Tc∗ Q 1 + c − d) . =− Tc∗ c 0 Q 1+ c 11 (9) The Durbin Amendment to the Dodd-Frank Act requires the Federal Reserve to regulate debit card interchange fees in the U.S. market based on issuers’ incremental costs on a specific transaction. While the costs were found to vary with issuers’ size and type, the Federal Reserve determined that 21 cents serve as a reasonable cap. In addition, a five basis point adjustment is allowed to cover issuers’ potential fraud losses and an additional 1 cent per transaction to cover fraud prevention costs if the issuer is eligible (See Federal Reserve Board (2011)). 14 Since the right-hand-side of (9) is the inverse hazard rate for F evaluated at 1 + Tc∗ /c, we can infer a unique family of distributions and its corresponding demand functions that justify a platform’s optimal fee schedule being an increasing affine function of the scale (as well as the price) of each good. The findings are established formally in the following proposition. Proposition 3 (Profit maximization) Suppose d > 0. The platform’s optimal fee schedule is an increasing affine function of the price of each good with slope less than unity, that is, T (pc ) = pc λd + , 1 + λ (2 − σ) 1 + λ (2 − σ) (10) where λ > 0 and σ < 2 if and only if one of the following equivalent conditions hold: (i) the distribution of buyers’ benefits F has an affine inverse hazard rate (including the special case the inverse hazard rate is constant) with the condition h0 + h2 > 0 (i.e., the hazard rate h never decreases too fast); (ii) the distribution of buyers’ benefits F is the generalized Pareto distribution 1 F (x) = 1 − (1 + λ (σ − 1) (x − 1)) 1−σ , (11) where λ > 0 is the scale parameter and σ < 2 is the shape parameter; (iii) the corresponding demand functions for sellers on the platform are defined by the class of demands with constant curvature of inverse demand Qc (pc ) = 1 − F p c c = λ (σ − 1) (pc − c) 1+ c 1 1−σ . (12) Proposition 3 focuses on the platform’s profit maximizing fee schedule. It characterizes the class of demand specifications that rationalizes a platform’s optimal fee schedule being an increasing affine function of the price of the good. For the demand faced by sellers, this turns out to be from the broad class of demand functions that has constant curvature of inverse demand defined by (12), where the curvature of inverse demand σ is the elasticity of the slope of the inverse demand with respect to quantity.12 As shown in Proposition 3, this class of demands can be derived from the 12 See Aguirre et al. (2010) for a discussion of demand functions that have constant curvature of 15 condition that buyers’ valuation follows a generalized Pareto distribution, so we will refer to it as the generalized Pareto demand in our following analysis.13 Proposition 3 suggests that the proportional component of the optimal platform fee hinges on the constant pass-through of c, which is a demand shifter. Also, consistent with Bulow and Pfleiderer (1983), we can show generalized Pareto demand functions feature constant pass-through of cost, which is d in our setting. Both results can be verified by rewriting (10) as Tc∗ = λd + c , λ (2 − σ) (13) 1 1 which implies a pass-through rate of λ(2−σ) for c and a pass-through rate of (2−σ) for d. For the class of generalized Pareto distributions and implied demand functions, σ is the key parameter. When σ < 1, the support of F is [1, 1 + 1/λ (1 − σ)] and it has increasing hazard. Accordingly, the implied demand functions Qc (pc ) are log-concave and include the linear demand function (σ = 0) as a special case. Alternatively, when 1 < σ < 2, the support of F is [1, ∞) and it has decreasing hazard. The implied demand functions are log-convex and include the constant elasticity demand function (σ = 1 + 1/λ) as a special case. When σ = 1, F captures the left-truncated exponential distribution F (x) = 1 − e−λ(x−1) on the support [1, ∞), with a constant λ(pc −c) hazard rate λ. This implies the exponential (or log-linear) demand Qc (pc ) = e− c . In our following analysis, we will consider the entire class of the generalized Pareto demand functions for the full range of σ < 2. Note the platform’s optimal fee has a fixed per-transaction component only if there is a positive cost to the platform of handling each transaction. For cases where we observe platforms using a fixed per-transaction component in their affine fee schedules, it justifies the focus on the generalized Pareto demand specification. Moreover, these results show that not only does an increasing affine fee schedule allow a platform to achieve its optimal outcome when there is heterogeneity in the goods that are traded (i.e., in c), but as can be seen from (10) it does so without requiring the platform to know the distribution G of goods that are traded. inverse demand. 13 This class of demands is also considered by Bulow and Klemperer (2012), who note some of its other properties. 16 4.2 Ramsey regulation Given that the demand belongs to the generalized Pareto class, we now consider the case where a social planner wants to maximize total welfare conditional on allowing the platform to just recover its costs, including a fixed cost K plus the per-transaction cost d. This is again the Ramsey problem but now allowing for the fact d > 0. We are interested in knowing whether such a planner will also want to use an affine fee schedule to do so. The short answer is yes, and we summarize our findings in the following proposition. Proposition 4 (Ramsey regulation): Given that the demand functions for sellers on the platform belong to the generalized Pareto class and the platform incurs a fixed cost K and per-transaction cost d, a social planner who maximizes welfare subject to recovering the platform’s costs should (i) choose an affine fee schedule, which involves a proportional component plus a fixed per-transaction component; (ii) set the proportional component lower than that chosen by the platform, and also set the fixed per-transaction component lower than the platform when demand is more convex than constant elasticity demand. As in Section 3, the platform’s problem and the social planner’s problem are aligned other than that the optimal demand elasticity with regard to Tc is equal to a lower level under the planner’s problem. That is, comparing with (8), the first-order condition for social planner’s problem is 0 eTcr = − Tcr c Tcr η Tcr Q 1 + = , r r T c Q 1+ c (Tc − d) 1 + η (14) c where η > 0 again is the Lagrangian multiplier on the revenue constraint. We note from (14) that when d > 0, the generalized Pareto demand requires the optimal solution η λd + 1+η c eTcr = , (15) λd(σ − 1) + c η where 1+η equals one in the monopoly platform’s solution. Given that σ − 1 < 1 and η ≤ 1, (15) implies ∂eTcr /∂c < 0. Therefore, for both the platform and the social 1+η 17 planner, fees or taxes should be set so that the demand elasticity with regard to Tc is higher for goods with low costs and values, but lower for goods with high costs and values. This is achieved by setting a fixed per-transaction fee as part of the optimal fee schedule. Given generalized Pareto demand, Proposition 4 implies welfare is always higher when the social planner uses affine fees to maximize welfare subject to a breakeven constraint. This justifies using ad-valorem fees in this fully regulated setting. Since the break-even constraint is binding at the welfare optimum, the platform just covers its costs and welfare only consists of consumer surplus. As a result, the social planner’s choice of affine fee schedule also maximizes consumer surplus subject to the same break-even constraint. 5 Banning ad-valorem fees We have already shown that in a regulated environment, the social planner would want to make use of ad-valorem fees given that sellers on the platform face the generalized Pareto demand. In this section, we consider whether banning the use of ad-valorem fees (i.e. any fees that depend on transaction prices) would harm welfare in an otherwise unregulated environment. Without any restrictions on its pricing, the platform chooses a fee schedule to maximize (7). We know from Section 4 that the resulting optimal fee schedule is an affine function given by (10). If instead the platform cannot condition on transaction prices in any way and so can only choose a single fixed per-transaction fee T across all goods, it will choose T to maximize Π= X gc (T − d) Qc (T ) , (16) c∈C where Qc (T ) = (1 − F (1 + T /c)). Our problem of interest is thus what happens to total welfare in going from the platform’s optimal fee schedule (10), which maximizes (7), to the single fee Tb, which maximizes (16). In other words, is banning ad-valorem fees desirable? The solution to this problem can be found by solving a dual problem, which amounts to the welfare effects of banning third-degree price discrimination. The dual problem involves considering a standard monopolist firm that sells distinct and 18 identifiable goods. It sets Tc for each good to maximize profit Πc = (Tc − d) Qc (Tc ) . (17) In our context, Qc (Tc ) can be interpreted as the demand function that the platform faces, where Tc is the relevant price for each good and c is a parameter which shifts demand across different identifiable goods. The expression of Qc (Tc ) is given in (25) in the proof of Proposition 3, which also belongs to the generalized Pareto class. If third-degree price discrimination is banned, the monopolist will instead choose a uniform price T across all goods to maximize (16). Given our demand specification, the resulting T (denoted Tb) is between Tc∗L and Tc∗H .14 The following duality result follows from the proof of Proposition 3 which shows the equivalence between solving for the optimal Tc for each observed good c and solving for the optimal fee schedule in case the latter is an increasing affine function of the transaction price with slope less than one. Proposition 5 (Duality): Given that the optimal fee schedule is an increasing affine function of the transaction price with slope less than one, the welfare effect of banning a platform from using an ad-valorem fee is identical to the welfare effect of banning third-degree price discrimination in the dual problem in which a monopolist can observe the costs and valuations of different goods and charge different (optimal) prices accordingly. This duality result allows us to draw on the existing literature on the welfare effects of monopolistic third-degree price discrimination. We will also obtain some new results of our own, given that we are interested in a particular class of demand functions and settings where there are many goods of widely different values. 5.1 Two goods Aguirre et al. (2010) focus on the case in which the monopolist sells two goods that will continue to be sold even with uniform pricing, and provide general conditions to Given σ < 2, it can be shown that dΠc /dT < 0 if T > Tc∗ and dΠc /dT > 0 if T < Tc∗ . Since Tc∗ is increasing in c, Tb would never be below Tc∗L (above Tc∗H ) since a higher T (lower T ) can increase profit on every good traded on the platform. 14 19 sign the output and welfare effects of price discrimination under non-linear demand. When the curvature of inverse demand function σ is common across goods, a sufficient condition for total output to increase is that σ is positive and constant for each good. This implies that in our setting with generalized Pareto demand and two goods, price discrimination will always expand output provided σ > 0 (so demand is more convex than linear demand). Aguirre et al. (2010) also show that when σ > 0, price discrimination raises (reduces) welfare when demand is log-convex (log-concave) if the discriminatory prices are not far apart. This applies to our setting, but we need to consider cases where discriminatory prices could be quite large. Moreover, if instead σ = 0, so demand is linear, it is well known (e.g., Schmalensee, 1981) that if both goods are still sold with uniform pricing then price discrimination leaves output unchanged while lowering welfare due to the misallocation of goods given that relative prices do not reflect relative costs under price discrimination. The case with only two goods is not suitable for studying platforms that typically deal with thousands of different goods, but we will briefly comment on it here since it provides some intuition for the results in our calibration and since we can obtain some new analytical results by considering the generalized Pareto demand family. Our analysis shows that in the special case in which d = 0, welfare is higher whenever σ ≥ 1 (i.e., when demand is exponential or log-convex), and also when σ < 1 (i.e., when demand is log-concave) provided the two goods are sufficiently dispersed (i.e., cL and cH are sufficiently far apart).15 The latter result is perhaps not surprising, since we know for log-concave demand there is always a choke price at which demand becomes zero. As a result, provided the goods are sufficiently dispersed, a monopolist that can only set a single price will want to stop selling good cL (i.e., the low-demand good) by setting its profit maximizing price at the monopoly level for good cH (i.e., the high-demand good). This is because setting a low enough price to continue selling the low-demand good will sacrifice too much of the monopolist’s profit from the high-demand good. Allowing it to price discriminate will not only increase the monopolist’s profit but also consumer surplus and social welfare since the low-demand good is then offered for sale. For example, the condition for this to arise with linear demand is k ≡ cH /cL > 3, so that the 15 The analytical results discussed in Sections 5.1 and 5.2 are established formally in an online appendix available at http://profile.nus.edu.sg/fass/ecsjkdw/. 20 dispersion of goods does not have to be very high for the result to hold. Figure 1: Welfare comparison for two goods For the exponential and log-convex case, the logic is actually similar even though there is no choke price at which demand becomes zero. When the two goods are sufficiently dispersed, a monopolist that can only set a single price will set its price very close to the monopoly price for the high-demand good, thereby almost ruling out all sales of the low-demand good, and implying a welfare gain of price discrimination. On the other hand, when the two goods are not dispersed very much, we know already from Aguirre et al. (2010) that when σ > 1, price discrimination also raises welfare. Figure 1 illustrates these results by showing the welfare differences between allowing and not allowing price discrimination for three different values of d and for a range 21 of values of k and σ.16 The dark blue area in the figure indicates a welfare loss due to price discrimination, while the light orange area indicates a welfare gain. In the log-concave demand case (σ < 1), there is a discrete jump between these two areas when k becomes sufficiently large, reflecting that the low-demand good gets shut off from the platform if price discrimination is banned. Figure 1 suggests the analytical findings established with d = 0 in the online appendix extend to d > 0. In the case of log-concave demands, the critical value of k for which welfare is higher under price discrimination than under uniform pricing is k > 3.5 when d = 0. As d increases, Figure 1 shows the critical value of k declines. We therefore do not require goods to be dispersed very much to show that banning price discrimination (i.e., banning ad-valorem fees) can harm welfare. In the exponential or log-convex case, welfare is always higher under price discrimination (i.e., with ad-valorem fees) regardless of the level of k, and Figure 1 suggests this result extends to d > 0. 5.2 Continuum of uniformly distributed goods Figure 2 illustrates the welfare differences between allowing and not allowing price discrimination when there is a continuum of goods with c uniformly distributed between cL and cH , with k ≡ cH /cL . The values of the other parameters remain the same as in Figure 1. Figure 2 again shows that allowing for price discrimination (i.e., an ad-valorem fee) increases welfare, at least when k is high enough. Again, the dispersion of goods does not have to be very high for price discrimination to increase welfare. In the case of log-concave demands, this is true for k > 5 when d = 0. As d increases, the figure shows the critical value of k declines. In the exponential or log-convex case, Figure 2 suggests welfare is always higher under price discrimination regardless of the level of k and d. It is worth noting that the special case with d = 0 and σ = 0, so demand is linear, corresponds exactly to the third-degree price discrimination problem analyzed by Malueg and Schwartz (1994). Reinterpreting the relevant part of their Proposition 16 The values of λ and cL just scale the results. In the figure, we normalize them to 4.5 and 1 respectively. 22 1 to our setting, it implies that provided p p k > 3k0 − 3k0 (4 − k0 ) / k0 − 4 + 3k0 (4 − k0 ) ' 4.651, where k0 ' 3.513 solves 1 + 2 ln k0 = k0 , then welfare is higher under price discrimination. The analytical results we establish in the online appendix show that the positive welfare effects of price discrimination when goods are sufficiently dispersed in this linear demand example extend to log-concave generalized Pareto demands, thereby generalizing Malueg and Schwartz’s earlier findings. Figure 2: Welfare comparison for a continuum of goods 23 5.3 Calibrated model In practice, platforms deal with thousands of different goods, and c will not be uniformly distributed across these goods. For example, there are typically many more transactions using Visa debit cards in the $10-$40 range than in the $60-$90 range. The same is true for goods traded on Amazon’s marketplace, where the sales distribution is highly skewed. The welfare effects of price discrimination in such settings are largely unexplored in the literature. In this section, we wish to work out the welfare effects of banning ad-valorem fees (i.e., banning price discrimination) using realistic sales distributions. The data we use are from Visa signature debit card transactions and DVD listings on Amazon’s marketplace. The fact that affine fee schedules are used in both cases (as shown in Table 1) justifies our focus on the class of generalized Pareto demands. 5.3.1 Methodology First, we illustrate how our theoretical model can be calibrated to real world data. Rather than assume there is a unit mass of buyers for each good, we allow for the possibility that for each good there can be a different number of potential buyers so that even goods with the same c can sell different amounts. The number of transactions for a distinct good i with cost c is Qi,c = ni,c Qc and the platform makes a profit Πi,c = ni,c Πc , where Qc and Πc are the quantity and profit expressions from Section 4 based on a unit mass of potential buyers, and ni,c is the number of potential buyers for good i with cost c. We denote the number of distinct goods with cost c as nc . A platform’s total profit is therefore Π= nc XX ni,c Πc . (18) c∈C i=1 Given (18), all our previous analysis holds except that we need to change the P c mass gc to ni=1 ni,c . This follows because the optimal platform fee does not depend on gc . Accordingly, an affine fee schedule can be rationalized by the platform facing generalized Pareto demands, and the profit maximizing platform fee is still given by (10). Given an observed platform fee schedule T (pc ) = a0 + a1 pc , (10) implies that 24 we can uniquely identify the values of λ and d for a given value of σ. Our welfare comparisons will then consider all the possible values of σ.17 Note that 1 = a1 , 1 + λ (2 − σ) λd = a0 , 1 + λ (2 − σ) so λ = (1/a1 − 1)/(2 − σ) and d = a0 /λ + a0 (2 − σ). Given the value of λ, and the observed price pc and quantity Qi,c for each good traded on the platform, we can then identify the number of potential buyers ni,c for each good. Substituting T (pc ) = a0 + a1 pc into Qi,c 1 λ (σ − 1) Tc 1−σ = ni,c 1 + , c we derive ni,c = h Qi,c 1+ λ(σ−1)(a0 +a1 pc ) (1−a1 )pc −a0 1 . i 1−σ (19) P c ni,c in place of gc when calculatWith ni,c determined, we can use the weight ni=1 ing profit and welfare. Our theory also allows us to identify the lower bound of σ from the data. Recall when σ < 1, the generalized Pareto demand has finite support on 1 [1, 1+ λ(1−σ) ]. This means that if we observe any good with positive sales, its price has ∗ p(c) 1 1 c) . Since p(c) = c + T ∗ (pc ), this requires that T (p < λ(1−σ) . to satisfy c < 1 + λ(1−σ) c Substituting in that T ∗ (pc ) = a0 + a1 pc , c = (1 − a1 )pc − a0 and the expression for λ 1 )pc . Thus, from above, the equivalent inequality can be written as σ > 1 + a1 − a1 (1−a a0 the minimum price we observe in the data pins down the minimum value of σ that our model permits. 5.3.2 Visa debit cards We use data from the Diary of Consumer Payment Choice (DCPC), conducted in October 2012 by the Boston, Richmond, and San Francisco Federal Reserve Banks to calibrate the model. The DCPC collects consumer payments data on the dollar value 17 Alternatively, we could pin down d and σ for a given value of λ, and then conduct welfare comparisons based on different values of λ. However, given that σ determines the shape of demand functions, deriving welfare results based on different values of σ is more informative. It allows us to compare our results with those in the literature that typically assume specific demand functions (e.g., linear demand). 25 of purchases, the payment instrument used, and the category of expense. A national representative sample of 2,468 U.S. respondents were selected, who each recorded all their payments over a three-day period. Since respondents were spread over the entire month of October 2012, this sampling methodology provides reasonable probability estimates of all consumers. For transactions made with payment cards, respondents were asked to report the dollar amount, the exact card type and the card network’s brand name. 0 0 .01 density .02 .04 density .02 .03 .06 Restaurant .04 Retail 0 100 200 transaction price 300 0 600 Small Ticket 0 0 .02 density .04 .06 density .02 .04 .06 .08 .1 .08 Gas Station 200 400 transaction price 0 20 40 60 transaction price 80 100 0 20 40 transaction price 60 Figure 3: Visa Signature Debit Card Transaction Distributions Based on the DCPC data, we identify 1,048 Visa signature debit card transactions in four distinct market categories, namely retail, restaurant, gas station, and small ticket, to form our empirical transaction distributions. For each market category, we use the interchange fee schedule published by Visa (shown in Table 1) to infer its platform pricing. Given merchant acquirers are highly competitive in the U.S. market, the interchange fee schedules posted by Visa mirror very closely the actual fee schedules passed onto sellers. Figure 3 plots the raw density distribution of transaction prices in each market cat26 egory. The distributions are quite skewed. Based on the raw transaction distributions and the fee schedules, we then numerically calculate percentage welfare gains under the observed affine fee schedule and the counterfactual optimal fixed per-transaction fee for each possible value of σ assuming the underlying demand takes the generalized Pareto form. The results are presented in Figure 4. Restaurant 3 4.5 2.5 4 welfare gain (%) welfare gain (%) Retail 5 3.5 3 2.5 2 1.5 1.5 1 0.5 0 -0.5 1 0.5 0.8 2 1 1.2 1.4 1.6 1.8 -1 0.8 2 1 1.2 3 2.5 2.5 2 1.5 1 0.5 0 0.8 1.6 1.8 2 Small Ticket 3 welfare gain (%) welfare gain (%) Gas Station 1.4 2 1.5 1 0.5 1 1.2 1.4 1.6 1.8 0 2 0.8 1 1.2 1.4 1.6 1.8 2 Figure 4: Visa Signature Debit Cards: Welfare Gain from Ad-valorem Fees Figure 4 shows that in three out of the four markets, welfare is consistently higher when ad-valorem fees are allowed for any possible value of σ.18 The only exception is for the Restaurant market for lower values of σ. However, this result is driven 18 Note that the implied value of d from our calibrated model varies from zero in the limit as σ tends to its highest allowed value (i.e., 2) up to 20.8 cents as σ tends to its lowest allowed value as determined by the lowest observed price. This is broadly consistent with Federal Reserve’s recent study based on comprehensive cost surveys on Visa and MasterCard debit card issuers, as mandated by the Durbin Amendment to the Dodd-Frank Act. Based on the study, the Federal Reserve initially proposed an issuer cost-based debit card interchange fee cap between 7 cents and 12 cents, and eventually set the cap at 21 cents per transaction (See Federal Reserve Board (2010) and Federal Reserve Board (2011)). 27 by a single outlier which has an unusually large transaction price, as can be seen from Figure 3. If that outlier is removed from the sample, welfare would also be consistently higher in the Restaurant market when ad-valorem fees are allowed. Note that the percentage welfare gain (or loss) from allowing ad-valorem fees is calculated by assuming the platform fixed cost K = 0. The percentage change in welfare would be even higher once a positive level of K is taken into account.19 Moreover, the absolute level of welfare change is likely to be substantial given the size of the payment card industry. In 2011, debit cards were used in 49 billion transactions for a total value of $1.8 trillion in the U.S. market, in which 60 percent were signature debit card transactions, with Visa’s share of these being about 75 percent. 5.3.3 Amazon marketplace We also calibrate our model using data from Amazon’s marketplace. We focus on DVDs given that it is a well defined market category for which we can be sure all the goods identified are subject to the same fee schedule, and also since we can collect consistent sales ranks for this category. Using a web robot, we collected data on every DVD that was listed under “Movies & TV” on Amazon’s marketplace in January 2014. We selected “New” under “Condition” and de-selected the “Out of Stock” option, and ended up with a total of 295,171 distinct items. The data collected include the title, unique ASIN number identifying the DVD, the price, and sales rank of each DVD.20 Given shipping fees are often not included in the listed price, we also separately collected data on only those items where the listed price included free shipping, resulting in a sample with 191,280 distinct items. Since some DVDs are listed with extreme prices, we restrict our sample to DVDs selling for under $1,000, which includes around 99% of the items collected. For robustness, we also tried alternative price limits, including $500 and $2,000, and the results are very similar.21 Given we do not directly observe the sales of each DVD, we use a power law to 19 It is easy to see for any W1 > W2 , (W1 − K)/(W2 − K) is an increasing function of K. The price is taken as the price posted at Amazon’s marketplace for the DVD. It is the price a buyer will face when they add the item to their cart and go to the checkout – i.e., the “buy-box” price. 21 A concern with extreme DVD prices is that the prices listed are unlikely to reflect the prices at which transactions actually take place. For instance, some sellers post extreme prices as placeholders to avoid a temporary delisting when they are out of stock or away for vacation. Others may be errors in the seller’s entry of its prices. 20 28 −φ infer it from the sales rank, so Qi,c = aRi,c , where Qi,c is the estimated sales of an item and Ri,c is the corresponding sales rank.22 The parameter a does not affect our results, so we normalize it by setting a = 1. We try different values for the parameter φ, including φ = 0 (where sales rank is assumed to be irrelevant), φ = 1 (Zipf’s law) and φ = 1.7 (which is the number suggested by Smith and Telang (2009) in an experimental study on DVD sales on Amazon, although it implies very little weight is placed on items with sales ranks below the top ten). Figure 5 plots the density of items listed at each price which corresponds to the sales distribution under the assumption that φ = 0. The distributions are highly skewed with a majority of items listed at prices below $50. With φ = 1 or φ = 1.7, the distributions become even more skewed. DVDs with Free Shipping 0 0 .02 .02 density .04 .06 density .04 .06 .08 .08 All DVDs 0 200 400 600 transaction price 800 1000 0 400 600 transaction price 800 1000 DVDs with Free Shipping .08 Figure 5: DVD Sales Distribution density .04 .06 density .04 .06 .08 All DVDs 200 0 0 .02 .02 Based on each of the sales distributions and the fee schedule from Table 1, we numerically calculate percentage welfare gains under the observed affine fee schedule and the counterfactual optimal fixed per-transaction fee for each possible value of σ assuming the underlying demand takes the generalized Pareto form. The results 0 200 400 600 800 1000 0 200 400 600 800 1000 transaction price transaction are presented in Figure 6, which shows that once sales ranks areprice taken into account, welfare is consistently higher under an affine fee schedule than under a fixed pertransaction fee.23 22 Power law distributions are widely used to describe rank data, with the well-known “Zipf’s law” being a special case. See Chevalier and Goolsbee (2003) for detailed discussions as well as an application to online sales data. 23 In the Amazon case, the implied value of d from our calibrated model varies from zero up to $1.65 as σ varies from its highest possible value to its lowest. 29 All DVDs (=0) Free Shipping DVDs (=0) 3 welfare gain (%) welfare gain (%) 10 5 0 -5 0.8 1 1.2 1.4 1.6 1.8 2 1 0 -1 0.8 2 All DVDs (=1) welfare gain (%) welfare gain (%) 2 1.5 1 0.5 1 1.2 1.4 1.6 1.8 1.6 1.8 2 2 1.5 1 0.5 0 0.8 2 All DVDs (=1.7) 1 1.2 1.4 1.6 1.8 2 Free Shipping DVDs(=1.7) 2.5 1.5 2 welfare gain (%) welfare gain (%) 1.4 2.5 2.5 1.5 1 0.5 0 0.8 1.2 Free Shipping DVDs (=1) 3 0 0.8 1 1 1.2 1.4 1.6 1.8 1 0.5 0 0.8 2 1 1.2 1.4 1.6 1.8 2 Figure 6: DVDs: Welfare Gain from Ad-valorem Fees 6 Concluding remarks In this paper, we investigated a puzzle and possible policy concern. Anecdotal evidence suggests that usually there is no obvious relationship between the total value of an exchange and the cost of the intermediation. So why do platforms often charge a fee proportional to the value of the exchange? We suggested that price discrimination provides a natural explanation for the widespread use of ad-valorem fees. A platform has to deal with trade in multiple goods with widely different costs and valuations. If only a fixed per-transaction fee is used, a disproportionate amount will be charged to the low-cost and low-value goods, resulting in their price elasticity of demand being too high, while high-cost and high-value goods will end up with their price elasticity of demand being too low. Ad-valorem fees and taxes ensure revenue extraction is more efficient by removing this distortion, resulting in better outcomes for platforms and planners, and by the same logic, for tax authorities. 30 Under the class of generalized Pareto demand functions and when platforms face costs of handling each transaction, we showed the use of an affine fee schedule (a proportional fee plus a fixed per-transaction fee) is optimal for the platform, allowing it to achieve the same profit that it would obtain under third-degree price discrimination. We also showed such an affine fee schedule would be chosen by a planner who wants to maximize social welfare (or consumer surplus) subject to the platform covering its costs including fixed costs. Finally, we showed that in an unregulated environment, allowing the platform to charge an ad-valorem fee as part of its fee schedule also often increases welfare, including when we calibrate our model to data on sales of DVDs on Amazon’s marketplace and data for Visa signature debit card transactions. Therefore, caution should be taken when policymakers consider restricting platforms from using such ad-valorem pricing. The idea that ad-valorem fees provide a simple and efficient way for price discrimination is likely to apply more generally. While we focused on two examples where ad-valorem fees are prevalent—online marketplaces and payment cards—there are many other potential applications of our theory including to auction houses, booking agencies, employment agencies, insurance brokers, investment banks, real estate agencies, etc. It would be interesting to explore the extent to which our theory applies to these different markets. Our theory can usefully be extended in several directions. First, one may consider sellers engaging in imperfect competition in our model environment. As shown in Shy and Wang (2011), imperfect competition of sellers together with a monopoly platform causes a double marginalization problem, which may add to the welfare gain of using ad-valorem fees by the platform. In this sense, our present result complements theirs. Second, it would be interesting to extend our analysis to the setting where platforms sometimes use non-linear fee schedules, in which the proportional component is constant up to some limit but is set at a lower level if the sale price increases beyond this limit. 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[28] Wang, Zhu (2010). “Market Structure and Payment Card Pricing: What Drives the Interchange?” International Journal of Industrial Organization, 28, 86–98. [29] Wright, Julian (2003). “Optimal Card Payment Systems.” European Economic Review, 47, 587–612. [30] Wright, Julian (2012). “Why Payment Card Fees Are Biased Against Retailers.” RAND Journal of Economics, 43, 761–780. 34 Appendix A: Alternative demand specifications Alternative trading environment In this appendix, we consider an alternative setting which gives rise to the same demand specification (1) used in the paper and for which all of the results of the paper continue to hold. Suppose buyers can buy from sellers through an efficient trading environment which involves a “tax” or through an alternative less efficient trading environment. In case of a tax authority, the alternative trading environment may be to trade in the underground economy, thereby avoiding any tax. In the case of a platform, the alternative trading environment may be trading directly or through a different means (e.g., from the seller’s own website rather than through eBay or Amazon, or using cash instead of Visa or MasterCard payment cards), thereby avoiding any platform fees. Different goods within a market are indexed by c. For good c, sellers all incur a unit cost c to provide the good, and buyers all value the good at vc > c, which is assumed sufficiently large such that all buyers want to buy the good regardless of the trading environment. Note the buyers’ valuation for each different good need not be scaled by c. There are assumed to be sufficient identical sellers so that there is homogenous Bertrand competition in both trading environments. Since there are no fees involved, the price sellers set in the alternative trading environment will exactly equal c. Buyers purchasing through the efficient trading environment avoid an inconvenience or loss that is expected when purchasing through the alternative means, an expected loss which is equal to bc, where b ≥ 0. Buyers have heterogenous valuations of this loss, so draw b from a distribution as in the model in the paper. Thus, the key assumption in this alternative setting is that the loss of using the less efficient trading environment is proportional to the cost or price of the goods traded. In case of a tax collected by authorities, the assumption requires that the expected loss from making a trade at the price c in the underground economy is proportional to c, perhaps reflecting that this is the amount at risk to each party in any such trade. In case of a payment card platform such as that offered by Visa, the assumption requires that the inconvenience of having to use cash for a payment equal to c is proportional to c, perhaps reflecting the opportunity cost of the cash involved. 35 Suppose as in the main text, taxes (or platform fees) are only on the seller side. Thus, when a transaction takes place, a buyer of good c that draws b and faces a price for good c through the efficient trading environment of pc receives a surplus of vc − pc using the efficient trading environment and vc − (1 + b) c using the inefficient trading environment. The assumptions on 1 + b and other parameters then follow as in Section 2. The number of transactions on the platform Qc for a particular good c is the measure of buyers whose surplus from using the efficient trading environment is greater than using the alternative, i.e., Pr 1 + b ≥ pcc , which gives rise to the same demand function (1) as in the benchmark model. The rest of the analysis follows as before. Demand heterogeneity Suppose demand can be written as ξc Q 1 + Tcc + εc , where ξc is randomly drawn from the distribution Ξ with E (ξc ) = 1 and εc is drawn (independently) from the distribution with E (εc ) = 0. To maximize revenue for good c, we choose Tc to maximize Z Z Tc Tc ξc Q 1 + + εc d (εc ) dΞ (ξc ) . c ξc εc The first-order condition is Z Z Tc 0 Tc Tc + εc + ξc Q 1 + d (εc ) dΞ (ξc ) = 0, ξc Q 1 + c c c ξc εc which given E (εc ) = 0 can be rewritten as Tc Tc Tc 0 ξc dΞ (ξc ) Q 1 + + Q 1+ = 0. c c c ξc Z Given E (ξc ) = 1, the first-order condition for determining optimal fees for good c becomes the same as in (2). Moreover, equilibrium prices are not affected by the demand heterogeneity since prices are determined entirely by costs so the optimal fee schedule still satisfies (4). The same logic applies to profit maximization and the Ramsey regulation problem. Thus, we can easily allow for random heterogeneity in demand. 36 Appendix B: Proof of Propositions Proof of Proposition 1. Note that at Tc = 0, we have ∂Rc /∂Tc = Q (1) = 1 > 0. Given ∂Rc /∂Tc < 0 for some Tc at which Q 1 + Tcc > 0 and that the revenue function is strictly concave, (2) characterizes the global maximum. The solution exists and is unique for each c, which is Tc∗ = ωc, with ω > 0 given by (3). The equilibrium price is characterized by the unique solution to pc = c + T (pc ) for each good c under Bertrand competition. Given (4), the fee set for good c will indeed be Tc∗ = ωc. Proof of Proposition 2. First, consider the hypothetical situation in which the regulator observes each of the goods and solves the following problem: max T~c X gc Wc − K s.t. c∈C X gc Rc ≥ K, c∈C where T~c denotes the vector of taxes Tc , one for each good c. Given the objective funcP tion Wc is strictly declining in Tc for each market, the objective function c∈C gc Wc is quasi-concave in T~c . Also, since the revenue function Rc is strictly concave in Tc , P the weighted sum c∈C gc Rc is strictly concave in T~c . As a result, the usual KuhnTucker maximum conditions give the global maximum of the problem (see Arrow and Enthoven, 1961). Solving the constrained maximization problem, the first order condition for each c yields (5), where the Lagrangian multiplier η > 0 so the right hand side lies strictly between 0 and 1. The rest of the proof follows the revenue-maximizing case. The existence and uniqueness of the solution to (6) follows from the existence and uniqueness of the solution to (3) given the right hand side of (6) is still positive but now less than unity. The optimal tax schedule is therefore the proportional fee schedule given in (4), but with ω > 0 solving (6). Finally, note that the value of ω solving (6) will be lower than that solving (3). = Q (1 + ω ∗ ) + ω ∗ Q0 (1 + ω ∗ ) = 0. To see this, denote the solution to (3) as ω ∗ , so ∂R ∂ω Denote the solution to (6) as ω r . Given the demand elasticity with regard to Tc must be less than unity in magnitude at ω r , we have ∂R = Q (1 + ω r ) + ω r Q0 (1 + ω r ) > 0. ∂ω 2 ωr ω∗ Given that ∂∂ωR2 < 0, we have ω r < ω ∗ , and so 1+ω r < 1+ω ∗ . 37 Proof of Proposition 3. We proceed in four steps. (1) We first show the result in the “if” direction. Suppose F has an affine inverse hazard rate ρ (x) = ρ0 + ρ1 x, where the hazard rate h satisfies h0 (x) + h2 (x) > 0 for x ∈ [1, 1 + b̄]. Given F 0 (1) > 0 we have ρ0 + ρ1 > 0. The property h0 (x) + h2 (x) > 0 implies ρ1 < 1. If we can observe c and choose a corresponding Tc∗ , the first-order condition for optimality can be written as (Tc∗ Tc∗ Q 1 + c − d) . =− Tc∗ c 0 Q 1+ c (20) (That Tc∗ satisfying this gives the global maximum when F has an affine inverse hazard rate is shown in step 4 below.) Since the right-hand-side of (20) is the inverse hazard rate for F evaluated at 1 + Tc∗ /c, we must have Tc∗ = d ρ0 + ρ1 + c. 1 − ρ1 1 − ρ1 (21) This can be implemented by the affine fee schedule T (pc ) = d ρ0 + ρ1 + pc , 1 + ρ0 1 + ρ0 (22) where ρ0 + ρ1 > 0 and ρ1 < 1 implies ρ0 > −1, so that the intercept is positive and the slope is between 0 and 1. Note implementation is possible since sellers of type c have to cover their costs c + T (pc ) and so equilibrium prices for good c are given by the lowest pc such that pc ≥ c + T (pc ). The equilibrium price is characterized by the 0) unique solution to pc = c + T (pc ) for each good c, which is pc = d+c(1+ρ . Given (22), 1−ρ1 the fee set for good c will indeed be (21) as required. (2) We next consider the “only if” part of the proof. Without loss of generality, suppose we start with (22) being the optimal fee schedule, with ρ0 + ρ1 > 0 and 0) ρ1 < 1. This implies the equilibrium price for good c will be given by pc = d+c(1+ρ . 1−ρ1 Given the slope of the fee schedule is between 0 and 1, a seller cannot make more profit by lowering its price below this level to enjoy a lower fee. Substituting this price into (22) implies the optimal fee for good c is given by (21). Substituting this 38 into (20) implies ρ0 + ρ1 1 + 1 − ρ1 d c =− Q 1+ Q0 1 + ρ0 +ρ1 + dc 1−ρ1 ρ0 +ρ1 + dc 1−ρ1 , (23) which given our definition of optimality must hold for any c in the interval cL ≤ c ≤ cH . This can only be true if the inverse hazard rate (the function −Q (·) /Q0 (·)) is the increasing affine function with intercept ρ0 and slope ρ1 , where h0 (x) + h2 (x) > 0 follows from ρ1 < 1. (3) The equivalence between (i), (ii) and (iii) is shown as follows. Without loss of generality, the requirement that the inverse hazard rate be linear is equivalent to requiring λ F 0 (x) = (24) 1 − F (x) 1 + λ (σ − 1) (x − 1) for some constant parameters λ and σ. Evaluating (24) at x = 1, implies that λ > 0 and σ < 2 given the requirements that ρ0 + ρ1 > 0 and ρ1 < 1. Solving the differential equation, the solution requires (11), with the special case F (x) = 1 − e−λ(x−1) when σ → 1. Accordingly, the demand faced by sellers is the broad class of demand functions that has a constant curvature of inverse demand σ. It is specified by (12), which includes linear demand (σ = 0) and constant-elasticity demand (σ = 1 + 1/λ), as well as exponential demand (σ → 1). The demand faced by the platform for good c is given by 1 λ (σ − 1) Tc 1−σ Tc , (25) = 1+ Qc (Tc ) = 1 − F 1 + c c which also belongs to this constant-curvature class of inverse demand functions, though it does not cover the full class (e.g., the platform does not face a constantelasticity demand). Given the platform demand function (25), the optimal fee satisfying (20) is Tc∗ = λd + c . λ (2 − σ) (26) Substituting pc = c + Tc∗ , the optimal fee can be written as (10), which is equivalent to the affine fee schedule (22) derived above with ρ0 = 1 − σ + 1/λ and ρ1 = σ − 1. (4) We can verify that Tc∗ satisfying (26) is indeed the global maximum for the 39 class of demand functions identified by (25), and equivalently is the global maximum when F has an affine inverse hazard rate. According to (11), when σ < 1, F has a finite support [1, 1 + 1/λ (1 − σ)], so the valid domain for Tc is [0, c/λ (1 − σ)]. Given the assumption that b > d/cL , we can verify that (26) implies 0 < Tc∗ < c/λ (1 − σ) for any c ∈ C. When σ ≥ 1, F has the support [1, ∞), and the domain for Tc is [0, ∞). In either case, Tc∗ given by (26) is the only interior solution in the domain of Tc that satisfies ∂Πc /∂Tc = 0. And ∂Πc /∂Tc > 0 for any Tc < Tc∗ , while ∂Πc /∂Tc < 0 for any Tc > Tc∗ . Therefore, Tc∗ is the unique global maximizer of Πc . Proof of Proposition 4. (i) Since the generalized Pareto demand functions do not necessarily ensure a globally concave profit function with regard to price, we prove the results using quantities. In the case σ 6= 1, the inverse demand function for good c can be derived from (12) as pc (Qc ) = c 1 − 1 λ (σ − 1) + c Q1−σ . λ (σ − 1) c (27) The corresponding welfare function for good c is Z Wc = 0 Qc c (pc (x) − c − d) dx = λ (σ − 1) Q2−σ c − Qc − dQc , 2−σ and the platform profit for good c is Πc = (pc − c − d)Qc = c Qc2−σ − Qc − dQc . λ (σ − 1) Given goods c ∈ C and the associated distribution function G, the social planner solves 2−σ X Qc c − Qc − dQc − K (28) M ax W = gc Qc ,∀c∈C λ (σ − 1) 2 − σ c∈C X c 2−σ s.t. Π = gc Qc − Qc − dQc ≥ K. (29) λ (σ − 1) c∈C Solving the constrained maximization problem, the first-order condition requires 40 that for any good c, c (Q1−σ − 1) c ((2 − σ) Qc1−σ − 1) c −d +η −d =0 λ (σ − 1) λ (σ − 1) (30) for all c ∈ C, where η > 0 is the Lagrangian multiplier associated with the constraint (29). Solving (30) yields the optimal output Qrc for any good c such that (Qrc )1−σ = d+ c λ(σ−1) c λ(σ−1) (1 + η (2 − σ)) (1 + η) . (31) Correspondingly, (27) and (31) imply the optimal regulated platform fee Tcr = prc − c = c ηc (1 + η) d + . (Qrc )1−σ − 1 = λ (σ − 1) 1 + η (2 − σ) λ (1 + η (2 − σ)) (32) Given that c = prc − Tcr , (32) implies the optimal regulated platform fee is affine in pc : T (pc ) = ηpc (1 + η) λd + , λ + (1 + (2 − σ) λ) η λ + (1 + (2 − σ) λ) η (33) η with the ad-valorem term λ+(1+(2−σ)λ)η ∈ (0, 1) given η > 0, λ > 0 and σ < 2. Equation (33) is a necessary condition for the above constrained maximization problem. Moreover, W and Π are both strictly concave in Qc for any c: c ∂ 2W = − Q−σ < 0; 2 ∂Qc λ c ∂ 2Π c = − (2 − σ) Q−σ c < 0. 2 ∂Qc λ Therefore, according to the Kuhn-Tucker theorem, (33) corresponds to the global maximum of the above constrained maximization problem. The case of exponential demand, σ = 1, can be proved similarly as a special case. (ii) Comparing the platform’s optimal fee schedule (10) with that set by the social planner (33), we find that the proportional component chosen by the planner is always lower than that chosen by the platform given η > 0, λ > 0 and σ < 2. The fixed per-transaction component chosen by the social planner is also lower if and only if σ > 1 + 1/λ. 41