Skewed
Skewed Pricing in Two-Sided Markets:
An IO approach?
Wilko Bolt?and Alexander F.Tieman?
January2005
Abstract
In two-sided markets,one widely observes skewed pricing strategies,in which the price
mark-up is much higher on one side of the market than the https://www.wendangku.net/doc/1611694339.html,ing a simple model
of two-sided markets,we show that,under constant elasticity of demand,skewed pricing
is indeed pro?t maximizing.The most elastic side of the market is used to generate
maximum demand by providing it with platform services at the lowest possible price.
Through the positive network externality,full participation of the high-elasticity,low-
price side of the market increases market participation of the other side.As this side is
less price elastic,the platform is able to extract high prices.Our skewed pricing result
also carries over when analyzing the socially optimal prices.Interestingly,this leads to
below-marginal cost pricing in the social optimum.We motivate the analysis by looking
at the Dutch debit card system.
Keywords:Two-sided markets,skewed pricing,corner solution,social optimum.
JEL Code:G21,L10,L41
?The views expressed in this article are those of the authors and do not necessarily represent those of De Nederlandsche Bank,the European System of Central Banks,the IMF or IMF policy.The authors thank Peter van Els and conference and seminar participants at the European Finance Association2003in Glasgow, the IMF,and the Federal Reserve Bank of New York for comments.
?Research Department,De Nederlandsche Bank,P.O.Box98,1000AB Amsterdam,The Netherlands,tel: +31205243524,fax:+31205242529,e-mail corresponding author:w.bolt@dnb.nl
?Monetary and Financial Systems Department,International Monetary Fund,70019th Street,N.W., Washington,D.C.20431,U.S.A,tel:+1-202-623-4434,fax:+1-202-589-4434,e-mail:atieman@https://www.wendangku.net/doc/1611694339.html,
1Introduction
Credit and debit card schemes,computer operating systems,shopping malls,television net-works,and nightclubs represent at?rst glance an odd shortlist of very di?erent industries. Indeed,these industries serve di?erent types of consumers,use very di?erent technologies, and maintain quite dissimilar business arrangements and standards.Yet,?rms in these so-called“two-sided”industries have adopted similar pricing strategies for solving the common problem they face-getting and keeping two sides of the market on board.To balance the demand for their product or service,these?rms tend to heavily“skew”prices towards one side of the market in the sense that the price mark-up is much larger on one side than the other1.The main contribution of our paper is to rationalize this widely observed skewed pricing behaviour in two-sided markets,which has thus far been problematic in economic theory.
The recognition that many markets are two-sided(or multi-sided)has recently triggered a surge in the theoretic literature on the economics of two-sided markets.Two-sided markets involve two distinct types of end-users,each of whom obtains value from“transacting”or “interacting”with end-users of the opposite type.In these markets,one or several platforms enable these transactions by appropriately charging both sides of the market.As Rochet and Tirole(2004)say,platforms“court”each side of the market while attempting to make money overall.In view of our shortlist,credit and debit card payment schemes court cardholders and retailers,computer operating systems court users and application developers,shopping malls court buyers and sellers,television networks court viewers and advertisers,and nightclubs court men and women.
A key aspect of research focuses on price determination in two-sided markets.Under two-sidedness,platforms need not only choose a total price for their services,but must also choose an optimal pricing structure:the division of the total price between the two sides of the market,so as to induce participation from both sides of the market.The need for both a pricing level and a pricing structure is one of the de?ning characteristics that distinguishes two-sided markets from industries ordinarily studied by economists2.In practice,in many two-sided markets platforms often treat one side of the market as a“pro?t center”with high prices and the other side as a“loss leader”with low prices.
Referring again to our shortlist,credit card company American Express earned82percent of its revenues from merchants in2001.In the Dutch debit card system,but also in many other European debit card schemes,consumers face zero transaction fees for using their debit cards,while retailers pay relatively high usage fees.Microsoft earns the bulk of its pro?ts from Windows from licensing Windows to computer manufacturers or other end-users.At shopping malls,shoppers are usually not charged an entrance fee and often receive cheap car parking,while shop owners heavily pay for available rental space.Television networks,but also other media platforms like newspapers,magazines,and websites,earn a disproportionate share of their revenues from the advertisers’side of the market.For example,during the2002 Super Bowl,the average price for a thirty-second commercial ran up to US$1.9million.The fees that the media platforms collect from advertisers pay for the content that the media present to the audience.And?nally,where it is legal,nightclubs often set much lower entrance fees for women than for men.Nightclub owners presumably believe that,while the 1See Evans(2003)for a list of examples of two-sided markets and corresponding skewed pricing strategies, see also Rochet and Tirole(2003)for existing business models in two-sided markets.
2Rochet and Tirole(2004)provide a formal de?nition of two-sidedness,and discuss conditions that make a market two-sided.
cost of servicing an additional male or an additional female is likely to be quite similar,at equal prices the bar attracts too many men,or too few women.All in all,skewed pricing is so commonly observed that it legitimates the question whether we can formalise this pricing behaviour within a standard model of two-sided markets?
Economic theory has di?culties explaining the observed skewed pricing behaviour in two-sided markets.In an elegant article,Rochet and Tirole(2003)derive a“interior”pricing result for a monopolistic two-sided market.They show that under log-concavity of the demand functions,the optimal total price is governed by a variant of the well-known monopolistic Lerner condition,and that the optimal pricing structure depends on the relative magnitude of the individual market sides’price elasticities of demand.To explain skewed pricing within their model,it must be assumed that one side of the market is far less price-elastic(or totally inelastic for a zero fee)than the other.Then,surprisingly,their interior pricing result predicts that the least elastic side of the market will be charged the lowest price.This result seems at odds with standard economic intuition where one would instead expect a lower fee for the more elastic side.In this paper we will show that under constant elasticity of demand,the side of the market which is su?ciently more elastic than the other side is kept to a minimum fee,while the other side pays a relatively high fee.The economic intuition underlying our skewed pricing result is that the most elastic side of the market is e?ectively subsidized by the other side so as to boost the demand for services supplied by the platform.Indeed, every agent on the high elasticity,low price side of the market will connect to the platform. Bene?tting from full participation on one side,the other side is therefore also encouraged to join.However,since this side is more price-inelastic,the platform is able to extract higher prices.
Our skewed pricing result,which is mathematically characterized by a corner solution, carries over in a qualitative way when studying socially optimal prices.We prove that a central planner would choose the same corner solution,in terms of full participation of the su?ciently more elastic side of the market and recovering almost all costs from the price-inelastic side.It is further shown that when socially optimal prices are implemented,the platform operates at a loss.Not surprisingly,since socially optimal prices are in general lower than monopolistic prices,the monopolistic platform tends to undersupply its services.
Our?ndings may have important bearings on antitrust issues which we will brie?y touch upon.We will further illustrate our general results by analyzing the Dutch debit card system and arguing that our theoretic results better?t the observed properties of debit card pricing in the Netherlands than the results in previous theoretical papers.In fact,recent empirical studies on payment systems have also shown that consumers are quite sensitive to price changes in payment instruments.As these apparent price-elastic consumers hardly face any price for making card payments,these empirical facts support our theoretical?ndings.
The remainder of the paper is organized as follows.Section2motivates our paper by analyzing the Dutch debit card model,which features a monopolistic platform processing the payments and setting the prices for both the consumers and retailers.Section3describes the model and derives the skewed pricing result.In section4we analyse socially optimal prices of the platform network.Finally,section5concludes.
2Motivation:Debit Cards in the Netherlands
To motivate our analysis about skewed pricing in two-sided markets,we use the Dutch debit card system as an illustration.Given that we observe skewed pricing in many di?erent
Figure 1:The “Dutch”Debit Card Model T c E d d d d d s d d d d d s
Bank R 'Bank C Retailer Interpay Consumer sale of good at price p cost c
pro?t πbank C debits consumer’s account for amount p bank R credits retailer’s account for amount p bank C transfers amount p cost of crediting cost of authorising and debiting fee t c :0euro fee t r :0.07euro pro?t sharing rule:λπ(1?λ)πretailer bene?ts of accepting debt cards
consumer bene?ts of using debit cards two-sided markets,many features of this speci?c illustration carry over to various other situations.Over the last decade the Netherlands,and many other western economies,have shown a rapid increase in the usage of debit cards.In particular,debit card payments in the Netherlands exceeded EUR 50billion in 2002(around 11percent of GDP)-with a volume of over one billions debit card transactions,more than 40times higher than in 1991-and are still growing rapidly 3.Consumer participation is complete,with virtually all consumers over age 18carrying a debit https://www.wendangku.net/doc/1611694339.html,age density on the retailers’side is lower,with about 56percent of all Dutch retailers accepting debit card payments.On average,retailers pay some 7eurocents per debit card transaction,while consumers do not pay any such usage fee.In this sense,debit card pricing in the Netherlands is completely skewed to the retailers’side,as is the case in most other countries with a developed debit card payment infrastructure 4.
To avoid an abundance of costly bilateral agreements and procedures between ?nancial institutions and banks,Dutch commercial banks have opted for a business model with only one debit card platform,called Interpay.Interpay acts as a central network switch facilitating the exchange of highly secured electronic payment data between end-users (consumers and
3
See Bolt (2003)for a more elaborate description of Dutch retail payment systems.
4For a theoretic analysis of the Dutch debit card market,see Bolt and Tieman (2003).Usually,cardholders in the Netherlands pay a small annual fee.However,these costs may also be attributed to having checking accounts at banks,using debit cards for stored value also,and using them abroad.In our analysis,for simplicity we abstract from membership fees.See Rochet and Tirole (2003)for dealing with ?xed costs in a theoretic model of two-sided markets.
retailers)and their respective banks5.Moreover,in the Netherlands,this platform signs up retailers to accept debit cards,and in principle it could set the prices of debit card transactions for both consumers and retailers,although currently it only charges retailers.Interpay is a private limited liability?rm jointly owned by the banks.Banks get reimbursed for incurred authorisation and other costs by Interpay through a speci?c pro?t sharing rule.Interestingly, and in contrast with most other countries,the Dutch debit card system does not implement any interchange fee6.Figure1depicts a schematic representation of a debit card payment in the Dutch system and the corresponding?ow of funds and fees.
Recently,Dutch retailers expressed their dissatisfaction with the current pricing strategy of Interpay,perceiving the high retailer fee as a form of abuse of market power.Indeed, the skewness of prices in the debit card industry triggered antitrust scrutiny and led to an in-depth investigation by the Dutch competition authority NMa.As a result,Interpay and the participating commercial banks were?ned some50million euro.In addition,retailers went to court to reclaim foregone revenues in the amount of200million euro7.Both cases are still pending.
Obviously,the debit card model as described in Figure1is quite limited.For instance, Rochet and Tirole(2002)and Wright(2003b)have extended the model by incorporating competition among consumer and retailer banks,and strategic behaviour by retailers.The basic model as described in the next section concentrates explicitly on the links between both groups of end-users and the monopolistic platform.
3The Basic Model:Monopoly Platform
Our model setup is similar to Schmalensee(2002)and Rochet and Tirole(2003).Potential gains from trade are created by transactions between two end-users,whom we will call buyers (subscript b)and sellers(subscript s).Such transactions are mediated and processed by the monopoly platform.To provide these(network)services,the platform charges buyers and sellers a transaction fee,denoted by t b≥0and t s≥0.For simplicity,we abstract from any?xed periodic fees for end-users to connect to the platform.In performing its tasks,the platform incurs joint marginal costs,which are set at c≥0per transaction.It is convenient to introduce a distinction between the pricing level,de?ned as the the total price t T=t b+t s by the platform to the two sides,and the pricing structure,referring to the allocation between t b and t s of the total price to buyers and sellers.As Evans(2003)shows,in two-sided industries the product may not exist at all if the business does not get the price structure right.
Buyers and sellers enjoy bene?ts when transacting on the platform.We assume that buy-ers are heterogeneous in the bene?ts b b,b b∈[b b,ˉb b],ˉb b≤∞,they receive from a transaction. The probability density function of these bene?ts is labelled h b(·)with cumulative density H b(·).Similarly,sellers di?er in the bene?ts b s associated with transacting on the platform, b s∈[b s,ˉb s],ˉb s≤∞,with probability density function h s(·)and cumulative density H s(·). To illustrate,in case of a debit card transaction,a buyer(or consumer)who wants to buy a good or service from a seller(or retailer)at price p,prefers to use his debit card whenever he gets positive bene?ts from using his card relative to other payment instruments,say cash.A 5Also in other countries such as Belgium,France,Norway,and Switzerland a nation-wide debit card network exists with only one central routing switch.
6In much of the theoretic analysis about card payment systems,the optimality of interchange fees has played a central role,see e.g.Gans and King(2001),Rochet and Tirole(2002),Wright(2003b).
7The NMa report on the Dutch debit card system can be downloaded from www.nmanet.nl
Figure 2:The monopoly platform
d d d d d s 'E Seller Platform Buyer “transaction”cost c
pro?t πt b t s bene?t b s bene?t b b transaction using the debit card takes place if at the same time the seller prefers accepting the debit card payment to accepting cash.
Only buyers with bene?ts b b larger than incurred fees t b will transact on the platform.Formally,the fraction of buyers connecting to the platform is given by
q b =D b (t b )=Pr(b b ≥t b )=1?H b (t b ).
(1)
Analogously,the fraction of sellers which connects to the platform is equal to
q s =D s (t s )=Pr(b s ≥t s )=1?H s (t s ).(2)Assuming independence between b b and b s ,the total expected fraction of transactions pro-cessed by the platform amounts to
q =D (t b ,t s )=D b (t b )D s (t s ).(3)
Further,we assume that the platform operates in a price region such that the price elasticities of (quasi-)demand exceed 1for both sides of the market.That is,we de?ne
i (t )=??D i ?t t
D i (4)
and assume that i (t )>1,i =b,s ,for every feasible fee t ≥0.As will be shown below,these elasticities are of critical importance for setting optimal prices in two-sided markets.It is also convenient to make a distinction between the joint price mark-up,de?ned by the ratio (t T ?c )/t T ,and the individual (or own)price mark-up,de?ned by the ratio (t i ?c )/t i ,i =b,s .
Throughout the paper,we assume that one side of the market is more price-elastic than the other.
Assumption3.1.Without loss of generality we assume that the buyers’side of the market is more price-elastic.That is,
b(t)≥ s(t)for every feasible fee t≥0.
Finally,for simplicity,we exogenously?x the total number of transactions,both on and o?the platform,at N.So,total demand for network services on the platform is given by N·D(t b,t s).Figure2schematically depicts the model.
3.1Interior platform pricing under log-concave demand
In a two-sided market the monopoly platform must make sure that both sides of the mar-ket get on board by appropriately setting the transaction fees.Two-sidedness has strong implications for the pricing strategy of the platform,which will depend on both sides’price elasticities of demand.A key?nding of price determination in two-sided markets is that the optimal prices for the two groups of end-users must balance the demand among these groups. This is seen as follows.
The monopoly platform sets its transaction fees so as to maximize total pro?ts
π(t b,t s,c)=N(t b+t s?c)D(t b,t s).(5) Under log-concavity of the demand functions D b and D s,the maximum is determined by the?rst-order conditions
?(logπ)?t b =
1
t b+t s?c
+
(D b)
D b
=0(6)
and
?(logπ)?t s =
1
t b+t s?c
+
(D s)
D s
=0.(7)
Consider the following theorem by Rochet and Tirole(2003).
Theorem3.2.(Interior pricing)
i)Under log-concavity of the demand functions D b and D s,the monopoly platform earns maximum pro?ts if it sets fees
(t?b,t?s)= c ?b ?T?1,c ?s ?T?1 ,(8) where ?i= i(t?i),i=b,s,and ?T= ?b+ ?s.
ii)The total fee t?T=t?b+t?s is determined by the‘inverse elasticity rule’for total elasticity ?T= ?b+ ?s.That is,
t?T?c t?T =
1
?T
,(9)
iii)The corresponding price structure is characterized by
t?b t?s =
?b
?s
.(10)
Equation(9)illustrates that the pro?t-maximising price level in a two-sided market is governed by the well-known monopolistic Lerner pricing rule when demand is log-concave. However,the Lerner rule applies to the total price t T and total elasticity T:The joint price mark-up is inversely related to the total elasticity(de?ned by the sum of the two individual elasticities).Hence,if the market as a whole gets less elastic,the total price will rise.For the two seperate sides of the market a di?erent but related result holds.Rewriting(10)using (8)-(9)yields
t?i?(c?t?j)
t?i =
1
?i
,i,j=b,s,i=j,(11)
which can be seen as a modi?ed Lerner rule,where the individual price mark-up needs to be corrected for the contribution in terms of revenues from the opposite side.In addition, (10)shows that the distribution of the optimal total fee over the two sides of the market is determined by the relative magnitude of the individual market sides’elasticities.This gives rise to the next corollary that is directly derived from Theorem3.2.
Corollary3.3.Under log-concavity,the market side with the highest price elasticity of de-mand is charged the highest fee.So,given Assumption3.1,buyers pay a higher price than sellers,that is,t?b≥t?s.
Interior platform pricing gives rise to some counterintuitive results.First,as seen from Corollary3.3,interior pricing in two-sided markets means that the most price-elastic side of the market pays the highest price,while standard economic intuition would predict the opposite8.Second,interior pricing cannot easily explain the widely observed skewness in pricing towards one side of the market,in the sense that in many real world settings the price mark-up is far less on one side of the market than on the other.Theorem3.2can only explain this empirical fact by assuming that the side which get charged the least,is far less elastic than the other side of the market.To illustrate,consumers in the Netherlands face a zero transaction fee when using their debit cards,while retailers get charged.Interior pricing would imply that consumers would be either totally inelastic, ?b=0in absolute terms,or much less elastic relative to retailers,since t?b→0as ?s→∞.However,recent empirical studies have found the opposite,concluding that consumers are quite sensitive to price changes of payment services,probably more so than retailers,contradicting the results of the above theorem9.
Together,these observations seem to indicate that log-concavity of the demand function may not be an appropriate assumption.
3.2Skewed platform pricing under constant elasticity of demand
The choice of the probability distribution describing the heterogeneity among both groups of end-users determines the curvature of the demand functions.If the resulting demand 8For instance,in the case of multiproduct pricing with complementary products,the product with relatively inelastic demand sells for a higher price,see e.g.Tirole(1989)for optimal pricing strategies in multiproduct markets.
9Humphrey,Kim,and Vale(2001)reach this conclusion by empirically estimating consumers’price elas-ticities with respect to various payment instruments that are used in Norway.
functions are not log-concave then solving the?rst order conditions do not yield a global maximum but instead induce a saddlepoint solution.In this case maximum pro?ts are found in one of the corner solutions.In a corner solution,participation of one side of the market is complete,while the pricing structure is completely“skewed”towards the other side of the market.This side of the market is then charged a high fee above joint marginal costs,whereas the side of the market with complete participation pays a low price below joint marginal costs.
Formally,a corner solution is characterized by player i being tied down to its minimum bene?t level b i and player j being charged a“two-sided”monopoly price m j.Observe that in the low-price market i,demand is at its maximum,i.e.D i(b i)=1.Moreover,as we will see below,the two-sided monopoly price can even be higher than the“normal”one-sided monopoly price,as the platform needs to recover(almost)all costs from only one side of the market.
We will de?ne the“buyers’corner”solution in which the buyer is“cornered”to his minimum bene?t level and the pricing structure completely skewed towards the sellers’side of the market by
(b b,m s),where m s=arg max
t s
π(b b,t s,c),(12) and the“sellers’corner”solution by
(m b,b s),where m b=arg max
t b
π(t b,b s,c).(13)
We will illustrate skewed pricing in a two-sided market by imposing a density function that induces demand functions with constant elasticity.Therefore,let us assume the follow-ing probability density function describing the bene?ts from a transaction
f b
i , i
(x)=b i i i x? i?1,x∈[b i,∞],b i>0, i>1,i=b,s.(14)
This density function yields a demand D i(t)that is not log-concave
D i(t)=b i i t? i,i=b,s.(15) and features a constant elasticity of demand i.The induced pro?t function is given by
π(t b,t s,c)=Nb b
b b s s t? b
b
t? s
s
(t b+t s?c).(16)
Similar to(8),solving the?rst-order conditions yields
t?b=
c b
b+ s?1
,t?s=
c s
b+ s?1
.(17)
However,the second-order conditions show that this interior solution is a saddlepoint,as graphically depicted in Figure3for a speci?c numerical example.Since the interior solu-tion does not yield maximum pro?ts,the global maximum can be found in one of the two corner solutions.Following de?nitions(12)-(13),and given the constant elasticity density function(14),the buyers’and sellers’corner solutions respectively yield
(b b,m s)= b b,(c?b b) s
s?1 and(m b,b s)= (c?b s) b
b?1
,b s .(18)
Hence,which of the two corner solutions yields maximum pro?ts,depends on the relative magnitude of the individual market sides’elasticities.The next proposition states that for
su?ciently low minimum bene?t levels,pro?ts are maximized by charging the low-elastic side of the market the highest price,whereas the high-elastic side is kept to its minimum bene?t level.In particular,given Assumption3.1,for a su?ciently high elasticity of buyers’demand, buyers pay the lowest possible price so as to ensure their complete participation.In other words,buyers’demand for platform services is at its maximum.Sellers,who are less price elastic,pay high prices10.Consider the following proposition.
Proposition3.4.(Skewed pricing)
Assume a constant elasticity distribution as speci?ed in(14)b b+b s≤c.Then there exists anˉ ≥ s,such that for all b>ˉ the monopoly platform earns maximum pro?ts by setting fees
(t M b,t M s)=(b b,m s)= b b,(c?b b) s
s?1 .(19) At these fees,buyers’participation is complete,that is,D b(t M b)=1.
Proof:
First,one may verify that if a buyer is charged b b and a seller is charged m s then platform pro?ts are given by
π(b b,m s,c)=N b s s( s(c?b b)
s?1
)1? s
s
.(20)
Likewise,the sellers’corner solution yields
π(m b,b s,c)=N b b b( b(c?b s)
b?1
)1? b
b
.(21)
These values exist and are positive for su?ciently small minimum bene?t levels,i.e if b i≤c, i=b,s.Second,observe thatπ(b b,m s,c)is constant in b.In the limit,as b→∞and holding
s constant,we computeˉπb=lim
b→∞π(b b,m s,c)>0,andˉπs=lim
b→∞
π(t M b,b s,c)=0.
Third,we derive
?π(m b,b s,c)
? b =?N
(c?b s) b(c?b s)b b( b?1) ? b log b(c?b s)b b( b?1)
b?1
<0(22)
if and only if log b(c?b s)b b b >0.Some algebraic manipulations show thatπ(m b,b s,c)is de-creasing for every b>1if and only if b b+b s≤c.Sinceπ(m b,b s,c)is decreasing in b and π(b b,m s,c)is constant,andˉπs<ˉπb,this implies that there exists anˉ ≥ s such that for all b>ˉ it holds thatπ(b b,m s,c)>π(m b,b s,c).Hence,(t M b,t M s)=(b b,m s)11.Existence of derivative(22)requires b s≤c which is automatically guaranteed if b b+b s≤c.Finally,by de?nition,D b(t M b)=D b(b b)=1,that is,complete buyers’participation.
10In a model of competition among intermediaries,Caillaud and Jullien(2003)show that providing low prices or transfers to one side of the market helps to solve the initial“chicken-and-egg”problem.They refer to this strategy as“divide-and-conquer”.
11Note that,due to symmetry,if b
b =b s thenπ(b b,m s,c)=π(m b,b s,c)for b= s.Further,one can show
thatˉ > s for b b>b s;otherwise,for b b≤b s one may chooseˉ = s.
The result of Proposition3.4provides an explanation for the earlier noted empirical fact that in many two-sided markets businesses tend to settle on pricing structures that are heavily skewed towards one side of the market.It shows that,under constant elasticity of demand, skewed pricing structures are indeed pro?t-maximising.The mechanism at work is that the high-elastic buyers’side of the market is used to generate demand by charging buyers the lowest possible price.Every buyer will now participate and connect to the platform.The sellers’side bene?ts from full buyer participation through positive network externalities and is therefore also encouraged to participate.However,since sellers are less price elastic,the platform is able to extract higher prices from them.This mechanism,which results in the most inelastic side of the market paying higher prices,con?rms standard economic intuition. By contrast,the interior pricing theorem of the previous section predicts the opposite.
Clearly,under skewed pricing,buyers are charged below joint marginal cost,whereas sellers face a high own price mark-up.In particular,the two-sided monopoly price can even be higher than the“normal”one-sided monopoly price which would occur if the inelastic side of the market is treated in isolation.To see this,let us assume a simple additive cost structure c=c b+c s,where c i,i=b,s,denotes the individual costs on the buyers’and sellers’side of the market when processing a single transaction.Then,the one-sided monopoly price for sellers would equal
t N s=arg max
t s N(t s?c s)D s(t s)=
c s s
s?1
.(23)
We state the following proposition without proof.
Proposition3.5.Under constant elasticity of demand,the two-sided monopoly fee t M s for sellers is higher than the one-sided monopoly price t N s if the buyers’minimum bene?t level b b is su?ciently small.That is,t M s≥t N s if and only if b b≤c b.
Hence,if in the low-price(buyers)market the contribution to total revenues is less than the own individual marginal costs c b,then,in order to compensate for this negative margin on one side of the market,the two-sided monopoly price exceeds the normal one-sided monopoly price on the other,high-price(sellers)side of the market.
Our results potentially have important bearings on antitrust issues.In antitrust analysis, high mark-ups raise concerns of abuse of market power.However,this traditional antitrust logic ceases to hold in two-sided markets12.The fact that bene?ts and costs arise jointly in the two sides of the market e?ectively means that there is no direct economic relation between price and cost on either side of the market.Any change in demand or cost on either side of the market will a?ect both the total pricing level and pricing structure.Hence,it is generally not possible to examine price e?ects on one side of a market without considering the corresponding e?ect on the other side and the resulting feedback e?ects between them. Therefore,in antitrust matters,one cannot examine prices on either side of the market in isolation.Rather,to analyze market power one should investigate whether total price is signi?cantly above joint marginal cost.
In particular,the skewed pricing result of Proposition3.4allows a nice interpretation of the current pricing practice in the Dutch debit card market where consumers pay no transaction fee for using their debit cards but retailers do.The reasoning is as follows.First,recent empirical analysis has shown that consumers are quite sensitive to price changes of payment 12Wright(2003a)discusses eight fallacies that can arise from using convential wisdom from one-sided markets in two-sided industries.See Evans(2003)for antitrust implications of pricing in two-sided markets.
Table1:Monopoly pro?ts under a constant elasticity distribution
Interior Corner solutions
Buyer Seller
(t?b,t?s)(b b,m s)(m b,b s)
Price:
buyer fee t b0.0460.0200.069
seller fee t s0.0340.0810.018
total fee t T0.0800.1010.087
Demand(in%):
buyer demand D b(t b,t s)8.4100.0 2.4
seller demand D s(t b,t s)25.5 3.7100.0
total demand D(t b,t s) 2.1 3.7 2.4
Pro?t:
totalπ(t b,t s,c)0.32 1.350.56
per transaction(×10?3)πq(t b,t s,c)15.236.723.0
Welfare:W(t b,t s,c) 1.41 4.20 1.77 Parameters: b=3, s=2.2,b b=0.02,b s=0.018,c=0.064(c b=c s=0.032),N=1000. services,see e.g.Humphrey,Kim,and Vale(2001).Second,retailers often complain that due to competitive pressures they are“forced”to facilitate debit card services.Retailers cannot a?ord to sell“no”to their customers.At the same time,they do not see many payment alternatives.Hence,retailers may be assumed to be much less price elastic in their demand for debit card services than consumers.Following Proposition3.4,this might explain why the pricing structure in the debit card market is so heavily skewed towards the retailers’side of the market13.Indeed,as mentioned previously,the high retailer fee triggered antitrust controversy,resulting in an in-depth investigation by the Dutch competition authorities.On the basis of this investigation a heavy penalty of about50million euro was imposed on the Dutch debit card platform and participating banks for setting excessive tari?s for retailers. However,it is questionable whether the Dutch antitrust authority fully took notice of the two-sided nature of the debit cards market and its economic consequences.
For the monopoly platform,the recouping of costs from the sellers’side of the market actually leads to positive pro?ts,which are increasing when the minimum bene?t level on the buyers’side of the market is lowered.To see this,we look at total net pro?ts per transaction πq(·,·,·),which amount to
πq(t M b,t M s,c)=t M b+t M s?c=(c?b b) s
s?1
?(c?b b).(24)
Hence,positive pro?t margins are realized for su?ciently small minimum bene?t levels in the low-price buyers’market,i.e.
πq(t M b,t M s,c)≥0if and only if b b≤c,(25) which always holds under the conditions of Proposition3.4.Moreover,?πq(t M b,t M s,c)/?b b< 0.That is,lower minimum bene?t levels on one side of the market dampen the fee on this 13It is noted however that our model speci?cation does not support a zero price,since b
i
>0,i=1,2.
Figure 3:Pro?ts under a constant elasticity distribution
π b π
s ,c )b s ,c )
Explanatory note :The left panel shows that the interior solution (0.0457,0.0335)is a saddlepoint (pro?ts below 0.32485are surpressed);the right panel shows both “corner”pro?t functions as a function of b ,and veri?es ˉ ≈2.33.
side of the market,but increase the fee levied on the other side of the market.This latter e?ect dominates,so that pro?ts per transaction increase when minimum bene?t level in the low-price market decreases.
The following example illustrates our ?ndings.
Example :Monopoly pro?ts
We set the following parameter values:c =0.064(c b =c s =0.032),b b =0.02,b s =0.018, b =3, s =2.2,and N =1000,in accordance with the assumptions in Proposition 3.4.Prices,demans,and pro?ts for the parameter values can be found in Table 1.
The table indeed shows that the interior solution of Rochet and Tirole (2003)does not yield maximum pro?ts for the monopoly platform when demand derives from a constant-elasticity distribution.The interior solution (0.046,0.034)is a saddlepoint as clearly depicted in the left panel of Figure 3.The right panel of Figure 3shows both “corner”pro?t functions π(b b ,m s ,c )and π(m b ,b s ,c )as a function of b .We know that π(b b ,m s ,c )is constant in b whereas π(m b ,b s ,c )is decreasing in b .We numerically verify that ˉ ≈2.33.Hence,following Proposition 3.4,for an b =3.0>2.33,pro?ts are maximized when buyers are tied down to their minimum bene?t level b b =0.02and sellers being charged a high two-sided monopoly price t M s =m s =0.081.
Also,from the table,total demand is highest in the buyers’corner solution even though it has the highest total price.This stresses the importance of the pricing structure in two-sided https://www.wendangku.net/doc/1611694339.html,plete buyers’participation boosts total demand,which at the same time allows a higher price to be extracted from the relatively inelastic sellers.As a result,around 4%of all payment transactions are executed by debit card.Moreover,from (25)it follows that positive pro?t margins are realised,since b b =0.02≤c =0.064.But note that on the buyers’side operational losses are incurred since the charged fee cannot make up for the individual costs,that is,t M b =b b =0.02 is higher than the one-sided monopoly price,that is,t M s =0.081>0.059=t N s . 4Social Welfare In the model,the network platform enjoys monopolistic power in setting prices for both buyers sellers.It is well known that a monopolistic price-setter distorts the market by reducing supply to raise prices,inducing a dead weight loss.In a two-sided market where network externalities play an important role,the underlying economics get more complicated. In our model,total(expected)social welfare that is generated from platform services is equal to consumer plus retailer(expected)bene?ts minus costs,conditional upon buyers’and sellers’participation in the network.Formally,total social welfare is described by W(t b,t s,c)=N(βb(t b)+βs(t s)?c)D(t b,t s),(26) whereβb(t b)denotes the(conditional)expected bene?t to a buyer,de?ned by βb(t b)=E(b b|b b≥t b)= ˉb b t b xh b(x)dx 1?H b(t b) ,(27) and,similarly for a seller, βs(t s)=E(b s|b s≥t s)= ˉb s t s xh s(x)dx 1?H s(t s) .(28) We will examine the social welfare function(26)for the constant elasticity probability distribution used in the previous section.Two questions of particular interest arise.First, how do socially optimal prices and demand compare to the prices set by the monopolistic platform?Second,under socially optimal prices,is the platform pro?table or does it operate at a loss?As it turns out that the platform will lose money in the social optimum,we subse-quently investigate optimal social welfare under a balanced budget constraint(i.e.Ramsey pricing). Substituting the density function f b b , b (x)=b b b b x? b?1,x∈[b b,∞],i=b,s,in social welfare function(26)leads to W(t b,t r,c)=N b b b b r r t? b b t? s s ( b( s?1)t b+ r( b?1)t r?( b?1)( s?1)c) ( b?1)( s?1) .(29) As was the case in the previous section,W(t b,t s,c)exhibits an interior saddle point solution. Analogous to the result in Proposition3.4,the next proposition states that,for a su?ciently high elasticity of buyers’demand,the global social optimum is found at the corner solution where buyers are charged the lowest admissible fee so that the whole buyers’market is captured.In contrast,sellers,who are more price-inelastic,are confronted with higher socially optimal prices. To see this,let us?rst de?ne both corner solutions (b b,w s),where w s=arg max w W(b b,w,c)=c? b b b b?1 ,(30) and (w b,b b),where w b=arg max w W(w,b s,c)=c? b s s s?1 .(31) Consider the following proposition. Proposition4.1. Assume a constant elasticity distribution as speci?ed in(14)and b s≤1 s ( s?1)(c?b b).Then there exists an? ≥ s such that for all b>? social welfare is maximized when the platform implements fees (t W b,t W s)=(b b,w s)= b b,c?b b b b?1 .(32) At these fees,buyers’participation is complete,i.e.D b(t W b)=1. Proof: First,one may verify that14 W(b b,w s,c)=N b s s(c? b b b b?1 )? s( b(c?b b)?c) ( s?1)( b?1) (33) and W(w b,b s,c)=N b b b (c? s b s s?1 )? b( s(c?b s)?c) ( s?1)( b?1) .(34) These values exist and are positive for su?ciently small minimum bene?t levels,i.e if b i< 1 i ( b?1)c,i=b,s.In the limit,as b→∞and holding s constant,we compute ˉW b =lim b→∞ W(b b,w s,c)=N b s s(c?b b)1? s s?1 >0(35) and,if b s≤1 s ( s?1)(c?b b), ˉW s=lim b→∞ W(w b,b s,c)=0.(36) Third,it is straightforward to show that ?W(b b,w s,c) ? b =?N b s s(c? b b b b?1 )? s ( b?1) <0(37) and ?W(w b,b s,c) ? b =N b b b(c? s b s s?1 )? b( s(c?b s)?c) ( s?1)log ( s?1)b s s(c?b s)?c ?1 ( s?1)( b?1) <0,(38) for b s≤1 s ( s?1)(c?b b).So,both value functions are monotonically decreasing,but converge to di?erent limit points.Hence,becauseˉW b>ˉW s,there exists an? ≥ s such that for every b>? we have that W(b b,w s,c)>W(w b,b s,c).Therefore,maximum social welfare is realized at fees(t W b,t W s)=(b b,w s)for su?ciently large b.Finally,by de?nition,D b(b b)=1. 14Note that due to symmetry W(b b ,w s,c)=W(w b,b s ,c)for b b =b s and b= s. Proposition4.1shows that the socially optimal price structure does not qualitatively di?er from the optimal price structure in the monopolistic case.A central planner would choose the same corner solution as the monopoly platform.So,in a social optimum,pricing is completely skewed towards the sellers’side of the market and hence all costs(net from the buyers’minimum bene?t contribution)are recovered from this side.Buyers are tied down to their minimum bene?t levels,so that participation is complete on their side.Again,the antitrust implications are apparent:To balance the demand for platform services in a socially optimal way,one will always observe a non-neglible price gap between the sellers’and buyers’price.Sellers might mistakenly perceive the resulting own price mark-up as a consequence of abuse of market power by the platform. Quantitatively,it is easy to show that for su?ciently small buyers’minimum bene?t levels, the socially optimal sellers’fee is lower than the monopolistic sellers’fee.More precise, t W s≤t M s if and only if b b≤c.(39) So,although the skewed pricing structure is optimal from both a social welfare and a monopoly point of view,a monopolistic platform will nevertheless distort the market by charging suboptimally high tari?s to sellers.This unambiguously leads to underprovision of network services,i.e.D(b b,t M s) To assess the pro?tability of the platform under social optimal pricing,we focus again on the net pro?ts per transaction πq(t W b,t W s,c)=t W b+t W s?c=? b b b?1 ,(40) which is smaller than zero,as we have assumed that b>1.That is, Proposition4.2. Under socially optimal fees(t W b,t W s),the platform sets prices below marginal costs and hence runs at a loss,i.e. t W b+t W s The high social bene?ts of sellers’participation stem from the positive network externality induced by complete buyers’participation.This can easily be seen from substituting t b=b b,βb(b b)=E(b b),and D b(b b)=1,into social welfare function(26),which then reduces to N E(b b)D s(t s)+ ∞t s xh s(x)dx?cD s(t s) .(42) 15Bolt and Tieman(2003)show that this loss under socially optimal fees also occurs under log-concave demand functions,when applying a uniform distribution.In a somewhat di?erent setup of two-sided markets, see Armstrong(2004)for a similar result where socially optimal prices are set below marginal costs. Table2:Social welfare under a constant elasticity distribution Interior Solution Corner solutions Buyer Seller (b b,m s)(m b,b s) Price: buyer fee t b0.0300.0200.031 seller fee t s0.0180.0340.018 total fee t0.0480.0540.049 Demand(in%): buyer demand D b(t b,t s)28.3100.026.9 seller demand D s(t b,t s)96.624.7100.0 total demand D(t b,t s)27.324.726.9 Pro?t: totalπ(t b,t s,c)-4.2-2.5-4.0 per transaction(×10?3)πq(t b,t s,c)-15.2-10.0-15.0 Welfare:W(t b,t s,c) 4.27.0 4.2 Parameters: b=3, s=2.2,b b=0.020,b s=0.018,c=0.064(c b=c s=0.032),N=1000. Optimizing the sellers’price by taking the derivative with respect to t s and equating to zero yields N(?E(b b)h s(t s)?t s h s(t s)+ch s(t s))=0.(43) or t s+E(b b)=c.(44) Hence,in the case of two-sided markets,the standard“price equals marginal cost equation”is augmented by a term which arises from the positive network externality16.In other words, equating marginal costs and marginal revenues on the sellers’side,requires adjusting the socially optimal sellers’fee to re?ect the positive externality of complete buyers’participation, as measured by their average bene?t of platform services.However,as this average bene?t which accrues to sellers is larger than the price buyers pay,that is,E(b b)= b b b/( b?1)>b b, costs cannot be fully recovered.The resulting negative pro?t per transaction is equal to t W b+t W s?c=b b?E(b b)=? b b b?1 <0. These losses incurred by the platform in the social optimum could warrant compensation through a goverment subsidy or cross subsidization from other sources of income.In the case of payment networks,imposing an interchange fee may also alleviate the problem(see e.g. Wright(2003b)and Rochet and Tirole(2002)). To illustrate the social welfare e?ects quantitatively,we revisit the example. 16It is straightforward to show that in the corresponding one-sided speci?cation of our model,the expectation term E(b b)drops out,so that prices equal marginal costs again in the one-sided market. Figure 4:Social welfare in the corner solutions 3456789 24 6 8 10 b W ˉW b =5.1ˉW s =0Explanatory note:The dotted line represents W (b b ,w s ,c )and the straight line represents W (w b ,b s ,c )as a function of b . Example (continued):Social welfare As before,we set the following parameter values:c =0.064(c b =c s =0.032),b b =0.020,b s =0.018, b =3, s =2.2,and N =1000.Table 2summarizes the results for the diferent solutions. First,from the table it is immediately clear that the interior solution does not yield max-imum social welfare.Maximum social welfare is achieved at the corner solution in which the buyers’fee is set at the minimum bene?t level.Figure 4shows social welfare in both corner solutions W (b b ,w s ,c )and W (w b ,b s ,c )as a function of b .The limit points are calculated as ˉW b =Nb s s (c ?b b ) 1? s /( s ?1)=5.1and ˉW s =0.Numerically,we verify that ? ≈2.32,so that for every b >2.32we have that the buyers’corner solution leads to maximum pro?ts.Second,following condition (39),as b b =0.020≤c =0.064,socially optimal prices will be lower than the prices set by the monopolistic platform,inducing higher demand.In fact,so-cially optimal demand for network services is almost seven times higher than demand in the monopolistic case.Clearly,network services are undersupplied in the latter case.Third,by Proposition 4.2,the platform incurs operational losses by implementing the socially optimal fees. 4.1Ramsey pricing:Balanced budget fees As shown above,for plausible parameter values,the network will operate at a loss by imple-menting socially optimal prices.Alternatively,we could optimize social welfare W (t b ,t s ,c )under a balanced budget condition t b +t s =c .In this constrained Ramsey setting,it is no surprise that maximum social welfare with a balanced budget is again found at one of the corner solutions,depending on the relative magnitude of the elasticities.In particular,it can be shown that there exists an ? ≥ s such that for all b >? ,it holds that t BB b =b b ,t BB s =c ?b b .(45) Table3:Social welfare and balanced budget prices Social Optimum Ramsey (Unconstrained)(Balanced budget) Price: buyer fee t b0.0200.020 seller fee t s0.0340.044 total fee t T0.0540.064 Demand(in%): buyer demand D b(t b,t s)100.0100.0 seller demand D s(t b,t s)24.714.0 total demand D(t b,t s)24.714.0 Pro?t: totalπ(t b,t s,c)-2.50.0 per transaction(×10?3)πq(t b,t s,c)-10.00.0 Welfare:W(t b,t s,c)7.0 6.5 Parameters:b b=0.020,b s=0.018,c=0.064(c b=c s=0.032),N=1000; c=3, s=2.2. Hence,if buyers are su?ciently price elastic,then sellers end up paying for all the cost of the network.Again,a result similar to(39)holds in the sense that the Ramsey price for the seller is lower than the monopoly fee if buyers’minimum bene?ts levels are su?ciently small. That is,in the constant elasticity case, t BB ≤t M s if and only if b b≤c.(46) s The continued example illustrates the impact of implementing balanced budget fees com-pared to socially optimal fees. Example(continued):Balanced budget As before,the parameter values are c=0.064(c b=c s=0.032),b b=0.020,b s=0.018, N=1000.We impose b=3and s=2.2.Table3gives the results for the constant elasticity distribution. As the total fee is higher than in the unconstrained social optimum,implementing bal-anced budget fees induces a drop in total demand for platform services,in this case from 24.7%to14%.By de?nition,there is also a drop in social welfare,but in this example rather limited one:welfare drops just7%from7.0to6.5.By de?nition,the platform breaks even when implementing balanced budget fees.Further,observe that the balanced budget fee for the seller is lower than the two-sided monopolistic fee,analogous to(46),but higher than the sellers’fee in the unconstrained social optimum,in order to balance the budget. 5Conclusions Many real world two-sided markets have adopted skewed pricing strategies,in which the price mark-up is much higher on one side of the market than the https://www.wendangku.net/doc/1611694339.html,ing a simple of two-sided markets,we show that under constant elasticity of demand these skewed pricing strategies are indeed pro?t maximizing.The underlying mechanism is a positive network externality: By keeping prices low,demand is stimulated on the more elastic side of the market,thus creating increased bene?ts of market participation on the other,less elastic,side.On this side,the platform can subsequently extract pro?ts by setting high prices. In a qualitative sense,this skewed pricing result carries over to the social optimum. We prove that a central planner would choose the same corner solution,in terms of full participation of the more elastic buyers’side of the market and recovering costs from the price-inelastic sellers’side.However,we show that the social planner will price below marginal costs,leading to an underrecovery of costs and hence an operational loss for the platfom.This result is driven by the positive externality on the sellers,exerted by full buyers’participation. The contribution of this positive externality to social welfare leads the social planner to increase sellers’participation by setting the sellers’price below marginal cost.As this low price is not compensated on the buyers’side,where the lowest admissible price is set to induce full participation,the result is an operational loss for the platform.As an alternative to a loss-making social optimum,we investigate Ramsey pricing,i.e.social welfare optimization under a balanced budget constraint.By de?nition,Ramsey pricing results in a zero pro?t. In addition,our analysis seems to indicate that having a platform operate under a balanced budget restriction results in only a limited social welfare loss. We illustrated our general results by analysing the Dutch debit card system and argue that our theoretic results?t the observed properties of debit card pricing in the Netherlands well.Dutch retailers pay a positive transaction fee for accepting debit card payments,while the consumers are not charged per transaction.When we assume that retailers are much less price elastic in their demand for debit card services than consumers,as was found in recent empirical studies on payment systems,this may explain the pricing structure in the debit card market.Hence,these empirical?ndings support our theoretical results. In our view,the antitrust consequences of skewed pricing in two-sided markets is a chal-lenging avenue for further research.In particular,the e?ects on skewed pricing strategies of platform competition,when more than one platform is active in the market,seems a natural extension of our current research.We plan to go down this road of inquiry in the near future.