Resale price maintenance in two-sided markets
Tommy S. Gabrielsen, Bjørn Olav Johansen, Teis L. Lømoy
Department of Economics, University of Bergen, Norway
July 13, 2016
Abstract
We consider a model with two competing two-sided platforms that sell directly
to one side of the market (the “direct side”), and indirectly through an agent to the
other side (the “retail side”). The platforms o¤er non-linear tari¤s to the agent,
and they are allowed to endogenously choose whether to contract with the same
or di¤erent agents. In this setting we study the platforms’ incentives to impose
minimum or maximum RPM, and the e¤ect on the …nal market outcome. We …nd
that, even if customers on both sides value each other’s participation, the …rms
may want to impose minimum RPM on the agent if the competition between the
platforms is su¢ ciently strong, in order to raise prices on both sides of the market
simultaneously. In a linear demand example we show that RPM reduces overall
welfare when the equilibrium involves the use of minimum prices, and increases
welfare when the equilibrium involves maximum prices.
1
Introduction
A two-sided market is characterized by a set of platforms and two distinct consumer
groups, sometimes referred to as “buyers”and “sellers,”that value each other’s participation. Many industries— particularly within “the new economy”— …t this description; the
We would like to thank Joe Farrell and the participants at the Sather Conference on Industrial
Organization at UC Berkeley (2013), Markus Reisinger, Patrick Rey, Greg Sha¤er and Thibaud Vergé for
valuable comments on an earlier draft of this paper. Thanks also to participants at the ANR workshop
“Competition and Bargaining in Vertical Chains” in Rennes (2014).
y
Department of Economics, University of Bergen, Fosswinckels Gate 14, N-5007 Bergen, Norway
([email protected], [email protected], [email protected]).
1
markets for video game consoles, payment cards and online auctions, are typical examples
of two-sided markets, and companies such as Sony, Microsoft, MasterCard and eBay are
well-known examples of platforms.
We often observe that two-sided platforms impose restrictions on the prices that can
be charged by their sellers, sometimes by deciding the seller’s prices directly, and other
times (more indirectly) by imposing contract clauses like resale price maintenance (RPM)
or minimum advertised prices (MAP). Newspapers and magazines, for example, who
sell space to advertisers and sign distribution contracts with retailers, usually impose
…xed RPM on their news vendors (“cover pricing”). Online streaming platforms and
marketplaces, such as Google Play Music, Spotify and Net‡ix, who sell services and
subscriptions to consumers, and write contracts with sellers (e.g., music publishers and
studios), frequently impose restrictions on the sellers’prices –or even dictate the prices
directly. Software and hardware companies (e.g., Microsoft, Apple, Sony ...), who sell
licenses and kits to software developers and sign contracts with retailers, often impose
minimum advertised prices when distributing their products (Windows, Xbox, iOS/Mac,
iPhone, Playstation, ...) to retailers.1
RPM is a much disputed practice in antitrust policy. Minimum or …xed RPM was
for a long time (until very recently) a banned practice in the U.S., through the Sherman
Act, and in Europe it is still on the black list over vertical restraints.2 Policy makers have
generally been much more lenient toward the use of maximum RPM, both in the U.S.
and in Europe. The economic literature, however, is less clear on the subject. Several
papers highlight that there may be an e¢ ciency rationale for the use of minimum RPM,
to facilitate valuable services at the retail level.3 Another branch of the literature (presented below) shows that, in the absence of any e¢ ciency bene…ts, minimum RPM, and
even maximum RPM, may cause higher prices and thus harm consumers. To complicate
matters further, several authors have warned that two-sided markets require a di¤erent
antitrust approach from ordinary markets (Posner, 2001; Evans, 2002; Wright, 2004).
In this paper we analyze how the use of RPM may a¤ect outcomes in a two-sided
market. To maximize pro…ts, a platform must take into account the indirect network
externalities between its sellers and buyers, as well as the pricing strategies of rival plat1
Sony was recently under investigation for imposing resale prices on PlayStation software for wholesalers and retailers in Japan (Sony Computer Entertainment Inc. v. JFTC ).
2
A recent case –Leegin Creative Leather Products, Inc. v. PSKS, Inc., 551 U.S. 877 (2007) –involved
a softer treatment of minimum and …xed RPM in the U.S. These practices are still considered “hard-core”
infringements of EU law.
3
See for example Telser (1960) and Mathewson and Winter (1984), as well as the quality certi…cation
argument preseted by Marvel and McCa¤erty (1984).
2
forms. An optimal pricing strategy will balance these concerns. However, if the platform
does not directly control the prices of its sellers, an incentive problem may arise. The
simplest example of this is perhaps media markets, with newspapers, magazines or TV
channels who earn a high share of their pro…t by selling ads. Media …rms may fear that
their distributors charge too high prices, as this will limit the number of readers or viewers; in turn this will cause the advertisers to reduce their participation as well.4 To induce
a lower price, the media …rm cannot simply o¤er a discount to its distributor, as this also
a¤ects the …rm’s incentive when deciding on how much space or time to allocate to the
advertisers. This therefore seems like an obvious case for imposing maximum RPM, which
would be an e¢ cient instrument for preventing high vendor prices.
To study the situation formally, we consider a model with two competing platforms
that sell directly to one side of the market (we refer to this as the “direct side”), and
indirectly through an agent (a “seller”) to the other side (we refer to this as the “retail
side”). We allow the platforms to sign general non-linear contracts with the agent, and
we allow them to endogenously choose whether to contract with the same seller (common
agency) or di¤erent sellers (exclusive agency). This setting is very similar to the situation
studied in Bernheim and Whinston (1985), with the important di¤erence that the …rms
in our model are two-sided platforms.
In this setting we investigate the platforms’ incentives to impose either maximum
or minimum RPM on the agent, and the e¤ect of this on the …nal market outcome.
Intuitively, we should expect the intuition from above to carry over, i.e., that the platforms
may want to resort to maximum RPM in order to prevent the agent from charging too high
prices to consumers. We con…rm this intuition for the case with a monopoly platform, and
also for the case with competing platforms –provided that the degree of competition is not
too strong. However, with su¢ cient competition we …nd that the platforms instead want
to impose minimum prices on the agent, in order to prevent too low prices to consumers.
Surprisingly, this holds even though both sides of the market highly value each other’s
participation. The intuition for this is as follows. If the platform earns a high margin
on each transaction with the agent, then the platform may want to charge a low price
to the buyers on the direct side of the market, in order to boost the agent’s sales. This
may be the appropriate strategy if the platform is a monopolist, or if the competition
between platforms is very weak. In this case a maximum price is su¢ cient to prevent
the agent from imposing a high margin on its customers. Yet, when the platforms are
4
A related issue arises if platforms deal sequentially with the two sides (Hagiu, 2006) or if one side
lacks information about prices on the other side (Belle‡amme and Peitz, 2014; Hagiu and Halaburda,
2014).
3
close substitutes, competition may have already caused the prices on the direct side to
become “too low,” as seen from the perspective of a fully integrated (horizontally and
vertically) monopolist. To dampen competition and thus increase prices, the platforms
should therefore charge a smaller margin on each transaction with the agent. However, in
turn this may cause the agent to charge too low prices from its customers, again seen from
the perspective of the fully integrated …rm. The platforms can prevent this by imposing
minimum prices. A minimum price on the retail side may therefore help the platforms to
increase prices on both sides of the market simultaneously, we …nd.
We also show that, if RPM is not feasible, then the platforms may sometimes resort
to using exclusive agents (di¤erent sellers) instead. The inutition is the following. For
the case when the platforms would like to impose maximum prices, but are prevented
from doing so, the agent will tend to set prices that are too high in equilibrium, due
to double marginalization. All else being equal, by using exclusive agents the platforms
introduce some competition between the agents, and in turn this will reduce the double
marginalization problem.
Our results are derived with a general demand speci…cation. To get a sense for the
potential welfare implications, we also present a linear demand example. Here we …nd
that for the cases when the platforms would like to impose maximum RPM, overall welfare
always increases when RPM is allowed compared to when it is not. On the other hand,
for the cases when the platforms would like to impose minimum prices, we …nd that RPM
reduces overall welfare. Given that this is just an example, we should be careful about
the interpretation. Yet, the general intuition for why minimum RPM (as opposed to
maximum RPM) may be harmful in our setting, is strong: In our framework, maximum
prices are used by the platforms to prevent double margins on the retail side of the market,
and this may help to increase participation on both sides of the market simultaneously.
This should normally increase welfare –at least for the cases when both sides value each
other’s participation. Minimum prices, on the other hand, are implemented in order to
dampen platform competition on the direct side of the market, while still facilitating high
prices on the retail side. Hence, minimum RPM causes prices to increase on both sides
of the market simultaneously, which normally will harm welfare by reducing the overall
participation on both platforms.
Our paper extends the industrial organization literature on two-sided markets, in
which Caillaud and Jullien (2003), Armstong (2006) and Rochet and Tirole (2006) are
the seminal contributions. In particular, our article is related to an emerging strand of
literature that studies how two-sided platforms can make use of vertical restraints. Re-
4
search in this area, surveyed by Evans (2013), has shown for example that tying can both
induce consumers to multi-home (Choi, 2010) and soften platform competition (Amelio
and Jullien, 2012), and that vertical integration and exclusivity contracts can facilitate
platform entry (Lee, 2013). Our contribution is to consider RPM, which, to our knowledge, has not been done before. However, a related issue is studied by Hagiu and Lee
(2011). In a model of the videogame industry, they distinguish between two business
strategies: Under “outright sale,” a content provider lets his content distributor set the
content price, whereas the provider retains price control under “a¢ liation.”These regimes
correspond to the cases with and without RPM in our paper. Hagiu and Lee (2011) show
that content providers will tend to contract with one distributor exclusively whenever
they give up price control. A di¤erent yet related result arises in our model, as competing
platforms are more likely to enter a into exclusive relations with agents (di¤erent sellers)
in equilibrium, if they cannot impose RPM.5
Outside the two-sided markets …eld, our article relates to the general literature on
RPM. One strand of this literature assumes that contracts between a monopolist manufacturer and its retailers are secretly negotiated (O’Brien and Sha¤er, 1992; Rey and
Vergé, 2004; Montez, 2015; Gabrielsen and Johansen, 2016). These papers illustrate how
either maximum or minimum RPM can be used to increase …nal prices, by helping the
monopolist overcome an opportunism problem when contracting with retailers. In addition, recent work has found that minimum RPM can be used to sustain monopoly prices
in markets where both manufacturers and retailers compete (Innes and Hamilton, 2009;
Rey and Vergé, 2010).6
Our benchmark case, with a monopoly platform, also resembles the situation studied
in part of the double-moral hazard literature, where a principal contracts with an agent,
and where both need to make costly investments. Romano (1994), for example, considers
vertical relations between a manufacturer and a retailer, and where both …rms need to
make investments. He …nds that either maximum or minimum RPM may be used by the
manufacturer to increase the industry pro…ts, but without fully restoring the …rst-best
outcome.
The rest of the article is structured as follows. Section 2 outlines the formal model and
presents a benchmark with a monoply platform. Section 3 contains our main analysis of
the case with competing platforms and an extension with quantity competition between
5
Note that Hagiu and Lee (2011) label …rms in a di¤erent way to us. In particular, our platforms
correspond to their content providers, and our agents correspond to their content distributors.
6
See also Dobson and Waterson (2007), who di¤ers from the literature cited above by assuming that
the manufacturers use (ine¢ cient) linear wholesale prices.
5
platforms. Section 4 conducts a welfare analysis with a linear demand system. Section 5
brie‡y discusses some of our assumptions. Section 6 then concludes. Most of the formal
proofs are in the appendix.
2
The model and a benchmark
We analyze a market with either a single monopolist platform, or two similar but differentiated platforms i 2 f1; 2g that sell their products in a two-sided market. On one
side of the market each platform sells directly to its customers. We will refer to this as
“the direct side,” denoted by d. On the other side of the market, we will assume that
the platform selects one among many homogenous and equally e¢ cient agents, which will
resell the product on the platforms’behalf to the …nal customer. We will refer to this as
“the retail side,” denoted by r. The platform may for example be a game console, like
Sony’s PlayStation, or a newspaper, like the New York Times, while the agent may be a
traditional retailer, like Expert or Costco.7
Each platform incurs a constant (and symmetric) marginal cost, equal to cd when
selling to side d and cr when selling to side r. Fixed costs are normalized to zero. The
agents incur no costs except the prospective payments they make to the platform(s).
We will refer to the platforms’…nal customers on each side of the market interchangeably as “buyers” “customers” or “consumers.” We assume throughout that the buyers
on each side pay linear prices (no …xed fees). We denote by pis the price charged to the
customers on side s 2 fd; rg for platform i 2 f1; 2g, and we denote by qsi = Dsi (ps ; q s )
the resulting quantity demanded and consumed at side s for platform i, as a function of
the price(s) charged on side s, ps = (p1s ; p2s ), and the consumption or participation on the
opposite side
s, q
s
= q1 s; q2 s .
For the case of two platforms our demand system comprises four products. For this
system to be invertible and stable, it is required that the feedback loops between the two
sides are convergent. It can be shown that this holds as long as the cross-group network
7
Note that the model may also …t for online marketplaces and streaming platforms, like Google Play
Music, where the buyers on the direct side are the subscribers and buyers of music tracks, while the
agents on the “retail” side are traditional sellers, like Universal Music Group. The essential feature of
the setting that we are studying, is that the participants on the direct side of the market (e.g., game
developers on PlayStation or subscribers on Google Play Music), have a preference for low prices from
the agent or seller (e.g., Expert or Universal Music Group), while the agent may value either high or low
participation (low or high prices) on the opposite side of the market, depending on the situation. In the
case of media …rms, for example, we know that some consumers may dislike advertisements, in which
case the readers and viewers, as well as the distributors, should have a preference for high prices to the
advertisers on the direct side of the market, all else being equal.
6
externalities are not too strong.8 We will assume that this is the case, and that our
e (p) = (e
demand system therefore has a unique solution q
qsi ) in quantities demanded as a
function of the prices set on each side. In the following, we will omit the tildes and denote
these reduced-form quantities simply by qsi (ps ; p s ) for i 2 f1; 2g and s 2 fd; rg.
To ensure that the reduced-form demands are well behaved, we will impose some
additional regularity conditions. First, we will assume that they are continuously differentiable in all prices almost everywhere, and that the partial derivatives of qsi have
the signs that we would expect. Speci…cally, we will assume that the goods consumed
on side s 2 fd; rg are “gross” substitutes as de…ned by Vives (1999, p. 145), i.e., that
we have both @qsi =@pis < 0 and @qsi =@pjs > 0 for i 6= j 2 f1; 2g, and that direct e¤ects
P
dominate on each side separately but also overall, i.e., we have k @qsk =@pis < 0 as well
P P
as h k @qhk =@pis < 0, for i 2 f1; 2g and s 2 fd; rg. Moreover, we will assume that
s to side s is negative (@Dsi =@q i s < 0), then
P
< 0 and k @qsi =@pk s > 0; and that if the indi-
if the indirect network e¤ect from side
this implies @qsi =@pi
s
> 0, @qsi =@pj s
s to side s is positive (@Dsi =@q i s > 0), then this implies
P
> 0 and k @qsi =@pk s < 0. Our assumptions are common in
rect network e¤ect from side
@qsi =@pi
s
< 0, @qsi =@pj s
models of two-sided markets. It also turns out that, for the case of linear demand, our
assumptions are not more restrictive than needed; for the linear demand system we use
in Section 4, for example, it can be shown that all of our assumptions hold as long as the
restriction @qsi =@pis < 0 holds.
Throughout the analysis we will assume that buyers on side d always attach a positive
value to participation on side r, i.e., that @Ddi =@qri > 0, and that the platform cannot be
active only on the direct side, i.e., if qri = 0 then qdi = 0 as well. On the other hand, we
will allow buyers on side r to either value or dislike participation on side d, i.e., we can
have either @Dri =@qdi > 0 or @Dri =@qdi < 0.9
2.1
Timing of the game
We will consider a game that proceeds in two main stages: At stage 1, each platform i 2
f1; 2g o¤ers a contract to one of the agents. A contract o¤er may consist of a (nonlinear)
tari¤ T i (qri ), which is a function only of the quantity qri of i’s product distributed by the
agent to side r, and (if feasible) a …xed, minimum or maximum resale price, v i , which
8
See Filistrucchi and Klein (2013) for a formal analysis of this problem.
9
These assumptions correspond to many real life markets. For example, advertisers on TV, in newspapers or in magazines always attach a positive value to consumption on the other side — without
viewers/readers, the platform will not attract any advertisers at all. Readers and TV viewers, on the
other hand, can be found to either like or dislike advertising, depending on the context.
7
constrains the agent’s price pir for i’s product.10 After having observed its own contract
o¤er(s), each agent decides whether or not to accept each o¤er. If a platform’s o¤er is
rejected at this stage, it is allowed to make another o¤er, but this time to a di¤erent
agent, who in turn may accept or reject. Accepted contract terms are then observed by
everyone.
We will assume that the …rms can make their terms (v i ; T i (qri )) at stage 1 conditional
on whether the agent serves one or two platforms. This is a natural assumption, as the
optimal contract terms generally will depend on whether the agent serves more than one
platform. If the contract terms were not conditional on market con…gurations, then the
platform and its agent may sometimes be stuck with an ine¢ cient contract, in which
case they would like to renegotiate. Naturally, we will therefore require these conditional
terms to be “renegotiation-proof”— i.e., we will not permit equilibria to be sustained by
“implausible”o¤-equilibrium terms.
At stage 2, each platform …rst decides whether or not to be active. If platform i decides
not to be active, then we simply have qri = qdi = 0 in the continuation game. Next, all
active …rms set prices, which are observed by all customers before demand is realized.
Finally, payments are completed according to the terms of trade accepted at stage 1.
P
Given our assumptions, we can write the pro…t of any agent a as a = k2f1;2g fpkr qrk
T k qrk g if it sells the goods of both platforms, and simply
a
= pir qri T i (qri ) if it sells the
goods of platform i only. Similarly, platform i’s pro…t can be written as
+ T i (qri )
i
= (pid
cd ) qdi
cr qri . The tari¤ T i (qri ) can take a wide variety of forms but is assumed to be
di¤erentiable almost everywhere.
We continue to solve the game in the usual way, looking for a subgame-perfect equilibrium (v ; T ; p ). We will focus on symmetric equilibria that are Pareto undominated
for the platforms.
2.2
Benchmark: A monopoly platform
We start by analyzing the benchmark situation with a monopoly platform. Because we
are dealing with only one platform, we will simply drop the superscripts i and j for the
remainder of this section.
We …rst consider the fully integrated (industry-pro…t maximizing) outcome, i.e., the
situation in which the platform is able to sell directly to both sides of the market siP
multaneously. Industry pro…ts are
= s2fd;rg (ps cs ) qs , and reach a maximum at
10
In principle, one could also allow the tari¤ to depend on the price and quantity on side d. We will
assume that this is not possible. See the discussion in Section 5.
8
M
some price vector that we will denote by pM = pM
d ; pr . We let
M
=
pM denote
industry pro…ts when prices are set equal to pM .
The fully integrated …rm’s …rst-order conditions, evaluated at pM , are then given by
qdM +
X
pM
s
cs
s2fd;rg
@qs
@pd
pM
@qs
@pr
pM
= 0;
(1)
= 0;
(2)
for the price on side d, pd , and analogously,
qrM +
X
pM
s
cs
s2fd;rg
for the price on side r, pr . Here, qdM and qrM represent the participation on each side when
prices are equal to pM . We assume that the monopoly markup on side d is nonnegative,
pM
d
cd
0.11 On the other hand, we will assume that the monopoly markup on the
cr S 0.12
retail side can take any value, positive or negative, pM
r
Now suppose that the platform must use an agent in order to sell to side r: First we
consider the case when RPM is not allowed. At stage 1 the platform o¤ers a tari¤ T to
one of the agents. Suppose the agent accepts the o¤er. Given the contract T , and given
that the platform decides to be active, the platform sets its price to side d while the agent
B
sets the price to side r. We let pB
r ; pd be the equilibrium prices at stage 2 as functions of
the accepted contracts T . Given that it is the platform that dictates the contract terms,
we know that T has to maximize the platform’s pro…t subject to the agent’s zero-pro…t
condition.
max
T (qr )
n
pB
d
cd
B
qd p B
d ; pr
cr q r
B
pB
r ; pd
+ T (qr )
B
B
subject to pB
r qr pr ; pd
o
(3)
T (qr ) = 0
We can then state the following lemma.
Lemma 1. If (T ; pr ; pd ) forms a subgame-perfect equilibrium, then the tari¤ T
is
continuous and di¤erentiable at the quantity qr induced by (pr ; pd ).
11
This assumption is made here for convenience. The case pM
d
12
cd < 0 is covered in Appendix B.
Platforms sometimes incur a loss on one side of the market while making a pro…t on the other side.
It is well known that Sony, for example, often makes a loss when selling videogame consoles to their …nal
customers, while they make up for it by charging their videogame developers. In the same way, many
newspapers make a loss when distributing their papers to readers (e.g., we see many free newspapers)
while they make up for it by charging their advertisers.
9
Proof. See Appendix A.
In addition to simplifying the rest of the analysis, Lemma 1 also provides valuable
insights into which contract arrangements do not occur in any equilibrium. Lemma 1
says that, in equilibrium, a slight increase or decrease in the quantity qr sold to side r
cannot induce a discontinuous change in the payment from the agent to the platform.
The accepted tari¤ may have discontinuities, but the point of discontinuity (the threshold
value) of, say, qr cannot be equal to the equilibrium quantity qr = qr (p ), because then p
would not be immune to pro…table deviations. The intuition for this is straightforward:
In a two-sided market, the quantity sold to side r is a function of the quantity sold to the
buyers on the direct side of the market. Hence, if for example T were to “jump up” at
the equilibrium quantity qr , then the platform could induce a discrete increase in its pro…t
by slightly adjusting its price or the quantity sold to side d (either up or down depending
on the sign of the cross-group e¤ect from side d to side r), so as to cause a slight increase
in the quantity sold to side r. Obviously, the payment T cannot “jump down,”otherwise
it would be pro…table for the agent to increase the quantity sold by charging a slightly
lower price to side r.13 Note that this result gives us a reason to focus on price restraints,
as we do here, and not, for example, quantity restraints or sales-forcing contracts.
Given that the contract T is accepted, the …rms proceed to set their prices at stage
2. We may note that the platform and the agent will not be able to achieve the fully
integrated outcome, even if they are both monopolists. To see this, notice that Lemma 1
together with our assumptions on demand imply that the agent’s and the platform’s pro…t
functions are di¤erentiable at the equilibrium point. The agent’s …rst-order condition for
pro…t maximization is therefore
(pr
T 0)
@qr
+ qr = 0;
@pr
(4)
while the platform’s …rst-order condition is
(pd
cd )
@qd
+ (T 0
@pd
cr )
@qr
+ qd = 0:
@pd
(5)
M
In order for pM
to form an equilibrium, (2) and (4) have to be aligned when evald ; pr
uated at the optimal prices. By using the implicit function theorem14 , we obtain the
13
Note that the intuition for this result resembles the intuition for Lemma 1 in O’Brien and Sha¤er
(1992).
14
We can use the implicit function theorem because our demand system is invertible under the assump-
10
condition
T0
cr =
pM
d
cd
@Dd
< 0;
@qr
(6)
which says that the platform’s markup on the retail side should be negative in order for
the agent to fully take into account the positive feedback on the platform’s sales on the
direct side. On the other hand, it is easy to see that to induce the optimal price to side d,
the platform needs to earn the full monopoly rent on the last unit sold to its agent. That
is, we require that T 0
cr = p M
r
cr when qr = qrM . These two conditions generally do
not hold simultaneously. We can therefore state the following result.
Proposition 1. If RPM is not feasible, a nonlinear tari¤ T (qr ) is not su¢ cient to induce
the fully integrated outcome
M
.
An important insight from the literature on vertical restraints is that a successive
monopoly, and also common agency settings with competing suppliers, can achieve the
…rst-best level of pro…t by using simple nonlinear contracts, e.g., a two-part tari¤ with
a marginal wholesale price equal to the manufacturer’s marginal cost. Such a sellout
contract will avoid double marginalization and allow the agent to maximize industry
pro…ts. The monopoly pro…t can then be shared or collected through a positive …xed fee.
Proposition 1 shows that in a two-sided market this does not work. The reason is that the
marginal wholesale price needs to be set high enough for the platform to fully take into
account the indirect network e¤ects when setting the price on side d— and a high marginal
wholesale price will cause the agent to set the price on side r too high. The second-best
contract (without RPM) therefore involves setting the marginal wholesale price below the
monopoly price to side r, yet above the level that would secure the monopoly price to side
r. All else being equal, this will cause too few sales to the retail side of the market, and
either too many (in the case of a negative indirect network e¤ect @Dr =@qd < 0) or too few
(in the case of a positive indirect network e¤ect @Dr =@qd > 0) sales to the direct side of
the market. The platform is therefore left unable to extract its full monopoly pro…t. This
suggests that there might be some scope for improving pro…ts by letting the platform
impose an RPM clause at stage 1 of the game, which leads us to the following result.
Proposition 2. If RPM is allowed, then a subgame-perfect equilibrium exists in which
the platform imposes a …xed or maximum resale price v = pM
r at stage 1 and collects the
fully integrated pro…t
M
at stage 2.
tion that cross-group externalities are not too strong.
11
Note that the platform always imposes (a …xed or) maximum price at stage 1, never a
minimum price. A maximum resale price imposed at stage 1 solves the problem fully for
the platform by committing its agent not to set the price too high on the retail side. This
implies that the platform is free to set the marginal wholesale price equal to T 0 = pM
r at
stage 1, which in turn will induce it to set the correct price pM
d on the direct side of the
market at stage 2.
In our our linear demand example in Section 4 below, we …nd that maintained prices
(weakly) increases the overall surplus for both the platforms and the buyers when platforms do not compete. This indicates that the use of maximum prices may improve
welfare in two-sided markets.
3
Competing platforms
Our benchmark above shows that there is a rationale for the use of maximum RPM in
two-sided markets, and that maintained prices may be good for the platform’s customers.
One should perhaps expect these insights to carry directly over to the case of competing
platforms. Yet, as we now demonstrate, this is not necessarily the case.
We start again with the fully integrated (industry-pro…t maximizing) outcome as a
benchmark, i.e., the situation with a single …rm that is fully integrated both horizontally
P
P
and vertically. The overall pro…t is now given by = s2fd;rg k2f1;2g pks cs qsk , and
we will again assume that it reaches its maximum for some unique price vector, which
M
M
M
we will denote by pM = pM
d ; pd ; pr ; pr . In the same way as before, we will denote by
M
=
pM the industry pro…ts when the prices are set equal to pM .
The fully integrated …rm’s …rst-order conditions, evaluated at the optimal prices pM ;
are now equal to
M
r
:= qrM +
X
X
pM
s
X
X
pM
s
cs
s2fd;rg k2f1;2g
@qsk
@pir
pM
@qsk
@pid
pM
= 0;
(7)
= 0;
(8)
for the price on side r, and
M
d
:= qdM +
cs
s2fd;rg k2f1;2g
for the price on side d, for i 2 f1; 2g. We will continue to assume that the fully integrated
monopoly markups are nonnegative on side d, pM
d
12
cd
0, and that they can take any
value on side r, pM
r
cr S 0.15
Suppose next that the platforms must use an agent in order to sell their goods to side
r. We solve the game …rst for the case without RPM. We take advantage of the fact that
Lemma 1 easily extends to the case of competing platforms.16
Lemma 2. If (T ; p ) forms a subgame-perfect equilibrium in the game with competing
platforms, then for each platform i 2 f1; 2g the accepted tari¤ T i is continuous and
di¤erentiable at the quantity qri induced by p .
Proof. See Appendix A.
Lemma 2 allows us to restrict the analysis to tari¤s that have a constant marginal wholesale price for all qri > 0, without loss of generality. Hence, in the following we will assume
that the …rms use tari¤s on the form
(
T
i
qri
=
wi qri + F i if qri > 0
fi
if qri = 0
for i 2 f1; 2g, where wi is a (constant) marginal wholesale price, and F i S 0 and f i S 0
are …xed transfers. Note that if either f i = 0 or f i = F i , this amounts to a typical
two-part tari¤.
In the game with competing platforms, the outcome at stage 2 depends on whether
the platforms adopt a common agent or di¤erent exclusive agents at stage 1 of the game.
Suppose …rst that the platforms choose a common agent. It is then possible to show that,
in any common agency equilibrium that is Pareto undominated for the platforms, the
equilibrium contract terms (w ; F ; f ) will have to maximize the sum of the platforms’
pro…ts, subject to the agent’s zero-pro…t condition, i.e.,
max
w;F
X
pBi
wi ; wj
d
i2f1;2g
subject to
cd qdi pB + wi
X
pBi
wi ; wj
r
cr qri pB + F i
wi qri pB
Fi
(9)
= 0:
i2f1;2g
i
j
where pB = pBi
s (w ; w ) is the vector of equilibrium prices at stage 2, as functions of the
contract terms signed at stage 1. However, just like in the case with a monopoly platform,
a pair of nonlinear tari¤s is generally not su¢ cient to induce the overall …rst-best outcome
15
The case pM
d
16
This holds for any subgame, whether the platforms use a common agent or exclusive agents.
cd < 0 is again covered in Appendix B.
13
for the platforms and the agent. To see this, note that, when both platforms are active,
the agent’s …rst-order condition with respect to retail prices is
i
r
:= qri +
X
pkr
wk
k2f1;2g
@qrk
= 0;
@pir
(10)
for i 2 f1; 2g, while for the platform i the …rst-order condition is
i
d
:= qdi + pid
cd
@qdi
+ wi
@pid
cr
@qri
= 0.
@pid
(11)
To induce the fully integrated prices on side r, we know that (7) and (10) have to be
aligned. Evaluating the two expressions at the prices pM , and using the implicit function
theorem, we obtain the condition
wi
cr =
pM
d
cd
X @Dk
d
@qri
k2f1;2g
(12)
< 0.
pM
On the other hand, to induce the fully integrated prices on side d, (8) and (11) have
to be aligned. When evaluating the two expressions at the optimal prices pM , we obtain
the condition
wi
cr = p M
r
cr +
X
pM
s
cs
s2fd;rg
@qri
@pid
@qsj
@pid
7 0:
(13)
pM
(12) and (13) are generally not the same. Hence, we can conclude that it is not possible
to induce the fully integrated outcome with only nonlinear tari¤s.17
De…ne pC = (pC;r ; pC;r ; pC;d ; pC;d ) as the prices in a common agency situation that we
obtain by 1) simultaneously solving the …rms’…rst-order conditions (10) and (11) at stage
2, and then 2) solving the maximization problem (9) at stage 1. De…ne as
pro…t when prices are equal to pC . Next, de…ne as
N
C
the industry
the equilibrium industry pro…t if
each platform is simply assigned an exclusive agent before contracting takes place (i.e., in
a game of “intrinsic exclusive agency”). We can then state the following result.
Proposition 3. If RPM is not feasible, a nonlinear tari¤ T i (qri ) for each platform
i 2 f1; 2g is generally not su¢ cient to induce the fully integrated outcome
M
. As a
17
A similar insight appears in an earlier version of Kind et al. (2016), who use a linear demand model
to analyze a TV industry with viewers that dislike advertising.
14
consequence, a common agency equilibrium may not always exist: If
C
N,
then
a Pareto undominated subgame-perfect equilibrium exists in which the platforms adopt a
common agent at stage 1 and collect the pro…t
C
at stage 2. If
C
<
N,
then a
common agency equilibrium does not exist.
Proof. See Appendix A.
Proposition 3 is an extension of Proposition 1 to the case of competing platforms.
To induce the agent to take into account the positive indirect network e¤ects exerted on
the direct side of the market, the platforms should sell their goods at a wholesale price
below cost according to (12). On the other hand, the wholesale margin should also take
into account the platform’s incentives when selling to the direct side of the market, which
(because of platform competition) now includes the incentive to set low prices in order to
steal customers from the rival. The latter implies that the appropriate wholesale margin
(13) could be either positive or negative, depending on the indirect network externalities
and the degree of competition between the platforms. Obviously, the wholesale margin
cannot (on a general basis) achieve both goals simultaneously, and the platforms are
therefore unable to extract monopoly rents.
Note …nally that because the platforms are unable to achieve the fully integrated
pro…t when using a common agent, this opens the possibility that the platforms will
opt to use exclusive agents instead. The intuition for this is as follows: In a one-sided
market, e.g., if the platforms were to sell to side r only, we know that the platforms
would always i) select a common agent and ii) supply this agent at cost (wi = cr ), hence
eliminating any upstream margins and maintaining retail prices at the monopoly level,
as for example in Bernheim and Whinston (1985). The platforms could then collect
the monopoly pro…ts through …xed fees. In a two-sided market, however, supplying the
distributor at cost is not necessarily optimal, because, in the presence of indirect network
externalities, the wholesale margin wi
cr also a¤ects the platforms’optimal quantities
and prices on the direct side of the market. Hence, without RPM, when choosing the
wholesale margin, there is a trade-o¤ between the concerns about the prices on the two
sides of the market. This trade-o¤ can sometimes, e.g., if the indirect network e¤ect on the
retail side is negative, and platforms are close substitutes, lead to excessively high prices
on the retail side, and to such an extent that the industry pro…t would be greater with
exclusive distribution. All else being equal, exclusive distribution would reduce double
marginalization and induce lower prices on the retail side, due to the competition between
the two agents.
15
Proposition 3 suggests again that there is scope for improving pro…ts by letting the
platforms impose RPM clauses, which leads us to our main result.
Proposition 4. If RPM is feasible, then a Pareto undominated subgame-perfect equilibrium exists in which the platforms adopt a common agent and set the marginal wholesale
price according to (13) at stage 1, and collect the fully integrated pro…t
M
at stage 2.
If the indirect network e¤ect on side r is negative (@Dri =@qdi < 0), then the appropriate RPM clause is always a (…xed or) maximum price.
If the indirect network e¤ect on side r is positive (@Dri =@qdi > 0), then the appropriate RPM clause is a (…xed or) minimum price i¤ the diversion ratio between the
@qdj @qdi
platforms on side d ij
:=
=
is su¢ ciently large, and a (…xed or) maximum
dd
@pid @pid
price otherwise.
Proof. See Appendix A.
Proposition 4 says that, even when platforms compete, an equilibrium exists in which
the platforms impose RPM clauses at the …rst stage of the game and extract the fully
integrated pro…t at the …nal stage. Moreover, depending on the signs of the indirect network externalities and the degree of substitution between the platforms, the appropriate
RPM clause may now turn out to be a minimum price. Recall from Proposition 2 that if
the platform is a monopolist the appropriate RPM clause is always a (…xed or) maximum
price. To understand this result, consider …rst the case with @Dri =@qdi > 0, and recall that
the equilibrium wholesale price wi is set according to (13).18 By substituting (13) into
i
r
i
r
M
r and evaluating the resulting expression at the fully integrated prices,
M
r < 0— in which case minimum RPM is appropriate— as long as
ij
dd
In (14),
ij
ss
>
ii
rd
1
+
ij
rd
ij
rr
ii
dr
pM
r
pM
d
cr
cd
ii
dr
+
ij
dr
:
we …nd that
(14)
= @qsj =@pis = ( @qsi =@pis ) is the (same-side) diversion ratio between the
platforms on side s 2 fd; rg;
ii
dr
= @qri =@pid = ( @qdi =@pid ) and
18
ii
rd
= @qdi =@pir = ( @qri =@pir )
Note that, depending the per-unit costs, the degree of substitution between the platforms and the
size and signs of the indirect network externalities, the marginal wholesale price in (13) could, in theory,
become either positive or negative. In the latter case, we will assume that the platforms can require the
agent to sell all that it purchases or, alternatively, that it can require the return of unsold units at the
same marginal wholesale price.
16
are the cross-side diversion ratios from side d to r and side r to d, respectively, for
platform i 2 f1; 2g— i.e., the fraction of the quantity lost by platform i on one side
that is captured by platform i on the opposite side; and
ij
rd
ij
dr
= @qrj =@pid = ( @qdi =@pid ) and
= @qdj =@pir = ( @qri =@pir ) are the cross-side diversion ratios between platforms, from
side d to r and side r to d, respectively— i.e., the fraction of the quantity lost by platform
i on one side that is captured by platform j on the opposite side.
Condition (14) says that, for the case of positive indirect network e¤ects, the resale
price has to be imposed as a (…xed or) minimum price whenever the diversion ratio between
the platforms is su¢ ciently large on side d.19 The intuition for this is straightforward.
Suppose the platforms’ wholesale markups are positive, i.e., wi
cr > 0 for i 2 f1; 2g.
Given that the indirect network externality on the retail side of the market is positive, for
each new customer captured on side d the platform gains @Dri =@qdi new customers on side
r, and each of them will net the platform wi
cr . This creates an additional incentive
for the platform to set a low price on side d, and this incentive grows stronger the larger
the indirect network externality @Dri =@qdi becomes and the stronger the diversion between
the two platforms becomes. To counter this incentive, the marginal wholesale price wi
should therefore be reduced. However, for a su¢ ciently low marginal wholesale price wi ,
the agent would like to set the retail price pir below pM
r . The platform can prevent this
by imposing a minimum price at stage 1 of the game.
A note on the structure of the equilibrium tari¤s Di¤erent from in a one-sided
market, the platforms’upstream margins, wi
cr and pid
cd , are typically non-zero in
the equilibria described in Proposition 3 and 4. If platform i were to adopt a “normal”
two-part tari¤ (i.e., with f i = 0 or f i = F i ), the rival platform j would not internalize
all the upstream margins when making his o¤er to the agent, and pro…table deviations
would then exist.20 This would mean that the equilibrium contract terms do not solve
the problem (9), as we have claimed above.
However, if only a part of the …xed transfer was contingent on actual trade taking
place (i.e., if F i 6= f i > 0), the platform could secure its pro…t “up front,” which in turn
would deter any deviations by the rival at the contracting stage. To see this, suppose that
19
Note that the diversion ratio in a two-sided market is a more complex variable than the diversion
ratio in a one-sided market. The reason is that the diversion ratio in a two-sided market not only re‡ects
the degree of substitution between the products on one side of the market; it also re‡ects feedback e¤ects
from demand shifting to the other side of the market.
20
In contrast, in the one-sided market analyzed by Bernheim and Whinston (1985), for example, the
jointly optimal upstream margins are always equal to zero. Normal two-part tari¤s are then su¢ cient to
sustain the candidate equilibrium.
17
RPM is not feasible, that f i =
pC;d
C =2,
and that F i is set according to
cd qdi (pC ) + (wC
cr ) qri (pC ) + F i =
C
2
Given these terms, and provided that platform j did not deviate, platform i earns
C =2
whether it decides to trade at stage 2 or not. Any deviation by the rival would induce
platform i not to trade at the …nal stage of the game. In turn this would harm the joint
pro…t of the agent and platform j, making the deviation unpro…table. The equilibrium
contract terms therefore need to have F i 6= f i > 0 in order to induce either the outcome
C
without RPM (Proposition 3) or the outcome
M
with RPM (Proposition 4).
Note that these tari¤s are structurally similar to and have the same strategic value
as the non-linear tari¤s analyzed by Miklós-Thal et al. (2011) and Rey and Whinston
(2013), both based on the “three-part tari¤s”introduced by Marx and Sha¤er (2007). In
the latter papers the market is one-sided, and the upstream margins are positive instead
because of the need to counter the downstream rivalry between the retailers, who are
making contract o¤ers to a monopolist supplier.21
3.1
The case of quantity competition
An assumption sometimes imposed in the two-sided literature, either explicitly or implicitly, is that the platforms choose quantities or participation rates on one side of the market
and prices on the other. This assumption guarantees the uniqueness of equilibria in the
customer market, and hence the platform can achieve any desired allocation of goods to
its customers on both sides of the market. See, e.g., Anderson and Coate (2005), Weyl
(2010) and White and Weyl (2012).22
To see how the assumption of quantity competition a¤ects our results, imagine that
each platform i 2 f1; 2g chooses a quantity qdi on the direct side of the market, selling it
at the market clearing price pid = Pdi (qd ; qr ), while the agent still charges prices, facing
demand functions qri = Dri (pr ; qd ) for i 2 f1; 2g on the retail side of the market. We
will assume that @Pdi =@qdi < 0, @Pdi =@qdj < 0, @Pdi =@qri > 0 and @Pdi =@qrj = 0.23 The
21
Also di¤erent from our paper, RPM is not needed to induce the fully integrated outcome.
22
Alternatively, one has to rely on the invertibility of the full demand system (comprising four products
in our setting) and the contraction mapping approach used by A¤eldt et al. (2013) and Filistrucchi and
Klein (2013).
23
In a newspaper advertising market, for example, @Pdi =@qrj = 0 amounts to the assumption that the
marginal pro…t of advertising in newspaper i is independent of the number of readers of newspaper 2,
for given advertising volumes qd1 ; qd2 and for a given number of readers qri of newspaper i, which seems
18
overall industry pro…t is now
=
P
k2f1;2g
Pdk (q)
cd qdk +
P
k2f1;2g
pkr
cr Drk . We
M
denote the pro…t-maximizing prices and quantities for the fully integrated
let pM
r ; qd
…rm. The …rst-order conditions are
M
r
X @P k @Dk
X
r k
d
q
+
@qrk @pir d
:=
k2f1;2g
pkr
cr
k2f1;2g
@Drk
+ Dri = 0;
@pir
(15)
with respect to the price pir , and
M
d
:= Pdi
cd +
X
@Pdk @Pdk @Drk
+
@qdi
@qrk @qdi
k2f1;2g
+
X
pkr
cr
k2f1;2g
qdk
(16)
@Drk
= 0;
@qdi
with respect to the quantity qdi , for i 2 f1; 2g.
Consider a candidate equilibrium in which the platforms adopt a common agent at
B
stage 1, and let pB
be the …rms’ equilibrium prices and quantities at stage 2, as
r ; qd
functions of the contract terms (wi ; wj ) accepted at stage 1. Suppose again that the
accepted o¤ers maximize the sum of the platforms’pro…ts, subject to the agent’s zeropro…t condition.
max
w;F
X
B
Pdi qB
d ; qr
cd qdBi wi ; wj + wi
i2f1;2g
subject to
X
pBi
wi ; wj
r
B
i
cr Dri pB
r ; qd + F
B
wi Dri pB
r ; qd
Fi
(17)
= 0
i2f1;2g
Assuming that the tari¤s are both continuous and di¤erentiable at the equilibrium point,
the agent’s …rst-order conditions at stage 2 of the game are24
i
r
:=
X
pkr
wk
k2f1;2g
@Drk
+ Dri = 0;
@pir
(18)
for i 2 f1; 2g, while the …rst-order condition for platform i is
i
d
:= Pdi
cd +
@Pdi @Pdi @Dri
+ i
@qdi
@qr @qdi
qdi + wi
cr
@Dri
= 0;
@qdi
(19)
reasonable.
24
The proof that the tari¤s are in fact di¤erentiable at the equilibrium point is analogous to the proof
of Lemma 2 above. It is omitted here for the sake of brevity.
19
M
for i 2 f1; 2g. We may then note that for pBi
r = pr to hold, the condition
i
r
M
r
=
has to hold when evaluated at the quantities and prices of the fully integrated …rm. By
symmetry, we can rewrite this condition as25
w
i
cr =
qdM
@Pdi
< 0:
@qri (pM
M
r ;qd )
(20)
Again we may note that as long as @Pdi =@qri > 0, the marginal wholesale price should be
set below the platform’s marginal cost in order to induce the agent not to set the price
too high.
On the other hand, we may note that for qdBi = qdM to hold, the condition
i
d
=
M
d
has to hold when evaluated at the quantities and prices of the fully integrated …rm. We
can write this condition as26
wi
cr = p M
r
cr +
@Pdj @Pdj @Drj
+
@qdi
@qrj @qdi
!
qdM + pM
r
cr
@Drj
@qdi
@Dri
@qdi
7 0:
(21)
M
(pM
r ;qd )
We can see that the conditions (20) and (21) are generally not the same, and we
may therefore conclude that simple nonlinear tari¤s are not su¢ cient to sustain the fully
integrated prices and quantities. We may then state the following result.
Proposition 5. (Quantity competition) A pair of imposed resale prices equal to pM
r ,
combined with a pair of marginal wholesale prices set according to condition (21), fully
restores the integrated outcome
M
at the …nal stage of the game.
If the indirect network e¤ect on side r is negative (@Dri =@qdi < 0), then the appropriate RPM clause is always a (…xed or) maximum price.
If the indirect network e¤ect on side r is positive (@Dri =@qdi > 0), then the appropriate RPM clause is a (…xed or) minimum price i¤ the degree of substitution between
@Pdi @Pdi
the platforms on side d, 'd :=
=
, is su¢ ciently large, and a (…xed or)
@qdj @qdi
maximum price otherwise.
25
Note that (20) is equivalent to condition (12), which we obtained for the case of price competition.
26
Note that condition (21) is analogous to condition (13), which we obtained for the case of price
competition.
20
Proof. Given that the indirect network externality is positive, @Dri =@qdi > 0, that the
marginal wholesale price is set according to (21) and that the platforms’ quantities on
side d are equal to qdM , the condition that the agent would like to set the price pir < pM
r
is equivalent to the condition that (21) < (20). With some rewriting we can express this
condition as follows:
pM
r
cr + qdM
'd >
where 'd :=
@Pdi @Pdi
=
@qdj @qdi
@Pdi
@qri
@Pdi M
q
@qdi d
X @Dk
r
@qdi
k2f1;2g
;
(22)
M
(pM
r ;qd )
represents the degree of substitutability between the two
)
(
M
platforms’services on side d when the prices and quantities are equal to pM
r ; qd . The
M
pM
r ;qd
condition says that the appropriate price restraint is a minimum price as long as the
degree of substitution between the platforms is su¢ ciently high on side d:
If the indirect network externality is negative, @Dri =@qdi < 0, condition (22) instead
becomes the condition that the agent would like to set the price above the integrated
price, pir > pM
r . Using symmetry, we may note that the fully integrated …rm’s …rst-order
condition with respect to pir , when evaluated at the optimum, can be expressed as
pM
r
cr + qdM
@Pdi
=
@qri
Dri
X @Dk > 0:
r
@pir
(23)
k2f1;2g
According to this, the right-hand side of condition (22) is always negative whenever
X
@Drk =@qdi < 0. Hence, the appropriate price restraint is then always a maximum
k2f1;2g
price. Q.E.D.
To show that the outcome described in Proposition 5 also is the outcome of a subgameperfect equilibrium of the full game when RPM is allowed, the proof follows the same
structure and is analogous to the proof of Proposition 4 above. It is omitted here for the
sake of brevity.
We may note, similar to the case of price competition, that a positive network externality from side d to side r; @Dri =@qdi > 0; means that the appropriate RPM clause
is sometimes a …xed or minimum price. The intuition for this result is the same as the
21
intuition under price competition: with a positive indirect network externality, for each
additional unit sold on side d the platform sells @Dri =@qdi > 0 additional units on side r. If
the platform has a positive markup wi
cr > 0 on each unit sold to side r, then this may
create an incentive to set the participation rate qdi on side d of the market too high, and
this incentive grows stronger the larger the degree of substitution between the platforms
becomes. This incentive can be dampened by reducing the upstream margin wi
cr , and
by imposing a minimum resale price to prevent the agent from then setting the …nal price
pir too low.
4
A linear demand example
To get a sense of the potential implications of allowing RPM for the platforms’customers,
we will consider an example with a single representative customer on each side of the
market s 2 fd; rg, each maximizing a surplus function equal to
Vs =
X
i2f1;2g
qsi
1
2 (1 + 's )
X
2
qsi
+ 2's qs1 qs2
2ns
X
q i s qsi
i21;2
i21;2
!
X
pis qsi :
(24)
i2f1;2g
From Vs we obtain the inverse demand functions
pis = Psi (qs ; q s ) = 1
1
qsi + 's qsj
1 + 's
ns q i s ;
(25)
for i 2 f1; 2g and s 2 fd; rg. In (24) and (25), ns 7 0 is a measure of the indirect network
e¤ect and 's 2 (0; 1) measures the degree of substitution between the platforms on side
s.
When inverting the system (25) for side s, we obtain the following direct demand
function as a function of prices on side s and quantities on side
Dsi (ps ; q s ) = 1 + ns
q i s 's q j s
(1 's ) (1 + 's )
s:
pis 's pjs
;
1 's
(26)
for i 2 f1; 2g and s 2 fd; rg. To make the analysis analytically tractable, we have
to impose some additional symmetry conditions. We therefore consider the case with
positive and symmetric network externalities nd = nr = n > 0, and we will assume that
the platforms are equally di¤erentiated on both sides of the market, 'd = 'r = ': To
ensure a unique and economically meaningful solution for the reduced form quantities,
our demand parameters must then satisfy n + ' < 1.
22
Indirect network effect n
Fixed or
max RPM
WM  WC  0
Fixed or min RPM
WM  WC  0
Degree of platform substitution 
Figure 1: RPM regimes and their welfare e¤ects when platforms compete by setting prices.
We focus on common agency equilibria in this example. This restriction does not affect our results in any signi…cant way, but makes it easier to derive analytically tractable
expressions for the whole parameter space. Equilibria with exclusive agents (see Proposition 3) could be ruled out formally by introducing small economies of scale on the retail
side.27 We also assume that the platforms’ marginal production costs are the same on
each side of the market, cd = cr = c.
For the case without RPM, we know that a unique Nash equilibrium wi = wj =
wC exists at stage 2 of the game, which is the solution to the maximization problems
(9) and (17) under price and quantity competition, respectively. We de…ne W M :=
P
P
M
M
and WC := s2fd;rg VC;s + C as the overall welfare, with and without
s2fd;rg Vs +
RPM, respectively. In Figures 1 and 2 we have plotted the loci for which maximum or
minimum RPM is appropriate, and for which W M
WC > 0 and W M
WC < 0, for
price and quantity competition between the platforms, respectively.
Notice that when the appropriate RPM clause is a minimum price, then the e¤ect
of RPM on overall welfare is negative, whereas when the appropriate RPM clause is a
maximum price, the e¤ect is positive. Banning …xed and minimum RPM clauses would
therefore be bene…cial in our example, whereas banning maximum prices would be detri27
Allowing for equilibra with exclusive agents would only increase the welfare gains from allowing
platforms to impose maximum prices, for the cases when the degree of substitution between platforms is
very weak but positive.
23
Indirect network effect n
Fixed or
max RPM
WM  WC  0
Fixed or min RPM
WM  WC  0
Degree of platform substitution 
Figure 2: RPM regimes and their welfare e¤ects when platforms compete by setting
quantities.
mental.
Note that this is just an example, but the main intuition should apply more generally:
When platform competition is strong, all else being equal, the platforms tend to set prices
on the direct side that are too low compared with the prices a monopolist would set. When
the wholesale prices are reduced, competition between the platforms is softened (given
that the indirect network e¤ects are positive) and the platforms will respond by increasing
their prices. A reduction in the wholesale prices, however, will cause the agent to reduce
his retail prices as well. This prevents the …rms from fully eliminating competition, and
as a result the equilibrium prices will end up below the monopoly level on both sides of
the market. By imposing a minimum resale price, the platforms can prevent the agent
from reducing his retail prices, and hence the prices can be increased to the monopoly
level on both sides of the market simultaneously.
On the other hand, when platform competition is weak, the platforms tend to set
prices on the direct side that are too high compared with what a monopolist would do.
When the wholesale prices are increased, competition between the platforms is increased
(given that the indirect network e¤ects are positive) and the platforms therefore respond
by reducing their prices. An increase in the wholesale prices, however, will cause the agent
to increase his retail prices as well. This prevents the platforms and the agent from fully
internalizing the positive network e¤ects, and as a result the equilibrium prices will end
24
up above the monopoly level on both sides of the market. By imposing a maximum resale
price, the platforms can prevent the agent from increasing his retail prices, and hence the
prices can be reduced on both sides of the market simultaneously.
Finally, we may note that the range for which minimum prices should be used is smaller
when the platforms set quantities, compared with the range when they compete in prices.
This is natural, given that competition is tougher when …rms set prices compared with
when they set quantities, all else being equal. The prices on both sides of the market
are therefore generally lower in the Nash equilibrium without RPM when the platforms
compete in prices, and a minimum price is therefore appropriate for a wider range of
parameter values.
5
Discussion
Our analysis demonstrates how two rival two-sided platforms operating with a common
agent may impose RPM to achieve the fully integrated outcome. We believe that the
main mechanism through which the platforms are able to achieve this in our model, may
provide valuable insights for antitrust authorities and regulators. In the following we will
brie‡y discuss the robustness of our results with respect to our assumptions about the
contracts and the structure of the market.
We have assumed that the contract signed between a platform and an agent may
depend only on the quantity distributed by the agent and (in the case of RPM) on the
agent’s price. This assumption is not innocuous, although we feel that it is both natural
and probably in line with what we observe in reality. As an alternative, we could have
assumed that the contracts depend also on the quantities and prices set by the platforms
on the direct side of the market. Such contracts could, in theory, enable the platforms
to induce the fully integrated outcome, as an alternative to imposing RPM. In practice,
however, it may be costly to use such contracts. First, it may be prohibitively costly for an
agent (or for a court) to monitor and verify the prices set on the direct side of the market,
especially if these prices are privately negotiated, as they probably sometimes are. The
quantities on the direct side, on the other hand, are sometimes directly observed — e.g.,
it is possible to observe the amount of ads in newspapers and commercials on TV. Yet,
they would still need to be counted and veri…ed by the agent, which is likely to be costly
compared to the cost for the platform of monitoring the agent’s retail price (or compared
to the cost of simply printing the retail price on the product cover). Finally, we may
note that in some cases not even the quantity on the direct side is directly observed by
25
the agent.28 In sum, we believe that these factors are likely to explain why, for example,
newspapers and magazines choose to use fairly simple wholesale contracts in combination
with cover pricing (RPM) — instead of …xing the prices for ads, or, alternatively, giving
wholesale discounts based on the amount of ads they sell.
On the subject of downstream competition, note …rst that real life retail markets often
comprise imperfectly substitutable agents or outlets, and not isolated retail monopolies
such as those in our model. Another layer of complexity would arise if we were to introduce
multiple imperfectly substitutable retail locations in our model, as is done, for example,
in Rey and Vergé (2010). As in Rey and Vergé (2010), however, we believe that not
much would change in our results if we were to simply include a second retail location
with its own set of agents. The fact that the second location is imperfectly substitutable
for the …rst matters very little: The o¤ers to the agents should still satisfy their zeropro…t condition; it would still be in the platforms’best interest to coordinate and use a
common agent at each location (either because there are economies of scale or because
the platforms can use RPM); and RPM would make it possible for the platforms to secure
the …rst-best “collusive” outcome for exactly the same reasons as in our original model.
However, because retail locations are competing, equilibrium prices will tend to be lower,
which may imply that minimum resale prices will be appropriate over a wider range of
parameter values.
Another contentious assumption is the premise that the agents are perfectly substitutable and therefore earn zero pro…t in equilibrium. The platforms in our model are
therefore able to appropriate all of the pro…ts created in equilibrium. Yet, we will argue
that our results would not change drastically if there were, for example, a single agent
with monopoly power downstream: An equilibrium would still exist in which the platforms imposed RPM clauses at the …rst stage of the game, and the fully integrated prices
and pro…ts were maintained at the second stage. The only di¤erence from our original
framework would be that each platform may have to leave the agent some rents, in order
to keep the rival from inducing the agent to serve it exclusively (assuming the platforms
cannot bypass the agent).
Finally, a third possibility is, of course, a combination of the market conditions listed
above: The downstream market may comprise imperfectly competing “bottlenecks,” as
in the one-sided market studied in Rey and Vergé (2010, pp. 945–951). It is beyond the
scope of this paper to describe what would happen in such a market. However, based on
28
As an example, it is not possible to directly observe the number of software developer kits sold by
Sony and Microsoft for their video game consoles. This is observed only indirectly, through the amount
of software and games released for each system.
26
the results in Rey and Vergé (2010), in such a market an equilibrium may not exist in
which all agents and platforms are active at the same time.
6
Concluding remarks
The existing literature on two-sided markets holds that rival two-sided platforms have
little to gain by coordinating their prices at a high level on one side of the market only,
as this will only induce them to compete more …ercely when selling to the other side. In
this paper we have shown that this reasoning does not hold when platforms sell to one
of the sides through an agent. More speci…cally, we have shown that two rival platforms
can induce prices at the fully integrated level on both sides of the market by using RPM
to …x their agent’s prices on the retail side of the market. Moreover, we have shown that
the appropriate RPM clause, i.e., whether it should be a minimum or maximum price,
will depend on i) the sign of the indirect network e¤ects on each side of the market, and
ii) the degree of substitution between the two platforms.
Finally, we have analyzed how the use of RPM in our setting may a¤ect overall welfare
in a linear demand example. Using a speci…c utility function, and assuming that the
indirect network e¤ects are positive, we …nd that when minimum RPM is used, the e¤ect
on welfare is negative, whereas when maximum RPM is used, the e¤ect is positive. This
suggests that the logic behind the current antitrust policy towards RPM in ordinary
markets, also applies in two-sided markets.
27
Appendix A
Proof of Lemma 1 (Monopoly). The proof is structurally identical to the proof of
Lemma 2 (see below) and is proved by deleting all superscrips i in the proof of Lemma 2.
Q.E.D.
Proof of Lemma 2 (Duopoly). The proof is comprised of the following three steps:
Step 1. We show that T i is continuous at the quantities induced by p . To see this,
assume that T i is not continuous at the quantities induced by p . Then, a marginal
deviation, either positive or negative, from qri (p ) = qr would cause a discrete change in
T i . If such a deviation causes T i to “jump up,”then, since @qri =@pid 6= 0, platform i could
adjust pid slightly to change qri , causing a discrete increase in his pro…ts through a larger
payment from the distributor. Since the jump up can be caused by both a positive and
a negative marginal deviation from qri , the appropriate adjustment of pid depends on how
qri varies with pid . For instance, to get qri > qr , platform i should reduce pid slightly when
qri falls in pid and increase pid slightly when qri rises in pid . The opposite adjustments are
needed to get qri < qr . If a marginal deviation causes T i to “jump down,”platform j and
the distributor could change T j , i.e. qrj , slightly, resulting in a discrete increase in their
bilateral pro…ts. Thus, in both cases of discontinuity, at least one player has a pro…table
deviation. Hence, T i must be continuous at the quantities induced by p .
By step 1., T i has both a right-hand (+) and a left-hand ( ) partial derivative wrt.
dT i
dT i
qr ,
and
respectively. For T i to be di¤erentiable in equilibrium, we
dqri +
dqri
dT i
dT i
require that
=
. We show this in two steps, as follows:
dqri +
dqri
Step 2. We show that
that
dT i
dqri
dT i
dqri
+
@ a
@pir
=
. For the agent, pro…t maximization requires
@qri i
p + qri
@pir r
@qri
@pir
dT i
dqri
@qri i
p + qri
@pir r
@qri
@pir
0
(A1)
+
which we can rewrite as
dT i
dqri
and
@ a
@pir
+
=
+
@qri i
p + qri
@pir r
28
@qri
@pir
dT i
dqri
1
(A2)
0
(A3)
which we can rewrite as
dT i
dqri
@qri i
p + qri
@pir r
for i 2 f1; 2g. The left hand-side derivative of
dT i
of T i because qri is decreasing in pir .
dqri
the rearranged inequalities.
a
+
@qri
@pir
1
(A4)
includes the right-hand side derivative
dT i
then follows directly from
dqri
dT i
dT i
. For the platform(s), two cases for the
dqri +
dqri
cross-group externality from side d to side r must be considered.
Step 3. We show that
(i) : With a positive cross-group externality (@qri =@pid < 0), pro…t maximization requires that
@ i
@pid
=
@qdi i
p
@pid d
cd + qdi +
dT i
dqri
cr
+
@qri
@pid
0
(A5)
which we can rewrite as
dT i
dqri
@q i
cr ri
@pd
+
@qdi i
p
@pid d
qdi
cd
@qri
@pid
1
(A6)
;
and
@ i
@pid
=
+
@qdi i
p
@pid d
cd + qdi +
dT i
dqri
cr
@qri
@pid
0
(A7)
which we can rewrite as
dT i
dqri
for i 2 f1; 2g. Hence,
@q i
cr ri
@pd
dT i
dqri
+
@qdi i
p
@pid d
qdi
dT i
dqri
cd
@qri
@pid
1
(A8)
:
by the same logic as in step 2.
(ii) : With a negative cross-group externality (@qri =@pid > 0), pro…t maximization
requires that
@ i
@pid
@qdi i
= i pd
@pd
cd +
qdi
+
29
dT i
dqri
cr
@qri
@pid
0
(A9)
which we can rewrite as
dT i
dqri
cr
@qri
@pid
@qdi i
p
@pid d
qdi
1
@qri
@pid
cd
(A10)
;
and
@ i
@pid
=
+
@qdi i
p
@pid d
cd + qdi +
dT i
dqri
cr
+
@qri
@pid
0
(A11)
which we can rewrite as
dT i
dqri
+
@q i
cr ri
@pd
@qdi i
p
@pid d
qdi
cd
for i 2 f1; 2g. Note that left hand-side derivative of
side derivative of T i
dT i
dqri
1
@qri
@pid
(A12)
i
now includes the left-hand
dT i
because qri is increasing in pid . Again it follows that
dqri +
.
dT i
dqri
is di¤erentiable at the quantity qri induced by p . Q.E.D.
Together step 2 and step 3 therefore implies that
=
+
dT i
dqri
, and thus T i
Bi
i
j
Proof of Proposition 3. De…ne pB
C (w) = pC;s (w ; w ) as the equilibrium prices that
simulatenously solve the system of …rst-order conditions represented by (10) and (11),
i
j
Bi
and let q pB
C (w) = qC;s (w ; w ) be the resulting quantities. De…ne
B
C
pB
C (w) =
(wi ; wj ) as the equilibrium overall industry pro…t in a common agency situation, as
a function of the marginal wholesale prices (wi ; wj ) accepted at stage 1. Let wC be
the symmetric wholesale price that solves d
B
C
(w; w) =dw = 0 — i.e., the (symmetric)
wholesale price that solves the maximization problem (9) for the case without RPM.
Bi
De…ne pBi
C;s (wC ; wC ) = pC;s and qC;s (wC ; wC ) = qC;s , as the equilibrium price and quantity
respectively for each …rm on side s, when the wholesale prices are equal to wC , and let
B
C
(wC ; wC ) =
C
be the resulting industry pro…t. Finally, de…ne
N
as the equilibrium
industry pro…t in a version of the game where each platform is assigned an exclusive agent
before the game starts (i.e., “intrinsic exclusive agency”). Note that this version of the
game is similar to the set-up in Bonanno and Vickers (1988), with the di¤erence that the
two manufacturers (platforms) in our model sell to two sides of the market (and through
a distributor to only one side).
Consider a candidate equilibrium, where the platforms adopt a common agent, and
30
where each platform o¤ers the agent a wholesale price equal to wC , and …xed fees fC =
C =2;
and FC , such that each platform earns the pro…t
(wC
cr ) qC;r + pC;d
cd qC;d + FC =
C
2
if it decides to be active. We may consider three possible deviations for platforms i:
At stage 2, as an alternative to setting the price pC;d , the platform could choose not
to trade with the agent (qri = 0). In this case the platform earns the pro…t
C =2,
and this is therefore not a pro…table deviation.
At stage 1, platform i could o¤er (to the same agent) a di¤erent wholesale price
wi = w0 6= wC , and then adjust F i so that the agent’s zero-pro…t condition still
holds. However, because wj = wC , any deviation by i to a di¤erent wholesale price
will reduce the overall industry pro…t, and thus either reduce the pro…t of platform
j, or reduce the pro…t of platform i. We may then note that a deviation to wi = w0
cannot reduce the pro…t of platform j, because j could always opt not to trade at
stage 2 (qrj = 0). Hence, any deviation by i at this stage will have to reduce the
pro…t of platform i.
At stage 1, platform i could o¤er a contract to a di¤erent agent instead. Given
our assumption that platform j’s conditional (on exclusivity) o¤er to the agent is
renegotiation proof, the maximum pro…t that i could obtain by o¤ering a contract
to a di¤erent agent, is
C
<
N =2.
Hence, this deviation is pro…table only as long as
N.
We can therefore conclude that a Pareto undominated common agency equilibrium
exists i¤
C
N.
Q.E.D.
Proof of Proposition 4. The …rst part of the proof follows the same structure as the
proof of Proposition 3.
Similar to the proof of Proposition 3, we de…ne as
N
the equilibrium industry pro…t
in a version of the game where each platform is assigned an exclusive agent before the
game starts (i.e., “intrinsic exclusive agency”), but assuming they are allowed to use RPM
this time. Note also that we have
M
N,
per de…nition.
Suppose that each platform o¤ers the agent a contract that leaves the agent with zero
pro…t should the agent distribute the platform’s product exclusively. Suppose next that
each platform i 2 f1; 2g o¤ers the agent a contract that (in case of common agency) has
31
a marginal wholesale price equal to wi = wM , where wM is set according to (13), together
with a binding (…xed, minimum or maximum) resale price v i = v M = pM
r ; and …xed fees
fi = fM =
M
=2 and f i = F M such that each platform earns the pro…t
M
pM
d
cd qdM + wM
cr qrM + F M =
2
if it decides to be active. Given that the agent accepts, we then have that the agent sets
M
prices equal to pM
r and the platforms set prices equal to pd . Moreover, the agent’s pro…t
is zero and each platform earns
M
=2. We may consider three possible deviations for
platforms i:
At stage 2, as an alternative to setting the price pM
d , the platform could choose not
to trade with the agent (qri = 0). In this case the platform earns the pro…t
M
=2,
and this is therefore not a pro…table deviation.
At stage 1, platform i could o¤er (to the same agent) a di¤erent wholesale price
wi = w0 6= wM , and/or impose a di¤erent resale price v i = v 0 6= v M , and then adjust
F i so that the agent’s zero-pro…t condition still holds. However, because wj = wM
and v j = v M , any deviation by i will reduce the overall industry pro…t, and thus
either reduce the pro…t of platform j, or reduce the pro…t of platform i. We may
then note that a deviation cannot reduce the pro…t of platform j, because j could
always opt not to trade at stage 2 (qrj = 0). Hence, any deviation by i at this stage
will have to reduce the pro…t of platform i.
At stage 1, platform i could o¤er a contract to a di¤erent agent instead. Given
our assumption that platform j’s conditional (on exclusivity) o¤er to the agent is
renegotiation proof, the maximum pro…t that i could obtain by o¤ering a contract
to a di¤erent agent, is
N =2.
This deviation cannot be pro…table, as
M
N.
We can therefore conclude than a Pareto undominated common agency equilibrium
exists, in which the platforms adopt a common agent and set the marginal wholesale price
according to (13) at stage 1, and collect the fully integrated pro…t
M
at stage 2.
Next, we may note that, when the wholesale prices are set according to (13), at the
fully integrated prices pM the condition that the agent would wish to reduce the price pir
is that (13) < (12). We have two cases to consider, depending on whether the indirect
network e¤ect @Dri =@qdi is negative or positive.
32
Case 1. Suppose the indirect network e¤ect is negative (@Dri =@qdi < 0): The condition
that (13) > (12), and therefore that the appropriate resale price is a maximum price, is
@qri X @qrk M
p
@pid
@pir r
X @q k X
r
@pir
cr
k2f1;2g
k2f1;2g
pM
s
cs
s2fd;rg
pM
d
cd
@qsj
@pid
(A18)
@qri X @qdk
> 0
@pid
@pir
k2f1;2g
From the fully integrated monopolist’s …rst-order condition with respect to pir , we have
that
qrM + pM
d
pM
r
cd
X @q k
d
@pir
k2f1;2g
cr =
(A19)
X @q k
r
@pir
k2f1;2g
which is negative i¤ qrM <
pM
d
cd
X @q k
d
. Substituting (A19) into (A18) and
@pir
k2f1;2g
rearranging, we obtain the condition
qrM
X @q k
r
+ pM
d
@pid
cd
pM
d
k2f1;2g
k2f1;2g
Given that
@qrj X @qdk
@pid
@pir
X @q k
@qrj
r
>
0
and
@pid
@pid
cd
@qdj X @qrk
>0
@pid
@pir
(A20)
k2f1;2g
0 (a negative indirect network e¤ect), the left-hand
k2f1;2g
side of (A20) is positive. Hence, the condition always holds and the appropriate resale
price is a maximum price.
Case 2. Suppose the indirect network e¤ect is positive (@Dri =@qdi > 0): The condition that
(13) < (12) and therefore that the appropriate resale price is a minimum price, is the
X @q k X @q k
r
r
condition (A18). Adding pM
c
to both sides of the inequality
r
r
@pir
@pid
k2f1;2g
sign and dividing through by pM
d
@qdj X @qrk
@pid
@pir
k2f1;2g
Dividing through by
k2f1;2g
cd , we obtain
@qri X @qdk
pM
r
>
i
M
i
@pd
@pr
pd
k2f1;2g
@qdi X @qrk
, setting
@pid
@pir
ij
ss
k2f1;2g
33
cr X @qrk X @qrk
@pir
@pid
cd
k2f1;2g
(A21)
k2f1;2g
= @qsj =@pis = ( @qsi =@pis ),
ii
dr
= @qri =@pid = ( @qdi =@pid )
and
ij
dr
= @qrj =@pid = ( @qdi =@pid ), and rewriting, we get
ij
dd
>
X @q k
d
@pir
k2f1;2g
X @q k
r
@pir
k2f1;2g
Finally, note that if we de…ne
ii
rd
pM
r
pM
d
ii
dr
cr
cd
ii
dr
+
ij
dr
ij
rd
= @qdi =@pir = ( @qri =@pir ) and
(A22)
= @qdj =@pir = ( @qri =@pir ),
we can rewrite condition (A22) as follows:
ij
dd
>
ii
rd
+
1
ij
rd
ij
rr
ii
dr
pM
r
M
pd
cr
cd
ii
dr
ij
dr
+
(A23)
;
The condition says that the diversion ratio between the platforms on the direct side must
be su¢ ciently high. Q.E.D.
Appendix B
In the following we will briefely consider the case where pM
r
cr > 0 and pM
d
cd < 0.
Note that this only has implications for the consideration of whether the appropriate
resale price is a maximum or minimum price. There are two cases to consider, depending
on whether @Dri =@qdi < 0 or @Dri =@qdi > 0. We now show that the …rst case is already
covered by Case 1 in the proof of Proposition 4 in Appendix A.
Case 1. Suppose the indirect network e¤ect is negative (@Dri =@qdi < 0). We may note
that the fully integrated monopolist’s …rst-order condition with respect to pid is
qdM +
pM
d
cd =
X @q k
r
pM
@pid r
cr
k2f1;2g
(B1)
X @q k
d
@pid
k2f1;2g
The right-hand side of (B1) is positive as long as pM
r
cr > 0, qdM > 0 and
X @q k
d
< 0.
@pid
k2f1;2g
Hence, pM
d
cd > 0 always, and this case is therefore covered by Case 1 in the proof of
Proposition 4 in Appendix A. In practice we therefore only have one case to consider,
which we do next.
34
Case 2. Suppose the indirect network e¤ect is positive (@Dri =@qdi > 0). Same as before,
we …nd that the condition that (13) < (12) and therefore that the appropriate resale price
is a minimum price, is the condition (A18). Rewriting the condition, keeping in mind
that pM
d
cd < 0, we …nd that the inequality (A23) is reversed. Hence, the condition that
the appropriate resale price is a minimum price now becomes
ij
dd
<
ii
rd
ij
rd
ij
rr
+
1
ii
dr
pM
r
pM
d
cr
cd
ii
dr
+
ij
dr
(B2)
This condition is a bit harder to interpret than (A23). Yet, we can still show that, for
a minimum price to be appropriate, one needs some degree of substitution between the
platforms. To see this, consider the case with no substitution on either side of the market,
ij
ss
=
ij
rd
=
ij
dr
= 0. The condition (B2) is reduced to
pM
r
pM
d
ii ii
rd dr
0<
cr
cd
ii
dr
(B3)
Using the …rst-order condition with respect to pid for the fully integrated monopolist, we
can rewrite condition (B3) as
0<
ii ii
rd dr
pM
r
0
B
B(pM
@ r
which never holds given that pM
r
cr
cr ) +
1
(B4)
qdM C
C
@qri A
@pid
cr > 0, qdM > 0, @qri =@pid < 0 and
ii ii
rd dr
< 1: Hence,
for a minimum price to be appropriate, we need some substitution between the platforms.
However, di¤erent from the case pM
d
cd > 0, condition (B2) tells us that we now also
need that the diversion ratio on the direct side,
diversion ratio on the retail side,
ij
rr .
35
ij
dd ,
is not too high compared to the
References
[1] A¤eldt, P., Filistrucchi, L., and Klein, T. J. (2013). “Upward pricing pressure in
two-sided markets.”Economic Journal, 123(572), F505–F523.
[2] Amelio, A., and Jullien, B. (2012). “Tying and freebies in two-sided markets.”International Journal of Industrial Organization, 30(5), 436–446.
[3] Anderson, S. P., and Coate, S. (2005). “Market provision of broadcasting: A welfare
analysis.” Review of Economic Studies, 72(4), 947–972.
[4] Armstrong, M. (2006). “Competition in two-sided markets.” RAND Journal of Economics, 37(3), 668–691.
[5] Armstrong, M., and Wright, J. (2007). “Two-sided markets, competitive bottlenecks
and exclusive contracts.” Economic Theory, 32(2), 353–380.
[6] Belle‡amme, P., and Peitz, M. (2014). “Price Information in Two-sided Markets.”
Unpublished Manuscript.
[7] Bernheim, B. D., and Whinston, M. D. (1985). “Common marketing agency as a
device for facilitating collusion.”RAND Journal of Economics, 16(2), 269–281.
[8] Bonanno, G., and Vickers, J. (1988). “Vertical separation.” Journal of Industrial
Economics, 36(3), 257–265.
[9] Caillaud, B., and Jullien, B. (2003). “Chicken & egg: Competition among intermediation service providers.”RAND Journal of Economics, 34(2), 309–328.
[10] Choi, J. P. (2010). “Tying in two-sided markets with multi-homing.” Journal of
Industrial Economics, 58(3), 607-626.
[11] Dobson, P. W., and Waterson, M. (2007). “The competition e¤ects of industrywide vertical price …xing in bilateral oligopoly.” International Journal of Industrial
Organization, 25(5), 935-962.
[12] Evans, D. S. (2002). “The antitrust economics of two-sided markets.”Working paper,
available at SSRN 332022.
[13] Evans, D. S. (2013). “Economics of Vertical Relations for Multi-sided Platforms.”
Working paper.
36
[14] Evans, D. S., and Noel, M. (2005). “De…ning antitrust markets when …rms operate
two-sided platforms.”Columbia Business Law Review, 2005(3), 667–702.
[15] Filistrucchi, L. and Klein, T. (2013). “Price competition in two-sided markets with
heterogeneous consumers and network e¤ects.”Working paper.
[16] Gabrielsen, T. and Johansen, B. O. (2016). “Resale Price Maintenance with Secret
Contracts and Retail Service Externalities.”American Economic Journal: Microeconomics (forthcoming).
[17] Hagiu, A. (2006). “Pricing and commitment by two-sided platforms.”RAND Journal
of Economics, 37(3), 720–737.
[18] Hagiu, A., and Halaburda, H. (2014). “Information and two-sided platform pro…ts.”
International Journal of Industrial Organization, 34(C), 25–35.
[19] Hagiu, A., and Lee, R. S. (2011). “Exclusivity and control.” Journal of Economics
& Management Strategy, 20(3), 679–708.
[20] Innes, R., and Hamilton, S. F. (2009). “Vertical restraints and horizontal control.”
RAND Journal of Economics, 40(1), 120–143.
[21] Kind, H. J., Nilssen, T., & Sørgard, L. (2016). “Inter-…rm price coordination in a
two-sided market.”International Journal of Industrial Organization, 44, 101-112.
[22] Lee, R. S. (2013). “Vertical integration and exclusivity in platform and two-sided
markets.”American Economic Review, 103(7), 2960–3000.
[23] Marvel, H. P., and McCa¤erty, S. (1984). “Resale price maintenance and quality
certi…cation.” RAND Journal of Economics, 15(3), 346-359.
[24] Marx, L. and Sha¤er, G. (2007). “Upfront Payments and Exclusion in Downstream
Markets.”RAND Journal of Economics, 38(3), 823–843.
[25] Mathewson, G. F., and Winter, R. A. (1984). “An economic theory of vertical restraints.” RAND Journal of Economics, 15(1), 27-38.
[26] Miklós-Thal, J., Rey, P. and Vergé, T. (2011). “Buyer Power and Intrabrand Coordination.”Journal of the European Economic Association, 9(4), 721–741.
[27] Montez, J. (2015). “Controlling opportunism in vertical contracting when production
precedes sales.”RAND Journal of Economics, 46(3), 650-670.
37
[28] O’Brien, D. P., and Sha¤er, G. (1992). “Vertical control with bilateral contracts.”
RAND Journal of Economics, 23(3), 299–308.
[29] Posner, R. A. (2001). “Antitrust in the new economy.”Antitrust Law Journal, 68(3),
925–943.
[30] Rey, P., and Vergé, T. (2004). “Bilateral control with vertical contracts.” RAND
Journal of Economics, 35(4), 728-746.
[31] Rey, P., and Vergé, T. (2010). “Resale price maintenance and interlocking relationships.”Journal of Industrial Economics, 58(4), 928–961.
[32] Rey, P. and Whinston, M. D. (2013). “Does retailer power lead to exclusion?”RAND
Journal of Economics, 44(1), 75–81.
[33] Rochet, J. C., and Tirole, J. (2006). “Two-sided markets: a progress report.”RAND
Journal of Economics, 37(3), 645–667.
[34] Romano, R. E. (1994). “Double moral hazard and resale price maintenance.”RAND
Journal of Economics, 25(3), 455-466.
[35] Telser, L. G. (1960). “Why should manufacturers want fair trade?” The Journal of
Law & Economics, 3, 86-105.
[36] Vives, X. (1999). “Oligopoly Pricing: Old Ideas and New Tools.” The MIT press,
Cambridge, Massachusetts.
[37] Weyl, E. G. (2010). “A price theory of multi-sided platforms.” American Economic
Review, 100(4), 1642–1672.
[38] White, A., and Weyl, E. G. (2012). “Insulated platform competition.”Working paper,
available at SSRN 1694317.
[39] Wright, J. (2004). “One-sided logic in two-sided markets.” Review of Network Economics, 3(1), 44–64.
38