Resale price maintenance with secret contracts and
retail service externalities
Tommy Staahl Gabrielsen and Bjørn Olav Johansen
Department of Economics, University of Bergen, Norway
August 5, 2015
Abstract
We analyze a setting where a monopolist sells through retailers that set prices
and provide valuable but non-contractible services to customers. We assume that
contracts are private. We …nd that purely bilateral price restraints have no e¤ect on
the equilibrium outcome and that the standard Bertrand prices and service levels
prevail. We also show that if manufacturers can commit to industry-wide resale
prices, they can obtain higher prices and service levels but will generally not be able
to achieve the fully integrated outcome. Using a speci…c linear demand system, we
…nd that industry-wide price ‡oors are harmful to consumers.
Department of Economics, University of Bergen, Fosswinckels Gate 14, N-5007 Bergen, Norway
([email protected], [email protected]). We would like to thank the editor John
Asker, three anonymous referees, and Patrick Rey, Mario Monti, Gordon Klein, Greg Sha¤er, Joao
Montez, and Andy Skrzypacz, as well as seminar and conference participants at DICE, CRESSE 2014,
EARIE 2014, the Danish Competition Authority, the University of Bergen and the University of Valencia
for their comments. This research has bene…ted from …nancial support from the Norwegian Research
Council.
1
1
Introduction
Resale price maintenance (RPM) describes the practice whereby manufacturers limit the
freedom of their retailers to determine resale prices and is much debated in the economics
literature and in policy circles. Until recently, competition authorities and courts in both
the EU and the US frowned on the practice, but after the so-called Leegin-case, the US
has adopted a softer approach.1 In general, competition policy in many countries tends
to be more hostile to minimum and …xed resale restrictions than to maximum RPM
restraints. The predictions from the economic literature on RPM are ambiguous and
partly con‡icting. As with other forms of vertical restraints, there are models that show
that the use of RPM may be anticompetitive as it will raise prices, and others that claim
that the use of RPM may be e¢ ciency enhancing because it encourages value-creating
retail services.
In the economics literature, the counteracting anticompetitive and e¢ ciency arguments for RPM have been analyzed within two broad classes of models. Some models
assume that contracts o¤ered by manufacturers to their retailers are public information
(e.g., Innes and Hamilton, 2009; Gabrielsen and Johansen, 2013; Rey and Vergé, 2010;
Mathewson and Winter, 1984), and other models have secret contracts (e.g., O’Brien and
Sha¤er, 1992; Rey and Vergé, 2004; Montez, 2012). In addition, the models developed in
the economics literature di¤er with respect to what kind of decisions retailers make. In
most of this literature, the retailers’sole decision is to set the retail price, but in Mathewson and Winter (1984), retailers both set retail prices and choose a (noncontractible)
retail service e¤ort, and there may be externalities between retailers in e¤ort provision.
In most real life markets, it is clear that retailers make many di¤erent decisions that
a¤ect demand. This includes setting the retail price for the products but also a variety of retail service e¤orts ranging from advertising to di¤erent point-of-sale services to
customers. In fact, it is di¢ cult to provide a single example where retail service e¤ort
is of no importance at all. Moreover, many of the retail services o¤ered to consumers
are noncontractible or at least imperfectly contractible. Our position is that in real life
markets, contracts o¤ered to retailers are predominantly secret (as argued by Katz, 1991).
Surprisingly, we have not been able to …nd a study of RPM in a setting with secret contracts and noncontractible retail service e¤ort. The key question that we address in this
paper is how a single variable restraint such as RPM will work in an environment where
retailers make many decisions that a¤ect demand and where some are noncontractible.
The tension between the di¤erent approaches in the literature regarding price restraints
1
See Leegin Creative Leather Products, Inc. v. PSKS, Inc. Slip Op. no. 06-480.
2
can be illustrated by O’Brien and Sha¤er (1992) (OS) and Mathewson and Winter (1984)
(MW). In a model where the retailers only decide the retail price, OS show that RPM may
be used by an upstream monopolist to extract monopoly rent. In essence, RPM allows
the monopolist to escape from an opportunism problem arising from secret contracting
with competing downstream retailers.2 If RPM is not used, the …rst retailer correctly
anticipates the monopolist’s incentive to o¤er a sweetheart deal to a second rival retailer,
which in turn allows the monopolist and the second retailer to free-ride on the margin of
the …rst retailer, and vice versa. In turn, this incentive reduces the retailers’willingness
to pay for the product and leaves the monopolist unable to exert market power fully. OS
show that two types of price restraints would allow the manufacturer to realize the fully
integrated outcome. The …rst solution is that the manufacturer should o¤er each retailer
a maximum resale price and then squeeze retail margins with a high wholesale price.
This will eliminate retail margins and thereby also the incentive for opportunism. This
result from OS challenges current competition policy, as in general, a maximum RPM is
considered to be unproblematic in most jurisdictions. Secondly, OS analyze the case where
a manufacturer can commit to an industry-wide price ‡oor for all of its retailers. In that
case, OS show that, as long as the retailers are symmetric, an industry-wide minimum
RPM is su¢ cient to realize the fully integrated outcome.
MW, on the other hand, formalize the classic Telser argument (Telser, 1960) saying
that RPM may be e¢ ciency enhancing. In a model with observable contracts, MW show
that RPM may solve vertical externalities both when there are spillovers between retailers
from retail sales e¤ort and when there are no such spillovers. Intuitively, the solution is
to …x retail prices at the integrated level with a minimum RPM and then to adjust retail
margins so as to give the retailers an optimal incentive to exert retail sales e¤ort.
Both arguments are convincing and intuitive given the setting in which they are developed. OS have secret contracts, and retail prices are the only decision made by the
retailers (i.e., there is no role for retail sales e¤ort); and MW have public contracts, but
retailers choose both retail prices and noncontractible retail sales e¤ort. Because secret
contracts and retail services are both likely to be important, there is a need for a model
that considers the e¤ects of RPM when both factors are present.
We therefore analyze a model with secret contracts (where opportunism is an issue)
and where noncontractible retail sales e¤ort is at least of some importance. This model
allows us to develop and understand better the arguments and e¤ects of RPM presented
by OS and MW. Speci…cally, we do this by introducing retail sales e¤ort with spillovers
2
The term was …rst used by Williamson (1975), and many examples of the problem arise in industrial
organization.
3
in the framework of OS (or unobservable contracts in the framework of MW). By doing
this, new insights emerge.
First, we show that bilaterally negotiated RPM clauses operate di¤erently in our model
than in the frameworks of both OS and MW. We …nd that the RPM mechanisms and
e¤ects identi…ed by OS and MW will not arise in our model (where demand depends on
the retailers’sales e¤ort and contracts are secret). In fact, we …nd that purely bilateral
price restrictions, irrespective of type, have no e¤ect on the equilibrium outcome. The
monopolist manufacturer will sell at marginal cost, and the equilibrium outcome at the
retail level involves Bertrand prices and e¤ort levels, as opposed to fully integrated prices
in OS and fully integrated prices and e¤ort levels in MW. This result holds irrespective
of how (un)important the retailers’e¤ort levels are for demand, as long as the e¤ect is
strictly positive, and irrespective of the type of spillovers in e¤ort (negative or positive).
Hence, our results suggest that allowing the retailers to use minimum or maximum RPM
clauses yields no anticompetitive e¤ects and no e¢ ciency e¤ects, as long as they are
negotiated bilaterally.
The intuition for this result is as follows. When an upstream monopolist negotiates
secret contracts with competing retailers and price is the only decision made by the retailers, the bilateral and joint pro…t maximizing transfer prices di¤er whenever retailers earn
positive quasi-rents. In order to achieve the joint maximizing outcome the manufacturer
can either eliminate the retail margins, by combining high transfer prices with maximum
RPM, or make a commitment not to engage in bilateral optimization by implementing
industry-wide RPM, as in O’Brien and Sha¤er (1992). Alternatively, the manufacturer
could impose closed territory distribution. However, allowing the retailers to make other
noncontractible decisions that a¤ect consumer demand— e.g., retail service e¤ort, as we
do in this paper— the strategy of squeezing the retail margins with maximum RPM is no
longer successful. The reason is that the manufacturer will then have an incentive to cut
its transfer price secretly in order to induce some e¤ort at the retail level, and if so, the
positive margins earned by the retailers open the door for opportunism again. We show
that in such a case, the unique contract equilibrium again involves setting the marginal
transfer prices equal to the manufacturer’s marginal cost.
Our second result is that industry-wide RPM— i.e., the practice where the manufacturer commits to a common RPM clause for all its retailers— may mitigate both problems.
Although a commitment to a minimum industry-wide RPM clause restores the integrated
outcome with symmetric retailers when price is the retailer’s only choice variable, we
…nd that this is no longer the case with noncontractible retail e¤ort. We show that
with industry-wide RPM, equilibrium retail prices and e¤ort levels can be lifted above
4
the Bertrand level— but the integrated outcome can be achieved only when there are no
spillovers. Importantly, we …nd that only a minimum price restraint can either elicit a
price increase or induce more retail e¤ort (or both).
Our third and …nal result concerns the implications for consumer and social welfare of
minimum pricing in our setting. For consumers, industry-wide minimum pricing will have
one negative e¤ect— it allows the monopolist to induce retail prices above the Bertrand
level— and one positive e¤ect— it may spur sales e¤ort at the retail level. This suggests
that industry-wide minimum prices should be more bene…cial for consumers (or less harmful) when sales e¤ort produces positive spillovers at the retail level, compared with when
there are no such spillovers. However, we …nd that this is not necessarily the case. To
evaluate the e¤ect of RPM, we use a speci…c utility function. In our example, we show
that even if consumers value sales e¤ort, and even if retailers are free-riding on each other’s
sales e¤ort, consumers are always hurt by the implementation of an industry-wide price
‡oor. Moreover, we …nd that if a retailer’s sales e¤ort produces positive spillovers, allowing the monopolist to implement a common minimum price will cause the equilibrium
retail prices to be higher than the fully integrated prices, given the same e¤ort levels.
Hence, in some cases, a minimum RPM clause may in fact turn out to cause more harm
to consumers when there are positive service externalities at the retail level, compared
with when there are no service externalities. This result shows that the presence of service externalities does not necessarily imply that price ‡oors are more bene…cial or less
harmful for consumers. The implications for overall welfare, on the other hand, are less
clear cut. We …nd that under certain conditions, such as when the cost of service provision is su¢ ciently low, even if consumer welfare decreases under industry-wide RPM, the
increase in producer surplus may be su¢ cient for total welfare to increase.
The rest of the paper is organized as follows. The next section brie‡y reviews the
literature related to the opportunism problem under secret contracting and RPM. Then,
in Section 3, we present our model, our basic assumptions, and our benchmark. Section 4
contains the analysis and presents our main results assuming individual RPM contracts,
and in Section 5, we derive our results with industry-wide RPM contracts. Section 6
concludes.
2
The opportunism problem
The opportunism problem in OS that arises when a manufacturer contracts secretly with
downstream retailers has long been recognized in the literature. An upstream manufac-
5
turer with market power has an interest in restricting supply to its retailers to exploit its
monopoly power, which in turn can be shared with the retailers. However, because of its
secret contracts with each retailer, the manufacturer has an incentive to free-ride on the
margins earned by its other retailers. This incentive has been demonstrated and analyzed
by several authors in di¤erent settings.3 In general, the problem prevents a manufacturer
with market power upstream from realizing its market power when selling to downstream
retailers. The ‡avor of the problem is similar to that of the durable goods monopolist:
the monopolist cannot avoid reducing his price.
There is evidence suggesting that the problem is perceived as real by retailers. For
instance, in a recent EU merger case concerning Unilever and a smaller upstream competitor, the European Commission presented evidence where retailers expressed explicit
concerns about the opportunism problem. As usual, in merger cases in upstream markets, the fear was that the merger would allow the merged entity to increase its prices
to retailers. The European Commission found evidence indicating that retailers across
the a¤ected EU member states “would accept (input) price increases if applied generally in the market,” and Unilever presented evidence where “retailers expressed doubts
on how they can be sure that Unilever indeed would uniformly increase prices across all
customers,”indicating an awareness of the opportunism problem among retailers.4
The essence of the problem can be illustrated using downstream price competition.
In this case, when negotiating with each retailer, the manufacturer and each retailer are
maximizing their bilateral pro…ts, and thus they ignore quasi-rents earned by the other
retailers. This induces each retailer and the manufacturer to free-ride on these rents, and
in equilibrium, they end up setting transfer prices at the manufacturer’s marginal cost.
There have been several proposals in the literature suggesting how the manufacturer may
circumvent the problem. Hart and Tirole (1990) argue that vertical integration may be
a way to remonopolize the market. If an upstream monopolistic …rm can vertically integrate with one of several homogeneous retailers, he will have no incentive to supply the
unintegrated retailers, and the manufacturer can restore the monopoly outcome. Furthermore, as noted by Hart and Tirole (1990) and Rey and Tirole (2006), signing an exclusive
dealing contract with one retailer may also solve the problem. However, if retailers serve
partially overlapping markets, there will be a loss of potential pro…t from selling through a
single retailer in the downstream market. In these cases, vertical integration and exclusive
contracts will generally not be enough to solve the problem fully. As noted above, OS
3
Hart and Tirole (1990) with downstream Cournot competition, O’Brien and Sha¤er (1992) with
Bertrand competition, and also McAfee and Schwartz (1994) and Marx and Sha¤er (2004).
4
See case M.5658 UNILEVER / SARA LEE BODY CARE (2010), #216 and 219.
6
examine how vertical restraints can restore the e¢ cient outcome for the manufacturer.
The result that maximum RPM can eliminate opportunism has since been rea¢ rmed
by Rey and Vergé (2004) and Montez (2012), albeit in di¤erent settings. Montez (2012)
shows that a monopolist producer may eliminate opportunism by using buybacks and
(sometimes) a price ceiling. In a similar setting to OS, Rey and Vergé (2004) show
that equilibria with wary beliefs (as opposed to passive beliefs in OS) exist and re‡ect
opportunism, and that maximum RPM with a price squeeze will also eliminate the scope
for opportunism in this case. In sum, these papers suggest that a maximum price may
raise prices to consumers because it eliminates the scope for opportunism. We now develop
our model.
3
The model
We follow OS and consider the classic setup for the opportunism problem with downstream
price competition. In a vertically related industry, the upstream monopolist produces an
intermediate good at constant marginal cost, c. This good is sold to two di¤erentiated
downstream retailers, R1 and R2 , using secret nonlinear contracts. The two retailers
transform the manufacturer’s good on a one-to-one basis into two di¤erentiated …nal
goods and sell them to consumers.
In contrast to OS, but as in MW, retailers may exert sales e¤ort that enhances demand.
We denote retailer Ri ’s demand by Di (e; p), where e = (e1 ; e2 ) denotes the vector of the
retailers’e¤ort levels, and p = (p1 ; p2 ) denotes the vector of retail prices. We will assume
that for all Di (:) > 0 the quantity demanded is decreasing in own price pi while demand
is increasing in own e¤ort ei , with @ei Di > 0 and @ei ei Di 0.5 For some of our results, we
will invoke the following set of assumptions about the retailers’demand (assuming both
Di and Dj are positive):
A1. All else equal, a uniform increase in p1 and p2 causes Di to fall, which implies that
@pi Di + @pj Di < 0.
A2. All else equal, a uniform increase in e1 and e2 causes Di to rise, which implies that
@ei Di + @ej Di > 0.
A3. All else equal, a marginal increase in pi causes total demand to fall, @pi Di +@pi Dj < 0.
A4. All else equal, a marginal increase in ei causes total demand to rise, @ei Di +@ei Dj > 0.
5
We will sometimes denote by @xi f the partial derivative of f with respect to xi , @xi xi f the second
partial derivative, @xi xj f = @ 2 f =@xi @xj the cross-partial derivative, and so on.
7
For any pj , we also assume that there is a choke price pi (pj ) implicitly de…ned by
Di (pi (pj ) ; pj ) = 0.
We make no speci…c assumption about the sign of the e¤ect of Ri ’s e¤ort on the
rival’s demand, @ei Dj . Hence, we allow for both positive, negative, and no spillovers in
retail e¤ort. We denote the retailer’s e¤ort cost by Ci (ei ), which is assumed to be twice
continuously di¤erentiable, with Ci (0) = 0, and Ci0 (ei ) > 0 and Ci00 (ei ) > 0 8ei > 0.
We will denote Ri ’s e¤ort cost per unit sold by i (ei ) := Ci =Di : All other retailing
costs are assumed to be zero. The overall industry pro…t can then be written simply as
P
= i (pi c
i ) Di :
We will assume that when the retailers buy the monopolist’s product “at cost”, maximization by the retailers will lead them to choose prices and e¤ort levels pB ; eB =
B
pB
, characterized by
i ; ei
B
pB
= arg max (pi
i ; ei
pi ;ei
c
i ) Di
B
pi ; ei ; pB
j ; ej
(1)
for i 6= j 2 f1; 2g. In the following, we will refer to pB ; eB as the “standard Bertrand”
levels of prices and e¤ort. We will also denote by B
i the standard Bertrand pro…t of
retailer i 2 f1; 2g; i.e., the retailer’s equilibrium pro…t when buying the monopolist’s
product “at cost”.
Similarly, we will refer to pI ; eI = pIi ; eIi as the fully integrated levels of prices
and e¤ort; i.e., the levels of prices and e¤ort that simultaneously solve
@pi
=
2
X
(pk
c) @pi Dk + Di = 0
(2)
(pk
c) @ei Dk
Ci0 = 0
(3)
k=1
and
@ei
=
2
X
k=1
P B
for i 2 f1; 2g. We will denote by I
i i the resulting fully integrated pro…t; i.e.,
the overall pro…t when the retailers’prices and e¤ort levels are equal to pI ; eI .
We will assume throughout the analysis that a retailer’s sales e¤ort is nonveri…able
and hence also noncontractible.
We consider the following simple two-stage game played between the manufacturer
and the two retailers. At stage 1 (the contracting stage), the manufacturer makes take-itor-leave-it contract o¤ers simultaneously and secretly to each retailer, which the retailers
either accept or reject. A supply contract Ti may depend on the quantity bought by the
8
retailer (a nonlinear contract) and the price set by the retailer (resale price maintenance).
The resale price restriction may be purely bilateral or industry wide, and in either case,
it may be a …xed price, a price ‡oor, or a price ceiling. A nonlinear contract Ti to retailer
i with a bilateral RPM clause only restricts the retail price of Ri . A nonlinear contract
Ti to retailer i with an industry-wide price restriction will commit M to the same price
restriction for both retailers.
A retailer never observes its rival’s nonlinear tari¤ and bilateral RPM clause. If the
RPM clause is industry wide, then the price restraint is observed by both retailers, and
only the nonlinear tari¤s are secret. At stage two (the competition stage), accepted
contracts are implemented, and retailers compete by simultaneously choosing their prices
P
and e¤ort levels. We let M ’s pro…t be given by M = i fTi cDi g, and let Ri ’s pro…t
be given by i = (pi
Ti :
i ) Di
To keep the analysis tractable, we use the “contract equilibrium”concept introduced
by Cremér and Riordan (1987), which was also adopted by OS.
De…nition 1. Let s = (si ) be the vector of retailers strategies in the downstream market.
A contract equilibrium with unobservable contracts is then a vector of supply contracts
T = (T1 ; T2 ) and a Nash equilibrium in the downstream market s induced by these
contracts, such that 8i, Ti is a contract that maximizes the bilateral joint pro…t of M
and Ri , M + i , taking Tj ;sj as given.
This equilibrium concept is both tractable and intuitively appealing. It says that in a contract equilibrium, there is no room for a manufacturer–retailer pair M Ri to revise their
contract and to increase their bilateral joint pro…t, holding …xed M ’s contract with Rj ,
and holding …xed Rj ’s choice of e¤ort and price. A contract equilibrium’s de…ning characteristic is therefore that it survives bilateral deviations; i.e., where a pair M Ri decides
to renegotiate their contract terms secretly. As noted by Rey and Vergé (2004), a weakness with contract equilibria is that they do not always survive multilateral deviations,
where the manufacturer revises its o¤ers and deviates (secretly) with both retailers simultaneously. Hence, a contract equilibrium does not always constitute a perfect Bayesian
equilibrium with passive beliefs.6 In Appendix B, we provide the conditions for when
6
One interpretation of contract equilibrium is a contracting game where the manufacturer uses a pair of
agents that simultaneously and independently negotiate contracts with the retailers on the manufacturer’s
behalf. If communication between the agents is costly, multilateral deviations may not be possible.
Another interpretation is that the manufacturer is facing strong buyers and has to bargain over the
contracts; in the extreme case where the retailers make take-it-or-leave-it o¤ers to the manufacturer, for
example, multilateral deviations would not be possible.
9
the contract equilibria identi…ed in our results below also constitute perfect Bayesian
equilibria with passive beliefs.7
4
Nonlinear contracts with bilateral price restraints
In this section, we analyze the equilibrium outcome when the manufacturer may use
general nonlinear contracts and bilateral RPM. Before we move on, we state the following
Lemma, which we have adopted from OS’s original paper and generalized to a setting
that allows for retailers’sales e¤ort.
Lemma 1. If (T ; s ) forms a contract equilibrium with general nonlinear contracts (and
possibly using bilateral RPM clauses), then 8j, Tj is continuous and di¤erentiable at the
quantity Dj induced by (T ; p ). If the contracts entail a commitment to industry-wide
price …xing, then the same result holds, as long as there are spillovers in retailers’ sales
e¤ort.
Proof. See Appendix A.
Lemma 1 greatly simpli…es the rest of the analysis, and the intuition for the result is
straightforward. If the contract was not continuous at the equilibrium quantity, either a
pairing between a manufacturer and a retailer could deviate by inducing a change in the
retail price or the e¤ort level, or the rival retailer could increase its pro…t by doing the
same.
We now show that it is impossible for the manufacturer to induce the fully integrated
pro…t I when using general nonlinear contracts and bilateral RPM.
Lemma 2. (Bilateral RPM) In equilibrium, it is not possible for the manufacturer to
induce the fully integrated pro…t I .
Proof: See Appendix A.
The intuition for this result is as follows. To overcome the opportunism problem, the
manufacturer has to take into account Rj ’s quasi-rents when making its contract o¤er
to Ri , and vice versa. As shown by OS, one way to do this is to eliminate the retailers’
quasi-rents completely; e.g., by …xing the retail prices and then squeezing the retailers’
7
Interestingly, in Appendix B, we show that multilateral deviations may be less pro…table and hence
less of a concern when the retailers provide demand-enhancing services in addition to competing in prices,
as is assumed in our model.
10
mark-ups through high marginal transfer prices. However, to induce the retailers to exert
some e¤ort, the retailers have to earn strictly positive quasi-rents on the margin, to cover
their marginal e¤ort cost. Because it is not possible for the manufacturer to achieve both
simultaneously, the fully integrated outcome is unattainable.
We now show our …rst main result; namely, that general nonlinear contracts and
bilateral RPM in fact yield the Bertrand level of prices and e¤ort.
Proposition 1. (Bilateral RPM) In all contract equilibria, we have that i) the marginal
transfer prices are the same for each retailer and equal to the manufacturer’s marginal
cost c, ii) retail prices are equal to pB , and iii) the retailer’s e¤ort levels are equal to eB .
Proof: See Appendix A.
Proposition 1 shows that when the retailers exert sales e¤ort, the manufacturer’s opportunism problem is restored with full force— and hence maximum RPM no longer yields the
integrated outcome in a contract equilibrium. The intuition is as follows. To overcome the
temptation to o¤er the retailers sweetheart deals, which would allow a retailer to charge
a lower price at its rival’s expense, the manufacturer can impose individual price ceilings
equal to pI and then squeeze the retailers’sales margins by charging high marginal transfer prices, Ti 0 ! pIi for i 2 f1; 2g. However, this cannot arise in any contract equilibrium
if retailers can also exert some sales e¤ort that would in‡uence demand. The reason is
that given that the retailers’mark-ups are squeezed, the manufacturer can pro…tably deviate with either retailer and charge it a slightly lower marginal transfer price. The reason
is that this would induce the retailer to exert (greater) sales e¤ort at the last stage of
the game. This means that a strategy of squeezing the retailers’margins cannot arise in
any contract equilibrium and that each retailer has to earn strictly positive quasi-rents.
This reopens the door for opportunism— and to such an extent that the outcome involves
Bertrand prices and e¤ort levels.
The intuition for this is identical to that for opportunism when retailer sales e¤ort
is unimportant: a positive margin drives the equilibrium all the way down to Bertrand.
Given that the marginal transfer price for the rival retailer is at marginal cost, it is bilaterally optimal for the manufacturer and a retailer to behave as if they were an integrated
Bertrand duopolist. This is because the unobservable action of the manufacturer and one
retailer does not a¤ect the rents that the manufacturer earns from the rival retailer, given
that the manufacturer’s marginal transfer price is at marginal cost. To behave like an
integrated Bertrand duopolist, the manufacturer and the retailer should transfer the good
at marginal cost, making the retailer the residual claimant to their bilateral pro…t. Thus,
11
marginal cost transfer pricing constitutes bilateral contracting’s best responses. Note also
that Proposition 1 implies that the equilibrium with RPM derived by MW no longer
emerges when contracts are secret.
A striking feature of the result in Proposition 1 is that even allowing for a small e¤ect
of retail service on consumer demand shifts the equilibrium outcome from one extreme
to the other; i.e., from the fully integrated outcome as in OS to the Bertrand outcome.
This is surprising, as one might have expected a more gradual transition from the fully
integrated outcome to the Bertrand outcome.
5
Nonlinear contracts and industry-wide RPM
We now analyze whether a commitment to industry-wide RPM can help the manufacturer
to restore the fully integrated outcome. The potential bene…t for the manufacturer is that
industry-wide RPM restricts the set of contracts that the manufacturer can establish with
rival retailers, therefore potentially limiting opportunism per construction.
Industry-wide vertical price …xing describes a situation where the manufacturer is
able to commit to adopting a common (uniform) resale price throughout the downstream
market.8 We can model this by incorporating a stage prior to the contracting stage,
where the manufacturer commits publicly to a mandatory industry-wide resale price,
before negotiating transfer prices privately and secretly with each retailer at stage 2.9
De…nition 2. Let pS and eSi := ei pS be the (…xed) industry-wide resale price and the
e¤ort level induced by this price for retailer i respectively, where
ei (p) := arg max (p
ei
and
pS := arg max
p
X
i
[p
c
c
i
i
(ei )) Di p; ei ; p; ej (p)
(ei (p))] Di p; ei (p) ; p; ej (p)
8
The assumption that the manufacturer can commit to industry-wide RPM but is unable to commit
to industry-wide wholesale prices can be motivated by several factors. First, retail prices are generally
easier to observe than wholesale prices. Therefore, if a manufacturer should defect from an industry-wide
RPM, retailers can observe this immediately. Second, when industry-wide RPM is adopted, products
often carry preprinted prices on the product itself. Finally, industry-wide RPM may be part of a trade
agreement. This is the case in, for example, some European book markets.
9
This resembles the setup in Dobson and Waterson (2007), who analyze the use of observable linear
tari¤s and industry-wide RPM in a bilateral oligopoly setting.
12
S
for i 6= j 2 f1; 2g. Finally, let
S
:=
X
i
be the resulting industry pro…t
pS
c
i
eS
Di pS ; eSi ; pS ; eSj :
We now show that the use of industry-wide price …xing may allow the manufacturer
to induce the fully integrated pro…t I but only as long as there are no spillovers in sales
e¤ort. First, we show that in all contract equilibria, the marginal transfer prices are at
the manufacturer’s marginal cost.
Lemma 3. (Industry-wide RPM) In all contract equilibria, the marginal transfer prices
are the same for each retailer and equal to the manufacturer’s marginal cost c.
Proof: See Appendix A.
With an industry-wide …xed resale price equal to p set by the manufacturer at the …rst
stage of the game, the unique Nash equilibrium at the …nal stage has retailer i 2 f1; 2g
exerting sales e¤ort equal to ei (p) (De…nition 2). The manufacturer’s optimal industrywide resale price is therefore characterized by
p = arg max
p
X
[p
c
i
(ei (p))] Di p; ei (p) ; p; ej (p)
(4)
i
given that it is optimal for the monopolist to impose a price restraint that binds for both
retailers.10 We then have the following result.
Proposition 2. (Industry-wide RPM) If the manufacturer can commit to an industrywide (minimum, maximum or …xed) resale price, he will be able to induce a pro…t of at
I
least S ; where S
as de…ned in De…nition 2. The monopolist is able to induce the
I
I
I
pro…t
i¤ i) p1 = p2 , and ii) there are no spillovers in retailers’sales e¤ort. As long as
retail prices are strategic complements, to induce a price increase for at least one retailer,
the price restraint will have to be introduced either as a …xed price or as a minimum price.
Proof: See Appendix A.
The intuition for this result is as follows. Assume that pI1 = pI2 = pI . Each retailer only
takes into account the e¤ect of its sales e¤ort on its own demand. Hence, when facing
10
Of course, if pI1 and pI2 are su¢ ciently di¤erent, it may not be optimal for the manufacturer to impose
a uniform price that binds simultaneously for both retailers. For example, if pI1 > pI2 , it may be optimal
to impose a minimum price that is low enough such that it binds for R2 but not for R1 .
13
a marginal transfer price equal to the true marginal cost of the manufacturer, and when
the manufacturer …xes the retail price at the fully integrated level pI ; each retailer will
provide too little e¤ort when spillovers are positive and too much e¤ort when spillovers
are negative. Hence, pS = pI and eS = eI cannot both hold when there are spillovers in
e¤ort.
Without spillovers in sales e¤ort, on the other hand, allowing for an industry-wide
(uniform) RPM clause may fully restore the manufacturer’s ability to induce the fully
integrated outcome but only if the integrated prices are uniform. The manufacturer can
then commit to the integrated price pI at the …rst stage of the game, and a marginal
transfer price equal to c— which characterizes the unique equilibrium at the contracting
stage— is then su¢ cient to induce the retailer to exert the integrated level of sales e¤ort
eIi at the …nal stage.
Because the manufacturer could always replicate the Bertrand outcome by setting a
B
su¢ ciently low minimum price, p min pB
1 ; p2 , at the …rst stage, the manufacturer’s
P B
pro…t will always be weakly higher than
i i when industry-wide resale prices are
allowed. Moreover, we can also note that given that prices are strategic complements, to
elicit a price increase relative to the Bertrand outcome, the industry-wide resale price will
have to be introduced either as a …xed price or as a price ‡oor. The reason is that all
contract equilibria are again characterized by marginal cost pricing for the manufacturer’s
B
product. A minimum price p > min pB
is therefore needed to induce at least one
1 ; p2
of the retailers to charge a price above the standard Bertrand price.
The reason that spillovers create a problem for the manufacturer is related to MW’s
…nding that closed territory distribution only maximizes joint pro…ts when there are no
spillovers. To achieve the fully integrated outcome, the manufacturer has two necessary
objectives: to eliminate retail competition and to provide optimal margins that induce
retailers to provide optimal e¤ort levels. This holds both in MW and here. In MW
without spillovers, the fully integrated pro…t is achieved by making the retailers residual claimants with marginal cost transfer pricing under closed territories. In our model
without spillovers, the manufacturer eliminates competition by setting the industry-wide
RPM at the fully integrated level pI . With secret contracts, the equilibrium marginal
transfer prices are at marginal cost, and the retailers will supply the fully integrated level
of e¤ort.
On the other hand, when there are spillovers in both models, the results are di¤erent.
In MW with spillovers, the fully integrated outcome cannot be achieved by only using
exclusive territories to restrict competition. With positive spillovers, for instance, the
manufacturer wants to induce more e¤ort compared with when there are no spillovers. To
14
induce more e¤ort, he should lower the marginal transfer price below marginal cost, but
in MW, this has the side e¤ect that each retailer would also lower its price; hence, the
manufacturer cannot achieve the fully integrated pro…t. In our model, the device to control
for competition is industry-wide minimum RPM. The manufacturer can achieve the fully
integrated prices by using industry-wide RPM (given symmetry), but he cannot achieve
the fully integrated e¤ort levels at the same time. When contracts are secret, equilibrium
transfer prices will be at marginal cost. If the manufacturer wants to induce more e¤ort—
for instance, because of positive spillovers— he will have to increase the industry-wide
minimum price above the fully integrated prices; hence, the fully integrated pro…t cannot
be achieved. Hence, there is a relation between the result in MW with exclusive territories
and spillovers, and our result with industry-wide RPM, but the underlying problem is very
di¤erent.
However, MW show that even in the case with spillovers, the manufacturer can achieve
the fully integrated pro…t if he can use RPM. If spillovers are positive, MW show that
marginal transfer prices below marginal cost and RPM at the fully integrated level will
enable the manufacturer to realize the fully integrated pro…t. In our model, this mechanism will not work. The reason is that with secret contracts, equilibrium transfer prices
are always at marginal cost.
To provide a sense of the potential welfare implications of allowing for an industrywide price ‡oor in our setting, we will evaluate consumer welfare using a commonly used
representative utility function V = y + U (q1 ; q2 ),11 where
U (q1 ; q2 ) = v
2
X
i=1
and
qi
1
1+
8
2
<1 X
:2
i=1
(2qi
A i ) qi +
2
2
X
i=1
qi
!2 9
=
;
(5)
Ai = (2 + (1 + )) ei + (2 + (1 + )) ej ; i 6= j 2 f1; 2g
and 2 [ 1; 1] is a measure of spillovers in e¤ort provision. That is, we allow for positive
or negative spillovers, and of a varying degree. y represents a composite good with a
price normalized to py = 1, qi is the quantity purchased by the consumer from retailer
i 2 f1; 2g ; and 2 [0; 1) is a measure of the substitutability between retailers. We
P
assume that the consumer’s budget constraint binds, y + i pi qi = I. Note that when
e1 = e2 = 0; U yields Shubik–Levitan (1980) demand functions. Finally, we assume that
the retailers’e¤ort cost is given by Ci = e2i =2, i 2 f1; 2g ; where > 1.
11
See for instance Motta (2004, pp 326–331).
15
By comparing the representative consumer’s net utility when retailers set the standard
Bertrand prices and e¤ort levels pB ; eB , with the consumer’s net utility under industrywide RPM and e¤ort levels pS ; eS , we get the following result.
Proposition 3. Given the utility function U in (5), the manufacturer would like to
implement an industry-wide price ‡oor [ceiling] when the service spillovers are positive
[negative] or [and] the degree of substitutability between the retailers is su¢ ciently high
[low]. The e¤ect on consumers is always negative [positive] in this case, while the e¤ect
on total welfare is ambiguous [positive].
Proof: See Appendix A.
Proposition 3 is potentially important because it challenges the view that in a setting
where retailers free-ride on each other’s service provisions, price ‡oors create e¢ ciencies
that ultimately bene…t consumers.12 Often behind this claim is an implicit assumption
that the manufacturer can commit to a set of public wholesale contracts. Contrary to
commonly held beliefs, our example suggests that when wholesale contracts are unobservable, instead of providing an e¢ ciency defense, service externalities may in fact constitute
an additional reason for not allowing …xed or minimum RPM clauses.
The intuition for this is easiest to explain when there is little or no price competition
between the retailers. In this case, if there are no spillovers from sales e¤ort, the manufacturer will be able to achieve the fully integrated outcome without price restraints. Hence,
allowing the manufacturer to use minimum RPM clauses will have no e¤ect on the equilibrium outcome. With positive spillovers (and no price restraints), however, each retailer
will provide too little e¤ort in equilibrium. One way to correct for this externality would
be to implement a minimum or …xed RPM clause to prevent the retailers from cutting
their prices, and then subsidizing the retailers’sales by setting marginal wholesale prices
below cost, as shown by MW. The problem is that when contracts are secret, the manufacturer is unable to convince the …rst retailer that he will set the marginal wholesale price
below marginal cost for the second retailer, and vice versa. Therefore, if the manufacturer
is allowed to implement …xed or minimum RPM, in order to induce its retailers to exert
more e¤ort, he will distort the retail price upward instead. In fact, the manufacturer will
increase the retail price above what would constitute the fully integrated price given the
same levels of e¤ort— i.e. pS > pI eS — and ultimately will hurt the consumers. The
conclusion is therefore that when we allow the manufacturer to implement price ‡oors, we
12
This view is not uncommon in the debate about the pros and cons of RPM. See, for example, the
discussions in Marvel and McCa¤erty (1985), Telser (1960, 1990), and Elzinga and Mills (2008).
16
may end up causing more harm to consumers if there are service externalities downstream,
compared with the case without such externalities.
Motta (2004) uses the same utility function as we do but with observable contracts.
Motta shows that price ‡oors in that case always increase both consumer and total welfare.13 On the other hand, our proposition states that if the manufacturer can commit
to a common price ‡oor for both retailers, and if retailers do not have other information
about rivals’ contract terms, consumers will always lose. Obviously, in our setting, the
monopolist will also bene…t from being allowed to implement price ‡oors, and hence we
…nd that the pro…t gain to the monopolist may sometimes dominate the harm to the
consumers. Typically this can happen when the cost of service provision is su¢ ciently
low. Moreover, on an intuitive level, the welfare implications for consumers will depend
on how demand reacts to retailers’ e¤ort. For example, if service e¤ort is necessary to
generate any positive demand (unlike the utility function speci…ed above), and if very
little or no e¤ort would be provided without price restraints (e.g., when price competition
is …erce), then we should be open to the possibility that both consumer and total welfare
could increase with public price ‡oors.
6
Conclusion
The literature on the use of resale price restraints has shown that RPM may be anticompetitive and may lead to increased prices to consumers but also that RPM may be
bene…cial because it may induce greater retail sales e¤ort. These results have been derived
either in models with secret contracts and no retail sales e¤ort or with public contracts
and where retail sales e¤ort is important. We have argued that in most real-life markets,
retail sales e¤ort is of at least some importance and that most contracts are secret. The
model that we present in this paper contains both of these key ingredients and generates
new insights.
We have shown that purely bilateral price restraints, irrespective of type, have no e¤ect
on the equilibrium outcome. Bilateral price restraints can be used neither for anticompetitive nor for e¢ ciency purposes, such as to induce retail sales e¤ort. In equilibrium,
the standard Bertrand prices and e¤ort levels prevail. Next, we have shown that if manufacturers can commit to industry-wide price ‡oors, a manufacturer can obtain higher
prices and e¤ort levels, but he will not generally be able to achieve the fully integrated
outcome. Using a linear demand example, we …nd that retail prices in fact will be higher
13
Motta (2004) restricts his analysis to positive spillovers in service,
17
2 [0; 1].
than the fully integrated prices would be for the same level of sales e¤ort, and consumer
welfare may decrease. In fact, our results suggest that by allowing the use of minimum
RPM, we may sometimes cause more harm to consumers if there are service externalities
downstream, compared with the case without such externalities.
The anticompetitive argument for RPM with secret contracts is based on the idea that
a maximum RPM coupled with high marginal wholesale prices can solve the manufacturer’s opportunism problem. On a general level, we have shown that the opportunism
problem arising from contract unobservability in vertical relations may be signi…cantly
harder to solve than has been recognized in the literature previously. Speci…cally, the result that maximum RPM mitigates opportunism, as demonstrated by O’Brien and Sha¤er
(1992) for the case when price is the only choice variable, does not arise when retail demand also depends on retail sales e¤ort. As the basic problem stems from positive margins
at the retail level and the intrinsic temptation to free-ride on these margins, a solution is
to eliminate these margins by using maximum RPM and high transfer prices, as shown by
OS. This works if retail service has no impact on retail demand. However, if the retailers
exert sales e¤ort, which in turn a¤ects demand, maximum RPM no longer supports the
integrated outcome, as the manufacturer would wish to lower its transfer price to each
retailer, inducing higher sales. Positive margins, in turn, completely reopens the door
for opportunism and returns the outcome to Bertrand prices and e¤ort levels. We have
shown that when retail service has any positive impact on demand, irrespective of the
size and sign of spillovers from such activity, no nonlinear contract with bilateral price
restraints can solve the opportunism problem.
To restore the integrated outcome fully, the manufacturer’s contract with one retailer
would have to be (indirectly) contingent on the price set, and the quantity sold, by rival
retailers, and vice versa. That is, the contracts need to include a credible horizontal
commitment from the manufacturer, and this may be di¢ cult to implement in practice.
Other industry-wide practices, such as price …xing agreements, may be a more viable
solution; e.g., when facilitated through industry trade agreements. We have seen the
latter implemented in European book markets; e.g., in Spain, France and Germany. Yet,
in general, even these types of horizontal agreements will not su¢ ce to implement the
fully integrated outcome as long as the rest of the contract terms are individually (and
secretly) negotiated.
As noted above, competition policy in many countries tends to be more hostile against
minimum or …xed RPM than maximum RPM. In fact, in most jurisdictions, maximum
RPM is regarded as unproblematic. Several recent articles have challenged this view by
showing that maximum RPM can also raise prices to consumers (Montez, 2012; O’Brien
18
and Sha¤er, 1992; Rey and Vergé, 2004). They all show that maximum RPM may be
detrimental to consumers because it is an instrument for solving the opportunism problem.
An important policy implication from our analysis is that this result is not very robust.
Another important policy implication is that bilateral vertical price restraints, irrespective
of their type, are harmless for consumers. Industry-wide price ‡oors on the other hand,
may be detrimental for consumers.
19
Appendix A
Proof of Lemma 1.
The proof follows closely the proof in O’Brien and Sha¤er (1992), which we have modi…ed
to encompass both retailers’sales e¤ort and RPM.
The proof consists of three steps.
Step 1. For all j 2 f1; 2g, Tj (Dj ) is continuous at the quantity induced by T .
Proof. Let Dj be the quantity induced by T , and suppose that Tj (Dj ) were not continuous at Dj . Then, for some in…nitesimal change in Dj , Tj would either jump up or jump
down. However, it cannot jump down, because retailer j could then adjust its e¤ort by
a small amount and induce a discrete reduction in its payment. Furthermore, it cannot
jump up, for then M and Ri could jointly adjust pi and/or ei , and induce a discrete jump
in their bilateral pro…ts. Hence, Tj must be continuous at Dj . Q.E.D.
Step 2. The function Tj (Dj ) satis…es Tj+0
Tj 0 ; for all j 6= i 2 f1; 2g, where Tj+0 and
Tj 0 are the right-hand (+) and left-hand (-) derivatives of Tj , respectively, evaluated at
Dj .
Proof. From step 1, we know that Tj has both a left-hand (-) and a right-hand (+)
derivative at Dj . Retailer j’s …rst-order conditions for optimal e¤ort then require that
@ej
j
= @ej pj
Tj 0 Dj
Cj0 (ej )
0
(A1)
@ej
j +
= @ej pj
Tj+0 Dj
Cj0 (ej )
0
(A2)
and
using the fact that @ej Dj > 0. Together, (A1) and (A2) yield Tj+0
condition for retailer optimality. Q.E.D.
Step 3. The function Tj (Dj ) satis…es Tj 0
Dj = Dj .
Tj 0 as a necessary
Tj+0 for all j 6= i 2 f1; 2g ; when evaluated at
Proof. In every contract equilibrium with general nonlinear contracts and RPM, M ’s
contract with Ri by de…nition solves
n
max (pi
pi ;ei
c) Di + Tj
20
cDj
o
Ci (ei ) :
(A3)
This yields the following …rst-order conditions for the price pi
(pi
c) @pi Di + Di
c@pi Dj
@pi Dj Tj 0
(A4)
(pi
c) @pi Di + Di
c@pi Dj
@pi Dj Tj+0
(A5)
and
using the fact that @pi Dj > 0. Together, (A4) and (A5) imply that @pi Dj Tj 0
Tj+0 because @pi Dj < 0. Together with step 2, this implies that
@pi Dj Tj+0 , or Tj 0
Tj 0 = Tj+0 for the contracts to be bilateral best responses with general nonlinear contracts
and bilateral RPM.
The rest of the proof considers the case where deviations on the price are not possible;
i.e., the case of a binding industry-wide resale price. Note …rst that even though M cannot
deviate with Ri on the price pi , as is the case with an industry-wide resale price p , M ’s
contract with Ri still has to solve
n
max (p
ei
c) Di + Tj
cDj
o
Ci (ei ) :
(A6)
The case without spillovers is trivial and not important for our results. In the following,
we therefore consider only the cases with positive and negative spillovers.
With positive spillovers in e¤ort, the left-hand (right-hand) derivative of the joint pro…t
function (A6) with respect to ei translates into the left-hand (right-hand) derivative of
the tari¤ Tj with respect to the quantity Dj (the same as when considering maximization
w.r.t. prices above). The …rst-order conditions for (A6) are therefore
(p
c) @ei Di
c@ei Dj
Ci0 (ei )
@ei Dj Tj 0
(A7)
(p
c) @ei Di
c@ei Dj
Ci0 (ei )
@ei Dj Tj+0 :
(A8)
and
Together, (A7) and (A8) imply that @ei Dj Tj+0
@ei Dj Tj 0 ; or Tj 0
Tj+0 , because
@ei Dj < 0 in this case.
With negative spillovers in e¤ort, the left-hand (right-hand) derivative of the joint
pro…t function (A6) with respect to ei translates into the right-hand (left-hand) derivative
of the tari¤ Tj with respect to the quantity Dj . The …rst-order conditions for (A6) are
therefore
(A9)
(p
c) @ei Di c@ei Dj Ci0 (ei )
@ei Dj Tj+0
21
and
(p
c) @ei Di
Ci0 (ei )
c@ei Dj
Together, (A9) and (A10) imply that
@ei Dj > 0 in this case.
@ei Dj Tj 0
@ei Dj Tj 0 :
@ei Dj Tj+0 ; or Tj 0
(A10)
Tj+0 , because
The proof is completed by noting that steps 2 and 3 together imply that Tj 0 = Tj+0
when evaluated at Dj = Dj , both for the case with general nonlinear contracts and
bilateral RPM, and for the case with general nonlinear contracts, industry-wide price
…xing, and spillovers in e¤ort. Q.E.D.
Proof of Lemma 2.
Because the manufacturer can use RPM, he is free to use Ti to induce the right level of
e¤ort ei . Hence, we can think of the pair M Ri as choosing both pi and ei directly at
the contracting stage. In any contract equilibrium (T ; s ), pi and ei must solve
(N =2
X
(pk
(N =2
X
(pk
@pi pj
Tj 0 Dj = 0
(A11)
)
@ei pj
Tj 0 Dj = 0:
(A12)
c) @pi Dk + Di
k=1
and
)
c) @ei Dk
k=1
Ci0
Note that the terms in the curly brackets are equal to zero when both p = pI and e = eI .
Hence, given that pj = pIj and ej = eIj , for it to be optimal for the pair M Ri to induce
pi = pIi and ei = eIi , at the quantity Dj , the marginal transfer price Tj 0 would have to
be equal to the integrated price, pIj . However, Rj ’s …rst-order condition for optimal sales
e¤ort at the …nal stage is pj Tj 0 @ei Dj Cj0 = 0. At Tj 0 = pIj , Rj ’s pro…t on the last
unit sold when exerting sales e¤ort ej = eIj > 0, is negative. Hence, p = pI and e = eI
cannot both hold in equilibrium. Q.E.D.
Proof of Proposition 1.
Note that the …rst-order maximizing condition for Ri at the …nal stage is (pi Ti 0 ) @ei Di
Ci0 = 0. Substituting this into (A12) from the proof of Lemma 2, and simplifying, yields
the following necessary conditions for (T ; s ) to form a contract equilibrium:
(N =2
X
k=1
(pk
c) @pi Dk + Di
)
@pi pj
22
Tj 0 Dj = 0; i 6= j 2 f1; 2g
(A13)
and
N
=2
X
(Tk 0
c) @ei Dk = 0; i 2 f1; 2g :
k=1
(A14)
Condition (A14) can be rewritten in matrix notation as (T0 c) De = 0; where T0 =
(T1 0 ; T2 0 ) and De is the 2-by-2 matrix of demand derivatives with respect to retailer sales
e¤ort. By assumption A2, De is always invertible. Q.E.D.
Proof of Lemma 3.
Note …rst that in any contract equilibrium (T ; s ) with (a binding) industry-wide p , ei
would have to solve the condition
(N =2
)
X
(A15)
@ei p
Tj 0 Dj = 0; i 6= j 2 f1; 2g :
(p
c) @ei Dk Ci0
k=1
Substituting in retailer i’s condition for optimal sales e¤ort, (p
Ti 0 ) @ei Di Ci0 = 0, we
obtain the following necessary condition for (T ; s ) to arise as a contract equilibrium
N
=2
X
(Tk 0
c) @ei Dk = 0; i 2 f1; 2g
k=1
(A16)
which is identical to condition (A14) above. Hence, in all contract equilibria, the marginal
transfer prices are again equal to the manufacturer’s marginal cost c. Importantly, this
result is independent of the industry-wide resale price chosen by M at the …rst stage of
the game. Q.E.D.
Proof of Proposition 2.
From Lemma 3, we have that the marginal transfer prices will be set equal to manufacturer
marginal cost c: Suppose that pI1 = pI2 = pI and that the industry-wide resale price is set
equal to pI : Then the …rst-order condition for retailer i is
pI
c @ei Di
Ci0 = 0
whereas for the fully integrated …rm it would be
N
=2
X
pI
c @ei Dk
Ci0 = 0:
k=1
These both hold if and only if @ei Dj = 0: Hence, p = pI and e = eI are mutually
exclusive when there are spillovers in e¤ort. Q.E.D.
23
Proof Proposition 3.
U yields the following direct demands
1
v + ei + ej
2
q i = Di =
(1 + ) pi +
2
(p1 + p2 )
; i 6= j 2 f1; 2g :
We can set = 2 = (1
), where 2 [0; 1] represents the degree of substitution between
the retailers. Without (industry-wide) RPM, each retailer solves
max (pi
pi ;ei
1
c)
2
pi
1
v + ei + e j
(ei )2
2
pj
yielding symmetric prices and e¤ort levels
pB
i = c+
2 (v c) (1
(1 + ) (1
4
)
)
2
; eB
i =
(v
4 +
c) (1
+
)
2
1
:
Substituting these into the consumer’s utility function U , we get consumers’surplus under
Bertrand levels of prices and e¤ort levels, SB
SB = I +
(2 (2
2 2 (v c)2
) (1 + ) (1
))2
and when substituting into the industry pro…t, we obtain
B
=
X
i
B
i
(1
=
(4
) (4 +
+ +
1) (v
2
c)2
:
1)2
With an industry-wide resale price p, each retailer chooses ei to solve
max (p
ei
c)
1
(v + ei + ej
2
which yields
ei (p) =
p
(ei )2
2
p)
c
2
for i 6= j 2 f1; 2g, and substituting these into the expression for industry pro…t, the
manufacturer chooses p to maximize
max
p
X
i
(p
1
c) v + ei (p) + ej (p)
2
yielding pS ; eSi
24
p
(ei (p))2
2
!
pS =
2v
c + 2c (
4
2
1
)
; eSi =
v
4
c
2
1
for i 2 f1; 2g. Substituting this into the utility function U , we can derive consumers’
surplus under industry-wide RPM: S RP M
S
RP M
(v c)2 (2
=I+
2 (4
2
)2
:
1)2
Substituting into the industry pro…t, we obtain
RP M
=
S
=
c)2
:
2
1
(v
4
The price pS has to be implemented as a (…xed or) minimum price as long as
pS
pB =
which always holds if
2 ((1
)
+ +
(4
+2
2
) (v c)
1) (4
2
>0
1)
> 0, and otherwise as long as
>
= e:
2
Let S = S B S RP M be the loss to consumers when allowing the manufacturer to
impose industry-wide RPM
(v
S=
2 (4
c)2 (2
(1 + )) ((1
2
2
1) (4 +
+
)
+2
2
)
1)2
where
= 16
2
10
+
4
2
4 +
2
+2
2
+4
:
We have @
= (2
) (2
(1 + )) < 0 and lim !1 = 2 (6
(3 + 1)) > 0.
We can therefore conclude that
> 0. The consumers are therefore hurt when we
allow the manufacturer to impose an industry-wide minimum or maximum RPM clause
( S > 0) as long as > e; i.e., when it is a minimum price. The consumers always
bene…t, on the other hand, when < e; i.e., when it is a maximum price. We can see
that the consumers are always hurt by RPM when the spillovers in retailers’sales e¤ort
are positive, > 0. To see that the consumers’loss from RPM can be increasing in the
25
spillovers , consider the example c = 0; v = 1,
@ ( S) =
2
4116 + 1113
(7
= 0, and
3773
103
3
) (7 2 )3
= 2. We then have that
3
4
>0
for all 2 [ 1; 1] : The reason for this is that when spillovers are positive, in order to
boost retailers’ sales e¤ort, the manufacturer will impose an industry-wide retail price
above what would constitute the fully integrated price given the same levels of e¤ort; i.e.,
pS > pI eS
(v c)
>0
pS p I e S =
2 (4
2
1)
and will end up hurting the consumers.
The industry pro…t always increases when we allow the manufacturer to impose
industry-wide RPM
S
B
=
(v
(4 +
+
c)2 (2 +
)2
2
1)2 (4
2
1)
0:
We can conclude that consumers’surplus as well as overall welfare increase when we allow
the manufacturer to impose industry-wide RPM, given that < e (i.e., as long as it is
a maximum price). However, when > e, we have both a net loss for the consumers
together with a net gain for the …rms; hence, overall welfare may go either up or down
in this case. Letting W be the change in overall welfare, we can write
c)2 (2
(1 + )) ((1
) +2 )
W =
2
2
1) (4 +
+
2
1)2
(v c)2 (2 +
)2
+
:
(4 + +
2
1)2 (4
2
1)
(v
2 (4
Even if > e, we may have W > 0 sometimes when the costs of sales e¤ort
enough. To see this, consider the example = 0 and = 1=2. We then have
W =
which is positive when 136
(4 + 1) (v
368
2
c)2
+ 256
are low
196
528 2 + 320 3 9
288 (2
1)2 (4
1)2
3
26
9 < 0; i.e., when
2 [1; 1:13). Q.E.D.
Appendix B
Proposition 1 in our paper demonstrates that any contract equilibrium (CE) with bilateral
RPM, in which the retailers charge the fully integrated prices, breaks down whenever the
retailers exert sales e¤ort. Obviously, this means that any corresponding passive-beliefs
equilibria (PBE) break down as well. In every CE, the manufacturer therefore distributes
its product for a marginal wholesale price equal to marginal cost (“cost-based contracts”),
and the retailers set the standard Bertrand prices and e¤ort levels at the …nal stage of
the game. The question that remains is if and when the latter also constitutes a PBE.
Rey and Vergé (2004) (RV hereafter) demonstrate in a model without sales e¤ort that
a PBE with cost-based contracts exists but only as long as the degree of substitution
between the retailers is not too strong. Hence, we should expect their result to carry over
to our model when the e¤ect of sales e¤ort on demand tends to zero. If the e¤ect of sales
e¤ort is nonnegligible, however, their condition is modi…ed, as we will now demonstrate.
Without loss of generality, we are going to focus on two-part tari¤s Ti (qi ) = wi qi + Fi
for i 2 f1; 2g : To simplify the exposition, we will also assume that demands and e¤ort
costs are symmetric.14 First note that in every PBE, the …xed fee for retailer i 2 f1; 2g
is equal to
Fi (wi ) = (pi (wi )
wi ) Di pi (wi ) ; ei (wi ) ; pB ; eB
Ci (ei (wi ))
where
wi ) Di pi ; ei ; pB ; eB
(pi (wi ) ; ei (wi )) = arg max (pi
pi ;ei
Ci (ei ) :
The manufacturer’s equilibrium pro…t as a function of wholesale prices can therefore be
written as
M (wi ; wj ) =
X
i
n
(wi
c) Di pi (wi ) ; ei (wi ) ; pj (wj ) ; ej (wj )
+ (pi (wi )
wi ) Di pi (wi ) ; ei (wi ) ; pB ; eB
(B1)
o
Ci (ei (wi )) :
We need to know if, when wi = wj = c, the manufacturer would like to deviate with both
retailers at once. If so, M (wi ; wj ) is not concave, and a PBE does not exist. Because of
14
RV focus on two-part tari¤s and assume symmetric demand. We make the same assumptions here
for the sake of brevity. The analysis could be generalized to asymmetric demands and cost functions but
at the cost of a signi…cantly longer exposition and heavier notation.
27
symmetry, we can write the Hessian determinant of
H (wi ; wj ) =
@wi wi
M
@wi wi
M
(wi ; wj ) as
(wi ; wj ) + @wi wj
(wi ; wj )
M
M
@wi wj
(B2)
(wi ; wj )
M
(wi ; wj ) :
Note then that a deviation of the type wi = wj = c + " is pro…table whenever @wi wi M +
@wi wj M > 0 for wi = wj = c, in which case, the monopolist’s pro…t function is not
concave. From (B1) we have that
@wi wj
M
= 2 [pi 0 (c) @pi Dj + ei 0 (c) @ei Dj ]
(B3)
when wi = wj = c. Next, de…ne retailer i’s maximized expected pro…t, gross of …xed fees,
as
wi ) Di pi (wi ) ; ei (wi ) ; pB ; eB
Ri (pi (wi ) ; ei (wi ) ; wi ) = (pi (wi )
Ci (ei (wi )) : (B4)
Note then that by using (B4), we can write
M
(wi ; c) = Ri (pi (wi ) ; ei (wi ) ; c) + Fj (c) ;
where Fj (c) is independent of wi . To …nd @wi wi
@wi
M
M
(B5)
we use the fact that
(wi ; c) = pi 0 (wi ) @pi Ri (pi (wi ) ; ei (wi ) ; c)
(B6)
+ei 0 (wi ) @ei Ri (pi (wi ) ; ei (wi ) ; c) :
Taking the second partial derivative and evaluating it at wi = c, we get
@wi wi
2
M
2
= @pi pi Ri [pi 0 (c)] + @ei ei Ri [ei 0 (c)] + 2@pi ei Ri pi 0 (c) ei 0 (c) :
(B7)
+@pi Ri pi 00 (c) + @ei Ri ei 00 (c)
From the retailers’maximization problems, we know that @pi Ri = @ei Ri = 0; hence, (B7)
is equal to
@wi wi
2
2
M
= @pi pi Ri [pi 0 (c)] + @ei ei Ri [ei 0 (c)] + 2@pi ei Ri pi 0 (c) ei 0 (c) :
28
(B8)
The condition that a multilateral deviation of the type c + " is pro…table is therefore that
2
2
@pi pi Ri [pi 0 (c)] + @ei ei Ri [ei 0 (c)] + 2@pi ei Ri pi 0 (c) ei 0 (c)
(B9)
+2 [pi 0 (c) @pi Dj + ei 0 (c) @ei Dj ] > 0:
From the maximization problem of retailer i 2 f1; 2g, we have that
"
@pi pi Ri @pi ei Ri
@ei pi Ri @ei ei Ri
#"
pi 0 (wi )
ei 0 (wi )
#
=
"
@pi Di
@ei Di
#
:
(B10)
Applying (B10) to condition (B9) using Cramer’s rule, and then simplyfying, we get
[@pi Di + 2@pi Dj ] pi 0 (c) + [@ei Di + 2@ei Dj ] ei 0 (c) > 0:
|
{z
}
RV (2004)
(B11)
Assuming that ei 0 (wi ) < 0 and pi 0 (wi ) > 0,15 we can see that compared with the condition
in RV, a deviation of the type c + " is less likely to be pro…table when the retailers exert
sales e¤ort, given that @ei Di +2@ei Dj > 0. The latter always holds as long as the spillovers
in sales e¤ort are positive or not too strong. The intuition that a multilateral deviation
c + " may be less pro…table now is that an increase in the wholesale prices will cause the
retailers to exert less sales e¤ort, which all else equal is bad for pro…ts when spillovers are
positive or not too strong.
Even if a multilateral deviation c+" is not pro…table, a deviation of the type wi = c+"
and wj = c " can be. This is the case if @wi wi M @wi wj M > 0, which, given our
calculations above, can be written as
[@pi Di
|
2@pi Dj ] pi 0 (c) + [@ei Di
{z
}
2@ei Dj ] ei 0 (c) > 0:
(B12)
negative
We can see that this condition never holds as long as the “service elasticity”of demand is
not too high compared with the price elasticity of demand. A necessary but not su¢ cient
condition for the deviation to be pro…table is that @ei Di < 2@ei Dj . For the utility function
used in Section 5, for example, the only relevant condition is (B11).
In Proposition 2 (industry-wide RPM), it is assumed that the manufacturer can …x
15
Given that the demand functions are additively separable, it can be shown that a su¢ cient condition
@pi Di
for this is that
< 1.
@pi ;pi i
29
the retailers’prices before negotiating supply terms (wi ; Fi ) bilaterally and privately with
the retailers at stage 2. It is easy to check that a PBE with wi = wj = c then exists at
stage 2 as long as
1 1
@ei Dj
2
;
(B13)
@ei Di
2 2
i.e., as long as the service spillovers are not too strong.
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