The Competition Effects of Industry-wide Vertical
Price Fixing in Bilateral Oligopoly
Paul W. Dobson a,*, Michael Waterson b
a
Business School, Loughborough University, Loughborough, LE11 3TU, United Kingdom.
b
Department of Economics, University of Warwick, Coventry CV4 7AL, United Kingdom.
March 2007
Abstract
This paper examines the competition and welfare effects of vertical price fixing through
industry-wide resale price maintenance (RPM) arrangements, such as those benefiting from exemption
from a general prohibition against RPM. A bilateral oligopoly framework is employed incorporating
differentiation between manufacturer products and between retailer services. Transactions between the
stages involve prices being determined through bargaining. We do not find RPM to be universally
undesirable. However where retailer power is strong, the social effects of RPM are likely to be adverse,
since the practice can assist in coordinating final price levels and prevent socially desirable
countervailing power arising.
JEL classification: L13, L41
Keywords: Resale price maintenance; bilateral oligopoly; bargaining; countervailing power
* Corresponding author
E-mail address: [email protected]
Acknowledgements
We thank two anonymous referees and the co-editor, Roman Inderst, for their very helpful comments and
suggestions. We are also grateful for comments from Robert Steiner.
1. Introduction
Negotiations between players at successive vertical levels in a market commonly result in agreements
that differ considerably from pure linear arms-length pricing. Arguably, the main effect of these
cooperative arrangements is to ensure efficient transfer between vertical levels.
EC Article 81
recognises this through a Block Exemption, Regulation 2790/99, which exempts various types of
vertical agreements from the operation of Article 81(1) (of the Amsterdam Treaty) on agreements
between firms. However, under Article 4 of the Block Exemption, certain “hardcore restrictions”
render the vertical agreement, as a whole, void. Prime amongst these are contractual provisions or
concerted practices that have the direct or indirect effect or object of fixing retail price, or minimum
retail price, i.e. resale price maintenance (“RPM”) agreements.1 Is this justified by the likely impact
of RPM? Our paper examines the effect of RPM within the context of a model of bilateral oligopoly.
This extends previous literature (e.g. O’Brien and Shaffer, 1992) that has examined monopoly
producers seeking vertical control to the increasingly relevant case where significant market power
exists at both levels.2 It is complementary to related work by Rey and Vergé (2004a) examining RPM
in multiple common agency relationships and to studies of non-price restraints, such as those by Rey
and Stiglitz (1995) and Besanko and Perry (1994), on exclusivity arrangements.
We focus on market power and competition effects, but still find that RPM may on occasion
have a beneficial impact.3 Essentially, there are two forces involved in the market power effects, with
opposing impacts. RPM means double marginalisation (i.e. manufacturers and retailers applying
successive mark-ups) is avoided, a beneficial effect. But RPM may also have an anticompetitive
purpose by dampening interbrand competition. In the absence of this, retailers seeking to remain
competitive would be likely to put pressure on manufacturers to cut margins, potentially spreading
intense competition to both levels.
Our model involves bilateral oligopoly in which we allow different degrees of differentiation
both of manufacturer products and retailer services. We incorporate negotiation over intermediate
prices.
We find that firms’ collective interests commonly conflict with the public’s interest,
particularly when competition between retailers has a socially beneficial countervailing power effect
in lowering prices to consumers. Such outcomes arise in our model when retail services are highly
substitutable and retailers dominate negotiations with manufacturers. Then, with retailers free to set
retail prices, intense retail competition strengthens retailers’ bargaining positions (since with thin
1
This tough stance against vertical price fixing, as opposed to a more lenient treatment towards most non-price
vertical restraints, is common in other jurisdictions. For example, fixed RPM has been per se illegal in the US
since 1975 (when Congress repealed the Miller-Tydings Act and the McGuire Act). But see also Leegin v PSKS
(2007).
2
Related papers examining other aspects of bilateral oligopoly include Björnerstedt and Stennek (2006), Inderst
and Wey (2003), de Fontenay and Gans (2005), Hendricks and McAfee (2005), and Antelo and Bru (2006).
3
A different strand of the literature (e.g. Telser, 1960; Marvel and McCafferty, 1984) centres on dealer freeriding effects, concerning pre-sales service levels, quality certification, etc. This is likely to yield additional
positive arguments for RPM’s use in vertical linkages, but we wish to examine market power effects directly.
1
retail margins they can ill afford to make concessions) resulting in both low transfer prices and low
retail prices (and thus low profits for both manufacturers and retailers). RPM can be used as a way of
avoiding this problem for the firms, but with the consequence of higher prices for consumers.
The paper’s focus is on industry-wide RPM arrangements, given that it is instances of these
which have received special exemptions in the past and in some cases continue to do so presently. In
particular, while fixed RPM is generally prohibited in almost all OECD countries, some special
exemptions continue to exist in certain countries (OECD, 1997; Holmes and Cerdan, 2005). Recent
years have witnessed many of these RPM exemptions being rescinded (e.g. books and over-thecounter medicines in the UK in the late 1990s), yet a number continue (e.g. for published goods in
Germany, copyrighted works in Japan, and medicines and books in Spain, amongst others). This
raises an important welfare issue for authorities considering granting or withdrawing an exemption.
Would consumers gain or lose by a change in regime from one governed by RPM to one of
unrestricted trading, or vice versa?
Some empirical evidence on particular cases has been offered
(e.g. OFT, 2003; Davies et al., 2004 (§4); Beck, 2006). This paper, though, examines the issue
theoretically to see if there are particular market circumstances that are likely to favour one regime
over the other from the consumer’s perspective, with a view to providing useful insights for
competition authorities treating these cases.
The plan of the paper is as follows. The next section details our analytical framework. Here,
we develop a model where there are differentiated manufacturers offering goods to differentiated
retailers, such that there is a trading opportunity (and thus an incentive) for each manufacturer to trade
with each retailer. If the trading parties are successful in pursuing all trading opportunities then
consumers face a full set of feasible product-service combinations, such they can choose between all
retailers selling the full range of available products.4 In this setting, we consider two different
regimes: one governed by RPM; the other by unrestricted trading. The critical difference between the
two regimes is which trading party sets the corresponding retail price to consumers. In the RPM
regime, it is the manufacturers that unilaterally and independently dictate the retail prices of their own
products sold through the retailers.
In the unrestricted trading regime, the retailers may take
suggested retail prices from the manufacturers but ultimately they are free to set retail prices for the
products sold in their own stores. In both regimes, we allow for bilateral bargaining over the transfer
price between each and every trading pair. Section 3 reports and discussed the outcomes for the RPM
regime. Section 4 reports and discusses the results for the unrestricted trading (“UT”) situation, and
contrasts the horizontal and vertical competitive differences in this situation with that of the RPM
regime. Section 5 concludes the paper with some remarks about the scope of the analysis and
possible policy implications.
4
For example, consider booksellers selling books from all major publishers, pharmacies selling over-the-counter
medicines from all major pharmaceutical companies, or grocery retailers selling branded goods from all major
fast-moving-consumer-goods manufacturers.
2
2. Analytical Framework
We consider the situation where the supply and distribution of particular consumer goods are controlled
by a very limited number of powerful parties at successive stages of the supply chain; characterised by a
tight oligopoly existing in both manufacturing and retailing these goods. We assume that each player
has successfully built up a distinct franchise with consumers in respect of its product or service
proposition as, respectively, a manufacturer or retailer. Thus, while the manufacturers may offer
broadly substitutable products or brands, there may be differentiation between these in the minds of
consumers, with each holding different attractions. Similarly, with retailers, consumers may view their
services as broadly substitutable yet see them as (potentially) differentiated in what they represent (e.g.
in regard to store location, in-store amenities, sales service, store layout or ambience), even when
offering the same range of manufacturer goods.
In this situation, and in the absence of credible threats of new entry or replacement (say, due to
the intellectual property rights held by the incumbent manufacturers or the scale and quality of the store
networks owned by the incumbent retailers), the players on both sides of the market may be able to earn
significant profits. However, they are likely to be individually constrained in doing so by the horizontal
competition (at both levels) and vertical competition (in regard to the relative bargaining power held by
the trading parties). Indeed, it may not only be individual profits which are influenced by how
horizontal and vertical competition operates, but the total level of profits available in the sector as well.
While it is conceivable that the scope of bargaining and the nature of contracting amongst the parties
eliminate all horizontal and vertical externality effects so that sector profits are maximised as a grand
cartel, in practice frictions may exist which result in sub-optimal outcomes. Such frictions can arise for
several reasons, the most significant of which for our purposes are antitrust policy constraints and
established “custom and practice” dictating the way that business is conducted.5
From a competition authority’s perspective, concern would arise where contracts are such that
final market competition is deliberately prevented. This theoretical possibility, Rey and Vergé (2004a)
show, exists in common agency franchise systems, where manufacturers extract profits from retailers
with a franchise fee while fixing retail prices with RPM contracts, as this eliminates both interbrand and
intrabrand competition by leaving no room for manoeuvre by the retailers other than to sell the joint
profit maximising quantities.6 Björnerstedt and Stennek (2001; 2006) show it is equally possible in a
general bilateral oligopoly, when bilateral negotiations between trading parties cover the traded
5
An interesting instance of the latter concerns two-part tariffs (i.e. a lump sum fee plus a uniform wholesale price)
levied by manufacturers, extensively modelled theoretically as a means to avoid vertical externality effects
associated with successive market power but in practice rare outside franchise relationships in retail markets.
Indeed, the only lump sum fees witnessed in most consumer goods markets are those paid to retailers by
manufacturers (i.e. “negative franchise fees”), in the form of slotting allowances, listing fees, or marketing support
payments. In relation to groceries, see Competition Commission (2000) and Clarke et al. (2002). Concerning
book retailing, see “£50,000 to get a book on recommended list”, Sunday Times (London), May 28, 2006.
6
In this setting, Rey and Vergé (2004a) show that two-part tariffs on their own are insufficient to ensure optimal
outcomes due to free-riding effects where each producer sets its wholesale prices insufficiently high, resulting in
retail prices being set below the industry-wide profit maximising level.
3
quantity (which maximises bilateral surpluses) as well as the transfer price (which divides the bilateral
surpluses between the contracting parties). Similarly, it can be shown that if bilateral bargains cover
both the retail price and the transfer price then again resulting market outcomes would be equivalent
to those arising under joint monopoly. Thus, by different means, it is conceivable that competition
(except over the division of the surplus) may be entirely eliminated in bilateral oligopoly.
In contrast to these approaches, and in keeping with a view that frictions may necessitate very
simple contracts even when they might not offer fully efficient outcomes, we allow for the possibility
of downstream competition. We do this by restricting bargains to cover only the transfer price
between the trading parties while allowing final prices to be set unilaterally, either by the
manufacturers when RPM applies or by the retailers when RPM is not enforced.
While the
assumption of linear contracts is made for convenience, we have argued (Dobson and Waterson,
1997) that there are practical reasons why trading parties might adopt such uncomplicated contracts.7
For example, it may be due to retailers demanding flexibility over order sizes when demand or
competitive circumstances are uncertain, especially when negotiations occur infrequently.8
In
addition, simple linear contracts have the benefit that they are generally renegotiation proof, since at
least one party to the contract is likely to have no incentive to renegotiate (a lower/higher transfer
price would leave the manufacturer/retailer worse off).9 Finally, recall that the bargaining agenda
itself may be contentious since the scope of any bargained settlement may affect the relative levels of
profit achieved by the parties (e.g. Dobson, 1997; Milliou et al., 2006). For instance, retailers may
gain a relative bargaining advantage by retaining control over retail market variables, potentially
offering them higher profits (and thus making them reluctant to extend agreements beyond transfer
prices without additional side payments) even though the joint surplus generated may be less than
when bargaining is extended to cover retail market aspects (e.g. quantity traded or retail price).10
Given unilateral decisions on the key final market variable (here, retail prices), there is an
interesting and potentially complex set of horizontal and vertical effects to consider. In particular, we
can expect horizontal and vertical externality effects to work in opposite directions, with the former
7
We recognise, though, that different market contexts may support broader bargaining agendas that extend
beyond flat unit rates for variable quantities of goods. For example, this is the inference drawn by the
econometric analysis of Villas-Boas (2004) on yogurt and Bonnet et al. (2005) on bottled water. However,
Smith and Thanassoulis (2006) use more direct evidence on contract details and interviews with relevant
management to conclude that linear contracts are used in trade in the supply of milk to UK supermarkets.
8
For example, the Competition Commission (2000) reports that negotiations between major UK supermarket
retailers and major brand producers may take place as infrequently as annually for a product line, while
retailers’ orders may be daily or weekly, with constant monitoring and adjustment in line with scanner data
analysis from recorded sales and inventory management linked to just-in-time store replenishment systems.
9
Specifically, more general contracts may require additional contractual arrangements such as enforceable
contingency clauses or most-favoured-customer (MFC) clauses to avoid opportunism problems (e.g. McAfee
and Schwartz, 1994). See also Hart and Tirole (1990) and O’Brien and Shaffer (1992).
10
In other words, bargaining over the bargaining agenda and the range of variables to be covered in an
agreement may be a part of an otherwise complex set of manufacturer-retailer negotiations to establish
contracts. These aspects are sidestepped here by our assumption that bargaining only covers the transfer price,
the minimum that trading parties could be expected to bargain over.
4
tending to increase quantities (as horizontal rivals compete) but the latter tending to lower quantities
(as a consequence of double marginalization).11 The relative strength and influence of these effects on
total sales (and thus available surplus to the parties and consumers) may be influenced by the degree
of rivalry amongst manufacturers and amongst retailers, which in turn may depend on how
substitutable their respective products and services are perceived to be by consumers. Essentially,
increased differentiation in the products and retail services weakens the horizontal externality effects,
but potentially strengthens the vertical externality effects. Nevertheless, the magnitude of these
effects may also be influenced by the relative bargaining power of the parties, double marginalization
effects being most pronounced when manufacturers hold all the bargaining power and retailers set
final prices by adding their own mark-up to (already) high transfer prices.
In this context, vertical restraints may take on particular significance by influencing the balance
of effects in a way that benefits the parties (but not necessarily consumers). For instance, RPM may be
attractive to the parties when it ameliorates vertical effects (by tackling double marginalization
problems) and lessens horizontal effects (e.g. when it coordinates prices and reduces competition in the
market). However, this is by no means assured. As a result, the competition effects of RPM in such a
bilateral oligopoly may depend intricately on the inherent degree of rivalry at each vertical level as well
as the relative bargaining power of the trading parties. The analysis in this paper seeks to disentangle
these elements and obtain clearer insights in a bilateral oligopoly setting where we can separately
represent the degree of intrabrand rivalry at the retail level, the degree of interbrand rivalry at the
manufacturer level, and the relative bargaining strength of retailers compared to manufacturers.
Taking the manufacturer product and retailer service propositions as fixed, we adopt a nonaddress framework whereby consumers’ preferences are fixed in terms of the available goods. In our
case, “goods” are represented by product-service combinations, where for example “good ih” is
manufacturer i’s product sold through retailer h, with differentiation between these product-service
combinations measured in regard to the extent of their perceived substitutability. We limit our attention
to a bilateral duopoly situation (but one that has scope to be extended to more players on each side of the
market), where players possess complete information regarding their own and each other’s demand and
cost positions. The market under consideration comprises two symmetric upstream manufacturers, M1
and M2, indexed by i,j = 1,2, i≠j, each producing their own branded product, together with two
symmetric downstream retailers, R1 and R2, indexed by h,k = 1,2, h≠k, capable of selling these
products.12 The manufacturers supply the products to the retailers at a constant unit price which is
11
Indeed, the opposing effects could be completely offsetting. For example, this can arise where a monopoly
supplier trades with a pair of symmetric homogeneous downstream Cournot oligopolists facing linear demand,
constant unit costs, and where negotiating parties possess equal relative bargaining power (i.e. where symmetric
Nash bargains apply). Here, negotiation over either linear (price-only) contracts or price-and-quantity contracts
can yield the same joint profit maximising quantities, but the downstream firms may earn a greater share of the
surplus under the former contract type (e.g. Dobson, 1990).
12
Symmetry makes the analysis substantially more tractable while ensuring that no firm plays a particular role (e.g.
as a natural leader). We assume that differentiation between the manufacturer brands and retailer services is
5
determined by bargaining, where the negotiated transfer price between Mi and Rh is wih for quantity qih,
which is sold to consumers at the unit price pih. The manufacturers produce under constant returns to
scale, incurring a common unit cost cM, which to aid exposition, and without further loss of generality, is
set to zero. Thus the profit for Mi when it supplies both retailers is πMi = wihqih + wikqik. Similarly, the
retailers incur common retailing costs at constant per unit level cR, which is again for convenience set to
zero. Thus the profit for Rh when it sells both manufacturers’ products is πRh = (pih – wih)qih + (pjh –
wjh)qjh.
We assume that consumer utility functions and budget constraints give rise to symmetric and
invertible indirect demand functions for the four goods, pih(q) ≡ pih(qih,qik,qjh,qjk) ∀i≠j ∀h≠k, that are
bounded, continuous and twice differentiable with ∂pih/∂qih < 0 (so inverse demand is downward
sloping). It is further assumed that available product-service combinations are imperfect substitutes in
the sense that ∂pih/∂qih < ∂pih/∂qjh ≤ ∂pih/∂qjk ≤ 0 and ∂pih/∂qih < ∂pih/∂qik ≤ ∂pih/∂qjk ≤ 0, so own-sales
effects dominate cross-sales effects. If ∂pih/∂qjh < (>) ∂pih/∂qik then consumers view the alternative
product from the same retailer as more (respectively, less) of a substitute than the same product from an
alternative retailer; we allow either possibility. The corresponding direct demand functions, qih(p) ≡
qih(pih,pik,pjh,pjk) ∀i≠j ∀h≠k, hold the same general properties of continuity, decreasing, and own-price
effects dominating cross-price effects. With this general demand structure we are able to provide some
results but in order to provide additional insights and make comparisons we will use a linear demand
specification that incorporates these features and allows for interbrand and intrabrand rivalry to be
parameterised in a convenient form.
In regard to market behaviour, the model assumes that the firms at each vertical level behave
non-cooperatively in setting prices, i.e. there are no manufacturer or dealer cartels. In addition it is
assumed that the interaction between the manufacturers and retailers when bargaining over transfer
prices still allows scope for effective competition in the market. Specifically, even though transfer
prices are negotiated, our assumption is that final prices are determined unilaterally either by the
manufacturers when RPM applies or by the retailers when RPM is not enforced. This independent
price-setting behaviour means that the outcome may depend on the degree of differentiation between
both the retailer services and the manufacturers’ products. Allowing the parties to bargain over both
transfer and final prices (with contingent contracts) in the present framework would amount in practice
to assuming a joint-profit-maximizing cartel situation in which the retail price is at the monopoly level
regardless of the degree of substitutability between different goods.13
essentially horizontal rather than vertical in nature. Thus, the manufacturers can be interpreted as well-known
brand producers competing in the same product category. The retailers can be viewed as nationally competing
rivals, or as competitors in a local market where consumer search costs effectively limit the market.
13
We acknowledge that other bargaining agendas are conceivable. For example, a referee has suggested that
parties might bargain over bilateral transfers. Conceivably, this might eliminate vertical externality effects
(associated with double marginalization) between trading parties but still allow horizontal competition at both
vertical levels. In a similar vein, Rey and Vergé’s (2004a) analysis avoids s double marginalization effects
through the use of franchise or royalty fees payable to manufacturers. In principle, these fees could be
6
Within each regime, we assume a two-stage procedure of setting prices, with parties informed
and aware of actions in the first stage prior to making decisions in the second stage.14 When the market
is characterized by RPM, the manufacturers each set and commit to retail prices and then negotiate
transfer prices (perhaps expressed in the form of discounts off the retail price) with the retailers on a
one-to-one basis.15 Thus RPM effectively removes intra-brand competition, but still allows for interbrand competition and efficient transfers between the trading parties. In contrast, when RPM is ruled
out, after negotiating transfer prices with the manufacturers, the retailers are free to compete on setting
final prices to consumers (treating each negotiated transfer price as a per unit cost). Here there is both
interbrand and intrabrand competition but, with successive (wholesale-then-retail) price determination,
an element of double-marginalisation is present.16 The outcomes from these two cases are derived in the
next two sections, highlighting the key differences in the regimes.
bargained over as a means to distribute profits, with transfer prices then set at the level that maximises bilateral
profits. Alternatively, following Kolay et al. (2004), bargaining could focus on non-linear transfer prices, with
differential rates over certain quantity thresholds (e.g. so-called “over-riders” as discounts to retailers for higher
levels of sales – see Competition Commission, 2000).
14
As will become evident, this is an important assumption because the first-stage decisions play a strategic role
in influencing second-stage outcomes. It might be viewed as a strong assumption given that one of the two
stages involves bilateral bargaining, where outcomes may be hidden from (or at least not revealed to) other
parties (e.g. O’Brien and Shaffer, 1992; Rey and Vergé, 2004b). We assume that in a close-knit industry with
few players and complete knowledge of each other’s demand, cost and bargaining power positions, there is
sufficient information leakage and insight that even with ostensibly secret bilateral trading the details of
concluded deals become known or can be worked out (i.e. rationally expected if not directly observable). For
example, the mechanisms in practice that might facilitate this could include industry gossip or staff changing
employers and taking information with them, or through deliberate leakage by negotiators demanding
concessions. In any event, we may observe that in practice, industry participants (at least in the consumer goods
sector) are quite well informed about rivals’ negotiated contracts or at least relative bargaining positions, even in
the absence of public announcements on concluded deals.
15
Thus retail prices under RPM entail some commitment to being fixed for some time. For example, it may
take the form of a price printed on a book or a label of a packaged grocery item. As Jullien and Rey (2001)
demonstrate, such commitments with high price transparency may be devices to facilitate collusion.
16
In principle, we could extend the analysis to consider simultaneous determination of transfer and retail prices.
However, underlying our set up is the notion that each regime is intrinsically characterised by a strategic
element, where a commitment of one form does have implications for competition. In the RPM regime, it is a
commitment by the manufacturers to a fixed retail price (around which manufacturers may base their advertising
or packaging). In the UT regime, it is a commitment to a negotiated transfer price, where manufacturer-retailer
negotiations are less frequent than retail price adjustments by competing retailers.
7
3. Resale Price Maintenance
Outcomes in this regime are determined in two stages. In the first stage, the manufacturers compete by
setting retail prices independently, i.e. Bertrand-Nash duopoly competition.
Transfer prices are
determined by negotiation in the second stage. As usual with complete information two-stage games of
this type, we solve for the sub-game perfect equilibrium in a recursive manner.
Bargaining in the RPM regime is assumed to occur on a one-to-one (i.e. bilateral) basis between
each manufacturer and each retailer over a (constant) unit transfer price. Negotiations are conducted
simultaneously so that during bargaining the firms’ (separate) negotiators treat the transfer prices from
other bargains as given and base their disagreement payoffs in each bargain on expected trade with the
remaining trading party (i.e. for Mi, its trade with Rk, and for Rh, its trade with Mj when wih is being
negotiated) under the (exogenous) risk of breakdown.17 The results from bargaining are modelled in
terms of the two-person generalised Nash bargaining solution with the outcomes represented by a set of
transfer prices which in essence represent a Nash equilibrium in Nash bargains (e.g. Davidson (1988),
Horn and Wolinsky (1988), Dobson (1994), Dobson and Waterson (1997) and O’Brien (2002) in the
context of monopoly-oligopoly bargaining, and the extension to bilateral oligopoly by Björnerstedt and
Stennek (2001; 2006)).18
When transfer prices are negotiated there are then four separate bargains. The outcome from
bargaining between Mi and Rh over the transfer price wih is characterised by
wih = arg max [ π Mi ( wih , w -ih ) − δ Mih ( w -ih ) ]1-α [ π Rh ( wih , w -ih ) − δ Rih ( w -ih ) ]α
wih
(1)
where w-ih ≡ (wik, wjh,wjk) denotes the set of the other three negotiated transfer prices. The parameter α ∈
[0,1], common to all four bargains, represents the retailer’s relative bargaining strength with respect to
the manufacturer, i.e. capturing retailers’ relative eagerness to settle, which may be accounted for by
differences in the bargaining parties’ discount rates or degrees of risk aversion (Binmore et al., 1986).
The term δMih is the disagreement payoff to Mi when it has to rely on trading with the other retailer in the
17
Here we assume simultaneous bargaining whereby each manufacturer has a separate agent to bargain with
each retailer, and vice versa. An alternative procedure, suggested by a referee, would be for sequential
negotiation of contracts. In situations where negotiated terms become publicly known, such as union-employer
pay agreements, then early settlements can be expected to influence later settlements (with disagreement payoffs
known with certainty) as shown by Horn and Wolinsky (1988) and Dobson (1994), inter alia. However, we
maintain that our assumption of simultaneous bargaining appears reasonable for at least three reasons. First, as
a practical observation, manufacturers negotiate with retailers on a broadly simultaneous periodic basis with no
set sequence (e.g. the annual negotiations between major brand producers and leading grocery retailers;
Competition Commission (2000)). Secondly, with symmetrically positioned firms, it is not clear in what order
the sequential negotiations would be or how a particular sequence would be co-ordinated (in contrast to
established union-employer pattern bargaining). Thirdly, in the absence of public announcements of negotiated
terms, all bargaining parties would not in practice be immediately aware of the terms concluded in other
bargains with different parties but instead would take a view on expected outcomes, an aspect captured in our
analysis of simultaneous bargains but not in sequential bargaining with determinative outcomes.
18
On the underlying non-cooperative bargaining foundations to this Nash-equilibrium-in-Nash-bargains
approach to multi-party interdependent bilateral bargaining, see Davidson (1988) and Jun (1989) on monopoly vs.
duopoly situations and Björnerstedt and Stennek’s (2001; 2006) extension to the case of bilateral oligopoly with
outcomes determined as a sequential equilibrium.
8
event of negotiations breaking down. Similarly, δRih represents the disagreement payoff to Rh when it
relies on selling the other manufacturer’s product.
The first order condition (FOC) for (1) reduces to
(1 − α)( π Rh − δ Rih )
∂π Mi
∂π
+ α( π Mi − δ Mih ) Rh = 0
∂wih
∂wih
(2)
Disagreement, with a breakdown in negotiations, is assumed to result in the two parties
involved not trading and so qih = 0, which can alter market outcomes for the other three goods (as the
absence of a good may encourage consumers to buy more of the available goods). Let us denote φik,
φjh and φjk as the three final (retail) prices and zik, zjh and zjk as the three corresponding quantities when
Mi and Rh are unable to conclude an agreement, so that the respective manufacturer and retailer
disagreement payoffs are δMih = wikzik and δRih = (φjh – wjh)zjh. In this RPM regime, though, final retail
prices are fixed prior to the bargaining round, and thus φik = pik, φjh = pjh, and φjk = pjk.19 Then, for
given anticipated outcomes in the other bargains (i.e. wik = wi*k, wjh = wj*h and wjk = wi*k ) and with these
goods as imperfect substitutes, we expect zik > qik, zjh > qjh and zjk > qjk, yet (zik + zjh + zjk) < (qih + qik +
qjh + qjk) (i.e. sales of the alternative goods rise but the absence of a good without any perfect
substitutes reduces the total quantity of goods sold).
Next, denote the respective disagreement quantities for the two bargaining parties as ziikh
representing Mi’s disagreement quantity when it only trades with Rk, and zjihh as Rh’s disagreement
quantity when it only trades with Mj. Then, we can define the terms λih ≡ (ziikh – qik)/qih and υih ≡ (zjihh –
qjh)/qih. In the present context, we can consider these two terms as separately indicating the intensity of
intrabrand rivalry between the retailers and interbrand rivalry between the manufacturers (with
respectively λ – “lambda” – referring to rivalry in the lower vertical level, and υ – “upsilon” – referring
to rivalry in the upper vertical level). Here, λih shows the extent to which consumers faced with the
absence of a particular product from one retailer would buy additional amounts of the same product
from the rival retailer. The higher the value of this term then, ceteris paribus, the closer consumers
perceive the retailers as substitutes. In contrast, υih shows the extent to which, for given prices,
consumers faced with the absence of one manufacturer’s good in a store would buy additional amounts
of the alternative manufacturer’s product from the same store. Similarly, the higher the value of this
term then, ceteris paribus, the closer consumers perceive the manufacturers’ products as substitutes.
However, in order for there to be an incentive for the manufacturers to deal with both retailers, and vice
versa, we assume for given (symmetric) retail prices that λih < 1 and υih < 1 (i.e. that the two retailers
and the two products are, respectively, at most less than perfect substitutes).
19
Here it is assumed that retail prices are fixed irrespective of subsequent agreements on transfer prices, i.e.
contingency arrangements in the first stage are ruled out and instead there is a clear upfront commitment to the
retail prices before negotiating intermediate prices. For example, in the case of books or magazines, a commitment
to a retail price could arise through the publisher printing the price on the cover. Similarly, RPM on groceries,
when it existed, often took the form of products with the price clearly marked on the packaging.
9
Returning to the first order condition (2), and noting that ∂πMi/∂wih = qih and ∂πRh/∂wih = – qih
(given that retail prices are already fixed), we can re-express this condition as
(1 − α)[ pih − wih − ( p jh − w jh )υih ] + α[ wih − wik λ ih ] = 0
(3)
The optimality conditions then reveal the negotiated transfer price as a function of the other negotiated
transfer prices involving the two parties and the final market variables in the following form:
wih ( w jh , wik ,.) = (1 − α)[ pih − ( p jh − w jh )υih ] + αwik λ ih
(4)
Thus, the negotiated transfer price can be expressed as a positive linear function of the transfer prices
from the other bargains made by the two parties. Specifically, this optimising functional relationship
(akin to a best-response function in oligopoly) yields ∂wih(wjh,wik)/∂wjh > 0, ∂2wih(wjh,wik)/(∂wjh)2 = 0,
∂wih(wjh,wik)/∂wik > 0 and ∂2wih(wjh,wik)/(∂wik)2 = 0.20
Taking into account all four first order conditions, we can establish the following proposition:
Proposition 1. In the RPM regime, where manufacturers fix retail prices prior to transfer price
bargaining, for given retail prices there exists a unique equilibrium set of transfer prices (w1*1(p), w1*2(p),
w2*1(p), w2*2(p)) when all direct alternative goods are imperfect substitutes such that λih < 1 and υih < 1,
∀i, ∀h.
Proof. Proofs for all propositions and corollaries are contained in the Appendix.
The optimising functions represented by (4) are thus continuous, linear, and upward sloping in
transfer price space, with uniqueness guaranteed if the slope with gradient (1 – α)υih + αλih is less than
unity. Given uniqueness, monotonicity and continuity, the following Corollary is immediate from (2):
Corollary 1. (a) All negotiated transfer prices are zero when α = 1. (b) When α = 0, the equilibrium
negotiated transfer prices are equal to the corresponding retail prices set by the manufacturers, i.e.
wi*h = pih ∀i ∀h. (c) When α < 1 and pih > pjhυih ≥ 0 all equilibrium transfer prices are positive. (d)
Transfer prices are decreasing in retailer bargaining strength, α.
Relative bargaining strength affects the weighting afforded to one party’s net profits over the
other party’s net profits. Accordingly, as retailers’ relative bargaining strength increases, more weight
in the Nash maximand is given to maximising their net profits, resulting in lower negotiated transfer
prices.
20
In a similar vein to Davidson’s (1988) analysis, this function represents the outcome of a 4-player noncooperative game when a set of particular subgames is reached. Here, in each subgame the strategies of the players
involved in the other bargains are fixed since it is assumed that they have settled on a transfer price.
10
The linearity of the optimality functions allows us to solve directly for the equilibrium transfer prices
(conditioned on final prices):
*
wih
(p,.) = (1 − α)[ pih A + αp jh B + αpik C + αp jk D]/E
∀i ≠ j, ∀h ≠ k
where
A ≡ 1 + (1 − α)3 υihυ jhυik υ jk − (1 − α)2 υik υ jk − (1 − α)υihυ jh − α2λ jhλ jk − α2 (1 − α)υ jhυik λihλ jk
B ≡ −(1 − αλ jhλ jk )υih + (1 − α)2 λihλik λ jk + α(1 − α)υik λihλ jk
(5)
C ≡ (1 − α2λ jhλ jk )λih − (1 − α)υ jk (υik λih + αυihλ jh )
D ≡ (1 − α)υihλ jh − (1 − α)2 υihυik υ jk λ jh − α(1 − αλ jhλ jk )υik λih
E ≡ (1 − α2λihλik )(1 − α2λ jhλ jk ) + (1 − α)4 υihυ jhυik υ jk − (1 − α)2 (υihυ jh + υik υ jk )
− α2 (1 − α)2 (υihυ jk λ jhλik + υ jhυik λihλ jk )
If, α ∈ (0,1), λih ∈ (0,1) and υih ∈ (0,1), ∀i ∀h, then with symmetric demand conditions we would
expect the identities in (5) would be signed such that A > 0, B < 0, C > 0 and E > 0, with the sign of D
dependent on the value of α and the other terms (tending to be positive for high α and negative for low
α). For example, this is evident under full symmetry of the intrabrand and interbrand variables.21 Since
these two variables may in general be expected to depend on the retail prices selected, we cannot be
certain of the effect of the individual retail prices on the equilibrium transfer prices. However, if direct
effects of the retail prices dominate the indirect ones arising through the intrabrand and interbrand
variables then, from (5), the net effect of the retail prices chosen by the manufacturers would (at least
under symmetry) be such that each equilibrium transfer price would be increasing in its own
corresponding retail price (i.e. ∂wi*h(p,.)/∂pih > 0), decreasing in the retail price set by the rival
manufacturer to the same retailer (i.e. ∂wi*h(p,.)/∂pjh < 0), and increasing in the retail price set by the
same manufacturer to rival retailer (i.e. ∂wi*h(p,.)/∂pik > 0).
With second-stage equilibrium transfer prices thus characterised, consider next each
manufacturer’s problem in the first stage of setting retail prices for their products to maximise their own
profits while anticipating outcomes from subsequent transfer pricing with the retailers. Mi’s problem is:
max π Mi = wih ( p) qih ( p) + wik ( p) qik ( p)
(6)
p ih , p ik
This yields two symmetric first-order conditions, taking the following form:
∂π Mi
∂q ( p)
∂q ( p)
∂w ( p )
∂w ( p)
+ wik ( p ) ik
= wih ( p) ih
+ qih ( p) ih
+ qik ( p) ik
=0
∂pih
∂pih
∂pih
∂pih
∂pih
21
∀h
(7)
Under full symmetry, such that λih = λ < 1 and υih = υ < 1, ∀i ∀h, then A = 1+(1–α)3υ4–α2λ2–υ2(1–α)(2–
α+α2λ2) > 0 (observing that A decreases in value as λ increases such that even if λ = 1 then A = (1–α)(1–υ2)(1+α–
υ2(1–α)2) > 0), B = – υ(1–υ2+α(2–α)(υ2–λ2)) < 0, C = λ(1–υ2+α2(υ2–λ2)) > 0, D = λυ((1–υ2)(1–2α)–α2(υ2–λ2))
11
Sufficient conditions for a unique equilibrium set of symmetric retail prices to exist are the
manufacturers’ profit functions being concave in respect of own prices (i.e. ∂2πMi(p)/(∂pih)2 < 0), retail
prices of immediate substitutes22 being strategic complements (where cross-price demand effects
dominate cross-price transfer price effects) and own-price profit effects exceeding the sum of crosspartial profit effects. We assume these conditions apply and we accordingly denote this equilibrium set
of retail prices as (p1R1PM, p2R1PM, p1R2PM, p2R2PM) = pRPM.
With symmetric equilibrium retail prices giving rise to symmetric equilibrium quantities,
implying λih = λ < 1 and υih = υ < 1, ∀i ∀h, we find that (5) reduces to:
wRPM ( p RPM ) =
(1 − α)(1 − υ) p RPM
1 − (1 − α)υ − αλ
(8)
Note the denominator reflects the uniqueness condition, which is satisfied when the goods are not
perfect substitutes.
At this point, and in order to provide further insights, it will be useful to consider cases where λ
and υ are independent of the final prices selected, thus allowing us to separate out effects arising from
the choice of retail prices and those arising from the inherent intensity of intrabrand and interbrand
rivalry. For instance, following Ziss (1995) and Dobson and Waterson (1996a,b) consider the following
(inverse) linear demand structure that allows for independent parameters to capture different intensities
of intrabrand and interbrand rivalry:23
pih = 1 − ( qih + β qik ) − γ ( q jh + β q jk )
0 ≤ β, γ < 1
(9)
Here, the parameter β represents how similar the retailers’ services are perceived by consumers to be
when selling the same product (i.e. intrabrand rivalry), such that when β = 0 the retailers’ services are
viewed as independent, but as β → 1 their services become closer substitutes. In contrast, γ represents
how similar the products are perceived to be when sold in the same store (i.e. in-store interbrand
rivalry), such that the products are demand independent when γ = 0 and become closer substitutes as γ
→ 1. In addition, the perceived similarity of the products when sold in different stores, i.e. betweenstore interbrand rivalry, is captured by the interactive term βγ.24 Moreover, it can be readily checked
that this linear demand system offers the convenient relation that, under given symmetric retail prices, λ
= β = – [(∂qik/∂pih)⏐p ]/[(∂qih/∂pih)⏐p ] and υ = γ = [(∂qjh/∂pih)⏐p]/[(∂qih/∂pih)⏐p ], thus offering respective
which is value dependent, and E = [1–(1–α)υ–αλ][1+(1–α)υ–αλ][1+(1–α)υ+αλ][1–(1–α)υ+αλ] > 0.
22
By this term we mean that for good ih its immediate substitutes are taken as the same product from a different
retailer (i.e. good ik) or a different product but from the same retailer (i.e. good jh).
23
Note that this is not the only demand structure to exhibit this feature of independent values of λ and υ. For
instance, this can apply to linear demand specifications with further parameters (e.g. for the demand intercept
and own-quantity/price effect). It can also apply to non-linear variants of (9), e.g. pih(q) = 1 – (qih1/α + βqik1/α) –
γ(qjh1/α + βqjk1/α), yielding concave demand functions for α > 1 and convex demand for α < 1. With this demand
specification, we find λ = (1+β)1/α – 1 and υ = (1+γ)1/α – 1 (both of which are necessarily non-negative and less
than unity if α > 1).
24
Thus our assumption is that as an influence on the price of product 1 from retailer 1, the ratio of product 2’s
effect is the same between retailers.
12
independent parameters for intrabrand and (in-store) interbrand rivalry that both range from zero
towards unity.
With λ and υ expressed as independent parameters, we can now draw on (8) to establish the
following proposition regarding the equilibrium symmetric negotiated transfer prices:
Proposition 2. In the RPM regime and for given symmetric retail prices, each symmetric equilibrium
transfer price is (i) increasing in terms of the equilibrium retail prices, (ii) decreasing in intrabrand
rivalry, and (iv) increasing in interbrand rivalry.
Retail prices have a positive effect on negotiated transfer prices since the higher the former,
for given shares of bilateral surplus created, the higher the latter. Higher intrabrand rivalry raises each
manufacturer’s disagreement payoff relative to its anticipated profit level, as consumers more readily
switch to the alternative retailer to buy its product (rather than the alternative in-store product), so
allowing for each manufacturer to negotiate a higher transfer price. In contrast, higher interbrand
rivalry raises each retailer’s disagreement payoff relative to its anticipated profit level, because
consumers will more readily switch to the alternative in-store product (rather than switch to a rival
store to buy the unavailable product), so allowing each retailer to negotiate a lower transfer price.
Within this parameterised framework, we can also establish the following three key results that
characterise retail prices:
Proposition 3. The equilibrium retail prices fixed by the manufacturers in the RPM regime are
increasing in the degree of retailer bargaining strength.
Proposition 4. (a) The maximum symmetric equilibrium retail prices under RPM are equal to the levels
that would be set by a single fully integrated monopoly supplier (i.e. corresponding to the joint industry
profit maximising price levels). (b) These maximum levels are attained if (i) interbrand rivalry is absent
with manufacturers’ products demand independent, and/or (ii) retailers have complete bargaining
power to unilaterally dictate transfer prices (i.e. α = 1).
Proposition 5. Where the retailers do not have complete bargaining power (i.e. α ∈ [0,1)), the
equilibrium retail prices under RPM (i) are declining in both intrabrand and interbrand rivalry with
respect to the fully integrated monopoly prices, and (ii) in the limit approach zero as both retail
services and manufacturer products respectively approach being perfect substitutes.
In effect, when the manufacturers set the retail prices, anticipating transfer prices to follow, they
take into account two key opposing effects. First, setting higher retail prices means they can potentially
13
create more bilateral surplus, which in their subsequent negotiations (and sharing the bilateral surplus
generated in each of their bargains) allows for an increase in the negotiated transfer price. This is a
positive surplus sharing effect working to induce manufacturers to raise RPM prices, which becomes
stronger the greater is the relative bargaining strength held by the retailers, as manufacturers seek to
counter retailers’ bargaining power (pushing transfer prices down) by raising retail prices to stretch the
negotiation range (explaining the finding in Proposition 3). Second, though, in setting retail prices the
manufacturers are indirectly competing with each other; a unilateral increase in the retail prices set by
one manufacturer would reduce demand for its products while raising demand for its rival, thereby
potentially hurting its own profits – the more so, the greater degree of substitutability between the
manufacturers’ products. Accordingly, there is a competition effect working in the opposite direction;
serving to dampen the manufacturers’ desire for setting high retail prices.
The combination of the two effects means that the retail prices will never be set higher than the
fully integrated monopoly levels (i.e. Proposition 4a). The surplus sharing effect is positive only so long
as retail prices are set no higher than the joint-surplus-maximising levels (where all vertical and
horizontal externality effects are internalised). The competition effect provides a negative cross-price
demand effect, pushing down retail prices below this maximum level.
The relative strength of these opposing effects will depend critically on the degree of relative
bargaining strength and the intensity of interbrand rivalry. To see this, begin by considering the case
where the manufacturers’ products are demand independent and so interbrand rivalry is wholly absent.
In this case, manufacturers’ sole concern will be negotiating the highest amount of surplus possible,
which firstly requires maximising bilateral surplus so that retail prices are set at the level equivalent to
those that would be set by a fully integrated monopolist (i.e. as claimed in Proposition 4b(i)). However,
with substitutable products, each manufacturer would need to take into account the negative demand
effect of its rival setting lower retail prices. This induces the manufacturers to compete through their
retail prices. This competition is at its most direct when the manufacturers hold all the bargaining
power, since at this point negotiated transfer prices will be at the same level as retail prices, so
competing through setting retail prices is equivalent to competing through setting transfer prices. Hence
there is a direct relation between the intensity of interbrand rivalry and the level of retail prices chosen –
i.e. at the fully integrated monopoly levels when such rivalry is absent and approaching zero as the
manufacturers’ products approach being perfect substitutes (Proposition 5).
With increasing retailer bargaining strength, manufacturers’ competition becomes less direct
when setting retail prices, since the negotiated transfer price is more removed from the retail price. In
these circumstances, the manufacturers will be less concerned with the negative competition effect and
more concerned with the surplus sharing effect. In the limit, as the retailers hold all the bargaining
power, the manufacturers’ emphasis shifts entirely towards the surplus sharing effect since they perceive
anticipated transfer prices being negotiated downwards (by retailers’ bargaining strength) and so the
only way to increase profits is by increasing retail prices to stretch the negotiable range of transfer prices
14
and in the process generate more bilateral surplus. Thus, when retailers holding all the bargaining power
they can rely on the manufacturers to set retail prices at the industry-wide surplus-maximising level (as
the manufacturers seek to enhance their prospects for negotiating higher transfer price levels), but the
retailers can then use their complete bargaining power to extract the entire surplus (Proposition 4b(ii)).25
Finally, increasing intrabrand rivalry weakens retailers’ bargaining position in negotiations over
transfer prices, by raising the manufacturers’ disagreement payoffs, so bringing transfer prices closer
into line with retail prices. In consequence the manufacturers’ emphasis shifts towards competitive
effects (rather than surplus sharing) in setting retail prices, and so works together with interbrand rivalry
in inducing the manufacturers to set lower retail prices – which in the limit approach zero as all productservice combinations approach being perfect substitutes (Proposition 5(ii)).
4. Unrestricted Trading and Comparisons
In the absence of agreements specifying RPM, we assume that any suggested or recommended retail
prices set by manufacturers have no significance (i.e. they do not have a market coordinating role) and
that, after negotiating transfer prices, the retailers are free to determine their prices for the
manufacturers’ products so as to maximise their own individual profits. In other words, the market is
characterised by Bertrand-Nash competition at the retail level after the transfer prices have been
negotiated. But of course the transfer prices are determined in the first stage bearing in mind the
anticipated final price outcome.26
In the second stage, having negotiated the transfer prices, retailers independently set final prices.
With all four agreements successfully concluded, the problem for Rh is:
max π Rh = ( pih − wih ) qih ( p) + ( p jh − w jh ) q jh ( p)
p ih , p jh
i ≠ j, h ≠ k
(10)
Following our assumptions on the nature of demand, we take the retailers’ profit functions as being
strictly quasiconcave in respect of own prices (with ∂2πRh(p)/(∂pih)2 < 0 for positive quantities), so
ensuring existence of price equilibria, and that own effects dominate cross effects (with
∂2πRh(p)/(∂pih)2+∑-ih |∂2πRh(p)/(∂pih∂p-ih )| < 0, where subscript –ih refers to the available product-service
combinations other than good ih), so ensuring uniqueness. On this basis, we assume that the retailers’
25
A possible interpretation is that when α = 1 then a market-wide commitment allowing manufacturers to set retail
prices represents a convenient form of delegation by the retailers as it removes competitive pressures and protects
against entry by an aggressive price cutter, while providing a result equivalent to joint profit maximization. This
formalises a traditional explanation for RPM in terms of organised dealer pressure where RPM is used to facilitate
collusion - e.g. see Yamey (1954) and Pickering (1966).
26
For example, this sequence of events appears to be borne out in the book market. Here, publishers (in the
absence of RPM) set recommended retail prices (RRPs), and then booksellers negotiate discounts off the RRP
level, and set retail prices to consumers usually expressed as a discount to the RRP. For details, see
Competition Commission (2006), where it is reported that the average selling prices for all UK retailers were
around 60% of RRP for the nine best-selling books in 2005, rising to about 90% of RRP for “deep-range” titles,
with average discounts off RRP of 21% in 2005.
15
profit maximisation problems yield a unique retail price equilibrium for any set of negotiated transfer
prices w = (w11,w12,w21,w22), which we denote by p**(w) = (p1*1*(w), p1*2*(w), p2*1*(w), p2*2*(w)) with
corresponding equilibrium quantities q**(w) = (q1*1*(w), q1*2*(w), q2*1*(w), q2*2*(w)).
In this particular scenario, competition effects can arise both directly and indirectly, potentially
reinforcing or opposing each other. In particular, the intensity of retail rivalry and the nature of
substitutability across the available product-service combinations may directly influence final market
outcomes but also may be expected to influence the prior negotiated transfer prices (given that
bargaining parties will anticipate the retailers’ subsequent pricing behaviour). Equally, the negotiated
transfer prices will impact the subsequent retail prices selected (since they then represent the retailers’
marginal costs), and consequently affect quantities and profits. Thus, there are a number of competition
and strategic effects to take into account when determining outcomes.
In respect of these effects, we assume that the negotiated transfer prices have the usual effects
on second-stage Bertrand-Nash equilibrium outcomes. Specifically, we assume that the retailers will
always want lower transfer prices, regardless of the intensity of intrabrand and interbrand competition
(as this raises retail margins and boosts sales), while manufacturers will only want higher transfer prices
so long as the resulting loss of sales does not become so great as to offset the direct benefit from a higher
unit price (i.e. up to the point where the positive direct effect from an increase in price received is
matched by the negative strategic effect of reduced demand arising from induced higher retail prices –
with this latter effect likely becoming stronger the greater the intensity of interbrand rivalry).
As well as the various effects relating to the anticipated retail equilibrium outcomes when all
four transfer prices are agreed, to analyse the bargaining we also need to consider outcomes when there
is a disagreement (with a breakdown in negotiations) over wih. In the situation where Rh fails to
conclude an agreement with Mi while all three other agreements are made, then qih = 0. The retailers’
profit maximising problems characterised by (10) are accordingly amended to reflect the availability
of only three product-service combinations. Again, for any set of negotiated transfer prices, here w-ih
= (wik,wjh,wjk), we assume there is a unique retail price equilibrium denoted by φ-*i*h(w-ih) = (φ*i*k(w-ih),
φ*j*h(w-ih),φ*j*k(w-ih)) and corresponding equilibrium quantities are z-*i*h(w-ih) = (z*i*k(w-ih),z*j*h(w-ih),z*j*k(w-ih)).
In this case, the respective manufacturer and retailer disagreement payoffs when they bargain over the
transfer price wih are δMih(w-ih) = wikz*i*k(w-ih) and δRih(w-ih) = (φ*j*h(w-ih) – wjh)z*j*h(w-ih). For Mi and Rh to
reach a worthwhile agreement, we assume that both parties weakly benefit, i.e. for a range of
negotiable transfer prices, πM*i*(w) – δM*h*i (w-ih) ≥ 0 and πR*h*(w) – δR*h*i (w-ih) ≥ 0.
Finally, for symmetric negotiated transfer price levels, where w = w, we assume that all retail
prices and manufacturer profits are increasing and all quantities and retailer profits decreasing in a
common transfer price.
Having characterised the second-stage equilibrium outcomes, we now move on to consider
the first stage where bargaining over transfer prices takes place. Following the analysis from the
previous section relating to the RPM regime, the bargaining problem between Mi and Rh when all
16
other agreements have been struck is again expressed by equation (1). For this unrestricted trading
setting, though, the first-order condition satisfying the Nash maximand takes account of the strategic
effects of transfer prices agreements on subsequent outcomes from retail competition in regard to
retail price choices. Specifically, we have the first-order condition as follows:
UT
(1 − α)( π Rh ( wih , wUT
−ih ) − δ Rih ( w−ih ))
∂π Mi ( wih , wUT
−ih )
∂wih
UT
+ α( π Mi ( wih , wUT
−ih ) − δ Mih ( w−ih ))
∂π Rh ( wih , wUT
−ih )
=0
∂wih
(11)
where w-UiTh denotes the symmetric level agreed in the other three bargains for this “unrestricted
trading” (“UT”) regime.
Taking the Nash maximand, which we denote ΓiUhT(w), as being quasiconcave in the negotiated
price with second-order cross-effects weakly positive but less than the second-order own-effects (so
∂2ΓiUhT (wih,w-UiTh)/(∂wih)2 < 0 ≤ ∂2ΓiUhT (wih,w-UiTh)/(∂wih∂w-UiTh) < ⏐∂2ΓiUhT (wih,w-UiTh)/(∂wih)2⏐) then we can
obtain the following result:
Proposition 6. In the Unrestricted Trading regime, where retailers unilaterally set retail prices after
transfer price bargaining, there exists a unique equilibrium set of symmetric negotiated transfer prices,
which we denote by wU1T1 = wU1T2 = wU2T1 = wU2T2 = wUT.
For each of the four interdependent bargains, the first-order condition (11) implicitly defines
an optimising functional relationship between the transfer price level being negotiated and the
symmetric transfer price arising from the other three bargains, where wih = f(w-UiTh). Given our
assumptions on the nature of the Nash maximand, each bargaining best-response function is (weakly)
upward sloping in (wih,w-ih) space with slope less than unity – where, using the implicit function
theorem, ∂wih/∂w-UiTh = – [∂2ΓiUhT (wih,w-UiTh)/(∂wih∂w-UiTh)]/[∂2ΓiUhT (wih,w-UiTh)/(∂wih)2] ∈ [0,1). Noting the
symmetry between the bargains, the equilibrium symmetric agreed transfer price is determined where
wUT = f(wUT).
Given uniqueness, monotonicity and continuity, the following corollary is immediate from
(11):
Corollary 2. In the UT case: (a) All negotiated transfer prices are zero when α = 1. (b) Transfer prices
are non-negative when α = 0. (c) Agreed transfer prices are decreasing in retailer bargaining strength.
These results are in line with the RPM case. We can observe that a reduction in α reduces the
magnitude of the first term while increasing it for the second term in (11). With the first term
positively signed and the second term negatively signed then, in order to maintain equalization of the
17
two terms, the retailer’s net profits need to be raised and the manufacturer’s net profits need to be
lowered, which is achieved through a lower negotiated transfer price level.
Corollary 2 in turn has the following consequences for the equilibrium outcomes arising from
the subsequent retail price competition between the retailers:
Corollary 3. In the UT case, increasing retailer bargaining strength, ceteris paribus, has the effect of
(i) reducing equilibrium retail prices, (ii) increasing equilibrium quantities, (iii) increasing retailer
profits, and (iv) reducing manufacturer profits.
These results follow directly from our assumptions regarding the impact of a change in the
symmetric transfer level. They provide a striking contrast with those from the RPM regime, and it
will be useful at this stage to turn to some comparative statements as between RPM and UT regimes.
From Proposition 3 we observed equilibrium retail prices under RPM increasing (and by association
equilibrium quantities decreasing) in retailer bargaining strength. Under UT, the sub-game perfect
equilibrium outcomes allow for transfer price levels to influence final prices (rather than vice versa as
in the RPM regime), with the consequence that lower transfer prices act as lower marginal costs for
the retailers, encouraging them to set lower retail prices.
In contrast, in the RPM regime the
bargaining strength parameter primarily determines the distribution of profits (for given retail prices),
though its secondary role is to serve as a means to coordinate market prices, which in the limit with α
= 1 results in retail prices equivalent to the levels that would be set by single-stage monopoly supplier
(i.e. Proposition 4(a)). This permits the following comparison regarding equilibrium retail prices
between the two regimes:
Corollary 4. If retailers hold all the bargaining power (i.e. for α = 1) then (i) in the presence of
intrabrand rivalry (i.e. with competing retail services), pUT < pRPM, and (ii) in the limit when the retail
services are perfectly substitutable, pUT = 0.
We have thus established that retail prices can be higher in the RPM regime than the UT
regime. Yet, this result is not universal. We can readily observe market conditions where the reverse
applies:
Corollary 5. If manufacturers hold all the bargaining power (i.e. for α = 0) and all four productservice combinations are demand independent then pUT > pRPM.
Corollary 4 points to the potential benefits of the UT regime when it allows for sociallybeneficial countervailing power, whereby high relative retailer bargaining strength means that supply
prices are kept down with the benefits being (partly) passed on in the form of lower retail prices. In
contrast, Corollary 5 highlights the problem of successive power, arising from the inefficiency of the
bargaining arrangement in the UT giving rise to a double marginalization problem when horizontal
18
competition is weak (or, as in this case, absent) at both vertical levels and when high transfer prices
translate into high retailer marginal costs which in turn result in retailers setting high retail prices.
We can further observe that the intensity of rivalry at each level is likely to enhance
countervailing power effects under UT – especially intense intrabrand rivalry as this can be expected to
keep retail margins low and thereby ensure that the benefits of retailer bargaining power are passed on to
final consumers. At the same time, intense interbrand and intrabrand rivalry can both be expected to
reduce successive power effects – reducing the profit margins at each successive vertical level. Taken
together, increased rivalry of both forms may be expected to lead to lower retail prices – just as it would
under the RPM regime, if for slightly different reasons. In the RPM regime, interbrand rivalry has the
double effect of encouraging manufacturers to compete directly through the retail prices they set, while
at the same time weakening their bargaining position when it comes to sharing the resulting pie with
retailers through transfer price bargaining. In the UT regime, the effect of interbrand rivalry can mainly
be expected to arise through its effect on transfer price bargaining – weakening manufacturers’
bargaining positions (by enhancing retailers’ disagreement payoffs), resulting in lower agreed transfer
prices and hence lower retail prices.
With intrabrand rivalry, a prime effect in both regimes is through retail prices being dampened.
In the RPM regime, this is because when setting the retail prices the manufacturers take into account the
closer substitutability of retail services, intensifying their own rivalry. In the UT regime, it is more
directly because more intense retail price setting competition amongst the retailers reduces retail
margins. Yet, we may expect intrabrand rivalry to work in opposite directions on agreed transfer prices.
In the RPM case, the greater is the substitutability of retailer services, the greater is the disagreement
payoff to the manufacturers, and thus the weaker is the retailer’s bargaining position with the
consequence that higher transfer prices are agreed (as verified by Corollary 1(e)). However, in the UT
regime, we might expect that as the retail services become very close substitutes, the retailers’ overall
bargaining position becomes very strong since not only is each manufacturer’s disagreement position
based on the limited sales to a monopoly retailer of that particular product, but the intensity of product
market competition means that each retailer has an extremely strong individual incentive to resist giving
bargaining concessions in order not to carry forward a cost disadvantage which would reduce its sales
and profitability to the benefit of its rival. If so, the important consequence is that the bargaining effect
of reducing transfer prices would reinforce the direct effect of reduced retail margins, suggesting that
intense intrabrand rivalry might lead to very low retail prices regardless of the retailers’ bargaining
strength (at least for α > 0) and the intensity of interbrand rivalry.
In order to consider such competition effects in the UT regime more formally and make
further comparisons with the outcomes from the RPM regime, it is worthwhile again separating out
the respective effects by treating interbrand and intrabrand rivalry parametrically. Here, we can make
explicit use of the linear demand system given in equation (9) above, enabling us to derive
straightforward closed-form solutions. Applying this demand system, we can establish the following
19
result regarding the impact of intrabrand and interbrand rivalry on negotiated transfer prices:
Proposition 7. In the UT case with linear demand, negotiated transfer prices are decreasing in both
intrabrand and interbrand rivalry and in the limit approach zero as products and/or retailer services
approach perfect substitutability.
While the demand system considered is clearly a special case, the underlying intuition for the
results suggests that other differentiated-goods demand structures might elicit similar outcomes. To
see why, it is useful to follow O’Brien (2002) and re-write the first-order condition (10) as follows:
α[ −∂π Rh ( wih , wUT
−ih ) / ∂wih ]
UT
π Rh ( wih , wUT
−ih ) − δ Rih ( w−ih )
=
(1 − α)[∂π Mi ( wih , wUT
−ih ) / ∂wih ]
UT
π Mi ( wih , wUT
−ih ) − δ Mih ( w−ih )
(12)
or expressed in words as:
Retailer h's weighted concession cost Manufactur er i's weighted concession cost
=
Retailer h's net profits
Manufactur er i's net profits
(13)
Thus, in a bargaining equilibrium, the transfer price negotiated by Mi and Rh equalises their weighted
concession costs as a percentage of their gains from trade, where the weights are the parties’ relative
bargaining strength weights. As O’Brien explains, the intuitive interpretation of this condition is that the
party with the lower percentage concession cost loses less when improving its offer and thus should do
so to facilitate reaching agreement.
With increasing intrabrand rivalry, the most pronounced effect is likely to be on retail profit
margins, which directly reduce the retailer’s net profit, i.e. the denominator on the left hand side (LHS)
of (13). The manufacturer’s net profits and also its weighted concession cost are likely to be relatively
unaffected (as explained above regarding opposing direct and strategic effects on product demand) by
the degree of substitutability between the retailer services, so the right hand side (RHS) of (13) is likely
to be relatively little changed. The only other effect to consider is the retailer’s weighted concession
cost which is likely to be decreasing in intrabrand rivalry, but not so significantly as to outweigh the
reduction of the retailer’s net profit. Accordingly, the LHS side of (13) can be expected to be increasing
in intrabrand rivalry. Then, the only way for the two sides to remain equalised is for the manufacturer’s
net profits to fall (while the retailer’s rise), which requires a reduction in the negotiated transfer price.
In the limit, with the retailer’s net profits collapsing to zero while all other values are non-zero then
equalization of the two sides is only satisfied if the manufacturer’s net profit falls to zero.
In respect of intense interbrand rivalry, we can consider two effects working in tandem to drive
lower negotiated transfer prices. First, the retailer’s net profits are likely to be declining in interbrand
competition, and in the limit approach zero, as increased product substitutability reduces the gap
between the retailer’s disagreement profits and its anticipated profits from making an agreement with
the manufacturer. Second, the manufacturer’s concession cost is likely to approach zero as a marginal
20
increase in the negotiated transfer price can translate into a very significant loss of sales for the
manufacturer (in favour of its rival). When the other two terms are bounded, both the denominator on
the LHS and the numerator on the RHS tend to zero as the products approach being perfect substitutes.
In this case, and in order for both sides to remain equalised, the negotiated transfer price must fall – in
the limit to zero as these two terms approach zero.
Thus, while using the linear demand system allows us to demonstrate formally that both intense
intrabrand and interbrand rivalry drive down negotiated transfer prices in the UT case, it is conceivable
that both limit results might apply more generally. All that is required is for retailer net profits to decline
to zero with intense intrabrand or interbrand rivalry while the other terms remain non-zero. If the
manufacturer’s weighted concession cost also declines to zero (as may occur under intense interbrand
rivalry) then this only adds to the effect of the intensity of rivalry pushing down negotiated transfer
levels to zero in the limit.
Intuitively, we might have expected strong interbrand rivalry to weaken manufacturers’
bargaining positions (just as it did in the RPM case). However, the finding that the same applies with
strong intrabrand rivalry may be somewhat more surprising (and, of course, is the opposite of what we
found in the RPM case; as stated in Proposition 2(iv)). Here, rather than a weakness that manufacturers
can exploit, it turns out in the unrestricted trading case that strong intrabrand rivalry can undermine the
manufacturers’ bargaining position. As O’Brien again observes, a firm’s bargaining power comes partly
from its ability to inflict a loss on the firm it is negotiating with by delaying an agreement (and here
risking a breakdown in negotiations). When the retail market is highly competitive, the retailers earn
little profit (and thus have little to lose), so the manufacturers can inflict only a small loss on each
retailer by delaying agreement. The limited impact of the manufacturers’ bargaining power falls to zero
as the retail market becomes perfectly competitive, so that in the limit as retail services become (very
near) perfect substitutes then the manufacturers do no better than earning their disagreement profits.
With negotiated transfer prices driven towards zero as a consequence of retailer services being
(near) perfect substitutes, manufacturer profits are also in turn driven towards zero. However, the ultra
intense intrabrand rivalry means that the retailers compete away their profits when setting retail prices.
Thus, intense intrabrand rivalry destroys all profits in the market – both manufacturers’ and retailers’
alike. The beneficiaries are consumers who obtain perfectly competitive prices. Moreover, because the
range of both products and services is unchanged across the two regimes, lower retail prices translate
directly into higher social welfare.
Having derived these results in our model, we should now examine some limitations. First, we
have employed a logical separation between buying power of retailers and their selling power.
Although buying power and selling power may in practice be correlated (e.g. when relative size affects
both types), there are counter examples, as where buying groups negotiate on behalf of small sellers.
Second, as Dobson and Waterson (1997) show, the socially beneficial countervailing power
effect in the UT case is likely to be less intense when there are more than two players in the retail
21
market. If, say, there were three retailers then each manufacturer’s disagreement profits would be based
on trade with a retail duopoly rather than a retail monopoly. With intense intrabrand rivalry, we may
then expect to find both the LHS and RHS denominator of (13) approaching zero (as both the retailer’s
and the manufacturer’s net gains from striking an agreement decline towards zero), meaning that the
negotiated transfer prices would be more dependent on the respective weighted concession costs.
Third, we should recognise that what we have called here Unrestricted Trading might more
accurately be called Non-restrictive Trading. The difference in wording is subtle, but important. We
rule out, by assumption, any possibility of a more general “take-it-or-leave-it” offer. For example, we
have not allowed negotiation of exclusivity/refusal to supply to another player, subject to favourable
terms, see e.g. Dobson and Waterson (1996a, 1997). We also abstracted from any scope for recovery of
profits made at another stage. In the strategic delegation modelling framework (e.g. Bonanno and
Vickers, 1988) there might be an incentive for a retailer to agree a wholesale price above marginal cost
in order to soften competition at the final stage, subsequently clawing back profit through a fixed (e.g.
shelf-space) contribution. Naturally, expanding the contract space (as in Rey and Vergé, 2004a) leads to
additional possible solutions.
In the present context, firms facing the prospect of intense retail rivalry destroying profits for all
industry members will have a joint interest in avoiding such destructive competition. An industry-wide
move to RPM or exclusivity would be a way of avoiding this outcome when it allows for higher
combined profits. However either move would be against consumer interests. A move to RPM would
maintain the same level of variety in the market, but lead to higher prices. A move to exclusive trading
would likely entail both reduced choice and higher prices.
What is interesting is that consumers’ interests are not always best served by unrestricted
trading in the form modelled here. As established above with Corollary 5, when intrabrand and
interbrand rivalry is weak and manufacturers dominate negotiations, successive power effects will
dominate countervailing power effects and consumers will end up paying higher prices than they would
under RPM. In the circumstances where the successive power effect is so strong as to entail retail prices
higher than the single-stage monopoly level then both the industry participants (at least collectively) and
consumers would benefit from a move to RPM.
22
5. Conclusion
The key purpose of the paper has been to assess welfare implications across regimes rather than to
explain how one or other regime emerges in a retail sector.27 In practice, the history and evolution of
market development and competition law appear to play crucial roles in determining where RPM
governs particular industries (e.g. Yamey, 1954; Pickering, 1966; Overstreet, 1983). In view of the
general prohibition against RPM in most countries, most competition authorities now focus attention
on whether specific industry exemptions should be allowed or rescinded. Some of these cases involve
considerations beyond the scope of our paper, such as cultural and accessibility concerns. Our focus
has been on competition effects within a given market situation when both sides of the market are
highly concentrated, which we contend increasingly typifies many consumer goods markets.
Within this context, our bilateral duopoly model allows for different degrees of market power
on each side of the market, encapsulated in the degree of differentiation between the firms at each
vertical level and their relative bargaining power. It encompasses conditions ranging from the market
operating in a perfectly competitive manner when both manufacturers’ products and retailer services are
perfect substitutes to situations of monopoly on the manufacturers’ and/or on the retailers’ side when
products and/or services are demand independent. In addition, the bargaining analysis concerning
transfer price determination includes bargaining conditions ranging from complete domination by
manufacturers (α = 0) to complete domination by retailers (α = 1).
Our main conclusion is that RPM can be socially preferable to unrestricted trading when the
retailers are in a weak bargaining position yet their services are significantly differentiated, because it is
in such circumstances that the double-marginalisation (successive power) problem is at its most severe
with unrestricted trading. Moreover, in contrast to Rey and Vergé (2004a), we do not find that RPM
completely eliminates competition. In our model with linear transfer prices (as opposed to the two-part
tariffs used in Rey and Vergé), intense interbrand rivalry can materialise indirectly in an RPM regime
through the retail prices manufacturers set for their products, such that even perfectly competitive
outcomes are possible. Nevertheless, when interbrand rivalry is weak or retailers dominate negotiations,
we find that retail prices will tend towards the (integrated) monopoly level when RPM applies. In such
circumstances, unrestricted trading with retailers setting their own retail prices can offer consumers
lower prices. This is especially so if intrabrand rivalry is intense since, absent RPM, retailers would be
able to use their greater bargaining power to negotiate low transfer prices, but be forced by intense retail
competition to set low mark-ups and thus pass on the benefits of their reduced costs to consumers.
This latter finding supports the common view that RPM can prevent socially beneficial
27
One possible extension would endogenise the choice of RPM in the present framework, looking from the
perspective of individual trading parties and their willingness to accept RPM arrangements. Unfortunately, the
present model’s complexity means we cannot determine outcomes in asymmetric cases, for example where one
pair agree to RPM, but the other does not. However, see Rey and Vergé (2004a) for analysis along these lines
within a different framework (where RPM prevents any effective competition and always yields the monopoly
outcome).
23
countervailing power arising both through eliminating intrabrand competition at the retail level and
dampening vertical competition, feeding into a more general reduction in the intensity of competition
and raised prices for consumers. The following pertinent remarks made by S. Robson Walton (Senior
Vice President, Wal-Mart Stores Inc) in 1982, as quoted by Steiner (1997, p.442) illustrate the point:
I think it’s probably true that manufacturers would get higher factory prices under resale
price maintenance – especially where you have a limited number of manufacturers – and
even if you rule out collusion. The manufacturer’s price is something that’s determined
largely by negotiating power of the retailers that carry his merchandise. The more these
retailers have to gain from that negotiating process, the harder they are going to bargain
and in our case we’ll negotiate harder for price concessions from manufacturers where
these savings will enable us to sell more of the merchandise as contrasted with where
there’s a fixed retail price that would simply generate a larger margin.
We may also observe parallels between the analysis here and work on exclusive trading
arrangements in successive differentiated oligopoly (e.g. Besanko and Perry, 1994; Dobson and
Waterson, 1996a). In both cases, it is remarkable how private and social desirability are so commonly
in conflict. Only when inter-firm and inter-product connections are slight is there clear agreement
between them (as then successive power effects harm industry profits and consumer welfare alike).
Indeed, one potential extension of the present work would be to consider what alternative strategies,
such as exclusive dealing or exclusive/selective distribution, suppliers might adopt if RPM were
removed as an instrument and to examine the resulting effects on social welfare. Here, as Dobson and
Waterson (1997) show, when successive power effects are strong then exclusive trading arrangements
may offer the benefit of lower prices (when contracts are more efficient) but at the cost of reduced
consumer choice.
However, when countervailing effects are strong then a move to exclusivity
arrangements that dampens competition may be doubly detrimental to consumers if it serves both to
raise prices and reduce choice.
In sum, our framework suggests that the critical determinant of the public interest implications
of RPM lie largely in the relative position of the retailers in terms of their bargaining power and the
extent of differentiation between their services.
In Europe, North America, and elsewhere,
concentration in retailing has increased markedly over recent years, whilst manufacturing industry
concentration levels have been relatively stable (e.g. OECD, 1999; Clarke et al., 2002). This may
suggest that the relative bargaining position of retailers has been improving over recent years.28 At the
same time, we have witnessed the widespread development across many retail sectors of discount store
chains and big-box “category killers” along with online retailers seeking to use their buying muscle to
obtain low prices and then compete aggressively on retail prices. Cast in this light, our analysis suggests
that although the impact of RPM on consumers is not necessarily adverse, now would not be a
propitious time to re-affirm the legitimacy of RPM.
28
Evidence supporting this assertion is the increasing level of gross and net margins for many retailing activities
while concentration has been increasing (e.g. Dobson and Waterson, 1995).
24
Appendix: Proofs of Propositions and Corollaries
Proposition 1. Proof. As a direct analogy to existence and uniqueness conditions in differentiatedgoods Bertrand-Nash oligopoly (e.g. Vives, 1999), a sufficient condition for existence and uniqueness
for the set of equilibrium transfer prices is that each Nash maximand represented in (1) is concave in its
own negotiated transfer price and that own-effects dominate cross-partial effects. First, observe that
each Nash maximand in the RPM regime, which we can denote as ΓiRhPM ≡ [πMi – δMih]1-α[πRh – δRih]α, is
concave since ∂2(ΓiRhPM)/(∂wih)2 = –1. Next, note that the two cross-partial effects are ∂2(ΓiRhPM)/(∂wih∂wjh)
= (1 – α)υih ≥ 0 and ∂2(ΓiRhPM)/(∂wih∂wik) = αλih ≥ 0. A sufficient condition for existence and uniqueness
is then ∂2(ΓiRhPM)/(∂wih)2 + ⏐∂2(ΓiRhPM)/(∂wih∂wjh) + ∂2(ΓiRhPM)/(∂wih∂wik)⏐ < 0. This condition is satisfied if
(1 – α)υih + αλih < 1, which necessarily follows if λih < 1 and υih < 1, ∀i, ∀h.
F
Corollary 1. Proof. (a), (b) By direct inspection of (2). (c), (d) By continuity and monotonicity from
the above.
F
Proposition 2. Proof. From (8), direct evaluation of respective comparative static effects reveals that
(i) ∂wRPM/∂pRPM = [(1–α)(1–υ)]/[1–(1–α)υ–αλ] > 0; (ii) ∂wRPM/∂λ = [α(1–α)(1–υ)pRPM]/[(1–(1–α)υ–
αλ)2] > 0; and (iii) wRPM/∂υ = –[α(1–α)(1–λ)pRPM]/[(1–(1–α)υ–αλ)2] < 0.
F
Proposition 3. Proof. A sufficient condition for this result to hold is that the first-stage optimality
conditions represented by (7) are increasing in α when evaluated at the equilibrium prices.
Differentiating (7) with respect to α, taking account of the effect of the variable arising through the
anticipated transfer prices, shows:
∂ 2 π Mi ∂wih ∂qih ∂wik ∂qik
∂ 2 wih
∂ 2 wik
.
.
=
+
+ qih
+ qik
∂pih ∂α
∂α ∂pih
∂α ∂pih
∂pih ∂α
∂pih ∂α
(A1)
With symmetric equilibrium outcomes then wih = w and qih = q, ∀i ∀h. We can then re-express (7) for
the case of symmetric equilibria as:
⎛ ∂q
q = − w ⎜ ih
⎜ ∂pih
⎝
+
pih = p
∂qik
∂pih
⎞
⎟
⎟
pih = p ⎠
⎛ ∂w
⎜ ih
⎜ ∂pih
⎝
+
pih = p
∂wik
∂pih
⎞
⎟
⎟
pih = p ⎠
(A2)
Substituting (A2) into (A1), re-arranging and evaluating at symmetric levels shows that the impact of α
on the optimality conditions is positive:
25
⎛ 2
∂ π Mi
SIGN ⎜⎜
⎜ ∂pih ∂α
⎝
⎡
⎞
⎛ 2
2
⎟ = SIGN ⎢1 - w ⎜ ∂ wih + ∂ wik
⎟⎟
⎜⎜ ∂p ∂α
⎢
∂pih ∂α
pih = p ⎠
⎝ ih p
⎣
⎞
⎟
⎟⎟
p⎠
⎛
⎜ ∂wih
⎜⎜ ∂α
⎝
⎛ ∂w
⎜ ih
⎜
p ⎝ ∂pih
∂w
+ ik
∂pih
p
⎞ ⎞⎤
⎟ ⎟⎥
⎟ ⎟⎥
p ⎠⎟
⎠⎦ (A3)
⎡ (1 − α)υ(1 + υ)(1 − (1 − α)υ − αλ) ⎤
= SIGN ⎢
⎥>0
2
⎣ [1 + (1 − α)υ − αλ][1 − (1 − α)υ − αλ] ⎦
F
Proposition 4. Proof. (a) First note that a fully integrated monopoly supplier would set retail prices
(with transfer prices redundant) to satisfy the following re-arranged first-order conditions evaluated at
symmetric equilibrium levels (pimh = pm, qmi h = qm ∀i ∀h):
p m = − q /( X + Y )
where
(A4)
⎛ ∂q
X ≡ ⎜ ih
⎜ ∂pih
⎝
+
pih = p
∂qik
∂pih
⎛
⎞
⎟ < 0, Y ≡ ⎜ ∂q jh
⎜ ∂p
⎟
pih = p ⎠
⎝ ih
+
pih = p
∂q jk
∂pih
⎞
⎟ ≥ 0, X + Y < 0
⎟
pih = p ⎠
The conditions on X and Y follow from our assumptions about the nature of the symmetric demand
functions and (imperfect) substitutability between the available product-service combinations. Next,
with the first-order condition (7) for retail price setting under RPM, evaluating at the symmetric
equilibrium levels, using the expressions for wRPM given in (5) and (8), and then re-arranging reveals that
the first-stage optimality condition can be expressed as
p RPM = − qZ / X
(A5)
where
Z≡
1 − (1 − α)υ 2 − αλ
(1 − υ)(1 + (1 − α)υ − αλ)
In the terms of expressions (A4) and (A5), the retail price under RPM can only exceed the retail price set
by a single-stage monopolist (i.e. pRPM > pm), if and only if Z > X/(X+Y). However, observe that given
the symmetric demand structure and symmetric equilibrium outcomes, then ⏐Y/X⏐∈ [0,1) represents
the degree to which the manufacturers’ products are viewed as substitutes, i.e. the value of the
interbrand rivalry. In the context of our assumed parameterisation regarding interbrand rivarly, this
implies Y/X = –υ and thus X/(X+Y) = 1/(1–υ). Accordingly, pRPM > pm if and only if Z(1–υ) = [1–(1–
α)υ2–αλ)]/[1+(1–α)υ–αλ)] > 1. Yet, [1–(1–α)υ2–αλ)]–[1+(1–α)υ–αλ)] = –(1–α)υ ≤ 0, thus, with the
numerator less than or equal to the denominator, this condition cannot be satisfied and so the
equilibrium retail prices under RPM cannot exceed the levels that would be set by a single-stage
monopolist. (b) The required conditions for pRPM = pm (and equally for qRPM = qm) are thus revealed as υ
= 0 and/or α = 1, which allows Z = X/(X+Y).
F
26
Proposition 5. Proof. Observe from (A4) and (A5), and again taking X/(X+Y) = 1/(1–υ), that pRPM/pm
= (qRPM/qm)Z(1–υ), where Z(1–υ) → 0 as both λ → 1 and υ → 1 for α ∈ [0,1). Then, since price-ratio
effects dominate quantity-ratio effects (as determined by our assumption of non-convex demand
together equilibrium outcomes occurring where absolute demand elasticity is greater than unity) this
implies that the RPM prices collapses towards zero as both retailer services and manufacturer products
respectively approach being perfect substitutes. Furthermore, on the same basis we can observe that
∂(Z(1–υ))/∂λ = –[αυ(1–α)(1+υ)]/[1+(1–α)υ–αλ)]2 < 0 for α ∈ (0,1), ∂(Z(1–υ))/∂υ = –(1–α)[(1–
α)υ2+(1+2υ)(1–αλ)]/[1+(1–α)υ–αλ)]2 < 0 for α ∈ [0,1), which implies the ratio of prices is declining in
respect of both intrabrand and interbrand rivalry. Finally, in accordance with Proposition 3, ∂(Z(1–
υ))/∂α = [υ(1+υ)(1–λ)]/[1+(1–α)υ–αλ)]2 > 0 for υ > 0, i.e. the ratio of prices is increasing in respect of
retailers’ bargaining power.
F
Proposition 6. Proof. On the same basis as the proof for Proposition 1, observing that quasiconcavity
of the Nash maximand ensures uniqueness, and second-order own effects dominating cross effects along
with symmetry of the interdependent Nash bargains ensures uniqueness and stability of symmetric
equilibrium levels.
F
Corollary 2. Proof. (a), (b) By direct inspection of (11), given the stated assumptions. (c) By
continuity and monotonicity given the above.
F
Corollary 3. Proof. The results follow trivially from the assumptions made about the effects of a
change in a common transfer price on final stage outcomes.
F
Corollary 4. Proof. (i) From Proposition 4b, pRPM = pm when α = 1. From Corollary 2, wUT = 0 when
α = 1, which implies that the retailers’ marginal cost is zero when setting retail prices. Intrabrand
competition in the UT regime then ensures that the Bertrand-Nash equilibrium results in pUT < pm.
Only when retailer services are demand independent can pUT = pm = pRPM, which is ruled out here by
competing retail services. (ii) Follows directly.
F
Corollary 5. Proof. From Proposition 4b, the absence of interbrand competition means that pRPM = pUT
(even with α = 0). However, in this situation, in the UT regime the manufacturers negotiate positive
transfer prices (i.e. wUT > 0 based on ∂πM*i*(wih)/∂wih = 0 with ∂2πM*i*(wih)/(∂wih)2 < 0), which means that
when retailers add their monopoly-mark-ups to retail costs then it results in pUT > pm.
27
F
Proposition 7. Proof. On inverting the indirect demand equation system, given by (9), to solve for the
direct demands, the demand for good i at retailer h when both retailers sell both products is
qih (p) =
(1 - β)(1 - γ ) - pih + β pik + γ p jh - βγ p jk
(A6)
(1 - β2)(1 - γ 2)
In the second stage, having negotiated the transfer prices, retailer h’s profit function is then
2
⎡ (1 - β)(1 - γ ) - pih + β pik + γ p jh - βγ p jk ⎤
π Rh (p) = ∑ ( pih - wih ) ⎢
⎥
(1 - β2)(1 - γ 2)]
i =1
⎦
⎣
i ≠ j, h ≠ k
(A7)
Determining the FOCs for profit maximization under independent price setting behaviour and then
solving for the equilibrium price set by retailer h for good i in terms of the transfer prices shows
pih ( wih , wik ) =
(1 - β)(2 + β) + 2 wih + βwik
4 - β2
(A8)
and substituting back into (A6), implies that the corresponding quantity sold is
qih (w ) =
(1 - β)(2 + β)(1 - γ ) - (2 - β2 ) wih + βwik + γ (2 - β2 ) w jh - βγw jk
(A9)
(1 - β2 )(4 - β2 )(1 - γ 2 )
In the first-stage where the transfer prices are negotiated, and anticipating the subsequent final
retail prices, the profit function for Rh is
2
π Rh (w ) = ∑
i =1
⎡ (1 - β)(2 + β) - (2 - β2) wih + βwik ⎤
⎢
⎥
(4 - β2)
⎣
⎦
(A10)
⎡ (1 - β)(2 + β)(1 - γ ) - (2 - β2 ) wih + βwik + γ (2 - β2) w jh - βγw jk ⎤
× ⎢
⎥
(1 - β2 )(4 - β2 )(1 - γ 2 )
⎥⎦
⎢⎣
while the profit function for Mi is
2
⎡ (1 - β)(2 + β)(1 - γ ) - (2 - β2 ) wih + β wik + γ (2 - β2 ) w jh - βγ w jk
π Mi (w ) = ∑ wih ⎢
2
2
2
(1 - β )(4 - β )(1 - γ )
h =1
⎢⎣
⎤
⎥
⎥⎦
(A11)
Disagreement is assumed to result in the two parties involved not trading and so qih = 0. Then,
with only three manufacturer-retailer trades taking place, from (9), the structure of inverse demands is
φik = 1 − zik − βγz jh − βz jk
(A12a)
φ jh = 1 − βγzik − z jh − γz jk
(A12b)
φ jk = 1 − γzik − βz jh − z jk
(A12c)
where φik, φjh and φjk represent the three final prices, and zik, zjh and zjk represent the three corresponding
quantities, when Mi and Rh are unable to reach an agreement. On inverting this system, the direct
demands are accordingly
28
zik (φik , φ jk ) =
1 − γ − φik + γ φ jk
z jh (φ jh , φ jk ) =
(A13a)
1 − γ2
1 − β − φ jh + β φ jk
(A13b)
1 − β2
z jk (φ jk , φik , φ jh ) =
(1 − β)(1 − γ )(1 − βγ ) − (1 − β2 γ 2 ) φ jk + γ (1 − β2 ) φik + β(1 − γ 2 ) φ jh
(1 − β2 )(1 − γ 2 )
(A13c)
From (A12b) and (A13b), the disagreement payoff for Rh when it is unable to strike an
agreement with Mi is
δ Rih = (φ jh − w jh ) z jh =
[(1 − β)(2 + β ) − (2 − β2) w jh + βw jk ]2
(A14)
(1 − β2)(4 − β )
2 2
Similarly, using (A13a), the disagreement payoff for Mi in its bargain with Rh is
δ Mih = wik z ik =
wik (1 − γ − wik + γw jk )
(A15)
2(1 − γ 2)
Observing the symmetry between the bargains, we can let wih = w, ∀i and ∀h, to solve for the
equilibrium symmetric agreed transfer price where wUT = f(wUT). Evaluating the terms in (11) for the
symmetric level shows
| w =w = (1 − β)(1 − γ )(12 − w )
UT
[ π Rh ( wih , wUT
-ih ) − D Rih ( w-ih )]
∂ π Mi ( wih , wUT
-ih )
∂ wih
=
wih = w
(1 + β)(2 − β ) (1 + γ )
1 − γ − (2 − γ ) w
(1 + β)(2 − β)(1 − γ 2)
UT
[πMi ( wih , wUT
-ih ) − D Mih ( w-ih )]
∂ π Rih ( wih , wUT
-ih )
∂ wih
w
ih
2
= −
ih = w
| w =w =
ih
w(1 − w)(2 − β + β2)
2(1 + β)(2 − β )2 (1 + γ )
2(2 − β2)(1 − w)
(1 + β)(2 + β)(2 − β )2 (1 + γ )
(A16a)
(A16b)
(A16c)
(A16d)
where w-ihUT is the (symmetric) transfer price level assumed by the representatives of Mi and Rh to have
been agreed in the other three bargains.
Substituting (A16a)-(A16d) into (11) and solving for the perfect Nash equilibrium transfer price
under unrestricted trading yields
wUT =
(1 − α)(1 − β)(2 + β)(1 − γ )
(2 − β − β2)(2 + αγ − γ ) + α β2 (2 + β − β2)
29
(A17)
This ranges from 0 (for α,β,γ = 1) to 1/2 (for α = γ = 0) with ∂wUT/∂β < 0 and ∂wUT/∂γ < 0 implying that
the equilibrium negotiated transfer price is respectively declining in intrabrand rivalry and interbrand
rivalry as claimed in the Proposition:
UT
2αβ(1 − α)(1 − γ )(4 + 2β - 5β 2 + β3 + β 4 )
∂w
< 0 for α, γ < 1
=−
∂β
[(2 − β − β2)(2 + αγ − γ ) + α β2 (2 + β − β2)]2
(A18)
UT
(1 − α)(1 − β)(2 + β)[2 - β - β 2 + α(2 − β + β 2 + β 3 − β 4 )]
∂w
=−
< 0 for α, β < 1
∂γ
[(2 − β − β2)(2 + αγ − γ ) + α β2 (2 + β − β2)]2
(A19)
F
30
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