Download Why variable inventory costs intensify retail competition, and a case

Why variable inventory costs intensify retail competition, and a
case for manufacturer returns
Abstract
We study how the extent to which the inventory cost is sunk or remains variable a¤ects
retail competition when retailers do not observe each others’ stocks before choosing their
prices. We …nd intense price competition when inventory costs remain variable, and that
markups increase as the sunk fraction increases. We then study whether a manufacturer
should endogenously reduce the extent to which retailers’ inventory cost is sunk by o¤ering
the possibility of returning unsold stocks for a fraction of the wholesale price. The answer
solves a trade-o¤: returns intensify retail competition by giving retailers the incentive to order
excess stocks, but such ine¢ cient over-production must be paid up-front by the manufacturer.
Therefore the optimal return decreases as the manufacturing marginal cost increases. We
…nally study how the inability to observe retail stocks a¤ects manufacturer pro…tability relative
to a situation where stocks are observed (where returns do not change pro…ts).
1
Introduction
More often than not retailers must order stocks in advance to make them readily available to
consumers on their shelves. Yet the extent to which retailers’inventory cost is sunk or remains
variable varies signi…cantly from one product to another as unsold inventory may some scrap value.
Variations in the value to retailers of unsold stocks can be either exogenous or arise endogenously.
As an example of the latter, manufacturers often let retailers return unsold goods for a fraction
of the wholesale price— a practice that makes inventory cost less sunk and that has been widely
documented in a variety of contexts and for a multitude of products. Several other tools can have
a similar e¤ect, when they require the retailer to pay a price per unit sold and a lower price per
unit that remains unsold— such as a wholesale price charged per unit delivered and an additional
royalty paid per unit actually sold (the analysis of this paper naturally extends to all such tools).
1
This paper carries two main messages. First, that the extent to which inventory costs are
sunk or remain variable has a signi…cant a¤ect on retail market outcomes if realistically retailers
do not observe rivals’inventory before choosing prices (while, as supported by the currently held
view discussed below, it has no e¤ect if instead retailers observe inventory choices). Second, we
show how a return policy can be pro…tably used by an upstream manufacturer (and by extension
other policies, such as the above mentioned per unit royalties).
A current view is that the extent to which inventory costs are sunk or remain variable has no
e¤ect on retail market outcomes if retailers’inventory choices precede price competition. As Kreps
and Scheinkman (1983) found in their study of what has became a workhorse model, if retailers
choose prices after having observed the stock choices of their rivals even “situations that look
very Bertrand-like”, i.e. where most of the cost remains variable at the pricing stage, “will still
give the Cournot outcome”(Kreps and Scheinkman, 1983). When it was introduced, their model
intended to describe industries where …rms make long-term capacity choices, such as the size of
a plant, that are either observed or can be accurately inferred by competitors over time— such as
cement or chemicals.
The assumption that stocks are observable seems however hard to justify in many retail
settings, which provides the motivation for the present analysis. In practice a retailer is unlikely
to know the stocks other retailers have of any particular good when choosing his price. Unlike the
size of a plant, retail stocks are not only hard to observe but also, given their transient nature,
historical data cannot be used to provide a reliable estimate. This view seems to be shared by the
operations research literature. For example, standard newsvendor settings assume that retailers
choose prices and stocks without knowing the choices of rival retailers.
Our …rst result challenges the above mentioned irrelevance result by showing that if retailers
do not observe each other’s stocks before choosing prices then the extent to which inventory cost
is sunk a¤ects substantially the level of stocks ordered and retail markups. When the whole
inventory cost remains variable both retailers charge a price equal to the wholesale price, as in
Bertrand competition, yet each retailer also orders enough stocks to serve on its own the whole
market. This is intuitive since if each retailer believes the other retailer is following such a strategy
then he has no incentive to deviate: he cannot expect to make a larger pro…t by setting a higher
price (as all consumers would purchase from the other retailer) but also gains or loses nothing if
he orders less than the prescribed equilibrium inventory.
When part of the cost is sunk each retailer selects instead a price that is above the wholesale
price and uses that margin to cover the cost of any stock that remains unsold. While each
2
retailer still orders enough inventory to serve the market at his chosen price, his exact price choice
cannot be predictable (i.e., a pure-strategy equilibrium does not exist) since otherwise he would
be undercut by his rival and sell nothing (while still incurring the sunk part of the cost and thus
making a loss). Therefore when part of the cost is sunk, retailers must use a mixed-strategy,
sometimes charging a high price (and accordingly choosing a low stock) and other times charging
a low price as if holding a sale (and accordingly ordering a large stock). Fortunately in the
absence of retailer di¤erentiation the unique “regular” equilibrium is elegantly characterized. In
equilibrium the expected price and retail margin of each retailer increases as the fraction of the
wholesale price that is sunk increases, as this increases the downside of being left with unsold
stock— which leads retailers to order smaller stocks more often and results in less intense price
competition. This naturally creates retail price dispersion and, like in Varian’s (1980) theory of
sales, even if there is no uncertainty about aggregate demand the retailers face a residual demand
that is endogenously uncertain (a consequence of the mixed-strategy of his rival).
Padmanabhan and Png (1997) (PP hereafter) have found empirically that retail margins were
signi…cantly reduced when an upstream manufacturer o¤ered retailers the possibility to return
unsold stocks. PP used the following argument to explain their …ndings: If retailers buy from
the manufacturer at a linear wholesale price and observe the stocks of each other before choosing
their prices, without returns, retail competition is of the Cournot style— following Kreps and
Scheinkman (1983). They then conjectured that o¤ering retailers the possibility to return unsold
stock at the wholesale price, i.e. full returns, would create intense Bertrand price competition
among retailers since it eliminates retailers’cost of holding excessive stocks. However this insight
“cannot be reached if the (PP’s) model is solved correctly”, wrote Wang (2004), while showing
that when retailers observe each others’stocks then in equilibrium they do not order excess stocks
but instead use their stock choice as a precommitment to soften price competition and the Cournot
outcome is still obtained— therefore returns do not change the equilibrium strategies or pro…ts of
that game.1
Our second result shows that PP’s original idea can be successfully formalized if instead,
but realistically, we drop that rather standard assumption that retailers observe each others’
stocks before choosing prices. Naturally, if stocks are not observed then they cannot provide
a precommitment to soften price competition. In that case if a producer makes the retailers’
1
Reintroducing demand uncertainty in their model, Padmanabhan and Png (2004) concluded that returns could
help to "manage price competition between retailers but that this e¤ect holds only in the presence of demand
uncertainty" under particular parameter conditions. The magnitude of that e¤ect remains however small.
3
inventory cost variable by o¤ering them a full return policy then retailers set prices at the wholesale
price level, like in Bertrand, since as PP conjectured the retailers “choose not be constrained by
their stocks.”This is an important …nding since, judging by the number of citations, the original
idea proposed by PP remains compelling and has had a profound impact on how academics
think about return policies. Yet, and unlike what PP had conjectured, full returns may still be
unpro…table (relative to a situation with no returns). This is also intuitive, because returns can
only result in intense price competition when they induce retailers to order excess stocks, but
the manufacturer must then pay up-front for the production cost of stocks that are eventually
returned.
Our third result is to characterize the optimal distribution policy, i.e. not only the wholesale
price but also the fraction of the wholesale price a manufacturer should o¤er as a return. This
requires solving the manufacturer’s trade-o¤ between the bene…t of more intense retail competition, which leaves him with more room to increase the wholesale price, and the higher cost of
more excess stocks. We …nd that the optimal wholesale price increases with the manufacturing
marginal cost, while the optimal fraction of the wholesale price that is returned decreases with
that cost. For example, consistent with this result, generous return policies are common in the
distribution of books, CDs and software, all products that have a low cost relative to the price
paid by consumers.
Finally we ask if a manufacturer bene…ts from unobservable stocks. Our fourth and …nal result
is that a manufacturer bene…ts when the manufacturing marginal cost is low, but unobservable
stocks hurt him when that cost is high— a result which also holds in the presence of downstream
di¤erentiation. This follows from directly comparing our results with those of Wang (2004), for
which we summarize the main di¤erences and …ndings in picture 1.
Wang (2004) showed that when retailers’ stock levels are observed before prices are chosen
then the retail outcome is always Cournot (independently of the return policy). In that case
in equilibrium all stocks that are ordered are also eventually sold to consumers but the manufacturer cannot squeeze the retail margins— since retailers use their stock choices as a quantity
precommitment to soften price competition and protect those margins. Here we have found that
when stocks levels are unobservable then the manufacturer may use a return policy to intensify
retail competition but at the cost of excess stocks. An observable stocks regime should eventually become more pro…table to a manufacturer when the number of retailers is large. In that
case Cournot retail competition with many retailers already drives retail markups close to zero
without the need of producing excessive stocks.
4
2
Literature review
In addition to the work described above, there is a large literature in economics and marketing
that explains the use of returns by an upstream monopolist. Yet all those theories crucially rely
on demand uncertainty. We brie‡y review this large literature below.
With a single retailer, if demand is uncertain, it has been shown that returns can for example
transfer risk from a retailer to the manufacturer (e.g. Kandel, 1996, Marvel and Peck, 1995).
In that case a return policy and a linear wholesale price can also coordinate a supply chain or
improve pro…ts over outright sales (e.g. Pasternack, 1985, Emmons and Gilbert, 1998 and Marvel
and Peck, 1995). If demand is realized after production takes place but before the wholesale price
is determined, then returns allow a manufacturer to achieve the same pro…t whether she remains
the owner of the stock until demand is realized or transfers that ownership to the retailer— which
would otherwise decrease the manufacturer’s pro…t, despite increasing the industry total surplus
(Biyalogorsky and Koenigsberg, 2010). If a retailer needs to invest on a marketing-mix activity
that a¤ects consumer demand but the terms of trade may be renegotiated once demand is realized,
then transferring the ownership of the stock to the retailer and allowing for returns can reduce
double marginalization by converting what would be an ex-post negotiation over the wholesale
price into a negotiation over the return price (Iyer and Villas-Boas, 2003).
When retailing is perfectly competitive, but demand is still uncertain, buybacks can also lead
to optimal levels of inventory and price dispersion (Marvel and Wang, 2007). They may also
be pro…tably used as a substitute for a price ‡oor. By inhibiting the price cutting that would
otherwise occur when demand is low, returns protect the retailers margins in those states. This
increases the retailers’incentives to hold larger inventories in equilibrium, even if retail margins
are lower when demand is high (Deneckere, Marvel and Peck, 1996). Such logic, that returns
policies attenuate retail price competition when demand is low but intensify competition when
demand is high, also extends to a retail oligopoly (Butz, 1997, Padmanabhan and Png, 2004). In
a retail oligopoly, contracts with a linear wholesale and return prices can coordinate the supply
chain only if the manufacturer retains controls over the retail price (Pasternack, 1985). Yet supply
chain coordination can still be achieved if the wholesale and return price depend on the prices
chosen by retailers (Bernstein and Federgruen, 2005), or if in addition a …xed fee is charged and
demand uncertainty or retailer di¤erentiation are large (Narayanan, Raman and Singh, 2005 and
Krishnan and Winter, 2007).2
2
In Narayanan et al. (2005) the uncertainty is over the level of demand, i.e. any level of demand between zero
5
There is also work in the context of asymmetric information on an uncertain demand. For
example, returns may be used by a manufacturer to signal to retailers private information on
demand (Kandel, 1996), or elicit from retailers their private information on demand or provide
them with incentives to acquire such information (Arya and Mittendorf, 2004 and Taylor and Xiao,
2009). The role of returns in learning demand of a new product has also been studied, assuming
that the retail price is …xed and demand is random but identical across periods (Sarvary and
Padmanabhan, 2001).
Finally, as mentioned in the opening paragraph, our results also apply for example to per
unit royalties. While we refrain from providing here a review of the large literature on royalties,
we note that most of the work in that literature with a single manufacturer has focused on
either agency problems with a single buyer or on situations where there are multiple buyers
assuming that the outcome of downstream competition is either Bertrand or Cournot. In the
former case the focus is on how royalties can help control moral hazard, adverse selection and
risk-sharing (see e.g. Bousquet et al., 1998 and Cachon and Lariviere, 2005). In the later the
focus has been in understanding the choice of di¤erent licensing mechanisms in the presence of
competition for example when the seller’s innovation reduces the costs of downstream buyers
(Kamien and Tauman, 1986, Wang , 1998 and San Martin and Saracho, 2010). This paper
suggests that Bertrand and Cournot forms of competition may not provide good benchmarks if
realistically we assume that retailers order stocks before choosing their prices and cannot observe
each others’stocks. Our work would therefore also provide new tools to advance our analysis and
understanding of royalties. However we leave further extensions to future research.
3
A retail model with unobservable stocks
There are two risk neutral retailers that purchase stocks at a constant per unit wholesale price w,
a part r of which is recoverable if a unit is not sold. We consider a two stage setting similar to
Kreps and Scheinkman (1983) (they are equivalent when r = 0):
Stage one: Each Retailer i, knowing w and r, simultaneously chooses his stock si . The only
cost of retailing is the payment of the wholesale price w, so i pays wsi .
Stage two: Each retailer i chooses a price pi . Consumers observe both prices and purchase or
and some higher level is equally likely. In Krishnan and Winter (2007) there is uncertainty over both the level of
demand and consumers’ preferences over retailers while using a …rst-order approach— which means the results do
not apply to cases without su¢ cient retail di¤erentiation.
6
not. The value of sold and unsold stock is realized at a value of pi and r respectively per unit
sold and not sold.
One important element of the information structure still needs to be speci…ed: do retailers observe or not the stocks ordered by the other retailer before choosing prices? Kreps and Scheinkman
(1983) assumed that retailers observe the stocks of each other before stage two. Instead we assume
here that each retailer does not observe the stock of his rival before choosing his price. When
retailers do not observe stocks before choosing prices, stage one and stage two of the game become
a simultaneous move game where retailers choose a pair (si ; pi ). As we argued in the introduction,
in many retail settings both the observability and variability of cost assumptions may be more
realistic than the alternatives. This approach builds on Levitan and Shubik (1978) and Gertner
(1985), but in addition we allow for a part of retailers’costs to be variable: for each unit a part
of the cost w
r is sunk and a part r remains variable.
Demand assumptions are also similar to those in Kreps and Scheinkman (1983). The consumer
demand D(p), with D0 < 0, has a choke price c < p < 1 where p = inf(p jD(p) = 0 ). Market
revenue pD(p) is strictly concave on [0; p], so …rst-order conditions on the market pro…t (p c)D(p)
determine the market monopoly price pm (c). Retailers are undi¤erentiated and therefore all
consumers would like to purchase from the retailer with the lowest price— and do so if the stock
of that retailer is su¢ ciently large. When both charge the same price, they share the market
equally. If prices are di¤erent then, for any given pairs (s1 ; p1 ) and (s2 ; p2 ), if pi < pj retailer
i faces the total consumer demand, while if pi > pj then i faces a residual demand from those
consumers who were unable to buy from retailer j because of a stockout, i.e. since D(pj ) > sj .
Formally the sales of retailer i are given by3
8
>
min fsi ; D(pi )g if pi < pj
>
>
<
qi =
min fsi ; max f0; D(pi ) sj gg if pi > pj
>
n
n
oo
>
>
: min s ; D(p ) min s ; D(pi )
if pi = pj
i
i
j
2
A pure-strategy for retailer i is a pair (si ; pi ), where pi
0 and si
0. To focus on the
payo¤ relevant strategies, and make it possible to de…ne uniquely the equilibrium, we make the
innocuous assumption that each retailer i can choose a price for which there is positive demand
3
Here we have e¢ cient rationing of demand but this is not essential as our results hold under a large family
of demand rationing rules introduced by Davidson and Deneckere (1986) that range from e¢ cient to proportional
rationing. E¢ cient rationing can be interpreted as a situation where if there is price dispersion then those consumers
with the highest valuation enter the market …rst and purchase from the lowest priced retailer.
7
only if he has some stock to sell, i.e. retailer i can choose a pair (si ; pi ) for which D(pi ) > 0 only
if si > 0. The pro…t of retailer i is then
i
= (pi
r)qi
(w
r)si .
(1)
This game has no pure-strategy equilibria (unless r = w), since if retailer i were to choose an
exact price and stock then either his rival j would …nd it pro…table to just undercut i and sell
to the whole market (in which case i would still incur the sunk part of the cost and thus make
a loss), or if it did not then it would be i …nding it pro…table to expand its capacity (in which
case it would instead be j that makes a loss). The absence of a pure-strategy equilibrium means
that retailers are unable to predict rival’s choices, even if there is complete information about
aggregate demand and all costs involved.
In a mixed strategy equilibrium each retailer i must be indi¤erent between ordering a low
quantity and choosing a high price (in which case i is likely to be undercut by his rival j and not
sell his stock, but he will make a high margin on this small quantity if he sells), and ordering a
large quantity and hold a sales promotion (in which case i is likely to undercut j and sell all his
stock, but then make a low margin on this large quantity). Retailer i predicts the frequency and
intensity of j’s promotions, but he cannot predict if retailer j will hold a promotion or set a high
price.
This retail game has a unique regular Nash equilibrium, i.e. where strategies have no gaps or
mass points inside the support of prices (although they could have mass points at the boundaries).4
To characterize the equilibrium, we …rst show that retail competition always drives retail pro…ts
to zero: even if each retailer charges a positive markup over the wholesale price, the amount he
makes when he is the lowest priced retailer exactly o¤sets the amount needed to cover the sunk
part of his inventory cost when he is the higher priced retailer. Second, in a regular equilibrium
each retailer must order enough to serve the market at the price he has chosen and therefore the
lowest priced retailer covers the whole market, i.e. si (pi ) = D(pi ). These two properties identify
a single distribution of prices: when retailer i sets a price above w there is a probability F (pi )
that the other retailer j chooses a lower price and leaves retailer i with unsold stock. From the
4
While it is possible that other equilibria could exist, we have found none and this equilibrium is the unique
symmetric equilibrium, and also the unique equilibrium if demand is linear.
8
zero pro…t condition and since si (pi ) = D(pi ) we have that5
(1
F (pi ))(pi
w)D(pi ) = F (pi )(w
r)D(pi ) , F (pi ) =
pi
pi
w
.
r
We have the following formal result:
Proposition 1. For a given w and r, the unique regular equilibrium outcome of retail competition
has each retailer i choosing his price using a cumulative distribution function (with a pdf f )
F (pi ) =
pi
pi
w
for all pi 2 [w; p] and F (pi ) = 1 for pi > p,
r
and for each chosen price pi he orders the stock si (pi ) = D(pi ). An increase in r intensi…es retail
competition since, by shifting F leftward, it leads retailers to order larger stocks and charge (on
average) lower margins. As r ! w the price of both retailers converges to the wholesale price w
and therefore retailers’ mark ups converge to zero.
Proof. See all proofs in the Appendix.
When most of the cost is sunk the retailers buy on average small stocks and charge larger
margins to make up for the loss they make when they are unable to sell their stocks. As the
fraction of the cost that is sunk decreases, the downside of being left with unsold stock also
decreases. This leads retailers to order (on average) larger stocks and charge lower margins—
which is formally captured by a shift in F leftward.
4
Endogenous use of returns by a manufacturer
Given the previous result, we would expect manufacturers to use such marketing tools that change
retail competition to their advantage. One tool that achieves this are returns, but as we also
mentioned in the introduction the insights apply to other tools with a similar structure (such as
per unit royalties). This idea was …rst suggested by Padmananabhan and Png (1997).
We study a model similar to theirs but with a di¤erent information structure on stocks.6 Like
Kreps and Scheinkman (1983), but unlike PP, we assume no retail di¤erentiation (see however
section 5 on how the …ndings extend to di¤erentiation). PP assumed that retailers observe the
5
Note that retailer i orders si = 0 with probability
w r
,
p r
understood as the probability that a particular retailer
delists the good. Here this is represented by a mass point at p.
6
Also, like Kreps and Scheinkman (1983) we assume no retail di¤erentiation, which PP had in their model. See
section 5 on how the …ndings extend to di¤erentiation.
9
stocks of each other before choosing prices, following Kreps and Scheinkman (1983). Wang (2004)
has however shown formally that in such setting returns have no e¤ect on strategies and pro…ts
(that result holds with and without di¤erentiation). Here we will instead assume, like we did
above, that stocks are unobserved.
There are now two retailers and a manufacturer with a constant marginal cost c. Like PP we
consider a game with three stages, where the manufacturer’s choice of a public retail policy, w
and r, precedes stages one and two of the game described in section 3.7
Stage zero: The manufacturer sets a distribution policy with a wholesale price w and return
price r at which retailers can buy and return any quantity.
Stage one: Each retailer i simultaneously decides how much to order and pays wsi .
Stage two: Each retailer i chooses a price pi and consumers, after observing both prices, decide
to purchase or not. Unsold stocks are returned to the manufacturer and refunded at the return
price r per unit.
A pure-strategy for the manufacturer is a pair (w; r), with w
strategy for retailer i is a pair (si ; pi ), where pi
0 and si
0 and 0
r < w. A pure-
0. The manufacturer’s pro…t
is
=
P
[(w
c)si
r(si
qi )]
i
and the retailers’pro…t remain given by (1).
4.1
Full returns versus no returns
With full returns, as r ! w, the cost of holding an excessive stock is zero and therefore each
retailer i orders si = D(w), enough stock to serve the whole market at the wholesale price, and
as F (w) ! 1 both prices converge to the wholesale price. It follows that with full returns (F R)
the manufacturer’s pro…t becomes
F R (w; c)
= (w
2c)D(w) =
m
(2c)
Note the equivalence to the pro…t of a monopolist that sells directly to consumers at a price of
w but has a marginal cost 2c instead of c. Thus the optimal wholesale price, wF R (c) = pm (2c),
exceeds the monopoly price. The bene…t from intense retail competition created by full returns
7
Discriminatory distributions policies can, for example, attract antitrust scrutiny. Also, in the ten product
categories empirically studied by PP, no manufacturer used non-linear pricing. One explanation is that arbitrage
by retailers across multiple markets served by the same manufacturer can restrict the use of non-linear policies.
10
is countered by the production cost of excess stocks, so only when c = 0 does the manufacturer
extract the vertically integrated monopoly pro…t.
In the case of no returns (N R) we have r = 0 and the manufacturer’s (expected) pro…t is
given by the wholesale margin multiplied by the expected stocks to be sold to both retailers since
all sales are …nal, i.e.,
N R (w; c)
= 2(w
c)
Z
p
f (p)D(p)dp,
(2)
w
In this case retail competition is less intense (as the prices exceed the wholesale price) but there
is no cost associated with excess stocks. We …nd that:
Proposition 2. In the unique regular equilibrium the manufacturer pro…t is higher with full
returns than with no returns if the manufacturing marginal cost c is lower than some threshold
value c > 0 and the opposite holds if c > c
c. Moreover the optimal wholesale price with no
returns is lower than with full returns.
A prediction of this model is that if a manufacturer switches from a policy without returns to a
policy with full returns then the wholesale price should increase. Moreover a manufacturer should
only do this once c becomes su¢ ciently low. So this predicts a positive relationship between such
transitions with simultaneous reductions in the manufacturing cost and increases in the wholesale
price.
To illustrate these results, suppose that demand is described by a general linear form D(p) =
a
bp (in this case the regular equilibrium is also the unique equilibrium). Then a is the measure
of market size, p = a=b is the choke price, and the monopoly price lies at half the distance between
c and p, i.e. pm (c) = (p + c)=2. In the case of no returns the pro…t in (2) simpli…es to
N R (w; c)
and wN R (c)
if c <
p
2
p + (1
and p if c
p
2.
)c with
= 2 (w
c) p
w(1
(ln
w
) ,
p
= 0:4=1:4. In the case of full returns wF R = pm (2c) is
p
2
+c
Therefore wF R > wN R since the former is slightly below a third of the
distance between the marginal cost and the choke price.
The comparison between the equilibrium pro…ts when demand is linear is illustrated in …g. 2
below, where the horizontal axis represents c as a share of the choke price, i.e., k = c=p, and the
vertical axis represents the percentage of the monopoly pro…t captured by the manufacturer.
11
Profit
100
80
60
40
20
0
0.0
Fig 2.
FR
0.1
0.2
0.3
(grey) and
NR
0.4
0.5
0.6
0.7
0.8
0.9
1.0
k
(black) as a percentage of
m.
Note that a change in policy from full to no-returns (and vice-versa) can have a signi…cant
e¤ect on pro…t when k is either high or low. Also, reductions in the marginal cost c and increases
in the consumers’willingness to pay— the latter captured by an increase in p that reduces k— are
likely to make full returns more pro…table than no returns.
The dashed line in …gure 2 represents the manufacturer’s expected pro…t when she uses the
optimal wholesale and return prices. Naturally this pro…t is larger than both the pro…t with full
and no returns. We derive it in the next subsection.
4.2
The optimal wholesale and return prices
Finding the optimal return level requires solving a trade-o¤ between the bene…t of more intense
retail competition and the production cost of excess stocks associated to it.
Given the retailers’optimal responses, each retailer orders enough to serve the market at the
price he chooses, and the equilibrium price paid by consumers is therefore the lowest of the two
retail prices, denoted by pl = min fpi ; pj g. The manufacturer makes (w
is purchased by consumers and gets (w
c
r) on each unit that
r) on each unit that is produced but ends being
returned. So for any pl the quantity expected to be returned is
Z
p
pl
f (p0 )
D(p0 )dp0 .
1 F (pl )
The probability the lowest of the two prices is pl is the …rst (or smallest) order statistic of
a sample of size 2 from the continuos probability distribution F , which is exactly 2f (pl )(1
F (pl )). Therefore the manufacturer’s (expected) pro…t with unobservable stocks (U N ) for a
12
general distribution policy (w; r) is
Z p
=
2f (pl )(1 F (pl )) (w
UN
c)D(pl ) + (w
c
r)
w
Z
p
pl
f (p0 )
D(p0 )dp0 dpl ,
1 F (pl )
For tractability we consider the situation where demand is linear.
Proposition 3. If demand is linear, the optimal w and r with unobservable stocks are
w (c) =
p+c
p
= pm (c) and r (c) = = pm (0).
2
2
From this result it follows that the returned fraction of the wholesale price is8
p
r (c)
,
=
p+c
w (c)
and since the optimal wholesale price is the integrated monopoly price we have
(w
c) = (p
c)=2 and
(w
c
r )=
c=2
and therefore the manufacturer’s (expected) pro…t simpli…es to
UN
=
p
c
2
ha
|2
k)2
(1
{z
i
}
Expect. quantity sold
c ha
(1 + 2k ln k
2| 2
{z
i
c
k 2 ) where k = .
p
}
(3)
Expect. quantity returned
When the manufacturer’s marginal cost is low the optimal distribution policy involves large
returns and therefore a large fraction of the total production is also returned. As the marginal
cost increases, the manufacturer …nds it optimal to reduce the percentage of the wholesale price
that is returned to reduce the unsold fraction of the total quantity produced— note from (3) that
the ratio of the quantity returned to the quantity sold decreases as k increases, converging to zero
as c converges to the choke price, i.e. as k ! 1.
5
Observable versus unobservable stocks
Comparing the situation where stocks are observed before price competition to the situation where
they are not will help us understand the incentives of a manufacturer to promote or hinder the
exchange of this information.
Wang (2004) showed that if retailers observe each others’stocks before choosing prices then
the market outcome is independent of returns and coincides with the Cournot outcome with
8
The ratio decreases with c, from 100% to 50%. As discussed in the conclusion, in practice these percentage
should be lower due to additional retailing costs.
13
wholesale price w. When demand is linear the optimal wholesale price is pm (c), and thus the
optimal wholesale price is the same whether stocks are observed or not. Moreover, when stocks
are observable and demand is linear, each retailer purchases one third of the monopoly quantity.
With public information on stock there will be no excess stocking but the manufacturer cannot
squeeze the retail margins since retailers use stocks as a quantity precommitment to soften price
competition and protect their margins. The manufacturer’s equilibrium pro…t with observable
stocks (OB) becomes
OB
=
2p ca
(1
3 2 3
k) =
2
3
m
(c).
(4)
As we have seen, unobservable stocks allow the manufacturer to squeeze retail margins at
the expense of overstocking. If the marginal cost is low the manufacturer uses a generous return
policy to intensify retail competition without su¤ering much from the cost of producing excess
stocks. Naturally the net e¤ect of stock information on the manufacturer’s pro…t depends on the
cost of producing the excess stocks. Comparing (4) with (3)(…g. 3 illustrates this e¤ect):
Proposition 4. When demand is linear and the manufacturer uses the optimal wholesale and
return prices, there is a critical value of the normalized cost e
k such that for k < e
k the manu-
facturer’s pro…t when stocks are not observable is larger than when they are observable, and the
opposite holds for k > e
k (where k = c=p).
a
0.25
0.20
0.15
0.10
0.05
0.00
0.0
Fig 3.
UN
0.1
0.2
0.3
0.4
(black line) and
0.5
OB
0.6
0.7
0.8
0.9
k
(grey line) as a fraction of a.
Suppose now that the manufacturer must commit to make stocks observable or not before
learning his marginal cost, as for example costs change from period to period. Can we still tell
14
which information regime is preferred? Again we can answer this question in the speci…c case
where demand is linear. Suppose c is a random variable drawn from a continuous probability
distribution with density g and full support in [0; b
c]. It can be shown that if g is uniformly
distributed and b
c < p the manufacturer is better o¤ if stocks are not observable, and is indi¤erent
if b
c
p (this result if of course dependent on a linear demand). More generally, it can be further
shown that if g has full support in [0; p] then the manufacturer is better o¤ with observable stocks
if g is monotone increasing, she is better o¤ with unobservable stocks if g is monotone decreasing
(and thus indi¤erent between the two regimes if c is uniformly distributed, i.e. g is constant).
This last observation reinforces Proposition 4, showing that it extends to situations with
uncertain marginal costs: if lower marginal costs are relatively more likely than higher marginal
costs then the manufacturer is better o¤ if stocks are unobservable — and the opposite result holds
when high manufacturing marginal costs are relatively more likely.
5.1
An extension to retail di¤erentiation
The model we have studied so far has the advantage of being directly comparable with Kreps
and Scheinkman’s (1983), an important workhorse model in marketing, economics and operations
research. Yet Both PP (1997) and Wang (2004) considered a model where retailers are di¤erentiated. While we have been unable to provide a closed form solution to the mixed strategy
equilibria in their speci…c model with di¤erentiated retailers, we were able to con…rm that several
main …ndings extend to that particular setting: when stocks are unobservable then i) with full
returns there exist equilibria with Bertrand prices but necessarily excess stocks, ii) retail competition is changed by the introduction of a full return policy, and iii) a manufacturer is better o¤
than when stocks are observable if the manufacturing cost is su¢ ciently low.
Consider a model which is similar in all aspects to the one studied above with the exception
that the demand directed to each retailer i is like in PP (1997) and Wang (2004) (in that case
their and our models are similar, except for the information on stocks). They considered the
demand system, derived in Dixit (1979) from a representative consumer, which gives rise to
inverse demands given by pi = a
qi
This leads to the direct demands qi =
and
= d=(1
dq2 , with d 2 (0; 1) and substitutability increasing in d.
pi + p2 , with
= a(1
d)=(1
d2 ),
= 1=(1
d2 )
d2 ). To ensure that strategies spaces are compact, we have that retailers cannot
order stocks larger than some value S and prices larger than some value P (with both chosen to
be su¢ ciently large). We …nd:
15
Proposition 5. For any r < w the retail stage game with product di¤ erentiation only has
equilibria in (non-degenerate) mixed strategies. For r = w there exist pure-strategy equilibria with
prices equal to, and stocks necessarily larger than, in the retail Bertrand outcome with a wholesale
price w.
From this result we conclude that when stocks are not observed before retailers choose their
prices the irrelevance result in Wang (2004) no longer holds: First, the retail outcome is never
Cournot. Second, o¤ering full-returns can change the equilibrium outcome.
Unfortunately the method used in section 3 to characterize the mixed strategies equilibrium
does not apply in this model with di¤erentiation. There we used the fact that we could collapse the joint randomization of prices and quantities to a randomization over a single variable
(interchangeably price or quantity). Here this cannot be done. To characterize the equilibria
we would need to …nd not only a distribution of prices but in addition a distribution of stocks
for each price in the support of that distribution— which we have not been able to do. Yet, by
revealed preferences and continuity, we obtain the analogous to Proposition 4 in the context of
retail di¤erentiation.
Corollary 1. In the game with product di¤ erentiation, if with unobservable stocks when
retailers are o¤ ered full returns they play an equilibrium with Bertrand prices and su¢ ciently
large stocks (which are invariant in w), then for c su¢ ciently low the manufacturer’s pro…t when
stocks are not observable must be larger than when they are observable.
6
Conclusion
In practice retailers are unlikely to perfectly observe the stocks of their rivals before choosing
their prices. The two main messages of this paper are that in this situation i) retail competition
will be a¤ected by the extent to which inventory costs remain variable and ii) a manufacturer
can pro…tably use policies that change the fraction of the retailers’cost that is variable to induce
retailers to compete more aggressively in prices, thus squeezing the retail margins. In our model
some level of returns always increases manufacturer pro…tability if retailers do not observe the
stocks of their rivals, while this would have no e¤ect if instead stocks were observable.
We have assumed that there were no retailing costs in addition to the wholesale price. We
could introduce two types of retail costs: a cost of selling the product (e.g. sales and post-sales
service) and a cost of stocking the product. With the former type of cost the analysis and intuition
16
remain unchanged, since it can be thought as a inward shift of the demand curve— changing only
the critical values. On the other hand a cost of holding stock would make the manufacturer choose
to o¤er less generous return policies since using returns to intensify retail competition becomes
then a less e¤ective tool.9
For the manufacturer the returned stock may also be a source of additional revenue (e.g.
manufacturer scrap value or recycling) or cost (e.g. transportation and disposal). We have
omitted these from the present version, but it is simple to introduce those elements. If the scrap
value of excess stock is a fraction of the manufacturing cost then the manufacturer will …nd it
pro…table to o¤er a more generous return policy to intensify retail competition. The opposite
holds in the case where returned stocks create additional costs.
The latter can motivate a manufacturer to develop secondary uses for her product, such as
for example export returned apparel to developing countries. Even if those products were sold
below their original production cost, and therefore those alternative markets are in isolation
unpro…table, the additional revenue they generate makes a return policy less onerous. This allows
the manufacturer to o¤er more generous return policies in the main markets to intensify retail
competition, which in turn increases her overall pro…tability.
9
Note that the wholesale price is then only a fraction of a retailer’s cost, which means that even a full return
policy cannot completely eliminate the retailer’s cost of holding excessive stocks.
17
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Proofs
Proof of Proposition 1. In step 1 we show that there cannot exist a pure-strategy equilibrium.
In step 2 we show that there cannot either exist a mixed strategy equilibrium where the expected
pro…t of both retailers is strictly positive. In step 3 we show that in any equilibrium the expected
pro…t of both retailers must be zero. In step 4 we show that the proposed strategies form the
unique regular equilibrium. We perform the comparative statics in step 5.
For a given distribution policy (w; r), the Nash equilibrium of the retail subgame is a pair
(s1 ; p1 ) and (s2 ; p2 ) such that for each retailer i we have that (si ; pi ) maximizes
i
given (sj ; pj )
(for simplicity we have dropped the dependence of (si ; pi ) on w and r). A mixed strategy of
retailer i is a c.d.f. Fi of prices and a c.d.f. Gpi of stocks when i’s price is p. Given the strategy
of retailer j, and since each retailer never orders more than D(p) when setting p, the expected
19
pro…t of retailer i if he chooses a pair (si ; pi ) is
e
i (si ; pi )
=
Z
0
p Z D(p)
0
p
i (pi ; si ; pj ; sj )dGj ( sj j pj )dFj (pj ).
If i uses a mixed strategy, then his expected pro…t must be equal for all pairs (si ; pi ) that are in
the support of his strategy.
Step 1. By contradiction there is no pure-strategy equilibrium. (The formal proof is not
essential for the remainder of the argument, therefore we here only provided the intuition, but
everything can be stated formally.) If both retailers charge the same price and make positive
pro…ts, then one retailer would …nd it pro…table to deviate by slightly undercutting his rival and
order a stock equal to the market demand at that price. Also, there cannot be an equilibrium
where both retailers earn zero pro…ts and charge the same price because, with the assumption
that si > 0 if D(pi ) > 0, one retailer could deviate to a slightly higher price and make a positive
pro…t (since in any equilibrium each retailer must order less than D(pi )). It can be shown in a
similar way that pure-strategy equilibria with di¤erent prices also cannot exist because a similar
pro…table deviation always exists for one retailer.
Step 2. We now show that there cannot exist a mixed strategy equilibrium where the expected
e > 0. De…ne p as the
j
i
o
n
lowest price that retailer i uses with a positive probability, and p = min pi ; pj . If both pro…ts
pro…t of both retailers is strictly positive, i.e. where both
e
i
> 0 and
were positive then p > w. We …rst must show that p = pi = pj and then that both retailers
choose to serve the whole market at the bottom of their price distribution, i.e. at p both order
D(p).
Both retailers cannot have a mass point at p. Suppose they do. If both pro…ts are positive,
and (p
p
w) > 0, both retailers should order at least D( 2 ) when charging p (since the pro…t is
p
strictly increasing in the stock at least up to D( 2 ), as they can be sure to sell any lower amount
for p independently of the stock choice of the other retailer). But then j could increase his pro…t
by deviating from p to p
" (with " arbitrarily small) and serve the whole demand at that price
for sure (instead of sharing it with i with a strictly positive probability), which means retailer j
would have no mass at p.
We also have that pm (w) > p. Suppose that pm (w) < p. In equilibrium the sum of the pro…t
of both retailers cannot exceed the monopoly pro…t, so one retailer could pro…tably deviate to
pm (w) and D(pm (w)) to make monopoly pro…t for sure. Moreover if pm (w) = p and j does not
have a mass point at p (since both cannot have one), then the probability i ties with j at p is zero.
But then it would be optimal for retailer i to have Fi (p) = 1 with si (p) = D(pm (w)) and obtain
20
the monopoly pro…t. However this leaves retailer j with no pro…ts (and therefore a contradiction).
It follows then that pi = pj . Suppose not, that pi < pj while pm (w) > p. Then i increases
n
o
his pro…t by raising its price from pi to any price below min pj ; pm (w) and sell to the whole
market at that price. However we have seen that pi = p < pm (w), thus it must be that pi = pj .
Neither retailer has a mass point at p. If there is then the one with a mass point (say i, as
both cannot have a mass point) would face a zero probability of a tie at p while p
w > 0 and
thus at that price his pro…t increases up to D(p). But if i chooses D(p) at p, then j can make a
higher pro…t if instead of charging p he charges p
" and orders D(p
") and avoids the strictly
positive probability of a tie with i (as in case of a tie he would not sell more than D(p)=2, while
it sells D(p) for sure for any price p < p).
Since neither retailer has a mass point at p (and therefore the probability of a tie at p is zero),
the pro…t of a retailer i at p is (p
since p
w)si (p), which is strictly increasing in the stock up to D(p)
w > 0. Thus at p both retailers order D(p).
Further, suppose i is the retailer that charges the highest price with a positive probability,
and therefore the highest price at which any of the two retailers orders a strictly positive stock
and still makes a strictly positive pro…t (if it is the same for both then i can be either). Call that
highest price pi . By the de…nition of pi we have pj
pi < p. Thus the probability i’s demand
at pi is D(pi ) must be zero if j makes a strictly positive pro…t, and therefore chooses a strictly
positive quantity for every price in his strategy). It follows that the stock chosen by i if he charges
pi must be less than D(pi ). Given that at p both retailers order D(p) and at pi we have that
si (pi ) < D(pi ), there must exist some p < p0
pi which is the minimum price such that at least
one of the two retailers orders with some positive probability an amount 0 < s < D(p0 ).
Both retailers cannot have a mass point at this price p0 . If they did then both must be
ordering at least D(p0 )=2 at this price. The reason is that a retailer only makes a sale when he
chooses p0 if he is the lowest priced retailer (as both order D(p) for lower prices), and therefore if
the expected pro…t at p0 is positive then it must be strictly increasing up to D(p0 )=2. Then one
retailer, say i, would …nd some price p0
" at which j does not have a mass (since the number
of points of positive mass in any probability distribution must be countable) and at which he is
able to sell D(p0
") > D(p0 ) with a probability that is strictly higher by at least the value of
the mass point of j at p0 but for a price that is only marginally lower than p0 . This provides a
pro…table deviation for i from p0 and we obtained a contradiction. For a similar reason we cannot
have one retailer, say player i, having a mass point at p0 while j does not and yet fj (p0 ) > 0 as
j would have a similar pro…table deviation.
21
By the de…nition of p0 we must have that p0 is in the support of strategies of at least one
retailer. We are therefore left with two cases: either p0 is in the support of the strategy of only
one retailer, say i but not j, or it is in the support of both but neither has a mass point at p0 . In
both cases the pro…t of retailer i if he chooses p0 is then
(1
Fj (p0 ))(p0
w)
Fj (p0 )(w
r) si (p0 ) for all si (p0 )
since by the de…nition of p0 retailer j chooses a stock D(p) for any price p
D(p0 )
p < p0 and j has no
mass point at p0 (and therefore the probability of a tie with j at p0 is zero). If the pro…t of i is
positive then his pro…t is increasing in si (p0 ) up to D(p0 ) and i should therefore choose D(p0 ).
A similar argument holds for j when p0 is in his strategy support but neither has a mass point
at p0 . This implies that choosing 0 < s < D(p0 ) when choosing p0 is not part of the strategy of
either retailer, which contradicts the de…nition of p0 . This concludes the proof that an equilibrium
where the pro…t of both retailers is strictly positive cannot exist.
Step 3. To show that both retailers must earn zero pro…ts we need in addition to show that
we cannot have an equilibrium where retailer i makes a strictly positive pro…t and retailer j has
zero pro…t. This holds because retailer i must then earn a positive pro…t also when he charges pi
(the lowest price in the support of his price distribution). For i’s pro…t to be positive it must be
that pi > w. But then retailer j would …nd a deviation that gives him a strictly positive pro…t by
undercutting pi and order stock equal to the whole demand at that price. This contradicts the
statement that i can make a positive pro…t if j is making zero pro…t. Together with step 2, this
shows that in any equilibrium it must be the case that the pro…t of both retailers is zero. This
implies that pi = pj = p = w as otherwise one retailer could charge a price below the support of
his rival and make a strictly positive pro…t.
Step 4. We have now shown that in any equilibrium the pro…ts of both retailers are zero and
that the lowest price in the support is w. We now can make use of the proof in the Appendix
of Levitan and Shubik (1978) (starting at middle of their page 10, by "Hence
i
= 0 and pl = 0
is the remaining case", corresponding to our zero pro…t and p = w) to …nd that the proposed
strategies form the unique Nash equilibrium of the game for strategies which have no gaps nor
mass points inside the support of prices. A important di¤erence is that they consider a model
where …rms have no cost of production but only a cost for unsold inventory. A second important
di¤erence is that we consider a general demand function D(p), while they consider the linear case.
In their linear model they have shown that equilibrium strategies must have no gaps or mass
22
points, from which follows uniqueness. Here we have shown for a general demand that retailers
must make zero pro…ts but must add the restriction to regular strategies to obtain uniqueness (as
there could remain additional mixed strategy equilibria with other types of strategies, but they
would also make zero pro…ts). The proof that in the current setting the proposed equilibrium
is the unique regular equilibrium then becomes isomorphic to Levitan and Shubik (1978), once
we notice and account for the following equivalences (and change pro…t functions accordingly):
(w r) is equivalent to their , a cost to the retailer of each unit that is ordered but remains unsold
w) is equivalent to their price p0 (as they
at the end of the game, the bene…t of a unit sold (p
assume retailers have no cost of stocks, as if w = 0). Therefore p0 +
= (p
w) + (w
r) = p
r.
Finally, after having shown the zero pro…t condition holds generally for D(p), we replacing a
p
with D(p).
Step 5. Since si (pi ) = D(pi ), the expected stock and retail margin of i for a given pair (w; r)
are respectively
Z
p
D(x)f (x)dx and
w
Note that
@F (pi )
@r
Z
p
(x
w)f (x)dx.
w
> 0 for all p 2 (w; p], and D(p) decreases with p while (p
w) increases in p.
Thus, for a given p, we have that the expected stock and retail margin respectively decrease and
increase with r. In the limit as r ! w we have the expected stock and retail margin respectively
converging to D(w) and 0.
Proof of Proposition 2. In step 1 we show the …rst part of the Proposition, in step 2 we explain
that wN R (c) is unique, and in step 3 we show that wN R (c)
Step 1.
F R (wF R ; c)
and
N R (wN R ; c)
wF R (c).
are continuous in c, and by revealed preferences both
are strictly decreasing in c. As
F R (wF R ; 0)
=
m
(0) >
N R (wN R ; 0),
and
F R (wF R ;
we have that there must exist 0 < c
p
)=0<
2
N R (wN R ;
p
),
2
c < p such that the manufacturer pro…t is higher with full
returns than with no returns if c < c, and the opposite holds if c > c.
Step 2. Notice that
N R (c; c)
=
N R (p; c)
NR
is di¤erentiable and strictly positive for all w 2 (c; p) and that
= 0. We can therefore use …rst-order conditions to determine wN R (c).
Moreover wN R (c) is unique since
third derivative of
N R (w; c)
N R (w; c)
has a single in‡ection point— using the second and
, we …nd that the third derivative with respect to w is positive for
23
every w 2 [c; p] and that the second derivative with respect to w is negative for w close to c and
positive for w close to p.
Step 3. With the FOC we …nd that wN R (c) must satisfy
Z p
f (p)D(p) (wN R c)D(wN R ) = 0.
(2wN R c)
wN R
So wN R is the single w where the revenue made with the stock expected to be sold to both retailers
minus the cost of producing the expected stock sold to one retailer is equal to the pro…t made by
selling directly to consumers at wN R .
Suppose that wN R > wF R . Then we should have that at wF R the FOC of
positive, i.e.,
c)
(2wF R
Z
is
p
f (p)D(p)
c)D(wF R ) > 0,
(wF R
wF R
in which case its second derivative must be negative, i.e.,
Z p
2
f (p)D(p) 2D(wF R ) (wF R
c)D0 (wF R ) < 0,
wF R
and …nally that the FOC of
N R (w; c)
F R (w; c)
is zero, i.e.,
2c)D0 (wF R ) = 0.
D(wF R ) + (wF R
From the second and third of these conditions we get that
Z p
f (p)D(p).
D0 (wF R )(wF R 3c) > 2
wF R
Replacing in the …rst condition we get that
Z p
2(2wF R c)
f (p)D(p) > 2(wF R
wF R
D0 (wF R )(wF R
3c)(2wF R
c) >
(wF R
3c)(2wF R
c) > 2(wF R
2(wF R
c)D(wF R ) )
2c)D0 (wF R ) ,
c)(wF R
c)(wF R
2c) ,
c (c + wF R ) > 0,
which is an impossibility. Thus we get a contradiction, and therefore it must be that wN R
wF R .
Proof of Proposition 3. In step 1 we determine the expected pro…t. In step 2 we take the
FOC with respect to r and w. In step 3 we discuss second order and corner conditions.
Step 1. As explained in the text, the manufacturer’s (expected) pro…t for a general distribution
policy (w; r) is
Z
UN =
p
2f (pl )(1
F (pl )) (w
c)D(pl ) + (w
w
c
r)
Z
p
pl
24
f (p0 )
D(p0 )dp0 dpl
1 F (pl )
Suppose D(p) = a
f (p) =
w r
(p r)2
and 1
bp and the choke price is therefore p = a=b. From Lemma 1, F (p) =
F (p) =
Z
w r
p r.
p w
p r,
Replacing, we have that
f (p0 )D(p0 )dp0 =
(w
r)b(
p
p0
r
+ ln p0
r
r )
and thus
Z
p
pl
f (p0 )
D(p0 )dp0 =
1 F (pl )
pl
w
r
)(w
r
r)b (1 + ln(p
r))
(pl
r)b(1
p
pl
r
)).
r
(
p r
pl r
Using below the fact that y =
Z
=
2f (pl )(1
=
In addition, setting z =
2b
r)2
(w
(pl
(w
2b
p
b(p
w r
p r
F (pl ))
Z
p
pl
w
=
p r
(pl r)2
=
p
w
p
Z
dy
dpl
so
r
p
+ ln(
r
pl
r)2
r)2
r
w
r)((
p
(
Z
p
pl
r
r
or dy =
(
p
pl
r
+ ln(pl
r
p r
dpl ,
(pl r)2
r)) =
we will have that
f (p0 )
D(p0 )dp0 dpl
1 F (pl )
1
ln(
p
pl
r
))dpl
r
p=p
(y
1
ln y)dy
1
w
2(
p
pl =w
r 2
)
r
r
w
) ln(
r
p
r
))
r
and since p = a=b, we also have that
Z
p
2f (pl )(1 F (pl ))(w
Z p
(w r)2
= (w c)
(a
2
(pl r)3
w
c)D(pl )dpl
w
= (w
= b(1
(p w)2
(p r)
2z + z 2 )(p r)((p
bpl )dpl
c)b
r)z
(c
r))
So we …nally have
UN
= b(1
2z + z 2 )(p
= b(p
r)(2(1
r)((p
r)z
z + z ln z)(z(p
(c
r)
r))
(w
c) + r(1
c
r)b(p
r)(z 2
1
2z ln z)
z)2 )
Step 2. To simplify further but without loss of generality, we normalize the problem by setting
p = 1, so k = c and
UN
= b(1
r)(2(1
z + z ln z)(z(1
25
r)
k) + r(1
z)2 )
We take FOC with respect to r and solve it to …nd
r(z; k) =
8z
1
2k
4z 2 ln z + 6z 2 + 2
6z
4z 2 ln z
2kz + 5z 2 + 2kz ln z + 1
Given the length of the expressions, we only report here the remainder of the reasoning leading to
the result: Take the FOC with respect to z and replace r(z; k) in that FOC. Solve the expression
and …nd that the only solution that lies in the admissible interval z 2 (0; 1) is z = k. Replace
back in r(z; k) and simplify to …nd r = 1=2. Since we previously normalized the units with p = 1
and z =
w r
p r,
p
we multiply the previous solution by p to conclude that the pair (w ; r ) = ( p+c
2 ; 2)
is the solution of the …rst-order approach.
Step 3. We now show that the solution above is the global maximum of
UN .
(Again, given the
length of the expressions, we have not included the expressions and only report here the procedure
and result.) To do this we …rst take the SOC and compute the Hessian matrix (in the normalized
case). Evaluate the Hessian matrix at z = k and r = 1=2 to …nd that it is negative-de…nite and
therefore this point is a local maximum. Since this is the only interior solution that satis…es FOC,
to check that it is a global maximum we only need to look at the corner solutions. Solutions along
the plane boundary where z = 1 and r 2 (0; 1) generate zero pro…t and are therefore dominated.
Solutions along that boundary where r = w correspond to z = 0 and therefore to the case of full
returns, which is also dominated (see Fig. 1)— only equal if k = 0. Finally those solutions on
the boundary where z 2 (0; 1) and r = 0 correspond to the case of no returns also studied above,
and are also dominated (see Fig. 1). Therefore the solution z = k and r = 1=2 for a global
p
maximum of the normalized problem, and therefore the pair (w ; r ) = ( p+c
2 ; 2 ) is indeed a global
maximum of
UN .
Proof of Proposition 4. Follows directly from comparing (4) with (3). We solve the inequalities
to …nd that there exists a e
k ' 0:228 47 such that
8
>
>
> U N if 1 > k > e
k
>
<
= U N if k = e
k
OB
>
>
>
: <
e
U N if k < k
Proof of Proposition 5. Given the stated assumptions, the retailers’strategies sets are compact and, unlike the non-di¤erentiation case, retailers’ payo¤s are continuous. The existence of
a (mixed-strategy) equilibrium is therefore insured by the theorem of Glicksberg (1952), which
extends Nash equilibrium existence theorem from a …nite set of strategies to a (non-empty) com26
pact set. In step 1 we show that when r < w there are no pure-strategy equilibria, while in step
2 we show that when r = w there exist pure-strategy equilibria with Bertrand prices.
Step 1. In the case where r < w there are no pure-strategy equilibria. To see this suppose
that j uses a candidate equilibrium pure strategy (sj ; pj ). In the relevant ranges where quantity
is non-negative, the demand direct at i, given the proposed strategy of j, is
8
sj
+ pj
<
pi + pj if pi
qi (sj ; pj )
:
a pi dsj otherwise
Thus i’s demand is determined by pj but not sj if pi is su¢ ciently low and by sj but not pj if
pi is su¢ ciently high (and the same applies to j). Thus if i charges a su¢ ciently low price and
chooses a su¢ ciently high stock then j will not sell all his stock, while if i charges a too high price
then j must be capacity constrained.
A necessary condition of a pure-strategy equilibrium is that for both retailers the demand at
the price each chooses to charge, given the price and quantity of the rival, is exactly their chosen
equilibrium stock pair— otherwise at those prices there would be a pro…table deviation of ordering
either less (if part of the stock is unsold) or more (if capacity is binding). A necessary condition
to have this satis…ed for j is that i chooses to set his price at exactly pi =
chooses a stock si =
sj
+ pj
, and thus
pi + pj (and likewise for i). Solving the system of these necessary
conditions we get (and likewise for j)
pi =
1
2
2
+
si
sj = a
si
dsj .
(5)
Replacing in the pro…t functions, and evaluating the derivative of i’s pro…t at these same pairs
(si ; pi ) and (sj ; pj ), we have that right-hand derivative of the pro…t with respect to the price (as
the stock is reduced optimally to avoid unsold stock) is
a
2pi
dsj + w = w
a + 2si + dsj
while the left-hand derivative of the pro…t is (as the stock is optimally increased to serve all
demand)
2 pi + pj + w =
w
a + 2si + dsj
(1 d2 )
d2 si
To avoid a pro…table deviation it must be that the right-hand derivative is non-positive and
the left-hand is non-negative. However this condition cannot be satis…ed, since if the former is
non-positive then the latter must be strictly negative. Thus i always has at least one pro…table
deviation from any candidate pure-strategy equilibrium.
27
Given that equilibria exist, we conclude that when r < w all equilibria must have at least
one player using a non-degenerate mixed-strategy. Unfortunately, as also explained in the text,
here we cannot collapse the joint randomization of prices and quantities to a randomization over
a single variable, like we did in the absence of di¤erentiation. To characterize the equilibria we
would in addition need to …nd not only a distribution of prices but also a distribution of stocks
for each price in the support of that distribution— which we have not been able to do.
Step 2. Consider now the case where r = w. In that case the necessary condition for a purestrategy equilibrium we identi…ed above does not apply: it is no longer the case that for both
retailers the demand at the price each chooses to charge, given the price and quantity of the rival,
is exactly the stock chosen since there is no downside to order a stock which is strictly larger
than the quantity actually sold. We now show by construction that for a given wholesale price w
an equilibrium with the Bertrand price pB (w) and stocks that exceed the Bertrand level sB (w)
always exists. The Bertrand price when retailers face a wholesale price equal to w is
pB (w) =
+ w
2
.
Suppose that each retailer i sets pi = pB (w) and orders a stock si such that, even if j would
charge the choke price of his demand when i is charging pB (w), i would always have enough stock
to satisfy all his demand. Speci…cally, with the choke price of retailer j being
0=
pj + pB (w) , pj =
+ pB (w)
the stock of i would need to be
si = (1
)( + pB (w)).
This is a safe strategy for i since, conditional on pi = pB (w), we have that i ensures he does not
face a stockout regardless of the price and stock choice of retailer j. It is checked directly that
)( + pB (w)); pB (w)) for both retailers, form a
these stock and prices pairs, (s ; p ) = ((1
Nash equilibrium of the retail (sub)game, and that the same applies for any pair of quantities
such that si
(1
)( + pB (w)) for each retailer i. Therefore in the case of full returns there
is a continuum of equilibria with Bertrand prices, provided both stocks remain su¢ ciently large.
Finally, for a given w the Bertrand quantity of each retailer is
sB (w) =
(
w(
2
))
.
Can an equilibrium where both retailers set their prices at pB (w) and one retailer chooses a stock
equal (or smaller than) sB (w) be sustained? The answer is no. Having a stocks smaller than
28
sB (w) is not possible since there would always be a pro…table deviation to sB (w). Retailer j
having a stock equal to sB (w) is also not possible since it would then be pro…table for i to slightly
increase the price above pB (w). The reason is that in the case where j is not constrained by his
stock (as in the standard Bertrand formulation) then the derivative of i’s pro…t would be exactly
zero (at pi = pj = pB (w)). But then if i increases his price and j has no more than sB (w) it must
be that j becomes capacity constrained and, as there can be no diversion of purchases from i to
j due to that constraint, it must be that i increases his pro…t by raising his price above pB (w)
and choosing his stock accordingly.
Proof of Corollary 1. Suppose with unobservable stocks the retailers choose an equilibrium
with Bertrand prices when o¤ered full returns, where si > sB (w). Then by revealed preferences
the expected pro…t of the manufacturer cannot be less than the full return pro…t, which is
c)sB (w)
2(w
c
X
sB (w)),
(si
i
as she makes w
c on each unit sold,
c on each unit not sold and si is invariant with w. If
retailers remain unconstrained by their stocks (as si is large and invariant in w), the derivation
of the optimal wholesale price, obtained from the FOC, gives
w =
2(
)
For c = 0, the obtained maximized pro…t, call it
UN ,
.
coincides with those of the manufacturer
under standard Bertrand competition , as obtained in PP (1997) and denoted by
lower than
PP
for c > 0. Both
PP (1997) reported that
PP
and
UN
>
W
PP
PP ,
and is
are di¤erentiable and decreasing in c.
for all c, where
W
is the equilibrium pro…t under observ-
ability (i.e., when retailers play Cournot following the optimal choice of w by the manufacturer,
where W stands for Wang, 2004, due to his irrelevance result). It follows by the continuity and
diferentiability of
UN ,
PP
and
W
that
UN
>
29
W
for c su¢ ciently low.