The Role of Returns Policies on Manufacturers’ Competition
Thierno Diallo
Département des sciences économiques et administratives,
Université du Québec à Chicoutimi (UQAC)
January 2008
(Preliminary draft. Please, do not quote)
Abstract
We investigate an oligopoly to determine the strategic effects of manufacturers’ returns
policies on market competition. Each competing manufacturer produces and distributes a
product through retailers in competition. The demand function is subject to random
shocks. Manufacturers observe these shocks only after choosing simultaneously
wholesale prices and returns policies. For each manufacturer, we derive the optimal
wholesale and buy-back prices. In a comparative static analysis, we determine how prices
(wholesale, retail, and buy-back), inventories, and welfare change with respect to returns
policies. We show that returns policies decrease manufacturers profitability by increasing
price competition.
1
1.
Introduction
Returns policies are commitment by sellers to take back excess inventories or
unsatisfactory products from buyers. The format of returns policies varies across
industries and across stages of the distribution channel. In upstream distribution channel
the returns option is offered by the manufacturer (manufacturer’s returns policy), while in
downstream distribution channel the returns option is offered by the retailer (retailer’s
returns policy). Under returns policies, unsold goods are accepted for full or partial
refunds of their purchase price. Most generous policies promise to refund all returned
products with full wholesale price, while less generous policies offer credits against
future orders.
What explains the returns policies adopted by the retailers is the fact that costumers are
often uncertain about their valuation for a product at the time of purchase. Therefore to
induce them to purchase, most retailers guarantee satisfaction of their products and offer
an option to return in case consumers are not satisfied1.
Manufacturers allow returns by retailers because they face a problem of inducing retailers
to optimally stock their products when these are subject to unpredictable demand. E.g. if
products have one of the following characteristics: uncertain demand, finite selling
season or finite lifetime, retail overstock is possible. Thus retailers may stock less of
products than manufacturers would like. Therefore to coordinate the distribution channel,
the manufacturers can offer to retailers a return option for their unsold products. These
characteristics are present in a wide category of product market. The markets where
returns policies are most prevalent are markets for books, newspapers, CDs, perishable
goods, computer hardware and software, and pharmaceuticals…………………………….
1
There are many works on that subject. The important contributions are : Che (1996), who argues that the
rationale of retailer’ returns policies is to help consumers to learn about their uncertain utility levels and he
showed that a return policy cannot be optimal unless either consumers are sufficiently risk-averse or retail
costs are sufficiently high, with the effect on welfare being ambiguous. Mixon (1999) provides an
extension of the previous theoretical work of Che (1996) by way of empirical test regarding seller-provided
information on return policies in advertising for search and experience goods. Davis et al. (1998) construct
a model to identify potential causes for variation among retailers’ return policies.
2
In this literature review, we look more specifically at the role of manufacturer’s returns
policy. What does literature say about reasons for manufacturer’s returns policy? The
details follow bellow.
2.
Manufacturer’s Returns Policy
In the literature of manufacturer returns policies two approaches have been used to
model a distribution channel facing uncertain demand. The first approach use the
classical “newsboy” or “newsvendor” problem2 under uncertain demand and derive the
retailer’s optimal order product and the retail price before the resolution of the demand
and salvages any overstock at some value less than the procurement cost. The second
approach is the one of Padmanabhan and Png (1997), which uses a linear demand with
random shift parameter. Here the demand is observed before the retailer sets the
equilibrium price. We will review this latter methodology later in this paper.
There are many studies into the reasons why and when manufacturers should offer to
accept unsold products from retailers instead of outright sales. The principal reasons
given are related to three broad categories of objectives. The first objective is related to
information issue, i.e. risk sharing, asymmetric and elicit information between
manufacturers and retailers. The second objective is the alignments of incentives between
manufacturers and retailers. These alignments of incentives are characterizing by an
inventories and non inventories issue, i.e. returns policy are offered by manufacturers to
ensure adequate inventories of their products in the inventories issue or to provide better
representation in the non inventories issue. Finally, the third objective is the use of
returns policies as a strategic device to induce retailers to compete more aggressively.
2
The newsboy or newsvendor problem, a stochastic inventory replenishment problem can be describe as
follow: given a known stochastic distribution for the demand of a product, what is the optimal order
quantity if only one order can be placed before actual demand is observed. This is a problem a newspaper
vendor faces: how many newspaper to buy before the actual number of buyers is known.
3
Information Issue
Risk sharing
The difference of attitude toward risk may exist between manufacturers and retailers. The
risk arises because it is necessary to make partially irreversible investment prior to
demand realization: production and distribution costs for the manufacturer, the inventory
carrying cost for the retailer. Depending on the risk aversion of each side, the return
policy will be offered fully, partially or not at all. When both sides are risk averse, the
optimal risk sharing contract must allocate risk to both sides. High risk tolerance of the
manufacturer induces him to bear higher proportion of risk under the return policy by
offering higher buyback price. The opposite is true for the risk tolerance of the retailerthe higher it is, the lower should be the buyback price set by the manufacturer.
Asymmetric information
Examples of potential asymmetric information between manufacturers and retailers can
be found in the market for newly introduced products, which are usually sold on
consignment. Indeed before introducing a new product, every manufacturer conducts
market tests to forecast sales. The forecasting should be optimistic for the manufacturer
before taking the risk of introducing the product. However, if the retailer is not integrated
to the manufacturer, he does not have access to the manufacturer’s data. Given the low
success rate of new products, we can say that the manufacturers are inevitably more
optimistic about their product than retailers. Therefore the returns policies can serve as a
warranty for retailers and induce them to order more of the manufacturers’ products. As
noted by Kandel (1996), This situation apply well to the industry of book publishing,
where every new title is a new product and where the understandable optimism of the
author and the editor is frequently not shared by bookstores.
4
Elicit Information
Another objective of returns policy is to better exploit the retailer’s private information
regarding the demand. This role is analyzed by Arya and Mittendorf (2002). They
showed that the manufacturer can use returns policy to elicit retailer information. They
study a distribution channel in which the retailer is privately informed about the retail
market conditions. To obtain more favorable price from the manufacturer, the retailer is
inclined to underreport his assessment of market condition. Anticipating this behavior,
the manufacturer responds by tailoring returns term to the retailer’s report. In particular,
by offering high return allowances (instead of low prices) for unfavorable market report,
the manufacturer dissuades the retailer from understating market conditions. Besides
altering retailer reporting, returns policy can induce retailer behavior that conveys
information to consumers. By limiting the retailer’s downside risk, generous returns
policy can encourage retailers to focus product promotion to the high valuable consumers
or dissuade them from offering product on clearance. [IS THIS STILL THE SAME
ARGUMENT] These activities may boost consumer perception on manufacture’s
product quality.
Alignement of Incentives
Non Inventories Related
The non inventories related are the incentives for product quality, service and promotion.
Indeed consumers demand can be affected by actions taken by either retailers or
manufacturers. The manufacturer can affect consumer demand by their choice of product
quality, retailers can achieve the same purpose by provision of service, and both parties
can promote the product let say by ads. These interventions are achieved optimally in the
vertically integrated firm. When the two levels of a distribution channel are separately
owned, the contract between them affects the incentives to increase the consumer
demand. [THIS ALSO APPLIES TO ISSUE OF INVENTORIES] There is a double
moral hazard problem: if the manufacturer supports all the risk, the retailer has less
5
incentive to provide better service or promotion. This is the case in the consignment
contract. If the retailer has no return options for unsold products, the manufacturer has
less incentive to provide high quality or to advertise the product. [WHY] Therefore the
benefit from trade is not fully obtained in both cases. The provision of partial return
options is clear: the two levels of the distribution channel must bear the risk for the
provision of high product quality, high service and good promotion.
Inventories Related
This is the most popular objective of returns policy. Here, manufacturers want retailers to
optimally stock their products or ensure adequate inventories of their products under
uncertain demands. However, Marvel and Peck (1995) showed that not all demand
uncertainty matters. Indeed they show that manufacturers’ decision to accept returns
depend crucially on the nature of the demand uncertainty. Uncertainty over the valuation
consumers will place on a manufacturer’s product leads the manufacturer to avoid returns
in favor of outright sales, while uncertainty over the number of customers who arrive at
the retailer favors returns systems. [EXPLAIN THE DIFFERENCE IN TYPES OF
UNCERTAINTY] In other words manufacturers offer returns policies if uncertainty
enters simply as a random arrival process for customers, where it is quantity uncertainty,
not uncertainty over valuation that matters. For quantity uncertainty the manufacturer is
able to extract the entire retailer’s surplus buy choosing a wholesale price equal to
consumer’s valuation and ensuring that the optimal inventory is held by paying for
unsold goods by accepting full returns. On the other hand, for valuation uncertainty, a
returns policy eliminates the cost to retailer to stock excessively. [EXPLAIN WHY
THERE IS A DIFFRENCE IN THE EFFECT OF RETURNS POLICIES]. Thus the
returns options provided by manufacturers can be exploited by the retailers to push retail
prices above the manufacturer’s desired level. In other words the returns policies distort
pricing without offsetting inventory effects, and are therefore not employed when there is
uncertainty only over the consumers’ valuation. For the general case where both
uncertainties are present, they show that the manufacturer must offer to reimburse
partially unsold products. The manufacturer must balance the effect of reducing the level
6
of reimbursement since that reduces the quantity ordered from the manufacturer’s desired
level.
Kandel (1996) and Marvel and Wang (2000) take account an additional factor which
affect the decision of the manufacturers to offer returns policies for inducing retailers to
optimally stock their products. Kandel (1996) showed that the difference in opportunities
for the disposal of unsold products, i.e. the different scrap values of unsold products is an
important factor, while Marvel and Wang (2000) demonstrated that returns policies can
be used to achieve optimal price dispersion, i.e. to provide better representation of the
product in all state of consumers demand.
Different scrap values
In the vertically integrated firm, the unsold product is placed at the disposal of the party
which obtains its highest scrap value. [EXPLAIN] Thus when the two levels of the
distribution channel are separately owned, the choice of the disposal of the overstock is
important for determining the nature of the contract between the manufacturer and the
retailer. Examples of products with similar characteristics of demand uncertainty yet
different in terms of their scrap values: milk, flowers and apparel. There is not much
supermarket can do with unsold milk. However, the manufacturer of milk can use it to
make ice cream or other dairy products. This explains the prevalence of returns policies
or consignment contracts in milk distribution. Collecting and reprocessing unsold flowers
is not economical; selling them at a discount at the same store offers a better scrap value.
Thus return options are not widely observe in flower distribution contract. Unsold
apparel, especially fashion items, can be discounted as well, rather than being sent back
to manufacturers, which involves lost time, damage, and high transportation costs. The
scrap value at the store is higher than at the manufacturer’s location indicating why return
options is not so much offer in these type industries.
7
Optimal price dispersion
Marvel and Wang (2002) proposed a model that shows that returns policies can be used
to achieve optimal price dispersion if the manufacturer does not know the distribution of
the demand uncertainty. They found that only retailers need to have knowledge about
demand uncertainty for returns policies to induce optimal inventory holding from the
manufacturer’s desired level without requiring the manufacturer to have the same level of
information. The result of imposing returns policy in this setting is price dispersion. This
explains why manufacturers prefer to control inventories through returns policies as
opposed to price maintenance which require more information on market conditions for
the manufacturer. [WHAT DOES THIS PAPER DO THAT PREVIOUS PAPER DID
NOT DO] These results are proved for a monopolist manufacturer and for two
differentiated but competing manufacturers selling to consumers through competitive
retailers. First, they show that if the marginal production cost is zero, the manufacturer’s
unique optimal strategy is to accept full returns and retailers choose a retail price equal to
the wholesale price. The optimality of full returns is easy to understand when marginal
production cost is zero. Since the product can be offered at no cost, the manufacturer
wants to induce the retailer to order an inventory as high as possible in order to satisfy all
orders. Also the manufacturer wants to charge consumers the optimal retail prices. To
achieve both objectives, a policy offering full credit for returns is necessary. Second, with
constant marginal cost of production and linear demand, the manufacturer splits the
production cost of unsold inventory equally with retailers. Consequently only partial
refund is offered. They also found the presence of the upstream competitor does influence
the manufacturer’s buy-back price. Precisely they found that the returns allowances are
less generous with the oligopoly than the monopoly.
Strategic
Pasternack (1985) examines how a manufacturer can use returns policies to induce
multiple retailers to carry the optimal level of stocks. He shows first that the two
extremes cases of returns policies which are full return for all unsold products and no
8
returns of unsold products are suboptimal. He proved that full credit, i.e. buy-back price
equal wholesale price for partial returns achieves channel coordination even if the
optimal buy-back price will be function of retailer demand. [WHY] This implies that in a
multi-retailers environment this policy is also suboptimal. His main result is to prove in a
single-period setting with the manufacturer as a channel leader, i.e. the returns policy is
decided by the manufacturer, that a designed policy allowing complete returns at price
less than the wholesale price, i.e. offering partial credit for all unsold products can
coordinate the channel to the benefit of both sides of the industry in a multi-retailers
environment. Donohue (2000) extends this latter result for a two period-model. Pellegrini
(1986) shows that by providing a returns policy, a manufacturer can encourage retailers
to carry larger stocks and thereby improve sales of its brand relative to competing
products.
Intensify competition at retail market
Padmanabhan and Png (1997) point out that insurance does not seem a plausible
explanation for returns schemes. Manufacturers who accept returns are often much
smaller and less diversified than the retailers who ship them unsold goods. They give the
following examples: in books, Ten Speed Press (revenues of $ 2 million) accepts returns
from Barnes and Noble (revenues of $ 1087 million); in record music, Windham Hill
(revenues of $25 million) accepts returns from Wherehouse Entertainments (revenues of
$457 million); in clothing apparel, Aris Isotoner (revenues of $6449 million) accepts
returns from Macy’s (revenues of $6163 million). They propose another explanation for
returns policies. They first consider the benchmark where there is a single retailer and a
single manufacturer and show that a returns policy makes no difference in that case. Then
they consider the case of a single manufacturer and two competing retailers. They show
that returns policies can be used strategically to increase the level of competition between
retailers and consequently the profits of the manufacturer. [EXPLAIN INITUTIVELY
WHY] They developed a framework that explains when and how a manufacturer should
adopt returns policies and analyzed the costs and benefits of returns policies. They found
that when retailing is competitive and there is no uncertainty in demand, a return policy
9
induces retailer to compete more intensively. Thus the provision of a return option
reduces retail prices without affecting the manufacturer’s prices, thereby enhancing its
profits. When however, demand is uncertain and retailing is a monopoly, they show that a
returns policy encourages the retailer to overstock and so decrease the manufacturer
profits. When both markets’ characteristics are present, i.e. competing retailers and
demand is uncertain, there is a trade-off for the manufacturer between the benefits
provided by more intense retail competition and the costs of excessive stocking due to the
return options.
However Wang (2004) show that if retailers face stock constraints, returns policies do not
change manufacturer profitability when demand is certain and retailing competitive. The
reason is that Padmanabhan and Png (1997) unreasonably assume that the retailers would
never face stock constraints, thus change the retail competition from a Cournot-like
competition to a Bertrand competition.
Recently Padmanabhan and Png (2004) show that when there is end-user demand
uncertainty and retailing competitive, returns policies do increase manufacturer
profitability if the marginal cost of production is sufficiently low and demand parameters
satisfy particular conditions by attenuating retailer price competition when demand is low
and intensifying competition when demand is high. Indeed by accepting returns, the buyback price will serve as a floor to the retail price when demand is low. Therefore in case
of low demand this price floor attenuates price competition and thus raises the retailer’s
profits. Manufacturer can consequently charge a higher wholesale price. But when
demand is high, the returns policy induces more intense price competition because by
providing a returns policy the manufacturer eliminate the retailers’ cost of excess
inventory and through its impact on retail competition, the returns policy encourages
retailers to order larger stocks. [IT IS IMPORTANT TO GIVE THE SEQUENCE OF
EVENTS IN THE GAME] Thus when offering a returns policy, the manufacturer must
balance between the cost of products returned in the even that demand is low
(disadvantage) to the possible higher wholesale price when demand is low and the
increase of the stock when demand is high (advantage). Given that the production cost is
10
sufficiently low and the high demand is not too much larger than the low demand, the
advantages are greater than the disadvantage when the manufacturer offers a returns
policy. From the manufacturer’s standpoint, since its profitability conditions are weaker
when retailing is a duopoly than when retailing is a monopoly they conclude that returns
policies serve not only to resolve or reduce demand uncertainty but also to manage
competition and mitigate demand uncertainty by showing that manufacturer’s
profitability conditions are weaker when retailing is a duopoly than when retailing is a
monopoly.
To sum up note that the options to return unsold products are relevant only when there is
a significant degree of uncertainty about retail demand, products are identifiable and the
marginal cost of production is a relative small portion of retail price (low marginal cost).
Now, we turn to another popular method the manufacturers use in order to support
adequate inventory holdings by the retailers. This method refers to resale price
maintenance (RPM). Since both marketing strategies can help to achieve adequate
inventories holdings, which reasons guide the manufacturers’ choice between them? We
define first resale price maintenance (RPM), and then we compare it to returns policies.
3.
Returns Policy Vs Resale Price Maintenance
Resale price maintenance (RPM) is the specification of the final price imposed by a
manufacturer on wholesale or retail resellers of its own products. Types of this restriction
include specifying only a price ceiling, i.e. maximum resale price stipulation, or price
floor, i.e. minimum resale price stipulation.
Different theories have been developed to explain the use of RPM by manufacturers. The
usual justifications are: eliminating double mark-up problems, preventing free riding,
preventing discounting, and collusive effects. One line of justifications is the free riding
theories. These theories assume that the demand for a manufacturer’s product depends on
some informational services provided by the retailers. RPM assures the retailers to
11
capture the demand generated by the services and thus provides incentive for them to
invest in those services. RPM then improves social efficiency by enhancing demand.
However, for the products that do not need extensive sale services, Deneckere, Marvel
and Peck (1996, 1997) showed that the use of RPM or more precisely the use of
minimum resale price stipulation by a manufacturer facing uncertain demand can help
like returns policies to support adequate inventory holdings by the retailers. They
analyzed a model with a monopoly manufacturer selling to competitive retailers in
market where demand is uncertain. They find that with RPM, the monopoly manufacturer
has higher wholesale demand and makes more profits.
We have already mentioned that when the marginal product cost is positive full returns
policy is merely a second best way of achieving minimum resale price stipulation (RPM).
It does not achieve the manufacturer’s first-best. Nevertheless, there is wide category of
industrial products where a returns policy instead of minimum resale price stipulation is
offered (books, newspapers, CDs, perishable goods, computer hardware and software,
and pharmaceuticals). [WHY]
First, the use of RPM in many countries has not always been allowed since they restrict
free trade. It is prohibited per se with possible exemptions. [WHERE] Whereas, the
returns policies is always allowed when they are offered without any form of
discrimination.
Second the problems of enforcement, the allocation of risk between the parties, and the
economic incentives under the two alternative vertical agreements are different. For
example under RPM, to enforce the contract the manufacturer must monitor the actual
details of transaction that the parties involved might wish to hide. While under returns
policy, the physical return of the good or proof of good’s destruction is enough to
enforce, even if they are some possibilities of retailers moral hazard or high cost of
administrating the system of returns. On these facts, minimum resale price stipulation
(RPM) actually seems more costly than returns policy for the manufacturer to administer
and to enforce.
12
As noted by Faith and Nairu (2000), besides the enforcement, full and partial returns and
RPM also differ in how they divide risk between manufacturers and retailers and in how
they structure economic incentives. They showed that under a full return system
manufacturer bears all the risk and the retailer none. A partial return system divides the
risk between the manufacturer and the retailer. [DID THEY HAVE TO SHOW THIS]
The economic incentives reason is to better exploit the manufacturer’s private
information regarding the demand. This reason is explained by Faith and Nairu (2000).
They demonstrated that only the one that bear the risk of unsold goods has an economic
incentive to collect private information about demand and act appropriately. If the
manufacturer’s information regarding the likely future demand is superior to the
retailer’s, then a full return system is indicated, but if retailer’s information is superior
then minimum retail price stipulation might be indicated.
To sum, the advantages of returns policy over resale price maintenance (RPM) are first
the legality of returns policy and second the less information required in returns policy to
achieve the quasi same purposes.
4.
Our Contribution to The Literature on Returns Policy
To our knowledge, the literature on manufacturer’s returns policies, except the one of
Marvel and Wang (2002), considers a monopolist manufacturer which offers returns
policies in their analysis. However, Marvel and Wang (2002) have looked for the
strategic effects of returns policies on the retail market. We know that relative to
monopolists, oligopolists have additional strategic incentives to use selling strategies like
advertising, most- favored customer clauses or raincheck [Cooper (1996), Salop (1986),
Hamilton (2000)]. [EXPLAIN WHY] These selling strategies are instruments for each
firm for being more competitive. Since returns policies also have this latter characteristic,
we investigate a duopoly market structure for determining the strategic effects of returns
policies on manufacturer’s competition. We consider two competing manufacturers each
13
producing and distributing a product through a single retailer. Demand will be taken to be
a stochastic linear function of price and affected by the degree of product differentiation.
[EXPLAIN] We assume that each manufacturer sets his quantity and its returns policy
strategy to maximize its profit subject to the response of the competing manufacturer and
the retailer. We derive for each manufacturer the optimal return option and obtain the
industry equilibrium. Then we analyze this equilibrium to see how prices (wholesale,
retail, and buy-back), inventories, and welfare evolved.
We follow the framework of Padmanabhan and Png (2004) [WHY] to develop and
analyze this model of price competition with manufacturer’s returns policies. In this
model, the manufacturers have identical cost functions and they produce differentiated
products. To allow a role for manufacturer’s returns policies, we introduce exogenous
uncertainty about consumer’s demand.
Four types of price subgames can result in this setting: The first type of subgame (N, N),
results when no manufacturer offers a return policy on their products. The second subgame (R, R) results when both manufacturers offer a return policy on their products. The
third and fourth type of sub-game (R, N) or (N, R), result when one manufacturer offers a
return policy and the other do not.
The rest of the paper is organized as follow: we present a benchmark model with
monopolist retailer in section 2. We present the model with oligopolistic manufacturer
with differentiated products and provide some results in section 4. Section 5 presents
results to describe when firms choose to offer returns policies. Section 6 contains some
conclusion and discusses the welfare effect of manufacturer’s return policies.
14
4.1 Monopolist manufacturer and perfectly competitive
retailers
We consider a market structure with a monopolist manufacturer and perfectly
competitive retailers. We assume that the demand function is subject to random shocks.
This introduces the possibility of desiring to overstock products by the retailers. The
Manufacturer observes these shocks only after choosing wholesale price and returns
policies. The information structure and sequence of actions will be as follows.
First the state of the demand which could be high or low (θ = H or L respectively ) is
uncertain for all parties. The probability of being high is λ . In the first stage the
manufacturer sets a distribution policy comprising a wholesale price w and whether to
accept returns.
In the second stage, the retailers order stock s to the manufacturer. We assume that the
true state of the demand is reveled to all parties after the second stage.
Then in the third stage the retailers set the price pθ in state θ . The direct demand
function is given by:
qθ = α θ − β pθ , with β > 0
4.1.1 No Returns Policies of the manufacturer (N R)
In this case, the manufacturer sets a wholesale price w and does not accept returns of
unsold products. Since retailers are in perfect competition, they price such that they have
zero profit. For constant average and marginal production costs, this yields the unique
equilibrium price equals marginal cost (Bertrand equilibrium). Here the marginal cost is
the wholesale price, if we assume that retail costs are negligible. Formally, we have:
w = λp H + (1 − λ ) p L
(1)
15
Two cases arise with no returns (graphic 1).
Case 1: the retailers can order a stock greater than α L ( s ≥ α L )
Case 2: the retailers can order a stock smaller that α L ( s < α L )
If s ≥ α L , then p L = 0 . Therefore, by (1) w1 = λp H = λ (
s1 = α H −
α H − s1
) . This implies that:
β
β
w1
λ
(2)
If s < α L , then p L ≠ 0 . Therefore by (1)
w2 = λp H + (1 − λ ) p L = λ (
α H − s2
α − s2
) + (1 − λ )( L
) . This implies that:
β
β
s 2 = λα H + (1 − λ )α L − β w2
(3)
In each case, the manufacturer sets its price w to maximize its profits:
π M = ( w − c) s ( w) .
The first order condition with respect to the wholesale price yields:
dπ M
= s ( w) + ( w − c) s ′( w) = 0
dw
In case 1, w1 satisfies s ( w1 ) + ( w1 − c) s ′( w1 ) = 0 . This implies that:
αH −
β
β
w1 − ( w1 − c) = 0
λ
λ
Therefore
w1NR = λ
s1NR =
αH c
+
2β 2
αH
2
−
π 1M ( NR ) =
(4)
βc
2λ
(5)
β λ
( α H − c) 2
4λ β
(6)
16
In case 2, w2 satisfies s ( w2 ) + ( w2 − c) s ′( w2 ) = 0 . This implies that:
λα H − (1 − λ )α L − βw2 − β ( w2 − c) = 0
Therefore
w2NR =
s 2NR =
λα H + (1 − λ )α L + β c
2β
λα H + (1 − λ )α L
π 2M ( NR ) =
2
−
(7)
βc
(8)
2
β λα H + (1 − λ )α L
(
− c) 2
4
β
(9)
Note that when α H = α L , i.e. λ = 1 (no uncertainty) we have π 1M = π 2M
p
p H2
p 1H
w
p L2
s2
αL
s1
αH
Graphic (1): no returns accepted by the manufacturer (NR)
17
q
4.1.2 Returns Policies of the manufacturer (R)
Here the manufacturer sets its wholesale price w and gives to retailers a refund r (the
buy back price), with 0 < r ≤ w , for its unsold products. Also the perfect competitive
retailers price such that price equal marginal cost. Two cases arise with returns policies
(grahpic 2).
Case 1: the retailers can order a stock greater than ŝ ( s ≥ sˆ )
Case 2: the retailers can order a stock smaller that ŝ ( s < sˆ )
Where ŝ is defined as the stock level such that p L = r , i.e. such that the retailers are
indifferent between selling the product when demand is low at price p L and to return it to
manufacturer to obtain r .
If s ≥ sˆ , then r ≥ p L . Therefore, by (1) w1 = λp H + (1 − λ )r = λ (
α H − s1
) + (1 − λ )r .
β
This implies that:
s1 =
λα H + (1 − λ )rβ − βw
λ
(10)
If s < sˆ , then r < p L . Therefore by (1)
w2 = λp H + (1 − λ ) p L = λ (
α H − s2
α − s2
) + (1 − λ )( L
)
β
β
(11)
In each case, the manufacturer sets its price w to maximize its profits:
π M = ( w − c) s ( w) − (1 − λ )rz (r , w)
(12)
The first orders conditions with respect to the wholesale price and buy back price yield:
dπ M
ds
∂z
= s ( w) + ( w − c)
− (1 − λ )r
=0
dw
dw
∂w
18
(13)
dπ M
ds
∂z
= ( w − c) − (1 − λ ) z − (1 − λ )r
=0
dr
dr
∂r
In case 1, z (r , w1 ) ≠ 0 and w1 satisfies s ( w1 ) + ( w1 − c)
(14)
ds1
∂z ∂s1
− (1 − λ )r
= 0 . This
dw1
∂s1 ∂w1
implies that:
w1 = λ
r* =
αH
c
+ (1 − λ )r +
2β
2
λα L
(3λ − 1) β
(15)
Note that the buy back price is defined for λ >
1
.
3
Consequently we have:
w1R =
λα h λ (1 − λ )α L c
+
+
2β
(3λ − 1) β
2
π 1M ( R ) =
(16)
(1 − λ )λα L2
β λ
( α H − c) 2 +
(2λ − 1)
4λ β
(3λ − 1) 2 β
(17)
In case 2, z (r , w2 ) = 0 and p L ≠ 0 . This is similar to the problem of case 2 in the no
returns policies in part 1. Therefore we deduce immediately that:
w2R = w2NR =
s 2R = s 2NR =
λα H + (1 − λ )α L + βc
2β
λα H + (1 − λ )α L
2
π 2M ( R ) = π 2M ( NR ) =
−
βc
2
β λα H + (1 − λ )α L
(
− c) 2
4
β
19
(18)
(19)
(20)
p
p H2
p 1H
w
p
r
2
L
s2
ŝ
αL
s1
αH
q
Graphic (2): returns accepted by the manufacturer (R)
4.1.3 Analysis and comparison of returns policies v. no returns policies
To compare the profitability of returns policies to no returns policies for the
manufacturer, we must compare its profits under each retailers’stock ordering possibility.
The possibilities are:
(a)
αL ≤ s
(b)
sˆ ≤ s < α L
(c)
0 < s < sˆ
For α L ≤ s , the manufacturer’s profits with returns policies are greater than those with no
returns options if:
(1 − λ )λα L2
β λ
β λ
2
π ( R) =
( α H − c) +
(2λ − 1) > π 1M ( NR ) =
( α H − c) 2
2
4λ β
4λ β
(3λ − 1) β
M
1
This inequality is true for λ >
1
.
2
20
For sˆ ≤ s < α L , the manufacturer’s profits with returns policies are greater than those with
no returns options if:
π 1M ( R) =
(1 − λ )λα L2
β λ
1− λ
β λ
α L − c) 2
( α H − c) 2 +
(2λ − 1) > π 2M ( NR ) = ( α H +
2
4 β
β
4λ β
(3λ − 1) β
This inequality is true for λ >
1
.
2
For 0 < s < sˆ , the manufacturer’s profits are equals under both regimes of returns options.
π 2M ( R ) = π 2M ( NR )
In summary, we derive lemma 1.
Lemma 1: The manufacturer’s profits with returns policies are greater than those with
no returns options.
4.2 Oligopolist manufacturers and perfectly competitive
retailers
21
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