Download Quantity Discounts in Single Period Supply Contracts

Quantity Discounts in Single Period Supply
Contracts with Asymmetric Demand
Information ∗
Apostolos Burnetas†, Stephen M. Gilbert ‡, Craig Smith
§
August 13, 2005
Abstract
We investigate how supplier can use a quantity discount schedule to influence the stocking
decisions of a downstream buyer that faces a single period of stochastic demand. In contrast to
much of the work that has been done on single-period supply contracts, we assume that there
are no interactions between the supplier and the buyer after demand information is revealed and
that the buyer has better information about the distribution of demand than does the supplier.
We characterize the structure of the optimal discount schedule for both all-unit and incremental
discounts and show that the supplier can earn larger profits with an all-unit discount.
(Supply Chain Management, Channel Coordination, Channels of Distribution, Asymmetric Information)
∗
The first author acknowledges support by the University of Athens Research Committee via program Kapodistrias
†
University of Athens, Department of Mathematics, Athens 15784, Greece, [email protected]
‡
The University of Texas at Austin,
[email protected]
§
Management Department,
CBA 4.202,
Diageo, plc, 9 W. Broad St., 3rd Floor,Stamford, CT 06902 [email protected]
Austin,
TX 78712,
1
Introduction
In many industries, such as fashion apparel, popular toys, etc., the combination of long lead times
and short product life cycles forces retailers to make procurement decisions while there is still a
great deal of uncertainty regarding demand. To make these newsvendor procurement decisions,
the retailers attempt to maximize their own profits by balancing the potential opportunity costs
associated with unsatisfied demand and excess stock. Unfortunately, there are a variety of reasons
why the quantities the retailers choose fail to maximize either the profits of their suppliers or of the
supply chain as a whole.
In recent years, a large amount of attention has been devoted to studying mechanisms for
addressing agency issues in newsvendor environments. However, most of this attention has been
directed toward overcoming the effects of double marginalization with various forms of returns
policies, quantity flexibility, price protection and revenue sharing. A quantity discount schedule is
another mechanism that is often used in practice. One of the reasons that small independent retailers
have struggled to match the prices of the big box retailers is that they lack the volume necessary to
obtain quantity discounts from manufacturers. Order sizes for retailers (and quantity based pricing
schemes for suppliers) in an EOQ environment are primarily tied to operational efficiencies. For
a retailer, they naturally focus on their EOQ while CPG (Consumer Packaged Goods) suppliers
routinely offer incentives to order in quantities based on factors such as their production batch
size and efficient transportation sizes (full ocean container, full truckload, etc.) In a newsvendor
environment, the pricing is based on the quantity for an entire selling season which relegates the
importance of operational issues to a simple matter of how product is released and delivered against
that contracted quantity. Although there is a wide literature on the role of quantity discounts in
EOQ (long product life cycle) environments, little attention has been paid to understanding how
such discount schedules can be used in newsvendor environments.
In this paper, we address the mechanism design problem faced by a supplier in her interactions with a buyer who faces a single period of uncertain demand. We assume that there is
asymmetric information between the supplier and the buyer with respect to demand. This asymmetry can be represented in terms of the supplier’s uncertainty over which probability distribution,
from a discrete or continuous set, characterizes the buyer’s demand. In this environment, a fixed
price contract is unable to achieve the maximum profit for the supply chain, nor extract the maximum profit for the supplier. We explore several approaches that a supplier might take that would
dominate a single-price contract. One approach, to which we refer as fixed package pricing, would
be where the supplier offers the buyer a schedule of specific quantity-price pairs. From the revelation principle, we know that the supplier would need to offer no more than one quantity-price pair
for each buyer type in order to maximize her profits. Although we confirm that this fixed package
pricing approach is indeed the optimal solution to the supplier’s mechanism design problem, we also
note that in practice, we often see suppliers offering a discrete set of quantity discounts instead of
fixed package pricing. In contrast to fixed package pricing which restricts the buyer to only as many
quantity choices as there are options in the pricing schedule, a quantity discount always allows a
buyer a full continuum of quantity choices, regardless of how many options are in the price schedule.
Thus, a quantity discount may represent a compromise between the need to provide the buyer with
the flexibility to choose from a continuum of order quantities and the simplicity of a discrete set of
menu options. However, our intention is not to explain why quantity discounts are used in practice,
but rather to evaluate how well they allow a supplier to deal with asymmetric demand information
and whether one form of quantity discount is more effective than another.
The remainder of our paper is organized as follows: In Section 2 we review the literature on
quantity discounts, returns policies, and back-up agreements. In Section 3,we develop a model of
asymmetric demand information in a supply chain involving a supplier and a newsvending retailer.
After introducing the mechanism design problem, we then examine two common forms of quantity
discount: the all-unit and the incremental. In an all-unit discount, when the buyer’s quantity
exceeds a given threshold, the corresponding price decrease applies to all of the units that he buys.
In an incremental discount, when the buyer’s quantity exceeds a given threshold, the corresponding
price decrease applies to only the additional, i.e. incremental, units that he buys beyond the
threshold. Our analysis shows that the supplier will always prefer an optimal all-unit discount
over an incremental discount, but that either form of quantity discount places more restrictions
on the supplier’s pricing schedule than would a fixed package pricing approach. After presenting
numerical results in Section 4, we discuss our results and draw conclusions. Throughout the paper,
we arbitrarily adopt the convention of referring to the supplier with female pronouns and to the
buyer(s) with male pronouns.
2
Related Literature
A major objective of our research is to understand how asymmetry with respect to demand information can be addressed in a decentralized supply chain. Typically, research on principal-agency
issues in supply chains has tended to either ignore information asymmetry or to consider it with
3
respect to the buyer’s costs. For example, Corbett & deGroote (2000) and Corbett & Tang (1999)
develop an EOQ-based model in which the buyer and the supplier have asymmetric information
regarding the buyer’s holding costs, and develop an optimal non-linear pricing policy. Recently,
Ha (2001) considers asymmetric cost information between a supplier and retailer in a newsvendor
context.
Several authors have addressed the issue of designing a quantity discount scheme for heterogeneous buyer types or asymmetric information in EOQ environments, including: Lal & Staelin
(1984), Dolan (1987), Drezner & Wesolowsky (1989), and Martin (1993).
Less work has been done on asymmetric demand information, in spite of the fact that it
is arguably a more common practical problem than is asymmetric cost information. Atkinson
(1979) was among the first to recognize and model the issue of asymmetric demand information in
a stochastic demand environment. His approach is consistent with the traditional principal-agent
framework in which the objective is to allow an owner to provide proper incentives to a manager
who has better demand information. Specifically, he studies a contract in which a newsvending
manager receives a share of the additional profits that are generated as a result of using his order
quantity instead of the one that would have been chosen by the owner.
Porteus & Whang (1999) develop a model of informational asymmetry regarding demand in
the context of long term contracting and stochastic demand. Specifically, they assume that there
are two stochastically ordered demand distributions for the buyer, and that the supplier knows only
the relative probability with which the buyer’s demand is from either distribution. In their model,
the supplier offers a discrete set of prices, each one corresponding to an amount of installed capacity.
Cachon & Lariviere (2001) use a similar representation of information asymmetry in a single period
model of a capacity reservation contract.
Much of our research addresses the issue of how quantity discounts can be used to address
asymmetric demand information. Although there is a large literature on quantity discounts, nearly
all of it is been based on situations in which there is on-going, deterministic demand. In the marketing literature, Jeuland & Shugan (1983) argue that quantity discounts can be used to overcome
double marginalization, while Moorthy (1987) argues that they can be used for price discrimination.
Ingene & Parry (1995) extended this work to multiple independent retailers. The operations literature on quantity discounts has tended to focus on Economic Order Quantity (EOQ) environments
where it can be desirable to provide incentives for retailers to order in larger quantities to control
the supplier’s setup costs. Heskett & Ballou (1967), Crowther (1967), and Monahan (1984) were
4
among the first to consider quantity discounts in EOQ environments. A good review of the operations literature on quantity discounts can be found in Viswanthan & Wang (2003), who classify the
literature according to whether there is one or multiple buyers and whether there is price sensitive
demand. Recent contributions to the quantity discount literature include Boyaci & Gallego (2002),
who introduce the issue of inventory ownership into the determination of the quantity discount
that coordinates the channel, and Viswanthan & Wang (2003) who allow for a quantity discount
and a total volume discount to be used in combination. Chen, Federgruen, & Zheng (2001) show
that although a traditional quantity discount cannot be used to achieve the first best solution in an
environment in which a single supplier serves heterogenous retailers, coordination can be achieved
with a discount is based on a retailer’s annual sales volume, order quantity, and order frequency.
Shin & Benton (2004) perform and extensive simulation analysis to test the effectiveness of quantity
discounts under a variety of environmental conditions.
Since our research involves a comparison between all-unit and incremental discounts, it is
worth pointing out that, with two notable exceptions, nearly all of the work on quantity discounts
has focused on all-unit discounts. Although both Kim & Hwang (1988) and Weng (1995) consider
incremental discounts, neither one examines the question of whether a supplier should prefer one to
the other as we do. In fact, Weng (1995) shows that in an EOQ context with symmetric information,
either an all-unit and incremental discount can be used to achieve channel coordination, and that the
supplier would be indifferent between the two. Our work complements this result by demonstrating
that, in a newsvendor context with asymmetric information, a supplier will prefer the all-unit
discount.
Our work is also related to the potential incentive problem that exists between a supplier and
a newsvending retailer that results in understocking. This issue was first identified by Pasternack
(1985), and has received wide-spread attention. Several excellent review papers have been published.
Lariviere (1999) provides a general overview of research that addresses the relationship between a
supplier and a newsvending retailer. Tsay, Nahmias, & Agrawal (1999) focus on papers that seek
to offer guidance to a supplier regarding the negotiation of the terms of trade with a newsvending
retailer. Another review paper, Cachon (1998), addresses the issues related to the interactions that
may occur among independent newsvendors that are supplied by a single supplier. A more recent,
and very thorough review of the literature in this area is provided by Cachon (2004).
The contribution of our paper is that we examine how a supplier can employ a quantity
discount to best extract rents from a newsvending retailer who possesses better demand information
5
than does the supplier. In contrast to returns policies, back-up agreements, or quantity flexibility,
quantity discounts require no additional flow of information or logistics between the supplier and
the retailer following the initial transaction. Since this can be a significant operational advantage
for some products, we feel that this is a topic that merits attention.
3
The Model
We consider the situation faced by a supplier who wants to design a pricing schedule that will
be offered to a buyer who is procuring a product in order to satisfy a single period of uncertain
demand. The buyer is an intermediary, either a retailer or value-adding manufacturer, who sells to
end consumers. For each unit sold, the buyer receives an exogenously specified revenue of r. We
assume that any additional costs incurred by the buyer, e.g. for shipping material handling, etc.,
are linear in the number of units sold, and these costs have been normalized to zero.
We assume that the supplier faces a constant marginal cost of production, denoted c, and
has less information about the distribution of end-demand for this product than does the buyer.
We model this informational asymmetry by considering n buyer types. A buyer of type i = 1, ..., n
faces uncertain demand that follows a continuous probability distribution with density fi (x) and
cumulative distribution function Fi (x). Although the buyer knows his type, the supplier has only
probabilistic information about the buyer’s type. Let pi be the probability that the buyer is of
type i = 1, ..., n. Aside from having different demand distributions, the different buyer types
are identical. We assume that the demand distributions have support only for x ≥ 0 and that
distribution i stochastically dominates distribution j for j < i, i.e. Fi (x) ≤ Fj (x) for all x ≥ 0.
This assumption of first-order stochastic dominance is similar to the one made by Porteus & Whang
(1999) and Cachon & Lariviere (2001).
To represent the fact that buyers are typically not captive to specific suppliers, we assume
that the buyer has access to an alternative source of supply for the product at an exogenous per-unit
price of u where c < u < r. Thus, the buyer will not purchase from the quantity discount schedule
that is offered unless he can earn at least as large a profit as he could by purchasing from the
alternative supplier at a constant wholesale price of u.
After the supplier announces her pricing schedule, the buyer evaluates his purchasing options, including both the supplier’s pricing schedule and the alternative source, and responds by
purchasing the quantity that maximizes his own expected profits. The supplier then produces the
6
quantity that has been ordered at a cost-per unit of c, and the buyer receives his entire order
quantity prior to the start of the selling season. During the selling season, demand is realized
and the buyer sells the product at the exogenous retail price r. To simplify the presentation, we
assume that the product has zero salvage value at the conclusion of the selling season. We do not
allow for returns, back-up agreements, or other mechanisms that might be used in conjunction with
quantity discounts. This can be justified by the fact that it is often desirable to avoid additional
informational and material flows following the initial transaction.
3.1
Preliminaries
We first oberve that if a buyer of type i orders quantity q, then his expected revenue can be expressed
as:
Ri (q) = r
Z q
0
xfi (x)dx + rq
Z ∞
q
fi (x)dx = r
Z q
0
(1 − Fi (x))dx
(3.1)
Lemma 3.1 For buyer types i < j, the expected incremental revenue earned by a buyer of type i
from having T 0 > T instead of T units satisfies the following:
0
Ri (T ) − Ri (T ) = r
Z T0
T
(1 − Fi (x))dx ≤ r
Z T0
T
(1 − Fj (x))dx = Rj (T 0 ) − Rj (T )
(3.2)
Let us define πi (q, w) to be the expected gross profit earned by buyer i if he orders quantity q
at a per-unit price of w, i.e. πi (q, w) = Ri (q) − qw. Thus, with no restrictions on the quantity
ordered, the buyer faces a simple newsvendor problem. Let also qi (w) be the optimal solution to
this unconstrained newsvendor problem, given a per-unit procurement cost of w. Thus:
µ
qi (w) = Fi−1 1 −
w
r
¶
(3.3)
Let ²i represent the maximum profit that buyer i could earn if he procured the product from the
external market at price u. It follows that ²i = πi (qi (u), u), and that ²i represents the threshold
level of profits that a buyer of type i must be able to earn under a quantity discount scheme in
order to be induced to participate.
From the perspective of the total supply chain (the supplier and the buyer together), the
marginal cost of production is c and the marginal revenue from an additional sale is r. It follows
that the total expected supply chain profits that can be obtained from the market served by buyer
type i are maximized when the order quantity is qi (c). If the buyer orders quantity qi (c), then the
7
total expected gross profit in the channel is:
πi (qi (c), c) = Ri (qi (c)) − cqi (c) = r
Z qi (c)
0
xfi (x)dx
(3.4)
where the final equality follows from the definition of qi () that is given in (3.3). Because u > c, we
must have qi (c) > qi (u) and πi (qi (c), c) > ²i . Clearly, if the supplier could induce the buyer to order
quantity qi (c) and extract all additional rents, this would maximize the supplier’s profit. Let Ti be
the number of units that the supplier offers so that the buyer must purchase all Ti units or none of
them. By using (3.4), it can be shown that the per-unit price corresponding to quantity Ti = qi (c)
that allows the buyer to earn expected gross profit equal to his threshold profit ²i is:
Ri (qi (c)) − ²i
r
w =
=
qi (c)
∗
R qi (c)
0
xf (x)dx − ²i
+c
qi (c)
(3.5)
If there were no asymmetry of information, i.e. the buyer is of type i = 1 with probability p1 = 1,
then the supplier optimally offer the buyer an opportunity to order Ti = qi (c) for a total price of
w∗ qi (c). This would coordinate the channel as well as insure that the buyer would weakly prefer
our supplier over the alternative source. Thus, the supplier could maximize the total channel profits
and extract all but the buyer’s threshold level of profits for herself.
3.2
Mechanism Design
When the supplier is not certain about the buyer’s distribution of demand, then she may want to
offer more than one purchasing option to the buyer. In fact, the problem can be framed as a classic
mechanism design problem where, according to the revelation principle, the supplier can maximize
her profits by establishing no more than n different quantity-price pairs to the buyer, i.e. one for
each buyer type. Let Ti and wi be the quantity and per-unit price intended for buyer type i. Thus,
the supplier’s problem can be expressed as:
(MD)
P
max{wi ,Ti } { ni=1 pi (wi − c)Ti }
s.t. πi (Ti , wi ) ≥ ²i
πi (Ti , wi ) ≥ πi (Tj , wj )
wi ≥ 0 , Ti ≥ 0
i = 1, ..., n
i = 1, ..., n; j 6= i
i = 1, ..., n
(IRi )
(ICi,j )
The first set of constraints, (IRi ), insure that each buyer type prefers the purchasing option that is
intended for him over purchasing options designed for other types. The second set of constraints,
ICi,j , insure that each buyer type earns at least as much under the supplier’s intended purchasing
option as he would if he used his alternative procurement source. These are known as the Incentive
8
Compatibility (IC) and the Individual Rationality (IR) constraints respectively. (See Fudenberg &
Tirole (1998) for more on these issues.) Because the solution to this mechanism design problem
involves the design of a pricing schedule that makes a specific quantity-price pair attractive to each
buyer type, let us refer to it as the fixed-package pricing problem.
This mechanism design problem can be expressed for a continuum of buyer types, so long
as their demand distributions are differentiated according to a single parameter and adhere to
the assumption of first-order stochastic ordering. In the appendix we present the solution to a
continuous version of this mechanism design problem in which each buyer type is assumed to have
a uniform demand distribution, but higher indexed buyer types have larger mean demand.
It is interesting to observe that, if we interpret quantity as a measure of product quality, and
the buyer’s expected incremental revenue as the consumer’s marginal utility, then the structure of
our problem is somewhat similar to that for the product-line design problem that has been studied
by Mussa & Rosen (1978) and Moorthy (1984), among others. In that problem, the supplier must
design a product-line in an environment in which each product offering is characterized by a unidimensional measure of quality. For each specific product offering, the supplier also specifies a
price. The basic idea in these product-line design problems is that, by offering a product-line that
is differentiated, it is possible for the supplier to exploit differences in consumers’ marginal utilities
for quality. However, while it is quite natural for firm to offer consumers a discrete set of product
variants, there may be less willingness among buyers to accept a discrete set of quantities, and this
may be one reason why we see quantity discounts offered in practice.
As previously mentioned, quantity discounts that are observed in practice generally come
in two general forms: all-unit and incremental. Under either form of discount, multiple quantity
thresholds can be offered. Because both of these types of discount are common in practice, it is of
interest to see how well they work with respect to one another as well as with respect to the fixed
package pricing problem.
3.3
An all-unit Discount Under Asymmetric Information
In our original mechanism design problem, the supplier can offer a discrete set of price-quantity pairs
from which the buyer must choose. Under a quantity discount, the supplier can offer a discrete set
of prices (either all-unit or incremental), but she must allow the buyer to purchase from a continuum
of quantities. Thus, a quantity discount allows the buyer greater flexibility in how he will respond.
As we will show, this additional flexibility benefits the buyer at the expense of the supplier.
9
An all-unit quantity discount schedule can be represented by a series of threshold quantity
breakpoints (T1 ≤ T2 ≤ ... ≤ Tn ), and per-unit wholesale prices (w1 ≥ w2 ≥ ... ≥ wn ) such that
price wi is available to the buyer only if he orders a quantity at or above breakpoint Ti . According
to the revelation principle, the supplier can maximize her profits from using an all-unit discount by
establishing no more than n different break-points and prices so that the buyer self-selects the policy
that has been designed for his type. Because πi (q, w) is concave in q, it follows that the optimal
quantity for a type i buyer to order under menu option j is the maximum of Tj and qi (wj ). Let
us denote this quantity by QAU
i (Tj , wj ) = M ax{qi (wj ), Tj }. To satisfy the individual rationality
constraint for a type i buyer, the supplier must set wi and Ti to insure that if a type i buyer
purchases according to menu option i, his gross profit is at least ²i , the profit that he would earn
by procuring the product from the external market at per-unit price u. In addition, the supplier
must make sure that the incentive compatibility constraints are satisfied. Recall that, to satisfy
the incentive compatibility constraints, the supplier must eliminate any incentive for the buyer to
purchase QAU
i (Tj , wj ) for any j 6= i.
The supplier’s problem is to set w1 , ..., wn ; T1 , ..., Tn in order to maximize her own profits
subject to the (IR) and (IC) constraints. Formally, the supplier’s problem can be written as follows:
(S.1)
P
max{wi ,Ti } { ni=1 pi (wi − c)QAU
i (Ti , wi )}
s.t. πi (QAU
(T
,
w
),
w
)
≥
²i
i
i
i
i
AU
πi (Qi (Ti , wi ), wi ) ≥ πi (QAU
i (Tj , wj ), wj )
wi−1 ≥ wi ≥ 0 , Ti ≥ Ti−1 ≥ 0
i = 1, ..., n
i = 1, ..., n; j 6= i
i = 1, ..., n
(IRi )
(ICi,j )
Note that, although the solution to the above model can have T1 > 0, i.e. a minimum purchase
requirement, the buyer has the option of procuring any quantity he likes from the external source at
a per-unit price of u. Because constraint (IR1 ) guarantees that the buyer weakly prefers quantity T1
at a per-unit price of w1 over buying any quantity at a price of u, the above model is equivalent to
one in which our supplier also offers the buyer the opportunity to purchase an unrestricted quantity
at a per-unit price of u.
Theorem 3.1 There exists an optimal solution to problem (S.1) in which, for each j = 1, ..., n:
QAU
j (Tj , wj ) = Tj , i.e. the buyer’s order quantity is exactly equal to the quantity breakpoint specified
for his type.
AU
∗
∗
∗
Proof: From the definition of QAU
j (Tj , wj ), we must have Qj (Tj , wj ) ≥ Tj . Suppose that for some
∗
∗
∗
∗
AU
∗
∗
j, the optimal Tj∗ and wj∗ are such that QAU
j (Tj , wj ) > Tj . Note that for any Tj < Qj (Tj , wj ),
10
the profits of buyer type j are independent of Tj∗ . Thus, we can increase the quantity breakpoint
associated with price wj∗ until the point where buyer type j’s optimal order quantity is equal to the
threshold without affecting any of the constraints involving buyer type j’s preferences. Moreover,
each of the ICij constraints for i 6= j continues to be satisfied since their right-hand sides are
non-increasing in the breakpoint associated with price wj∗ . ♦
We can apply this result to problem (S.1) by substituting Ti for QAU
i (Ti , wi ) in the objective function
and in the left-hand-sides of the IR and IC constraints. Based on Theorem 3.1, this substitution
affects neither the feasibility nor the objective value of any optimal solution to problem (3.1).
Moreover, since the proposed substitution makes the IR and IC constraints more restrictive, the
set of feasible solutions after making the substitutions will be a subset of the feasible solutions to
problem (3.1). Therefore, we propose the following alternative formulation of the supplier’s problem:
(S.2)
P
max{wi ,Ti } { ni=1 pi (wi − c)Ti }
s.t.
πi (Ti , wi ) ≥ ²i
πi (Ti , wi ) ≥ πi (QAU
i (Tj , wj ), wj )
wi−1 ≥ wi ≥ 0 , Ti ≥ Ti−1 ≥ 0
i = 1, ..., n
i = 1, ..., n; j 6= i
i = 1, ..., n
(IRi )
(ICi,j )
It is of interest to compare the right-hand-sides of the incentive compatibility constraints in the
above all-unit quantity discount problem to those in the original mechanism design problem. In
problem (S.2), the right-hand side of (ICi,j ) is QAU
i (T, w) which is the quantity greater than or equal
to T that maximizes the expected profit of buyer i at a per unit price of w. In contrast, in problem
(MD) the right-hand side of this constraint is just T . This is a non-trivial distinction between the
two problems, and it alters the analysis considerably. We can now compare the quantity discount
pricing problem to the fixed-package pricing problem.
Theorem 3.2 For a given set of parameters (r, c, ², f1 (), ..., fn ()), the optimal value of the objective
function to the mechanism design, i.e. fixed-package pricing, problem is at least as large as the
optimal value of the objective function to (S.2).
Proof: The fixed-package pricing problem differs from (S.2) only in the right-hand sides of the
incentive compatibility constraints, where Tj is substituted for QAU
i (Tj , wj ). From the definition
AU
of QAU
i (T, w), we have that: πi (Qi (Tj , wj ), wj ) ≥ πi (Tj , wj ). Thus, the incentive compatibility
constraints are less restrictive in the fixed-package pricing problem than they are in the quantity
discount pricing problem as expressed in (S.2). ♦
It is interesting to note that, in the all-unit quantity discount problem, buyer i purchases
11
exactly the threshold quantity that is intended for his type by the supplier, just as in the fixedpackage problem. However, it is the buyer’s increased flexibility to do otherwise under the quantity
discount that prevents the supplier from extracting as much of the total profits. From the results
of Theorem 3.1, we can now focus our attention on solving the alternative representation of the
quantity-discount-pricing problem in (S.2).
Theorem 3.3 a) A necessary and sufficient condition for a set of prices and breakpoints, (w1 , ..., wn ;
T1 , ..., Tn ), to be a feasible solution to (S.2) is for the following constraints to be satisfied: IR1 ;
ICj,j−1 for j = 2, ..., n; and ICj,j+1 for j = 1, ..., n − 1.
b) A sufficient condition for a set of prices and breakpoints, (w1 , ..., wn ; T1 , ..., Tn ), to be a feasible
solution to (S.2) is for: Tj = QAU
j (Tj , wj ) for j = 1, ..., n, and for IR1 and ICj,j−1 , for j = 2, ..., n
to be satisfied at equality.
c) If there is a feasible solution to (S.2), then there must be an optimal solution in which:
i) IR1 and ICj,j−1 , for j = 2, ..., n, are binding, and
ii) qj (wj ) ≤ Tj = QAU
j (Tj , wj ) ≤ qj (c) for j = 1, ..., n, and Tj = qj (c) for j = n.
The proof of this result is based on the approach outlined in Fudenberg & Tirole (1998).
Because its details are quite tedious, we have not included it in this paper. However it is available
upon request from the authors. The above result implies that to identify an optimal solution to
(S.2), we can restrict our attention to solutions in which IR1 and ICj,j−1 , for j = 1, ..., n are binding.
Moreover, as shown in Theorem 3.1, an optimal solution to (S.2) must also be an optimal solution
to (S.1). Restricting our attention to this subset of the solution space has the following advantage:
For any given set of quantity breakpoints (T1 ≤ T2 ≤ ... ≤ Tn ), the set of corresponding prices
w1 , ..., wn can be determined as the solution to the set of equations consisting of IR1 and ICj,j−1
for j = 2, ..., n. Note also that the value of wj depends only on Tj−1 , Tj and wj−1 .
Thus, if the quantities that buyers can request are limited to or can be approximated by
a finite discrete set, then we can solve the problem using a dynamic programming approach. Let
Wj (Tj , Tj−1 , wj−1 ) be the value of wj that solves:
AU
Rj (Tj ) − wj Tj = Rj (QAU
j (Tj−1 , wj−1 )) − wj−1 Qj (Tj−1 , wj−1 )
(3.6)
for j = 2, ..., n, and let W1 (T1 , 0, u) be the value of w1 that solves:
R1 (T1 ) − w1 T1 = ²1 = R1 (q1 (u)) − uq1 (u)
12
(3.7)
Define uj (T ) to be the conditionally optimal expected profits that the supplier can earn from buyer
types 1, ..., j given that the quantity breakpoint for buyer type j is T . Define ωj (T ) to be the
corresponding (conditionally) optimal per-unit price offered to buyer type j. These functions can
be represented by the following recursive relationship:
uj (T ) = M axτ ≤T {pj T (Wj (T, τ, ωj−1 (τ )) − c) + uj−1 (τ )}
(3.8)
where u0 (T ) = 0 for all T . Let τj−1 (T ) be the optimal quantity breakpoint offered to buyer type
j − 1 given that the quantity designed for buyer type j is equal to T . Thus, τj−1 (T ) is the value of
τ maximizing the right hand side in (3.8), and
ωj (T ) = Wj (T, τj−1 (T ), ωj−1 (τj−1 (T ))) for j = 2, ..., n
(3.9)
ω1 (T ) = W1 (T, 0, u).
The optimal solution to the n buyer-type problem is then:
vn = M axT {un (T )}.
(3.10)
Note that if there is a finite number m of potential quantities that can be ordered, then vn can
be solved in O(m2 n) time. In cases in which quantities must be ordered in discrete units, this
assumption is reasonable. In other instances where the product can be produced in continuous
quantities, we can use a discrete quantity space as an approximation to the continuous one.
It is also worth noting that the above approach can also be used to solve the fixed-package
pricing problem. The only thing that would be different is that we would substitute Tj−1 for
QAU
j (Tj−1 , wj−1 ) in equation (3.6) to represent the fact that in fixed-package pricing, the buyer
must purchase quantity Tj−1 to obtain per-unit price wj−1 .
3.4
An Incremental Quantity Discount Under Asymmetric Information
An incremental discount scheme can be represented by a series of thresholds (t1 , t2 , ..., tn ) and
marginal prices (v0 , v1 , ..., vn ) such that the marginal price for units in the interval (ti , ti+1 ] is vi for
i = 0, ..., n − 1, and the marginal price for units beyond tn is vn . Thus, the total cost of buying
q ∈ (tk−1 , tk ] is
Pi
j=1 vj−1 (tj
− tj−1 ) + vi (q − ti ), where t0 = 0.
It is important to recognize that, in contrast to the all-unit problem, buyers will not tend
to order at the thresholds since the marginal cost decreases at each threshold. Note that in the
all-unit problem, the buyer’s cost function is not generally concave. For example, consider a single
13
breakpoint T , where the per-unit price is u for quantities less than T and is w1 for quantities q ≥ T .
Thus, the buyer’s cost function, C(q), is equal to uq for q < T , and is equal to w1 q otherwise. If
there exists q > T for which w1 q < u(T − 1), then for 0 < α < 1, we have αC(T − 1) + (1 − α)C(q) >
C(α(T − 1) + (1 − α)q). With an incremental discount, the buyer’s cost function is concave so long
as the vi ’s are decreasing. Moreover, in order to prevent buyers from pretending to be several
small buyers, it is reasonable to assume that the vi ’s are decreasing. This guarantees a concave
cost function for the buyer. The implication of this is that the buyer’s first-order condition will
be satisfied for the purchasing option that he chooses. Thus, if buyer i purchases according to the
conditions defined for buyer j, he will order the following quantity: QIi (tj , vj ) = M ax{tj , qi (vj )}.
To simplify the notation, let us define the following function to represent the average per-unit price
paid by a buyer purchasing quantity q ≥ ti under the purchasing option designed for buyer i:


i
1 X
ACi (q) = 
vj−1 (tj − tj−1 ) + vi (q − ti )
q j=1
(3.11)
where t0 = 0 since there is no minimum quantity required to purchase from the exogenous source.
The most obvious formulation of this problem is as follows:
Incremental Discount Problem (I.1)
P
max{vi ,ti } { ni=1 pi (ACi (Qi (ti , vi )) − c)Qi (ti , vi )}
s.t. πi (Qi (ti , vi ), ACi (Qi (ti , vi ))) ≥ πi (Qi (0, u), u)
πi (Qi (ti , vi ), ACi (Qi (ti , vi ))) ≥ πi (Qi (tj , vj ), ACj (Qi (tj , vj )))
vi−1 ≥ vi ≥ 0 , ti ≥ ti−1 ≥ 0
i = 1, ..., n
i = 1, ..., n; j 6= i
i = 1, ..., n
(IRi )
(ICi,j )
Note that because the supplier knows the demand distributions for each buyer type, he can anticipate the optimal response, QIi (tj , vj ), of any buyer type i to any menu option j, and design the
quantity discount to target specific purchase quantities from each buyer type. Borrowing notation
from the all-unit problem, the buyer can target purchase quantities of T1 , T2 , ..., Tn subject to the
constraint that: Ti = qi (vi ). We will also borrow the notation wi to represent the average price per
unit that would be paid by a buyer of type i. Using this notation, we can express the incremental
discount problem in a form that lends itself to comparison with the all-unit problem:
(I.2)
P
max{wi ,Ti ,vi ,ti } { ni=1 pi (wi Ti − c)Ti }
s.t. πi (Ti , ACi (Ti )) ≥ πi (Qi (0, u), u)
πi (Ti , ACi (Ti )) ≥ πi (Qi (tj , vj ), ACj (Qi (tj , vj )))
wi = ACi (Ti )
Ti = qi (vi )
Ti ≥ ti
vi−1 ≥ vi ≥ 0 , ti ≥ ti−1 ≥ 0
14
i = 1, ..., n
i = 1, ..., n; j 6= i
i = 1, ..., n
i = 1, ..., n
i = 1, ..., n
i = 1, ..., n
(IRi )
(ICi,j )
Note that for a given set of marginal prices and breakpoints, (v1 , ..., vn ; t1 , ..., tn ), the wi and
Ti variables in the above problem are uniquely determined.
Theorem 3.4 a) A necessary and sufficient condition for a set of marginal prices and breakpoints,
(v1 , ..., vn ; t1 , ..., tn ), to be a feasible solution to (I.2) is for either vj = vj−1 or Tj−1 ≤ tj for
j = 2, ..., n, and the following constraints to be satisfied: IR1 ; ICj,j−1 for j = 2, ..., n; and ICj,j+1
for j = 1, ..., n − 1.
b) A sufficient condition for a set of prices and breakpoints, (v1 , ..., vn ; t1 , ..., tn ), to be a feasible
solution to (I.2) is for IR1 and ICj,j−1 , for j = 2, ..., n to be satisfied at equality.
c) If there is a feasible solution to (I.2), then there must be an optimal solution in which: i) IR1
and ICj,j−1 , for j = 2, ..., n, are binding, and ii) Tj ≤ qj (c) for j = 1, ..., n, and Tj = qj (c) for
j = n.
The proof of this result is similar to that for Theorem 3.3, and is available upon request. Based on
this result, we can adopt a dynamic programming approach similar to the one used for the all-unit
discount to solve the incremental discount problem. Because we have converted the marginal costs
into average costs per unit at various quantities, the buyer profit functions are identical to the ones
in problem S.2:
πi (q, ACj (q)) = Ri (q) − qACj (q)
(3.12)
Define πsAU to be the optimal solution to the all-unit quantity discount problem, and let πsI be the
optimal solution to the incremental quantity discount problem.
Theorem 3.5 For a given set of problem parameters, F1 , ...Fn , u, c, the supplier can earn at least
as much under an all-unit discount than under an incremental discount. Specifically, πsI ≤ πsAU .
Proof: Observe, that the two objective functions are identical. To prove the claim, it suffices to
show that problem I.2 is more tightly constrained than problem S.2. First note in that any solution
(w, T, v, t) that satisfies the last four constraints of I.2, (w, T ) must satisfy the constraints in S.2
requiring that wi ≤ wi−1 and Ti ≥ Ti−1 . Now consider the ICji constraints for i < j: From the
assumption of stochastic dominance, we have that:
Qj (ti , vi ) ≥ Qi (ti , vi ) = qi (vi )
(3.13)
where the final equality follows from the requirement that Ti = qi (vi ), which implies that ti ≤ Ti .
It follows from the definition of ACi (q) that ACi (Qj (ti , vi )) ≤ ACi (Ti ). From the constraint in I.2
15
requiring that wi = ACi (Ti ). It follows that:
πj (Qj (ti , vi ), ACi (Qj (ti , vi ))) ≥ πj (Qj (ti , vi ), ACi (Ti )) ≥ πj (Ti , ACi (Ti ))
(3.14)
and because the right-hand-side of the above inequality is identical to the right hand side of the
ICji constraint in problem S.2, the corresponding constraint in problem I.2 is more restrictive. Let
us now consider the ICij constraints for i < j. From stochastic dominance, we have:
Qi (tj , vj ) ≤ Qj (tj , vj ) = Tj
(3.15)
where the final equality follows from the definition of Qj (tj , vj ) and the requirements Tj = qj (vj )
and tj ≤ Tj . By the definition of Qi (tj , vj ) as buyer i’s optimal response to (tj , vj ), it follows that:
πi (Qi (tj , vj ), ACj (Qi (tj , vj ))) ≥ πi (Tj , ACj (Tj , vj ))
(3.16)
and this demonstrates that the right hand side of the upward IC constraints in problem I.2 are no
smaller than the right hand sides of the corresponding constraints in problem S.2. Since for given
values of (w, T ), the left hand sides of these constraints are identical, the upward IC constraints
are more difficult in I.2 than in S.2. ♦
Intuitively, the above result can be explained as follows. In the all-unit discount, the downward ICji constraint requires the buyer to prefer buying quantity Tj at average per-unit cost wj
over buying any quantity q ≥ Ti at average per-unit price wi . In the incremental discount problem,
this constraint is more severe because the buyer’s average per-unit price under menu option i falls
below wi when q > Ti . ( Recall that, by construction, ACi (q) = wi when q = Ti , and is decreasing
in q.) Similar intuition exists for the upward ICij constraints. In the all-unit discount, buyer i
purchases exactly Tj under menu option j, since by stochastic dominance, qi (wj ) ≤ qj (wj ) and
qj (wj ) ≤ Tj . In the incremental discount, tj ≤ Tj , and this allows the buyer more flexibility to
purchase less than the targeted amount under menu option j.
3.5
Exclusion of Buyer Types
In many situations where a firm offers a menu of purchasing options to customers who differ in
their marginal valuation for a product, there is an issue as to whether the firm should attempt to
serve all segments of the market. For example, Moorthy (1984), Mussa & Rosen (1978), Porteus &
Whang (1999) all provide analysis that demonstrates that, by serving the low end of the market,
the firm cannot extract as much rent from customers with the highest valuation. All three of these
papers provide conditions in which, by ignoring the low end of the market, the firm can increase
16
the amount that it earns from the high end consumers by more than it gives up by not serving the
low end consumers.
In a paper that is more closely related to ours, Ha (2001) shows that in a newsvendor
environment in which there is asymmetric information about the buyer’s costs, it is optimal for the
supplier to not service buyers whose costs exceed a threshold. The main reason for this is that it is
assumed that the buyer’s participation constraint is independent of his type, i.e. a high cost buyer
requires just as much profit as does a low cost buyer. However, in our model this is not the case,
and the supplier has no reason to exclude the low end segment of the market. To see why this is the
case, recall that we assume that the buyer has an alternative source of supply at market price u (u).
Our individual rationality (participation) constraints require that each buyer type weakly prefer
purchasing from the supplier over purchasing from the market. However, by offering the low end
buyer type(s) a per-unit price of u (u), the supplier can trivially satisfy this requirement without
adversely affecting its ability to extract rents from the higher indexed buyer types.
4
Numerical Experiment
We consider several numerical examples in order to assess the benefit of quantity discounts to the
supplier as well as to explore the sensitivity of the optimal discount policies with respect to the
model parameters.
To estimate the benefit of quantity discounts to the supplier we first must analyze her
optimal pricing policy without quantity discounts, i.e., when a single common price is offered to
all buyer types. We refer to this case as the single-price problem. This problem was analyzed
in Lariviere & Porteus (2001). Because we apply this to examples in which demand is normally
distributed, let us briefly review the solution to the single-price problem for this special case of
normally distributed demand: Assume that the demand distribution of buyer i is normal N (µi , σi2 ).
Let w be the single per-unit price that the buyer offers to all buyer types. Thus, buyer i will solve
a newsvendor problem and order:
Qi (w) = Fi−1 (1 − w/r) = µi + σi Φ−1 (1 −
w
), i = 1, . . . , n.
r
where P hi() denotes the unit-normal cumulative distribution function. The supplier’s expected
profit as a function of w is:
ΠPS (w) =
n
X
pi (w − c)Qi (w) = (w − c)(µ̄ + σ̄Φ−1 (1 −
i=1
17
w
)),
r
where µ̄ =
Pn
i=1 pi µi
and σ̄ =
Pn
i=1 pi σi .
By making the transformation z = Φ−1 (1 −
w
r ),
i.e.,
w = rΦ̄(z), the supplier’s profit can be written as:
ΠPS (z) = (rΦ̄(z) − c)(µ̄ + σ̄z).
Because a normal distribution has an increasing generalized failure rate (IGFR), ΠPS (z) is unimodal
and is maximized at the point where the first-order conditions are satisfied. (See Lariviere & Porteus
(2001) for details.) Thus, it is easy to determine the maximizing value z ∗ numerically. The optimal
price w∗ and profit ΠPS ∗ can then be obtained using the above transformation.
So far, we have not considered the possibility that buyers may have an alternative source of
the product at an exogenous price of w0 per-unit. When such an opportunity exists, the optimal
price for the supplier to charge becomes:
w̃ = min(w∗ , w0 )
and the supplier’s optimal profit is:
Π̃PS ∗ = ΠpS (w̃).
In our first numerical experiment we consider a case where there are five different buyer
types, all perceived by the supplier as equally likely. Each buyer type has normally distributed
demand with common standard deviation σ = 10, and the following means: µ1 = 50, µ2 = 60, µ3 =
70, µ4 = 80, µ5 = 90. The selling price is r = 50 and the supplier’s cost c = 10.
Table 1 shows the optimal price and profit under the single-price policy for w0 varying
from 25 to 50, the optimal profits under all-unit and incremental discounts, as well as the profit
improvement for each quantity discount policy with respect to the single-price standard.
Table 2 presents the optimal pricing policy for all-unit discounts. Tables 3 and 4 present the
optimal pricing policy for the incremental discount case, in terms of the order quantities/average
prices and breakpoints/marginal prices, respectively. Note that when the alternative price (w0 or
x0 ) equals the retail price of 50, the buyer’s profit from using the alternative supply is equal to zero.
In this case the individual rationality constraints require only that the buyer earn non-negative
profits. As this price for the alternative source of supply decreases, the intensified competition
from the alternative source of supply makes it more difficult for our supplier to extract profits
from the buyer, and this is confirmed by the supplier’s profits in these two tables. Note that as
w0 (x0 ) increases, the quantities decrease and the average per-unit prices increase for both types
of discount. It can also be observed in Table 1 that, when external pressure on price intensifies,
18
i.e. w0 (x0 ) decreases, the benefit from using either an all-unit or an incremental quantity discount
instead of a single-price policy becomes very small. However, when external pressure on price is
non-existent, i.e. w0 (x0 ) = 50, then the two types of quantity discount are most advantageous,
respectively allowing for 14.68% and 12.27% increases in profit over a single price policy.
From Tables 2 and 3, it can be observed that for the lowest indexed buyer type (i = 1),
the incremental discount results in smaller quantities and higher average per-unit prices than does
the all-unit discount. Moreover, the magnitude of the gap between the quantity (average per-unit
price) from the all-unit discount versus that for the incremental discount is increasing in the external
price w0 = x0 . These two tables also confirm that, regardless of which type of discount is used, the
quantity sold to the highest indexed buyer type is the one that maximizes channel profits for this
buyer type’s distribution of demand, i.e. q5 (c) = 98.5.
In the second numerical experiment we explore the sensitivity of the profits with respect to
variability of demand. There are various ways we could change demand variability. For example
we could assume that all types of buyers have normally distributed demands with different means
but a common variance σ 2 , and then investigate how the policies are affected by different values
of σ. Alternatively, we assume that all types of buyers have a common coefficient of variation, i.e.
σj /µj = C for all j = 1, ..., n, and then investigate how the policies are affected by different values
of C. We use an approach that lies between these two alternatives. In our approach, we assume
that the difference in demand distributions among buyer types is due to the different market sizes
they serve. Specifically, buyer j serves a market of Nj individual customers. Each customer in every
market has demand that is independent and identically distributed with mean µ0 and variance σ02 .
Therefore, when the market sizes Nj are large, the demand that is faced by a buyer of type j is
approximately normally distributed with mean µj = Nj µ0 , variance σj2 = Nj σ02 , and coefficient of
variation Cj =
p
Nj (σ0 /µ0 ). In this approach, the differentiation of variance across buyer types is
justified on the basis of our assumptions about the behavior of individual consumers.
For the numerical experiment, we consider a situation with n = 4 buyers, r = 50, c =
10, w0 = v0 = 30. The market sizes are N1 = 9, N2 = 25, N3 = 36, N4 = 49 and the mean demand
for each customer is equal to µ0 = 2. Tables 5 to 8 show the optimal quantity discount policies
and supplier profits as well as the comparison to the single price benchmark case for various values
of σ0 . From these results it can be observed that, regardless of the pricing policy employed, the
supplier’s profit decreases with σ. This is consistent with the fact that, in any newsvendor setting,
an increase in demand uncertainty will increase the total expected cost of over and under-stocking.
19
However, it can be observed that increasing the standard deviation of demand does not have the
same effect upon the all-unit discount that it does on the incremental discount. In Table 6, it can
be seen that, under the all-unit discount, increasing σ causes the quantities to increase and the
average per-unit prices to decrease. However, in Table 7, it can be seen that, under the incremental
discount, the quantities actually decrease for the lower indexed buyer types (i = 1, 2, 3).
Although these numerical computations confirm our analytical result that the supplier’s
expected profit is higher under an all-unit discount than under an incremental discount, it is interesting to examine the magnitude of the difference which is shown in the far right column of Table 7.
It can be observed that the incremental discount generally results in a loss of no more than about
1.5% compared to the all-unit discount.
In our last numerical experiment, we investigate the sensitivity of the profits of the supplier and of the total channel to the parameters of normally distributed demand when there are
two equally likely buyer types. Specifically, we investigate how the parameters of the normally
distributed demands for the two buyer types affect the percentage gap between the expected profits
earned by the supplier (total channel) under each form of quantity discount and the expected profits
that would be earned under symmetric information. Recall that under symmetric information, the
supplier would know the buyer’s type and could use either form of discount to induce the (channel) profit maximizing quantity, leaving the buyer with only enough profit to satisfy his individual
rationality constraint. Since we assume that the two buyer types are equally likely, the expected
SI profits is the average of the amounts that the supplier could extract from each type of buyer
with perfect information about his types. Obviously, the SI profits represent an upper bound on
the supplier’s and the channel expected profits, respectively.
To perform this last experiment, we consider two different cases for the means of the two
buyer types: one in which the buyer types have means µ1 = 53 and µ2 = 58, and another in which
they have means µ1 = 40 and µ2 = 70. We consider a range of standard deviations, σ, and assume
that this is the same for both buyer types. We assume that the retail price r = 40, the production
cost c = 10, and the price from the alternative source of supply w0 = x0 = 25.
Figure 1 shows the percentage gap between the supplier’s expected profits under symmetric
information versus those under either the incremental (IN vs. SI) or the all-unit (AU vs. SI)
discount. Similarly, Figure 2 shows the percentage gap between the channel expected profits under
symmetric information versus those under either the incremental (IN vs. SI) or the all-unit (AU vs.
SI) discount. The lines are labelled to reflect the percent profit gap that they represent ( IN vs SI
20
or AU vs SI ), and the assumption about the means for the two buyer types (53 − 58 or 40 − 70).
Figure 1 confirms our result that the supplier can earn more under an all all-unit discount
than under an incremental discount, since the profit gap with (with respect to symmetric information) is higher for the all-unit than the incremental discount. Moreover, the gap between the
supplier’s symmetric information profits and her profits under either type of discount increases with
the magnitude of the difference between the means for the two buyer types, and the size of the
standard deviation. It is also interesting to observe how the standard deviation affects the relative
performance of the all-unit and incremental discounts when the buyer types have mean demands of
µ1 = 53 and µ2 = 58, versus when their mean demands are µ1 = 40 and µ2 = 70. When the buyer
types have means that are close together, i.e. 53 and 58, the relative gap between the incremental
and the all-unit discount decreases with demand uncertainty. However, when the buyer types’ demands are more differentiated, i.e. means of 40 and 70, the relative gap between the two types of
discount increases with demand uncertainty. In fact, for this case, when the standard deviation of
demand is 25, the gap between the incremental discount and the symmetric information profit is
67% larger than that between the all-unit discount and the symmetric information profit.
For the channel profits, we first observe from Figure 2 that both quantity discount policies
yield lower channel profits than the symmetric information case, which is the result of the supplier
trading off channel efficiency against the information rents paid to the buyer. However, it is worth
noting that the gaps in Figure 2 are smaller than those in Figure 1, indicating that the lack of
symmetric information hurts the supplier more than the supply chain as a whole. Finally, we can
observe that when the buyer types differ significantly, i.e. demand means (40,70), the all-units
discount results in greater channel efficiency (lower profit gaps) than does the incremental. On
the other hand, when the buyer types are more similar, i.e. demand means (53, 58), the channel
efficiency is very high (profit gaps are less that 1% ), and the specific type of discount makes very
little difference.
5
Conclusions and Extensions
In this paper, we have considered how incremental and all-unit discounts can be used to increase a
supplier’s profits when selling to a newsvending downstream buyer who has more information about
the distribution of demand than does the supplier. Understanding how quantity discounts can be
used in such environments is important because they are commonly used in practice and because
they are a natural mechanism for a supplier to use to overcome informational asymmetry regarding
21
12
10
Profit Gap (%)
8
6
IN vs SI 53-58
4
AU vs SI 53-58
IN vs SI 40-70
AU vs SI 40-70
2
8
10
12
14
16
18
20
22
24
26
Standard Deviation
Figure 1: Supplier Profit Gap as a Function of Standard Deviation and Difference in Means
7
6
Profit Gap (%)
5
4
3
2
IN vs SI 53-58
AU vs SI 53-58
1
IN vs SI 40-70
0
AU vs SI 40-70
8
10
12
14
16
18
20
22
24
26
Standard Deviation
Figure 2: Channel Profit Gap as a Function of Standard Deviation and Difference in Means
22
the buyer’s distribution of demand. One important feature associated with a quantity discount is
the fact that, after the initial transaction, there is no need for additional flows of information or
logistics between the buyer and seller. Certainly this could be advantageous in many environments.
Our main results include characterizing the structure of the all-unit and incremental discounts and demonstrating that the all-unit discount always dominates the incremental discount
from the perspective of the supplier. This compliments previous results in EOQ settings with symmetric information in which the supplier has been shown to be indifferent between incremental and
all-unit discounts.
In our model, the quantity discount is used to counter the informational asymmetry between
the supplier and the buyer. This is different than the role that is typically played by returns policies,
quantity flexibility, price protection, and revenue sharing in the existing literature. Most of the
results for these other mechanisms have been obtained in symmetric information environments in
which the primary purpose of the mechanism is to overcome double marginalization. It is worth
noting that when the retail price r is exogenous and demand information is symmetric, a quantity
discount is equivalent to a two-part tariff and allows a supplier to simultaneously induce the order
quantity that maximizes total profits and extract all but the participation profits from buyer. The
fact that this is inconsistent with the reality that newsvendor-like buyers do earn excess profits may
be one of the reasons why so little attention has been paid to quantity discounts in the literature.
However, when there is asymmetric demand information, the supplier can no longer extract all of
the profits from the buyer with a quantity discount, even if the buyer cannot control the retail price.
Our purpose is to recognize the importance of quantity discounts in environments in which
there is asymmetric demand information, and show that one form of quantity discount, i.e. the allunit discount, dominates the incremental discount. Although there are many environments in which
it is desirable to avoid continued interactions between the supplier and the buyer, making quantity
discounts particularly attractive, it is possible for quantity discounts to be used in conjunction with
other mechanisms, such as returns policies, quantity flexibility, and price protection. Clearly, this
is a potential direction for future research. In addition, we believe that it may be worthwhile to
consider the role played by quantity discounts when the buyer can control the retail price.
23
References
Atkinson, A. (1979). Incentives, uncertainty, and risk in the newsboy problem. Decision Sciences 10, 341–353.
Boyaci, T. & G. Gallego (2002). Coordinating pricing and inventory replenishment policies for
one whlesaler and one or more geographically dispersed retailers. International Journal of
Production Economics 77, 95–111.
Cachon, G. P. (1998). Competitive supply chain inventory management. In S. Tayur, R. Ganeshan;, & M. Magazine (Eds.), Quantitative Models for Supply Chain Management. Kluwer
Academic Publishers.
Cachon, G. P. (2004). Supply chain coordination with contracts. In S. Graves & T. de Kok (Eds.),
Handbooks in Operations Research and Management Science. North Holland Press.
Cachon, G. P. & M. A. Lariviere (2001). Contracting to assure supply: How to share demand
forecasts in a supply chain. Management Science 47 (5), 629–646.
Chen, F., A. Federgruen, & Y. Zheng (2001). Coordination mechanisms for a distribution system
with one supplier and multiple retailers. Management Science 47 (5), 693–708.
Corbett, C. & X. deGroote (2000). A supplier’s optimal quantity discount policy under asymmetric information. Management Science 46 (3), 444–450.
Corbett, C. & C. Tang (1999). Designing supply contracts: Contract type and information asymmetry. In S. R. G. Tayur & M. Magazine (Eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers.
Crowther, J. F. (1967). Rationale for quantity discounts. Harvard Business Review 42, 121–127.
Dolan, R. J. (1987). Quantity discounts: Managerial issues and research opportunities. Marketing
Science 6, 1–22.
Drezner, Z. & G. O. Wesolowsky (1989). Multi-buyer discount pricing. European Journal of
Operational Research 40 (1), 38–42.
Fudenberg, D. & J. Tirole (1998). Game Theory, Chapter 7: Bayesian Games and Mechanism
Design. The MIT Press.
Ha, A. Y. (2001). Supplier-buyer contracting: Asymmetric cost information and cutoff level policy
for buyer participation. Naval Research Logistics 48, 41–64.
24
Heskett, J. L. & R. H. Ballou (1967). Logistical planning in inter-organizational systems. In
M. P. Hottenstein & R. W. Millman (Eds.), Research Toward the Development of Management
Thought. Academy of Management.
Ingene, C. A. & M. Parry (1995). Coordination and manufacturer profit maximization. Journal
of Retailing 71, 129–151.
Jeuland, A. & S. Shugan (1983). Managing channel profits. Marketing Science 2, 239–272.
Kim, K. & H. Hwang (1988). An incremental discount pricing schedule with multiple customers
and single price break. European Journal of Operational Research 35, 71–79.
Lal, R. & R. Staelin (1984). An approach for developing an optimal discount pricing policy.
Management Science 30 (12), 1524–1539.
Lariviere, M. A. (1999). Supply chain contracting and coordination with stochastic demand. In
S. R. G. Tayur & M. Magazine (Eds.), Quantitative Models for Supply Chain Management.
Kluwer Academic Publishers.
Lariviere, M. A. & E. Porteus (2001). Selling to the newsvendor: An analysis of price-only
contracts. Manufacturing and Service Operations Management 3 (4), 293–305.
Martin, G. E. (1993, Sept.). A buyer-independent quantity discount pricing alternative.
Omega 21 (5), 567–572.
Monahan, J. P. (1984). A quantity discount pricing model to increase vendor profits. Management
Science 30 (6), 720–726.
Moorthy, K. S. (1984). Market segmentation, self-selection, and product line design. Marketing
Science 3 (4), 288–307.
Moorthy, K. S. (1987). Managing channel profits: Comment. Marketing Science 6 (4), 375–379.
Mussa, M. & S. Rosen (1978). Monopoly and product quality. Journal of Economic Theory 18,
301–317.
Pasternack, B. A. (1985). Optimal pricing and return policies for perishable commodities. Marketing Science 4 (2), 166–176.
Porteus, E. L. & S. Whang (1999). Supply chain contracting: Non-recurring engineering charge,
minimum order quantity and boilerplate contracts. Technical report, Stanford GSB, Research
Paper No. 1589.
Shin, H. & W. Benton (2004). Quantity discount-based inventory coordination: Effectiveness and
critical environmental factors. POM 13, 63–76.
25
Tsay, A., S. Nahmias, & N. Agrawal (1999). Modeling supply chain contracts: A review. In
S. R. G. Tayur & M. Magazine (Eds.), Quantitative Models for Supply Chain Management.
Kluwer Academic Publishers.
Viswanthan, S. & Q. Wang (2003). Discount pricing decisions in distribution channels with price
sensitive demand. European Journal of Operational Research 149, 571–587.
Weng, Z. K. (1995). Channel coordination and quantity discounts. Management Science 41 (9),
1509–1522.
26
Appendix
Continuum of Buyer Types with Uniform Demand Distributions
Let us consider a situation in which all consumers have demand that is uniformly distributed,
but they differ in terms of one of the parameters of the Uniform distribution. Specifically, let us
index consumer types by θ, and assume that a consumer of type θ has demand that is distributed
U [θ, θ + a]. Thus, fθ (x) =
x
a
, and Fθ (x) =
x−θ
a for
x ∈ [θ, θ + a]. The density of the parameter θis
g(θ), and assume that g(θ) has positive support on the interval [θ, θ̄].
The expected revenue function for a consumer of type θ is the following function of Q:
Ã
R(Q, θ) = r θ +
Z Q
θ+a−x
a
θ
!
¡
r 2Q(θ + a) − θ2 − Q2
dx =
2a
¢
(A.1)
and the consumer’s profit is equal to the total expected revenue less the total amount that he pays
to receive Q units, which we denote by w(Q). Thus, if the supplier offers a menu of contracts,
(Q(θ̂), w(θ̂)), then a retailer of type θ earns profits of:
πR (θ) =
o
M ax n
R(Q(θ̂), θ) − w(Q(θ̂))
θ̂
(A.2)
Differentiating with respect to θ, we have:
dπR (θ)
dR(Q, θ)
r(Q − θ)
=
=
dθ
dθ
a
Integrating, we can obtain an alternative expression for the profit of a type θ retailer:
πR (θ) =
r
a
Z θ
θ
(Q − µ)dµ − πR (θ) =
r
a
Z θ
θ
(Q − µ)dµ
(A.3)
By definition, w(Q(θ)) = R(Q(θ), θ) − πR (θ). The supplier’s profit can now be expressed as follows:
πM
=
=
Z θ
θ
Z θ
θ
(w(Q(θ) − cQ(θ))g(θ)dθ
(R(Q(θ), θ) − πR (θ) − cQ(θ))g(θ)dθ
where the last equality was obtained by substituting (A.1). Substituting (A.3) for πR (θ) into the
above, we have:
πM
=
Z θÃ
θ
r
R(Q(θ), θ) −
a
Z θ
θ
!
(Q − µ)dµ − cQ(θ) g(θ)dθ
We can apply integration by parts to the middle term in this integrand to obtain:
πM
=
=
Z θ·
θ
¸
r
(R(Q(θ), θ) − cQ(θ)) g(θ) − (1 − G(θ)) (Q − θ) dθ
a
Z θ ·µ
r
θ
¶
¸
r
(2Q(θ + a) − Q2 − θ2 − cQ(θ) g(θ) − (1 − G(θ)) (Q − θ) dθ
2a
a
27
The manufacturer wants to choose Q(θ) to maximize the integrand in the above expression for every
value of θ. Differentiating the integrand and setting the rusult equal to zero implies:
Q(θ) = θ + a −
ca 1 − G(θ)
−
r
g(θ)
Note that this has the desirable properties that: 1) Q(θ) is increasing in θ(So long as g() is IFR),
and 2) when θ = θ, we have Q(θ) = θ + a −
ca
r
so that 1 − Fθ (Q(θ) = rc .
28
29
w̃
25.00
30.00
35.00
40.00
45.00
46.56
Π˜PS
1050.00
1349.33
1618.90
1847.51
2001.46
2016.36
All-Unit
1062.44
1374.32
1658.77
1918.05
2145.60
2312.46
Improvement (%)
1.18
1.85
2.46
3.82
7.20
14.68
Incremental
1055.58
1365.43
1646.50
1903.72
2101.49
2263.74
Improvement (%)
0.53
1.19
1.70
3.04
5.00
12.27
T1
53.75
49.25
51.25
50.25
49.25
49.25
T2
63.75
57.50
61.50
60.50
59.50
59.25
T3
73.75
67.50
71.50
70.50
69.75
69.25
T4
84.00
77.75
81.75
80.75
80.00
79.50
T5
98.50
98.50
98.50
98.50
98.50
98.50
w1
24.74
29.94
34.23
38.74
43.02
46.32
w2
24.54
29.94
33.61
37.81
41.69
44.49
w3
24.37
29.94
33.13
37.11
40.67
43.19
w4
24.22
29.94
32.72
36.52
39.86
42.16
w5
23.59
28.76
31.21
34.48
37.29
39.19
Profit
1062.44
1374.32
1658.77
1918.05
2145.60
2312.46
50.13
47.50
45.00
42.00
42.00
42.00
25.00
30.00
35.00
40.00
45.00
50.00
60.25
57.75
55.25
52.25
52.25
52.25
T2
70.38
68.00
65.50
62.50
62.50
62.50
T3
80.50
78.25
75.75
77.75
77.75
77.75
T4
98.50
98.50
98.50
98.50
98.50
98.50
T5
25.00
30.00
35.00
40.00
44.71
48.57
w1
24.96
29.99
34.92
39.88
43.67
46.77
w2
24.89
29.91
34.79
39.74
42.91
45.50
w3
24.81
29.78
34.64
39.23
41.78
43.86
w4
24.06
28.49
32.74
36.05
38.06
39.70
w5
1055.58
1365.43
1646.50
1903.72
2101.49
2263.74
Profit
Profit
Loss(%)
.65
.65
.74
.75
2.06
2.11
Table 3: Incremental Discount (Quantities and Average Prices-per-unit) for Normally Distributed Demands and Varying External Price;
r = 50; c = 10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90
T1
x0
Table 2: All-Unit Discount for Normally Distributed Demands and Varying External Price; r = 50; c = 10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ;
µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90
w0
25.00
30.00
35.00
40.00
45.00
50.00
Table 1: Profits Under All-Unit and Incremental Quantity Discounts vs. Single Price for Normally Distributed Demands and Varying External
Price; r = 50; c = 10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90
w0
25.00
30.00
35.00
40.00
45.00
50.00
30
t1
50.06
47.48
44.88
41.79
39.79
36.33
t2
60.19
57.63
55.13
52.13
52.13
52.13
t3
70.31
67.88
65.38
62.38
62.38
62.38
t4
80.44
78.13
75.63
75.24
75.24
75.24
t5
94.27
93.09
91.85
92.84
92.84
92.84
v1
24.75
29.94
34.57
39.41
39.41
39.41
v2
24.50
29.45
34.13
39.04
39.04
39.04
v3
24.25
28.96
33.68
38.67
38.67
38.67
v4
24.00
28.47
33.23
29.45
29.45
29.45
v5
9.88
9.88
9.88
9.88
9.88
9.88
Profit
1055.58
1365.43
1646.50
1903.72
2101.49
2263.74
w̃
30.00
30.00
30.00
30.00
Π˜PS
1176.70
1163.40
1150.10
1136.80
All-Unit
1188.13
1186.76
1183.09
1182.90
Improvement (%)
.97
2.01
2.87
4.06
Incremental
1178.44
1170.50
1169.98
1167.68
Improvement (%)
.15
.61
1.73
2.72
T1
18.00
18.00
18.25
18.25
T2
50.50
51.00
51.50
52.00
T3
72.50
73.25
74.00
74.50
T4
101.00
104.00
107.00
110.00
w1
29.95
29.90
29.77
29.72
w2
29.85
29.70
29.49
29.35
w3
29.78
29.54
29.25
29.06
w4
29.38
28.78
28.17
27.69
Profit
1188.13
1186.76
1183.09
1182.90
Table 6: All-Unit Discount for Normally Distributed Demands and Varying Standard Deviation; r = 50; c = 10; w0 = v0 = 30; N1 = 9, N2 =
25, N3 = 36, N4 = 49; µ0 = 2
σ0
.50
1.00
1.50
2.00
Table 5: Profits Under All-Unit and Incremental Quantity Discounts vs. Single Price for Normally Distributed Demands and Varying Standard
Deviation; r = 50; c = 10; w0 = v0 = 30; N1 = 9, N2 = 25, N3 = 36, N4 = 49; µ0 = 2
σ0
.50
1.00
1.50
2.00
Table 4: Incremental Discount (Breakpoints and Marginal Prices) for Normally Distributed Demands and Varying External Price; r = 50; c =
10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90
x0
25.00
30.00
35.00
40.00
45.00
50.00
31
17.63
17.25
16.88
16.50
.50
1.00
1.50
2.00
49.41
48.88
48.25
47.63
T2
71.31
70.75
70.00
69.25
T3
100.97
104.00
106.88
109.88
T4
30.00
30.00
30.00
30.00
w1
29.96
29.96
29.96
29.96
w2
29.88
29.80
29.85
29.87
w3
29.39
28.85
28.62
28.33
w4
1178.44
1170.50
1169.98
1167.68
Profit
Profit
Loss(%)
.82
1.37
1.11
1.29
t1
17.62
17.24
16.87
16.49
t2
49.39
48.81
48.19
47.56
t3
71.30
70.70
69.95
69.20
t4
98.98
100.06
100.97
101.93
v1
29.94
29.94
29.94
29.94
v2
29.69
29.45
29.61
29.69
v3
29.53
29.13
29.40
29.53
v4
9.91
9.78
9.95
9.91
Profit
1178.44
1170.50
1169.98
1167.68
Table 8: Incremental Discount (Breakpoints and Marginal Prices) for Normally Distributed Demands and Varying Standard Deviation;
r = 50; c = 10; w0 = v0 = 30; N1 = 9, N2 = 25, N3 = 36, N4 = 49; µ0 = 2
σ0
.50
1.00
1.50
2.00
Table 7: Incremental Discount (Quantities and Average Prices-per-unit) for Normally Distributed Demands and Varying Standard Deviation;
r = 50; c = 10; w0 = v0 = 30; N1 = 9, N2 = 25, N3 = 36, N4 = 49; µ0 = 2
T1
σ0