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Transcript
Quantity Discounts in Single Period Supply
Contracts with Asymmetric Demand
Information
Apostolos Burnetas∗, Stephen M. Gilbert †,Craig Smith
‡
November 25, 2003
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-units and incremental
discounts and show that the supplier can earn larger profits with an all-units discount.
(Supply Chain Management, Channel Coordination, Channels of Distribution, Asymmetric Information)
∗
Case Western Reserve University, Department of OR and OM, 10900 Euclid Ave., Cleveland, OH 44106
[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 that 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 big box retailers is that they lack the volume necessary
to obtain quantity discounts from manufacturers. 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 investigate how a supplier can best use a quantity discount schedule 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 by allowing for different types of demand distribution for the buyer
and the supplier is uncertain as to which one represents the buyer’s demand. In this environment, the
quantity discount is a natural mechanism for the manufacturer to use for screening or differentiating
among buyer types. It is worth noting that if the information were symmetric, i.e. the supplier
knew the buyer’s distribution of demand, then a quantity discount would be a form of two-part tariff
and it would allow the supplier to induce the buyer to order the first-best quantity and extract all
of the expected profits from him. As a result, in a symmetric information environment, a quantity
discount can completely mitigate double-marginalization.
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 and
characterize an optimal all-units and incremental quantity discount schedules from the perspective
of the supplier. This analysis yields the important insight that the supplier will always prefer
to offer an all-units discount than an incremental discount. 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
The potential incentive problem between a supplier and a newsvending retailer that results in
understocking has been widely recognized in the literature. In his seminal paper, Pasternack (1985)
demonstrates that a returns policy can be used to insure that the retailer will order the first-best
quantity when the retail price is exogenous. The approach that he takes is to determine that a
contract that includes a returns policy can be used to induce the first-best order quantity, avoiding
the issue of whether the supplier will want to offer such a contract.
Since Pasternack’s paper was published, a number of other authors have examined issues
related to the incentive conflict between a supplier and retailer in single period stochastic demand
environments, and several review papers have been published. Lariviere (1999) provides an excellent
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. Note that, by adopting
the perspective of the supplier, the papers in this subset of the literature do not necessarily address
the question of maximizing the combined profits of the supplier and the retailer. Moreover, this
approach is consistent with the one that we have taken in this paper. Another review paper,
Cachon (1999), 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 literature on interactions between a supplier and a retailer in single period, stochastic
environments has tended to focus on four types of contractual mechanisms that can be used by a
supplier: returns policies, quantity flexibility, price protection, and revenue sharing. Among these
four types of mechanisms, returns policies have received perhaps the most attention. A returns
policy allows the retailer to receive a credit from the supplier for items that are returned at the end
of the selling season. Recent additions to this literature include Webster & Weng (2000), Donohue
(2000), and Tsay (2001). Webster & Weng (2000) consider a returns policy that provides a returns
3
credit to the retailer on only those units that would not have been ordered without a returns policy,
thereby assuring that the policy cannot reduce the supplier’s profits. In addition to a returns
credit, Donohue (2000) allows for a second opportunity for the retailer to order after additional
information about demand is revealed. Tsay (2001) compares the performance of returns policies
against markdown money that the supplier pays to the retailer for unsold items without requiring
that these items be returned.
Quantity flexibility allows the retailer to adjust its order quantity as more information about
demand becomes available. In fact, a returns policy that allows for the retailer to receive full-credit
for a limited number of returns is one way to implement a quantity flexibility contract. Tsay
(1999), Donohue (2000), and Taylor (2001) all consider variations on quantity flexibility in either
one or two period models in which some demand information is revealed between the retailer’s two
ordering opportunities. Lariviere (2002) considers both returns policies and quantity flexibility in
an environment in which the retailer can obtain improved demand information at some cost.
Price protection generally involves the supplier promising the retailer some form of compensation in the event that either retail or wholesale prices should fall. Lee, Padmanabhan, Taylor,
& Whang (2000) assume that the supplier protects the retailer against declines in an exogenous
retail price, while Taylor (2001) assumes that the protection is offered with respect to the wholesale
price. Both papers are based on a two-period model, and the latter one also includes for returns at
the end of both periods. The issue of retail price maintenance (RPM) is somewhat related to price
protection and has been studied in the economics literature by Gal-Or (1991a) and Gal-Or (1991b)
among others.
Revenue sharing involves splitting the total sales revenue that is collected between the
retailer and the supplier. This mechanism has received increased attention following its high profile
adoption by Blockbuster Video, and others. Examples of research on revenue sharing in single
period stochastic environments include Cachon & Lariviere (2002), Pasternack (2002), and Dana &
Spier (2001).
Another mechanism that is prevalent in practice is a quantity discount. However, nearly all
of the work that has been done on quantity discounts has 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 explained as a mechanism for overcoming double-marginalization, while
Moorthy (1987) argues that price discrimination provides a more plausible justification for the use
of quantity discounts. Ingene & Parry (1995) later extended this work to multiple independent
4
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 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 constant or price sensitive demand. Since our research involves a comparison between allunits 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-units 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-units 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 incremental discount.
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 both a quantity discount
as well as a total volume discount to be used in combination.
One of the significant features of our research is the fact that we consider asymmetry of
information between the supplier and the retailer regarding the distribution of demand. The research
on principal-agency issues associated with informational asymmetry in supply chains has tended
to focus on either informational asymmetry with respect to costs or with respect to demand. 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).
Atkinson (1979) was among the first to recognize and model the issue of asymmetric de5
mand information in a stochastic demand environment. His approach is consistent with 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.
With the recent exception of Lariviere & Porteus (2001), who examine the coordination /
agency issues associated with a simpler, price only contract in a newsvendor setting, the literature
has paid little attention to contractual mechanisms that can be completed in a single interaction.
Most of the work in this area focuses on returns policies, back-up agreements, or quantity flexibility
contracts as mechanisms for improving coordination. 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 than does the supplier. In contrast to the
contractual mechanisms that have received the most attention in newsvendor contexts, a quantity
discount requires no continuation of the relationship 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 quantity discount 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 costs that the buyer incurs are a linear function of the number of units that he
sells, and these costs have been normalized to zero.
6
We assume that the supplier has less information about the distribution of end-demand for
this product than does the buyer, and 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).
Prior to the selling season, the supplier announces a pricing schedule that can be characterized as either an all-units or an incremental quantity discount. An all-units discount can be
represented by a series of quantity breakpoints (T1 ≤ T2 ≤ ... ≤ Tn ), and per-unit wholesale prices
(w1 ≥ w2 ≥ ... ≥ wn ) such that price wi is available only to buyers purchasing quantities at or
above breakpoint Ti . An incremental discount can be represented by a series of quantity breakpoints
(t1 ≤ t2 ≤ ... ≤ tn ), and marginal prices (v1 ≥ v2 ≥ ... ≥ vn ) so that the buyer’s marginal cost of
units in the interval (ti , ti+1 ) is vi .
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 perunit price of w0 , (v0 ) where c < w0 = v0 < 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 w0 = v0 .
After the quantity discount schedule is announced, the buyer selects among his purchasing
options, including both the quantity discount schedule and the alternative source, and purchases the
quantity that maximizes his own expected profits. We denote by QAU
and QIi the optimal quantity
i
that a buyer of type i would purchase under a particular menu option for an All-Units or Incremental
discount. The supplier then produces the 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. There are two reasons for this: First, there are many
7
situations in practice in which it is desirable to avoid ongoing interactions between the supplier and
the seller. Second, by treating quantity discounts in isolation, we can clarify our understanding of
the role that they play.
3.1
Preliminaries
Before we begin our analysis of the all-units and incremental discounts, we must establish some
preliminary results. We first note that if a buyer of type i orders quantity q, then his expected
revenue can be expressed as:
Z
q
Z
∞
Z
fi (x)dx = r
xfi (x)dx + rq
Ri (q) = r
(1 − Fi (x))dx
(3.1)
0
q
0
q
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:
Ri (T 0 ) − Ri (T ) = r
≤ r
Z
T0
(1 − Fi (x))dx
T
Z T0
(1 − Fj (x))dx = Rj (T 0 ) − Rj (T )
(3.2)
T
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 news-vendor problem. Let also qi (w) be the optimal solution to
this unconstrained newsvendor problem, given a per-unit procurement cost of w. Thus:
qi (w) =
Fi−1
w
1−
r
If the buyer procures the product from the external market at the exogenous per-unit price of w0 ,
his optimal profits will be πi (qi (w0 ), w0 ). Let us denote this level of profits by i . Note that i
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.
Note that, 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 total expected gross profit in the channel is:
πi (qi (c), c) = Ri (qi (c)) − cqi (c) = r
Z
xfi (x)dx
0
8
qi (c)
(3.3)
Note that since w0 > c, we must have qi (c) > qi (w0 ) and πi (qi (c), c) > i . Clearly, if the supplier
can induce the buyer to order quantity qi (c) and extract all additional rents, this would maximize
the supplier’s profit. By substituting qi (c) for Ti in (3.6) it can be shown that the per-unit price
corresponding to quantity qi (c) that allows the buyer to earn expected gross profit equal to his
threshold profit i is:
Ri (qi (c)) − i
r
w =
=
qi (c)
∗
R qi (c)
0
xf (x)dx − i
+c
qi (c)
(3.4)
If there were no asymmetry of information, i.e. the buyer is of type i = 1 with probability p1 = 1,
then the supplier could induce him to order the channel optimal quantity qi (c) by offering an allunits quantity discount in which a per-unit price of w∗ is available on all orders for at least T = qi (c)
units. From the expression for w∗ in (3.4), it can be seen that the supplier would effectively be
selling the product at her own marginal cost (c) plus an adjustment to extract all of the profits
from the buyer, leaving him just enough that he weakly prefers the quantity discount to purchasing
from the external market. Thus, this arrangement results in the same order quantity and division
of profits as would an arrangement in which the supplier allows the buyer to order any quantity for
a per-unit price of w = c, and charges an additional franchise fee that extracts all of the profits.
Under an incremental quantity discount framework, we could accomplish the same thing by
making sure that the buyer’s first order condition was satisfied at quantity qi (c) by setting v1∗ = c.
Assuming that the highest marginal cost corresponds to the external market price, i.e. v0 = w0 ,
the threshold t1 would be set to make sure that the buyer weakly prefers the incremental discount
to ordering from the external market at per-unit price w0 .
t∗1 =
Ri (qi (c)) − i − cqi (c)
w0 − c
(3.5)
Note that this incremental quantity discount is equivalent to an arrangement in which the supplier
charges a per-unit price of c plus an additional franchise fee equal to t∗1 (w0 − c). It is worth noting
that in both the all-units and the incremental discounts that are discussed here, the buyer is induced
to order the quantity qi (c) and pays an average price per unit as specified in (3.4).
3.2
An All-Units Discount Under Asymmetric Information
We now turn to the more complex, and interesting, scenario in which the supplier has less information about demand than does the buyer. In this setting, the supplier must trade-off channel
coordination, i.e. inducing the buyer to order the quantity that would maximize the total profits in
the channel, against her own self-interests. Note that by setting w = c and placing no restrictions on
9
the buyer’s quantity, the supplier could induce him to order the quantity that that would maximize
channel profits, but the supplier would earn zero profits by making this offer.
Recall that we represent an all-units quantity discount schedule 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 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. Recall that wholesale price wj is available to the buyer only if he orders a
quantity q ≥ Tj . 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 }. A necessary condition for a type i buyer to purchase according
to the restrictions of menu option i is that wi and Ti be set in such a way that the buyer’s expected
gross profit is at least as large as what it would be if he procured the product from the external
market at per-unit price w0 :
πi (QAU
i (Ti , wi ), wi ) ≥ i = πi (qi (w0 ), w0 )
(3.6)
To make sure that the buyer self-selects the appropriate purchasing policy, the supplier must make
sure that when a buyer of type i purchases quantity QAU
i (Ti , wi ), he earns an expected gross profit
that at least exceeds his reservation profit, i , and that he has no incentive to purchase QAU
i (Tj , wj )
for any j 6= i. These are known as the Individual Rationality (IR) and the Incentive Compatibility
(IC) constraints respectively. (See Fudenberg & Tirole (1998) for more on these issues.)
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)
max{wi ,Ti } { ni=1 pi (wi − c)QAU
i (Ti , wi )}
s.t. πi (QAU
(T
,
w
),
w
)
≥
i
i
i
i
i
AU
πi (Qi (Ti , wi ), wi ) ≥ πi (QAU
i (Tj , wj ), wj )
wi−1 ≥ wi ≥ 0 , Ti ≥ Ti−1 ≥ 0
P
i = 1, ..., n
i = 1, ..., n; j 6= i
i = 1, ..., n
(IRi )
(ICi,j )
We now present the following result for the supplier’s problem (S.1) in order to obtain an alternative
representation of it.
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.
10
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 ),
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)
max{wi ,Ti } { ni=1 pi (wi − c)Ti }
s.t.
πi (Ti , wi ) ≥ i
πi (Ti , wi ) ≥ πi (QAU
i (Tj , wj ), wj )
wi−1 ≥ wi ≥ 0 , Ti ≥ Ti−1 ≥ 0
P
i = 1, ..., n
i = 1, ..., n; j 6= i
i = 1, ..., n
(IRi )
(ICi,j )
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, a fundamental assumption in the product-line design literature is that each
consumer purchases at most one item, and can select only among the discrete set of quality-price
pairs announced by the supplier. Note that such an assumption could be imposed upon our problem
by adding the following constraints: QAU
i (T, w) = T , for i = 1, ..., n to problem (S.2). Let us define
the resulting pricing problem as the fixed-package pricing problem.
Note that, in comparison to a schedule of fixed-package prices, a quantity discount affords
the buyer considerably more flexibility since it allows him to purchase any quantity that is at or
11
above an announced breakpoint. This can be seen in the incentive compatibility constraints (ICi,j )
in problem (S.2), where QAU
i (T, w) is the quantity greater than or equal to T that maximizes the
expected profit of buyer i at a per unit price of w. This is a non-trivial distinction between the two
problems, and it alters the analysis considerably.
With the problem specification in (S.2), it is now possible to compare the quantity discount pricing
problem to the fixed-package pricing problem.
Theorem 3.2 For a given set of parameters (r, c, , f1 (), ..., fn ()), the optimal value of the objective
function to the 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-units quantity discount problem, buyer i purchases
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.
12
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 )
(3.7)
for j = 2, ..., n, and let W1 (T1 , 0, w0 ) be the value of w1 that solves:
R1 (T1 ) − w1 T1 = 1 = R1 (q1 (w0 )) − w0 q1 (w0 )
(3.8)
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 (W (T, τ, ωj−1 (τ )) − c) + uj−1 (τ )}
(3.9)
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 (3.9), and
ωj (T ) = Wj (T, τj−1 (T ), ωj−1 (τj−1 (T ))) for j = 2, ..., n
(3.10)
ω1 (T ) = W1 (T, 0, w0 ).
The optimal solution to the n buyer-type problem is then:
vn = M axT {un (T )}.
(3.11)
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.
13
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 (3.7) to represent the fact that in fixed-package pricing, the buyer
must buyer quantity Tj−1 to obtain per-unit price wj−1 .
3.3
An Incremental Quantity Discount Under Asymmetric Information
Recall that an incremental discount scheme can be represented by a series of thresholds (t1 , t2 , ..., tn )
at which the marginal price changes and the marginal price to the buyer for units in the interval
(ti , ti+1 ) is vi .
It is important to recognize that, in contrast to the all-units problem, buyers will not tend
to order at the thresholds since the marginal cost decreases at each threshold. Note that in the
all-units problem, the buyer’s cost function is not generally concave. For example, consider a single
breakpoint T , where the per-unit price is w0 for quantities less than T and is w1 for quantities
q ≥ T . Thus, the buyer’s cost function, C(q), is equal to w0 q for q < T , and is equal to w1 q
otherwise. If there exists q > T for which w1 q < w0 (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:

ACi (q) =

i
X
1
vj−1 (tj − tj−1 ) + vi (q − ti )
q j=1
(3.12)
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)
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, w0 ), w0 )
πi (Qi (ti , vi ), ACi (Qi (ti , vi ))) ≥ πi (Qi (tj , vj ), ACj (Qi (tj , vj )))
vi−1 ≥ vi ≥ 0 , ti ≥ ti−1 ≥ 0
P
14
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-units 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-units problem:
(I.2)
max{wi ,Ti ,vi ,ti } { ni=1 pi (wi Ti − c)Ti }
s.t. πi (Ti , ACi (Ti )) ≥ πi (Qi (0, w0 ), w0 )
πi (Ti , ACi (Ti )) ≥ πi (Qi (tj , vj ), ACj (Qi (tj , vj )))
wi = ACi (Ti )
Ti = qi (vi )
T i ≥ ti
vi−1 ≥ vi ≥ 0 , ti ≥ ti−1 ≥ 0
P
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.
Based on this result, we can adopt a dynamic programming approach similar to the one used for
the all-units discount to solve the incremental discount problem.
Note that, 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)
15
(3.13)
Define πsAU to be the optimal solution to the all-units 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 , w0 , c, the supplier can earn at least
as much under an all-units 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 is 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 )
(3.14)
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
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 ))
(3.15)
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
(3.16)
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 ))
(3.17)
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-units 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
16
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-units 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.
3.4
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
the amount that he earns from the high end consumers by more than he 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
w0 (v0 ). 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 w0 (v0 ), the supplier can trivially satisfy this requirement
without adversely affecting its ability to extract rents from the higher indexed buyer types.
17
4
Numerical Experiment
In this section, we consider several numerical examples for the purpose of showing how the optimal
all-unit and incremental discounts are affected by the parameters. We first consider a case in which
there are five different buyer types, each of which are perceived by the supplier as being 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 = 40
and the supplier’s cost c = 10.
In tables 1, 2, and 3, the alternative wholesale price varies from 15 to 40. Table 1 shows the
results for all unit discounts. Tables 2 and 3 present the incremental discount case, in terms of the
breakpoints/marginal prices and order quantities/average prices, respectively. Note that when the
alternative price (w0 or v0 ) equals the retail price of 40, 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 (v0 ) increases, the quantities decrease and the average per-unit prices increase for
both types of discount.
From Tables 1 and 2, 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-units discount. Moreover, the magnitude of the gap between the quantity (average perunit price) from the all-units discount versus that for the incremental discount is increasing in the
external price (w0 = v0 ). 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) = 96.75. Finally, it can be observed
that, with the exception of when v0 = w0 = 30, the incremental discount results in a lower average
per-unit price for buyer type i = 5 than does the all-units discount.
In Tables 4, 5, and 6, we hold the alternative wholesale price at w0 = v0 = 25 and allow the
standard deviation of demand to vary from 5 to 25. As in the first three tables, we assume that the
retail selling price is r = 40, the production cost is c = 10, and the means of demand for the five
buyer types are µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90.
In both Tables 4 and 5, it can be observed that the supplier’s profit decreases with σ. This
18
is consistent with the idea that a given change in the critical ratio that governs the buyer’s quantity
has a bigger impact upon profit when the standard deviation of demand is large. It is also of interest
to observe 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 the all-unit case, increasing σ causes
the quantities to increase and the average per-unit prices to decrease. However in the incremental
case, 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-units discount than under an incremental discount, it is
interesting to note the magnitude of the difference which is shown in the far right column of Table
2 and Table 5. It can be noted that the incremental discount generally results in a loss of no more
than about 2 percent compared to the all-units discount.
To further investigate the magnitude of the difference in the supplier’s expected profits under
the two types of discount, we experimented with the parameters of normally distributed demand for
two equally likely types of buyer as shown in Table 7. In this table, we have computed the optimal
expected supplier profits under all-units (AU) and incremental (Inc.) discounts. For purposes
of comparison, we have also computed the supplier’s optimal expected profits under symmetric
information (SI). 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 and
leave the buyer with only enough profit to satisfy his individual rationality constraint. In this case,
the two buyer types are equally likely, so the reported expected SI profits represent an average of the
amounts that the supplier could extract from type one and type two buyers with perfect information
about their types. Obviously, the SI profits represent an upper bound on the supplier’s expected
profits under any mechanism. As before, we assume that the retail price r = 40, the production
cost c = 10, and the price form the alternative source of supply w0 = v0 = 25.
In Table 7, it can be seen that the expected profits from the all-units (AU) discount are
within 2.15 - 6.15 % of those with symmetric information (SI), while the incremental (Inc.) discount
profits are within 3.31 - 10.22 %. For both types of discount, the size of the profit gap (with respect
to symmetric information) increases with the magnitude of the difference between the means of the
two buyer types, and the size of the standard deviation. In the far right column of the table, it can be
seen that, although the size of the gap between the all-units and the incremental discount is relatively
small (varying from .043 to 4.56 %), these gaps are significant relative to the gap between the allunits discount profits and symmetric information profits. In fact, for µ1 = 45, µ2 = 65, σ = 25,
19
the profit gap between an incremental discount and symmetric information is 77 % larger than
that between the all-units discount profits and symmetric information profits. It is also interesting
to observe that that when the mean demands of the two buyer types are relatively similar (e.g.
µ1 = 53, µ2 = 58), the percentage profit gap between all-units and incremental discounts decreases
in the standard deviation of demand. However, when the mean demands of the two buyer types are
further apart (e.g. µ1 = 70, µ2 = 40), increased standard deviation tends to increase the magnitude
of this profit gap.
As a final experiment, we consider the following situation: Both the supplier and the buyer
know that a given consumer will purchase the product with probability α = .1, however the supplier
believes that it is equally likely that the buyer has N1 or N2 consumers, where N1 < N2 . Note
that this is simple though plausible explanation of one possible source of informational asymmetry
between the supplier and the buyer. Note also that it results in demand for each buyer type that
follows a binomial distribution with parameters Ni and α, such that our assumption of stochastic
dominance is satisfied. In contrast to our previous examples, this one implies a specific relationship
between the mean and standard deviation for a buyer type. As before, we assume that the retail
selling price is r = 40 and the production cost is c = 10. The results of the computations for this
scenario are presented in Tables 8 and 9. It can be observed that in all cases, buyer type i = 1
orders a smaller quantity (T1 ) and pays a higher average price per-unit (w1 ) under the incremental
discount than under the all-unit discount. Although buyer type i = 2 also pays a larger average
price per-unit under the incremental discount, his quantity is identical under both forms of discount
(as predicted by our analytical results).
5
Conclusions and Extensions
In this paper, we have considered how incremental and all-units discounts can be used to increase
a supplier’s profits when selling to a 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 overome informational asymmetry
regarding the buyer’s distribution of demand.
Our main results include characterizing the structure of the all-units and incremental discounts and demonstrating that the all-units discount always dominates the incremental discount
from the perspective of the supplier. This compliments previous results in EOQ settings with sym20
metric information in which the supplier has been shown to be indifferent between incremental and
all-units discounts.
One important feature associated with a quantity discount is the fact that it requires no
ongoing interactions between the buyer and seller. That is, after the initial transfer of units and
money, there is no need to monitor sales, collect unsold items, etc. Certainly this could be advantageous in certain environments in which a supplier may be able to benefit from selling an item
through a small buyer with which there is no reason to invest the fixed costs of establishing the
infrastructure that is necessary for continued interactions.
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
is 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 has no control over 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 allunits 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.
21
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Appendix
Proof of Theorem 3.3 a) Since these constraints are a subset of the ones specified in problem (S.2),
it is obvious that they are necessary. To see that they are sufficient, we will first demonstrate
that all of the downward incentive compatability constraints must be satisfied by showing that, for
i < j < k, if ICkj and ICji are both satisfied, then ICki must also be satisfied. From ICkj :
AU
Rk (Tk ) − wk Tk ≥ Rk (QAU
k (Tj , wj )) − wj Qk (Tj , wj ) ≥ Rk (Tj ) − wj Tj
(5.18)
where the latter inequality follows from the definition of QAU
k (T, w) as the order quantity Q ≥ T
that maximizes the profits of buyer type k at price w. From ICji and the definition of QAU
j (T, w):
AU
Rj (Tj ) − wj Tj ≥ Rj (QAU
j (Ti , wi )) − wi Qj (Ti , wi )
(5.19)
AU
≥ Rj (QAU
k (Ti , wi )) − wi Qk (Ti , wi )
(5.20)
AU
There are two possibilities for QAU
k (Ti , wi ). If Qk (Ti , wi ) ≥ Tj , then since wj ≤ wi it must also
AU
be true that QAU
k (Tj , wj ) = qk (Tj , wj ) ≥ Qk (Ti , wi ) = qk (Ti , wi ) ≥ Tj , and by the cancavity of
πk (q, wj ) with respect to q, we have:
AU
AU
AU
Rk (QAU
k (Tj , wj )) − wj Qk (Tj , wj ) ≥ Rk (Qk (Ti , wi )) − wj Qk (Ti , wi )
AU
≥ Rk (QAU
k (Ti , wi )) − wi Qk (Ti , wi )
(5.21)
Together with (5.18), this implies that ICki must be satisfied. Alternatively, suppose that QAU
k (Ti , wi ) <
Tj . From Lemma 3.1, we know that: Rk (Tj ) − Rk (Qk (ti , wi ) ≥ Rj (Tj ) − Rj (Qk (ti , wi ). Substituting
this into (5.19), we have:
AU
Rk (Tj ) − wj Tj ≥ Rk (QAU
k (Ti , wi )) − wi Qk (Ti , wi )
Constraint ICki follows from combining this expression with (5.18).
To see that the upward incentive compatability constraints must also be satisfied, it suffices
to show that for i < j < k, if ICij and ICjk are both satisfied, then ICik must also be satisfied.
AU
Note that, from stochastic dominance, QAU
l (T, w) ≤ Qm (T, w) for any indices l < m. Therefore,
AU
if QAU
j (Tj , wj ) = Tj , then Qi (Tj , wj ) = Tj , so that we can write ICij and ICjk as follows:
Ri (Ti ) − wi Ti ≥ Ri (Tj ) − wj Tj
(ICij )
Rj (Tj ) − wj Tj ≥ Rj (Tk ) − wk Tk (ICjk )
From Lemma 3.1, Rj (Tk ) − Rj (Tj ) ≥ Ri (Tk ) − Ri (Tj ). Thus, by re-arranging ICjk and substituting
for Rj (Tk ) − Rj (Tj ), we have:
Ri (Tk ) − Ri (Tj ) ≤ wk Tk − wj Tj
25
Together with ICij , this implies ICik . We have now shown that all of the incentive compatibility
(IC) constraints must be satisfied. To see that all of the individual rationality (IR) constraints are
also satisfied, we will show that ICj1 and IR1 together imply IRj for j = 2, ..., n. We must consider
two possibilities for buyer type j’s optimal response to the menu option for buyer type 1: Either
qj (w1 ) ≥ T1 or qj (w1 ) < T1 . Suppose first that qj (w1 ) ≥ T1 , so that QAU
j (T1 , w1 ) = qj (w1 ) and
ICj1 can be expressed as:
Rj (Tj ) − wj Tj ≥ Rj (qj (w1 )) − w1 qj (w1 )
It follows from the assumption that w0 > w1 that the right hand side of the above expression
is strictly greater than j = Rj (qj (w0 )) − w0 qj (w0 ). Now suppose that qj (w1 ) < T1 , so that
QAU
j (T1 , w1 ) = T1 . By rearranging constraint IR1 and dividing each side by the number of additional units that a buyer of type 1 would order under the discount versus from the external market,
we have:
w1 T1 − w0 q1 (w0 )
R1 (T1 ) − R1 (q1 (w0 ))
≥
T1 − q1 (w0 )
T1 − q1 (w0 )
(5.22)
The left hand side represents the average incremental revenue that a buyer of type i = 1 earns for
each unit that he buys under menu option 1 that he would not have purchased from the external
market. The right hand side represents the average incremental cost that this buyer incurs for
these units. We will show that a buyer of type j > 1 would also have greater average incremental
revenue than average incremental cost by purchasing under menu option 1 than by buying from the
external market. We first consider the average incremental revenue earned by buyer j for each unit
purchased under option 1 that he would not have purchased at price w0 , and note that:
Rj (T1 ) − Rj (qj (w0 )) = r
Z
T1
(1 − Fj (q))dq
(5.23)
qj (w0 )
Since 1 − Fj (qj (w0 )) =
w0
r
for all j = 1, ..., n, and Fj (q) < F1 (q) for all q, it follows that:
R1 (T1 ) − R1 (q1 (w0 ))
Rj (T1 ) − Rj (qj (w0 ))
≤
T1 − q1 (w0 )
T1 − qj (w0 )
(5.24)
Let us now consider the average incremental cost to buyer type j for the additional T1 − qj (w0 )
units that he would purchase under menu option 1. It follows from the fact that qj (w0 ) ≥ q1 (w0 )
that:
w1 T1 − w0 qj (w0 )
w1 T1 − w0 q1 (w0 )
≤
T1 − qj (w0 )
T1 − q1 (w0 )
(5.25)
By combining (5.22), (5.24), and (5.25), it follows that:
Rj (T1 ) − Rj (qj (w0 ))
w1 T1 − w0 qj (w0 )
≥
T1 − qj (w0 )
T1 − qj (w0 )
26
(5.26)
By multiplying both sides of (5.26) by T1 −qj (w0 ) and using the assumptions that QAU
j (T1 , w1 ) = T1
and constraint ICj1 is satisfied, it follows that constraint IRj must also be satisfied. b) If ICj,j−1
is satisfied at equality, then:
AU
Rj (Tj ) − Rj (QAU
j (Tj−1 , wj−1 )) = wj Tj − wj−1 Qj (Tj−1 , wj−1 )
(5.27)
AU
Lemma 3.1 implies that Rj (Tj ) − Rj (QAU
j (Tj−1 , wj−1 )) ≥ Rj−1 (Tj ) − Rj−1 (Qj (Tj−1 , wj−1 )). By
substituting into (5.27), we have:
AU
Rj−1 (Tj ) − Rj−1 (QAU
j (Tj−1 , wj−1 )) ≤ wj Tj − wj−1 Qj (Tj−1 , wj−1 )
(5.28)
AU
By re-arranging (5.28), the definition of QAU
j−1 (T, w), and the assumption that Qj−1 (Tj−1 , wj−1 ) =
Tj−1 , we have:
Rj−1 (Tj ) − wj Tj
AU
≤ Rj−1 (QAU
j (Tj−1 , wj−1 )) − wj−1 Qj (Tj−1 , wj−1 )
≤ Rj−1 (Tj−1 ) − wj−1 Tj−1
(5.29)
AU
Recall that stochastic dominance implies that if QAU
j (T, w) = T , then Qj−1 (T, w) = T . Thus, it
follows immediately from (5.29) that ICj−1,j must be satisfied. Having established that all of the
ICj−1,j constraints are satisfied, it follows from part a) that all of the other constraints must also
be satisfied.
c) Suppose first that (w1 , ..., wn ; T1 , ..., Tn ) is an optimal solution to (S.2) in which at least one of
the ICj,j−1 constraints are not binding. Let k be the largest index for which ICk,k−1 is not binding.
Now, consider the following iterative procedure for adjusting the prices indexed j = k, ..., n: Starting
with j = k, for each j = k, ..., n, increase wj to wj + δj where δj be the value of δ that satisfies:
AU
Rj (Tj ) − (wj + δ)Tj = Rj (QAU
j (Tj−1 , wj−1 )) − wj−1 Qj (Tj−1 , wj−1 )
(5.30)
Note that price wj is increased before determining the increment δj+1 for price wj+1 . By applying
this approach iteratively to the largest index k for which ICk,k−1 is not binding, we will eventually
arrive at a solution in which all ICj,j−1 constraints are binding. If IR1 is satisfied, then from part
(b), this modified solution must be feasible. Thus, it remains to be shown that our procedure does
not reduce the value of the objective function. We will do this by showing that in each iteration,
our procedure does not decrease any of the prices, i.e. δj ≥ 0 for j = k, ..., n. In a given iteration,
let us denote the adjusted prices by wj0 = wj + δj . From the definition of k as the largest index for
which ICk,k−1 is not binding, we have δk > 0. We will now show that, for j > k, if wj−1 is replaced
27
0
by wj−1
≥ wj−1 then the increment δj ≥ 0 so that wj0 ≥ wj . The definition of k also implies that,
for each j > k, constraint ICj,j−1 was binding prior to the adjustment of price wj−1 . Thus:
AU
Rj (Tj ) − wj Tj = Rj (QAU
j (Tj−1 , wj−1 )) − wj−1 Qj (Tj−1 , wj−1 )
0
0
AU
0
≥ Rj (QAU
j (Tj−1 , wj−1 )) − wj−1 Qj (Tj−1 , wj−1 ),
(5.31)
AU
where the inequality follows from the fact that πj (T, w) = Rj (QAU
j (T, w), w) − wQj (T, w) is
decreasing in w. It follows from (5.31) that there must be δj ≥ 0 such that:
0
0
AU
0
Rj (Tj ) − (wj + δj )Tj = Rj (QAU
j (Tj−1 , wj−1 )) − wj−1 Qj (Tj−1 , wj−1 ).
Thus, in each iteration of adjusting the prices, we must have wj0 ≥ wj for all j = k, ..., n. Since
the objective function of (S.2) is linearly increasing in the wj , our procedure cannot result in a
decrease in the objective value. If IR1 is not binding, then after adjusting the prices to insure that
all ICj,j−1 constraints are binding, we can increase w1 by δ1 , where δ1 satisfies:
R1 (T1 ) − (w1 + δ1 )T1 = .
and then adjusting prices wj , ..., wn according to the procedure above.
It remains to be shown that qj (wj ) ≤ Tj ≤ qj (c) for j = 1, ..., n, and Tj = qj (c) for j = n.
From Theorem 3.1 and the definition of QAU
j (Tj , wj ), it is obvious that qj (wj ) ≤ Tj . Suppose that,
in some optimal solution,w1∗ , ..., wn∗ ; T1∗ , ..., Tn∗ , we have Tn∗ 6= qn (c). Let w0 be the solution to:
πn (Tn∗ , wn∗ ) = πn (qn (c), w0 )
Note that by changing Tn∗ to qn (c), and wn∗ to w0 , we have left the expected profit of buyer n
unchanged. Hence, none of our IC or IR constraints have been violated. Recall that, by definition,
qn (c) is the quantity that maximizes the channel profit associated with buyer type n. Therefore,
Rn (qn (c)) − cqn (c) > Rn (Tn∗ ) − cTn∗
Since the supplier’s share of this profit is w0 qn (c), and the buyer’s share continues to be πn (qn (c), w0 ) =
πn (Tn∗ , wn∗ ), it follows that
w0 qn (c) = Rn (qn (c)) − cqn (c) − πn (qn (c), w0 )
> Rn (Tn∗ ) − cTn∗ − πn (Tn∗ , wn∗ ) = wn∗ Tn∗
(5.32)
(5.33)
(Note that these inequalities are strict so long as qn (c) is unique.) Hence the proposed substitution
has increased the value of the nth term in the supplier’s objective function, without affecting any
of the other terms. This contradicts the optimality of the solution w1∗ , ..., wn∗ ; T1∗ , ..., Tn∗ .
28
Now suppose that we do not have Tj∗ ≤ qj (c) for all j = 1, ..., n − 1, and that k is the largest
buyer type index for which Tk∗ > qk (c). Let w0 be the solution to:
πk (Tk∗ , wk∗ ) = πk (qk (c), w0 )
If we now change Tk∗ to qk (c) and wk∗ to w0 , we have left the profit of buyer type k unchanged, so
that ICk,j continues to hold for all j 6= k. Moreover, ICk,k−1 continues to be binding, so that ICj,k
continues to hold for all buyer types j < k. As before, the definition of qk (c) implies that:
Rk (qk (c)) − cqk (c) > Rk (Tk∗ ) − cTk∗
(5.34)
By applying the same reasoning as above, it can be shown that w0 qk (c) > wk∗ Tk∗ . Thus, we have
increased the value of the k th term in the objective function without affecting the first k − 1 terms.
By substituting qk (c) for Tk∗ and w0 for wk∗ , we may have created a violation of ICk+1,k so that:
∗
∗
0
πk+1 (Tk+1
, wk+1
) < πk+1 (QAU
k+1 (qk (c), w )
∗ , ..., w ∗ to restore all of the IC
If so, we can iteratively increment wk+1
j+1,j constraints to binding
n
for j = k, ..., n. Note that these proposed changes increase the value of the k th term in the objective
function, and do not decrease the value of any other terms. Since we have insured that all of
the ICj,j−1 constraints continue to be binding, we have contradicted the optimality of solution
w1∗ , ..., wn∗ ; T1∗ , ..., Tn∗ . ♦
The proof of Theorem 3.4 is similar to that for Theorem 3.3. It is available upon request from the
authors.
29
w0
15.00
20.00
25.00
30.00
35.00
40.00
T1
54.75
53.00
51.50
50.00
48.75
48.50
T2
64.75
63.00
61.50
60.25
59.00
58.50
T3
74.75
73.25
71.75
70.25
69.25
68.50
T4
84.75
83.25
81.75
80.50
79.25
78.75
T5
96.75
96.75
96.75
96.75
96.75
96.75
w1
14.97
19.87
24.67
29.35
33.77
37.29
w2
14.94
19.76
24.41
28.84
32.91
35.93
w3
14.92
19.66
24.18
28.45
32.24
34.97
w4
14.90
19.57
24.00
28.11
31.72
34.22
w5
14.81
19.24
23.30
26.97
30.08
32.15
Profit
367.83
706.99
1018.17
1299.68
1542.95
1720.36
Table 1: All-Unit Discount for Normally Distributed Demands; r = 40; c = 10; σ1 = σ2 = σ3 = σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 =
80, µ5 = 90
v0
T1
T2
T3
T4
T5
w1
w2
w3
w4
w5
15.00
20.00
25.00
30.00
35.00
40.00
53.25
50.25
47.00
43.50
41.25
41.25
63.50
60.50
57.25
53.75
51.50
51.50
73.75
70.75
67.50
64.00
61.75
61.75
84.00
81.00
77.75
74.25
76.75
76.75
96.75
96.75
96.75
96.75
96.75
96.75
15.00
20.00
25.00
30.00
34.92
38.98
14.98
19.93
24.95
29.94
34.41
37.66
14.92
19.82
24.85
29.85
34.02
36.74
14.83
19.69
24.73
29.73
33.38
35.57
14.63
19.23
23.85
28.27
30.87
32.60
Profit
359.82
695.25
1009.82
1288.92
1516.38
1684.04
Profit
Loss(%)
2.18
1.66
0.82
0.83
1.72
2.11
Table 2: Incremental Discount (Quantities and Avg. Prices-per-unit) for Normally Distributed Demands; r = 40; c = 10; σ1 = σ2 = σ3 =
σ4 = σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90
v0
15.00
20.00
25.00
30.00
35.00
40.00
t1
53.22
50.12
46.91
43.38
39.94
35.74
t2
63.37
60.37
57.13
53.63
51.38
51.38
t3
73.62
70.62
67.38
63.88
61.63
61.63
t4
83.87
80.87
77.63
74.13
74.37
74.37
t5
95.34
93.77
92.10
90.44
91.61
91.61
v1
14.90
19.60
24.72
29.69
32.37
32.37
v2
14.53
19.20
24.33
29.36
32.09
32.09
v3
14.15
18.80
23.95
29.03
31.81
31.81
v4
13.78
18.41
23.56
28.69
25.10
25.10
v5
10.00
10.00
10.00
10.00
10.00
10.00
Profit
359.82
695.25
1009.82
1288.92
1516.38
1684.04
Table 3: Incremental Discount (Breakpoints and Marginal Prices) for Normally Distributed Demands; r = 40; c = 10; σ1 = σ2 = σ3 = σ4 =
σ5 = 10 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90
σ
5.00
10.00
15.00
20.00
T1
50.75
51.50
52.50
53.50
T2
60.75
61.50
62.50
63.75
T3
70.75
71.75
72.75
74.00
T4
80.75
81.75
83.00
84.25
T5
93.25
96.75
100.00
103.50
w1
24.83
24.67
24.47
24.29
w2
24.70
24.41
24.09
23.76
w3
24.58
24.18
23.77
23.35
w4
24.49
24.00
23.50
23.02
w5
24.12
23.30
22.58
21.89
Profit
1032.86
1018.17
1004.02
991.48
Table 4: All-Unit Discount for Normally Distributed Demands; r = 40; c = 10; w0 = 25 ; µ1 = 50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90
σ
T1
T2
T3
T4
T5
w1
w2
w3
w4
w5
Profit
5.00
10.00
15.00
20.00
48.50
47.00
45.25
43.75
58.75
57.25
55.50
54.00
69.00
67.50
65.75
64.25
79.25
77.75
76.00
82.75
93.25
96.75
100.00
103.50
25.00
25.00
25.00
25.00
24.95
24.95
24.99
24.98
24.80
24.85
24.95
24.94
24.59
24.73
24.88
24.50
23.99
23.85
23.70
22.74
1017.46
1009.82
999.09
988.80
Profit
Loss(%)
1.49
0.82
0.49
0.27
Table 5: Incremental Discount (Quantities and Avg. Prices-per-unit) for Normally Distributed Demands; r = 40; c = 10; v0 = 25 ; µ1 =
50, µ2 = 60, µ3 = 70, µ4 = 80, µ5 = 90
σ
5.00
10.00
15.00
20.00
t1
48.45
46.91
45.24
43.69
t2
58.63
57.13
55.38
53.88
t3
68.88
67.38
65.63
64.13
t4
79.13
77.63
75.88
78.52
t5
91.18
92.10
92.79
97.93
v1
24.72
24.72
24.97
24.91
v2
23.95
24.33
24.72
24.72
v3
23.17
23.95
24.46
24.53
v4
22.38
23.56
24.21
17.81
v5
10.00
10.00
10.00
10.00
Profit
1017.46
1009.82
999.09
988.80
Table 6: Incremental Discount (Breakpoints and Marginal Prices) for Normally Distributed Demands;; r = 40; c = 10; v0 = 25 ; µ1 = 50, µ2 =
60, µ3 = 70, µ4 = 80, µ5 = 90
Symmetric
Information
µ1
53
53
53
53
50
50
50
50
45
45
45
45
40
40
40
40
µ2
58
58
58
58
60
60
60
60
65
65
65
65
70
70
70
70
σ
10
15
20
25
10
15
20
25
10
15
20
25
10
15
20
25
Profit (SI)
849.57
861.85
874.13
886.42
849.57
861.85
874.13
886.42
849.57
861.85
874.13
886.42
849.57
861.85
874.13
886.42
Asymmetric
Information
All Unit
Incremental
Profit (AU) Profit (Inc)
831.30
821.49
836.88
830.38
845.76
841.03
856.06
852.37
830.44
813.38
833.73
812.60
837.43
817.98
841.96
826.15
828.72
812.91
831.07
807.50
833.79
801.76
836.89
798.71
826.98
812.45
828.37
807.35
830.06
801.29
832.09
795.82
Percentage
Profit Gaps
AU vs. SI
2.15
2.90
3.25
3.43
2.25
3.26
4.20
5.02
2.45
3.57
4.61
5.59
2.66
3.88
5.04
6.13
Inc. vs. SI
3.31
3.65
3.79
3.84
4.26
5.71
6.42
6.80
4.32
6.31
8.28
9.89
4.37
6.32
8.33
10.22
Inc. vs. AU
1.18
0.78
0.56
0.43
2.05
2.53
2.32
1.88
1.91
2.84
3.84
4.56
1.76
2.54
3.47
4.36
Table 7: Sensitivity of profit gap to normally distributed demand for two buyer types
N1
500
600
700
800
900
N2
1000
1000
1000
1000
1000
T1
50.13
60.38
70.75
81.00
91.38
T2
106.50
106.50
106.50
106.50
106.50
w1
24.88
24.87
24.85
24.84
24.83
w2
24.20
24.19
24.17
24.16
24.15
Profit
1128.87
1204.25
1279.73
1355.26
1430.82
Table 8: All-Unit Discounts for binomial (0.1, Ni ) demand. r = 40, c = 10, w0 = 25
N1
500
600
700
800
900
N2
1000
1000
1000
1000
1000
t1
47.87
57.70
67.49
77.34
87.19
t2
101.61
101.66
101.61
101.64
101.66
v1
24.97
24.81
24.94
24.86
24.8
v2
10.00
10.00
10.00
10.00
10.00
T1
47.88
57.75
67.50
77.38
87.25
T2
106.50
106.50
106.50
106.50
106.50
w1
25.00
25.00
25.00
25.00
25.00
w2
24.29
24.23
24.29
24.28
24.28
Profit
1120.02
1191.05
1267.08
1340.57
1415.04
Percent
Profit Gap
.78
1.10
.99
1.08
1.10
Table 9: Incremental Discounts for binomial (0.1, Ni ) demand. r = 40, c = 10, v0 = 25