1.
SOME PROPERITES OF BUYBACK AND OTHER RELATED SCHEMES IN A
NEWSVENDOR-PRODUCT SUPPLY CHAIN WITH PRICE-SENSITIVE DEMAND
Keywords: Supply Chain. Newsvendor Product. Buyback. Resale Price Maintenance.
by
Amy Hing Ling LAU
School of Business
The University of Hong Kong
Pokfulam Road, HONG KONG
E-mail: ahlau@business.hku.hk
Hon-Shiang LAU (corresponding author)
Spears School of Business
Oklahoma State University,
Stillwater, OK 74074, USA
E-mail: wendy@okstate.edu
and
Jian-Cai WANG
School of Business
The University of Hong Kong
Pokfulam Road, HONG KONG
E-mail: wangjc@business.hku.hk
(authors’ names arranged in alphabetical order)
July 2005
2.
ABSTRACT
A dominant manufacturer supplies a newsvendor-type product to a retailer, whose market
volume varies with the unit retail price according to a stochastic demand curve. We study the
design and performance of “price-only,” “buyback” and “manufacturer-imposed retail price”
schemes. All these schemes have been considered in earlier papers. The first part of this paper
studies some important but previously overlooked aspects of price-only and buyback schemes.
We show that the performance of these schemes are strongly and somewhat counter-intuitively
affected by the specific form of demand curve and demand randomization. These results are
important for designing price-only and buyback schemes for actual implementation. The
second part of this paper demonstrates the practicality and merit of using buyback in
conjunction with a manufacturer-imposed retail price – an arrangement overlooked in the
literature because it is widely mistaken as illegal. Overall, the paper provides new insights on
how a manufacturer should design practical buyback schemes to improve her profit.
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3.
SOME PROPERITES OF BUYBACK AND OTHER RELATED SCHEMES IN A
NEWSVENDOR-PRODUCT SUPPLY CHAIN WITH PRICE-SENSITIVE DEMAND
§1. INTRODUCTION
§1.1 Brief Problem Statement
A dominant “manufacturer” wholesales a newsvendor-type (or a “single-period”) product to
a “retailer,” who then retails the product to the consumer market at $p/unit. The expected value
of the retail-market demand varies with p according to a known “demand curve” Dp, while the
stochastic demand at any given p-value follows a known probability distribution. How should
or would the “players” (i.e., the manufacturer and the retailer) make their pricing and ordering
decisions? The current literature on this widely studied problem is briefly summarized in §1.3.
This paper concentrates on one aspect of this problem: the design and performance of “price-
only”, “buyback” and “manufacturer-imposed retail price” schemes. All these schemes have
been considered by earlier papers. Our purpose is to supplement the earlier works by presenting
additional insights and information on the performance of these schemes. As will be seen, these
additional insights are important for the practical implementation of these schemes.
§1.2 Definition of Basic Terms and Symbols
The random demand per period of the “single-period” or “newsvendor” product is D, with
probability density function (pdf) f(•), cumulative distribution function (cdf) F(•), mean µD,
standard deviation σD, and finite support (Dmin, Dmax). The manufacturer incurs a manufacturing
cost of $k/unit; she wholesales to the retailer at $w/unit. Without loss of generality, we assume k
= 1 throughout this paper. The retailer buys VR units from the manufacturer and sets the retail
price at $p/unit. The retailer incurs a loss-of-goodwill cost of $π/unit for demand not satisfied
during the period. At the end of the period, the retailer’s unsold units can be salvaged in the
open market for $s/unit. Typically, s < k ≤ w ≤ p. The manufacturer may also offer a “buyback”
scheme [w,β], under which the manufacturer “buys back” the unsold product for $β/unit. We
assume β < w; i.e., we consider only the “full return with partial credit” version of buyback (see,
e.g., Pasternack 1985 for other buyback variations). For β to be meaningful, obviously either “β
> s” or “β = 0”. The retailer’s and manufacturer’s random profits are θR and θM, respectively.
Generally, we denote the expected value of a random variable x as a bold capitalized X.
Define:
ΘR and ΘM : the expected values of θR and θM, respectively;
ΘC = ΘM+ΘR (subscript “C” denotes “channel”);
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ΘI = the expected profit of a vertically integrated firm, i.e., a single manufacture/ retailer
entity (subscript “I” denotes “integrated”);
CE = channel efficiency = (ΘM+ΘR)/ΘI.
In the newsvendor literature, the most common price-demand relationship consists of two
components: (i) the “demand-curve” function Dp(p) specifying how µD varies with p; and (ii) the
“randomization process,” which determines how σD varies with p. Regarding the first
component, this paper will consider the two most widely assumed demand-curve functions in
the related literature (see, e.g., Arcelus & Srinivasan 1987, Parlar & Wang 1994, Li & Huang
1995, Urban & Baker 1997, Weng 1999, Ertek & Griffin 2002, among numerous others):
(i) Linear demand curve: Dpl(p) = a–bp, where (a/b) > k (k = unit manufacturing cost);
(ii) Iso-elastic demand curve with constant elasticity: Dpc(p) = K/pα, α > 1.
There are two common forms of modeling randomness (or σD) of price-sensitive demands:
the “multiplicative” and the “additive” form; for a comparison between them, see, e.g., Petruzzi
& Dada (1999) and Arcelus et al. (2005). This paper will consider both forms. Under the
multiplicative form (used in, e.g., Emmons & Gilbert 1999), given a specified retail-price level
p0, the mean demand is µD = Dp(p0), and this demand is then randomized by multiplying µD with
a random term ε; i.e.,
D = µD•ε = [Dp(p0)]•ε ; where µε = 1. (1a)
Under this structure σD (= µD•σε ) varies with p, but D’s coefficient of variation cv(D) remains
constant as p varies; i.e.,
cv(D) = σD/µD = σε. (1b)
Following earlier studies, we assume that ε is uniformly distributed. Hence the range of D’s
finite support is:
Dmin = µD[1–σε√3]; and Dmax = µD[1+σε√3]. (2)
Thus, as long as we restrict σε’s magnitude to no more than (1/√3) or 0.577 (an assumption
made in, e.g., Emmons and Gilbert 1999), Dmin is positive; i.e., no negative realized-D value will
arise.
The additive form is used in, e.g., Ha (2001) and Lau & Lau (2002). Under this form µD
(i.e., Dp(p0)) is randomized by the additive relationship
D = µD +ε = Dp(p0) + ε ; where µε = 0. (3)
In contrast to the multiplicative form, here σD = σε remains constant as p varies, but cv(D) varies
with p. Under the additive form, D’s finite support is:
Dmin = µD–σD√3; and Dmax = µD + σD√3. (4)
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5.
In contrast to (2) for the multiplicative form, the Dmin in (4) will become negative when p is
sufficiently large and hence µD becomes sufficiently small ― this is true no matter how smallσD
is. The significance of this factor will become clear in §4.1.2.
In many related earlier works either a linear or an iso-elastic demand curve is chosen for
illustration, typically with little justification. The implicit assumption appears to be that either
curve is reasonable and the main results/insights obtained under one curve can be generalized to
other curves. For perhaps similar reasons, in most earlier works either the multiplicative or the
additive form is chosen to randomize demand, again with little justification. This paper will
reveal the substantial effects of arbitrarily selecting a demand-curve form and/or a demand
randomization process in modeling a newsvendor-product supply chain.
§1.3 Brief Literature Review
Among others, Silver, Pyke and Peterson (1998) and Khouja (1999) provided comprehensive
reviews to the huge literature on newsvendor-type products. Most earlier studies considered a
one-echelon scenario; i.e., an “integrated” firm doing both manufacturing and retailing.
Pasternack’s (1985) pioneering paper on a two-echelon scenario considered a fixed-p scenario;
his model assumes that:
(i) all parameters are symmetrically and perfectly known to both players, who are expected-
profit maximizers;
(ii) the manufacturer is the dominant player in a manufacturer Stackleberg game.
In contrast to Pasternack’s (1985) fixed-p scenario, this paper considers the situation in
which p can be varied by the retailer, albeit subject to a demand curve Dp. We summarize below
the known results relevant to this variable-p problem:
● Pasternack (1985) considered a buyback contract[w,β] under which the retailer can return
unsold units to the manufacturer for a credit of $β/unit. When p is a fixed exogenously,
Pasternack (1985) showed that a price-only [w] contract does not coordinate a channel, but there
exist an infinite number of channel-coordinating (“cc”) buyback policies [wcc,βcc]. By selecting
an appropriate pair of [wcc,βcc]-values, the manufacturer possesses absolute power in deciding
what proportion of the optimal channel profit ΘI* the retailer can earn.
● When p can be varied by the retailer while the manufacturer is the dominant player, many
studies have shown that a [w,β] scheme cannot perfectly coordinate the channel. See, e.g.,
Kandel (1996), Emmons & Gilbert (1998), Weng (1999), Ha (2001), Lau & Lau (2002), and
Bernstein & Federgruen (2005). Various schemes that can coordinate perfectly such a channel
have been studied in the literature, among them are two-part tariffs (Weng 1999, Ha 2001),
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quantity fixing (Ha 2001) and revenue sharing (Cachon & Lariviere 2005). Bernstein &
Federgruen (2005) proposed a “price discount sharing” (PDS) scheme (see also Cachon 2003),
which is a more sophisticated buyback scheme in which both the unit wholesale price w and the
buyback price β are set as linear functions of the retailer-imposed p. Ha (2001) pointed out that
a dominant manufacturer can also perfectly coordinate the channel by imposing simultaneously
a [w,β] scheme and an appropriate retail price (hereafter the “{pm} scheme”), but he then
indicated that such a manufacturer-imposed retail price might be illegal. Tsay (2001) considered
the use of “markdown money” and pointed out that this device does not perfectly coordinate the
channel.
● While the above-mentioned studies assumed players to be expected-profit maximizers, Weng
(1999) and Tsay (2002) showed that one should not overlook the players’ risk aversion. Weng
(1999) showed that a two-part-tariffs scheme can perfectly coordinate a channel with risk-averse
players. In another direction of extension, Ha (2001) considered a situation in which the
dominant manufacturer does not have perfect information on the retailer’s “unit preparation
cost” (which the retailer has to incur in addition to w), and he studied how the manufacturer
could design a “contract menu” to maximize her expected profit.
For a two-echelon newsvendor-product channel, beyond the basic price-only ([w]) scheme,
the most commonly observed scheme in the real world is buyback ([w,β]). Revenue sharing is
relatively new and implemented by only a small number of firms/industries, while some other
theoretical schemes considered in the academic literature have not yet been implemented in the
real world. Therefore, although the [w] and [w,β] schemes do not perfectly coordinate the
channel, it is worthwhile to take a closer look at their various characteristics. Looking at another
angle, one notes that there are some real-world cases where a manufacturer-mandated retail
price is imposed (not necessarily for a newsvendor-type product or in conjunction with
buyback), but this is widely perceived as a practice of questionable legality and propriety. Thus,
the fact that Bernstein & Federgruen (2005) proposed “PDS” (which is a more sophisticated
buyback scheme) implies that Ha’s simpler {pm} scheme is perceived to be unacceptable due to
its manufacturer-imposed-p component. In view of the above background, the purpose of this
paper is to present new results/insights on the [w], [w,β] and {pm} schemes. Our results reveal a
number of important factors that must not be overlooked when designing[w] and [w,β] schemes.
We also show that {pm} is much more attractive than what has been widely perceived.
§1.4 Overview of the Paper and Summary of Findings
The mathematical formulations of our two-echelon newsvendor problems are stated in
Section 2. In Section 3 a multiplicative form of random demand is assumed, and we compare
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the performance of the [w] and [w,β] schemes under the two demand-curve forms: Dpl and Dpc.
As a counterpart to Section 3, we assume in Section 4 an additive form of random demand, and
again compare the performance of [w] and [w,β] under Dpl and Dpc. These investigations reveal
that the two demand randomization processes and the two demand-function forms lead to
significantly different performance patterns. The implications of these differences are discussed
in §4.3. Section 5 presents information for correcting some common misconceptions on using a
manufacturer-imposed retail price pM, leading to the conclusion that {pm} is a very attractive
channel coordinating device. The concluding §6 includes brief suggestions of future extensions.
§2. MATHEMATICAL FORMULATIONS OF OUR PROBLEMS
§2.1 Basic Expressions for the Newsvendor Problem
Because this paper uses extensive numerical investigations, we summarize in (5) to (9)
below the expressions given in Lau & Lau (2002). These expressions simplify considerably
newsvendor-model computations.
Define Ex(q) as x’s “partial expectation with upper limit q” (Winkler, Roodman & Britney
1972), i.e.,
q
Ex(q) = ∫ xg ( x )dx ,
x
where g(x) is x’s density function. (5)
Simple formulas for computing Ex(q) for various x-distributions can be found in, e.g., Winkler et
al. (1972) or Lau & Lau (2002).
Let SL denote “service level” (the probability of meeting all demand). For an integrated
firm and under a given (or fixed) p-value, it is known that:
θI = −kVI+p•min(VI, D)+s•(VI−D)+−π(D−VI)+ (for any given production quantity VI) (6)
SLI* = (p+π−k)/(p+π−s) (7a)
VI* = (integrated firm’s optimal production quantity) = F−1(SL*) (7b)
ΘI* = (p+π−s)•ED(VI*)−πµD (ED(•) defined in (5)) (8)
When µD varies with p according to a given demand curve Dp, the integrated-firm’s problem
can be stated as, using (8) and (1):
P0. 1Find pI* that maximizes ΘI = (pI+π−s)•ED(VI)−π•µD,
(9)
where for a given pI-value; µD = Dpl(pI), and VI = F−1[(pI+π−k)/(pI+π−s)].
§2.2. Formulations for the Problems Considered in this Paper
We consider a dominant manufacturer implementing a manufacturer-Stackelberg game for a
newsvendor product in a two-echelon channel. In a price-only ([w]) scheme the manufacturer
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announces a unit wholesale price w, then the retailer responds with unit retail price p and order
volume VR. One way of stating this supply-chain problem is (see, e.g., Lau & Lau 2002):
P1. 1Find w* that maximizes ΘM = (w−k)•F−1[(pw*+π−w)/(pw*+π−s)] ;
where pw* is the p-value that maximizes ΘR = (p+π−s)•ED(VRw*)−π•µD
(10)
for a given w-value; and
VRw* = F−1[(p+π−w)/(p+π−s)].
In a buyback ([w,β]) scheme the manufacturer announces a unit buyback price β in addition
to w; the retailer then responds with p and VR. This problem can be stated as (see, e.g., Lau &
Lau 2002):
P2. 1Find [w*, β*) that maximize ΘM =
[w−k−(β−s)(p2*+π−w)/(p2*+π−β)]•F−1[(p2*+π−w)/(p2*+π−β)]+(β−s)•ED(VRw*),
where p2* is the p-value that maximizes ΘR = (p+π−β)•ED(VRw)−π•µD for (11)
given values of (w,β); and
VRw* = F−1[(p2*+π−w)/( p2*+π −β)].
Since it is recognized from the literature that neither P1 nor P2 can be solved analytically,
we study their behavior by solving them for a very large number of different combinations of
parameter values. However, for practicality sake, the representative characteristics revealed by
observing these numerous solutions are reported in this paper via a very small number of
examples. Also, P1 and P2 are stated in the forms shown in (10) and (11) because they lead to
simpler computational procedures.
The optimizations for solving P1 and P2 are performed using the IMSL (1994) subroutine
BCPOL. This subroutine executes Nelder-Mead’s (1965) algorithm, which does not assume a
smooth function. Furthermore, to ensure that the global (instead of a local) optimum is found,
for each problem the subroutine BCPOL is executed with 100 different initial points ― noting
that P1 and P2 have no more than three decision variables. These 100 different initial points are
generated with the IMSL subroutine GGUES, which uses Aird and Rice’s (1977) procedure to
systematically disperse a given number of initial points over a multi-dimensional space.
§3. ASSUMING A MULTIPLICATIVE FORM OF RANDOM DEMAND
In this section we assume that the demand D is randomized multiplicatively by ε, as in (1).
Recall that in this paper we assume that k (unit manufacturing cost) = 1 and ε is uniformly
distributed.
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§3.1 Linear Demand Curve Dpl = a−bp
§3.1.1. A Price-Only (No Buyback) Scheme [w]
As will become clear later, in §3.1.1 we will consider only situations where π = s = 0. The
remaining free parameters under the current scenario are then a, b and σε. Among the numerous
combinations of (a, b, σε)-values we considered, Table 1 presents a very small subset of
representative solutions to problem P1 (defined in (10)). As in other similar numerical
illustrations in the literature (e.g., Emmons & Gilbert 1998, Lau & Lau 2002), the solutions in
Table 1 illustrate two well-known phenomena:
(i) (ΘM[w]*+ΘR[w]*) < ΘI*; i.e., the channel is not coordinated; and
(ii) ΘM[w]* > ΘR[w]*, which appears to be intuitively expected, since the manufacturer is the
dominant player and the leader.
However, regarding phenomenon (ii), we pose the following question:
Q1: Will ΘM[w]* always be larger than ΘR[w]*
(given that the manufacturer is the dominant leader)? (12)
Although this has not been explicitly investigated in the literature, results from earlier works on
newsvendor-product supply chains suggest that the answer is “yes.” In fact, it appears
intuitively obvious that the dominant and leading player (the manufacturer) would earn a higher
profit than the dominated follower. However, we will show in §3.2 that the answer is “no.”
§3.1.2. A Buyback Scheme [w,β]
The current scenario is the same as the scenario considered in the preceding §3.1.1, except
that the manufacturer now implements [w,β]. As stated in §1.3, there exists an infinite number
of [w,β] schemes that perfectly coordinate the channel with CE = 1 when p is a constant. In
contrast, when the retailer can vary p, a [w,β] scheme cannot perfectly coordinate the channel.
Table 2 presents a very small subset of the large number of representative solutions we have
obtained for problem P2 (defined in (11)). Recall that the integrated-firm solutions are the same
as those given in Table 1.
Note that although CE < 1 in Table 2 (as emphasized in the literature), but ΘM[wβ]* in Table 2
is greater than the corresponding ΘM[β]* in Table 1; e.g., for (a, b, σε) = (100, 50, 0.55), ΘM[wβ]* (=
2.4, one of the underlined-italicized entries in Table 2) is greater than ΘM[β]* (= 2.1, Table 1).
That is, although a [w,β] scheme does not coordinate the channel, it does give the manufacturer
a higher profit. Thus, the dominant manufacturer may still want to introduce a [w,β] scheme,
noting that in many cases a manufacturer is probably more interested in maximizing her own
profit than CE. In the real world, buyback is a much more widely implemented channel-
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10.
coordinating device for newsvendor products than such alternatives as revenue sharing or two-
part tariffs. Therefore, beyond the recognition that a 100%-CE cannot be attained (as
emphasized in the earlier works), it is worthwhile to look closer at the solutions of the buyback
formulation P2 (i.e., (11)). This is one of the major purposes of this paper.
Tables 1 and 2 depict the following characteristics (confirmed by a much larger set of
solutions not shown):
(a) Not only is ΘM[wβ]* always greater than the corresponding ΘM[β]*, but the CE attainable with a
[w,β] scheme is also always larger than the corresponding CE of a [w] scheme. In contrast,
the retailer’s ΘR[wβ]* is always smaller than the corresponding ΘR[β]*. That is, buyback
enables the manufacturer to increase her profit via two sources: slightly higher CE and
cannibalization of ΘR. It has been well established in the literature that under a fixed p, a
dominated retailer may actually lose when the dominant manufacturer offers buyback
schemes. We show here that even when the retailer has the new power of setting p, he may
still lose under a buyback scheme.
(b) While Table 1 shows that ratio (ΘM[w]*/ΘR[w]*) for a newsvendor product is not a constant
under a [w] scheme, Table 2 shows that the ratio (ΘM[wβ]*/ΘR[wβ]*) under [w,β] remains
constant at 2. We now need to digress temporarily and refer to a larger and more established
part of the game-theoretic two-echelon supply chain literature regarding a “regular” (for lack
of a better name) or “non-newsvendor-type” product ― i.e., a product whose demand at a
given p0-value is a deterministic value Dpl(p0). For such a “regular” product, it is well known
that the ratio (ΘM*/ΘR*) under a manufacturer-Stackelberg game is a constant of 2 (e.g.,
Tirole 1988). Thus, we now see an unexpected and interesting equivalence between a
regular-product channel and a newsvendor-product [w,β] channel (see, however, §4.1.2 for
partial refutation). However, since P2 cannot be solved analytically, we are only able to
demonstrate numerically that the ratio (ΘM*/ΘR*) remains constant at 2 when buyback is
implemented.
(c) Similarly, while Table 1 shows that the CE attainable for a newsvendor product under a [w]
scheme is a variable value less than 0.75, Table 2 shows that the CE attainable for a
newsvendor product under a [w,β] scheme is a constant of 0.75. Again, for a “regular”
product it is known that the CE of a manufacturer-Stackelberg game is also a constant of
0.75 (see, e.g. Bresnahan and Reiss 1985, Tirole 1988), and we now see another equivalence
between a regular-product channel and a newsvendor-product [w,β] channel (see, however,
§4.1.2 for partial refutation).
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11.
Table 2 also depicts the following unexpected “neat” characteristics for a [w,β] scheme under a
linear Dpl:
w* = (a+bk)/(2b), and β* = (a+bs)/(2b). (13)
However, we are only able to “prove” (13) numerically, but not analytically. An analytical proof
of these relationships is probably not worthwhile anyway because, by contrasting the last two
rows of w* and β* figures in Table 2 with those in the preceding rows, it can be seen that the
above relationships (13) do NOT hold when π ≠ 0. Incidentally, the w*-formula in (13) is
identical to the w*-formula for a regular product in a manufacturer-Stackelberg two-echelon
channel (see, e.g., Lau & Lau 2003).
We are unable to obtain “neat” relationships similar to (13) that incorporate a non-zero “π.”
We will return to this point later in §4, where additively-randomized demands are considered.
§3.2. Iso-elastic Demand Curve Dpc = K/pα
On the issue of channel coordination, earlier works (Ha 2001, Bernstein and Federgruen
2005, among others) have already shown that, regardless of the demand-curve form, neither a
price-only [w] nor a buyback [w,β] scheme can coordinate the channel. However, we consider
below other aspects of the problem, among which is: how does the demand-curve form affect
the players’ profit ratio (ΘM*/ΘR*).
§3.2.1. A Price-Only (No Buyback) Scheme [w]
Among the numerous combinations of (π, s, K, α, σε)-values we considered, Table 3
presents a very small subset of representative solutions to problem P1 (see (10)).
As a preliminary answer to our earlier “Question Q1,” Table 3 illustrates a situation in which
the dominant manufacturer-leader’s profit ΘM[w]* is less than the retailer’s ΘR[w]*. This
contradicts the expectation one might surmise from the newsvendor supply chain literature, and
this phenomenon is elaborated below.
In the literature on two-echelon “regular” products mentioned earlier in §3.1.2, it has been
shown analytically (see, e.g., Lau & Lau 2003) that the ratio of the players’ profit in a
manufacturer-Stackelberg game under an iso-elastic Dpc is
ΘM*/ΘR* = (α−1)/α ; (14)
i.e., ΘM[w]* will always be less than ΘR[w]*. One can easily verify from Table 3 that the values of
(ΘM[w]*/ΘR[w]*) very closely approximate the equation-(14) values derived for a regular product.
Recall from §3.1 that under a linear Dpl , only under [w,β] do the profit ratios (ΘM[wβ]*/ΘR[wβ]*)
match the regular-product value of (ΘM*/ΘR*) = 2, whereas the (ΘM[w]*/ΘR[w]*) ratios under [w] do
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12.
not. In contrast, here under an iso-elastic Dpc even the (ΘM[w]*/ΘR[w]*) ratios under [w] closely
match the regular-product values of (ΘM*/ΘR*) = (α−1)/α.
From the literature on two-echelon regular products it is also known that (see, e.g., Lau &
Lau 2003) that the CE in a manufacturer-Stackelberg game under an iso-elastic Dpc is
CE = (2α−1)(α−1)(α−1)/αα. (15)
With the first 5 rows in Table 3 where π = s = 0, one can verify that the CE values match the
equation-(15) CE-values derived for a regular product. Again, recall from §3.1 (characteristics
(c)) that, under a linear Dpl, the newsvendor-product CE-values match the regular-product value
of 0.75 only under [w,β], but not under [w]. However, the last 3 rows of Table 3 illustrate that
(15) is no more applicable when either π or s is non-zero.
§3.2.2. A Buyback Scheme [w,β]
We first consider situations where π = s = 0. Two examples of such solutions, with β* = 0,
are depicted in the first two row of Table 4. They are identical to the [w]-only solutions ― i.e.,
the 1st and 4th examples in Table 3. Recall from §3.1 that under a linear Dpl the manufacturer’s
ΘM[w]* already exceeds ΘR[w]* under a [w] scheme, and via a buyback scheme the manufacturer
increases the gap (ΘM*−ΘR*) even further. In stark contrast, under an iso-elastic Dpc not only is
the dominant manufacturer’s ΘM[w]* less than ΘR[w]* under a [w] scheme, but furthermore the
dominant manufacturer cannot improve her situation via a buyback scheme. Thus, under an iso-
elastic Dpc the dominant manufacturer should determine the optimal w*-value for a [w]-scheme
but need not be bothered with determining [w*,β*] (see, however, §5). This counter-intuitive
characteristic contradicts what one would expect on the basis of Pasternack’s (1985) paper ―
which of course considers only the fixed-p scenario. Appendix 1 shows in detail that this
seemingly incorrect solution is indeed correct; i.e., the manufacturer cannot improve her ΘM by
using any non-zero β-value.
The bottom 3 rows in Table TX4 with either s ≠ 0 and/or π ≠ 0 do show a non-zero β*. That
is, under an iso-elastic Dpc a buyback scheme is useful to a dominant manufacturer only when s
and π are not both zero. However, even in these cases buyback’s usefulness to the manufacturer
is limited, because ΘM[wβ]* is only slightly higher than ΘM[w]*. Thus, in the bottom-most row with
(π, s, K, α, σε) = (1, .3, 800, 3, .55), ΘM[wβ]* =15.5 in Table 4 is only slightly larger than ΘM[w]* =15.3 in
Table 3.
From a very large set of solutions, we have also found empirically that the following
relationships hold:
w* = αk/(α-1), and β* = αs/(α-1). (16)
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13.
However, the last row of Table 4 shows that the “neat” formulas of (16) become invalid when π
≠ 0. Incidentally, similar to the w*-formula in (13) for a linear Dpl, the w*-formula in (16) is
again identical to the w*-formula for a regular product in a manufacturer-Stackelberg two-
echelon channel under an iso-elastic Dpc (see, e.g., Lau & Lau 2003).
§3.2.3. Discussion of the Phenomenon “ΘM* < ΘR*”
Most earlier numerical illustrations to the variable-p two-echelon newsvendor problem have
been for the case of a linear Dpl , as in, e.g., Emmons & Gilbert 1998 (the iso-elastic Dpc scenario
considered in Weng 1997 is not directly comparable). Their numerical answers showed that the
Stackelberg dominant leader’s ΘM* exceeds the dominated-followers ΘR* ― an intuitively
reasonable outcome. However, in the “regular” product two-echelon literature, it is well known
that when the demand curve is not linear but iso-elastic, in a manufacturer-Stackelberg game
ΘM* will become less than ΘR* ― which the literature recognizes as counter-intuitive. In the
context of a regular product, some authors (e.g., Dowrick 1986, Gal-Or 1985) suggested that
under an iso-elastic Dpc the dominant manufacturer could ask the dominated retailer to act as the
leader in a retailer-Stackelberg game (i.e., the retailer leads by announcing a desired profit
margin). Under this arrangement, the dominant manufacturer becomes the follower and hence
will earn a larger profit than the retailer ― thus satisfying the intuitive expectation that the
dominant player should earn a higher profit than the dominated player. However, it is unclear
how the dominant manufacturer can get the retailer to act as the leader, given that the retailer
knows that he will be considerably better off by staying put as a follower. We have now shown
in §3.2.1 and §3.2.2 that with a newsvendor product the same dilemma exists; i.e., the dominant
manufacturer appears to be trapped in the implausible situation of earning a lower (expected)
profit than the dominated follower. We will, however, offer a solution to this dilemma in §5.
§4. ASSUMING AN ADDITIVE FORM OF RANDOM DEMAND
In this section we assume that the demand D is randomized additively by ε, as in (3).
§4.1 Linear Demand Curve Dpl = a−bp
§4.1.1. A Price-Only (No Buyback) Scheme [w]
Table 5 is the counterpart of Table 1 for the case where D’s is randomized additively. It
offers no new insights beyond Table 1, recalling that the solutions in Table 1 exhibited very little
meaningful pattern beyond the observation that CE < 1.
11
14.
§4.1.2. A Buyback Scheme [w,β]
The first two rows of solution values in Tables 5 and 6 show that although CE is variable
(but below 0.75) under the optimal [w] scheme (see Table 5), but under the optimal [w,β]
scheme (Table 6) the CE is a constant of 0.75 and matches the CE-value of a “regular” product.
This matches the characteristic pointed out in §3.1.2 for the case where D was randomized
multiplicatively.
However, the same solution values in Table 6 show that the ratio (ΘM[wβ]*/ΘR[wβ]*) under [w,β]
does NOT remain constant at 2. This contradicts the characteristic observed in§3.1.2, where
(ΘM[wβ]*/ΘR[wβ]*) was a constant of 2 (depicted in Table 2). The significance of this contradiction
will be discussed in §4.3.
We now explain why no solution value is given in the last row of Table 6, where σε has been
increased from 3 (in row 1) through 6 (row 2) to 9. Recall from (4) that, if D is randomized
additively (as in Table 6), Dmin will become negative when p exceeds a certain critical value
pcritical, where
Dmin = 0; or µD−σD√3 = (a−b•pcritical)−σD√3 = 0; or pcritical = (a−σD√3)/b. (17)
Since a negative demand is meaningless, this means that the random-demand model becomes
inoperative beyond pcritical. However, when σε is sufficiently low, as in the first 2 rows of Table
6, the optimal [w*, β*] decisions correspond to p*-values that are below pcritical, therefore the
existence of an inoperative region of the random-demand model is irrelevant. For example, in
the second row of Table 6, at p* = 14.9761,
Dmin = µD −σD√3 = (100−5×14.9761)−6×√3 = 14.7272.
Thus, the p*-answer and hence the associated [w*,β*]-answer are valid. However, for the last
row of Table 6 where σD = σε = 9, (17) shows that Dmin reaches 0 at
pcritical = (a−σD√3)/b = (100−9×√3)/5 = 16.8823.
However, the p*-value associated with the “optimal” [w*,β*] is higher than this pcritical-value of
16.8823, therefore the [w*,β*]-answers are meaningless. In other words, the additively-
randomized demand model is unable to handle the current situation. In general, for any given
set of (a,b) values, the additively-randomized demand model will fail to operate when σε (or,
equivalently, σD) becomes sufficiently high.
§4.2 Iso-elastic Demand Curve Dpc = K/pα
§4.2.1. A Price-Only (No Buyback) Scheme [w]
Table 7 is the counterpart of Table 3 for the case of additively-randomized demand.
12
15.
Recall from §3.2.1 that under a multiplicatively-randomized demand, the (ΘM*/ΘR*) values
from a [w*] scheme follow the regular-product formula (14). It also follows that ΘR[w]* always
exceeds ΘM[w]*. However, the solutions in Table 7 for additively-randomized demands illustrate
that the (ΘM*/ΘR*) and CE values do not follow the respective regular-product formulas
anymore. Furthermore, in Table 7, ΘR[w]*exceeds ΘM[w]* only when σD is sufficiently small. At
higher σD-levels, ΘM[w]* overtakes ΘR[w]*. Incidentally, over the large number of solutions we
examined, the fact that ΘM[w]* overtakes ΘR[w]* when σD is sufficiently large is the only
generalizable pattern we are able to surmise for the case of [w]-schemes under iso-elastic Dpc
and additively-randomized demand. In other words, solutions for the additively-randomized-
demand model shown in Tables 5 to 7 exhibit much less simple patterns than their respective
counterparts (Tables 1 to 3).
§4.2.2. A Buyback Scheme [w,β]
Table 8 is the counterpart of Table 4 for the case of additively-randomized demand. Similar
to Table TX4, the first three rows of solutions in Table 8 illustrate that, when π = s = 0, the
dominant manufacturer cannot use a buyback scheme to improve her profit ΘM[w]*. Of course, as
pointed out in §4.2.1, in contrast to the case of multiplicatively-randomized demand, here the
dominant manufacturer’s ΘM* could be greater than ΘR* (when σD is sufficiently large) without
the help of a buyback scheme.
The solution for (π, s, σε) = (0, 0, 21) given in Table 8’s 4th row illustrates the same situation
explained earlier in §4.1.2; i.e., the additively-randomized demand model is unable to handle the
current situation because the p*-value associated with the “optimal” [w*,β*] is higher than pcritical.
Table-8’s last two rows of solutions provide illustrations for situations where s ≠ 0 and/or π ≠ 0;
they do not provide additional insights.
§4.3 Intermediate Discussion on the Implications of the Presented Results
The significant effects of assuming different demand curve (i.e., Dp) forms can be seen by
comparing §3.1 with §3.2 and by comparing §4.1 with §4.2. For example, we see that a
dominant manufacturer can always increase her profit by switching from a [w] to a [w,β] scheme
under a linear Dpl, but under an iso-elastic Dpc it is often futile for the manufacturer to try to
“improve” to a [w,β] scheme. Also, the manufacturer’s profit is larger than the retailer’s under
Dpl, but very often the reverse is true under Dpc. However, both Dpl and Dpc are widely adopted
in theoretical modeling not because they accurately represent an actual price-vs.-mean-demand
relationship (i.e., Dp), but because they are mathematically convenient. Both appear to be
equally “reasonable” or “plausible.” Nevertheless, most actual Dps are probably neither exactly
13
16.
linear nor exactly iso-elastic, but somewhere “in between.” Our numerical solutions illustrate
that it is dangerous to generalize any characteristic observed from one or two Dp-forms to
another Dp-form.
Consider now the multiplicatively-randomized versus the additively-randomized demand
model. The former assumes a constant cv(D) while the latter assumes a constant σD; and neither
appears prima facie to be less plausible than the other. By comparing §3.1 with §4.1 and §3.2
with §4.2, we again see that the two different assumptions produced significantly different
results. The multiplicatively-randomized model produces “neater” results with more discernible
simple patterns (e.g., some results follow the simple relationships stated in (13) and (16)), while
the additively-randomized model not only produce solutions that exhibit hardly any simple
pattern, the model may also break down ― as illustrated in the rows with indeterminate
solutions in Tables 6 and 8. From the perspective of producing theoretically well-behaved
models and numerically clean results, the multiplicatively-randomized model is therefore
superior. However, this conclusion becomes debatable from the standpoint of obtaining reliable
answers for a real-life problem. Very often a demand curve needs to be estimated empirically,
and the process is likely to involve regression analyses; see, e.g., Crouch 1994, Stavins 1997,
Weingarten & Stuck 2001. Many regression models involve the assumption of homoscedastic
error term; thus, an empirically-estimated demand curve corresponds closer to additive-
randomization than multiplicative-randomization. On the other hand, if the demand curve is to
be estimated subjectively, then it is likely that demands in the central p-range can be estimated
more accurately than the demands at the two ends of the p-range. Thus, the demand curve
would have a smaller cv(D) or σD in the central p-range and a larger cv(D) or σD at both ends of
the p-range; in other words, it has neither a constant cv(D) nor a constant σD.
Our numerical results suggest that, given an actual situation, the only prudent thing to do is
to model as accurately as possible both the Dp-form and the price-demand relationship, then
compute the actual numerical solutions.
§5. THE MANUFACTURER IMPOSES
A MAXIMUM PERMISSIBLE UNIT RETAIL PRICE pM
§5.1 The Legality and Feasibility of Imposing a Maximum Retail Price
Ha (2001) pointed out that in the two-echelon newsvendor-product supply chain where the
retailer can vary the retail price p, the manufacturer can theoretically coordinate the channel by
offering a buyback scheme in conjunction with a manufacturer-imposed retail price pM; i.e., the
{pm} contract defined by a 3-tuple [pM,w,β]. However, he then noted (on his pg. 48) that “price
14
17.
fixing may be illegal.” It appears that a manufacturer-imposed retail price pM is widely
perceived to be illegal, which explains why it is seldom suggested or studied in the supply chain
literature. This subsection supplements Ha’s work by showing that a {pm} contract [pM,w,β] is
not only perfectly legal but also a very convenient channel-coordinating device; it should
therefore receive much more attention than it does now.
Stipulating a retail price by a “supplier” (or “manufacturer”), popularly known as “resale
price maintenance” or “RPM,” is often perceived in the form of minimum price maintenance ―
a widely known and debated practice explicitly prohibited by anti-trust laws in many countries
(see, e.g., the references in Deneckere, Marvel and Peck 1997, Flath and Nariu 2000, among
numerous others). This leads many to assume that “price maintenance” is illegal per se.
However, actually it is not illegal for a supplier to fix a maximum retail price (say) pM ― which
is in effect what we are considering in our context, since the p* that the ΘR-maximizing retailer
wants to set will always be higher than the channel-profit maximizing pI* (the so-called “double
marginalization” principle). For the United States, in the 1997 “State Oil Co. v. Khan” case, a
service station owner (Khan) litigated with his supplier State Oil Company over the legality of a
contract that incorporates a maximum permissible resale price. The U.S. Supreme Court
unanimously and explicitly held that suppliers do not violate antitrust laws by implementing
“maximum RPM” (hereafter “{pm}”). This judgment occurred because by that time many came
to recognize that, while minimum RPM is often harmful to society, maximum RPM is often
beneficial to society. See, e.g., U.S. Federal Trade Commission website http://www.ftc.gov/ogc/
briefs/khan.htm, or Blair and Lafontaine (1998). In the European Union, Regulation 2790/99 of
the European Commission (see, e.g., Gogeshvili 2002) explicitly exempts maximum RPM from
antitrust prohibitions. Similarly, most developed Asian economies (e.g., Hong Kong,
Singapore) do not prohibit maximum RPM.
Given that [pM,w,β] is legal, one can easily see that it can perfectly coordinate the channel
considered in this paper. To illustrate, consider the second example in Tables 1 and 2, where (π,
s, a, b, σε) = (0, 0, 100, 5, 0.55). Noting that pI* = 10.90 and VI* = 80.92 (italicized-underlined
entries under the panel “Integrated-Firm Optimal Solution” in Table 1), we saw that the optimal
buyback solution of [w*,β*] = (10.50, 10.00) in Table 2 is unable to bring the retailer-controlled
p*-value (= 15.45) down to pI* and the retailer-controlled VR*-value (40.47) up to VI*. However,
by imposing a maximum retail price pM (= pI*), or pM = 10.90, the manufacturer transforms the
variable-p problem into the fixed-p problem of Pasternack (1985), who showed that channel-
coordinating [wcc, βcc] values can be determined using the relationship:
15
18.
(pI*+π–k)•(pI*+π–βcc) = (pI*+π–wcc)•(pI*+π–s). (18)
For the current scenario, two numerical examples are:
(i) Offer a buyback contract of [wcc,βcc] = (10.00, 9.90). In other words, impose the channel
coordinating scheme:
[pM,wcc,βcc] = (10.90, 10.00, 9.90); which gives ΘM*= 373.7 and ΘR* = 37.4. (19)
(ii) Impose:
[pM,wcc,βcc] = (10.90, 5.95, 5.45); which gives ΘM* = 205.5 and ΘR* = 205.5. (20)
The ratio (ΘM*/ΘR*) is 9.989 in (19), which is much higher than the (ΘM*/ΘR*) of 1.0 in (20).
The results in (19) and (20) illustrate that the format [pM,w,β] not only enables the manufacturer
to achieve a CE of 1, it also returns to the manufacturer the complete power to control profit
allocation between the players ― the same situation with a [w,β] contract under Pasternack’s
fixed-p environment.
Nevertheless, although maximum RPM exists in the real world, as exemplified in “State Oil
Co. v. Khan” and in its explicit recognition by the European Union legal code, it is much less
well known than and often confused with “minimum RPM,” hence it is often assumed to be
illegal – again as exemplified by the lower courts’ decisions on “State Oil Co. v. Khan” before it
reached the U.S. Supreme Court. This is perhaps why {pm} is largely overlooked in the supply
chain literature. Regarding the manufacturer’s cost of enforcing/validating {pm}, we submit
that in many situations this cost should be no higher than that of, say, a simple [w,β] buyback
scheme. Thus, under a simple [w,β] scheme, an unsold bulky/perishable item often is not
actually shipped “back” to the supplier, but it is disposed of locally and the retailer merely
returns something like a proof-of-purchase label for refund – a procedure obviously susceptible
to fraud. On the other hand, for many products, simply printing a “maximum allowed retail
price” on the packaging will enlist the consumers as enforcers. Note that currently many
displayed “suggested retail prices” or “list prices” are set at levels not only higher than pI*, but
also higher than what the retailer would actually want to charge. Thus, in situations where a
simple [w,β] scheme is feasible, a [pM,w,β] scheme should also be feasible.
§5.2 Comparing {pm} with Other Schemes
The “price discount sharing” (PDS) scheme (described in, e.g., Bernstein & Federgruen
2005) is essentially a more complicated variation of “buyback” under which the manufacturer
must specify non-constant w and β as a function of p. Therefore, given that the [pM,w,β] scheme
is legal, there is little reason to implement the more complicated PDS scheme. The amount of
“trust” required between the players is lower in {pm} than in a two-part-tariffs scheme, under
16
19.
which the retailer must pay the manufacturer a considerable sum in advance solely on the basis
of anticipated but unrealized channel profit. Compared with revenue sharing, it should be noted
that revenue sharing is shown to be perfectly channel-coordinating (in, e.g., Cachon & Lariviere
2005) only under the assumption that the revenue-sharing proportion has already been
determined exogenously. In practice, the revenue-sharing proportion needs to be negotiated
between the players, and little has been said about how this proportion is determined. We will
show in a subsequent paper that when this revenue-sharing proportion is explicitly recognized to
be another decision variable, revenue sharing will not perfectly coordinate a channel in most
realistic situations. Of course, buyback (and hence {pm}) also has many shortcomings, as
discussed in, e.g., Tsay (2001). Nevertheless, we submit that in many situations {pm} is less
difficult to implement than such alternatives as PDS, two-part tariffs and revenue sharing.
With a [pM,w,β] scheme, the counter-intuitive phenomenon depicted in §3.2 (i.e., ΘM* < ΘR*
under an iso-elastic Dpc) also becomes irrelevant. That is, the dominant manufacturer avoids
earning a lower profit than the retailer by simply implementing a [pM,w,β] scheme.
Incidentally, some pre-1997 (and hence pre-State Oil vs. Khan) papers have studied the
effectiveness of minimum RPM as an alternative to buyback for the manufacturer to increase her
profit (see, e.g., Flath and Nariu 1989, pp. 52-55, on Japanese practice). Referring to minimum
RPM, Kandel (1996, pg. 344, lines 11 to 13)) concluded that “… an RPM contract does not
solve (the channel-coordination and manufacturer-product-maximization) problem(s) …” This
section presents a different perspective.
§6. CONCLUSION
§6.1 Summary
This paper considers a newsvendor-type product whose expected retail-sales volume varies
with the unit retail price p according to a known demand curve Dp. The supply chain consists of
one dominant manufacturer supplying one retailer; both players are expected profit maximizers.
For this system, beyond the basic price-only ([w]) scheme, buyback ([w,β]) is by far the most
common pricing scheme in the real world. The first part of this paper shows that:
(i) The solutions for the optimal [w*] and [w*,β*] schemes are quite sensitive to the demand-
curve form and the demand randomization process; hence these factors must not be
arbitrarily assumed. Although assuming a multiplicatively randomized demand leads to
“cleaner” solution values, sometimes an additively randomized demand provides a closer
fit to the actual situation.
17
20.
(ii) Buyback can improve the manufacturer’s expected profit when the demand curve is
linear, but not when the demand curve is iso-elastic. Under a linear Dpl it is not unlikely
that the dominant manufacturer can be satisfied with a [w*,β*] scheme; in contrast, under
an iso-elastic Dpc the dominant manufacturer will be highly motivated to seek an
alternative to a [w,β] scheme because the scheme often gives her a lower expected profit
than the retailer’s.
The second part of the paper shows that buyback in conjunction with a manufacturer-
imposed maximum retail price is a legal, practical and relatively simple scheme for a dominant
manufacturer to perfectly coordinate the channel. The scheme should receive more attention
than it has in the past.
§6.2 Extension
Among the many standard assumptions made in this paper are information asymmetry and a
dominant manufacturer. The fact that the manufacturer uses a retailer implies that the retailer
has better local information – most likely better information on the retail market demand curve.
Schemes such as two-part tariffs and revenue sharing involve a profit/revenue sharing parameter
– which has often been assumed to be exogenously fixed but in reality is probably the result of
negotiation. This in turn implies that neither player completely dominates the other. In contrast,
under the Pasternack-type [w,β] scheme or the Ha-type [pM,w,β]-scheme the manufacturer is
clearly assumed to be dominant and there is no negotiation parameter. Our subsequent research
will consider the modification and performance of [w,β] and [pM,w,β] schemes and compare
them with such alternatives as revenue sharing when retail-market information is asymmetric
and/or when neither player dominates the other.
APPENDIX 1: Demonstrating that β*=0 under an Iso-elastic Dpc
Figure A1 is 3-dimensional plot of ΘM[wβ] (vertical axis) as a function of w and β ― note that
[w,β] are the only decision variables for the manufacturer. It shows an arched dome with an
“entrance” for an observer standing on the diagram’s right (or standing on the right hand side of
the page, looking left (see arrow A). Starting from the right-side top point, the “ridge” of the
dome slants downwards as β increases. Figure A2 shows a series of ΘM[wβ] -vs-w curves for
different fixed-β levels. These curves are cross-sectional views of the Figure-A1 dome, “cut” at
different β-levels. It shows clearly that the height of the dome’s ridge” (or the peak ΘM[wβ]
-value) decreases as β increase from 0. For each of the Figure A2-curves, the left side of the
arch rises like a vertical because we only consider schemes where β ≤ w.
18
21.
FIGURE A1. 3-dimensional plot of ΘM[wβ] versus w and β
Multiplicatively randomized demand, iso-elastic Dpc. K = 800, α = 3, π = s = 0, σε = 0.3
ΘM[wβ]
β
w A
FIGURE A2. Graphs of ΘM[wβ] versus w at selected β-values
Multiplicatively randomized demand, iso-elastic Dpc. K = 800, α = 3, π = s = 0, σε = 0.3
ΘM[wβ]
β = 0.0
β = 1.0
β = 2.0
β = 3.0
β = 4.0
β = 5.0
w
19
27.
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