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 LAUSchool of Business

The University of Hong KongPokfulam Road, HONG KONGE-mail: ahlau@business.hku.hk

Hon-Shiang LAU (corresponding author)Spears School of Business

Oklahoma State University,Stillwater, OK 74074, USAE-mail: wendy@okstate.edu

and

Jian-Cai WANGSchool of Business

The University of Hong KongPokfulam Road, HONG KONG

E-mail: wangjc@business.hku.hk

(authors’ names arranged in alphabetical order)

July 2005

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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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 D,

standard deviation D, 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 $/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,], under which the manufacturer “buys back” the unsold product for

$/unit. We assume < 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 to be meaningful,

obviously either “ > s” or “ = 0”. The retailer’s and manufacturer’s random profits are R

and M, 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 R and M, 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 D varies with p; and (ii) the

“randomization process,” which determines how D 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/p, > 1.

There are two common forms of modeling randomness (or D) 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 D = Dp(p0), and this demand is then randomized by multiplying D with

a random term ε; i.e.,

D = D• = [Dp(p0)]• ; where = 1. (1a)

Under this structure D (= D• ) varies with p, but D’s coefficient of variation cv(D)

remains constant as p varies; i.e.,

cv(D) = D/D = . (1b)

Following earlier studies, we assume that is uniformly distributed. Hence the range of D’s

finite support is:

Dmin = D[1–3]; and Dmax = D[1+3]. (2)

Thus, as long as we restrict ’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 D

(i.e., Dp(p0)) is randomized by the additive relationship

D = D + = Dp(p0) + ; where = 0. (3)

In contrast to the multiplicative form, here D = remains constant as p varies, but cv(D)

varies with p. Under the additive form, D’s finite support is:

Dmin = D–D3; and Dmax = D + D3. (4)

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In contrast to (2) for the multiplicative form, the Dmin in (4) will become negative when p is

sufficiently large and hence D becomes sufficiently small ― this is true no matter how smallD

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,] 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,cc]. By selecting

an appropriate pair of [wcc,cc]-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,] 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),

quantity fixing (Ha 2001) and revenue sharing (Cachon & Lariviere 2005). Bernstein &

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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,] 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

the performance of the [w] and [w,β] schemes under the two demand-curve forms: Dpl and Dpc.

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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.,

Ex(q) = , 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:

I = kVI+p•min(VI, D)+s•(VID)+(DVI)+ (for any given production quantity VI)

(6)

SLI* = (p+k)/(p+s) (7a)

VI* = (integrated firm’s optimal production quantity) = F1(SL*) (7b)

ΘI* = (p+s)•ED(VI

*)D (ED(•) defined in (5)) (8)

When D varies with p according to a given demand curve Dp, the integrated-firm’s problem

can be stated as, using (8) and (1):

P0. Find pI* that maximizes ΘI = (pI+s)•ED(VI)•D,

where for a given pI-value; D = Dpl(pI), and VI = F1[(pI+k)/(pI+s)].

(9)

§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. Find w* that maximizes ΘM = (wk)•F1[(pw*+w)/(pw

*+s)] ;

where pw* is the p-value that maximizes ΘR = (p+s)•ED(VRw

*)•D

for a given w-value; and

VRw* = F1[(p+w)/(p+s)].

(10)

In a buyback ([w,]) scheme the manufacturer announces a unit buyback price 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. Find [w*, *) that maximize ΘM =

[wk(s)(p2*+w)/(p2

*+)]•F1[(p2*+w)/(p2

*+)]+(s)•ED(VRw*),

where p2* is the p-value that maximizes ΘR = (p+)•ED(VRw)•D for

given values of (w,); and

VRw* = F1[(p2

*+w)/( p2*+ )].

(11)

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 , as in

(1). Recall that in this paper we assume that k (unit manufacturing cost) = 1 and is uniformly

distributed.

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§3.1 Linear Demand Curve Dpl = abp

§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 = s = 0. The

remaining free parameters under the current scenario are then a, b and . Among the numerous

combinations of (a, b, )-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,]

The current scenario is the same as the scenario considered in the preceding §3.1.1, except

that the manufacturer now implements [w,]. As stated in §1.3, there exists an infinite number

of [w,] schemes that perfectly coordinate the channel with CE = 1 when p is a constant. In

contrast, when the retailer can vary p, a [w,] 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[w]* in Table 2

is greater than the corresponding ΘM[]* in Table 1; e.g., for (a, b, ) = (100, 50, 0.55), ΘM

[w]* (=

2.4, one of the underlined-italicized entries in Table 2) is greater than ΘM[]* (= 2.1, Table 1).

That is, although a [w,] 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,] 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-

7

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[w]* always greater than the corresponding ΘM

[]*, but the CE attainable with a

[w,] scheme is also always larger than the corresponding CE of a [w] scheme. In contrast,

the retailer’s ΘR[w]* is always smaller than the corresponding ΘR

[]*. 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[w]*/ΘR

[w]*) under [w,] 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,] 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,] 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,] channel (see, however,

§4.1.2 for partial refutation).

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Table 2 also depicts the following unexpected “neat” characteristics for a [w,] scheme under a

linear Dpl:

w* = (a+bk)/(2b), and * = (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 * 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/p

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,] 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 (, s, K, , )-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

* = (1)/ ; (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,] do the profit ratios (ΘM[w]*/ΘR

[w]*)

match the regular-product value of (ΘM*/ΘR

*) = 2, whereas the (ΘM[w]*/ΘR

[w]*) ratios under [w] do

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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

*) = (1)/.

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 = (2α1)(α1)(α1)/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,], 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,]

We first consider situations where = s = 0. Two examples of such solutions, with * = 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*,*] (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 -value.

The bottom 3 rows in Table TX4 with either s 0 and/or 0 do show a non-zero *.

That is, under an iso-elastic Dpc a buyback scheme is useful to a dominant manufacturer only

when s and are not both zero. However, even in these cases buyback’s usefulness to the

manufacturer is limited, because ΘM[w]* is only slightly higher than ΘM

[w]*. Thus, in the bottom-

most row with (, s, K, , ) = (1, .3, 800, 3, .55), ΘM[w]* =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* = k/(-1), and β* = αs/(α-1). (16)

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However, the last row of Table 4 shows that the “neat” formulas of (16) become invalid when

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 , as in (3).

§4.1 Linear Demand Curve Dpl = abp

§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

§4.1.2. A Buyback Scheme [w,]

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,]

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[w]*/ΘR

[w]*) under [w,]

does NOT remain constant at 2. This contradicts the characteristic observed in§3.1.2, where

(ΘM[w]*/ΘR

[w]*) 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 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 DD3 = (ab•pcritical)D3 = 0; or pcritical = (aD3)/b. (17)

Since a negative demand is meaningless, this means that the random-demand model becomes

inoperative beyond pcritical. However, when is sufficiently low, as in the first 2 rows of Table

6, the optimal [w*, *] 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 = D D3 = (100514.9761)63 = 14.7272.

Thus, the p*-answer and hence the associated [w*,*]-answer are valid. However, for the last

row of Table 6 where D = = 9, (17) shows that Dmin reaches 0 at

pcritical = (aD3)/b = (10093)/5 = 16.8823.

However, the p*-value associated with the “optimal” [w*,*] is higher than this pcritical-value of

16.8823, therefore the [w*,*]-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 (or,

equivalently, D) becomes sufficiently high.

§4.2 Iso-elastic Demand Curve Dpc = K/p

§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

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 D is sufficiently small. At

higher D-levels, ΘM[w]* overtakes ΘR

[w]*. Incidentally, over the large number of solutions we

examined, the fact that ΘM[w]* overtakes ΘR

[w]* when D 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,]

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 = 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 D is sufficiently large) without

the help of a buyback scheme.

The solution for (, s, ) = (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*,*] is higher than pcritical.

Table-8’s last two rows of solutions provide illustrations for situations where s 0 and/or

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,] 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,] 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

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 D; 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 D in the central p-range and a larger cv(D) or D at both ends

of the p-range; in other words, it has neither a constant cv(D) nor a constant D.

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 IMPOSESA 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,]. However, he then noted (on his pg. 48) that “price

14

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,] 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,] 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 (,

s, a, b, ) = (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*,*] = (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, cc] values can be determined using the relationship:

(pI*+–k)•(pI

*+–cc) = (pI*+–wcc)•(pI

*+–s). (18)

15

For the current scenario, two numerical examples are:

(i) Offer a buyback contract of [wcc,cc] = (10.00, 9.90). In other words, impose the channel

coordinating scheme:

[pM,wcc,cc] = (10.90, 10.00, 9.90); which gives ΘM*= 373.7 and ΘR

* = 37.4. (19)

(ii) Impose:

[pM,wcc,cc] = (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,] 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,] 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,] 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,] 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

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

16

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,] 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,] 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.

(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-

17

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,]-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,] 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[w] (vertical axis) as a function of w and ― note that

[w,] 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 increases. Figure A2 shows a series of ΘM[w] -vs-w curves for

different fixed- levels. These curves are cross-sectional views of the Figure-A1 dome, “cut” at

different -levels. It shows clearly that the height of the dome’s ridge” (or the peak ΘM[w] -

value) decreases as 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 ≤ w.

18

FIGURE A1 . 3-dimensional plot of Θ M[ w ] versus w and

Multiplicatively randomized demand, iso-elastic D pc. K = 800, α = 3, π = s = 0, σ ε = 0.3

FIGURE A2 . Graphs of Θ M[ w ] versus w at selected -values

Multiplicatively randomized demand, iso-elastic D pc. K = 800, α = 3, π = s = 0, σ ε = 0.3

ΘM

[w ]

= 0.0

= 1.0

= 2.0

= 3.0

= 4.0

= 5.0

ΘM

[w ]

w

βw A

19

TABLE 1Optimal Price-Only Solutions to Problem P1 for Different ( a , b , )- Values; π = s = 0 Parameter

ValuesPrice-Only (w) Optimal Solution to P1

Integrated-Firm Optimal Solution CE

a b p[w]* w[w]* VR[w]* ΘM

[w]* ΘR[w]* pI

* VI* ΘI

*

100 5 .05 15.14 10.17 23.60 216.4 113.7 10.54 50.64 447.5 .738100 5 .55 14.36 6.41 31.09 168.1 128.9 10.90 80.92 411.1 .723100 50 .05 1.75 1.49 11.81 5.8 3.0 1.51 24.04 11.8 .747100 50 .55 1.74 1.33 6.44 2.1 1.5 1.61 15.05 4.9 .736

TABLE 2Optimal Buyback Solutions to Problem P2 for Different ( , s , a , b , )- Values

Parameter Values Buyback (w,) Optimal Solution to P2CE

s a b p[w]* w[w]* β* VR[w]* ΘM

[w]* ΘR[w]*

0 0 100 5 .05 15.27 10.50 10.00 25.33 223.8 111.9 .750 0 100 5 .55 15.45 10.50 10.00 40.47 205.5 102.8 .750 0 100 50 .05 1.75 1.50 1.00 12.02 5.9 2.9 .750 0 100 50 .55 1.80 1.50 1.00 7.53 2.4 1.2 .750 .3 100 5 .05 15.26 10.50 10.15 25.45 224.3 112.1 .750 .3 100 5 .55 15.40 10.50 10.15 42.03 211.1 105.6 .751 0 100 50 .55 1.90 1.24 0.32 6.57 0.9 0.5 .751 .3 100 50 .55 1.86 1.31 0.68 9.60 1.7 0.9 .75

TABLE 3Optimal Price-Only Solutions to Problem P1 for Different ( , s, K, , )- Values

Parameter Values Price-Only (w) Optimal Solution to P1Integrated-Firm

Optimal SolutionCE

s K p[w]

*w[w]

*VR

[w]

*ΘM

[w]

*ΘR

w]

*ΘM

*/ΘR

* pI* VI

* ΘI*

0 0400 2

.05

4.09

2.00

23.9 23.9 47.9.499

.5002.0

595.7 95.7

.7500

0 0400 2

.55

5.86

2.00

15.2 15.2 30.3.502

.5002.9

360.7 60.7

.7500

0 0800 3

.05

2.27

1.50

66.2 33.1 49.7.666

.6671.5

2223.

4111.

7.740

7

0 0800 3

.30

2.47

1.50

47.0 23.5 35.2.668

.6671.6

5158.

679.3

.7407

0 0800 3

.55

2.93

1.50

31.1 15.5 23.3.665

.6671.9

5104.

852.4

.7407

0.6

800 3

.55

2.60

1.41

53.9 22.0 33.4.659

.6671.7

8185.

274.9

.7393

1 0800 3

.55

3.56

1.53

23.3 12.4 18.8.660

.6672.4

179.5 42.1

.7404

1.3

800 3

.55

3.24

1.47

32.8 15.3 23.3.657

.6672.2

0112.

552.2

.7403

20

21

22

TABLE 4Optimal Buyback Solutions to Problem P2 for Different ( , s, K, , )-Values

Parameter Values Buyback (w,) Optimal Solution to P2

CE s K p[w]* w[w]* β* VR

[w]* ΘM[w]* ΘR

[w]*

0 0 400 2 .05 4.09 2.00 0.00 23.9 23.9 47.9 0.50 .75000 0 800 3 .30 2.47 1.50 0.00 47.0 23.5 35.2 0.66 .74070 .6 800 3 .55 2.67 1.50 0.90 54.9 22.2 33.3 0.66 .74081 0 800 3 .55 3.62 1.69 0.48 23.6 12.5 18.7 0.66 .74071 .3 800 3 .55 3.30 1.65 0.81 33.3 15.5 23.2 0.66 .7408

p

TABLE 5Optimal Price-Only Solutions to Problem P1 for Different ( a , b , )- Values; π = s = 0

Parameter Values

Price-Only (w) Optimal Solution to P1Integrated-Firm

Optimal Solution CEa b p[w]* w[w]* VR

[w]* ΘM[w]* ΘR

[w]* pI* VI

* ΘI*

100 5 3 14.67 9.81 24.88 219.2 112.6 10.50 51.73 446.5 .743100 5 6 14.14 9.15 26.23 213.9 112.5 10.49 55.96 441.8 .739100 5 9 13.66 8.54 27.78 209.6 112.3 10.49 60.19 437.1 .736100 5 18 12.54 7.03 33.51 202.2 109.1 10.47 72.86 423.0 .736

TABLE 6Optimal Buyback Solutions to Problem P2 for Different ( a , b , )- Values, π = s = 0

Parameter Values Buyback (w,) Optimal Solution to P2CE

s a b p[w]* Dmin w[w]* β* VR[w]* ΘM

[w]* ΘR[w]*

0 0 100 5 3 15.11 19.24 10.33 8.07 26.30 226.1 108.8 .75000 0 100 5 6 14.98 14.73 10.17 7.90 28.85 226.5 104.8 .74990 0 100 5 9 ?? ?? ?? ?? ?? ?? ?? ??

23

TABLE 7Optimal Price-Only Solutions to Problem P1 for Different ( , s, )- Values; K= 800, =3

Parameter Values Price-Only (w) Optimal Solution to P1Integrated-Firm

Optimal Solution CE s p[w]* w[w]* VR

[w]* ΘM[w]* ΘR

[w]* ΘM[w]*/ΘR

[w]* pI* VI

* ΘI*

0 0 3 2.24 1.52 69.5 35.94 48.94 0.734 1.493 238.7 116.8 .7270 0 9 2.19 1.54 69.5 37.29 42.61 0.875 1.479 241.9 113.4 .7050 0 15 2.13 1.54 71.1 38.21 37.95 1.007 1.465 244.9 110.1 .6920 0 21 2.06 1.52 73.9 38.72 34.62 1.119 1.452 247.6 106.8 .6870 0 30 1.96 1.49 78.9 38.90 31.12 1.250 1.433 251.2 102.0 .6860 0 48 1.80 1.43 88.8 37.90 26.58 1.426 1.398 256.8 92.9 .6940 .3 9 2.21 1.54 69.2 37.42 42.85 0.873 1.483 242.2 114.0 .7041 .3 9 2.30 1.57 68.1 38.57 36.85 1.047 1.495 245.0 111.1 .679

TABLE 8Optimal Buyback Solutions to Problem P2 for Different ( , s, )- Values; K= 800, =3

Parameter Values Price-Only (w) Optimal Solution to P1CE

s p[w]* w[w]* β* VR[w]* ΘM

[w]* ΘR[w]*

0 0 3 2.24 1.52 0.00 69.5 35.94 48.94 .7270 0 9 2.19 1.54 0.00 69.5 37.29 42.61 .7050 0 15 2.13 1.54 0.00 71.1 38.21 37.95 .6920 0 21 ?? ?? ?? ?? ?? ?? ??0 .3 9 2.21 1.54 0.00 69.2 37.42 42.85 .7041 .3 9 2.30 1.57 0.00 68.1 38.57 36.85 .679

24

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