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  1. 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: Hon-Shiang LAU (corresponding author) Spears School of Business Oklahoma State University, Stillwater, OK 74074, USA E-mail: and Jian-Cai WANG School of Business The University of Hong Kong Pokfulam Road, HONG KONG E-mail: (authors’ names arranged in alphabetical order) July 2005
  2. 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. 2
  3. 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 θR and θ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 θR and θM, respectively; ΘC = ΘM+ΘR (subscript “C” denotes “channel”); 1
  4. 4. 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) 2
  5. 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), 3
  6. 6. 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 4
  7. 7. 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: θI = −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 5
  8. 8. 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. 6
  9. 9. §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- 7
  10. 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). 8
  11. 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 9
  12. 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) 10
  13. 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. 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. 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. 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. 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 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. 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. 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. 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. 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
  22. 22. TABLE 1 Optimal Price-Only Solutions to Problem P1 for Different (a, b, σε)-Values; π = s = 0 Parameter Integrated-Firm Price-Only (w) Optimal Solution to P1 Values Optimal Solution CE a b σε 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 .738 10.9 80.9 100 5 .55 14.36 6.41 31.09 168.1 128.9 0 2 411.1 .723 100 50 .05 1.75 1.49 11.81 5.8 3.0 1.51 24.04 11.8 .747 100 50 .55 1.74 1.33 6.44 2.1 1.5 1.61 15.05 4.9 .736 TABLE 2 Optimal Buyback Solutions to Problem P2 for Different (π, s, a, b, σε)-Values Parameter Values Buyback (w,β) Optimal Solution to P2 CE π s a b σε p[wβ]* w[wβ]* β* VR[wβ]* ΘM[wβ]* ΘR[wβ]* 10 . 15.2 10.5 10.0 0 0 0 5 05 7 0 0 25.33 223.8 111.9 .75 10 . 15.4 10.5 10.0 0 0 0 5 55 5 0 0 40.47 205.5 102.8 .75 10 5 . 0 0 0 0 05 1.75 1.50 1.00 12.02 5.9 2.9 .75 10 5 . 1.80 1.50 1.00 7.53 2.4 1.2 0 0 0 0 55 .75 . 10 . 15.2 10.5 10.1 25.45 224.3 112.1 0 3 0 5 05 6 0 5 .75 . 10 . 15.4 10.5 10.1 42.03 211.1 105.6 0 3 0 5 55 0 0 5 .75 10 5 . 1.90 1.24 0.32 6.57 0.9 0.5 1 0 0 0 55 .75 . 10 5 . 1.86 1.31 0.68 9.60 1.7 0.9 1 3 0 0 55 .75 TABLE 3 Optimal Price-Only Solutions to Problem P1 for Different (π, s, K, α, σε)-Values Integrated-Firm Parameter Values Price-Only (w) Optimal Solution to P1 Optimal Solution CE  α −1 π s K α σε p[w]* w[w]* VR[w]* ΘM[w]* ΘRw]* ΘM*/ΘR*   pI* VI* ΘI*  α  40 . 4.0 2.0 . 2.00 23.9 23.9 47.9 .500 95.7 95.7 0 0 0 2 05 9 .499 5 7500 40 . 5.8 2.9 . 2.00 15.2 15.2 30.3 .500 60.7 60.7 0 0 0 2 55 6 .502 3 7500 80 . 2.2 1.5 223. 111. . 1.50 66.2 33.1 49.7 .667 0 0 0 3 05 7 .666 2 4 7 7407 20
  23. 23. 80 . 2.4 1.6 158. . 1.50 47.0 23.5 35.2 .667 79.3 0 0 0 3 30 7 .668 5 6 7407 80 . 2.9 1.9 104. . 1.50 31.1 15.5 23.3 .667 52.4 0 0 0 3 55 3 .665 5 8 7407 . 80 . 2.6 1.7 185. . 1.41 53.9 22.0 33.4 .667 74.9 0 6 0 3 55 0 .659 8 2 7393 80 . 3.5 2.4 . 1.53 23.3 12.4 18.8 .667 79.5 42.1 1 0 0 3 55 6 .660 1 7404 . 80 . 3.2 2.2 112. . 1.47 32.8 15.3 23.3 .667 52.2 1 3 0 3 55 4 .657 0 5 7403 21
  24. 24. 22
  25. 25. TABLE 4 Optimal Buyback Solutions to Problem P2 for Different (π, s, K, α, σε)-Values Parameter Values Buyback (w,β) Optimal Solution to P2 CE  α −1 π s K α σε p[wβ]* w[wβ]* β* VR[wβ]* ΘM[wβ]* ΘR[wβ]*    α  . 4.09 2.00 0.00 23.9 23.9 47.9 0.50 0 0 400 2 .05 7500 . 2.47 1.50 0.00 47.0 23.5 35.2 0.66 0 0 800 3 .30 7407 . 2.67 1.50 0.90 54.9 22.2 33.3 0.66 0 .6 800 3 .55 7408 . 3.62 1.69 0.48 23.6 12.5 18.7 0.66 1 0 800 3 .55 7407 . 3.30 1.65 0.81 33.3 15.5 23.2 0.66 1 .3 800 3 .55 7408 p TABLE 5 Optimal Price-Only Solutions to Problem P1 for Different (a, b, σε)-Values; π = s = 0 Parameter Integrated-Firm Price-Only (w) Optimal Solution to P1 Values Optimal Solution CE a b σε 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 .743 100 5 6 14.14 9.15 26.23 213.9 112.5 10.49 55.96 441.8 .739 100 5 9 13.66 8.54 27.78 209.6 112.3 10.49 60.19 437.1 .736 100 5 18 12.54 7.03 33.51 202.2 109.1 10.47 72.86 423.0 .736 TABLE 6 Optimal Buyback Solutions to Problem P2 for Different (a, b, σε)-Values, π = s = 0 Parameter Values Buyback (w,β) Optimal Solution to P2 σ CE π s a b p[wβ]* Dmin w[wβ]* β* VR[wβ]* ΘM[wβ]* ΘR[wβ]* ε 10 15.1 19.2 10.3 8.0 . 5 3 26.30 226.1 108.8 0 0 0 1 4 3 7 7500 10 14.9 14.7 10.1 7.9 . 5 6 28.85 226.5 104.8 0 0 0 8 3 7 0 7499 10 ?? ?? ?? ?? ?? ?? ?? 5 9 ?? 0 0 0 23
  26. 26. TABLE 7 Optimal Price-Only Solutions to Problem P1 for Different (π, s, σε)-Values; K=800, α=3 Integrated-Firm Parameter Values Price-Only (w) Optimal Solution to P1 Optimal Solution CE π s σε 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 .727 0 0 9 2.19 1.54 69.5 37.29 42.61 0.875 1.479 241.9 113.4 .705 0 0 15 2.13 1.54 71.1 38.21 37.95 1.007 1.465 244.9 110.1 .692 0 0 21 2.06 1.52 73.9 38.72 34.62 1.119 1.452 247.6 106.8 .687 0 0 30 1.96 1.49 78.9 38.90 31.12 1.250 1.433 251.2 102.0 .686 0 0 48 1.80 1.43 88.8 37.90 26.58 1.426 1.398 256.8 92.9 .694 0 .3 9 2.21 1.54 69.2 37.42 42.85 0.873 1.483 242.2 114.0 .704 1 .3 9 2.30 1.57 68.1 38.57 36.85 1.047 1.495 245.0 111.1 .679 TABLE 8 Optimal Buyback Solutions to Problem P2 for Different (π, s, σε)-Values; K=800, α=3 Parameter Price-Only (w) Optimal Solution to P1 Values CE π s σε p [wβ]* w[wβ]* β * V R [wβ]* ΘM [wβ]* ΘR [wβ]* 0.0 . 0 0 3 2.24 1.52 69.5 35.94 48.94 0 727 0.0 . 0 0 9 2.19 1.54 69.5 37.29 42.61 0 705 0.0 . 0 0 15 2.13 1.54 71.1 38.21 37.95 0 692 0 0 21 ?? ?? ?? ?? ?? ?? ?? 0.0 . 0 .3 9 2.21 1.54 69.2 37.42 42.85 0 704 0.0 . 1 .3 9 2.30 1.57 68.1 38.57 36.85 0 679 24
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