Manufacturer Dominant SCM

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Production, Manufacturing and Logistics

A stochastic and asymmetric-information frameworkfor a dominant-manufacturer supply chain

Amy Hing Ling Lau a,1, Hon-Shiang Lau b,c,*, Yong-Wu Zhou d

a School of Business, University of Hong Kong, Pokfulam, Hong Kongb Department of Management Sciences, City University of Hong Kong, Kowloon, Hong Kong

c College of Business, Oklahoma State University, Stillwater, OK 74078, USAd Institute of Logistics and Supply Chain Management, School of Management, Hefei University of Technology,

Hefei, Anhui 230009, P.R. China

Received 15 July 2004; accepted 21 June 2005Available online 21 October 2005

Abstract

Consider a dominant manufacturer wholesaling a product to a retailer, who in turn retails it to the consumers at $p/unit. The retail-market demand volume varies with p according to a given demand curve. This basic system is com-

monly modeled as a manufacturer-Stackelberg ([mS]) game under a deterministic and symmetric-information(det-sym-i) framework. We first explain the logical flaws of this framework, which are (i) the dominant manufac-turer-leader will have a lower profit than the retailer under an iso-elastic demand curve; (ii) in some situations the sys-tems correct solution can be hyper-sensitive to minute changes in the demand curve; (iii) applying volumediscounting while keeping the original [mS] profit-maximizing objective leads to an implausible degenerate solutionin which the manufacturer has dictatorial power over the channel. We then present an extension of the stochasticand asymmetric-information (sto-asy-i) framework proposed in Lau and Lau [Lau, A., Lau, H.-S., 2005. Sometwo-echelon supply-chain games: Improving from deterministicsymmetric-information to stochastic-asymmetric-information models. European Journal of Operational Research 161 (1), 203223], coupled with the notion that aprofit-maximizing dominant manufacturer may implement not only [mS] but also [pm]i.e., using a manufac-turer-imposed maximum retail price. We show that this new framework resolves all the logical flaws stated above.Along the way, we also present a procedure for the dominant manufacturer to design a profit-maximizing volume-dis-count scheme using stochastic and asymmetric demand information.

0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2005.06.054

* Corresponding author. Tel.: +1 405 744 5105; fax: +1 405 744 5180.E-mail addresses: [email protected] (A.H.L. Lau), [email protected] (H.-S. Lau), [email protected] (Y.-W.

Zhou).1 Fax: +852 2858 5614.

European Journal of Operational Research 176 (2007) 295316

www.elsevier.com/locate/ejor
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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Using our sto-asy-i framework to resolve the logical flaws of the det-sym-i framework also reveals two noteworthypoints: (i) the attractiveness of the perfectly legal but overlooked channel-coordination mechanism [pm]; and (ii) volumediscounting as a means for the dominant manufacturer to benefit from information known only to the retailer. 2005 Elsevier B.V. All rights reserved.

Keywords: Supply chain; Stackelberg game; Information asymmetry

1. Introduction

Consider the following basic building block of many two-echelon supply-chain and market channelmodels: a manufacturer wholesales a product at $w/unit to a retailer, who in turn retails it at $p/unit tothe consumers. The retail-market demand volume V varies with retail price p according to a givendeterministic demand curve Dp. How would or should the players (i.e., the manufacturer and the

retailer) set their w and p? An increasingly popular approach in answering this question is to assumethat the players interact in the form of a non-cooperative game, and the most popular game assumedin the supply-chain literature is the manufacturer-Stackelberg (hereafter [mS]) game, where themanufacturer is assumed to be the games leader, see, e.g., Arcelus and Srinivasan (1987), Ertek and Grif-fin (2002), Li et al. (2002), Parlar and Wang (1994), Weng (1995a,b) , among numerous others. Thispaper extends two earlier papers by Lau and Lau (hereafter LL, 2003, 2005) in demonstrating thefollowing:

(i) The popular two-echelon framework based on a deterministic and symmetric-information manufac-turer-Stackelberg ([mS]) game has some very basic logical flaws.

(ii) These logical flaws are resolved by using our stochastic and asymmetric-information (hereafter

sto-asy-i) framework, under which the dominant manufacturer can dictate a maximum permissibleretail price instead of playing the [mS] game. We will show that our framework leads to much moreplausible behavior and results.

This paper also reveals the strength and practicality of a legitimate channel-coordinating device that hasbeen largely overlooked hitherto; namely, using a manufacturer-imposed maximum permissible retail price

pM.This paper assumes that the reader is familiar with Lau and Lau (LL, 2003, 2005). Section 2 briefly

summarizes the problematic characteristics of the deterministic and symmetric-information (hereafterdet-sym-i) framework as pointed out in LL, while Section 3 points out yet another logical dilemmaof this framework. Section 4.1 summarizes the sto-asy-i framework presented in LL (2003), while Section4.2 presents a numerical illustration that motivates and facilitates the extensions presented in this paperon the sto-asy-i framework. Section 5 presents results on applying LLs (2003) sto-asy-i framework to aniso-elastic curve; these results show how our sto-asy-i framework can resolve the two problematic char-acteristics summarized in Section 2. Sections 6 and 7 present results on applying the sto-asy-i frameworkto designing a volume (or quantity) discounting scheme; these results show that our sto-asy-i frame-work can resolve the logical dilemma we pointed out in Section 3 of this paper. These results also dem-onstrate clearly the strength of using a manufacturer-prescribed pM as a channel-coordinating device;hence, we discuss briefly in Appendix A the legal aspects of this pM arrangement. In the concluding Sec-tion 8 we summarize the main differences between our sto-asy-i framework and the classical det-sym-iframework. Review of much of the relevant literature has been done in LL (2003, 2005) and will notbe repeated here.

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2. Summary of the system and the basic known results

2.1. Basic definitions of the system to be considered

Define:

P Deterministic profit. P may have a subscript, which may be either M (for manufacturer), R (forretailer), C (for channel, i.e., manufacturers plus retailers), or I (for the Integrated firm doingboth manufacturing and retailing). The same subscripting convention is also used (when needed)for the other variables defined below;eH stochastic profit, with the same subscripting convention as P above;

c unit manufacturing cost incurred by the manufacturer;w or wM unit wholesale price charged by the manufacturer to the retailer;

p or pR retailer-determined unit retail price that the retailer charges the end consumers;pM a manufacturer-imposed maximum allowable unit retail price;V volume (i.e., quantity) per unit period (say, per year) that the retailer orders from the manufac-

turer and sells to the consumers.

For any random variable ~x, its expected value is the bolded x and its standard deviation is rx. Thus, H de-notes E eH, or expected profit.

Dp or Dp() denote a demand function of p. We will consider three Dp-forms:

(a) the linear demand curve Dpl = a bp, where a and b are positive constants, p 6 a/b;(b) the iso-elastic curve Dpc = Kp

a, where K and a are positive constants (we consider only cases witha > 1, otherwise situations with infinite PC would arise);

(c) a hybrid curve Dph that mixes a linear Dpl with an iso-elastic Dpc, using parameter s as the mix-ture-ratio, as explained in LL (2003, 2005):

Dph sa bp 1 sKpa. 1Notice that Dps optional second subscript denotes the demand-function type: l, c or h.

This paper considers the case of a dominant manufacturer (a she) selling to one retailer (a he). Themanufacturers and the retailers deterministic profit functions are, respectively:

PM w cV and PR p wV. 2As justified in LL (2003, 2005), the typical logistic cost components (i.e., ordering and carrying costs) areignored in (2) in order to reveal more effectively the differences between the traditional det-sym-i and theproposed sto-asy-i systems.

Formulas of optimal solutions for the one-echelon (integrated firm) are given in Table 1 (adapted fromTable 1 in LL, 2005).

Table 1Summary of one-echelon (integrated firm) formulas

Linear Dpl = a bp; c < (a/b) Iso-elastic Dpc = Kpa; a > 1pI (a + bc)/(2b) ac/(a 1)VI (a bc)/2 K[(a 1)/(ac)]aP

I (a bc)2/(4b) K(a 1)a1/(aaca1)

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2.2. Brief summary of known problematic behavior of the deterministic-symmetric-information (det-sym-i)

manufacturer-Stackelberg ([mS]) framework

Problematic Behavior 1 (PB1): Retailer-Followers Profit Exceeds Manufacturer-Leader

s Profit Un-der an Iso-elastic Dpc. E.g., in Table 2 ofLL (2005), one sees in the Iso-elastic Dpc column that P

M < P

R.

Implications of this anomaly are discussed in LL (2005, Sections 3.1 and 4). To resolve this anomaly, someearlier authors suggested that under an iso-elastic Dpc the dominant manufacturer should implement a re-tailer-Stackelberg game by making the retailer act as the leader (see, e.g., Gal-Or, 1985; Dowrick, 1986).However, it is unclear exactly how this can be implemented, since the retailer knows that if he acts as aleader his profit will be less.

Problematic Behavior 2 (PB2): Instability of the Two-Echelon [mS] Process. LL (2003, Section 6,2005, Section 3) showed that, for some very plausible Dp forms, a very minute change in the Dps curvaturecan lead to a drastic change in the theoretically correct optimal decision for an [mS] game. In practice,this means that the dominant manufacturer will not know how to make a decision in such a situation.

We will show in Section 5 that our proposed sto-asy-i framework resolves both PB1 and PB2.

3. Another logical problem in manufacturer-Stackelberg: When a volume discount can be offered

Although the justification behind assuming a non-cooperative game such as manufacturer Stackelberg([mS]) is that the players will not voluntarily cooperate, the literature using deterministic-and-symmetric-information (det-sym-i) models has accepted volume discounting as a legitimate channel coordination de-vice. However, we show in this section that the volume-discount device reveals another inherent logicalinconsistency of the det-sym-i [mS] framework. Incidentally, this paper adopts the convention used in,e.g., Viswanathan and Wang (2003) to differentiate between two types of quantity/volume discounting:(i) quantity discount for a discount given on the basis of the amount ordered in each batch (regardless

of the total amount per period); and (ii) volume discount for a discount given on the basis of the totalamount ordered over a period (in this paper, this corresponds to the period over which Vis specified).Since our profit functions as stated in (2) do not contain logistics costs, only volume discounting is rel-evant in our model. This is somewhat fortunate because volume discounting is preferable when one takesinto consideration the bullwhip effect.

The simplest and most widely taught volume-discount format is the following: the wholesale price/unit isreduced from the regular level of w to the discounted level of wd if the order volume equals or exceeds thequalifying volume Vdis. For a known linear Dpl = a bp, consider now the following degenerate vol-ume-discount schedule (the VI and p

I formulas for the linear-Dp case are given in Table 1):

Define M as an arbitrarily large number satisfying M> a=b; then

a set Vdis VI ; and charge w M when V< Vdis;b charge wd pI e when VP Vdis e arbitrarily small.9>=>; 3

To illustrate, assume the following numerical parameters:

Dp a bp 100 10p i.e., a 100 and b 10; c 1. 4For these parameters, the Table-1 formulas give pI 5:5 and VI 45. Therefore, an example of a degen-erate schedule according to (3) might be:

a charge w 10:1 when V< 45;b charge wd 5:45 i.e., set e 0:05 when VP 45.

5



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Since V = Dp(p) = 0 ifp > a/b (i.e., ifp > 10 in this case), one can easily see that the above schedule forcesthe retailer to coordinate perfectly; i.e., the optimal alternative for the retailer is to purchase VI units at wd/unit and sell it at pI=unit. However, the retailer now makes an arbitrarily small profit ofPR eVIpre-sumably a subsistence-level profit. That is, the manufacturer simultaneously attains a 100% C.E. and lion-izes the channels profitconsistent with the [mS] attitude defined in Section 2.2. It can be easily shownthat, for any other Dp-form, the manufacturer can design similar degenerate discount schedules that notonly attain a 100% C.E. but also empower her to allocate an arbitrarily small subsistence PR to the retailer.Therefore, under the det-sym-i framework, there is no reason why the (explicitly assumed) profit-maximiz-ing and dominant manufacturer would not always implement this degenerate volume-discount scheme.

The preceding discussion shows that in a world where no discounting is allowed, assuming an [mS]profit-maximizing self-centered manufacturer leads to a seemingly meaningful problem and a good-looking standard textbook solution. Thus, at equilibrium the retailer gets a reasonable share of the chan-nels profit through his own market power (whatever it might mean); e.g., under a linear Dpl the retailersP

R is assured of one-third of the channels P

C (see Table 2 in LL, 2005). However, when discounting is

allowed, if one uses consistently the original [mS] assumption made in the no-discount situation, a degen-

erate solution ensuesthe retailer is unable to get any profit through his own market power. Yet, theabove outcome and the discounting scheme (i.e., (3)) that produces this outcome are entirely consistent withthe prevalent framework of a two-echelon [mS] game in a det-sym-i environment.

In a det-sym-i system both players know exactly what it takes to maximize the channel s profit. Imposingan [mS] (or other kindred non-cooperative) game here means that the players are: (i) on one hand suffi-ciently rational and smart to play the non-cooperative (say) [mS] game under very definite rules; but (ii)on the other hand also sufficiently irrational to opt for playing a non-cooperative game instead of simplycollaborating and sharing the maximized channel profit through negotiation. However, by setting w atessentially a ridiculously high value of M, the degenerate discounting scheme (3) is for all practicalpurposes equivalent to the following manufacturers statement to the retailer: If you want to do businesswith me, you must buy exactly VI units, no more and no less, and at the unit wholesale price of

pI

e

;

take it or leave it. As long as e is sufficiently large (as presumably determined through negotiation), theretailer will submit. Thus, firstly, assuming a game means assuming explicitly that the players would notsimply coordinate and negotiate; but then secondly, discounting is allowed by the gaming framework inthe current literature, and this discounting dimension in effect opens a backdoor way to return to the out-right coordinate and negotiate process which the gaming framework explicitly assumed away.

The above volume-discount degeneracy constitutes not only an additional example of the logical circu-larity of assuming a det-sym-i non-cooperative game (pointed out in LL, 2005); it will also establish thedesirability (as will be shown in Sections 6 and 7) of using a sto-asy-i framework when volume discountingis to be considered.

Note, incidentally, that the implicit assumption here is that the retailer wants to carry the product andthat no other manufacturer offers the identical productwhich are the same assumptions made in the stan-

dard non-degenerate discounting schemes.

4. The stochastic-asymmetric-information framework

4.1. Summary of the framework as presented in Lau and Lau (2005)

The sto-asy-i framework presented in LL (2005) assume a linear demand curve. At a most likely (or arecently implemented) retail price p0, the most likely (or observed) demand is D0. The rate of change indemand with retail price is known to be a constant, but the value of this constant rate is stochastic to themanufacturer at the time when she has to set w; i.e., the manufacturer perceives Dp as:



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eDpl D0 ~bpp0; where ~b is a random variable 6or, equivalently,

eDpl D0 ~bp0 ~bp ~a ~bp; where ~a D0 ~bp0. 7In contrast, the retailer knows the actual b-value, and the manufacturer is aware that the retailer has thisknowledge. Thus, knowledge of market demand parameter b is asymmetric between players and also sto-chastic to the manufacturer.

LL (2005) pointed out that, under their sto-asy-i framework, the dominant manufacturer may considerplaying one the following three alternative games (listed in decreasing order of the level of channel coor-dination/integration): (i) Replicate the integrated firm (symbol [ri]); (ii) Enforce a manufacturer-imposedretail price pM before knowing the actual ~b-value (symbol [pm]); and (iii) Play the [mS] game. Table 5 inLL (2005) gives the formulas for the players profits for each of these three games.

4.2. Numerical illustration of the analytical results presented in LL (2005)

LL (2005) argued analytically that their Table 5 results generalize and rationalize the [mS] game. To bet-ter understand their claim and (more importantly) to demonstrate the necessity of the extensions to be pre-sented in this paper, we present in Table 2 a numerical illustration ofLLs (2005) sto-asy-i framework usingthe following parameter values:

a 100; b 10 for eDpl ~a ~bc; ~b uniformly distributed;rb b jb jb set to vary from 0 to 0:5; c 1; p0 3; and D0 70.

)8

Since the range of a uniformly distributed ~b is rbp

(12) and occurrence of a negative b-value must beavoided, the largest rb-value in Table 2 is 5.

First, recall that it was explained in Section 3 that it is unreasonable to assume that the players in a det-

sym-i environment will not implement Alternative (i)i.e., [ri], the highest level of coordination. In con-trast, as explained in Section 7.2 of LL (2005), the sto-asy-i framework provides a genuine reason (namely,impracticality) for not implementing [ri].

Second, when stochasticity (namely, ~bs uncertainty) is still sufficiently low, the dominant manufacturershould implement Alternative (ii) (i.e., [pm], the medium level of coordination). Thus, Table 2 shows thatwhen jb (see column 1) is small, [pm]s H

C (see column 4) exceeds [mS]s H

C (see column 7).

Table 2Optimal profits in the [ri], [pm] and [ms] processes; linear D(p)

Col. #1 Col. #2 Col. #3 Col. #4 Col. #5 Col. #6 Col. #7 Col. #8jb rb Integrated firms

or [ri]s HI

[pm]s channelprofit HC

[mS] process

Manuf.s HM Retailers HR H

C HM HR C.E. HC=HI

(Col. 7/Col. 3)

0 0 202.50 202.50 101.25 50.625 151.875 0.750.1 1 203.75 202.50 101.25 51.87 153.12 0.750.2 2 207.79 202.50 101.25 55.91 157.16 0.760.3 3 215.75 202.50 101.25 63.87 165.12 0.770.4 4 230.89 202.50 101.25 79.01 180.26 0.780.5 5 266.29 202.50 101.25 114.41 215.66 0.81

H denotes profit, the subscripts I, C, M and R stand for, respectively, integrated firm, channel, manufacturer and retailer.



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Third, when stochasticity becomes sufficiently high, it becomes worthwhile for the dominant manufac-turer to play [mS]i.e., operate at the lowest level of coordination. In Table 2, when jb (column 1) rises toaround 0.47, [mS]s HC (column 7) overtakes [pm]s H

C (column 4). Note also that at the same time [mS] s

HR (column 6) overtakes [mS]sHM (column 5). The fact that the retailer

sHR (column 6) increases with

rb

follows the principle that a retailer earns his keep by dealing with retail-markets uncertainty (explained in,e.g., Lau and Lau, 1999).

Thus, the dominant manufacturer is not restricted to playing only one game (i.e., [mS]) in our sto-asy-iframework. Instead, she has three perfectly legal and practical choices: (i) play [ri] when rb is zero; (ii) play[pm] when rb is non-zero but small; (iii) play [mS] when rb is sufficiently large. Choice (iii) also illustrates anoft-observed phenomenon: the advantage of tighter coordination decreases as uncertainty increases.

However, the analytical results ofLLs (2005, Table 5), as numerically illustrated here in Table 2, pertainonly to a linear eDpl, hence they do not address the flaws pointed out in Section 2.2; i.e., flaws revealed whennon-linear Dps are used. The single-w scheme considered in Table 2 also cannot address the logical flawpointed out in Section 3 when discount schemes are considered. Section 5 will present results of the sto-asy-i framework under an iso-elastic eDpc, while Sections 6 and 7 will present results on how volume-dis-count schemes might be designed under the sto-asy-i framework. The results in Sections 57 demonstratethe superiority of the sto-asy-i framework by showing how it can resolve the difficulties explained in Sec-tions 2.2 and 3 for the det-sym-i framework.

5. The stochastic-asymmetric-information framework under an iso-elastic curve

5.1. Analytical developments

In order to address problematic behavior PB1 as stated in Section 2.2, it is necessary to consider aniso-elastic eDpc. However, LL (2005) considered only a linear eDpl. Unfortunately, when eDp is iso-elastic,it is not possible anymore to obtain closed-form profit expressions such as those shown in LLs (2005,Table 5). We sketch below the procedure for determining the players optimal profit for an [mS] gameunder eDpc. Procedures for determining the optimal profits for the [ri] and [pm] processes are similarbut considerably simpler. The remainder of this subsection (Section 5.1) can be skipped without loss ofcontinuity.

Using the same approach for expressing the linear eDpl in (7), the stochastic iso-elastic demand function isnow eDpc D0p0=p^~a. Note that an exponential expression (xk) is written as (x^k) here so that the ran-dom-variable symbol $ of the exponent ~k can be clearly shown. Under an [mS] game, without know-ing the actual ~a-value, the manufacturer sets w. She keeps in mind that for any given w and the actual~a-value known to the retailer, he will set a p-value that maximizes

eHR, which is (adapting Table-1s p

I -

formula):

~pw ~aw=~a 1. 9

Substituting the above ~pw into eDpc gives the corresponding sales volume aseVw D0 fp0~a 1=w~a^~ag. 10

Therefore, the profit of an [mS]-implementing manufacturer is:

eHM w c

eVw w cD0 p0~a 1

w~a

^~a

. 11



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The manufacturers problem is to determine the value of w that maximizes E eHM, or HM. Thus, under[mS], the manufacturers problem can be stated as:

Find w that maximizes Z1

1w cD0p0

x

1

wx x

faxdx; 12

where fa(x) is the density function of eDpcs exponent ~a. Although (12) does not have a closed-form analytical

solution, it can be easily solved numerically. In this study the IMSL Math/Library (1994) of FORTRANsubroutines are used to perform the integrations and optimizations needed to solve (12) and other similarproblems. After w is determined, eHRs expected value HR can be computed by substituting w into (9) and(10) and the relationship eHR eV~p w.

Consider now the [pm] arrangement. Combining the approach explained above with the approach ex-plained in LL (2005) for deriving the HC-expression in Alternative (ii) (see LLs (2005, Table 5)), itcan be easily shown that the problem under [pm] can be stated as:

Find pM that maximizes HC pM c Z11

D0p0=pMxfaxdx. 13

Thus, the following needs to be emphasized: Under a stochastic linear eDpl, referring to Table 1s pI -formulagives pM as simply (a + bc)/(2b)this was stated as expression (26) in LL (2003). In contrast, under a sto-chastic iso-elastic eDpc, pM can only be determined numerically through solving (13).5.2. Numerical examples and the resolution of problematic behavior

Table 3 presents the relevant answers for the following numerical example:

Iso-elastic demand curve eDpc D0p0=p^~a;where ~a is uniformlydistributed with mean of a 2:5. Also; p0 3; c 1;D0 15. ) 14Given a = 2.5, since (i) situations with a < 1 produce infinite PC and must be avoided; and (ii) the range ofa uniformly distributed ~a is r

a

p(12); we only consider r

a-values up to 0.8 so that the realized ~a-values will

always exceed 1.

Table 3Optimal profits in the [ri], [pm] and [mS] processes; iso-elastic D(p)

#1 #2 #3 #4 #5 #6 #7 #8

ra

Integrated firm. [ri]s HI [pm] process [mS] process fpms HCgf mSs HCgH

C C.E. H

M H

R H

C HM HR

0 43.47 43.47 1.000 12.12 20.20 32.32 1.340.1 43.60 43.54 0.999 12.14 20.29 32.43 1.340.2 44.01 43.77 0.995 12.21 20.56 32.77 1.340.3 44.69 44.16 0.988 12.33 21.00 33.34 1.320.4 45.66 44.73 0.980 12.51 21.65 34.16 1.310.5 46.95 45.50 0.969 12.75 22.51 35.26 1.290.6 48.58 46.49 0.957 13.06 23.60 36.66 1.270.7 50.62 47.74 0.943 13.45 24.97 38.42 1.240.8 53.17 49.29 0.927 13.92 26.70 40.62 1.21



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Comparing Table 3 with Table 2, notable observations are:

(i) As in Table 2, the following Table 3 values increase as ra

increases: [ri]s HI (column 2), [mS]s HR

(column 6). However, the values of [pm]sHC (column 3) and [mS]

sHM (column 5) also increase with

ra

in Table 3, whereas these values do not change with rb in Table 2 (columns 4 and 5).(ii) In both Tables 2 and 3, as r

a(or rb) increases, one observes a decrement in the ratio between [pm] s

HC and [mS]s H

C; i.e., (column 4 column 7) in Table 2 and (column 3 column 7 ! column 8) in

Table 3. This follows the principle that the advantage of tighter coordination decreases as uncer-tainty increases, as stated earlier in Section 4.2.

However, in contrast to Table 2, [pm]s HC never falls below [mS]s HC in Table 3. This simply means that

the manufacturer should choose to implement [pm] for all levels of a-uncertainty.Consider now PB1 stated in Section 2.2. First, Table 3 shows that when Dp is iso-elastic, [mS]s H

M

(column 5) is still always smaller than HR (column 6) in our sto-asy-i model; i.e., we are still stuck withthe phenomenon that the dominant manufacturer-leader makes less than the retailer-follower (the both-

ersome characteristic stated in Section 2.2.1). However, this characteristic does not pose a problem any-more under our sto-asy-i framework. This is because, as explained in Section 4, even under a linear eDp,the dominant manufacturer should only switch from [pm] to [mS] when eDpls uncertainty (or ~bs uncer-tainty) is sufficiently high. Table 3 simply depicts a situation where the dominant manufacturer shouldalways implement [pm]. Since the manufacturer is explicitly assumed as dominant in the classical[mS] framework, such a dominant manufacturer is allowed to implement [pm] if she wishes to (seeAppendix 1 for additional justifications). In contrast, it is much less obvious how she can actually imple-ment the alternative of making the retailer act as the leader when the demand curve is iso-elastic (asimplied in the current literature). When implementing [pm], column 4 of Table 3 shows that under an iso-elastic Dpc the C.E. remains very high as uncertainty (ra) increases, but columns 3 and 4 of Table 2indicate that the same cannot be said for a linear Dplanother advantage of using [pm] under Dpc. Inci-

dentally, one might note that after one recognizes the advantage of [pm] by looking at the row ra = 0,its use in resolving PB1 can be invoked independent of our sto-asy-i framework. However, the sto-asy-iframework covers a wider perspective that includes the linear Dpl as well as the stochastic forms of Dpcand Dpl.

The instability problem stated as PB2 in Section 2.2 can also be resolved by using [pm]. Table 3 in LL(2005) shows that, while the values ofwM and p

R are extremely sensitive to the very small differences be-

tween the iso-elastic Dpc and the hybrid Dph considered in their illustration, the values of pI remain very

stable. Thus, faced with the uncertainty that the markets demand curve may be either Dpc or Dph, the dom-inant manufacturer should simply implement [pm] instead of [mS] and achieve much better results not onlyfor herself but also for the channel. Recall that the Dph illustrated in Table 3 of LL (2005) is primarily aniso-elastic Dpc slightly contaminated by a linear-Dpl component; thus, had the manufacturer not been told

that the actual demand curve may be Dph instead ofDpc, she would have implemented [pm] as explained inthe preceding paragraph. One might also perceive the current scenario, where the actual Dp may be eitherDpc or Dph, as another form of demand stochasticity.

Although the preceding discussions are based on the numerical answers for the set of parameters spec-ified in (14), we need to emphasize the following:

(i) A grid system has been used to generate many different combinations of parameter values, and sets ofnumerical answers corresponding to those presented in Table 3 have been examined to ensure that ourpreceding discussions are valid for other combinations of parameter values.

(ii) The same approach is used in the following sections when patterns of numerical solutions arediscussed.

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Since [pm] has turned out to be a very desirable arrangement, we briefly discuss its legal aspect inAppendix 1.

6. A stochastic and asymmetric-information manufacturer-Stackelberg game with volume discount,

linear eDpl6.1. Statement of the problem

We will consider the simplest and most widely taught volume-discount scheme, defined by the 3-tupel(w, wd, Qd); i.e., the wholesale price/unit is reduced from the regular level of w to the discounted level ofwd if the (periodic, or annual) order volume V equals or exceeds the qualifying volume Qd. This sectionconsiders the case of a linear eDpl D0 ~bpp0 ~a ~bp; recall also that ~b is assumed to be known tothe manufacturer only as a random variable, but the retailer knows the actual b-value (and hence the a-va-lue). We show in (A.16) and (A.17) of Appendix 2 that, for any given set (w, wd, Qd)-values, the expected

profit of the manufacturer in a [mS] game is:

HM a bw2

w c ZL3

Qdwd c D0 xp0 w

2w c

fbxdx

wd w

Z1L1

D0 xp0 w wd c2

fbxdx

for p0 P wd; 15a

HM a bw2

w c ZL5

Qdwd c D0 xp0 w

2w c

fbxdx

wd w

ZL1

1

D0 xp0 w wd c2

fbxdx

for p0 < wd; 15b

where the integral limits L1, L3 and L5 are defined in (A.11)(A.13) in Appendix 2.

6.2. Numerical results for a manufacturer-Stackelberg game

The manufacturers problem of finding the optimal volume-discount schedule (w, wd, Qd) that maximizes

HM in (15) is a straightforward nonlinear programming problem with three decision variables and can beeasily solved numerically. To illustrate, we will assume the same parameter values defined in (8) for con-structing Table 2, i.e., a = 100, b = 10, ~b uniformly distributed, c = 1, p0 = 3, and D0 = 70. Considerthe case with rb = 2. The discount schedule that maximizes HM in (15) is found to be:

Regular wholesale price w 8:186 if order volume is less than Qd 48:545;Discount wholesale price wd 4:593 if order volume is not less than Qd 48:545.

With this discount schedule the expected profits of the manufacturers and the retailers are, respectively,174.45 and 31.29.

Table 4 depicts the effect of bs uncertainty (i.e., rb) on the optimal solutions; its rb = 2 row corre-sponds to the solution in the preceding paragraph. Note that the HI values here are the same as the Table2 column-3 HI values, hence they are not repeated in Table 4. The rb = 0 row repeats the conclusionreached in Section 3 and highlights the fact that the deterministic-b scenario is a special case of our general-ized sto-asy-i model. However, when rb is non-zero the manufacturer can no longer design a (w, wd, Qd)-type volume-discount schedule that gives herself an arbitrarily large share of the channels profit. Also,as rb increases, the retailers profit share on the basis of his own market power (i.e., H

R in column

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#7) increases, which of course merely echoes the principle reviewed earlier in Section 4.2: a retailer earnshis keep by dealing with the retail-markets uncertainty. Note that when jb reaches 0.5, the column-8 ratioHM=HR is 1.68; i.e., it actually drops below the HM=HR of a deterministic [mS] processwhich is 2 (seeTable 2 in LL, 2005).

Thus, under our sto-asy-i framework, when a [mS] manufacturer designs a volume-discount scheme tomaximize her profit (consistent with [mS]s basic premises); a degenerate solution will not ensuein con-trast to the det-sym-i framework explained in Section 3.

We summarize what we have shown so far:

(A) When volume discounting is NOT to be used (for whatever reason), the dominant manufacturer in asto-asy-i environment may implement either [mS] or [pm].

(B) Given the volume-discount option, the dominant manufacturer should choose to play an [mS] gamethat is coupled with a volume-discount scheme.

Point (B) relates partially to the currently well-known notion of the advantage of volume discount. How-ever, in the existing det-sym-i literature the volume discount can be introduced only with the simultaneousalteration of the dominant manufacturers objective (i.e., she does not maximize her own profit anymore).With our sto-asy-i framework the original objective need not be altered.

Another important characteristic depicted in Table 4 is the very high C.E.s (compared to the C.E.sshown in Table 2 for the no-discount-[mS] counterpart). It is of course already well known that a vol-ume-discount scheme can attain a 100% C.E. in the det-sym-i case. Table 4 now shows that even whenthe manufacturer faces considerable uncertainty in ~b, she can still achieve a close-to-100% C.E. by designinga proper (w, wd, Qd) schedule.

Table 5 summarizes the channels and the players expected profits under the three viable sto-asy-i pro-

cesses considered so far; namely, the [pm] game and the [mS] without discount game considered in Section4 and Table 2, plus the [mS] with discount game considered in this section. The threeHC columns underTotal Channel Profit HC in Table 5 show that (as expected intuitively), the [mS] with discount processgives consistently the highest HC.

Comparing Tables 2 and 4 also reveals an important function of volume discounting hitherto over-looked in the academic literature, because this function is only revealed under a sto-asy-i framework.The function is: volume discounting serves as a mechanism for the manufacturer to benefit from the re-tailers b-knowledge, even though the manufacturer does not share the knowledge itself. Note that in adet-sym-i context, the manufacturer benefits from a volume-discount scheme by merely forcing the retailerto operate at a single desired levelthe channel-optimal VI level (which is deterministically and a prioriknown to both players). In contrast, in a sto-asy-i context the same simple volume-discount format

Table 4Effects ofrb on the optimal solutions; H

M-maximizing volume-discounting manufacturer; linear D(p)

a

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10jb rb w

wd Qd H

M H

R H

M=H

R H

C

H

M

H

R C.E.

0 0 >10 5.5 e 45 202.545e 45eb Arbitrarily large 202.50 10.1 1 8.917 4.982 46.75 186.15 17.18 10.83 203.33 0.9980.2 2 8.186 4.593 48.55 174.45 31.29 5.57 205.74 0.9900.3 3 7.591 4.303 50.20 165.81 44.74 3.71 210.55 0.9760.4 4 7.220 4.015 52.20 160.09 59.03 2.71 219.12 0.9490.5 5 7.195 3.811 54.56 158.04 94.28 1.68 252.32 0.948

aH

I same as in Table 2.

be can be arbitrarily small.



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(w, wd, Qd) does not force the retailer to operate at a pre-determined level. The ignorant manufacturerlets the retailer make the right decision with his b knowledge (this mechanism can be better appreciated byfollowing HMs mathematical derivation given in Appendix 2). If the actual-b value (known to the retailer)is relatively low (Situation 1, defined clearer in Appendix 2), he will opt to take advantage of the dis-count and in effect move the channel towards a higher volume of operation (by ordering at least Qd).However, if the known b-value is relatively high (Situation 2, defined clearer in Appendix 2), he willopt to forgo the discount (and pay $w/unit) and thus move the channel towards a lower volume of oper-ation. Moreover, even within each of the two Situations, the volume of operation (i.e., the V value) isstill not single-valued; it varies with the retailers observed b-value. Tables 2 and 4 show that volume dis-counting increases HM not only through cannibalizing the retailers H

R (i.e., Table 4s column-7 entries are

less than the Table 2s column-6 entries), but also through enlarging HC (i.e., through the retailers use ofhis b-knowledge in ordering).

Closely related to the mechanism explained above is the observation that a volume-discount schedule s w

needs to be conscientiously set only under the sto-asy-i framework. Referring to Section 3 and 3, it can beseen that under the det-sym-i framework, w can be arbitrarily set at any value above a/b, because w is notmeant to be actually used.

6.3. A better looking volume-discount scheme

Noting that 1 (wd/w) gives the discount rate under a discount scheme, the Table-4 volume-discountschemes (for the non-degenerate cases with rb5 0) call for a discount of more than 40%. A manufacturerquoting such a large discount may either appear to be somewhat capricious or be betraying her large profitmargins. Assume that the manufacturer wants to offer only a (say) 20% volume discount. Imposing thisconstraint actually leads to a simpler mathematical problem, because in maximizing the HM-functions given

in (15), the number of free decision variables is now reduced from 3 in (w, wd, Qd) to 2. The answers (for thesame parameters given in (8)) are given in Table 6. Understandably, the HM-values in Table 6 are less thantheirHM-counterparts in Table 4, however, they are still consistently higher than the no-discount H

M-coun-

terparts given in Table 2. As for the C.E., the C.E. figures in Table 6 may or may not be higher than theircounterparts in Table 4, but they are consistently higher than their no-discount C.E.-counterparts given inTable 2. The retailer is the major beneficiarythe HRs in Table 6 are consistently higher than their coun-terparts in Table 4. Theoretically, one recognizes that imposing an additional constraint (in this case a dis-count-magnitude constraint) can never benefit the optimizer (i.e., the manufacturer); nevertheless, thepreceding finding may be perceived in a form that appears counter-intuitivei.e., the retailer benefits(and the manufacturer loses) when the manufacturer is restricted from offering too high a discount rateto her retailer!

Table 5Comparison ofHC;H

M and H

R in [pm], [mS] without discount and [mS] with discount; linear D(p)

rb Total channel profit HC H

Ma

HR

[pm]a

[mS],no discount [mS],with discount [mS],no discount [mS],with discount [mS],no discount [mS],with discount

0 202.50 151.875 202.5 101.25 202.545e 50.625 45e1 202.50 153.12 203.33 101.25 186.15 51.87 17.182 202.50 157.16 205.74 101.25 174.45 55.91 31.293 202.50 165.12 210.55 101.25 165.81 63.87 44.744 202.50 180.26 219.12 101.25 160.09 79.01 59.035 202.50 215.66 252.32 101.25 158.04 114.41 94.28

aH

M and H

R under [pm] are negotiated (see Table 5 in LL (2005)), hence they are not listed.

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Table 6 depicts a relatively more plausible world. First, when one moves from a manufacturer obliviousto the volume discount tool to one that recognizes that this tool is at her disposal, there is no need tosimultaneously assume the sudden conversion of this manufacturer from a selfish [mS] profit maximizerto a caring player concerned about her opponents welfare in order to avoid degeneracy (explained inSection 3). Second, it is not necessary for the manufacturer to offer suspicion-arousing discount rates de-picted in Table 4. Table 6 depicts schemes with realistic discount rates; both players retain their originaland realistic [mS] objective and are able to obtain plausible shares of the channel profit. Furthermore,the C.E.s are very respectable and significantly higher than the 75% level for the det-sym-i counterpart(see Table 2 in LL, 2005).

This section demonstrated the merits of our sto-asy-i framework when one begins to consider the man-ufacturers volume-discounting option. However, while the examples in this section agree with the currentliteratures notion that a manufacturer can design a volume-discount scheme (w, wd, Qd) that performsbetter than a single-price scheme, we will show in the following Section 7 that this notion cannot be safelygeneralized to situations where the demand curve is non-lineara phenomenon that further illustrates themerit of our sto-asy-i framework.

7. A stochastic and asymmetric-information manufacturer-Stackelberg game with volume discount,

iso-elastic D(p)

We again consider the simple volume-discount scheme defined by the 3-tupel ( w, wd, Qd). Eqs. (A.26) inAppendix 3 shows that, for any given set of (w, wd, Qd)-values, the expected profit of the manufacturer in a[mS] game is:

Condition 1: when p0 > wd

HM

Z1

L0

HM1fa

x

dx

ZL1 HMXfaxdx ZL2 HMYfaxdx for p0 > w; 16:1HM

Z1L0

HM1faxdx ZL3

HMXfaxdx ZL4

HMYfaxdx for wd < p0 6 w. 16:2

Condition 2: when p0 = wd

HM Z1

1

HMXfaxdx for D0=Qd < e e natural constant; 16:3

HM Z1L0

HM1faxdx ZL0

1

HMXfaxdx for D0=QdP e. 16:4

Table 6Effects ofrb on the optimal solutions; H

M-maximizing manufacturer, 20% discount permitted, linear D(p)

a

rb w wd Q

d H

M H

R H

M=H

R H

C HM HR C.E.

0 Irrelevant for the current situation1 6.997 5.598 35.79 164.57 30.70 5.36 195.27 0.9582 6.425 5.140 37.37 154.71 47.23 3.28 201.94 0.9723 5.989 4.791 38.90 147.47 64.39 2.29 211.86 0.9824 5.641 4.513 40.44 142.04 86.08 1.65 228.12 0.9885 5.341 4.273 42.14 138.13 125.08 1.10 263.21 0.988

aH

I same as in Table 2.

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Condition 3: when p0 < wd

HM Z1

1

HMXfaxdx for D0=Qd < ha1; 16:5

HM Za3a2

HM1faxdx Za2

1

HMXfaxdxZ1a3

HMYfaxdx for D0=QdP ha1. 16:6

See Appendix 3 for the definitions of the integrand-components HM1, HMX and HMY; the quantity h(a1) aswell as the integral limits a2, a3 and Li (i = 04) used in (16).

The manufacturers problem is to find the set of optimal set of (w, wd, Qd)-values that maximize HM. Asin Section 5, this problem can be solved numerically. To illustrate, we will assume that the elasticity param-eter ~a in Kpa is uniformly distributed with a mean of a = 2.5 and r

aranging from 0.1 to 0.8 (the same

parameter values used in Table 3 of Section 5). The answers are presented in Table 7. Perhaps the mosteye-catching characteristic is that now HM is consistently larger than H

R; i.e., even when D(p) is iso-elastic,

the dominant manufacturer is now able to get a larger profit share than the retailer without resorting to the

dubious mechanism of somehow inducing the retailer to be the leader. However, scrutinizing the numbersin Tables 3 and 7 reveals something much more importanti.e., the unexpected attractiveness of the [pm]arrangement when the demand curve is iso-elastic. To elaborate:

(i) As summarized in Section 4.1, LL (2005) pointed out that under a sto-asy-i system it is impractical toreplicate an integrated firm (even if the players would like to do it) and achieve a 100% C.E.

(ii) Under a linear eDpl, [pm]s HC (=202.5, at column 4 of Table 2) is always less than the HC in Table 4(column 9); it is also less that theHC in Table 6 when rb is sufficiently high. In other words, under eDpl,volume discount continues to work better than [pm] in most situations. In contrast, if the eDp is iso-elastic, then for all jb-values, [pm]s H

C given in column 3 of Table 3 is always higher than the cor-

responding HC of the optimal volume-discount scheme given in Table 7. Thus, while discounting issuperior to the plain (single-w) [mS] under both eDpl and eDpc, discounting is superior to [pm] onlyunder eDpl but NOT under eDpc.

This phenomenon addresses the question raised at the end of Section 6. Note that volume discounting isthe most widely studied channel-coordinating and manufacturer-benefiting device in the literature. The pre-ceding sections have suggested [pm] as a viable alternative channel-coordinating device, and this sectionshows that, under an iso-elastic Dpc, a single-price scheme (albeit using pM under [pm] instead of a w

under[mS]) can be a considerably better channel-coordinating device than volume discounting.

Table 7Effects ofr

aon the optimal solutions; HM-maximizing volume-discounting manufacturer; iso-elastic D(p)

ra

w wd Qd H

M H

R H

C HM HR HI C.E.

0 Degenerate volume-discounting scheme (see Section 3)0.1 14.84 1.62 61.28 38.02 5.33 43.35 43.60 0.9940.2 15.74 1.85 40.87 34.61 6.46 41.07 44.00 0.9330.3 18.23 2.01 31.31 31.53 6.89 38.42 44.69 0.8600.4 18.30 2.15 24.83 28.51 7.26 35.77 45.66 0.7830.5 17.85 2.15 24.63 25.87 7.26 33.13 46.95 0.7060.6 15.94 2.01 30.24 24.15 8.13 32.28 48.58 0.6640.7 12.10 1.83 41.47 23.09 9.58 32.67 50.62 0.6460.8 10.31 1.68 59.11 22.73 11.47 34.2 53.17 0.643

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8. Summary and conclusion

8.1. Summary

We considered a dominant manufacturer wholesaling a product to a retailer, who in turn retails it at $p/unitto the consumers. The retail-market demand volume Vvaries withp according to a given demand curve Dp. Avery common approach is to assume that the players will play a manufacturer-Stackelberg ([mS]) game un-der a deterministic and symmetric-information (det-sym-i) framework. In Sections 2 a n d 3 we explainedthe logical flaws of this approach, which are (i) the dominant manufacturer-leader will have a lower profit thanthe retailer under an iso-elastic Dpc; (ii) in some situations the systems correct solution can be hyper-sen-sitive to minute changes in the demand curve; (iii) applying volume discounting while keeping the original [mS]profit-maximizing objective leads to an implausible degenerate solution in which the manufacturer has dicta-torial power over the channel. This paper extends the stochastic and asymmetric-information (sto-asy-i)framework proposed in Lau and Lau (2005), coupled with the notion that a profit-maximizing dominant man-ufacturer may implement not only [mS] but also [pm] (i.e., using a manufacturer-imposed maximum retail

price). We show that this new framework resolves all the logical flaws stated above. Specifically:

(i) The dominant manufacturers problematic inferior profit under an iso-elastic Dpc (i.e., PB1) isresolved by recognizing that she should implement [pm] instead of [mS] under Dpc; in contrast, recallthat she should implement [mS] instead of [pm] under a linear Dpl with high parameter uncertainty.

(ii) Implementing [pm] also resolves the hyper-sensitive problem that may arise with some non-lineardemand curves, because in those situations a [pm] solution is not only stable but also more profitableto the dominant player.

(iii) Under the sto-asy-i environment, if Dp is linear, the optimal [mS]-discounting scheme available to adominant manufacturer will no longer be degenerate. However, if Dp is iso-elastic, the dominant man-ufacturer should use [pm] instead of a discounting scheme.

Using our sto-asy-i framework to resolve the logical flaws of the det-sym-i framework revealed two note-worthy points: (i) the attractiveness of the perfectly legal but overlooked channel-coordination mechanism[pm]; and (ii) volume discounting as a means for the dominant manufacturer to benefit from informationknown only to the retailer.

Along the way, we also presented a procedure for the dominant manufacturer to design a profit-maxi-mizing volume-discount scheme.

8.2. Future extensions

A large number of studies have considered two-echelon supply chains with asymmetric information; see,

e.g., Ha (2001), Corbett et al. (2004) and their references. However, many of these models consider infor-mation asymmetry with respect to either logistic-cost-component parameters or the retailers variable han-dling costs; these components are explicitly excluded in our profit functions stated in (2) because we wantedto concentrate on the effects of stochasticity and asymmetry in the demand curve. The effects of includinglogistic/variable costs and their information asymmetry into our model remain to be studied.

In Sections 6 and 7 we have considered only the simplest discounting format. Our results in Section 7indicate that, ifDp is iso-elastic, then [pm] is superior to this simple discount scheme for a dominant man-ufacturer. One might investigate whether a more sophisticated discounting scheme or other channel-coor-dinating devices may surpass [pm].

As shown in Appendix 1, although our legitimate maximum-price-maintenance [pm] does occur in thereal world, it is admittedly rare. In contrast, the illegal minimum-price-maintenance variation of [pm] is

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much more well-known. The practical issues of implementing maximum-price-maintenance needs to beexamined. In the past decades, many new channel-coordinating devices have been proposed and/or usedby a small number of firms; among them are revenue sharing (see, e.g., Pasternack, 2002) and quan-

tity-flexibility contracts (see, e.g., Tsay, 1999).Section 7 shows that the relative merits of [pm] versus discounting is reversed when one switches fromeDpl to eDpc; it is unclear what their relative merits would be under other eDp functional forms. The need toconsider the effect of different eDp-assumption has of course been raised by LL (2003).Acknowledgement

This research is partially supported by a Hong Kong Research Grants Council Competitive EarmarkedResearch Grant (project number CityU 1128/02H).

Appendix 1. Some legal aspects of the [pm] arrangement that are overlooked in the quantitative

supply chain literature

Fixing a retail price at pM by a supplier (or manufacturer), popularly known as resale price main-tenance, is often perceived in the form of minimum price maintenancean illegal practice in many coun-tries (see, e.g., Deneckere et al., 1997; Flath and Nariu, 2000, among numerous others). In contrast, it is notillegal to fix a maximum retail pricewhich is in effect our [pm] arrangement, since the optimal retail priceset by the (non-cooperative) game-playing retailer will always be higher than the retail price needed to max-imize the channel profit (the double marginalization principle). For the United States, in State Oil Co. v.Khan, in which a retailer sued a supplier for imposing a maximum gasoline retail price, an unanimous 1997Supreme Court explicitly held that suppliers do not violate antitrust laws by setting a maximum allowable

retail price; see, e.g., US Federal Trade Commission website http://www.ftc.gov/ogc/briefs/khan.htm, orLevy and Weitz (2001, p. 483 and p. 714, note #38). In the European Union, Regulation 2790/99 of theEuropean Commission (see, e.g., Gogeshvili, 2002) explicitly exempts maximum retail price maintenancefrom antitrust prohibitions. Similarly, most developed Asian economies (e.g., Japan, Hong Kong) donot prohibit maximum price maintenance.

Nevertheless, although maximum resale price restriction (i.e., [pm]) exists in the real world, as exempli-fied in State Oil Co. v. Khan, it is not very common. Also, although minimum resale price maintenancehas been the subject of many theoretical analyses, relatively few theoretical studies of [pm] exist (see, e.g.,the references in Reiffen (1999) and http://www.ftc.gov/ogc/briefs/khan.htm). Sections 57 of this paperdemonstrate: (i) the merits of [pm] as a normative channel-coordination arrangement; and (ii) its abilityto rationalize the non-cooperative gaming structure. Therefore, [pm] should receive more attention from

theoreticians, practitioners and government regulators.Besides imposing [pm] rigidly, for many products (though not for, say, the gasoline involved in StateOil Co. v. Khan), the manufacturer can print a suggested retailer price or a list price on the product.In many case this can effectively pressure the retailer to charge no more than the labeled price.

Appendix 2. Deriving the HM (Eqs. (15)) and HR expressions in an [mS] quantity-discounting

system; linear demand curves

The quantity discount scheme is defined by the 3-tupel (w, wd, Qd); i.e., the wholesale price/unit is reducedfrom the regular level of w to the discounted level of wd if the order quantity equals or exceeds the qual-

http://www.ftc.gov/ogc/briefs/khan.htmhttp://www.ftc.gov/ogc/briefs/khan.htmhttp://www.ftc.gov/ogc/briefs/khan.htmhttp://www.ftc.gov/ogc/briefs/khan.htm


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ifying quantity Qd. The demand curve is eDpl D0 ~bpp0 ~a ~bp; while ~b is known to the manufac-turer only as a random variable, the retailer acts after the actual b-value (and hence the a-value) is observed.

For any given values of (w, wd, Qd) declared by the manufacturer, the profit-maximizing retailer will re-

spond as follows (after having observed the actual b-value).First compute QT = (a bwd)/2. QT is the profit-maximizing order quantity for the retailer; assumingthat no qualifying order quantity Qd has been imposed; the formula is adapted from the V

I -formula

in Table 1. IfQTP Qd, handle it as Situation 1 (explained below), otherwise handle it as Situation 2.Situation 1. If QTP Qd, the retailers order quantity and retail price will be

Q1 a bwd=2 and p1 a bwd=2 adapted from Table 1s pI -formula. A:1Hence, the profits of the two players would be

HR1 Q1p1 wd a bwd2=4b; HM1 a bwdwd c=2. A:2Situation 2. If QT < Qd, the retailer has to consider two alternatives:Alternative X: forgo the discount-price opportunity; the relevant expressions are then

QX a bw=2; pX a bw=2b; A:3HRX QXpX w; HMX QXw c. A:4

Alternative Y: order Qd units to take advantage of the lower wholesale price; then set the retail price pYsuch that the Qdunits can be sold; i.e., set pY such that a b pY = Qd. The order-quantity, price and prof-it expressions are then

QY Qd; pY p0 Qd D0=b; A:5HRY QdpY wd; HMY Qdwd c. A:6

Obviously, the retailer implements Alternative X ifHRXP HRY.

Straightforward manipulations with (A.3)(A.5) and (A.6) show that the conditions for Alternatives Xand Y can be stated as follows:

Implement Alternative X if fb > Bg or fb < Cg;Implement Alternative Y if fC< b < Bg;

A:7

where

B1 2Qdw wdD0w wd 2p0 w wd2Qd D0; A:8B fD0w wd 2Qd D0p0 wd

pB1g=p0 w2; A:9

C fD0w wd 2Qd D0p0 wd pB1g=p0 w2. A:10

Furthermore, the condition QTP Qd (or [(a bwd)/2] P Qd) is equivalent to the conditions:bP L1 if p0 > wd; b 6 L1 if p0 < wd; where L1 2Qd D0=p0 wd. A:11

To summarize, the manufacturers profit HM will be

(i) When p0 > wd, HM is

HM1 if f1 > bP L1g;HMX if b 2 fb > Bg[f1 < b < Cg \fb < L1g L2;HMY if b 2 fC< b < Bg \ fb < L1g L3.

9>>=>>;

A:12



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(ii) When p0 < wd, HM is

HM1 if f1 < b 6 L1g; or;H

MX if b 2 fb < Cg [ fB < b < 1g\fb > L1g L4;HMY if b 2 fC< b < Bg \ fb > L1g L5.

9>>=>>; A:13Consider now the situation where p0 = wd. The condition QTP Qd (or [(a bwd)/2] P Qd) is now

equivalent to Qd6 D0/2 or (2Qd D0) 6 0. Thus, if the condition (2Qd D0) 6 0 holds, the retai-ler implants (A.1) in Situation 1. Otherwise, Situation 2 applies, and the retailer implements Alterna-tive X if {b > B} or {b < C}, but he implements Alternative Y if {C < b < B}. The manufacturers profitHM is therefore

HM1 if 2Qd D0 6 0; otherwiseH

MX if b 2 fb > Bg[f1 < b < Cg; and;HMY if b 2 fC< b < Bg.

However, it can be seen that the preceding expression is equivalent to (A.12) because: (i) L1 = 1 if(2Qd D0) 6 0 and p0 ! wd + 0; and (ii) L1 = +1 if (2Qd D0)P 0. Hence, the validity region of(A.12) can be rewritten as When p0P wd. Incidentally, it can be similarly shown that the validity regionof (A.13) can be written as When p0 6 wd.

Therefore, for any given wholesale-pricing scheme (w, wd, Qd), the manufacturers expected profit is

HM Z

1

L1

HM1fbxdxZL2

HMXfbxdxZL3

HMYfbxdx if p0 P wd; A:14

HM ZL1

1HM1fbxdx

ZL4

HMXfbxdx ZL5

HMYfbxdx if p0 < wd. A:15

Substituting (A.2), (A.4) and (A.6) into the above formulas, we have

HM 1=2a bww c ZL3

Qdwd c D0 xp0 w

2w c

fbxdx

wd wZ1L1

D0 xp0 w wd c2

fbxdx; for p0 P wd; A:16

HM 1=2a bww c ZL5

Qdwd c D0 xp0 w

2w c

fbxdx

wd wZL1

1

D0 xp0 w wd c2

fbxdx; for p0 < wd. A:17

These are stated in the main paper as (15). The manufacturers problem is to find the values ofw, wd, and Qdthat will maximize HM.

Following similar arguments, it can be shown that retailers expected profit under a given manufacturersoptimal pricing scheme w;wd;Qd is



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HR Z1L1

HR1fbxdxZL2

HRXfbxdx ZL3

HRYfbxdx a bw2=4b D20

4E

1~b

1

b

Z1L1

D0w

wd2

w

wd

2p0w

wd4 x fbxdx

ZL3

Qdp0 wd 2Qd D02

4xD0p0 w

2

p0 w2

4x

" #fbxdx if p0 P wd; A:18

HR ZL1

1HR1fbxdx

ZL4

HRXfbxdx ZL5

HRYfbxdx a bw2=4b D20

4E

1~b

1

b

ZL1

1

D0w wd2

w wd2p0 w wd

4x

fbxdx

ZL5

Qdp0 wd 2Qd D02

4xD0p0 w

2 p0 w

2

4x" #fbxdx if p0 < wd. A:19

In (A.18) and (A.19), note incidentally that E 1~b

is not the same as 1

b(which is 1=E~b); see, e.g., further

derivations in LL (2005), Eqs. (35) to (36).

Appendix 3. Deriving the expressions ofHM (Eqs. (16)) and HR in a [mS] quantity-discounting system;

iso-elastic demand curve

For any value of (w, wd, Qd) declared by the manufacturer, the retailer will, after having observed theactual value of~a, respond as follows:

First computeQt

=D0

[p0

(a

1)/(wda)]a.

Qtis the profit-maximizing order quantity for the retailer

assuming that no qualifying order quantity Qd has been imposed; the formula is adapted from the eVw-for-mula of (10). If QtP Qd, handle it as Situation 1, otherwise handle it as Situation 2.

Situation 1. If QtP Qd, the retailers order quantity and retail price will be

Q1 D0p0a 1=wdaa and p1 awd=a 1 adapted from Table 1s pI -formula. A:20Hence, the profits of the two players would be

HR1 Q1p1 wd D0wdp0a 1=wdaa=a 1;HM1 D0p0a 1=wdaawd c. A:21

Situation 2. If Qt < Qd, the retailer has to consider two alternatives:Alternative X: forgo the discount-price opportunity; the relevant expressions are then

QX D0p0a 1=waa; pX aw=a 1; A:22HRX QXpX w; HMX QXw c. A:23

Alternative Y: order Qd units to take advantage of the lower wholesale price; then set the retail price pYsuch that Qd can be sold; i.e., set pY such that a b pY = Qd. The order-quantity, price and profit expres-sions are then

QY Qd; pY p0D0=Qd1=a; A:24HRY QdpY wd; HMY Qdwd c. A:25

Obviously, the retailer implements Alternative X ifHRXP HRY.



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How the retailer will react ultimately for any given manufacturer-set (w, wd, Qd) is stated below:First define the following symbols and parameters:

h

x

fwdx=

x

1

p0

gx

; F

x

D0p0

p0

x

1

=

wx

x1=x;G

x

Qd

p0

D0=Qd

1=x

wd

;

L0 is the root to the equation: h(x) = D0/Qd;

a1 is the root to the equation: Loge{wdx/[p0(x 1)]} 1/(x 1) = 0;a2 and a3 are two roots to the equation h(x) D0/Qd = 0 when D0/Qd > h(a1);a4 is an unique root to the equation F(x) G(x) = 0 when wd < p0 6 w;a5 and a6 are two roots to the equation F(x) G(x) = 0 when p0 > w.

Then it can be shown that (detailed proofs are available from the authors):

(1) Ifp0 > wd, the retailer will implement Situation 1 for aP L0; otherwise, (i) when p0 > w, the retailer

implements Alternative Y (Situation 2) if a 2 {(1,(L0) \ (a5,a6)} and Alternative X if a 2 (1, L0)n(a5,a6); (ii) when wd < p0 6 w, the retailer will use Alternative X ifa 2 {(1, L0) \ (1, a4]} and Alterna-tive Y ifa 2 {(1, L0) \ (a4, +1)}.

(2) Ifp0 = wd, (i) the retailer will implement Alternative X if D0/Qd < e (e is the natural constant); (ii) ifD0/QdP e, the retailer will implement Situation 1 for aP L0 and Alternative X for a < L0.

(3) If p0 < wd, (i) the retailer will implement Alternative X if D0/Qd6 h(a1); (ii) if D0/Qd > h(a1), theretailer will implement Situation 1 for a2 6 a 6 a3 and Alternative X for other a values.

As the Stackelberg leader, the manufacturer knows the retailers reaction for any given quantity-discountpricing scheme set by her. Therefore, the manufacturers expected profit will be

(1) If p0 > wd,

HM Z1L0

HM1faxdxZL1

HMXfaxdxZL2

HMYfaxdx if p0 > w; A:26:1

HM Z1L0

HM1faxdxZL3

HMXfaxdxZL4

HMYfaxdx if wd < p0 6 w. A:26:2

(2) If p0 = wd,

HM Z1

1

HMXfaxdx if D0=Qd < e; A:26:3

HM Z1L0

HM1faxdx ZL01

HMXfaxdx if D0=QdP e. A:26:4

(3) If p0 < wd,

HM Z1

1

HMXfaxdx if D0=Qd < ha1; A:26:5

HM Z

a3

a2

HM1faxdx Z

a2

1

HMXfaxdxZ1a3

HMYfaxdx if D0=QdP ha1; A:26:6



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where L1 = (1, L0)n(a5,a6), L2 = (1, L0) \ (a5,a6), L3 = (1, L0) \ (1, a4] and L4 = (1, L0) \ (a4, +1).Eqs. (A.26) are reproduced as Eqs. (16) in the main paper.

Under the optimal pricing scheme of the manufacturer, the retailers expected profit is

(1) If p0 > wd,

HR Z1L0

HR1faxdxZL1

HRXfaxdxZL2

HRYfaxdx if p0 > w; A:27:1

HR Z1L0

HR1faxdxZL3

HRXfaxdxZL4

HRYfaxdx if wd < p0 6 w. A:27:2

(2) If p0 = wd,

HR Z

1

1

HRXfa

x

dx if D0=Q

d< e;

A:27:3

HR Z1L0

HR1faxdxZL0

1

HRXfaxdx if D0=QdP e. A:27:4

(3) If p0 < wd,

HR Z1

1

HRXfaxdx if D0=Qd < ha1; A:27:5

HR Z

a3

a2

HR1faxdxZ

a2

1

HRXfaxdxZ1a3

HRYfaxdx if D0=QdP ha1. A:27:6

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