Supply Chain Dynamics and Channel Efficiency .....

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MANUFACTURING & SERVICE OPERATIONS MANAGEMENT

Vol. 14, No. 2, Spring 2012, pp. 327343ISSN 1523-4614 (print) ISSN 1526-5498 (online) http://dx.doi.org/10.1287/msom.1110.0370

2012 INFORMS

Supply Chain Dynamics and Channel Efciency

in Durable Product Pricing and DistributionWei-yu Kevin Chiang

College of Business, City University of Hong Kong, Kowloon, Hong Kong,[email protected]

This study extends the single-period vertical price interaction in a manufacturerretailer dyad to a multi-period setting. A manufacturer distributes a durable product through an exclusive retailer to an exhaustiblepopulation of consumers with heterogeneous reservation prices. In each period, the manufacturer and retailerin turn set wholesale and retail prices, respectively, and customers with valuation above the retail price adoptthe product at a constant (hazard) rate. We derive the open-loop, feedback, and myopic equilibria for thisdynamic pricing game and compare it to the centralized solution. Although in an integrated supply chain aforward-looking dynamic pricing strategy is always desirable, we show that this is not the case in a decentral-ized setting, because of vertical competition. Our main result is that both supply chain entities are better off inthe long run when they ignore the impact of current prices on future demand and focus on immediate-termprots. A numerical study conrms that this insight is robust under various supply- and demand-side effects.We use the channel efciency corresponding to various pricing rules to further derive insights into decisions ondecentralization and disintermediation.

Key words: supply chain management; dynamic pricing; differential game; channels of distribution;new product diffusion; operationsmarketing interface

History : Received: October 17, 2010; accepted: November 6, 2011. Published online in Articles in AdvanceMarch 28, 2012.

1. IntroductionRetailers such as Best Buy must decide periodicallyhow to set prices for durable goods, such as a newSony 3D TV, in order to extract the most prot fromthe potential market for this product. When makingpricing decisions over time, Best Buy might want toaccount for the opportunity of offering subsequentmarkdowns. Indeed, the forward-looking paradigmpredicts that the retailer would be better off consid-ering future revenue streams in its pricing decisionsas opposed to focusing on short-term prots. At thesame time, Best Buy must also consider that the man-ufacturer, Sony in this example, may also dynamicallyadjust the wholesale price at its own best interest.In this setting, how should Best Buy and Sony makepricing decisions over time? In particular, are both par-ties still better off pricing dynamically, in equilibrium,or are there advantages to focusing on short-term prof-its in a decentralized setting?

This paper analyzes the pricing problem in a distribution channel with intertemporal demand. This prob-lem is challenging because channel members have todeal with the dynamic pricing problem and doublemarginalization at the same time. Double marginaliza-tion (Spengler 1950) occurs when an upstream rm,as a result of charging a wholesale price above themarginal cost, induces its intermediary to set a price

above the optimal level. The joint effect of these twoproblems creates both current and future competition,making the price decisions perplexing. Because of ana-lytical intricacies, prior work typically examines thesetwo problems separately. This study combines theminto a single model with an objective of unveilingthe impact of different pricing rules on the distribu-tion efciency. Specically, based on diffusion theory,we develop a dynamic pricing game in a setting of amanufacturerretailer dyad facing an exhaustible pop-ulation of customers who differ in their valuationsof purchasing. The two independent channel par-ties sequentially set wholesale and retail prices overtime to maximize their own benets. We analyticallyderive the open-loop, feedback, and myopic equilib-rium prices for such a game. Our result indicates thatalthough a forward-looking dynamic pricing maxi-mizes the net discounted prot for a monopolist, it isnot an efcient one in a decentralized supply chain.Regardless of which forward-looking equilibrium con-cept is applied, the manufacturer and the retailer regis-ter a higher prot when they both act myopically, basing their price decisions on immediate-term prof-itability. Ironically, the overpricing result induced bydouble marginalization can be mitigated by the myo-pic behavior because of its ignorance of using time todiscriminate heterogeneous customers.

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Chiang: Supply Chain Dynamics and Channel Efciency328 Manufacturing & Service Operations Management 14(2), pp. 327343, 2012 INFORMS

Within the context of a distribution channel, a recentempirical study by Che et al. (2007) reveals that sim-pler shorter-horizon games explain the behavior of channel members better than more complex longer-horizon games. Managerial actions that discountthe future and emphasize short-term performance

are usually condemned as economic short-termismor myopia, the existence of which has been extensi-vely observed by numerous studies (e.g., Mizik and Jacobson 2007). As reected by the remark below,concerns over economic short-termism have gainedserious attention from academics and practitioners:

All employees (managers, product designers, serviceproviders, production workers, etc.) allocate theireffort between actions that inuence current periodsales and actions that inuence sales in future periods.Unfortunately employees are generally more focusedon the short term than the rm would like.

(Hauser et al. 1994, p. 328)

There are various explanations for the existenceof economic short-termism. For example, behavioralresearch suggests that managers frequently nd it dif-cult to accommodate intertemporal effects correctlyin their decision making (e.g., Chakravarti et al. 1979,Meyer and Hutchinson 2001). In addition, employ-ees have incentives to behave myopically when theirperformance evaluation depends on a current-termoutcome measure, when they feel pressured to meetearnings expectations, and when their compensationand job security are tied to market reactions (Stein1989). Although economic short-termism is typicallyconsidered harmful, our results paradoxically indicatethat rms in a supply chain may benet from such behavior. In a particular sense, the employees inabil-ity or failure to look ahead could turn out to be a bless-ing in disguise.

The explicit solutions of our analysis allow us tofurther explore relevant insights and implicationsfor managing a supply chain. For example, supplychain decentralization is typically considered inef-cient because it does not allow for rational allocationof resources based on a central plan. In contrast, weshow that when managers are myopic in pricing,decentralization can possibly improve channel efcie-ncy by alleviating intertemporal competition. Anothernotable contribution of this study is its investigation of the interaction between the pricing rule and the inter-mediation decision. In selling through an intermedi-ary over time, it is often challenging to align the bestinterests of independent channel members. We estab-lish conditions under which it is more protable forthe manufacturer to eliminate the retailer.

The remainder of this paper is organized as fol-lows. After a review of the relevant literature in 2, 3describes the process of demand dynamics and inves-tigates the optimal monopolist pricing to establish a

performance benchmark. Section 4 analyzes the equi-librium results of forward-looking and myopic pricinggames. Section 5 explores the impact of myopic pricingon channel efciency. Section 6 provides insights intothe conditions for disintermediation. Section 7 numer-ically tests the robustness of the major nding. The

paper is concluded in 8, where the results are sum-marized with a discussion on limitations and futureresearch directions.

2. Literature ReviewThe primary research streams underlying our workare studies of diffusion-based pricing of durablegoods, intertemporal pricing with heterogeneous cus-tomers, supply chain coordination, and channel dyna-mics. The relevant work and knowledge are reviewed below to clarify our contribution.

The Bass diffusion model (Bass 1969) has proven

to be seminal for characterizing the adoption processof a new durable product among a group of poten-tial buyers. This epidemic model describes the adop-tion rate over time as the product of the likelihoodof adoption, determined by the innovation and theimitation effects, and the remaining market poten-tial, specied as the difference between a xed mar-ket potential and the time-dependent installed base.Two approaches have been used to incorporate theprice effect into the Bass model: one that multiplies theadoption rate by a decreasing function of price, andanother where the xed market potential is a price-dependent function. Postulating that the likelihood

of adoption is negatively correlated to price, the rstapproach is initiated by Robinson and Lakhani (1975)and followed by numerous pricing models in bothmonopolistic (e.g., Dolan and Jeuland 1981, Bass andBultez 1982, Krishnan et al. 1999) and oligopolistic set-tings (e.g., Thompson and Teng 1984, Eliashberg and Jeuland 1986). Although this approach is empiricallysupported by Jain and Rao (1990), it is potentiallyproblematic for modeling durable goods because theresulting demand elasticity is independent of theinstalled base, as pointed out by Kalish (1983).

Recognized as more appropriate for studyingdurable goods pricing, and thus adopted in this study,the second approach follows Mahajan and Peterson(1978), who argue with data that the market poten-tial over time is more likely to be dynamically inu-enced by marketing mix variables. With the rationalethat customers decision to purchase depends on theirreservation price, this approach replaces the xed mar-ket potential in the Bass model with a price-dependentfunction so that whenever the price goes down, thepotential goes up, and vice versa. If the price goes up,some of those who have purchased may nd out thatthe price is above their valuation. In this case, it is


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Chiang: Supply Chain Dynamics and Channel EfciencyManufacturing & Service Operations Management 14(2), pp. 327343, 2012 INFORMS 329

presumed that the utility maximization principle willlead them to resell to those with higher valuations inthe secondary market (see Kalish 1983, 1985; Xie andSirbu 1995). Besides rendering the analysis tractable,as will be illustrated in 3, this stylization is appealingin the sense that it captures a crucial market behav-

ior: price increase induces arbitrage/resale behaviorsthat in turn decelerate diffusion. With this stylization,the formulation of price-dependent market potentialhas been empirically validated by multiple studies(e.g., Kalish 1985, Jain and Rao 1990, Horsky 1990,Mesak and Berg 1995) for a variety of durable goods,such as televisions, air conditioners, and dishwashers.

Relevant studies examining diffusion-based pric-ing with the price-dependent market potential includethose of Feichtinger (1982), Jrgensen (1983), andKalish (1983), who, respectively, apply concave, lin-ear, and general price functions in their monopolisticmodels. In addition, Kalish (1985) and Horsky (1990)

derive more complicated price functions to gener-ate descriptive and normative implications of optimalpricing. Using linear price functions similar to that of Jrgensen (1983), Raman and Chatterjee (1995) inves-tigate the effect of uncertainty on monopolist pric-ing, and Xie and Sirbu (1995) examine the impact of demand externalities on the open-loop equilibriumof duopolistic competition. The price function of ourmodel, in which the rationale is more explicitly elu-cidated, also appears in a linear form. Although themodels and the contexts vary across studies, unlessunder some special circumstances where the imita-tion effect dominates the innovation effect, the result-ing pricing patterns typically follow a decreasing path,known as price skimming.

The fundamental issue of price skimming can betraced back to the economics and marketing litera-ture (e.g., Stokey 1979, Conlisk et al. 1984, Sobel 1984,Besanko and Winston 1990) regarding ways to inter-temporarily price discriminate consumers. This issuecontinues to motivate recent studies to develop vari-ous dynamic models from a joint perspective of oper-ations and marketing (e.g., Klastorin and Tsai 2004,Su 2007, Aviv and Pazgal 2008, Elmaghraby et al. 2008,Ahn et al. 2009). None of the studies along this line hasconsidered diffusion-based pricing in a supply chaincontext, which concerns our work. When supply chainentities vertically compete for prot margins in sellingto reactive customers, we show that forward-lookingdynamic pricing is undesirable. This result resonateswith that of Liu and Zhang (2011), who extend themodel by Besanko and Winston (1990) to a duopolysetting and discover that dynamic pricing is often aninferior strategy when rms horizontally compete forstrategic customers with quality-differentiated prod-ucts. Our nding also complements to other explana-tions for adverse effects of dynamic pricing, including

the presence of price-adjustment costs (e.g., Gallegoand van Ryzin 1994, elik et al. 2009) and behav-ioral regularities of consumer learning (e.g., Nasiryand Popescu 2011).

The literature in supply chain coordination hasdocumented various mechanisms through which the

incentives of independent channel members can bealigned to prevent pricing breakdowns caused by double marginalization. These mechanisms include protsharing (Jeuland and Shugan 1983), quantity discounts(Monahan 1984, Weng 1995), and quantity exibility(Tsay and Lovejoy 1999). For a thorough review of therelevant models, refer to Cachon (1998). Most studieson this subject hold a static view of pricing. As ourresult suggests, failing to consider dynamic effectsmay leave such short-term static analyses unable toprovide effective guidance. Among the rst to exam-ine manufacturerretailer interactions in dynamic set-tings was Shugan (1985), who shows that implicit

understandings from repeated interactions can leadto increased channel protability. Subsequently, Jrgensen (1986) derives the open-loop equilibrium fora dynamic production, purchasing, and pricing game between a manufacturer and its retailer. Eliashbergand Steinberg (1987) take an integrated view of pric-ing and operations decisions to explore the dynamicnature of coordination in an unstable demand pattern.These studies, however, do not consider the diffusion- based pricing effect investigated in this paper. Otherstudies on channel dynamics (e.g., Chintagunta and Jain 1992) are not directly related to our work, as theirmodels involve different control variables, such asadvertising.

3. Model Formulation and theIntegrated Supply Chain

Consider a supply chain in which a manufacturerdistributes a new durable product through a retailerover an innite time horizon. By durability, we meanthat each consumer adopts one unit of the product atmost. The product, for which there are no substitutesor complements, is available to a population of con-sumers who are price takers and whose product valu-ations are heterogeneous. A price must be set for eachperiod, within which an untapped customer whosevaluation is weakly higher than the price will adoptwith a given likelihood. To concentrate our analysison dynamic price interactions, we ignore capacity con-straints and inventory-related costs. In particular, weassume that the manufacturers production quantityaccords with the retailers order quantity, which fol-lows the demand rate. This stylization applies directlyto those products with short production/deliverylead times or with insignicant unit production costs(e.g., books, software, or other digital products). It is


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also justiable in make-to-order settings where theproduct is built once an order with payment is con-rmed, or in chasing demand settings where theproduction plan matches the demand rate as closelyas possible to minimize inventory holding costs.

3.1. Demand DynamicsTo better explain the demand dynamics, we start witha discrete-time uid model in which the rm setsprices at the time epochs t i =i t , where i =1 2 3 ,and t 0 1 is a xed length of time for eachperiod. Let di be the demand, or the amount of adopters, in period i (during time interval ti ti+1 .As explicated below, the demand di is affected notonly by the price in period i, denoted by pi, but also by the previous prices if i > 1. Following a conven-tional approach to capturing heterogeneity in con-sumer tastes and preferences, we assume that thecustomers product valuations are spread uniformly

between 0 and N , where the market density is normal-ized to unity without loss of generality. Let y a bi be theamount of customers with valuations between a andb who have adopted by the end of period i; a b0 N . Denote the cumulative demand by the end of period i by xi ; xi =y

0 Ni . As y

a bi is a subset of xi ,

we call it the segmented installed base. There is noadopter prior to the initial period; i.e., x0 =y

0 N0 =0.During any period in the selling horizon, say

period n, n 1, all customers with a valuation inpn N who have not yet adopted may possibly adoptat the price pn . Those customers are referred to aslikely adopters. With the segmented installed basedened above, the amount of likely adopters in periodn, denoted by Ln , can be specied as Ln =N pn y pn Nn1 . Let 0 1 be the likelihood per time unitthat a likely adopter will adopt. This hazard rate, alsoknown as the trial rate (Fourt and Woodlock 1960),reects the speed of adoption. It applies at any valu-ation level and is assumed to be constant over time.This modeling assumption is empirically supported by Horsky (1990), who nds that whereas the priceeffect on the market potential is signicant, the imita-tion effect in the Bass model does not coexist (or is veryweak) with the price effect for all product classes in hisstudy. Because the period length is t , the percentageof likely adopters who will actually purchase in eachperiod is t 0 1 . Multiplying this percentage byLn yields the demand in period n; that is,

dn = t N pn ypn N

n1 (1)

After dn amount of customers have adopted inperiod n, the diffusion process proceeds to periodn +1, within which the price can be lower than, equalto, or higher than pn . To see how price change affectsthe demand dynamics, according to the relationship

between the prices for the two consecutive periods,we can specify the segmented installed base pertinentto deriving dn+1 as

y pn+1 Nn =dn +y pn+1 Nn1 if pn+1 pny pn Nn

1

zn otherwise,

(2)

where zn =ypn pn+1n > 0. If the price is nonincreasing

over time, the segmented installed base on the left-hand side of Equation (2) recursively converges to thecumulative demand, i.e., y pn+1 Nn = ni=1 di =xn . In thiscase, as depicted in Figure 1(a), the amount of likelyadopters in period n +1 can be characterized by

Ln+1 =N pn+1 xn (3)We next argue that this equation remains valid even

if the retailer considers the possibility of increasingprices at a given time. We justify this fact below based on the existence of efcient secondary markets.

Figure 1 Likely Adopters in Period n +1

1

M a r

k e t d e n s i

t y

M a r

k e t d e n s

i t y

Likelyadopters

Likelyadopters

x n

Ln+1

Ln+1

Product valuation

Product valuation

1

(b) Initial price increase occurs

yn[ pn+1, N ]

0

0

pn

pn

pn+1

pn+1

zn

N

N

(a) Price keeps nonincreasing


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An alternative justication, which does not involveresale markets, is provided in Appendix A, where weshow that optimal prices must follow a decreasingpath, so (3) does indeed describe the relevant dynam-ics. We focus the presentation on the model withresale, because this approach is consistent with related

literature (e.g., Kalish 1983, Xie and Sirbu 1995) andit makes it easier to accommodate additional exten-sions (such as the imitation effect, see 7) where ananalytical proof of decreasing prices is difcult. Nev-ertheless, all our analytical results in this paper holdwithout resale markets.

So, what happens if the price increases, and resalemarkets are allowed? Suppose that the initial increaseoccurs in period n +1 (i.e., pn+1 > p n , and pi+1 pi if 1 i < n . Equation (2) then implies that y

pn+1 Nn =xn zn , as illustrated in Figure 1(b). Because zn is theamount of prior adopters with valuations below thecurrent price, the utility maximization principle will

rationalize them to resell to those with higher valua-tions at a competitive price. As reviewed previously,the literature on durable goods pricing argues thatresale is behaviorally and technically justiable in thesecondary market. As a result of resale, the eligibleamount of likely adopters diminishes by zn and turnsout to be

Ln+1 = N pn+1 ypn+1 Nn zn (4)

which, after substituting y pn+1 Nn with xn zn , is iden-tical to (3). A similar rationale continues to applywhenever the price increases thereafter. The modelthus captures a sensible phenomenon that arbitrage behaviors induced by a price increase decelerate dif-fusion, and is meanwhile rendered tractable as itreduces to having only one state variable, xi .

Now let x ti =xi and p ti =pi. Then the cumu-lative demand in period n + 1 can be expressedas x tn+1 =x tn + tL n+1, where, based on Equa-tion (3), Ln+1 =N p tn+1 x tn . Accordingly, afterdenoting tn+1 by t, we can specify the rate at whichdemand increases during time interval t t t asx t x t t

t = N p t x t t (5)

This allows a continuous approach to characteriz-ing the diffusion process with an innitesimal lengthof time for each period. Specically, as t 0, thedemand rate on the left-hand side of Equation (5) becomes the time derivative of x t . Rewriting it withthe dot notation for the time derivative leads to thefollowing differential equation of demand dynamics:

x t = N p t x t (6)Obviously, the model captures the saturation effect

(the shrinkage in potential demand with increasing

market penetration) and is endogenous with respectto price. It corresponds to a traditional diffusionmodel with linear price-dependent market potential,as reviewed in 2.

3.2. Monopoly Benchmark: The OptimalPricing Strategy

We rst establish a performance benchmark by analyz-ing the problem of a vertically integrated supply chainto clarify the intuition. Assume that the supply chainincurs a constant marginal production cost c. Let bethe discount rate, which is exogenously determined by the cost of capital. Given the dynamic demand pro-cess in (6), the objective for an integrated supply chain(monopolistic seller) who follows a forward-lookingpricing rule is to nd p t that will maximize thenet discounted prot, specied below, over an innitehorizon:

F p=

0e t p t

c

x t dt (7)

Applying standard control theory (see Kamienand Schwartz 1991), we dene the current valueHamiltonian for the dynamic optimization problem as

H x p= p t c + t x t (8)where t is the shadow price (also known as theadjoint or costate variable) associated with the statevariable x t , which can be interpreted as the impactof selling an additional unit on future prots. Ceterisparibus, a positive shadow price implies lowering thecurrent price to sacrice prot now for future bene-t, and vice versa. The following proposition presentsthe optimal pricing strategy for the integrated sup-ply chain.

Proposition 1 (Optimal Forward-Looking Pric-ing). The optimal pricing strategy and the correspondingaccumulated sales over time, respectively, are given by

pF t =c + N c 1 / e t andxF t = N c 1e t (9)

and the shadow price can be expressed as

t = N c 12 / e t (10)where

= 2 +2 /2 > 0 (11)As expected, the optimal strategy is that of priceskimming, which consists of a monotonically decreas-ing price over time. One observation that makes intu-itive sense is that a higher speed of adoption causesthe initial price to be higher and to decrease morerapidly early in the selling process. Clearly, when thespeed of adoption is low, a high price results in a


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slow selling speed, preventing the rm from reapingmore cash in early periods. As money now is preferredwith a positive discount rate, a lower pressures theseller to lower the initial price to speed up sales andincrease early cash ow, as it is too costly to start witha higher price and wait for more high-valuation cus-tomers to adopt. On the other hand, when the speed of adoption is high, most customers are willing to adoptat the outset as long as the price is right for them.In this case, because the price can always be droppedin the near future before discounting makes futurerevenues less valuable, the rm will set higher pricesinitially and then cut prices over time to sell to thelow-valuation consumers remaining in the market.

After plugging (9) into (6) and then into (7), we canspecify the optimal discounted prot as

F = + 2 +22

N c 2 (12)It is straightforward to show that the prot increaseswith the speed of adoption but decreases withthe discount rate . Now if the integrated supplychain is myopic when setting prices, it then disregardsthe evolution of cumulative sales and maximizes theinstantaneous prot given by p tc x t . With thesame dynamic demand process in (6), it can be veri-ed that the myopic prices over time and the resultingnet discounted prot, respectively, are

pM =c +N c

2e /2 t and

M = 4 +N c 2 (13)

Consistent with the result that the shadow price tin (10) is uniformly negative, comparing the two dis-tinct pricing rules reveals that the myopic pricingin (13) is lower than the forward-looking pricing in(9) at any time. This conforms to our intuition thata forward-looking rm will sacrice current protsfor future benets by setting higher current pricesto price discriminate high-valuation customers acrosstime. Not surprisingly, from a long-term perspective,the resulting net discounted prot in (12) is higherthan that in (13) with myopic pricing.

4. Dynamic Pricing in theDecentralized Supply Chain

When the supply chain is decentralized, the manu-facturer and the retailer independently set wholesaleand retail prices, ignoring the collective impact of theirdecisions on the supply chain prot as a whole. Withthe manufacturer being the Stackelberg leader, we con-sider three separate games to analyze the strategicprice interactions over time: the open-loop, the feedback, and the myopic pricing games. In what fol-lows, we detail the equilibrium concept and result of each game.

4.1. Open-Loop EquilibriumWith the open-loop equilibrium concept, both channelmembers are forward looking and plan ex ante theirprice decisions, which depend only on time. Specif-ically, at the outset of the selling horizon, the man-ufacturer announces a schedule of wholesale prices,

denoted by w t , before the retailer makes the retailprice decision p t for each time instance t . Note thatwherever there is no confusion, we will omit the func-tion argument t for brevity. To obtain the equilibriumprices, we start by solving the retailers problem andthen recursively solve that of the manufacturer, tak-ing into account the retailers rational decision behav-ior. Subject to the dynamic demand process in (6),the retailer reacts to the manufacturers wholesaleprice decision by maximizing its net discounted prot,which can be specied as

r p =

0

e t pw xdt (14)Anticipating the retailers price choice, the manufac-turer determines a wholesale price path that maxi-mizes its net discounted prot given by

m w = 0 e t wc xdt (15)The open-loop equilibrium of the pricing game is pre-sented in Proposition 2 below, with the proof rele-gated to Appendix B.

Proposition 2 (Open-Loop Equilibrium). Whenthe manufacturer and retailer are both forward looking,

the open-loop wholesale and retail prices, respectively, are given by

wOL t =N +c

2and

pOL t =wOL t +N c

21 e t (16)

the resulting cumulative sales over time can then be char-acterized by

xOL t =N c

21e t (17)

where is specied in (11).Unlike the integrated supply chain wherein the

retail price continues to drop until driven down to theunit production cost, the result in Proposition 2 indi-cates that the decentralized supply chain stops exploit-ing the remaining demand when the price is slashed tothe stationary wholesale price. Whereas the retail pricedecreases over time, the wholesale price is time invari-ant. One immediate interpretation of this result is thatif the wholesale price decreases over time, the retailer,anticipating its future reduction, will then slow down


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sales by setting higher prices early on to prot fromthe lower wholesale price in the future. This in turnhurts the manufacturers protability. By endeavoringto set a constant wholesale price to induce the retailerto sell more rapidly, the manufacturer, who faces thediscounting effect, can siphon off a more protable

revenue stream in the early periods. Another way tolook at it is that the manufacturer is unable to directlyprice discriminate customers because it must stickto the preannounced wholesale price schedule in theopen-loop case. Specically, if the retailer knows thatthe wholesale price will drop tomorrow, it will holdsome sales today (when the discount rate is negligi- ble, sales will halt). Compared with maintaining thesame wholesale price, this implies that a chunk of revenue that could have been generated by the man-ufacturer today will be deferred to tomorrow, whenit will become less valuable on account of the lowerwholesale price and the time cost of money.

Note that the open-loop equilibrium is time-incon-sistent (i.e., the original best decision for some futureperiod is inconsistent with what is preferred whenthat future period arrives). Thus, to sustain the equilibrium, an implicit assumption is that the manufactu-rer is able to commit creditably to its preannouncedwholesale price schedule. In reality, this can beachieved practically through a contract when the legalsystem is effective in remedying a breach of con-tractual obligations. Nevertheless, because periodi-cally revisiting the wholesale price helps to exhaustthe residual market, modifying the initial agreementis likely to be ex post mutually benecial. A ques-tion arises in this context: Would the manufacturer betempted to deviate from the original schedule as timeevolves and decrease the wholesale price over timeuntil it reaches the level of unit cost? Besides the phys-ical costs of recontracting or renegotiation, this behav-ior is undesirable for two reasons. First, when theretailer anticipates such behavior and does not con-sider the manufacturers price threat ex ante to becredible, the latters protability may erode. Second,as mentioned above, implementing a constant whole-sale price prevents the retailer from behaving oppor-tunistically so that the manufacturer can prot earlier.Thus, even if recontracting in the future brings inadditional revenue, by the time the remainingdemand is exploited, that revenue may lose its value because of the time cost of money. In general, whenthe manufacturer has no incentive to deviate from theoriginal plan and has a reputation for refraining fromrenegotiation, the open-loop equilibrium is still sus-tainable even without resorting to a contract.

4.2. Feedback EquilibriumAlthough the open-loop equilibrium reects someobservations of actual channel practice likely due to

its relatively easier tractability in deriving strategicactions, it may unravel if the manufacturer cannotcredibly commit to its decisions. We now examineanother strategic concept, the feedback equilibrium,which is known to be time consistent and is thusrenegotiation proof. Unlike the open-loop concept in

which the manufacturer commits to the entire sequ-ence of price decisions through time, when the feedback concept is applied, the pricing decision at eachpoint in time is made on the basis of the status of cumulative demand at that particular time. Because of its intricacy and complexity, the derivation of the feedback equilibrium is generally intractable, even numer-ically. Yet, with the quadratic prot functions speciedin (14) and (15) for this particular problem, the equi-librium can be analytically derived through solvingtwo partial differential equations simultaneously; thisis detailed in Appendix C.

Proposition 3 (Feedback Equilibrium). When themanufacturer and retailer are both forward looking, the feedback wholesale and retail prices, respectively, are char-acterized by

wFB t =c +23

N c 1 e t and

pFB t =c + N c 1 e t (18)

and the corresponding sales volume over time is given by

xFB t

=N

c 1

e t (19)

where = 6 +4 2 2 /6 and 0 < < .In contrast to the static wholesale price in the open-

loop equilibrium, the feedback wholesale price wFB tdecreases over time with a relatively higher initiallevel and converges to the unit cost of production c.Accordingly, the corresponding retail pricing pFB talso demonstrates a falling path over time and con-verges to c, which is similar to the monopolist pricing.This result is not surprising because the feedbackequilibrium, which takes into account strategic inter-actions between the channel members through the

evolution of cumulative demand over time, is sub-game perfect and time consistent.

With the equilibrium prices in Propositions 2 and 3,the net discounted prots for the manufacturer withthe open-loop and feedback pricing strategies, respec-tively, can be spelled out as

OLm =12

12

N c 2 and

FBm =13

14

N c 2 (20)


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Moreover, it is straightforward to verify that theretailer appropriates only half of what the manufac-turer makes in either equilibrium; that is, OLr = OLm / 2and FBr = FBm / 2, which coincides with the results inthe typical static analysis of vertical price competi-tion. Comparing the net discounted prots with thetwo forward-looking equilibrium concepts, we ndthat both supply chain members are better off withthe feedback than with the open-loop equilibrium;that is, FBm > OLm and FBr > OLr . Consistent with thetraditional analysis of the static supply chain inter-action, the total net discounted prot generated bythe two independent channel members is lower thanthat of the integrated supply chain (i.e., OLm + OLr < F and FBm + FBr < F , regardless of which equilib-rium concept is applied. The reason behind the lossin gains is that the equilibrium prices specied in (16)and (18) are always higher than the optimal pricing,given in (9), over time. That is, the inefciency of double marginalization with the static analysis alsoexists with the dynamic analysis in a multiple-periodmanner.

4.3. Myopic EquilibriumNow we explore the pricing behaviors in the myopicsupply chain, wherein the manufacturer and theretailer emphasize on immediate-term prots whensetting prices, ignoring the future impact of their deci-sions on the dynamics describing the demand evolu-tion. By adopting a myopic pricing rule, both channelmembers act as if the planning horizon is reduced toone period; therefore, they solve a static optimizationproblem for each time instance. It is straightforwardto show that the retailers best reaction to the manu-facturers wholesale price decision is

p= N +w x /2 (21)Backward substitution implies that the equilibriumof the myopic pricing game corresponds to the solu-tion of the manufacturers control problem speciedas follows:

maxw

wc N px subject to (6) and (21).(22)

Following the standard optimization approach, weobtain the equilibrium result below.

Proposition 4 (Myopic Equilibrium). When themanufacturer and retailer are both myopic, the equilibriumwholesale and retail prices, respectively, can be specied as

wM t =c +N c

2e /4 t and

pM t =c +3N c

4e /4 t (23)

with the following accumulated sales over time:

xM t = N c 1e /4 t (24)

We can make a few observations after comparingthe myopic equilibrium pricings in Proposition 4 withthe forward-looking ones in Propositions 2 and 3.First, the wholesale price decreases over time in themyopic supply chain, with an initial level equal to thestatic wholesale price in the open-loop equilibrium.

Second, regardless of which forward-looking equilib-rium concept is applied, over time the myopic pricesare uniformly lower than the forward-looking prices.Next, although a higher initial price is associated witha higher speed of adoption in the forward-lookingsupply chain, the speed of adoption has no impact onthe initial price in the myopic supply chain. Finally,unlike the forward-looking prices that decrease in thediscount rate regardless of which equilibrium conceptis adopted, the myopic prices are insensitive to the dis-count rate.

5. Myopic Pricing andChannel Efciency

In this section, we investigate the impact of myopicpricing on supply chain protability and channel ef-ciency. In line with the myopic equilibrium in Propo-sition 4, it can be veried that the net discountedprots for the manufacturer and the retailer, respec-tively, are

M m =N c 2

4 +2and M r =

12

M m (25)

Comparing these prots to those with the open-loopand feedback forward-looking pricing rules leads tothe following proposition.

Proposition 5 (Benet from Myopic Pricing). Interms of long-run protability, both the manufacturer andthe retailer are better off when they are myopic instead of forward looking in deciding prices. Moreover, with myopic pricing, the decentralized supply chain performs at the opti-mal level (i.e., M m + M r = F when =4 .

To explain the intuition underlying this surprisingresult, Figure 2 plots the pricing trajectories underdifferent scenarios. On one hand, because of thesaturation effect, the future marginal benet of sell-ing one more unit is negative. As Figure 2(a) shows,in failing to accommodate this effect, myopic behav-ior places a downward pressure on prices and resultsin the equilibrium wholesale price wM being lowerthan the optimal pricing path. On the other hand, Fig-ure 2(b) illustrates that the effect of double margi-nalization forces the myopic price path pM to moveupward above wM . The two effects, counteracting eachother, cause pM to remain in the vicinity of the optimalpricing path in equilibrium. Consequently, the myopicdecentralized supply chain is economically more ef-cient than the forward-looking one. Figure 3 illustrates


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Figure 2 Pricing Trajectories

w M

2015105

0.8

0.6

0.4

0.2

0 25

Open-loop wOL

F e e d b a c k w F B M y o p i c

(a) Effect of pricing myopia

O p t i ma l p *

P r i c e

t

2015105

0.8

0.6

0.4

0.2

0 25

O p t i m a l p *

M y o p i c

w M

O p en-lo o p p O L

F e e d b a c k p F B

p M M y o p i c

P r i c e

t

~

~

~

(b) Effect of double marginalization

the impact of the speed of adoption on supply chainperformance under different supply chain structuresand pricing rules. Interestingly, in some special situa-tions where =4 , the myopic decentralized supplychain performs at the systems optimal level in termsof long-run protability.

There is another interesting result we can explore by comparing performance under different supplychain structures: If the speed of adoption is higherthan the discount rate, the myopic decentralized sup-

ply chain is economically more efcient than themyopic integrated supply chain. This result yields animportant managerial implication highlighted in theproposition below.

Proposition 6 (Strategic Decentralization).With myopic pricing, the net discounted prot of the decen-tralized supply chain is higher than that of the integratedone when the speed of adoption is higher than the discountrate (i.e., M m + M r > M if > . That is, decentral-ization might improve protability if the integrated supplychain acts myopically.

Figure 3 Impact of Speed of Adoption on Net Discounted Prots

0

T o

t a l d i s c o u n

t e d p r o

f i t s

S y s t e m o p t i m a l

M y o p i c e q u i l i b

r i u m

F e e d b a c k

e q u i l i b r i u m

M yo p ic

i n teg ra ted

42 3 + 2 3

O p e n - l o

o p

e q u i l i b r

i u m

1

Speed of adoption

As mentioned in 1, in many situations managersare compelled to engage in myopic management.Proposition 6 suggests that when managers are morefocused on short-term prots than the rm wouldlike, strategic decentralization (by establishing a trans-fer pricing mechanism in the context of a multi-divisional organization) may enhance supply chainprotability. If focusing on the short term is a reac-tive behavior, one may conjecture that after the supplychain is strategically decentralized, managers mightnot have incentives to remain short-term focused. Ourresults show that even if the decision makers becomelong-term focused after decentralization, the myopicintegrated supply chain can still benet from decen-tralization. As Figure 3 illustrates, this occurs when

> 3+2 3 in the open-loop equilibrium, andwhen > 2 in the feedback equilibrium. In gen-eral, past research has shown how strategic decentral-ization can mitigate interbrand (McGuire and Staelin1983) and intrabrand (Arya and Mittendorf 2006)competition. We add another upside of decentraliza-tion by showing that it can also mitigate intertemporalcompetition.

6. To Intermediate orDisintermediate?

The primary role of an intermediary is to facilitateproduct diffusion by offering specic sales skills (inc-luding brand building and product promotion), pro-viding service, gathering market information, makingaccess available to customers with special locations,and so forth. For example, as a major retailer of consumer electronics, Best Buy has the marketingexpertise and service competency to help Sony bet-ter demonstrate and promote the unique features of its 3D TV. Best Buy can also enhance customers per-ceived value of the TV by integrating it into a hometheater system. Nevertheless, in selling through an


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intermediary, besides the fact that the intermediaryappropriates a certain portion of market revenue, itis typically challenging to align the best interests of independent channel members. Whether it would bemore lucrative for a manufacturer to disintermediateis a pertinent question. In this section, we address

this question by providing insights into the conditionsfor disintermediation based on the speed of adop-tion associated with the manufacturers competenceto engage in direct sales.

Indeed, manufacturers may bypass intermediariesand engage in direct sales through their own outletstores or via Internet marketing. However, if they can-not reach a reasonable speed of sales, removing inter-mediaries will lead to an erosion of prots. Recall thatin our model, the speed of adoption is the likeli-hood that a customer will purchase the product in agiven period. Essentially this parameter reects thecompetence of a given channel to distribute a partic-

ular product to a market. Given that the value of may vary depending on which sales channel the prod-uct is distributed through, we now distinctly dene

m and r as the respective speeds of adoption whenthe product is distributed through the manufacturersdirect sales channel and the retail store. In terms of mand r , the conditions under which the manufactureris better off eliminating the retailer can be analyticallyestablished.

Proposition 7 (Disintermediation Conditions).There exists a threshold i j such that if m > i j,it is better off for the manufacturer with pricing rule i

to disintermediate its retailer with pricing rule j, wherei j F M; F denotes forward looking and M symbol-izes myopic. The relevant thresholds are characterized by(i) OLM F <

OLF F < r / 2 and

OLF M = M M = r / 2 in

the case of open-loop equilibrium; and(ii) FBF F 0, then, for any pn pn+1 p, pn pn+1 > p p+ implies

at least one of the followings is true:(i) p p+pn pn+1> p p+p p+ ;

(ii) p+ Npn pn+1> p+ Np p+ .

Proof. Let p= p1 p2 pn =p pn+1 =p + pn+2 be the optimal pricing. Consider a price vector p with pi =pi if i < n and pi =pi if i n, and

p argmax n+2 p p1 pn+1 = p1 p2 pn pn+1

If pn pn+1> p p+ , then from (A1) we must have

n

+2 p > n

+2 p . The result follows with pn , pn

+1

p, since n+2 p n+2 p if p p+pn pn+1 p p+p p+ and

p+ Npn pn+1 p+ Np p+

Suppose that the initial price increase occurs in periodn +1 for some n 1. We must then have pn < pn+1, whichimplies that pn pn+1 = p p+ for some p > 0. To con-clude the result, we now prove by contradiction that this isnot possible. Based on (A2), it can be veried that

p+ p+ p p+ = 1 2 (A3) p p p p+ = 1 1 2 (A4)


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where 1 = N p + yp+ N

n1 > 0 and 2 =p y

p p+n1 > 0. Given that

p+ Np+ p+

p+ Np p+ =0 and

p p+p+ p+ p p+p p+ = y

p p+n1 < 0, by Lemma A1, inorder for p p+ to be optimal, we must have p p+ >

p+ p+ . Accordingly, (A3) implies that 1 < 2 and it fol-lows from (A4) that p p > p p+ . Since p p > p p+ >

p+ p+ , there exists pp p+ such that p p p p+ .Based on (A2) again, we can show that p p p+ p+ = 1 +1 1 3 (A5)

p+ p p p = 1 3 + 2 3 (A6)where 3 = p y

p p+n1 > 0. Since p p p p+ > p+ p+ , (A5) implies 1 3. Accordingly, from (A6),

we must have p+ p > p p , which, given that p p p p+ , leads to p+ p > p p+ . Since

p+ Np+ p X

p+ Np p+ =

0 and p p+p+ p p p+p p+ = pp y

p pn1 < 0, Lemma A1implies that pn pn+1 = p p+ cannot possibly be

optimal.

Appendix B. Proof of Proposition 2(Open-Loop Equilibrium)Based on Pontryagins maximum principle, we will rstderive the retailers best price response to the manu-facturers price decision. After backward substituting theretailers response into the manufacturers control problem,the equilibrium will then correspond to the solution of theproblem. Specically, the current value Hamiltonian for theretailers problem is H r x p= pw + x, which yields

x = x N +w /2 and =x + (B1)The optimality conditions in (B1), obtained with a similarapproach in Online Appendix I, characterize the retailers best reaction to the manufacturers wholesale price decision.Treating the shadow price as a state variable, the cur-rent value Hamiltonian for the manufacturers optimizationproblem is dened as

H m w x = wc + x + (B2)where and are the shadow prices associated with xand , respectively. Plugging (B1) into (B2), we obtain H m =

wc + x + +N w /2 + x + 1 +2 / +N w /2, which yields

w = N +c x + /2 (B3)

and the corresponding optimality conditions = /4+ + /4 x + +N c (B4)

= /4 + x + +N c (B5)x = /4 + x + +N c (B6)

= /4 + x +N c + + / 4 (B7)It can be inferred from (B5) and (B6) that = x. Thus, thesolution must have =x+k, where k is an arbitrary con-stant. Substituting it into (B4), (B6), and (B7), we obtain thefollowing system of differential equations whose Jacobian

matrix is nonsingular:

x

=A

x +b where

A = 4

+4 0

2 1

1 1 1 11 0 2 11 0 2 +

4

and

b = 4k+N cN +c

k+N ck+N c(B8)

The eigenvalues of A and the corresponding matrix of eigenvectors of A are given by

r 1r 2r 3r 4

=

12

12

2

+2

12 + 12 2 +2

14and

H =

+2r 1 +2r 2 1 0

1 1 0 11 1 0 0

+2r 1 +2r 2 1 0

(B9)

Following the same method found in Online Appendix I,we have

x =Her 1t 0 0 0

0 er 2t

0 00 0 er 3t 00 0 0 er 4t

k1

k2k3k0

A1b

=

k1 +2r 1 etr 1 +k2 +2r 2 etr 2 k3etr 3

k1etr 1 k2etr 2 +k0etr 4 +k2

V c2

k1etr 1 +k2etr 2 +k2 +

V c2

k1 +2r 1 etr 1 +k2 +2r 2 etr 2 +k3etr 3

(B10)

Since = x +k, we must have k0 =0. Let k4 =k/ 2, thenthe general solution for (B10) is given by

x =+

2r 1e

tr 1

+2r 2

etr 2

etr

3 0etr 1 etr 2 0 1etr 1 etr 2 0 1

+2r 1 etr 1 +2r 2 etr 2 etr 3 0

k1k2k3k4

+

0

N c2

N c20

(B11)


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The four boundary conditions x 0 = 0, 0 = 0,lim t e

t t x t= 0 and lim t e t t t= 0 implyk1 = N c /2 and k2 =k3 =k4 =0. Substituting in (B11),it follows that

t = t =N c

21

2 a

e t and

t = x t =N c2 1e t (B12)where =r 1. The result in (16) follows immediately afterplugging (B12) into (B3) and the retailer reaction functionp w = N +w x /2.Appendix C. Proof of Proposition 3(Feedback Equilibrium)With the feedback equilibrium concept, the two channelmembers make decisions at each time instance, taking intoaccount the status of the current installed base. Because thenature of the decisions is in the spirit of dynamic program-ming, as detailed below, the proof will follow the principlesof optimality of dynamic programming. Let V

r t x and

V m t x denote the value functions for the retailer and themanufacturer, respectively. Given the manufacturers pricedecision w, the retailers HamiltonJacobiBellman (HJB)equation can be specied as

V r =maxp pw +V r x

N x p (C1)The maximization with respect to p yields the retailersinstantaneous reaction function:

p = N +w x V r / x /2 (C2)Anticipating the retailers response in (C2), the manufac-

turers HJB equation is given by

V m =maxw wc +V mx

N x p (C3)

which yields below the optimal feedback wholesale pricedecision for the manufacturer:

w = N x +c + V r / x V m/ x 2 (C4)Substituting (C4) into (C2) produces

p = 3 N x +c V r / x V m/ x 4 (C5)Subject to (C4) and (C5), the feedback equilibrium corre-

sponds to the solution of the system of the partial differen-tial equations in (C1) and (C3). Substituting (C4) and (C5)into (C1) and (C3) yields

V r = 16 N c x +V r x +

V mx

2

and

V m = 8 N c x +V r x +

V mx

2

(C6)

Conjecture the following quadratic functions as the solutionto (C6):

V i = Ai/ 2 x2 +Bix +K i (C7)

where ir m and the values of Ai , Bi , and K i are to bedetermined. It follows from (C7) that

V i/ x =Aix +Bi (C8)After substituting (C8) into (C5) and then into (2), the stateequation can be specied as

x

=/4 Ai

1 x

+/4 Bi

+N

c

where A i =Ar +Am and Bi =Br +Bm (C9)Solving the differential equation in (C9) with the initial con-dition x 0 =0 results in

x t =Bi +N cA i 1

1e /4 Ai1 t (C10)Plugging (C7) and (C8) into (C6) yields

2Ar x2 + Br x + K r

=16 Ai 12x2 + 8 Ai 1 Bi +N c x

+16 Bi +N c2 (C11)

2Amx2 + Bmx + K m

= 8 Ai 12x2 + 4 Ai 1 Bi +N c x

+ 8 Bi +N c2 (C12)

Equating the coefcients of x2 on both sides of (C11) and(C12) gives

Ai 1 2 8 Ar =0Ai 1 2 4 Am =0

(C13)

which leads to two possible solutions of Ar and Am. Because

x t specied in (C10) has to converge in t , the solution to(C13) must satisfy A i 1 < 0. Only one of the two solutionsis eligible. The unique one isAm =2Ar = 2/ 9 3 +4 2 6 +4 2 / (C14)Similarly, equating the coefcients of x on both sides of

(C11) and (C12) yields a unique solution

Bm =2Br =2 Ai 1 N c

8 3 Ai 1

= 23

2 6 +4 22 + 6 +4 2

N c (C15)It follows from (C14) and (C15) that

Ai =

1

4 / and B

i =1

4 / N

c

where = 6 +4 2 2 6 (C16)Substituting (C16) into (C10) yields the result in (19). Theresult in (18) can be obtained by substituting (C14) and(C15) into (C8), and then plugging (C8) and (19) into (C4)and (C5).

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