Price Postponement in a Newsvendor Model with Wholesale Price-Only Contracts

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1 Prde University Prde e-pbs Prde CIER Working Papers Krannert Gradate School of anagement Price Postponement in a Newsvendor odel with Wholesale Price-Only Contracts Yanyi X Shanghai University Arnab isi Prde University Follow this and additional works at: X, Yanyi and isi, Arnab, "Price Postponement in a Newsvendor odel with Wholesale Price-Only Contracts" (011). Prde CIER Working Papers. Paper This docment has been made available throgh Prde e-pbs, a service of the Prde University Libraries. Please contact epbs@prde.ed for additional information.

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3 Price Postponement in a Newsvendor odel with Wholesale Price-Only Contracts Yanyi X 1 and Arnab isi 1 School of anagement, Shanghai University, Shanghai 00444, China Krannert School of anagement, Prde University, West Lafayette, IN 47907, USA Abstract: We consider a variant of the wholesale price-only contract in a simple spply chain consisting of one manfactrer and one retailer, where the manfactrer is the Stackelberg leader and the retailer is the follower. In or model, the manfactrer decides the wholesale price first, and then the retailer chooses his order qantity before the stochastic demand is realized bt postpones his pricing decision ntil after the realization of demand. The existing literatre on this model has established strctral reslts nder restrictive conditions. In this stdy, we show that the optimal policies are niqe and profit fnctions are nimodal for both manfactrer and retailer nder mild conditions on the demand distribtion. We consider both mltiplicative and additive demand models. Insights are developed from analyzing the strctres of the optimal policies. Or reslts contribte as well as generalize the existing reslts in the literatre. Key words: wholesale price-only contract; price-postponement 1. Introdction Understanding the mechanisms of different contracts is one of the main aspects of spply chain management in today s bsinesses. any of these contracts are aiming to coordinate the actions of different partners to achieve higher spply chain performance; the examples are byback contract, revene sharing contract, qantity flexibility contract, etc. However, some contracts that do not coordinate the channel are also widely stdied in spply chain literatre wholesale price-only contract is one of those. For example, Lariviere and Portes (001) point ot, Given the complexity of spply chains, price-only contracts may owe their poplarity to their simplicity. Similarly, on page 38, Cachon (003) states that, Even thogh the wholesale-price contract does not coordinate the spply chain, the wholesale-price contract is worth stdying becase it is commonly observed in practice. In this paper, we consider a variant of the wholesale price-only contract in a simple spply chain consisting of one manfactrer and one retailer, wherein the manfactrer is the Stackelberg leader and 1

4 the retailer is the follower. In or model, the manfactrer decides her wholesale price first, and then the retailer chooses his order qantity before the stochastic demand is realized bt postpones his pricing decision ntil after the realization of demand. In the literatre (e.g., Van ieghem and Dada 1999, Granot and Yin 008), this is coined the price-postponement model. To mitigate the effect of demand ncertainty and be flexible enogh to offer a broad range of prodcts, postponement strategies have recently become a poplar bsiness concept that are sed in many spply chains. There are for types of postponement strategies, namely, pll postponement, logistics postponement, form postponement and price postponement (see Cheng et al. (010) for the details on each of these strategies). The first three strategies are also referred to as the prodction postponement (Van ieghem and Dada 1999). ased on the observations of Van ieghem and Dada, one advantage of the price postponement strategy over the prodction postponement strategy is that it makes capacity investment and prodction (inventory) decisions relatively insensitive to demand ncertainty, since profit margin can be covered by sitably charging the selling price after demand is realized. Ths, price postponement can help to redce market risks while taking other strategic and tactical decisions before the realization of stochastic demand. In some manfactring contexts, marginal costs of prodction may increase nder prodction postponement de to the reqirement of faster response time once prodction starts. In sch a case, price postponement may be preferred over the prodction postponement. Another advantage of price postponement is its ease of implementation. Unlike the prodction postponement strategy, that reqires re-engineering techniqes sch as operations reversal and standardization of prodct and process, price postponement is a managerial decision that is determined by marketing managers. Realistic variations of price postponement strategies are sed by Greatodels.com, an online retail store selling many items sch as books, accessories, decals, airbrsh, compressors, and scale models of aircrafts, helicopters, missiles, ships, cars, etc. (Granot and Yin 008). This retailer determines or readjsts the price of an item after demand information from pre-orders is observed. As an example of price postponement strategy, Van ieghem and Dada (1999) mentions the case of an atomobile

5 dealership who mst decide on the nmber of cars to stock before market demand is known. Then, the selling price can be negotiated with the cstomers dring the sales. Another realistic variation of price postponement strategy was sed by ank of China (OC) Hong Kong in Jly 00. In the face of high demand ncertainty, OC set an initial pblic offering price range, between HK$6.93 and HK$9.5 per share, for investors to sbscribe to its shares. Since the pblic offering was over-sbscribed by 6 times, OC finally allotted at least 500 shares to each investor at a price of HK$8.5 (Cheng et al. 010). It is qite possible that similar strategy of offering a price range initially and then, after observing the market demand, adjsting the final price later, cold very well be sed for expensive prodcts sch as aircrafts, ships, missiles, cars, etc. These examples show the potential applications of the price postponement strategy. As the postponement strategies are becoming more and more poplar day by day with the advancement of information technology, and since the price postponement strategy has mltiple benefits as mentioned above, it is worthwhile to stdy this strategy in some detail. oreover, wholesale price-only contract is the one which is most poplar and commonly sed in practice (Lariviere and Portes 001, Cachon 003), yet theoretical reslts are far from complete for this case as can be seen from the limited reslts available in the literatre so far (see, e.g., Granot and Yin 008). Therefore, it is important for researchers to investigate the combined model of price postponement strategy nder the wholesale price-only contract. In or stdy of the price-postponement model, for the mltiplicative demand case, we establish the nimodality of the expected profit fnctions of both manfactrer and retailer, and derive the niqe optimal wholesale price (w) and retail order qantity () nder mild conditions on the demand distribtion. Specifically, for the linear demand fnction, the condition on the probability distribtion is either IGFR (as defined in Lariviere and Portes 001) or a generalized version of a condition stdied in Ziya et al. (004); for the exponential demand fnction, the condition is very general and incldes all distribtions for which generalized failre rate is monotone; and for the isoelastic demand fnction, no condition is needed on the distribtion. The reslts on the first two cases provide a significant generalization over Granot and Yin (008, see Table 3) who, for the analogos models with wholesale 3

6 price-only contracts, established nimodality of the manfactrer s expected profit fnction nder the restriction that ε f ( ε ) is increasing in ε, where f ( ε ) represents the probability density fnction of ε [ A, ] with A 0 and <. Apparently, one distribtion that satisfies the above condition of Proposition in Granot and Yin (008) is the beta distribtion with restricted vales ( 1) for one of its two parameters that inclde the power and niform distribtions. These distribtions have bonded spport. Note that other commonly sed distribtions, sch as gamma, Weibll, normal, lognormal, Pareto, etc., do not sally satisfy the condition ε f ( ε ) is increasing inε, nless, as the athors mentioned, the domain of ε is relatively small. However, the market demand or the optimal stocking level may not always correspond to the vales of ε in this small domain. In this paper, for both linear and exponential demand fnctions with mltiplicative model, we are able to generalize Granot and Yin (008) s reslt on the nimodality of the manfactrer s expected profit fnction for classes of distribtions that inclde all IGFR distribtions which, as noted by Lariviere and Portes (001), inclde most of the common distribtions. We also provide the example of a DGFR distribtion (defined in Lariviere 006) for which or nimodality reslts will apply. The condition for the case of exponential demand fnction is even weaker than reqiring that the generalized failre rate is monotone; hence all IGFR and DGFR distribtions will satisfy this condition. Sch generalizations make the price postponement model more viable to se, since nder nimodality, both the optimal w and are niqe and can be expressed in simple analytical forms that are easily comptable, as can be seen from or propositions. Similar to the mltiplicative demand model, the case of additive model also contines to interest the researchers. However, no reslt is available for this model nder the wholesale price-only contract and is recognized to be a difficlt case in the literatre (Wang et al. 004, Granot and Yin 008 and Song et al. 008). For the linear demand fnction, here we prove the nimodality of the manfactrer s and retailer s expected profit fnctions and derive the niqe optimal wholesale price and retail order qantity for IFR demand distribtions. We also provide conter-examples to show how the nimodality 4

7 of the manfactrer s expected profit fnction may fail for some DFR distribtions that are IGFR. Note that we do not consider the cases of exponential and isoelastic fnctions becase they are not meaningfl when the demand model is additive (Petrzzi and Dada 00). Finally, we compare the optimal policies for the for models we stdy here to gain insights into the strctres of the soltions and their implications. In particlar, we explore why, to establish the nimodality of the manfactrer s expected profit fnction, different sfficient conditions on the distribtions are needed for different demand models. The remaining part of the paper is organized as follows. In Section, we describe the model. The analysis of both mltiplicative and additive demand models and insights into their soltions are provided in Section 3. Proofs of all the reslts are inclded in the Appendix.. odel Description We consider the mltiplicative and additive demand models for the form X = D( p) and X = D( p) + respectively, where D( p ) is the deterministic part of demand X that decreases in the retail price p, and is the random part of X with a spport on ( A, ), where A 0,. The events, contract and cost parameters in or model occr in the following seqence. A manfactrer with nlimited capacity prodces items at a cost of c per nit and acting as a Stackelberg leader offers to charge a per nit wholesale price w from a retailer. The retailer then decides his optimal order qantity for the selling season. Sbseqently, the demand is realized. ased on the demand realization, retailer chooses his retail price p that maximizes his profit. For simplicity, we assme that the salvage vale of any nsold inventory is zero and that any nsatisfied demand is lost withot any shortage penalty. To avoid trivial soltions, we let c< w< p. Note that, the analysis of or reslts wold not change if we allow for positive salvage vale and shortage penalty. For the convenience of analysis, let F( ) and f ( ) respectively denote the distribtion and density 5

8 fnctions of, the random component of the demand. To avoid technicalities, nless stated otherwise, we will assme the following: Assmption 1. The probability density fnction f ( ) exists and is differentiable. Now, letting F ( ) = 1 F ( ), we denote the failre rate of as f ( ξ ) h( ξ ) = and the generalized F( ξ ) failre rate of as g( ξ ) = ξh( ξ). To clearly specify the probability distribtions to be sed in or analysis, let s now define the following classes of distribtions: Definition 1. (Ross 1996) The random variable is said to have an increasing failre rate (IFR) distribtion if h( ξ ) is non-decreasing in ( A), and a decreasing failre rate (DFR) distribtion if h( ξ ) is non-increasing in ( A, ). Definition. (Lariviere and Portes 001, Lariviere 006) The random variable is said to have an increasing generalized failre rate (IGFR) distribtion if g( ξ ) is non-decreasing in ( A, ) and a decreasing generalized failre rate (DGFR) distribtion if g( ξ ) is non-increasing in ( A),. ost common distribtions are IGFR (Lariviere and Portes 001). IFR is a large sbset of IGFR and incldes normal, trncated normal, niform, gamma with shape parameter 1, Weibll with shape parameter parameter 1, etc. (Portes 00). Examples of DFR distribtions are gamma and Weibll with shape 1. While DGFR distribtions may be rare, later we will show an example of a DGFR distribtion. Finally, we write the profit fnctions of the channel members. In or decentralized system with the wholesale price-only contract, the retailer s expected profit fnction is given by Π ( p, ) = E R [ p min(, X )] w, and the manfactrer s expected profit fnction is given by Π ( w, ) = ( w c ). The reslts and insights of or analysis are provided next. 6

9 3. The Analysis, Reslts and Insights In this section, we derive or reslts for the price-postponement model with wholesale price-only contracts. The linear, exponential and isoelastic demand fnctions for the mltiplicative model and the linear demand fnction for the additive model are discssed in the following three sbsections. 3.1 The ltiplicative Demand odel For the mltiplicative demand model X = D( p), we show that when the demand fnction is linear, that is, D( p) = 1 p, the manfactrer s profit fnction is nimodal in the retailer s order qantity for IGFR demand distribtions as well as for demand distribtions satisfying another very general condition which holds for most of the common distribtions. Next, we establish nimodality of the profit fnctions and derive niqe optimal soltions respectively for the exponential demand fnction nder a very mild condition on the generalized failre rate of and for the isoelastic demand fnction nder no condition on the distribtion of. To proceed, we start with the linear demand fnction. The Case of Linear Demand Fnction When D( p) =1 p, by Lemma 1 of Granot and Yin (008), we know that the optimal retail price is given by p 1/ if = 1 / ˆ if ˆ, ˆ (1) where û is the realized vale of, the random part of the demand. Also, from (13) and (14) of Granot and Yin (008), the retailer s optimal order qantity is given by f( ) w= F( ) d, and the manfactrer s expected profit fnction is given by f( ) Π ( ) = ( w c) = F( ) d c. () 7

10 While the concavity of the retailer s expected profit fnction has been established in Lemma of Granot and Yin (008) for general demand distribtions, the nimodality of the manfactrer s expected profit fnction Π ( ) and the existence of niqe optimal soltions have only been established for demands for which f() is increasing in [ A, ] (see their Proposition ). We significantly generalize this later reslt in the following proposition. y sitably analyzing the behaviors of the first, second and third derivatives of Π ( ) and their limits as 0 + and +, we show that Proposition 1. Under Assmption 1, the manfactrer s expected profit fnction Π ( ) is nimodal in if the distribtion of is either (i) IGFR, or ' f () (ii) +, as a fnction of, changes sign at most once. f () Therefore, the optimal order qantity of the retailer and the optimal wholesale price w of the manfactrer are niqe and given by f( ) = and c F( ) 4 d w = c+ F( ). (3) 1 The first condition (IGFR) is the same as that of Condition C3 in Ziya et al. (004, Proposition 4.1). The second condition is a generalization of Condition C in Ziya et al., where their C is eqivalent to f () or case a) that + 0 f () ' (which is a sb-case of or condition (ii), see the proof of Proposition 1 in the Appendix). To see how general the condition (ii) is, consider the conter-example given in Section 6 of Ziya et al. (004) with the probability distribtion F( x) 1 ( x v) =, x [1 + v, + ), 0 < v <, which satisfies neither their Condition C nor C3 (IGFR); however, this distribtion satisfies or case b) that ' f () + 0 f () and moreover it s a DGFR distribtion. For this distribtion, by or Proposition 1, the manfactrer s expected profit fnction is nimodal in. 8

11 Frther note that, as also mentioned in Lariviere and Portes (001), Condition C can fail for many common distribtions, sch as exponential, normal and Gamma distribtions for which Condition C3 as well as or condition (ii) hold. For example, with the exponential density f ( x) = λe λx, we get ' xf ( x) + = λx, which is not nonnegative for all x. However, it satisfies condition (ii) that f ( x) ' xf ( x) + f ( x) changes sign at most once. This shows or condition (ii) is a significant generalization of Condition C. Ziya et al. (004) have pointed ot that neither of their Condition C nor C3 is more restrictive than the other. The same is also tre with or conditions (i) and (ii) in Proposition 1. While we cold not prove whether condition (ii) is more general than the IGFR condition (i), it is worth noting that or condition (ii) is satisfied by the long list of distribtions provided in agnoli and ergstrom (005), which incldes the commonly sed log-concave and log-convex densities. If we let S 1 be the class of probability distribtions that satisfies either condition (i) or condition (ii) of Proposition 1, then it is evident that, with or previos example of the DGFR distribtion, the class S 1 is strictly bigger than the class of all IGFR distribtions. Ths, Proposition 1 establishes the nimodality of the manfactrer s expected profit fnction and provides the niqe optimal soltions for a wide class of distribtions. The Case of Exponential Demand Fnction When D( p) = e p, the optimal retail price is given by (a derivation is shown in the Appendix) p l if ˆ e = ln ( ˆ/ ) if ˆ e, (4) where û is the realization of and ln represents the natral logarithm. Then, the expected profit of the retailer is given by 9

12 Π ( ) = E[ pmin(, X)] w R e 1 = [1 e ] f( ) d+ [(ln ln ) ] f( ) d w A e e 1 e f d f d F e w A e = ( ) + ln ( ) ln ( ). (5) R Sbseqently, from (5), the first order condition = 0 gives the optimal wholesale price w as w= ln f( ) d [ln + 1] F( e ). (6) e Therefore, the expected profit fnction of the manfactrer is given by Π ( ) = ( w c) = ln f( ) d ln e F( e) c. e (7) Now, by analyzing the first and second derivatives of Π ( + ) and their limits as 0 and +, we prove that Proposition. Under Assmption 1, (i) The retailer s expected profit fnction is strictly concave in, and (ii) The manfactrer s expected profit fnction Π ( ) is nimodal in if the distribtion of satisfies the following condition C: (C) ef ( e) 1, as a fnction of, changes sign at most once. Fe ( ) Therefore, the optimal order qantity of the retailer and the optimal wholesale price w of the manfactrer are niqe and given by ln. ( ) [ ln ] ( ) e c= f d + F e and w = c+ F( e ). (8) Once again, the above reslt overcomes the restriction of f() is increasing in in Granot and Yin (008, see Table 3). Condition C is weaker than reqiring that ef ( e) Fe ( ) S is monotone. Let be the class of probability distribtions satisfying condition C. Clearly, S incldes all IGFR and DGFR 10

13 distribtions. Ths, by Proposition, for the distribtion F( x) = 1 ( x v), x [1 + v, + ), 0 < v <, which is DGFR, the manfactrer s expected profit fnction is nimodal in and the optimal soltions are niqely obtained from (8). The Case of Isoelastic Demand Fnction For the isoelastic demand fnction D( p) = p, q > 1, the optimal retail price is given by p = ( ) 1/q q ˆ /, (9) where û is the realized vale of. The following general reslt is shown in Granot and Yin (008). A proof is inclded in the Appendix for facilitating some of or explanations in sbsection 3.3. Proposition 3. For any distribtion of, the expected profit fnctions of both the manfactrer and retailer are strictly concave in the retailer s order qantity. Therefore, the optimal order qantity and the optimal wholesale price w are niqe and given by ( q 1) 1/ q 1/ q c= ( ) f( ) d q and A qc w =. (10) q 1 From the proof we can observe that Proposition 3 holds even for discrete distribtions, bt with a continos decision variable. The nderlying reason will be explained in sbsection 3.3. Now that we have analyzed the mltiplicative demand model, we discss the case of additive demand model next. 3. The Additive Demand odel We now analyze the additive demand model X = D( p) +. While an interest in the additive demand case is always shown in the literatre (e.g., Wang et al. 004, Granot and Yin 008, Song et al. 008), no detail discssion or preliminary reslts are available for this model nder the wholesale price-only contract. Here, for the linear demand fnction, with IFR demand distribtions, we show that the retailer s expected profit fnction is strictly concave and the manfactrer s expected profit fnction is nimodal in the retail order qantity. Sbseqently, we derive the niqe optimal wholesale price of the manfactrer, and the optimal order qantity and price of the retailer. Note that, as discssed in Petrzzi 11

14 and Dada (00), if one considers the exponential and isoelastic demand fnctions for the additive model, then demand X > 0 even when p if the realized > 0, so that infinite profit is possible. Therefore, we do not analyze these cases since they are not natral models for demand. The Case of Linear Demand Fnction When D( p) =1 p, the optimal retail price is given by (the derivation is provided in the Appendix) p 1+ ˆ if ˆ 1 = 1 + ˆ if ˆ 1. (11) where û is the realized vale of. Then, the expected profit of the retailer is given by Π R ( ) = f( ) ) ( ) d w A + d (1 + f. (1) 1 R From (1), solving the first order condition = 0, we get the optimal wholesale price w as w= (1 + ) f( ) d F( 1). (13) 1 Ths, the manfactrer s expected profit fnction is given by, (14) Π ( ) = ( w c) = (1 ) f( ) d F( 1) c + 1 so that = (1 ) f( ) d 4 F( 1) c +, (15) 1 and 4 ( 1) 4 ( 1) = f F f ( 1) = F( 1) F( 1) ( 1) f( 1) f( 1) = F( 1) +. F( 1) F( 1) (16) 1

15 With these, we establish the following reslt: Proposition 4. Under Assmption 1, (i) The retailer s expected profit fnction is strictly concave in, and (ii) The manfactrer s expected profit fnction Π ( ) is nimodal in if the distribtion of is IFR. Therefore, the optimal order qantity of the retailer and the optimal wholesale price w of the manfactrer are niqe and given by (1 ) ( ) 4 ( 1) 1 c= + f d F and w c F( 1) = +. (17) Note that, nlike Proposition 1, we cannot relax the assmption of IFR distribtion on to IGFR distribtion. The reason is that IGFR distribtions also inclde DFR distribtions (Lariviere 006); and if that is the case, then in ( 1) f( 1) f( 1) + F( 1) F( 1) ( 1) f( 1) term of (16) we have, F( 1) is non-decreasing in, while f( 1) F( 1) is non-increasing in, so that the monotonicity of ( 1) f( 1) f( 1) + F( 1) F( 1) and hence, is not garanteed. Sbseqently, may have mltiple zeroes. We actally observe these behaviors for some gamma and Weibll distribtions with shape parameter < 1, for which these distribtions are both DFR as well as IGFR. For the gamma distribtion with shape parameter /3 and scale parameter 0.6, with the manfactrer s prodction cost c = 0.1, we plot the graphs of and in Figres 1a-1b and a-b, respectively (shown in the Appendix). From Figres 1a and 1b, we observe that, in the interval [0.495, 0.540] for, has three zeroes, namely, , and oreover, from Figres a and b, we observe that, in the above interval, is not monotone. Very similar behaviors are also observed for the Weibll distribtion with shape parameter 0.8 and scale parameter 1, and with the manfactrer s prodction 13

16 cost c = For this case, in the interval [0.498, 0.508] for, has three zeroes at , and A detailed explanation for these behaviors of mltiple zeroes for and nonmonotonicity of is provided in the Appendix. These examples show that Proposition 4 does not hold for some DFR distribtions that are also IGFR. Therefore, we preclde establishing nimodality of Π ( )for all IGFR distribtions and restrict or reslts in Proposition 4 to IFR distribtions. 3.3 Insights and Conclsions We now compare the reslts for the above for models to garner some insights. Let s first consider the cases of linear and exponential demand fnctions for the mltiplicative model. From (1) and (4) we observe that the optimal retail price in both cases has similar strctre, namely, for relatively high realizations of the random component of the demand (i.e., ˆ and û e, respectively for linear and exponential demand fnctions), the retailer charges the price that clears the market so that there is no nsatisfied demand. On the other hand, for relatively low realizations of (i.e., ˆ and û e, respectively), the retailer charges the price at which his profit achieves the maximm vale. For both demand fnctions, it s a constant price (1/ and 1, respectively) and the retailer may have some nsold stock. While the manfactrer s optimal wholesale prices also have similar strctre for both these demand fnctions (see (3) and (8)), the strctres of the retailer s optimal order qantities are slightly different and it is primarily de to the specific form of the corresponding demand fnction. For the isoelastic demand fnction with mltiplicative model, the strctres of the soltions are qite different. The optimal retail price in this case has a niqe form for all vales of the realization û (see (9)). This allows the manfactrer s wholesale price to be initially written in terms of a moment of the distribtion (see (3)) which, conseqently, after sbstitting ot the expression for the optimal order qantity, leads to a distribtion-free expression for the optimal wholesale price. oreover, notice from (10) that the retailer s optimal order qantity is written in terms of a moment of the distribtion. These special forms 14

17 of the optimal soltions lead to the nimodality of the manfactrer s profit fnction for all distribtions, both continos as well as discrete. On the other hand, for the linear and exponential demand fnctions, the optimal retail prices change from a constant vale to an increasing fnction of û, once û exceeds and e, respectively. This makes the corresponding wholesale prices to depend on the tail probabilities F( ) and Fe ( ), respectively (see (3) and (8)), which, sbseqently, prompt s to find the reglarity conditions that will lead to the nimodality of the manfactrer s expected profit fnction. Sch exploration reslts in sfficient conditions based on the generalized failre rate of the distribtions, as we observe in Propositions 1 and (except condition (ii) of Proposition 1 that come from stdying the reglarity properties of the second and third derivatives of the manfactrer s expected profit fnction). Next, for the additive model with linear demand fnction, from (11) we observe that the optimal retail price for relatively high realizations of (i.e., ˆ 1) is the price that clears the market. And for relatively low realizations of (i.e., ˆ 1), the retailer charges the price that maximizes his profit. Notice that this price is 1+ ˆ which depends on the realized vale of and is not a constant as was the case for the mltiplicative model with linear demand fnction. This sensitivity of the optimal retail price for lower vales of the realization of inflences the wholesale price (see (1) and (13)) in sch a way that it can lead to non-monotonicity for the second derivative and mltiple optima for the manfactrer s expected profit fnction if the distribtion is DFR, as illstrated with two examples in the previos sbsection. From the explanation provided in the Appendix, we observe that one of the reasons for the above non-monotonicity and mlti-modality is the sharp change in the density fnction for smaller vales of when its distribtion is DFR (see Johnson et al. 004, p. 341 and 631, respectively for the gamma and Weibll densities with shape parameter < 1). For this reason, to establish the nimodality of the manfactrer s expected profit fnction, we had to restrict to IFR distribtions. 15

18 Finally, let s compare the optimal retail prices between the mltiplicative and additive demand models. When the realizations of are relatively high so that the retailer charges the market clearing prices, the optimal retail prices for all three mltiplicative demand models are expressed as a fnction of the ratio between û and, whereas for the additive model it is expressed as a fnction of the difference between û and. Clearly, these forms come directly from the mltiplicative or additive strctre of the model itself. However, when the realizations of are relatively low in which case the retailer charges the price that maximizes his profit, the characteristics of the optimal retail prices are qite different between the mltiplicative and additive demand models. For the mltiplicative model with linear demand fnction, the optimal retail price is 1/, a fixed nmber. In contrast, for the additive model, it is 1+ ˆ which can take any vale between (1/, ) since û can vary between 0 and 1 in this case. This shows that not only the vales of the optimal retail prices differ between the mltiplicative and additive demand models, bt also their strctres are different. Conseqently, this creates some strctral differences in the optimal soltions and w between these two models (see (3) and (17)) as a reslt of which we get different sfficient conditions on the distribtions to garantee the nimodality of the manfactrer s expected profit fnction. To conclde, here we have stdied the price-postponement model for a newsvendor problem with wholesale price-only contract. For both mltiplicative and additive demand models, we have shown that the optimal policies and expected profit fnctions of the manfactrer and retailer are well-behaved nder reasonably mild conditions on the demand distribtion. Extension of this research to the nopostponement model in which the optimal retail price is also decided before the realization of demand eldes s at the moment. While a complete set of reslts for this model nder byback contracts is provided in Song et al. (008) and Granot and Yin (008) for the mltiplicative demand case, the corresponding reslts nder the wholesale price-only contract do not follow from there and most probably wold reqire a different techniqe to solve the problem. 16

19 Appendix Proof of Proposition 1. In (), we have the manfactrer s expected profit fnction as f( ) Π ( ) = F( ) d c, so that the first derivative of Π is given by f( ) = F( ) 4 d c. (18) In (18), we can write f( ) 1 F( ) 1 4 d = 4 df( ) 4 F( ) d = which implies that F ( ) f( ) F( ) = d+ d. Hence, (18) can be written as f( ) = F( ) 4 d c F( ) F( ) = 4 + d F ( ) = F( ) + 4 d, F ( ) f( ) f ( ) = d+ d 4 d c F ( ) f( ) c = d d. F ( ) f( ) c If we define: L( ) = d d, then = L( ), and the first derivative of L() eqals dl( ) F( ) f ( ) c F( ) f ( ) c = + + = 1. + F( ) F() (19) 17

20 c Notice that in (19), is non-decreasing in ; and if the demand distribtion is IGFR, then F ( ) f () is also non-decreasing in. oreover, F () f( ) c lim c 1 p = + < F( ) F( ), f( ) c lim =+ 1 0 F( ) F( ) >, dl ( ) which means that changes sign only once, from negative to positive. Now, note that + = lim + L( ) = 1 c> 0, (0) 0 0 and = lim L( ) c < 0, (1) + + which implies that L (0) > 0 and L ( + ) < 0. Since dl ( ) changes sign only once from negative to positive, L() changes sign exactly once from positive to negative, and so is. Therefore, we can conclde that if the demand distribtion is IGFR, then Π () is nimodal and there exists a niqe which maximizes Π (). Next, we will show that nder condition (ii), Π For this prpose, we need to analyze the second derivative of () is again nimodal. Π : f( ) ( ) 4 = f d f( ) f( ) = d f( ). d () We will analyze the sign of + = 0 separately in three cases below: 1a) If f ( 0) = 0, then f( ) lim + 0 lim = 0 d <. 18

21 1b) If f ( 0) > 0 and f (0) is finite, then lim + 0 f( ) = lim + f( ) lim + 4 d f( ) f( ) = lim + f ( ) lim + 4 d 4 d f( ) = lim + f ( ) 4 f ( ) lim + d 4 d (by mean vale theorem of integrals) f( ) = lim + f( ) 4 f( ) lim + ln d 1 f( ) = lim + f( ) ( + ) 4 d 0 1 =, (3) where ~ ( ) 0 < < 1. From (3) we also have: lim 4 f + 0 d = +. 1c) If lim 0 f ( = +, then sing L Hospital rle, from () we get + ) f( ) f( ) lim + = lim lim 0 + d f( ) d df ( ) f( ) = lim + d lim d f( ) d ' f( ) f ( ) = lim + d lim f( ) (4) Now, we are ready to analyze lim +. From the theory of probability, we know that lim f () = 0 + for any distribtion. esides, f( ) lim + 4 d < lim + 4 f ( ) d = lim + 4 F ( ) 0 =. Hence, from () we get 19

22 f( ) f( ) lim + = lim lim + d + f( ) d df ( ) f( ) = lim + d lim + d f( ) d ' f( ) f ( ) = lim + d lim +. f( ) And, the third derivative d 3 Π 3 eqals 3 ' ' f( ) f( ) f ( ) 4 f ( ) 4. 3 = + = + f ( ) ' f () Now, assme that + crosses the horizontal zero line (or changes sign) at most once f () (notice that this is a weaker condition than saying Then, there are for cases to consider: ' f () is monotone). f () (5) (6) ' f () a) + 0 f () ' f () b) + 0 f () ' for all ; for all ; f () f () c) lim + + < 0 and lim > f () f () ' ' ; f () f () d) lim + + > 0 and lim < f () f () '. 3 In cases a) and b), from (6) we have 0 and 0, respectively. Therefore, 3 3 is monotone increasing or decreasing. Since, from (0) and (1), 3 = 1 c > and c < 0, therefore, + changes sign exactly once in the domain. In case c), again from (6) we observe that d 3 Π 3 changes sign from negative to positive once in the 0

23 domain, so that decreases initially and then increases. Here we need to analyze three sb-cases: for cases 1a) and 1b), that is, f ( 0) = 0 and f ( 0) > 0 with finite f (0), we have lim + < 0 0 Since, by (5), in case c) we have ' ( ) f ( ) lim + lim 0 + d + f = lim, f (). we conclde that 0 for all. Therefore, sing (0) and (1) it follows that changes sign exactly once in the domain. For case 1c), we have ' ( ) f f ( ) lim + = lim d 0 > 0 lim +, and f () ' ( ) f ( ) lim + lim 0 + d + f = lim, f () which implies that decreases and crosses the horizontal zero line once (since 3 changes sign once) and remains negative thereafter. Since, from (0) and (1), + = 1 c > 0 and 0 + c < 0, hence, changes sign exactly once. In case d), similarly to case c) we can show that changes sign exactly once. This completes the proof of the proposition. 3 Derivation of p in Eqation (4): p When D( p) = e, with û as the realized vale of, and z = / D ( p), the retailer chooses his retail price p to maximize p Π ( ) min{, ˆ} min{, ˆ R = p D p z w= pe z } w. If z ˆ, then e p ˆ, that is, ( ˆ ) p ln / = ln ˆ ln. In this case, Π R = p w, which is increasing in p; as p increases, z = increases as well, ntil we have e p 1

24 z = ˆ p e. This implies that the optimal stocking factor z cannot be less than ˆ. Hence, we only need to look for the optimal z with z ˆ. For z ˆ, we have e p ˆ p ln ˆ/ = ln ˆ ln., that is, ( ) p Then, the profit of the retailer becomes Π ˆ R = pe w. In this case, the first derivative of Π ~ ~ R eqals R p = e ˆ ( 1 p), which attains its local maximm at dp p = 1. Since p ln ˆ ln in this case, we have p = max{1,l n ln }. Letting ln ˆ ln = 1, we ˆ get the break-even point ˆ = e, and hence, we obtain p l if ˆ e, = ln ( ˆ/ ) if ˆ e. Proof of Proposition. (i) From (5) we get R = 1 e e e f( e) + ln f ) d e ln( ) f( e) ( e e [ln + 1] F( e) + e ln f ( e) w = ln f( ) d [ln + 1] F( e) w. e Therefore, R 1 1 = eln e f ( e) F( e) + eln( e) f ( e) = F( e) < 0, that is, the retailer s expected profit fnction is strictly concave in. (ii) Taking derivative of the manfactrer s expected profit fnction Π ( ) given in (7), we obtain 1 = ln f( ) d ln( e) F( e) c F( e) e (7) so that = ln f( ) d [ + ln ] F( e) c, e

25 = ln f( ) d [+ ln( e)] F( e) c + 0 e +, (8) and ~ + = 0 [ + lne] F ( e) c < 0. (9) oreover, 1 ln ( ) ( ) [ ln ] ( ) = e e f e F e + e + f e 1 = ef ( e) F( e) 1 ef ( e) = Fe ( ) 1. F( e) Now, note that condition C in Proposition corresponds to the following cases: ef ( e) (i) 1 0 for all ; Fe ( ) ef ( e) (ii) 1< 0 for all ; Fe ( ) ef ( e) ef ( e) (iii) lim + 1 < 0 and lim > 0 ; Fe ( ) Fe ( ) (iv) ef ( e) lim 1 0 Fe ( ) + > and 0 ef ( e) lim + 1 < 0. Fe ( ) (30) According to (8) and (9), case (i) can be rled ot since by (30), Π is convex and hence, its first derivative shold be increasing. For case (ii), 0 < for all, so that Π is strictly concave and hence, optimal is niqe. For case (iii), changes sign exactly once from negative to positive. Therefore, from (8) and (9), it follows that changes sign exactly once from positive to negative. Ths, optimal is niqe. For case (iv), the argment is similar to that of case (iii) above. Therefore, we conclde that nder condition C, Π ( ) is nimodal in. 3

26 Now, from (7), the optimal is obtained by solving ln. ( ) [ ln ] ( ), e c= f d + F e and therefore, from (6), the optimal wholesale price is given by w = ln f ( ) d [ln + 1] F( e ) = c + F( e ). e Derivation of p in Eqation (9): With û as the realized vale of, the retailer chooses his retail price p to maximize ~ Π = p D( p) min{ z, ˆ } w, R q where z = / D( p). When D( p) = p, q > 1, if z ˆ, then p q ˆ, that is, 1/ q ˆ p, ~ ˆ Π R = p D( p) min{ z, ˆ } = p w, which increases in p and attains the maximm at p = ; if z ˆ, then ˆ p 1/ q, and q Π ~ R = p D( p) min{ z, ˆ} = p 1 maximm at 1/ q Proof of Proposition 3. ˆ w, which decreases in p, so that Π ~ R ˆ p =. Hence, whatever the realization û is, The retailer s expected profit fnction is given by 1/ q Π R = f d w = f d w A Therefore, d Π R = 1/ 1/ (1 q 1/ ) q ( ) A Hence, R 11/ q 1/ q ( ) ( ). A p 1/ q ˆ =. still reaches the q f d w, (31) q 1 11/ q 1/ q = f( ) d 0 q <, A Π ~ R is strictly concave in, and from (31), the optimal w is given by 1/ q 1/ (1 1/ ) q ( ) A w= q f d. (3) 1/ q 4

27 Therefore, the expected profit fnction of the manfactrer is 1/ q 1/ ( ) (1 1/ ) q Π = w c = q f( ) d c, A so that and 1/ 1/ 1 q q q 1/ q 1/ q = (1 1/ q) f( ) d c f( ) d A q A (33) ( q 1) 1/ q 1/ q = f( ) d c, q A ( q 1) 11/ q 1/ q = f( ) d 0 q <, A 3 which implies that Π ~ is strictly concave in. Therefore, is niqe and from (33) it is obtained by solving: ( q 1) 1/ q 1/ q c= f( ) d q. A And, therefore, from (3) the optimal wholesale price is given as qc w =. q 1 Derivation of p in Eqation (11): When D( p) =1 p, the retailer s profit with û as the realized vale of is given by Π ( ) = p min{1 p+ ˆ R, } w. If < 1 p+ˆ, then Π R ( ) = p w, which is increasing in p, so that the retailer is better-off to increase p ntil 1 p + ˆ, that is, p 1+ ˆ ; this means the optimal p does not exist if < 1 p+ˆ. Therefore, for the optimal p we consider 1 p+ ˆ. Then, Π ( ) = p (1 p+ ˆ R ) w, which attains maximm at 1+ ˆ p =. Since p 1+ ˆ in this case, we have 1+ ˆ 1+ ˆ p = max,1+ ˆ. Letting 1+ ˆ =, we get the optimal p as 5

28 p 1+ ˆ if ˆ 1 = 1 + ˆ if ˆ 1. Proof of Proposition 4. (i) From (1) we get R = (1 ) f( ) d F( 1) w +, 1 so that R = F( 1) < 0. Hence, Π is strictly concave in. R (ii) Since the spport of is ( A, ), where A 0, we have, for 0 < 1/, f( 1) = 0 and F( 1) = 1. Now, if has an IFR distribtion, then in (16) we have that f( 1) F( 1) is non-decreasing in and so f ( 1) F( 1) is strictly increasing in. oreover, ( 1) f( 1) f( 1) lim + 0, 0 + = F( 1) F( 1) < and ( 1) f( 1) f( 1) lim + + =+ 0, F( 1) F( 1) > which means that changes sign only once, from negative to positive. Now, we look at in (15). Since the expectation of demand X is positive, we have that = (1 + ) f( ) d 0 c= 1 c+ E( ) > 1 p+ E( ) > , and = lim (1 + ) f( ) d 4 F( 1) c = c< 0. Ths, changes sign exactly once, from positive to negative. Therefore, we can conclde that if the demand distribtion is IFR, then maximizes Π (). Π () is nimodal and there exists a niqe which 6

29 Conter-examples of DFR/IGFR distribtions for which Π ( ) is not monotone: is not nimodal and Consider the following gamma density fnction with shape parameter /3 and scale parameter 0.6, (0.6) f( ) = e Γ(/3) /3 1/3 (0.6) Let the manfactrer s prodction cost c = 0.1. In Figres 1a and 1b of the next page, we plot respectively. Also, in Figres a and b, we plot. d Π in the intervals [0.5, 0.54] and [0.495, 0.5] of,, respectively. We have made separate plots for 0.5 and 0.5 in the intervals [0.5, 0.54] and [0.495, 0.5] of becase of the following reason: For the gamma distribtion, if 0 < 1/, then we have f( 1) = 0 and F( 1) = 1. These make d Π as piece-wise continos with different expressions for 0 1 / and 1/. Similar behaviors are also observed for the Weibll density fnction f ( ) (0.8) e =, with shape parameter 0.8 and scale parameter 1, and with the manfactrer s prodction cost c =

30 Figre 1a: Plot of in the interval [0.50, 0.54] of (for gamma distribtion). Figre 1b: Plot of in the interval [0.495, 0.50] of (for gamma distribtion). 8

31 Figre a: Plot of in the interval [0.50, 0.54] of (for gamma distribtion). Figre b: Plot of in the interval [0.495, 0.5] of (for gamma distribtion). 9

32 Explanation for the mlti-modality of Π ( ) and non-monotonicity of (gamma & Weibll with shape parameter < 1) distribtions: for DFR/IGFR From (15) and (16) we have = (1 ) f( ) d 4 F( 1) c +, 1 = 4 f ( 1) 4 F( 1). Now, for gamma and Weibll distribtions, for 0 < 1/, we have f( 1) = 0 and F( 1) = 1. d Therefore, for 0 < 1/, Π = 1 + E( ) 4 c; and 0 = 1 c+ E( ) > 1 p+ E( ) >0, and = 1/ = E( ) 1 c. = If c> E( ) 1, then 1/ = E ( ) 1 c<0, which implies that = changes sign once in [0, 1/] (see Figre 1b). oreover, for 0 < 1/, 4 =, as shown in Figre b. Next, for >1/, = 4 f ( 1) 4 F( 1), and lim + lim ( 1) 4 lim ( ) 4 1/ + f + f = 1/ = 0. α α 1 λ For the gamma density with shape parameter 0< α < 1, lim ( ) λ e + f = =+. 0 Γ( α) Therefore, for slightly bigger than 1/, will be a very large positive nmber (see Figre a). d This implies that Π will increase dramatically and may become positive for >1/ (see Figre 1a). Then as increases, will decrease and may become negative, as can be seen from Figre a; conseqently, will also decrease and may become negative (see Figre 1a). Therefore, Π ( ) can have mltiple optima and can be non-monotone for some gamma distribtion with shape parameter 0< α < 1. The same is also tre for some Weibll distribtion with shape parameter 0< α < 1. 30

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