Dynami Priing of Di erentiated Produts Rossitsa Kotseva and Nikolaos Vettas August 6, 006 Abstrat We examine the dynami priing deision of a rm faing random demand while selling a xed stok of two di erentiated produts over an in nite horizon. Pries in eah period depend on the available stok of both produts (varieties). In addition to the standard tradeo between a higher revenue and the probability of selling the produt, a higher prie for one produt also a ets the probability of a sale for the other produt. We haraterize the optimal prie paths of the two varieties and nd that the market may be optimally overed or not overed. With a positive stok of only one variety, the prie path is nondereasing. The same holds for the prie paths with two varieties, with respet to the own stok level and as long as the probability of no sale is positive. When this probability equals one, the prie of a given variety is dereasing in its own stok and inreasing in the stok of the other variety. The optimal paths re et the property that the stream of expeted future pro ts is higher when the inventory levels of the two produts tend to be loser to eah other. Kotseva: Department of Eonomis, Athens University of Eonomis and Business, Patision 76, Athens, 10434, Greee; kotseva@aueb.gr. Vettas (orresponding author): Department of Eonomis, Athens University of Eonomis and Business, Patision 76, Athens, 10434, Greee and CEPR, UK; nvettas@aueb.gr. Finanial support from the European Commission and the Greek Ministry of Eduation ( Herakleitos program) is gratefully aknowledged. 1
1 Introdution How should a seller prie di erent varieties of a produt when he has di erent inventories of eah? This problem appears important for many rms: situations where rms are multi-produt, selling di erent varieties of a produt, with these varieties viewed by onsumers as imperfet substitutes to one another, appear to be the norm in markets, rather than an exeption. At the same time, the ability of rms to sell eah partiular variety is restrited by the urrent inventory (stok) that eah rm holds. How available inventories evolve over time depends on the sales that take plae, whih in turn depend at least partly on the pries set for eah variety. Importantly, sine the varieties are viewed as substitutes by the onsumers, the pries of all varieties in uene the evolution of the available stok of eah variety. Dereasing, say, the prie of one variety while keeping onstant the prie of another will tend to a et the sales of the seond variety (and not only of the rst one) and, thus, also a et the pries the seller would like to set in the future for both varieties. Motivated by the above-desribed problems, in this paper, we set up a simple dynami model to study optimal dynami priing of di erentiated produts when there are apaity onstraints (or inventories) for eah produt. Spei ally, our objetive is to examine how the available stok of two produts, di erentiated horizontally along a single dimension, and its redution over time in uene the optimal intertemporal prie paths of these produts. The analysis of the behaviour of multiprodut rms is both of theoretial interest and of pratial importane. There are several ways in whih the deisions of a seller who o ers more than one produt about one partiular produt may in uene the pro t from selling the rest of the produts. In general, suh an interdependeny may arise either through demand or through osts. 1 In the presene of demand dependenies, the posted prie for one of the produts a ets the demand for the rest. With ost dependenies, the issue of eonomies of sope arises. These two general onsiderations may be viewed either in a stati or in a dynami setting. In the present paper we fous on a distint mehanism that reates an important intertemporal link between the priing deisions of a multiplrodut monopolist, that of inventory redution. To larify the fous of our analysis, onsider the following simple example. An automobile retailer has a given number of ars in stok. All the ars are the same model of a ertain brand and di er only with respet to their olor. Suppose that there are only two varieties, blue and red ars. Every time that a onsumer enters the store, the retailer announes the prie of the 1 For a disussion of the basi issues related to the multiprodut rm see e.g. Tirole (1986) and the referenes therein. In the marketing literature, the negative e et of one produt in the produt line on the rest is known as annibalization.
ar, possibly depending on its olor, not knowing the ustomer s preferenes about the olor of the ar. A number of questions arise in this set-up. First, how do optimal pries depend on the total stok of ars? When would the seller serve a buyer irrespetive of his preferenes and when would he let a buyer with ertain preferenes leave empty-handed? Seond, how do optimal pries depend on the relative availability of the two varieties? In other words, would the rm set a higher or a lower prie for the variety of whih it has less stok? Third, independent of the relative stok of red and blue ars, would the seller inrease, derease or leave unhanged the next-period prie of the variety he has just sold? How would the prie of the other variety hange? These questions have remained relatively unexplored in the literature, although, asual evidene suggests that they are relevant in pratie. 3 While the above example refers to di erentiation in the produt harateristis spae, the model applies to spatial di erentiation too. Consider, for example, a rm that sells the same produt at two stores loated opposite to eah other in the outskirts of some ity. The nature of the produt is suh that it is very di ult (oslty) to transfer stok between the two loations. 4 In eah period a onsumer living in the ity deides whih store to purhase from, depending on the pries posted and the transportation ost he inurs and the goal of the rm mis to maximize its total revenue. In an attempt to o er answers to the above questions, we examine the joint e et of three fators on the dynami priing deisions of the rm: nonreplenishable stok (or apaity onstraints), produt di erentiation and demand unertainty. We use as a base of our model a horizontal-di erentiation framework, roughly along the lines of Hotelling. A monopoly rm holds xed stok of two distint varieties of some (non-perishable) produt. Consumers have unit demands and arrive sequentially, one in eah period, with preferenes that are independently and identially distributed and are represented by their loation on the Hotelling line. At the beginning of the period the rm sets two pries, one for eah variety, not knowing the onsumers preferenes over varieties. The objetive of the rm is to maximize the disounted value of expeted future pro ts over an in nite horizon. The role of pries is threefold and, as a result, the monopolist faes a trade-o between posting low and high pries. Firstly, the two pries taken together determine the probability of a sale within a given period. The lower the 3 For instane, in a reent artile about the ar industry in China, it is mentioned that...oddly, the nished ars then sit in parking lots for up to 90 days before they are sold, usually at a disount beause they are not the olour or do not have the optional extras that the buyer wants. ( Ripe for revolution, The Eonomist, September 4th 004). Also, Bitran and Caldentey (003) disuss the possibility of a rm prie disriminating when ustomers preferenes di er and state that Retailers, on the other hand, are muh more ative in this way, harging di erent pries for a blue shirt and a red shirt (same model, brand and size). 4 Otherwize only total apaity would matter and not its initial alloation or subsequent utuations aross the two loations. 3
pries are, the higher the probability the ustomer will buy one of the two varieties. Seondly, onditional on a sale taking plae, the variety with the lower prie has a higher probability to be sold. Finally, the higher is the prie of the produt sold, the higher is the per-period revenue obtained. As we will see, it is important in that setting that the optimal prie of eah variety depends not only on its available stok but also on the stok of the other variety. Within this set-up we haraterize the optimal prie paths. Our main results are as follows. In a given period, two possible states of the market arise in equilibrium: either it is overed so that there is a purhase with probability one, or it is not overed, so that the probability that the buyer leaves empty-handed is positive. Depending on the initial stok of the two varieties and on the parameters values, the market may be overed or not overed in all periods, or it may be overed initially and not overed in later periods (when less units are left in stok). We nd that when the market is overed, the prie of the variety whih has been sold in the immediately previous period inreases, while that of the other variety dereases. We an show this parametrially, for small and symmetri initial stoks, and numerially, for a larger range of initial stok levels. We also prove that expeted pro ts inrease when total apaity is distributed more evenly between the two varieties. When the market is not overed, the prie of the variety sold in the previous period inreases, while the prie of the other variety remains unhanged. When the stok of one of the varieties has been sold out, the prie path of the other variety is nondereasing over time. More spei ally, as the available stok of the produt with positive stok dereases, its prie inreases, if the market is not overed, and remains onstant, as long as the monopolist wants to ensure that a sale is made with probability one. Our paper ontributes to the multi-produt rm literature by proposing a dynami model of priing di erentiated produts under apaity onstraints. Regarding produt di erentiation, an important literature has haraterized the optimal priing and produt line deisions of a multi-produt rm, as well as some issues that emerge in this setting, like the possibility of entry deterrene. This literature inludes Mussa and Rosen (1978), Shmalensee (1978), Eaton and Lipsey (1979) and some important work following on suh ontributions. However, the behavior of a rm selling multiple produts has been studied in a stati setting. To the best of our knowledge, the literature studying the optimal priing of substitutable produts in a dynami ontext is very limited. Our paper also ontributes to the literature studying the relation between apaity onstraints or inventories and optimal priing. Somewhat surprising, this issue has not reeived the attention it deserves in the ore of eonomi theory; however, it is at the heart of many studies in operation researh. 5 Talluri and van Ryzin (005), Elmaghraby and Keskinoak (003) and Bitran and Caldentey (003) are among the reent studies that o er reviews of the 5 The strand of operations researh literature that examines the problem of a rm maximizing the revenue from selling a given quantity over time, is known as revenue, yield or demand management. 4
revenue management literature. The problems examined by this literature fall into two broad ategories depending on whether rms manage diretly the available apaity or set pries over time. Our work is related to the seond group of models, where pries are the ontrol used to manage sales over time. Most of these models analyze single-produt nite-horizon problems. Depending on the assumptions, the optimal pries are either dereasing over the entire time horizon or utuate up and down. 6 Nonmonotoni prie paths are obtained, for example, by Gallego and van Ryzin (1994) who study the dynami priing of a xed inventory over a nite selling horizon. 7 In their model, the optimal prie is a funtion both of the number of units in stok and of the length of the horizon. They show that, at a given point in time the prie dereases as the stok inreases, and that, for a given level of stok the prie is higher if there is more selling time left. As a result, the prie path exhibits ups and downs, being dereasing between sales (i.e. in the intervals of time where there is no purhase) and jumping upwards immediately after a sale. In ontrast, Das Varma and Vettas (001) allow for an in nite selling horizon, so optimal pries are a funtion of the available stok only. Time is disrete and buyers, with unit demand and random valuations (unknown to the monopolist at the time of the prie setting), arrive sequentially. In that model, beause of disounting of future pro ts, the option value of a unit of the produt inreases as the available stok dereases, whih leads to a prie path inreasing in the number of units sold. The present paper builds on that work, with the main important di erene that we extend the analysis to a di erentiated-produt ontext. In ontrast to the single-produt problem, the multi-produt dynami priing problem has remained relatively unexplored. Most of the revenue management literature related to multiprodut problems studies the alloation of a xed apaity of a single resoure to multiple produts, where it is assumed that ustomers demand for a given produt is independent of the availablitiy of other produts (and the pries assoiated to them). 8 Talluri and van Ryzin (004) extend this framework by onsidering a single-resoure, multi-produt set-up where the 6 The monotoniity of the prie paths may also depend on learning about the demand, whih is not onsidered in our model. For instane, in Lazear (1986), pries are delining over time beause onsumers are idential and there is demand learning. 7 Our assumption of an in nite horizon may be interpreted as apturing existing unertainty with respet to the exat end date of a nite selling horizon. Then the disount fator embodies the probability with whih the market loses. 8 In these problems the pries of the di erent produts are predetermined. Customers arrive sequentially and request a produt. The seller observes the request and deides whether to aept or rejet it. Sine this kind of a problem has been historially rst analyzed in the ontext of the airline industry, the term usually employed to denote a produt and its assoiated prie is fare produt. For instane, a business lass seat and an eonomy lass seat (with the orresponding pries) onstitute two di erent fare produts. 5
probability of a onsumer hoosing a given produt depends on the other produts available. 9 The objetive of the seller then beomes to hoose whih fare produts to o er in eah period. They nd that, the greater the available apaity or the more time left for the sale, a larger subset (haraterized by a higher probability of a purhase and a higher revenue) will be hosen. The study of stohasti multi-produt priing problems have been very limited due to the omplexity of suh problems. 10 One approah has been to approximate these problems by their deterministi ounterparts. For instane, Gallego and van Ryzin (1997) study the joint problem of alloating many resoures over many produts and their optimal priing over a nite horizon. They show that the solutions of the deterministi problem are asymptotially optimal, suggest heuristi poliies and present appliations to networks. An alternative approah has been proposed by Maglaras and Meissner (003), who redue the multiprodut dynami priing problem to one of hoosing, rst, the optimal aggregate rate at whih apaity is used and, subsequently, determining the optimal per-period pries subjet to the onstraint that the optimal a[aity utilization rate is satis ed. We di er from the papers mentioned above in that we assume an in nite selling horizon and fous expliitly on horizontally di erentiated produts. The struture of the paper is the following. Setion presents the model. Setion 3 disusses the ase where one of the varieties has been sold out and we haraterize the optimal dynami priing of the one remaining variety. In Setion 4 we examine our main ase, where there is positive stok of both varieties. We onlude in Setion 5. Some proofs and numerial examples are presented in the Appendix. The Model We onsider a (single) seller who holds a xed stok of two varieties of some produt. 11 In eah period the seller determines the per-unit prie of eah variety, not knowing the buyers preferenes for one variety or the other. Then a single buyer arrives who either buys one unit (of one of the two varieties) or leaves empty-handed. A buyer that does not buy never omes bak. The same sequene of ations is repeated in eah period until the stok of both varieties is sold out. Time is disrete, denoted by t = 1; ; 3::: and the horizon is in nite. In order to analyze the above problem we set up a dynami model, building on a properly 9 See Anderson, de Palma and Thisse (1996) for a presentation of disrete hoie models and their appliations (their h. 4 disusses the relationship between the disrete hoie and the address approah). 10 The multiple-resoure problem is sometimes referred to as network revenue management. 11 The analysis in the remainder of the paper will be made with expliit referene to di erentiation in a harateristis spae. However, it applies equally well to a spatial framework, if one onsiders loation instead of preferene for variety. 6
modi ed horizontal-produt-di erentiation framework. More spei ally, normalizing the degree of di erentiation, we assume that the two varieties are situated at the two ends of an interval with lenght equal to unity. We denote eah buyer s preferene for variety (loation) by a parameter v t [0; 1]. Then, the utility of a buyer haraterized by a preferene parameter v t is 8 < U t s p t 1 v t ; if he buys variety 1 = : s p t (1 v t ); if he buys variety. The parameter s aptures the surplus from buying a unit (no matter of whih variety) of the produt and is ommon to all buyers. For instane, returning to our ars example in the Introdution, s is the gross utility a buyer obtains from buying a ertain ar irrespetive of its olor. 1 Importantly, and in ontrast to the main body of the horizontal-produt-di erentiation literature, we do not impose the assumption of a high enough gross surplus, whih ensures that the market is overed. 13 The parameter an be thought as apturing both the transportation ost per unit of distane, say ; and the degree of produt di erentiation, say bv; that is bv. 14 Consequently, the generalized ost that a buyer inurs if he buys a unit of produt one, equals p t 1 + vt, while, if he buys a unit of produt two, this ost is p t + (1 the pries of the two produts posted by the seller in period t. 15 (1) vt ), where p t 1 and pt are Having desribed the buyers side of the model, we now turn to the seller. We denote by K i the initial stok of produt i (i = 1; ) and by k t i the available stok (i.e the units left unsold) of variety i in period t, where k t i = 0; 1; ::::K i, i = 1;. As we have mentioned above, in eah period the seller sets pries p t 1 and pt, not knowing the value of vt, the reservation prie of the buyer in the period. Thus, from the seller s point of view, the preferene parameter v is a random variable, whih, we assume, is distributed uniformly on [0; 1]. 16 Given the unertainty that the seller faes, the role of prie determination in eah period is threefold. By determining 1 The assumption of a ommon s represents a situation where there is no di erentiation along some other, non horizontal, dimension. For instane, quality di erentiation ould be inorporated into the model by assuming a di erent gross surplus s i for eah variety. not. 13 Our following analysis shows that the prie dynamis are di erent when the market is overed and when it is 14 As in a standard Hotelling model, denotes the disutility a buyer obtains from not purhasing his most preferred variety and it is ommon to all buyers. The degree of di erentiation is aptured by the distane between the produts, whih equals bv: Beause of the linearity of the transportation ost and without loss of generality, we have normalized the distane between the produts to be equal to a unit. 15 As we have mentioned above, the model may be equally well interpreted in terms of spatial di erentiation. In this ase, v t would denote the address (loation) of the buyer who arrives in period t; would apture the transportation ost per unit of distane and bv would be the distane between the two stores. 16 Respetively, the realizations of the preferene parameter in eah period, v t, are independent and random draws from this distribution. 7
pries, the seller ontrols: rst, the probability that there is a sale in the urrent period (the lower the pries of both produts, the higher this probability); seond, given that there is a sale, the relative probability of eah variety being sold (given the prie of one variety, the lower the prie of the other variety is, the higher the probability that it will be bought) and third, the urrent-period payo (the higher the prie of the variety sold, the higher the urrent revenue onditional on a sale). Clearly there are opposite fores in work, whih would jointly determine the optimal pries in eah period. The buyer s problem In order to determine the demand for eah variety in every period we need to onsider two possibilities depending on the pries posted by the seller: either the optimal pries are suh that they ensure a sale of one of the two varieties or they are suh that a buyer, with a valuation in a ertain range, would possibly leave empty-handed. Let us rst assume that pries in period t are low enough that the market is overed, in the sense that ertainly there will be a sale in this period. Whether the buyer will purhase one unit of variety 1 or of variety depends on pries and on his preferenes, i.e. on the value of v t. Therefore, as in a standard spatial model, we have to determine rst the preferene ev t of the buyer indi erent between the two varieties. By equalizing U1 t and U t from expression 1, we nd that ev t = (p t p t 1 + )=: () Note that, sine ev t [0; 1], optimal pries should satisfy p t 1 p t : (3) Moreover, there will be a sale with probability one only if the net utility of the buyer with a preferene parameter ev t (the indi erent buyer) is nonnegative. By substituting ev t in either one of the equations in (1) we obtain the following ondition: p t 1 + p t s : (4) Consequently, when the seller posts pries that ensure a sale of one unit, a buyer will purhase variety 1, if he is haraterized by a preferene parameter v t [0; ev t ], and will purhase variety, if v t (ev t ; 1]. 17 If we let p t i > pt j ; it follows from (3) and (4) that pt i s; i; j = 1; : We, therefore, have that when the market is overed the higher of the two pries should not exeed s. 18 This ondition ensures that the probabilities of a sale we employ in our analysis are well 17 As a onvention, we assume that, given pries, if a buyer is indi erent between the two varieties he buys produt 1. 18 This onstraint is without loss of generality. Both a prie equal to s and a prie higher than s imply a zero probability of a sale for the produt pried at this level and, thus, have exatly the same impliation for the analysis. 8
de ned. The other possibility to be onsidered is that the seller has announed pries suh that buyers with ertain preferenes would hoose not to buy at all. This is the ase when, given pries, the indi erent buyer (and, by ontinuity, some buyers in his neighborhood) reeives stritly negative net utility, whih will happen only if pries satisfy p t 1 + p t > s : Then, only a buyer, haraterized by a preferene parameter suh that his utility from a purhase is positive, will buy a unit of the produt. More spei ally, a buyer will buy variety 1 only if U1 t 0, or equivalently if vt (s p t 1 )=, and will buy variety only if U t 0, or equivalently if v t 1 (s p t )=. Again, sine vt [0; 1], we obtain p t 1 ; pt s; whih ensures that the probability of a sale when the market is not overed is well de ned. The seller s problem By hoosing the pries he will set, the seller aims at maximizing the present expeted value of future pro ts with an one-period disount fator equal to (0; 1): We next determine the probabilities of eah variety being bought, onditional on a sale taking plae. Assume rst that, in some period, the seller optimally wants to ensure a sale from either one of the varieties. The probability of variety 1 being sold equals the probability that the urrent-period valuation, v t, is not higher than the valuation of the indi erent buyer, ev t : Given that v t U[0; 1], this probability equals F (ev t ) = ev t. After substituting for ev t ; from expression () we obtain F (ev t ) = 1= + (p t p t 1)=: The respetive probability of variety being sold is 1 F (ev t ) = 1= (p t p t 1)=: We an see from the above expressions that a higher prie of a given variety redues the probability of this variety being bought, while a higher prie of the other variety inreases this probability. Further, we know that there will be a sale with probability one whenever pries are suh that the indi erent buyer reeives non-negative utility. However, leaving the indi erent buyer with stritly positive utility is never optimal for the seller. The reason is the following. Imagine that pries are suh that U(ev t ) > 0. Then the seller an inrease both pries by the same amount, keeping in this way ev t unhanged and, onsequently, the probability F (ev t ) unhanged, but inreasing his urrent period expeted payo. 19 Therefore, the optimal pries that 19 From (), the loation of the indi erent buyer, ev t ; depends only on the di erene between the two pries and therefore hanging them by the same amount does not alter the value of ev t. Moreover, given that one of the produts will be sold for sure, by inreasing both pries, the seller inreases his urrent-period payo. 9
ensure a sale with probability one should satisfy U(ev t ) = 0, i.e. should leave zero utility to the indi erent buyer, whih, after substituting for ev t, leads to the following optimality ondition p t 1 + p t = s : (5) Clearly, the above reasoning would not hold if the two varieties were o ered by two separate rms. However, in the ase of a single seller that we onsider here, it aptures the fat that in determining optimal pries the monopolist internalizes the e et of the prie of the one variety on the other, being able in this way to extrat more of the surplus. What if, now,in the presene of positive stok of both varieties, the Seller wanted to ensure that a partiular variety, any of the two, is sold with probability one? He would set the prie of this variety equal to p t i = s probability of a sale remains the same, but the pro t dereases. : Again, posting a lower prie annot be optimal, sine the Further, we onsider the possibility that the market is not overed, i.e optimal pries satisfy p t 1 + pt > s, so there is a positive probability that the buyer will not purhase at all in that partiular period. If pries satisfy this ondition, from the analysis of the buyer s problem it follows that the probability of a sale of variety 1 is equal to s p t F 1 = s pt 1 ; while that of a sale of variety is equal to 1 F 1 s p t = s pt : Diret alulations give us the probability of no sale, whih equals s p t s p t 1 1 F 1 F 1 = 1 s p t 1 p t Note that this probability equals zero (that is, the probability of a sale equals one) exatly when p t 1 +pt = s : In addition, our requirement that the higher of the two pries should not exeed the valuation s and the fat that the seller will never harge a prie lower than s : ; ensure that the above probability funtions are well de ned. Equivalently, one may de ne the probabilities of a sale as above for p t 1 ; pt s; and equal to zero for pt 1 ; pt > s: We summarize below: Lemma 1 Optimal monopoly pries in period t satisfy p t 1 + pt s sale is positive, if p t 1 + pt > s, and equals zero, otherwise. : The probability of no Having derived the probabilities of a sale of the two produts in eah period, we an now formulate the seller s problem. Let V t (k 1 ; k ) denote the ontinuation expeted payo (disounted sum of future pro ts), at time t, when there are k1 t and kt units in stok from the respetive varieties (equivalently, when (K 1 k 1 ) and (K k ) units of eah variety have been sold). Given 10
that the buyers preferenes are independent aross time, the only link between the periods is the stok of unsold goods (k 1 ; k ), whih represents the state of the problem. This allows us to simplify the notation by dropping the time indexes. Following standard arguments, we an write the Bellman s equation of the value funtion, when there is positive stok of both varieties, as follows: 8 9 (s p 1 ) (p 1 + V (k 1 1; k ))+ >< >= (s p V (k 1 ; k ) = max ) (p p 1 +p s + V (k 1 ; k 1))+ (6) p t 1 ;pt s >: s p 1 p (1 )V (k 1 ; k ) >; The rst term in the maximand onstitutes the pro t from selling a unit of the rst variety multiplied by the respetive probability. The pro t itself onsists of the urrent period pro t, whih equals the prie of that variety and the ontinuation payo with one unit less in stok, V (k 1 1; k ): The seond term represents the pro t from selling a unit of the seond variety and an be interpreted in a similar way. The last term represents the pro t if there is no sale in the urrent period, whih equals the ontinuation payo with the same number of units in stok, times the probability of no sale. In the following Setions we haraterize the solution of the above problem. In the following Setion, we examine rst the ase where there is a positive stok of only one variety left. After this is done, we will turn to the ase of two varieties. 3 The ase of only one variety Reasoning bakwards, we rst analyze the ase where the stok of the one variety has been sold out and there is only one variety o ered. To simplify the notation and sine the problem is symmetri, we drop the indexes denoting variety. Then, modifying (6) aordingly, the seller s problem an be written as: V (k) = max sps ( s p )(p + V (k 1)) + (1 s p )V (k) ; (7) where V (k) is the value funtion with k units of one of the varieties (or V (k) = V (k; 0) = V (0; k)). In addition, sine in our model apaity is limited while the selling horizon is in nite, we have the following boundary ondition: V (0) = 0. Clearly, the probability of a sale equals one, if p = s ; and zero, if p (k) = s. Taking the relevant rst-order ondition with respet to p; we obtain: 0 p(k) = s + (V (k) V (k 1)) : (8) 0 In orresponding one-produt models, where no substitute produt exists, the di erene V (k) V (k 1) represents the opportunity ost of selling the unit today, or equivalently, the bid prie (see Bitran and Caldentey, 003). 11
and, after solving the system of equations (7) and (8), we obtain the optimal prie in state k: q p(k) = (1 ) + s (1 )((1 ) + s + V (k 1)). (9) Given that V (0) = 0 one ould solve reursively for the optimal prie path using expression (9). From Proposition 1 in Das Varma and Vettas (001), we know that V (k) > V (k 1) and that the prie sequene fp k g 1 k=k is inreasing. However, the analysis in Das Varma and Vettas (001) refers to one variety only, where buyers valuations are distributed on some interval [0; v] : In this ase, only a prie equal to zero would ensure a sale with probability one, whih is equivalent to throwing away a unit and annot be optimal given that the number of units in stok is nite. In the present set-up, where the buyers are aware of the (potential) existene of more than one varieties and have ertain preferenes among them, the seller ould ensure a sale of one unit of a given variety by setting a prie for that variety equal to s : Hene, optimal pries annot be lower than s and it follows, from (9), that p (k) (s ) only if the following ondition holds: 1 (1 )s + (1 )V (k 1) g k (): (10) Given that p (k) > 0; for every k; and by the reasoning in Das Varma and Vettas (001), we have that V (k) > V (k 1). It follows that g k () > g k 1 (). Furthermore, V (k) is dereasing in, therefore g k () is also dereasing in, with g k (0) > 0. Consequently, = g k () will have a unique solution e k, for every k = 1; ; ::; K, suh that, if e k, the prie that the seller posts when he has a stok k is p(k) = s, whih ensures that, irrespetive of the preferenes of the urrent-period buyer, a unit of the produt will be sold for sure (i.e. the market is overed). If > e k, the prie is p(k) > s and there is positive probability that the buyer leaves empty-handed. Moreover, it is easy to hek that the sequene of the threshold values fe k g 1 k=k is dereasing. Hene, we extend the result in Das Varma and Vettas (001) to take into aount the (potential) existene of a substitute produt, the stok of whih has already been sold out. Proposition 1 When there is only one variety left to be sold, the optimal prie path fp k g 1 k=k is non dereasing. Spei ally, if e k is the solution of = g k () (as given by 10) and i) if > e K ; then p(k) > s and the market will not be overed for every k = 1; :::; K ii) if e 1, then p(k) = s and the market will be overed for all k = 1; :::; K iii) if e j > e j 1 ; then p(k) > s ; for k < j; and p(k) = s ; for k j, where j = 1; ; :::K: 1 In addition, the optimal pries given by (9) are always smaller than s, whih ensures a positive probability of a sale. The argument is as follows. When the rm has k units in stok, that is, in state (k) it ould always mimi the prie sequene fp k 1 g till the rst k 1 units are sold. But, then the Seller will have one more unit for a sale and, sine > 0; the disounted pro ts, V (k), should be stritly higher than V (k 1) : 1
The above Proposition states that the prie sequene of the variety, from whih there is positive stok left, is nondereasing. The intuition for this result is as follows. As mentioned above, the prie posted by the seller in a ertain period determines, on the one hand, the probability of a sale and, on the other hand, the urrent-period pro t. So, given that buyers valuations are unknown and that the stok is nite while the number of buyers is in nite, there is an option value of keeping a unit and selling it in the future. If the stok is large, this option value is small and the rm is better o by setting a relatively low prie, thus, selling today with high probability and, therefore, realizing positive pro ts sooner. 3 The opposite holds, if the number of units in stok is relatively small. The di erene from Proposition 1 in Das Varma and Vettas (001) is that in our set-up we have to distinguish between three ases, depending on the value of the parameter, i.e. depending on the degree of di erentiation and/or the level of the transportation ost. First, if is relatively small ( e 1 ), there will be a purhase with probability one in every period until the stok is sold out. Pries will be onstant at (s after substituting into (7) we obtain the disounted expeted pro t from selling a stok k as ) and V (k) = (s )(1 k ) : (11) 1 Seond, if there is a high degree of di erentiation (or a high transportation ost), namely, if > e K, then the seller will always post pries suh that there is some positive probability of not selling. In this ase, the prie in eah period is given by (9), the prie sequene is stritly inreasing and the value funtion satis es: V (k) = q (1 ) + s + V (k 1) p (1 ) (1) Finally, the most interesting ase is the one where the market is overed for some levels of the stok and not overed for other. Sine fe k g 1 k=k the stok is large, pries are relatively low (p(k) = s is dereasing, the only possibility is that, when ), so that the market is overed, but they start to inrease (p(k 1) > p(k) > s ) as the stok dereases. In this ase, we an establish a simple threshold value for the ontinuation payo, suh that, when solving bakwards for the optimum pries, we an determine the stok size when the optimal prie falls and stays at (s ) so that the market will be overed for any larger stok size. More spei ally, when the value funtion for some stok level (k 1) beomes larger than b V (k 1) = ( )=(1 ) s=, then we know that for a stok of k units and for any larger stok the prie will equal (s ) and a sale will take plae for sure. Further, if we denote by b k the stok level, whih denotes the period when a swith from a overed to a not-overed market ours, it follows that, for k b k, 3 Beause of disounting, the present value of units sold farther in the future is small and, therefore, it is not pro table to forego a sale today and delay the realization of the stream of future pro ts. 13
Table 1: Optimal prie paths and expeted pro ts for s = 8 and (a) = 6, = 0:9; (b) = 3, = 0:9; () = 6, = 0:8 the value funtion is given by equation (1), while diret omputations show that for K k > b k it satis es: where b k + n = k and n = 1; ; :::; K V ( b k + n) = (s )(1 n ) 1 + n V ( b k); (13) b k denotes the units sold, while the market is overed. We an hek that, if the market is overed for all available stok levels, i.e. if b k = 0, then V ( b k) = 0 and (13) equals (11), i.e. we are bak to the ase where in every period optimal pries ensure that there is a purhase with probability one. We onlude this Setion with some numerial examples that illustrate the properties of the solution. In Table (1) we present the optimal pries and the expeted pro ts for stok levels k = 1; ; :::0: We have set s = 8; = 0:9, while varies aross Tables (1a) and (1b). Spei ally, we have set = 6 for alulating the results presented in Table (1a), whih ensures that p(k) > s ; for all k: Hene, the prie sequene inreases as the available stok dereases (by Proposition 1). The optimal pries and pro ts presented in Table (1b) are alulated for = 3: The latter results show that, when the transportation ost or the degree di erentiation between the two varieties are dereased substantially (here, delines from 6 to 3), for any stok level k 9, p(k) = s = 5 and a unit will be sold for sure. We an see from the last olumn of Table (1b) that this holds as soon as e k beomes higher than (whih happens for k = 9): When there are only 8 units left in stok, the seller starts to inrease the prie, hene, leaving some positive probability that a ustomer with not so strong preferene for the partiular variety does not buy. We present diagrammatially the inreasing pro le of optimal pries as the available stok dereases (see Figure 1). Further, we examine 14
Figure 1: Optimal prie paths in the ase of one variety. how the optimal prie path hanges in response to a derease in the disount fator,. Table (1) shows that (with a lower disount fator) for every stok level the prie dereases. 4 This is very intuitive sine, as the seller beomes more impatient he is expeted to post lower pries in order to realize the stream of pro ts earlier. From the above analysis we onlude that, depending on the parameters values, the seller may initially post pries that ensure that a unit of the variety left unsold is purhased even by a buyer with a strong preferene for the other variety, while after a ertain number of periods the seller starts inreasing steadily the prie after eah period when a sale takes plae. We next ontinue the analysis by examining our main ase of interest, where the seller holds positive stok of both varieties. 4 Positive stok of both varieties In ase there is a positive stok (k 1 ; k ) of both varieties, optimal pries should satisfy the Bellman s equation given by (6). Following some slight modi ations this an be written as 8 ( (s p 1) p 1 + (s p 9 ) p ) >< V (k 1 ; k ) = max +( (s p 1) V (k 1 1; k ) + (s p >= ) V (k 1 ; k 1) p 1 +p s p 1 ;p s >: +(1 s p 1 p )V (k 1 ; k )) >; ; (14) where the terms in the rst braket denote the urrent-period expeted pro t and the terms in the seond braket denote the expeted ontinuation payo. 4 In the ases where the market is overed the optimal prie remains p = s : 15
4.1 Positive probability of no sale: the market is not overed. First, let the optimal pries satisfy p 1 + p > s ; meaning that there is a positive probability of no sale. We obtain the optimal pries, from the relevant rst-order onditions, as follows: p 1 (k 1 ; k ) = s + (V (k 1; k ) V (k 1 1; k )) (15) p (k 1 ; k ) = s + (V (k 1; k ) V (k 1 ; k 1)) ; (16) from whih it follows that p (k 1 ; k ) = p 1 (k 1 ; k )+(V (k 1 1; k ) V (k 1 ; k 1))=: Substituting for V (k 1 ; k ) in (15) and (16) we obtain the following expressions for the optimal pair of pries: p 1 (k 1 ; k ) = p (k 1 ; k ) = (1 ) + s (1 ) + s (V (k 1 1; k ) V (k 1 ; k 1)) 4 + (V (k 1 1; k ) V (k 1 ; k 1)) 4 Z 4 (17) Z 4 ; (18) where v u Z t 4(1 )[(1 ) + s + (V (k 1 1; k ) + V (k 1 ; k 1))] : (19) 4 (V (k 1 1; k ) V (k 1 ; k 1)) Using expressions (17) and (18) we obtain the value funtion in state (k 1 ; k ); V (k 1 ; k ) = (1 ) + s + V (k 1 1; k ) + V (k 1 ; k 1) Z : (0) The optimal prie paths of the two varieties an be omputed reursively using the relations V (k 1 ; 0) = V (k 1 ), V (0; k ) = V (k ) and V (0) = 0: Furthermore, note that when the probability of no sale is positive given the pries that have been optimally set, the two priing optimality onditions are independent from eah other, sine the probability of a sale of eah of the produts is independent of the probability of a sale of the other produt (see expression 14). This means that the seller s priing deision regarding one variety is not onstrained by the existene of the other variety. In this ase, learly the optimal prie of produt i, p i (k 1 ; k ); has to be equal to p i (k i ); the optimal prie de ned above for the ase when only one of the varieties remains to be sold, i = 1;. 5 We, therefore, obtain the following result Proposition When the probability of no sale is positive, the two priing problems are independent and optimal the pries satisfy p 1 (k 1 ; k ) = p(k 1 ) and p (k 1 ; k ) = p(k ): 5 Remember that p (k) is the prie that maximizes the seller s pro t from the sale of the k th unit of one of the produts when the stok of the other produt has been sold out. 16
From the above analysis we obtain that the properties of the prie paths, in the ases where there is positive stok of the two varieties and the market is not overed, follow diretly from the properties of the prie sequene fp k g 1 k=k de ned in Setion 3. In Proposition 1 we have established that the latter is nondereasing. We now show that the sequene of pries p i (k 1 ; k ) stritly inreases as k i dereases, for eah i = 1;. This is beause the optimal pries p(k), when only one variety is available, remain unhanged as apaity dereases in time only if p(k) = s ; that is, only if the market is overed in every period. But this annot hold in the present ase, sine optimal pries are suh that the market is not overed with positive probability. Consequently, p i (k 1 ; k ) 6= s, so p i (k 1 ; k ) inreases after a unit of variety i is sold, while the prie of the other variety remains unhanged. Hene, the following holds. Corollary 1 If the optimal pries are set so that the probability of no sale is positive, apaities are (k 1 ; k ) and one unit of variety 1 is sold, then p 1 (k 1 1; k ) > p 1 (k 1 ; k ) and p (k 1 1; k ) = p (k 1 ; k ). Respetively, if one unit of variety is sold, p 1 (k 1 ; k 1) = p 1 (k 1 ; k ) and p (k 1 ; k 1) > p (k 1 ; k ): A diret onsequene of the above result is that, if it is optimal to have a positive probability of no sale for ertain stok levels, then this probability should inrease as the stok dereases. This is beause the prie of the variety from whih a unit has been sold inreases and onsequently the probability of selling a unit of that variety dereases. At the same time, the prie of the other variety remains unhanged and so does the probability of selling a unit of it. Hene, we obtain that Corollary If optimal pries at some t are suh that the probability of no sale is positive, then this probability will be positive in all subsequent periods and nondereasing. The above result implies that, if the market is not overed in some period, it will not be overed in the subsequent periods. Consequently, it also holds that, if optimal pries with stok (k 1 ; k ) are suh that the market is not overed, then it will not be overed when there is positive stok of only one variety left. Proposition and Corollary also imply that Proposition 3 V (k 1 ; k ) = V (k 1 ) + V (k ) when the probability of no sale is positive: Sketh of the proof. We have established that, if the market is not overed in some period, it will not be overed in any subsequent period and that for any stok level p i (k 1 ; k ) = p i (k i ) : It follows that V (k 1 ; k ) = V (k 1 )+V (k ) (for a detailed proof see the Appendix). Finally, we show that the seller is better o when his remaining apaity is more symmetri. Given a total number of units available, the seller s expeted pro t is inreasing if these units beome more evenly distributed aross the two varieties. Let k 1 > k : Then, sine p 1 (k 1 ) p (k ) 17
and by Proposition, we have that p 1 (k 1 ; k ) = p 1 (k 1 ) < p (k ) = p (k 1 ; k ) (see Corollary 1 for why the inequality is strit). Substituting with the respetive expressions given by (15) and (16), we obtain that Corollary 3 If the probability of no sale is positive and the number of units in the stok of eah variety satisfy k 1 > k ; then V (k 1 1; k ) > V (k 1 ; k 1): So far, we have analyzed the ase where the seller optimally leaves some positive probability that there is no purhase. Then the priing deisions regarding one of the produts are independent of the priing deisions regarding the other produt. In addition, if for some stok levels the market is not overed, it will not be overed for any lower stok levels. However, the market may be optimally overed for higher stok levels. In the next Setion we examine this possibility. 4. Zero probability of no sale: the market is overed. From Lemma 1, we know that when the seller ensures that a buyer purhases one of the varieties with probability one, optimal pries should satisfy p 1 (k 1 ; k )+p (k 1 ; k ) = s : Therefore, the problem is redued to that of determining only one of the pries, while the other follows diretly from the above equation. Given that, when the onstraint p 1 (:) + p (:) s probability of no sale is zero, we an write (6) for this ase as follows V (k 1 ; k ) = s max p 1 s 8 >< >: (s p 1 ) (p 1 + V (k 1 1; k ))+ + 1 (s p 1 ) From the rst order onditions with respet to p 1 we obtain (s p 1 + V (k 1 ; k 1)) is binding, the 9 >= : (1) >; p 1 (k 1 ; k ) = s (V (k 1 1; k ) V (k 1 ; k 1)) : () 4 From p (k 1 ; k ) = s p 1 (k 1 ; k ) it follows that p (k 1 ; k ) = s + (V (k 1 1; k ) V (k 1 ; k 1)) : (3) 4 Substituting the pries bak into (1), we obtain V (k 1 ; k ) = s + (V (k 1 1; k ) + V (k 1 ; k 1)) + (V (k 1 ; k 1) V (k 1 1; k )) : 8 + (4) As before, the optimal prie path for eah variety an be omputed reursively using equations (), (3) and (4). Sine the sum of the two pries within a period (that is, for given levels of inventories) is onstant, it is lear that, when the prie of one variety inreases, the prie 18
of the other variety should derease and vie versa (as long as the market is overed in the next period). We examine the properties of the optimal prie paths for small symmetri initial stok levels and then (due to the ompliation of the implied relations) we present numerial examples for a wider range of stok levels. First, we establish a result, whih shows that having a wider available variety of produts is pro table. This is expeted sine, when two di erentiated produts are o ered for sale instead of one, the transportation ost for some of the buyers is redued, whih allows the monopolist to harge higher pries. Lemma V (k 1 ; k ) V (k 1 + k ): Proof. It annot be that V (k 1 ; k ) < V (k 1 + k ) sine, with k 1 units of produt 1 and k units of produt, the seller ould always mimi the optimal prie path of a single produt the stok of whih is (k 1 + k ): He ould do this by setting p 1 (k 1 ; k ) = p(k 1 + k ) for, say, the rst variety and a prie p (k 1 ; k ) = s; for the seond variety (whih ensures that no buyer would buy the latter), as long as there is positive stok of the rst variety. One this stok has been depleted, the seller sets p (k 1 ; k ) = p(k ). Therefore, the seller annot be worse-o when he has a positive stok of both varieties than when he has the same (total) number of units of one variety only. 6 Now, suppose that the seller has two units of eah variety in stok and parameter values are suh that it is optimal to sell a unit in the rst period with probability one. Hene, the optimal rst-period pries equal p 1 (; ) = p (; ) = (s )= (by equation ()). Sine the market is overed, the rst-period buyer will buy a unit of his preferred variety (also taking pries into aount) for sure. Assume that the buyer s preferenes end up being suh that he purhases one unit of variety 1. Therefore, the seond-period stok is (1; ) and the prie of the rst variety in that period equals p 1 (1; ) = (s )= (V () V (1; 1))=4: By Lemma, V (1; 1) V () and, onsequently, p 1 (1; ) p 1 (; ) and p (1; ) p (; ): We, therefore, obtain that, after a sale of one unit of a ertain variety, the next-period prie of that variety inreases, while the prie of the other variety dereases (as long as the market is overed). We an show by diret alulations that this is also the ase if the rm holds initially 3 units of eah variety. The rst period pries are p 1 (3; 3) = p (3; 3) = (s )=: Assume again that a unit of the rst variety is sold in eah of the following periods. Then the pries of this variety will beome p 1 (; 3) = (s )= (V (1; 3) V (; ))=4 and p 1 (1; 3) = (s )= (V (3) V (1; ))=4; respetively. In the Appendix we show that p 1 (3; 3) p 1 (; 3) p 1 (1; 3) and therefore the prie of variety, the stok of whih remains unhanged, is noninreasing, p (3; 3) p (; 3) p (1; 3): 6 Note that for the ase where the market is not always overed (for the optimal pries) then V (k 1; k ) = V (k 1) + V (k ) and thus this property implies that V (k 1) + V (k ) V (k 1 + k ). 19
Remark 1 When the initial stok is (; ) or (3; 3) ; and the probability of a sale is one, the prie of the variety whih is sold does not derease in the next period, while the prie of the other variety does not inrease. The intuition from the ases onsidered regarding the monotoniity of pries an be extended to ases of higher and arbitrary apaity levels. However, the formal derivations appear overly ompliated (as the number of possible states for eah future period is inreasing very quikly). Thus, in the remainder of this Setion we turn to numerial alulations whih illustrate that the above ndings hold for a wider range of stok levels. 4.3 Numerial Examples First, we ompute the optimal prie paths assuming that = 6; s = 8, = 0:9 and letting k 1 = 1; :::0 and k = 1; :::10. Details about the proedure we follow and the Tables with the respetive results are given in the Appendix. Figure?? summarizes the results in two diagrams showing the evolution of optimal pries as the stok of eah variety hanges. Part (a) depits the optimal prie path of variety 1, while part (b), the optimal path of variety. We observe that, for eah variety, dereasing the stok level of this variety (and keeping onstant the level of the other variety) inreases the optimal prie of this variety. Moreover, holding the stok level of a given variety onstant and dereasing that of the other variety implies a derease in the prie of the rst variety, if the market is overed, while this prie remains onstant when there is a positive probability of no sale.further, we examine the e et of a derease of the disount Figure : Optimal prie paths for = 6; s = 8; = 0:9: (a) p 1 (k 1 ; k ) and (b) p (k 1 ; k ) fator,, on the optimal pries. The results, for = 6; s = 8 and = 0:8, are presented in the Appendix. As expeted, when the rm beomes more impatient ( dereases), it sets pries that ensure a sale with probability one for lower stok levels. In other words, for any stok level of variety i; the market is overed for lower stok levels of variety j (i; j = 1; ): For instane, when 0
= 0:9 and k i = 1; optimal pries are suh that there is a positive probability of no sale for every k j = 1; :::10: When the disount fator dereases to = 0:8; optimal pries are suh that a sale with probability one is ensured for every k j 6: In partiular, we nd that, in the ases where the probability of no sale remains positive after has dropped to 0:8, this probability dereases. It also dereases for these stok levels, for whih the market is not overed when = 0:9; but beomes overed when = 0:8: In all the remaining ases, the market is overed both for = 0:9 and for = 0:8, therefore, the probability of no sale remains equal to zero when the disount fator dereases. It follows that the probability of no sale does not inrease when the disount dereases from 0:9 to 0:8. This is beause this probability depends positively on the sum of the two pries, whih either dereases or remains unhanged as dereases. However, regarding the e et on eah one of the pries, we nd that, in ontrast to the one-variety ase where the optimal pries annot inrease as dereases, there is no suh a monotoni prie response in the two-varieties ase. We have formed the prie di erene between p i (k 1 ; k ), for = 0:9, and p i (k 1 ; k ), for = 0:8, in order to examine the prie hanges, i = 1;. We denote this di erene by 4p i : We present the results for k 1 ; k = 1; :::10 in Table, where a positive entry indiates that a prie dereases when the disount fator dereases from 0:9 to 0:8 and a negative entry indiates that a prie inreases. It is lear from Table that there are three possibilities regarding the response of pries to a derease of the disount fator: i) pries remain unhanged, ii) both pries derease and iii) the prie of the variety with higher stok inreases while the prie of the variety with lower stok dereases. Whih of these will be observed, depends jointly on the absolute stok levels and the asymmetry between them. 7 However, note that, for the stok pairs for whih the market remains not overed after the derease of ; both pries derease. 5 Conlusion In this paper we have examined the dynami priing of a monopolist who sells a nite stok of two distint varieties of some produt over an in nite selling horizon. The seller s optimal priing deisions are derived based on a stohasti dynami programming formulation. In eah period, the seller faes unit demand from a ustomer with random preferene for one variety or the other. The objetive of the rm is to maximize the present value of future expeted pro ts and in doing so it has to take into aount the trade-o between inreasing the probability of a sale and, thus, obtaining the stream of pro ts earlier in time and inreasing the revenue from selling a given unit. The existene of two varieties signi antly enrihes the problem sine in setting the optimal prie for one variety the rm should onsider the available stok of both varieties. Hene, in ontrast to the single-produt setting, the question that the seller faes is 7 Detailed disussion of how pries hange depending on the apaity levels is presented in the Appendix. 1
Table : Prie di erenes 4p i (n; m) = p =0:9 i p =0:8 i, i = 1; ; for s = 8; = 6. not only whether it is more pro table to sell a unit today or not, but in addition whih variety he would rather sell. Consequently, optimal pries determine, in addition to the instantaneous revenue and the overall probability of a sale, the probability of a purhase of a given variety. The evolution of observed pries in the market will depend jointly on the optimal priing rule and also on the (random) sequene of onsumers preferenes. For the ase where there is positive stok of only one variety left, we extend the results of Das Varma and Vettas (001) by establishing that the optimal prie path is nondereasing. Spei ally, for ertain parameter values (e.g. relatively low transportation ost or large stok), the seller may optimally wish to ensure a purhase even from a buyer with a strong preferene for the other variety. Then, the seller harges a prie that leads to a sale with probability one and, as long as it is optimal to over the whole market, the prie path is onstant. This path beomes inreasing in the number of units that have been sold, when the probability of a sale is lower than one. With a positive stok of both varieties, the market may also be either overed or not. We nd that, all else kept equal, the market may be overed (zero probability of no sale) for high stok levels and beome not overed (positive probability of no sale) as the available stok dereases.