Should a monopolist sell before or after buyers know their demands?

Transcription

1 Shold a monopolist sell before or after byers know their demands? Marc Möller Makoto Watanabe Abstract This paper explains why some goods (e.g. airline tickets) are sold cheap to early byers, while others (e.g. seasonal prodcts) offer disconts to those who by late. We derive the profit maximizing selling strategy both for a monopolist who is capacity constrained and for one who can choose his capacity ex ante. We assme that (1) byers face ncertainty abot their individal demand and (2) byers are heterogeneos. So far the literatre on dynamic monopolistic pricing has considered these two aspects in separation reslting in clear ct predictions. We show that none of these predictions srvives in a model that combines both aspects. JEL classification: D42, D82, L12 Keywords: Dynamic monopolistic pricing, individal demand ncertainty, intertemporal price discrimination We thank Jeremy Blow, Harrison Chen, Gido Friebel, Dan Kovenock, Diego Moreno, Volker Nocke, Katharine Rockett, Manel Santos, Alan Sorensen, and participants at varios seminars and conferences for valable discssions and sggestions. The second athor acknowledges spport from the Spanish government research grant SEJ C02. Department of Economics, University Carlos III Madrid, Calle Madrid 126, Getafe Madrid, Spain. mmoller@eco.c3m.es, Tel.: , Fax: Department of Economics, University Carlos III Madrid, Calle Madrid 126, Getafe Madrid, Spain. mwatanab@eco.c3m.es, Tel.: , Fax:

2 1 Introdction Many goods and services can be prchased in advance, i.e. long before their actal date of consmption. Common examples are airline travel, theater performances, the right to participate in sports events or trade fairs, and seasonal prodts like the newest skiing eqipment. In these examples consmers often face a trade off between bying early and bying late. When capacity is restricted, bying early minimizes the consmers risk to become rationed. On the other hand, when individal demands are ncertain, bying late maximizes the consmers information. In this paper we stdy the implications of this trade off for a monopolist s optimal selling strategy. We are motivated by the fact that examples which share the above characteristics often show systematic differences in pricing patterns. For instance, while airline ticket prices increase steadily ntil departre, theater tickets can often be boght at half price on the day of the performance. For some prodcts introdctory offers are employed whereas others are sold by se of a clearance sale. In this paper we explain these differences in pricing patterns sing a model with two main ingredients; (1) individal demand ncertainty and (2) byers heterogeneity. So far the literatre on dynamic monopolistic pricing has considered these two aspects in separation. One branch of the literatre assmes that byers are heterogeneos bt know their individal demands, see for example Conlisk, Gerstner and Sobel (1984), Harris and Raviv (1981), Lazear (1986), Nocke and Peitz (2007), and Van Cayseele (1991). In all of these papers the monopolist s profit maximizing price schedle is non increasing, i.e. introdctory offers are never optimal. When consmers know their demands in advance the above trade off is absent. Consmers have a clear preference for bying early as it lowers their risk to become rationed. As a conseqence the above papers are nable to explain the existence of increasing price paths. A second branch of the literatre allows for individal demand ncertainty bt assmes that byers are homogeneos ex ante, see for example Corty (2003b), and DeGraba (1995). These papers find that the monopolist maximizes his profits by selling exclsively either before or after individal demand ncertainty has been resolved. As cstomers are homogeneos ex ante, intertemporal price discrimination has no bite. 2

3 In this paper we arge that none of these predictions srvives in the presence of both, individal demand ncertainty and byers heterogeneity. For this prpose we consider a simple two period model. In the first period byers face individal demand ncertainty whereas in the second period all demand ncertainty has been resolved. Byers differ in their expected valations and valations might trn ot to be high or low. A monopolist faces ex ante ncertainty abot the byers types and ex post ncertainty abot their realised valations. As the byers trade-off between bying early and bying late hinges on their risk to become rationed, the monopolist s capacity is a crcial variable in or model. In some of the above examples exogeneos restrictions prevent the seller from adjsting his capacity. For instance, de to logistic restrictions, organizers of the London Marathon have been nable to increase their capacity for several years even thogh demand has soared. Similarly, the capacity of airline travel between two specific destinations is often restricted by the nmber of landing slots available. In Section 3 we therefore start or analysis by considering the case of a capacity constrained monopolist. In Section 4 we then extend or analysis to the case of a monopolist who can choose his capacity ex ante. Or reslts are as follows. When the monopolist is capacity constrained his profit maximizing selling strategy depends crcially on his capacity. When capacity is small the monopolist s profits are maximized by selling exclsively after byers have learned their demands. When capacity is large and the fraction of good types is small, it is optimal for the monopolist to sell exclsively before byers have learned their demands. In all remaining cases selling in both periods at either increasing or decreasing prices maximizes the monopolist s profits. However, a necessary condition for decreasing prices to be optimal is that capacity is smaller than potential demand so that consmers face a positive risk to become rationed. With respect to the monopolist s optimal choice of capacity we find that the monopolist never has an incentive to restrict his capacity in order to be able to implement a clearance sale. If the monopolist wishes to screen byers, he does so by se of an introdctory offer as it comes withot the implied cost of a capacity restriction. The main insight of this paper is that in the presence of individal demand ncer- 3

4 tainty and byers heterogeneity a monopolistic seller faces a trade off. On the one hand the existence of individal demand ncertainty gives the monopolist an incentive to sell exclsively either before or after demand ncertainty has been resolved. On the other hand, acconting for byers heterogeneity reqires some form of intertemporal price discrimination and hence the sale in both periods. We are the first to stdy this trade off and its implications for the monopolist s optimal selling strategy. The only papers that combine individal demand ncertainty with byers heterogeneity are Corty and Li (2000) and Gale and Holmes (1993). Corty and Li (2000) allow the monopolist to screen types by offering a men of refnd contracts before demand ncertainty has been resolved. Offering one contract with zero refnd and price P 1 and another contract with fll refnd and price P 2 is eqivalent to selling the prodct at price P 1 before demand ncertainty has been resolved and at price P 2 after. Note that in order to screen types it has to hold that P 1 < P 2 as otherwise all types wold prefer the contract with fll refnd. Hence sing refnd contracts the monopolist is nable to implement decreasing price paths. However, as in Corty and Li (2000) the monopolist s capacity is nconstrained, this inability has no conseqence. In the absence of rationing risk, byers have a clear preference for bying late which implies that decreasing price paths can never be optimal. In this paper we abstract from the possibility of refnd contracts. Moreover, in order to cover the above examples we allow the monopolist to be capacity constrained and show that in this case decreasing price paths may become optimal. Gale and Holmes (1993) consider a monopolistic airline which aims to divert demand from a peak time flight to an off peak time flight. Cstomers learn over time which flight they prefer and differ in their distility from taking the wrong flight. Gale and Holmes offer the insight that the airline shold offer a discont for the off peak flight together with an advance prchase reqirement in order to benefit from cstomers demand ncertainty. As Gale and Holmes restrict the degree of individal demand ncertainty and byers heterogeneity, intertemporal price discrimination is always optimal. In or model we allow for an arbitrary level of demand ncertainty and heterogeneity and derive the conditions nder which intertemporal price discrimination is dominated by selling exclsively before or after byers know their demands. 4

5 Finally or work is also related to Lewis and Sappington (1994) who stdy a monopolist s incentive to spply potential cstomers with information (e.g. a test drive). They identify general settings in which it is optimal for the monopolist to spply either perfect information or no information at all. Translated into or setp this amonts to selling exclsively either before or after byers have learned their demands. As we allow byers to be heterogeneos ex ante the monopolist may find it optimal to spply intermediate levels of (aggregate) information by selling to some byers before they know their demands and to others after. The plan of the paper is as follows. Section 2 describes or setp. In Section 3 we take capacity as exogeneosly given and characterize the monopolist s profit maximizing selling strategy in dependence of the size of individal demand ncertainty and the monopolist s capacity. We distingish between the case where the monopolist s capacity constraint is binding (i.e. capacity is smaller than potential demand) and the case where it is slack. Section 4 considers the monopolist s optimal choice of capacity. In Section 5 we show that or reslts remain valid when the monopolist is nable to commit to price schedles ex ante. Section 6 concldes. All proofs are contained in the Appendix. 2 The setp We consider a monopolistic seller who faces a continm of byers with mass normalized to one. 1 Byers have nit demands and trade might take place in two periods; in an advance prchase period 1 and in the consmption period 2. We assme that demand exhibits the following two characteristics. Individal demand ncertainty: Ex ante, byers are ncertain abot the consmption vale they will derive from the good. A byer s valation might take two vales; a high vale normalized to 1 and a low vale (0, 1). 2 Valations are distribted independently and byers privately learn their own valation between period 1 and period 2. 1 A model with N byers leads to similar reslts bt is less tractable. 2 An alternative interpretation is that ex ante byers vale the good at 1 bt face a risk that their consmption vale is redced to. 5

6 Heterogeneity: There are two types of byers. A fraction g (0, 1) has good type and a fraction 1 g has bad type. Bad types face a greater risk to have a low consmption vale than good types. In particlar, we assme that a bad type s likelihood of having a low valation is r (0, 1) while for good types it is ar with a (0, 1). We let U G 1 ar + ar and U B 1 r + r denote the expected valations of good and bad types respectively. Byers types are private knowledge. Note that or assmptions on demand imply that a bad type s consmption vale might trn ot to be higher than a good type s expected valation. We will see that this assmption is crcial for the optimality of increasing price schedles. Selling strategies We assme that the monopolist is able to commit to a selling strategy ex ante. We consider two possible cases. In Section 3 we focs on the case of a capacity constrained monopolist by assming that capacity k > 0 is fixed exogeneosly. We assme that the monopolist optimizes over selling strategies of the form (P 1, P 2, k 1 ). A selling strategy specifies the prices P 1 and P 2 at which the prodct can be prchased in period 1 and 2 respectively, as well as a first period capacity limit k 1 [0, k]. 3 In Section 4 we remove the monopolist s capacity constraint and assme instead that capacity can be chosen endogeneosly by allowing the monopolist to commit to selling strategies of the form (P 1, P 2, k 1, k). Every selling strategy together with the implied byers behavior belongs to one of five categories which we define as follows: Definition 1 We say that a selling strategy implements Clearance Sales if P 1 > P 2 and sales are positive in both periods. Introdctory Offers if P 1 < P 2 and sales are positive in both periods. Constant Prices if P 1 = P 2 and sales are positive in both periods. Advance Selling if sales are positive in period 1 and zero in period 2. Spot Selling if sales are positive in period 2 and zero in period 1. 3 Capacity limits are freqently employed in the sale of airline tickets. Airlines partition their available capacity into fare classes and different fares are released into the electronic booking systems at different pre determined points in time. 6

7 Timing. The timing of events is as follows. First natre determines the byers types and the monopolist commits to a selling strategy. In period 1, all byers decide simltaneosly whether to by the good at price P 1. If demand exceeds the capacity limit k 1 byers are rationed randomly. After period 1 and before period 2 byers learn their valations. Finally, all capacity that has remained nsold is offered in period 2. Byers who have not boght in period 1 decide simltaneosly whether to by the good at price P 2. Again, if demand exceeds the remaining capacity byers are rationed randomly. Note that we do not allow for the possibility of resale. Payoffs. We assme that prodction and capacity costs are zero so that the monopolist s payoff, Π, is identical to his revene. Byers are risk netral and have qasi linear preferences. In particlar, when a byer prchases the good his payoff is eqal to the difference between his (expected) consmption vale and the price. Otherwise his payoff is zero. 4 3 Capacity constraints We begin or analysis by considering the case of a capacity constrained monopolist. In particlar, we take capacity k > 0 as given exogenosly and derive the monopolist s profit maximizing selling strategy (P1, P 2, k 1 ). The monopolist s optimal capacity choice will be the topic of the next section. As capacity is sally harder to adjst than prices, or reslts in this section might be interpreted as a description of the monopolist s behavior in the short rn. However, in the introdction we gave some examples in which exogeneos restrictions prevent a monopolist from adjsting his capacity even in the long rn. In the presence of sch exogenos capacity restrictions or reslts in this section are relevant also in the long rn. It is immediate that for k > 1 the monopolist s profit maximizing selling strategy is identical to the one for the case k = 1. We can therefore restrict or attention to the case where k (0, 1) and attain the monopolist s optimal selling strategy for the 4 We assme throghot that a byer demands the good when indifferent between bying and not bying. 7

8 remaining cases by considering the limit as k 1. We start by discssing optimality for the five categories of selling strategies defined above. Spot Selling Consider the possibility that the monopolist sells exclsively after demand ncertainty has been resolved by choosing k 1 = 0. Note first that Spot Selling with P 2 = cannot maximize the monopolist s profits as it is dominated by (P 1, P 2, k 1 ) = (U B, 1, k). For Spot Selling to be optimal it therefore has to hold that P 2 = 1. If the monopolist s capacity k does not exceed the expected second period demand, g(1 ar)+(1 g)(1 r), then Spot Selling with P 2 = 1 clearly maximizes the monopolist s profits. Otherwise, the monopolist can increase his profits by setting (P 1, P 2, k 1 ) = (U B, 1, k g(1 ar) (1 g)(1 r)). Defining r S 1 k 1 g(1 a) (1) we therefore have the following reslt: Lemma 1 Sppose that k (0, 1). Spot Selling maximizes the monopolist s profits if and only if k g(1 ar)+(1 g)(1 r) r r S. When Spot Selling is optimal, the monopolist sells his entire capacity k at price P 2 = 1 and his profits are Π S k. Clearance Sales Clearance Sales are employed for a broad variety of prodcts ranging from bargain holidays to theater tickets. By definition a Clearance Sale reqires that P 2 < P 1 and that sales are positive in both periods. A Clearance Sale therefore reqires that P 2 < 1. Also note that a Clearance Sale (P 1, P 2, k 1 ) with P 2 (, 1) is dominated by the selling strategy (P 1, 1, k 1 ) which increases P 2 withot lowering neither second nor first period demand. Hence for a Clearance Sale to be optimal it has to hold that P 2 =. Now consider the first period price. Given that P 2 = implies a positive option vale to byers who postpone their prchase ntil period 2, a P 1 < U B is necessary to indce first period demand from both types of byers. Hence a Clearance Sale 8

9 that indces first period demand from both types is dominated by the selling strategy (P 1, P 2, k 1 ) = (U B, 1, k) which sells the same qantity bt at strictly higher prices. This implies that for Clearance Sales to be optimal, the monopolist has to se the two selling periods to screen his cstomers. In particlar, P 1 has to be chosen sch that good types prchase before and bad types prchase after demand ncertainty has been resolved. 5 In order to indce first period demand from only good types given P 2 = and k 1 min(g, k) the monopolist optimally sets P 1 = P C 1 (k 1) U G k k 1 1 k 1 (1 ar)(1 ). The term k k 1 1 k 1 (1 ar)(1 ) represents the good types option vale of delaying the prchase ntil period 2. Note that P C 1 = (1 ar)(1 ) 1 k 1 k 1 which implies that in order to implement a Clearance Sale it has to hold that k < 1. When deciding whether to by in advance or not, byers trade off a lower risk to be rationed when bying early, with a price discont and better information when bying late. For k 1 the risk to be rationed is absent so that first period demand is zero for any decreasing price schedle. This implies that the monopolist might have an incentive to restrict his overall capacity in order to be able to screen byers throgh a Clearance Sale. We will consider this incentive in Section 4 where we allow the monopolist to choose his capacity, k. Finally, consider the monopolist s optimal choice of k 1. For any k 1 min(g, k) the monopolist s profits nder the selling strategy (P C 1 (k 1 ),, k 1 ) are given by Π = k 1 P1 C(k 1)+(k k 1 ). As P1 C > and P 1 C k 1 = 1 k (1 k 1 (1 ar)(1 ) > 0 the monopolist ) 2 optimally chooses k 1 = min(g, k). Note that this implies that for k g Clearance Sales are dominated by the Advance Selling strategy (P C 1 (k),, k) which leads to the profits Π = ku G. The next lemma smmarizes these findings. Lemma 2 Sppose that k (0, 1). For Clearance Sales to maximize the monopolist s profits it has to hold that k (g, 1). When Clearance Sales are optimal, the monopolist sets k 1 = g and P 1 = U G k g 1 g (1 ar)(1 ) > = P 2 and his profits are Π C gp 1 + (k g)p 2. Using a Clearance Sale, the monopolist sells his entire capacity bt pays information 5 Note that good types have a stronger propensity to by in advance than bad types. In particlar there is no selling strategy which indces bad types to by in advance and good types to wait. 9

10 rents. He leaves the option vale k g (1 ar)(1 ) to good types and a rent of 1 1 g to bad types who trn ot to have a high consmption vale. Introdctory offers Empirical evidence for the se of introdctory offers stems from ticket pricing of low cost airlines and early registration disconts for academic conferences and sports events. By definition an Introdctory Offer reqires that P 1 < P 2 and that sales are positive in both periods. It is straight forward to see that for an Introdctory Offer to be optimal it has to hold that P 2 = 1. However, for the optimal first period price there are two possible candidates, P 1 = U B and P 1 = U G. Given Lemma 1 we can restrict attention to the case where r > r S. In this case the monopolist cannot expect to sell his entire capacity at a price of P 2 = 1. However, by implementing an Introdctory Offer with P 1 = U B and k 1 k i k r(1 g(1 a)) (0, k) the monopolist can ensre to sell all remaining capacity k k 1 at P 2 = 1 in period 2. Under the selling strategy (U B, 1, k 1 ) profits are given by Π = { k1 U B + (1 k 1 )(g(1 ar) + (1 g)(1 r)) for k 1 < k i 1 k 1 U B + k k 1 for k 1 k i 1. (2) Note that Π k 1 = { r( g(1 a)) for k1 < k i 1 r(1 ) for k 1 k i 1. (3) When the fraction of good types is sfficiently large, i.e. when g >, the price discont P 1 = U B < 1 = P 2 that is necessary to ensre sell ot is too large in 1 a comparison to the gain in sales and profits are maximized by setting k 1 = 0. Hence for g > 1 a an Introdctory Offer with P 1 = U B cannot maximize the monopolist s profits. For g the optimal Introdctory Offer with P 1 a 1 = U B sells as mch in period 1 as is necessary to ensre sell ot, by setting k 1 = k1 i and the reslting profits are Π i k (1 )(k 1 + r(1 g(1 a))). (4) 1 g(1 a) An Introdctory Offer with P 1 = U G offers a smaller price discont than an Introdctoty Offer with P 1 = U B, bt as it only appeals to good types, expected demand in 10

11 the second period is smaller. As a conseqence, in order to ensre sell ot at P 2 = 1, the monopolist has to sell a larger qantity k1 I k 1+r(1 g(1 a)) (k i ra 1, k) in period 1. Moreover, Introdctory Offers with P 1 = U G offer less flexibility than Introdctory Offers with P 1 = U B as only a maximm of k 1 g can be sold in the first period. This threshold becomes binding when k > g and r > 1 k in which case 1 g ki 1 k 1 [0, min(g, k)], profits nder the selling strategy (U G, 1, k 1 ) are given by > g. For Π = { k1 U G + (g k 1 )(1 ar) + (1 g)(1 r) if k 1 < k I 1 k 1 U G + k k 1 if k 1 k I 1. (5) As profits are increasing in k 1 for all k 1 < k I 1 and decreasing in k 1 for all k 1 > k I 1 the monopolist optimally ensres sell ot whenever he can by setting k 1 = { g if k g and r 1 k 1 g k I 1 (0, min(g, k)) otherwise. (6) The reslting profits are Π I { gu G + (1 g)(1 r) if k g and r 1 k 1 g k (1 )(k 1 + r(1 g(1 a))) otherwise. (7) Comparing the profits Π I and Π i and defining r ii (1 k)( g(1 a)) (1 g(1 a))( g(1 a(1 ))) (8) we get the following reslt: Lemma 3 Sppose that k (0, 1). When Introdctory Offers are optimal the monopolist sets P 1 = U B, P 2 = 1, and k 1 = k i (0, k) if g < and r > r ii. 1 a(1 ) Otherwise he sets P 1 = U G, P 2 = 1, and k 1 = min(g, k1 I ). The reslting profits are Π = max(π i, Π I ). Note that when feasible, the monopolist adjsts first period capacity to ensre sell ot. By implementing an Introdctory Offer the monopolist avoids to pay information rents to those byers whose valation trns ot to be high. Moreover, byers option vale of postponing their prchase is zero. However, the elimination of information rents 11

12 and option vales comes at a cost when the monopolist s capacity is sfficiently large and byers are sfficiently likely to have low valations. In this case the monopolist faces a high risk to become rationed and in order to ensre sell ot he has to offer a large capacity with a large introdctory discont. Advance Selling Consider the possibility that the monopolist sells exclsively before demand ncertainty has been resolved. 6 He can achieve this by setting P 2 > 1 and k 1 = k. For the optimal first period price there are two candidates, P 1 = U B and P 1 = U G. Choosing P 1 = U B the monopolist receives the profits Π = ku B. Note however, that this Advance Selling strategy is dominated by an Introdctory Offer with P 1 = U B which sells the same qantity bt at higher prices. Similarly, Advance Selling with P 1 = U G leads to the profits Π = gu G bt is dominated by an Introdctory Offer with P 1 = U G. We therefore have the following reslt: Lemma 4 Sppose that k (0, 1). Advance Selling cannot maximize the monopolist s profits. Constant Prices Consider a selling strategy that implements positive sales in both periods at constant prices P 1 = P 2 = P. Since in period 1 no byer is willing to pay a price P 1 > U G sch a strategy reqires that P U G. While Constant Prices with P = are dominated by (P 1, P 2, k 1 ) = (U B, 1, k), for P (, U G ] Constant Prices are dominated by the selling strategy (P 1, P 2, k 1 ) = (P, 1, k) which increases prices withot decreasing sales. We therefore have the following reslt: Lemma 5 Sppose that k (0, 1). Constant Prices cannot maximize the monopolist s profits. 6 A recent example of Advance Selling is the 2006 FIFA Football Worldcp. Ticket sale started on Febrary 1st 2005 bt the composition of the 32 finalists was not known ntil November 30th. Many tickets were sold before byers knew whether they wold be able to watch their team. 12

13 3.1 Characterization As the monopolist s optimal selling strategy has to fall in one of the five categories above, Lemmas 1 to 5 together with a direct comparison of the profits nder Clearance Sales, Π C, with the profits nder Introdctory Offers, max(π i, Π I ) lead to a characterization of the monopolist s profit maximizing strategy. Defining the thresholds r Ci (1 g) 2 + (k g)ga (1 g) 2 + ga(k(1 g) ga(1 k)) r CI (1 g)2 + (k g)(g ) (1 g) 2 + (k g)ga(1 ) (9) (10) we have: Proposition 1 Sppose the monopolist s capacity constraint is binding, i.e. k < 1. Spot Selling maximizes the monopolist s profits for all (k, r) S {(k, r) (0, 1) 2 r r S }. Clearance Sales are optimal for all (k, r) C {(k, r) (0, 1) 2 r max(r Ci, r CI )}. For all remaining (k, r) I (0, 1) 2 C S, Introdctory Offers constitte the monopolist s optimal selling strategy. Note that S and I while C if and only if g <. 1 a(1 ) Figre 1 provides a graphical representation of the monopolist s optimal selling policy. The intition for this reslt has two parts. First, the monopolist faces the sal negative relation between price and demand. Spot Selling implements the highest price P 2 = 1 bt reslts in the lowest expected demand g(1 ar) + (1 g)(1 r). Introdctory Offers with P 1 = U G decrease the price for good types thereby increasing expected demand to g +(1 g)(1 r). Introdctory Offers with P 1 = U B decrease the price even frther in order to match expected demand with the monopolist s capacity. Finally, Clearance Sales implement the lowest prices reslting in demand from the entire nit mass of byers. It is intitive that when byers expectations are high already, boosting demand throgh price disconts cannot be profitable. Only if the fraction of good byers is sfficiently small, i.e. g <, the monopolist s capacity is sfficiently large, 1 a(1 ) i.e. k > g, and byers face a relatively high risk r of having a low consmption 13

14 1 g(1-a) g r S r CI C Ci r r S I r ii 0 1 k Figre 1: Characterization of the monopolist s profit maximizing selling strategy (P1, P2, k1) in dependence of the monopolist s capacity k (0, 1) and the byers likelihood r (0, 1) of having a low consmption vale. The bonds r S, r ii, r Ci, and r CI are defined by (1), (8), (9), and (10) respectively. The areas for which Spot selling, Introdctory Offers, or Clearance Sales are optimal are separated by bold lines. The figre shows the case where g <. For g 1 a(1 ) 1 a(1 ) the set C is empty. vale, Clearance Sales or Introdctory Offers with P 1 = U B maximize the monopolist s profits. In contrast, when capacity is sfficiently small, i.e. k g(1 a), or r is relatively low, Spot Selling is optimal. In an intermediate range, Introdctory Offers with P 1 = U G maximize the monopolist s profits. Second, the monopolist s choice between Clearance Sales and Introdctory Offers with P 1 = U B hinges on the comparison of ex ante verss ex post information rents. Under Introdctory Offers with P 1 = U B, information rents stem from the monopolist s inability to distingish between byers of distinct types ex ante. The good types of those k i 1 byers who by in advance receive an information rent of U G U B. In contrast, the information rents nder Clearance Sales originate from the nobservability of byers consmption vales ex post. Bad types have an expected information rent of (1 1 k 1 k )(1 r)(1 ) while good types receive (1 )(1 ar)(1 ). 1 g 1 g 14

15 Under both selling strategies, information rents increase with k. Note however, that ex post information rents depend crcially on the byers risk to become rationed in the second period, 1 k. For k g byers expect to be rationed with certainty. 1 g Hence ex post information rents become neglegible and Clearance Sales are the most profitable way to implement a sell ot. On the other hand, for k 1 the byers risk to become rationed in a Clearance Sale converges to zero. Hence byers prefer to postpone their prchase ntil period 2 and the monopolist has to let P1 C in order to implement a Clearance Sale. When capacity is sfficiently high, Clearance Sales are therefore dominated by Introdctory Offers with P 1 = U B. By considering the limit as k 1 we can derive the monopolist s optimal selling strategy for the remaining case k 1. For k 1 Clearance Sales are no longer feasible. Moreover, the optimal Introdctory Offers with P 1 = U B converges to Advance Selling since the capacity k i 1 that is sold in advance strictly increases in k with lim k 1 k i 1 = 1. We therefore get the following reslt: Proposition 2 Sppose that the monopolist s capacity constraint is slack, i.e. k 1. If g < then Advance Selling with P 1 a(1 ) 1 = U B, P 2 > 1, and k 1 = 1 maximizes the monopolist s profits. Otherwise an Introdctory Offer with P 1 = U G, P 2 = 1, and k 1 = g is optimal. One can smmarize or reslts so far by saying that whether a monopolist will sell before or after byers have learned their demands, depends crcially on his capacity. When capacity is low, the monopolist will sell exclsively after byers have learned their demands. When capacity is high, the monopolist will sell exclsively before byers have learned their demands if the fraction of good byers is sfficiently low. For intermediate vales of capacity the monopolist s will employ some form of intertemporal screening by selling in both periods at either increasing or decreasing prices depending on the byers distribtion of types and valations. 15

16 4 Capacity choice In the last section we have seen that for Clearance Sales to be feasible, byers mst face a positive probability to become rationed. For k 1 the threat to be rationed is absent and a Clearance Sale can no longer be implemented. It is therefore important to nderstand whether the monopolist has an incentive to restrict his overall capacity in order to be able to screen byers by se of a Clearance Sale. In order to answer this qestion sing or framework we now follow Nocke and Peitz (2007) by assming that the monopolist can choose his capacity k ex ante at zero cost. We find the following reslt: Proposition 3 If the monopolist can choose his capacity ex ante at zero cost then his profits are maximized by setting k = 1. His optimal selling strategy (P1, P 2, k 1 ) is as stated in Proposition 2. It is immediate that except for Clearance Sales the monopolist has no incentive to restrict his capacity to some k < 1. Moreover, restricting capacity in order to implement a Clearance Sale can only be optimal when the increase in the price paid by good types, otweighs the loss in sales to bad types. This is the case when the fraction of good types is sfficiently high, i.e. when g. It trns ot however, that in this 1 ar(1 ) case the optimal capacity choice nder Clearance Sales is smaller than the expected sales nder an Introdctory Offer with P 1 = U G and k = 1. As prices are strictly higher nder the Introdctory Offer, Clearance Sales are never optimal. Proposition 3 differs from the main reslt of Nocke and Peitz (2007). In their setp Introdctory Offers can never be optimal and the monopolist s profits are maximized by selling at a niform price (as nder Advance Selling) or by implementing a Clearance Sale. This difference stems from the fact that in Nocke and Peitz (2007) byers know their individal demands ex ante bt aggregate demand is ncertain. When byers know their demands ex ante, they prefer to by in period 1 in order to minimize their risk to become rationed. As a conseqence it cannot be optimal to offer a price discont to those who by early. In or setp, byers do not know their demands ex ante and, given the risk to be rationed is absent, i.e. for k = 1, prefer to by in period 2. Hence in or model it cannot be optimal to give a price discont to those who by 16

17 late. The comparison of or reslts with those of Nocke and Peitz (2007) sggests that Introdctory Offers are likely to be sed for prodcts whose vale is ncertain to byers at the time of prchase (e.g. airline tickets) while Clearance Sales can be expected to be employed when byers know their valations bt aggregate demand is ncertain ex ante (e.g. for seasonal prodcts). In order to relate or reslts to the literatre which assmes that byers are homogeneos ex ante (Corty (2003b) and DeGraba (1995)) we now consider the limiting case where a 1. We find the following: Corollary 1 Sppose that byers are homogeneos. If the monopolist is capacity constrained and k < 1 then profits are maximized by Spot Selling if r 1 k and by Introdctory Offers with k 1 = 1 1 k (0, k) otherwise. If the monopolist can choose r his capacity he sets k = 1 and implements Advance Selling. The second part of Corrolary 1 is similar to the bying frenzies of DeGraba (1995) and Corty s (2003b) reslt that selling both before and after byers have learned their demands can never be optimal. The intition for this reslt is that in the absence of a capacity constraint the monopolist prefers to sell ex ante in order to avoid the information rents he wold have to pay ex post. Corollary 1 however shows that the monopolist s profit maximizing selling strategy differs from the one obtained by DeGraba (1995) and Corty (2003b) when the monopolist is capacity constrained. In this case the monopolist avoids information rents by setting P 2 = 1 and optimally sells either in both periods or exclsively after byers have learned their valations. Selling in both periods does not serve the prpose of sreening byers (as byers are homogeneos) bt enables the monopolist to avoid rationing risk in period 2. He ses his first period capacity restriction to make second period capacity match second period demand. 5 Price commitment In this section we discss the robstness of or reslts with respect to an important assmption. As in Corty (2003b), Harris and Raviv (1981), and Van Cayseele (1991) 17

18 we have assmed that the monopolist can commit to a price schedle in advance. 7 In the absence of sch commitment the monopolist faces a time consistency problem similar to the one stdied in the drable goods literatre (see Blow (1982), Coase (1972), Gl, Sonnenschein and Wilson (1986), and Stokey (1981)). After selling to good types in the first period, the monopolist might have an incentive to adjst his second period price to second period demand. Under any screeing policy, the monopolist s lack of commitment therefore reqires second period prices to maximize second period revene. In Section 3 we have shown that for an Introdctory Offer with P 1 = U B to be optimal the monopolist chooses the first period capacity k i 1 (0, k) so as to ensre the sale of his remaining capacity k k i 1 at price P 2 = 1. Hence in the parameter range for which an Introdctory Offer with P 1 = U B is optimal, it does not reqire any commitment power. However, Introdctory Offers with P 1 = U G fail to sell ot when k g and r 1 k. In this case 1 g an Introdctory Offer with P 1 = U G is feasible in the absence of commitment if and only if P 2 = 1 maximizes second period revene, i.e. when (1 g)(1 r) (k g) r r 1 k g. In trn, a Clearance Sale does not reqire commitment if and only 1 g if r r. Note that 1 k < 1 g r < r CI for all k (g, 1). This implies that in the parameter range where Clearance Sales are optimal they do not reqire commitment power. The monopolist s optimal pricing strategy only changes for parameter vales in a strict sbset of I which is depicted as the shaded area in Figre 2. For all remaining parameter vales the monopolist s optimal pricing strategy withot commitment is identical to the optimal strategy with commitment as stated in Proposition 1. As for k = 1 commitment power is not reqired, Propositions 2 and 3 hold even if the monopolist cannot commit to prices in advance. 7 This practise is common in the organisation of events sch as sports or msic contests and trade exhibitions. For example, the Madrid Marathon states on its website that participation in the race taking place in April will cost 55 Eros if prchased ntil the end of December, 60 Eros if prchased from Janary till March, and 85 Eros if prchased thereafter. Althogh sch explicit commitment is absent in the sale of airline or theater tickets it is implicitely achieved throgh the repeated natre of the transactions. 18

19 1 g(1-a) g r S r CI C r Ci r I S r r ii 0 1 k Figre 2: The monopolist s profit maximizing selling strategy withot price commitment. The shaded area depicts the parameter set for which the optimal selling policy withot commitment differs from the one with commitment. For parameter vales in this area, Introdctory Offers with P 1 = U G are not feasible in the absence of commitment and will be sbstitted by a selling policy that implements sell ot. 6 Conclsion In this paper we have considered a monopolist s profit maximizing selling strategy when byers are heterogeneos and ncertain abot their individal demands. Or analysis fills an important gap in the literatre on dynamic monopolistic pricing. So far this literatre has concentrated exclsively on the effects of either individal demand ncertainty or byers heterogeneity. As a conseqence athors have either fond that it is optimal to sell exclsively before or after demand ncertainty has been resolved or that prices shold be decreasing. We have shown that in a model which combines the two effects, none of these predictions srvives. From an applied viewpoint or model is particlarly sitable to stdy the trade off between increasing and decreasing price schedles. The model s simplicity sggests the introdction of frther ingredients that may help to explain why for instance airline 19

20 tickets are sold cheap to early byers, while theater tickets offer disconts to those who by late. For example, an important isse concerning advance sales problems is the presence of third parties, so called brokers or middlemen. Brokers stock an inventory in advance and stand ready for byers to sell close to the consmption date. The existence of brokers will therefore inflence the consmers trade off between bying early verss bying late and hence the monopolist s optimal selling strategy. A closer look at this isse is left for ftre research. Appendix Proof of Lemma 1 In text. Note that r S decreases linearly in k with lim k 1 r S = 0 and r S = 1 for k = g(1 a) (0,g). Proof of Lemma 2 In text. Proof of Lemma 3 Compare the profits in (4) with those in (7). As 1 g(1 a) (0,1) it is immediate that for Π i > Π I to hold it is necessary that k g and r 1 k 1 g so that ΠI = gu G + (1 g)(1 r). Π i > Π I is therefore eqivalent to r > r ii. Moreover r ii < 1 for some k (0,1) if and only if g < 1 a(1 ). Note that rii decreases linearly in k with lim k 1 r ii = 0 and r ii = 1 for k = g g(1 a)(1 a(1 )) g(1 a) (g,1). Proof of Lemma 4 and 5 In text. Proof of Proposition 1 Follows directly from Lemmas 1 5 and a direct comparison of Π C with max(π i,π I ). We start by comparing Π C with Π I. As P1 I = 1 ar + ar 1 ar + ar V = P 1 C and P2 I = 1 > = P 2 C a necessary condition for Π C > Π I is that the monopolist is rationed sing an Introdctory Offer which happens only if k g and r 1 k 1 g. In this case ΠI = 20

21 g(1 ar + ar) + (1 g)(1 r) and Π C = g(1 ar + ar k g 1 g that Π C > Π I r > r CI. Note that r CI < 1 for some k (0,1) if and only if g < In this case (1 ar)(1 )) + (k g) so 1 a(1 ). r CI k = (1 g)2 [g(1 a(1 )) ] [(1 g) 2 + (k g)ga(1 )] 2 < 0 (11) and lim k 1 r CI 1 = 1 g(1 a(1 )) (0,1) and rci = 1 for k = g. Also note that 2 r CI > 0 2 k implies that there exists a niqe k (g g(1 a)(1 a(1 )) g(1 a),1) sch that r ii (k ) = r CI (k ) and r ii (k) > r CI (k) if and only if k < k. Now compare Π C with Π i. It holds that Π i > Π C if and only if r < r Ci. Note that r Ci k = (1 a)ag 2 (1 g)(1 g(1 a)) [(1 g) 2 + ga(k(1 g) ga(1 k))] 2 > 0 (12) and 2 r Ci 2 k < 0 with lim k 1 r Ci = 1. Also note that r Ci (k ) = r ii (k ) as for (k,r) = (k,r ii (k )) it holds that Π i = Π I and Π C = Π I which implies that Π C = Π i. We have therefore shown that the monopolist s profit maximizing selling strategy looks as depicted in Figre 1. Proof of Proposition 2 Note first that the monopolist s profit maximizing selling strategy for k > 1 is identical to the one for the case k = 1 as total demand is normalized and never larger than 1. De to continity we can attain the optimal strategy for k = 1 from Proposition 1 by letting k 1. lim k 1 r S = 0 and lim k 1 r Ci = 1 imply that for k = 1 neither Spot Selling nor Clearance Sales can be optimal. Moreover lim k 1 r ii = 0 together with Lemma 3 implies that the optimal selling strategy depends on g. If g < 1 k 1 a(1 ), Lemma 3 implies that selling with P 1 = U B, P 2 = 1 and k 1 = k1 i = 1 r(1 g(1 a)) has to be optimal. As lim k 1 k1 i = 1 this selling strategy falls into the class of Advance Selling. For g, Lemma 3 implies that 1 a(1 ) selling with P 1 = U G, P 2 = 1 and k 1 = min(g,k1 I) is optimal. As lim k 1 k1 I = 1 g(1 a) a > 1 it holds that k 1 = g so that this selling strategy constittes an Introdctory Offer. Proof of Proposition 3 The only reason why choosing a k < 1 might be optimal for the monopolist is that by restricting his capacity the monopolist increases the consmers risk to be rationed which redces the consmers option vale of waiting ntil period 2. Except for Clearance Sales this effect is absent and the monopolist optimally chooses k = 1 as profits are increasing in k. 21

22 Under Clearance Sales, profits are Π C = g(1 ar + ar k g 1 g (1 ar)(1 )) + (k g) and for k (g,1) Π C k = g (1 ar)(1 ). (13) 1 g If < g 1 g (1 ar)(1 ) the monopolist therefore has an incentive to restrict his capacity. However as Π C gu G Clearance Sales are dominated by Introdctory Offers with (P 1,P 2,k 1,k) = (U G,1,g,1). If instead g 1 g (1 ar)(1 ) the monopolist has no incentive to restrict his capacity in a Clearance Sale. It holds that Π C so that Clearance Sales are dominated by Advance Selling with (P 1,P 2,k 1,k) = (U B,1,1,1). This shows that the monopolist s optimal capacity choice is k = 1 and his optimal selling strategy (P1,P 2,k 1 ) is therefore as described in Proposition 2. Proof of Corollary 1 We consider the limiting case a 1 for which byers become homogeneos. lim a 1 r S = 1 k implies that for r 1 k Spot Selling is optimal. Moreover, for a 1 the two types of Introdctory Offers converge to one which sets P 1 = 1 r + r, P 2 = 1, and k 1 = 1 1 k r. Finally, lim a 1 r Ci = 1 implies that Clearance Sales are dominated by Introdctory Offers. This shows that when the monopolist is capacity constrained with k < 1 his profits are maximized by Spot Selling for r 1 k and by Introdctory Offers with k 1 = 1 1 k r otherwise. When the monopolist can choose k, Propositions 2 and 3 and the fact that for a 1 it always holds that g < implements Advance Selling. References 1 a(1 ) imply that the monopolist chooses k = 1 and [1] Blow, J. I. (1982) Drable Goods Monopolists. Jornal of Political Economy 90(2), [2] Coase, R. H. (1972) Drability and Monopoly. Jornal of Law and Economics 15, [3] Conlisk, J., Gerstner, E., Sobel, J. (1984) Cyclic Pricing by a Drable Goods Monopolist. Qarterly Jornal of Economics 99(3), [4] Corty, P., Li, H. (2000) Seqential Screening. Review of Economic Stdies 67(4), [5] Corty, P. (2003a) Some Economics of Ticket Resale. Jornal of Economic Perspectives 17(2),

23 [6] Corty, P. (2003b) Ticket Pricing nder Demand Uncertainty. Jornal of Law and Economics 46, [7] DeGraba, P. (1995) Bying Frencies and Seller Indced Excess Demand. RAND Jornal of Economics 26(2), [8] Gale, I. L., Holmes, T. J. (1993) Advance Prchase Disconts and Monopoly Allocation of Capacity. American Economic Review 83(1), [9] Gl, F., Sonnenschein, H., Wilson, R. (1986) Fondations of Dynamic Monopoly and the Coase Conjectre. Jornal of Economic Theory 39(1), [10] Harris, M., Raviv, A. (1981) A Theory of Monopoly Pricing Schemes with Demand Uncertainty. American Economic Review 71(3), [11] Lazear, E. (1986) Retail Pricing and Clearance Sales. American Economic Review 76(1), [12] Lewis, T. R., Sappington, D. E. M. (1994) Spplying Information to Facilitate Price Discrimination. International Economic Review 35(2), [13] Nocke, V., Peitz, M. (2007) A Theory of Clearance Sales. Economic Jornal, 117, [14] Stokey, N. L. (1981) Rational Expectations and Drable Goods Pricing. Bell Jornal of Economics 12(1), [15] Van Cayseele, P. (1991) Consmer Rationing and the Possibility of Intertemporal Price Discrimination. Eropean Economic Review 35,