Centre de Referènia en Eonomia Analítia Barelona Eonomis Working Paper Series Working Paper nº 229 Priing ylial goods Ramon Caminal July, 2005
Priing ylial goods Ramon Caminal Institut danàlisi Eonòmia, CSIC, and CEPR 08193 Bellaterra, Barelona, Spain e-mail: ramon.aminal@uab.es July 27, 2005 Barelona Eonomis WP nº 229 Abstrat Consumption of ertain ommodities auses transitory satiation, in the sense that potential instantaneous utility from an additional unit is very low immediately after a onsumption episode, but it inreases over time. Suh ylial pattern of preferenes has important impliations: (i) If the monopolist annot ommmit to long-run pries, then some equilibria are Pareto dominated (both buyers and the seller would rather play a different equilibrium involving a lower prie), (ii) introdution of loyalty-rewarding shemes may benet both buyers and sellers, (iii) restritions on the timing of purhases (purhase deadlines, sales, et) are likely to hurt onsumers and inrease effiieny, and (iv) ollusion may involve a prie redution. Keywords: ylial preferenes, repeat purhases, monopoly priing, loyalty-rewarding shemes JEL Classiation numbers :D42,L14 I am grateful to Mari Paz Espinosa, Clara Ponsatí, Patrik Rey, Mihael Riordan, and speially Roberto Burguet, for helpful disussions. The usual dislaimer applies. I also thank the support of the Barelona Eonomis Program of CREA and the Spanish MCyT (grant SEC2002-02506). 1
1 Introdution Consumption of ertain ommodities auses transitory satiation. The day after visiting an amusement park, the utility derived from another visit is likely to be very low, but tends to inrease over time. Most people also experiene a similar hange in preferenes after dining at an ethni restaurant, getting a hairut, or attending a onert by their favorite pianist. One may even argue that transitory satiation is assoiated with the onsumption of most types of goods, at least at some level of disaggregation 1 and on different time sales. In this paper I fous on markets in whih (i) onsumer satiation is suffiiently persistent over time to the extent that an potentially interat with prie dynamis, and (ii) sellers enjoy a signiant amount of market power. Thus, amusement parks would be a suitable example, but other examples will also be disussed. I propose a very simple haraterization of this type of preferenes. I 2 dene a ylial good as a perishable good for whih onsumers have ylial preferenes, and a new yle starts every time individuals onsume 3 the good. More speially, onsumers potential instantaneous utility, R, measured in monetary units, inreases monotonially with the time elapsed sine the last onsumption episode, s, and falls disontinuously when onsumption takes plae. Thus, if a onsumer pays a prie p after s units of time sine the last purhase, then she obtains an instantaneous net surplus of R() s p. Suh a representation does not take aount of two potentially important issues. First, various types of random shoks may play a role in many real world situations, whereas the above haraterization is purely deterministi. Seond, preferenes for some ylial goods may exhibit long-run dereasing marginal utility, in the sense that the funtion R() s may shift downwards after every purhase. In ontrast, I assume station- 1 To the best of my knowledge Harmann (2004) is the only empirial paper that has approahed this issue. In partiular, he estimates a dynami model of onsumption deisions for rounds of golf, and onludes that it takes 32 days for the median onsumers willingness to pay to return to pre-onsumption levels. He shows that this fat has important impliations for the estimation of the long-run own-prie elastiity. 2In Setion 2 I disuss the analogies between (non-durable) ylial goods and durable goods subjet to depretiation. 3Cylial goods are quite different from seasonal goods. In the latter ase, the time pattern of preferenes is exogenously given, and thus unrelated to the history of purhases. On the other hand, existing models of addition and habit persistene apture the opposite phenomenon: urrent onsumption raises future marginal utility. 2
ary preferenes throughout the paper. Thus, the urrent approah should be interpreted as a rst step in modelling ylial preferenes. The main goal of this paper is to study the priing of ylial goods. In the monopoly ase, the main insights are the following. First, if the monopolist annot ommit to long-run pries, then some equilibria are Pareto dominated (buyers and the seller would rather play a different equilibrium involving a lower prie). This is a rather unusual result. Typially, when we ompare different equilibria, a lower prie implies higher onsumer surplus but lower prots. Seond, the model offers a natural justiation for the introdution of loyalty-rewarding priing shemes and restritions on the timing of purhases (purhase deadlines, sales, et.), whih omplements the existing literature. In the urrent set up, loyalty shemes may benet both buyers and sellers, whilst restritions on the timing of purhases are likely to hurt onsumers and inrease total surplus. Finally, in a ompetitive environment with produt differentiation, and if transitory satiation is relatively stronger for the variety onsumed, it is shown that equilibrium pries may be above prot maximizing levels. In other words, ollusion may involve prie redutions. A ruial impliation of the ylial pattern of preferenes is that expeted future pries inuene urrent demand. When onsumers hoose the optimal timing of a purhase they balane the benets of waiting, higher instantaneous utility, and the osts of waiting, delaying the realization of the net surplus assoiated with the next and all other future purhases. Thus, if buyers expet higher future pries (lower expeted surpluses), then the osts of waiting are redued and hene the next purhase is delayed. Consequently, equilibrium pries depend on the sellers ommitment apaity. In Setion 3 I deal with the monopoly priing problem when pries annot depend on the history of purhases of individual onsumers (anonymous markets). Whenever the seller ommits to a onstant prie forever, then the prie has a large effet on the timing of purhases beause it affets not only the net surplus assoiated to the next purhase but also to the surplus assoiatedtoallsubsequentpurhases;and,asaresult,theeffetonthe osts of waiting is large. In the opposite extreme, if onsumers pereive the urrent prie as being relevant only to the next purhase and believe that future pries are independent of the urrent prie, then urrent demand is not very sensitive to the urrent prie, sine it only affets the surplus from the next purhase, i.e., the effet on the osts of waiting is small. In other words, in a game where the monopolist an only ommit to the prie of 3
the next purhase, the equilibrium in Markov strategies features a higher prie than in ase the monopolist ommits to a onstant prie forever. If we introdue the possibility of developing a reputation (by allowing for trigger strategies) then the range of pries than an be sustained as a subgame perfet equilibrium expands signiantly. Moreover, some of these equilibria are Pareto dominated by others. Thus, players may nd themselves stuk in a bad equilibrium with high pries, and instead they would rather be playing a different equilibrium with lower pries. If sellers keep trak of the history of purhases of individual onsumers then they might also be able to ommit to priing poliies that reward onsumer loyalty. In the urrent set up if we allow the seller to ommit to a sequene of pries (Setion 4.1), with the nth prie orresponding to the nth purhase, then the equilibrium poliy inludes a high prie for the initial purhase and a prie equal to marginal ost for the following ones. The introdution of suh loyalty rewarding shemes obviously benets the monopolist, in omparison to the ase of ommitment to a onstant prie, but its effet on onsumer welfare is ambiguous. Sellers of ylial goods have strong inentives to restrit the timing of purhases. In some (properly dened) markets, produts are available only oasionally. This is the ase, for instane, of a live performane of an artist in a partiular ity. More often, sellers may nd the way of restriting the availability of the produt, or, more generally, of affeting the timing of purhases. This is the aim of some ommon marketing tehniques, like subsriptions, oasional sales and purhase deadlines. In my set up, the seller always nds it optimal to restrit the timing of purhases (Setion 4.2). Typially, onsumers are worse off under restrited timing but total welfare is higher. In Setion 5 I embed the ylial pattern of preferenes in a model of produt differentiation. If transitory satiation is stronger for the last variety onsumed, i.e., if onsumers prefer a diversied onsumption prole, then ompetition is relaxed. The reason is that rm-ustomer relations are only oasional and hene rms pay less attention to the effet of their pries on onsumers future purhases. As a result, pries are determined by the net balane of two ountervailing effets: the standard business stealing effet (stati ompetition) and the frequeny of purhases from the same supplier. It is shown that if stati ompetition is not suffiiently intense then equilibrium pries might be above joint prot maximizing levels. In otherwords,ollusionouldleadtoaredutioninpries. 4
The sellers ability to ommit to future pries is also ruial in many other ontexts. A well known example is the Coase onjeture. In the absene of ommitment, a durable goods monopolist may end up setting 4 pries very lose to marginal ost. In ontrast to this literature, I am not so muh interested in the (short-run) opportunisti behavior of the monopolist in reation to endogenous hanges in the distribution of onsumers. Instead, I emphasize the effet of various priing shemes on the long-run pattern of repeat purhases. Under asymmetri information and onsumer learning, pries have also 5 been shown to follow various dynami patterns. In the urrent model quality is ommon knowledge; the time dimension matters only beause the preferenes of individuals vary systematially over time aording to the endogenous timing of onsumption. The literature on repeated games has improved our understanding of how tait ollusion an be sustained. In standard dynami models of oligopolisti ompetition multiple equilibria arise and players disagree about their ranking. Firms prefer equilibria with high pries while onsumers are better off in low prie equilibria. Similar arguments have been applied in 6 situations where the monopolist faes a time inonsisteny problem. Reputation an replae ommitment as a rent-extrating devie. One again, if we ompare different equilibria an inrease in onsumer surplus is neessarily assoiated to a redution in rms prots. In ontrast, in the urrent model equilibria an be Pareto-ranked (players fae a oordination problem). In a model of experiene goods Crémer (1984) showed that the monopolist would like to ommit to a priing sheme that inludes pries equal to 4 Bulow (1982) formalized these ideas in a nite horizon model, and Stokey (1981) and Gul et al.(1986) in an innite-horizon model. 5If onsumers repeat purhases, monopolists set an introdutory prie to indue experimentation, and later inrease the prie as onsumers beome better informed (Milgrom and Roberts, 1986). When onsumers purhase the good only oasionally, if the prie is a positive signal of quality then a dereasing sequene of pries may be obtained in equilibrium (Bagwell and Riordan, 1991), although if onsumers learn the quality of the good from market shares then the prie sequene will, on average, inrease (Caminal and Vives, 1996). 6For instane, in the durable goods ase, Ausubel and Denekere (1989) have shown that ompetitive pries are not the only possible outome. When the disount rate approahes zero all seller payoffs between zero and stati monopoly prots an be obtained as the outome of subgame perfet equilibria. 5
marginal osts for all repeat purhases. Thus, two apparently very different environments (experiene and ylial goods) deliver similar preditions regarding the monopoly priing of repeat purhases. However, there is a differene. In Crémers framework the prie of the rst purhase is equal to total gains from trade and, as a result, the sellers ommitment apaity hurts onsumers, whereas in the ase of ylial goods the prie of the rst purhase must be relatively moderate in order to indue onsumers to make their rst purhase relatively soon. As a result, onsumers may atually 7 benet from the introdution of suh loyalty rewarding shemes. 2 The baseline model 2.1 Desription Time is a ontinuous variable that runs from 0 to innity. There is a single seller and an arbitrary number of onsumers with idential preferenes. In some of the ases onsidered in the paper pries are ustomer-spei and heneitwillbeagamebetweenonebuyerandoneseller,wheretheformer retains the power of unilaterally setting pries. The monopolist an instantaneously produe a homogeneous perishable good at zero ost. Immediately after a purhase, onsumers potential instantaneous utility from an additional purhase is equal to L< 0 and evolves deterministially aording to R() s, where s is the time elapsed sine the last purhase. More speially, Rs () is a three times ontinuously differentiable funtion, from R++ into R, satisfying (See Figure 1 for an illustration): A.1. R() s > 0,R () s < 0,R () s > 0. A.2. lims 0R()= s L < 0, lims R()= s M. All agents disount the future at the rate r> 0. For simpliity, let us suppose that at time 0 the onsumer has just purhased the good. If the buyer expets to pay a prie pn in the nth purhase, n =12,,..., and to spend sn units of time between the ( n 1) th andthenth purhase,thenthebuyersexpetedpayoffattime 0 isgiven by: 7 In oligopolisti markets with random onsumer preferenes, loyalty-rewarding shemes reate onsumer swithing osts. Consumers tend to lose when sellers use oupons to reward loyalty (Banerjee and Summers, 1987), but they may gain if sellers ommit to pries for repeat purhases (Caminal and Matutes, 1990). 6
r U0 = [ R( sn) pn] e n=1 Similarly, the sellers payoff is given by: n r s j=1 j 0 = pe n (2) n=1 Finally, total welfare is the sum of the onsumers utility and the rms prots: 2.2 Effiieny r W0 = R( sn) e n=1 n s j=1 j The only variables that affet total welfare are the length of the time intervals between purhases. Thus, the effiient outome is a sequene of time intervals with length { sn } n=1 that maximize 3. We an set up the optimization problem as nding the optimal timing of the next purhase, s1, that maximizes: rs1 0 1 W = e [ R( s )+ W ] 1 1 R ( s ) r[ R( s )+ W ] = 0 o rs e o = o Rs ( ) 1 e rs n s j=1 j where W is the maximum surplus that an be obtained after the rst purhase (whih is independent of s1 ). The solution is given by the rst order ondition: Thus, the optimal timing is obtained by balaning the gains from waiting, i.e., the inrease in instantaneous utility, and the osts of waiting, i.e., the interest on the apitalized gains from trade. The latter is the sum of the instantaneous utility plus the net present value of future gains from trade. Note that the short-run optimal timing, s1, depends on the long-run surplus, W. Sine the optimization problem is stationary, the optimal time intervals are onstant and the maximum surplus after a purhase is given by: W 7 (1) (3)
where s o is given by: o o rr( s ) R( s ) o =0. (4) 1 e rs o Note that s inreases with r and is invariant to multipliative transformations of Rs. () 2.3 Analogies with durable goods The above haraterization is meant to represent the ase of non-durable goods with transitory satiation. Nevertheless, there are lose analogies to 8 the ase of durable goods subjet to depreiation. Let us onsider a durable good that generates a ow of servies equal to,where istheageofthegoodand 0 istherateofdepreiation. Suppose that the durable good an be produed under a onstant returns to sale tehnology with no apaity limits. Let denote the unit ost of produing the durable good. For simpliity, suppose that at time 0 onsumers have just purhased a new unit. If onsumers expet to pay a prie pn in their nth purhase, then their payoff funtion an be written 9 as: s qe s > U 8 0 [ ] [ ] q ( r+ ) s q ( + ) n = 1 + + + 1 r s r e e pn e r r n 1 +1 s j=1 j n=1 Similarly, the sellers payoff an be written as: = ( p ) e 0 n=1 n n r s j=1 j The analogy an be easily grasped by onsidering one of the examples mentioned in the introdution. A hair ut an perhaps be interpreted as an instantaneous servie that produes a level of utility whih depends on the length of time sine the last hair ut. However, a hair ut is better haraterized as a durable servie that deteriorates over time. 9A model with these harateristis has been analyzed in a ompanion paper (Caminal, 2004). In that paper I study the inentives to innovate and the rationale behind priing poliies that aim at affeting the timing of purhases, like trade-in allowanes, whih are ontingent on the age of the old unit. (5) (6) 8
In this ase I annot normalize variable osts to zero, otherwise in the effiient alloation onsumers would ontinuously buy new units. In the above formulation of non-durable ylial goods, ontinuous purhases were ruled out by the assumption that potential utility right after eah purhase fell below zero (marginal ost). Thus, we must keep in mind that p represents the prie-ost margin in one ase, but the absolute prie in the other. Exept for this, 6 and 2 are idential. The analogies between 5 and 1 are a bit less obvious. However, we an dene: 10 10 [ ] ( r+ ) s q R()= s 1 e (7) + r and note that this funtion satises assumptions A.1 and A.2, where q L = and M =. If we plug 7 into 5 then the only differene with + r respet to 1 is that in the latter ase R( sn) is enjoyed by onsumers at the timeofthe nth purhase,whileintheformerisenjoyedatthetimeofthe ( n 1) th purhase. It turns out that suh a differene has no eonomi signiane. Thus, the priing of some non-durable goods (like a visit to an amusement park) is in fat subjet to similar onsiderations than the priing of durable goods that depreiate over time (like automobiles). Nevertheless, physial harateristis do play a role in some ases. For instane, in what Fudenberg and Tirole (1988) all semianonymous markets, sellers of durable goods an offer disounts to those buyers that trade in their old units. Obviously, this is not possible in markets for (non-durable) ylial goods. Another important onsideration is that in most durable goods markets (hardware, software) tehnologial innovations are ruial. In suh highly non-stationary (and stohasti) environment is diffiult to think about ommitment to future pries, or about ontrating on the frequeny of purhases. In ontrast, many (non-durable) ylial goods markets are fairly stationary and omplex intertemporal priing poliies are more likely to be feasible. For all these reasons in the rest of the paper I stik to the non-durable ylial good interpretation. Some of the issues analyzed in this paper resemble those studied by Fishman and Rob (2000). In partiular, both papers attempt to haraterize the equilibrium timing of purhases under monopoly. However, they fous on produt innovation and onsumers adoption deisions are trivial: onsumers purhase the good as soon as it beomes available. 9
3 Monopoly priing and ommitment apaity In this setion I onsider the ase in whih the seller annot keep trak of the history of purhases of individual onsumers (anonymous market). At the end of the setion I will disuss the problems involved in handling posted pries and asynhronized onsumers. For the moment, I onsider a simple set up that illustrates very learly the role of the sellers ommitment apaity on monopoly pries. First, I present the benhmark ase in whih the seller an ommit to a onstant prie forever. Seond, I onsider the onsequenes of time-limited ommitment power. In partiular, I assume that the seller sets ustomer-spei pries and ommits to maintain the announed prie until the buyer makes the next purhase. I start haraterizing the equilibrium in Markov strategies and next I onsider more general strategies (to onsider reputation effets). I also disuss the ase of intermediate ommitment apaities and alternative modelling approahes. 3.1 Commitment to a onstant prie Suppose the monopolist ommits to a onstant prie forever. At time 0 the seller sets a prie p and onsumers hoose the timing of purhases. For a given prie p, the onsumer hooses s 1, s 2,... in order to maximize 1. The rst order ondition that haraterizes the optimal s is given by: 1 1 R( s ) r[ R( s ) p+ U ] = 0 where U is the onsumers ontinuation value. Thus, the onsumers short-run optimal timing depends not only on the urrent prie but also on the long-run surplus. However, in this subsetion the sellers prie is onstant and hene U also depends on p. More speially, given the stationarity of the problem, the buyers ontinuation value an be written as: U rs e = [ R( s) p] 1 e rs where s = s1 is also given by equation 8. Thus, the relationship between theoptimallengthofinterpurhasetimeperiodandtheonstantprie, s () p, is impliitly given by: 1 (8) 10
rrs [ ( ) p] R() s =0 (9) 1 ers From the above expression we an ompute the sensitivity of s to hanges in the (onstant) prie. In partiular: ds dp = Gs ( ) 1 e rs where r Gs () 0 R ()+ s rr () s > Thus, a higher prie inreases the length of the time intervals between purhases (dereases frequeny). Also note that G() s > 0, and that s (0)= o s. The monopolist antiipates that onsumers behavior is given by s () p and hooses p in order to maximize: 0 = 1 rs () p e rs () p The rst order ondition haraterizes the equilibrium prie : rp ds 1 =0 (10) 1 e rs dp Thus, equation 10 shows the trade-off faed by the monopolist: a higher prie inreases the margin but redues the frequeny of purhases. The ds size of the latter effet depends on. Combining equations 9 and 10 we dp an haraterize the equilibrium value of s, denoted by s ( stands for ommitment): rs rr( s ) ( ) rs + 1 e R s 1 e Gs ( ) e p =0 (11) Seond order onditions imply that the left hand side of equation 11 dereases with s. Also, from equation 4, we know that the left hand side, o evaluated at s, is positive. Hene, we obtain the following (straightforward) result. 11
Proposition 1 Under ommitment to a onstant prie, interpurhase time periods are ineffiiently long:. o s < s The intuition is straightforward. A monopolist harges a prie above marginal ost, whih redues the onsumers osts of waiting, and as a result the frequeny of purhases dereases. Finally, I denote the equilibrium prie as p, i.e., the value of p suh 11 that s( p ) = s. 3.2 Short-run ommitment Suppose now that at time 0 the monopolist sets ustomer-spei pries and an only ommit to keeping those pries unhanged until the next purhase. Immediately after the onsumer purhases the good then the seller an set a different prie. The idea is to rule out short-run priing poliies that restrit de fato onsumers timing of purhases, while allowing for some disretionary power in the medium and long-run. In this ase a Markov strategy for the seller is simply a prie, sine every time the seller sets a new prie all payoff relevant variables take the same value. A Markov strategy for the buyer an be expressed as a reservation prie, whih depends on the urrent state of preferenes, p() s ; or, more onveniently, as a hoie of the timing of next purhase as a funtion of the urrent prie, s( p ). The onsumers optimization problem is similar to that of the previous setion and thus s( p) is also given by equation 8. The ruial differene is that now her ontinuation value, U, does not depend on the urrent prie but only on expeted future pries. In fat, the relationship between the timing of the rst purhase and the urrent prie, s () p, is different than in the previous subsetion. In partiular: ds = Gs ( ) dp Hene, in this ase, sis less sensitive to pthan in the ase of ommitment to a onstant prie. The reason is that in the latter ase a hange in the prieisexpetedtobepermanent,whileinthemarkovequilibriumofthe urrent game any deviation from the equilibrium prie is expeted to be transitory. 11 Note that the equilibrium prie is independent of the initial distribution of onsumers, provided no onsumer starts with an shigher than s. 12
Given onsumer behavior, the sellers best response is the value of whih maximizes: rs () p 0 = e [ p + ] where is independent of. Thus, the rst order ondition an be written as: p ds 1 rp ( + ) = 0 (12) dp Sine the game is stationary, from equations 8 and 12 the equilibrium d timing of purhases, s ( d stands for disretion), is given by: p d d rr( s ) R( s ) 1 e rsd + 1 G s ( d) =0 Also, from equation 9 we have that the equilibrium prie, d d that s =ˆ s p. Next, let us ompare equations 11 and 13. p d (13), is suh Proposition 2 In the Markov Equilibrium of the short-run ommitment game the prie is higher and the average interpurhase time interval longer d d than under ommitment to a onstant prie, i.e., p > p, and s > s. The driving fore of this result is the effet of ommitment power on onsumers expetations. In the Markov equilibrium of the short-run ommitment game any deviation from the equilibrium prie is interpreted by onsumers as a transitory deviation and hene the impat on the timing of the next purhase is relatively small. In ontrast, whenever the seller an ommit to a onstant prie, a deviation is pereived as permanent, whih has a larger effet on the frequeny of purhases. As a result, in the Markov equilibrium of the short-run ommitment game the monopolist has 12 inentives to harge a higher prie than under long-run ommitment. 3.3 Disussion 3.3.1 Reputation Suppose expetations about future pries are formed aording to the following rule: If past and urrent pries have been equal to q then onsumers 12 Again, the equilibrium prie is independent of initial onditions, provided the initial d value of sis lower than s. 13
expet future pries to be equal to q, otherwise they expet future pries to d be equal to p. This is reminisent of trigger strategies, in the sense that if the seller deviates from the presribed prie then the punishment onsists on onsumers playing their strategy in the Markov equilibrium from [ then ] d onwards. In the Appendix it is shown that any prie in the interval p,p l, d where p l < p < p an be supported as a subgame perfet equilibrium. If we ompare the players payoffs aross these equilibria then a higher prie in the interval [ p,p l ] is assoiated with lower onsumer [ surplus ] and higher d rm prots. However, a higher prie in the interval p,p is assoiated with lower payoffs for both the buyer and the seller (See Figure 2.) In other words, there exist equilibria that are Pareto dominated: both buyers and sellers ould benet from swithing to a different equilibrium with lower pries. 3.3.2 Intermediate ommitment apaity Suppose that the seller an ommit to a (onstant) prie for the next N purhases. For eah N we obtain a prie and a length of interpurhase time periods { p N,sN} prevailing in the Markov equilibrium. It is immediate (although the algebra is quite messy) that as the degree of ommitment, N, inreases the sensitivity of demand to a prie ut also inreases, whih implies that the equilibrium prie falls and the frequeny of purhases inreases. More speially, p N < p N1, s N < s N1. Also, d d p1 = p,s1 = s, lim N pn = p, lim N sn = s. In fat, for any nite N we an talk about a double margin. For instane, d if N =1we an split the Markov equilibrium margin, p, into the margin aused by monopoly power, p, and the margin assoiated to the lak of d ommitment, p p. As the degree of ommitment inreases both buyer and seller payoffs inrease. This result suggests that sine sellers benet from any inrease in ommitment apaity they ould be willing to invest in various ommitment devies. Intheurrentmodel,iftheselleranhoosethedegreeofprie ommitmentatthebeginningofthegamethenhewillhoosetoommitto a xed prie forever. However, in a riher model sellers might fae a tradeoff between the benets from prie ommitment analyzed in this paper and the osts of prie rigidity. For instane, marginal osts may be random. If the variane of these osts is suffiiently large then the osts of prie rigidity may overome the benets from ommitment. Thus, higher ost volatility 14
wouldbeassoiatedwithmoreprieexibilityandhigheraveragemargins. 3.3.3 Modelling short-run ommitment The game studied in Setion 3.2 is highly stylized, but nevertheless it provides some useful insights on the effets of ommitment on equilibrium pries through onsumers prie sensitivity. Two assumptions seem partiularly ontroversial: ustomer-spei pries and the sellers open-ended ommitment apaity (prie is maintained until the buyer shows up). A natural alternative would be a game where the seller posts a prie that an only be oasionally hanged. In this ase the length of prie rigidity would parametrize the degree of ommitment. More speially, pries ould be xed for a time interval of length T (the length of the period) but trade an take plae at any time within that interval. In suh a set up we ould even think of dealing with asynhronized onsumers. Unfortunately, suh a game involves formidable analytial hallenges, even in the ase of a single onsumer. First of all, stationary equilibria do not generially exist. The reason is that the number of purhases in a given period will tend to utuate along the equilibrium path, sine generially T will not be a multiple of the time between two purhases. As a result, pries will also utuate. In partiular, the larger the number of purhases in a given period, the lower the equilibrium prie. The intuition is analogous to the one disussed in the previous subsetion. A possible solution to the non-stationarity problem would be to restrit ourselves to those values of T that generate a stationary pattern of purhases and pries. For instane, the ase T = s d would appear to be a partiularly interesting ase, sine we may hope that in suh a ase the d equilibrium prie might be p, whih in turn would indue a stationary pattern of purhases. Unfortunately, even in this partiular example the haraterization of equilibria with Markov strategies is rather umbersome beause of the existene of a deadline effet. Consumers willingness to pay disontinuously inreases right at the end of the period when a higher prie is expeted to replae the urrent one. Thus, for some initial onditions, the seller might have inentives to deviate and set a prie below d p in order to indue the buyer to purhase twie over the period (the seond purhase right before a new prie is quoted). Therefore, the players ontinuation value at the time of setting a new prie will in general depend on the time elapsed sine the last purhase. As a result, any attempt to 15
obtain a tratable haraterization of stationary strategies looks hopeless. In spite of these analytial ompliations it is not lear at all that suh an alternative model ould provide substantial additional insights. In partiular, it is very unlikely that suh a deadline effet ould offset the driving fore behind the main result of setion 3.2. In the alternative game where the prie is posted for T units of time, onsumers prie sensitivity will also depend on the number of purhases to be made at the urrent prie and, hene, it seems reasonable to onjeture that average pries along a Markov equilibrium will also derease with T. 4 Alternative priing shemes SofarIhavefousedontheaseinwhihthesellerannotkeeptrakof the history of purhases of individual onsumers and annot restrit the timing of purhases. However, in some ylial goods markets it might be feasible to keep reords of individual transations or at least to disriminate between old ustomers and newomers, through oupons and similar devies. Similarly, sellers might be able to ommit to supply some ylial goods only at spei points in time. In this setion I onsider rst the ase where the seller an set pries onditional to the number of previous purhases but, as above, annot diretly restrit the timing of those purhases. Next, I onsider the opposite ase: the seller an hoose in advane the timing of the next purhase but annot ondition the prie on previous transations. Finally, I briey disuss the possibility of writing long-run ontrats speifying both the prie and the frequeny of purhases, like in the ase of subsriptions to magazines. 4.1 Commitment to a sequene of pries Suppose that the seller an keep trak of the individual history of purhases and is able to ommit to a sequene of pries { pn } where n refers to the order of purhases, n =12,,... Now the seller an reward or penalize onsumer loyalty by setting a dereasing or an inreasing prie sequene. Given the sequene { p n}, onsumers hoose the timing of purhases { sn} in order to maximize U0 (equation 1). The optimality ondition for the timing of the rst purhase is well known by now and given by equation 8. Thus, as in the Markov Equilibrium of the game of Setion 3.2, the effet of p on s is 1 1 16
given by: s p 1 1 = G( s ) 1 However, the monopolist an inuene s1 not only through p1 but also through p,p,.... By the envelope theorem we have that for all n> 1: n 1 U1 r s j=2 j = G( s1 ) = G( s1 ) e n pn Notethatahangeinp1 hasthesameeffetons1 asahangeofthesame size in the present value of p. n However, the hange in pn has additional effets on ( s, 2...,s n). The monopolist hooses { pn } in order to maximize 2, antiipating how the timing of purhases is affeted by the prie sequene. The umulative effet of future pries drives the following result: Proposition 3 The equilibrium prie sequene inludes a positive margin in the rst purhase and zero margin in the following purhases, i.e., p 1 > o o 0,p = 0 for all n> 1. As a result, s >s, s = s for all n> 1. n 2 3 s p 1 FortheproofseetheAppendix. Theintuitiongoesasfollows. Therst prie of the sequene only has an effet on the timing of the rst purhase. However, suessive pries affet not only the timing of the orresponding purhases but also the timing of the previous ones. Consider a sequene of pries that involves a positive margin in the nth purhase. The monopolist an make higher prots by raising the rst prie and lowering the nth prie in suh a way that the present value of pries (evaluated at the timing of purhases assoiated with the original prie sequene) remains unhanged. The reason is that the new prie sequene does not have any rst round effet on the timing of the rst purhase, but it does bring forward the seond,third,...,and nth purhases. Next, I haraterize p1 and s 1. Sine, the onsumer appropriates all the surplus after the rst purhase the optimality ondition for s1 is an adaptation of equation 9: n rso e o R( s1) r[ R( s1) p1 + 1 o R( s )] = 0 e rs (14) 17
Sine the monopolist makes zero prots after the rst purhase, the optimality ondition for p is an adaptation of equation 12: 1 1 1 rp ds 1 = 1 rp1g( s1) = 0 dp Combining equations 14 and 15 we obtain: 1 (15) 13 rso e o R( s1) r[ R( s1)+ o R( s )]+ 1 1 e rs G( s1 ) =0 (16) Thus, the optimal priing poliy rewards onsumer loyalty. In fat, the result of marginal ost priing after the rst purhase is analogous to that of Crémer (1984) in a different ontext (onsumer learning in a twoperiod model). The mehanism behind suh a result is different although both models share the same priniple. In both ases the rst best an be implemented through a two-part tariff, and the monopolist an apture the entire surplus. In Crémers two period model, the rst period prie is analogous to an upfront fee. In my model if the monopolist an harge a fee upfront (before the game starts and thus unrelated to any purhase) and a prie per purhase then the prot maximizing poliy also inludes a prie equal to marginal ost in all purhases and a fee equal to the present value of total surplus. In most ases payment of an upfront fee is not feasible. Whenever seller and buyer are ready to sign a ontrat it is very likely that 13 the buyers potential utility hanges over time. In this ase, the buyer is willing to pay the upfront fee only at the moment of the rst purhase. Hene, the prie of the rst purhase is the instrument that the monopolist uses to ollet rents, although the size of these rents is moderated by the inentives to indue onsumers to make the rst purhase relatively soon. The equilibrium poliy haraterized in this setion may look somewhat unrealisti. First, onsumers ould be liquidity onstrained and unable to pay at the rst purhase an amount equivalent to a signiant fration ofthepresentvalueofallfuturegainsfromtrade. Seond,themonopolists inentives to default on her promises are very powerful and therefore her ommitment apaity must also be very strong. If we assume that onsumers are unable to pay at a single purhase a prie above a ertain For instane, when a new variety is introdued onsumers potential utility is likely to be affeted by the time period elapsed sine the last purhase of a different variety. 18
threshold, and/or that the monopolist is only subjet to a nite (and relatively small) penalty if he defaults on the priing poliy announed at time 0, then the slope of the time prole of equilibrium pries is redued, although the main qualitative features remain. Do onsumers benet from suh loyalty rewarding poliies? Let us ompare onsumer payoffs in the equilibrium where the monopolist ommits to p,s a onstant prie ( ) with the equilibrium where the monopolist an ommit to a (dereasing) prie sequene. In priniple, there are two ountervailing effets. In the latter ase, on the one hand, the prie harged after the rst purhase is lower but, on the other hand, the prie of the rst purhase is higher than in the onstant prie equilibrium. The examples disussed in the Appendix suggest that onsumers may atually lose or gain from loyalty rewarding poliies, depending on parameter values. In partiular, in Example 1 onsumers lose if the monopolist an ommit to a variable prie poliy, and in Example 2 onsumers gain. Thus, the introdution of loyalty rewarding shemes in ylial goods markets may be a Pareto improvement, whih ontrasts with the results of Crémer (1984). 4.2 Restriting the timing of purhases Sellers may be able to restrit the atual timing of purhases. For instane, they might redibly announe a very high regular prie with oasional and predetermined periods of sales. Similarly, sellers ould restrit the length of the time period for whih new varieties of the same produt are available (purhase deadlines). Finally, sellers ould offer long-term ontrats that inlude the prie and the frequeny of purhases. Real world examples of suh praties are not hard to nd. For instane, Disney video tapes are usually marketed under purhase deadlines, and subsriptions to magazines inlude a prie and a frequeny. Moreover, some produts an only be offered oasionally at a partiular loation. For instane, a live onert of Brue Springsteen is available in Barelona only from time to time. 4.2.1 Oasional purhasing periods Suppose that the monopolist wishes to indue onsumers to purhase every s unitsoftime. Inprinipleheoulddothateitherbymakingtheprodut available only at time s, 2s,..., or by setting a very high prie for purhases made at other points in time. Suppose that the monopolist annot refuse 19
to serve a onsumer at time ns just beause she has not purhased at time ( n 1) s. In this ase, the monopolist will be able to implement a prie, p, and a time interval between purhases, s, provided: rs R() s p+ U e [ R(2) s p+ U ] (17) In other words, the onsumer purhases at time s only if it is not worthwhiletowaituntilthenexttradingperiod, 2s. Thegainsfromwaitinghave to do with the inrease in the instantaneous utility, and the osts are due to the disounting. Sine I only onsider stationary poliies, the ontinuation utility, U, is given by: rs e U = [ R( s) p] (18) 1 ers Plugging equation 18 into ondition 17 we obtain the highest prie that the monopolist an harge for a given frequeny: rs rs p = 1+ e R( s) e R(2 s) (19) Thus, the optimal poliy onsists of hoosing a pair ( p, s) that maximizes 2 subjet to onstraint 19. By restriting the timing of purhases the monopolist faes a more favorable trade-off between pries and frequeny. r The optimal value of s, denoted by s, is given by: r rsr r rsr rsr r r rr ( s ) 1 e R ( s ) = e 1 e [2 R (2 s ) R ( s )]+ (20) rsr rsr r r + re 2 e [ R (2 s ) R ( s )] r and the optimal prie is given by ondition 19 evaluated at s. In order to ompare the outome of the urrent game with the ase in whih the monopolists sets either a onstant prie (Setion 3.1) and a sequene of pries (Setion 4.1) we must turn to a partiular example. Consider the following funtional form: ) rs e R()= s M 1 1 z 20
In this ase we an atually ompute the payoffs under the various priing poliies (See Appendix for details). The following table reports the 14 results for the limiting ase of z =0, and M =100. U0 0 W 0 (1) (2) (3) 25 25 16.1 25 50 38.3 50 75 54.4 Columns (1) and (2) ontain the payoffs under a onstant prie and a sequene of pries, respetively. Column (3) presents the payoffs under the stationary poliy with restrited timing analyzed in this setion. Comparison between olumns (1) and (2) illustrates the disussion of the previous subsetion: Setting a prie sequene signiantly inreases the sellers payoff without neessarily hurting onsumers. However, the sellers ability to restrit the timing of purhases inreases rm prots in omparison with the ase of ommitment to a onstant prie (olumns (1) and (3)), but it hurtsonsumers. Nevertheless,totalsurplusishigher. Thisisbeauseby restriting the timing of purhases the seller is able to indue more frequent o r onsumption at a higher prie. That is, s < s < s. Finally, if we ompare olumns (2) and (3) we realize that restriting the timing of purhases is Pareto dominated by the ommitment to a sequene of pries. In other words, the sellers ability to ommit to trading exlusively at ertain periods of time has only a modest impat on total surplus and the sellers ability to appropriate rents. This is beause the seller annot refuse onsumers that did not purhase the good in the previous trading period. Therefore, the prie annot be too high otherwise onsumers will nd it optimal to wait until the next trading period. 4.2.2 Contrating prie and frequeny Clearly, the monopolist ould implement the rst best and appropriate the entiresurplusifheanontratex-anteboththeprieandthefrequeny of purhases. Subsription to magazines is one example of this type of ontrat. Other servies suh as house leaning, maintenane of equipment, and so on, are sometimes marketed under ontrats that inlude a prie 14 Payoffs turn out to be proportional to M, thus setting M =100is only a normalization. However, the hoie of z is not at all irrelevant. 21
If we allow onsumers to buy from both neighbouring rms then individual demand funtions are onvex and have a kink, and no symmetri prie equilibria exist. Alternaand a frequeny, although usually the arrangement an be breahed at no peuniary ost. In partiular, if the monopolist an ommit to serving only those onsumers that buy a ontrat that inludes ( p, s ), then the optimal o o ontrat onsists of p = R( s ) and s = s. Notie that if breahing the ontrat involves no ost, onsumers will prefer to purhase at the prie 0 R( s ) at a lower frequeny, and hene we are bak to the ase analyzed in the previous subsetion. 5 Competition So far I have only onsidered an homogeneous good produed by a single rm. Introduing produt differentiation in a ylial goods framework involve non-trivial modelling hoies. In partiular, the potential utility derived from the onsumption of a partiular variety may depend not only on the time elapsed sine the last onsumption episode, but also on whih varieties have been onsumed reently. More speially, the relative valuation of two partiular varieties may hange after onsuming one of them. For instane, after dining at the loal Chinese restaurant a onsumer may value relatively more an additional meal at the loal Italian restaurant vis à vis the Chinese. That is, temporary satiation of the good may in fat be stronger for the variety that was atually onsumed. This implies that onsumers may pursue a diversied onsumption pattern. If different varieties are produed by independent rms, then diversiation is likely to have important impliations for their optimal priing strategies. In order to onsider these issues let us embed the ylial pattern of preferenes into Salops irular market model. There are n equally distant loations in the unit irumferene. In eah loation there is a single rm. Consumers are uniformly distributed over these loations (no onsumer is loated between two rms). Thus, this is a model of n ities sattered in a irumferene. If a onsumer purhases from the loal rm at prie p then she obtains an additional utility equal to R() s p, where s is the time elapsed sine the last onsumption episode. Instead if she purhases from the lokwise neighboring rm she obtains R() s p t, where t an be interpreted the transportation ost. Consuming from neighboring rms 15 loated ounterlokwise or more distant loations is assumed to be pro- 15 22
hivitevely ostly. Finally, onsumers are heterogeneous with respet to the transportation osts. More speially, t is uniformly distributed in the interval [0,t]. We onsider two extreme patterns of preferene dynamis. In the benhmark ase, onsumer preferenes are xed. In other words, the relative valuation does not hange with the history of purhases. In the seond ase, onsumers randomly realloate after eah purhase. The interpretation is that onsumers start up with ertain preferenes over all possible varieties. After eah onsumption episode, satiation is stronger for the variety that has been reently onsumed and as well as for other similar varieties. As a result, onsumers will only onsider relatively distant varieties in the next purhase. 5.1 Fixed relative preferenes Let us rst onsider the ase in whih onsumer loation remains xed over time. In order to fous on the effets of ompetition it seems reasonable to disregard the problems assoiated with limited ommitment apaity and 16 letrmstoommittoaonstantprie. In a symmetri equilibrium onsumers purhase always from their loal suppliers. Thus, their behavior an be summarized by s () p, whih is given by equation 9. Let us denote by p the symmetri equilibrium prie. If a partiular rm sets p>p, then it will sell only to those loal onsumes with t pp. Similarly, if p p then it attrats those onsumers in the lokwise neighboring loation with t p p. Therefore, given that other rms are playing p the payoff funtion of a partiular rm that harges p an be written as: rs () p = 1 pe p p 0 if p p 1 ers () p t rs () p = + 1 p p rs(+) p t pe pe dt otherwise 1 ers () p t 0 1 ers (+) p t tively, we ould have onsidered the two-rm Hotelling set up. The message would have been very similar, but the presentation would have beome less transparent. 16Also, for simpliity, at time 0 all onsumers have just purhased one of the varieties, i.e., they have initially s =0. 23
In a symmetri equilibrium, the rst order ondition evaluated at must be zero: p = p p rp ds 1 =0 (21) ( t 1 e rs p ) dp Ifweompare21withequation10thenitislearthatas t goestoinnity then p goes to p (the monopoly prie under long-run ommitment). Also, if t is equal to zero then p =0. Finally, using seond order onditions, p inreases with t. Hene, as usual, ompetition redues pries below the joint prot maximizing level, p, beause of the business stealing effet. 5.2 Variable relative preferenes Let us now onsider the other extreme ase. Suppose that after onsuming variety i onsumer swithes loation and the probability of every other loation is the same. This assumption aptures the idea that relatively satiation is stronger for the variety onsumed. Under variable relative preferenes, ompetition has two different effets on pries: (a) business stealing, as rms may have inentives to underut their neighbors pries, and (b) less frequent purhases from the same supplier, whih implies that rms pay less attention to the effet of their pries on the long-run behavior of their urrent ustomers. It is onvenient to assume that n is arbitrarily large so that we an disregard the probability that a onsumer revisits the urrent supplier. In this ase, the effet (b) is magnied, as rms ompletely disregard the effet of their pries on the long-run behavior of their urrent ustomers. Sine onsumers swith suppliers after eah purhase, their behavior is given by s () p as dened in Subsetion 3.2, sine they expet the urrent prie to be effetive in their next purhase only. Also, the rm takes as exogenous the long-run behavior of onsumers in any loation. Hene, given that other rms are playing p the payoff funtion of a partiular rm that harges p an be written as: rs () p pe p p 0 = 1 if p p 1 ers ( p ) t pe rs() p (+) = + 2 p p rs p t 1 rs ( p ) t 0 1 rs( p + t) e pe e dt otherwise 24
The rst order ondition at p = p is given by: 2 p ( ) 1 rp ds p =0 (22) t dp Now the omparison between the equilibrium prie and the joint prot maximizing prie is less straight forward. One again if t =0 then p =0. Also, using seond order onditions p inreases with t. Finally, as tgoes h to innity then p goes to p. By omparing equations 22 and 12 then h d we have that p > p > p. In words, if the business stealing effet is not present ( t equal to innity) then the only effet of ompetition omes from shortening rm-ustomer relationships and as a result, demand is less sensitive to pries and rms have lower inentives to ut pries in order to bring purhases forward. In this ase, the equilibrium prie is above the joint prot maximizing level (ollusion would involve a prie ut). Thus, in general, the sign of ( p p ) is ambiguous. If t is very small, then the stati ompetition effet dominates and p is below p. However, if t is relatively large, then again the shortening of the rm-ustomer relationship dominates and p is above p. 6 Referenes Ausubel, L. and R. Denekere (1989), Reputation in Bargaining and Durable Goods Monopoly, Eonometria, 57, 511-531. Bagwell, K., and M. Riordan (1991), High and Delining Pries Signal Produt Quality, Amerian Eonomi Review, 81, 224-239. Banerjee, A. and L. Summers (1987), On Frequent Flyer Programs and Other Loyalty -Induing Arrangements, H.I.E.R. DP No. 1337. Bulow, J. (1982), Durable Goods Monopolists, Journal of Politial Eonomy, 90, 314-332. Caminal, R. (2004), Tehnologial and physial obsolesene and the timing of adoption, mimeo Institut danàlisi Eonòmia, CSIC. Caminal, R. and C. Matutes (1990), Endogenous Swithing Costs in a Duopoly Model, International Journal of Industrial Organization, 8, 353-373. Caminal, R. and X. Vives (1996), Why Market shares Matter: an Information-Based Theory, Rand Journal of Eonomis, 27, 221-239. 25