Social learning and monopoly pricing with forward looking buyers

Transcription

1 Social leaning and monopoly picing with fowad looking buyes JOB MARKET PAPER Click hee fo the most ecent vesion Tuomas Laiho and Julia Salmi Januay 11, 217 Abstact We study monopoly picing in a dynamic model of social leaning. The quality of the poduct is uncetain and is evealed though consumes expeiences. Heteogeneous buyes aive to the maket ove time and have the option of delaying thei puchases. The key question fo the monopolist is whethe and when to sell to low valuation buyes: selling ealy facilitates leaning but postponing enables sales to take place when the maket paticipants ae bette infomed. We solve the monopolist s poblem unde commitment and constuct Makov pefect equilibia. We show that unde commitment the monopolist neve holds low pice sales to accumulated buyes but may keep a lowe pice in the beginning to geneate leaning. In contast, we constuct equilibia in which the monopolist fist accumulates and then sells to low valuation buyes. Because commitment powe can impove the efficiency of leaning, we show that the commitment solution can povide moe welfae than the equilibia. The effect of commitment powe on leaning efficiency is ambiguous, howeve, and depends on the the leaning envionment. JEL classification: D42, D83 Keywods: Picing, social leaning, stategic expeimentation, Poisson bandits 1 Intoduction Fims often launch poducts whose quality is not fully known befoehand. Instead, the quality of the poduct is evealed ove time as consumes buy it. The pupose of this pape is to undestand how this influences a monopolist s pice setting. How does Aalto Univesity School of Business s: tuomas.laiho@aalto.fi and julia.salmi@aalto.fi). We ae extemely gateful to Pauli Muto and Juuso Välimäki fo thei continuous suppot and encouagement fo this poject. We also thank Attui Bjök, Fancesc Dilme, Mia Fick, Johannes Höne, Matti Liski, Shunya Noda, Aniko Öy and semina audiences at HECER fo thei comments. Laiho acknowledges funding fom Jenny and Antti Wihui Foundation. Salmi is gateful to Emil Aaltonen Foundation, OP Goup Reseach Foundation, and Yjö Jahnsson Foundation fo financial suppot. 1

2 a fim pice its poduct while consumes ae still leaning how valuable it is? How well do we lean the quality of the poduct? What ae the welfae implications of the monopolists behavio? Imagine a fim stating to sell a new poduct of uncetain quality. The poduce of an expeience good, say a theate show o a musical, might be unsue if a wide audience likes it befoe the opening night. O if the poduct is a duable good, say a new phone, thee may be uncetainty about its eliability. 1 As the fim stats selling the poduct, the quality is evealed though social leaning when moe and moe consumes buy the poduct. The main question is how a fim takes the uncetainty and the dynamics of leaning into account in its picing. Two cental featues distinguish ou model fom the pevious liteatue on picing and consume leaning: the heteogeneous consumes ae fowad looking and they cae about a common state of the wold so that they lean fom each othe. Afte aiving to the maket, the consumes decide when to buy the poduct and exit when they do so. As buyes ae heteogeneous and make one time puchases befoe exiting, they can accumulate to the maket. Ou main contibution is to incopoate picing into this social leaning envionment. The possible accumulation of buyes to the maket pesents a dilemma to the monopolist: is it bette to sell now as much as possible in ode to geneate leaning o to wait and sell to lowe valuation buyes late, when quality has been leaned. Because the consumes ae fowad looking, the monopolist also has to manage buyes expectations about futue pices. In the latte aspect, the monopolist s poblem shaes many featues with duable goods models. One implication of having leaning in the model is that we can show that the commitment solution can povide moe welfae than the equilibia. This is in contast to the complete infomation case in which the commitment solution always povides less welfae. The esult essentially goes though because the commitment solution can be moe efficient in the sense how well we lean the quality of the poduct. In tems of picing, we show that when the monopolist has commitment powe it is neve optimal to hold sales fo accumulated low type buyes. This esult stems fom the need to manage the buyes pice expectations and leads to an inefficiency. The selle does not decease the pice to attact moe buyes to test the poduct even when expeimentation becomes moe valuable. We then constuct equilibia, which diffe stakly fom the commitment solution in this sense. In the equilibia, the monopolist accumulates low valuation buyes so that she can sell to them late, when moe infomation about the poduct quality is available. The fist best solution also exhibits this featue: it is socially efficient sometimes to let buyes wait and only sell to a pat of the maket. In ou model, individual buyes impose an infomational extenality on othe buyes: when they consume, othe buyes lean about the quality of the poduct. Though he picing, the monopolist can intenalize this extenality and povide incentives fo consumes to buy the good so that the quality is leaned. Since the monopolist is a 1 See New Yok Times June 8, 216): Hamilton Raises Ticket Pices: The Best Seats Will Now Cost $849, and The Wall Steet Jounal Sept. 2, 216): Samsung to Recall Galaxy Note 7 Smatphone Ove Repots of Fies, smatphone

3 esidual claimant to the benefits of leaning, she has to balance between infomational and evenue motives. The decision to sell is simultaneously a decision to expeiment - we lean something about the poduct. Fom the pespective of infomation poduction, the pesence of a monopolist is welfae enhancing as in a puely competitive envionment the pice matches the maginal cost and hence the fims cannot intenalize any gains fom leaning. Fo a monopolist, it is pofit maximizing to sell the poduct even below the maginal cost povided that she is optimistic enough about the quality of the poduct. The monopolist faces a maket with a constant enty of new consumes, whose valuation fo the poduct depends on thei belief about the quality of the poduct and thei own type. Simila to many duable goods papes, we model consume heteogeneity with two types. Upon aival, the consumes decide at each point of time whethe to buy o wait given thei valuation, the cuent pice, and thei expectation about futue pices. Once they buy they exit the maket. Thee ae two points of inteest hee. Fist, the monopolist influences the buyes behavio both though cuent and futue pices, which intoduces time inconsistency to the monopolist s poblem simila to complete infomation duable goods models. Essentially, lowe futue pices cannibalize sales today as today s pice has to eflect futue pices. Second, the buyes option to wait ceates a esouce poblem fo the monopolist: she needs to decide whethe she wants to accumulate buyes and when to sell to them. The leaning model we use is based on the Poisson bandits by Kelle, Rady and Cipps 25) and Kelle and Rady 215). The value of the poduct can eithe be low o high. Consumes who buy the poduct discove the value and might epot this publicly. Leaning happens though pefectly evealing signals that tell eithe that the poduct is of low quality o that the poduct is of high quality. We analyze models with high beakthoughs) and low signals beakdowns) sepaately and call them in accodance with the liteatue as pefect good news and pefect bad news envionments. The ate at which these signals aive is popotional to the amount of buyes who buy the good. The monopolist s pice setting entes into the leaning model as it contols the ate at which the buyes buy the good. We solve the commitment solution to the monopolist s poblem and constuct Makov pefect equilibia that have the belief about poduct quality and the amount of consumes in the maket as state vaiables. We do this analysis sepaately fo the good news envionment and the bad news envionment. With full commitment powe the monopolist neve holds sales: she neve fist sells to high valuation buyes and then to low valuation buyes. In a bad news envionment, the commitment solution is emakably simple: the monopolist only sells to high valuation buyes. Unde good news, the monopolist can fist sell to all types in ode to geneate infomation. The solution is also histoy dependent: who we sell today depends on who we sold in the past. We show that both unde bad and good news thee ae multiple Makov pefect equilibia with the following stuctue: fo low beliefs the monopolist only sells to high valuation buyes and fo high beliefs she sells to all buyes in the manne of the Coase conjectue. We can suppot these equilibia by diffeent self-fulfilling buyes expectations. To undestand why suppose the following: buyes have to decide between buying today and tomoow. If the buyes expect that tomoow s pice is low, they 3

4 will not buy at a high pice today. Thus the pice has to be low today as well. It is woth noting is that the multiplicity of equilibia emains in a finite game with a long enough time hoizon. Both unde bad and good news, we have equilibia in which the monopolist accumulates low valuation buyes and then depending on the news sells to them late. Although quantitatively diffeent, this stuctue is simila to fist best. We show that unde bad news thee is always moe expeimentation in the commitment solution than in the equilibia we constuct. The esult follows because the amount of expeimentation is detemined by how much value of infomation the monopolist can intenalize in the beginning of the game. Clealy, this has to be lage in the commitment solution as it maximizes the monopolist s value in the stat. Supisingly, this is not tue unde good news. We show that the accumulation of buyes in an equilibium can be so lage that it ceates incentives fo the monopolist to expeiment longe than in the commitment solution. It is also easy to see that wheneve the monopolist expeiments moe in the commitment solution than in an equilibium, the commitment solution can povide moe welfae. Thus the effect of commitment powe on welfae is ambiguous unlike in the complete infomation vesion of ou model. Ou appoach to using the Poisson bandits is simila to a few othe papes studying social leaning. The most elated ae two ecent papes by Fick and Ishii 216) and Che and Höne 215). Uncetainty about poduct quality and the model of leaning they employ is vey simila to ous, but both look at situations whee pices cannot adjust in ode to facilitate expeimentation. One majo motivation fo ou pape is the need to undestand if the lack of monetay incentives is the only souce of inefficient leaning in the existing models. Fick and Ishii 216) show how infomational incentives shape the adaptation of innovations in the absence of adjusting pices. The cental theme of that pape is consumes infomational fee-iding by delaying puchases: equilibium adoption ate must balance the value of futue infomation and cuent adoption. Ou pape adds to thei analysis by intoducing picing and heteogeneous consumes into the model. We can epoduce thei adoption pattens, and thus aive to a diffeent explanation fo the same obseved adoption behavio. Che and Höne 215) study social leaning fom the point of view of a social planne who designs a ecommendation system fo the consumes. As they ae inteested in nonmonetay mechanisms, the tade-offs facing the social planne ae vey diffeent to ous. They show that the planne s solution geneally exhibits egions of ove-ecommendation to consumes to induce them to expeiment. This aises fom the same easons as the monopolist selling to a pice lowe than the maginal cost in ou model. Social efficiency calls fo inefficient individual consumption. A numbe of papes have looked at picing and leaning. Papes by Begemann and Välimäki 1997, 2 and 22) use Bownian bandits due to Bolton and Hais 1999) and myopic buyes to model leaning and picing in a duopoly setting, in which a new fim entes the maket with a poduct of uncetain quality. Buyes myopia in these papes esults fom epeat puchases and the fact that they ae infinitesimally small. Bose et al 26) look at monopoly picing in a heding model with exogenous timing of puchases. Bonatti 211) studies menu picing in the pesence of social leaning. Begemann and Välimäki 1996 and 26) look at idiosyncatic leaning and picing. The main diffeence to these papes is that ou model has a combination 4

5 of social leaning and heteogeneous fowad looking consumes, who exit the maket when they buy. This leads to vey diffeent dynamics. Ou model shaes many chaacteistics with the liteatue elated to the Coase conjectue and duable goods models. In fact, the model can natually be thought of as a duable goods model with leaning about poduct quality. Ou commitment solution shaes featues with the solution when poduct quality is known as then the dynamic monopoly pice is equal to the static monopoly pice Stokey 1979 and Conlisk 1984). The equilibia of a game nealy identical to ous but with complete infomation is analyzed in depth by Sobel 1991). Unlike in the continuous time limit of his model, we have Makov pefect equilibia whee the monopolist can cedibly commit to not selling to lowe valuation buyes immediately afte highe valuation buyes have bought. Sales ae an impotant pat of the monopolist s behavio in ou model. This finding is not unique to a leaning envionment. Conlisk et al. 1984) show that in the discete time and complete infomation vesion of ou model, pices ae cyclic in an equilibium with the monopolist holding peiodic sales. Sales in thei model ae not caused by new infomation but the patial commitment powe that follows fom the combination of discete time and influx of new consumes. Öy 216) builds a model whee the monopolist s stategy involves peiodic sales. Thee the monopolist s ability to commit to peiodic high pices stems fom an advetising cost: it is costly to sell to the accumulated buyes. Both Conlisk et al. 1984) and Öy 216) find pice fluctuations because thee ae fictions in thei models ceating some but not complete commitment powe. Gaett 216) solves the commitment solution of a duable goods monopolist who faces consumes whose valuation fo the good vaies stochastically and shows the optimal stategy can involve sales. Stochasticity is fundamentally diffeent fom ous: pefeence changes ae idiosyncatic and thee is no aggegate uncetainty. The est of the pape is oganized as follows. Section 2 intoduces the model and the monopolist s and buyes poblems. Section 3 deives the solution to the monopolist s poblem unde bad news. Section 4 deives the solution to the poblem unde good news. Section 5 compaes the bad and good news cases and the diffeences between commitment and equilibium behavio. Section 6 discusses what happens if we change some assumptions of the model and pesents conclusions. 2 Model 2.1 Payoffs A monopolist is selling a poduct, whose quality is unknown both to the monopolist and the buyes. Define ω to denote the quality of the good and let ω {1, }, whee 1 denotes high quality and low quality. While neithe the monopolist no the buyes obseve ω they have beliefs ove it based on a common pio and past expeiences of buyes. Let x denote the common belief that ω = 1. Time, t, is continuous and goes to infinity. The buyes have binay types: they eithe have a high valuation o a low valuation fo the poduct in the good state of the wold, θ {θ H, θ L }. Valuation in the bad state is equal to zeo fo both types. The buyes aive to the maket at ates λ H > and λ L > espectively. The total aival 5

6 ate of buyes is λ A = λ H + λ L. The inflow of buyes stops if the monopolist stops selling: exit is ievesible. We assume that individual buyes ae infinitesimally small, so that an individual buye has no effect on the aggegates. 2 Each consume wants to buy one unit of the good and then exit the maket. We nomalize the outside option of the buyes to zeo. The buye s utility fom buying the good with pice p t can then be witten as U i ω, θ i ) = θ i ω p t. This means that the expected utility fom buying the good is simply whee x t is the belief at time t. EU i ω, θ i ) = θ i x t p t, Once the buyes aive, they make a decision whethe to buy o wait depending on thei valuation and the pice path, p t ) t [, ), the monopolist sets. If a buye desies to wait, he stays in the maket and can puchase wheneve he wants to. We assume that if buyes ae indiffeent between buying and waiting, they buy. We also estict the buyes to use pue symmetic stategies: all willing buyes buy. The monopolist sets a pice fo each t and has to pice anonymously so that the pice does not depend on the buye s type. She faces a constant maginal cost c, θ L ) fom poducing the good. He flow pofit is thus λ t p t c), whee λ t is the amount of buyes buying the good. We assume that the static monopoly pice is high. That is, we let λ H θ H c) > λ A θ L c). We futhe assume that both the monopolist and the buyes discount the futue with the same discount ate >. 2.2 Leaning The monopolist and the buyes obseve public signals about the poduct quality ove time as buyes buy the poduct. We model the leaning pocess with an exponential bandit model simila to models employed by Kelle, Rady, and Cipps 25) and Kelle and Rady 215). Cental to the model is that signals aive popotional to sales: the maket paticipants lean about quality though andom communication by the consumes who bought the good. To make the model tactable, we focus on pefect good news and pefect bad news leaning envionments. This means that a single signal news) is conclusive evidence that the poduct is eithe high quality o low quality. Suppose the monopolist is selling the poduct with ate λ. Then unde pefect good news a pefectly infomative public signal about the poduct quality aives at ate ωλ. Afte the signal aives the maket paticipants know fo sue that the poduct is of high quality. If the monopolist is selling to a discete mass of buyes, say a >, then the signal aives accoding to the exponential distibution with pobability ω1 e a ). Equivalently, we can wite this as an integal ove vitual time and handle similaly 2 This assumption is the same e.g. in Sobel 1991) and Gul, Sonnenschein and Wilson 1986). All in all ou model is almost identical to Sobel 1991) except fo leaning. 6

7 as selling to a flow. If the signal does not aive, the playes become moe pessimistic about the quality of the poduct. We can use Bayesian updating to deive the posteio conditional on the signal not aiving. With a slight abuse of notation we let this posteio be x t. Given a pio x, we can wite the no news posteio hencefoth the belief) unde pefect good news as x t = x e Qt) x e Qt) x ), whee Qt) is the amount of sales up to t. If the monopolist sells to a mass of consumes i.e. at) > fo some t), thee is a jump in the belief at that point. Howeve, if a = until t and the ate of sales λ t is continuous, the above implies the following law of motion fo the beliefs fo the good news belief ẋ = λ t x1 x). Unde pefect bad news if the public signal aives the maket paticipants know fo sue that the poduct is of low quality. If the monopolist is selling the poduct with ate λ, unde bad news the signal aives at ate 1 ω)λ. If the monopolist is selling to a mass of consumes, say a >, the signal aives with pobability 1 ω)1 e a ). If the signal does not aive, the playes become moe optimistic about the quality of the good. Given a pio x, we can wite the no news posteio belief) unde bad news as x t = x x x )e Qt). Supposing a continuous ate of sales λ t, the law of motion unde bad news is then ẋ = λ t x1 x), To summaize, unde good news the belief is deceasing in the absence of news, wheeas unde bad news it is inceasing. It is useful to note hee that the likelihood pocess q t = x t /1 x t ) possesses nice popeties. Fom above, we get that the law of motion fo the likelihood unde good news is q = λq. This implies that the likelihood evolves accoding to qt) = q e Qt). Similaly, unde bad news the likelihood evolves accoding to qt) = q e Qt). As the laws of motion ae simple fo the likelihood, we make fequent use of the likelihood pocess in the equations. 2.3 Buyes poblem At each point of time buyes decide whethe to buy o not, which is equivalent to setting a esevation pice. A buye exits when he buys and othewise stays in the maket. We estict to symmetic pue stategies so that all indiffeent buyes buy. 3 3 Fo futhe discussion on this assumption see section 6. 7

8 A buye s willingness to pay depends on the common belief and expectation of futue pices as well as his own type. If the buye does not buy, he gets his expected value of waiting, denoted by W t, θ i ). The buye s esevation pice p t solves θ i x t p t = W t, θ). Fo low types, W t, θ L ) =, and theefoe we denote high type s waiting value by W t) thoughout the pape. 2.4 Monopolist s poblem The monopolist sets the pice at each point of time and decides whethe to be in the maket. Since all indiffeent buyes buy, he decision can be simplified to a choice between thee actions: selling to both types action A), selling only to high types H), and exiting the maket E). Exit is ievesible. She caes about two state vaiables: the belief, x o the likelihood q), and the mass of buyes waiting. The assumption that the monopolist contols only pices guaantees that high type buyes always buy upon enteing the maket: the monopolist must be selling to high types o she has exited the maket. We denote the mass of waiting low type buyes by m. The monopolist maximizes a discounted flow of pe peiod payoffs. We state the maximization poblems sepaately fo each case when analyzing the commitment and equilibium solutions unde bad and good news. 2.5 Solution concepts Befoe solving the monopolist s poblem, we intoduce the solution concepts we use. Commitment solution. The commitment solution to the monopolist s poblem maximizes he value at time t =. An impotant featue of the commitment solution is time dependency: the monopolist s policy today depends on what she did yesteday. The commitment solution povides an uppe bound fo the equilibium value set. Equilibium. We estict to Makovian stategies and look fo Makov pefect equilibia MPE) of the game. That is, we let monopolist s and the buyes equilibium stategies to be functions of the state x, m) only. The monopolist s stategy maps states to pices. Thus the monopolist s stategy is s M : [, 1] R + R +, whee s M x, m) = p is the pice the monopolist sets in state x, m). Since the monopolist is always willing to pice at the maginal buye s valuation, the decision is equivalent to choosing between selling to high types, selling to all types o exiting. Selling to high types implies that the pice is p t = θ H x t W t and selling to all types implies that the pice is p t = θ L x t. The buyes decide whethe to buy o not. Thei action depends on the cuent state and thei own type. A buye s stategy is s B : [, 1] R + Θ R +, 8

9 whee s B x, m, θ) = p means that the buye is willing to buy if the pice is at most p. We estict to symmetic pue stategies fo the buyes. We define a Makov pefect equilibium of the game the following way. Definition 1. A Makov pefect equilibium is a pai s M, s B ), whee s M maximizes the monopolist s value given the state x, m) and the buyes stategy s B, s B maximizes the buye s expected payoff given the state x, m) and s M. The equilibium is simple afte the signal has aived. Unde bad news, the belief jumps to zeo and the monopolist exits. In the good news envionment, afte the news aives the belief jumps to one and we have a standad complete infomation game analyzed in depth by the existing liteatue. Sobel 1991) shows that in the continuous time limit of the game the monopolist pices at most at low type s valuation in the Makov pefect equilibium. Hence, when the good news signal has aived the monopolist sells to both types in ou model, too. We make a futhe assumption to simplify the equilibium at x = 1: the pice is at least θ L x t. We then have the following esult fo the complete infomation equilibium. Lemma 1. Assume that the pice is always at least θ L x t. Then the equilibium at x = 1 is such that the monopolist sells to all types at pice θ L. Poof: Sobel 1991 Theoem 2, Coollay). 3 Bad news envionment 3.1 Fist best A natual place to stat the analysis of the monopolist s poblem is the fist best. That is we solve the solution to the monopolist s poblem, when she can intenalize all value fom sales and thus is not esticted to using a unifom pice. This is equivalent to maximizing the total suplus, which we can define as the discounted sum of the buyes valuations. The fist best solution is what would happen if the monopolist was able to use fist degee pice discimination. We have elegated the fomal teatment to the appendix and instead simply pesent the qualitative featues of the solution hee. The initial state is x, 1) and m =. The monopolist has to decide thee things: when to sell to all types, when to sell only to high types and when to exit. Selling to all types means faste leaning than selling only to high types, but can be moe costly if belief is low. Futhemoe, the monopolist has option value fom waiting - absent bad news she can sell to buyes at a bette pice late. The monopolist also has to decide when to exit, which happens when the value of staying in the maket is zeo. In essence, then the maginal cost of poducing the good is too high compaed to the belief that the poduct is high quality. Figue 1 illustates the fist best solution as a function of the belief. Recall that absent news the belief is inceasing ove time unde bad news so we ae moving fom 9

10 E H A x L x U x ) x Figue 1: Fist best solution unde bad news as a function of the belief x left to ight ove time in the figue. When the belief is too low, below a cutoff x L, the monopolist exits the maket. We denote this inteval with E. Between x L and x U x ) the monopolist only sells to high types and accumulates low types. We denote this inteval with H. The uppe cutoff, x U x ), depends on the initial belief, x, because it tells us how many low types the monopolist has accumulated when she eaches the uppe cutoff. Above the uppe cutoff x U x ) the monopolist sells to all types. This is inteval A in the figue. 3.2 Commitment solution The next task is to deive the commitment solution to the monopolist s poblem unde bad news. The initial state is again x, ). In the commitment solution the monopolist only makes one decision, at t =, to which path of pices she commits to. With full commitment powe the monopolist can manage buyes expectations. This is impotant fo he because the pice today eflects futue oppotunities to buy the poduct. The incentive to manage buyes expectations is vey stong as becomes clea fom the next esult. Poposition 1 No sales unde commitment and bad news). The monopolist neve sells to all types if she has peviously sold only to high types. Poof: Appendix B. We omit the poof hee and efe the eade to the appendix. The main idea is simply that afte selling only to high types, the monopolist would lose too much in the pice fo it eve to be optimal to sell at the low types valuation. That is, the value of waiting fo high types inceases so much that it does not payoff fo the monopolist to hold a sale. Impotant fo this esult is that the buyes and monopolist discount the futue with same ate. The esult is vey impotant. It implies that in the commitment solution the monopolist neve sells to the mass of waiting buyes and that the buyes value of waiting is always zeo. It also simplifies the analysis consideably, as we can wite the monopolist s optimal policy to be dependent only on the initial belief likelihood) and time. One way of intepeting poposition 1 is that it always pays off fo the monopolist to manage high valuation buyes expectations so that they do not ean positive ents. This esult is in line with what Stokey 1979) and Conlisk 1984) find in the complete infomation model: selling to low types cannibalizes sales to high types and so is neve pofitable. Intoducing leaning into the poblem does not fundamentally change this tade-off, since the monopolist and the buyes have identical beliefs about the state of the wold. 1

11 Poposition 1 only allows two possibilities fo the monopolist s stategy: i) eithe the monopolist only sells to high types o ii) she fist sells to all types until an uppe cutoff and then switches to selling only to high types. We can theefoe wite the monopolist s poblem as TA V x ) = max e t +λa1 x))ds λ A θ L x c)dt T A + e T A +λ A 1 x))ds e t +λ H1 x))ds λ H θ H x c)dt, whee T A is when the monopolist stops selling to low types. We have also substituted in the pices θ L x t and θ H x t fom selling to all types and high types espectively. We simplify the discounting tem hee by using the law of motion fo the belief: λ1 x) = ẋ/x, which gives that the discounting tem is e t x /x). Thus we can ewite the monopolist s poblem with the help of the invese likelihood as TA V x ) = max x e t λ A θ L 1 + q 1 )c)dt T A + e T A x e t λ H θ H 1 + q 1 )c)dt, whee q 1 = q 1 e Qt) is the invese likelihood ove which we ae integating. The monopolist has to make two decisions: i) when to stat selling and ii) how long to sell to all types befoe selling to high types, which is given by the deadline T A. We can solve this poblem using backwad induction. Lemma 2. The monopolist neve sells to all types. Poof. Taking the fist ode condition with egads to T A yields x e T A λ A θ L c) cq 1 T A ) λ H θ H c) + λ Hc λ H λ H + q 1 T A + λ A λ H + cq 1 T A ), whee q 1 = 1 x)/x is the invese likelihood. Ignoing the common multiplie we can simplify the above to H λ H + cq 1 T A < Since this is always negative, it is optimal to set T A =. Unde bad news, the value of leaning fom low type buyes is neve geat enough to offset the loss in evenue. Lemma 2 implies that unde bad news the commitment solution is emakably simple: the monopolist sells only to high types o exits immediately. We still need to solve, when the monopolist wants to sell the poduct. This happens when the monopolist is indiffeent between selling and exiting, i.e. the value staying in the maket equals the outside option. We expess this decision with the help of the likelihood q. 11

12 Lemma 3. The monopolist sells the poduct if q q L, whee q L is q L = λ H c λ H + ) λ Hθ H c) Poof. Solving fo V x) and letting it equal zeo yields x L λ Hθ H c) 1 x L ) λ Hc λ H + = Dividing with 1 x L and solving fo q L = x L /1 x L ) yields the expession in the lemma. Since the value is linealy) inceasing in x it is stictly less than zeo fo all q less than q L and stictly geate than zeo fo all q geate than q L. We can now summaize the monopolists optimal selling stategy, when she has commitment powe and the leaning envionment is bad news. Poposition 2 Commitment solution unde bad news). The monopolist sells to high types fo q q L and othewise exits the maket. Poof. The poposition follows diectly fom poposition 1, lemma 2 and lemma 3. In tems of quantities, the commitment solution unde bad news is identical to the solution without leaning Conlisk 1984) except fo the enty and exit decision. This is supising, because the tade-offs the monopolist faces ae clealy diffeent. The esult eally stems the way beliefs affect discounting: thee is no clea gain to sell at a highe ate, because the monopolist does not expect the belief to change. That is, the monopolist expects bad news at a ate that makes he not willing to incease the ate of sales. Futhemoe, the monopolist neve wants to sell to the accumulated mass of low type buyes, because that would mean lowe pices befoehand. Managing buyes expectations tumps possible benefits of leaning. In tems of pices, the solution poduces a pice path that is inceasing ove time. This is of couse diffeent to what we get without leaning, but it just eflects the inceasing belief. Othe than keeping the maginal buye indiffeent, the monopolist s policy plays no ole hee. We will late see that this is diffeent to what we get unde good news. 3.3 Equilibium analysis We now look how the maket outcome changes if the monopolist does not have commitment powe. We estict to Makovian stategies and look fo Makov pefect equilibia of the game. As in the commitment case, we ae inteested in the monopolist s poblem stating fom m = and x, 1). Recall that the equilibium of the complete infomation game when the poduct is known to be high quality is that the monopolist sells to all types Sobel 1991). Unde bad news beliefs dift upwads absent news so we get close and close to the complete infomation game. This inspies us to make the following guess fo the equilibium 12

13 stuctue: fo low beliefs the monopolist sells only to high types and fo high beliefs she sells to all types. Supposing ou intial belief is low enough, we can wite the value fo the monopolist as details in the appendix) V H x, m ) =x TH e t λ H θ H W 1 + q 1 )c)dt + e T x x T V A x TH, m TH ), 1) whee we have simplified discounting using the law of motion, T H is the time left at which the monopolist stats selling to all types, W is the value of waiting fo high type buyes and V A x, m) is the value fom selling to all types. Note that until t = T H, the monopolist accumulates low types at ate λ L. We can wite V A x, m) as mt V A x t, m t ) =x t θ L 1 + q 1 )c)ds + x t e t λ A θ L 1 + q 1 )c)dt. 2) Hee, the monopolist fist sells to the mass of waiting buyes as expessed by the fist integal and then sells to all aiving buyes until bad news aives as expessed by the second integal. We have witten value fom the sale as an integal ove vitual time so that the monopolist has the ability to adjust pices accoding to news. That is, we can think of the sale happening vey quickly instead at one instance. Unde bad news, this assumption inceases the value fom the sale, but plays no ole unde good news. Given the monopolist s value we can poceed to constucting the equilibia. While we have witten the value in tems of the deadline T H, we can invet this back to the state x, m). Doing this esults in two cutoffs: x L m) below which the monopolist exits and x U m) above which the monopolist sells to all types. Between x L m) and x U m) the monopolist sells only to high types. This gives us the following egions in the state space: H :={x, m) [, 1] R + : x L m) x x U m)} A :={x, m) [, 1] R + : x U m) < x} E :={x, m) [, 1] R + : x x L m)} Buyes expectations Expectations about futue pices ae impotant fo the behavio of the buyes. They ae an integal pat of the equilibium chaacteization. To get an intuitive undestanding how expectations wok in ou model, suppose the following. The buyes expect the pice tomoow to be low and that thee is no discounting between tomoow and today. At what pice ae buyes willing to buy today? The pice today must also be low, because othewise the buyes might as well wait fo tomoow and buy the good at a low pice. This self-fulfilling aspect of expectations is a cental pat of the equilibium analysis. We now define the buyes expectations so that they ae consistent with the cutoff equilibia we want to build. Fo all x, m) A, the buyes believe that the pice will be low. That is p t = θ L x, the low type s valuation. Similaly fo all x, m) H, the buyes believe that the pice will be high: the high type s valuation minus his value of waiting. The value of waiting 13

14 fo high type buyes depends on the time they have to wait fo the low pice: W x t, m t ) = x t e T t) θ H θ L ), whee T is the time at which the monopolist stats selling to both types. Note that it will depend on m. The pice in H egion is simply p t = θ H x W x, m). We now tun to constucting two diffeent cutoff equilibia. Fist we constuct the minimal cutoff equilibium, which is such that the buyes expect low pices to the point at which the monopolist would athe exit the maket than to sell at a low pice. Second, we constuct the maximal cutoff equilibium in which the buyes expect high pices to the point at which it is no longe incentive compatible fo the monopolist to sell only to high types Minimal cutoff equilibium The minimal cutoff equilibium is such that the buyes expect low pices until the monopolist s exit is a cedible theat, i.e. best esponse to the buyes decision not buy at high pices. This is detemined by a cutoff at which the monopolist is indiffeent between staying in the maket o exiting. We can define this cutoff, say xm), as the one at which monopolist s value fom selling to all types is zeo, V A x, m) =. This condition then yields the next esult. Lemma 4. If buyes expect low pices fo all x, m) such that V A x, m), thee cannot be an equilibium whee the monopolist fist sells to high types and then to both types. Poof. Assume in the contay that the monopolist sells only to high types befoe the cutoff xm) : V A xm), m) =. Fom 2), it follows that the flow benefit is stictly inceasing in sales. Since the total value is zeo at the cutoff, the flow benefit must be negative at the cutoffs and we get xθ L c <. Since V A xm), m) =, the monopolist s value in H egion is T x e t λ H θ H e T t) θ H θ L ) 1 + q 1 )c)dt. The integand is λ H xθ L c) at T, which is negative. The integand is a continuous function of time, and hence thee is a neighbohood below T whee it is always negative. This diectly implies that the integal, the monopolist s value, is negative in that neighbohood. Theefoe, she is bette off by staying out. To chaacteize the minimal cutoff equilibium we need to find the belief at which the monopolist is willing to stay in the maket while selling to all types. This happens at V A x, m). The solution takes a closed fom see the appendix fo details): xm) := cλ A + 1 e m )) cλ A + 1 e m )) + + λ A )θ L c)m + λ A ). 3) 14

15 When the game stats, the elevant condition is to evaluate the above at m =. The following lemma follows by plugging m = into the fist ode condition. Lemma 5. The monopolist sells to both types in the minimal cutoff equilibium, if the initial belief satisfies x c c + + λ A )θ L c). We still need to check that the monopolist does not a have pofitable deviation below xm). Lemma 4 is not enough to guaantee the existence of an equilibium whee the monopolist neve sells only to high types. She still might want to deviate by selling to high types below xm) given that she exits at xm). The deviation is not pofitable if the flow benefit fom high types is negative below the bounday. The necessay and sufficient condition fo this is θ H whee V 1 = λ Aθ L c). c c + + λ A )θ L c) c V 1 θ H θ L, Poposition 3 Minimal cutoff equilibium unde bad news). Suppose V 1 ) θ H θ L. Then thee is a minimal cutoff equilibium whee the monopolist sells to both types when x xm) and othewise exits the maket. Poof. We can veify that this is an equilibium by using the one step deviation pinciple. Fo the buyes, this is staightfowad since the pice is always low type s valuation and hence they ae indiffeent between buying and waiting. Theefoe they buy. Fo the monopolist, since the buyes expect a pice θ L x they ae not willing to buy at highe pices. Thus the monopolist faces a choice between selling at θ L x o exiting the maket. The cutoff point in the poposition is the lowest belief when the monopolist s value is positive at m = and above it is stictly positive so she pefes to sell. The assumption V 1 ) θ H θ L makes sue that the monopolist does not want to deviate below the cutoff Maximal cutoff equilibium We now poceed to solving the equilibium in which the pice expectations of the buyes ae detemined by the monopolist s incentive compatibility condition. That is, we solve the highest possible cutoff x U m) at which the monopolist is indiffeent between selling to the waiting mass of buyes and selling only to high types. Hee we will have both an uppe, x U m), and a lowe cutoff, x L m). The fist step is to analyze the uppe cutoff, which detemines the state at which the monopolist switches selling to both types. The monopolist s exit decision, the lowe cutoff, depends on the value of the whole game and hence the uppe cutoff. We can find the maximal cutoff x U m) at the point at which the monopolist cannot esist the temptation to sell to the waiting buyes even if the buyes expect high pices. We fomalize this intuition with the following lemma. 15

16 Lemma 6. In any Makov pefect cutoff equilibium, the monopolist stats selling to both types at the latest when the belief eaches x U m): x U c1 e m ) m) := mθ L c) + c1 e m, fo m >. ) x U m) := c fo m =. θ L Poof. The cutoff x U m) is the point at which the monopolist is indiffeent between selling to the waiting buyes now o slightly late. At that point the fist ode condition of V H with espect to T has to hold at T : λ H xθ L 1 + q 1 )c) V A λ H 1 x)v A + λ L V A m + λ H x1 x)v A x =. By plugging in the value function and its deivatives, the left hand side becomes λ H xθ L c) x θ L c)m + θ L c) λ A + c1 ) e m λ A + ) c1 e m ) λ A + ) + λ H 1 x)c1 e m λ A + ) + λ Lx θ L c1 e m ) λ A + ) ce m λ A + ) ) = x λ A θ L c) θ L c)m + λ A ) c1 e m ) + c1 e m ). Cutoff x U m) solves the fist ode condition. The fist ode condition is stictly deceasing in x fo all m > and hence positive fo all x below the cutoff implying that the monopolist is willing to wait. Notice that the cutoff is deceasing in m: lage the mass of the waiting buyes, the lowe the belief needs to be fo the monopolist to be willing to sell to them. It is exactly the condition if the diect payoff fom selling to m is positive. This is inteesting, because it means that the monopolist does not accumulate low types beyond the point she stats getting a positive flow pofit fom them. In a sense, this tells us that bad news does not delive much commitment powe to the monopolist: she cannot ean a positive pofit - except fo the leaning effect - fom the mass of buyes. This is vey simila to the complete infomation duable goods models, whee we get the Coase conjectue. In any Makov pefect cutoff equilibium, the monopolist sells to the waiting low type buyes wheneve she gets positive evenue fom the sales. We now tun to the poblem of detemining the lowe cutoff. At the maximal uppe cutoff, the monopolist s value is stictly positive. This implies that the value must also be positive just befoe the cutoff and thus we must have a egion whee the monopolist only sells to high types. The selle must be willing to ente at a belief lowe than x U m). Lemma 7. In the maximal cutoff equilibium, the lowe cutoff x L m) solves V H x L, m) =, whee the time at which the monopolist stats selling to both types is detemined by x U and the laws of motion. 16

17 While we cannot solve the lowe cutoff, x L m), in closed fom, we can show that it is below the maximal uppe cutoff, x U m) fo small m. The easiest way to see this is to compae the uppe cutoff with the enty condition in the minimal equilibium. The lowe cutoff in the maximal equilibium must be lowe than in the minimal equilibium. This is guaanteed by the optimality of T H - ou deadline until which we sell to high types. Because x U ) > x) and because x is deceasing, we know that thee is a H egion whee the pocess neve hits the lowe cutoff. Similaly to the minimal cutoff equilibium, the maximal cutoff equilibium exists only if the monopolist does not want to sell to high types fo lowe beliefs conditional on exiting at the lowe cutoff x L m). This is the same issue we an into with the minimal equilibium. Since the lowe cutoff is lowe in the maximal equilibium, the existence condition is easie to satisfy. The necessay and sufficient condition is x L )θ H c, whee x L ) comes fom lemma 6 and lemma 7. We can now chaacteize the maximal cutoff equilibium. Poposition 4 Maximal cutoff equilibium unde bad news). Suppose x L )θ H c. Then thee is a maximal cutoff equilibium, whee given m the monopolist sells to both types fo x x U m), she sells only to high types fo x L m) x x U m)), and exits fo x x L m). Poof. We veify that this is an equilibium again by using the one step deviation pinciple. Fo the buyes, in A low valuation buyes ae indiffeent and in H high valuation buyes ae indiffeent and thus both pefe to buy accoding to the equilibium stategy. Fo the monopolist, in A egion the buyes expect a pice θ L x and ae not willing to buy at highe pices and thus the monopolist faces a choice between selling at θ L x o exiting the maket. Since he value is stictly positive in A, the monopolist does not want to exit. In H egion, fom lemma 6 T is sequentially optimal) we get that the monopolist must pefe to sell to only to high types athe than to all types and as he value is positive she wants to stay in the maket. The assumption x L )θ H c guaantees that the monopolist does not want to deviate below the lowe cutoff. We make a note of a poblem hee that does not show up unde bad news but will be appaent when we conside the good news envionment: we need to check that afte cossing the uppe cutoff the monopolist does not want to switch he action again. Unde bad news this means selling only to high types again. This is taken cae of by the buyes expecting low pices afte the uppe cutoff. In fact, we can show that the monopolist heself does not want to switch independent of the expectations. Thus thee can only be one sale unde bad news. This will not be tue fo good news. Figue 2 illustates what happens in the maximal cutoff equilibium, when we have x < x. The uppe and lowe boundaies ae dawn with solid lines and the line with aows depicts the equilibium path fom some abitay initial belief x L < x < x U and m =. We stat fom H egion and accumulate low types until the belief hits the uppe bounday x U m). At that point the monopolist holds a sale and sells to the 17

18 m x L x U H A x x Figue 2: Maximal cutoff equilibium unde bad news accumulated mass of buyes. Afte that the monopolist finds heself in egion A and sells to all types until bad news aives. 4 Good news envionment 4.1 Fist best Similaly as unde bad news we stat ou good news analysis with fist best. Recall that the fist best is equivalent to the monopolist being able to use fist degee pice discimination so that she captues all suplus fom the buyes. We have elegated the details of the solution to the appendix and pesent only the qualitative featues hee. The initial state is again x, 1) and m =. We ae eally afte two things fom the monopolist s fist best solution: when the monopolist wants to stop selling to all types and when to exit fom the maket. These decisions have to balance the evenue with the value of leaning and the option value of postponing sales. E H A x L x ) x U x Figue 3: Fist best solution unde good news as a function of the belief x Figue 3 illustates the fist best solution as a function of the belief. Recall that absent news the belief is deceasing ove time unde good news. Thus we move fom ight to left ove time in the figue. Above the uppe cutoff x U the monopolist sells to all types. Between the lowe cutoff x L x ) and x U she sells only to high types. The lowe cutoff depends on how many low types the monopolist accumulates and so it depends on the initial belief. At x L x ) and below the monopolist exits the maket. 18

19 4.2 Commitment solution In this section, we deive the commitment solution to the monopolist s poblem unde good news. As befoe, the commitment solution maximizes the monopolist s value at t =. We stat with a vey simila esult to what we had unde bad news: the monopolist neve holds sales. Poposition 5 No sales unde commitment and good news). The monopolist neve sells to all types if she has peviously sold only to the high types. Futhemoe, the monopolist neve sells to all types at x = 1. Poof: Appendix E. This is identical to poposition 1 unde bad news. We omit the poof fo poposition 5. The intuition is exactly the same as unde bad news: afte selling only to high types the monopolist loses moe fom the high types in the pice than she gains fom the low types. Poposition 5 shows that the news envionment does not matte fo the monopolist s incentives to manage the high valuation buyes expectations: the optimal solution is such that the high types have no value in waiting. Similaly as unde bad news, poposition 5 means that if the monopolist is eve selling to all types she must do so in the beginning. Recall that unde good news beliefs ae deceasing absent news. We can thus wite the monopolist s poblem as two sequential stopping poblems V x ) = max T A TA + e T A e t +λ Ax)ds λ A θ L + V 1 )q 1 + q)c)dt TH +λ A x)ds max T H e t +λ Hx)ds λ H θ H + V 1 )q 1 + q)c)dt, whee V 1 = λ H θ H c)/ is the value when good news aive, T A is when the monopolist stops selling to all types and T A + T H is when she exits the maket. We simplify discounting by noting fom the law of motion that λx = ẋ/1 x), which gives that the discounting tem equals e t 1 x )/1 x). Rewiting the monopolist s poblem using the likelihood q we get V x ) =1 x ) max T A TA + e T A 1 x ) max T H e t λ A θ L + V 1 )q 1 + q)c)dt TH e t λ H θ H + V 1 )q 1 + q)c)dt, 4) We can solve the monopolist s poblem using backwad induction. That is, we fist solve fo how long the monopolist wants to sell only to the high types, T H. Afte this, we solve fo how long she sells to all types, T A. The next lemma establishes the optimal T H with the help of the likelihood q. Lemma 8. The likelihood at which the monopolist wants to exit the maket is q L c = θ H + V 1 c. Given q > q L and that selling to only high types is optimal, the exit deadline T H is given by qe λ HT H = q L. 19

20 Poof. The optimal T H is detemined by taking the fist ode condition of 4) with egads to T H 1 x )e T A e T H λ H θ H + V 1 )q L 1 + q L )c + f H )) =. Solving this fo q yields q L. The deadline T H is then detemined by using the law of motion q e λ AT A e λ HT H = q L. We have elegated the details to the appendix E. The exit likelihood is simply the point at which the flow cost, c, exceeds the flow benefit, xθ H + V 1 ). Now we move on to solve fo the uppe cutoff, whee the monopolist switches selling only to high types. Taking the fist ode condition of 4) with egads to T A solving fist the latte integal and using the envelope theoem) we get that 1 x )e T A λ A θ L + V 1 c)e λ AT A q c) 1 e T H )λ H c λ A + )1 e +λ H)T H )V 1 e λ AT A q ). Ignoing the common multiplie and futhe simplifying we get H + e +λ H)T H λ A V 1 )q λ A c 1 e T H )λ H c + e +λ H)T H λ H θ H c)q, 5) whee H = λ H θ H c) λ A θ L c). This is the condition that detemines whethe it is optimal fo the monopolist to sell to all types. If this is positive at T A =, then it is pofitable fo the monopolist to sell to all types in the beginning. Lemma 9. Given that 5) is positive fo q = q, we can define the likelihood at which the monopolist stops selling to all types, q U, as a solution to H + e +λ H)T H λ A V 1 )q λ A c 1 e T H )λ H c + e +λ H)T H λ H θ H c)q =, such that q U < q. Given q U the deadline T A is detemined by q e λ AT A = q U. Poof. The optimal stopping likelihood has to be such that the fist ode condition 5) equal is equal to zeo. Since T A we must have q U < q. The deadline T A then is detemined by the law of motion, which implies q e λ AT A = q U. Appendix E has the details. The fist ode condition 5) eflects two consideations fo the monopolist. Fist, the monopolist caes whethe the diect benefit of selling to all types, that is the makup and the value of leaning, is geate than fom selling to high types. Second, it also depends on the option value of selling to all types the monopolist fogoes if she sells only to high types. This is because she has to sell to all types in the beginning if at all due to poposition 5. 2