HIAS-E-96

Quality and Price Personalization under Customer Recognition:a Dynamic Monopoly Model with Contrasting Equilibria
Didier Laussel(a),(b),(c) , Ngo Van Long(d),(e) and Joana Resende(f )(a) Aix-Marseille University(b) CNRS(c) EHESS(d) Department of Economics, McGill University(e) Hitotsubashi Institute for Advaned Study, Hitotsubashi University(f ) Cef.up, Economics Department, University of Porto

February, 2020

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Quality and Price Personalization under Customer
Recognition: a Dynamic Monopoly Model with
Contrasting Equilibria
Didier Laussel!, Ngo Van Longyand Joana Resendez

19th February 2020

Abstract
We present a model of market hyper-segmentation, where a monopolist acquires
within a short time all information about the preferences of consumers who purchaseits vertically di§erentiated products. The Örm o§ers a new price/quality schedule
after each commitment period. Lower consumer types may have an incentive todelay their purchases until next period to obtain a better introductory o§er. The
monopolist counters this incentive by o§ering higher informational rents. Considering the dynamic game played by the monopolist and its customers, we Önd that
there is always a Markov perfect equilibrium (MPE) in which the Örm immediately sells the good to all customers, o§ering the Mussa-Rosen static equilibriumschedule to Örst time customers (and getting full commitment proÖts). However,if the commitment period between two o§ers is long enough, there is another MPE
with gradual market expansion. Contrary to the Coasian result for a durable-goodmonopoly, we Önd that in both equilibria the proÖt of the monopolist increases (andthe aggregate consumers surplus decreases) as the interval of commitment shrinks.
!

Aix-Marseille University (Aix-Marseille School of Economics), CNRS & EHESS.
Corresponding Author. Department of Economics, McGill University, and Hitotsubashi Institute forAdvanced Study, Hitotsubashi University. E-mail: ngo.long@mcgill.ca.
z
Cef.up, Economics Department, University of Porto. Email: jresende@fep.up.pt. This research has
been Önanced by project NORTE-01-0145-FEDER-028540 and POCI-01-0145-FEDER-006890.
y1

The model yields policy implications for regulations on collection and storage ofcustomers information.
JEL codes: L12; L15.
Key words: monopoly; product quality; customer information; intertemporal
price discrimination.

2

1

Introduction

In recent years, the rapid growth of new-generation digital technologies has been reshapingrms business models in many economic sectors, such as software, retail, energy, automobiles, fashion and apparels, media and entertainment, and so on. On the one hand,these technologies allow rms to collect and treat very large volumes of (big) data about
their customers, which enables them to get very accurate information on consumers true
willingness to pay for their products. On the other hand, new-generation digital technologies (including machine learning, robotics, digital platforms, 3D printing, etc) also enable
rms to engage in innovative personalization strategies, through which they tailor theirproduct/price o¤ers to suit consumers individual preferences and needs, which the rms
uncover within a short time.
The ability to process unprecedented amounts of real time data about their customers indeed permits rms achieve what is now known as hyper-segmentation of markets,
increasing the scope of price and quality discrimination.1 As a result, in recent years,
many

rms are committing themselves to deliver personalized product-price o¤ers totheir customers at scale.2 Among a plethora of examples of this new trend, one can citeo¤ers of personalized clothing items (e.g. Nike, Adidas, Longchamp, Gucci, Louis Vuitton
and other luxury brands), the customizable Tesla dashboard, personalized TV scheduling (through on-demand video systems), personalized tourism experiences, customizable
executive education programmes, and so on...
In this paper, we investigate the market dynamics that arise when a monopoly rm
is able to: (i) collect Big Data about its customers (after their rst purchase) and (ii)engage in hyper-segmentation strategies, making personalized quality-price o¤ers to its
returning customers. More precisely, we propose a dynamic extension of the static modelof Mussa-Rosen (1978) on monopoly and product quality, extending the seminal Mussa1

As Zinsmeister (2016) puts it, both prioritization and personalization are key when it comes toyour hyper-segmentation strategy. Rather than creating a few rigid personas or a large list based onbroad rmographic characteristics, hyper-segmentation allows you to use all of your customer data to
pinpoint speci c marketing problems you can solve for smaller customer groups, aka a segment ofone. https://martechtoday.com/achieving-hyper-segmentation-reach-personalization-scale-165097 [Access date: 10/02/2019]2
As reported in the New York Times (2018), For some brands, a future of made-to-order everything is
on its way to becoming a reality. Nike, for instance, is experimenting with new customization and personalization concepts in its stores, where consumers can help to create products, said John R. Hoke III, Nikeschief design o¢cer. https://www.nytimes.com/2018/11/19/fashion/luxury-retail-personalization.html[Access date: 10/02/2019]3

Rosen model in order to account for the consumers repeated purchases of a non-durable
good for instantaneous consumption and the monopolists gradual information collection
on the preferences of its set of heterogeneous customers. We assume that the monopolist
cannot modify the price-quality schedules o¤ered to its customers during an exogenous
period of nite length, called the commitment period (which corresponds to the usual
contractual commitment period in dynamic models investigating the Coasian conjecture).
While in our formal model, for expositional reasons, the length of the commitment period
is treated as a parameter, as we shall show, the monopolist has an incentive to makethis length as close as possible to an irreducible minimum consistent with the state of
information technology and the legal requirements that limit rms ability to collect andstore consumers information. This result allows us to discuss the policy implications ofour model.3
At the beginning of each commitment period, the monopolist proposes to new customers a period-speci c price-quality schedule intended to induce them to reveal their true
type. Once a consumer has made her rst purchase, the rm is able to collect "Big Data"
to uncover her exact type, so that any possible misreporting (of consumers willingness
to pay for the good) has no consequence beyond the period of the rst purchase.4
In other words, quality-price menus and data collection constitute complementary tools
to implement personalization strategies within our model: the rm relies on quality-pricemenus to screen new customers, and as soon as they have made their rst purchase, it exploits Big Data to uncover customers true willingness to pay. This allows the monopolist
to o¤er to each returning customer the personalized quality-price deal that maximizes her
potential surplus (which is fully extracted by the rm).
Thanks to accurate data collection (after the rst purchase), former customers will be
unable to misreport their type in subsequent periods.5 This means that consumers cancheat and misreport their types only in the period in which they purchase the good for the3

See subsection 4.3.2 for details.
For example, in the context of digital markets, such information could be collected through cookies
and other tracking devices that allow rms to recover the consumers digital footprint.
5
By complementing price-quality menus with Big Data, the rm does not need to account for theold customers incentive compatibility constraints in subsequent periods since their true preferences havebecome known after their rst purchase. (If this were not the case, our dynamic game would be much more
complex since we would have to account for the incentive compatibility constraints of former customers).
Of course, in other contexts, data collection and the provision of personalized quality-price menus canbe regarded as substitutes, given that both of them can be useful intruments that help the rm identify
consumer preferences.
4

4

rst time. As a result, only new consumers (who are buying the good for the rst time)
may earn information rents. Hence, the model combines second-degree discrimination
with respect to new customers (whose preferences are unknown to the monopolist) with
rst-degree discrimination on former customers (whose preferences have become fully
known to the monopolist after their rst purchase). Consequently, the monopolist only
needs to distort downward the quality o¤ered to the successive rungs of the customers
who buy the good for the rst time.6
Our main focus is on the case where the monopolist is unable to commit to future
sales decisions. Under this scenario, we characterize the Markov Perfect Equilibria of the
dynamic game played by the rm and its customer base (which consists of a continuum of
consumer types). The rm makes decisions on (a) the quality-price schedules to be o¤eredto new customers, and (b) personalized quality-price o¤ers to old customers. Knowingthat the rm will be able to track their true willingness to pay immediately after their
rst purchase, the targeted new consumers must decide on whether to make their rst
purchase now, or to delay it until the next period.
In a Markov Perfect Equilibrium, the monopolist quality/price strategy maximizes its
discounted lifetime pro t, given consumers expectations; and, given the monopolists sales
strategy, consumers expectations are rational. Our rst result for this model is that there
exists a Markov-Perfect Equilibrium (which we will refer to as Equilibrium D from nowon) in which the monopolist simply o¤ers the static Mussa-Rosen equilibrium schedule in
the initial period and all customers purchase the good for the rst time in that period7 .
Afterwards, each consumer is o¤ered a take-it-or-leave-it personalized price/quality deal
which fully extracts her surplus (due to the rms data collection ability). At this Equilibrium D, the monopolist obtains the Mussa-Rosen pro ts in the initial period, whereas
it gets the maximum pro t of a rst-degree (perfectly discriminating) monopolist in all
subsequent periods. This explains why the monopolist chooses to serve all the consumers
in one go, o¤ering the lowest quality at zero price to a subset of (bunched) consumers:by selling them the good immediately in the rst period, the monopolist is able to collect data on their tastes, subsequently engaging in rst-degree price discrimination (and6

If we were to assume the absence of tracking devices and big data collection, misreporting would
impact the options o¤ered to old consumers as well. In that case, they would also be o¤ered distorted
quality levels in subsequent periods.
7
In the Mussa-Rosen static model, a subset of customers are pooled together and o¤ered a good atzero quality at the price s = 0. These customers are deemed to purchase the lowest quality good (zeroquality).

5

collecting the corresponding pro ts).8
Our second result is that there is generically another Markov-Perfect Equilibrium(which, from now on, we will refer to as Equilibrium E) with completely di¤erent properties. In Equilibrium E, successive rungs of customers purchase the good for the rst timein consecutive periods, meaning that the monopolist is expanding the market gradually(in small steps) from high-type to low-type consumers. While it remains true that once a
consumer has bought the good for the rst time, she will be o¤ered the personalized rstdegree price discriminatory deal in subsequent periods, in Equilibrium E, some customer
types end up delaying their rst-time purchase, which means that the monopolist must
wait to discover their exact preferences (which can only be uncovered after consumers
rst-time purchase period, which is up to their rational decision).
The intuition behind the multiplicity of MPEs in this model is similar to the one
underlying self-ful lling expectations equilibria. When all customers expect that all ofthem or almost everyone (except possibly a set of measure zero) will buy in the initialperiod, no low-type customer can hope to gain anything from unilaterally deviating bydelaying her rst purchase: she anticipates that there will be no new o¤ers to rst-timebuyers in any period in the future. Consequently, low-type customers do not require any
informational rent to induce them to buy in the rst period. Facing such expectations,the monopolist covers immediately the whole market. Thus, customers expectations are
ful lled.
In contrast, when consumers expect that the rm will not be able to sell the good in
the initial period to a subset of consumers with a positive measure, they contemplate thepossibility of gaining by delaying their rst purchase in order to take advantage of therms future attractive o¤ers to recruit new customers. Therefore, in any given period,
some set of consumers could be persuaded to become rst-time customers only if they areo¤ered strictly positive informational rents, and this applies even to the marginal customerof that subset. In turn, the consumers aspiration to get more informational rents makes
the rm unwilling to sell immediately to all new customers. Thus, in Equilibrium E,
consumers expectations are also ful lled.9
8

To some extent the characteristics of this equilibrium mimic the so-called "freemium business model"
in which low-quality variants of the good are o¤ered, free of charges, and at the same time positivelypriced, high-quality versions of the good are available.
9
Notice that this argument does not work when the commitment period between two o¤ers is too
short since, in that case, the monopolists cost of o¤ering marginal customers rents over such a short6

Interestingly, the model exhibits non-Coasian dynamics. We nd that as the lengthof the commitment period shrinks, for each of the two equilibria, the rms aggregate
pro t increases, contrary to the well-known vanishing pro t result obtained in thestandard Coasian model of durable good monopoly. The intuition behind our result is
as follows. The monopolists discounted stream of pro ts is higher, the sooner it hassu¢cient information and ability to practice rst-degree discrimination over consumers.
An almost immediate consequence of this result is that the discounted stream of aggregate
consumers surplus falls when the monopolists length of commitment period decreases.
Moreover we show that, when the length of the commitment period falls below a thresholdvalue, Equilibrium E (which is less favorable to the rm and more favorable to customers
relative to Equilibrium D) disappears, and only Equilibrium D remains. Accordingly, ouranalysis suggests that rms and customers interests are diametrically opposed regarding
public policies aiming at reducing rms ability to collect information on consumers (e.g.,
the EUs General Data Protection Regulation (GDPR) policy or the California Consumer
Privacy Act of 201810 ). Such policies tend to increase the length of time needed tocompletely identify consumers tastes, thus resulting in greater consumer surplus andlower monopoly pro t. We shall argue that our model may provide some hints as towhy, relative to the USA, the European Union is more inclined to institute and enforce
restrictive data protection laws which tend to hurt rms and bene t consumers.

2

Related literature

In recent years, the analysis of rms (price) personalization based on consumer recognition has attracted a great deal of attention among scholars and professionals in the eldsof economics and management. In particular, the recent literature on behavior- based
price discrimination has investigated, using mostly two-period models, the implications
period is not su¢ciently important to outweight the monopolists prospective gain from obtaining perfect
information on all customers for immediate rst-degree price discrimination.
10
Several states (in cluding Alabama, Arizona, California, Colorado, Iowa, Louisana, Nebraska, Nevada, Oregon, South Carolina, South Dakota, Vermont, Virginia) have recently passed
privacy laws.
See Serrato, Cwalina, Rudawski, Coughlin and Fardelmann (2018) for details.
https://www.dataprotectionreport.com/2018/07/u-s-states-pass-data-protection-laws-on-theheels-of-the-gdpr/.

7

of endogenous acquisition of information on consumers using their purchase history.11(For a survey see Fudenberg and Villas-Boas (2006)). For example, the pioneering workby Fudenberg and Tirole (2000) has focused on the simple case of a two-period duopoly
model with horizontal product di¤erentiation, where in the second period, rms are able
to engage in (third degree) price discrimination between old and new customers (depending on whether these consumers belong to the rms own turf or to the rivals one). Morerecent papers (see for instance Choe et al., 2017, or Laussel and Resende, 2019) havegone further by assuming that rms data collection capabilities (e.g. cookies and othertracking devices) allow them to identify the consumers exact preferences (i.e., rms know
the each returning customers exact location on the Hotelling line).
In these models, the possibility of price personalization ends up hurting rms pro t
because it intensi es duopolistic competition. In contrast, when targetability is imperfect(which amounts to say that rst degree price discrimination is not possible), the literature has pointed out that rms may actually bene t from price discrimination basedon customer recognition. In such cases, Chen et al. (2001, p. 25) show that individual
marketing with imperfect targetability can lead to a win-win outcome if rms exchangewith each other information about their customers.12
The present paper also focuses on how rms price (and product speci cation)
strategies may be reshaped by the possibility to implement price and product personalization strategies. We depart from previous literature by looking at price discrimination
through customer recognition in a fully dynamic set-up in which a monopoly rm makes
hyper-segmented quality speci cation o¤ers to its customers.
In building our model, we draw from two distinct lines of literature.13 The rst stream
addresses the issue of the losses that arise from an agents inability to commit (which may
not be related to the issue of private information). The second stream deals with private
11

For the conditioning of prices on purchase history see for instance Acquisti and Varian (2005), Chen
(2005) or Chen and Iyer (2002).
12
The reason why imperfect targetability may lead to a win-win outcome in Chen et al. (2001) isthat their model allows for misperception. They wrote that due to imperfect targetability, each rm
mistakenly perceives some price-sensitive switchers as price-insensitive loyal customers and charge them
all a high price. These misperceptions thus allow its competitors to acquire those mistargeted customers
without lowering their prices and hence, reduces the rival rms incentive to cut prices. This e¤ect softens
price competition (p. 23).
13
From a methological point of view, our paper is also related to the literature on Markov-perfectequilibrium in games involving rms with market power that interact with in nitely-lived consumers who
have Markovian rational expectations (see, e.g., Driskill and McCa¤erty, 2001; Laussel et al. 2004). For
a recent survey of this literature, see Long (2015).

8

information issues and the resulting informational rents: consumers are heterogeneous interms of their individual values represented by a preference parameter, which are private
information until the moment they have completed their rst purchase.
The rst stream of literature mentioned above has its origin from Coase (1972). Assuming that consumers have rational expectations, Coase (1972) argued that, in continuous time, if a monopolist selling a durable good is not able to make a commitment about
its future prices and outputs, it will lose all its monopoly power: in equilibrium, the pricemust be equal to the constant marginal cost, and the monopolist must serve all its customers immediately in one go. This conjecture has been subsequently proved rigorously.
In particular, using a model with heterogeneous valuations where each consumer buys
at most one unit, Bulow (1982) con rms this result for the "No Gap" case, de ned as
the situation in which the constant marginal cost is higher than the willingness to pay
for the good of the (unserved) consumer with the lowest valuation. There are a number of exceptions (see, e.g. Kahn, 1987, for non-constant marginal cost, Karp, 1996, fordurable goods subject to depreciation, and Mason, 2000 or Laussel, Long and Resende,
2015 for the case where the durable good generates consumption network e¤ects). In thecontext of non-renewable resource markets, various authors have shown that the lack ofability to commit may reduce monopoly pro t, or make monopoly power disadvantageous (Kemp and Long, 1980; Maskin and Newbery, 1990). Similarly, the macroeconomic
literature is abound with the disadvantages of governmental discretion as compared torule (Kydland and Prescott, 1977). The literature addressing the conditions leading
to Coasian dynamics, has highlighted the role of the length of the period of commitment
between any two successive contract o¤ers. We assume the rm can make commitment
over a short time interval, and then we take the limit as the time interval goes to zero.
This modelling feature is related to the earlier literature on the length of commitment
period (Reinganum and Stokey, 1985; Bond and Samuelson, 1987; Mason, 2000).
The second stream of literature, which deals with informational rents when there
is a continuum of types, arguably received the greatest impetus from Mirrlees (1971),
who analyzed optimal income tax under asymmetric information. This was followed bya formal analysis of the revelation principle (e.g., Holmstrom, 1977, Myerson, 1979).
This principle has been applied to models of regulation (La¤ont and Tirole,1986) andmodels of incentive contracts (see, for example, La¤ont and Tirole, 1988, La¤ont andMartimort, 2002). The theory has been extended to the multi-dimensional case (e.g.,9Martimort, 2006), as well as the multi-period case (e.g., the extension of Mirrlees model
to a multi-period setting, as in Kocherlakota, 2005; Golosov et al., 2003, 2006, 2007,2016a,b; Stantcheva, 2017).
By combining these two streams of literature, we are able to shed light on rms incentives to o¤er personalized quality-price schedules to successively new sets of customers,
deriving novel results concerning the market dynamics and the magnitude of consumers
informational rents in the context of markets for non-durable goods where rms are able
to gather accurate information on their customers preferences. Our results enrich thescarce literature that investigates dynamic monopoly models with multiple varieties. To
the best of our knowledge, the contributions that account for the interplay between the
Coasian conjecture and the private information issues due to asymmetric information on
consumers exact willingness to pay for the good (à la Mussa-Rosen, 1978) are relatively
scarce and they are mostly focused in the case of durable goods.14 For example, theinteresting contribution by Nava and Schiraldi (2019) looks at a dynamic game where themonopolist o¤ers multiple varieties. Considering the case of durable goods, the authorsnd that in any perfect Bayesian equilibrium of the dynamic game: (1) there is skimming, as the measure of buyers in the market at any point in time is a truncation of theoriginal measure; and (2) the market clears instantaneously whenever the seller sets static
market-clearing prices. Our set-up di¤ers from the Nava and Schiraldi (2019) framework
since we focus on non-durable goods pricing and consumption, under the possibility ofaccurate data collection.
Board and Pycia (2014) also look at a dynamic monopoly with durable-goods but
they consider that buyers have an outside option they may exercise each period. Such
possibility leaves low-type consumers out of the market as the authorsnd that themodel displays a unique equilibrium in which the rm charges the monopolist price every
period. Other recent papers looking at dynamic monopoly set-ups with multiple varieties
include the works of Inderst (2008), Takeyama (2002) and Kumar (2006). In particular,
the latter also looks at a monopoly set-up with durable goods, where the rm may vary
product quality over time (also engaging in inter-temporal price discrimination). In thisframework, there are equilibria where prices and qualities are non-monotone with respect
14

In a static set-up, the issue of optimal product design has been widely studied, following the seminal
contribution by Mussa and Rosen (1978). Recent static contributions on this include Deneckere and
McAfee (1996) and Johnson and Myatt (2002). In competitive set-ups, a recent contribution related tothe problem of optimal product design is Johnson and Myatt (2018).

10

to time and randomization may take place.
As mentioned earlier, all these papers focus on the case of durable goods. The recent
contribution by Maestri (2017) constitutes an exception. In line with the present paper,
the author also looks at the case of non-durable goods. However, di¤erently from us,
Maestri (2017) assumes that the monopolist does not have access to tracking technologies that allow the rm to uncover the consumers true willingness to pay. The author
concludes that there is a bound on the quality distortion of low types (and also on thehigh types rent). When the commitment period vanishes, the monopolist o¤ers menus
to quickly separate types, which allows the author to show that a version of the Coase
conjecture holds for the nondurable goods monopoly model. Our paper complements thisanalysis by looking how results are a¤ected by the possibility of collecting accurate dataon consumers preferences after the rst purchase in a set-up without commitment.
An important result we obtain is that the game has multiple equilibria. In one of thoseequilibria (the Markov Perfect Equilibrium E), we nd that price-quality o¤ers vary not
only across individuals (due to di¤erences in their type-speci c taste parameter) but, fora given individual, they also di¤er from the rst-period to the subsequent periods. In
particular, after their rst purchase, individuals bene t from a quality upgrade. However,consumers end up worse-o¤ as the additional surplus generated by such quality upgrade
is extracted by the rm, given its ability to implement rst-degree price discrimination.

3

The Model

A monopolist produces a continuum of varieties of a non-durable good. The varieties
di¤er in terms of their quality level, which we denote by t. The set of feasible quality
levels is the real interval [0> tmax ] where 0 is the lowest quality, and tmax is the highest
feasible quality. The unit cost of each variety depends on its quality level. The unit costis denoted by f(t).
Assumption A1: f(0) = 0> f0 (t) ¸ 0> f0 (0) = 0, and f00 (t) A 0 for all t A 0.
There is a continuum of consumer types. Let  be a variable denoting the type of a
£ ¤consumer. Assume that  is distributed over a closed interval >  , where  A  ¸ 0.
The density function and the cumulative distribution function are denoted by i() and

11

£ ¤
I () respectively, where i () A 0 for all  2 >  . Let k() denote the inverse hazard()rate, k() = 1¡I.
i ()Assumption A2: k0 () ? 0=
Assumption A2 is a standard assumption to ensure that there is no bunching in a(static) second-degree discrimination model. This assumption is satis ed by many familiar distributions, such as the uniform, exponential, normal, binomial, and Poisson
distributions.15 However, as we show below, bunching will occur in our dynamic model,
because the participation constraint for new customers in any period must take accountof the fact that they can delay their rst purchase until next period.
Assumption A3: k() A =
For the uniform distribution case, A3 is equivalent to 2 ? . Taken together, Assumptions A2 and A3 ensure that, in a static second-degree discrimination framework,
the market is not covered. More precisely, there must exist a cut-o¤ type f A , de ned
by f ¡ k(f ) = 0 such that all types  in the interval [>  f ] are not served by the rm.

Herein, we investigate the monopolists optimal quality-price decisions within an in nite horizon framework with customer recognition. Time is a continuous variable, w 2 [0> 1)

and a consumer at each time w buys either one or zero units of the good. Consumers live
forever and the good is non-durable: it is consumed instantaneously, yielding the individual a net instantaneous utility equal to:x(tw ()> sw ()> ) = tw() ¡ sw ()=We refer to  as the individuals marginal valuation of quality. Each individuals type isconstant over time.16 The consumers type is initially unknown to the rm. However,as soon as a consumer makes her rst purchase, the rm can use tracking devices (e.g.
cookies in the case of on-line shoppers, or delity cards in the case of more conventional
brick-and-mortar stores) to collect data about the consumers. Such data can then be¡¢
For the uniform distribution, I () = ( ¡ )@  ¡  and k() =  ¡ .
16
In our model, goods sold to customers who have the same  have the same characteristics. In reality,
customers who have the same valuation of quality may have di¤erent personal tastes (e.g., concerning
design, as in clothes). Extending the model to account for such di¤erences and for the cost of quality
personalization would be an interesting topic for future research.
15

12treated by powerful algorithms to uncover the relevant information for the monopolist toaccurately identify the consumers type.17
Assumption A4: f(tmax ) A tmax .
Assumption A4 implies the rm would never produce the good at quality level tmax ,because the unit cost f(tmax ) is higher than the highest-type consumers maximum willingness to pay for a unit of the good at that quality level.
To the assumption that consumption takes place instantaneously, we add the assumption that the monopolist cannot modify the price-quality schedules o¤ered to its customers
during a period of nite length ¢. For expositional simplicity, we treat ¢ as exogenous
in the model. This allows us to evaluate the welfare consequences of varying ¢. We willshow later that the rm has an incentive to bring ¢ to the minimum length which allows
it to collect the data necessary to discover the customers types. This minimum length,
denoted by ¢, is dependent on both (i) the state of information collection technology and(ii) on the strength of legislations on consumers data protection. For the moment, we
simply treat ¢ as contractual, which may a priori exceed ¢.
There is an in nite sequence of periods, indexed by q, where q = 0> 1> 2> 3> ===. The
rst period (denoted by q = 0) corresponds to the subset [0> ¢) of the time line. The
second period (q = 1) corresponds to the subset [¢> 2¢), and so on. At the beginningof each period q, the monopolist o¤ers to all potential new customers (those who havenot bought the good in some earlier period) a price-quality schedule, which applies for all
points of time in the interval [q¢> (q + 1)¢),
sq = !q (t)=where !q is a mapping from the quality domain [0> tmax ] to the space of non-negative
prices, R2+ = [0> 1). During each period (whose length ¢ is the time that elapses between

two di¤erent contract o¤ers) the price and the quality selected by a consumer at the17

The possibility of using tracking technologies to accurately uncover consumers preferences is a distinctive feature of our model vis-à-vis the previous literature based on the revelation principle. Herein theinformation on consumers true type is always uncovered after the consumers rst-purchase (regardlessof the action taken by the buyer). Di¤erently, in previous works (e.g., La¤ont and Tirole, 1986, 88) theseller learns the valuation of a buyer with whom he interacted in the past because her actions reveals
her private information. We are grateful to an anonymous reviewer for drawing our attention to thisimportant di¤erence between the two settings.

13

beginning of the period are contractually xed. Moreover, ¢ cannot be smaller than theminimum length of time it takes to the rm, given the current regulation on consumers
data protection, to collect and process the information about the consumers tastes inorder to get accurate information on the new consumers type (so that the rm canthen start implementing personalized quality-price o¤ers in the subsequent periods). As
consumers tastes are assumed to be unchanged during their life, the price-quality o¤ers
proposed by the rm to each customer will change only once in her lifetime (after the endof the period in which the customers made their rst purchase). After each commitment
period, the rm possesses information on the exact type of its old customers and each old
consumer always receives the same o¤er (which corresponds to the quality-price o¤er that
expropriates her entire potential surplus). Accordingly, the length of ¢ does not a¤ect
old customers, since they always receive the same contract o¤er. This is not the casefor new customers (who may get some informational rents during the rst commitment
period after their rst purchase). Taking this into consideration (and also the fact thatthe length of ¢ is not relevant for the case of old customers), one may argue that ¢ must
be equal to or greater than the length of time the monopolist takes to collect information
on customers after their rst purchase.
We assume positive discounting, such that, within each period, a dollar paid at the endof the period is worth only as much as a fraction h¡u¢ of a dollar paid at the beginningof the period, where u A 0 denotes the instantaneous discount rate (which is constantand independent of ¢). The same discounting applies to utility. Then a consumer oftype  who selects the quality tq () in period q and pays the price sq () ´ !q (tq ()) at
each w 2 [q¢> (q + 1)¢) achieves the following net utility (properly discounted) for theperiod:18

yq () ´Z ¢x(sq ()> tq ()> )h¡u0g = [tq () ¡ sq ()]Z ¢h¡u g0¢[tq () ¡ sq ()] ¡1¡
=1 ¡ h¡u¢ ´ [tq () ¡ sq ()]=uuNote that u is independent of ¢, whereas  is the discount factor between periods,
i.e.,  ´ h¡u¢ . Clearly  ! 1 when ¢ ! 0=18

De ning t ¡ s as the utility over period q would have the undesirable consequence that the utility
per unit of time would increase when the length ¢ of the period would decrease, tending to 1 as ¢ tends
to 0=

14

At the beginning of any period q (q = 0> 1> 2> ===), the rm faces two disjoint subsets of¡¤customers: those whose type belongs to q >  are customers who have already bought
the good in some previous periods (and therefore the monopolist is fully informed about

their preferences) and those whose types belong to [>  q ] have never purchased the good(and therefore they are potentially new consumers, whose preferences remain unknown
to the rm). Among the latter, the consumers whose types  belong to (q+1 > q ] > with
q+1 ? q will be induced to buy the good for the rst time in period q. Accordingly,
type q+1 consumers are called the marginal consumers in period q. In equilibrium they
are indi¤erent between (a) being a rst-time consumer in period q, at the bottom of therung, and (b) being a rst-time consumer in period q + 1, at the top of the rung.19 It is
not di¢cult to show that in equilibrium (i) if a type- consumer has already bought thegood, so have all type-0 customers such that 0 A > and that (ii) if a type- consumer
has not purchased the good, all type-0 customers such that 0 ?  also have not done so.
We assume that initially nobody has purchased the good, i.e., 0 = .
In the following section, we will look at the monopolists optimal price-quality decisions
in a dynamic set-up in which the monopolist is able to fully track consumers tastes after
their rst purchase (and uses this information to implement price/quality discrimination
based on consumer recognition). First, we consider the case where the monopolist cannot
commit to future contract o¤ers. Afterwards, we will compare equilibrium outcomes
described in the next section to the ones arising in a model where the monopolist fullycommits to a sequence of price/quality o¤ers.

4

Equilibrium under non-commitment

The monopolists inability to commit to future prices and qualities has two aspects. First,
when selling to rst-time consumers in period q, the rm cannot commit to o¤er them inperiods q + m (where m = 1> 2> 3===) the same contract as the initial one when they comeback to purchase the good. Moreover, in any period q, the monopolist cannot commit
to o¤er pre-determined contracts to new customers in periods q + m, m = 1> 2> 3==> eventhough such o¤ers could pro tably induce potential rst-time consumers in period q to
19

This implies that they obtain a higher surplus per unit consumed in period q + 1 (if they choose tobe rst-time consumers in period q + 1) than the surplus per unit consumed in period q (if they choose
to be rst-time consumers in period q), but this di¤erence is fully o¤set by the fact that the surplus isdelayed by ¢ units of time. See eq. (12) below.

15

purchase in that period, in preference to delaying their purchase until the next period orlater periods.
In addition, due to data collection by the rm, customers type becomes known after
their rst purchase so that the monopolist always uses rst-degree discrimination withrespect to old customers (this amounts to say that even if the monopolist could commit
to future contracts, it does not have any incentive to distort the quality o¤ered to former
customers).
Let us study the implications of these considerations.

4.1

Old customers: quality and price personalization

Contrary to the one-shot problem studied by Mussa and Rosen (1978), a consumer does
not purchase the good only once. At the beginning of period q> all consumers whose¡¤types belong to q >  have already purchased the good at least once in previous periods.
Because the rm maintains a dataset of former customers, when they return in subsequent
periods, the rm will not o¤er them contracts designed for new customers. Instead, it
will make them personalized price-quality o¤ers, extracting all their consumers surplus
t ¡ f(t). More precisely, the rm exclusively o¤ers to a type- returning customer agood with quality level t ¤() which equates the consumers marginal valuation of quality,
, with the marginal cost of quality upgrading, ¡ f0 (t¤ ()) = 0=(1)

and charges the corresponding surplus-eliminating price s¤() = t ¤()=
Under Assumption A1, t¤ () is uniquely de ned by (1). Since f0 (0) · > it is clearthat in any period q the monopolist earns a positive pro t from each former customer oftype  by making the take-it-or-leave-it o¤er (t ¤ ()> t¤ ()).

Let us denote by  I () the pro t that the monopolist obtains with a sale to a former
customer of type I () ´ t ¤() ¡ f(t ¤())>(2)where the superscript I stands for former customers. The aggregate pro t earned in¡¤period q from selling to all former customers (those whose types are in the interval q >  )is then given by

16

1¡
¦Iq =
u

4.2

Z 

 I ()i ()g=

(3)

q

New customers: screening mechanism with possible bunching

The type of new customers is unknown to the monopolist. Thus, in order to discriminate
among new customers, the monopolist uses a screening mechanism. At the beginningof period q, the targeted new customers are those whose type  belongs to the interval(q+1> q ]. We will refer to members of this group as vintage-q customers. By o¤ering a
price-quality schedule sq (t) for these rst-time consumers20 , it is indeed possible to inducethem to select a couple (tq ()> sq ()), provided that an incentive compatibility condition(ICC) is satis ed. The nature of the ICC is explained below.
Being rational, these rst-time consumers fully expect that the monopolist, in later
periods, will o¤er them a take-it-or-leave-it o¤er that extracts all their surplus. The
consumer knows that the rm will gather data about her tastes, identifying her true typeafter her rst purchase. Hence, any consumer who has reported a type b (regardless ofwhether it is equal to her true  or not) knows that, in all subsequent periods, her true 
will be known, and thus she will be proposed a quality t ¤() as de ned by (1) and a pricet¤ () such that the surplus of the consumer is fully extracted by the monopolist.
Truthful reporting is optimal for a type -customer (when she purchases the good forthe rst time, say in period q) if and only if pretending to be a di¤erent type (say b 6= )would make her worse o¤ than telling the truth:

´
1¡
1¡ ³
(tq () ¡ sq ()) ¸tq (b) ¡ sq (b
) >
u
u

where the right-hand side (RHS) term is the net utility earned in period q by misreportingones type (pretending to be of type b instead of type ). The left-hand side (LHS) is thenet utility earned in period q if the consumer is truthful. The above ICC condition may
be written as³´
Or, equivalently, a direct mechanism t(b)> s(b) depending on the type b which the consumer chooses
to report.
20

17 = arg max
e


´
1¡ ³
tq (b) ¡ sq (b
) =
u

(4)

Let us denote the net utility of a rst-time buyer of type  2 [q+1 > q ] in period q by
Xq () ´ max
e


´
1¡ ³
tq (b) ¡ sq (b
) =
u

(5)

The envelope theorem implies that the pair (tq ()> sq ()) is incentive compatible i¤
Xq0 () =1¡tq ()=u(6)

Accordingly, by integrating (6), we nd that for all  2 [ q+1 > q ], it holds that
Xq () = Xq (q+1 ) +

Z  µ
q+1

¶
1¡
tq (v) gv>
u(7)

where the integral on the RHS is the di¤erence between the informational rent of a type-
consumer, where q+1 ·  · q , over the informational rent of a marginal type q+1 , who
is indi¤erent between (a) being a rst-time customer in period q> at the bottom of therung, and (b) being a rst-time customer in period q + 1> at the top of the rung. For
later use, let us denote by U(q > q+1 ) the di¤erence between the informational rent of
rst-time customer at the top rung in period q and that of the rst-time customer at thebottom rung in period q:
Z q

1¡
tq (v)gv=
u
q+1

(8)

1¡
(tq () ¡ sq ()) =u(9)U(q > q+1 ) ´Equation (5) gives:
Xq () =Moreover, a familiar revealed preference argument yields the following inequality (second
incentive-compatibility constraint):1¡ 0
t () ¸ 0=
u q

(10)

This implies, via (6), that Xq () is convex in .
A pair (tq ()> sq ()) is incentive-feasible if, in addition to the incentive compatibility
18condition (6) (or, equivalently, (7)), it also satis es the following participation constraint(PC) for rst-time buyers in period q:·¸¢
1¡ ¡ ee
Xq () ¡ 
tq+1() ¡ sq+1() ¸ 0> 8 2 [q+1 > q ] >
(11)
u
¡ e
¢
where tq+1()> seq+1() denote the best price-quality pair that a type- consumer would
choose in period q + 1 if she delays her rst purchase to period q + 1=

To understand the participation constraint (11), note that, for  2 [q+1 > q ] > a type -

consumer that is targeted by the monopolist as a potential rst-time customer in period q
can deviate by delaying her rst purchase to the next period. The rm, wishing to prevent

such a deviation, must ensure that her net utility from buying as a rst-time customer inperiod q is greater than or equal to her net utility from buying as a rst-time customer inperiod q + 1. Since the monopolist is able to identify the true type of all those customers
who have made their rst purchase and will extract their whole surplus in subsequent
periods, the life-time net utility of a rst-time customer in period q is simply equal to
the net utility obtained in period q itself (the net utility from her purchases in periodsl A q being zero). The (discounted) life-time net utility obtained by a consumer of type
 2 [q+1 > q ] who delays her rst purchase to period q + 1 is then:·¸¢
1¡ ¡ ee
tq+1 () ¡ sq+1 () =uHence, such type- consumer will choose to buy in period q i¤ Xq () ¸
£
¡ e
¢¤
e
 1¡
t
()
¡
s
()
. The RHS of this inequality is the reservation-utility level
q+1
q+1
u

of the targeted type- consumer. It turns out that this reservation level can be easily
characterized. This is the essence of Claim 1 below.
Claim 1. A type- customer, where  2 [q+1> q ], when evaluating the merit ofdelaying her rst-time purchase to period q + 1 instead of buying in period q> will nd it
optimal (conditional on deviating) to report the highest type among consumers who buy
in period q + 1, i.e., b = q+1=
Proof.

See the Appendix.

Given Claim 1, the participation constraint (11) for any type  2 [ q+1 > q ] may be

written as19¢q () ´ Xq () ¡ µ1¡(tq+1 (q+1 ) ¡ sq+1 (q+1))
u

¶

¸ 0>

(12)

i.e., the price-quality schedule o¤ered to new customers in period q and the scheduleo¤ered to new customers in period q + 1 must be such that, for all  2 [q+1 > q ], buying
for the rst time in period q is at least as advantageous as doing so in period q + 1.
Since q+1 is de ned as the marginal type who is indi¤erent between buying for the
rst time at q and buying at q + 1> it holds that, condition (12), when evaluated at = q+1 , is satis ed with equality, i.e., we have ¢q (q+1 ) = 0= How can we ensure thatthe participation constraint (12) is satis ed also for all the inframarginal types,  A q+1?
Lemma 1 below provides a necessary and su¢cient condition.
Lemma 1. The participation constraint (12) is satis ed for all  2 [q+1 > q ] i¤,in addition to the condition that the marginal rst-time customer is indi¤erent, i.e.,¢q (q+1) = 0> the following inequality is met
tq (q+1) ¡ tq+1 (q+1 ) ¸ 0=

(13)

Proof. See the Appendix.
Condition (13) requires that the lowest quality o¤ered to new customers in period q
is greater than the (discounted) highest quality o¤ered to new customers in period q + 1.
Recall that tq+1(q+1 ) is the quality level intended for the highest consumer type
among those who purchase for the rst time in period q + 1= One may expect that thefamiliar property that there isno distortion at the top (which holds in the static setting)also applies in our dynamic model, so that tq+1(q+1 ) = t¤ (q+1 )> which turns out to be
the case in equilibrium, as it will be veri ed afterwards. When this is indeed the case,constraint (13) reduces to:tq (q+1) ¸ t¤ (q+1 )=(14)that is, the quality o¤ered to the lowest-type rst-time customer in period q is greaterthan the (discounted) rst-best quality for that type.
Finally, from (9) and (11), the condition ¢q (q+1 ) = 0 implies that
Xq (q+1) = Xq+1 (q+1)=
20

(15)

Equation (15) indicates that marginal consumers in period q are indi¤erent between
being the lowest type among all new consumers in period q and being the highest type
among all new consumers in period q + 1 (the higher surplus being exactly o¤set by
the discount factor). With the help of equation (7) evaluated at q + 1> this indi¤erence
condition implies the following dynamic relationship between the net utilities obtained by
the marginal rst-time customers in period q and period q + 1:µ¸ ¶Z q+1 ·1¡
Xq (q+1 ) =  Xq+1 (q+2 ) +tq+1 () g =
u
 q+2

(16)

In period q, the monopolists pro t from sales to new customers, ¦Q
q > is equal to
1¡
¦Q
q =
u

Z q

q+1

[sq () ¡ f(tq ())] i()g(17)where the superscript Q in ¦Q
q stands for new customers.
Re-arranging terms, we get
1¡
¦Q
q =
u

Z q

 q+1

f[tq () ¡ f(tq ())] ¡ [tq () ¡ sq ()]g i ()g=(18)

Using the de nition of Xq (), we can rewrite (18) as:
¦Q
q =

µ
¶
¾
Z q ½
1¡[(tq () ¡ f(tq ())]¡ Xq () i()g=
u
 q+1

(19)

Equation (19) shows that the pro ts obtained from sales to new customers in period q
is equal to the vintage q total virtual surplus (i.e., gross utility minus production costminus the life-time rent of all rst-time consumers of vintage q).
It is shown in the Appendix that the RHS of equation (19) is equivalent to
8 h³
´
i¡
¢ 9
Z q <I ( q )¡I ()1¡=¡tq () ¡ f(tq ())i ()ui()g=;q+1 :¡Xq (q+1 )(20)

For each  2 [q+1 > q ], let tqQ () denote the quality level that maximizes the term

21

inside the square brackets [===] in eq. (20), subject to the participation constraint thattq () ¸ t ¤(q+1 ) for all  2 [q+1 >  q ]It it is easy to see that tqQ () is given by

where tq# () is de ned by©
ªtqQ () = max t ¤(q+1 )> tq# ()¡µI (q ) ¡ I ()
i()

i.e.
¡1
tq# () = (f0 )¶¡ f0 (tq# ()) = 0

(21)

·
µ
¶¸
I (q ) ¡ I ()¡i()Using the speci cation that the distribution i() is uniform, and the cost function isquadratic, i.e.,

1f(t) = t 2
2

and
¡

µ

I ( q ) ¡ I ()i()¶= 2 ¡ q

we nd thattq# () = 2 ¡ q =Then, for a uniform distribution i () and a quadratic cost function, it follows that:tqQ () =(2 ¡ q
q+1

h
i
q+1
if 2 ¡ q ¸ q+1 i.e., if  2  q +
>

q
h
i 2
q + q+1
if  2 q+1 >2

We state this result as Lemma 2.
Lemma 2: A screening mechanism is o¤ered to customers with  2

h

(22)

q + q+1
> q
2

i

at

the top of the rung. Bunching occurs at the bottom of the rung as a consequence of themonopolists optimization under the participation constraint tq (q+1 ) ¸ t¤ (q+1 ): it is
optimal for the monopolist to o¤er these consumers the same low quality level, so as toreduce the informational rents granted to the higher types.
22

Remark 1: Note that the set½¾
 q + q+1
VW ´ j?  · q2
is non-empty, because q+1 ? q and 0 ?  ? 1. The set
VE ´½

 q + q+1
j q+1 ·  ·2¾

is a non-empty if and only if q+1 is much further to the left of q , such that
q+1 ?

q=(2 ¡ )(23)

If inequality (23) holds, we say that the monopolist expands the market by large steps.
If q+1 A q @(2 ¡ ), we say he expands the market in small steps. We will show in
Section 4.3.1 that under the uniform distribution and quadratic cost, in equilibrium it

is never optimal for the rm to expand the market in small steps given that consumers
expectations of future rents to marginal customers is quadratic.
Using Remark 1 and eq. 22, we derive Lemma 3, where we assume a quadratic quality
cost speci cation and a uniform distribution of consumers willingness to pay:
Lemma 3: Thefrom selling to the set of screened new customers - they
³ pro t obtainediare of types  2

q + q+1
> q
2

Z q

- over the period in which they are rst-time customers is q +q+1
2

8 h
9
i
< (2¡ q )2 ¡ 1¡ ¢ =u2:¡Xq ( q+1 )

1
g
;¡

and hthe pro t fromifrom selling to the set of pooled (i.e. bunched) new customers of typesq+1
 2 q+1 > q +
, if this set is non-empty, is2

Z q +q+1 ( £
2

 q+1

(2 ¡ q ) (q+1 ) ¡ 12 (q+1 )2¡Xq (q+1)¤ ¡ 1¡ ¢ )
u

1
g=
¡

(24)

Remark 2: Since I (q ) ¡ I () A 0 for all  A q , condition (21) implies that there23

is quality distortion for all rst-time customers except for type q . This con rms that thefamiliar property called no-distortion at the top in static models also applies to eachgroup of new customers in our dynamic model.
Remark 3: Given Assumptions A1 and A2, even when i () is not uniform, we still
can prove that tq# () is a strictly increasing function of  provided thati 0 ()@i () · i()@ [1 ¡ I (q )] =The latter condition ensures that higher-type consumers are o¤ered strictly higher
quality levels. To see this, rewrite (21) as ¡ k() +Then
1 ¡ k0 () ¡[1 ¡ I (q )]= f0 (tq# ())=i()[1 ¡ I (q )] i 0 ()gtq# ()
00 #
=
f
(t
())
>
q
(i())2g

where ¡k0 () ¸ 0> given Assumption A2.

#

It follows that gt g() A 0 provided thati 0 ()@i() · i()@ [1 ¡ I (q )].

4.3

Markov Perfect Equilibria

In this subsection we focus on the characterization of the Markov Perfect Equilibria (MPEfor short). In a MPE, the strategy of the monopolist is expressed as function of a statevariable, say £(q) (to be de ned), and consumers have a Markovian expectations function, denoted by ­(=), which is a function of the same state variable, such that (i) given
the consumers expectations function, starting from any (date, state) pair (q> £(q)), themonopolists strategy maximizes its pro t, and (ii) given the monopolists strategy, theconsumers expectations are rational, i.e., their expectations are correct. Let us now de ne
the state variable, the monopolists strategy, and the consumers expectation function.
In any period q, q = 0> 1> 2> 3===> let [(q) 2 [0> 1] denote the fraction of the total
population (the customer base) that has purchased the product prior to that period, with[(0) = 0. It may seem natural to use [(q) as the state variable. However, it turns
24

out to be more convenient to use as state variable £(q)>which corresponds to a monotone
decreasing transformation of [(q), being de ned as follows:£(q) ´ I ¡1 (1 ¡ [(q))>£ ¤where I (=) is the cumulative distribution of . Then £(0) =  ´ 0 , and £(q) 2 >  .

The rms Markovian strategy is a pair (#> ), that consists of two components: (a) aMarkovian cut-o¤ rule #(=), specifying, at the beginning of period q, given  q , what is the
cut-o¤ (on the distribution of consumers types) that de nes the targeted group of rsttime customers for period q; and (b) a Markovian quality-schedule rule (=), specifying
the monopolists type-dependent quality o¤ers to rst-time consumers in period q.
To be more precise, in each period q, the rms Markovian cut-o¤ rule is a function #(=)
that determines, for any currently-observed value £(q), a value £(q + 1) = #(£(q)) ·£(q), where #(=) is non-increasing and bounded below by . We interpret £(q + 1) =# (£(q)) as the lowest-type of rst-time consumers in period q that the rm intends toserve.
A quality-schedule rule is a mapping  (=) that determines for any given £(q) an
associated quality schedule tq (=j£(q) ), which is itself a function that assigns to each  2[£(q + 1)> £(q)] a value in [0> tmax ], where £(q + 1) = # (£(q)). We interpret tq (j£(q))
as the quality level o¤ered to a rst-time customer of type  2 [£(q + 1)> £(q)] in periodq, given that the value of the state variable is £(q).

On the consumers side, their Markovian expectations rule ­(=) predicts, given £(q),
the life-time rent of the marginal rst-time customer in period q, i.e., Xq (q+1 ). When
consumers have the ability to perfectly anticipate future market outcomes, the expectations function ­(=) must reect rational expectations, i.e.,­(£(q)) =

1
X
m=1

m

ÃZ ¤
£ (q+m) ·£¤ (q+m+1)¸ !
1¡tq+m (v) gv = Xq (q+1 )>u(25)

where f£¤ (=)g1
q is the path of the state variable £ induced by the strategic behavior of

the monopolist from period q, when the state variable takes the value £(q), and wheretq+m (v) = tq+m (vj£¤(q + m)), i.e., the quality schedule that the consumers expect to beo¤ered in period q + m is the same as the schedule that the monopolists equilibrium

25

strategy would select. Note that, due to (16), ­(£(q)) satis es the following equation:µ­(£(q)) =  ­(£(q + 1)) +Z  q+1 ·
 q+2

¸ ¶
1¡
tq+1 (v) gv =u(26)

A Markovian strategy ((=)> #(=)) chosen by the monopolist is called a best reply tothe consumer expectations function ­(=) if (a) it yields a sequence of schedules tq (=) andcut-o¤ values q+1 that maximize the monopolists expected pro ts from any starting(date, state) pair (q> £(q)), and (b) the rational expectations condition (25) is satis edby such a sequence.
In what follows, we analyze in more detail the rms optimal Markovian sales andquality strategies and we provide an analytical characterization of the consumers optimalMarkovian expectation rules, under the assumptions that the distribution of  is uniform
and the production cost is quadratic.
4.3.1

Equilibrium strategies under quadratic cost and uniform distribution

In order to obtain a closed form solution for the rms optimal strategy, let us now supposethat the distribution of types is uniform on [0> 1] 21 and that f(t) is quadratic: f(t) = 12 t2 .
Because of the quadratic cost, the rst best quality o¤ered to old customers of type ¤

is t () = . Then, using (2) and (3), the pro t obtained in period q from old customers
is:
1¡
¦Iq ´
u

Z 

I

 ()i()g =

q

µ

¶
1¡ 3
3
( ¡ £(q) ) =
6u

(27)

It follows that the Bellman equation for the monopolist (where we have set  = 1 and = 0) isY (£(q)) =max

tq (=)>£(q+1)((1¡)(1¡£(q)3 )+ L [£(q)> £(q + 1)> tq (=j£(q))]
6u+Y (£(q + 1))

)

>

(28)

where L [£(q)> £(q + 1)> tq (=j£(q))] denotes the pro t obtained from rst-time customers
in period q (this is speci ed in appendix, before the proof of Lemma 4). In the Bellman
21£ ¤Assuming that the distribution is uniform on 0>  would not make any important di¤erence.

26equation (28), the choice of tq (=) is subject to the participation constraint (13), which istq (q+1 j£(q)) ¡ tq+1(q+1 j£(q + 1)) ¸ 0=

(29)

and the choice of £(q + 1) is subject to the non-negativity constraint, £(q + 1) ¸ 0.

Turning to the consumers expectations of the life-time rents of marginal rst-time

customers in period q, they depend on whether consumers expect the market to expandin large steps or in small steps. More precisely, we obtain from (26) and (22) the following:­(£(q)) = Ã1¡­(£(q + 1)) +uÃZ£(q+1)£(q+2)(2 ¡ £(q + 1)) g

!!

>

(30)

if £(q + 2) ¸ £(q+1), i.e., if they expect the market to expand in small steps; and2¡0­(£(q + 1))

1

0 R
1 C
B
£(q+1)
B
C£(q+1)+£(q+2) (2 ¡ £(q + 1)) g+­(£(q)) =  B 1¡C>
2
A@ + u @
AR £(q+1)+£(q+2)
2
£(q + 2)g£(q+2)(31)

if £(q + 2) · £(q+1), i.e., if they expect the market to expand in large steps.
2¡Given the linear-quadratic structure of the model, let us focus on Markov-perfectequilibria with a linear cut-o¤ rule by the monopolist£(q + 1) = #(£(q)) = £(q), with  2 [0> 1](32)and, for consumers, a quadratic expectation function,­(£(q)) = £(q)2 >  A 0(33)

where the parameters  and  are to be determined.
³
i Note that  = 0 means that the1market is fully covered in one period, and  2 0> 2¡means that the monopolist chooses

to expand market in large steps (such³ that somebunching occurs in every periods among´
1the new customers with low ),  2 2¡ > 1 means market expansion in small steps (nobunching in any period), and nally  = 1 means that £(q + 1) = £(q) = === = £(1) =£(0) = 1, i.e., the monopolist does not o¤er the good to anyone in any period.
27

Substituting the quadratic expectations function and the linear cut-o¤ rule into eq.
(30) and (31), we deduce that for the quadratic expectations function to be a best reply
to the linear cuto¤ rule , the parameter  must be such that:(1 ¡ )(1 ¡ ) 3>
1 ¡  2(34)(1 ¡ ) 2 (1 +  (2 + ( ¡ 4)))>4(1 ¡  2)

(35)

u =

1
for  ¸ 2¡(i.e., for slower rates of market expansion), andu =1for  · 2¡(i.e., for faster rates of market expansion). Notice that u is continuous with(1¡)
1
respect to  at  = 2¡where it equals (4¡)(2¡)2=Let us now investigate what is the monopolists best reply if the consumers expectations function is quadratic, such that ­(£(q)) = £(q)2 . The following Lemma showsthat, facing consumers expectations rule ­(£(q)) = £(q)2 , the monopolists optimal
response to any strictly positive  consists of choosing a small cut-o¤ parameter , which
implies that the market is expanded by large steps (and thus in each period there isbunching of the bottom rung of that periods rst-time customers).
Lemma 4. The monopolists best reply to the consumers expectations must satisfy
the rst-order condition that24u(1 +  2 (¡3 + 2)) ¡ (1 ¡ )£3

7£(q)2 66 (1 + (2 ¡ 8) + (13 ¡ 2 +  2 ) 2 7 +  = 0>5
4u 4+2 3 + ( ¡ 4) 4 )(36)

where  ¸ 0 is the Kuhn-Tucker multiplier associated with the constraint that £(q + 1) ¸
0. This equation implies that, facing customers quadratic expectations rule
h with´ strictly
1
positive , the monopolists optimal response,  ¤, must lie in the interval 0> 2¡, whichmeans either the monopolist serves all³ customers
in the initial period (  = 0) or he´
1
expands the market by large steps,  2 0> 2¡.
Proof. See the Appendix.

It follows from Lemma 4 that a Markov-Perfect Equilibrium must be fully characterized by (i) an expectations parameter  that satis es equation (35), which describes the
28

h
´
1
consumers rational expectations for given  2 0> 2¡> and (ii) a cut-o¤ parameter that satis es equation (36), which describes the monopolists best reply to the customers
expectations function. We characterize below two contrasting Markov-perfect Equilibria,
which we call Equilibrium D and Equilibrium E, respectively. As we shall see later, the two
equilibria have substantially di¤erent properties regarding the way the market is covered.
In particular, in equilibrium D, the market is immediately covered in a "twinkle of aneye" (with some customers being bunched to enter the market immediately by getting thelowest-quality version at zero price). In contrast, in equilibrium E, the market is covered
gradually. Also, as reported in Propositions 1 and 2 below, an equilibrium type D always
exists for any  2 [0> 1], whereas equilibrium E exists if and only if  is small enough inthe sense that  · 0=525 (approximately). If  2 (0=525> 1], the two reaction³ functions´

1(35) and (36) fail to intersect in the relevant region where  A 0 and  2 0> 2¡. To

see intuitively why an equilibrium of type E does not exist if  2 (0=525> 1], recall that
 ! 1 means that the period of commitment tends to in nity. With a long period ofcommitment between any two consecutive o¤ers, consumers do not have a strong incentive
to delay their purchases, so they expect immediate market coverage, and the future rentof any potential future marginal customer must be zero. And the monopolist, guessingthat consumers expectation parameter is ¤ = 0, will choose the optimal reply,  = 0.

Let us now turn our attention to Equilibrium D. As described in Proposition 1, themonopolist chooses to serve all the customers from the very beginning. In this case, themonopolists optimal strategy partitions the rst-period market into two segments. For£ ¤the higher types of consumers, i.e.,  2 12 > 1 > the rm o¤ers the Mussa-Rosen price£ ¢quality schedule. In contrast, for the lower types,  2 0> 12 it induces them to choose
a low quality (t = 0) at price 0 (leading to bunching at the bottom rung of rst-timebuyers).
While the pro t earned from this lower segment is zero in the rst period, it is worth
noting that the possibility of full customer recognition after the rst purchase (using themonopolists Big Data) means that the immediate full market coverage allows the rm
to immediately get information on all the customers exact willingness to pay for the
good, including those consumers who would not be willing to get the good at a positive
price. In all subsequent periods, such information will allow the monopolist to make thequality-price o¤er that extracts from each customer the maximum possible surplus.
Accordingly, the equilibrium type D provides a theoretical foundation for the adoption29

of the so-called "freemium business model " when a new good or service is brought to themarket. Indeed, under this business strategy, the rm o¤ers some low-quality variant ofa good or service for free and, simultaneously, o¤ers higher-quality variant(s) of the goodor service at a strictly positive price. In line with our model, the latter o¤ers intend toscreen consumers with high willingness to pay for this good or service.
Proposition 1 (Equilibrium D):There always exists a Markov-Perfect Equilibrium(called Equilibrium A) such that  ¤ () = 0 and ¤ () = 0> 8 2 [0> 1] = In this MPE,
all consumers buy the good for the rst time in the initial period, in which the monopolisto¤ers them the static Mussa-Rosen price-quality schedule, i.e., a quality 2 ¡ 1 at a price£ ¤2 ¡ 14 (screening), for all types  2 12 > 1 and a quality 0 at a price 0 (bunching) for all£ ¤types  2 0> 12 =Proof. See Appendix.

As it is stated, Proposition 1 depends on the (implicit) assumption, embodied in theway participation constraints are written, that the low types (bunched) customers buy
the good even though this does not dominate staying out. Suppose, on the contrary,that a strictly positive measure of these low-type customers would choose to stay out of
the market and the monopolist were aware of that. It would then be optimal for themonopolist to o¤er them a contract in the next period, granting the top ones among
them strictly positive rents. This would mean that ­(£(1)) = ¤ £ (£(0))2 = 0 would notmeet the rational expectations requirement, and thus the Proposed Equilibrium D would
not be an MPE. However, this di¢culty is resolved if one replaces the assumption of acontinuum of consumer types with a ne grid with, say, a di¤erence % between any two
adjacent types, % being a very small positive number. Then a cut-o¤ rule #(£(q)) =  +%>8£(q) and an expectations rule ­(£(q)) = 0> 8£(q) would form an MPE. Indeed themonopolist would optimally o¤er to type  + % customers a rst best quality t ¤( + %) A 0
and a price that leaves them with no surplus. Given constraint (14), a strictly positive

quality level t¤ ( + %) would then be o¤ered to bunched customers. This would provide
them with a strictly positive surplus, breaking indi¤erence. By appropriately choosing %,this MPE can be made as close as desired to the full commitment equilibrium.
Notice that the monopolists pro ts at Equilibrium D equals¦D () =

1¡

1+
+
=
12u
6u
12u
30

(37)

Over the rst period of length ¢, aggregate consumer surplus, in present value terms,
is

Z ¢

1¡
24u

(38)

1+ 1¡
3+
+
=
=
12u
24u
24u

(39)

1
FV () =24
Dh¡uw gw =0

It follows that social welfare at Equilibrium D is
Z D () =We now show in Proposition 2 that there may also exist another MPE at which market
coverage occurs only gradually.
Proposition 2 (Equilibrium E): I¤  ·  ¤ ' 0=525> there exists a Markov-Perfect

Equilibrium(called Equilibrium B) such that  ¤ () is the greatest real root in the intervalhi
10> 2¡
of the seventh-order polynomial
0 = ¡1 + (8 ¡ 2) + (¡12 + 3 ¡  2 ) 2

(40)+(¡8 + 2 2 ) 3 + (10 ¡ 2 2 +  3 ) 4+(2 ¡ 4 2 ) 5 + (12 2 ¡ 2 3) 6 + (¡8 2 + 2 3 ) 7
while ¤() satis es (35) and is strictly positive for all  2 (0> 1). At this equilibrium
the market is only gradually covered. In each period, the higher-type rst-time buyers arescreened but the lower types of customers are bunched.
Proof. See Appendix.
Remark 4: Let us de ne ¢¤ by¤h¡u¢ ´  ¤ =Then the condition  ·  ¤ ' 0=525 is equivalent to the condition¢ ¸ ¢¤ '0=644 36=u

In light of Remark 4, the above Proposition asserts that for any given ¢ 2 [¢¤ > 1),
there exists a corresponding Equilibrium E such that the monopolist chooses to expand
31

market in large steps (implying that some bunching occurs in all periods among the newcustomers with lowh). In isuch equilibrium, we obtain that  ¤ () is the the greatest real1
root in the interval 0> 2¡of the above seventh-order polynomial in . Correspondingly,

consumers expectations ¤ () satis es condition (35).
From Propositions 1 and 2, it follows that there exist two contrasting Markov-Perfectequilibria when the length of time between two di¤erent contractual o¤ers is not too
small. The intuition behind this multiplicity of equilibria is as follows. Why is it notalways optimal for the monopolist to cover instantaneously the market by o¤ering theMussa-Rosen static equilibrium schedule in the initial period?

It is indeed optimal

when the consumers expect the rm to do so because, in that case,  = 0 (and thus
marginal customers do not require any rents to buy immediately). In that case, the rms
pro t is equal to its full commitment pro t. But when the customers, on the contrary,
expect market expansion to be gradual, they always ask for positive rents to buy thegood for the rst time ( A 0). This may dissuade the rm to expand the market tooquickly. This is indeed the case when  is not too large (i.e., ¢ is not too small). Notice
that, at Equilibrium B, ¤ () is a hump-shaped function of : when  is too high, thecorresponding low value of  is not su¢cient to dissuade the monopolist from selling
immediately to all customers (because the commitment period is very short in this case,making it pro table to get full information on the customers exact willingness to pay
from the very beginning).
4.3.2

Pro ts, Consumer Surplus and Welfare: Equilibrium Comparisons andPolicy Implications

Since, for su¢ciently low values of > there exist two MPE, it may be interesting to compare the pro ts, the welfare and the consumers surplus at these equilibria. Investigating
how these values are a¤ected by a decrease in the length of the commitment period ¢ will
enable us to derive some interesting policy implications regarding the recent discussions
on legislations that regulate the collection, storage, and trading of data on consumers
information.
The rst step, however, is to compare the equilibrium values of , since it may help
to understand the multiplicity of equilibria. At Equilibrium D, we have that ¤ () is
identically equal to zero> 8 2 [0> 1] = In contrast, at Equilibrium E, which exists for all in the interval [0>  ¤ ], we have that ¤ () varies with , as pictured below in Figure 1.
32

As mentioned earlier, in the case of Equilibrium E, ¤() is a hump-shaped function of where  2 [0>  ¤ ].

Insert Figure 1 here with the caption below
Figure 1. ¤() at Equilibrium E
From straightforward computations, we can express pro ts as a function of > with¦() being equal to
1 +  + [¡3 + 3 + 8 + (¡13 + 3)+(¡11 + (12 + ( ¡ 4))) 2 + 6(1 ¡ ) 3¡
(¡2 + ( ¡ 3)(¡1 + )) 4 ]
12u(1 ¡  2 )(1 ¡  3 )

>

(41)

where  = 0 at Equilibrium D and  is the solution of (40) at Equilibrium E.
Figure 2 shows that the monopolists pro ts are increasing in  (i.e., decreasing in ¢)at both equilibria: pro ts are greater, the smaller is the length of time, ¢, between twocontractual o¤ers. Since the rm is eager to be able to quickly extract all old customers
surplus at the end of their initial period of purchase, this negative relationship between
pro t and ¢ is not a surprising result, but it is worth pointing out that it is clearly a
non-Coasian feature. We also learn from Figure 2 that the pro t is greater at Equilibrium
D than at Equilibrium E. This is to be expected, since the former corresponds themonopolists full commitment pro t.22

Insert Figure 2 here with the caption below
Figure 2. Equilibrium pro ts (Eq. A vs Eq. B)
The monotone decreasing relationship between pro t and ¢ shows that the monopolist
has strong incentives to set the value of ¢, the length of the commitment period, to its
minimum feasible value, which we will denote as ¢. The parameter ¢ can be interpreted
as the time needed, given the state of the information collection technology and the22

Notice however that the di¤erence between the two is always smaller than 10%.

33

strength of the legislation on consumers data protection, to uncover customers exact
WTPs after their rst purchases. Arguably, legislations on the protection of consumers
privacy have the e¤ect of increasing the minimum feasible value ¢, thus hurting rms
pro ts. Our theoretical result on the negative relationship between ¦ and ¢ seems tobe supported by the empirical nding that many US rms23 would be so much hurt by
the E.U.s General Data Protection Regulation (GDPR) policy that they prefer to block
European users entirely from accessing their web sites rather than to comply with it.24
The next step is to consider aggregate consumer surplus (FV). From straightforward
computations, we can express it as a function of > with FV() being equal to:
1 +  f9 +  [6 ¡ 19 +  [3 + (3 ¡  + 3 ( ¡ 4)  + (12 + ( ¡ 6) )  2)]]g£ (1 ¡ )
24u(1 ¡  2)(1 ¡  3 )where  = 0 at Equilibrium D and  is the solution of (40) at Equilibrium E. Alternatively, FV() may be written as
1 + 9 + (3 ¡ 19) 2 + ((3 ¡ ) + 6) 3 + 3( ¡ 4) 2 4 + (12 + ( ¡ 6) )  2 5(1¡)24u(1 ¡  2 )(1 ¡  3 )(42)
For both MPEs, we nd that the aggregate consumer surplus is a decreasing functionof  (i.e., an increasing function of ¢). This is not surprising since the consumers areable to derive a positive surplus only as new customers. Due to the monopolists ability
to track consumers willingness to pay for the good, the entire surplus of old customers
is completely extracted by the monopolist. Moreover, as shown in Figure 3 below, theaggregate consumer surplus is always greater at Equilibrium E (depicted in green). Thisis because at Equilibrium D, the bottom 50% of customers never have any surplus.
23

See for instance Lanxon, Nate (25 May 2018). "Blocking 500 Million Users Is Easier Than Complying
With Europes New Rules" https://www.bloomberg.com/news/articles/2018-05-25/blocking-500-millionusers-is-easier-than-complying-with-gdpr
24
Additionally, recall that, according to Proposition 2, there is a unique threshold value ¢¤ A 0 suchthat if ¢ ? ¢¤ then Equilibrium E ceases to exist. Since rms prefer equilibrium D to equilibrium E,
any relaxation of regulations which enable ¢ to fall below the threshold ¢¤ would be most welcome by
rms.

34

Insert Figure 3 here with the caption below
Figure 3. Expected consumers surplus (Eq. A vs Eq. B)

Our above ndings concerning pro ts and consumer surplus show that the monopolists
and the consumers interests are diametrically opposed. First, the rm is better o¤ at
Equilibrium D than at Equilibrium E while the opposite ranking holds for the consumers.
Second, the monopolists pro t is decreasing in ¢ whereas the aggregate consumer surplus
is increasing in ¢. Any regulation policy on rms collection of data on consumers, suchas the EUs General Data Protection Regulation (GDPR) policy, which ends up a¤ecting
in an important way the length of time over which rms remain unable to completely
identify their consumers tastes, in sofar as it raises ¢, is clearly bene cial to customers
but detrimental to the rms.
As is well known, while the EU has passed a rather restrictive regulation policy onrms data pertaining to consumers, in the USA, there is no equivalent federal law andexperts seem to agree that, if a similar law is to be passed, it will certainly have to bemuch more lenient than the European one.25 Apart from the well known philosophicaldi¤erences between the two political landscapes, our model o¤ers an economic rationale
behind these diverging legislations. Namely, as online industries and tech-companies (suchthe GAFA), are much more often US-based than European-based, it is natural that, on
matters relating to customers information, the EU cares more about consumers interests
while the US politicians care more about rms interests. Accordingly, consistent with ouranalysis, the EU is more readily inclined to pass restrictive data protection legislations.
Consider nally social welfare. For the moment, let us temporarily adopt the usualde nition of welfare, namely, the sum of the discounted streams of consumers surplus
and pro t. We can then express welfare as a function of > with Z () being equal to:25

A U.S. law wouldnt simply copy the GDPR, according to Eduardo Ustaran, codirector of the privacy
practice at law rm Hogan Lovells. Privacy and data protection are fundamental rights from the EU
perspective but not in the U.S., he says. That is a major philosophical di¤erence between the twojurisdictions, and that will be reected in the law. (See David Meyer (2018), "In the Wake of GDPR,
Will the U.S. Embrace Data Privacy?" Fortune, November 29, 2018.)

35

£
¤
3 +  + 3(1 ¡ ) ¡ 3(1 ¡ ) 2 2 + 24 + 3 2( ¡ 5) ¡ 16  3
24u(1 ¡  3 )(43)

where  = 0 at Equilibrium D and  is the solution of (40) at Equilibrium E.
Using our eq. for welfare, we nd that, over the range of values of  for which both
equilibria exist, i.e., 0 ·  ·  ¤ ' 0=525, the social welfare under Equilibrium D isgreater than that under Equilibrium E only if the length of time between two contractualo¤ers is very long (i.e., if  is close to zero). Indeed, Z D () is greater than Z E ()for  2 [0> 0=0765631] but it is smaller than Z E () for  2 [0=0765631> 0=525].26 Thisis pictured in Figure 4 below for u = 1, where we can see that for  2 [0> 0=0765631],
the welfare di¤erential is quite small. However, as is quite obvious looking at Figure 4
below, comparing the social welfare levels at the two equilibria over the range of parametervalues over which both equilibria exist is not the whole story. When  is greater than0=525 (approximately), only equilibrium D exists and for all values of  2 [0=652607> 1]>
it turns out that Z D () is greater than the highest possible welfare level for Equilibrium
E, for  2 [0> 0=525].27

Insert Figure 4 here with the caption below
Figure 4. Equilibrium Social Welfare (Eq. A vs Eq. B)
Clearly, the above ndings about welfare are subject to several quali cations. First, theabove de nition of welfare assumes that consumers do not attach any value to privacy per
se. Second, in the absence of non-distortionary transfers between gainers and losers
(consumers versus monopolist or vice-versa), the procedure of treating a dollar as adollar, no matter to whom it accrues has been criticized (see, e.g., Hendren, 2014; Saez
26

At  = 0=0765631, we have  = 0=488 55, and Z D = Z E ' 0=128 19@u.
Suppose we accept this welfare de nition, then it follows that if in the absence of data protection
regulation, the minimum period length (given the state of technology), say ¢, that permits the monopolist to uncover consumers tastes corresponds to some  in the interval [0=652607> 1] then any proposed
legislation that has the e¤ect of making ¢ somewhat longer (i.e., making  somewhat smaller) would
reduce welfare, and thus it should not be adopted. If the corresponding  is the interval [0> 0=525], it also
follows that ¢ should not be lengthened, as welfare would be reduced. However, if the corresponding  isin the interval (0=525> 0=652607), then passing a law that makes ¢ a bit longer, so that the coresponding 
falls just below 0=525 might be welfare improving if it induces a switch from Equilibrium D to Equilibrium
E.
27

36

and Stantcheva, 2013). This criticism gains even more weight when the owners of therm do not reside in the same tax jurisdiction (e.g. in our global economy, it is often
the case that multinational rms with headquarters in the US serve a substantial fractionof consumers in other countries, which patronize di¤erent levels of consumers privacy
protection rules).
Indeed, one may argue that the fact that there are proportionately more US than
European rms which may be hurt by legislations restricting access to consumer information is a good explanation of why potential consumer protection laws similar to the EUs
General Data Protection Regulation (GDPR) policy are less likely to be passed in theUS.

5Commitment to Future Sales Plans

Suppose now that, contrary to what has been assumed in the preceding sections, the monopolist is able to commit, right from the beginning, to o¤er a sequence of pre-determined
contracts.28 This commitment capability is supposed to include the ability to commit
both with regard to new customers and to former customers. Thus, at time w = 0, themonopolist announces (i) the old-customer contracts it will o¤er at any future period q+m(m = 1> 2> 3===) to consumers who rst purchase the good in period q; and (ii) the newcustomer contracts to consumers who have not yet purchased the good (and consequently
the rm cannot identify them since it has not yet had the possibility to collect data about
their tastes).
The aggregate discounted pro t (made by the monopolist who can commit) fromperiod 0 is the sum of discounted pro ts from all periods:F¦ ´
where, given Lemma 3,

1
X q q >q=0

28

The contracts proposed to a given consumer are, as in the non-commitment case,announced at time w = 0 and irrevocably committed to) for periods of xed length ¢=37

xed (but now

q =(1 ¡ )(1 ¡ 3q )
6u ½·
¸µ
¶
¾
Z q
(2 ¡  q )21¡
+  +
¡ Xq (q+1) g +
q
q+1
2
u
2
¤¡
¢ )
Z q +q+1 ( £2(2 ¡ q ) ( q+1 ) ¡ 12 (q+1 )2 1¡ug=¡Xq (q+1 )q+1

Proposition 3 When the monopolist is able to commit to future contractual o¤ers:£ ¤(i) new customers with  2 12 > 1 , are o¤ered distorted qualities tq () = 2 ¡ 1> and£ ¤prices sq () = 2 ¡ 14 > 8q ¸ 0> while new customers with  2 0> 12 are bunched and o¤ered
quality tq () = 0 and price sq () = 0, 8q ¸ 0> i.e; in all periods, new consumers face thesame Mussa-Rosen static equilibrium schedule,(ii) old consumers are o¤ered personalized qualities tq () =  and prices sq () = 2 >8q ¸ 0> i.e; they face rst-degree price discrimination.
Proof. See Appendix.
Accordingly, the full commitment equilibrium mimics Equilibrium A described above.
Hence, when comparing the equilibrium with commitment to the no-commitment equilibria, either the results are the same (in the case of equilibrium A) or the monopolist
bene ts with the possibility of commitment (this is so, when we compare the commitment
equilibrium to equilibrium B). In contrast, consumers either are indi¤erent or better-o¤when the monopolist loses the ability to commit to future prices.

6

Conclusion

We have developed a dynamic version of Mussa and Rosen (1978) product quality model,
in which a monopolist sells over an in nite number of periods a vertically di¤erentiatednon-durable good to consumers. The monopolist has access to modern data collection
technologies (e.g., cookies in online websites or datasets with consumers previous purchaserecords), which allow him to track the exact preferences of consumers after their rst
purchase and subsequently use such information to make price-quality personalized o¤ers
that allow the monopolist to fully extract the consumers surplus.
38

We have shown that there always exists a Markov-Perfect Equilibrium in which themonopolist o¤ers to new customers the static Mussa-Rosen price/quality schedule (seconddegree price discrimination) and all consumers purchase the good when it is launched (i.e.,the market is immediately covered). Thereafter, in subsequent periods, old customers receive personalized price-quality o¤ers, which result in a quality upgrade for all customers.
Nonetheless, the personalized price-quality o¤ers made to old customers leave them withno surplus, since the quality upgrade is accompanied by an increase in the price.
We have also proved that, if the length of time between two di¤erent contractual offers is not too small, there exists a second Markov-Perfect Equilibrium with substantiallydi¤erent economic properties. In particular, in this equilibrium, the market is only gradually covered. Instead of all consumers buying the good in the moment it is launched,in this second equilibrium, we get that subsequent cohorts of consumers are sequentiallyadded each period. For each cohort of consumers entering the market, the highest types
select a screening price-quality schedule, whereas the types at the bottom of the rung arebunched.
The intuition behind the multiplicity of equilibria follows from the fact that we are
dealing with equilibria with self-ful lling expectations. When consumers expect the monopolist to expand the market immediately, they conclude that they would not gain by delaying their rst purchase, even though a subset of low-type consumers get no rent in theinitial period. Given such consumers expectations and the consumers low informational
rents, it is clearly in the interest of the monopolist to expand the market immediately, andthus consumers expectations are ful lled. An argument along the same line applies when
consumers anticipate gradual expansion of the market. Though the two MPEs di¤er in a
number of respects, they share a common feature: the monopolists pro ts increase when
the length of the commitment period decreases. This is a non-Coasian outcome but very
intuitive: since the rms pro ts are greater under rst-degree than under second-degree
price discrimination, the sooner it can exercise rst-degree price discrimination, the bettero¤ the monopolist is. On the contrary, for equally intuitive reasons, the consumers are
hurt by a reduction of the length of time between di¤erent o¤ers.
Finally, we have argued that our model may shed some light on the regulatory divergence between the USA and the EU regarding consumers data protection. As the latter
hurts rms and bene t consumers, and given that a disproportionate number of high-tech
rms are US, it is natural that the EU is more inclined to defend its consumers interests.
39

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APPENDIX
Proof of Claim 1

44Incentive compatibility at q + 1 implies that if a consumer is of type q+1 , thenreporting any 0 ? q+1 would not be optimal, i.e.,1¡(q+1tq+1 (q+1) ¡ sq+1 (q+1 ))u1¡¸(q+1tq+1 (0 ) ¡ sq+1(0 )) > 80 · q+1
u
And any type- consumer such that  A q+1 who deviates to be rst-time consumers inperiod q + 1 would obtain a net utility at least as high as that of a consumer of type q+1
who is a rst-time customer in period q + 1:1¡(tq+1 (q+1 ) ¡ sq+1 (q+1))
u1¡
¸(q+1tq+1 (q+1) ¡ sq+1 (q+1 )) > 8 ¸ q+1
u
Combining the two inequalities, we nd that 8 ¸ q+1 and 80 · q+1 >1¡(tq+1 ( q+1 ) ¡ sq+1 (q+1 ))u1¡¸(q+1tq+1( 0 ) ¡ sq+1 (0 )) =u
Consequently, e() = q+1 for all  2 [q+1 >  q ] =¥Proof of Lemma 1

From (6), (10) and (12), ¢q () is a convex function. Accordingly, since ¢q (q+1 ) = 0>
if follows that ¢q () ¸ 0 8 2 [q+1> q ] i¤ ¢0q (q+1 ) ¸ 0. Let us determine the sign of¢0q (q+1). From (6) and (12),¶1¡
tq+1 (q+1 )
u
µ
¶
1¡
1¡
=
tq () ¡ 
tq+1 (q+1 )uu¢0q () = Xq0 () ¡ µ45

Evaluating ¢0q () at  = q+1 , we obtain¢0q (q+1 ) =1¡(tq (q+1 ) ¡ tq+1 (q+1 ))=u

It follows that condition stated in Lemma 1 is necessary and su¢cient for the participation
constraint ¢q () ¸ 0 (for all  2 [q+1> q ]) to be satis ed.¥
Derivation of the equation (20) for ¦Qq
By integrating the last terms in eq (19) by parts, we obtain
¡
=

Z q

Xq ()i ()g q+1¡ [Xq (q )I (q ) ¡ Xq (q+1 )I (q+1 )] +Z q

Xq0 ()I ()g q+1

where the integral on the RHS is equal toZ q ·
 q+11¡
tq ()
u

¸µ

¶
I ()
i ()gi()Thus ¦Q
q can be expressed as¦Qq =Z q ·
 q+1

1¡
u

µ

I ()
tq () ¡ f(tq ()) +tq ()i ()¶¸i ()g¡ [Xq (q )I (q ) ¡ Xq (q+1)I (q+1)](44)

Now, the second term on the RHS of (44) can be written as[Xq (q+1 )I (q+1 ) ¡ Xq ( q+1 )I (q )]+ [Xq (q+1)I (q ) ¡ Xq (q )I ( q )]
R q
R q
which is equal to ¡Xq (q+1)  q+1
i()g ¡ I (q )  q+1
Xq0 ()g.
This is, in turn, equal to
¡

Z q

 q+1

Xq ( q+1 )i()g ¡Z q µ q+1

46

I (q )
i()

¶·

¸
1¡
tq () i()g=uAfter substitution, we obtain the equation (20) for ¦Qq .¥Speci cation of the pro t from new customers L [£(q)> £(q + 1)]
From Lemma 3, this pro t equals

L [£(q)> £(q + 1)] =8 h
9i
Z £(q) < (2¡£(q))2 ¡ 1¡ ¢ =2£(q+1) :¡­(£(q))

u

;

g>

(45)

if £(q + 1) ¸ £(q)
A £(q)> (i.e., if the market expansion proceeds by small steps), and it2¡2
equalsL [£(q)> £(q + 1)]
½·
¸µ
¶
¾
Z £(q)(2 ¡ £(q))21¡=¡ ­(£(q)) g£(q)+£(q+1)
2
u
2
8 "
9
#
>
>
¡
¢
(2
¡
£(q))£(q+
1)>
Z £(q)+£(q+1) >
1¡
<
=
2
2
u
(£(q+1))
+
g>
¡
2
>
>
£(q+1)>>:;¡­(£(q))

(46)

(47)

if £(q + 1) · £(q)
, (i.e., if the market expansion proceeds by large steps). Note that we2¡have substituted ­(£(q)) for Xq ( q+1 ) because of the rational expectations requirement.¥
Proof of Lemma 4.
The rst-order condition for maximizing the right-hand side of (28) with respect to£(q + 1)> subject to the constraint £(q + 1) ¸ 0> is obtained as
CL [£(q)> £(q + 1)> tq (=j£(q))]+ Y 0 (£(q + 1)) +  = 0>
C£(q + 1)(48)

where  is the Kuhn-Tucker multiplier associated with the constraint £(q + 1) ¸ 0, andwhere

CL [=]
1¡=¡(2£(q + 1) ¡ £(q))2 + ­(£(q))>
C£(q + 1)
2u

47

(49)

1
if £(q + 1) ¸ 2¡£(q)> and,

8
9
>
>
>
< (2£(q + 1) ¡ £(q)) (£(q + 1)) >=CL [=](1 ¡ )= ¡¡ 12 (£(q + 1))2>>
C£(q + 1)u>: +  (£(q) ¡ (2 ¡ ) £(q + 1))2 >;4+­(£(q))>1
if £(q + 1) · 2¡
£(q)=

CL[=]
Notice that C£(q+1)
is continuous at £(q + 1) =hi(1¡)  2 £(q)2¡ u+ ­(£(q))=
2(2¡)2

1
£(q)>
2¡

(50)
(51)

where it is equal to

Finally, di¤erentiating the Bellman equation with respect to £(q) and using the en-velope theorem, we obtain the value of Y 0 (£(q)), from which we deduce thatY 0 (£(q + 1)) = ¡­(£(q + 1))¡¢ #Z £(q+1) "¡(2 ¡ £(q + 1)) 1¡
u
+
g>
0
¡­
(£(q
+
1))
£(q+2)

(52)
(53)

1
1
when £(q + 2) ¸ 2¡£(q + 1) (i.e.,  ¸ 2¡> the market expansion proceeds by small

steps) andY 0 (£(q + 1)) = ¡­(£(q + 1))"¡¢ #Z £(q+1)¡(2 ¡ £(q + 1)) 1¡
u
+
g +
0
£(q+1)+£(q+2)
¡­
(£(q
+
1))
2
¡
¢ #
Z £(q+1)+£(q+2) "2(¡£(q + 2)) 1¡ug>0¡­ (£(q + 1))£(q+2)

(54)
(55)

1
1
when £(q + 2) · 2¡£(q + 1) (i.e.,  · 2¡, the market expansion proceeds by large
steps).
1
Notice that the two terms coincide when £(q + 2) = 2¡£(q + 1).

We now show that it is not optimal to expand the market by small steps. To show
this, let us substitute for Y 0 (£(q + 1)) in (48) using its value from (52). Then using the48

conjectured cut-o¤ rule, we obtain the derivative of the right-hand side of the Bellman
equation (28) wrt £(q + 1) as¡¢¤£(q)2 £u 2 (4 ¡ 6) + (1 ¡ ) ¡1 + 4 ¡ 4 2 ¡ 2 3 + 2 4 >
2u

(56)

1
1
when  ¸ 2¡= This expression is negative for all  ¸ 0 and  2 [ 2¡> 1]> which implies that1
to select a cut-o¤ rule such that  ¸ 2¡is never a best reply to a quadratic customers
29
expectations rule. We conclude that the monopolists best reply, , must belong to
1[0> 2¡)= Accordingly, the FOC of the Bellman equation is obtained by substituting forY 0 (£(q + 1)) in (48) using its value from (54). Then, using the conjectured cut-o¤ rule,
we obtain the monopolists best reply to the customers expectations function:
2

¥

3

2

4u(1 +  (¡3 + 2)) ¡ (1 ¡ )
7£(q)2 66 (1 + (2 ¡ 8) + (13 ¡ 2 +  2 ) 2 7 +  = 0=
4
5
4u
3
4
+2 + ( ¡ 4) )(57)

Proof of Proposition 1(i) Let  = 0> 8 2 [0> 1] = From (35),  = 0=(ii) Let  = 0= (57) then becomes2

3¡(1 ¡ )(1 + (2 ¡ 8)
7£(q)2 6
6 +(13 ¡ 2 +  2 ) 2 + 2 3 7 +  = 0=4
5
4u4
+( ¡ 4) )(58)

At  = 0> the bracketed term is negative, implying that  = 0 corresponds at least toa local maximum. In order to show that this is also a global maximum, it is enough to
notice that the monopolists aggregate pro t equals the one under full commitment whichis the maximum that the rm may obtain. ¥Proof of Proposition 2
In the bracketed term of the LHS of (57), let us substitute for  its value from (35); at
any interior equilibrium30  = 0> so that condition (40) must hold. This equation has two
29

More precisely, the negative value of (56) would imply, via (48), that  A 0, i.e., £(q + 1) = 0, which
contradicts with £(q + 1) = £(q) ¸ £@(2 ¡ ).
30
The only possible corner equilibrium is the one characterized in Proposition 1.

49

1
solutions  1() ¸  2 () between 0 and 2¡i¤  · 0=648628> implying that no interior

MPE can exist for values of  A 0=648628= The two solutions are pictured below in Figure
5.

Insert Figure 5 here with the caption below
Figure 5. Candidate interior equilibrium cut-o¤ parameters
Only the greatest of the two,  1 () may correspond to a MPE. Let K(> > u) denotethe bracketed term in the LHS of (57). Denote U(> ) the RHS of (35). We obviously
verify thatK( l ()> > U( l ()> )) = 0> l = 1> 2=
In order for  l () to correspond to a local maximum of the monopolists problem, it mustbe31 in addition that
CK( l ()> > U( l ()> ))· 0=CThis never holds at  2 () where the derivative is always strictly positive, indicating that
this solution corresponds to a local minimum. This always holds at  1 () where thederivative is negative, indicating that this solution corresponds to a local maximum.
Below we picture as an illustration K(> > U( 2 ()> )) in the case for  = 0=4 (and thus 2(0=4) = 0=206063) and then K(> > U( 1 ()> )) also for the case  = 0=4= See Figures
6 and 7, respectively.

Insert Figure 6 here with the caption below
Figure 6. K(> 0=4> 0=00286051)

Insert Figure 7 here with the caption below
Figure 7. K(> 0=4> 0=0123984)
31

A local concavity condition.

50

That  1() corresponds to a local maximum of the monopolists problem does not
ensure that it corresponds to a global maximum. Indeed, as shown for instance above for = 0=4> there is another local maximum at  = 0> where K0 (0> > > U( 1 ()> )) ? 0= We
need to compare the rms pro ts at these two values.
When the monopolist selects  = 0> its pro t is equal to
¦D =

1¡ 1
1
+
¡ U( 1()> )=
u 12 u 6(59)

When it chooses  =  1 ()> its pro t is
¡1 +  + (¡3 + 3 + 8 + (¡13 + 3)+(¡11 + (12 + ( ¡ 4))) 2 + 6(1 ¡ ) 3 ¡¦E =(¡2 + ( ¡ 3)(¡1 + )) 4
12u(1 ¡  2 )(1 ¡  3)

=

(60)

Computations show that the second is greater than the rst i¤  is approximately
greater than 0.525 and that, accordingly, candidate equilibrium B is indeed an MPE
under this condition. ¥Proof of Lemma 3
1. Point-wise maximization of ¦F with respect to the quality levels t(), subject tot() ¸ 0, leads to
tqQ () = t ¤¤ ()> 8q ¸ 0> where t ¤¤ () is the solution of ¡ k() ¡ f0 (t¤¤()) · 0, with equality holding if t¤¤() A 0=
That is, all new customers are o¤ered distorted quality levels (except the customersof type ).
2. tqR () = t ¤()> 8q ¸ 0> where t¤ () is the solution of (1), i.e., to each type, themonopolist o¤ers the corresponding rst-best quality level.

Proof of Proposition 351

1. It is always optimal to o¤er quality  and price 2 to old customers types  inorder to maximize the gross surplus out of these customers and leave them withzero net surplus ( rst-degree discrimination). The old customers types are knownwith certainty by the rm. There is no bene t after the initial purchasing period
to consumers from misreporting their type. Accordingly the monopolist has no
incentive whatsoever to leave positive net rents to old customers.
2. Point-wise maximization of ¦F with respect to the quality levels tq () for new£ ¤customers, subject to tq () ¸ 0, leads to tq () = 2 ¡ 1> 8 2 12 > 1 > 8q ¸ 0>
£ ¤and tq () = 0> 8 2 0> 12 > 8q ¸ 0>that is, all new customers are o¤ered distorted
quality levels (except the customers of type ).
3. Equilibrium new customers prices follow from the fact that the monopolist does
not leave any rent to the lowest types customers.

52

FIGURES

Figure 1. ¤() at Equilibrium E

Figure 2: Equilibrium pro ts (Eq. A vs Eq B)

53

0.050
0.045
0.040
0.035
0.030
0.025

0.0

0.1

0.2

0.3

0.4

0.5

Figure 3: Expected consumers surplus (Eq. A vs Eq B)

0.16

0.15

WA

WB
0.14

0.13
0.0

0.525
0.2

0.4

0.6

0.8

1.0

Figure 4: Social Welfare at the two MPE (Eq. A vs Eq. B)

54

Figure 5. Candidate interior equilibrium cut-o¤ parameters

Figure 6. K(> 0=4> 0=00286051)

55

Figure 7. K(> 0=4> 0=0123984)

56