HIAS-E-93

The Curse of Knowledge: Having Access to Customer
Information Can be Detrimental to Monopoly’s Profit
Didier Laussel(a) , Ngo Van Long(b),(c) and Joana Resende(d)(a) Aix-Marseilles University(b) McGill University(c) Hitotsubashi Institute for Advanced Study, Hitotsubashi University(d) University of Porto, CEF.UP

December, 2019

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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The curse of knowledge: Having access to customer
information can be detrimental to monopoly’s
profitDidier Laussel (Aix-Marseilles University)
Ngo Van Long (McGill University and Hitotsubashi Institute for Advanced Study,
Hitotsubashi University)
Joana Resende (University of Porto, CEF.UP)
Abstract
We show that a monopolist's profit is higher if he refrains from collecting coarse
information on his customers, sticking to constant uniform pricing rather thanrecognizing customers' segments through their purchase history. In the Markov-perfectequilibrium with coarse information collection, after each commitment period, a new
introductory price is offered to attract new customers, creating a new market segment
for price discrimination. Eventually, the whole market is covered. Shortening thecommitment period results in lower profits. These results sharply differ from the ones
obtained when the firm can uncover the exact willingness-to-pay of each previous
customer.
JEL Classification codes: L12, L15
Keywords: curse of knowledge; customers information; intertemporal pricediscrimination; Coase conjecture; price personalization.

1

Introduction

According to the conventional theory of price discrimination, if a monopolist can partition the customer base into segmented markets, he will be able to reap larger proÖts,
even when. in each market segment, only linear pricing is used. Indeed, the standard
model of third-degree price discrimination has shed light on some frequently observed
phenomena, such as discounted prices for senior citizens on train travels, studentsí discounts for admissions to movies, and so on. Market segmentation in these examples ishowever rather crude, because ages or schooling status are only rough proxies for more
relevant characteristics such as income and preferences.
With the advance in digital technology, Örms are increasingly able to collect andprocess huge amounts of consumer-speciÖc data, allowing them to classify consumers on
the basis of their purchase histories and browsing histories, their location, what they like or
dislike on social networks, their preferred sites and so on (EOP, 2015). This enables Örmsto implement strategies that exploit the "consumer addressability" features described by
Blattberg and Deighton (1991).
Thanks to the Invisible Digital Hand (Mazlkiel, 2016), price personalization, the monopolistís old dream, has now become the norm in many sectors (Petrison et al., 1997;
Mohammed, 2017). Depending on the Örmsí technology options and on the regulatory
restrictions (such as the General Data Protection Regulation (GDPR) in the European
Union), the information Örms are able to collect on consumers may be more or less Öne.
In the literature on third-degree price discrimination, there is a presumption thatwith a more reÖned market segmentation, the Örm can tailor di§erent prices to di§erent
consumer groups, thus increasing its proÖt. However, this literature has overlooked an
important issue: when customers are heterogeneous with respect to their willingness topay (WTP), their purchase histories are endogenously determined by the Örmís dynamic
pricing policy. Indeed, the number of distinct market segments based on the monopolistísgrouping of customer types may well depend on his current, past, and future pricing
policies. Anticipating the Örmís future prices and grouping strategy, lower-type customers
may have an incentive to defer their purchases until later periods in order to receive a
better deal. The Örm may have to counter this incentive by o§ering higher informational
rents to the new customers it wishes to serve in each period. Under these circumstances,a Örmís ability to acquire customer information might well be detrimental to its proÖt.

2

In other words, the ability to use information on customersí WTP may be a curse to themonopolist.
Our article demonstrates that the ìCurse of Knowledgeî may arise within a coarse
information setting.1 We propose a model where a monopolist is able to recognize former
customers only on the basis of the moment of their Örst purchase and uses this coarse
information to engage in third-degree price discrimination. We show that his use ofcustomer information for intertemporal price discrimination reduces his aggregate proÖt
below the level he would get if such information were not available.
The equilibrium dynamics arising in this model with coarse information on customersí
preferences are later compared to the ones arising in the polar case of full informationacquisition (FIA, for short), in which the monopolist is able to use consumersí purchasehistory to uncover their exact WTP, which enables him to engage in price personalization.
These two cases lead to diametrically opposed conclusions concerning the equilibrium
dynamics and its Coasian or non-Coasian features. Under FIA, the monopolist gains fromhis ability to acquire full information, and his proÖt is even greater than that obtainedby a full commitment monopolist under the coarse information scenario. Interestingly,
comparing proÖts across three scenarios (the FIA case, the coarse information collection
case, and the case of complete absence of customer records) shows that Örmís proÖt canbe non-monotonic in the degree of precision of information.
To derive the above mentioned results, we set up a model of a monopolist producing
a homogenous good (or service) that must be consumed instantaneously. Consumers
may purchase and consume at each instant of time, whereas the monopolist is committed
to making pricing decisions at discrete points in time. The length of the time interval
between two consecutive price o§ers is called the commitment period. There is a continuumof consumers with heterogeneous WTP for the good. The (type-dependent) consumersí
WTP is initially private information. In the base-line model with coarse information,
we posit that, as time goes by, the monopolist can collect some imperfect informationon the consumersí WTP and use it to implement third-degree price discrimination. If a
consumer makes her Örst-time purchase in a given period n, the monopolist will label her
as a vintage-n consumer, clustering her with other consumers who have chosen to buy the1

As argued later, our model sheds light on optimal dynamic pricing in several real-world set-ups,
including the subscription-based business models, the telecommunications industry, the streaming musicindustry, the online video industry, the online betting sector and the pricing of football/baseball tickets.

3

good for the Örst time in the same period.2 In all its future dealings with the consumer,the Örm will be able to recognize her as a vintage-n consumer and price discriminate
accordingly.
After having been in the market for n periods, the Örm faces n groups of former
customers and one group of new customers. Facing these n + 1 market segments, themonopolist announces n + 1 contractually Öxed prices for the period (one price for eachsegment). Note that whereas the size of each of the n groups of former customers is
known, the size of the new market segment is endogenously determined by a number offactors. The Örst factor is the introductory price that the monopolist o§ers to this group.
The second one is the decision of each member of the new targeted group concerningwhether to make her Örst purchase in that period, or delay it until the next period. Thisdecision depends not only on the consumersí WTP and the current introductory price but
also on her expectations regarding next periodís introductory price and the price she will
have to face in the future as an old customer who bought the good for the Örst time inperiod n + 1.
We suppose that consumers have rational expectations about future prices, and that
their decisions concerning when to make their Örst purchase are rationally made. The
Örm is aware that consumers are rational. We are thus dealing with a dynamic gamebetween the Örm and the potential new customers. We characterize the Markov Perfect
Equilibrium (MPE) of this game, considering Örst the coarsest information acquisition
scenario.
In the MPE with coarse information, the dynamic pricing strategy combines elementsof both introductory pricing and price skimming (see Kotler and Armstrong, 2012, formore details on these concepts). More precisely, when adding a new market segment,
the monopolist adopts an introductory pricing strategy: new customers beneÖt from aprice discount when they buy the good for the Örst time.3 At the same time, whenserving di§erent cohorts of consumers, the monopolist charges di§erent prices to di§erent
consumersí cohorts: he will set higher prices to early buyers, knowing that they havehigher WTP in comparison to late buyers. This means that if we look at the prices paidby successive groups of new customers, the pricing dynamics are consistent with a priceskimming strategy (more precisely, the equilibrium price dynamics feature a decreasing2Di§erently from the FIA case, in our base-line model, we suppose that the monopolist can neverknow for sure a former consumerís exact WTP for the good.
3
Conitzer et al. (2012) present a plethora of evidence on Örmsí introductory pricing strategies.

4

sequence of introductory prices).
The results of our base-line model capture some of the ingredients of the complex
price-discrimination strategies shaping the new subscription-based business models thathave become prevalent in many markets, such as software (e.g., Adobe Creative Cloud,Mathematica licenses), Apps (e.g., auto-renewable apps on iOS through iTunes), telecommunications or e-commerce (e.g., Amazon Subscribe and Save).4 For example, thepioneering business model used by Adobe to market Creative Cloud has quite interesting
features. On the one hand, consistent with our theoretical Öndings, Adobe provides anintroductory plan for new customers.5 On the other hand, it tries to implement price discrimination based on clustering customers according to their WTP.6 Although the driversof market segmentation are clearly not the same as in our model, the Adobe example is agood illustration that Örms may be interested in combining introductory pricing strategies
with third-degree price discrimination based upon an imperfect market segmentation.
Our characterization of the MPE under coarse information also unveils that (i) eventually, the whole market is covered (in sharp contrast with the static equilibrium outcome),and (ii) a shortening of the commitment interval will result in a fall in the Örmís aggregate proÖt. In the limit, as the commitment period tends to zero, the proÖt vanishes, aCoasian feature, even though the good is non-durable. We also Önd that the monopolistís
proÖt under the MPE is strictly lower than that obtained under the full commitment
scenario, where he is able to commit, right from time t = 0, to a speciÖed sequence of
prices for new and old customers for each vintage.7 Thus, the Örm is paradoxically hurtwhen it can recognize its consumersí vintage and use this information to implement pricediscrimination.
This result illustrates the "curse-of-knowledge" and it is consistent with business practices observed in some markets, where Örms avoid this curse by committing to very simple
4
Indeed, the subscription-based business model is often associated with the repeated purchases, afeature that is well captured by our non-durable goods framework. Success stories include Amazon
Subscribe and Save, Dollar Shave Club, Blue Apron, among others.
5
Adobe provides trial versions to new customers. Moreover, when lauching the Creative Cloudsubscription-based product, Adobe o§ered an introductory plan to encourage the adoption of Creative
Cloud for current Creative Suit users.
6
According to Chen (2015) Adobeís plans are segmented into four user segments: individuals, businesses, students and teachers, and schools and universities.
7
In the full commitment scenario, the monopolist Önds it optimal to treat old customers the same
way as new customers. In fact, the Örm is optimally committed to the policy that there will be no new
customers after the initial period.

5

pricing strategies even though they could potentially gather information to engage inbehavior-based price discrimination. Indeed, as pointed out by Rao (2015),ìApple andAmazon, two of the leading online video retailers, maintain a relatively uniform pricing
policy - $14.99/$3.99 to purchase/rent their new movies and $9.99/$2.99 to purchase/rent
their catalog titles. This policy has the semblance of a commitment device, because prices
are held Öxed over a relatively long period of time.î
However, there are also real-world markets in which Örms exert great e§orts to gather
information on their customers.8 To show how, under some circumstances, a monopolist
may proÖt from customer recognition, thus overcoming the curse of knowledge, we develop,
at the end of this paper, an extension of our base-line model to the case of full informationacquisition (FIA). In this extension, the monopolist can uncover each customerís exact
WTP after their Örst purchase. This allows us to contrast the result of vanishing proÖt fora non-committed monopolist (who collects coarsest information) with what happens in thepolar FIA scenario: we show that under FIA, the monopolist beneÖts from intertemporal
price discrimination and Coasian-like outcomes are not obtained. Indeed, we Önd thatin the FIA model, shortening the period of commitment would lead to increased proÖts
rather than vanishing proÖts.
Our theoretical investigations suggest that proÖt and welfare e§ects of dynamic pricing
strategies based on customersí purchase history may be ambiguous. Concerning proÖts,
when the accuracy of information on consumersí WTP is not too precise, there seems
to be such a thing as blessed ignorance, but when the monopolist is able to uncover theexact WTP of his customers, he actually proÖts from engaging in price-personalization
strategies, building up a hyper-segmented market. Concerning welfare, our analysis suggests that economists need to be cautions when assessing the e§ects of public policies
that aim at limiting Örmsí ability to collect and keep data on their customers (such as
GDPR in the European Union). Interestingly, if a monopolist is only able to collect coarse
information on his customers, he may proÖt from legal or technological restrictions thatprevent him to use such data.
The rest of the paper is organized as follows: Section 2 presents an overview of the related literature. Section 3 introduces the model. Section 4 characterizes the MPE. Section
5 investigates equilibrium outcomes arising in the full commitment scenario, comparing
them to the MPE. Section 6 presents the welfare analysis. Section 7 analyzes the case of
8

We are grateful to a reviewer for encouraging us to analyze this issue.

6

full information acquisition. Finally, Section 8 concludes.

2

Related literature

Our model is related to three strands of literature. The Örst strand deals with third-degree
price discrimination in a static framework, where a monopolist partitions his customer
base into several market segments and charges di§erent unit prices for di§erent segments.
For an exposition, see Tirole (1988).
The second strand of literature deals with price discrimination when Örms are able
to learn about consumersí purchase history and/or the personal tastes of individual consumers. More precisely, our base-line model enriches the literature on ìbehavior-based
price discriminationî (e.g., Fudenberg and Tirole, 2000; Chen, 1997; Choe et al., 2018),9whereas our full information acquisition model adds to the literature on price personalization.
The behavior-based price discrimination literature shares with our model the featurethat Örms are able to learn about consumersí preferences by accessing data on their histories of purchase and use this information to practice third degree price discrimination (seeFudenberg and Villas-Boas, 2006, for a survey). Most papers in this strand of literatureassume a two-period model and focus on duopoly competition.10 A general conclusion isthat history-based pricing intensiÖes oligopolistic competition and erodes Örmsí proÖts.
Indeed, following the seminal work by Stokey (1979), many scholars have stressed thatintertemporal price discrimination could hurt Örms.11 Our article arrives at a similar
conclusion on proÖt erosion. In our base-line model, a consumer faces a trade-o§ between
buying today and buying tomorrow given that there is a price advantage associated withdelay. Competition exists this time between the di§erent selves of the monopolist insteadof between oligopolists. Although in both settings it is the pressure of competition that
9

Chen (1997) focuses on duopolistic behavior-based price discrimination (BBPD) with consumersíswitching costs. Fudenberg and Tirole (2000) analyze BBPD in a Hotelling duopoly model with two
period. In Choe et al. (2018), there are multiple equilibria when the duopolists have di§erent information.
10
Following the seminal works by Fudenberg and Tirole (2000) and Chen (1996), a number of authors
have investigated the competitive e§ects of behavior-based price discrimination. See, for example, Chen
and Pearcy (2010), Esteves (2009, 2010), Gherig et al. (2011, 2012), and Villas Boas (1999).
11
In contrast, Akan et al. (2015) found that, if consumers learn about their own WTP under a timedependent process, then intertemporal price discrimination may result in higher proÖts. Ata and Dana
(2015) examined the robustness of this result.

7

reduces the Örmsí proÖts, the two market structures yield remarkable di§erences in termsof market coverage. In the monopoly case, to exert some competitive pressure on the current monopolist, some consumers need to stay out of the market (until the period when
they make their Örst purchase). Di§erently, in standard duopoly models of behavior-based
price discrimination (such as Fudenberg and Tirole, 2000), the market is fully covered ineach period, and the competitive pressures follow from the possibility of switching to therival Örm.
The recent literature on price personalization concentrates on the competitive and
proÖt e§ects associated with Örst-degree price discrimination in oligopoly set-ups. Following the pioneering paper by Thisse and Vives (1988), several authors (e.g., Choe etal., 2017, Anderson et al., 2015), have shown that Örms may be harmed by the possibilityof setting personalized prices, as this leads to asymmetric Bertrand competition for eachconsumer in the market.12 Their results on price personalization are di§erent from what
we obtain in section 7 for our polar case, the FIA model: we show that the opportunity
to practice Örst degree price discrimination among old customers actually beneÖts theinformation-seeking monopolist, even though he must o§er introductory prices in eachperiod to attract new clients.
The third strand of literature studies the market dynamics of a durable-good monopoly,and shows that a monopolist who is unable to fully commit to future prices would lose
all his monopoly power if the time interval between two consecutive price o§ers goes tozero (Coase, 1972; Stokey, 1981; Bulow, 1982; Gul et al., 1986; Bond and Samuelson,
1987; Kahn, 1987; Hart and Tirole, 1988; Karp, 1996; Laussel et al., 2015; Correia-daSilva, 2019; see Long, 2015, for a survey.) The logic behind this result is that early
customers would not accept monopoly prices because they anticipate that future prices
will be lower. As in our base-line model, it is the competition between the current andthe future incarnations of the monopolist that reduces his market power.
While our model has the Coasian áavor, it is quite novel, and is not mathematicallyisomorphic to Coaseís durable good monopoly model. To see this, we would like to draw
the readerís attention to the following points.13
12

There are some exceptions. Anderson et al. (2019) Önd that price personalisation may beneÖt the
duopolists if the demand curve is strongly convex. Anderson and Dana (2009) Önd that intertemporal
price discrimination is proÖtable when the relative change in surplus related to product upgrading increases with consumersí WTP.
13
We are very grateful to Editor Kathryn E. Spier for suggesting these points.

8

First, our model brings some novel insights to the durable good monopoly problem
itself. It is well known that short-term leasing can solve this problem (Tirole, 1988). Withleasing, the monopolist can charge the consumers just for using the product for a shortperiod of time. In the leasing mechanism discussed in the literature, the Örm cannot
distinguish between customers who leased in earlier periods and those who did not lease.
Consequently, it charges the monopoly price for the lease, the same for all customers ineach period, and proÖts are maximized. Our model is similar to a durable goods monopoly
problem with leasing, except we allow the monopolist to discriminate directly based on
previous lease adoption. In other words, the classical leasing solution is predicated on
the assumption of customer anonymity, while in our model, information collection allows
the monopolist to identify previous customers, leading to substantially di§erent pricing
strategies (as each cohort of consumers is charged a di§erent price).
Second, in the durable goods monopoly model without commitment, the monopolist
grabs successive slivers of consumers from the top of the demand curve. In our model,the Örm does grab the Örst sliver from the top, but then in successive rounds it reÖnes
its information, raising the price charged to former consumers. Thus, for example, themarginal customer in period n = 1, whose type is #2 , enjoys, if she chooses to be a Örsttime customer in period 1, the introductory price p(1; 1) that is strictly lower than #2 ,but in all later periods j ! 2, being a former customer, she must pay a higher pricep(1; j) = #2 , leaving her with zero surplus.
Third, our model and the durable goods monopoly model have di§erent empiricalimplications. In our model, at any point of time, the monopolist posts di§erent prices
to di§erent segments of consumers. In the classic durable goods monopoly model, themonopolist o§ers a single price at each point of time.
Our article is also related to the literature on Markov-Perfect Equilibrium in games
involving interactions between Örms and inÖnitely-lived consumers with rational expectations (see, e.g., Driskill and McCa§erty, 2001; Laussel, Montmarin and Long, 2004;
Laussel, Long and Resende, 2015; Long, 2015).
Finally, there is some connection between some of this articleís conclusions and the real
options literature (Dixit and Pindyck, 1994; Abel et al., 1996). In our model, consumers
have an incentive to delay purchase: with delayed entry, they could face a lower price as
they would be classiÖed into a market segment of consumers with a lower valuation of thegood. By waiting, a consumer is transmitting the information that she has low valuation,9thereby anticipating greater surplus in future periods. Thus, the value of waiting is
associated with information transmission. In this sense, there is a connection between
our model and the real options literature, which has identiÖed the option value associatedwith delaying investment decisions when facing uncertainty. This value should be taken
into account for the characterization of optimal investment timing: delayed investment
facilitates information acquisition. Although in both set-ups, the value of waiting is
associated with information transmission, there are important di§erences between thetwo problems. In particular, in our case, by waiting, the consumer will certainly gethigher future surplus in the future (at the cost of forgoing current surplus), whereas in
the real options theory, the expected beneÖt of waiting arises from reduced uncertainty.

3

The model

A monopolist produces a non-durable good (or a service) at a constant marginal cost,normalized to zero. Although the good is homogeneous, consumers have heterogeneous
WTP for the good.
Time is a continuous variable. The market is populated with inÖnitely-lived consumers.
Each consumer buys and instantaneously consumes at most one unit of the good at each
instant of time. There is a continuum of consumer types. A consumer of type # derives #units of utility for consuming one unit of the good per unit of time. If she pays the unit
price p for the good, her instantaneous net utility is # " p. The support of the distribution! "of # is a closed set #; # .
The Örm enters the market at time t = 0. It partitions the (non-negative) timeline [0; 1) into a sequence of ìcommitment periodsî of equal length ). Thus, period 0

corresponds to the interval [0; )) of the time line and period n corresponds to the interval[n); (n + 1))), where n = 0; 1; 2; 3; :::.
In period n = 0, there are no former customers, and the monopolist o§ers a price!
"p(0; 0) to the top sliver of the customer base, i.e., customers with # 2 #1 ; # , where#1 and p(0; 0) are to be optimally determined. A customer who buys the good for theÖrst time in period i is called a vintage-i consumer. The collection of all customers of

the same vintage constitutes a market segment. At the beginning of any period n; themonopolist o§ers to all former consumers of vintage i (where i < n) a vintage-speciÖc
price p(i; n) at which they can purchase a unit of the good at each instant of time in the10

time interval [n); (n + 1))). We assume that, thanks to the Örmís ìbig-dataî capability,
the monopolist can identify the vintage of all former customers and use that information
to engage in third degree price discrimination: a consumer of vintage i < n can acquire
the good in period n only at the price p(i; n) and not at any other prices. Customers who
buy the good for the Örst time in period n are o§ered an introductory price p(n; n).
For all i = 0; 1; 2; ::; n, a consumer of vintage i who purchases and consumes a unitof the good at each point of time in period n ! i enjoys the instantaneous net utility# " p(i; n). Her net utility over the whole period n (discounted back to the beginning of
period n) isv(#; i; n) % (# " p(i; n))Z #0e!r" d- % (# " p(i; n))1".
, where . % e!r# :rHere r > 0 denotes the instantaneous discount rate, and . is the discount factor
between periods. Note that whereas r is exogenous and independent of ), . depends on). Clearly . ! 1 when ) ! 0:
Let us consider the case where the monopolist cannot commit to future prices. SpeciÖcally, when selling to Örst-time consumers in period n, the Örm cannot commit to o§erthem, in future periods, the same price as the introductory price. In addition, in any
period n, the monopolist cannot commit to o§er pre-determined introductory prices tofuture new customers. In the absence of such a commitment capability, we assume themonopolist uses third-degree price discrimination with respect to old market segments:
they are discriminated on the basis of the monopolistís endogenous market segmentation(which in our set-up relates to the consumersí vintage clustering).
The cumulative distribution of # is denoted by F (#) and the density function is denotedby f (#). We make the following assumptions:! "Assumption A1. f (#) " #f 0 (#) ' 0 for all # 2 #; # .
Assumption A2. 1 " #f (#) > 0:

Assumption A1 is su¢cient to ensure that the monopolistís proÖt function for period nfor each market segment i < n is concave in the price p(i; n). This assumption is satisÖedby the uniform distribution and, more generally, by all distributions such that F is not
11too convex. Assumption A2 ensures that a static monopolist would not serve the wholemarket. In the uniform distribution case, it is equivalent to # > 2#:

4

Equilibrium under non-commitment

We start our analysis of the model by scrutinizing the monopolistís third degree pricediscrimination among former customers. This will be followed by an examination of the
optimal pricing for new customers.
! Third-degree price discrimination among former customers. The following
observation is useful in what follows. As consumers are rational, we can deduce that, inequilibrium, if a consumer of type #0 Önds it optimal to be a Örst-time customer in period
j ! 0, then any consumer of type #00 > #0 must Önd it optimal to be a Örst-time customer

in some period i, with i ' j. At the beginning of any period j, there are already j known
cut-o§ types #1 ! #2 ! #3 ! ::: ! #j , and the Örm, through its price decisions, determines"$#j+1 . New customers in period 0 are of type # 2 #1 ; # , new customers in period 1 areof type # 2 (#2 ; #1 ], and so on. In any period n > 1, with j + 1 ' n; customers of type
# 2 (#j+1 ; #j ] are called vintage-j customers. A customer of type #j+1 is called a vintage-j

marginal customer. In equilibrium, the vintage-j marginal customer #j+1 is indi§erent
between making her Örst purchase in period j or in period j + 1. Thanks to his ability tokeep its data records registering the moment in which each consumer enters the market,
the Örm is able to identify the vintage of a former customer when she comes back insubsequent periods. In our base-line model with coarsest information, the Örm cannot
tell the di§erence among former customers that belong to the same vintage. Later on,
we will relax this assumption, by looking at the polar case of full information acquisition(the FIA case, studied in Section 7).

$"
At the beginning of period j, all consumers whose types belong to #j ; # have alreadypurchased the good at least once in previous periods. The monopolist would not o§er aformer customer the introductory price that it o§ers to new customers. In any period j,
the Örm o§ers former customers a vintage-dependent price, p(i; j), for each vintage i < j(i = 0; 1; 2; ::; j " 1) so as to maximize the proÖts it makes from them. The population
share of customers that belong to vintage i is F (#i ) " F (#i+1 ). Clearly, o§ering thesecustomers any p(i; j) > #i would be a dominated strategy because it would result in zero12demand from that market segment. Thus, in period j, for former customers that belong
to vintage-i (where i < j), the monopolist will o§er a price p(i; j) ' #i . The quantity soldin period j to this market segment is
Qi;j =(

F (#i ) " F (p(i; j))
F (#i ) " F (#i+1 )if #i+1 ' p(i; j) ' #i
if 0 ' p(i; j) ' #i+1

Thus, the proÖt obtained in period j from this market segment is4(i; j) =1".
[F (#i ) " maxfF (#i+1 ); F (p(i; j))g] p(i; j):r

DeÖne p(#i ) as the solution of the unconstrained problem maxp [F (#i ) " F (p)] p: Thenp(#i ) satisÖes the Örst order condition that F (#i ) " F (p(#i )) " f (p(#i ))p(#i ) = 0:14
Clearly there is no point in o§ering former customers of vintage i a price that is below
the lowest valuation among them, #j+1 . Thus, the monopolistís optimal choice mustsatisfy the condition p# (i; j) ! #i+1 . Accordingly, the optimal price is
p# (i; j) = maxfp(#i ); #i+1 g for all i ' j:(1)

It follows that if p(#i ) ' #i+1 , then the monopolistís optimal price for former vintage-i

customers is exactly equal to #i+1 , i.e., the price is equal to the maximum WTP of thelowest valuation customers in vintage i. We will show later that along the optimal path,
the property p(#i ) ' #i+1 is indeed satisÖed. For future reference, we record the following
result as Lemma 1:

Lemma 1 . Under the assumption that p(#i ) ' #i+1 (which will be showed to be satisÖedin equilibrium), in all periods j > i, the monopolistís optimal price for market segment i
is equal the maximum willingness to pay of the lowest type of that segment:p# (i; j) = #i+1 :(2)

In view of equation (2), the monopolistís optimal aggregate proÖt in period n over all
14

The second order condition is always satisÖed thanks to Assumption A1.

13

the former customers is
1". X[F (#i!1 ) " F (#i )] #i :
r i=1
i=n

/Fn =

(3)

where the superscript F stands for ìformerî customers.
! Monopoly pricing for Örst-time customers. Let us now turn to new customers
in period n: Let Un (#n+1 ) denote the intertemporal net utility of period-n marginal new
customer. By marginal, we mean that she is indi§erent between (i) being a Örst-time
customer in period n (paying the price p(n; n) for the good), and (ii) being a Örst-time
customer in period n + 1, paying a lower price p(n + 1; n + 1) < p(n; n), but at the cost of
having to forgo her net utility #n+1 " p(n; n) at each instant of time over the time intervalof length ).
Recall that Lemma 1 stated that marginal customers in period n will be charged an
instantaneous price p# (n; j) = #n+1 in all later periods j > n. Thus, they get zero utility
in all later periods. It follows that their intertemporal utility from period n onwards is
simply equal to their utility in period n; that is Un (#n+1 ) = 1!)(#n+1 " p(n; n)) : It isruseful to re-write this equality asp(n; n) = #n+1 "r
Un (#n+1 ):
1".

(4)

We will refer to Un (#n+1 ) as the informational rent of period-n marginal new customer.
The period-n proÖt over all new customers may then be written as

/N
n =

'

1".
r

(

[F (#n ) " F (#n+1 )] p(n; n) = [F (#n ) " F (#n+1 )]'((1 " .)#n+1" Un (#n+1 ) :r(5)Now, any vintage-n customer of type # > #n+1 will face the same present and future
prices as the ones that the marginal customer #n+1 faces, but values the good more.
Therefore, the di§erence in their intertemporal net utility is
1
Un (#) " Un (#n+1 ) = (# " #n+1 ):r(6)

If the Örm wants to induce a new customer with # 2 [#n+1 ; #n ] to buy in period n; it
14

must ensure that she would not be better o§ waiting until period n+1. This participation
constraint may be written as)*
1
1(# " #n+1 ) + Un (#n+1 ) ! . (# " #n+2 ) + Un+1 (#n+2 ) for all # 2 [#n+1 ; #n ] :
r
r

(7)

From (6), the LHS of (7) is simply Un (#), the intertemporal utility obtained by a
type-# customer who chooses to be a period-n new customer. The RHS is the alternativeintertemporal utility, measured from period n; that she would obtain if she chose to be a
new customer in period n + 1. Notice that if the participation constraint (7) is satisÖed
for type # = #n+1 (the marginal customer), it will also be satisÖed for all customers oftype # 2 (#n+1 ; #]: Thus, the constraint (7) is redundant if the following simpler constraint!
"
is satisÖed: Un (#n+1 ) ! . 1r (#n+1 " #n+2 ) + Un+1 (#n+2 ) . In fact, the latter constraint issatisÖed with equality,
Un (#n+1 ) = .

'(1(#n+1 " #n+2 ) + Un+1 (#n+2 ) ;r(8)because, by deÖnition, a marginal customer is indi§erent between being a Örst-time customer in period n and being a Örst-time customer in period n+1: By repeated substitution,
the di§erence equation (8) yields the solution
Un (#n+1 ) =1X j. [#n+j " #n+j+1 ] ;
r j=1
1

(9)

where we have used the fact that limj!1 . j Un+j (#n+j+1 ) = 0 (as U is bounded).
! The Markov perfect equilibrium. In any period n, let X(n) 2 [0; 1] denotethe fraction of the total population that has purchased the good prior to that period.

Given the nature of our problem, X (n) would be a natural state variable for our dynamic
problem. However, it turns out to be more convenient to use as state variable the following
transformation of X(n): 2(n) % F !1 (1 " X(n)); where 2(n) is our state variable and! "F (:) is the cumulative distribution of #: Then 2(0) = # % #0 , and 2(n) 2 #; # . Clearly,$"
the real interval 2(n); # corresponds to the set of customers who have not purchased
the good prior to period n.
15

As we have solved for the Örmís optimal pricing for former customers, its dynamic
optimization problem reduces to determining, at the beginning of each period n, given
the observed value of the state variable 2(n), the optimal size of the new market segment.
It follows that the Örmís Markovian strategy consists of a cut-o§ rule , such that 2(n+1)
is determined by 2(n + 1) = (2(n)) ' 2(n) .

Also, at the beginning of period n, given the observation 2(n), consumers who havenot purchased the good need to decide on the optimal period to enter the market. To
perform this calculation, they must form their expectations on the informational rents ofthe marginal customers for all successive future periods. When characterizing the MPE,
we assume that consumers have a Markovian expectations rule 3(:) that predicts thelife-time rent of the marginal Örst-time customer in period n, i.e., Un (#n+1 ). Rational
expectations require that, given (9), we have3(2(n)) =!

+1X j$ !
!
. 2 (n + j) " 2 (n + j + 1) = Un (#n+1 );
r j=1
1

(10)

where f2 (:)g1
n is the path of the state variable 2 induced by the strategy of the monopolist.

A Markovian strategy (:) chosen by the monopolist is called a best reply to theconsumer expectations function 3(:) if (a) it yields a sequence of cut-o§ values #n+1 thatmaximizes proÖts, starting from any pair (n; 2(n)), and (b) the rational expectations
condition (10) is satisÖed by such a sequence. In order to obtain a sharp characterizationof MPE, in this section we replace Assumptions A1 and A2 with a stronger assumption:
Assumption B. The distribution of # is uniform, with # = 1 and # = 0.
In what follows, some Propositions rely on Assumption B, while others rely only on
the weaker Assumptions A1 and A2. We will state explicitly which proposition relies on
Assumption B and which one relies only on the weaker Assumptions A1-A2.
Using Assumption B, with the help of equation (3), the proÖt obtained in period nfrom old customers is
1".
/Fn =
r

i=n
X
i=1

(#i!1 " #i ) #i :(11)

Corresponding to eq. (5), the proÖt obtained in period n from Örst-time customers inthat period is16

/N
n = (# n " # n+1 )

'

(
1".
#n+1
" 3(2(n)) ;r

where we have substituted 3(2(n)) for Un (#n+1 ) because of the rational expectations
requirement.
It follows that, under Assumption B, the Bellman equation for the monopolist is15
V (2(n)) =max((n+1)

(

1".
r

i=n
X
i=1

!

(2(i " 1) " 2(i)) 2(i) +

'

(
.
1".
(2(n) " 2(n + 1)) 2(n + 1)
" 3(2(n)) + .V (2(n + 1)) :r
Given Assumption B and the general structure of the problem, we conjecture the that
the MPE consists of a linear cut-o§ rule2(n + 1) = (2(n)) = 92(n)(12)

and a linear expectations function16
3(2(n)) = :2(n)(13)

where both 9 and : are to be determined. We must show that these rules are best replies
to each other if : and 9 are appropriately chosen.
Using our conjectured expectations rule (13), the rational expectations requirement,with the help of equation (8), can be rewritten as'(1:#n = . :#n+1 + (#n+1 " #n+2 ) :r(14)

Using the conjectured equilibrium cut-o§ rule, we have 2(n + 1) = 92(n) and 2(n +2) = 9 2 2(n), and equation (14) becomes :#n = . [:9#n + (1=r)(9#n " 9 2 #n )] ;from which
we obtain

15
V ("(n)) depends also on the values of "(n " 1); "(n " 2); ::; "(0). This is omitted for the sake ofnotational simplicity. Note that in period n these given values have no relevance for the decision on the
optimal "(n + 1).
16
Starting from a more general conjectured linear-quadratic expectations functions would lead to asimple linear expectations function, so we take this shortcut for the sake of simplicity.

17

r: =

.(9 " 9 2 ):1 " .9(15)

This equation says that, given the monopolistís cut-o§ rule, represented by 9, the consumers expectations are rational (best reply) if and only if: = :(9; .; r) =.(9 " 9 2 ):r(1 " .9)

(16)

Maximizing the Bellman equation with respect to 2(n + 1) yields(1 " .)

2(n) " 22(n + 1)
+ :2(n) + .V 0 (2(n + 1)) = 0:r

(17)Di§erentiating the Bellman equation with respect to 2(n) and using the Envelope
Theorem, we obtain

V 0 (2(n)) =1".
[2(n " 1) " 22(n) + 2(n + 1)] " :2(n) " : [2(n) " 2(n + 1)] : (18)r

From (18), we can obtain an expression for .V 0 (2(n + 1)) and substitute it into (17).
We Önally obtain the Euler equation,(1 " . 2 + r:)2(n) " 2(1 " . 2 + .r:)2(n + 1) + .(1 " . + r:)2(n + 2) = 0:(19)

If the monopolist uses a linear cut-o§ rule, then 2(n + 2) = 92(n + 1) = 9 2 2(n) andequation (19) becomes(1 " . 2 + r:) " 2(1 " . 2 + .r:)9 + .(1 " . + r:)9 2 = 0(20)

Equation (20) says that, taking the parameter 9 of the consumersí expectations ruleas given, the monopolistís linear cut-o§ rule is an optimal response (best reply) if andonly if

18

2

9=

2(1 " . + .r:) *q4(1 " . 2 + .r:)2 " 4(1 " . 2 + r:).(1 " . + r:)
2.(1 " . + r:)

:

(21)

The two best-response functions (16) and (21) yield a unique Öxed point (9 # ; :# ) whichcharacterizes the Markov-perfect Nash equilibrium. Indeed, we can show that, for any
given . 2 (0; 1), 9 # is the unique positive real root of the equation
E(9; .) = 1 " . 2 + (2. 2 + . 3 " .)9 + (2. " 3. 2 " 2. 3 )9 2 + (2. 2 + . 3 )9 3 " . 2 9 4 = 0: (22)This is a polynomial of degree 4 in 9: With 0 < . < 1, we Önd that the polynomial
has a unique positive real root 9 # . In the limiting case where . = 0, we have 9 # = 1=2,
and when . = 1, we have 9 # = 1.17 Figure 1 shows that 9 # is strictly increasing in ..
In the Ögure, the horizontal axis represents .; whereas the vertical axis represents theequilibrium value of 9 as a function of .:Insert Figure 1
The following proposition characterizes the MPE.
Proposition 1. Under Assumption B, there exists a MPE in which the monopolistís
optimal cuto§ rule and consumersí expectations rule are both linear in the state variable,! "where the equilibrium parameters (9 # ; :# ) are such that 9 # (.) 2 12 ; 1 is the solution ofequation (22) and :# (.) is given by:# (.) =.(9 # (.) " 9 # (.)2 ):r(1 " .9 # (.))

(23)

Proof See the Appendix."
Proposition 1 identiÖes the equilibrium parameter :# of the consumersí expectations
rule, 3(2(n)) = :# 2(n) and the parameter 9 # of the monopolistís optimal market expansion rule, 2(n + 1) = 9 # 2(n). In the MPE characterized in Proposition 1, the monopolist
17

When % = 1, & = 0 is also a real root. It is the limit of a negative real root (when % ! 1) and assuch may be ruled out.

19

adopts a simple market expansion rule: in any period n, the fraction of the customer basethat is not served is (9 # )n+1 . The measure of customers that have been served by the endof period n is (1 " (9 # )n+1 ). Hence the market is gradually fully covered, although fullmarket coverage is only reached asymptotically, as n tends to inÖnity. The result that fullmarket coverage is eventually reached provides a sharp contrast to the static equilibrium
case and, as we shall see, also to the dynamic case with commitment, as in both of thesecases only partial coverage occurs.
Proposition 1 also allows us to shed light on the prices targeted to successive cohortsof new customers and the magnitude of the rent of the marginal customer that chooses topurchase for the Örst-time in period n. Using eq. (9) and the linear cut-o§ rule, so that#n+j " #n+j+1 = (9 # )n (1 " 9 # ) (9 # )j , the informational rent of the marginal customer thatrationally enters the market in period n is equal to:
9 n (1 " 9)Un (#n+1 ) =r'.9
1 " .9(:

Accordingly, the monopolistís introductory price to Örst-time customers in period n isp(n; n) = 9n+1

9 n (1 " 9)
"
1".

'

.9
1 " .9(;n = 0; 1; 2; 3; :::To give a concrete áavor to Proposition 1, let time be measured in days, and suppose
the instantaneous interest rate is r = 0:01, then, if ) = 160 days (i.e., if . % e!r# ' 0:20),
using eq. (22), we obtain the monopolistís equilibrium cut-o§ parameter, 9 # = 0:55, i.e.,the monopolist initially serves only the top 45% of the customer base (during the Örst160 day period). Thus, in period n = 0, the bottom 55% of the potential market is notserved. In the second round of o§ers (n = 1), the monopolist expands the market, so thatthe fraction of unserved customers shrinks to (9 # )2 , i.e., around 30%. In period n = 1, themonopolist serves two market segments: all the vintage-0 customers (which consist of the
top 45% of the customer base) must pay a higher price, p(0; 1) = 9 = 0:55 (higher than the
introductory price p(0; 0) that they paid in the previous period, which was approximatelyp(0; 0) ' 0:48). The period 1ís Örst-time customers are asked to pay only the introductory
price p(1; 1), which is approximately p(1; 1) ' 0:264 26 < p(0; 0) < p(0; 1). It is easy toverify that period 1ís marginal Örst-time customers (whose type is #1 = 9 # ) are indi§erent
between (a) making their Örst purchase at the beginning period 1 and (b) delaying their
20Örst purchase until the beginning of period 2. Indeed, condition (22) ensures that theywould be indi§erent, as expectations rule is given by (23). And their expectations are infact rational.
Now, suppose the period length is only 7 days () = 7). Then . ' 0:93 and 9 # = 0:86.

That is, the monopolist serves only 14% of the customer base during the Örst 7 day
period. However, after 23 periods (i.e., after 161 days), the measure of customers thathave been served by the end of period n = 22 is (1 " (9 # )23 ) = (1 " 0:8623 ), that is 97%
of the customer base. This numerical example suggests that as the length of the periodof commitment shrinks, the market coverage at any given date increases.
Let us scrutinize more formally the role of the length of the commitment period, ). A
natural enquiry is whether a shortening of the commitment period will make the market
coverage larger at any given point of time. Let t be a continuous variable denoting time,
where t = 0 is the beginning of period 0, and, more generally, t = n) ! ) at the beginningof period n ! 1. Recall that 9 # is a function of . % e!r# ; so that 9 # (.) % 9(e!r# ).

Therefore the fraction hof the consumer basei that is served by the monopolist at time t is
t
+1
!r# !
given by M (t; ); r) % 1 " (9(e)):

Let us investigate how, at any given t, the market coverage M (t; ); r) depends on). A shortening of ) generates two opposite e§ects. On one hand, since 9 0 (.) > 0,
9 # increases as ) gets shorter, i.e., the market expansion from one period to the next
is smaller (meaning the monopolist is doing a Öner partition of its customers). We callthis the ìstep size e§ectî. On the other hand, the interval between two periods becomes
smaller and thus over any time interval [0; t] market expansions occur more often. We callthis the ìfrequency e§ectî. The two e§ects work in opposite direction. When ) shrinks,
the frequency e§ect tends to increase M at any given t, while the step size e§ect tends
to decrease M at any given t. Numerical simulations indicate that the frequency e§ect
dominates the step size e§ect if ) is not too large. As an illustration, please refer toFigure 2 below.
Insert Figure 2
Figure 2 shows that, given a future date, t = 10, and letting r = 10%, the frequencye§ect outweighs the step size e§ect: starting from any ) ' 10, a shortening of the commit-

ment period ) speeds up market expansion. Such dynamics are clearly Coasian. While
this result seems robust in our simulations, analytical results of the e§ect of a decrease in
21) on market coverage can only be obtained in the limiting case where ) tends to zero,
as reported in Claim 1 below.
We prove in Claim 1 below that full coverage occurs instantaneously as ) tends tozero, meaning that all consumers will be served in a twinkle of an eye when ) becomesinÖnitesimal.
Claim 1.

Under Assumption B, when the length of time ) between two di§erent

proposals to two consecutive sets of new customers becomes inÖnitesimal, the market
is covered instantaneously. For any given time t > 0, as the length of the period ofcommitment ) tends to 0; the fraction of the customer base that has been served up to0

that time tends to (1 " e!+ (1)rt ); which equals 1 because 9 0 (.) ! +1 as . ! 1:$
+ t +10
lim 9(e!r# ) ! = e!+ (1)rt = 0:

#!0

(24)

Proof See the Appendix."

What happens to the consumer equilibrium expectations coe¢cient, :, when )shrinks? A numerical illustration is presented in Figure 3 below, where . is measured
along the horizontal axis, and r: is measured along the vertical axis (the picture is drawn
for r = 1).
Insert Figure 3
Notice that r: is increasing in ..18 Interestingly, starting at any strictly positive ., amarginal increase in . implies that for any given number of previous customers, the rent
which is to be left to marginal customers in any period n (the type #n+1 , i.e., the lowest
type among the Örst-time consumers in period n) increases.19 In other words, the rent
increases as the length of the period of commitment shrinks. The reason is that as ) getssmaller, the marginal Örst-time customers have a stronger incentive to delay their Örstpurchase to the following period, unless this incentive is countered by giving them more
rent. In the limit as ) ! 0, r: ! 1 (this is shown easily by using LíHospitalís rule: theratio of the Örst-order derivatives of the numerator and denominator of (23) equals in the18
As % ! 0, we see & ! 1=2 and ( ! 0. This means! that" when the commitment period is inÖnite, theÖrm serves only customers whose type ) belongs to 12 ; 1 , and the rent of the lowest type served by themonopolist is zero.
19
Note, however, the type )n+1 (when % increases) is not the same as the type )n+1 at the initial %.

22

0

+ (1)
0
limit (1++0 (1))r = 1 as 9 (1) = +1). In other words, in the limit, the consumers capture

all the beneÖts of the relationship and no proÖt is left to the monopolist.
Let us conÖrm that aggregate proÖt vanishes as the length of the commitment period
tends to zero. The monopolistís equilibrium aggregate proÖt, or the value of the Örm (i.e.,the sum of all future discounted proÖts), as viewed at time t = 0, can be easily expressed
as a function of ., as given below:1/(.) =
r'(1 " .)9 # (.) [1 " 9 # (.)]
[1 " .9 # (.)] [1 " .9 # (.)2 ]

(

:

(25)

The relationship between / and . is pictured in Figure 4 below.20 The x-axis represents .; whereas the y-axis depicts equilibrium aggregate proÖt.
Insert Figure 4

The Ögure shows that the aggregate proÖt is decreasing as . increases, i.e., as thelength of the period ) of commitment shrinks. This result is intuitively plausible: one
expects that aggregate proÖt is greatest when the monopolist can fully commit right fromthe beginning to a sequence of contracts. (As we shall show in the next section, this
implies sticking forever to the monopolistís initial price.) Intuitively, the opportunity of
reoptimizing repeatedly very soon ends up being detrimental to the Örm, because rationalcustomers would then expect that they do not have to wait very long to beneÖt from thefuture surplus, and accordingly they would buy in the current period only if they areo§ered greater current rents (lower prices) to purchase immediately.
1
When ) tends toward inÖnity (so that . ! 0), /(0) = 4r
:This is equivalent to an

inÖnite repetition of the static equilibrium. For the other polar case, where . ! 1 (i.e.,) ! 0), Claim 2 below states formally that the limit of /(.) when ) ! 0 is equal to0. This is again a Coasian result but in our set-up it applies to a monopolist producing
a non-durable good (instead of the durable goods case already studied in the literature).
The Coase Conjecture indeed states that in the limiting case where the time interval thatelapses between two o§ers tends to zero, the durable-good monopolistís equilibrium priceis equal to the constant marginal cost, i.e., the proÖt is zero.
20

The picture is drawn for r = 1 (or, alternatively, the Ögure depicts r)(%) for any value of r).

23

Claim 2. Under Assumption B, the aggregate proÖt falls as ) decreases, and tends
toward zero as ) tends toward zero.
Proof. See the Appendix.

5

Comparison with the case of commitment

Suppose now that, contrary to what has been assumed in the preceding sections, themonopolist is able to commit, right from the beginning, to a sequence of pre-determined
prices. This commitment capability is supposed to include the ability to commit bothwith regard to the price charged to new customers and to the prices charged the di§erent
groups of former customers, who are segmented according to the date of Örst purchase.
Thus, at the initial time t0 = 0, the monopolist announces (i) the old-customer pricesp(n; n + j) that it will o§er at any future period n + j (j = 1; 2; 3:::) to consumers whochoose to make their Örst purchase of the good in period n and (ii) the new-customer
prices p(n; n) that it will o§er at each period n to consumers who have not bought thegood before. When solving for the optimal sequence of pre-determined contracts, themonopolist has no knowledge of any consumer. The following analysis does not rely on
Assumption B; instead, we only need the weaker Assumptions A1-A2.
Let us consider consumers who are Örst-time buyers in period n: They face (a) a priceOp(n; n) = pN
n , when buying for the Örst time in period n and (b) a price p(n; n + j) = pn ;8j ! 1 for purchasing in all the remaining periods n + j (the superscripts N and O standfor new customers and old customers, respectively).21 These contract o§ers are announced
from the outset, i.e., at time t = 0.
For heuristic reasons, we Örst consider the Öctitious case when not only the monopolist
but also the consumers commit right from the beginning, i.e., we suppose provisionallythat when a customer purchases the good for the Örst time (say in period n), she not only
commits to purchase the good at all points of time during period n (paying during thisperiod the new-customer price pNn ), but she also commits to purchase the good in subsequent periods (at the old-customer price pO
n ; that was initially agreed on). Technically,
this means that we only need to consider the consumersí initial participation constraints:
they are not allowed to renege later on their contractual commitments. (Afterwards,
21
Allowing contract o§ers made to old customers to be (possibly) di§erent across periods would notchange the results.

24

we will show that when we remove this supposition of commitment by customers, themonopolistís optimal path and consumersí equilibrium choices are not a§ected).
Under this assumption of commitment, let Un (#n+1 ) denote the marginal customersíintertemporal net utility in period n (that results from all purchases in period n and inall subsequent periods):$+)*. #n+1 " pO
(1 " .)(#n+1 " pN
#n+1
(1 " .)pN
.pO
n
n)
n
n
Un (#n+1 ) =+="+: (26)
r
r
r
r
r
!
"
O
Then, for all # 2 [#n ; #n+1 ], we have Un (#) = #=r + (1 " .)pN
n + .pn =r. This impliesthat1
Un0 (#) = for # 2 [#n ; #n+1 ] :(27)r
By a familiar argument, it follows from (27) thatUn (#n+1 ) =1X j. (#n+j " #n+j+1 ) :
r j=1
1

(28)

Clearly, equation (28) is the same as equation (9).
The aggregate proÖt (discounted to the beginning of period n) which the monopolist
makes from vintage-n consumers over their whole life time is then equal to:)*)*1". N . O
#n+1
4 n = [F (#n ) " F (#n+1 )]pn + pn = [F (#n ) " F (#n+1 )]" Un (#n+1 ) :r
rr
The aggregate discounted proÖt is the sum of discounted proÖts from all vintages,
P
n
/ % 1
n=0 . 4 n . Using (28), this can be rewritten asC/C %(1 " .) X n. (1 " F (#n+1 ))#n+1 :
r
n=0
1

(29)

Point-wise maximization with respect to #n+1 leads to #n+1 = ### ; 8n ! 0; where ###is the solution of(1 " F (### )) " f (### )### = 0:

(30)
$ +
Thanks to Assumptions A1 and A2 such a solution exists and is interior, i.e., ### 2 #; # .
25

This implies that in the initial period (n = 0), the monopolist sells only to those!
"
consumers with # 2 ### ; # , and in subsequent periods, n = 1; 2; 3:::; only these oldcustomers are served. In other words, #1 = ### and the monopolist does not expand themarket coverage after the initial period. This result, together with (28), implies thatU0 (#1 ) = 0, i.e., the marginal customers are left with no surplus. Now, from (26), itO
follows that the equilibrium prices pN
n and pn must satisfy the condition

1 " . N . O ###pn + pn =; 8n ! 0:
r
r
r

(31)

Up to now, we supposed that the customers cannot renege on their initial contractual
commitments: when vintage n customers become ìold customersî (in periods n+j, wherej ! 1) they are forced to buy at the contractual price pOn even if they would be better o§O
not buying. Notice that equation (31) does not determine uniquely pNn and pn : only their##weighted sum is determined. It is easy to see that the monopolist can set any pO
n ' #for all n ! 0 to guarantee that all former customers always purchase the good, even whenallowed to renege on initial agreements. Therefore, there is a continuum of commitment

equilibrium prices, but they all correspond to the same extent of market coverage, thesame consumersí intertemporal utilities and the same aggregate proÖt for the Örm. At
these equilibria, the consumers are o§ered the same weighted price for purchasing thegood, independently of the period of their Örst purchase.
Proposition 2. Assume A1 and A2 hold. If the monopolist can commit right from thestart to any sequence of contract o§ers, then only consumers whose type # belongs to the!
"interval ### ; # will be served, and they all make their Örst purchase in the initial period,n = 0. Moreover,(i) in all periods, they are o§ered the same time-invariant weighted price ( 1!)
pN
n +
r
) O
##
p ) = 1r ### ; with pO
n ' # ;
r n(ii) the Örm does not use the information it has acquired on Örst-time customers.
The equilibrium described in Proposition 2 (which does not rely on Assumption B) is

the inÖnite repetition of the static non-discriminatory monopoly equilibrium. The Örm
deÖnes its optimal sequence of contracts at the beginning of the game, at a time when
it only knows the distribution of types and, therefore, it is unable to attribute a type
neither to any speciÖc individual customer nor to any group of customers. The monopolist
thus optimally commits to refrain from selling to a new set of consumers in subsequent26

periods, denying his future selves the ìopportunityî to exploit, in subsequent periods, theinformation that he has acquired on his customersí preferences. Such a commitment has
the e§ect of reducing the high type customersí informational rents, thus diminishing their
incentives to delay their Örst purchase.
The monopolistís aggregate proÖt under commitment is then, from (29) and (30),/C =(### )2 f (### )
:
r

(32)

In order to compare numerically the proÖt under full commitment and that obtained
under non-commitment (the MPE proÖt), in what follows, we use Assumption B, which1
implies that eq. (32) reduces to /C = 4r
. Comparing with (25), we see that /C > /(.)

if 0 < . ' 1; which is of course to be expected. Our comparison of aggregate proÖts withand without price commitment reveals our curse of knowledge result. The monopolist
would be better o§ in the absence of records of the period in which each consumer entersthe market. Indeed, if the monopolist were unable to collect data about consumers, hewould neither segment the market (according to the time of consumersí Örst purchase)
nor price discriminate among di§erent cohorts of consumers, sticking to a uniform pricing
strategy instead.
Obviously the optimal strategy under commitment is not time-consistent. In otherwords, if at some future period m > 0 the Örm is released from its commitment, it will
have an incentive to deviate from its commitment. There are two reasons for this. First,
if allowed to reoptimize, the monopolist would always beneÖt from using his data on
the dates of old consumersí Örst purchase to carry out third-order price discrimination
among them. Second, when allowed in period m > 0 to make new o§ers to potential
new customers, he would beneÖt from selling to at least some of them in order to identify
them (and make proÖts from them in all subsequent periods). Rational customers who
expect to beneÖt from better o§ers in the future would then delay their purchase to thenext period unless they are granted lower present prices (higher rents).22
22

A monopolist that can commit is in e§ect an ìopen-loop Stackelberg leader,î and it is well-known that
open-loop Stackelberg leaders are typically beset by the time-inconsistency problem. For more detailed
discussion, see Chapter 5 of Dockner et al. (2000), and Chapter 1 of Long (2010).

27

6

Welfare under MPE

In this section, our objective is two-fold: (i) to investigate whether the welfare in the MPE
increases or decreases when the period of commitment becomes longer; and (ii) to compare
the welfare outcome under the MPE with that under the commitment equilibrium. For
the sake of simplicity, in this section, we rely on Assumption B.
Let w(n) denote the social welfare at the MPE in period n. We deÖne social welfare
as the sum of consumer surplus and proÖt. Consumer surplus is gross utility minus thepayment made to the Örm. ProÖt is the sum of consumersí payments to the Örm (because
we assume that the cost of production is zero). Therefore welfare in any period is simply
the gross utility of consumers who are served in that period. Thus the social welfare that
accrues in period n is
w(n) =" 1".
1". !
1 " #2n+1 =(1 " (9 n+1 )2 ):
2r
2r

(33)

Notice that, as can be seen from eq. (33), for any given r, a shortening of the period ofcommitment ) leads to a decrease in both the terms (1 " .) and (1 " (9 n+1 )2 ). Thus adecrease in ) unambiguously reduces the social welfare that accrues in period n. This isbecause of two factors: (a) the length of the period itself decreases, and (b) the measureof customers who are not served in period n, 9 n+1 , becomes larger.
The overall aggregate social welfare is the sum of the present values of w(n); over all
n:

1
X

1
W (.) =
. n w(n) =2r
n=0

1 " 9 (.)2
1 " . (9 (.))2

!

;

(34)

where 9 = 9(.) is the solution of the Euler equation (22). Even though w(n) unambiguously decreases as . increases, the e§ect of a marginal increase in . on . n w(n) turns out
to be ambiguous.
A close look at the expression for aggregate social welfare, eq. (34), shows that fora given value of r, an increase in . (i.e., a decrease in )) has two opposite e§ects on
aggregate social welfare: a direct e§ect (i.e., keeping 9 constant) which is positive and an
indirect e§ect, through the induced increase of 9; which is negative. In this respect, it isworth noting that the welfare per unit of time, w(n); is a step-wise increasing function,
the length of each step being equal to ): Accordingly, for a given market expansion rule28

(i.e., for a given value of 9 > 0; or a given height (1 " (9 n+1 )2 ) of the steps), a decrease of) (i.e., shorter steps) leads to a greater aggregate intertemporal welfare. This is because
over any interval [0; t], there are more steps when ) decreases.
Working in the opposite direction, the indirect e§ect takes place because a greater. implies a slower market expansion from period to period (i.e., 9(.) becomes greater,
and thus the height (1 " (9 n+1 )2 ) of the steps falls, when ) shrinks). Does one e§ect

dominates the other? Yes, in a global sense, the direct e§ect dominates. One can indeed
3
show that lim)!1 W (.) = 1=(2r), which is greater than W (0) = 8r
, so that W is globallyan increasing function of . (a decreasing function of )).23 Despite being globally increasingin . over the interval 0 < . ' 1; it is worth noting that, as shown in Figure 5, W is

not a monotonic function of . within this sub-domain: starting from . = 0; aggregatesocial welfare is at Örst (slightly) decreasing as . increases marginally, before beginning
to increase with . (after . reaches approximately 0:20).
Insert Figure 5
When the length of time ) is nearly inÖnite (i.e., when . is su¢ciently close to 0), thenegative e§ect of a marginal reduction of ) on welfare (due to a slower market expansion)
dominates the positive one. To show this analytically, let us compute dW=d., and evaluate
it at . = 0:
)
*
dW
@W
@W d9
1
d9
2 2
=
+
=
(1 " 9 )9 " 29 (1 " .)
d.
@.
@9 d.
2r
d.
where

d9
E)
"2. + (4. + 3. 2 )9 + (2 " 6. " 6. 2 )9 2 + (4. + 3. 2 )9 3 " 2.9 4
="
="
:
d.
E+
("2 + 2. 2 + . 3 ) + 2(2. " 3. 2 " 2. 3 )9 + 3(2. 2 + . 3 )9 2 " 4. 2 9 3
We know that at . = 0, 9 takes the value 1=2. Hence, the sign of dW
at (.; 9) = (0; 1=2)d)is given by "0:062 5 < 0. This shows that, starting at . = 0, a marginal increase in .
indeed leads to a fall in welfare, consistent with Figure 5.

23
In evaluating lim"!1 W (%), we make use of LíHospitalís rule. The limit is ratio between the Örstorder derivatives of the numerator and the denominator of (34). The proof is completed by appealing to
the fact that & 0 (1) = +1:

29

These results provide useful insights for the welfare comparison between the full commitment and the non-commitment scenario. Recall that in the non-commitment case3(MPE), for . = 0; we have W (.) = 8r
: As described above, when . is su¢ciently close tozero, welfare decreases in ., and afterwards it starts to increase, with the highest welfare
1
3
occurring when . tends to 1; with lim W (.) = 2r
> 8r
: In the full commitment scenario,)!1

the Örm adopts the uniform pricing policy and the welfare is simply equal toWcommit(.) =1
X. n wcommit (n);

n=0

hR
i
R#
1
3
with wcommit (n) = 0 e!rt 1 #d# dt. Therefore W commit (.) = 8r
= W (0). In other2
words, the welfare outcome in the commitment scenario coincides with that under MPE
3when . tends to zero, W (0) = 8r
. This was to be expected, since as . tends to 0; the com-mitment period ) tends to inÖnity and therefore in that limiting case the MPE withoutcommitment is isomorphic to the commitment scenario. Accordingly, since W (1) > W (0);we can conclude that provided . is su¢ciently close to 1 (i.e., the commitment period issu¢ciently short), the equilibrium welfare under MPE will be greater than the equilibrium
welfare with commitment, because the monopolistís market power is eroded.

7

Comparison with the polar case of full informationacquisition

In the previous sections, although the monopolist can collect information about consumersí purchase history, he cannot perfectly identify consumersí type. The informationis coarse: consumers are classiÖed in terms of vintage, but not in terms of their exact
preferences. When only such coarse information is available, the monopolistís future
selves will have an incentive to segment the markets and practice third degree price discrimination, o§ering several prices, one for each vintage of former consumers, and an
introductory price to customers who have not purchased before. Consumers anticipate
this, and therefore they demand a higher informational rent to purchase earlier rather
than later. Under these conditions, we are able to demonstrate the curse of knowledge,
and show that Coasian dynamics obtain in the sense that, as the commitment interval

30shrinks to zero, the monopoly power vanishes.
However, casual observations suggest that some Örms exert great e§orts to gather
consumer information. This apparently suggests they would proÖt from such information.
It is then worth investigating conditions under which the collection of information beneÖts
the monopolist. In this section, we show that if information is su¢ciently Öne to let themonopolist identify precisely each consumerís WTP, the curse of knowledge will no longer
hold.
To this end, we distinguish two polar cases of information quality and we then compare
the MPE of the resulting games. In both cases, the monopolist is able to collect some
information about the customers in the Örst period in which they purchase the good. In
the Örst polar case, the monopolist is only able to recognize a previous customer and toremember in which period she Örst purchased the good (we call this the Purchase History
Information case, PHI for short, which is the scenario treated in sections 1 to 6 of thispaper). The second polar case corresponds to the Full Information Case (FIA for short),
in which at the end of each commitment period, the Örm becomes fully informed about
the exact WTP of each customer who has purchased during the period. Accordingly,
in the FIA case, the monopolist is capable of charging each returning old customer apersonalized price equal to her WTP, i.e., he exercises Örst-degree price discrimination.
Di§erently, in the PHI case, the monopolist only recognizes the customersí vintage andtherefore he can only exercise third-degree price discrimination.
What happens then in the FIA case? Using the same framework and equilibrium
concepts as in the main part of this model, we can brieáy characterize the resulting MPEof the FIA model in the linear-quadratic case. The main changes are the following.
First, in each period n; in the FIA case, the Örm is able to extract all the surplus fromold customers when they return to make their purchase, i.e., the monopolist can practiceÖrst-degree price discrimination among former customers. The proÖts made in period nfrom all former customers are
1".
/Fn =
r

Z 1

#d# =

4n

1".
(1 " #2n )2r

as compared with its counterpart in the PHI case, equation (3).
Second, instead of equation (8), the arbitrage equation becomes

31

(35)

'

(
1".
Un (#n+1 ) = . Un+1 (#n+2 ) +(#n+1 " #n+2 ) :r(36)

Applying then the same arguments as in Section 4 but using these di§erent functions
yields the following results:
Proposition 3. Suppose that Assumption B is satisÖed. The Markov Perfect Equilibrium under full information acquisition is such that (i) the monopolistís cut-o§ rule islinear, with 9 #F IA (.) being the unique real solution in (0; 1) of the following fourth-degree
polynomial in 9 F IA
1 " (2 + .)9 F IA + (2. " . 2 )9 2F IA + 2. 2 9 3F IA " . 2 9 4F IA = 0; with 0 ' . < 1;(37)

and (ii) the consumer expectations rule is linear, with :# (.) given by:#F IA (.) =1 .(1 " .)(9 #F IA (.) " 9 #F IA (.)2 );r
1 " .9 #F IA (.)

The resulting aggregate discounted proÖt is given by/F IA =

1 (1 " 9 F IA )(29 F IA + .(1 " (3 " .(1 " 9 F IA ))9 F IA ));r2(1 " .9 F IA )(1 " .9 2F IA )where 9 F IA = 9 #F IA (.).
Proof. The arguments are along the lines used in the derivation of the MPE in the
PHI case. The full proof is available upon request. "/F IA is an increasing function of . (i.e., a decreasing function of )) as shown in Figure1
6.24 The proÖt is lowest when . = 0, with /F IA = 4r
; whereas the highest proÖt obtains
1for . = 1, with /F IA = 2r.

Insert Figure 6
We have that 9 #F IA (.) tends to 12 ; when . tends to zero. Moreover 9 #F IA (.) is decreasing in., which means it is always smaller than 12 ; tending toward - 0:445042 as . tends toward

1. The equilibrium :#F IA (.) is pictured below in Figure 7, evolving non-monotonically24This picture is drawn for r = 1.

32

with ..
Insert Figure 7
Comparing the PHI case with the FIA case, we Önd that the two corresponding Markov
Perfect Equilibria are strikingly di§erent, with very contrasting implications for the monopolistís proÖts. Recalling that the equilibrium proÖts in the commitment scenario were1
/C = 4r; Proposition 3 above shows that full information acquisition is actually beneÖcial
to the Örm: it always makes more proÖt compared to the case where it cannot recognize
its old customers. In contrast, collecting and using coarsest information on customersípurchase history hurts the Örm: its proÖts are smaller than under no information acquisition. If the monopolist cannot prevent his future selves from exploiting the information tocreate additional market segments and rational consumers correctly anticipate the Örmís
future prices, the ability to engage in third-degree price discrimination hurts the monopolist, which would not be the case if the information was Öne enough to implement
Örst-degree price discrimination, as shown in our FIA extension of the model. Our results
suggest that if the degree of coarseness of information can be represented by a continuous
variable, say F 2 [0; 1] then one should be able to prove that proÖt is non-monotonic in

F. For instance one could suppose that after a consumer has Örst purchased the good inperiod n, then, with probability 1 " F; the monopolist is able to discover her exact WTP,
whereas with probability F he simply learns that # 2 [#n+1 ; #n ]. The parameter F is thena measure of the degree of coarseness of the monopolistís information.

Finally, it is interesting to note that under PHI, the monopolistís aggregate proÖt decreases when the interval of time ) needed for changing the price o§ered to new customers
itself decreases, consistent with standard Coasian dynamics. In contrast, under FIA, themonopolistís proÖts are greater the smaller is ), departing from Coasian dynamics.

8

Conclusion

We have shown that the acquisition of purchase history information about customers to
implement third degree price discrimination based on customer recognition can hurt amonopolist. Accordingly, the monopolist would be better o§ by committing to a policyof not keeping any information about customers, so as to prevent his future selves fromengaging in opportunistic behavior. In other words, it may pay to tie oneís own hands.

33

We have illustrated this general point by formulating and analyzing a dynamic extension of the static monopoly model, such that the monopolist may segment the set of
consumers according to the date of their Örst purchase. We Örst looked at the equilibriumoutcomes when the monopolist can only collect coarse information, i.e., only customersíÖrst date of purchase. Then, we compared such outcomes with the ones arising in thepolar case in which the monopolist is able to get full information on the consumersí exact
WTP after they have made their Örst purchase.
Starting with the coarsest information case, we characterized the equilibrium under
two alternative scenarios: the non-commitment scenario (MPE) and the full commitment
scenario. In the former scenario, we found that the lack of commitment on the part of themonopolist makes customers demand more informational rents, and this is detrimental tothe Örm. In addition, we found that the Markov-Perfect Equilibrium of the game exhibits
Coasian dynamics: when the length of time between two di§erent price o§ers shrinks
to zero, the market is covered instantaneously and the monopolistís proÖt vanishes. By
contrast, if the monopolist could credibly commit to any sequence of contract o§ers, hewould commit to a policy of non-discrimination between new and old customers and to
stick forever to his initial contract o§er.
Our analysis demonstrated that in the absence of commitment, the acquisition ofknowledge can be harmful to a monopolist. This is what we call the curse of knowledge.
This suggests that in some monopoly markets, Örms may actually beneÖt from public
policies designed to limit Örmsí ability to collect and keep data records about their customers. Moreover, comparing the theoretical results of the coarsest information case to
those obtained in the polar case of full information acquisition, we found that in the lattercase the curse of knowledge does not arise. Instead, in this polar case Örms are able toengage in price personalization upon recognizing their returning customers, thus achieving
more proÖt (higher than the one obtained under full commitment to refrain from coarse
knowledge acquisition, which, in turn, is higher than the MPE proÖt under the coarse
knowledge scenario). These results suggest that a Örmís proÖt can be non-monotonic in
any continuous measure of its ability to gather customersí information.
Our theoretical results are broadly consistent with pricing strategies in many markets,
such as telecommunication, software, electricity, TV and home video, etc., as reported in
Section 1. Moreover, as the subscription based economy starts to boom, the sophisticatedprice discrimination strategies considered in our model may apply to a wider range of34

sectors. Our comparison of the coarse information case and the full information caseallows us to explain two very distinct trends arising in these markets: in some markets (e.g.
online áights/ hotel booking platforms), Örms are exerting great e§orts to gather consumer
information and using it to implement increasingly sophisticated price strategies; while
in some other markets (e.g., iTunes, online videos), Örms stick to rather simple pricing
strategies. Both types of strategies may indeed make sense, since we showed that the
proÖtability of engaging in price discrimination critically depends on the accuracy of theÖrmís information on customers tastes.
Finally, let us note that in our simple setting, a consumer has only one choice: tobuy now, or to buy later. In practice, consumers may choose other actions, such as
frequency to visit a sellerís website, quantity and frequency of purchases, etc. If themonopolist can collect customer-speciÖc information on these actions, he will be able tohave a more reÖned segmentation of the customer base. These cases lie in between the twopolar cases, FIA and PHI, that we analyzed. A natural extension of the present analysis
would be to allow the degree of coarseness of information to take any intermediate value
between the two cases. We conjecture that the curse of knowledge result would hold if themonopolistís acquired information is su¢ciently coarse. Coasian-like results would fail if
acquired information is Öne enough.

35

APPENDIX
Proof of Proposition 1
Equation (22) follows directly from (19). It deÖnes the equilibrium value of 9 # (.) as
the solution of a fourth order polynomial. As shown in Figure 1, (22) has one positive
solution for all . > 0: It has also a negative one but only the former can be a valid
solution. Accordingly 9 # (.) is strictly increasing in ., from 9 = 12 when . = 0 to 9 = 1
when . = 1: "Proof of Claim 1
$
+ t
$
+ t
$
+ t
Since 9(1) = 1; lim 9(e!r# ) ! +1 = lim 9(e!r# ) ! : Notice that ln 9(e!r# ) ! =#!0#!0$ !r# +
$ !r#
+
t
t
ln
9(e
):
Since
9(1)
=
1;
ln
9(e
) is undeÖned when ) = 0: Then, in order##ln(+(e#r! ))to determine the limit value ofwhen ) ! 0, we have to use LíHospitalís rule#and evaluate the ratio of the derivatives of the numerator and the denominator at ) = 0:$
+ t0This ratio turns out to be equal to "r9 0 (1): It follows that lim 9(e!r# ) ! = e!r+ (1)t :#!0

Now let us determine 9 0 (1): Di§erentiating the identity (22), we obtain E)0 (.; 9(.)) +E+0 (.; 9(.))9 0 (.) = 0: Remembering that 9(1) = 1; it turns out that E)0 (1; 9(1)) =E 0 (1;+(1))

E+0 (1; 9(1)) = 0 so that the ratio " E$0 (1;+(1)) is undetermined. Again, we need to use%

LíHospitalís rule to Önd the value of 9 0 (1): Then, we have to di§erentiate the numerator and the denominator to Önally obtain 9 0 (.) ! +1 as . ! 1: It follows that$+ t
lim 9(e!r# ) ! = 0:"#!0

Proof of Claim 2

According to equation (25), the value of /(.) is undetermined when . = 1 sinceboth the numerator and the denominator equal zero. Let us then apply LíHospitalís rule.
The ratio of the Örst order derivatives of the numerator and the denominator is again
undetermined. So we consider the ratio of the second order derivatives. It is equal to
9 0 (1):(1 + 29 0 (1))(1 + 9 0 (1))
Using the fact that 9 0 (1) = +1; we obtain /(1) = 0:"
Derivation of equation (29)
Using the expression for 4 n , we obtain

36

)
*
)
*
! $ +
" #1
#2
1
/ = . F # " F (#1 )" U0 (#1 ) + . [F (#1 ) " F (#2 )]" U1 (#2 ) + :::etc
r
r
0

C

Substituting for

!4

1

r

)

"
!
"
" U0 (#1 ) and 4r2 " U1 (#2 ) etc., we obtain

*
'
(
'
(
#1
#1
#1 " #2#2 " #3
2" U0 (#1 ) =
".
".
+ :::
r
r
r
r

)

*
'
(
'
(
#2
#2
#2 " #3#3 " #42" U1 (#2 ) =
".
".
+ ::
r
r
r
r

Cancelling out terms, we Önally get/C($ + (1 " .)$ + (1 " .)1".
#1 + .F ##2 + . 2 F ##3 etc.
r
r
r
'
(
'
(
'
(
1".
1".
1".
3
"F (#1 )#1 " .F (#2 )#2 " . F (#2 )#3 etc.
rrr$ += . F #
0'This completes the proof. "37

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40

FIGURES

Figure 1. Equilibrium cut-o§ rule 9 # (.)

Figure 2. Market coverage as a function of )41

Figure 3. Equilibrium expectations rule : (.)

Figure 4. Equilibrium aggregate proÖt / (.)

Figure 5. Equilibrium aggregate social welfare, W (.) :42

Figure 6. /F IA as a function of . (r = 1)

Figure 7. :#F IA as a function of .(r = 1)43