History-based versus uniform pricing in growing and declining markets

Accepted Manuscript

History-based versus Uniform Pricing in Growing and Declining Markets Oz Shy, Rune Stenbacka, David Hao Zhang PII: DOI: Reference:

S0167-7187(16)30075-3 10.1016/j.ijindorg.2016.06.002 INDOR 2312

To appear in:

International Journal of Industrial Organization

Received date: Revised date: Accepted date:

22 October 2015 13 May 2016 1 June 2016

Please cite this article as: Oz Shy, Rune Stenbacka, David Hao Zhang, History-based versus Uniform Pricing in Growing and Declining Markets, International Journal of Industrial Organization (2016), doi: 10.1016/j.ijindorg.2016.06.002

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Highlights • The paper compares history-based with uniform pricing in growing and declining markets. • Under history-based pricing, firms charge higher prices to locked-in customers and lower prices to new customers.

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• In sufficiently declining markets, history-based pricing generates higher average prices, higher firm profits, and lower consumer surplus than uniform pricing.

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• In sufficiently growing markets, history-based pricing generates lower average prices, lower firm profits, and higher consumer surplus than uniform pricing.

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Oz Shy† MIT Sloan School of Management

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History-based versus Uniform Pricing in Growing and Declining Markets∗ Rune Stenbacka‡ Hanken School of Economics

David Hao Zhang§ Federal Reserve Bank of Boston

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June 14, 2016

Abstract

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We analyze the Markov Perfect Equilibria of an infinite-horizon overlapping generations model with consumer lock-in to compare the performance of historybased and uniform pricing in growing and declining markets. Under historybased pricing, firms charge higher prices to locked-in customers and lower prices to new customers. We show that a high exit rate of consumers (sufficiently declining market) constitutes a sufficient condition for history-based pricing to generate higher average prices than uniform pricing, thereby harming consumer welfare. In contrast, a high consumer entry rate (sufficiently growing market) ensures that history-based pricing intensifies competition compared with uniform pricing.

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Keywords: Growing market, declining market, history-based pricing, uniform pricing, consumer lock-in.

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JEL Classification Number: L1, L4 (Draft = INDOR2312-ms.tex 2016/06/14 09:46)

∗ The views expressed in this article are those of the authors and do not necessarily represent the views of the institutions they are affiliated with. We thank Andrew Rhodes, an editor, and an anonymous referee for valuable and constructive comments. † Corresponding author. E-mail: [email protected]. MIT Sloan School of Management, 100 Main Street, E62-613, Cambridge, MA 02142, U.S.A. ‡ E-mail: [email protected]. Hanken School of Economics, P.O. Box 479, 00101 Helsinki, Finland. § E-mail: [email protected]. Federal Reserve Bank of Boston, 600 Atlantic Ave., Boston, MA 02210, U.S.A.

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1. Introduction The literature on price discrimination with exogenous switching costs has facilitated comparisons between uniform pricing and history-based pricing within the framework of two-period models of duopoly competition. According to these comparisons, history-based pricing tends to intensify

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competition compared with uniform pricing, meaning that history-based pricing leads to lower prices and industry profits than uniform pricing, when evaluated over a two-period horizon (see, for example Chen (1997) or Esteves (2010)). In contrast, we show that in an infinite horizon overlapping generations (OLG) framework, history-based pricing can lead to higher average prices and industry profits than uniform pricing, particularly in sufficiently declining markets with a

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high consumer exit rate.

With uniform pricing the competing firms set a single price which is applied to all consumers. Such a uniform price has to balance the incentives to exploit locked-in customers with a high price (the harvesting effect) against the incentives to set a very competitive price in order to attract

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unattached new consumers so as to establish a customer base for the future (the investment effect). With history-based pricing in markets with lock-in, firms are able to disentangle the two incen-

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tives: they can charge higher prices to locked-in customers (the harvesting effect) and charge lower prices to new customers (the investment effect) at the same time. This mechanism implies a pricing

cost barrier.

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structure with an introductory offer succeeded by a phase of a high price to exploit the switching

In the present study we extend the restrictive two-period horizon commonly used in the liter-

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ature in order to conduct a comparison between history-based pricing and uniform pricing within the framework of an infinite horizon OLG model with market expansion and decline. In partic-

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ular, we address the following question: How do the rates of entry and exit of consumers affect the comparison between history-based and uniform pricing in markets with significant switching costs? Such an approach seems to particularly well capture many important features of subscription markets, with a subscription stage starting the customer relationship (with a possible “introductory discount”) and followed by repeated renewals at higher prices.1 1

Examples of these industries include cable TV and Internet, magazines and newspapers, colleges and universities,

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Our analysis reveals that the entry and exit rates of consumers play key roles for the comparison between history-based pricing and uniform pricing. In particular, we find that a sufficiently high exit rate of consumers (sufficiently declining markets) implies that history-based pricing yields higher average prices than uniform pricing. The mechanism is that firms have weak in-

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centives to invest in the acquisition of new customers if the horizon of the customer relationship is short (high exit rate of consumers) or if the firms discount future profits heavily (high discount rate). This result stands in sharp contrast with the perceived wisdom of the existing literature, which subscribes to the view that history-based pricing intensifies competition compared with uniform pricing.

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We also show that a sufficiently strong growth of the market, i.e., a sufficiently high entry rate of consumers, ensures that the uniform price exceeds the average price under history-based pricing. Thus, with sufficiently strong market growth history-based pricing intensifies competition compared with uniform pricing. This result confirms the common view that the effects of historybased price discrimination in two-period models can be extended to competition with an infinite

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horizon in markets with sufficiently high growth. Overall, our analysis supports the conclusion that with significant switching costs history-based pricing tends to be more problematic from the

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perspective of consumer welfare in a declining industry than in a strongly growing industry. Our results can be explained intuitively as follows. Compared with uniform pricing, history-

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based pricing has two effects: (i) it makes it easier for firms to harvest, leading to higher prices for locked-in customers; and (ii) it strengthens the incentives of firms to invest in order to acquire

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customers, leading to lower prices for new consumers. The magnitude of the investment effect relative to the harvesting effect then determines whether history-based pricing leads to higher average

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prices or not. The incentives to invest in order to acquire customers depend on the discount rate applied by firms and on the expected length of the customer relationships. With more myopic firms or with stronger industry decline due to a higher exit rate, the incentives to invest become weaker, which leads to higher average prices under history-based price discrimination. On the other hand, in a strongly growing industry where new consumers dominate the market due to a banking relationships, as well as insurance.

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high entry rate, the investment motive becomes more important, leading to lower average prices under history-based price discrimination. In the literature, comparisons between history-based and uniform pricing have typically been conducted within the framework of one- or two-period models. Chen (1997) focuses on a sym-

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metric duopoly model with a two-period horizon and heterogeneous switching costs and finds that competition with history-based pricing leads to more competitive outcomes than competition with uniform pricing, implying that consumers benefit from history-based pricing whereas firms lose. Similar qualitative conclusions are reached within the framework of a somewhat different two-period model by Esteves (2010). Focusing on an asymmetric duopoly model with in-

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herited market dominance Gehrig, Shy, and Stenbacka (2012) establish that consumers in such a configuration tend to benefit from history-based pricing. However, as Gehrig, Shy, and Stenbacka (2011) demonstrate, this conclusion does not carry over to a setting where the dominant firm has exclusive access to price discrimination.

Further, Chung (Forthcoming) presents a two-period duopoly model of experience goods with

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the feature that history-based pricing can lead to higher average prices than uniform pricing if consumers overestimate the quality of goods purchased at a more than moderate level. We focus

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on an infinite horizon OLG model where consumers correctly assess quality, and it shows that overestimation of quality is not required for history-based pricing to lead to higher prices than

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uniform pricing.

Overall, as Cabral (2016) emphasizes, two-period models of oligopoly competition with switch-

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ing costs tend to unrealistically create game-beginning and game-ending effects. Thus, he argues that many important features of dynamic competition with switching costs cannot be captured

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unless the analysis is conducted with an infinite horizon in which the state variable specifies the established customer relationships. Our model captures this aspect within the framework of an infinite OLG-model with the additional feature that at each point in time welfare has to be assessed by taking into account that the consumer population consists of both entering unattached consumers as well as consumers already locked-in with established customer relationships. In the literature, infinite horizon models have mostly been applied to study the effects of

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switching on competition with a focus exclusively on either uniform pricing or history-based pricing. For reasons of tractability we focus on an infinite OLG model with locked-in consumers, thereby adopting the same modeling choice as Beggs and Klemperer (1992) and To (1996) in their analysis of uniform pricing.2 Other studies analyzing the effects of switching costs with uniform

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pricing in an infinite-horizon model include Rosenthal (1986), Farrell and Shapiro (1988), Padilla (1995), Chen and Rosenthal (1996), Villas-Boas (2006), Arie and Grieco (2014), and Rhodes (2014). Cabral (2016) studies competition with history-based pricing in the presence of switching costs, but makes no comparison with uniform pricing.

Our study is organized as follows. Section 2 presents the continuous-time model with over-

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lapping generations of consumers who enter and exit the market randomly, and defines Markov perfect equilibrium price strategies. Section 3 analyzes competition with history-based pricing, whereas Section 4 focuses on uniform pricing. Section 5 compares the history-based with uniform prices and characterizes how entry and exit of consumers affect the price comparison. Section 6

2. The model Firms and consumers

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2.1

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compares consumer, producer, and social welfare under these regimes. Section 7 concludes.

There are two firms indexed by i = A, B. The firms produce homogeneous services (or goods).

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Consumers gain utility u ¯ from consumption, which means that u ¯ is the maximum amount that they are willing to pay for the service. Formally, we can refer to u ¯ as the reservation utility, or the

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willingness to pay. Our model is designed so that the history-based price discrimination is based on the lock-in effect and not on preference heterogeneity. Consumers face different adoption costs

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when initially choosing between brand A (τA ∈ [0, 1]) and brand B (τB ∈ [0, 1]). Therefore, each

consumer is indexed by τA and τB , that capture the adoption cost of brand A and B, respectively. The difference in adoption costs, given by (τA − τB ), is assumed to be distributed uniformly on the interval [−1, 1]. We define two “demographic” parameters to introduce growth (decline) in the population of

2 High switching costs and consumer lock-in are well documented in the literature. Shapiro and Varian (1998) provide many examples and discussions of consumer lock-in a wide variety of industries.

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consumers. (a) Consumers enter the market at a rate β ≥ 0 (birth rate or entry rate). (b) Existing consumers exit the market at a rate δ ≥ 0 (death rate or exit rate). D EFINITION 1. Let n denote the industry net growth rate, and N (t) the number of active consumers at

n=β−δ

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time t. Define,

N (t) = N (0)ent .

and

(1)

Then, we say that the market is growing if n > 0 and the market is declining if n < 0.

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New consumers purchase when they enter the market and thereafter become locked in with the firm they initially chose to purchase from. Existing customers purchase at a unit rate. More specifically, in a period of length dt, conditional on not exiting the market, an existing consumer purchases with probability 1dt. In aggregate, since there are N (t) existing (locked-in) consumers, the expected number of purchases by locked-in customers is N (t)dt. This means that, in a period

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of length dt, new consumers buy βN (t)dt units (since there are βN (t)dt entering consumers during dt) and existing consumers buy N (t)dt units, whereas consumers who exit the market do not buy

βN (t)dt βN (t)dt+N (t)dt

=

β β+1

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during the time interval dt. Thus, the proportion of products purchased by new consumers is ∈ [0, 1).

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Note that the exit rate δ has no effect on the proportion of products purchased by new consumers. This holds true because a decrease in N (t) induced by a higher exit rate δ reduces both

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the number of units bought by new consumers βN (t)dt and the number of units bought by existing consumers N (t)dt. Loosely speaking, a change in the exit rate affects the number of entering

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consumers and existing consumers at the same rate. To ensure that the market is fully covered, we assume that the utility rate u ¯ is sufficiently

high such that all consumers subscribe to the offered service (buy the offered good). A sufficient condition for the market to be covered under the two pricing regimes to be analyzed (historybased pricing and uniform pricing) is: A SSUMPTION 1. The rate of willingness to pay is bounded from below. Formally, u ¯ > 2 + β1 . 5

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We assume that consumers are locked in with the supplier they select when they enter the market, meaning that they are unable to switch providers after the initial purchase (or, alternatively, that they cannot be profitably poached due to high switching costs).

Dynamic profit maximization and the equilibrium concept

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2.2

Firm i has a market share of σiL (t) at time t among locked-in consumers and σiN (t) among new consumers with the shares for firm j being σjL = 1 − σiL and σjN = 1 − σiN , respectively. With L L L differentiated pricing firm i sets the price pN i (σi ) for new consumers and pi (σi ) for locked-in

consumers. These prices are contingent on the state variable σiL . Note that σiN is not a state

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variable since it can be changed instantaneously through current period prices. Under uniform L L L U L pricing, pN i (σi ) = pi (σi ) = pi (σi ).

Firm i’s instantaneous profit and time discount rate are denoted by πi (t) and ρf . Assuming L L L zero production cost, at each period t0 , firm i chooses a continuous time price strategy (pN i (σi ), pi (σi ))

to maximize the value function given by

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Vi (t0 ) =

Z∞

e−ρ

f (t−t ) 0

πi (t)dt,

i = A, B,

(2)

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t0

N L L where πi (t) = N (t0 )en(t−t0 ) [pN i (t)βσi (t) + pi (t)σi (t)]. For the profit maximization problems

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to be defined (Vi being finite), we impose the following condition. A SSUMPTION 2. Firms discount the future more heavily than the net growth rate of the consumer popu-

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lation. Formally, ρf > n.

At the time of adoption, the new consumers who are entering the market choose a brand to

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maximize a discounted stream of utilities with a discount rate ρc . Recall that consumers have heterogeneous initial adoption costs. Each new consumer entering at t0 chooses a brand i according to

argmax Ui = u ¯ − τi − i∈{A,B}

pN i (t0 )

+

Z∞

t0

e−(ρ

c +δ)(t−t ) 0

[¯ u − pL i (t)]dt.

(3)

The first three terms in (3) constitute the initial utility net of adoption cost and the introductory price. Combined with the last term, (3) implies that consumers choose the brand that yields the 6

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highest value net of brand adoption cost, current price, and net of all discounted future prices during the consumers’ lifetime when the consumers are locked in with the chosen brand. Consumers c +δ)(t−t ) 0

discount the future by e−(ρ

. As seen in (3), the discount rate ρc and the exit rate δ have

similar effects on the consumers’ brand selection. The parameter δ determines the expected dura-

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tion of customer relationships and thereby the industry growth rate n. This parameter influences firms’ return from an acquired customer relationship and thereby the intensity of competition for new customers.

It should be emphasized that our model is based on rational and forward-looking consumers with brand selection determined by (3). This feature distinguishes our analysis from Fabra and

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Garcia (2015) who also develop a continuous time uniform pricing model. But in their model, consumers “refresh” their preferences at each instant in time, and they do not incorporate full consumer rational expectations in the sense that consumers anticipate future prices based on current market shares.

We apply the simultaneous-move Markov Perfect Equilibrium (MPE) developed in Maskin

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and Tirole (1988) as the equilibrium concept of our study. A Markov pricing policy is a mapping L L L from the locked-in market share σiL to a vector of prices pi (σiL ) = (pN i (σi ), pi (σi )). As explained

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in Fabra and Garcia (2015), who also applied the same equilibrium concept in a continuous time L ), p∗ (σ L )i is a MPE if and only if at any instant context, a pair of Markov price strategies hp∗A (σA B B

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time t

hp∗i (σiL ),p∗j (σjL )i

Vi

hpi ,p∗j (σjL )i

(σiL (t)) ≥ Vi

(σiL (t))

(4)

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for any possible single-period deviation price pi and all market shares σiL (t) ∈ [0, 1], where σjL (t) = 1 − σiL (t), i = A, B and i 6= j. Intuitively, MPE strategies satisfy the property that, for any given

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market share of existing consumers, no firm should have any incentive to deviate at any instant point in time from its MPE price strategy, given that both firms adhere to their MPE strategies at any later point in time. For uniform pricing, as with the earlier literature, we look for a MPE in symmetric affine strate-

gies. Other equilibria may exist. For history-based pricing, our proposed MPE is unique.

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3. History-based pricing With history-based pricing (HBP), firms differentiate the price targeted for entering consumers from the price offered to locked-in consumers. More precisely, this pricing regime focuses on competition when firms charge two prices: pL (σiL ) to their existing (locked-in) customers, and

Result 1. With history-based pricing, the unique MPE is given by

yielding an average price of HBP

p¯

=p

N∗



β 1+β



L∗

+p



1 1+β

at each point in time.

ρf

u ¯ , +δ

for all σiL ∈ [0, 1],

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pL∗ = u ¯ and pN ∗ = 1 −

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pN (σiL ) to new customers. Appendix B proves the following result.



 = 1−

u ¯ f ρ +δ



β 1+β



+u ¯



1 1+β



(5)

(6)

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Our model focuses on a market where customers are locked-in with established customer relationships (high switching costs). This type of market competition does not impose any discipline

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on the ability of the firm to exploit the willingness to pay on behalf of its customers. Thus, the firm can benefit from established customer relationships by targeting the price pL∗ = u ¯ to this customer segment independently of competition. We can refer to this as the harvesting effect associated with

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history-based pricing.

With history-based pricing, the focus of competition is directed to the segment of entering,

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unattached consumers. Competition for new consumers is intense, because at that stage firms take the expected future value of an established customer relationship into account. The expected

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value of an established customer relationship is determined by the combination of two effects: the expected duration of the customer’s market presence (inverse of δ) and by the firms’ discount rate (ρf ). In this respect we can refer to the sum δ + ρf as the ”effective discount rate”, which

determines how valuable it is for a firm to acquire a customer relationship given that customers have the exit rate δ and the firms discount future profits by the discount rate ρf .

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Competition for unattached consumers therefore leads the firms to offer introductory discounts, which correspond to the value firms attach to the (lifetime) customer relationships. In particular, as (5) shows, the price charged to unattached consumers, pN ∗ = 1 −

u ¯ ρf +δ

, is strictly

increasing as a function of the exit rate of consumers δ and as a function of the firms’ discount rate

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ρf . This price captures the investment effect, whereby the firm builds up its stock of locked-in customers. Furthermore, from (5) we can conclude that competition for unattached consumers can be so intense as to generate short-term losses associated with the segment of entering consumers at the stage of competition for customer relationships. This happens when the price rate charged to locked-in consumers (¯ u) exceeds the effective discount rate (δ + ρf ), i.e. when the customer

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relationships are sufficiently valuable compared with the effective discount rate.

From (5) we can infer that the price targeted to existing customers, pL∗ = u ¯ , as well as that targeted to entering consumers without history, pN ∗ = 1 −

u ¯ ρf +δ

, are each independent of the rate of

consumer entry (β). In fact, with history-based pricing the rate of consumer entry only affects the proportions of consumers targeted by pL∗ and pN ∗ , respectively. Thus, the rate of consumer entry

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only affects the average price associated with history-based pricing. This average price is nevertheless important as it determines the profit rate of firms, firm value, and the rate of consumer

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surplus (we discuss this in more detail in Sections 5 and 6). An increased entry rate of consumers (higher β) implies a reduced average price, meaning that an increased entry rate makes the invest-

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ment effect stronger. On the other hand, a reduced β implies a higher average price as it makes the harvesting effect stronger.

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We formally summarize the effects of consumer entry and exit rates on the average price with history-based pricing in the following result (based on (5) and (6)).

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Result 2. With history-based pricing, the average price (¯ pHBP ) is (a) decreasing as a function of the customer entry rate β.

(b) increasing as a function of the customer exit rate δ. (c) increasing as a function of the firms’ time discount rate ρf . According to Result 2, average prices fall with an influx of new customers into the industry

(higher β) since a larger proportion of the consumers are offered introductory discounts. This is 9

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the investment effect. With an increased effective discount rate, due to either (i) a higher exit rate of existing customers (higher δ) or (ii) firms operating with a higher discount rate ρf , this investment effect is weakened. Further, for any fixed n (including the special case where n = 0), the prices approach the static

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Hotelling price of 1 when β → ∞ and δ → ∞, as the market under such circumstances no longer has any locked-in consumers.

4. Uniform prices

We next turn to uniform pricing (U) and assume that each firm is restricted to setting a single state-

the MPE price strategies as described in (4).

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contingent price applied to all consumers (entering as well as locked-in consumers). We look for

With uniform pricing and faced with a continuous flow of entering and exiting consumers, the firms have to determine the price as a compromise between the incentive to exploit locked-in customers and the incentive to attract new customers. In light of these considerations, intuition

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suggests that the uniform price rate should depend importantly on the entry and exit rates of consumers as well as on the firms’ discount rate.

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The derivation of an equilibrium with respect to uniform prices requires stronger restrictions on the parameters of the model than with history-based prices. The reason is that under uniform

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pricing and with a sufficiently low rate of consumer entry (low β), firms may be tempted to deviate to pU = u ¯ in order to extract maximum surplus from their locked-in consumers. Such an

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incentive would lead to a boundary solution.3 However, such a pricing strategy is not sustainable since no new consumer would buy at the price rate u ¯ due to adoption costs. For uniform pricing,

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we therefore assume that there is a sufficiently high rate of consumer entry, that is, a sufficiently high rate of price-sensitive customers so that the equilibrium strategy under uniform pricing constitutes an interior solution. This may require an addition Assumption on the lower bound of β if

u ¯ is greater than approximately 2.25. (otherwise, Assumption 1 is sufficient). Formally, 3

Some of the existing studies face exactly the same problem. Beggs and Klemperer (1992) and To (1996) address this issue by introducing an upper bound on u ¯.

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√
A SSUMPTION 3. The rate of consumer entry is bounded from below. Formally, if u ¯ > 17 9 + 46 ≈ √ 2.25, we assume that β > u ¯ − 31 + 31 9¯ u2 − 6¯ u + 4. Otherwise, β > (¯ u − 2)−1 which holds by Assumption 1. In combination with Assumption 1 the parameter restrictions define a non-empty set if the entry

purchasing in the market, which we earlier characterized to be

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rate of consumers satisfies β > 3.93. This means that the proportion of price-sensitive customers β 1+β ,

exceeds 79.7%.

Assumption 3 is merely a sufficient condition, and our proposed uniform pricing equilibrium may also apply for lower values of β, particularly if firms are forward-looking and the consumer exit rate is low. Indeed, Appendix C proves that, as ρf , δ → 0, the required bound on β for

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an interior equilibrium approaches zero. Nevertheless, we impose this condition so that we can allow the parameters ρf and δ to vary without additional restrictions.

Further, since Assumption 3 specifies an absolute boundary on the rate of consumer entry, the price comparisons conducted in Section 5 do not explore any results with small values of β.

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Despite this restriction, it should be emphasized that our model is very flexible as far as results regarding the implications of market growth are concerned. Because the exit rate δ is a free pa-

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rameter, we can still model any rate of growth (or decline) n despite the restriction on β. Appendix C proves the following result.

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∗ L Result 3. Under uniform pricing, there exists a unique MPE in symmetric affine strategies pU i (σi ) that

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is increasing in σiL , where the slope has the property that: 0<

∗ L 2 dpU i (σi ) < , L 3β dσi

for all σiL ∈ [0, 1].

(7)

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The unique globally stable steady stage market share among locked-in consumers is σiL = 21 . In the steady stage equilibrium, the uniform price lies between a lower (pU,L ) and an upper bound (pU,H ) defined by p

U,L

=



1 1+ β



δ + ρf 1 ∗ < pU i (σi = ) < 2 1 + δ + ρf

  1 13 + δ + ρf 1+ = pU,H . β 1 + δ + ρf

(8)

Furthermore, in steady stage the uniform equilibrium price converges towards the lower bound pU,L in 1 ∗ U,L . response to an increase in the discount rate of consumers (ρc ). Formally, clim pU i (σi = 2 ) = p ρ →∞

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With uniform pricing the firm has access to only one instrument with which to compete for a heterogeneous pool of consumers consisting of unattached consumers with no history as well as its inherited customers. The equilibrium with respect to uniform prices essentially balances two offsetting effects: a harvesting incentive to exploit the willingness to pay among locked-in

a stock of profitable customers to harvest in the future.

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customers and an investment incentive to acquire new, unattached consumers so as to accumulate

As Appendix C and the earlier literature (Beggs and Klemperer (1992) and To (1996)) make clear, the closed-form solution for the uniform price equilibrium with rational forward-looking consumers is very complex and cannot be easily compared with history-based pricing in general.

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However, as inequality (8) demonstrates, we can derive an interval, to which the steady stage value of the uniform equilibrium price belongs. The bounds of this interval, pU,L and pU,H , depend in a transparent way on the the parameters β, ρf and δ, thereby making economic interpretations possible. Another reason for why we apply the approach with bounds for the uniform price is that it facilitates a tractable comparison between the uniform price and the average history-based

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price.

An advantage of the continuous time approach is that market shares associated with the uni-

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form prices monotonically converge to the steady state. Thereby it is possible to avoid the feature with market share cycling that is characteristic of discrete time models, for example Beggs and

changes.

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Klemperer (1992) and To (1996), where firms face delays in their ability to react to market share

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It is a novel approach to characterize the bounds of an interval to which the uniform steady state price belongs. Such bounds facilitate the formulation of sufficient conditions for when the

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uniform price is lower (or higher) than the average price with history-based pricing. Furthermore, in the special case with n = 0 (neither growing nor declining markets), the bounds converge

towards the static Hotelling price of 1 when β → ∞ and δ → ∞, which means that the uniform

steady state price must also converge to the static Hotelling price.

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5. Uniform versus history-based pricing: A comparison of equilibrium prices and profits We now make use of the equilibrium characterizations derived in Sections 3 and 4 to analyze the effects on prices and profits of allowing firms to exercise history-based pricing compared with

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uniform pricing.

We first compare the dynamic paths of prices and market shares associated with the two pricing regimes. Because firms charge higher prices when they have a larger market share under uniform pricing, whereas the prices are invariant to the market share with history-based pricing, we can conclude that the acquired, locked-in market shares converge to

1 2

faster under uniform

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pricing than under history-based pricing. More formally, we obtain the following result. Result 4. Starting at any σiL (t0 ) ∈ [0, 1], σiL (t) →

1 2

as t → ∞ under both uniform and history-based

L = σL = pricing. Furthermore, the rate of convergence to the unique steady stage of σA B

1 2

is slower under

L = σL = Since σA B

1 2

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history-based pricing than with uniform pricing.

is the unique steady stage of both pricing regimes, we restrict all the

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comparisons in the remaining part of the study to satisfy the following: L (t) = σ L (t) = 1 , A SSUMPTION 4. Price competition is at the steady stage. Formally, σA B 2

Myopic consumers

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5.1

for all t.

We start our comparison between the average price with history-based pricing and the uniform

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price by focusing on myopic consumers in this subsection, and then proceed to show that the general pattern also holds for forward-looking consumers in Section 5.2. Based on a comparison

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between (6) and (8) we find the following result. Result 5. For myopic consumers (ρc → ∞), the average price with history-based pricing (¯ pHBP ) is higher than the uniform price (pU ∗ ) if and only if f

def

u ¯ > u ¯2 (β, δ + ρ ) =

β2 δ+ρf −β

13

−

β+1 δ+ρf +1

β

+ 2β + 1

.

(9)

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Result 5 specifies that the relationship between the average price with history-based pricing and the uniform price is determined by the parameters of the model according to (9). We can view the parameter relationship (9) from two different perspectives: (a) from the perspective of δ + ρf , which is the effective discount rate at the stage when firms compete with history-based pricing

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to acquire customers, or (b) the entry rate of new consumers β. Based on the strict monotonicity and limit properties of the function u ¯2 (β, δ + ρf ) defined in (9), Appendix D proves the following results.

Result 6. (a) There is a feasible effective discount rate δ + ρf such that the average price with historybased pricing is higher than the uniform price if and only if δ + ρf > δ + ρf .

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(b) The uniform price always exceeds the average price with history-based pricing if the entry rate of new consumers β is sufficiently high. Formally, there exists a β¯ with the feature that the uniform price ¯ exceeds the average history-based price for all β ≥ β.

Result 6 offers important insights regarding the effects of market growth or market decline

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on the relationship between the average price with history-based pricing and the uniform price. The first conclusion is that the exit rate δ is decisively important for the relationship between

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the average price with history-based pricing and the uniform price. A sufficiently high exit rate of consumers, i.e., a sufficiently declining market, implies that the average price with history-

PT

based pricing exceeds the uniform price. The exit rate is a significant determinant of the value of acquiring a new customer, therefore affecting the price targeted to new consumers. Furthermore,

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from (9), we see that δ and ρf are “perfect” substitutes for one another as far as comparisons between the uniform price and the average price under history-based pricing are concerned. For sufficiently high δ or ρf , the firms have very weak incentives to invest in the acquisition of new

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consumers, because the horizon of the customer relationship is short (high δ) or future profits are

heavily discounted (high ρf ). Weak incentives to offer aggressive introductory prices will increase the average price under history-based pricing so that it exceeds the uniform price. The second, important conclusion to draw from Result 6 concerns the effect of market growth on the relationship between the average price with history-based pricing and the uniform price. In this respect the lesson from Result 6 is that a sufficiently strong growth of the market, i.e., a 14

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sufficiently high entry rate of consumers (β), is a sufficient condition for the uniform price to exceed the average price with history-based pricing. As (6) shows, a sufficiently high entry rate of consumers means that the introductory offer is targeted to a sufficiently high proportion of consumers. And when a sufficiently high proportion of the consumers face the introductory offer the

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average price with history-based pricing will be lower than the uniform price, which is reached through the process of balancing the investment and harvesting effects with only one price instrument, as discussed in Section 4.

Rhodes (2012) also compares history-based pricing with uniform pricing in an OLG model. He finds that, similar to two-period models of Chen (1997) and Esteves (2010), history-based pricing

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leads to lower firm profits than uniform pricing. In contrast, we identify situations where historybased pricing can lead to higher average prices and higher profits than uniform pricing. The difference is that we evaluate the effects of history-based pricing in the presence of consumer entry and exit with high switching costs, such that the established customer relationship is based

further intensifies competition.

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on lock-in. In Rhodes (2012), the switching costs are low enough to facilitate “poaching” which

We finalize this section by conducting several numerical simulations in support of our results.

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The simulations are illustrated in Figures 1 and 2. These figures show the price equilibrium in the two pricing regimes for the following set of parameters: u ¯ = 4, ρf = 10, and ρc → ∞ (so that the

PT

uniform pricing equilibrium is equal to the lower bound pU ∗ = pU,L ). In addition, for Figure 1 we set δ = 10, and for Figure 2 we set β = 10.

CE

According to Result 2 and (8), the average price with history-based pricing (¯ pHBP ) as well as the uniform price (pU,L ) decrease as a function of β and increase as a function of δ. These

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properties are verified by Figures 1 and 2. Illustrating our Result 6, Figure 1 shows that the average price with history-based price discrimination, p¯HBP , is lower than the uniform price in markets with sufficiently high β (growing markets). Similarly, Figure 2 illustrates that the average price with history-based price discrimination, p¯HBP , is higher than the uniform price in markets with sufficiently high δ (declining markets).

15

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Price 1.15

Price 1.15

1.10

1.10

1.05

1.05

p HBP p

1.00

U

p HBP pU

1.00

0.95 0.90 10

11

12

13

0.90 0

β 15

14

2

Figure 1: The effect of the entry rate (β) on the rank- Figure 2: ing between the average history-based price and the uniform price for myopic consumers.

4

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0.95 6

8

10

12

14

δ

The effect of the exit rate (δ) on the ranking between the average history-based price and the uniform price for myopic consumers.

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While we know that the average price with history-based pricing will be lower than the uni¯ the effect of β at the lower end can be non-monotonic.4 Due form price for sufficiently high β > β, to our parameter restrictions, this non-monotonicity is only arises for large values of δ + ρf and small β, so that the selection of parameters in Figure 1 does not exemplify this possibility.

Forward-looking consumers

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5.2

We now turn our attention to the case with forward-looking consumers. Comparisons between the

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average price with history-based pricing in (6) and the lower and higher bounds of the uniform price, given by (8), respectively, make it possible for us to formulate the following result.

PT

Result 7. Consider forward-looking consumers (0 ≤ ρc < ∞). A sufficient condition for the average price

CE

with history-based pricing to exceed the uniform price (¯ pHBP > pU,H ) is that def

u ¯>u ¯1 =

1 2(β + 1) 4β + 1 + +2− . 3 [ρf − (β − δ)] β 3β(δ + ρf + 1)

(10)

AC

Conversely, a sufficient condition for the average price with history-based pricing to be lower than the uniform price (¯ pHBP < pL,H ) is that u ¯
(δ +ρf ) as well as the entry rate (β). Applying an approach similar to that associated with Result 6, 4

In particular, for β0 = when β > β0 .

p

(δ + ρf + 1)−1, u ¯2 (β, δ +ρf ) is strictly decreasing in β when β < β0 and strictly increasing

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Appendix E proves the following findings. Result 8. Consider forward-looking consumers (0 ≤ ρc < ∞). (a) The average price with history-based pricing exceeds the uniform price if the effective discount rate (δ + ρf ) is sufficiently high.

is sufficiently high.

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(b) The uniform price exceeds the average price with history-based pricing if the rate of consumer entry (β)

(c) If uniform pricing exceeds the average price with history-based pricing for myopic consumers (ρc → ∞), it also does so for forward-looking consumers (0 ≤ ρc < ∞).

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As emphasized in the introduction, the existing literature tends to subscribe to the view that history-based pricing leads to more intense competition than uniform pricing. In light of that view it is particularly interesting to explore Result 8(a) in greater detail in order to characterize those circumstances when history-based relaxes competition relative to uniform pricing. In other words, we are particularly interested in characterizing the circumstances specified in Result 8(a)

the literature typically specifies.

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where history-based pricing affects industry performance in a direction that is different from what

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Since our model assumes high switching costs (lock-in), the effective competition between firms takes place at the stage when consumers enter no matter whether we focus on history-based

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or uniform pricing. However, from the perspective of an entering consumer the intertemporal pricing profile associated with the two pricing systems looks very different. With history-based

CE

pricing the benefits of competition are realized immediately for the entering consumer, whereas these benefits are spread out over the whole customer’s horizon with uniform pricing. With a short expected horizon (high exit rate δ) the expected length of the phase with the lock-in price

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(pL∗ = u ¯), which is higher than the uniform price, is short. For a sufficiently high exit rate the equilibrium price targeted to entering consumers would then be increased to such an extent that the average price with history-based pricing exceeds the uniform price. Furthermore, as we have emphasized before, the effect of the firm’s discount rate (ρf ) is equivalent to that of the exit rate δ. For the special case where n = 0, whether uniform pricing leads to higher prices than history-

based pricing is ambiguous. Nevertheless, holding n constant, we can say that a high ρf is a 17

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sufficient condition for history-based pricing to yield a higher average price than the uniform price. Overall, a higher consumer discount rate (ρc ) means that consumers pay more attention to the current price and less considerations related to the phase of future lock-in. This mechanism

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would provide an argument for firms to lower their uniform price in response to an increase in the consumers’ discount rate, as shown in the models of Beggs and Klemperer (1992) and Rhodes (2014). But, in addition, a higher consumer discount rate may also make firms’ pricing decisions more responsive to the market share, and this may provide a countervailing force, thereby making it difficult to prove monotonicity with respect to ρc in our model. Our model does allow us to

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show, however, that if the uniform price pU,L exceeds the average price with history-based pricing for myopic consumers (ρc → ∞), it must do so for forward-looking consumers with any 0 ≤ ρc < ∞. This is documented in Result 8(c).

We illustrate Result 8 with Figures 3 and 4, using the same parameters as in the last section except that we now allow for 0 ≤ ρc < ∞. Figure 3 shows that both the upper and lower bounds

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of the uniform price are higher than the average price with history-based pricing for high values of β. Similarly, Figure 4 shows that the average price under history-based pricing rises above both

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the lower and upper bounds of the uniform price for sufficiently high values of δ. Price 1.15

PT

1.10 1.05

0.95

11

12

13

1.10

p HBP

p HBP

1.05

p U,H p

U,L

p U,H 1.00

p U,L

0.95

14

15

β

0

AC

10

CE

1.00

Price 1.15

2

Figure 3: The effect of the entry rate (β) on the rank- Figure 4: ing between the average history-based price and the bounds for the uniform price under rational expectations with forwardlooking consumers.

18

4

6

8

10

12

14

δ

The effect of the exit rate (δ) on the ranking between the average history-based price and the bounds for the uniform price under rational expectations with forwardlooking consumers.

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5.3

Equilibrium profits

Next, we explore the effects of history-based pricing on the discounted sum of profits. We let p¯k denote the average price rate associated with pricing regime k, where k = U with uniform pricing and k = HBP with history-based pricing. From an arbitrary point in time t0 , when evaluated

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over an infinite horizon the discounted value of equilibrium profits in pricing regime k is V = R ∞ −ρf (t−t ) 0 N (t)¯ pk dt. We can refer to this as the firm value. A comparison of the equilibrium t0 e

profit rates and firm values across the two pricing regimes is equivalent to a comparison between the associated average price rates. In light of Result 7, we can therefore formulate the following

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result.

Result 9. (a) The inequality u ¯>u ¯1 , where u ¯1 is defined in (10), is a sufficient condition for history-based pricing to generate higher equilibrium profit rates and firm value than uniform pricing. (b) The inequality u ¯ < u ¯2 , where u ¯2 is defined in (9), is a sufficient condition for uniform pricing to generate higher equilibrium profit rates and firm value than history-based pricing.

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Again, as emphasized in our comments regarding Result 7, even in steady state the relationship between the average history-based prices and uniform price is too complex to facilitate

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closed-form comparisons for the whole space of feasible parameters. But, Result 9 formulates sufficient conditions facilitating an unambiguous ranking between the equilibrium profits associated

PT

with the two pricing regimes.

CE

6. Uniform versus history-based pricing: Implications for consumer surplus and total welfare

AC

In this section we explore the effects of the pricing regime on consumer surplus and total welfare. The rate of consumer surplus associated with pricing regime k, k = U or k = HBP , is given by CS(t)k = (1 + β)N (t)(¯ u − p¯k −

β ¯k ), 1+β τ

where τ¯k is the average adoption cost per unit purchased

by new customers. At the steady state, the firms price symmetrically in both price regimes, so consumers make their product selection independently of the pricing regime, meaning that τ¯U = τ¯HBP . Therefore, the average price rate determines the effect of the pricing regime on the rate of 19

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consumer surplus. This means that we can directly apply Result 7 to reach conclusions regarding the comparison between the consumer surplus associated with history-based pricing and that associated with uniform pricing. Total welfare is defined as the sum of industry profits and consumer surplus. The rate of total

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welfare with pricing regime k, k = U or k = HBP , is given by  β W (t) = πA (t) + πB (t) + CS(t) = (1 + β)N (t) u ¯− τ¯ . 1+β k

k

k

k



(11)

The right-hand side of (11) is independent of the pricing regime, meaning that total welfare is invariant to whether the firms compete with history-based or uniform prices. Equation (11)

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formally captures the feature that a shift in the average prices across different pricing regimes represent only a redistribution between producers and consumers, thereby not affecting total welfare in a fully covered market.

We can summarize our findings in this section in the following way.

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Result 10. (a) The inequality u ¯>u ¯1 , where u ¯1 is defined in (10), is a sufficient condition for the rate of consumer surplus and for the discounted sum of consumer surplus to be higher under uniform pricing.

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(b) The inequality u ¯ < u ¯2 , where u ¯2 is defined in (9), is a sufficient condition for the rate of consumer surplus and for the discounted sum of consumer surplus to be higher under history-based pricing.

PT

(c) Total welfare is invariant to whether history-based or uniform pricing is applied. Making use of the same parameter values as in Section 5.1 (myopic consumers), we simulate

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the decomposition of total welfare into consumer surplus and industry profits as a function of the consumer entry and exit rate in Figure 5 and Figure 6, respectively, to illustrate Result 10. Such a decomposition can be undertaken because our model has the feature that total welfare is

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invariant to whether history-based or uniform pricing is applied. Thus, the share of total welfare not captured by consumer surplus constitutes the profits for firms. Figure 5 shows that, as a fraction of total welfare, consumer surplus is higher under history-based pricing in sufficiently growing industries (high β). Figure 6 demonstrates that the share of total welfare flowing into consumer surplus is lower under history-based pricing than under uniform pricing in sufficiently declining industries (high δ). 20

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0.78 π

Area equals profit as a fraction of welfare ( ) W

CSHBP W

0.74

CSU W

0.72 Area equals consumer surplus as a fraction of welfare (

0.70 10

11

12

13

CS W

14

π

0.76

Area equals profit as a fraction of welfare ( ) W

CSHBP W

0.74

CSU W

0.72

)

Area equals consumer surplus as a fraction of welfare (

0.70 0

15

β

2

4

6

8

10

CS W

)

12

14

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0.76

Fraction of W

Fraction of W

0.78

δ

Figure 5: The decomposition of total welfare into Figure 6: consumer surplus and industry profits as a function of the consumer entry rate (β) assuming myopic consumers.

The decomposition of total welfare into consumer surplus and industry profits as a function of the consumer exit rate (δ) assuming myopic consumer.

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It should be emphasized that our welfare conclusions depend on the fact that we focus on completely locked-in consumers so that there is no switching in equilibrium. A model with low switching costs would have to take additional considerations into account. Namely, with low switching costs, consumer surplus would suffer from switching and the aggregate switching costs would constitute a deadweight loss to the economy, thereby affecting total welfare.

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As shown in Result 1, the equilibrium with history-based pricing consists of an introductory offer followed by a phase of high prices when the firms exploit the locked-in consumers. By

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construction, the introductory price is sensitive to the average exit rate in the population of consumers. But, clearly, the different realizations of exit in the population of consumers will mean

PT

that some consumers face the lock-in price pL∗ = u ¯ for a long time, whereas the duration of this lock-in phase is short for others. Based on this feature the realized lifetime has consequences for

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whether an individual consumer is better off with history-based or uniform pricing. Formally, at a symmetric steady stage with σ L =

1 2

it holds true that pN ∗ = 1 −

u ¯ ρ+δ

< 1 < pU ∗

AC

and pL∗ = u ¯ > pU ∗ . Therefore, uniform pricing tends to become more favorable the longer the (ex-post) realized lifetime of an individual consumer. We formulate this finding in the following result.

Result 11. The realized individual duration of market presence determines the relative attractiveness of history-based pricing compared with uniform pricing for the consumer. More precisely, a consumer with longer market presence tends to benefit from uniform pricing. 21

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7. Concluding comments This study constructs an infinite horizon OLG model to demonstrate that the consumer entry and exit rates as well as the firms’ discount factor play key roles when evaluating the performance of history-based pricing compared with uniform pricing. In particular, challenging the conventional

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view in the literature, we characterize a sufficient condition for history-based pricing to generate higher average prices than uniform pricing. More precisely, we find that a sufficiently high effective discount rate, i.e., a sufficiently-high sum of the discount rate for firms and the exit rate of consumers, leads to higher average history-based prices than the uniform price, thereby harming consumer welfare. We also characterize a sufficient condition for uniform pricing to generate

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higher prices than the average price under history-based pricing. With a sufficiently high entry rate, meaning a sufficiently strong growth in the consumer population, uniform pricing generates higher prices than price discrimination. Thus, with significant switching costs history-based pricing tends to be more problematic from the perspective of consumer welfare in a declining industry

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than in a strongly growing industry.

We draw our conclusions from an analysis of the steady state in an infinite horizon OLG-

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model and we focus on the case of high switching costs. A modification of our model to capture low switching costs would most likely lead to more switching with history-based pricing. In models focusing on history-based pricing, the switching pattern is importantly determined by

PT

the poaching price, i.e., the price targeted to customers of a rival firm. Typically, this poaching price is higher than the price charged to new customers, but lower than the price charged to loyal

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customers.5 As we focus on consumers who are completely locked in once a customer relationship is formed, poaching prices play no role in our model. This feature makes it possible to analyze

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the interaction between the investment effect and the harvesting effect in a transparent way. It also facilitates our transparent characterization of the effects of consumer entry and exit on evaluations of history-based pricing compared with uniform pricing. Of course, it would be an important direction for future research to explore the effects of history-based pricing with low switching

5 Of course, the implementation of a poaching price requires that firms are able to separate customers of the rivals from new, entering consumers. In practice this could be challenging as switching consumers would have an incentive to claim that they are new consumers.

22

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costs within a dynamic framework with consumer entry and exit. Another possible extension of our approach would focus on the effects of history-based pricing for entry deterrence, where the potential entrant has no inherited locked-in consumers. Because our approach is dynamic, it can naturally be modified to capture such a configuration. Such an

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analysis would determine whether history-based pricing would be a more efficient instrument to exclude competitors than uniform pricing within the framework of an infinite horizon with consumer entry and exit. This issue has been analyzed by Gehrig, Shy, and Stenbacka (2011) within the framework of a static analysis and in the absence of considerations related to market expansion or decline.

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Our study is restricted to a comparison between history-based and uniform pricing with respect to prices, profits and welfare. But, we have not characterized which pricing system would emerge in equilibrium. It would be an interesting challenge for future research to also address this issue, because a characterization of the equilibrium pricing system could serve as a basis for testable empirical predictions about the circumstances under which we can expect to observe

A.1

Preliminaries for proofs in Appendix B and C

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Appendix A

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history-based pricing.

Derivation of market shares for locked-in consumers (σiL )

PT

We characterize an expression for σ˙ iL , which is useful for computing the first order conditions of the Hamiltonian. To do so, note that the instantaneous change in the number of consumers who

CE

are locked-in with firm i, N˙ iL (t), is equal to the difference between the “new” consumers who become locked in with that firm βN (t)σiN (t) and the number of that firm’s existing consumers

AC

who drop out of the market δN (t)σiL (t). Dropping t for clarity, we have: N˙ iL = βN σiN − δN σiL = βN σiN − (β − n)N σiL = βN (σiN − σiL ) + nN σiL , 23

(A.1)

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implying that N˙ iL = β(σiN − σiL ) + nσiL . N

d NiL N N˙ iL − NiL N˙ = dt N N2 L L ˙ ˙ N N N = i − i N N N

σ˙ iL =

= β(σiN − σiL ) + nσiL − nσiL

(A.2)

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= β(σiN − σiL ).

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Converting that to changes in market shares for locked-in consumers we find that

Therefore, we have the relationship σ˙ iL = β(σiN − σiL ). The rate of change of market shares for locked-in consumers, σ˙ iL , is proportional to the difference in market shares among new consumers and locked-in customers σiN − σiL with β as the factor of proportionality.

Derivation of market shares among entering consumers (σiN )

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A.2

We now find an expression for market shares among entering forward-looking consumers with

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rational expectations. Using consumers utility defined in (3), we have that for each brand i = A, B pN i

+

1 c ρ +δ



PT

Ui = u ¯ − τi −

CE

 = 1+

Z∞

e−(ρ

c +δ)(t−t ) 0

t0

u ¯ − τi −

pN i

−



Z∞

 u ¯ − pL i (t) dt c +δ)(t−t ) 0

e−(ρ

(A.3)

pL i (t)dt.

t0

The differences between utilities associated purchasing from different brands is given by

AC

Ui − Uj = (−τi + τj ) +

(−pN i

+

pN j )

+

Z∞

c +δ)(t−t ) 0

e−(ρ

t0

 L  −pi (t) + pL j (t) dt.

(A.4)

This is all we need for the price discrimination case. For the uniform pricing case, we solve

this in terms of symmetric affine strategies in Appendix C.1.

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Appendix B

Proof of Result 1

To find the equilibrium prices, we initially consider the optimization problems facing the firms.

Hi = πi + µi σ˙ iL L N N = N (t)(pL ˙ iL i σi + βpi σi ) + µi σ

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The Hamiltonian associated with the optimization problem of firm i is

L N N N L = N (0)ent (pL i σi + βpi σi ) + µi β(σi − σi ),

(B.1)

where µi is the shadow price associated with the market share. The first-order condition with

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respect to pL i is given by ∂Hi = N (0)ent σiL . ∂pL i Note that

∂Hi ∂pL i

(B.2)

> 0 for all σiL > 0. This holds true, because a single-period deviation in pL in-

L creases profit until pL ¯). This means i reaches a level where locked-in consumers do not buy (pi > u

that any Markov perfect equilibrium must satisfy pL∗ = u ¯. Otherwise, firms have an incentive to

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undertake single-period deviations that increase prices.

L∗ We next solve for pN ∗ . Substituting pL∗ ¯ into (A.4) implies that the integral term is i = pj = u

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zero. Thus, the expression characterizing the market share among new consumers is given by

PT

σiN =

1 1 N + (−pN i + pj ). 2 2

(B.3)

Substituting pL∗ = u ¯ into the first-order condition associated with the Hamiltonian with rei

AC

CE

spect to σiL is given by

∂Hi = N (0)ent u ¯ − µi β = ρf µi − µ˙ i . ∂σiL

Rearranging yields the following ordinary differential equation µ˙ i = (ρf + β)µ − N (0)ent u ¯.

25

(B.4)

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To solve this differential equation we substitute µ(t) = N (0)ent λ(t). This yields: ˙ nN (0)ent λ(t) + N (0)ent λ(t) = (ρf + β)N (0)ent λ(t) − N (0)ent u ¯.

˙ λ(t) − (ρf + δ)λ(t) = −¯ u.

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This can be simplified according to (B.5)

Applying the standard formula for the solution of a first-order ordinary differential equation with constant coefficients gives the solution

ρf

u ¯ f + Ae(ρ +δ)t , +δ

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λ(t) =

(B.6)

where A is a constant to be determined by the boundary conditions. Substituting this back into µ(t) = N (0)ent λ(t) yields µ(t) = N (0)ent

 u ¯ (ρf +δ)t + Ae . ρf + δ

(B.7)

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Therefore, the Hamiltonian is given by



ED

  N Hi = N (0)ent u ¯σiL + βpN σ + i i

  u ¯ (ρf +δ)t N L + Ae β(σ − σ ) . i i ρf + δ

(B.8)

Firm i chooses pN i in order to maximize this Hamiltonian. Taking the derivative with respect

PT

to pN i gives following the first-order condition: (B.9)

CE

  ∂Hi u ¯ nt 1 (ρf +δ)t N N = N (0)e β 1 − Ae − 2pi + pj − f = 0. 2 ρ +δ ∂pN i

By formulating the associated first-order condition for firm j, and solving the system of equations

AC

determined by the two reaction functions yields following price equilibrium: ∗ ∗ pN = pN i j =1−

ρf

u ¯ f − Ae(ρ +δ)t . +δ

(B.10)

Since prices are symmetric it must be the case that limt→∞ .σ L (t) = f

1 2

as long as β > 0.

To determine A we impose the transversality condition limt→∞ e−ρ t µ(t)σ L (t) = 0. We observe

26

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that lim e

t→∞

−ρt



L

(n−ρf )t

µ(t)σ (t) = lim N (0) e t→∞

 pL∗ (n+δ)t σ L (t) + Ae ρf + δ

= lim N (0)Ae(n+δ)t σ L (t). t→∞

A = 0. Substituting A = 0 into (B.10) establishes that ∗ ∗ pN = pN i j =1−

C.1

u ¯ . +δ

Proof of Result 3

(B.11)

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Appendix C

ρf

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Since limt→∞ σ L (t) = 12 , the only way for this limit to satisfy the transversality condition is that

Market shares of entering consumers

We characterize an equilibrium in symmetric affine strategies, meaning that consumers expect    
L − 1 , where dpL L = 2 dpL σ is the expected slope of the equilibrium price facpL − p i i j 2 dσ L dσ L e

e

function yields

+

pN j )

ED

Ui − Uj = (−τi + τj ) +

(−pN i

M

ing locked-in consumers with respect to market share. Substitution into the consumers’ utility

−

Z∞

2e

−(ρc +δ)(t−t0 )

t0



dpL dσ L

   1 L σi (t) − dt. 2 e

PT

In order to simplify this further we need an expression for σi (t). We observe that, under affine

pricing strategies, σiL (t) − 12 = e−c(t−t0 ) σiL (t0 ) − 21 for some constant c. We will later deter-

CE

mine c. Substituting this expression yields

   dpL 1 L Ui − Uj = (−τi + τj ) + + − e 2 σi (t0 ) − dt dσ L e 2 t0  L   1 dp 1 N N L , = (−τi + τj ) + (−pi + pj ) − 2 c σi (t0 ) − ρ + δ + c dσ L e 2 pN j )

Z∞

−(ρc +δ+c)(t−t0 )



AC

(−pN i

where (−τi + τj ) is uniformly distributed in [−1, 1]. From this we can conclude that firm i0 s market share in the segment of entering consumers is determined by the following market share equation:

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σiN

1 1 1 N = + (−pN i + pj ) − c 2 2 ρ +δ+c



dpL dσ L

   1 L σi − . 2 e

(C.1)

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We next determine c. For that purpose we also need to incorporate consumers’ expectations  
N = 2 dpN regarding future prices targeted to new customers, so let pN − p σiL − 12 , where L i j dσ e  N dp is the consumer expectation of the slope of how the price difference in the segment of dσ L e

entering consumers depend on the established market share. Substituting this into (C.1) yields     L   1 1 dp 1 dpN L L σ − − σ − i i dσ L e 2 ρc + δ + c dσ L e 2  N   L   1 dp 1 dp 1 = − + c σiL − . L L 2 dσ e ρ + δ + c dσ e 2 1 − 2



(C.2)

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σiN =

Substituting (C.2) into (A.2) defines the following linear ordinary differential equation with constant coefficients:

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σ˙ iL = β(σiN − σiL )  N   L     dp 1 dp 1 1 L L − + c σi − − σi =β 2 dσ L ρ + δ + c dσ L e 2   Ne  L    N   L   dp 1 dp 1 dp 1 dp L = −β 1 + + c σi + β 1 + + c . L L L dσ e ρ + δ + c dσ e 2 dσ e ρ + δ + c dσ L e Applying standard methods to solve this differential equation we find the following solution:       N  dpL 1 + ρc +δ+c (t−t0 ) −β 1+ dp L 1 1 L dσ dσ e e + , σiL (t) = σiL (t0 ) − e (C.3) 2 2 h  N  L i dp 1 from which we can conclude that c = β 1 + dp + (the unique solution with c L ρ +δ+c dσ L dσ

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CE

e

a positive value of c; a negative value of c requires that

e

1

ρc +δ+c

< 0, which leads to undefined

AC

utilities). Instead of expressing the solution in terms of c, we formulate it in terms of a more  L h  N i dp dp 1 relevant term C = ρc +δ+c satisfying the relationship c = β 1 + + C . In light of L L dσ dσ (C.1) we can then express firm

to

σiN

e 0 is

e

market share in the segment of entering consumers according

  1 1 1 N N L = + (−pi + pj ) − C σi − , 2 2 2

(C.4)

meaning that we can interpret C as a “consumer foresight” factor, i.e. a factor capturing how

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the firm’s market share among locked-in consumers affects the market share among entering con L  dp L  1 N  in closed form gives sumers. Solving for C = c dp dσ ρ +δ+β 1+

e

+C

e

r   2  L   N  dp dpN dp β dσiL + β + δ + ρc + 4β dσiL − β dσiL + β + δ + ρc e

i

i

e

i

e

2β

.

(C.5)

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C=

dσ L

It can be verified that the consumer foresight term C → 0 as ρc → ∞ or δ → ∞ and that C = 0  dpL if dσiL = 0. 

i

C.2

e

Equilibrium

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L U Under uniform pricing, pN i = pi = pi . The Hamiltonian associated with uniform pricing is given

by

L N Hi = N (0)ent pU ˙ iL i (σi + βσi ) + µi σ

L N N L = N (0)ent pU i (σi + βσi ) + µi β(σi − σi ).

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Substituting the expression for σiN into (C.4) we can formulate the first-order condition with

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respect to σiL according to

f N (0)ent (1 − βC)pU ˙ i. i + µi β(−1 − C) = ρ µi − µ

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To solve for µi , we apply the method of undetermined coefficients, substituting µi = N (0)ent (AσiL + N L B). This gives µ˙ i = N (0)ent Aσ˙ iL + nµi = N (0)ent i Aβ(σi − σi ) + nµi . Therefore, the first-order

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condition with respect to σiL can be expressed as:

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L f L N L (1 − βC)pU i + (Aσi + B)β(−1 − C) = (ρ − n)(Aσi + B) − Aβ(σi − σi ).

To solve for pU i , we make use of the first-order condition

expression:

pU i =

dHi dpU i

= 0, implying the following

h i + 1 + 2σiL β −2CσiL + C − (AσiL + B) + pU j 2β

29

(C.6)

.

(C.7)

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The second-order condition is ∂ 2 Hi = −βN (0)ent < 0. ∂p2i

(C.8)

Finding the associated first-order condition for firm j and solving the system of equations

pU i =

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defined by the two first-order conditions gives

2(σiL + 1) − β(AσiL + A + 3B + 2CσiL − C − 3) . 3β

(C.9)

Substituting (C.9) into (C.6) and matching coefficients allows us to solve for A and B. The solution for B is:

[β(A − C − 3) − 2)(Aβ − 2βC + 2] . 6β(β − n + ρf + 1)

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B=−

(C.10)

A is the lower root of the following quadratic equation:

h i 2(βC − 1)2 = 0. A2 β − 3A β(C + 2) − n + ρf + 1 + β

(C.11)

σiL (t0 ) > 12 , or

∂pi ∂σiL

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(The upper root is not feasible, because it would lead to ever-increasing market shares for < −1 − C, thereby violating the boundary condition defined by σiL ∈ [0, 1].)

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Substituting the expressions for A and B into (C.9) yields a characterization of pU i as a function dpU

of C. C in its turn is a function of the consumer expectations regarding the slope ( dσiL )e . Under i  U dp dpU rational expectations it holds true that dσiL = dσiL . We next verify the existence of a rational

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expectations equilibrium.

Focusing on the case with



dpU i dσiL

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the uniform price with respect to



i

e

i

= 0, (C.5) implies that C = 0. The slope of the derivative of  U dp conditional on dσiL = 0 is then bounded according

e dpU σiL , dσiL , i

i

e

AC

to the following inequalities: p  dpU  36β 2 + 36β + 1 − 6β + 1 dpU 2 i i 0< < =0< . 6β 3β dσiL dσiL e

(C.12)

Furthermore, in terms of how the slope of the uniform price with respect to σiL changes with

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dpU i dσiL



e

, we have the following results:

d 



d dpU i dσiL

lim 

dpU i dσ L i

e



e

→∞





dpU i dσiL dpU i dσiL

 

< 0,

(C.13)

= 0.

(C.14)

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respect to

(C.13) can be derived as follows. First, taking the derivative of C with respect to N L Equation (C.5), noting that under uniform pricing pU i = pi = pi , shows that

d

 dCU  dp i dσ L i

e



dpU i dσiL



e

in

> 0. Second,  U dpi d L dσ

dp

i d dσ L  Ui  dp i d L dσ

i

e

=

d

dpi dσ L i

dC



 dCU  dp i d L dσ

i

e

 < 0.

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taking the derivative of the slope of the uniform price with respect to C shows that dCi < q 2 f 2 f 2 2 U β (9(β(C+2)−n+ρ +1) −8(βC−1) )−β (7C+6)+β(3n−3ρ +1) dp 0. In particular, dσiL = . By the chain rule, 6β 2 i  U   U   

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Therefore, by the Intermediate Value Theorem, there exists a rational expectations equilibrium  U dp dpU ∗ 2 . This equilibrium is unique. To see this, starting at dσiL = 0, 0 < such that 0 < dσiL < 3β i i e  U  U  U U U U dpi dp dp dp dp dp 2 = 0 < 3β . As dσiL increases, dσiL falls, and as dσiL → ∞, dσiL falls to 0. dσiL dσiL i i i i e e  U i e U dp dp 2 Therefore, there exists a unique point of intersection where 0 < dσiL = dσiL < 3β . This finalizes

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i

e

i

the proof of the first part of Result 3.

We now turn to the bounds for the uniform price equilibrium with symmetric market shares. into (C.9) and taking the expression of A and B into account we find that

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1 2

CE

Substituting σi =

U∗

p

    1 1 βC + δ + ρf σi = = 1+ . 2 β 1 + δ + ρf

(C.15)

AC


To provide an upper and lower bound for pU ∗ σi = 12 , we use bounds for C. Taking into  U dp dpU 2 1 account that 0 < dσiL = dσiL < 3β , it follows (C.5) that 0 < C < 3β . Substituting these bounds i

e

i

into (C.15) establishes the second part of Result 3. Finally, we check some regularity conditions.

First, we verify that the solution leads to well-defined (feasible) market shares. This is immediate under Assumption 3, since 0 < 0 <

∗ dpU i dσiL

+C <

1 β

∗ dpU i L dσi

<

2 3β

and 0 < C <

1 3β ,

meaning that the sum satisfies

< 1. When substituted into (C.4) these inequalities ensure feasible market 31

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shares. Second, we check that the market is covered. Initially, we observe that the maximum price at σiL =

1 2

is equal to pU ∗ (σiL =

1 2)

< 1 + β1 , which can be found by taking the limit ρf → ∞.

∗ Our requirement for market coverage is satisfied if u ¯ > τ i + pU i . Observe that for the consumer

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to choose product i over product j, any difference between the prices must be made up for by ∗ U ∗ < −(τ − τ ). More specifically, for consumer to purtransportation costs, such that pU i j i − pj ∗ > pU ∗ , it must be the case that pU ∗ − pU ∗ < τ − τ , or that chase firm i’s product despite pU j i i j i j

the price difference of firm i’s and firm j’s products be less than the difference in adoption costs. This is a necessary and sufficient condition for myopic consumers. This is also a necessary (but

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not sufficient) condition for forward-looking consumers, because they consider future price differences in addition to present price differences which in equilibrium further tightens the require∗ U∗ ment on pU i − pj . Since the maximum possible τj is 1, rearranging this inequality it follows that 1 ∗ U∗ U∗ ∗ ¯ > τi + pU τi ≤ 1 − (piU ∗ − pU i is satisfied j ) = 1 − 2(pi − pi (σi = 2 )). Substitution shows that u 1 1 1 ∗ U∗ U∗ U∗ U∗ U∗ U∗ U∗ if u ¯ > 1 − 2(pU i − pi (σi = 2 )) + pi = 1 + p (σi = 2 ) − (pi − pi (σi = 2 )). Since pi > pj ,

1+

1 β

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the condition is satisfied if u ¯ > 1 + pU ∗ (σi = 12 ). Therefore, setting pU ∗ (σi = 21 ) at its maximum of gives a sufficient condition for market coverage, which is u ¯ > 2 + β1 . This condition is also 1 β

encounter a price of 1 +

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necessary if ρf → ∞ and ρc → ∞ since it is possible for consumers to have both τi , τj = 1 and from both firms. Our Assumption 1 displays this condition.

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Finally, we check that no firm would have any incentive to deviate to u ¯ from our claimed

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equilibrium. For this purpose we require that u ¯σiL

U∗


(σiL )(σiL

+

βσiN )

+

Z∞

βe−(ρ

f +δ)(t−t ) 0

σiN (t0 )pU ∗ σiL (t) dt.

(C.16)

t0

AC

for all σi ∈ [0, 1]. This condition ensures that switching to u ¯ induces a lower net present value of profits compared with the equilibrium price. To check that this non-deviation condition is satisfied under Assumption 3, we substitute pmin

for pU ∗ to make the condition more tractable as well as tighter. Rearranging yields β>


1 σiL min u ¯ − p . N min σi p 1 + ρf1+δ 32

(C.17)

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Condition (C.17) holds true for all positive values of β as ρf , δ → 0. To maximize the range of parameters for which deviation is ruled out, we focus on ρf , δ → ∞, which is the parameter configuration we applied as a basis for Assumption 3. The ratio

σiL σiN

is maximized at σiL = 1. The U∗

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smallest possible value for σiN can be found using the upper bounds for C and for dp , which dσiL
1 1 gives σiN (σiL = 1) > 12 − 3β − 6β . Furthermore, we have pU ∗ σiL = 12 > 1, which combined

with properties of the slope means that the smallest possible uniform equilibrium price satisfies pU ∗ (σiL = 0) > 1 −

1 3β .

Substitution of these properties ensures that (C.17) is satisfied by Assump-

tion 3.

Proof of Result 6

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Appendix D

We first explore the behavior of the threshold u ¯2 (β, δ + ρf ) viewed as a function of δ + ρf . Taking the derivative of u ¯2 (β, δ + ρf ), we find that it is strictly decreasing as a function of δ + ρf . Further, we find that limδ+ρf →∞ u ¯2 (β, δ + ρf ) = 2 + β1 , which is lower than the lower bound on u ¯ given by Assumption 1 and Assumption 3, ensuring that the average price with history-based

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pricing is higher than the uniform price for sufficiently high values of δ + ρf . On the other hand, limδ+ρf →β + u ¯2 (β, δ + ρf ) = ∞, which exceeds u ¯, implying that the uniform price exceeds the aver-

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age price with history-based pricing for sufficiently low values of δ + ρf (see, Assumption 2). As u ¯2 (β, δ + ρf ) is a continuous function the combination of these three properties implies that there

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is a uniquely determined value δ + ρf with the feature that the average price under history-based pricing is higher than the uniform price if and only if δ + ρf > δ + ρf . This finalizes the proof of

CE

Result 6(a).

f

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) Through ordinary differentiation it can be found that ∂ u¯2 (β,δ+ρ = 0 has one positive root, ∂β p given by β0 = (δ + ρf + 1) − 1. Furthermore u ¯2 (β, δ + ρf ) is strictly decreasing in β when β < β0

and strictly increasing when β > β0 . In addition, limβ→(δ+ρf )− u ¯2 (β, δ + ρf ) = ∞. This implies that there exists a β¯ > β0 with the feature that the average price with history-based pricing is lower

¯ In other words, for sufficiently high value of β the uniform price than the uniform price for β ≥ β. always exceeds the average price with history-based pricing. In other words, the uniform price exceeds the average price with history-based pricing in markets with a sufficiently high entry rate.

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This finalizes the proof of Result 6(b).

Appendix E

Proof of Result 8

To prove the Result 8(a), we check that: d¯ u1 d(δ+ρf )

< 0, which shows that the sufficient condition in Equation (10) admits a greater range

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(i)

of u ¯ for lower δ + ρf . (ii) limρf +δ→∞ u ¯1 =

1 β

+ 2, so that the inequality in Equation (10) is always satisfied under our

parameter restrictions.

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Next, the proof of Result 8(b) is the same as the myopic case. Finally the proof of Result 8(c) is as follows. If consumers are myopic (ρc → ∞), pU ∗ (σi =

1 2)

f

δ+ρ → (1 + β1 ) 1+δ+ρ f , which is the

lower bound for the uniform price pU,L , as characterized in Result 3. Further, the average price with history-based pricing is independent of ρc , as shown in Result 1. Therefore, if pU,L > p¯HBP it

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follows that also pU ∗ > p¯HBP .

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Beggs, Alan and Paul Klemperer. 1992. “Multi-period competition with switching costs.” Econometrica 60 (3):651–666.

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Fabra, Natalia and Alfredo Garcia. 2015. “Market structure and the competitive effects of switching costs.” Economics Letters 126:150–155. Farrell, Joseph and Carl Shapiro. 1988. “Dynamic competition with switching costs.” RAND Journal of Economics 19 (1):123–137.

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