Bargains Followed by Bargains: When Switching Costs Make
Markets More Competitive∗†
Jason Pearcy‡
September 13, 2015
Abstract
In markets where consumers have switching costs and firms cannot price discriminate, firms
have two conflicting strategies. A firm can either offer a low price to attract new consumers and
build future market share or a firm can offer a high price to exploit the partial lock-in of their
existing consumers. This paper develops a theory of competition when overlapping generations
of consumers have switching costs and firms produce differentiated products. Competition takes
place over an infinite horizon with any number of firms. This paper shows that the relationship
between the level of switching costs, firms’ discount rate, and the number of firms determines
whether firms offer low or high prices. Similar to previous duopoly studies, switching costs are
likely to facilitate lower (higher) equilibrium prices when switching costs are small (large) or
when a firm’s discount rate is large (small). Unlike previous studies this paper demonstrates
that the number of firms also determines whether switching costs are pro- or anti-competitive,
and with a sufficiently large (small) number of firms switching costs are pro- (anti-) competitive.
JEL Classification: D2, D4, L1.
Keywords: Switching Costs, Product Differentiation, Dynamic Competition, Discrete Choice
∗
I thank Daniel Spulber (the Editor), a Co-Editor, two reviewers, Stefano Barbieri, Michael Baye, Keith Finlay, Eric Rasmusen, Jay Shimshack, seminar participants at Tulane University, Montana State University, Purdue
University, Quinnipiac University, Indiana University (BEPP), IUPUI, the Spring 2009 Midwest Economic Theory
Conference and the 2010 IIOC for helpful comments and suggestions. All remaining errors are my own. Financial
support provided by the Committee on Research Summer Fellowship and the Research Enhancement Fund at Tulane
University is greatly appreciated.
†
c
Author Posting. Jason
Pearcy 2015. This is the authors’ version of the work. It is posted here for personal use,
not for redistribution. The definitive version is forthcoming in the Journal of Economics and Management Strategy.
‡
Montana State University; [email protected]
1
1
Introduction
When consumers have switching costs, a consumer’s previous purchase partially locks them in to
the same future purchase. Firms have two different strategies when their previous customers are
partially locked-in. A firm can offer a high price to exploit the partial lock-in of their existing
consumer base, but a high price will lead to more of its customers switching and a decrease in
future market share. Alternatively a firm can offer a low price to attract new consumers and invest
in market share with the hope of offering a high price in the future to more partially locked-in
customers. This paper presents conditions which determine whether equilibrium prices will be
higher or lower with switching costs and shows that the number of firms is an important factor
when considering how switching costs affect competition.
The previous literature has approached a firm’s use of pricing strategies with switching costs
in different ways.1 Competition usually takes place over two periods or infinitely many periods.
Two-period models dampen the exploitation/investment trade-off in any one period by allowing for
some inter-temporal substitution between the two different pricing strategies.2 This type of behavior
leads to the familiar result of bargains-then-ripoffs (Klemperer, 1995).3 If competition takes place
over infinitely many periods instead, firms must balance both investment and exploitation strategies
at once in each period.
Other approaches, such as dynamic pricing or subscription models, assume that firms can price
discriminate to offer a price for its previous customers and a different price for new customers.4 The
typical result is a scenario where bargains are followed by ripoffs; where new consumers without
switching costs are offered a low price and old customers with switching costs are offered a high
price (Farrell & Klemperer, 2007).5 Instead, if firms are unable to price discriminate, because they
cannot identify consumers by their past purchases, firms offer only one price to all consumers. In
this case, the exploitation and investment strategies conflict with one another and create an explicit
trade-off within each period.
1
See Villas-Boas (2015), Farrell and Klemperer (2007) and Klemperer (1995) for surveys of the switching cost
literature.
2
The end-game effects of two-period models are discussed in more detail by Cabral (2014) and Rhodes (2014).
3
Firms may have an incentive to lower prices in the second period if new consumers enter the market, but the
exploitation strategy still dominates.
4
See Armstrong (2006) and Chen (2005) for a survey of the dynamic pricing literature.
5
Note that this result is the opposite for dynamic pricing models without switching costs, see Chen and Pearcy
(2010).
2
This paper develops a switching cost model that exemplifies the trade-offs between the two
strategies since firms set one price for all consumers and competition takes place over an infinite
number of periods. The main purpose of this paper is to demonstrate under what conditions
switching costs lead to lower or higher prices in any steady-state symmetric equilibrium, and under
what conditions an increase in switching costs decreases the price. Both of these results depend
on the number of firms, the level of consumer switching costs, and firms’ discount factor. One of
the contributions of this paper is to show that switchings costs may be pro- or anti-competitive
depending on the number of firms in the market, and allowing for any number of firms in the model
provides the necessary insight into the main result.
The intuition of the main results and the reasoning why both results depend on the number
of firms in the market are explained by the following points. For any symmetric equilibrium, the
market share of each firm is the inverse of the number of firms. Under the duopoly case, each firm
commands half of the market and has a large consumer base partially locked-in to their firm. If
instead there are five firms in the market, each firm has only one fifth of consumers with switching
costs partially locked-in to their firm. The returns to exploitation are lower when a firm is only
able to exploit one fifth of the consumer base versus one half. Therefore it is more likely that the
investment effect dominates with many firms and the exploitation effect dominates with a duopoly.6
Recent empirical evidence is consistent with this intuition as well. Viard (2007) examines a
duopoly market and finds that a decrease in switching costs leads to a decrease in equilibrium
prices. Dubé, Hitsch, and Rossi (2009) examine two markets, one with six products and one with
four, and find that as switching costs increase equilibrium prices decrease. The results of Dubé et
al. also indicate that the number of firms and products needed to lower the price in a market with
switching costs may not be unreasonably large.
Existing theoretical studies of dynamic models with switching costs focus on various duopoly
outcomes. Earlier studies by Beggs and Klemperer (1992), Farrell and Shapiro (1988), and Padilla
(1995) find that switching costs lead to higher prices, while von Weizsäcker shows that prices may be
lower with switching costs. More recent dynamic duopoly studies including Arie and Grieco (2014),
Cabral (2014), Doganoglu (2010), Dubé et al. (2009), Fabra and Garcı́a (2015), Rhodes (2014), Shin
6
With many firms, prices with switching costs may be lower than prices when all consumers have no switching
costs. This result indicates that the investment effect dominates the exploitation effect even in the presence of more
intense competition.
3
and Sudhir (2008), and Villas-Boas (2015) have shown that switching costs may be either pro- or
anti-competitive depending on the nature of the competition. The growing consensus is that when
the level of switching costs is sufficiently low, or when a firm’s discount rate is sufficiently large,
switching costs are pro-competitive. Otherwise switching costs are anti-competitive. The findings
in this paper support this conclusion and also show that the nature of switching costs depends on
another dimension: the number of firms.
Other recent papers with switching costs allow for any number of firms. Wilson (2012) considers a model with search and switching costs with any number of firms where competition takes
place only over one period. While interesting, the analysis of the static one shot game lacks the
exploitation/investment trade-off which is the focus of this paper. Somaini and Einav (2013) focus
on antitrust issues and their paper is discussed in Section 3.1. In a considerably different setting,
Taylor (2003) finds that with more than two firms competition over switching consumers lowers
prices and firm profits. While similar results are obtained here, the number of firms required
for switching costs to be pro-competitive is not necessarily two, but is a function of the level of
switching costs and firms’ discount rate.
The remainder of the paper is organized as follows. Section 2 describes the setup of the model.
The symmetric equilibrium price is determined in Section 3. Section 3 also contains a discussion
relating the model and equilibrium price to the previous literature. Section 4 analyzes the equilibrium price from Section 3 and contains the main results. Section 5 tests the robustness of the main
results by considering a continuous distribution of switching costs and Section 6 concludes.
2
The Model
The model developed in this paper combines consumer switching costs with the discrete choice logit
model and allows for competition to take place over an infinite horizon. The logit model is typically
presented as a static discrete choice product differentiation model and is thoroughly discussed in
Train (2003, Chapter 3), Anderson, de Palma, and Thisse (1992) and elsewhere. Consumer switching costs and overlapping generations of consumers are modeled similar to Beggs and Klemperer
(1992). The model presented here is discussed in the context of the relevant literature at the end
of the next section.
4
There are N ≥ 2 single product firms. Each firm produces a unique variety of the differentiated
product and each consumer purchases exactly one differentiated product. Consumers choose the
product variety in each period which maximizes their utility. Firms choose their price to maximize
the discounted sum of profits over time. Competition takes place over an infinite horizon.
2.1
Consumers
i = y − pi − s0 I + εi .
The random utility specification of consumer z at time t for product i is Uzt
t
z
zt
y is the consumer’s income/baseline valuation of the product which is identical for all consumers.
It is assumed that y is sufficiently large so that all consumers purchase, but that y is not infinite.
pit is the price of Firm i’s product (i = 1, 2, . . . , N ) at time t. s0z indicates consumer z’s switching
cost and I = {0, 1} indicates whether consumer z switches. Consumers also have preferences for
the product varieties where εizt is consumer z’s product specific valuation of product i at time t.
The following assumptions are made concerning consumer preferences:
Assumption 1. Consumers are myopic with preferences described by the random utility function,
−ε
i , where εi is i.i.d. and follows a Gumbel distribution with density f (ε) = e−ε e−e , and s0 =
Uzt
zt
z
s0 ≥ 0.
The assumption that εizt is i.i.d. both over time and product varieties along with the preference
i is maintained throughout the paper. It is also assumed that consumers are
specification of Uzt
myopic and choose one product that maximizes their random utility in the current period. In
each period every consumer knows only their own realization of brand preferences (or their own
preference shock), εizt ∀ i = 1, 2, . . . , N . Alternatively, consumers always know their baseline
valuation y and switching cost s0z since these do not change over time. Prices, pit are known to
everyone.
Consumers have switching costs of s0z = s0 which they incur if a consumer switches products
i is an indicator function which takes a value of 1 when consumer
after an initial purchase. I in Uzt
z switches after an initial purchase and 0 otherwise. I = 1 if the purchase decision in period t − 1
is different than the purchase decision in period t regardless of the entire history of purchases.
Assumption 1 forces all consumers to have the same switching cost of s0 although the level of the
5
switching cost may change.7
At the beginning of period t, there are Mt consumers from the previous period. A fraction, α ∈
[0, 1], of these old consumers leave the market forever and the remaining (1−α)Mt consumers remain
in addition to ρ new consumers who have just entered the market. Both new and the remaining
old consumers choose one product to maximize their current period utility. New consumers do not
incur switching costs, but old consumers incur switching costs if they purchase a different product
than in the previous period. The number of consumers at the beginning of each period evolves
according to Mt+1 = ρ + (1 − α)Mt . Throughout most of the paper M0 = 1 and ρ = α so that
there is always a steady state unit mass of consumers and α is the renewal rate of consumers or
the fraction of new consumers in each period. Additionally, when M0 = 1 and ρ = α a firm’s share
is equivalent to their demand.
i .
Each consumer chooses a product i in period t to maximize current utility: maxi∈{1,2,...,N } Uzt
i = V i + εi . The
Let Vti ≡ y − pit − s0 I, so that the current utility can be expressed as Uzt
zt
t
product purchased in period t matters as it determines what product the consumer does not have
to pay switching costs for in period t + 1. In period t, consumer z chooses product i if Vti + εizt >
Vtj + εjzt ∀ j 6= i. The probability that consumer z chooses product i in period t is8
i
eVt
i
Pzt
= PN
j
Vt
j=1 e
2.2
.
(1)
Firms
Each firm serves potentially three different types of consumers simultaneously in every period: new
consumers and two types of old consumers. There are ρ new consumers who just entered the
market. Since new consumers have not made a previous purchase, they do not incur any switching
costs. There are (1−α)Mt old consumers with switching costs and each old consumer is categorized
by their purchase history. An old consumer is either a repeat customer and purchases from the
same firm as in the previous period, or an old consumer switches and purchases from a different
firm.
The probability that Firm i attracts each of the three different types of consumers is as follows.
7
8
This assumption is relaxed later on in the paper when a continuous distribution of switching costs is considered.
See Train (2003, p. 40 and pp. 78-79) or Anderson et al. (1992, pp. 39-40) for the derivation.
6
i , Firm i attracts an
With probability PNi t , Firm i attracts a new consumer, with probability POL
t
ki Firm
old (and loyal) consumer who previously purchased from Firm i and with probability POS
t
i attracts an old (and switching) consumer who previously purchased from Firm k 6= i. Firm i’s
market share in period t − 1 is σti . Firm i’s market demand in each period is
i
Qit (pt , σ t ) = ρPNi t + (1 − α)Mt σti POL
+ (1 − α)Mt
t
N
X
ki
σtk POS
t
(2)
k6=i
where each term on the right hand side from left to right is the demand from new consumers, old
and loyal consumers, and old and switching consumers. pt is the N × 1 vector of prices in period t
and σ t is the N × 1 vector of period t − 1 market shares.
In each period, firms choose their price to maximize their discounted sum of profits. The
following assumptions apply to the firm’s problem.
Assumption 2. Assume all firms have the same discount factor δ ∈ (0, 1) and that a firm’s price
in each period is an element of a nonempty, compact, and convex set P.
Consider the set of possible prices, P, to have an upper and lower bound, and in each period, a
firm chooses a price between the bounds. Bounding a firm’s price from below prevents a firm from
paying consumers an infinite sum to switch and bounding from above does not allow a firm to
receive an infinite sum from a captive consumer.
The period t profit for Firm i is πti (pt , σ t ) = Qit (pt , σ t )(pit −ci ) where ci is the constant marginal
cost of Firm i. For simplicity, let ci = 0. Firm i’s formal problem is
max
pit
∞
X
i
δ t πti (pt , σ t ) subject to σt+1
Mt+1 = Qit (pt , σ t ).
(3)
t=0
The solution to Firm i’s dynamic programing problem outlined in equation (3) is a pricing rule
pit (σ t ) which is a function of all firms previous shares. Given initial market shares for all firms σ 0 ,
a set of pricing rules satisfying equation (3) for each firm characterizes a competitive equilibrium
of the model.
7
3
Symmetric Equilibrium of the Model
In this section, the pure strategy pricing rule is determined for a symmetric Markov perfect equilibrium (MPE) of the model presented in Section 2. While a formal existence argument of a MPE
of the model is not presented, the intuition of why such an equilibrium exists is. Also, the pure
strategy pricing rule is not necessarily unique and a proof of uniqueness is left for future work.
The symmetric MPE determined considers both symmetry across firms within a period and
symmetry across time (a steady state). From the outset of the problem, we know that the perperiod symmetric subgame equilibrium of the model will involve all firms having identical market
shares of σti = 1/N and each firm charging the same price at an instance in time, pit = pjt ∀i, j. If the
size of the market is changing over time, then it is possible that pit 6= pit0 . Steady state conditions
of ρ = α along with M0 = 1 ensure that the size of the market does not change over time and a
firm’s market share is also equal to their demand.
When firms’ prices are symmetric within every period, pit = pjt ∀i, j and Lemmas 1 and 2
describe resulting equilibrium properties of the model.9 These properties are used to simplify the
symmetric pricing rule determined in Proposition 1, but only after the first order conditions of
the firm’s problem are determined. After the symmetric pricing rule is established, the model is
discussed in the context of the previous literature.
It is first shown how symmetric prices within each period influence the consumer’s problem
and how this effect works it’s way into the firm’s problem. The general probability that any one
consumer chooses any one product in a given period is defined by equation (1). The demand
segments for each firm are explicitly defined in the following lemma.10
0
Lemma 1. Let s = 1 − e−s . If pit = pjt ∀ i, j:
PN =
1
N
POL =
1
(1 − s)(N − 1) + 1
POS =
(1 − s)
(1 − s)(N − 1) + 1
(4)
The probability that a new consumer purchases from Firm i, PN , is equally likely for all firms
since prices are symmetric within periods and new consumers do not face any switching costs. For
old consumers with switching costs equal to s0 , switching costs are incurred if any of the N − 1
9
10
Since the focus is on an MPE, the time subscript is omitted when appropriate to avoid redundancy.
Omitted proofs are in the appendix.
8
products not purchased in the previous period are chosen. The difference between POL and POS
is that repeat consumers (POL ) do not incur switching costs, while consumers who switch from
Firm k to Firm i incur a switching cost of s0 . Notice that while s0 ∈ [0, ∞), s ∈ [0, 1]. Consumer
switching costs of zero, s0 = 0, correspond to s = 0 and infinitely high consumer switching costs
correspond to s = 1.
i M
i i
Market shares evolve according to equation (2) where σt+1
t+1 = Qt (σt ). Incorporating the
steady state conditions of, ρ = α and M0 = 1 results in a simplified version of demand and the
evolution of market share.
0
Lemma 2. Let s = 1 − e−s . If pit = pjt ∀ i, j, ρ = α and M0 = 1 Firm i’s demand/share is
Qi =
(1 − α)(1 − s(1 − σ i ))
α
+
N
(1 − s)(N − 1) + 1
(5)
and the market share evolves according to the following equation.
σti
t
1
1
(1 − α)s
i
=
+ σ0 −
.
N
N
(1 − s)(N − 1) + 1
(6)
Lemma 2 leads to an expected result concerning the symmetry of the model: when initial market
shares are symmetric, σ0i = 1/N , and prices are symmetric within periods, market shares are
symmetric across periods, σ i = 1/N . Although it seems dubious to expect firms’ prices to be
symmetric when initial market shares are not symmetric, note that if initial market shares are not
symmetric but firms’ prices are symmetric within every period, market shares are converging to
σ i = 1/N over time. Convergence occurs because the term on the right hand side of equation (6)
is between zero and one as 0 ≤ (1 − α)s ≤ (1 − s)(N − 1) + 1.
When initial market shares are symmetric, a pure strategy pricing rule of a MPE dictates that
all firms charge the same price, and when all firms charge the same price within a period, market
shares remain symmetric (Lemma 2). At the steady state symmetric equilibrium, each firm’s market
share is σti = 1/N and each firm announces a price of p∗ = pit (σti = 1/N ) in every period. Individual
firm profits are the price times market share, p∗ /N , and industry profits are equivalent to the price,
p∗ . The following proposition describes the pure strategy pricing rule of a symmetric MPE and is
the main result of this section.
9
Proposition 1. Given that a symmetric MPE exists, σ0i = 1/N for i = 1, . . . , N , ρ = α and
M0 = 1, the steady state symmetric pure strategy MPE price of the problem defined by equation (3)
is
p∗ =
N ((1 − s)(N − 1) + 1)((1 − s)(N − 1) + 1 − δ(1 − α)s)
(N − 1) (α((1 − s)(N − 1) + 1)2 + (1 − α)(1 − s)N ((1 − s)N + 2s))
(7)
0
where s = 1 − e−s .
The steady state symmetric price is a function of four parameters: α, δ, s and N . N ≥ 2 is
the number of identical firms in the market, α ∈ [0, 1] indicates the fraction of new consumers in
the market, and δ ∈ (0, 1) is the firm’s discount factor. To keep the denominator of equation (7)
non-zero, the degenerate case of infinite switching costs with no new consumers (s = 1 and α = 0)
is not considered.
It is important to note that Proposition 1 is based on the existence of a symmetric MPE, and
if a symmetric MPE fails to exist then the following analysis is vacuous. While a formal existence
argument is not provided, it is intuitive that such an equilibrium exists. The model presented is
symmetric as all firms have the same costs, all profit functions are constructed in a similar manner,
and the initial state is assumed to be symmetric. The firm’s payoff function is quasi-concave and
continuous – both properties utilized in formal existence arguments. Furthermore, the dynamics
of the model are relatively simple by construction as the state space and set of feasible strategies
do not change over time, and the set of feasible strategies are not constrained by the current state.
A simple analysis of the pricing rule in equation (7) and the choice probabilities of equation (4)
provides some intuitive validation. For instance, if switching costs are absent from the model we
expect the familiar symmetric logit price of N/(N − 1) and the probability of a repeat purchase
i
ki . Switching costs do not influence the model when
is equally as likely as switching, POL
= POS
t
t
either all consumers are new to the market, α = 1, or when all consumers have switching costs
equal to zero, s = 0. Either of these cases result in the symmetric logit price of N/(N − 1) and
i
ki = 1/N . Additionally, when all consumers have infinite switching costs, s = 1, no
POL
= POS
t
t
ki = 0 and P i
consumers switch (POS
OLt = 1).
t
10
3.1
Discussion & Relation to the Previous Literature
The model presented in Section 2 is compared to the earlier models presented by von Weizsäcker
(1984) and Beggs and Klemperer (1992) although von Weizsäcker’s equilibrium concept is different.
Beggs and Klemperer’s model has new consumers entering the market in each period with all
consumers having infinite switching costs (α 6= 0 and s = 1). Alternatively in von Weizsäcker’s
model, consumers do not have infinite switching costs and no new consumers enter the market since
old consumers live forever (α = 0 and s 6= 1). The model presented in Section 2 could be considered
more general than von Weizsäcker (1984) and Beggs and Klemperer (1992) as it accounts for both
scenarios and the effect of changing the amount of new consumers without switching costs, α, can
be distinguished from the effect of changing the level of switching costs s.
The most significant difference between the model in this paper and the relevant previous
literature is the use of a logit model to characterize the product differentiation.11 Beggs and
Klemperer (1992), Doganoglu (2010), Fabra and Garcı́a (2015), Rhodes (2014), Shin and Sudhir
(2008), Somaini and Einav (2013), Villas-Boas (2015), and von Weizsäcker (1984) characterize
the product differentiation in their models by utilizing a linear city Hotelling type model where
consumers are uniformly distributed over the line and the two firms are located at the opposite
ends of the product space. One notable exception is Cabral (2014) who allows for a general form
of product differentiation in a duopoly setting. The use of the logit model/Gumbel distribution in
this paper has two key advantages. First, the model is able to easily incorporate more than two
firms. The Hotelling framework is usually limited to two firms at the opposite end of the product
space. One exception in this regard is Somaini and Einav (2013) where they develop an extension
to the Hotelling framework allowing for any number of firms.12 Secondly, the logit model is not
subject to price under-cutting strategies present in the Hotelling model with linear transportation
costs.13 The absence of under-cutting strategies makes the analysis simpler because the parameter
space does not need to be restricted as in the aforementioned models.
11
Both Dubé et al. (2009) and Arie and Grieco (2014) examine a switching costs model with a logit model characterizing the product differentiation. The theoretical results of Dubé et al. (2009) are obtained by using numerical
methods on a restricted version of the model and Arie and Grieco (2014) only consider the case of no/small switching
costs.
12
Other switching cost models allow for more than two firms, but additional firms are regulated to the fringe
(see Biglaiser, Crémer, and Dobos (2013)) and/or a pure strategy price equilibrium does not exist (see Farrell and
Klemperer (2007, pp. 1985-1986 and footnote 31), Biglaiser et al. (2013) and Rosenthal (1980)).
13
See d’Aspremont, Gabszewicz, and Thisse (1979).
11
While the product differentiation assumptions made in this paper have their advantages, these
assumptions create differences in how brand preferences are correlated across products and over
time. In Hotelling type models, brand preferences across products are perfectly negatively correlated within periods, but here brand preferences across products are independent within periods.
This difference could be viewed as a consequence of allowing more than two products/firms or could
be seen as substituting one extreme case (correlation of −1) for another (correlation of 0).
Additionally, the model in this paper has brand preferences which are not correlated over time
and are independent across periods.14 Most other models also have brand preferences that are not
serially correlated over time. Two exceptions are von Weizsäcker’s (1984) model where individual
brand preferences change over time with some positive probability of an independent redraw, and
Cabral (2014) who considers the serial correlation of preferences as an extension. According to
Villas-Boas (2015), if consumer preferences are less stable over time, it is expected that prices will
be lower.
Besides assumptions regarding the correlation of consumer preferences, it is also assumed in
this paper that consumers are myopic. This is not an uncommon assumption in the literature,
and Villas-Boas (2015) indicates that prices are lower with myopic consumers in switching cost
models. Alternatives include allowing for two generations of overlapping consumers (Doganoglu,
2010; Rhodes, 2014; Somaini & Einav, 2013) or infinitely lived consumers with discounting (Cabral,
2014; Fabra & Garcı́a, 2015). One of the main difficulties with incorporating consumer discounting
in the current model is that the renewal rate of consumers is parameterized as α, and in other
studies α is fixed. With two generations of overlapping consumers, α = 0.5, and with infinitely
lived consumers, α = 0. The benefit of parameterizing α while having myopic consumers is the
direct connection made to the sign of
∂p∗
∂α
and whether switching costs are anti- or pro-competitive.15
Another important difference between the current analysis and the previous literature is with
the determination of the pricing rule. The focus in this paper is only on the symmetric steady state
pricing rule. A downside to this approach is that the current analysis does not address asymmetries
in market shares as considered by Cabral (2014), Fabra and Garcı́a (2015), and Rhodes (2014).
Cabral (2014) emphasizes that when market shares are more asymmetric, switching costs are anti14
The independence of brand preferences over time greatly simplifies the analysis and is critical in determining an
analytical solution to the pricing rule in equation (7).
15
See Corollary 1 and Corollary 4.
12
competitive. Even though the focus here is on symmetric outcomes, it is still found that switching
costs can be anti-competitive. Incorporating asymmetries into the analysis is not likely to change the
nature of the results, but it should change the pertinent parameter thresholds for which switching
costs are either pro- or anti-competitive. The same is true with consumer discounting. When
consumers are myopic, it is more likely that switching costs are pro-competitive, but the main
point is not that switching costs are either anti- or pro-competitive. The main point of the analysis
is to determine subsets of the parameter space for which switching costs are anti- or pro-competitive.
Upsides to the current approach include the ability to determine an analytical solution to a
closed form pricing rule and the MPE pricing rule not depending on a functional form restriction.
Doganoglu (2010), Rhodes (2014), and Somaini and Einav (2013) restrict their analysis to only
consider linear pricing rules in a MPE. In the context of the current model, a symmetric steady
state linear pricing rule takes the form pit = L1 + L2 σti with the pricing rule being an affine
transformation of the market share. In the current analysis, a linear pricing rule is inappropriate
when α 6= 1, s 6= 0 and N > 2.
4
Analysis
In this section, the symmetric steady state price, profits and consumers’ welfare are analyzed. The
comparative statics of the pricing rule are determined with respect to three of the four model
parameters: δ, α and s . Then it is determined under what conditions switching costs lead to lower
or higher prices when compared to a market without switching costs. The analysis is simplified by
rewriting the symmetric equilibrium price in equation (7) as follows. Let
X = (1 − s)(N − 1) + 1,
Y = X − δ(1 − α)s, and
Z = αX 2 + (1 − α)(1 − s)N (X + s).
Then the symmetric equilibrium price can be rewritten as
p∗ =
N XY
.
(N − 1)Z
(8)
Since δ, α and s are all bounded between 0 and 1 and N ≥ 2, it follows that X, Y and Z are all
greater than or equal to zero.
13
4.1
Comparative Statics
The symmetric steady state price of equation (7) is decreasing in δ. Using the simplified price in
equation (8),
∂p∗
NX
=
∂δ
(N − 1)Z
∂Y
∂δ
≤ 0.
As δ increases, firms care more about future profits and lower their price sacrificing current profits
to invest in future market share.
The partial derivative of the equilibrium price with respect to α is
∂p∗
NX
=
∂α
(N − 1)Z 2
∂Y
∂Z
Z
.
−Y
∂α
∂α
The sign of ∂p∗ /∂α is determined by the sign of (Z∂Y /∂α − Y ∂Z/∂α). Simplification yields
∂Y
∂Z
Z
−Y
= sX(δ(1 − s)N − (1 − δ)s)
∂α
∂α
(9)
and the sign of ∂p∗ /∂α depends on the relationship between N , s and δ. If δ(1−s)N > (1−δ)s, then
an increase in the renewal rate of consumers, α, will raise the equilibrium price. This result occurs
when the number of firms (N ) is large, switching costs (s) are small, or when the discount factor
(δ) is large. At the beginning of each period, the number of consumers partially locked in to Firm
i is (1 − α)/N and the number of consumers partially locked in to other firms is (1 − α)(N − 1)/N .
With a large number of firms, there are more consumers who are partially locked in to other firms
and this places downwards pressure on each firm’s price as they try to attract consumers who have
switching costs. An increase in the renewal rate of consumers α, relieves the downwards pressure
on the price as a firm can focus more on consumers without switching costs, and the increase in α
results in an increase in the price. Alternatively if δ(1 − s)N < (1 − δ)s, (small N , large s, or small
δ) an increase in the fraction of new consumers, α will decrease the equilibrium price.
A marginal change in the level of switching cost s influences the equilibrium price where
∂p∗
N
=
∂s
(N − 1)Z 2
∂X
∂Y
∂Z
YZ
+ XZ
− XY
∂s
∂s
∂s
and the sign of ∂p∗ /∂s is determined by term on the far right hand side. The term of interest which
14
determines the sign is
∂X
∂Y
∂Z
= (1 − α)N 2(1 − δ)sX + δαs2 − δ(1 − s)2 N 2
YZ
+ XZ
− XY
∂s
∂s
∂s
(10)
and may be either positive or negative depending on the relationship between the parameters.
Examining the bracketed term of equation (10), an increase in switching cost may lead to a decrease
in price (equation (10) < 0) if firms are sufficiently patient (large δ), switching costs are low, there
are more old consumers (small α), or there are many firms in the market.
4.2
When do Switching Costs Lead to Lower Prices?
The effects of marginal changes in the fraction of new consumers and the level of switching costs
on the equilibrium price determined in the previous section depend on the parameter specification.
The sign of ∂p∗ /∂α depends on the relationship between N , s and δ and the sign of ∂p∗ /∂s depends
on the relationship between N , s, δ and α. The interpretation of these results are better explained
when one compares the equilibrium price with switching costs to the equilibrium price without
switching costs.
With no switching costs, s = 0 and the equilibrium price is N/(N − 1). The difference in price
created by switching costs is
∆p∗ = p∗ −
N
N
=
(XY − Z)
N −1
(N − 1)Z
(11)
and the sign of ∆p∗ is determined by the sign of XY − Z. Expansion of this result yields XY − Z =
(1 − α)s((1 − δ)s − N δ(1 − s)) and the sign of XY − Z is determined by the sign of
(1 − δ)s − N δ(1 − s).
(12)
The sign of equation 12 can be either positive or negative so that switching costs can lead to higher
or lower prices.
Equation 12 provides additional insight to the comparative statics of the model. The sign of
∂p∗ /∂α is determined by the sign of equation (9). When switching costs lead to lower prices,
δ(1 − s)N > (1 − δ)s, and an increase in the fraction of new consumers, α, will raise the equilibrium
15
price. An increase in α effectively reduces the importance of switching costs because there will
be more new consumers in each period without switching costs. If switching costs lower prices
(∆p∗ < 0) and the importance of switching costs is reduced by increasing α, then the effect of an
increase in α will be to increase the price. Alternatively, if prices are higher with switching costs, a
marginal increase in α causes the equilibrium price to decrease. The following corollary summarizes
this relationship.
Corollary 1. With homogeneous switching costs, ∆p∗ < 0 if and only if ∂p∗ /∂α > 0.
Examining equation (11), switching costs do not influence the price for the two extreme cases of
no switching costs, ∆p∗ (s = 0) = 0, or no old consumers, ∆p∗ (α = 1) = 0. With infinite switching
costs,
∆p∗ (s = 1) =
N (1 − δ)(1 − α)
>0
(N − 1)α
and switching costs always lead to higher prices. This is the result obtained by Beggs and Klemperer
(1992). In their model, they assume that consumers have infinite switching costs, but allow the
fraction of new consumers in the market to vary. By changing α when s = 1, prices are always
higher with switching costs, but the magnitude of the price difference varies.
Avoiding the extreme cases of s = 0, s = 1 and α = 1, switching costs lead to a lower equilibrium
price when switching costs are sufficiently low, the discount rate is sufficiently large and there are
many firms. Surprisingly the direction of the price difference does not depend on the fraction of
new consumers, α, but α does influence the magnitude of the difference. The discount rate and the
number of firms influence the model by changing the relative returns to the conflicting investment
and exploitation strategies of the firm. As δ increases or N increases, the value to the firm of
exploiting its existing consumer base is diminished so firms lower prices to invest in future market
share.
The main result of the paper provides a more general result with respect to the parameters of
the model and is as follows.
Corollary 2. Assume s 6= {0, 1} and α 6= 1, then there exists an N 0 , s0 and δ 0 where
N0 =
(1 − δ)s
δ(1 − s)
s0 =
Nδ
N δ + (1 − δ)
16
δ0
s
=
1 − δ0
N (1 − s)
such that for any N > N 0 , s < s0 , or δ > δ 0 switching costs lead to lower prices (∆p∗ < 0).
Corollary 2 is a direct result of equation (12). With respect to N 0 , as long as switching costs are
not infinite and as long as some consumers might switch, switching costs lower the equilibrium
price if there are a sufficiently large number of firms in the market. The sufficiently large number
of firms required to lower prices is increasing in the level of switching costs and decreasing in the
discount rate, but irrespective of the level of switching costs and the discount rate there is always
some number of firms such that prices are lower with switching costs. Alternatively if the number
of firms is sufficiently small, then prices are higher with switching costs.
One can also interpret Corollary 2 as indicating that for any number of firms, as long as switching
costs are relatively low (s < s0 ) or the discount factor is relatively large (δ > δ 0 ), prices are lower with
switching costs. This interpretation gives more general support to previous studies which assume
N = 2. The numerical results obtained by Dubé et al. (2009) are consistent with Corollary 2 and
this interpretation also supports Cabral’s (2014) finding that switching costs may lower the price.
The contribution of Corollary 2 in the context of these studies is to provide analytical results with
a single price. Corollary 2 is also more general as the number of firms is not restricted to N = 2
which is important because the number of firms plays a major role in determining whether prices
are higher or lower with switching costs.
A similar story occurs with the sign of ∂p∗ /∂s which is determined by the sign of equation (10)
and results in the following corollary.
Corollary 3. Assume s 6= {0, 1} and α 6= 1, then there exists an N 00 , s00 , and δ 00 where
N 00 =
1 s 1 − δ + 1 − δ 2 (1 − α) 2
δ(1 − s)
s00 =
N N δ + (1 − δ) − 1 − δ 2 (1 − α)
1 2
δ(N − 1)2 + 2(N − 1) + δ(1 − α)
δ 00
2s((1 − s)(N − 1) + 1)
=
00
1−δ
(1 − s)2 N 2 − αs2
such that for any N > N 00 , s < s00 , or δ > δ 00 a marginal increase in switching costs lowers the
price (∂p∗ /∂s < 0).
Comparing the measures from Corollary 3 to Corollary 2 reveals that N 0 < N 00 , s00 < s0 , and
17
δ 0 < δ 00 . For a sufficiently large number of firms (N > N 00 ), sufficiently low switching costs
(s < s00 ), or sufficiently large discount rate (δ > δ 00 ), the equilibrium price is lower with switching
costs than without and a marginal increase in switching costs leads to a decrease in the equilibrium
price.16 Note that N 00 , s00 , and δ 00 are all functions of the other variables and in the case of N 00 , N 00
is increasing in s and decreasing in δ.
The price decreases with an increase in switching costs because the overall effect of switching
costs is to lower the price. Notice that while the fraction of new consumers, α, does not affect
whether prices are higher or lower with switching costs (∆p∗ ), α does partially determine the sign
of ∂p∗ /∂s.
The following proposition summarizes the main results thus far:
Proposition 2. Assume s 6= {0, 1} and α 6= 1, then:
1. For N < N 0 , s > s0 , or for δ < δ 0 , ∆p∗ > 0, ∂p∗ /∂α < 0 and ∂p∗ /∂s > 0.
2. For N 0 < N < N 00 , s00 < s < s0 , or for δ 0 < δ < δ 00 , ∆p∗ < 0, ∂p∗ /∂α > 0 and ∂p∗ /∂s > 0.
3. For N > N 00 , s < s00 , or for δ > δ 00 , ∆p∗ < 0, ∂p∗ /∂α > 0 and ∂p∗ /∂s < 0.
The sign of ∆p∗ determines whether prices are higher are lower with switching costs where if
∆p∗ < 0, prices are lower with switching costs. The sign of ∂p∗ /∂α determines whether a marginal
increase in the amount of new consumers (who do not have switching costs) leads to an increase
or decrease in price. Finally, the sign of ∂p∗ /∂s determines a marginal increase in switching costs
affects prices.
4.3
The Duopoly Case
While the main contribution of the paper is to examine how the number of firms influences the
price in a switching cost model, the analysis is also relevant to other duopoly models with switching
costs. In a duopoly setting, N = 2 and the effect of changes in α, δ, and s are examined. From
The result that ∂p∗ /∂s < 0 when s < s00 extends Proposition 2 of Doganoglu (2010) who finds that, in the
notation of the current model, ∂p∗ /∂s < 0 conditional on s = 0.
16
18
Section 4.1, the relevant comparative statics are
∂p∗ (N = 2)
−2s(1 − α)(2 − s)
≤0
=
∂δ
4(1 − s) + αs2
∂p∗ (N = 2)
−2s(2 − s)2 (s − δ(2 − s))
=
∂α
(4(1 − s) + αs2 )2
∂p∗ (N = 2)
4(1 − α)(2s(2 − s) − δ(4(1 − s) + s2 (2 − α)))
=
∂s
(4(1 − s) + αs2 )2
and from Section 4.2, the price difference is
∆p∗ (N = 2) =
2s(1 − α)(s − δ(2 − s))
.
4(1 − s) + αs2
Figure 1: Duopoly Parameter Space
1
δ
Price is Lower with Switching Cost
δ=
s
2−s
1
3
Price is Higher with Switching Cost
0
0
1
2
1
s
As before, a marginal increase in δ leads to a marginal decrease in the price. The sign of both
∆p∗ (N = 2) and
∂p∗ (N =2)
∂α
depend on the sign of s − δ(2 − s), where s − δ(2 − s) < 0 implies that
prices are lower with switching costs and a marginal increase in α results in a higher price. Figure 1
19
illustrates the parameter space over δ and s where switching costs lead to lower or higher prices. In
the duopoly case, switching costs lead to lower prices when the level of switching costs is low or the
discount factor is high. An examination of
∂p∗ (N =2)
∂s
is complicated as the sign of this comparative
static depends on α, δ and s. All things being equal,
∂p∗ (N =2)
∂s
< 0 when δ is relatively large, α is
relatively small, and s is relatively small.
4.4
Product Differentiation and Switching Costs
Also of interest is the level of switching costs (s0 ) compared to the extent of product differentiation
in the model (distribution of the εizt ’s). Because a logit model is used to characterize the product
differentiation in this paper, one may suspect that the logit model drives certain results rather than
switching costs. Implicitly assumed in the model is that the variance of εizt is equal to π 2 /6.17 If
instead the variance was π 2 β 2 /6, with β > 0, then s would be expressed as ŝ = 1 − e
−s0
β
. The level
of switching costs is determined by s0 , and β determines the extent of product differentiation.
As β approaches zero, the variance of εizt approaches zero and there is no product differentiation
because consumers have the same brand preferences for all products. As β approaches zero, ŝ
approaches 1 (for a fixed s0 ) and the same effect is achieved by increasing consumer switching costs
s0 (holding β constant). Alternatively as β gets large, consumers’ brand preferences are more varied
increasing the relative importance of product differentiation. When β gets large, ŝ approaches zero
and the same effect is achieved by decreasing consumer switching costs s0 . Examining changes in ŝ,
there is a direct trade-off between the importance of switching costs and product differentiation in
that when the magnitude of switching costs increases, the effect of product differentiation decreases.
A marginal increase in β results in more switching in equilibrium (POL decreases and POS
increases) as the marginal increase in β reduces ŝ. Examining the change in the price, let p∗ (s = ŝ)
be the price in equation (7) with s replaced by ŝ. The symmetric steady state equilibrium price is
p̂ = βp∗ (s = ŝ).18 As β approaches zero, consumers perceive the different products as homogenous,
and the price approaches zero (marginal cost) as consumers purchase from the firm with the lowest
price.
17
See Train (2003, p. 44) and Anderson et al. (1992, pp. 59-60). Note that π refers to the mathematical constant
and not profits.
18
p̂ is obtained using the procedure in the proof of Proposition 1. The major difference is that the partial derivative
of a logit probability, P , with respect to p̂ is P (P − 1)/β.
20
For a marginal increase in β,
∂ p̂
∂p∗ (s = ŝ) ∂ŝ
= p∗ (s = ŝ) + β
∂β
∂ŝ
∂β
where
∂ŝ
∂β
< 0 and the sign of
∂p∗ (s=ŝ)
∂ŝ
(13)
is determined by the sign of equation (10). As β increases,
there are two different effects on p̂ as indicated by the two different terms in equation (13). As β
increases, consumers have more diverse brand preferences and firms are able to raise their price by
p∗ (s = Ŝ). A change in β also affects p̂ through a change in ŝ. An increase in β decreases ŝ and
if
∂p∗
∂s
> 0 the overall effect of the second term in equation (13) is negative. Examining the sign
of the combined effect, when β is sufficiently small,
∂ p̂
∂β
> 0 as the second term in equation (13) is
negligible, and an increase in the dispersion of brand preferences results in an increase in the price.
When β is sufficiently large, ŝ < s00 so that
β, numerical simulations indicate that
∂ p̂
∂β
∂p∗ (s=ŝ)
∂ŝ
< 0, and
∂ p̂
∂β
> 0. For intermediate values of
< 0 as the second term in equation (13) is negative and
dominates the first term. This suggests that increases in the dispersion of brand preferences has a
non-monotonic effect on the price.
The dispersion of brand preferences also affects whether prices are higher or lower with switching
costs. In this case what matters is the ratio of the level of switching costs to the dispersion of brand
preferences: s0 /β. Similar to equation (11), the difference in prices created by switching costs is
∆p̂ = β p∗ (s = ŝ) −
N
N −1
.
While ∆p̂ is scaled by β, the sign of ∆p̂ is only influenced by β through ŝ. When the level of
switching costs is large compared to the dispersion of brand preferences, s0 /β is large, ŝ is large,
and it is more likely that ŝ > s0 so that ∆p̂ > 0. When brand preferences are more dispersed and
the level of switching costs are low, s0 /β is small, ŝ is small, and it is more likely that ŝ < s0 so
that the price is lower with switching costs. This result corresponds directly to the finding in Shin
and Sudhir (2008) except here ŝ is a function of N and δ and Shin and Sudhir’s (2008) version of
ŝ is such that ŝ = 1.
21
4.5
Welfare & Policy Implications
Missing from the analysis thus far is a discussion of how the model parameters influence firm
profits and consumer welfare. Individual firm profits are πi∗ = p∗ /N and total industry profits
are equivalent to the equilibrium price. The analysis of Sections 4.1 and 4.2 applies directly to
the analysis of profits. If a change in δ, α or s leads to a lower (higher) equilibrium price, then
equilibrium profits are lower (higher) as well.
Per-period consumer surplus/welfare is measured as the consumer’s expected level of utility.
The expected utility for each of the α new consumers is
CSN ew = y − p∗ + γ + ln(N )
and the expected utility for each of the 1 − α old consumers is
CSOld = y − p∗ + γ + ln((1 − s)(N − 1) + 1)
where γ is Euler’s constant. The overall level of per-period consumer surplus is CS = αCSN ew +
(1 − α)CSOld . Marginal changes in δ, α and s influence CSN ew and CSOld by way of a change
in the equilibrium price where an increase in price decreases welfare. The exception is ∂CSOld /∂s
because an increase in switching cost affects old consumers through the price and by the switching
that might occur. The change in welfare experienced by old consumers due to a marginal change
in the level of switching costs is
∂p∗
N −1
∂CSOld
=−
−
∂s
∂s
(1 − s)(N − 1) + 1
and the term on the right is the decrease in old consumers’ welfare attributed to switching.
From a social efficiency perspective, a change in δ only leads to a transfer between consumers
and firms and does not change the overall level of efficiency. An increase in the switching cost, s,
decreases social efficiency as switching costs are modeled to be socially wasteful. Related to the
switching cost, an increase in the amount of new consumers, α, increases social efficiency because
new consumers have no switching costs. With more firms, which have zero costs by assumption,
22
consumers have a larger variety to choose from and overall welfare increases.19
Policy makers have the ability to manipulate s and N . N can be manipulated through programs
which discourage or encourage entry and s, for example, has been lowered by the introduction
of phone number portability policies. If policy makers are only concerned about overall welfare
measures, the model indicates that switching costs should be lowered and/or firm entry/competition
should be encouraged.
If policy makers are more concerned with consumer welfare and if they can encourage sufficient
competition (large enough N ), prices will be lower with switching costs and consumers will have
more variety. The downside to encouraging entry/competition is that it results in more consumers
switching. If N < N 00 , δ < δ 00 and s > s00 policy makers should decrease the switching costs until
s = s00 which would lower the equilibrium price and the cost of switching.
If N > N 00 , s < s00 , or if δ > δ 00 policy makers might increase switching costs to lower the
price. This clearly benefits new consumers with no switching costs. An increase in switching costs
benefits old consumers if ∂CSOld /∂s > 0. Numerical simulations indicate that if α is small, δ is
large, s is small and N = 2 it is possible to have ∂CSOld /∂s > 0. Considering the extreme case of
α = 0 and s = 0 where ∂p∗ /∂s is minimized and ∂CSOld /∂s is maximized, equation (10) simplifies
and ∂p∗ /∂s = −δ/(N − 1). For this extreme case, ∂CSOld /∂s = ∂CS/∂s and
δ
N −1
∂CS
=
−
.
∂s
N −1
N
For more than 2.618 firms, its not possible for ∂CS/∂s > 0 as the upper bound of δ is one. Thus
in a duopoly setting with restrictions imposed upon α, δ and s, the welfare of all consumers may
increase due to a marginal increase in the switching cost. For more than 2.618 firms, it is not
possible to increase the welfare of all consumers with a marginal increase in switching costs.
5
A Continuous Distribution of Switching Costs
In the model outlined in Section 2, all old consumers have a switching cost of s0 so that switching
costs are homogeneous across consumers and over time. While the results listed in Proposition 2
are true for any level of positive finite switching cost, one might question the validity of the results
19
This result differs from Taylor (2003) where efficiency is maximized with only one firm.
23
since all consumers are forced to have the same level of switching cost. One of the contributions
of Biglaiser et al. (2013) is to show that switching cost heterogeneity has important implications
for firms’ strategies. This section evaluates the robustness of the homogeneous switching cost
assumption and allows for a continuous distribution of switching costs across consumers.
Switching cost heterogeneity is modeled similar to the formulation used by Taylor (2003). The
specification of s0z defined in Assumption 1 is modified by the following assumption which is maintained throughout this section.
Assumption 3. Let s0z = εszt where εszt is i.i.d. and follows an exponential distribution with density
g(ε) = e−ε .
εszt is independent across consumers and over time, similar to Taylor’s (2003) specification. The
exponential distribution is used because it incorporates more realistic switching cost heterogeneity
while still allowing for an analytical solution.
Allowing for switching cost heterogeneity does not significantly impact the determination of
the steady state symmetric pure strategy pricing rule. The biggest change is in the general logit
probability, equation (1), which is now
i
Pzt
=
Z
0
∞
i
eVzt
PN
j=1
j
eVzt
g(εszt )dεszt
and is the probability that consumer z chooses product i in period t. The difference between the
above equation and equation (1) results from the fact that the expected level of switching cost
must now be determined. This difference influences the results of Lemmas 1 and 2. Updating these
lemmas to include switching cost heterogeneity results in the following.
Lemma 3. If pit = pjt ∀ i, j,
PN =
1
N
POL =
ln(N )
N −1
POS =
N − 1 − ln(N )
(N − 1)2
(14)
are the demand functions for each consumer type.
As before, at the steady state symmetric equilibrium each firm’s market share is σti = 1/N
and each firm announces a price of p∗ = pit (σti = 1/N ) in every period. The following proposition
24
describes the pure strategy pricing rule supporting a symmetric MPE with heterogeneous switching
costs.20
Proposition 3. Given that a symmetric MPE exists, σ0i = 1/N for i = 1, . . . , N , ρ = α, M0 = 1,
and Assumption 3, the steady state symmetric pure strategy MPE price of the problem defined by
equation (3) is
p∗ =
N ((N − 1)2 − δ(1 − α)L)
(N − 1)3 − (1 − α)W
where
L = N ln(N ) − N + 1
and
W = N 2 − 1 − 2N ln(N ).
The steady state symmetric price is not a function of switching costs anymore since switching costs
are now treated as a random variable. The equilibrium price is now only a function of the number
of firms, firm’s discount factor and the fraction of new consumers in the market. For N ≥ 2, both
L and W are greater than zero.
Similar to the previous analysis, when α = 1 there are only new consumers in the model. New
consumers do not have any switching costs and the equilibrium price is the standard logit model
price of N/(N − 1). As before, the price is decreasing in the discount factor where
(1 − α)N L
∂p∗
=−
< 0.
∂δ
(N − 1)3 − (1 − α)W
As firms become more patient, the investment strategy takes precedent and firms offer lower prices
to build future market share.
A change in the fraction of new consumers affects the equilibrium price in the following way:
∂p∗
N (N − 1)2 (δ(N − 1)L − W )
=
.
∂α
((N − 1)3 − (1 − α)W )2
The sign of
∂p∗
∂α
is determined by the sign of δ(N −1)L−W which may be either positive or negative
depending upon the relationship between N and δ. Identical to the model with homogeneous
switching costs the sign of
∂p∗
∂α
is linked to the sign of the price difference when comparing the price
20
Similar to Proposition 1, Proposition 3 is based on the existence of a symmetric MPE. The intuition regarding
existence is identical to that provided for Proposition 1.
25
with switching costs to the price without. The difference in price created by switching costs is
∆p∗ = p∗ −
N
(1 − α)N (W − δ(N − 1)L)
=
N −1
(N − 1)((N − 1)3 − (1 − α)W )
(15)
and the sign of ∆p∗ is determined by the sign of W − δ(N − 1)L. Switching costs lead to lower
prices when W < δ(N − 1)L and this is more likely for large values of δ and large values of N .
When W < δ(N − 1)L, the price difference, ∆p∗ is negative and
∂p∗
∂α
> 0. This result leads to the
following corollary.
Corollary 4. With heterogeneous switching costs satisfying Assumption 3, ∆p∗ < 0 if and only if
∂p∗ /∂α > 0.
Note that the price difference does not depend on α as is the case with homogeneous switching
costs.
In Section 4.2, N 0 was determined such that for N > N 0 switching costs led to lower prices.
The analog in this section would be to solve W − δ(N − 1)L = 0 for N , but the analytical solution
does not exist. Instead, solve for δ where
δ̂ =
N 2 − 1 − 2N ln(N )
W
=
(N − 1)L
(N − 1)(N ln(N ) − N + 1)
and if δ > δ̂ switching costs lead to lower prices. For the minimum number of firms allowed (two),
δ̂(N = 2) = 0.5887. If δ > 0.5887 when there are two firms, equilibrium prices are lower with
switching costs. Since it is more likely that equilibrium prices are lower with switching costs as N
increases, if δ > 0.5887 equilibrium prices are always lower with switching costs regardless of the
number of firms. If instead δ is less than 0.5887, there is some N 0 such that for N > N 0 switching
costs lead to lower prices. Figure 2 illustrates the parameter space and indicates when prices are
higher or lower with switching costs. Note that the case of N = 10 is only provided as a reference
point.
The results from this section are collected in the following proposition.
Proposition 4. Assume α 6= 1 and consumer switching costs satisfy Assumption 3 then
1. if δ >
3−4 ln(2)
2 ln(2)−1 ,
switching costs always lead to lower prices (∆p∗ < 0) and
26
2. if δ <
3−4 ln(2)
2 ln(2)−1 ,
there exists some N 0 such that for N > N 0 ∆p∗ < 0.
This section shows that the results contained in Proposition 2 are quite robust. When a continuous distribution of switching costs is incorporated into the model, equilibrium prices are lower
with switching costs more than one might expect.
6
Conclusion
This paper examines the trade-offs firms face when consumers have switching costs and firms
cannot price discriminate. Each firm has two conflicting strategies in which they can offer a low
price to invest in future market share or offer a high price to exploit the partial lock-in of their
previous customers. The model developed in this paper combines the discrete choice logit model
with switching costs in a dynamic setting. Competition takes place over an infinite horizon where
overlapping generations of consumers have switching costs and any number of firms produce one
variety of a differentiated product.
The main results of this paper show under what conditions switching costs lead to lower or
higher prices and under what conditions an increase in switching costs decreases the price. These
results depend on the relationship between the number of firms, the level of consumer switching
costs, and firms’ discount factor. Similar to dynamic duopoly studies in the previous literature,
switching costs are pro-competitive when the level of switching costs is sufficiently small or firms’
discount factor is sufficiently large. Otherwise switching costs are anti-competitive. A main result
of the paper is to show that with a sufficiently large number of firms switching costs are procompetitive, and with a sufficiently small number of firms switching costs are anti-competitive.
The relevant threshold for the number of firms depends on firms’ discount rate and the level of
switching costs where this threshold is increasing in the level of switching costs and decreasing in
the discount rate. It is also shown that this result is robust to uniform and heterogeneous switching
costs.
27
A
A.1
Proofs and Derivations
Proof of Lemma 1
Proof. Equation (1) defines the probability that consumer z chooses product i in period t. V i is
either V i = y − pi or V i = y − pi − s0 depending on the consumer’s previous purchase. For a new
consumer with no switching costs,
PNi =
exp(y − pi )
N
X
exp(y − pj )
(16)
j=1
is the probability that a new consumer purchases from Firm i. An old consumer who purchased
from Firm i in the previous period purchases from Firm i again with probability
i
=
POL
exp(y − pi )
N
X
j
0
,
(17)
i
exp(y − p − s ) + exp(y − p )
j6=i
There are old and switching consumers who purchased from Firm k 6= i in period t − 1 and switch
to Firm i in period t with probability
ki
POS
=
exp(y − pi − s0 )
N
X
.
(18)
exp(y − pj − s0 ) + exp(y − pk )
j6=k
Price symmetry within periods allows exp(y − pi ) terms to be factored out of all the numerator
and denominators in the above probability expressions. Factoring out this term for Pni results in
Pni = 1/N . The summation of exp(−s0 ) in the denominator of both POL and POS occurs N − 1
times. After substituting in 1 − s for exp(−s0 ), the denominator for both probabilities becomes
(1 − s)(N − 1) + 1. The resulting probabilities are those listed in equation (4).
A.2
Proof of Lemma 2
Proof. Equation (5) is obtained by
substituting the symmetric probabilities from equation (4) into
PN
equation (2) and observing that k6=i σ k = 1 − σ i .
Iteration of equation (5) results in equation (6). Market shares evolve according to equation (5)
which can be rewritten as
i
σt+1
=
α
(1 − α)(1 − s)
(1 − α)sσti
+
+
.
N
(1 − s)(N − 1) + 1 (1 − s)(N − 1) + 1
Let
A=
α
(1 − α)(1 − s)
+
N
(1 − s)(N − 1) + 1
28
and
B=
(1 − α)s
(1 − s)(N − 1) + 1
i
so that σt+1
= A + Bσti . Iteration yields
σti
A
A
i
Bt
=
+ σ0 −
1−B
1−B
where A/(1 − B) = 1/N .
A.3
Proof of Proposition 1
Proof. Now suppose that all firms, except for Firm i, use a Markov pricing strategy of g(σ) ∈ P.
Below I specify Firm i’s problem conditional on all other players using g(σ), or g for simplicity.
When all other firms set their price to g, σ j = (1 − σ i )/(N − 1) and the state space of Firm
i’s problem is reduced from σ ∈ [0, 1]N to σ i ∈ [0, 1]. The Bellman equation from the problem
described in (3) reduces to
i
V (σti ) = max π i (pit , σti , g) + δV (σt+1
)
pit
where
π i (pit , σti , g) = pit Qit (pit , σti , g).
Dispensing with the i superscripts and using the given conditions of ρ = α and M0 = 1, the
first order condition of the Bellman with respect to pt is
∂Qt
∂V (σt+1 )
∂Qt
Qt + pt
+δ
= 0.
(19)
∂pt
∂pt
∂σt+1
This first order condition reflects the fact that a marginal change in a firm’s price affects it’s
current profits, but also changes its future market share affecting the continuation value. The
Euler equation is
∂V (σt )
∂Qt
∂Qt
∂V (σt+1 )
=
pt + δ
.
(20)
∂σt
∂σt
∂σt
∂σt+1
Qt is the demand for Firm i in period t as defined by equation (2). When all other firms use a
price of g
Qt (pt , σt , g) = αPNt + (1 − α)σt POLt + (1 − α)(1 − σt )POSt
where PNt , POLt , and POSt are functions of pt , g, s and N . Partially differentiate firm demand
with respect to prices and shares to get
∂Qt
= (1 − α)POLt − (1 − α)POSt
∂σt
which is substituted back into equation (20), and
∂Qt
∂PNt
∂POLt
∂POSt
=α
+ (1 − α)σt
+ (1 − α)(1 − σt )
∂pt
∂pt
∂pt
∂pt
29
(21)
(22)
which is substituted back into equation (19).
Equations (16) through (18) define the logit probabilities from the different types of consumers,
and the partial derivatives of the logit probabilities with respect to the price is determined. The
probability that a new consumer purchases from Firm i is
PNt =
exp(y − pt )
exp(−pt )
=
N
exp(−pt ) + (N − 1) exp(−g)
X
exp(y − pt ) +
exp(y − g)
j6=i
as defined in equation (16). The partial derivative of PNt with respect to pt is
∂PNt
exp(−pt )
(exp(−pt ))2
=−
= PNt (PNt − 1) .
+
∂pt
exp(−pt ) + (N − 1) exp(−g) (exp(−pt ) + (N − 1) exp(−g))2
The other partial derivatives are determined in a similar fashion and are listed below.
∂POLt
= POLt (POLt − 1) ,
∂pt
∂POSt
= POSt (POSt − 1)
∂pt
(23)
Any symmetric pure strategy MPE is characterized by price symmetry within each period. Now
impose g = pt so that prices are symmetric within each period to determine the symmetric pure
strategy price. Lemma 1 now applies and
PN =
1
N
POL =
1
(1 − s)(N − 1) + 1
POS =
(1 − s)
(1 − s)(N − 1) + 1
where the probabilities are now just functions of N and s. With price symmetry, the partial
derivatives in equation (23) are just functions of N and s, and the partial derivatives in equation (21)
are just functions of α, N , and s.
Since σ0i = 1/N , ρ = α and M0 = 1, if g = pt , Lemma 2 is also applicable. These symmetric
steady state conditions result in σ = Q = 1/N , and another condition of the symmetric steady
state is that
∂V (σt )
∂V (σt+1 )
=
∂σt
∂σt+1
for all t because in the steady state the marginal change in firm i’s continuation value over time is
independent of the period.
With the above conditions, equation (22) can now be expressed as
(1 − α)
(1 − α)(N − 1)
∂Q
= αPN (PN − 1) +
POL (POL − 1) +
POS (POS − 1)
∂p
N
N
where this partial is a function of α, N , and s. Equation (20) can be rewritten as
∂Q
∂σ p
∂V (σ)
=
∂σ
1 − δ ∂Q
∂σ
30
and first order condition, equation (19), simplifies to the following.
1
∂Q
∂V (σ)
∂Q
+δ
=0
+p
N
∂p
∂p
∂σ
After substitution, the first order condition is now a function of p, N , α, δ, and s and solving for
p yields the symmetric steady state price listed in equation (7).
Note that the method used to determine the price is unconventional. Typically equations (19)
and (20), are used to find a Markov pricing rule, pit (σ t ), characterizing firm i’s price in period t
as a function of all firms’ market shares in period t. Once the Markov pricing rule is determined,
simplifying assumptions can be made to determine the symmetric steady state pricing rule. The
approach taken here is to bypass the determination of the Markov pricing rule and just determine
the symmetric steady state price. Doing so requires the existence of a Markov pricing strategy
resulting in a symmetric steady state MPE in pure strategies for this problem.
A.4
Proof of Lemma 3
Proof. When switching costs are drawn from an exponential distribution, the general choice probability is
Z ∞
i
eVzt
i
s
s
Pzt =
PN V j exp(−εzt )dεzt .
zt
0
j=1 e
Similar to the proof of Lemma 1, Vzti is either Vzti = y − pit or Vzti = y − pit − εszt depending on the
consumer’s previous purchase. For a new consumer with no switching costs,
PNi t =
exp(y − pit )
N
X
exp(y − pjt )
(24)
j=1
is the probability that a new consumer purchases from Firm i. Since new consumers do not have
switching costs, equation (24) is identical to equation (16). An old consumer who purchased from
Firm i in the previous period purchases from Firm i again with probability
Z ∞
exp(y − pit )
i
POLt =
exp(−εszt )dεszt .
(25)
N
X
0
j
exp(y − pt − εszt ) + exp(y − pit )
j6=i
There are old and switching consumers who purchased from Firm k 6= i in period t − 1 and switch
to Firm i in period t with probability
Z ∞
exp(y − pit − εszt )
ki
POSt =
exp(−εszt )dεszt .
(26)
N
X
0
j
s
k
exp(y − pt − εzt ) + exp(y − pt )
j6=k
Price symmetry within periods allows exp(y − pit ) terms to be factored out of all the numerator
and denominators in the above probability expressions (note that exp(y − p − εszt ) = exp(y −
p) exp(−εszt )). Factoring out this term for PNi t results in PN = 1/N . The summation of exp(−εszt )
31
i
i
in the denominator of both POL
and POS
occurs N − 1 times. The choice probabilities for old
t
t
consumers are now
Z ∞
1
POL =
exp(−εs )dεs
s) + 1
(N
−
1)
exp(−ε
Z0 ∞
exp(−εs )
exp(−εs )dεs .
POS =
(N − 1) exp(−εs ) + 1
0
s
Applying a change of variables where r = e−ε the choice probabilities for old consumers are
transformed into
Z 1
1
POL =
dr
0 (N − 1)r + 1
Z 1
r
POS =
dr
0 (N − 1)r + 1
and integration yields the choice probabilities listed in equation 14.
Firm i’s market demand/share in each period is defined by equation (2). The choice probabilities
from equation (14) are substituted into equation (2) to yield
Qit =
(1 − α)((1 − σti )(N − 1 − ln(N )) + σti ln(N )(N − 1))
α
+
N
(N − 1)2
for the symmetric demand. Rearrange the demand equation to get
Qit =
α
(1 − α)(N − 1 − ln(N ))
(1 − α)(N ln(N ) − N + 1)
+
+ σti
.
2
N
(N − 1)
(N − 1)2
The evolution of steady state market share is determined in Lemma 2 for a homogeneous level
of switching costs. The technique in the proof to Lemma 2 is applied to the symmetric demand
equation above letting A equal the constant term without σti , and letting B equal the multiplicative
term on σti . Each firm’s market share evolves according to the following equation:
σti
1
(1 − α)(N ln(N ) − N + 1) t
1
i
+ σ0 −
.
=
N
N
(N − 1)2
Similar to the results of Lemma 2 if initial market shares are symmetric (σ0i = 1/N ) and prices are
symmetric within periods, market shares are symmetric across periods, σti = 1/N .
A.5
Proof of Proposition 3
Proof. The proof of Proposition 3 is directly related to the proof of Proposition 1. Suppose that
all firms, except for Firm i, use a Markov pricing strategy of g(σ) ∈ P. The first order condition
of the Bellman, equation (19), and the Euler equation, equation (20), from Proposition 1 still
apply. The partials of firm demand from Proposition 1 are still relevant as well: equation (21) and
equation (22).
∂P
Similar to Proposition 1, ∂pNt t = PNt (PNt − 1) since switching costs do not influence the choice
probability for new consumers. The remaining partial derivatives of choice probabilities,
∂POSt
∂pt ,
∂POLt
∂pt
and
are determined from equation (25) and equation (26). The partial derivative of POLt in
32
equation (25) with respect to a change in pt is
i
∂POL
t
=
∂pit
Z
∞
!2
ey−pt
N
X
0
e
−εs
s
Z
dε −
0
s
ey−g−ε + ey−pt
j6=i
∞
ey−pt
N
X
s
e−ε dεs .
s
ey−g−ε + ey−pt
j6=i
Now impose price symmetry within periods (g = pt ) to factor out ey−pt terms, and allow for a
s
change of variables where r = e−ε to get
∂POL
=
∂p
Z
0
1
1
(N − 1)r + 1
2
Z
dr −
0
1
1
1
ln(N )
dr =
−
.
(N − 1)r + 1
N
N −1
The remaining partial derivative is determined in a similar fashion and is listed below.
∂POS
2 + N ln(N ) − (N − 1)2
2 ln(N )
=
−
∂p
N (N − 1)2
(N − 1)3
As in the proof of Proposition 1, since σ0i = 1/N , ρ = α and M0 = 1, if g = pt we have a
symmetric steady state of the model. These symmetric steady state conditions result in σ = Q =
1/N , and another condition of the symmetric steady state is that
∂V (σt )
∂V (σt+1 )
=
∂σt
∂σt+1
for all t because in the steady state the marginal change in firm i’s continuation value over time
is independent of the period. After substitution, the first order condition of the Bellman and the
Euler equations are now just functions of p, N , α, and δ, and solving for p yields the symmetric
steady state price.
33
Figure 2: The Parameter Space with Switching Cost Heterogeneity
δ
1
Price is Lower with Switching Cost
0.5887
δ=
N 2 −1−2N ln(N )
(N −1)(N ln(N )−N +1)
0.4195
Price is Higher with Switching Cost
0
2
10
34
N
References
Anderson, S. P., de Palma, A., & Thisse, J.-F. (1992). Discrete choice theory of product differentiation. Cambridge, MA: MIT press.
Arie, G., & Grieco, P. L. E. (2014). Who pays for switching costs? Quantitative Marketing and
Economics, 12 , 379 – 419.
Armstrong, M. (2006). Recent developments in the economics of price discrimination. In Advances in economics and econometrics: Theory and applications: Ninth world congress of the
econometrics society (Vol. II, pp. 97 – 141). Cambridge, UK: Cambridge University Press.
Beggs, A., & Klemperer, P. (1992). Multi-period competition with switching costs. Econometrica,
60 (3), 651 – 666.
Biglaiser, G., Crémer, J., & Dobos, G. (2013). The value of switching costs. The Journal of
Economic Theory, 148 (3), 935 – 952.
Cabral, L. (2014). Dynamic pricing in customer markets with switching costs. Working Paper .
Retrieved from http://luiscabral.org//economics/workingpapers/scostsApril2014
.pdf
Chen, Y. (2005). Oligopoly price discrimination by purchase history. In Pros and cons of price
discrimination (p. 101-130). Stockholm: Swedish Competition Authority.
Chen, Y., & Pearcy, J. (2010). Dynamic pricing: When to entice brand switching and when to
reward consumer loyalty. The RAND Journal of Economics, 41 (4), 674 – 685.
Doganoglu, T. (2010). Switching costs, experience goods and dynamic price competition. Quantitative Marketing and Economics, 8 (2), 167 – 205.
Dubé, J.-P. H., Hitsch, G. J., & Rossi, P. E. (2009). Do switching costs make markets less
competitive? Journal of Marketing Research, 46 (4), 435 – 445.
d’Aspremont, C., Gabszewicz, J. J., & Thisse, J. F. (1979). On hotelling’s ”stability in competition”.
Econometrica: Journal of the Econometric Society, 47 (5), 1145 – 1150.
Fabra, N., & Garcı́a, A. (2015). Dynamic price competition with switching costs. Dynamic Games
and Applications, May.
Farrell, J., & Klemperer, P. (2007). Coordination and lock-in: Competition with switching costs and
network effects. In M. Armstrong & R. H. Porter (Eds.), Handbook of industrial organization
35
(Vol. 3, pp. 1967 – 2072). Elsevier.
Farrell, J., & Shapiro, C. (1988). Dynamic competition with switching costs. The RAND Journal
of Economics, 19 (1), 123 – 137.
Klemperer, P. (1995). Competition when consumers have switching costs: An overview with
applications to industrial organization, macroeconomics, and international trade. The Review
of Economic Studies, 62 (4), 515 – 539.
Padilla, A. J. (1995). Revisiting dynamic duopoly with consumer switching costs. Journal of
Economic Theory, 67 (2), 520 – 530.
Rhodes, A. (2014). Re-examining the effects of switching costs. Economic Theory, 57 , 161 – 194.
Rosenthal, R. W. (1980). A model in which an increase in the number of sellers leads to a higher
price. Econometrica, 48 (6), 1575 – 1579.
Shin, J., & Sudhir, K. (2008). Switching costs and market competitiveness: De-constructing
the relationship. Working Paper . Retrieved from http://faculty.som.yale.edu/ksudhir/
papers/Switching%20Cost Shin%20and%20Sudhir%202008.pdf
Somaini, P., & Einav, L. (2013). A model of market power in customer markets. The Journal of
Industrial Economics, 61 (4), 938 – 986.
Taylor, C. R. (2003). Supplier surfing: Competition and consumer behavior in subscription markets.
The RAND Journal of Economics, 34 (2), 223 – 246.
Train, K. E. (2003). Discrete choice methods with simulation. Cambridge, UK: Cambridge University Press.
Viard, V. B. (2007). Do switching costs make markets more or less competitive? the case of
800-number portability. The RAND Journal of Economics, 38 (1), 146 – 163.
Villas-Boas, J. M. (2015). A short survey on switching costs and dynamic competition. Working
Paper .
von Weizsäcker, C. C. (1984). The costs of substitution. Econometrica, 52 (5), 1085 – 1116.
Wilson, C. M. (2012). Market frictions: A unified model of search costs and switching costs.
European Economic Review , 56 (6), 1070 – 1086.
36