What Types of Switching Costs to Create?
Ki-Eun Rhee†
October 2009
Abstract
One of the perennial questions in the switching cost literature is whether switching costs
enhance or harm firms’ profits. In this paper, we ask what types of switching costs,
among those that are commonly observed, enhance firms’ profits provided that firms
can endogenously influence the magnitude of switching costs. We find that the timing
when firms can commit to the establishment of switching costs is one important
criterion that determines whether a particular type of switching costs will enhance or
harm firm’s profits. More specifically, we find that the types of switching costs that
must be established prior to price-competing stages have adverse effects on profits.
On the other hand, the types of switching costs that can be established in the midst of
price-competing stages are found to enhance profits. The results carry a strategic
implication that firms should generally try to minimize real or social switching costs
while maximize contractual or pecuniary switching costs.
Key words: endogenous switching costs, behavior-based price discrimination.
JEL Code: L11, L15.
†
Author’s affiliation: KDI School of Public Policy and Management, 87 Heogiro, Dongdaemun-gu, Seoul,
Korea, 130-868. Email: [email protected] I thank seminar participants at Sogang University,
Kookmin University and the KDI School of Public Policy and Management.
1
Introduction
Switching costs arise when buyers who purchase products repeatedly or who purchase
follow-on products find it costly to switch from one supplier to another. Examples are
numerous and there are many different types of switching costs that arise from different
sources depending on the nature of the product.
Klemperer (1995) and Farrell and
Klemperer (2007) group various types of switching costs into two broad categories —
real/social costs and contractual/pecuniary costs. Real or social costs of switching arise
when firms choose incompatible technologies between complementary products, when
firms create transaction costs of switching, or when there are high start-up or learning
costs for first-time users.1 Contractual or pecuniary costs are created when firms employ
loyalty contracts such as frequent-flyer programs or when firms offer on-pack coupons to
induce repeat purchases.
In this paper, we ask which types of switching costs, among those that are commonly
observed, enhance firms’ profits provided that firms can endogenously influence the magnitude of switching costs. We present a simple yet versatile model that treats switching
costs in a broad sense while being capable of reflecting many different types of switching costs with further specification.
The model thus allows us to compare switching
costs of different nature within one base model.
Our main finding is that the timing
when firms can commit to the establishment of switching costs is one important criterion
that determines whether a particular type of switching cost will enhance or harm firms’
profits.
To be more specific, we consider two timing structures — the early commitment and the
late commitment. In the early commitment model, we consider types of switching costs
that must be established at the product design or manufacturing stages before any price
competition begins. Many examples of real or social switching costs fit the description of
this model since choices regarding technologies are often made before engaging in price
1
See Section 2.1 of this paper or Section 2 of Klemperer (1995) for more detailed examples.
1
competition in the product market.
In the late commitment model, we consider the
types of switching costs that are usually created in the midst of price competing stages.
Many contractual or pecuniary switching costs can be categorized into this model as
most coupons or loyalty contracts are distributed or signed along with the purchase
of a product, at which point the price of the product has already been determined.
Comparing the two timing structures, we show that firms find it optimal to create minimal
switching costs in the early commitment model while finding maximal switching costs to
be optimal in the late commitment model. The results imply that firms should generally
try to minimize real or social switching costs and maximize contractual or pecuniary
switching costs.
The results are driven by how different timing structures interact with firms’ incentives in intertemporal price competition. Traditionally it has been found in the switching
cost literature that firms intertemporally adopt ’bargains-then-ripoff’ price trend. That
is, knowing that customers will face switching costs after the initial purchase, firms try
to get as large a market share as possible by charging a ’bargain’ price in the first period.
Afterwards, firms charge a high ’ripoff’ price to the locked-in customers. This leads to
a fierce competition in the first period and then firms become local monopolies in their
respective turfs.
We add two additional features to the typical two-period switching
cost model.2 First, we assume that firms can endogenously determine the magnitude of
switching costs either before or in between price competing stages. Second, we enable
firms to price discriminate between old and new customers.3 The second feature exacerbates second-period profits from locked-in customers because competition emerges in
2
Note that many switching costs models adopt two-period structure because switching costs, by
nature, link two markets either through time (e.g., cases of repeat purchases) or through product space
(e.g., cases of complementary products). Another branch of literature consider infinite time horizon
with overlapping customers as in Beggs and Klemperer (1992).
3
Early generation of switching cost models without this feature usually find that no consumer ends
up switching suppliers in equilibrium. But in reality, many consumers do switch suppliers. To account
for the discrepancy, Chen (1997) and Fudenberg and Tirole (2000) have shown that consumers do end
up switching in equilibrium if firms can discriminate between old and new consumers. We discuss this
literature later in the this section.
2
each firm’s turf separately. In the early commitment model, we find that the reduction
in second-period profits is large enough so that it is better for firms to commit to not
create any switching costs at all and prevent the first-period competition from becoming
too fierce. In sum, the absence of these types of switching costs softens intertemporal
price competition. The story is quite different in the late commitment model. The types
of switching costs to which firms commit in the midst of price competing stages do not
have intertemporal influence because first-period market shares are already determined
by the time switching costs can be established.
As a result, firms find it optimal to
maximize these types of switching costs in order to defend market shares and make it
unprofitable for competing firms to poach customers.
It is worth noting that studies on endogenous creation of switching costs are quite
limited compared to the vast amount of literature on switching costs.4 Moreover, most
of studies focus on specific types of switching costs to address the question whether
switching costs will enhance or harm firms’ profits.5
The literature so far suggests
that the answer varies from one type of switching costs to another.
We add to the
literature by comparing different types of switching costs within one unifying model and
by providing a simple criterion — the timing of commitment — that firms should consider
when endogenously establishing switching costs.
Our results regarding minimal or maximal degrees of switching costs being optimal
for different types of switching costs seem to fit nicely into the literature on endogenous
creation of switching costs. The following are studies that fit the description the early
commitment model.
Marie
noso (2001) studies system products6 and uses the price of
durable products that consumers must replace when switching suppliers as a proxy for
endogenously created switching costs. Firms can choose between compatible and incom4
See Section 2.8 of Farrell and Klemperer (2007).
Caminal and Matutes (1990) and Caminal and Claici (2007) may be considered as exceptions since
they compare different forms of price commitments, e.g., price commitment vs. coupons or full vs.
partial commitments. Our model includes these types as well as others mentioned in the text.
6
Marie
noso defines systems as goods comprising of durables that are sequentially updated with complements.
5
3
patible technologies and the choice of technologies, which act as a tool for establishing
switching costs, is made prior to price competing stages. She shows that firms opt for
compatible technologies thereby not creating any switching costs.
The result is thus
consistent with our finding that it is optimal to minimize the types of switching costs
that fall into the early commitment model. Also related is the ’mix-and-match’ literature
for systems which often find that firms desire fully compatible technologies.7 Another
example of the early commitment model is Bouckaert and Degryse (2004). They show
that banks share information on their customers to soften intertemporal price competition caused by switching costs when the decision to share information is made before
price competition begins. Therefore, both the timing structure and the results coincide
with our findings in the early commitment model.
Caminal and Matutes (1990), Caminal and Claici (2007) and Wallace (2004) are
studies on types of switching costs that fit the description of the late commitment model.
Caminal and Matutes (1990) endogenize switching costs through long-term contracts
where pre-commited second-period discounts act as switching costs.
Firms pre-comit
to second-period prices or coupons as they decide first-period prices and therefore the
timing structure coincides with our late commitment model.
Wallace (2004) studies
switching costs that are created when firms offer customization services.
The timing
structure in his model also corresponds to our late commitment model since firms decide
whether to customize their products or not after consumers make initial purchases (i.e.,
after learning consumer preferences). Both papers conclude that switching costs enhance
firms’ profits8 and thus the results are consistent with ours that the types of switching
costs that can be categorized into the late commitment model enhance firms’ profits.
This paper is also related to the recent literature on behavior-based price discrimination (hereafter BBPD), wherein firms discriminate their customers based upon past
7
See Marie
noso (2001), Matutes and Regibeau (1988) and Einhorn (1992).
While Caminal and Matutes (1990) show that equilibrium profits increase when firms precommit to
a discount, they also show that equilibrium profits decrease when firms precommit to a second period
price. Nonetheless, it is optimal for firms to create positive switching costs in both cases.
8
4
purchase behaviors.9 This type of price discrimination differs from the traditional first,
second or third degree price discrimination but it is becoming increasingly important with
the advancement of information technologies. BBPD is especially important in relation
to the switching cost literature because models with BBPD explain customer switching
as an equilibrium outcome whereas switching does not occur in equilibrium in many traditional switching cost models. Chen (1997) is the pioneering paper to show customer
switching as an equilibrium behavior using BBPD. He considers a homogeneous duopoly
market where customers are differentiated by switching costs.
Fudenberg and Tirole
(2000) study BBPD in an environment without switching costs. They study a differentiated duopoly model where past purchases of customers matter only for informational
values and call the practice of BBPD "poaching" since firms offer lower prices selectively
to customers of rival firms. Neither Chen (1997) nor Fudenberg and Tirole (2000) study
endogenous creation of switching costs although Fudenberg and Tirole (2000) extends
the firm’s strategy set to include long-term price commitments in a manner similar to
Caminal and Matutes (1990).
The remainder of the paper proceeds as follows.
Section 2.
The base model is introduced in
Subsection 2.1 describes various applications of the base model to many
different types of switching costs that are commonly observed.
In Subsection 2.2, we
present preliminary findings in the second-period competition.
In Section 3, we ana-
lyze the early commitment model which considers the types of switching costs that are
established prior to price-competing stages.
In Section 4, we alter the model to the
late commitment model in which types of switching costs that are created in between
the first- and the second-period price competitions are considered. Section 5 checks the
robustness of our results to two important variances — heterogeneous switching costs and
changing customer preferences. Section 6 concludes.
9
Pioneering contributions were made by Chen (1997), Fudenberg and Tirole (2000) and Villas-Boas
(1999) among many others. Also see Fudenberg and Villas-Boas (2006) for a recent survey.
5
2
Model and Preliminaries
Our base model is a two-period duopoly Hotelling model. Two firms, A and B, produce
products A and B, respectively, and are located at the extremes on the Hotelling line of
length l, where l represents the degree of product differentiation in the industry. The
price charged by firm j ∈ {A, B} in each period t ∈ {1, 2} is denoted by pjt . In period
two, firms can observe the purchase behavior of customers in the previous period and
can "poach" customers of rival firms by offering those customers a discount dj .10 The
marginal costs of production are assumed to be zero and we also assume that the discount
rate δ = 1 for analytical simplicity.11
On the demand side, a unit mass of consumers are uniformly distributed along the
Hotelling line and it is assumed that consumers have a unit demand for the product in
each period. Each consumer has a reservation value v for the product and is indexed
by type θ ∈ [0, l] which denotes a consumer’s relative preference for product B over
product A.12 Thus, the transportation cost incurred by a consumer located at θ when
she purchases from firm A is θ and that when she purchases from firm B is l − θ.
Let sj , j ∈ {A, B} denote the level of switching costs that are endogenously created
by firm j. When a consumer who purchased from firm j in period 1 switches its supplier
to firm −j in period 2, the consumer incurs switching costs in the amount of sj . We
assume that 0 ≤ sj ≤ l, implying that firms cannot artificially create switching costs in
the magnitude that is larger than the degree of differentiation between two products.13
In our model, repeat customers pay pj2 which is the nominal price of the product and new customers
pay pj2 − dj which is the discounted price. This is equivalent to a firm setting two separate prices pR
2 and
j
pN
2 , respectively representing prices to repeat and new customers. We use d to contrast the discount
amount to the amount of switching costs firms endogenously create.
11
Having 0 < δ < 1 will not cause qualitatively different results as long as firms and customers share
the same discount rate.
12
Our base model assumes that consumers’ types are stable over time. We extend the model to allow
for consumers whose preferences change over time in Section 5.2.
13
While our assumption on the upper bound of switching costs is an intuitive one, removing the upper
bound does not change our results. We show in Lemma 2 that sj = l is the minimal amount of
switching cost that is needed to prevent switching altogether. Thus even when there is no upper bound
on switching costs, firms do not have a need to create sj larger than l.
10
6
Lastly, we assume that costs of creating switching costs are negligible.14 The assumption, however, does not imply that firms cannot credibly commit to the establishment of
switching costs because revoking the decision to create switching costs may still be costly.
For example, designing and manufacturing a product to be less or more compatible with
a competing firm’s product may be costless. But once the product has been manufactured, reversing the decision may involve significant cost because some products must be
discarded or because a new product line must be established. In another example, costs
of issuing coupons may be small compared to manufacturing costs of a product. But
after coupons are distributed to consumers, not accepting those coupons may involve
significant legal and administrative costs.15 We provide how the model can be specified
further in the next subsection.
2.1
Types of Switching Costs
As stated in the Introduction, the main goal of this paper is to provide a criterion
regarding what types of switching costs are beneficial to firms.
Since we attempt to
compare switching costs of different nature within one unifying model, we leave the
sources of switching costs quite general in our model as presented above. This is to keep
the versatility of the model and we present below examples of how the model can be
specified further to reflect various types of switching costs without altering any results
we obtain from the general model. We provide literature review along with examples
to demonstrate that our model may encompass different models that study endogenous
creation of switching costs. In particular, we specify what sj and dj in our base model
capture within the context provided in each example.
Compatible vs. Incompatible Systems Firms can use incompatible technologies to cre14
Incorporating costs of creating switching costs will affect the qualitative results of the paper merely
by presenting the maximum amount of switching costs over which it is always not optimal to create
switching costs.
15
Note that, by costs of issuing coupons, we mean administrative costs of manufacturing and distributing coupons and not the discount amounts guaranteed on coupons. The latter cost is captured by dj
in our model.
7
ate switching costs in markets where firms produce a durable primary component and
then produce new consumables or complement components in the aftermarket. Examples
are video game consoles and cartridges, medical equipment and matching consumables,
razors and razor blades, and so on.
Marie
noso (2001) studies this problem and states
that firms can endogenously create switching costs through the price of durable parts
that a consumer must replace when there is an inter-brand incompatibility. Switching
costs sj in our base model is thus equivalent to the price of the primary component. The
discount amount dj is then the difference between prices of complement products that are
offered to old and new customers holding constant the price of the primary component
within the system.16
Product Differentiation Another way of creating switching costs is through product
differentiation. Consumers may perceive larger magnitude of learning costs the larger the
degree of product differentiation. For example, a consumer who is accustomed to using
Windows operating system may feel more reluctant to switch to a Mac operating system
when two systems are more different. Gehrig and Stenbacka (2004) consider this setup by
directly linking the size of switching costs to the degree of product differentiation. In our
model, the degree of product differentiation l is exogenously given and is separate from
switching costs sj . To fully incorporate the model presented in Gehrig and Stenbacka
(2004), we can modify switching costs to be dependent on differentiation, i.e., sj (l) with
∂sj (l)
∂l
> 0.
While we do not consider this case explicitly, we do discuss consequences
of separating/integrating the two variables in the Conclusion. The discount amount dj
is equivalent to the difference between prices charged to old and new customers in the
second period.
Coupons
Firms often use coupons to allure repeat customers thereby artificially
creating switching costs.
Common examples include on-pack coupons that apply to-
16
Note that because new customers usually purchase the "bundled" system, firms can implicitly price
discriminate old and new customers by charging different prices for complementary products. To be
more specific, dj = pjY − (pjXY − pjX ) where pXY is the price of the system offered to new customers while
pX and pY are prices of primary and complementary products, respectively, offered to old customers.
8
wards the next purchase or coupons that are issued after a customer signs up for a
membership. These coupons can be considered as defensive coupons that firms use to
retain own customers. On the other hand, firms can also use offensive coupons in order
to poach competing firms’ customers.
Offering rebate coupons when switching long-
distance phone service providers or automatically dispensing coupons when rival firms’
products are scanned at the grocery are both well-known examples of offensive coupons.
In our model, switching costs sj and discount amounts dj can respectively capture defensive and offensive coupons.
Our model thus encompasses the model with coupons
presented in Caminal and Matutes (1990) which is one of the pioneering works in the
analysis of endogenous switching costs.
Price Commitments Several papers model firms as simply committing to prices as
opposed to using coupons. Examples are Shaffer and Zhang (2000), Caminal and Claici
(2007) and a section in Caminal and Matutes (1990). Firms’ strategies in these models
are typically composed of a first-period price and two second-period prices — one to repeat
customers and another to new customers.17
For these models, we can easily interpret
the difference between the price offered to repeat customers and that offered to new
customers as the combined amount of sj and dj .18
Shaffer and Zhang (2000) calls it
the "pay to switch" strategy when the price offered to old customers is higher than that
offered to new customers and the "pay to stay" strategy when the reverse is true. In
our model, the "pay to switch" strategy is equivalent to the case when dj > sj and the
"pay to stay" strategy is equivalent to the case of the reverse inequality.
Loyalty Programs Another popular method of artificially creating switching costs is
through loyalty programs. Frequent flyer mileages offered by airlines, numerous benefits
provided by credit card companies based on accumulated charges that consumers put
17
Shaffer and Zhang (2000) only compares second-period prices, taking the initial market shares as
given. Also note that Shaffer and Zhang (2000) use the term "switching cost" quite differently from
ours in that switching costs include ex ante product differentiation in their paper.
18
In Caminal and Matutes (1990) and Caminal and Claici (2007), there is no behavior-based price
discrimination and thus there is no discount amount dj .
9
on their credit cards, or gift cards distributed by department stores at the end of the
year based on customers’ yearly spending are all examples of various loyalty programs.
Effects of these programs can be captured by switching costs sj in our model. Because
loyalty programs often offer non-cash rewards (as opposed to coupons that offer direct
cash rewards), consumers may perceive different amounts of switching costs even if they
are enrolled in the same loyalty program.
In such a case, it may be more realistic
to model switching costs as being heterogeneous across consumers.
We consider this
variation in Section 5.1 where we extend the model and check the robustness of our results.
The discount amount dj may take many different forms in this setting. Examples are
department stores offering 10-15% off everything that are purchased on the day customers
open accounts with them or airlines offering 5,000 miles free when customers first sign
up for their frequent flyer programs. These are all attempts to poach competing firms’
customers by reducing the size of switching costs that customers face.
Customization Firms can also make customization efforts to create switching costs
as studied by Wallace (2004). A couple of examples presented in his paper are designing
packing materials best suited for a special inventory tracking system and designing a
payment plan timed to coincide with the customer’s case flow.
Switching costs sj in
this setting are equivalent to the monetary value that consumers assign to customization. Naturally, firms may want to offer a one-time discount to those customers whose
preferences are unknown (i.e., customers who purchased from a rival firm in the previous
period) and this would constitute dj in our model.
Examples above demonstrate that we can easily assign specific contexts to our model
even though the sources of switching costs are left open. In fact, the general nature of the
model allows us to compare different types of switching costs instead of focusing on one
particular kind. In the remainder of the paper, we show that the timing of commitment
matters when determining why some types of switching costs enhance firms’ profits while
some other types do not.
10
2.2
Preliminary Findings : Second-period Competition
We look for subgame perfect equilibrium and thus solve the model backwards. We start
by analyzing the second period competition and the results we obtain here are common
for both early and late commitment models. Let θ1 denote the indifferent consumer in
period one. Then, consumers located along the line segment [0, θ1 ] are offered pA
2 from
B
firm A and pB
2 − d from firm B. Similarly, consumers located along [θ 1 , l] are offered
A
B
prices pA
2 − d and p2 from firms A and B, respectively. If a consumer who purchased
from firm j in period one decides to purchase from the other firm in period two, the
consumer incurs switching costs in the amount of sj , j ∈ {A, B}.
In period two, firm A will successfully poach some of firm B’s customers if and only
A
B
B
if θ + pA
2 − d + s < l − θ + p2 for θ ∈ [θ 1 , l]. Following Fudenberg and Tirole (2000),
let θB
2 ∈ [θ 1 , l] denote the marginal customer on firm B’s turf who switches to firm A in
period 2. Then,
B
A
A
B
B
θB
2 + p2 − d + s = l − θ 2 + p2
1
A
B
A
B
∴ θB
2 = (l − p2 + p2 + d − s )
2
(1)
Analogously, θA
2 ∈ [0, θ 1 ] is defined as the marginal customer on firm A’s turf who switches
to firm B in period 2:
A
A
B
B
A
θA
2 + p2 = l − θ 2 + p2 − d + s
1
A
B
B
A
∴ θA
2 = (l − p2 + p2 − d + s )
2
(2)
B
A
B
A
B
Note that θA
2 ≤ θ 1 ≤ θ 2 as long as s +s ≤ d +d . When the reverse inequality holds,
i.e., sA + sB > dA + dB , switching no longer occurs in both directions (i.e., either θA
2 = θ1
or θB
2 = θ 1 , or both) and the demand changes discontinuously to the one typically found
11
in the switching cost literature.19 We focus on the case where sA + sB ≤ dA + dB as the
analysis for the reverse case is analogous to those found in the literature and later compare
this case to the equilibrium we find here to make sure that the latter one is the global
equilibrium. We mention two important points. First, problems with the existence of
equilibrium is mild compared to models with exogenous switching costs.
Traditional
models with exogenous switching costs face problems with the existence of equilibrium
for some ranges of switching costs because demand is discontinuous. With endogenous
switching costs, however, we only need to confirm that there exists an equilibrium at the
value of switching costs that firms endogenously select. Second, we show in Lemma 2
B
that switching does not occur with θA
2 = θ 2 = θ 1 when firms create maximal switching
costs. It happens that sA + sB = dA + dB when firms create maximal switching costs.
Thus, it becomes unnecessary to separately analyze the case with sA + sB > dA + dB .
j
j
j
−j
−j
j
Let σj2R = σ j2R (pj2 , p−j
2 − d ) and σ 2S = σ 2S (p2 − d , p2 ) represent market shares
R θA2
θA
2
from repeat and switching customers, respectively. That is, σ A
2R = 0 dF (θ) = l ,
R θB2
Rl
R θ1
θB
l−θB
θ1 −θA
2 −θ 1
2
2
, σB
and σ B
.
σA
2S = θ1 dF (θ) =
2R = θB dF (θ) =
2S = θA dF (θ) =
l
l
l
2
Additionally, let
σ j1
=
σ j1 (pj1 , p−j
1 )
2
denote firm j’s market share in period one. The firms’
second-period profits are then:
π j2 (pj2 , dj ; θ1 ) = σ j2R pj2 + σ j2S (pj2 − dj )
(3)
19
θA
2
Specifically, the demand is characterized by a single threshold type θ̂ (instead of two threshold types
and θB
2 ):
⎧ 1£
¤
B
B
A
A
A
B
, if pB
⎨ 2 l − pA
2 + p2 − d + s
2 − p2 < 2θ 1 − l − s + d
A A
θ1 ,
if otherwise
θ̂(p2 , d ) =
¤
⎩ 1£
B
A
A
B
B
A
B
A
−
(p
−
d
)
−
s
−
p
l
+
p
,
if
p
2
2
2
2 > 2θ 1 − l + s − d
2
Note also that the condition sA + sB > dA + dB , i.e., switching costs are sufficiently large, coincides
with the assumption that is often found in the exogenous switching cost literature.
12
The first-order conditions result in following best response functions:
1 −j 1 −j 1 j 1 j
l
p2 − d + d + (s − s−j ) + σ −j
2
4
2
2 1
¶
µ4
1 −j
1
1
−j
p2 − pj2 − s−j + l
− σj1
dj (p−j
2 ,d ) =
2
2
2
−j
pj2 (p−j
2 ,d ) =
Solving the first-order conditions, the second-period equilibrium is specified as:
=
pj∗
2
dj∗ =
j∗
pj∗
=
2 −d
σ j∗
=
R
=
σ j∗
S
1 j
(s + l + 2lσ j1 )
3
1 j
(s + s−j − 2l) + 2lσ j1
3
1
4
l − s−j − lσ j1
3
3
1 j 1 sj
σ + +
3 1 6 6l
2 −j 1 s−j
σ − −
3 1
6
6l
(4)
Observations of the second-period equilibrium show that both an increase in switching
costs and an increase in first-period market shares result in higher prices charged to
repeat customers and lower prices offered to switching customers. Also, the proportion
P j
A
of customers that switch, θB
2 − θ 2 = 2l −
j s , shrinks as switching costs increase.
Therefore, switching costs reduce the intensity of poaching in terms of the number of
customers that switch but increase the intensity of poaching in terms of the aggressiveness
of poaching prices. In addition, firms retain more customers the larger the initial market
shares and switching costs.
3
Early Commitment Model
In this section, we consider types of switching costs that can be established prior to
price competing stages.
Many examples are real and social switching costs that arise
from technological incompatibility and/or product differentiation since decisions regard13
ing technology and/or product designs are typically made before firms engage in price
competition. Formally, we model firms as making their switching cost decisions in period zero. Then firms compete in prices for two consecutive periods where in the second
period, firms can price discriminate consumers based on their first period purchase behaviors as described in Section 2.2.
Since we solve for the subgame perfect Nash equilibrium, we take the second period
equilibrium characterized in Equations (4) as given and analyze the first-period price
equilibrium. We then study the optimal choices of switching costs in period zero.
3.1
Equilibrium in Period One
We start the analysis by characterizing the first-period demand. We assume that consumers have rational expectations as to second-period prices and thus make their purchase
decisions to maximize (minimize) the sum of utilities (payments) in two periods. A consumer who is located at θ ∈ [0, l] and purchases from firm A in the first period expects
a payment stream of:
A∗
B∗
B∗
+ sA }
(θ + pA
1 ) + min{θ + p2 , l − θ + p2 − d
⇐⇒
1
5
1 A 2
1
2
(θ + pA
l + θ + sA }
1 ) + min{ l + θ + s ,
3
3
3
3
3
3
Thus, a consumer who purchase from firm A in period one will purchase from A again
if θ < 14 l + 14 sA . On the other hand, if this consumer purchases from firm B in the first
period, she expects a payment stream of:
A∗
A∗
+ sB , l − θ + pB∗
(l − θ + pB
1 ) + min{θ + p2 − d
2 }
2 B
1 B
1
5
⇐⇒ (l − θ + pB
1 ) + min{l − θ + s , 2l − θ + s }
3
3
3
3
14
This consumer will purchase from B again if θ > 34 l − 14 sB . Note that 14 l + 14 sA ≤ 34 l − 14 sB
for all sj ∈ [0, l].
B
Thus, if first-period prices are such that θ1 (pA
1 , p1 ) <
1
l
4
+ 14 sA ,
there is a marginal consumer who is indifferent between buying from firm A in both
periods and buying from firm B in the first period and then switching to firm A in
B
In the reverse case where θ1 (pA
1 , p1 ) >
the second period.
3
l
4
− 14 sB , the marginal
consumer is indifferent between purchasing from firm B in both periods and buying from
firm A first and then from firm B in the second period. Lastly, there is a case where
1
l + 14 sA
4
3
1 B
B
≤ θ1 (pA
1 , p1 ) ≤ 4 l − 4 s . In this case, the marginal consumer is someone who is
indifferent between buying from firm A first and then switching to firm B in the second
period and vice versa. The first-period market shares σA
1 =
θ1
l
and σ B
1 =
l−θ1
l
can thus
B
be summarized by θ1 = θ1 (pA
1 , p1 ) as follows:
B
θ1 (pA
1 , p1 ) =
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
5
l
12
1
l
2
7
l
12
B
− 14 (pA
1 − p1 ) −
1 A
s
12
+ 16 sB ,
2
4 A
2 B
B
pA
1 − p1 > 3 l − 3 s + 3 s
1 A
1 B
B
− 38 (pA
1 − p1 ) − 4 s + 4 s ,
1 A
B
− 14 (pA
1 − p1 ) − 6 s +
1 B
s ,
12
otherwise
(5)
2
2 A
4 B
B
pA
1 − p1 < − 3 l − 3 s + 3 s
The profit function is then:
j j
−j j
j∗ j∗
j∗
j∗
j∗
π j (pj1 , p−j
1 ) = σ 1 (p1 , p1 )p1 + σ 2R p2 + σ 2S (p2 − d )
(6)
j∗
j∗
j∗
B
are all functions of θ1 (pA
where σ j∗
1 , p1 ) as specified in Equations (4).
2R , σ 2S , p2 and d
Solving the first-order conditions and verifying that second-order conditions are satisfied,
we obtain the following result.
Proposition 1 Equilibrium prices in the early commitment model given sj is characterP j j∗
P j j∗
4
1
2
1
j∗
= 13 l + 13 (sj − 2s−j ). Market
ized as: pj∗
1 = 3l − 3
j s , p2 = 3 l + 6
j s , p2 − d
P j
j∗
j∗
1
1
1
1
1
1
j
−j
j
−j
shares are: σ j∗
1 = 2 − 4l (s − s ), σ 2R = 3 + 12l
j s and σ 2S = 6 + 6l (s − 2s ).
The equilibrium first-period price decreases in switching costs, reflecting the fact that
switching costs intensify intertemporal price competition.
15
The second-period price to
repeat customers increase in switching costs while the price offered to switching customers increase only when own switching costs are twice as large as the competing firm’s
switching costs. Note that the bargains-then-ripoff price trend that is often found in the
literature is restored only when switching costs are sufficiently large. More specifically,
P j 4
j∗
j∗
j∗
j∗
j∗
j s > 3 l must hold for p1 < p2 to be true and it is not possible to have p1 < p2 − d
in the relevant range of s.
As for market shares, the first-period market share decreases in own switching costs.
This is because consumers are forward looking and predict that firms with high established switching costs will charge high prices in the second period.
Therefore, only
those customers with strong preferences purchase from a firm that has established high
switching costs.
3.2
Equilibrium in Period Zero
In this section, we analyze the choice regarding switching costs while taking as given
equilibrium prices specified in Proposition 1. We first establish that firms need to create
maximal switching costs to lock in all customers.
Lemma 2 If an equilibrium outcome in which no customer switches exists, then firms
must create maximal switching costs, i.e., sj = l in such an equilibrium
When firms create maximal switching costs, each firm will offer a zero price (i.e.,
pj2 − dj = 0 when sj = l) to customers of rival firms in the second period but no consumer
will end up switching suppliers due to high switching costs. We proceed the analysis by
finding the equilibrium choice of switching costs when poaching is successful, i.e., when
some consumers do switch, and compare it to the case when all customers are locked-in.
We first characterize profit in terms of switching costs alone by substituting equilibrium prices provided in Proposition 1 into the profit function in Equation (6):
πj (sj , s−j ) =
P
P
1
1
1
(2l − sj + s−j )(4l − j sj ) +
(4l + j sj )2 +
(l + sj − 2s−j )2 (7)
12l
36l
18l
16
Calculations of first and second order conditions show that profit function is convex with
respect to switching costs.20
or sj = l.
Thus, the maximum occurs at either boundaries, sj = 0
The following Proposition shows that firms find it best to not create any
switching costs at all.
Proposition 3 Equilibrium choice of switching costs in the early commitment model is
sj∗ = 0.
4
Corollary 4 Equilibrium prices and profits in the early commitment model are: pj∗
1 = 3 l,
j∗
2
1
j∗
j∗
=
pj∗
2 = 3 l and p2 −d = 3 l. Total profits are π
and π j∗
2S =
17
l
18
among which π j∗
1 =
12
l,
18
π j∗
2R =
4
l
18
1
Firms share the market equally in the first period σj∗
1 = 2 , and keep
1
l.
18
two-thirds of the initial market share with σ j∗
2R =
1
3
1
and σ j∗
2S = 6 .
The equilibrium outcome essentially coincides with the result in Fudenberg and Tirole
(2000) since firms decide not to create any switching costs at all. To understand why firms
find it beneficial to not create any switching costs, first consider standard switching cost
models without BBPD. The existence of switching costs in those models has two opposing
effects on profit.
First, there is the positive "Harvesting" effect: firms can charge a
high second-period price to repeat customers as they are locked in due to switching
costs.
Second, there is the negative "Investment" effect: the competition for market
shares is fierce in the first period because of high second-period profits. With BBPD,
the harvesting effect is exacerbated because a portion of customers who are targeted
with lower prices by competing firms will end up switching suppliers. Meanwhile, the
investment effect persists.
It turns out that the costs of securing large market shares
outweigh the benefits of locking-in customers since not all customers are locked in. Hence
firms find it better to commit to not create any switching costs at all in order to avoid
unnecessarily fierce competition for initial market shares. As stated in the Introduction,
20
∂π j
5
= − 18
l+
∂sj
11 j
36 s
−
7 −j
36 s
and
∂ 2 πj
=
∂(sj )2
11
36
> 0.
17
the same intuition is often found in studies for systems in which compatibility between
two competing system products softens intertemporal price competition.
4
Late Commitment Model
In this section, we study types of switching costs that can be established in between
price-competing stages. Formally, firms decide sj either simultaneously with pj1 , or after
deciding pj1 but before deciding pj2 . Either way, the mathematical analysis turns out to be
the same. Many strategies adopted by firms to create contractual or pecuniary switching costs, such as loyalty programs including frequent-flyer programs, on-pack coupons
to promote repeat purchases, or customization to personalize customer services fit this
description.
Note that this game is no different from the one analyzed in Section 3 as far as
the second-period price competition is concerned.
and applicable to this section.
Thus, Equations (4) are still valid
Substituting Equation (4) into Equation (3), firm’s
maximization problem becomes:
j
maxj π j (sj , pj1 ) = σ j1 (pj1 , p−j
1 )p1 +
sj , p1
1
1
−j
(l + sj + 2lσ j1 )2 +
(4lσ −j
− l)2
1 −s
18l
18l
(8)
The first term in the profit function is the profit from first-period sales. The second term
is the profit from repeat customers in the second period and it is positively related with
the firm’s first-period market share and switching costs. The last term is the secondperiod profit from switching customers and it is positively related with the competing
firm’s market share and negatively related with the competing firm’s switching costs.
Thus, it is easier and more profitable to poach rival firm’s customers the larger the
competing firm’s customer base or the smaller its switching costs.
B
Note that the optimal sj is determined separately from pj1 and thus from σ j1 (pA
1 , p1 ).
∂π j
> 0 for ∀sj . Hence firms wish to
It is evident from the above equation that
∂sj
18
create maximal switching costs, i.e., sj∗ = l.21
This is true whether sj is determined
P
P
simultaneously with or after the decision regarding pj1 . Note that j sj∗ = j dj∗ and
B∗
= θ1 when sj∗ = l.
that θA∗
2 = θ2
Thus, switching costs are at maximal levels and
customer switching does not occur in the second period.
What is left to determine are first-period prices. Since switching does not occur in the
second period, the marginal consumer in the first period is someone who is indifferent
between purchasing from firm A in both periods and purchasing from firm B in both
periods. Formally,
2
7
5
5
B
θ1 + pA
1 + l + θ 1 = l − θ 1 + p1 + l − θ 1
3
3
3
3
j
−j
j A B
1
3
j∗
This results in σ j1 (pj1 , p−j
1 ) = 2 − 16l (p1 − p1 ). Given σ 1 (p1 , p1 ) and s = l, profits firms
wish to maximize in the first period can be re-written into:
j j
−j j
π j (pj1 ; p−j
1 ) = σ 1 (p1 , p1 )p1 +
1
1
2
2
(2l + 2lσ j1 (pj1 , p−j
(4lσ j1 (pj1 , p−j
1 )) +
1 ) − 2l)
18l
18l
First-order conditions result in the best response function pj1 (p−j
1 ) =
7 −j
p
19 1
+
24
l.
19
(9)
Equi-
librium for the late commitment model is specified in the Proposition below.
Proposition 5 Equilibrium choice of switching costs in the late commitment model is
sj∗ = l.
j∗
j∗
j∗
Equilibrium prices are: pj∗
= 0.
1 = 2l, p2 = l and p2 − d
Equilibrium
j∗
j∗
j∗
3
1
profits are πj∗
2 = 2 l among which π 1 = l, π 2R = 2 l and π 2S = 0. No consumer switches
suppliers.
The result that firms find it optimal to create maximal switching costs is the opposite of the result we obtained in the early commitment model.
The crucial difference
between the two models is whether first-period prices are strategically dependent on
21
While s is bounded above by l by assumption, even without such an assumption, s = l is the minimal
amount of switching costs that are necessary to prevent switching altogether as stated in Lemma 2. Thus,
while assumed to be negligible in our model, with even a slight cost of creating switching costs, firms
will choose s = l.
19
switching costs or not. In the late commitment model, first-period prices are strategically independent of switching costs because the types of switching costs are the ones
to which firms commit along with price-setting stages. In short, switching costs do not
have intertemporal influence on first-period prices and firms only need to consider how
switching costs affect second-period prices.
As shown above, the larger the switching
costs, the higher the profits because switching costs lead to customer lock-in which in
turn lead to higher second-period prices and less customer switching.
An interesting aspect about the above result is the fact that prices decrease over
time even when all customers are locked in. This contrasts with both the literature on
BBPD and that on switching costs: the former typically finds some portion of customers
who switch suppliers and the latter finds an increasing (bargains-then-ripoff) price trend.
The intuition behind how firms can maintain high first-period prices is as follows. Suppose a firm deviates by charging a price slightly below the equilibrium first-period price.
That firm will gain extra first-period market shares at the cost of lowering prices to all
customers. However, marginal customers who are won over will switch to the rival firm
when those customers are offered a zero price (p∗2 − d∗ = 0) in the second period.22 It
turns out that gains from earning additional market shares are not worth the costs. In
a sense, firms use the threat of zero poaching prices to maintain high first-period prices
as any extra potential gain from deviating is taken away by BBPD.
5
Extensions
In this section, we consider two important variations of our base model.
The first
extension is to consider the case where consumers are differentiated with respect to
switching costs. That is, consumers perceive different amounts of switching costs that are
drawn from a certain distribution function. We endogenize switching costs by allowing
22
Note that, in reality, price need not be actually zero. A zero price simply represent firms pricing at
cost since we assume cost to be zero in our model.
20
firms to determine the support of the distribution. There are both intuitive and technical
advantages to this model. Intuitively, the model captures certain types of switching costs
more realistically.
For example, dependent on their familiarities with the technology,
consumers may perceive different levels of learning costs when switching products. In
another example, consumers with the same amount of coupons may find actually clipping
and using those coupons as a hassle with different degrees. Technically, the model gets
rid of problems associated with discontinuous demand and the existence of equilibrium
as shown in Chen (1997), Gehrig and Stenbacka (2004), and Bouckaert and Degryse
(2004).
Although we show that equilibrium does exist for levels of switching costs
that firms select in our base model, it still seems beneficial to confirm that qualitative
results obtained within the base model carry over to the modified model without technical
concerns regarding the existence of equilibrium.
The second extension involves a case where consumers’ preferences change over time.
This is the case where a consumer may prefer product A in the first period but prefer
product B in the second period (i.e., consumer’s type θ changes over time). This model
may apply to consumers’ preferences for different airlines. For example, a consumer may
prefer airline A for some trip in period one but may prefer airline B for a different trip
in the second period depending on the connecting flights and time schedules for different
routes of travel. In the literature, some papers assume persistent types (e.g., Fudenberg
and Tirole (2000)), some others assume varying types (e.g., Caminal and Matutes (1990)),
and some assume both (e.g., Klemperer (1987)).
Recently, Chen and Pearcy (2006)
focus on this issue and show that BBPD decreases profits as shown in Fudenberg and
Tirole (2000) only when preferences are highly persistent over time. It therefore seems
important to check the robustness of our results to this variation especially since our aim
is to capture various types of switching costs within one unifying model. We show that
qualitative results of the base model are invariant to both extensions, although precise
equilibrium values may differ.
21
5.1
Heterogeneous Switching Costs
In this extension, we assume that consumers incur heterogenous switching costs s that
are randomly drawn from a uniform distribution function F (s) on [0, s̄j ].
that firms can endogenously choose 0 ≤ s̄j ≤ l, for j ∈ {A, B}.
We assume
We use notations
that are similar to those used in the base model while mathematical specifications are
re-defined to fit the modified model. We start by examining the second period. Since
consumers incur heterogenous switching costs, we define threshold levels of switching
costs that induce consumers to switch rather than defining threshold types of consumers
B
θj2 (pA
2 , p2 ). For those consumers who purchased from firm j in the first period, they will
−j
and switch to firm −j otherwise.
buy from firm j again if only if sj > pj2 − p−j
2 +d
Firm’s second-period profit can thus be specified as:
B
π j2 (pA
2 , p2 )
= σ j1 pj2 ·
j
=
s̄ −
pj2
σ j1 pj2
Z
+ p−j
2
s̄j
s̄j
j
dF (s ) +
−j
pj2 −p−j
2 +d
−j
−d
j
σ −j
1 (p2
−d )
p−j
2
− pj2
s̄−j
j
j
+ σ −j
1 (p2 − d ) ·
j
Z
j
j
p−j
2 −p2 +d
dF (s−j ) (10)
0
+ dj
The first-order conditions are:23
1 −j
σ −j
σj
σj
p2 + −j1 dj − j1 d−j + 1
2
s̄ φ
2s̄ φ
2φ
1 −j
j
dj (p−j
2 ) = p2 − p2
2
pj2 (p−j
2 ) =
23
Note that
³ A
B
∂ 2 πj2 (pA
2 , p2 )
= −2 σs̄A +
³
´2
∂pj2
σB
s̄B
´
< 0 and
function is concave with respect to pj2 and dj .
22
B
∂ 2 π j2 (pA
2 , p2 )
2
(∂dj )
= −2
³
σ −j
s̄−j
´
< 0. Thus the profit
where φ ≡
σA
1
s̄A
+
σB
1
.
s̄B
Solving for the equilibrium, we get:
2 j
s̄
3
2 j 1 −j
s̄ − s̄
=
3
3
1 −j
=
s̄
3
=
pj∗
2
dj∗
j∗
pj∗
2 −d
(11)
−j
and the second-period profits are πj2 = 49 σ j1 s̄j + 19 σ −j
1 s̄ .
We solve backwards and specify the first-period demand using the second-period equilibrium specified in Equations (11). When a consumer buys from firm A, the expected
payment stream is:
pA
1
+θ+
Z
s̄A
B∗
B∗
pA∗
2 −p2 +d
= pA
1 +θ+
A
pA∗
2 dF (s )
+
Z
B∗
B∗
pA∗
2 −p2 +d
o
11 A
s̄
18
B∗
(pB∗
+ sA )dF (sA )
2 −d
The expected payment when a consumer buys from firm B is:
pB
1
+l−θ+
= pB
1 +l−θ+
Z
s̄B
A
A
pB
2 −p2 +d
B
pB
2 dF (s )
+
Z
A
A
pB
2 −p2 +d
0
11 B
s̄
18
A
B
B
(pA
2 − d + s )dF (s )
1
B
B
A
The indifferent consumer is thus specified as θ1 (pA
1 , p1 ) = 2 (l + p1 − p1 ) +
11 B
(s̄
36
− s̄A )
and the profits are:
4
1
s̄−j
π j = σ j1 pj1 + σ j1 s̄j + σ −j
9
9 1
A A B
where σ A
1 = σ 1 (p1 , p1 ) =
θ1
l
(12)
A
and σ B
1 = l − σ1 .
We use the above profit function to compare the early commitment model (firms
determine s̄j before deciding pj1 ) and the late commitment model (firms determine s̄j
either after or along with pj1 ).
23
Proposition 6 Suppose consumers have heterogeneous switching costs that are distributed uniformly on [0, s̄j ]. Firms find it optimal to choose minimal switching costs in the
early commitment model (i.e., s̄j∗ = 0) and maximal switching costs (i.e., s̄j∗ = l) in the
late commitment model.
Proposition 6 states that main results in the base model are robust to heterogeneous
switching costs. While it is valuable to artificially create types of switching costs to which
firms commit in the midst of price-competing stages, it is not beneficial to establish the
of types of switching costs that need to be created prior to all price-competing stages.
While the mathematical derivation of Proposition 6 is provided in Appendix A, we
elaborate on features of equilibrium specifications of the early and the late commitment
models here. Substituting s̄j∗ = 0 to the early commitment model, we find that equij∗
j∗
j∗
librium prices are pj∗
1E = l and p2E = p2E − dE = 0 where the subscript E stands for
the early commitment model.
When firms decide not to create any switching costs,
consumers in each firm’s turf are no longer differentiated in the second period. Thus we
get the Bertrand price equilibrium in the second period in which both firms lower prices
to cost. In the first period, we get the Hotelling price equilibrium.
j∗
j∗
j∗
2
2
In the late commitment model, we find that pj∗
1L = 3 l, p2L = 3 l and p2L − dL =
1
l,
3
where the subscript L stands for the late commitment model.
The second-period
equilibrium prices are equivalent to those found in Chen (1997) as the two models are
equivalent for the second period.24
However, first-period prices are different.
We
find that firms charge a positive first-period price while Chen (1997) finds that firms
charge a price that is below cost in the first period. In other words, we find a (weakly)
decreasing price trend as we did in Proposition 5 while Chen (1997) finds the typical
bargains-then-ripoff price trend.
The source of the difference is that switching costs
are endogenously determined in the midst of price competing stages in our model while
In Chen (1997), equilibrium second period prices are 23 θ to repeat customers and 13 θ to switching
customers where θ in his model stands for the upper bound on the distribution of switching costs. Since
we find in Proposition 6 that firms select s̄j = l, results for the second-period equilibrium are equivalent.
24
24
they are granted as given in Chen (1997)’s model. In a sense, Chen (1997)’s model is
analogous to the situation where firms take maximal switching costs as given in the early
commitment model. Therefore, the source of discrepancy relates again to how switching
costs influence intertemporal price competition and the explanation behind different price
trends are the same as those provided following Proposition 5.
5.2
Changing Preferences
We consider the case where consumers’ preferences (i.e., locations) change over time in
this subsection. More specifically, we make a change to the base model by assuming that
all consumers change their locations in the second period by randomly drawing locations
from a uniform distribution function H(θ).
Thus the base model is one extreme in
which all consumers have persistent preferences and here we consider another extreme in
which all consumers experience changes in preferences. A more general model would be
similar to the one presented in Klemperer (1987), in which a portion of customers change
their locations while another portion does not. In such a model, equilibrium prices are
dependent on exact portions of consumer groups.
In this subsection, we consider the
extreme case for simplicity and use the result to infer what would happen in the general
model.
As usual, we start by analyzing the second period. Since all consumers change their
locations in the second period, a consumer who purchased from firm A in the first period
B
B
A
will buy from firm A again if her new location is such that θ + pA
2 ≤ l − θ + p2 − d + s ,
£
¤
1
A
B
B
A
i.e., θ ≤ θA
Similarly, a consumer who purchased from
2 = 2 l − p2 + p2 − d + s .
firm B in the first period will switch and buy from firm A in the second period if the
£
¤
1
A
B
A
B
. The second-period profit
new location is such that θ ≤ θB
2 = 2 l − p2 + p2 + d − s
is therefore:
j
j
j
πj2 = σ j1 σ j2R pj2 + σ −j
1 σ 2S (p2 − d )
25
(13)
where σ A
2R =
θA
2
,
l
σA
2S =
θB
2
,
l
σB
2R =
l−θB
2
l
and σ B
2S =
l−θA
2
.
l
Note that first-period market
shares influence the profit function in a different way than it influenced the profit function
in the base model (Equation (3)).
In the base model, initial market shares divided
consumers into each firm’s turf locationally.
Therefore, the demand from switching
consumers were dependent on the initial market share as it influenced the location of the
market boundary, e.g. σ A
2S =
θB
2 −θ 1
.
l
Here, on the other hand, all consumers relocate and
thus initial market shares divide consumers only proportionally and play no locational
role. Thus the demand from switching consumers in Equation (13) is only a function of
θ−j
2 and not of θ 1 .
The best-response functions are:
1 −j l
1 j −j 1 j j 1 −j −j
j
+ σ −j
+ σ1s − σ1 s
pj2 (p−j
2 ) = p2 +
1 d − σ1d
2
2
2
2
2
1
1 −j
l
j
+ s−j
dj (p−j
2 ) = − p2 + p2 −
2
2 2
which results in equilibrium prices of:
1 j j∗
1 −j
j∗
pj∗
2 = l + s , p2 − d = l − s
3
3
(14)
There are two distinct features with the above equilibrium compared to those specified
in the base model.
First, prices to both repeat and switching customers are higher
than those in the base model (see Equation (4)).
To understand why, consider the
poaching price offered to switching customers. Because consumers have relocated, firms
are not locationally disadvantaged when poaching competing firms’ customers.
This
compares to the base model where rival firm’s customers were always the ones far away,
i.e., those with strong preferences for competing firm’s products. In this model, however,
firms simply need to attract only those customers who have relocated close to themselves
while providing discounts sufficient enough to compensate for switching costs.
26
As a
result, firms can charge higher poaching prices compared to the base model since firms
do not need to compensate for travelling costs as much as they did in the base model.
The second distinct feature is that equilibrium prices are not dependent on first-period
market shares.
The reason behind this feature is similar to the one discussed above.
Because all consumers relocate, initial market shares affect profits only by determining
proportions of consumers in each firm’s turf. However, competition arises in each firm’s
turf separately and thus equilibrium prices are not influenced by market shares.
B
Now we analyze the first-period. The indifferent consumer θ1 (pA
1 , p1 ) is determined
by rational consumers who know that their locations will be different in the second period.
The expected payment stream if a consumer buys from firm A is:
θ1 +
pA
1
+
Z
θA
2
pA
2 dH(θ)
+
l
θA
2
0
= θ1 + pA
1 +l+
Z
1 A 2
(s )
9l
B
(pB
2 − d )dH(θ)
Expected payment stream if a consumer buys from firm B is:
l − θ1 +
pB
1
+
Z
θB
2
0
= l − θ1 + pB
1 +l+
(pB
2
B
− d )dH(θ) +
1 B 2
(s )
9l
Z
l
θB
2
pB
2 dH(θ)
¡ B ¢2 ¡ A ¢2
l
1 B
1
B
A
The indifferent consumer can be found as θ1 (pA
− s ).
1 , p1 ) = 2 + 2 (p1 −p1 )+ 18l ( s
The profit function is then:
j
π =
σ j1 pj1
σj
+ 1
2l
¶2
¶2
µ
µ
1 j
1 −j
σ −j
1
l+ s
l− s
+
3
2l
3
(15)
Proposition 7 Suppose that consumers draw new preferences in each period according
to a uniform distribution function on [0, l]. In the early commitment model, firms are
dπ j
= 0. In the late commitment model,
indifferent to levels of switching costs, i.e.,
dsj
27
firms find it optimal to create maximal switching costs, i.e., sj∗ = l.
It is immediate from the profit function that firms want to create maximal switching
costs if equilibrium first-period prices and market shares are already determined. The
intuitive reason behind the result is analogous to that of previous results.
However,
if switching costs are created before the initial price competition, we see that amounts
of switching costs do not have marginal effects on profits.
harvesting and investment effects.
To see why, we compare
The harvesting effect is represented by the second
term in the profit function which increases with switching costs. The investment effect
is implicit in the first term in the profit function as high switching costs lead to low
first-period equilibrium prices. It turns out that the two effects cancel each other out
∂πj ∂pj
∂πj
when preferences of all consumers change over time. That is, j = − j · 1j . Thus,
∂s
∂p1 ∂s
whatever the amount of benefits from creating switching costs in the second period, it
all dissipates in the first period through competition for market share.
In comparison, the investment effect was larger in the base model. That is, costs of
fighting for market shares outweighed benefits of locking in customers and this led to the
result that it is better to not create any switching costs at all in the early commitment
model. The source of discrepancy in qualitative results comes from the fact that investment effect decreases when all consumers relocate. In the base model where consumers
did not relocate, initial market shares not only determined portions of customers in each
firm’s turf but also dictated the locational boundary of customers that are being poached.
This was an additional incentive for firms to behave more aggressively in the first period. When all consumers change locations in the second period, however, initial market
shares only proportionately determines customers in each firm’s turf. This mitigates the
incentive to fight for market shares in the first period.
We can take results of the two polar cases presented in Propositions 3, 5 and 7 and
apply to a generalized model that is similar to the one presented in Klemperer (1987).
Corollary 8 Suppose that a portion α ∈ [0, 1] of customers remain at their locations in
28
the second period while the other portion 1 − α of customers change locations. In the
early commitment model, firms find it best to create minimal switching costs as long as
α > 0. In the late commitment model, it is best for firms to create maximal switching
costs.
We therefore conclude that our finding that the timing of commitment matters in deciding which types of switching costs are beneficial to firms is robust to several important
variations of the base model.
6
Conclusion
We have presented a simple switching cost model that encompasses many different types
of switching costs and have analyzed which types of switching costs, among those that
are commonly observed, enhance firms’ profits. The timing when firms can commit to
the establishment of switching costs turns out to be one important criterion. Focusing
on two different timing structures, we have shown that the types of switching costs that
must be established prior to all price-competing stages harm firms’ profits (the early
commitment model) while those that are established in the midst of price-competing
stages enhance firms’ profits (the late commitment model). Many real or social costs
of switching that arise from technological incompatibilities, learning or start-up costs fit
the description of the early commitment model. On the other hand, many contractual
or pecuniary switching costs such as coupons and loyalty programs fit the description of
the late commitment model. Hence the results carry a strategic implication that firms
benefit from contractual/pecuniary switching costs and not from real/social switching
costs.
We emphasize that the results are closely linked to the way intertemporal price competition is influenced by switching costs. To be more specific, the crucial determinant
factor is whether first-period prices are strategically dependent on switching costs or not.
29
If they are, as is the case in the early commitment model, costs of fighting for market
shares outweigh the benefits of locking in customers.
It is thus better to not commit
to the establishment of switching costs. On the contrary, if first-period prices are determined independently from switching costs, it is better to establish switching costs as
they only enhance firms’ profits.
The intuition behind the results may provide insights beyond the scope of the twoperiod model presented in this paper. In particular, we can generalize the results and
argue that what matters is whether market shares are settled or not at the time firms
can commit to the establishment of switching costs.
If market shares are yet to be
determined, prematurely establishing switching costs will only harm profits as competition for market share becomes too severe. If market shares are determined and stable,
however, it is better for firms to establish switching costs so that poaching customers
becomes unprofitable for rival firms. In reality, this implies that firms should opt for low
switching costs in a new or expanding industry but should prefer to create high switching
costs once market shares are stabilized.
As discussed in the Introduction, our findings fit nicely into the literature on endogenous creation of switching costs.
One exception is Gehrig and Stenbacka (2004) who
study switching costs that are created through differentiation. They find that maximal
differentiation is optimal although this type of switching costs fits the description of the
early commitment model as the decision regarding the degree of differentiation is made
prior to all price-competing stages. We can intuitively explain the discrepancy behind
the two seemingly opposing results.
In our early commitment model, switching costs
exacerbates first-period profits as they stimulate intense competition for initial market
shares. Switching costs have an additional effect on first-period profits in Gehrig and
Stenbacka (2004). Because product differentiation is directly linked to switching costs,
high switching costs lead to more differentiation thereby mitigating the first-period price
competition. This effect is more immediate than the intertemporal effect and thus firms
30
find it better to create maximal switching costs, i.e., maximal differentiation.
While we have shown the robustness of our results to two important variations of
the base model, there are several other modifications that may or may not change the
qualitative outcomes of the paper. The first is the infinite-horizon model in the spirit
of Beggs and Klemperer (1992).
Comparing between the investment and harvesting
effects, Cabral (2008) has shown sufficient conditions under which the investment effect
dominates. The intuition behind his result is quite similar to the one discussed above —
that the effects of switching costs on intertemporal price competition and market shares
are important. The second is the increase in number of firms. As Taylor (2003) has
shown, if there are more than three firms in the industry, the profits from poaching customers plunges as Bertrand-type competition emerges among the firms that try to poach
customers. On a similar note, Caminal and Claici (2008) also consider the relationship
between the number of firms and the effects of switching costs. Lastly, we focus only on
symmetric cases while asymmetric demand structures may have influences on our results
in a manner similar to Shaffer and Zhang (2001).
31
7
Appendix
Proof of Proposition 1: Substituting Equations in (4) to Equation (6), we can re-write
the profit function as π j = σ j1 pj1 +
1
(2σ j1
18l
+ l + sj )2 +
1
(4lσ −j
1
18l
− l − s−j )2 , where we
2
2 A 4 B
suppress the notation σ j1 = σ j1 (pj1 , p−j
1 ). We first consider the case when − 3 l− 3 s + 3 s ≤
2
4 A
2 B
B
pA
1 − p1 ≤ 3 l − 3 s + 3 s , which corresponds to the middle case in Equation (5). Then
we show that there does not exist a subgame perfect Nash equilibrium in either the first
or the third case in Equation (5).
1 A
1 B
B
Suppose that θ1 = 12 l − 38 (pA
1 − p1 ) − 4 s + 4 s . The first-order condition is given
by:
!
!
Ã
Ã
j
−j
1
∂σ j1 j
1
∂σ
∂σ
∂π j
j
1
−j
(2σ j1 + l + sj ) 2 · j1 + (4lσ −j
=
1 −l−s ) 4·
j
j · p1 + σ 1 +
9l
9l
∂p1
∂p1
∂p1
∂pj1
7
1X j
1
1
= − pj1 + p−j
l
−
+
s =0
16
16 1
2
8 j
which leads to best-response function:
2X j
1 −j 8
p
l
−
)
=
+
s
pj1 (p−j
1
7 1
7
7 j
(A1)
Note that second-order condition is satisfied and the profit function is indeed concave in
pj1 .
4
Best-response functions lead to equilibrium prices: pj∗
1 = 3l −
1
3
P
j
sj and the rest
of the characterization is derived by substituting pj∗
1 into Equations in (4).
Finally,
B∗
pA∗
1 − p1 = 0 and thus the condition in Equation (5) is satisfied.
Now we consider the case when θ1 =
first case in Equation (5).
50
l
27
+
7 A
s
243
+
4 B
s
27
5
l
12
B
− 14 (pA
1 − p1 ) −
1 A
s
12
+ 16 sB , which is the
Going through the same procedure as above, we get pA
1 =
and pB
1 =
50
l
27
+
124 A
s
243
+
22 B
s .
27
117 A
B
Thus, pA
1 − p1 = − 243 s −
18 B
s
27
<0
which violates the condition provided in Equation (5). The third case in Equation (5)
is analogous to this one.
Q.E.D.
32
Proof of Lemma 2: From Proposition 1, given the amounts of switching costs sj ,
P
the equilibrium choice of discounts is dj∗ = 13 l + 13 j sj . Switching does not occur when
P j∗
P
P j
P
= 23 j sj + 13 l. Thus, j sj ≥ 2l must hold which is the case when
js ≥
jd
firms choose maximum level of switching costs, i.e., sA = sB = l.
Q.E.D.
Proof of Proposition 3: Since πj is convex in sj , maximum occurs either at sj = 0
or at sj = l.
Substituting sj = 0, π j (sj = 0; s−j ) =
substituting sj = l, we get πj (sj = l; s−j ) =
0; s−j ) − π j (sj = l; s−j ) =
7 −j
s
36
+
9
l
72
11
(s−j )2
72l
> 0 for ∀s−j .
−
11
72l
2
(s−j ) +
5 −j
s
36
+
1 −j
s
18
59
l.
72
+
17
l
18
and
Then, π j (sj =
Thus, sj∗ = 0 is the dominant
strategy. To show that sj∗ = 0 is a global equilibrium, we need to compare the profits
to the profits firms can earn when no switching occurs at all. By Lemma 2, this occurs
when sj = l and the profit is π j (sj = l, s−j = l) = 13 l + 12 l =
π j (sj = 0, s−j = 0) =
17
l.
18
15
l
18
which is less than
Q.E.D.
Proof of Proposition 6: Consider the early commitment model first.
We first
solve for period 1 equilibrium prices and then solve for optimal levels of switching costs.
Using the profit function in Equation (12), the first-order condition is:25
13 −j 19 j
1 −j 1
s̄ − s̄
pj1 (p−j
1 ) = p1 + l +
2
2
36
36
(A2)
7 −j
− 25
s̄j . Substituting equilibrium
Solving for the equilibrium price, we get pj∗
1 = l + 54 s̄
54
B∗
first-period prices to the initial market share, we get σ j1 (pA∗
1 , p1 ) =
1
2
+
1 −j
s̄
108
−
1 j
s̄ .
108
The choices of s̄j in period 0 can be found by substituting equilibrium prices and market
shares to the profit function:
1
1
π (s̄ , s̄ ) = s̄−j +
9
2l
j
25
Note again that
∂ 2 πj
2
∂ (pj1 )
j
−j
¶2
µ
1 j
1 −j
l − s̄ + s̄
54
54
= − 1l < 0 so the profit function is strictly concave with respect to pj1 .
33
(A3)
The first- and the second-order conditions are:
¶
µ
1
1 j
1
1 −j
∂π j
(−
l
−
s̄
s̄
)=0
=
+
∂s̄j
l
54
54
54
¶ µ
¶
µ
1
∂ 2 πj
1
· −
>0
=
−
54l
54
∂ (s̄j )2
(A4)
(A5)
Thus, profit function is convex with respect to s̄j and thus maximum occurs at either
s̄j = 0 or s̄j = l. Note that:
π j (s̄j = 0) − πj (s̄j = l)
¶2
¶2
µ
µ
1 −j
1 −j
1 −j
1 53
1
1
l + s̄
l + s̄
+ s̄ −
− s̄−j
=
2l
54
9
2l 54
54
9
¶µ ¶
µ
1
2
1 107
l + s̄−j
l >0
=
2l 54
54
54
(A6)
Thus, it is true that π j (s̄j = 0) > π j (s̄j = l) for all possible s̄−j . Therefore, s̄j∗ = 0.
Now consider the late commitment model. Since firms determine s̄j in the interim
period, the first-order condition is:
4
∂π j
= σj > 0
j
∂s̄
9
Thus, it is best to have maximal s̄j . Therefore, s̄j∗ = l.
(A7)
Q.E.D.
Proof of Proposition 7: From the profit function in Equation (15), we can immediately verify that
∂π j
∂sj
> 0. Thus, firms want to create maximal switching costs in the
late commitment model in which the decision is made after or along with the first-period
price decisions. We show next that the size of switching costs does not matter in the
early commitment model. The first-order condition with respect to prices is:
pj1 (p−j
1 )
¶
µ
1 j
1 −j l
1 j 1 −j
−j
= p1 + − (s + s ) l + s − s
2
2 6l
2
2
34
The equilibrium price is pj∗
1 = l −
1
(sj
18l
+ s−j )(6l + sj − s−j ).
Substituting pj∗
1 into
l
B
A B
θ1 (pA
1 , p1 ), we find that θ 1 (p1 , p1 ) = 2 . Thus initial market share is not dependent on
switching costs. The profits are then:
π
j
¶2
¶2
µ
µ
1 j
1 −j
1 j
1
1
p +
l+ s
l− s
=
+
2 1 4l
3
4l
3
¶2
¶2
µ
µ
µ
¶
1 j
1 j
1 −j
1
1
1
−j
j
−j
l−
(s + s )(6l + s − s ) +
l+ s
l− s
=
+
2
18l
4l
3
4l
3
Differentiating with respect to sj , we get
∂π j
∂sj
do not have marginal effects on profits.
= 0. Thus the amounts of switching costs
Q.E.D.
Proof of Proposition 8: Let π jC (pj1 , p−j
1 ) denote profits when all consumers change
their locations as presented in Equation (15) and let π jP (pj1 , p−j
1 ) denote profits when all
consumers remain at their locations are presented in Equation (6). Redefine the profits
j
j
−j
j
j
−j
For the
in the generalized model as π j (pj1 , p−j
1 ) = απ P (p1 , p1 ) + (1 − α)π C (p1 , p1 ).
j
j
j
dπ
dπ
dπ
early commitment model, j = α · Pj + (1 − α) · Cj ≤ 0 for ∀sj (with equality when
ds
ds
ds
dπ jP
dπ jC
α = 0) since
< 0 by Proposition 3 and
= 0 by Proposition 7. Similar logic
dsj
dsj
j
dπ
establishes that j ≥ 0 for ∀sj in the late commitment model.
Q.E.D.
ds
35
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