Small Switching Costs Lead to Lower Prices
Luı́s Cabral∗
New York University and CEPR
April 2007; revised September 2007
still incomplete draft
Abstract
I prove a general theoretical result based on the numerical result
obtained by Dubé, Hitsch and Rossi (2006). In a dynamic model of
price competition with switching costs, I show that average equilibrium
price decreases in the value of the switching cost for low values of the
switching cost.
∗
Stern School of Business, 44 West Forth Street, New York, NY 10012; lcabral@
stern.nyu.edu.
1
Introduction
Consumer switching costs have two effects on firm’s pricing incentives: the
harvesting effect and the investment effect. Firms with locked-in customers
are able to price higher without losing demand; this is the harvesting effect.
Firms without locked-in customers, in turn, are eager to cut prices in order to
attract new customers; this is the investment effect.
Do switching costs make markets more or less competitive? The harvesting
and the investment effects work in opposite ways (in terms of the market
average price). Which effect dominates? Conventional wisdom and previous
economics literature (such as Beggs and Klemperer, 1993) suggest that the
harvesting effect dominates. However, a recent paper by Dubé, Hitsch and
Rossi (2007) casts doubt on the assertion that switching cots increase market
power. In fact, Dubé, Hitsch and Rossi’s (2007) numerical simulations suggest
that, if switching costs are small, then the investment effect dominates, that
is, switching costs increase market competitiveness.
In this paper, I extend the analysis by Dubé, Hitsch and Rossi’s (2007).
I analytically solve a version of the model they consider. Analytical solution
has two advantages. First, it leads to more general results, that is, results
that are not dependent on specific assumptions regarding functional forms
and parameter values. Second, the process of solving the model leads to a
better understanding of mechanics underlying the result that market prices
decline when switching costs increase. It also shows why there is an important
difference between small and large switching costs; and between homogeneous
and differentiated products.
In addition to Beggs and Klemperer (1993), other authors have argued,
based on dynamic models of price competition, that switching costs lead to
higher equilibrium prices. The list includes Farrell and Shapiro (1988) and
Padilla (1995). In a recent paper, Doganoglu (2005) shows that such conclusion is not warranted. My approach is similar to Doganoglu (2005) in that
I assume product differentiation and consider the possibility of low values of
the switching cost. However, unlike Doganoglu (2005), I make very mild assumptions regarding the distribution of buyer preferences and the shape of the
seller’s pricing strategies.1
1
Doganoglu (2005) assumes uniformly distributed preferences and linear pricing strategies.
1
2
Model
Similarly to Dubé, Hitsch and Rossi (2007), I consider an infinite-period model
with two sellers and one buyer. In each period, sellers simultaneously set prices
and then the buyer chooses one of the sellers. The buyer’s valuation for seller
i’s good is given by zi , which I assume is stochastic and i.i.d. across firms and
periods. Moreover, if the buyer previously purchased from firm j, then his
utility from buying from seller i in the current period is reduced by s, the cost
of switching between sellers.
I focus on symmetric Markov equilibria where the state indicates which
firm made a sale in the previous period. I denote the firm that made a sale
in the previous period (the “incumbent” seller) with the subscript 1, and the
other firm (the “challenger” firm) with the subscript 0.
Symmetry implies that the buyer’s continuation values from being locked
in to seller i or seller j are the same. This greatly simplifies the analysis. In
particular, in each period the buyer chooses the incumbent seller if and only if
z1 − p1 ≥ z0 − p0 − s.
Define
x ≡ z1 − z0
P ≡ p1 − p0 − s
(1)
In words, x is the relative preference for the incumbent’s seller product, whereas
P is the price difference corrected for the switching cost. It follows that the
buyer chooses the incumbent if and only if x > P .
Define by q1 and q0 the probability that the buyer chooses the incumbent
or the entrant, respectively. If x is distributed according to F (x), then we
have
q1 = 1 − F (x)
q0 = F (x)
I make the following assumptions regarding the c.d.f. F and the corresponding
density f :
Assumption 1 (i) F (x) is continuously differentiable; (ii) f (x) = f (−x);
(iii) f (x) > 0, ∀x; (iv) f (x) is unimodal; (v) F (x)/f (x) is strictly increasing.
2
In the Appendix, I present a Lemma that proves a series of properties of F (x)
that are derived from Assumption 1.
My main goal in this paper is to characterize equilibrium pricing as a
function of switching costs. Before then, however, I show that, in the relevant
range of parameter values, there exists a unique Markov equilibrium. In this
result, I extend the model by allowing prices to be indexed by the firm’s name,
that is, I do not impose symmetry.
Proposition 1 If s is small, then there exists a unique Markov equilibrium.
This equilibrium is symmetric.
Proof: See the Appendix.
Notice that equilibrium symmetry is a derived result. This implies that
my focus on symmetric equilibria is with no additional loss of generality with
respect to the Markov assumption.
I now turn to the main result in the paper. Let p be the average price paid
by the buyer, that is,
p = q1 p 1 + q0 p 0 .
Proposition 2 If s is small, then average price is decreasing in s.
Proof: The firm value functions are given by
v1 =
1 − F (P )
p1 + δ v1 + F (P ) δ v0
v0 = F (P ) p0 + δ v1 + 1 − F (P ) δ v0
(2)
(3)
The corresponding first-order conditions are
−f (P ) p1 + δ v1 + 1 − F (P ) + f (P ) δ v0 = 0
−f (P ) p0 + δ v1 + F (P ) + f (P ) δ v0 = 0
(Recall that, from (1),
I get
dP
d p1
= 1 and
dP
d p0
= −1.) Solving for equilibrium prices,
1 − F (P )
−δ V
f (P )
F (P )
=
−δ V
f (P )
p1 =
(4)
p0
(5)
3
where
V ≡ v1 − v0
Substituting (4)–(5) for p1 , p0 in (2)–(3) and simplifying, I get
v1 =
2
1 − F (P )
f (P )
F (P )2
=
+ δ v0
f (P )
v0
+ δ v0
It follows that
P =
1 − 2 F (P )
1 − F (P ) F (P )
−
−s=
−s
f (P )
f (P )
f (P )
V
=
2
1 − F (P )
f (P )
−
F (P )2
1 − 2 F (P )
=
f (P )
f (P )
Average price is given by
p ≡ q1 p1 + q0 p0 = 1 − F (P ) p1 + F (P ) p0
Substituting (4)–(5) for p1 , p0 and (6) for V , and simplifying, I get
p =
1 − F (P )
=
2
1 − F (P )
+
f (P )
=
1 − F (P )
−δ V
f (P )
2
1 − F (P )
F (P )
+ F (P )
−δ V
f (P )
!
F (P )2
−δV
f (P )
+ F (P )2
f (P )
!
2 F (P ) − 1
+δ
f (P )
!
(6)
At this point, I introduce a lemma that characterizes several properties of
F that follow from Assumption 1:
Lemma 1 Under Assumption 1, the following are strictly increasing in x:
F (x)2
,
f (x)
F (x) − 1
,
f (x)
2 F (x) − 1
.
f (x)
Moreover, the following is increasing in x iff x > 0 (and constant in x at
x = 0):
2 2
1 − F (x) + F (x)
.
f (x)
4
The proof of Lemma 1 may be found in the Appendix. The lemma implies
that, at s = 0, the first term on the right-hand side of (6) is constant in P . It
also implies that the second term on the right-hand side of (6) is increasing in
P . It follows that, if s is small, then dd Pp > 0.
Equation (6) may be rewritten as
P+
2 F (P ) − 1
= s.
f (P )
By Lemma 1, the left-hand side is increasing in P . It follows by the implicit
function theorem that ddPs < 0. Finally, if s is small then
dp
=
ds
dp
dP
!
dP
ds
!
< 0,
which concludes the proof.
To understand the intuition for Proposition 2, it is useful to look at the
first-order condition. For the incumbent firm, it is given by
p1 =
1 − F (P )
−δ V
f (P )
(7)
where V ≡ v1 − v0 is the difference, in terms of continuation value, between
winning and losing the current sale (vi is firm value). In other words, −δ V is
the “cost,” in terms of discounted continuation value, of winning the current
sale.
Recall that q1 = 1 − F (P ) and. Since P = p1 − p0 − s, we have dd pq11 = f (P ).
It follows that (7) may be re-written as
p1 − (−δ V )
1
= ,
p1
1
where 1 ≡ dd pq11 pq11 . This is simply the “elasticity” rule of optimal pricing, with
one difference: the future discounted value from winning the sale appears as
a negative cost (or subsidy) on price.
We thus have two forces on optimal price, which might denoted by “harvesting” and “investing.” If the seller is myopic (δ = 0), then optimal price is
given by the first term in (7). The greater the value of s, the smaller the value
of P , and thus the greater the value of p1 . We thus have harvesting, that is, a
higher switching cost implies a higher price (by the incumbent firm, which is
the more likely seller).
5
Suppose however that δ > 0. Then we have a second effect, investing, which
leads to lower prices. The greater the value of s, the greater the difference
between being an incumbent and being an entrant, that is, the greater the
value of V .
What is the relative magnitude of the harvesting and the investment effects
on average price? First notice that harvesting leads to a higher price by the
incumbent but lower by the entrant. If fact, by symmetry, the effects are
approximately of the same absolute value of s close to zero. This implies
that, for s close to zero and in terms of average price, the harvesting effects
approximately cancel out, since for s = 0 incumbent and challenger sell with
equal probability.
Not so with the dynamic effect. In fact, the entrant’s first-order condition
is given by
F (P )
−δ V
p0 =
f (P )
that is, the “subsidy” resulting from the value of winning is the same as for
the incumbent. It follows that the effect on average price is unambiguously
negative, and of first order importance.
In other words, the harvesting effect is symmetric: the amount by which
the incumbent increases its price is the same as the amount by which the
outsider lowers its price. However, the investment effect, is equal for both
firms — and negative.
3
Discussion
• The proof of Proposition 2 suggests that there is an important difference
between the homogeneous product and the product differentiation cases.
If products are homogeneous, then the incumbent firm sells with probability 1 regardless of the level of switching cost. If the degree of product
differentiation is strictly greater than zero, then, as the level of switching costs goes to zero the incumbent firm sells with probability 50%. In
other words, fixing the value of s, there is an important discontinuity as
the degree of product differentiation converges to zero.
• I conjecture that both firms and consumers are strictly worse off as the
level of switching cost increases.
• The extension of the model to the case of multiple buyers is easy if we
allow for customer recognition (that is, different prices for locked-in and
6
unlocked-in buyers). If there is no customer recognition, however, then
the model becomes more complicated.
• I conjecture that a version of Proposition 2 holds in the case when there
are more than two sellers.
To be completed.
7
Appendix
Proof of Lemma 1: First notice that
F (x)2
F (x)
= F (x)
.
f (x)
f (x)
(x)
Since F (x) is increasing and Ff (x)
is strictly increasing (by Assumption 1), it
follows that the product is strictly increasing.
Next notice that, by part (ii) Assumption 1,
−F (−x)
−F (−x)
F (x) − 1
=
=
f (x)
f (x)
f (−x)
−F (−x)
f (−x)
(x)
Since Ff (x)
is strictly increasing,
Next notice that
is strictly increasing too.
2 F (x) − 1
F (x) − 1 F (x)
=
+
.
f (x)
f (x)
f (x)
I have just proved that F f(x)−1
is strictly increasing. We thus has the sum of
(x)
two strictly increasing functions, the result being a strictly increasing function.
Finally, taking the derivative of the fourth expression I get

2 
1 − F (x) + F (x) 
d 

=

dx 
f (x)
2
− 2 1 − F (x) f (x) + 2 F (x) f (x) f (x)
=
0
f (x)
2
1 − F (x)
−
−
2
f (x)
2 + F (x)
2
f (x)
!
1
= 4 F (x) −
− f 0 (x) ξ,
2
where ξ =
2
1 − F (x)
2 + F (x)
2
/ f (x)
follows from Assumption 1.
8
is positive. The result then
Proof of Proposition 1: Let v1i and v0i be firm i’s value when it is an
incumbent or an entrant, respectively. These value functions are recursively
given by
v1i =
1 − F (Pi )
p1i + δ v1i + F (Pi ) δ v0i
v0i = F (Pj ) p0i + δ v1i + 1 − F (Pj ) δ v0i
where Pi = p1i − p0i − s and δ is the discount factor.
The corresponding first-order conditions are
−f (Pi ) p1i + δ v1 + 1 − F (Pi ) + f (Pi ) δ v0i = 0
−f (Pj ) p0i + δ v1i + F (Pj ) + f (Pj ) δ v0i = 0
or simply
1 − F (Pi )
− δ Vi
f (Pi )
F (Pj )
=
− δ Vi
f (Pj )
p1i =
(8)
p0i
(9)
Plugging back into the value functions and simplifying, we get
v1i =
v0i
2
1 − F (Pi )
+ δ v0i
f (Pi )
F (Pj )2
+ δ v0i
=
f (Pj )
(10)
(11)
From (10)–(11), I get
Vi ≡ v1i − v0i =
2
1 − F (Pi )
f (Pi )
−
F (Pj )2
f (Pj )
(12)
From (8)–(9) and (12), I get
Pi ≡ p1i − p0j − s
!
1 − F (Pi )
F (Pi )
=
− δ Vi −
− δ Vj − s
f (Pi )
f (Pi )

=
 1 − F (Pi )
1 − 2 F (Pi )
−δ

f (Pi )
f (Pi )
9

2
−
F (Pj )2 

f (Pj ) 


2
 1 − F (Pj )
+δ

f (Pj )
−
2
F (Pi ) 
−s
f (Pi ) 
or simply

Pi +

2
2
 1 − F (Pi ) + F (Pi ) 
2 F (Pi ) − 1
=
+δ 


f (Pi )
f (Pi )

2 
 1 − F (Pj ) + F (Pj ) 
−s
=δ 


f (Pj )
2
(13)
The first two terms of the left-hand side are increasing in Pi , by Lemma 1.
Consequently, if the left-hand side is increasing in Pi for δ = 1, then it is
increasing in Pi for any value of δ. For δ = 1, the left-hand side reduces to
(Pi )2
Pi + 2 Ff (P
, which, by Lemma 1, is increasing in Pi .
i)
Equation (13) defines a pair of correspondences Pi = Pbi (Pj ), i 6= j. Figure 1 illustrates several properties of these correspondences, which I show next.
I first show that there exists no solution in the first quadrant, that is, when
both Pi and Pj are positive. By the implicit function theorem and Lemma 1,
and given that Pj > 0, (13) implies that d Pbi (Pj ) / d Pj > 0. Given this and
given symmetry of the system, the only possible equilibria are where Pi = Pj .
If Pi = Pj , then the implicit function theorem implies that d Pbi (Pj ) / d Pj < 1.
By continuity of the Pbi (Pj ) correspondence, the intercept of Pbi at Pj = 0 would
then be positive, that is, P̂i (0) > 0. But for Pj = 0 and Pi = 0, the left-hand
side of (13) is greater than the right-hand side. Moreover, the left-hand side
is decreasing in Pi (and the right-hand side does not depend on Pi ), which
contradicts Pbi (0) > 0.
By the implicit function theorem and Lemma 1, the correspondence Pbi (Pj )
is flat at Pj = 0 and negatively sloped for Pj < 0. By symmetry of the system and continuity of the equilibrium correspondences, there exists a unique
symmetric solution. I now show that, at this symmetric solution, the correspondences cross from below, that is, | d Pbi (Pj ) / d Pj | < 1. From the implicit
function theorem, a necessary and sufficient condition is


2
 1 − F (Pi ) + F (Pi ) 
2 F (Pi ) − 1
d 


Pi +
+2 
>0



d Pi 
f (Pi )
f (Pi )

10
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Pi ...................
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45◦
Pbi (Pj )
Pj
Figure 1: Correspondences Pbi (Pj ) when F is a standardized normal and s = .3.
If s is close to zero, then the intercept Pbi (0) is also close to zero. Since Pbi is decreasing (in the third quadrant), it follows that the symmetric solution has Pi
close to zero as well. Straightforward computation then show the above derivative is approximately equal to 3, which confirms that the correspondences cross
from below. Finally, this implies that there can be no other solution.
11
References
Beggs, Alan W, and Paul Klemperer (1992), “Multi-period Competition with Switching Costs,” Econometrica 60, 651–666.
Doganoglu, Toker (2005), “Switching Costs, Experience Goods and Dynamic Price Competition,” University of Munich.
Dubé, Jean-Piere, Günter J Hitsch, and Peter E Rossi (2007), “Do
Switching Costs Make Markets Less Competitive?,” Graduate School of
Business, University of Chicago.
Farrell, Joseph, and Carl Shapiro (1988), “Dynamic competition with
switching costs,” RAND Journal of Economics 19, 123-137.
Padilla, A Jorge (1992), “Revisiting Dynamic Duopoly with Consumer
Switching Costs,” Journal of Economic Theory 67, 520–530.
12