The Strength of the
Waterbed Effect
Depends on Tariff
Type
Steffen Hoernig
Working Paper
# 586
The Strength of the Waterbed E¤ect
Depends on Tari¤ Type
Ste¤en Hoernig
14 May 2014
Abstract
We show that the waterbed e¤ect, i.e. the pass-through of a change in one price of a …rm to its other prices, is much stronger if the latter include subscription rather than only usage fees. In particular, in mobile network competition with a …xed number of customers, the waterbed e¤ect is full under two-part tari¤s, while it is only partial under linear tari¤s.
Keywords: Waterbed e¤ect; two-part tari¤; linear tari¤; mobile termination; two-sided platforms.
JEL: D43, L13, L51
1
Introduction
The "waterbed e¤ect" describes the interdependence between prices at multiple-good …rms and multi-sided platforms. As much as a waterbed rises on one side if it is pressed down on the other, …rms may optimally change prices if some other price is forced to a di¤erent level, for example through regulatory interventions. The extent of the waterbed e¤ect can be a contentious issue when it would weaken the e¤ectiveness of the regulatory measures. In the debate about the downward regulation of the charges paid by …xed networks to mobile networks for routing calls from the former to their receivers on the latter, the so-called "mobile termination rates", mobile networks have claimed that the result would be higher retail prices for mobile customers,
while regulators argued there would be no e¤ect.1
Nova School of Business and Economics, Universidade Nova de Lisboa, Campus de Campolide, 1099-032 Lisboa, Portugal; email: shoernig@novasbe.pt.
1See Schi¤ (2008) for an introduction to the waterbed e¤ect and a discussion of these
In this note we show how the waterbed e¤ect depends on the type of tari¤ that is charged to the unregulated side of the market. On the regulated side, the …rm receives a …xed payment per customer of the unregulated side. This payment can be the pro…ts from …xed-to-mobile termination of calls, or advertising, or any other pro…ts that depend on the customer’s existence (rather than his usage). We determine the pass-through for two-part tari¤s, where customers pay for subscription and usage, and for linear tari¤s where
they only pay for usage.2 We show that the waterbed e¤ect is much stronger
under two-part than under linear tari¤s; in particular, under the assump-tion of a …xed number of mobile subscribers we show that under two-part tari¤s the waterbed e¤ect is full, while it is only partial with linear tari¤s. This implies that downward regulation of some price leads to a stronger rise negative e¤ect on clients of the other services if the latter are charged a multi-part tari¤. In multi-particular, this result contracts mobile networks’ contention that lower …xed-to-mobile termination rates would disproportionately hurt customers on pre-pay tari¤s.
The issue of the strength of the waterbed e¤ect has been studied in both in theoretical and empirical work. Wright (2002) remains the most important theoretical treatment of …xed-to-mobile interconnection. He shows, generi-cally, that if the pass-through of …xed costs to pro…ts is full (partial), net-works are indi¤erent about termination rates (jointly want to set them at the monopoly level). Below we show that these cases arise due to competition in
two-part or linear tari¤s, respectively.3
Genakos and Valletti (2011a, 2011b) provide an empirical study of the waterbed e¤ect with simultaneous …xed-to-mobile and mobile-to-mobile in-terconnection. They show that the waterbed e¤ect is signi…cantly stronger for post-pay (two-part) than for pre-pay (linear) tari¤s. They ascribe this di¤erence to how the regulation of mobile termination rates a¤ects the inter-connection of calls between mobile networks, and therefore indirectly changes how intensively networks compete for subscribers. While their argument is certainly correct, it is does not take into account that the actual direct pass-through of …xed-to-mobile termination pro…ts depends on the type of tari¤s in the mobile market. In this note, we isolate this factor by considering the two types of termination separately.
2In the market, these types of contract are normally denoted as "post-pay" or "pre-pay
/ pay-as-you-go" tari¤s.
3Armstrong (2002) discusses a model of perfect competition in two-part tari¤s. It
2
Model Setup
The model setup is a generalization of La¤ont, Rey and Tirole (1998) to many networks and general (instead of Hotelling) subscription demand. We
assume that there are n 2 symmetric mobile networks i = 1; :::; n who
compete in tari¤s. In the main text we consider linear and two-part tari¤s that do not discriminate between calls within the same network (on-net calls) and those to rival networks (o¤-net calls), while in the appendix we analyze tari¤s which price discriminate between these types of calls. Thus for now
we assume that network i charges a price pi for each call minute. In case
networks compete in two-part tari¤s it also charges a …xed fee Fi.
The marginal on-net cost of a call is c >0 and the cost of terminating a
call is c0 >0. Networks charge each other the access charge a per incoming
call minute. Thus the marginal cost of an o¤-net call isc+m, wherem =a c0
is the termination margin. There is a monthly …xed cost f per customer,
and networks receive further monthly pro…ts of Q per customer that do not
originate from payments for retail services o¤ered to them. Our focus will
be on how equilibrium pro…ts depend on Q.
From making a call of length q, a consumer obtains utility u(q), where
u(0) = 0, u0 > 0 and u00 < 0. For call price p, the indirect utility is
v(p) = maxqu(q) pq, call demand is q(p) = v0(p) with elasticity (p) =
pq0(p)=q(p). Receiving a call of length q yields utility u(q), where 0
indicates the strength of the call externality. Lettingvi =v(pi)and assuming
a balanced calling pattern (i.e. subscribers call any other subscriber with the
same probability) the surplus of a consumer on network i is given by
wi =vi+ n
X
j=1
juj Fi;
where Fi is zero for a linear tari¤. The market share of network i= 1; :::; n
is assumed to be
i =A(wi w1; :::; wi wn);
where An : Rn ! R is strictly increasing and symmetric in its arguments,
with 0 i 1, Pni=1 i = 1, from which follows thatA(0; :::;0) = 1=n. Let
= dA(x;0; :::;0)=dxjx=0.4
Denote the pro…ts from a pair of originated and terminated calls between
networksiandj asPij = (pi c m)qi+mqj,i; j = 1; :::; n(access payments
4This demand speci…cation is encapsulates both the generalized Hotelling model of
Hoernig (2014) and the logit model i = exp(wi)=
Pn
j=1exp(wj). We can allow for the
more general speci…cation i=Di(w), but in this case is no longer constant. Expression
cancel for on-net calls). Network i’s pro…ts are
i = i
n
X
j=1
jPij +Fi f+Q
!
:
3
Equilibrium Pro…ts and the Waterbed
Ef-fect
We will now derive equilibrium pro…ts and determine their dependence on
pro…ts Q, for both linear and two-part tari¤s. As for the latter, network i’s
…rst-order condition for a pro…t maximum is
0 = @ i
@Fi
= i
i
@ i
@Fi
+ i
n
X
j=1
@ j
@Fi
Pij + 1
!
:
In a symmetric Nash equilibrium we have i = 1=n, @ i=@Fi = (n 1) ,
and for allj 6=i,@ j=@Fi = andPij =Pii. Solving the …rst-order condition
for i we obtain
i =
1
(n 1)n2 : (1)
These pro…ts do not depend on Q, i.e. we have a full waterbed e¤ect. As for
linear tari¤s, consider the …rst-order condition for maximizing pro…ts with
respect to the call price pi:
0 = @ i
@pi
= i
i
@ i
@pi
+ i
n
X
j=1
@ j
@pi
Pij + n
X
j=1 j
@Pij
@pi
!
:
In a symmetric Nash equilibrium, we have pi = p and qi = q for all i =
1; :::; n, and thus@ i=@pi = (n 1) q and @ j=@pi = q , with
i =
1 L
(n 1)n2 ; (2)
whereL = (p c (n 1)m=n)=p is the Lerner index for the equilibrium
call price and the corresponding price elasticity of demand. Combining
both expressions for pro…ts shows that even in our more general framework
under two-part tari¤s the call price continues equal to average cost, i.e. L =
0 or p = c+ (n 1)m=n, i.e. does not depend on Q at all. On the other
combine (2) with the symmetric equilibrium pro…ts i = (P f +Q)=n,
P = (p c)q , to obtain
dp dQ =
(n 1)n
(n 1)n(P )0+ ( L = )0;
where apostrophes denote derivatives with respect to p . Since p is below
the monopoly price (P )0 is strictly positive, and the denominator is
pos-itive unless the demand elasticity decreases very strongly as the call price increases. The following assumption, common in the economic literature,
provides a simple su¢cient condition for ( L )0 >0.
Assumption 1: The price elasticity of demand (:)is non-decreasing.
Under this assumption, we conclude that under linear tari¤s higher Q
feeds through to lower call prices, dp =dQ <0. Finally, we obtain
d(n i)
dQ =
( L = )0
(n 1)n(P )0+ ( L = )0; (3)
which implies that only by chance the waterbed e¤ect is full (d(n i)=dQ=
0). Under Assumption 1, we obtain 0 < d(n i)=dQ < 1, i.e. higher Q is
translated into higher industry pro…ts, but only partially so.5
Summing up:
Proposition 1 In symmetric equilibrium, the waterbed e¤ect is
1. full under two-part tari¤s;
2. partial under linear tari¤s.
As shown in the Appendix, much the same results hold if networks price discriminate between on- and o¤-net calls, as originally discussed in Hoernig (2010).
Two conclusions follow from these results: First, in general terms the exact structure of tari¤s on one side of a market dictates how price changes on some other side are transmitted, even though di¤erent groups of customers are involved. Thus the design of regulation must take types of tari¤s in unregulated market segments into account. Second, for the speci…c case of regulation of …xed-to-mobile termination charges, our results show that concerns about reduced consumer welfare due to the waterbed e¤ect are less justi…ed for consumers on pre-pay (linear) tari¤s than those on post-pay (two-part) tari¤s, contrary to what networks have often publicly claimed.
5If Assumption 1 were to be strongly violated then industry pro…t would even decrease
References
[1] Armstrong, Mark, 2002. "The theory of access pricing and interconnec-tion," in Cave, M., Majumdar, S., Vogelsang, I., (Eds.), Handbook of Telecommunications Economics. North-Holland, Amsterdam.
[2] Genakos, Christos and Tommaso Valletti, 2011a. "Testing The “Wa-terbed” E¤ect In Mobile Telephony," Journal of the European Economic Association, 9(6), 1114-1142.
[3] Genakos, Christos and Valletti, Tommaso, 2011. "Seesaw in the air: In-terconnection regulation and the structure of mobile tari¤s," Information Economics and Policy, 23(2), 159-170.
[4] Hoernig, Ste¤en, 2010. "Competition Between Multiple Asymmetric Net-works: Theory and Applications," CEPR Discussion Paper 8060.
[5] Hoernig, Ste¤en, 2014. "Competition Between Multiple Asymmetric Net-works: Theory and Applications," International Journal of Industrial Or-ganization, 32(1), 57-69.
[6] La¤ont, Jean-Jacques, Patrick Rey and Jean Tirole, 1998. "Network Competition: I. Overview and Nondiscriminatory Pricing," RAND Jour-nal of Economics, 29(1), 1-37.
[7] Schi¤, Aaron, 2008. "The “Waterbed” E¤ect and Price Regulation," Re-view of Network Economics, 7(3), 392-414.
[8] Wright, Julian, 2002. "Access Pricing under Competition: An Applica-tion to Cellular Networks," Journal of Industrial Economics, 50(3), 289-315.
4
Appendix: Destination-Based Price
Discrim-ination
As in Hoernig (2010), we assume that network i charges a per-minute price
pi for calls within the same network (on-net calls) and a per-minute price
^
pi for calls to the other mobile network (o¤-net calls). Thus either networks
charge multi-part tari¤s(Fi; pi;p^i)or linear tari¤s(pi;p^i). Lettingvi =v(pi),
^
any other subscriber with the same probability) the surplus of a consumer
on network iis given by
wi = i(vi+ ui) +
X
j6=i
j(^vi+ ^uj) Fi = n
X
j=1
jhij Fi;
where hii = vi + ui and hij = ^vi+ ^uj for j 6= i, and Fi is equal to zero
under a linear tari¤.
Denote the pro…ts from one on-net call as Pii = (pii c)qii and those of
a pair of outgoing and incoming o¤-net calls as Pij = (^pi c m)^qi +mq^j,
j 6=i. Network i’s pro…ts are
i = i
n
X
j=1
jPij +Fi f+Q
!
:
Letting h be the n n-matrix of hij, and w and F then 1-vectors of wi
and Fi, we can write w= h F. Write market shares as i =Di(w) and
= D(h F) for a function D : Rn ! Rn with Jacobian W, then we
obtain the market share derivatives
d
dF = W h d
dF I () d
dF = G d
dp = W h d dp +
dh
dp () d
dp =G dh dp ;
where I is the identity matrix and G = (I W h) 1W, with elements Gij,
i; j = 1; :::; n. For the derivatives with respect to …xed fees, we obtain
d j
dFi
= Gji:
As for call prices, note …rst thatdh=dpiis ann n-matrix with entrydhii=pi =
qi+ u0(pi)qi0 = qi(1 + i)at position(i; i)and zeros otherwise; similarly,
forj 6=ithe matrixdh=dp^ihas entriesdhij=dp^i = q^ianddhji=dp^i = q^i ^i,
and is otherwise equal to zero. As a result, we have, for j = 1; :::; n,
d j
dpi
= qi(1 + i) iGji;
d j
dp^i
= q^i Gji(1 i) + ^i i
X
k6=i
Gjk
!
:
Since market shares sum to 1, we have Wii+
P
j6=iWij = 0 for all i. This
implies Pk6=iGjk = Gji,
6 and thus
d j
dp^i
= ^qiGji((1 + ^i) i 1):
In a symmetric equilibrium, Wij = for all i and j 6= i, and thus Wii =
(n 1) . Furthermore, in symmetric equilibrium all hii hon are identical,
and so are all hij hof, for j 6=i. After some computations, we …nd
Gii=Gon
(n 1)
1 n(hon hof)
; Gij =Gof
1 n(hon hof)
; 8j 6=i;
First we determine the equilibrium pro…ts under multi-part tari¤s, follow-ing Hoernig (2014): The …rst-order condition for pro…t-maximization with respect to …xed fees is
0 = @ i
@Fi
= d i
dFi i i + i n X j=1 d j dFi
Pij + 1
!
;
which can be solved for the symmetric equilibrium pro…ts ( i = 1=n, Pii =
Pmp
on , Pij =Pofmp),
mp i = 2 i n X j=1 Gji Gii Pij 1 Gii ! = 1 n2 1
(n 1) +
n(hof hon)
n 1 +P mp
of P
mp
on :
As is known (e.g. Hoernig 2014),7 under multi-part tari¤s with price
dis-crimination between on- and o¤-net calls the equilibrium call prices are
pmp = c=(1 + ) and p^mp = (c+m)=(1 =(n 1)). Call prices and h on,
hof,PofmpandPonmpdo not depend onQ. As a result, equilibrium pro…ts under
multi-part tari¤s are independent of Q, and the waterbed e¤ect is full.
As for linear tari¤s, the …rst-order condition for the pro…t-maximizing on-net price at the symmetric equilibrium is
0 = @ i
@pi
= d i
dpi i i + i n X j=1 d j dpi
Pij + i
dPii
dpi
!
;
with pro…ts under linear tari¤s of
lt on = 2 i n X j=1 Gji Gii Pij 1 (1 + i)Gii
1 pi c
pi i
!
= 1
n2 Pof Pon+
1 n(hon hof)
(n 1)
1 Lon
1 +
7This result can be derived from using the above …rst-order condition with respect to
and on-net Lerner index Lon = (p c)=p. Equally, by using the …rst-order condition for the pro…t-maximizing o¤-net price we obtain
0 = @ i
@p^i
= d i
dp^i i
i
+ i
n
X
j=1
d j
dp^i
Pij +
X
j6=i j
dPij
dp^i
!
;
or, with Lof = (^p c m)=p^,
lt
of =
2 i
n
X
j=1
Gji
Gii
Pij +
X
j6=i
j
((1 + ^i) i 1)Gii
1 p^i c m
^
pi ^i
!
= 1
n2 Pof Pon+
1 n(hon hof) 1 ^Lof
n 1 ^ :
Equating lt
on to ltof, we obtain
1 Lon
1 + =
(n 1) (1 ^Lof)
n 1 ^ :
This result implies that the Lerner indices tendLon and Lof tend to move in
lockstep, that is, if higherQleads to a lower on-net price then the o¤-net price
will decrease as well. While the exact comparative statics are too involved
to be discussed here, this implies that changes in Q are not compensated by
opposing shifts in on- and o¤-net call prices.
If call externalities and access margins are small ( ; m 0), then the
equilibrium condition implies p^ p, and similar computations as in the
main text lead to
d n lt
dQ
( Lon= )
0
(n 1)nP0
on+ ( Lon= )0
;
