DRAFT
Two-part interconnection tariffs
Paper Submitted to the ITS European Conference
Porto, September 2005
Jonathan Sandbach1
Head of Regulatory Economics, Vodafone Group
Tel:
+44 (0) 7795 300 653
Fax: +44 (0) 1635 238042
E-mail: jonathan.sandbach@vodafone.com
Abstract
This paper explores two-part tariffs for interconnection, assessing their implications
for regulation. Two-part interconnection tariffs sometimes refer to a per minute and
per call structure within network usage charges. However, this paper looks at a more
fundamental two-part interconnection tariff structure in which networks pay a
traditional usage charge (e.g. per minute or per call) for call termination, plus a fixed
(non-usage) contribution towards the fixed costs of other networks, assessed on the
number of subscribers (or market share) of the network.
This paper uses a theoretical model of the interaction between two mobile networks
and an incumbent fixed network to investigate the implications of the two-part
interconnection tariff. We assume the networks compete within a Bertrand pricing
framework, with Hotelling differentiation.
Under a policy of cost based regulation of termination charges, a two-part
interconnection tariff will result in lower usage based payments (per call or per
minute) between networks. This should give lower retail prices per call, which will
stimulate traffic between networks. Retail fixed fees will increase, but there will be an
overall gain in consumer welfare, and overall economic efficiency.
This paper does not consider practical challenges of introducing a two-part
interconnection tariff.
Introduction
Two-part interconnection tariffs exist in a number of ways:

Many interconnection charges are comprised of a call flag-fall charge, plus a per
minute charge;

Many interconnection arrangements require initial or on-going payments for the
dedicated interconnection capacity (both traffic and signalling) between networks,
plus usage related charges (per minute);
1
The views expressed in this paper are those of the author, and should not necessarily
be attributed to Vodafone.
1
DRAFT

In Australia, Optus has recently offered a two-part interconnection tariff for
termination on its mobile network in which interconnecting networks can opt to
pay a “fixed” sum (actually dependent on the number of end users), plus a lower
usage charge.2
This paper is stimulated by the last of these cases. The paper uses a theoretical
model of the interaction between an incumbent fixed network and two mobile
networks to investigate the impact of two-part interconnection tariffs consisting of
usage payments, plus fixed (non-usage) payments between networks covering a
proportion of the fixed costs incurred by each network.3
We assume the networks compete within a Bertrand pricing framework, with
Hotelling differentiation. The basic theory behind this model is laid out in the paper
by Laffont, Rey and Tirole (1998) for the case of two competing networks,
differentiated by a single metric. We will adopt essentially the same model, but
extend the model to include a third network, distinguished by a second metric. This
allows us to analyse the situation of two competing mobile networks (differentiated by
one metric), and a fixed network equally differentiated from both mobile networks by
a second metric (mobility). Thus the fixed network offers subscribers a different
(lower) level of utility to either of the two mobile networks (because of the lack of
mobility). For simplicity we further assume the mobile networks offer symmetrically
equal utility to consumers, thus ensuring that, in equilibrium, they will attract the
same number of subscribers.
Model
Let
pm
pf
Fm
Ff
cmt
cft
cm
cf
am
af
fm
ff
2
be the price of a call originated on a mobile network;
be the price of a call originated on the fixed network;
be the fixed monthly payment for subscription to a mobile network;
be the fixed monthly payment for subscription to the fixed network;
be the marginal cost of a call terminated on a mobile network;
be the marginal cost of a call terminated on a fixed network;
be the marginal cost of a call on a mobile network (origination, transport and
termination);
be the marginal cost of a call on the fixed network (origination, transport and
termination);
be the interconnection charge for terminating calls on a mobile network;
be the interconnection charge for terminating calls on the fixed network;
be the marginal cost of a mobile network subscription (excluding calls);
be the marginal cost of a fixed network subscription (excluding calls);
See
http://www.accc.gov.au/content/item.phtml?itemId=573237&nodeId=file425216ae5d14f&fn=O
ptus%20submission%20in%20support%20of%20MTAS%20access%20undertaking.pdf
And
http://www.accc.gov.au/content/item.phtml?itemId=573257&nodeId=file425216ddd9755&fn=
Optus%20MTAS%20access%20undertaking%E2%80%94December%202004.pdf
3
An extension of the analysis to a second fixed network, or third mobile network would be
laborious, and unlikely to provide any further useful insights.
2
DRAFT
gm be the fixed cost of a mobile network (e.g. geographical coverage);
g f be the fixed cost of the fixed network;
q ( p ) be the volume of calls per subscriber on a network, as a function of the
price ( p ), assumed to be the same for both fixed and mobile networks;
be the subscriber share of the fixed network;
sf
sm be the subscriber share of a mobile network;
be the subscriber share of the fixed network;
sf
m
f
be the profit earned by a mobile network;
be the profit earned by the fixed network.
In addition, we will sometimes use a second subscript on some variables relevant to
the two mobile networks to identify separate values for each individual network.
The way we model this is as follows. Subscribers initially decide whether they will
subscribe to the fixed or a mobile network. We assume that all consumers gain
sufficient benefit from making calls (at prices within a relevant range) to ensure that
they will join either a fixed or a mobile network – but not both.4 This assumption
implies that the total number of subscribers (to all networks) is independent of prices.
Subscribers are assumed to be distributed uniformly along a segment [0,1] in respect
to their preferences for mobility. At one extreme ( x  0 ) subscribers place no value
on mobility, whilst at the other extreme ( x  1) they place a value of t f on belonging
to a mobile network. Therefore, we can represent the consumer surplus that an
individual subscriber would receive from the fixed network as:
w f  tfx
where wf is the difference between the variable consumer surplus from calls, v ( pf ) ,
and the fixed monthly payment:
wf  v( pf )  Ff
Re-iterating, we assume that all consumers gain positive net surplus, and so
subscribe to a network at any call price in a range relevant to our analysis.
Meanwhile, the consumer surplus that an individual subscriber would enjoy from
each of the mobile networks is wm1  tf (1  x) and wm 2  tf (1  x) , where
wm1 and wm2 are the consumer surplus of subscribers to each of the mobile networks.
The market share of the fixed network is found by determining the value of x at which
subscribers are indifferent between the fixed and mobile networks. Thus:
sf 
1 wf  sm1wm1  sm 2 wm 2  / sm1  sm 2  1
sm1wm1  sm 2 wm 2

  f wf 
2
2tf
2
sm1  sm 2
(
) (1)
4
Clearly, in reality, many consumers will have both fixed and mobile subscriptions. In
principle the analysis could be adapted to include this situation, but for the purposes of this
paper we restrict our attention to cases where fixed and mobile subscriptions are alternatives.
This will highlight issues that arise in this particular case.
3
DRAFT
where f  1 / 2tf  is an index of substitutability between the fixed and mobile
networks.
The two mobile networks then compete for the remaining market share and, to the
extent that they offer different prices, will face market shares given by:
sm1  1  sf ½  m(wm1  wm2) and sm2  1  sf ½  m(wm2  wm1)
(2)
where m is an index of substitutability between the two mobile networks.
Substituting equations (2) into equation (1) gives explicit equations for the market
share of the fixed network:
sf 


1
wm1  wm 2
2
 f  wf 
 m wm1  wm 2  


2
2
(3)
It will be useful to write out the following partial derivatives:
sf
 f
wf
1

sm1
 f   m wm1  wm 2  
wf
 2



sm 2
1
 f   m  wm 2  wm1  
 2

wf
1

sf
 f   2m  wm1  wm 2  
wm1
 2



sm1
1
 m1  sf   f   2m wm1  wm 2   sm1
 2
 1 sf
wm1
1

sm 2
 m1  sf   f   2m  wm1  wm 2   sm 2
1sf
wm1
 2

(4a)
(4b)
(4c)
(4d)
(4e)
(4f)
Now turning to look specifically at the mobile networks, we are able to write the profit
function (for, say, network 1). For convenience, we take the profit to be a function of
price ( pm1 ) and consumer surplus ( wm1 ). The profit function is:
m1  sm1 pm1  cm  sm 2 am  cmt   sf  af  cmt  q  pm1  v pm1  wm1  fm
 sm1sm 2am  cmt q pm 2   sm1sf am  cmt q pf   gm    sf  sm 2  sm1  gm  sm1 gf 
(5)
Where  is the proportion of the network’s fixed cost that is covered from a fixed
payment between networks (pro-rata according to subscriber shares). Differentiating
with respect to price pm1 gives
m1
 sm1pm1  cm  sm2 am  cmt   sf  af  cmt q'  pm1  q pm1  v'  pm1 (6)
pm1
Noting v'  pm1  q pm1 , this simplifies to:
4
DRAFT
m1
 sm1 pm1  cm  sm 2 am  cmt   sf  af  cmt  q'  pm1
pm1
(7)
And so the first order condition gives:
pm1  cm  sm2 am  cmt   sf  af  cmt 
(8)
This simply states that networks will price outgoing calls at perceived cost, taking
account of outpayment expenses to other networks. The price depends on the
usage element of the interconnection tariff, but not the fixed element. Now
differentiating with respect to wm1 , and substituting from equation (8) gives:
m1
wm1
sm1  
v pm1  wm1  fm  sm 2am  cmt q pm 2   sf am  cmt q pf 

wm1  
 sm 2
am  cmt q pm 2   sf am  cmt qqf    sm 2  am  cmt   sf  af  cmt  q pm1  1
 sm1
wm1
wm1
 wm1

 wm1

sf
sm 2 sm1 
sm1 
 gm 


gf 
  wm1
wm1 wm1 
wm1 



  
Assuming symmetry between the two mobile networks, we expect in equilibrium:
wm1  wm2  wm , sm1  sm2  sm , pm1  pm 2  pm and m1  m2  m
(9)
Thus, equations (4) become
sf
 f
wf
sm1
f

wf
2
sm 2
f

wf
2
sf
f

wm1
2
sm1
f sm
f
 m1  sf  
 2msm 
wm1
2 1sf
4
sm 2
f sm
f
 m1  sf  
 2msm 
1

s
f
wm1
2
4
(10a)
(10b)
(10c)
(10d)
(10e)
(10f)
And so the first order condition becomes
m
wm
5
DRAFT
f 

  2msm  v pm   wm  fm  am  cmt smq pm   sfq pf    2 gm  gf 
4

f

 sm  am  cmt q pm   af  cmt q pf   1  0
2

Thus
wm  v pm   fm   2 gm  gf   am  cmt smq pm   sfq pf 
f

 sm  am  cmt q pf   af  cmt q pm   1
2

(11)
f 

 2msm  
4

Finally, substituting both first order conditions of equations (8) and (11) back into the
profit function of equation (5) gives
f am  cmt q pf   af  cmt q pm   1






sm 2 
m 


2

2msm 
f


 (1   ) gm
(12)
4
The same analysis for the fixed network yields the following.
f  sf  pf  cft  sm1  sm 2  am  cft  q  pf   v pf   wf  ff 
 sf af  cft sm1q pm1  sm 2 q pm 2   gf    sm1  sm 2 gf  gm 
(13)
Differentiating with respect to price pf gives
f
 sf pf  cf   sm1  sm2  am  cft q'  pf   q pf   v'  pf 
pf
(14)
Noting v'  pf   q pf  , this simplifies to:
f
 sf  pf  cf   sm1  sm 2  am  cft  q'  pf 
pf
(15)
And so the first order condition gives:
pf  cf   sm1  sm2  am  cft 
(16)
Now differentiating with respect to wf , and substituting from equation (16) gives:
f
wf
6
DRAFT

sf  
v pf   wf  ff  af  cft sm1q pm1  sm 2q pm 2 
wf  

sm 2
 sm1

 sm1 sm 2    
 sf af  cf 
q pm1 
q pm 2    am  cf 

q  pf   1
wf
 wf

 wf wf 




sm 2 
 sm1

  gf  gm 
  


 wf

wf 


Assuming symmetry between the two mobile networks, the first order condition
becomes
f

wf
f v pf   wf  ff  2smaf  cf q pm    gf  2gm   sf f af  cft q pm   am  cft q pf   1  0
Thus
wf  v pf   ff  2smaf  cf q pm    gf  2 gm  
sf f af  cft q pm   am  cft q pf   1
f
Finally, substituting both first order conditions of equations (8) and (11) back into the
profit function of equation (5) gives
f 
sf 2 f af  cft q pm   am  cft q pf   1
f
 (1   ) gf
(17)
Table 1 summaries the main results from the model.
7
DRAFT
Table 1:
Summary of Model Results
Mobile networks
pm
Retail call
charge
Retail
fixed
charge
Profit
Fm
m
One-part interconnection tariff
Two-part interconnection tariff
cm  sm am  cmt   sf  af  cmt 
cm  sm am  cmt   sf  af  cmt 
fm  am  cmt smq pm   sfq pf 
fm   2 gm  gf   am  cmt smq pm   sfq pf 
f

 sm  am  cmt q pf   af  cmt q pm   1
2

f

 sm  am  cmt q pf   af  cmt q pm   1
2

f am  cmt q pf   af  cmt q pm   1





2

2msm 
smwm
Welfare
s mw m
 m
f am  cmt q pf   af  cmt q pm   1






sm 2 


Consumer
surplus
f 

 2msm  
4

f



sm 2 

m



 gm
2

2msm 
4
v p  fm  am  cmt smq pm   sfq pf 

f


sm  sm  2 am  cmt q pf   af  cmt q pm   1



f


 2msm 

4

sm v pm   fm  am  cmt smq pm   sfq pf  gm


f 

 2msm  
4








f


 (1   ) gm
4
v p  fm   2 gm  gf   am  cmt smq pm   sfq pf 



 sm f am  cmt q pf   af  cmt q pm   1



sm 

2


f 


 2msm  

4 

smv pm   fm  am  cmt smq pm   sfq pf  gm   sfgm  smgf 



m

8
DRAFT
Table 1 (continued):
Summary of Model Results
One-part interconnection tariff
Two-part interconnection tariff
Fixed network
pf
Retail call
charge
cf  2sm am  cft 
cf  2sm am  cft 
Retail
fixed
charge
ff  2smaf  cf q pm  
Profit
Ff
f
sf f af  cft q pm   am  cft q pf   1
sfwf
Welfare
sfwf
 f
All networks
Welfare
f
2
f
Consumer
surplus
sf f af  cft q pm   am  cft q pf   1

sf f af  cft q pm   am  cft q pf    1
f
sf f af  cft q pm   am  cft q pf    1
2
 gf
v pf   ff  2 smaf  cft q  pm 



sf  sf f af  cft q pm   am  cft q  pf    1


f






sf v pf   ff  2smaf  cft q pm  gf
 s mw m 

2
  m 
 sf w f
 f
ff   gf  2 gm   2smaf  cf q pm 
2 smv  pm   fm  am  cmt smq pm   sfq pf 
 sf v  pf   ff  2 smaf  cft q  pm   2 gm  gf
f
 (1   ) gf
v pf   ff  2smaf  cft q  pm     gf  2 gm 


sf  sf f af  cft q pm   am  cft q  pf    1 


f






sf v pf   ff  2smaf  cft q pm  gf  2 smgf  sfgm 
2smv pm   fm  am  cmt smq pm   sfq pf 
 sf v pf   ff  2smaf  cft q  pm   2 gm  gf
9
DRAFT
Implications of the Model
A number of implications can be drawn from the results in Table 1:

In all cases (where there is a two-part retail tariff), each network’s retail call price
will equate to the cost incurred by that network, taking account of proportion of
calls that bear a usage termination charge on another network. Thus, call prices
will differ between fixed and mobile networks (to the extent that network costs,
usage based termination charges, and the proportion of calls terminating on other
fixed and mobile networks will differ). In general mobile networks will have higher
own network costs, but lower outpayments to other networks.

Retail fixed prices will be based on the fixed cost per subscriber, offset by the
profit made on terminating calls to that subscriber, plus a profit. The retail fixed
price is also increased by the contribution each network needs to make to the
fixed costs of other networks for increases in its subscriber share (and the loss of
contribution it receives itself).

The profit is related to the degree of differentiation between networks. The profit
of the fixed network is determined by its degree of differentiation from the mobile
networks ( f ). The profit of the mobile networks is determined by its degree of
differentiation from the fixed network and the differentiation between mobile
networks( f and m ), irrespective of whether or not there is a two-part
interconnection tariff.

The profits of both networks are determined by the quantity afq pm   amq pf  :
the “balance of payments” between the fixed and a mobile network. Fixed
network profits are positively related to this quantity, whilst mobile network profits
are negatively related to this same quantity. The extent to which this quantity
does affect profits of either fixed or mobile networks is dependent on the degree
of differentiation between fixed and mobile networks. Profits are not directly
affected by call costs, since these are fully passed on in retail tariffs, although
there will be an indirect impact via quantities.

If we set f  0 (i.e. the fixed is fully differentiated from mobile networks), the
model predicts that the fixed network can make infinite profit due to the lack of
competitive constraint and the insensitivity of overall subscription levels to prices.
Furthermore, the fixed network does not affect the profitability of the mobile
network and so we get the Laffont, Rey and Tirole (1998) result that profit of the
mobile networks is independent of the usage based interconnection charge.
Implications for Interconnection Tariffs
One-part Tariff
Interconnection tariffs for call termination are usually regulated. The regulated rates
are often (but not always) based on the cost of providing the interconnection service
(e.g. long run incremental cost), plus an allocation of fixed and common costs. In the
case of the simple one-part usage based interconnection tariff, this would lead to:
10
DRAFT
am  cmt 
af  cft 
gm
smq  pm 
gf
(18)
sfq  pf 
In this case, total welfare becomes:

  
smq  pm  
smq pm   sfq pf 


W 1 part  2 sm v pm   fm  gm

s
f
v
p
f

f
f

2

g
f

  
  2 gm  gf
smq  pm 
sfq  pf  



 smq pm   sfq pf  
 smq  pm  


W 1 part  2 smv p  fm   sf v p  ff   2 
 1 gm   2    1 gf


q
p
m
q  pf 









m




f

The intuition behind this result is simply that welfare is the total consumer surplus on
calls, plus a factor that accounts for the additional producer surplus from
interconnection termination rates being marked-up to recover a proportion of fixed
costs (resulting in call retail prices being above underlying cost), less subscriber and
network fixed costs.
Two-part Tariff
The situation is different with a two-part interconnection tariff. Here fixed costs are
recovered by the fixed component of the interconnection tariff, which is passed on to
consumers through the retail fixed price. Therefore
am  cmt
af  cft
(19)
And so welfare becomes
W 2 part  2smv pm   fm  sf v pf   ff   2gm  gf
Comparison of one-part and two-part interconnection tariff cases
In comparing these two welfare results ( W 1 part and W 2 part ), it is important to
distinguish the two different retail price levels ( pm , pf and Fm , Ff ) (and also, to a
lesser degree of importance, the different market shares) between the one-part and
two-part interconnection tariff cases. Compared to the one-part tariff case, the retail
call price will be lower but the fixed monthly price will be higher. However, the latter
does not feature in the total welfare, since it is simply a transfer from consumers to
networks, and we assume the participation in the networks is independent of the
fixed retail payment (although shares of individual networks are not). Therefore, we
would expect that, compared to the one-part tariff case, v pm  and v pf  will be
greater, reflecting both greater consumer surplus on the calls than would be made
under the one-part tariff, and additional consumer surplus from greater volume
( q pm  and q pf  ). Off setting this will be the loss of producer surplus that networks
make from terminating calls at a charge that allows some recovery of fixed costs.
However, there will be an unambiguous net gain in welfare, principally from the
stimulated calling volumes.
11
DRAFT
Conclusion
This paper has explored a model under which two mobile networks compete with a
fixed incumbent under a two-part interconnection tariff.
A two-part interconnection tariff will result in lower usage based payments (per call or
per minute) between networks. This should give networks greater flexibility in pricing
calls, and result in welfare enhancing increases of traffic between networks. It may
thus capture some of the consumer benefit normally associated with regulatory
intervention in reducing interconnection charges.
We have not discussed the significant practical challenges in implementing a twopart interconnection tariff of the type analysed in this paper. The Optus tariff
referenced in the introduction (which is similar in structure to the one discussed here)
has a number of auditing requirements (e.g. to verify subscriber number), and
excludes transit and international traffic. Nevertheless, two-part interconnection
tariffs do appear to have positive welfare features that justify further consideration.
References
Laffont, J., Rey, P. and Tirole, J. “Network competition: I. Overview and nondiscriminatory pricing” RAND Journal of Economics, Vol. 29, No. 1, Spring 1998, pp.
1-37
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