Part C
Pricing
Pricing Communication Networks: Economics, Technology and Modelling.
Costas Courcoubetis and Richard Weber
Copyright

2003 John Wiley & Sons, Ltd.
ISBN: 0-470-85130-9
7
Cost-based Pricing
This chapter is about prices that are directly related to cost. We begin with the
problem of finding cost-based prices that are fair or stable under potential competition
(Sections 7.1 –7.2). We look for types of prices that can protect an incumbent against entry
by potential competitors, or against bypass by customers who might find it cheaper to supply
themselves. We explain the notions of subsidy-free and sustainable prices. Such prices are
robust against bypass. Similar notions are addressed by the idea of the second-best core. The
aim now differs from that of maximizing economic efficiency. We see that Ramsey prices,
which are efficient subject to the constraint of cost recovery, may fail sustainability tests.
In Section 7.3 we take a different approach and look at practical issues of constructing
cost-based prices. Now we emphasize necessary and simplicity. Prices are to be computed
from quantities that can be easily measured and for which accounting data is readily
available. An approach that has found much favour with regulators is that of Fully
Distributed Cost pricing (FDC). This is a top-down approach, in which costs are attributed
to services using the firm’s existing cost accounting records. It ignores economic efficiency,
but has the great advantage of simplicity.
Section 7.3.5 concerns the Long-Run Incremental Cost approach (LRIC). This is a
bottom-up approach, in which the costs of the services are computed using an optimized
model for the network and the service production technologies. It can come close to
implementing subsidy-free prices. We compare FDC and LRIC in Section 7.4, from the
viewpoint of the regulator, who wishes to balance the aims of encouraging efficiency and
competition, and of the monopolist who would like to set sustainable prices. The regulator

This motivates the idea of subsidy-free prices. A second reasonable necessary condition is
that prices should be defensive against competition, discouraging the entry of competitors
who by posting lower prices could capture market share. This motivates the idea of
sustainable prices. If prices do not reflect actual costs or they hide costs of inefficient
production then they invite competition from other firms. Since customers will choose the
provider from whom they believe they get the best deal, a game takes place amongst
providers, as they seek to offer better deals to customers by deploying different cost
functions and operating at different production levels. Prices must be subsidy-free and
sustainable if they are to be stable prices, that is, if they are to survive the competition in
this game.
Interestingly, the set of necessary conditions that we might like to impose on prices can
be mutually incompatible. They can also be in conflict with the aim of maximizing social
welfare maximization, since they restrict the feasible set of operating points, sometimes
reducing it to a single point.
7.1.1 Fair Charges
Consider the problem of a single provider who wishes to price his services so that they
cover their production cost and are fair in the sense that no customer feels he is subsidizing
others. Unfair prices leave him susceptible to competition from another provider, who has
the same costs, but charges fairly. Customers might even become producers of their own
services.
Let N Df1; 2;:::;ng denote a set of n customers, each of whom wishes to buy some
services. For T that is a subset N,andletc.T / denote the minimal cost that could by
incurred by a facility that is optimized to provide precisely the services desired by the set
of customers T . We call this the stand-alone cost of providing services to the customers in
T . Assume that because of economies of scale and scope this cost function is subadditive.
That is, for all disjoint sets T and U ,
c.T [ U/ Ä c.T / C c.U/ (7.1)
In the terminology of cooperative games, c.Ð/ is called a characteristic function.
The service provider wants to share the total cost of providing the services amongst the
customers in a manner that they think is fair. Suppose he charges them amounts c

for these services than the incumbent firm. This happens because the incumbent uses part
of the revenue obtained from selling services to N n T to pay for some of the cost of the
services wanted by T . Next, we investigate certain variations and refinements of the above
concepts.
7.1.2 Subsidy-free, Support and Sustainable Prices
Let reformulate the ideas of the previous section to circumstances in which charges are
computed from prices. Suppose that a set of n services is N Df1;:::;ng and an incumbent
firm sells service i in quantity x
i
,atpricep
i
, for a total charge of p
i
x
i
. Suppose that x
i
is given and does not depend on p D . p
1
;:::;p
n
/. We call p a subsidy-free price if it
satisfies the two tests
X
i2T
p
i
x
i
Ä c.T /; for all T Â N (7.4)

2
.1000/ that are obtained from the video and the voice services
must satisfy
2 Ä r
1
.100/ Ä 12; 1 Ä r
2
.1000/ Ä 11; r
1
.100/ C r
2
.1000/ D 13
166 COST-BASED PRICING
Thus, assuming that there is enough demand for services, possible prices are 0:006 units
per voice service and 0:07 units per video service. Note that such prices are not unique
and they may not even exist for general cost functions. Suppose three services are pro-
duced in unit quantities with a symmetric cost function that satisfies (7.1). Let c.fi g/ D 2:5,
c.fi; j g/ D 3:5, and c.fi; j; kg/ D 5:5, where i; j; k are distinct members of f1; 2; 3g.Then
we must have 2 Ä p
i
Ä 2:5, for i D 1; 2; 3, but also p
1
C p
2
C p
3
D 5:5. So there
are no subsidy-free prices. The problem is that economies of scope are not increasing, i.e.
c.fi; j; kg/  c.fi; j g/>c.fi; j g/  c.fig/.
How can one determine if (7.4) and (7.5) are met in practice? Assume that a firm posts its

1
;:::;x
n
/. We say the vector p is a support price for c at x if it
satisfies the two conditions
X
i2N
p
i
y
i
Ä c.y/; for all y Ä x (7.6)
X
i2N
p
i
z
i
½ c.x/  c.x  z/; for all z Ä x (7.7)
Note these imply
P
i2N
p
i
x
i
D c.x/. We can compare them to (7.2) and (7.3). For example,
(7.6) implies that one cannot produce some of the demand for less than it is sold. They
imply (7.4) and (7.5) (but are more general since they deal with arbitrary sub-quantities of
the vector x, instead of looking just at subsets of service types), and hence a support price

0
i
D p
i
for i 2 N n T ),
X
i2N nT
p
0
i
x
i
. p
0
/>
X
i2N nT
p
i
x
i
. p/
i.e. when p
0
is replaced by p, the introduction of services in T reduces the demand for (and
revenue earned from) services in N n T . Noting that
P
i2N nT
p
i

x
i
. p/ 
X
i2N nT
p
0
i
x
i
. p
0
/
Here the incremental cost test (7.7) is passed (by the left hand inequality), but net additional
revenue does not cover additional costs (the right hand inequality). Thus, the additional
costs must be covered (at least in part) by increasing the charges levied on customers who
were happy when only services in N n T were offered, rather than only making charges to
customers who purchase services in T . These former set of customers may feel that they
are subsidizing the later set of customers, and that these new services decrease the overall
efficiency of the system. We conclude that, as a matter of fairness between customers, the
second test condition (7.7) should take account of demand, and reason in terms of the net
incremental revenue produced by an additional service, taking account of the reduction of
revenue from other services. In other words, services are fairly priced if when service i is
offered at price p
i
the customers of the other services feel that they benefit from service i.
They are happy because the prices of the services they want to buy decrease. This is called
the net incremental revenue test. Let us look at an example.
Example 7.2 (Net incremental revenue test) Suppose a facility costs C and there is no
variable cost. It initially produces a single service 1 in quantity x

condition may be achieved by reducing the other prices and hence benefiting the customers
of the other services. In our example, suppose that by setting p
2
D p
1
and keeping p
1
at
its initial value, x
1
becomes a=2andx
2
D a=2 C b=2. In other words, half the customers
of service 1 find service 2 to suit them better at the same price, and so switch. There are
also new customers that like to use service 2 at that price. Then the net revenue increase
becomes p
1
b=2 > 0; so it is possible to decrease p
1
and allow customers of service 1 to
benefit from the addition of service 2.
Finally, consider a model of potential competition. Imagine an incumbent firm sets prices
to cover costs at the demanded quantities, i.e.
X
i2N
p
i
x
i
. p/ ½ c.x . p// (7.8)

i
for some i ; and x
0
Ä x
E
. p; p
0
/ (7.9)
That is, there is no way that the potential entrant can post prices that are less than the
incumbent’s for some services and then serve all or part of the demand without incurring
loss. Prices satisfying this condition are called sustainable prices.Wehaveyetonemore
‘fairness test’ by which to judge a set of prices.
The above model motivates the use of sustainable prices in contestable markets. A
market is contestable when low cost ‘hit-and-run’ entry and exit are possible, without
giving enough time to the incumbent to react and adjust his prices or quantities he
sells. Such low barrier to entry is realized by using new technologies such as wireless,
or when the regulator prescribes that network elements can be leased from incumbents
at cost.
In the idea of sustainable prices we again see that price stability is related to efficiency.
If prices are sustainable, a new entrant cannot take away market share if his cost function
is greater than that of the incumbent. Hence sustainable prices discourage inefficient entry.
However, if a new entrant is more efficient than the incumbent, and so has a smaller
cost function, then he can always take away some of the incumbent’s market share by
posting lower prices. Thus an incumbent cannot post sustainable prices if he operates with
inefficient technologies.
It can be shown that for his prices to be sustainable, an incumbent firm must fulfil a
minimum of three necessary conditions:
FOUNDATIONS OF COST-BASED PRICING 169
1. He must operate with zero profits.
2. He must be a natural monopoly (exhibit economies of scale) and produce at minimum

more than cover his costs by electing not to produce service 1. After doing this, he can
slightly lower the prices of all the services that are priced above their marginal costs, so as
to obtain all that demand for himself and yet still cover his costs.
Example 7.3 (Ramsey prices may not be sustainable) Whether or not Ramsey prices
are sustainable can depend on how services share fixed costs, i.e., on the economies of
scope. Consider a market in which there are customers for two services. The producer’s
cost function and demand functions for the services are
c.x
1
; x
2
/ D 25x
1=2
1
C 20x
1=2
2
C F ; x
1
. p/ D x
2
. p/ D
10
4
.10 C p/
2
The Ramsey prices are shown in Table 7.1. When the fixed cost F is 6 the Ramsey prices
are not sustainable even though they exceed marginal cost. The revenue from service 2 is
169:45 and this is enough to cover the sum of its own variable cost and the entire fixed
cost, a total of 162:76. This means that a provider can offer service 2 at a price less than the

Variable cost C F 195.70 162.76 215.71 183.57
To show how the existence of common cost plays a vital role in the sustainability of
Ramsey prices, we can construct a simple example out of Figure 5.5.
Example 7.4 (Common cost and sustainability of Ramsey prices) Suppose that two
services are produced with same stand-alone cost function A C bx. First, consider the
case in which there is no economy of scope, and hence the total cost is the sum
of the stand-alone cost functions. Since both services are produced at equal quantities
x
i
D x
j
D x we have x. p
i
C p
j
/ D 2. A C bx/ which implies xp
i
< A C bx < xp
j
.
But A C bx is the stand-alone cost for service j, which violates the sustainability
conditions.
Now suppose that there are economies of scope and the fixed cost A is common to both
services. Then x. p
i
C p
j
/ D AC2bx, and since p
i
> b we obtain xp

j
.T nfig/>0,
then customer j pays more than he would pay if customer i were not being served. He
might argue this was unfair, unless customer i can counter-argue that he is at least as
disadvantaged because of customer j. But then if customer i is not to feel aggrieved then
he must see similarly that customer j is at least as much disadvantaged. Putting this all
together requires

i
.T /  
i
.T nfjg/ D 
j
.T /  
j
.T nfig/ (7.10)
On the other hand, if 
j
.T /  
j
.T nfig/<0, then customer j is better off because
customer i is also being served. Customer i might feel aggrieved unless he benefits at
least as much from the fact that customer j is present. But then customer j will feel
FOUNDATIONS OF COST-BASED PRICING 171
aggrieved unless he benefits at least as much from customer i’s presence. So again, we
must have (7.10).
Surprisingly, there is only one function  which satisfies (7.10) for all T Â N and
i; j 2 T . It is called the Shapley value, and its value for player i is the expected incremental
cost of providing his service when provision of the services accumulates in random order.
It is best to illustrate this with an example.

an airport runway and terminal is the cost sharing of the runway plus the cost sharing of
the terminal. The Shapley value also gives answers that are consistent with other efficiency
concepts such as Nash equilibrium.
The Shapley value need not satisfy the stand-alone and incremental cost tests, (7.2) and
(7.3). However, one can show that it does so if c is submodular,i.e.if
c.T \ U / C c.T [ U / Ä c.U / C c.T /; for all T ; U Â N (7.11)
The reader can prove this by looking at the definition of the Shapley value and using
an equivalent condition for submodularity, that taking the members of N in any order,
172 COST-BASED PRICING
say i; j; k;:::;`,wemusthave
c.fig/ ½ c.fi; jg/  c.f j g/ ½ c.fi; j; kg/  c.f j; kg/ ½ ÐÐÐ ½ c.N /  c.N fig/
Note that choosing T and U disjoint shows that submodularity is consistent with c.Ð/ being
subadditive, i.e. (7.1).
7.1.4 The Nucleolus
The Shapley value has given us one way to allocate charges and it is motivated by a nice
story of argument and counterargument. However, there are other stories we can tell. Let
us call c an imputation of cost (i.e., an assignment of cost) if
X
i2N
c
i
D c.N / and c
i
Ä c.fig/; for all i
That is, the provider exactly covers his costs and no customer is charged more than his
stand-alone cost.
We now suggest a reasonable condition that the imputation c should satisfy. Suppose
that for all imputations c
0
and subsets T Â N such that

i2T
c
i
 c.T /
So if a set of customers T prefers an imputation c
0
(because their total charge is less), then
there is always some other set of customers U who can object because
ž under c
0
the total charge they pay is more, i.e.
P
i2U
c
0
i
>
P
i2U
c
i
,and
ž they pay under c
0
a greater increment over their stand-alone cost, c.U /,thanT pays
under c over its stand-alone cost, c.T /.
so U argues that T should not have a cost-reduction at U’s expense.
Then c is said to be in the nucleolus (of the coalitional game). It is a theorem that
the nucleolus always exists and is a single point. Thus the nucleolus is a good candidate
for being the solution to the cost-sharing problem. In the runway-sharing example, the

. p/,andc.x / is the cost of producing x. The entrant targets a
subset of customers S who he wishes to woo. He chooses p
S
s.t. . p
S
/
>
x
S
 c.x
S
/ ½ 0, where
x
S
D
P
i2S
x
i
. p
S
/, and such that the incentive compatibility condition holds, CS
i
. p
S
i
/ ½ CS
i
. p/,
for all i 2 S.Wesay p is in the second-best core if an entrant has no such possibility.

such that both
P
i2S
P
j
p
0
j
x
i
j
. p
0
/ ½ c

P
i2S
x
i
. p
0
/
Ð
and
u
i
.x
i
. p
0

core prices may not exist.
There is a subtle difference in the assumptions underlying sustainable prices and second-
best core. In the second-best core model a customer who is a member of a coalition S must
buy all his services from the coalition and nothing from the outside. So a successful entrant
must be able to completely lure away a subset of customers, S. This is in contrast to the
sustainable price model, where a customer may buy services from both the monopolist and
the new entrant.
This difference means that sustainable prices are quite different to second-best core prices.
Prices that are stable in the sense of the second-best core may not be stable if a customer
is allowed to split his purchases. Also, prices that are not sustainable because a competitor
may be able to price a particular service at a lesser price may be stable in the second-best
core sense, since the net profit of customers that switch to the new entrant can be less. In
the second-best core model customers must buy bundles of services and the price of the
bundle offered by the entrant could be more.
In conclusion to this section, let us say that we have described a number of criteria by
which to judge whether customers will see a proposed set of costs as fair, and presenting
174 COST-BASED PRICING
no incentive for bypass or self-supply. Anonymously equitable prices are attractive, but
they may not exist. We would not like to claim that one of these many criteria is the
most practical or useful in all circumstances. Rather, the reader should think of using these
criteria as possible ways of checking what problems a proposed set of prices may or may
not be present.
7.2 Bargaining games
Another approach to cost-sharing is to let the customers bargain their way to a solution.
7.2.1 Nash’s Bargaining Game
Suppose that the cost of supplying x is c.x/, x 2 X. Here, x is the matrix x D .x
ij
/,
where x
ij

bargaining are to take place until a point in U is agreed. At the first round, player 1 proposes
that they settle for .u
1
; u
2
/ 2 U. Player 2 can accept this, or make a counterproposal
.v
1
;v
2
/ 2 U at the second round. Now, player 1 can accept that proposal, or make a new
proposal at the third round, and so on, until some proposal is accepted. We assume that
both players know U. Note that only proposals corresponding to points on the northeast
boundary of U need be considered, i.e. the players should restrict themselves to Pareto
efficient points of U .
Rounds are s minutes apart. Let us penalize procrastination by saying that if bargaining
concludes at the nth round, then the utility of player i is reduced by a multiplicative factor of
exp..n1/sÁ
i
/.IfÁ
1
and Á
2
differ then the players have different urgencies to settle. Note
that this game is stationary with respect to time, in the sense that at every odd numbered
round both players see the same game that they saw at round 1, and at every even numbered
round they see the same game that they saw at round 2. Thus player 1 can decide at round
1 what proposal he will make at every odd numbered round and make exactly the same
proposal every time, say .u
1

sÁ
1
u
1
.
In summary,
u
2
D e
sÁ
2
v
2
and v
1
D e
sÁ
1
u
1
(7.12)
Let u and v be the two points on the boundary of U for which (7.12) holds. A possible
strategy for player 2 is to propose .v
1
;v
2
/ and accept player 1’s proposal if and only if he
would get at least u
2
. A possible strategy for player 1 is to propose .u