Optimal Network Access Pricing for Natural Monopolies when Costs are Sunk
and Revenues are Uncertain
Ephraim Clark
Finance Department, Middlesex University Business School
The Burroughs, London NW4 4BT, UK
and
Joshy Z. Easaw
Department of Economics and International, University Of Bath,
Claverton Down, Bath, BA2 7AY, UK
15/05/2003
2
Optimal Network Access Pricing for Natural Monopolies when Costs are Sunk
and Revenues are Uncertain
Abstract
Based on the principle that competition in the final goods (services) market leads to a
situation of social welfare maximization, regulation has come to focus on network
provision and access pricing that can guarantee this competition. In this paper, we
study optimal access pricing for natural monopoly networks with large sunk costs
and uncertain revenues. Using techniques from the option pricing literature, we show
that the optimal access price corresponds to a risk-free form of the Efficiency
Component Pricing Rule (ECPR), that is, where the opportunity cost is based on the
risk free rate of return. We also show that at levels of revenue above the optimal level
that triggers entry, the entrant should pay a premium above risk-free ECPR that
rewards the incumbent for relinquishing his rights to the risky cash flows at the
higher revenue level.
JEL Classification: L51, D81
Keywords: Network Access Pricing, Real Options, Investment under Uncertainty.
3
I. Introduction
In this paper, we study optimal access pricing for networks that are natural
monopolies. Our study incorporates uncertainty about the future revenue to be
derived from providing the final good or service and the fact that many natural
monopoly network investments are intrinsically irreversible because they include a
large element of sunk costs. Using techniques from the option pricing literature, we
show that the optimal access price is determined by a certain level of revenue that
equates the position of incumbent and entrant and should be equal to the price
derived in the Efficiency Component Pricing Rule (ECPR), that is, operating costs
plus opportunity cost. The first contribution of this paper to the literature is that we
show that this opportunity cost should be based on the risk-free rate of return. The
second contribution is that we show that the access price can differ from one entrant
to another, depending on the level of revenue at the time of entry. At levels of
revenue above the optimal level that triggers entry, the entrant should pay a premium
above risk-free ECPR that rewards the incumbent for relinquishing his rights to the
risky cash flows at the higher revenue level.
Most economists agree that regulation should be applied only when
significant market power can lead to unregulated prices well above prices that would
prevail in conditions of full competition. Thus, natural monopolies are the natural
habitat for regulators, whose general goal should be to maximize present and future
social welfare. The tendency in regulation has been to liberalize the final goods
(service) market so that prices in this market are determined by the laws of supply
and demand and to concentrate regulation on the price to be paid by final good
(service) suppliers for access to the monopoly network. Traditional studies of
4
network access pricing, such as Averch and Johnson (1962) for rate of return
regulation (ROR), Laffont and Tirole (1994), Laffont, Rey and Tirole (1998a and
1998b) and Baumol and Sidak (1994) for marginal cost pricing and Laffont and
Tirole (2000) for price cap regulation, are carried out in conditions of certainty and
fail to consider sunk costs and the irreversiblity of many natural monopoly network
investments. Non-traditional studies based on real option theory, such as Economides
(1999) and Hausmann (1999) for the telecommunications industry and Clark and
Mondello (2000) for water management, explicitly incorporate the role of uncertainty
and sunk costs. Economides and Hausman deal with an oligopoly and underline the
role of depreciation in determining the optimal access price while Clark and
Mondello consider a natural monopoly. They point out the role of exit and re-entry in
determining a socially optimal revenue interval. The main premise of all these studies
is the need to incorporate the role of investment on the quality and supply of the final
good (service) and also consider the irreversible nature of many investments and the
effect of the uncertainty on the investment decision.
The problem is how to account for uncertainty and sunk costs while
maintaining incentives for final goods (service) suppliers. If the access price is too
low to cover sunk costs, the network provider will not have the incentive to maintain
the network, thereby reducing future social welfare by reducing the supply of future
final goods (services). If the access price is too high, final goods (service) providers
will not find it worthwhile to enter or stay in the market, thereby reducing current
social welfare through a reduction in competition.
5
In this paper, using standard techniques in stochastic calculus under a set of
mild assumptions1, we argue that the optimal access price is the certainty equivalent
of expected risky project cash flows that rewards the network provider for
undertaking the risky investment. We show that this price can be calculated as the
solution to a conventional control problem in capital budgeting where the access
price, rather than revenue, is the object of the optimisation. We also show that when
entry is an ongoing process, estimating the optimal access price in this way
guarantees that no advantage can be gained by timing the entry date. The optimal
access price rewards the incumbent for relinquishing his rights to the risky cash flows
at the higher income level. It also offsets what would otherwise be the entrant’s
windfall for entering at a moment when revenue is higher than the optimal entry
level. The situation is analagous to that of an entrant who enters by buying out an
existing supplier. The price he would have to pay would reflect the current level of
revenue. Thus, eliminating the windfall has the desirable effect of maintaining
competition in the final goods market, since it ensures that no entrant is advantaged
or disadvantaged with respect to other final goods suppliers as well as with respect to
the incumbent.
We begin our analysis by examining the network investment decision when
final goods (services) prices are welfare maximizing, that is, competitively
determined.2 In such a case the relevant issue is the timing of the investment and the
level of revenue that triggers it. In Section III we use this analysis to determine the
optimal access price when the entrant has to interconnect to the natural monopoly
network provided by the incumbent. Section IV concludes.
The assumptions refer to the application of Ito’s Lemma and the solution of the resulting differential equation which
require that the function be from C2 , that is a twice differentiable function with the second derivative continuous.
1
6
II. Real Options and Optimal Access Price: The Model
In the present section we analyze the monopolist's decision to invest, where
we assume that prices for final goods (services) are determined in a perfectly
competitive market. We consider such a scenario not as a representation of reality,
since we are dealing with a natural monopoly, but rather to establish a benchmark
that we then use as the basis to determine the optimal access price associated with the
risky investment.
We use the following notation:
I = the investment outlay
P = price per unit
D = total demand
C = c1  c2  c3 = operating cost where
c1 = operating costs for supplying the final good
c2 = operating costs for operating the network
c3
= depreciation defined as the loss of investment value due to usage and the
passage of time.3
We also make the following assumptions:
1. The investment outlay is constant, which means that the current period's cost of
undertaking the investment is equal to the cost of the original investment.
2. The investment scale is constant and optimal.
3. Costs are constant.4
Assumptions 1, 2 and 3 are made to facilitate the exposition and are not necessary for
the demonstration.
4. The final goods market is perfectly competitive. From Newberry (1997), we know
that competition in the final goods market leads to a welfare-maximizing price. This
assumption eliminates the possibility of competition stifling behaviour such as that
2
See Newberry (1997).
In accounting practice depreciation allowances are often influenced more by tax considerations than by the actual loss of
the asset’s value. In this paper we are interested in the economic cost associated with the reduction in the value of the
investment due to usage and the passage of time. See Economides (1999) and Hausmann (1999) for a discussion of this
point in the context of the telecommunications industry.
4
Stochastic operating costs can easily be included. However, it complicates the analysis without
enhancing the understanding of the issues.
3
7
embedded in first mover advantage and makes it possible to focus on the aspect of
network supply and the access price.5
5. Price evolves stochastically through time. Since prices cannot be negative, we let P
follow a process of geometric Brownian motion with drift to reflect the stochastic
innovations as well as any long-term trend in price evolution:
dP(t )  P(t )dt  P(t )dz (t )
where 
(1)
is the expected rate of price growth,  is the standard deviation of
dP(t ) / P(t ) and dz(t ) is a Wiener process with zero mean and variance equal to dt .
6. Since demand includes a stochastic element and cannot be negative, we let D
follow a process of geometric Brownian motion to reflect the stochastic innovations.
The drift term is equal to zero because the investment scale is constant:
dD(t )  D(t )dw(t )
(2)
The parameter w is the standard deviation of the expected percentage change in
demand. The variable dw is a standard Weiner process with zero mean and standard
deviation of dt. The relationship between D and P is reflected in dwdz  dt where
 1    0 is the instantaneous correlation between D and P. The negative
correlation implies a downward sloping demand curve.
II.A The Value of the Investment
Let x (t ) represent the revenue from the investment project calculated as the
unit price multiplied by the number of units sold:
x(t )  P(t ) D(t )
(3)
Using Ito’s lemma and equations 1 and 2, the evolution of x (t ) can then be written
as:
dx  xdt  xds
(4)
where the time arguments have been dropped for ease of notation and
  (   ) ,    2   2  2
5
and ds 
dw  dz
   2  2
2
.
As argued in Dixon and Easaw (2001) it is essential to remove any first mover advantage to enable a
fully liberalized final goods market that ensures welfare maximizing outcomes.
8
Market equilibrium is a prerequisite for optimal resource allocation, where
each asset just earns its risk adjusted required rate of return. The market equilibrium
risk adjusted required rate of return on x (t ) can be found by applying the continuous
time CAPM directly to x (t ) . The required rate of return will be given by
  r  xm
(5)
where r is the riskless rate of interest,  is the market price of risk,  x ,m is the
correlation coefficient of the percentage change in x (t ) with the market rate of return
and    . Let       0 , where  represents the known dividend rate, which
is given by policy.6
Net revenue is equal to x  C 7 and the value of the investment project,
V ( x (t )) , can then be found by setting up a hedge portfolio with a long position of one
unit of the investment and a short position in V ( x (t )) units of x (t ) . Using standard
methods in stochastic calculus gives the following differential equation:8
2
V ( x) x 2  (r   )V ( x) x  rV ( x)  x  C  0
2
(6)
whose general solution is9:
V ( x) 
x


C
 A1 x  1  A2 x  2
r
(7)
where  1  1 (because   0 ) and  2  0 (because r  0 ) are the roots to the quadratic
equation in  :
 1, 2 
 (r   
2
2
)  (r    ) 2  2 2 r
2
2
2
Using the boundary conditions in Appendix 1 gives
V ( x) 
6
x


C
r
(8)
An alternative, equivalent procedure involves applying the Girsanov theorem to obtain the risk
neutral martingale measure. In this case, if we assume that  is known or given by policy, 
becomes *  r   .
7
Note that the cash flow is net of depreciation, which is included in the cost. This implies that the cash
flows accruing to depreciation are reinvested, thereby guaranteeing the perennity of the investment.
8
We drop the time arguments to simplify the notation.
9
Notice that the second member of the left hand side of equation 6 includes the risk neutral growth rate *  r   ,
which was mentioned in footnote 7. This risk neutrality is the result of the hedging operations and requires that all
discounting be done at the risk-free rate r when there is no growth and at   r   * when there is growth.
9
Equation (8) says that the value of the investment project is equal to the
present value of the expected net cash flows where revenue (x) is discounted at the
rate   r   * and cost (C) is discounted at the rate r.
II.B The Value of the Option to Invest
We now turn to the investment decision and the value of the option to invest.
We want to know the value of the total investment opportunity of size I , given
assumptions 1 through 6. The investment is undertaken if the level of revenue
associated with the project warrants it, that is, when x reaches the threshold, noted as
x*, where more is to be gained from investing than from waiting. This is the wellknown option to defer (see for example, Dixit and Pindyck, 1994 and Trigeorgis,
1996), whose value, derived in Appendix 2 and noted as F ( x (t )) , is given as:
F ( x)  B1 x  1
(9)
Equation (9) represents the value of the option to undertake the investment if
it were possible for the firms supplying in the final goods market to provide their
own network at constant returns to scale. The option will be exercised and the
investment undertaken when the revenue to be realized from the investment reaches
x*, determined in Appendix 2 as equal to:
x* 
 1  C 
 I
 1  1  r

(10)
This, then, is a description of the monopolist's decision to invest if the prices
in the final goods (services) market are those that would prevail in conditions of
perfect competition. Equation (10) brings out the role of uncertainty in the
investment decision.10 If investment decisions do not account for uncertainty, the
investment would be undertaken according to equation (8) when V ( x)  I . We
denote x * * the level of x that just makes the investment acceptable, that is, where
V ( x * *)  I . From equation (8) this gives
C
  1
x **     I   1
x*
1
r

10
For an in-depth treatment of this point see Dixit and Pindyck (1994 pp 140-147).
(11)
10
Comparing equations (10) and (11) shows that x*  x * * by a factor due to
uncertainty of  1 ( 1  1) (since  1  1 )11. Thus, x** is the certainty equivalent of x*
and the ratio  1 ( 1  1) is the factor for risk adjustment.
We now turn to the case where network provision is a natural monopoly with
a fully deregulated and liberalized final goods market. The incumbent has already
undertaken the investment and the entrants interconnect by paying an access price.
III. Determining the Optimal Access Price
When there is a natural monopoly for network supply and the investment has
already been undertaken, the problem is similar to the one described above but the
focus changes from the optimal timing of the investment to determining the optimal
access price. Consider the position of the incumbent who has undertaken the
investment along with all the associated risks. If he cedes access to his investment,
the access price, which is a fixed amount, should include compensation for the risks
he has undertaken up to the time of cession. By the same token, the entrant will be
taking on these risks and should be rewarded for doing so. In effect, the incumbent is
exchanging his risky cash flows for a certainty equivalent, which was determined in
preceding section. Thus, the problem now is to find the amount the entrant must pay
that satisfies the certainty equivalent requirement for the incumbent but does not
deter entry.
The position of the entrant is analogous to that of the potential investor in the
preceding section. Since he is under no obligation to enter the market, he has an
option to defer entrance until it is in his best interest to do so. However, whereas the
investment decision described above depended on the minimum level of revenue (x =
x*) necessary to give up the option and undertake the investment, the entrant’s
decision depends on the access price he will have to pay if he decides to enter. If the
price is too high, he will not want to enter. On the other hand, if it is too low, the
11
Thus, when there is uncertainty, the level of revenue (and profit, since costs are assumed constant)
must be higher to justify the investment. Since  1   0 , the uncertainty factor increases with
volatility, which means that the required amount of revenue increases as uncertainty increases.
11
incumbent will not want to provide access.12 The problem is to determine the price
that satisfies both parties. To do this, we model the position of the entrant.
Suppose that there are n potential entrants, each one willing to purchase a i
percent of the network capacity with i  1,2 n and
n
a
i 1
i
 1 . In the absence of
access payments the ith entrant would have neither the costs associated with running
the network nor network depreciation costs. Thus, his revenue would be
ai x  C  (c2  c3 ) , which is equal to ai ( x  c1 ) . Let vi (x) represent the present
value of the entrant’s cash flows in the absence of access payments. Going through
the same steps as before gives
x c
vi ( x )  a i (  1 )
 r
(12)
Let f i (x) represent the value of the ith entrant’s option to enter and ai M T the
outlay or total access price to be paid by the entrant, where M T is the total payment
n
to the incumbent for ceding
a
i 1
i
of his network capacity. We can think of it as the
present value of all the instantaneous payments to be paid by the entrants. Going
through the same steps as before gives

f i ( x)  ai H 1 x  1  H 2 x  2

(13)
where  1  1 and  2  0 are the same as above.
The first boundary condition is based on the argument stated in Appendix 1
that when revenue is equal to zero, the investment opportunity has no value:
f i (0)  0
(14)
This implies that H 2  0 .
The second boundary condition, called the value matching condition, relates
to the fact that when the entrant contracts to pay the access price, he acquires the
rights to the associated cash flows. The present value of these cash flows is given in
equation (12). The present value of the instantaneous access payments is given as
ai M T . Thus, from the family of boundary conditions BC , the value matching and
smooth pasting conditions at ai M T are given as:
12
If the incumbent is forced to provide access at a price that is too low, he will not have the incentive
12
f i ( xT )  vi ( xT )  ai M T
BC M T :
(15a)
and
fi( xT )  vi( xT )
(15b)
Solving simultaneously gives the particular solution to equation (13) as
f i ( x)  ai H 1 x  1
(16)
where  1 is given above,
H1 
xT 
and
1
 1
xT
1 1
(17)
 1  c1

 MT 

 1 1  r

(18)
The level of revenue necessary to trigger entry depends on M T , the total cost
n
of access to be paid by the entrants to the incumbent for ceding
a
i 1
i
of his network
capacity. Rearranging equation (18) gives M T in terms of xT :
MT 
xT   1  1 c1

   1  r
(19)
In equation (10) we determined the minimum level of revenue to justify
undertaking the investment. Thus, at levels of xT  x * , the investment would not be
justified because although the entrants would be satisfied, the payout to the
incumbent would be insufficient. To see this start with M T  M * , the total access
payment that would trigger entry at x*. From equation (19) this gives
M* 
x *   1  1 c1

   1  r
(20)
Notice that M* includes the certainty equivalent of x* defined in equation (11) as x**
= x* ( 1  1) /  1 . This is as it should be since the incumbent is trading his risky cash
flows for a constant access payment. In fact, in exchange for ceding the right to the
to maintain the network.
13
risky cash flows to the entrant, the incumbent receives M* plus the present value of
the operating costs he saves: c1 / r . Substituting the value for x* from equation (10)
into equation (20) and adding the present value of savings in operating costs c1 / r
gives the incumbent's total value received from ceding operations to the entrant as
M *
c  c3 c1
c1
C
I 2
 I
r
r
r
r
(21)
We recognize this as the certainty equivalent for x * /  in equation (11), which
proves that M* satisfies both incumbent and entrants. The instantaneous payout to the
incumbent by the entrants is rM *  rI  c2  c3 , the return on the investment plus the
cost of operating the network plus depreciation. Since M* is a certainty equivalent,
the investment earns the riskless rate. Thus, this is the minimum payment that the
incumbent could accept; the riskless rate of return on his investment plus
depreciation and the costs of operating the network. It is easy to verify that at
xT  x * the payment to the incumbent would be insufficient by adding and
subtracting
x *   1  1
from equation (19) and rearranging. This gives
   1 
MT 
xT  x *   1  1 x *   1  1 c1





 1    1  r
(22)
which, because xT  x * is negative, is less than the minimum given in equation (20).
Now suppose that xT  x * , which could happen if entry can take place at
different points in time or if it is an ongoing process. An access cost of M * would
provide the entrant with a windfall profit. He would be willing to pay M T but would
only pay M * . Subtracting equation (20) from equation (19), gives a windfall profit
equal to
MT  M * 
xT  x *   1  1


  1 
(23)
The incumbent, however, would not accept M* because the certainty equivalent of
   1
   1
xT is xT  1  , which is higher than x *  1  the amount he would receive at
 1 
 1 
14
M*. In fact the incumbent would require a value of M T 
xT   1  1 c1
 . We can
   1  r
express this in terms of the certainty equivalent at M*
MT 
xT  x *   1  1 x *   1  1 c1





 1    1  r
(24)
The last two terms on the right hand side of equation (23) are equal to the
certainty equivalent M* estimated at x*. The first term is the present value of the
difference in revenue due to entry at xT weighted by the reciprocal of the uncertainty
factor 0  ( 1  1)  1  1 . The incremental payout rewards the incumbent for
relinquishing his rights to the risky cash flows at the higher revenue level. It also
offsets what would otherwise be the entrants’ windfall for entering at a moment when
revenue is higher than the optimal entry level. If the price were the same for all
entrants, regardless of the level of x at the time they entered, a higher entry level of x
for one entrant would present him with a windfall gain with respect to those who
entered at a lower level of x.
IV. Concluding Remarks
In this paper, we study optimal access pricing for natural monopoly networks
that incorporates uncertainty about the future revenue to be derived from providing
the final good or service and the fact that many natural monopoly network
investments include a large element of sunk costs. We show that the optimal access
price is the certainty equivalent of expected risky project cash flows that rewards the
network provider for undertaking the risky investment. It turns out that this price
corresponds to the risk free form of the Efficiency Component Pricing Rule, that is,
operating costs plus opportunity cost estimated at the risk-free rate of return. We also
show that if entry occurs at a level of revenue above the optimal level that triggers
entry, the entrant should pay a premium above risk-free ECPR that rewards the
incumbent for relinquishing his rights to the risky cash flows at the higher revenue
15
level. It also offsets what would otherwise be the entrant’s windfall for entering at a
moment when revenue is higher than the optimal entry level and has the desirable
effect of maintaining competition in the final goods market, since it ensures that no
entrant is advantaged or disadvantaged with respect to other final goods suppliers as
well as with respect to the incumbent.
16
REFERENCES:
Alleman, J and Noam, E, (eds), 1999, The New Investment Theory of Real Options
and its Implications for the Costs Models in Telecommunications, Kulwer.
Clark, E. and Mondello, G., (2000), "Water Management in France: Delegation and
Irreversibility", Journal of Applied Economics, Vol. 3, no. 2, pp. 325-352.
Dixit, A and Pindyck, R., 1994, Investment under Uncertainty. Princeton: Princeton
University Press.
Dixon, H. and Easaw, J. , 2001, “Strategic Responses to Regulation Policy: Are there
lessons to be learnt from the UK Contract Gas Market, 1986-1996”, Review of
Industrial Organization, Vol. 18, pp 379-396.
Economides, N., 1999, “Real Options and the Costs of the Local
Telecommunications Network,” in Alleman, J and Noam, E, (eds), The New
Investment Theory of Real Options and its Implications for the Costs Models in
Telecommunications, Kulwer.
Hausman, J., 1999, “The Effect of Sunk Costs in Telecommunications,” in Alleman,
J and Noam, E, (eds), The New Investment Theory of Real Options and its
Implications for the Costs Models in Telecommunications, Kulwer.
Newbery D., 1997, “Privatization and liberalization of network utilities”,
Presidential Address, European Economic Review, 41, pp 357-383.
Trigeorgis,
L.,
1996,
Real
Options,
Cambridge:
MIT
Press.
17
Appendix 1
The general solution to (6) is:
V 
x(t )


C
 A1 x(t )  1  A2 x(t )  2
r
(A1.1)
The constants A1 and A2 depend on the boundary conditions. The first boundary
condition in straightforward. When revenue is equal to zero, the investment has no
value:
V ( 0)  0
This condition implies
(A1.2)
A2 = 0. If we rule out speculative bubbles, the second
boundary condition is:
V ()  
(A1.3)
which implies that A1  0 . Thus, the solution to (A1.1) is that given in the text.
Appendix 2
In order to value this option, noted as F ( x (t )) , we follow Dixit and Pindyck
(1994) and set up a hedge portfolio with one unit of the option to invest and a short
position in F ( x (t )) units of x (t ) . Using standard techniques yields the following
differential equation:
2
F x 2  (r   ) F x  rF  0
2
(A2.1)
The general solution to (A2.1) is:
F  B1 x  1  B2 x  2
(A2.2)
When revenue is zero, the option has no value. Thus, the first boundary
condition is:
F (0)  0
(A2.3)
which implies that B2  0 .
The second boundary condition depends on revenue and the cost of exercise.
There will be a value of x (t ) , noted x *, where it will be optimal for the potential
investor to exercise the option. At this point he will receive the value of the
investment less the cost of exercising the option. This cost is equal to the investment
outlay. Thus, the value matching condition is
F ( x*)  V ( x*)  I
(A2.4)
18
where I is the exercise price, i.e. the investment outlay.
The smooth pasting
condition that makes it possible to find x * jointly with F ( x (t )) is:
F ( x*)  V ( x*)
(A2.5)
Solving equations (12) and (13) simultaneously gives the solution in the text where
x* 
and
 1  C 
 I
 1  1  r

B1 
1

x *1 1
(A2.6)
(A2.7)