WPS4198
The Pricing Dynamics of Utilities with
Underdeveloped Networks
Omar O. Chisari, Universidad Argentina de la Empresa
Ioannis N. Kessides, World Bank1
ABSTRACT
This paper employs an analytically tractable intertemporal framework for analyzing the dynamic
pricing of a utility with an underdeveloped network (a typical case in most developing countries)
facing a competitive fringe, short-run network adjustment costs, theft of service, and the threat of
a retaliatory regulatory review that is increasing with the price it charges. This simple dynamic
optimization model yields a number of powerful policy insights and conclusions. Under a variety
of plausible assumptions (in the context of developing countries) the utility will find its long-run
profits enhanced if it exercises restraint in the early stages of network development by holding
price below the limit defined by the unit costs of the fringe. The utility's optimal price gradually
converges toward the limit price as its network expands. Moreover, when the utility is threatened
with retaliatory regulatory intervention it will generally have incentives to restrain its pricing
behavior. These findings have important implications for the design of post-privatization
regulatory governance in developing countries.
World Bank Policy Research Working Paper 4198, April 2007
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the
exchange of ideas about development issues. An objective of the series is to get the findings out quickly,
even if the presentations are less than fully polished. The papers carry the names of the authors and should
be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely
those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors,
or the countries they represent. Policy Research Working Papers are available online at
http://econ.worldbank.org.
1We would like to thank Daniel Benitez, Eric Groom, Atsushi Iimi, Michael Klein, Luis
Serven, and Maria Vagliasindi for very helpful comments.
1. Introduction
An important post-reform issue confronting policy makers and regula-
tory authorities in developing countries is how much pricing flexibility to
accord to their privatized utilities to eliminate service backlogs. In many of
these countries, effective demand for infrastructure services at current prices
continues to substantially exceed supply, resulting in long waiting lists for
service. Under these conditions, the utility's profit maximization problem
is a dynamic optimization--to expand capacity at a profit-maximizing rate
given short-term adjustment costs and potential diseconomies of scale to ca-
pacity expansion (marginal costs of network expansion are likely to increase
as the utility reaches rural and low density areas). In formulating its vari-
ous policies, including those with respect to pricing, the utility has to take
into account both the current-profit and future-profit effects of those policy
determinations. The two effects are generally competing with each other.
For example, a price increase that is favorable to current profit will entail a
sacrifice of future profits because it is likely to slow network expansion which
in turn would generate future profits for the utility. Alternatively, the util-
ity might choose to price low initially and thus sacrifice current profits for
the sake of accelerating the expansion of its network and reap higher profits
in the future. At the optimum, the utility's policy choice must balance the
marginal increase (decrease) in current profits induced by its policy choice
against the marginal decrease (increase) in future profits that the policy will
entail. In the face of significant network expansion opportunities and under
a variety of plausible assumptions regarding the nature of technology, costs
and demand facing the infrastructure entities, dynamic optimization will
lead to low pricing (relative to the short-run profit maximizing monopoly
level) in the early stages of network development. Under those conditions
there would be a strong rationale for light-handed regulation--i.e., for grant-
ing the infrastructure entities substantial pricing flexibility during the early
stages of network development.
There is another strong rationale for light-handed regulation and for de-
centralizing pricing decisions to the firm level during the early post-reform
years. The pursuit of pricing and other regulations to elicit optimal in-
dustrial performance in many developing countries is hindered by the lack
of proper accounting systems and by the dearth of information on mar-
ginal costs, demand elasticities, and other pertinent attributes of demand
and cost relationships. Under the traditional command-and-control regula-
tory model, any prices calculated without such information are apt to be
3
inconsistent with economic efficiency and damaging to economic welfare.
The information available to the firms themselves is also highly imperfect
in many developing countries. Still, it seems likely that the firms will have
better and more up-to-date estimates of cost and demand conditions than
the regulators (Baumol and Sidak 1994). How can regulators in developing
countries acquire a realistic chance of becoming effective in the face of such
severe information problems? One promising policy direction would be to
decentralize the decisions on pricing and other key variables to the firms that
have the necessary information. The role of the regulator would be limited
to imposing floors and ceilings on prices (to protect against predation and
monopolistic pricing). The price-determining process then would be left to
the firm, which would be free to select prices within the floor-ceiling lim-
its. In the face of significant network expansion opportunities the firm's
self-interest would motivate it to adopt prices that best serve the public in-
terest. Such a framework could meet the dual policy objectives of giving the
infrastructure entities an opportunity to obtain adequate revenue to sup-
port needed investment (an important issue in many sectors in developing
countries) and protecting consumers from monopolistic pricing.
This paper presents a dynamic model of a utility in a sector with a low
coverage ratio, thus facing significant network expansion opportunities but
also the prospect of entry or expansion by a competitive fringe. The rate of
expansion of the utility's (or equivalently the fringe's) network is assumed
to vary continuously with the price set by the utility--the higher the utility's
price is the more customers will join the fringe and hence the slower will be
the expansion of the utility's network. Also, the utility faces the threat of
retaliatory regulatory intervention that is an increasing function of the price
it charges--if such regulatory intervention takes place the utility's profit is
suppressed to a level that is very low relative to its short-run profit maxi-
mizing level. Moreover, a portion of the customers that leave the utility's
network because of higher prices may engage in service theft--a plausible
assumption in the context of many developing countries. The utility max-
imizes its long-run profits by balancing the impact of its pricing policy on
current profits against the impact of such policy on the size of its network
and the probability of regulatory intervention and hence its future profits.
The utility's optimal price path depends upon its cost advantage relative to
the fringe, the speed with which existing and potential customers respond
to the price differential between the utility and the fringe, the discount rate,
the nature of the adjustment costs that it faces with respect to network ex-
pansion, and the character of regulation. When the utility has a unit cost
4
advantage over its fringe competitors, it sets low tariffs initially in order to
develop its network more rapidly and thus reap larger revenues and profits
in the future. As the size of the utility's network approaches its asymptotic
optimum, it raises its price toward the limit defined by the unit costs of the
fringe.
5
2. The Basic Model
A common characteristic of developing countries is that a significant por-
tion of their population, especially in periurban and rural areas, is without
access to basic services. Thus, their network utilities are characterized by
low coverage ratios. We consider an infrastructure sector that has not as yet
achieved full coverage. A single utility firm dominates the sector serving
N out of M potential customers. The remaining M - N consumers are
either being served by alternative service providers (fringe), have no access
at all, or a certain portion of them may actually steal the service from the
dominant utility through illegal connections. The services offered by the
dominant utility and the fringe are assumed to be interchangeable in terms
of their utility to the consumers.2
The service arrangements of the customers not connected to the utility's
formal network vary by sector and country. In Yemen's electricity sector,
for example, small generators supply rural towns and villages not served by
the public utility. Operations range from individual households generating
power for themselves and a few neighbors to units supplying up to 200
households (Ehrhardt and Burdon 1999). Electricity theft through illegal
connections is a serious problem in many developing countries, especially in
the inner city communities and urban areas. Electricity theft accounted
for over one-quarter of total production in 1992 when privatization reforms
were initiated in Argentina (Haselip 2004). In Jamaica, in excess of 30,000
illegal connections were removed each month during 2003, allowing for a
monthly reduction of 4.5 MWh of electricity production (Jamaica Public
Service Company 2004). A 1989 study of the water supply in the Nigerian
town of Onitsa found that water vendors delivered more water than the
public water utility and that revenues for vended water exceeded revenues for
water supplied by the public utility by a factor of more than 10 (Whittington
et al 1991). These results indicated significant opportunities for expansion
by the public utility.
2This assumption of service homogeneity is likely to be violated in many real-world
circumstances. Clearly, the characteristics of the services offered by the dominant utility
can differ from those offered by the fringe on several dimensions of quality. Still, in
the context of many developing countries where large segments of the population remain
without access to basic services, costs rather than product differentiation are likely to be
the main driving force. This paper, therefore, focuses more on the nature of production
technology and costs facing the fringe rather than the properties of the demand function.
6
Let P denote the average revenue generated by the utility from each
customer unit and (N) represent the minimum unit cost of the fringe ser-
vice providers. We assume that the dynamics of network expansion are
characterized by the differential equation
N = [ (N) - P] (1)
where N is the rate of the dominant utility's customer base expansion
and the coefficient reflects the speed at which existing and potential cus-
tomers respond to the price differential between the dominant utility and
the fringe. Equation (1) implies that the rate at which customers join (or
leave) the dominant utility's network varies continuously with its current
price-- (N) is the limit price, i.e. that price level at which the dominant
utility experiences zero network expansion. Charging less than the unit
cost of the fringe (N) represents an investment in creating a larger network
of customers which could yield dividends in terms of larger future revenues
and profits. Pricing above the unit cost of the fringe, on the other hand,
causes exit of customers from the utility's network and the loss of profitable
network expansion opportunities.3
Let C denote the utility's total cost of serving its network of N customers.
We allow for the possibility that a portion of the M-N unserved customers
will illegally connect to the utility's network. We further assume that
C = F + c(N,(M - N)) (2)
where F 0 represents the utility's fixed network costs and c, its variable
cost function, is twice differentiable with cN > 0 and cM-N 0.
Intertemporal profit maximization
The dominant utility's long-run profit-maximization problem can be
solved using the techniques of optimal control theory. The firm chooses
the price P(t)--control variable-- to maximize its profit performance func-
tional subject to some dynamic constraints. These constraints describe the
3The dynamics of network expansion described by (1) are similar to the entry equation
used by Gaskins (1971) and the vast literature that followed.
7
evolution of the structure of the utility's network, as represented by the state
variable N(t),over time. Thus, the utility's long-run profit-maximization
problem is given by
max (0) =
P(t) 0 e-rt PN - C(N) - N 2 dt (3)
subject to
(a) N = [ (N) - P],
(b) N(0) = N0, given,
(c) lim N(t)e-rt = 0.
t
where (0) is the value of the utility's profit function at time 0, r is the
discount rate which is assumed to be constant over time, and the term N 2
represents the adjustment costs the utility must incur in changing the size
of its network (Brock and Dechert 1885; Hamermesh and Pfann 1996).
The expansion constraint in (3a) is the transition equation or equation
of motion, and shows how the choice of the control variable, P(t), translates
into a pattern of movement for the state variable, N(t). Equation (3b) sim-
ply states that the state variable N(t), representing the size of the utility's
network, starts at time 0 with a given value N0.
The present value Hamiltonian of this problem is given by:
H(N,P,t,) = e-rt(P,N) + (t) [(N) - P] (4)
where (P,N) = PN - C(N) - 2 [(N) - P]2 and (t), is the La-
grange multiplier associated with the constraint in Eq. (1). The Lagrange
multiplier can be interpreted as a shadow price--(t) is the imputed value to
the utility of an extra customer at time t. The economic interpretation of
the Hamiltonian in this problem is quite straightforward. The first compo-
nent of the right hand side of (4) is simply the profit function at time t which
is dependent on the utility's pricing decision and the size of its network at
that time. The second component of (4) represents the rate of change of
network size which is converted to a monetary value when it is multiplied
8
by the shadow price . At every instant in time, the utility chooses a price
P(t) and has a network with N customers. These two variables affect the
utility's objective function (profit) through two channels. First, the direct
impact of P(t) and N on the utility's profit function is captured by the first
term in (4). Second, the price P(t) chosen by the utility affects the change
in the size of its network in accordance to the transition equation for N in
(1). The value of this change is captured by the second term in (4). Thus
for a given value of the shadow price , the Hamiltonian captures the total
impact on the utility's profit function from the choice of P(t).
In terms of the present value Hamiltonian, the first-order necessary con-
ditions for the utility's intertemporal maximization problem are given by:
HP = e-rt N + 22 [(N) - P] - = 0 (5)
HN = e-rt P - C (N) - 22 [(N) - P] (N) + (N) = - (6)
These first-order conditions represent Pontryagin's Maximum Principle
(Pontryagin et al 1962).
Lemma 1. If marginal network costs are increasing and (N) 0,
then the first-order conditions (5) and (6) are sufficient for maximization iff
22 C (N) - 2 (N) - 1 > 0.
PROOF. According to the Mangasarian Sufficiency Theorem (Man-
gasarian 1966), the necessary conditions of the maximum principle are also
sufficient for the global maximization of the objective function (0) if the
Hamiltonian H is concave in (N,P). From (5) and (6), differentiation with
respect to N and P yields:
HNN = e-rt -C (N) - 22 (N) - 22 [(N) - P] (N) + (N)
2
(7)
HPP = e-rt(-22) (8)
HPN = HNP = e-rt(1 + 22 (N)). (9)
9
Solving for in (5) and substituting into (7) yields:
HNN = e-rt -C (N) - 22 (N) + N (N)
2 e-rt -C (N) - 2(10)(N) .
2 2
If marginal network costs are increasing (i.e. C (N) > 0), then HNN <
0. Also, from (8) it is clear that HPP < 0.Moreover, from (8), (9), and (10)
we obtain
HNNHPP - HNP = e-2rt 22 C (N) - 2 (N) - 1 .
2 (11)
Thus, if 22 C (N) - 2 (N) - 1 > 0, then the Hessian of H is
negative definite (HNN < 0,HNNHPP - HNP > 0), i.e. the Hamiltonian H
2
is concave in (N,P). and the Mangasarian sufficiency conditions are met.
The Hamiltonian and the maximum principle that requires its maximiza-
tion with respect to P can obtain a more intuitively appealing economic
meaning if we define the current value Hamiltonian
Hc = (P,N) + c(t) [(N) - P] (12)
where c is the current value Lagrange multiplier. At the optimum, we
obtain from the first-order conditions for maximization of Hc
= 0 . (13)
P - r - (N) N
The first term in (13) measures the marginal increase in the current profit
of the utility that is induced by an increase in its price. The second term
represents the marginal decrease in future profits that such a price increase
will induce via the change in the size of the utility's network--an increase in
the utility's price reduces the price differential between the utility and the
fringe, leading to a loss of customers to the fringe and a slower expansion
in the utility's network. This effect will be larger the higher the speed
10
with which existing and potential customers respond to the price differential
between the utility and the fringe. If the utility is myopic and discounts
future profits very heavily (r is large), then it will not weigh very much in
its pricing decision the impact of such a decision on the future size of its
network. The utility will increase its price as long as the marginal gain in
current profit made possible by such an increase is greater than the marginal
decrease in future profit that its pricing decision will induce via its effect on
the size of its network. The utility's optimal choice P must balance these
two effects.
Explicit solutions for N,P, can be found by specifying C(N) and (N).
In general, it costs more to provide utility services to rural than to urban
communities. The higher rural investment costs--measured as capital in-
frastructure costs per unit of service (e.g. kilowatt hour in the electricity
sector) consumed--largely reflect the lower density of rural connections. In
the electricity sector, for example, these costs depend on the community's
distance from the existing medium voltage grid, and on the community's
size and potential demand pattern. Investment costs per unit of demand
are higher in rural areas because the bulk of demand in rural areas is for
lighting during the early evening--the ratio of average demand (which deter-
mines financial and economic benefits) to peak demand (which determines
investment cost) is much lower in rural systems than in urban, where there
is considerable daytime electricity use. Many, if not most, developing coun-
tries are characterized by low coverage ratios and relatively underdeveloped
networks with infrastructure services mainly being offered in the urban ar-
eas. In these countries, it can be reasonably assumed that the marginal costs
of network expansion are increasing with the size of the network. Thus, we
adopt a functional form for the dominant utility's cost function C(N) that
is quadratic in the size of the network N
C(N) = F + aN + c[M + (1 - )N]2 (14)
where a > 0 and c > 0.
We assume that the production function of the fringe exhibits increasing
returns. This can be captured by positing that the minimum unit cost of
the fringe service providers, (N), is increasing with N. Moreover, as the
size N of the formal network increases, the pool of the unserved customers
M - N becomes more costly to serve (assuming that the dominant utility
expands service to low-cost customers first). Thus we set
11
(N) = 0 + 1N (15)
where 1 > 0. We differ here from Gaskins (1971) where the fringe is
assumed to have constant average and marginal cost. Where the fringe rep-
resents a small company distributing water with a truck (like in Paraguay) to
periurban customers, it is reasonable to assume that because of the presence
of fixed costs (e.g. the cost of capital of the truck) unit costs will increase
as more customers migrate to the network of the dominant water utility--i.e.
the units costs of the fringe will increase with N. The same would apply
to a small distributed electricity company with a diesel generator offering
services to customers not connected to the grid. Thus, an increase in the
size of the formal network confers advantages to the dominant utility not
because of the presence of positive demand externalities [e.g. Arthur (1989)]
but rather because of the changes in the cost function of the fringe (which
has to distribute its fixed costs over a smaller customer base).
Differentiating (5) with respect to t and substituting for and into (6)
yields a second-order differential equation in N (Euler equation)
N - rN -
¨ r + 2c(1 - )2 - 21N 0 - a -22 c(1 - )M (16)
2 = -
whose solution is given by4
N(t) = (N0 - N)e-r(2 -1)t + N (17)
where N0 = N(0), = 1 + 2r+2 c(1-)2-21 1/2
, and
r2
N = r 0 - a - 2c(1 - )M (18)
+ 2c(1 - )2 - 21
4 The general solution of (16) is given by
N(t) = r 0 +Ae1t+Be2t where 1 and 2 are the roots of the quadratic
+2c(1-)2-21
-a-2c(1-)M
equation 2 -r- r+2c(1-)2-21 = 0. One of the two roots (1) is positive, and hence
2
it is rejected by the requirement that N(t) be bounded as t . The constant B is
determined by the initial condition N(0) = N0 as prescribed by (3b).
12
is the steady-state size of the network. It should be noted that if the
Mangasarian sufficiency conditions are met, then > 1.5
The optimal size of the utility's network is a function of its cost char-
acteristics and those of the fringe, the nature of its competitive interaction
with the fringe, the extent to which the utility can exclude customers for
nonpayment, and the discount rate. Simple comparative static analysis
indicates that the optimal size of the utility's network will be:
· bigger the larger is 0, the initial unit cost (price) of the fringe;
· smaller the larger is , its initial marginal cost;
· bigger the more rapidly the unit costs of the fringe increase as the
network absorbs more customers, i.e. the larger 1 is;
· smaller the higher is c, the utility's marginal cost;
· smaller the higher is the utility's discount rate r;
· bigger the higher is the portion of the unserved customers who ille-
gally connect to its network.
Equations (1) and (17) imply that the utility's optimal price path
is given by
P(t) = 0 + 1N + (N0 - N)(1 + - 1)e-r(2 -1 )t (19)
2
with a steady-state price level
P = 0 + 1 0 - a - 2c(1 - )M . (20)
r + 2c(1 - )2 - 21
5According to Lemma 1, the Mangasarian sufficiency conditions are met if
22 [C (N) - 2 (N)] - 1 > 0 = 1 < c(1 - )2
-42 which would also imply that
1
1 < c(1 - )2 + r since ,r and are positive. This would simply imply that
2
r+2c(1-)2-21 > 0 and thus = (1+)1/ > 1 since = 2r
2 +2c(1-)2-21 > 0.
r2
13
The utility's terminal price is an increasing function of 0 and 1 .
Thus, it sets a higher price as the unit and marginal costs of the fringe
increase.
Lemma 2. The steady state is a saddle point equilibrium.
PROOF. Differentiating (5) with respect to t and substituting for
and into (6) lead to the differential equation in P
P = rN + 22r [(N) - P] - [(N) - P] - 23 (N)[(N) - P] - P + C (N) - (N)N
(-22) (21)
which along with the differential equation for N in (1) forms a two-
equation system. To determine the properties of the steady-state equi-
librium we first form the Jacobian matrix of the two-equation system and
evaluate it at the steady-state point E =(N,P),
(N)
JE = r+22r (N)- (N)-23 (N)(-23 (N)+C (N)- (N)N- (N)
2 - .
-22) r - (N) (N,P)
(22)
The product of the two characteristic roots 1,2 is given by
12 = |JE| = - r + 2c(1 - )2 - 21. (23)
2
21 > 06 which implies that 12 < 0 and thus the steady state is locally
If the Mangasarian sufficiency conditions are met, then r+2c(1-)2 -
a saddle point (Chiang 1992).
The initial size N0 of the utility might reflect a variety of country-specific
characteristics and economic policies (e.g. the country's macroeconomic con-
dition, the socioeconomic characteristics of its population, policies related
to the structure of ownership and universal access). It can also be the
6Supra note 5.
14
unintended consequence of a host of misguided public policies (e.g. fail-
ure to prescribe cost-reflective tariffs, statutory restrictions on competition,
governmental restrictions on investment) towards the sector. In many de-
veloping countries, especially during the pre-reform era, coverage ratios were
very low. Large segments of the population remained with access to basic
infrastructural services despite willingness to pay for such services. If the
initial size N0 of the utility's network is small, then
N (0) = -r( - 1)(N0 (24)
2 - N) > 0
and
P0 = P(0) = 0 + 1N0 + (N0 - N) - 1 < 0 + 1N0. (25)
2
Thus, the utility's initial price P0 is lower than the limit price permitted
by the initial unit cost 0 +1N0 of the fringe--the utility prices initially low
(below the limit price afforded by the presence of the fringe) in order to
expand its network. The amount by which the utility sets its initial price
its initial and long-run optimal size) with a proportionality constant
below the limit price is proportional to N0 -N (i.e. the difference between. -1
2
Given that we defined above = 1 + 2r+2 c(1-)2-21 1/2
, it is easy to
r2
show that the amount by which the utility prices initially below the limit
price:
· increases with (0 -a)--when the dominant utility has a cost advantage
over its fringe rivals, is maximizes long-run profits by setting its initial
price below the limit price by an amount that is proportional to that
cost advantage
· decreases with the discount rate r--if the utility discounts future rev-
enues and profits very heavily then the strategy of charging less than
the unit cost of the fringe in order to create a larger network of cus-
tomers and reap the benefit of higher future revenues and profits is
clearly less appealing;
15
· decreases with , the speed with which existing and potential cus-
tomers respond to the price differential between the dominant utility
and the fringe--clearly the more responsive customers are to the price
differential the smaller will be the amount by which the utility will
need to underprice in order to attract a given number of customers to
its network;
· decreases with , the rate at which the utility's adjustment costs in-
crease with the speed with which it expands its network--the utility
will have less of an incentive to price aggressively below the fringe in
order to rapidly expand its network if the adjustment costs that it
incurs rise rapidly with the speed of expansion;
Through simple differentiation we obtain from (19)
P(t) = - - 1(N0 - 1)e-r(2 -1)t (26)
2 - N)(1 + 2
If N0 is small, then P (t) > 0, i.e. the utility follows a path of increasing
tariffs.
These results could have significant implications for the design of reg-
ulatory policy in developing countries where the utility networks are un-
derdeveloped and only a small portion of the population has access to basic
infrastructural services. To the extent that the utility's intertemporal profit-
maximization plan calls for a low initial price, there is no need to subject
it to a detailed regulatory scrutiny, at least with respect to pricing. This
is comforting because the pursuit of pricing and other regulations to elicit
optimal industrial performance in many developing countries is hindered, es-
pecially during the early stages of the reform process, by the lack of proper
accounting systems and by the dearth of information on marginal costs,
demand elasticities, and other pertinent attributes of demand and cost re-
lationships. Under the traditional command-and-control regulatory model,
any prices calculated without such information are apt to be inconsistent
with economic efficiency and be damaging to economic welfare. The infor-
mation available to the firms themselves is also highly imperfect in many
developing countries. Still, it seems likely that the firms will have better
16
and more up-to-date estimates of cost and demand conditions than the reg-
ulators. Therefore, under those circumstances it would be appropriate to
accord the utilities considerable pricing freedom to develop their networks
and expand service.
Pricing under endogenous regulatory threat
One of the most urgent tasks for policy towards infrastructure indus-
tries of developing countries, especially those that have implemented priva-
tization reforms, has been to improve the effectiveness of their regulatory
frameworks. Perhaps the most serious error besetting the process has been
the lack of organizational and financial independence of regulation from the
politicized arms of government. Indeed, many government entities in de-
veloping countries, such as sectoral ministries, have resisted giving up their
regulatory functions, especially those related to pricing. In the face of
political agendas that are endemically discretionary and subject to change
and reinterpretation, there can be no credible commitment to limit the dis-
cretionary powers of the regulatory apparatus. Under those institutional
conditions, there exists the danger of arbitrary administrative intervention
and unilateral changes in pricing policy or other regulatory directives that
can substantially diminish investor value. Thus, the privatized utilities
in developing countries are frequently confronted with regulatory regimes
that pose significant risks of political expropriation--regulatory intervention
which can substantially reduce their profits.
We analyze now the utility's pricing behavior in the face of such reg-
ulatory threat.7 Our underlying assumption is that there is a positive re-
lationship between the price charged by the utility and the probability of
retaliatory regulatory intervention. This supposition is especially relevant
in the context of developing countries which have a long history of price
controls, ostensibly imposed to protect the poor and to counter inflationary
pressures and macroeconomic imbalances. Price levels and profits that are
deemed unreasonably high by government policy makers can readily elicit
retaliatory regulatory responses in the form of tighter price controls, higher
profit taxes, and even threats of renationalization. In the face of endoge-
nous regulatory threat, the utility will constrain its pricing behavior and
7 The impact of regulatory threat on firm behavior has received considerable attention
in the literature. These contributions include Klevorick (1973), Bawa and Sibley (1980),
Logan et al (1989), Glazer and McMillan (1992), Acutt and Elliott (2001), Brunekreeft
(2004), Blum et al (2006).
17
forgo short-term profits to stave off the threatened regulation which could
reduce substantially its long-term profits and market value.
Let (t) the utility's assessment of the probability of regulatory inter-
vention at time t, with (0) = 0. The conditional probability density of
intervention at time t, given its nonoccurrence prior to t, is (t)/[1 - (t)] .
We assume that this conditional probability density of intervention is an in-
creasing convex function of the utility's price P(t) :
(t)
= h(P(t)) (27)
1 - (t)
where h(0) = 0, h (P) 0, h (P) 0 (Kamien and Schwartz 1991).
The firm's objective is to maximize its expected discounted profits
max (0) = e-rt (P,N) [1 - (t] + (t) dt ¯ (28)
P(t) 0
where (P,N) = PN - C(N) - N 2
subject to
(a) N = [ (N) - P],
(b) (t) = [1 - (t)]h(P(t))
(c) N(0) = N0,
(d) lim[N(t)e-rt] = 0.
t
Thus, P is the control variable and N and are the state variables. The
first component of the integrand is the utility's expected profit at t if there
is no regulatory intervention by t, while the second component is the firm's
¯
reduced profit in the event it is subjected to a retaliatory regulatory scrutiny.
In many developing countries, novice regulators who lack independence from
sectoral ministries and the politicized arms of government, may be prone to
unduly restrict the profits of privatized utilities. Under those conditions, ¯
could be substantially smaller than .
The current value Hamiltonian is
18
Hc = (P,N) [1 - (t)]+(t)+c(t) [ (N) - P]+µc(t)h(P(t))[1 -(29))]
¯ (t
where the costate variables c(t) and µc(t) are the current shadow prices
of an extra customer and the probability of regulatory intervention. The
sum of the first two terms in (29) represents the current-profit effect of
the utility's pricing policy: it realizes a profit if there is no regulatory
intervention by t (probability 1-(t))) and a profit if there is regulatory
¯
intervention by t (probability (t)). The third and fourth components of
the Hamiltonian can be viewed as the future-profit effects of P : the terms
c(t) [ (N) - P] and µc(t)h(P(t))[1 - (t)] represent the rate of change
in the value of the utility's network and the cost of potential regulatory
intervention caused by the utility's pricing decision P.
The maximum-principle conditions are
(30)
P [1 - (t] - c(t) + µc(t)h (P)) [1 - (t)] = 0
(31)
N [1 - (t)] - c(t) r - (N) = -c(t)
-( - ) - µc(t)[r + h(P)] = µc(t).
¯ (32)
It is easy to show that at the optimum c(t)
1-(t) and µc are constant.8
Thus, from (32) we obtain for the optimum value of µc
8 Differentiating (30) with respect to time yields
2 + 2
P2
µc(t)h (P))[1 - (t)] + µc(t)h (P)P [1 - (t)] - µc(t)h(P)h (P))[1 - (t)] = 0.
P [1 - (t)] PN N [1 - (t)] - P [1 - (t)] h(P) - c(t) +
In the steady state, P = 0 and N = 0. Substituting for c(t) and µc(t) from (31) and
(32) and for c(t) from (30) yields
c(t) ¯ c(t)
1-(t) 2 (N) = [r + h(P)]- -(-)h (P). Thus, in the steady state
P N 1-(t)
is stationary. Equation (30) then implies that µc(t) is also stationary.
19
-( - ). ¯
µc = (33)
r + h(P)
An increase in the state variable raises the probability of regulatory
intervention and thus reduces the optimal value at a rate proportional to
the difference in profits without and with regulatory intervention.
From (30), (31), and (32) we obtain at the optimum9
h (P)
( - ) = 0.
¯ (34)
P - r + h(P) - (N) N - r + h(P)
As in (13), the first term of (34) measures the marginal increase in the
current profit of the utility that is induced by an increase in its price, while
the second term represents the marginal decrease in future profits that such
a price increase will induce via the change in the size of the utility's net-
work. The third term measures the marginal decrease in future profits that
a price increase will induce via its impact on the probability of regulatory
intervention. The utility's optimal pricing policy must balance these com-
peting effects. An increase in the utility's price increases the conditional
probability of regulatory intervention at any moment which would for ever
reduce the utility's profit to . This effect is stronger the more sensitive is
¯
the probability of regulatory intervention to the utility's pricing behavior
(i.e. the larger h (P) is) and the more severe is the punishment imposed
by regulatory intervention, i.e., the bigger is the set-back in profits - . ¯
If the probability of intervention is constant (i.e. invariant with respect to
the utility's behavior), then h (P) = 0) and the third term in (34) becomes
zero. Thus, the mere existence of a fixed probability of regulatory inter-
vention does not necessarily affect the utility's behavior (Brunekreeft 2004).
The constant probability of intervention simply augments r, the utility's dis-
count rate. Moreover, if the utility is myopic and discounts future profits
very heavily (r is large), then it will not weigh very much in its pricing de-
cision the impact of such a decision on the probability of future regulatory
intervention and hence future profits.
9In the steady state c(t)
1-(t) = or equivalently c(t) = [1 - (t)] ,where is a
constant. This imples that c(t) = -c(t)h(P). Substituting for c(t) into (31) yields
c(t) = 1
r+h(P)- (N) N [1 - (t)] . Finally, substituting for c(t) and µc into (30) yields
(34).
20
Regulatory threat as an expected cost
The analysis of the utility's strategic response to regulatory threat and
the calculation of its optimal price path in particular are complicated by
significant nonlinearities. For the sake of analytic tractability, we choose
a different way of modelling the relationship between the utility's conduct
and the threat of retaliatory regulation. In the previous section we assumed
that the probability of regulatory intervention is endogenously determined--
it is an increasing function of the utility's price. However, the penalty
suffered by the utility in the case of intervention is exogenously determined--
the utility's profits are fixed after intervention at the low level . We now
¯
instead assume that the penalty of intervention is endogenously determined--
it is an increasing function of the price charged by the utility and a decreasing
function of the size of its network, i.e. the utility is penalized by the regulator
for charging high prices and rewarded for expanding its network.
Let (P,N) denote the regulatory penalty anticipated by the utility.
The present value Hamiltonian in the presence of these expected regulatory
penalties will be given by
H(N,P,t,) = e-rt PN - C(N) - N 2 - (P,N) + (t) [(N) - P]
(35)
We assume that (P,N) is given by
(P,N) = p(P - PR) + N(NR - N) (36)
where P > 0 and N > 0. Thus, the utility is anticipating that it
will be penalized in proportion to the amount by which its price exceeds
a given benchmark level PR and its coverage falls short a given level NR.
The benchmark levels PR and NR might be conjectural or they could be
signalled by the regulator.
Let's denote with N and P¯ the steady-state size of the utility's network
¯
and its corresponding terminal price in the presence of regulatory penalties,
as contrasted with Nand P, their values when there is no regulatory in-
tervention, given by equations (18) and (20). It is straightforward to show
that10
10We obtain the optimal path of N by solving the Euler equation which is given by
21
r
N =
¯ 0 - a - 2c(1 - )M + P( - 1) + N (37)
r + 2c(1 - )2 - 21
and
r
P¯ = 0 + 1 0 - a - 2c(1 - )M + P( - 1) + N . (38)
r + 2c(1 - )2 - 21
It is instructive to make a comparison with the case where the utility
faces no regulatory penalty. Since r
(37) and (38) with (18) and (20) indicates that N > N and P¯ > P,
- 1 > 0,11a comparison of equations
¯
i.e., the optimal size of the utility's network and its terminal price will be
larger when the utility is facing regulatory penalties. We also note that
r
N ¯ ( -1) > 0 and N¯ = > 0.12 Moreover,
P = r+2 N
[ N(t) c(1-1)2-¯21
e -1 r+2c(1-)2-211
N(t) N )t
2 > 0 and [ ¯ e-r(2 -1 > 0.
P t ] = r-2 P
N -r( )t N t ] = r-2 N
Thus, when the utility is faced with regulatory penalties, it expands its
network more rapidly.13 It is also easy to show that the utility chooses
a lower initial price P¯0 than the in the case where it faces no regulatory
intervention. That is, P¯0 = 0 + 1N0 + (N0 - N )-2 < P0 = 0 + 1N0 +
¯ 1
(N0 - N)-2 because, as it was noted above, N > N.14
1 ¯
FNN N +
¨ FN N + = 0 where F(N,N,t)
=
N FtN - FN
e-rt (-N + 0 + 1N)N - F - aN - c[M + (1 - )N]2r- N
1 2- p(P - PR) - N(NR - N). .This
leads to N -rN -
¨ r+2c(1-)2-21 N + 0-a-2c(1-)M+P ( -1)+N = 0 whose solution
2 2
0-a-2c(1-)M+P ( -1)+N
r
is given by N(t) = (N0 - N )e-r
¯ ( -1 )t
2 + N where N =
¯ ¯
r+2c(1-)2-21
and, as before, = 1 + 2r +2c(1-)2-21 1/2
.
r2
11 This can be seen from from equation (13) where = 1 .
r- (N) r
-1
12 This is because r +2c(1 - )2 - 21 > 0 from the Mangasarian sufficient conditions
(supra note 4).
13 As N increases, the penalty that the utility suffers for falling short the benchmark
network size will increase. It is intuitively appealing then that the utility will expand
its network to reduce the size of the penalty. However, we also find that the utility will
expand its network when P, the penalty that it suffers due its pricing behavior, increases.
This is because the utility in maximizing its long-run profits, needs a larger network size
to compensate for the loss in profit due to the pricing regulatory penalty.
14
N)2 then it is easy to show that the terminal size of the utility's network is
If we assume a quadratic penalty function (P,N) = p(P - PR)2 + N(NR -
22
The impact of the character of regulation on the utility's network ex-
pansion and price behavior should be an important consideration for policy
makers in developing countries with underdeveloped networks. The imposi-
tion of a regulatory penalty has a moderating influence on the pricing behav-
ior of a utility that pursues a long-run profit maximizing strategy by setting
a low initial price in order to accelerate the development of its network
and reduce the size of the regulatory penalty. Both the speed of network
expansion and the utility's terminal network size will be larger when it is
faced with regulatory penalties for not achieving certain network expansion
benchmarks. However, while the utility's initial price is lower, its terminal
price will be higher when it is faced with regulatory penalties. If we can
reasonably assume that the richer customers are served first, then we can
conclude that they will receive lower prices relative to the poor customers
we are likely to be served after the utility expands its network. Thus, an
unintended consequence of an excessively coercive regulatory policy (large
P and N) might be that it conflicts with distributional equity.
The role of the discount rate
Equations (13) and (34) illustrate the extent to which the discount rate
influences the utility's choice of pricing policy which in turn has important
feedback effects on the size of its network and on market structure. The
basic idea of the model presented is that the utility maximizes its long-run
profitability by adopting a pricing policy which at every instant balances any
marginal increase on current profits against the sacrifice of future profits that
such a policy might entail via its impact on the rate of the utility's network
expansion and the threat of regulatory intervention. The utility's rate of
time preference naturally affects the relative weights that it attaches to its
current and future profits and hence the time path of prices which maximize
the utility's discounted present value of profits. If the utility employs a very
high discount rate it will then exercise less restraint in holding price below
the fringe firm's unit cost or the short-run profit maximizing level--the value
given by N =
¯ 0-a-rc(1-)M+2P (0-PR)( -1)+2NNR
2 r
. In that case then N ¯ =
( -1)(1-2P 1)-1+2c(1-)2+2N P
2(P -PR)( -1)
¯ r r
N
and ¯ = 2(NR-N )( -1)
¯ where P¯
( -1)(1-2P 1)-1+2c(1-)2+2N
r N ( -1)(1-2P 1)-1+2c(1-)2+2N
r
is again the terminal price. If the firm expects to be regulated then it is sensible to
assume that P¯ > PR and NR > N . Since, as we noted above
¯ r N¯ > 0
¯ -1 > 0, then P
and N > 0. Thus, the basic conclusions about the impact of regulatory penalty on
N
the utility's network expansion and pricing are similar to those derived under the linear
penalty function specification.
23
of future profits that such a restraint will generate via its impact on the rate
of the utility's network expansion decreases as the discount rate increases.
Regulation that provides a credible commitment to safeguarding the in-
terest of both investors and customers--particularly when economic shocks
create political pressure to shift the balance of power among competing inter-
est groups--is an indispensable precondition for rational long-term planning
on the part of the utilities. Unfortunately, because of their long history
of arbitrary administrative intervention, governments in many developing
countries lack the capacity to credibly commit against political expropria-
tion of private value. Utilities in those countries feel especially vulnerable
to unilateral changes in policy or regulatory directives that tend to diminish
investor value. The fact that regulatory policy frequently lacks coherence,
stability and credibility causes the utilities to employ a high discount rate
in their planning. This tendency is reinforced by the uncertainty frequently
underlying the macroeconomic environment in developing countries.
The initial conditions in developing countries may generate conflicting
incentives on the part of their utilities with respect to pricing policy. The
fact that these utilities are operating underdeveloped networks with signif-
icant expansion opportunities may provide them with powerful incentives
to exercise restraint and price below the short-run profit maximizing level
during the early stages of network development. The high discount rates
that such utilities generally employ in the face of substantial policy uncer-
tainty, on the other hand, will cause them to assign greater weight to current
profits and consequently to set their price close to the limit defined by the
fringe firms' unit costs. Therefore, if network expansion is a key goal of
public policy in developing countries it is imperative that their governments
undertake steps to reduce policy risk.
The rate of discount may affect the utility's price path and the extent
to which it exercises pricing restraint via a different mechanism. During
periods of high economic uncertainty, when the utility's discount rate is
likely to increase, the regulatory agencies may be prone to intervene more
aggressively in order to protect consumers or to minimize the risks that
rapidly increasing utility prices may pose to a fragile economy in the midst
of an economic shock. In that case, the conditional probability density of
regulatory intervention might be assumed to be an increasing function of
both P and r
24
(t)
= h(P,r) (39)
1 - (t)
with h(0,0) = 0,hP(P,r) > 0, and hr(P,r) > 0.
To gain some analytic tractability we assume that
h(P,r) = Pr (40)
where > 0 and > 0. Equation (34) then becomes
1 1 P-1r
(41)
P - r - (N) N - r + Pr( 1 - 2) = 0.
It is easy to show that
P-1r
= P-1r( - 1) (42)
r r + Pr (r + Pr)2 0 if 1.
Thus the discount rate r has two competing effects on the utility's op-
timal pricing policy. A higher discount rate decreases the value of future
profits and thus it induces the utility to set a price that is closer to the
short-run profi-maximizing level. However, a rising discount rate also in-
creases the probability of regulatory intervention for a given choice of price.
Consequently, as the discount rate increases, the utility will have to set a
lower price if it is to avoid a retaliatory regulatory review. The net impact
of these two competing influences depends on the magnitude of , the con-
ditional probability density of regulatory intervention with respect to the
discount rate. If the probability of regulatory intervention rises more than
in proportion to the discount rate r then an increase in the discount rate
will cause the utility to exercise restraint with respect to pricing. This is
because the increased threat of regulatory intervention associated with the
higher discount rate creates an incentive for the utility to moderate its pric-
ing, an incentive that is more powerful than the opposite one according to
which a higher discount rate causes the utility to weigh its current profits
more and hence to a adopt a pricing policy that maximizes short-run profits.
25
3. Concluding Comments
We have presented in this paper an analytically tractable intertemporal
framework for analyzing the dynamic pricing of a utility with an underde-
veloped network (a typical case in most developing countries) facing a com-
petitive fringe, short-run network adjustment costs, theft of service, and the
threat of a regulatory review that is increasing with the price it charges.
This simple dynamic optimization model yields a number of powerful policy
insights and conclusions.
Under a variety of plausible assumptions (in the context of develop-
ing countries) concerning the utility's cost structure, the initial size of its
network, the speed of expansion of this network in response to the price
differential between the utility and the fringe, the cost structure of the com-
petitive fringe, the costs of rapid adjustment of capacity, the percentage
of customers that engage in service theft, and the probability of a retalia-
tory regulatory review, the utility will find its long-run profits enhanced if
it exercises restraint in the early stages of network development by holding
price below the limit defined by the unit costs of the fringe (or the short-run
profit-maximizing level). The utility's optimal price gradually converges
toward the limit price as its network expands.
These findings have important implications for the design of post pri-
vatization regulatory governance in developing countries. Many of these
countries lack several institutional prerequisites for well-functioning regula-
tory mechanisms, including: separation of powers, especially between those
of the executive and those of the judiciary; effective political and economic
institutions; a well-functioning legal system; good contract law and mecha-
nisms for resolving contract disputes; and a good supply of professional staff,
expert in the relevant economic, accounting, and legal principles. In the
face of scarce technical expertise, severe information problems, and lack of
well-developed accounting and auditing systems, the U.S.-U.K. regulation
models are likely to prove too challenging for many developing countries,
especially during the early stages of the reform process. Most develop-
ing countries are ill-suited to the traditional quasi-judicial, command-and-
control techniques of regulation, with their elaborate and complex technical
and procedural requirements. More appropriate regulatory mechanisms for
these countries would be simple, decentralized, flexible, and light-handed.
This policy prescription for light-handed regulation during the early stages
26
of the reform process is substantially reinforced by the analysis of this paper
highlighting the powerful economic incentives that utilities with underdevel-
oped networks are likely to face during the same period to exercise restraint
with respect to pricing.
The paper's intertemporal framework also highlights the critical impor-
tance of a stable, credible, and predictable policy environment for the util-
ities to expand their networks. If governments are able and willing to
impose appropriate limits on the discretionary exercise of their regulatory
powers, utilities will feel less vulnerable to unilateral changes in the policy
regime that could substantially undermine their profits. Under those cir-
cumstances, utilities will tend to employ a low discount rate which effectively
will cause them to set low initial prices so as to accelerate the expansion of
their networks and reap higher future revenues and profits.
27
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29