competing against a public service

Competing Against a Public Service

ANN VAN ACKERE*London Business School, Sussex Place, London NW1 4SA, U.K.

AbstractÐThe private sector provides a service also available from a public facility (where customersonly incur a congestion cost). The quality of the service is determined solely by the level of congestion.The total number of customers is ®xed. Examples include the provision of medical care and education.We study how optimal capacity, price and congestion level depend on market structure (monopolisticversus competitive), and on how the public facility is managed (®xed capacity or a maximum conges-tion level). We also discuss the impact of these elements on market share, customer congestion costsand total social cost. # 1998 Elsevier Science Ltd. All rights reserved

INTRODUCTION

Consider a situation where the state provides a service to a captive customer base. Customersonly incur a congestion cost, and service quality is di�erentiated solely by the congestion level.

We interpret congestion in its broadest sense, i.e. including not only delays but also reduced

levels of comfort. By captive customer base we mean that customers do not have the option to

not use the service. They must therefore use the service provided by the state (labelled statefacility) unless a private alternative (at which customers also incur a monetary cost) is available.

One example is the British National Health Service (NHS). Anyone is entitled to free medicaltreatment, but congestion costs are extremely high (see, for instance, [8]) causing people to `go

private'. Examples include waiting to see a practitioner, even when an appointment has been

made (see, for instance, [7]), long delays for elective surgery (spending two years on a waitinglist is not uncommon; see, for instance, [3]), frequent last-minute cancellation of surgical pro-

cedures, large crowded wards, stretched nursing sta�, etc.

Another example is education: children are required to attend school up to a certain age. Inmany countries parents can choose between schools run by the state or local authorities, where

no ®nancial contribution is required, and one or more private alternatives. For instance, in

U.K. state schools, the typical primary school class has 27 pupils [18], compared to 14 in privateschools.

The assumption of a captive customer base is very strong. Its aim is to represent the extreme

case where either not getting service results in such major disutility that it becomes an extremelyunlikely option, or where the service is imposed by law (e.g. education). Assuming that quality

di�erences across facilities are determined solely by congestion allows us to focus on one speci®c

aspect of the problem. In particular, it enables us to express willingness to pay for a privatealternative as the di�erence in congestion cost between the two facilities.

The debate about optimal funding of the NHS is as old as the NHS itself. The complexity of

the issue can be better understood when, following Musgrave's [15] classi®cation, the NHS isconsidered as a public good characterized by rival consumption (if one patient sees a doctor,

this time is not available to another patient) and the impossibility of exclusion. It is this latter

characteristic (everyone's right to free health care through the NHS) that makes the provisionof health care in the U.K. a public good. Impossibility of exclusion usually occurs for technical

or economic reasons. In the cases of the NHS and education, the exclusion problem has a

moral origin: no one should be excluded from health care and schooling.

Most of the literature on public goods focuses on goods with non-rival consumption. Laux-

Meiselbach [12] ascribes this to the di�culty of constructing rules for the optimal supply of rival

Socio-Econ. Plann. Sci. Vol. 32, No. 3, pp. 171±187, 1998# 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0038-0121/98 $19.00+0.00PII: S0038-0121(97)00036-0

*Fax: (44) (0) 171 724 7875, E-mail: [email protected]

171

goods when exclusion is not possible. He goes on to argue that the market cannot supply thegood (at least not the optimal quality) because of the impossibility to charge a price given theabsence of willingness to pay when exclusion is not possible. In the absence of a mechanismthat guarantees allocation according to preferences, the government is also unable to achieve op-timal allocation of the good.

It has been suggested that rather than providing the service for free, patients should bear partof the cost. This would reduce congestion, as potential customers would avoid using the servicewithout good cause. If this is indeed the case, McConnell's [13] study of a congested recreationfacility indicates an undesirable side-e�ect. He shows that under plausible assumptions aboutthe e�ects of income and price responsiveness, the use of a fee would tend to ration the servicetowards people who are less price responsive and more averse to congestion than the averageuser; typically, higher-income people. Such a result would be contrary to the aim of the NHS.Equally important, a fee could cause low-income individuals to postpone seeking medicaladvice, or to avoid preventive care (e.g. timely eye tests), leading to higher demand in the longrun.

Consequently, private companies have entered the market, providing a higher-quality alterna-tive (in terms of less congestion). In the context of health care, congestion is indeed the maindi�erentiating factor between the state and private facilities, as it is common for surgeons tocarry out both NHS and private work, while a number of hospitals provide both NHS and pri-vate beds. Di�erences include waiting time and comfort factors such as private rooms vs wards.

Our aim here is to investigate when and how a state facility and a private facility coexist,focusing on the following elements:

Ð the structure of the private market: monopoly vs perfect competition;Ð the management method of the state facility: resource constrained versus a quality require-

ment;Ð the resulting service quality (congestion level) provided by both facilities; andÐ the resulting implications in terms of total social cost and total customer cost.Empirical observation suggests that expecting a monotone relationship between market share

of the private sector and congestion level in the state sector would be overly optimistic. Toomany other variables play a role. For instance, using share of health expenditures as a proxy formarket share, the state sectors of the U.K. and Belgium were of the same order of magnitude(83% and 88%, respectively, [19]), in 1993. However, Belgium does not su�er from the endemicwaiting list problem faced by the UK. Some authors have argued that congestion is not necess-arily ine�cient. Kornai and Weibull [11], for example, consider an economy characterized bychronic shortage and queueing. Although this market does not satisfy the conditions of aWalrasian equilibrium (see, for instance, [21], p. 315), it is in a stationary state. Queueing is usedas a rationing device, leading to postponement of purchase or forced substitution. It is worthnoting that Kornai and Weibull deal with a deterministic model because ``This choice re¯ectsour belief that in situations characterized by chronic shortage, the stochastic element is second-ary in comparison to the inter-dependencies and feedback mechanisms regulating the system''.[11, p. 376]. Barro and Romer [4] consider the supply of leisure services such as ski-lifts andamusement rides. They argue that it may be rational to set prices such that queues occur regu-larly and are longer at peak times. They derive conditions under which congestion does not leadto a loss of e�ciency.

This work relates to several bodies of literature, including research on priority classes inqueueing systems, and peak-load pricing. Examples of the former include Alperstein [2] andMendelson and Whang [14]. Examples of the latter include Chao and Wilson [5], Viswanathanand Tse [22] and Crew et al. [6]. In these types of situations, customers can select to pay a higherprice to obtain a higher-quality service (e.g. less waiting time in a queueing environment, or alower probability of their electricity supply being interrupted). In each instance, when one custo-mer elects to pay more for a higher quality, he jumps the queue, making the less impatient custo-mers worse o� in terms of quality of service, assuming capacity is una�ected by his decision.

The situation we consider here is di�erent in this respect: when a customer selects to use theprivate facility, he reduces congestion in the state facility (unless capacity is reduced),therebyproviding the remaining customers with a better-quality service, while increasing congestion in

Ann van Ackere172

the private facility. In the context of the NHS example, a patient who elects to go private frees

up resources to treat other NHS patients, thereby reducing the delay these patients face.

Similarly, moving a child from a state school to a private school reduces the class size in the

state system.

A third body of literature that does incorporate this aspect relates to the work on public

transport: by travelling ®rst-class, for example, a customer reduces congestion in the second-

class carriages on a train, or makes an economy seat available for a customer who cannot a�ord

®rst-class travel. (See, for instance, [16].) This literature di�ers from our problem in that both

levels of service are provided by the same organization (the railway or airline) which attempts to

optimize the joint bene®ts (e.g. a measure of social welfare such as passenger miles, or pro®t)

from providing both qualities of service. In our work, the state facility and the private facility

each pursue their own objective, with no attempt at joint optimization. Alpern and Reyniers [1]

generalize the results of Reyniers [16], and also consider a duopoly where each ®rm provides one

class of service, attempting to maximize its pro®ts. Our work di�ers from theirs in several

respects: (i) while the private facility is a pro®t maximizer, the state facility is not; and (ii) the

state facility is required to provide service to all remaining customers.

The work most closely related to that done in this paper is found in Stenbacka and

Tombak [17]. They consider the co-existence of a state and a private sector in a duopoly context,

and compare this to the cases of state and private monopolies. The state sector in [17] is

assumed to choose a waiting time so as to optimize the sum of customer surplus and its own

surplus, while we consider the case where the state sector either has a ®xed budget or has a

quality target. Our work also di�ers in that Stenbacka and Tombak take price as given, focus

exclusively on the cost of waiting, and are mainly interested in market coverage. We, however,

interpret congestion costs in a much broader sense, and require that all customers be serviced.

The key conclusions can be summarized as follows. The di�erences between the four cases

considered are most noticeable when the state sector has limited resources (either low capacity

or a low quality target) and are summarized in Table 1.

The main di�erence between the competitive and monopolistic cases lies in the increased pri-

vate capacity and resulting dissipation of potential pro®ts, especially for low levels of state ca-

pacity or low quality targets. This leads to lower congestion costs but higher capacity costs.

Comparing the ®xed capacity and quality target cases, in the latter, it is optimal for the private

sector to capture the entire market when congestion in the state sector is high (low-quality

requirement). With low resources (low capacity or a low-quality target), the gain in total social

welfare from allowing a private sector is largest for a monopolist entrant combined with a qual-

ity target for the private sector, while it is lowest for a monopolist with ®xed capacity. This lat-

ter case also yields the highest social cost, making it particularly undesirable from a public

policy point of view. As the resources of the state sector increase, the gains from a private

entrant decrease.

The remainder of this paper is structured as follows. In the next section we describe the basic

model. We then consider a private monopolist, after which we deal with perfect competition. In

each section we consider two cases: the state facility is resource constrained, or there is a pre-

speci®ed quality level. Following this we illustrate that the qualitative conclusions are not depen-

Table 1. Overview of results

Di�erences between the di�erent scenarios are largest when state resources are limited(low capacity or low quality target).

Competition versus Monopoly:The more limited resources are, the more the monopolist over-invests in capacity, leading to a dissipation of pro®ts.

Quality target versus Fixed Capacity:Ð When the state sector has a low quality target, the private sector captures the entire market. This does not occur for low levels of

capacity.Ð A quality target leads to a larger private sector, lower state costs and higher social welfare.

When state resources are limitedÐ Gain in social welfare: maximised for the case of a quality target and a monopolist; minimized for the case of ®xed state capacityand a monopolist.Ð State costs: lowest for the case of a quality target and competitive entry; highest for the case of ®xed state capacity and amonopolist.

Competing against a public service 173

dent on the speci®c shape of the cost function and the heterogeneity assumption. In closing, wesummarize our conclusions and suggest some directions for further research.

THE MODEL

We standardize the rate of customer demand to equal 1 over a given time period. The ca-pacity of the state facility is denoted by ms, and the capacity of the private facility by mp. Let lpdenote the fraction of customers selecting the private facility. Customers select a facility basedon their expected total cost (the congestion cost, plus a fee, should they choose the private facil-ity), i.e. they do not observe the actual situation. This assumption is discussed at the end of thesection. As we assume that the total demand per period equals 1, lp also equals demand at theprivate facility. The state facility services the residual demand, i.e. ls=1ÿ lp.

Both facilities incur two types of costs: a variable cost per customer (cv), and a ®xed cost perunit of capacity per period (cf). The variable cost refers to expenditures per customer, such asmedication for a patient, or a pencil for each student. Capacity costs refer to resources that areshared between patients, more resources implying less congestion or more comfort. Examplesinclude number of teachers (a�ecting the teacher/pupil ratio), and space (determining the num-ber of patients per ward). We do not consider economies of scale.

The private facility charges a fee F per customer. Consequently, its pro®t per period equalsY�lp;mp;F� � lp�F ÿ cv� ÿ cf mp: �1�

We consider two management approaches for the state facility: (i) the state facility faces aresource constraint. We will represent this constraint by a ®xed capacity (ms), determined ex-ogenously. It would be more realistic to represent this constraint by a budget B. This case ismuch more complex to analyse, but yields similar results, and is therefore omitted. (See [20] fordetails.) As a reference capacity we choose ms=1. This could, for instance, represent the ca-pacity required to achieve an average class size of 25 pupils if the state facility serviced the wholemarket. (ii) The state facility installs the capacity required to achieve a speci®c quality level forits customers. This required quality level is expressed as a capacity per patient ratio Q. We use areference quality level equal to 1, again for a class size of 25. This standardization ensures that,without the private sector, total social costs are the same for the resource constraint and qualitytarget cases whenever ms=Q, thus enabling meaningful comparison of the various cases.

We also consider two competitive situations: (i) the private ®rm is a monopolist; and (ii) theprivate market is perfectly competitive, in which case lp and mp denote the total demand and ca-pacity, respectively, of the private market.

The maximum fee a customer is willing to pay equals the di�erence between the expected con-gestion cost incurred at the state facility and the private facility. The same fee must be chargedto all customers. Results depend critically on whether or not customer congestion costs arebounded. In the next two sections we consider the more realistic case of bounded congestioncosts. After that we brie¯y discuss unbounded congestion costs.

Customers are characterized by their tolerance for congestion. We assume that customer type(i) is distributed uniformly on the unit interval [0, 1], with i= 0 denoting the least tolerant cus-tomer and i= 1 the most tolerant customer. As the less-tolerant customers select the privatefacility, the marginal customer (i.e. that customer who is indi�erent between the two facilities)will have type lp.

Let C(l, m, i) denote the congestion cost of a type i customer when demand equals l and ca-pacity equals m. Any reasonable function C(l, m, i) should be increasing in the number of custo-mers l, and decreasing in capacity m. This function should also decrease in i: that is, at the samelevel of congestion, a more tolerant customer (i close to 1) incurs a lower congestion cost.

As discussed below, the qualitative results are not a�ected by the speci®c shape of C(l, m, i).This is the case as long as congestion costs are bounded for all customers under all circum-stances. In the next two sections we consider the function

Ann van Ackere174

C�l;m; i� � al

m

1

�1� i� �2�

where a is a parameter that translates units of delay or congestion into monetary units, l/mmeasures the congestion e�ect, and 1/(1 + i) captures the customer's type.

Readers familiar with queueing theory will react to the use of a cost function that is linear inutilization l/m, as it is well known that waiting time increases exponentially in utilization. Oneshould remember that we do not focus solely on waiting time, but on all aspects of congestionand, while some aspects (e.g. waiting time) increase exponentially, others increase at a decreasingrate. Consider, for instance, the discomfort related to the size of a ward: the increase in discom-fort when going from a private room to a double room far exceeds the increase resulting fromgoing from a ward of 9 to a ward of 10. Similarly, going from a class size of 10 to a class sizeof 15 has more impact than going from one with 45 to one with 50 pupils.

We assume that the customer selects a facility before observing the actual level of congestion.In the context of education, access to some private schools requires children to be registered sev-eral years before starting school. Within the context of the NHS, this assumption is realistic asthe vast majority of customers who elect to go private have subscribed to a medical insuranceplan before actually needing the service. This situation can be easily accommodated in ourmodel, assuming actuarially fair premiums and risk-neutral customers. Denoting the insurancepremium by I, and the probability of needing medical care by p, an actuarially fair premiumshould satisfy the equality I = pF. The remainder of the analysis can be carried out in a similarway, with a total population size of 1/p, of whom a fraction p will need the service, and with lpand ls denoting the actual arrival rates.

Koenigsberg [10] considers a duopoly situation where customers can observe the queue uponarrival and elect to switch facilities. Each competitor selects price and capacity, and is character-ized by an initial arrival rate, which is a function of price and expected quality at both facilities.Upon arrival, the customers observe their position in the line and may elect to switch facilities.All customers are served by one of the two ®rms.

MONOPOLISTIC MARKET STRUCTURE

The private facility selects parameters lp, mp, and F so as to maximize pro®t. The equilibriumcondition requires that the marginal customer be indi�erent between the two facilities. The lesstolerant customers (with types in the interval [0, lp]) prefer the private facility, while the moretolerant ones (types in the interval [lp, 1]), prefer the state facility. The marginal customer there-fore has type lp, and the equilibrium condition requires

F � C�lp;mp; lp� � C�ls;ms; lp�: �3�We consider two cases for the state facility: ®xed capacity, and a quality target.

FIXED CAPACITY FOR THE STATE FACILITY

In this case, the state facility's capacity ms is ®xed, and the private facility faces the followingproblem:

maxlp;mp;F

Y�lp;mp;F� �lp�F ÿ cv� ÿ cf mp

s:t: F � C�lp;mp; lp� �C�ls;ms; lp�ls �1ÿ lp: �4�

The ®rst constraint requires that the marginal customer be indi�erent between the two facilities:for the customer with type lp, fees plus congestion cost at the private facility should equal con-gestion cost at the state facility. The second constraint formalizes the assumption that total cus-tomer demand equals one per period.


After substituting the constraints into the objective function, the ®rst-order conditions yield

cf � ÿ @C�lp;mp; lp�@mp

lp; �5�

C�1ÿ lp;ms; lp� ÿ C�lp;mp; lp� ÿ cv � lp@C�1ÿ lp;ms; lp�

@lpÿ lp

@C�ip;mp; lp�@lp

� 0: �6�

eqn (5) states that the ®xed cost of a unit of capacity should equal the change in customer will-

ingness to pay as capacity changes, multiplied by the number of customers. Substituting the con-

straint F= C(1ÿ lp, ms, lp)ÿC(lp, mp, lp) in eqn (6) yields

F � cv ÿ lp@F

@lp�7�

with the following intuitive interpretation. The optimal fee equals the costs imposed on the facil-

ity by an additional customer, i.e. the variable cost (cv) and the reduction in fee income resulting

from the reduced willingness to pay (ÿ@F/@lp) multiplied by the existing customer base (lp).

Using the cost function of eqn (2) yields

mp � lp

��a

cf �1� lp�r

: �8�

This expression indicates that the optimal quality level equals mp/lp=Za/(cf(1 + lp)), which is

decreasing in the optimal market share of the private facility. As will be seen in the next section,

this relationship does not depend on how the state facility is managed. While closed-form sol-

utions can be obtained for lp and mp, they are too cumbersome to provide any insight, and we

therefore illustrate the results using a numerical example. All graphs are drawn for a = 10,

cv=1, and cf=1. These values were chosen such that, at the reference capacity (ms=1), total

congestion costs equal approximately two-thirds of the total cost, while the marginal customer's

congestion cost at the private facility equals about one-third of his total cost. Further, all this

occurs while the private facility remains pro®table until the state facility expands to about one-

third above the reference capacity. We have purposely given a high weight to congestion costs.

Where appropriate, we indicate the impact on the results of lowering this weight.

Figure 1 illustrates the results for state capacity, ms in the range 0.5 to 1.5. These results are

summarized in Table 2, column 1. As expected, the larger the capacity of the state facility, the

smaller the private facility [Fig. 1(A)]. As long as the private facility is active, total capacity

increases more slowly than state capacity, as increases in state capacity are partly o�-set by the

shrinking of the private facility. The monopolist exits the market when state capacity reaches

1.36, i.e. approximately a third more than the reference level.

Table 2. Comparative statistics

Monopolist Competition

Fixed capacity Quality target Fixed capacity Quality targetPrivate entry for ms<1.36 Q<1.36 ms<1.36 Q<1.36Private sector captureentire market for

Never Q<0.58 Never Q<0.82

SensitivityAs state capacity increases As the quality target

increasesAs state capacity increases As the quality target

increasesPrivate market share + + + +State capacity V (assumed) V V (assumed) VPrivate capacity + + + +Total capacity V + until lp=0, then V + until lp=0, then V + until lp=0, then VState utilization + + (assumed) + + (assumed)Private utilization + + V VCongestion cost + non-monotonic non-monotonic V until lp=0, then +Total social costs + non-monotonic + non-monotonicFee + + + +State cost V V V VPrivate cost + + + +Pro®t + + 0 0

Ann van Ackere176

Fig.1.Monopolist

and®xed

capacity.(a)Capacity

andmarket

shares;(b)utilization;(c)costsandfees;(d)costsandpro®ts.


As capacity increases, congestion decreases quite signi®cantly in the state sector, and margin-ally in the private sector [Fig. 1(B)]. The decrease in congestion in both facilities leads to lowercongestion costs for customers [Fig. 1(C)]. At the same time, the private facility is forced tolower its fee, as the congestion di�erential between the two facilities decreases. The decrease incongestion cost more than o�sets the increase in capacity costs, leading to lower total socialcosts. Further increases in capacity will ultimately lead to an increase in total cost, as the ad-ditional capacity yields little decrease in congestion costs. In our example, this occurs forms>2.7. Note that fees are a transfer and, as such, do not a�ect total social cost. Figure 1(C)also illustrates that the total social cost is reduced by the presence of the private facility.

Figure 1(D) shows the increasing costs incurred by the state facility (to be funded by taxincome) and the decreasing costs incurred by private facility (to be funded by fee income), aswell as the monopolist's decreasing pro®tability as the state sector expands.

What happens if we halve the weight of congestion costs (i.e. people become less willing topay for reduced congestion, as a is reduced from 10 to 5)? The qualitative results are unchanged,the main di�erence being that the private facility now exits when ms reaches approximately 0.9.Total social costs decrease until ms reaches 1.9.

A QUALITY TARGET FOR THE STATE FACILITY

In this section, we assume that the state facility must achieve a speci®c quality target withrespect to the congestion costs incurred by customers. Further, there is no budget constraint.Referring to the example of the NHS, the government has announced as a target that no custo-mer should spend more than two years on a waiting list [23].

The target is speci®ed as a minimum amount of capacity per customer: ms/lseQ. As the statefacility seeks to achieve this target at the lowest possible cost, and all variables are continuous,we can assume without loss of generality that the target will be satis®ed with equality at the op-timum.

The private facility's problem is therefore as in eqn (4), with an additional constraint: ms/ls=Q. The ®rst-order conditions are similar to those for the ®xed capacity case:

cf � ÿ @C�lp;mp; lp�@mp

lp �9�

C�1ÿ lp; �Q�1ÿ lp�; lp� � lp@C�1ÿ lp; �Q�1ÿ lp�; lp�

@lpÿ C�lp;mp; lp� ÿ cv ÿ lp

@C�ip;mp; lp�@lp

� 0: �10�

The interpretation is similar to that of eqns (5) and (6). Using the cost function of eqn (2) yieldsthe same relationship between optimal capacity and market share:

mp � lp

��a

cf �1� lp�r

: �11�

As for the ®xed capacity case, closed form expressions can be obtained for lp and mp, but theyare too cumbersome to yield any useful insights.

Figure 2 illustrates the results for parameter values a = 10,cv=cf=1. These are summarizedin Table 2, column 2. Let [Q1, Q2] denote the range of quality targets over which the two facili-ties co-exist. In our example, this range is [0.58, 1.36]. For targets below Q1, the monopolist cap-tures the entire market (lp=1), while for values above Q2, he is driven out (lp=0).

In the range [Q1, Q2], an increase in the state's quality target leads to lower overall capacity[Fig. 2(A)], as increases in state capacity are more than o�set by resulting decreases in privatecapacity. Congestion decreases in both facilities as the quality target rises [Fig. 2(B)], and the feeis driven down [Fig. 2(C)]. Congestion costs are not monotone: initially they increase, as custo-mers switch to the more congested state facility, but start to decrease for larger Q as theimproved quality of the state facility o�sets the switch from the private to the state facility.Total social costs re¯ect the congestion costs: they start to decrease slightly earlier than the con-gestion costs, due to the decrease in capacity costs. As in the ®xed capacity case, increases

Ann van Ackere178

Fig.2.Monopolist

andaquality

target.(a)Capacity

andmarket

shares;(b)utilization;(c)costsandfee;

(d)costsandpro®t.


beyond Q=2.7 will result in increased social costs, as little or no further reductions in conges-tion are achieved.

Again, total social costs are lower than they would be without the private sector, the discre-pancy being especially large for low quality targets.

Halving the congestion costs (a = 5) yields similar qualitative results, with the monopolistcapturing the entire market for Q<0.4, and exiting the market when Q reaches 0.9. The conges-tion and social costs remain non-monotone, while the social costs begin to increase again forQ>1.9.

The importance of assuming heterogenous customers is noteworthy. If customers are identical,an all-or-nothing situation arises: the monopolist either captures the entire market, or he staysout. In other words, the range [Q1, Q2] collapses to a single point. This does not occur in the®xed capacity case. For details, see [20].

Comparing these results to the ®xed capacity case (®rst two columns of Table 2), the maindi�erence lies in the decrease in total capacity as the quality target increases. This causes thecongestion costs to increase over part of the range. For low levels of state capacity (or low-qual-ity levels), the social costs, and the state costs, are lower in the quality target case. The impactof lowering the weight given to congestion costs is similar for both models.

PERFECT COMPETITION

In this section, we consider a situation where there is perfect competition in the private mar-ket. We make the simplifying assumption that all ®rms are identical and focus on the total ca-pacity and market share of the private sector. We again consider ®xed capacity and a qualitytarget for the state facility.

FIXED CAPACITY FOR THE STATE FACILITY

The optimization problem of the private sector can be analysed in two stages: (i) for a giventotal private capacity mp, the ®rms choose the optimal market share and corresponding fee; and(ii) entry occurs as long as pro®ts remain positive. The formulation is thus as follows:

Stage 1.

maxlp;F

Y�lp;mp;F� �lp�F ÿ cv� ÿ cf mp

s:t: F � C�lp;mp; lp� �C�ls;ms; lp�ls �1ÿ lp; �12�

i.e. compared to the section Fixed Capacity for the State Facility, mp is no longer a decisionvariable for the individual ®rm. The ®rst-order condition is as in eqn (5), and yields the optimalmarket share as a function of capacity, denoted l*p(mp). Using the ®rst constraint yields the opti-mal fee as a function of private capacity, F*(mp).

Stage 2.

m�p solvesY�l�p �mp�;mp;F

��mp�� 0

with optimality condition@Q�l�p �mp�;mp;F

��mp��@mp

< 0: �13�

Using the cost function of eqn (2) yields

l�p �mp� ��

a�2mp �ms�amp � ams �mpmscv

sÿ 1

m�p �3a�mscv ÿ 2

��a�2a� cf m2

s � 2m2cv�p

cf ms: �14�

Ann van Ackere180

Fig.3.Competitionand®xed

capacity.(a)Capacity

andmarket


(d)state

andprivate

costs.


Fig. 3 illustrates the main results, which are summarized in Table 2, column 3. As state capacityincreases, total capacity goes down, because the increase is more than o�set by the decrease inprivate capacity [Fig. 3(A)], in contrary to the monopolistic case. Congestion in the state facilitydecreases, while there is a deterioration in the private sector [Fig. 3(B)].

Especially for low levels of state capacity, total capacity is signi®cantly higher than in themonopolistic case (Fig. 1), as entry dissipates the potential for pro®ts. While the private marketshare has approximately doubled, capacity is up to three times higher, leading to lower conges-tion, while the fees are of comparable magnitude. Note that the monopolistic and competitivecases converge to each other as state capacity increases to the point where the private facilityexits (ms=1.36).

Congestion costs are no longer monotone [Fig. 3(C)]. Initially, the strongly decreasing conges-tion in the state facility dominates, despite the decrease in total capacity. Then, the increase incongestion in the private sector dominates. Once the private facility exits, the increase in totalcapacity leads to lower congestion costs, until ms exceeds 2.7.

Total social costs decrease monotonically, and are fairly similar to those in the monopolisticcase. Monopoly pro®ts have been spent on extra capacity: note the much increased private costin Fig. 3(D).

Halving the weight given to congestion costs (a = 5) again yields very similar results, with theprivate sector exciting when ms reaches 0.9, as in the monopolistic case.

A QUALITY TARGET FOR THE STATE FACILITY

The problem faced here by the private sector is as in the previous section, with the additionalconstraint ms/ls=Q (as earlier in A Quality Target for the State Facility). Using the cost func-tion of eqn (2) yields the following results:

l�p �mp� ��a�1�mp=�Q�a�mpcv

sÿ 1

m�p �a=�Q� cv ÿ 2

��a�cf � cv=�Q�

qcf

: �15�

Fig. 4 illustrates the main results, which are summarized in Table 2, column 4. It is interestingto compare the monopolistic and competitive cases. We again have a range of quality levels [Q1,Q2] for which the state and private facilities coexist. The private sector captures the entire mar-ket for a larger range of quality levels [Q1=0.82 compared to 0.58, Fig. 4(A)]. The free entryleads to signi®cantly larger private capacity, and thus lower congestion in the private sector[Fig. 4(B)]. The di�erence is most obvious when the private sector captures the whole market(Q<Q1): the monopolist held capacity (and quality) constant over this range [Fig. 2(A)], whilecapacity in the free entry case soars (and congestion decreases) as the state facility's quality tar-get decreases.

This is re¯ected in the larger social cost and low congestion costs, and in the steep shape ofthe fees over the range Q<Q1 [Fig. 4(C)]. Congestion costs increase monotonically, until the pri-vate facility exits, and decrease thereafter. Total social costs decrease when Q<Q1, increase overthe range [Q1, Q2], and decrease when Q>Q2, until Q>2.7. Figure 4(D) shows the costs of thestate and private sectors, which re¯ect the high levels of private capacity.

Again, total social costs are lower than in the case without a private facility [Fig. 4(C)].Halving the weight of the congestion costs yields similar results, with [Q1, Q2] = [0.54, 0.9].

COMPARISON

In this section, we brie¯y compare the four cases discussed above. The gain in social welfarefrom private entry (i.e. lower total social cost) is largest for the case where the state facility hasa quality target and the entrant is a monopolist. This is followed by a quality target with com-

Ann van Ackere182

Fig.4.Competitionandaquality

target.(a)Capacity

andmarket


(d)state

andprivate

costs.


petitive entry, ®xed capacity with competitive entry and, ®nally, ®xed capacity with a monopo-list. On the other hand, state costs are lowest for a quality target with competitive entry, fol-lowed by a quality target with a monopolist, ®xed capacity with competitive entry and, ®nally,®xed capacity with a monopolist. This makes the case of ®xed capacity and a monopolist theleast desirable.

A quality target leads to a larger private sector than does a ®xed capacity constraint. Thisresults in lower social costs and lower state costs. Focusing on total social costs hides the factthat the relative magnitude of the components of the total social cost do vary depending on thescenario being considered. Social costs consist mainly of congestion costs when the entrant is amonopolist (i.e. a relative under-investment in capacity), while they consist mainly of private ca-pacity costs (re¯ected in a high fee level) in the scenario combining a quality target with com-petitive entry.

SENSITIVITY

The purpose of this section is to illustrate the sensitivity of our results to the two mainassumptions:

Ð the cost function considered (and, speci®cally, its boundedness); andÐ the speci®c form of heterogeneity.Our sensitivity analysis indicates that the results are in¯uenced by whether or not the cost

functions are bounded, but not by their exact functional shape. More speci®cally, if the costfunction is `very steep' at low levels of quality for some or all customer types (e.g. if the focus issolely on waiting time), it may be optimal for a private monopolist to skim the market, i.e. o�era very high quality service to a very small fraction of the population, to keep congestion costshigh in the state sector.

The next section considers other bounded cost functions. We then look at an example ofunbounded costs, after which we consider alternative forms of heterogeneity, both bounded andunbounded.

OTHER BOUNDED COST FUNCTIONS

First consider the following family of cost functions, parameterized by b:

C�l;m; i� � a�1� l=m�b ÿ 1

1� i: �16�

The larger b, the steeper the cost function. For b = 1, we obtain the cost functions in earliersections.

The qualitative results are similar to those found previously. The only noticeable di�erenceoccurs for the competitive case with ®xed capacity: for very low levels of state capacity andb>1, market share, lp, decreases as state capacity, ms, decreases. This e�ect becomes more pro-nounced as b increases, i.e. as the cost function becomes `steeper'. We will elaborate on thisresult when discussing unbounded costs.

Next, consider a case of exponentially increasing costs; for instance

C�l;m; i� � ael=m

1� i: �17�

Note that this cost function is bounded for strictly positive values of capacity m. Results for thecase of the monopolist, and for the competitive case with a target, are again as in earlier sec-tions. For the competitive case with ®xed capacity for the state facility, the only di�erence isagain that, for very low levels of state capacity, ms, the market share of the private sectordecreases as ms decreases. As ms goes to zero, the cost function is no longer bounded, and theresults are as discussed in the next section: private market share goes to zero and private ca-pacity increases without bounds.

Ann van Ackere184

UNBOUNDED COST FUNCTIONS

We have thus far assumed that however inadequate the resources, congestion costs remain®nite. One could imagine a situation where a shortage of resources causes the number of peoplewaiting for service (e.g. people on a waiting list for elective surgery, or children waiting for aplace in kindergarten) to grow without bounds, leading to (theoretically) in®nite costs.

When considering unbounded costs, it is important to emphasize a crucial di�erence betweenthe cases of ®xed capacity and quality targets: changes in the size of the private sector impactthe quality of service in the state sector when the state sector has a ®xed capacity (or, more gen-erally, a ®xed budget), but not if the state facility has a quality target, and faces no budget con-straint. This implies that the case of unbounded costs is only relevant in the ®xed capacity case.This is so because in the quality target case, the unboundedness of the costs only comes intoe�ect if the quality target is so low that it implies in®nite congestion costs, a not particularlyinteresting case. (In this case, costs in the state sector remain in®nite, whatever the private sectordoes.) We therefore focus on the ®xed capacity case.

We consider a simple queueing model to illustrate what might happen in such a situation.Assume that customers arrive according to a Poisson process with arrival rate l, and that theamount of time required for the service they need follows a negative exponential distributionwith mean 1/m. In this case, the average number of people in the system (either receiving serviceor waiting for service) can be shown to equal l/(mÿ l) (see any standard queueing theoretic text-book, for instance, Gross and Harris [9], p. 71). Consider, therefore, the following cost function:

C�l;m; i� � a1

�mÿ l��1� i� for l < m; and 1 otherwise:

For the monopolistic case, we can derive an optimal relationship between mp and lp, as ineqn (8):

mp � lp � lp

��a

cf �1� lp�r

: �18�

First, consider the case where the state facility has su�cient capacity, i.e. mse1. When the statefacility is highly congested (ms close to 1), it is optimal for the private facility to `skim the mar-ket', i.e. o�er a high-quality service to a small fraction of the market. lp then tends to 0 as ms

approaches 1. As state capacity increases, this strategy becomes less pro®table, making it opti-mal for the private facility to expand and attract more customers. The service level of the pri-vate facility deteriorates slightly. Further increases in state capacity force the private facility toclose down gradually. lp and mp then go to zero, while quality increases slightly under the press-ure of the improving quality in the state facility. For inadequate state capacity (ms<1), it is op-timal for the monopolist to attract as many customers as possible, while keeping costs in thestate sector at in®nity, i.e. lp=1ÿms.

The optimal strategy in the competitive case is comparable to the monopolistic case. Themain di�erences are a larger private market share, and potential pro®ts being dissipated by newentrants. If state capacity is inadequate (ms<1), the private sector takes a share 1ÿms of themarket, and provides these customers with `in®nitely' high-quality service. (As there is a strictlypositive number of customers with in®nite willingness to pay, there is always space for an ad-ditional entrant.) For ms=1, private capacity equals about three times the capacity of the mono-polist. As ms increases, the private market share initially increases, before gradually decreasingto zero. Private capacity decreases monotonically. Quality of service in the private sectordecreases monotonically.

ALTERNATIVE HETEROGENEITY ASSUMPTIONS

Alternative forms of heterogeneity [for instance, C(l, m, i) = a(l/m)/(1 + i)2 or C(l, m,i) = a(l/m)*(1ÿ i)] yield similar results, as long as costs remain bounded for all customers. Inthe case of a monopolist and ®xed capacity, there is a minor di�erence in that total capacity


reaches a minimum, and then starts increasing, before the increased state capacity forces themonopolist to leave the market.

The picture does change if the costs of the least patient customers are in®nite. Consider, forinstance, C(l, m, i) = a(l/m)/i, which goes to in®nity as i goes to zero. In the case of a monopo-list, the e�ect is similar to that in the previous section, with ms=1: it is optimal to o�er a veryhigh quality service to an in®nitesimal fraction of customers, independently of whether the statefacility has ®xed capacity or a quality target.

A similar outcome is observed for the competitive case but, again, with much higher levels ofprivate capacity.

CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH

In this paper, we have attempted to gain a better understanding of how a state service and aprivate competitor co-exist, when providing a service whose quality is determined solely by thelevel of congestion, and all customers must be served. We considered both a monopolisticentrant, and a competitive market. We concluded that the way in which the state facility is man-aged plays an important role, as this will a�ect the behaviour of the private competitor, and hisreaction to a change in the state facility's resources.

To obtain tractable results, we made speci®c assumptions about customer costs and customerheterogeneity. In our sensitivity section, we illustrated that the qualitative results were reason-ably robust with respect to this assumption.

When considering the NHS example mentioned in the introduction, it seems reasonable toassume that this service is attempting to minimize congestion given a resource constraint, ratherthan attempting to minimize costs given a quality target. It is worth emphasizing the similarityof results when limited resources are modelled as a ®xed capacity or as a limited budget (see [20]for details). This similarity of results, together with our analysis in the sensitivity section, indi-cate that our main conclusions are quite robust with respect to the various modelling assump-tions.

We could also consider a scenario where the state facility allocates its capacity partly to astate service and partly to a private service, and then study the resulting optimal allocation.This situation is closely related to the transportation literature mentioned earlier in the paper.The key di�erence is that here all customers must be served.

We did not consider the issue of tax income resulting from the pro®ts realized by the privatefacility being used to improve the state facilities. This issue deserves further investigation. Theimpact of a tax allowance towards covering the fees charged by the private facility also deservesfurther investigation, with a special focus on the case where the state has a ®xed budget to be al-located between tax subsidies and improving the state facility.

We assumed throughout that both facilities face an identical cost structure. The impact of taxmeasures in a situation where these cost structures di�er would also be of interest.

AcknowledgementsÐI would like to thank participants at seminars at London Business School and Stanford GraduateSchool of Business, and the November 1992 ORSA/TIMS conference for useful comments on an earlier draft of thispaper.

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