On the private sector provision of public goods

Abstract

Generally, private provisions of public goods are considered as a case of market failure. Below some of the private sector mechanisms of providing public goods are examined. A Kolm triangle is used for voluntary contributions mechanism and the provision of public goods via dominant assurance contracts are modeled using game theory. Private provision mechanisms like lotteries or via dominant assurance contracts led to an efficient allocation of public goods. However, the efficient provision of public goods is limited to some categories of public goods, not all. Also, much depends on risk aversion and how altruistic a society or an individual is.

INTRODUCTION

Public goods exhibit two distinct characteristics; it is non excludable and it is non rival in consumption. Because of these two attributes of the public good it is difficult to find a private supplier to provide these goods to the market. Public goods for example defense, street lightning etc are generally provided by the state or by the public sector. An additional characteristic of public good is that they generate externalities i.e. the economic effects which flow from their production or use to other parties or economic units. In contrast private good is rivalry in consumption and is subject to the principle of exclusion. It can be priced and those who do not pay for it are deprived of it. However there are few goods which are considered as ‘pure’ public goods. Most of the public goods lie between the spectrum of ‘pure’ and ‘impure’ public goods. Despite possessing such characteristics, there are mechanisms by which the private sector provides such goods. In this paper we an assessment is made of the mechanisms of providing public goods by the private sector. The paper starts with a literature review. This is followed by the types of private sector mechanism of provision of public goods where an assessment is made of each type of mechanism.

LITERATURE REVIEW

A number of articles have been published on the private sector provision of public goods and services which is a very much a debated issue in the literature. Some of the authors have cited that private sector performs reasonably well and ensure optimal production of public goods. Whereas, critics of private or market mechanisms advocate the greater role of the government to intervene and provide public goods by means of taxation since public good in nature is non rival and non excludable hence it is a typical case of market failure.

Helsley and Strange (1991) examined the competitive provisions of public goods under two exclusion regimes and showed that under these two regimes the provision of public goods is constrained Pareto efficient.

Morgan and Sefton (1997) experimented lotteries as a means of providing public goods. The results of the experiments found that lottery mechanisms perform better than voluntary contribution (donations and charities) in providing the public goods. Moreover, lottery mechanisms also help to alleviate the problem of free riding.

Van Dun (1984) mentioned that technical progress will make it possible to

supply more public goods in the market by making separable benefits that can

be financed by pricing.

Blanchette and Tolley (1997) studied the involvement of private and public

sector in health care systems in a number of OECD countries. Private

Insurance and out of payments type of voluntary contributions were

considered. The authors concluded that the trend is likely to result an increase

in greater degree of private sector provision in the case of health care.

Carman (2003) provided evidence that the decision for the groups to make

voluntary contribution towards public goods were influenced by social

influences i.e. the participation of the peers. Thus greater is the number of

peers more the voluntary contributions likely to be made by individuals.

Using the data of local government service delivery Warner and Hefetz (2002) concludes that market solutions towards the provision of public goods leads to

greater inequality and uneven distribution and hence cites the need of the government to internalize the costs.

Itaya, Meza and Myles (1997) mentioned that if all individuals contribute towards the provision of a public good their utilities are equalized even if the income distribution is unequal. It may seem that there is then no role for redistributive policy, but it is proved that social welfare can be raised by creating sufficient income inequality that only the rich provide public goods.

A paper examined the consequences of increasing the size of the community in the standard model in the private provision of public goods when costs are variable, where increase in the number of agents leads to a decline in the provision of public goods (Vicary 2004). Furthermore contrasted to an economy with constant costs there is no simple relationship between provision of public good and individual utility.

PRIVATE PROVISION OF PUBLIC GOODS

Voluntary contributions and the theory of ‘clubs’

Voluntary contributions by the individual members of the society comprises of

donations, campaign funds, charities etc.

In the USA for example, private funding of US medical research accounted for

55.8% of total in 1995, the bulk being industrial research on drugs. Among

112 art museums surveyed over the period 1986 to 1988 private and

corporate contributions accounted for 33.55 of revenues while total

government support comprised 31.6% with remaining revenues from

endowment earning and direct earnings like entry fees (Epple and Romano

2003).

Assume that there are two individuals i = 1, 2 and each consumes a private

good xi and one shared public good, G. individual I has preference ordering

over the pairs (xi and G) that is represented by a differentiable and strictly

quasi concave utility function, ui ( xi and G). Both the goods are assumed to be

strictly normal goods. It is also assumed that public good is produced at

constant marginal cost (MC). By choosing units suitably, the Marginal rate of

substitution (MRS) can be made equal to one. The individuals choose their

private contributions, gi to the public good. The total amount of public good

provided is determined by the sum of the individual contributions, G = g1 + g2.

Each individual solves

Ui (xi, g1 + g2)

s.t xi + gi = wi

xi + gi 0

It can be written more compactly by using the budget constraint t eliminate xi

Max ui (wi – gi, g1 +g2)

s. t 0 gi wi

Figure 1a shows a Kolm triangle for the simple model economy. A Kolm

triangle is the analogue of the Edgeworth box in an economy with public good

(Ley 1996).

The height of the equilateral triangle is given by the total amount of resources

available, w1 + w2. Since it’s an equilateral triangle, we have from any point

inside the triangle; we have from any point inside the triangle

x 1 + x2 + G = w1 + w2

Therefore any point inside the triangle is associated with a feasible allocation.

In allocation z, individual’s private consumption is given by the distance form z

to 0i0o. The amount of public good, G, is associated with z is simply given by

the distance from z to the base of the triangle, 0102

Figure 1b represents the individual’s indifference curves. To the right of the

dashed line which is parallel to 0102, individual 1 has more of the public good

than that at z. it follows that any other allocation in the set B must be better

than z for agent 1 since in B she gets more of both the goods than in z. In w,

on the other hand, individual 1 gets less form both the goods so she must be

worse off. The direction of the preferences is shown where individual’s

indifference curves are convex to their origin whenever the preferences are

quasi concave (Ley 1996).

Nash equilibrium is a vector of contributions (g*1, g*2) which solves the

individual’s optimization program above. Figure 1c shows Nash equilibrium, E.

Let A = (w1, w2) represent the initial allocation. When g2 = 0, agent one’s

opportunity locus is given by the segment AC which is parallel to 020o i.e.

along AC we have that x2 = w2. When individual two is contributing g*2 = A’J,

individual 1’s opportunity locus shifts to A’C’. The Nash, equilibrium, E, is

individual’s 1 optimal choice on her budget line A’C’. She contributes g*1 = A’’I

and consumes EH = w1 – g*1 of the private good. When individual one

contributes g*1, individual’s 2 opportunity locus shifts from AB (where g1 = 0)

to A’’B’’. (Note that AB and A’’B’’ are parallel to 010o) On A’’B’’ individual 2’s

most preferred point is E, where he contributes g*2. Since the individual’s

indifference curves goes through E, the Nash equilibrium is not Pareto optimal

(Ley 1996)

In the above case voluntary contribution leads to inefficient provision of public

goods since each individual has the incentive to minimize his contribution and

is thus likely to engage in strategic behavior which will involve in preference

distortion. In general voluntary contributions like donation and charities is

unlikely to provide public goods efficiently without the strong assumption of

true revelation of preferences i.e. individuals have an incentive not to actually

reveal how much he values the public good in hope of free riding. In other

words given the nature of the public goods, complete market failure occurs

when neither individual places sufficient value on the public good. Therefore it

is argued that the state should engage in the provision of public good a use

coercive action to reveal the true preferences of the individuals and thus

provide public goods at the optimal level (Brown and Jackson 1998; Connolly

and Munro 1999).

However smaller the community, the less the problem of truthful revelation

and free riding since individuals see themselves as being part of the

community; they are less likely to free ride. But larger the community, more

likely that individuals will engage in free riding and understate their true

preferences (Connolly and Munro 1999). In figure 1c, voluntary contribution

leads to inefficient provision of public goods. However, since some individuals

care more about others and when individuals are not identical voluntary

contributions like charities and donations can lead to optimal provision of

public goods i.e. when preferences are differentiable a Pareto optimal Nash

equilibrium is possible in figure 1c, which again is redrawn in figure 1d to

show the optimal provision in a simplified version. At point E the sum of MRS

of each individual between private and a public good is equal to Marginal rate

of technical substitution which is the top level Pareto condition for public good.

Secondly some individuals may be altruistic and as per as the Warm glow

model, voluntary contributions can lead to efficient provisions which can make

individuals contributing more or deter them to make free riding attempts.

(Gruber 2005)

Another private solution for optimal provision of public goods is known as the

club solution; and provision is only possible when good is excludable in some

way for example swimming pool and/or golf course etc. A club is defined as a

voluntary group deriving mutual benefits from sharing one or more of the

following: production costs, the member’s characteristics or a good

characterized by excludable benefits (Brown and Jackson 1998).

It is important that the good is excludable, so that it is possible to charge a

price and alleviate the problem of free riding for example sport facilities can

exclude no members; they are non rival because they can be shared by a

limited number of users. However as the club becomes larger it becomes

difficult to reap the benefits of facilities because of over crowding. Therefore

the size of the club will be optimized at where marginal cost (overcrowding)

equals marginal benefit (lower costs) (Connolly and Munro 1999). Note that

not all public goods can be entirely provided my means of clubs where it is

very costly to exclude non payers; in such cases club provision may lead to

market failure. For example local public goods like roads, sewage disposal,

traffic control system etc which are highly non excludable and non rival.

The above model is a Buchanan’s model of club theory. Individuals are

assumed to maximize utility which is a function of private good, a common

impure public good, X, and s, which is the size of the club. The first quadrant

shows the benefit and the cost curves where the former shows diminishing

returns to consumption and latter constant returns to scale. For a given

membership, s1; the optimal level of X is X1 where the slope of the benefit

curve is equal to slope of the cost curve. For a giving quantity of X1, for

example sports facility, as the size of the consuming group increases from s1

to s2, the benefit curves moves down because of the effects of congestion

while the cost per person fall as the membership increases. In quadrant 1, a

set of optimal combinations and output of the public good are established i.e.

{s1, X2}, {s2, X3}, {s*, X*}. These are than plotted in quadrant 4as the locus of

the line Xopt. In quadrant 2, the optimal membership is shown for given facility

sizes X1, X2 and X*. The shape of the benefit curve shows the increasing

benefits of a number of people associating with one another and then the

costs of the congestion. The falling costs curve shows the increase in the

membership. Optimal membership exists where both the curves are equal.

Thus s1 is the optimal club size for X1 and likewise for s2 and s*. These are

optima form a locus sopt in quadrant 4. The club optimal provision of public

good exists where sopt and Xopt intersects (Brown and Jackson 1998)

Dominant assurance contracts

Many types of public good can be provided by the profit seeking firms from a

modified form of assurance contracts, called the dominant assurance

contracts in which there a single dominant strategy which ensures that the

public good is provided. The concept behind this type of provision is that

many public good problems are contribution problems rather than revelation

problems. Each agent lacks the ‘assurance’ that others will contribute in the

event when he contributes. Tabbarok (1998) modeled the provision of public

goods via dominant assurance contract using a game theory which is

explained below.

Following is a two stage game called the public good game: in first stage the

entrepreneur ends the game immediately or offers to each of N agents a

contract. In second stage each of the N agents can accept or reject the

contract. If an agent accepts the contract, she receives a pay off which is

conditional on the total number of agents who accept it. If fewer than K agents

accept, the contract fails and each accepting agent receives from the

entrepreneur a payoff F (for Fail). If K or more agents accept the contract is

said to succeed and each accepting agent must pay the entrepreneur S (for

Succeed) and the entrepreneur produces the public good which will be worth

Vi = V to each agent and costs C in total. Letting X be the number of agents

who accept the contract, the contract and the payoff are presented in table 1.

Payoffs to the entrepreneur are therefore (-XF) if the contact fails, (XS – C) if

the contract succeeds and 0 if the entrepreneur decides not to offer a

contract. Payoffs to the non accepting agent are zero if the contract fails and

V if it succeeds. The entrepreneur and the agent are both assumed to be risk

neutral. The game is voluntary and the entrepreneur cannot non-contractually

impose costs on the agents, thus, F 0. In order to recover its cost the firm

must charge S 0 for C 0 is Nash equilibria. If all agents accept than a

deviator earns V (V – S). If all agents reject than a deviator earns F 0.

There are two types of pure strategy equilibria; one in which the contract

succeeds and the other in which it fails. If V- S 0 then the following is a pure

strategy sub game equilibria; K agent s accepts and the reminder rejects. A

rejecter cannot increase his pay off by accepting since V > V-S. An acceptor

cannot increase his payoff by rejecting since V-S 0. There are ( ) of these

equilibria, one for each possible set of acceptors. In every one of these sub

games the public good is provided.

If V-S < 0 then the following are pure strategy sub game equilibria; K-1 agents

accept and the reminder reject. A rejecter cannot increase his payoff by

accepting since if he accepts the contract succeeds and 0> V-S. An acceptor

cannot increase his payoff by rejecting since F>0. There are ( ) of these

equilibria.

These are the only pure strategy equilibria (see appendix)

The entrepreneur can always earn a zero payoff by exiting the game at stage

one. Thus the entrepreneur will always set V>S so that the contract will

succeed in all the pure strategy equilibria of the full game and the

entrepreneur will earn positive profits.

The entrepreneur thus wishes to maximize the profits subject to the condition

V>S

E = SK – C > 0

s.t. V-S 0

Maximum profits are reached when the entrepreneurs sets K = N and S = V –

epsilon. Maximum profits are then given by VN – C. Note that VN is the total

value of the public good and C, the total cost. The profit maximizing decision

therefore implies that a necessary and sufficient condition for the entrepreneur

to produce the public good is that it be efficient to do so i.e. VN>C.

To review, in the first stage of the game entrepreneur sets K= N and S= V –

epsilon. In the second stage all agents accept the contract. The all accept

equilibrium in the public good game is the unique sub game perfect Nash

equilibrium. Moreover, accept is the dominant strategy. Every agent has the

incentive to accept the contract regardless of what he believes what others

will do. Dominance makes the all accept equilibrium very strong.

When public good is excludable the accept is a dominant strategy for each

agent regardless of K. Assume N agents and bridge which costs C to

produce. Let the entrepreneurs offer the agents an (F, S, K) contract with the

additional provision that only accepting agents can consume the public good if

it is produced. In contract because non accepting agents earn zero while

accepting agents earn either V- S >0 or F>0. In the above model the profit

maximizing entrepreneur or firm earns the entire consumer surplus from the

public good.

Offering a contract is a very low cost activity and thus contract provision

should be contestable market. Competition will push s down to C/N so that the

public good provision will be efficient and will benefit consumers (Tabbarok

1998).

However the criticism that has been leveled against the dominant assurance

contracts is that it assumes homogeneous preferences and second it

assumes complete information. With heterogeneous preferences and

asymmetric information dominant assurance contracts are unlikely to provide

the public goods efficiently (Tabbarok 1998).

Secondly it is sometimes impossible to write a contract that is anywhere near

being complete because one cannot specify in advance every possible

contingency. (Rosen 2005)

Thirdly since in this type of public good game, the profit maximizing

entrepreneur captures the entire consumer surplus; the allocation of the public

goods by such mechanism will not be Pareto or allocative efficient.

Lotteries

Lotteries like lotto and Keno games are a decentralized mechanism for

funding public goods (Morgan 2000). A charitable fund raiser (or a

government organization) can link contributions to the public goods with the

chance of winning a prize in a lottery). Lotteries have an advantage that is it

does not require tax or transfer power on part of the organizations conducting

the lottery (Lang, List and Price 2007). Lotteries are type of joint public –

private good; because wages lead to a private prize as well as contributions to

the public goods. Currently in Britain private charities raise 8 percent (or

£500m) of their income through lotteries. Lotteries as a means of providing

public goods are practiced by majority of the states in US (Morgan 2000).

Under lottery mechanism an individual’s contribution (xi) towards the lottery

ticket purchase are used to fund the provision of the public good and the

contribution gives him (xi) a chance to win a lottery. Aggregate public good

provision is determined by the difference between the sum of the contribution

and the lottery prize, R i.e.:-

G = xj - R

The probability that individual i win the lottery is simply x i/ xj. Individuals or

agents who purchased the lottery tickets choose now xi to maximize. The

group putting on the lottery has access to a small amount of deficit financing .

The lottery is called off and the xi refunded if total contributions sum less than

R - . If total contributions exceed this amount, the lottery is held. Allowing for

the possibility of a small amount of deficit financing eliminates a Nash

equilibrium in which zero contributions are made by the all individuals or

agents. By adjusting or increasing the arbitrary price the mechanism can

arbitrarily come close to efficient level of provision. For a pure public good for

example lighthouse, the lottery mechanisms performs quite well as public

good provision increases with group size, even when the lottery price is held

constant. This is because of the nature of the lottery mechanism as outlined

above (Pecorino and Temimi 2007). Moreover, lottery mechanism performs

better than voluntary contributions like charities or donations in larger groups;

the negative externality to the other participants which partially offsets the

positive externality provided by voluntary contributions to the public good.

Moreover, lotteries obtain higher level of public good provision than voluntary

contributions because the lottery rules introduce additional private benefits

which serve the gap with social marginal benefit thus mitigating the tendency

to free ride and understating preferences (Lang, List and Price 2007).

However in contrast to a rival public good for example education and/or fire

protection, per capita provision is found to decrease as group size increases

even when lottery prize is proportional to group size (Pecorino and Temimi

2007).

Another drawback occurs with the lottery mechanism is risk aversion. The

lottery mechanism model with n individuals performs well who are mostly risk

neutral, if not all. But more risk averse the people are, the more likely the

lottery mechanism to fail in the provision of the public goods. If agents or

individuals are risk averse the utility is less from the chance of winning the

lottery prize. However, the expected utility can be increased by flattening

payoffs i.e. splitting the single prize into two or several smaller prizes. Hence it

may be optimal for the fund raisers to provide more than one prize (Lang, List

and Price 2007). Public goods provided by means of lotteries may lack quality

hence giving rise to economic inefficiency which requires government step up

and intervene. In current economic research there is an extensive debate on

weather lottery mechanism regarding equity and efficiency of this mechanism

as a fund raising instrument. Much of the analysis of these questions

examines lotteries to tax instruments and by these criteria researchers have

largely concluded that lotteries does not appear to be a efficient and equitable

means of revenue generation (Morgan 2000).

Perfect price discrimination and concept of complementary

goods

Public goods are a typical case of market failure because it is non rival hence

marginal costs is zero and it is non excludable. A profit maximizing firm will

never supply public goods to the market since efficiency requires that price be

equal to MC and since MC is zero efficiency requires that price be zero as

well. If the firm charges price greater than zero; there will a loss of consumer

surplus (figure 3).

However by practicing perfect price discrimination profit maximizing firms to

some extent can provide public goods to the market. This requires that two

conditions hold: 1) the firm knows each person’s demand curve 2) it is

impossible to transfer the good from one person to another. Under these

conditions the firm could charge a price based on willingness to pay. In this

case the demand curve of the consumer is the marginal revenue curve for the

profit maximizing firm (figure 4)

However there is the problem of preference revelation where individuals have

the incentives to understate values and hence free ride (Rosen 2005). And

secondly since perfect price discrimination appropriates entire consumer

surplus in producer surplus (figure 4); the subject might invite government

intervention on the grounds of efficiency. Even if the firm is successful in

providing the public good through perfect price discrimination it would not be

Pareto optimal since the firm captures the entire consumer surplus.

A private firm may be willing to provide a public good when it is in joint supply

or is complementary with a private good in which charges a price for the

private good which is rival and excludable and cross subsidizes the provision

of a public good. For example a private firm engages in the provision of a

private good like sun loungers and the price they receive for it; they can use

that amount to provide safe and clean beaches (Connolly and Munro 1999).

The provision of public good under this mechanism raises the question that

weather the standard of the service is maintained as profit and cost cutting is

the driving force for private enterprises. Secondly, although some cases of

public goods may not be optimally provided under private sector because

there may exist large economies of scale which requires government to act as

a single supplier (Cannadi and Dollery 2005).

Furthermore, a private sector supplier may fail to supply the public good to

market if the transaction costs associated with it are very high.

Vouchers

Rather expressing the consumer purchasing power in terms of cash but in

voucher or tokens. The National Health Service (NHS) UK uses vouchers as

a means to promoting health care to the people. For example each individual

who uses impure public goods like health or education could be given a

voucher which can be kept, redeemed or donated to the group exchange

which then in turn becomes the basics to public goods provision. For each

token or a voucher that an individual has it represents a payoff to him. Since

donations to the group exchange provide a non rival and non excludable

payoff, the free rider theory suggests that the subjects might well decide to

make no contributions to the group exchange, so that they could benefit from

everyone else’s donations while putting nothing in themselves (Rosen 2005).

However, if the individuals are altruistic and size of the community is feeling,

they may have some sense of social responsibility which makes them

contribute to the group exchange. This may lead to the optimal provision of

public goods. But not public goods could be provided by means of vouchers

and secondly even if the voucher mechanisms are successful in provision of

certain types of public goods; the qualities of the service or good will remain in

doubt.

CONCLUSION

Public goods have two important characteristics; they are non rival and non

excludable. Because of these two distinct attribute of the public goods they

are either left unprovided by the markets or underprovided thus resulting in

economic inefficiency and thus citing the need of the government to intervene

and provide the public good by means of taxation where everyone contributes

to the public good.

However there are private sector or market mechanisms that can also ensure

the optimal provision of public good to the community. But there are problems

that undermine such mechanisms; the problem of preference revelation, high

costs and quality of the service of the public good and equity/fairness issues.

These problems are either specific or common to private mechanisms

mentioned above. The extent private sector will be successful in supplying

optimal quantities of public good depends on risk aversion and how altruistic

the society is. Nevertheless there are some special categories of public goods

where optimal private provision can occur.

APPENDIX

Pure strategy equilibria

For each possible relationship between V-S, F and 0 we check for pure

strategy equilibria. The necessary and sufficient condition for pure strategy

Nash equilibria are that neither acceptors nor rejecters can improve their

payoff by switching to the other strategy. The table illustrates weather the

condition hold under each of the parameter specifications. For example, in the

first table the acceptors earn V- S when accepting zero and zero when

rejecting thus V-S >0 is a necessary and a sufficient condition for K agents to

accept.

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