Private provision of a discrete public good with uncertain cost

Journal

of Public

PRIVATE

Economics

42 (1990) 357-370.

PROVISION

North-Holland

OF A DISCRETE PUBLIC UNCERTAIN COST

GOOD

WITH

Shmuel NITZAN Bar Ilan University, Ramat-Gan 52100, Israel

Richard E. ROMANO* University of Florida, Gainesville, FL 32611, USA Received

February

1989, revised version

received

November

1989

It is known that a discrete public good is efficiently provided in the subset of ‘undominated equilibria’ (those not Pareto dominated within the set of Nash equilibria). We make the cost of the discrete public good uncertain at the time the contribution game is played. This can lead to strikingly different results. Often, the public good is underprovided in any Nash equilibrium and there is a unique undominated equilibrium. These results hold for some distributions when there is arbitrarily little uncertainty and always when there is enough uncertainty.

1. Introduction

There is a growing literature that has been concerned with the analysis of voluntary contribution games with discrete public goods [Palfrey and Rosenthal (1984, 1988), Gradstein and Nitzan (forthcoming), Bagnoli and Lipman (1989)l.l Binary public goods, or threshold public goods, constitute an important class of such goods. A fixed level of a binary public good is provided if contributions are sufficient to pay for it. Palfrey and Rosenthal (1984) analyze the voluntary contribution game of complete information *This paper was written while Nitzan was a visitor at the Department of Economics at the University of Florida and at the Workshop in Political Theory and Policy Analysis at the University of Indiana. He is indebted to both institutions for their hospitality and financial support. Both authors are grateful to Mark Bagnoli, James Friedman, Mark Gradstein, Jonathan Hamilton, Eric Maskin, Steven Slutsky, Edward Zabel, the participants of the Joint University of Pittsburgh and Carnegie-Mellon Economics Theory Workshop and the University of Florida Industrial Organization and Public Finance Workshop, and an anonymous referee for their helpful comments and discussions regarding an earlier draft of this paper. Any remaining errors are ours. ‘Most work on private provision of Dubiic goods has dealt with a complete information setting where individuals’ strategies, the* payoff- functions and, in particular, the production function of the oublic good are continuous rolson (196% Chamberlin (1974). McGuire (1976L Cornes and Sanbler (1986), Bergstrom, Blum> and \iarian (1986)]. In the standard (continuous) model the equilibrium provision level of the public good is inefficient - due to the free-rider problem the public good is underprovided. 0047-2727/90/$3.50

0

1990, Elsevier Science Publishers

B.V. (North-Holland)

358

S. Nitzan and R.E. Romano, Private provision of a discrete public good

under the restriction that individuals can only adopt dichotomous strategies where they either contribute zero or some fixed positive amount. More recently, Bagnoli and Lipman (1989) have studied the analogous case but without the restriction on contributions, i.e. allowing for continuous contributions. Their game generally has multiple Nash equilibria in pure strategies and the subset of equilibria which are not Pareto dominated within the set of equilibria are efftcient.2 The attainment of efficiency runs counter to one’s intuition. Moreover, the model fails to predict a particular outcome because, in general, there are an infinite number of undominated equilibria. In this paper we extend Palfrey and Rosenthal’s and Bagnoli and Lipman’s game by introducing uncertainty regarding the cost of provision of the public good at the time contributions are committed. This extension may often be more realistic. Consider Palfrey and Rosenthal’s example of MCI’s successful case against AT&T wherein MCI was granted access to AT&T’s then local telephone loops. This established a precedent which made obtaining similar access less costly for other long-distance firms. In short, MCI provided a discrete public good. It is probably more realistic to assume that, at the time MCI was preparing the case, the probability of winning was an increasing function of the resources devoted to it, rather than Palfrey and Rosenthal’s assumption that winning required some known fixed expenditure. It is the former structure that we consider here. Another example is that where a community attempts to attract a national sports franchise by offering a package including some number of committed season tickets sales. In this example, our conjecture is that increasing the number of committed season ticket sales is better regarded as increasing the probability of success in attracting the sports franchise. The consideration of uncertainty is important since it can lead to strikingly different results. First, there is often a unique undominated equilibrium. This can be true even when there is ‘very little’ uncertainty and is generally true when there is ‘sufficiently much’ uncertainty. Second, this equilibrium is inefficient. Less is contributed than the first-best amount. This is manifested in the public good being produced less often than it should be produced. Although we make the case that equilibrium will often differ fundamentally under uncertainty, this will not always be true. Hence, the outcome can be analogous to that which is obtained in the certain case of a discrete public good in spite of the fact that payoff functions will be continuous. This raises the question as to what differs between the usual case of a nonstochastic continuous public good and that of an uncertain discrete public good. We will sort out this difference. ‘This is not exactly how Bagnoli and Lipman (1989) present their result. Our modified presentation is for a more cohesive presentation of our points. It is also notable, though obvious, that analogous results are obtained in the Palfrey-Rosenthal (1984) model where the constraints on contributions are inconsequential.

S. Nitzan and R.E. Romano, Private provision of a discrete public good

359

This paper proceeds as follows. In the next section we set out the model. In section 3, the main results are developed. A final section contains concluding remarks.

2. The model Consider a simple model with two commodities, a public good denoted s and a private good denoted y. The stochastic production function of the public good is given by:

I

1, if i

g(%...,cN)=

0,

cjzCc,

j=l

if i

(1) cj
j=l

where cj, j = 1,. . . , N, is individual j’s contribution (in terms of the private good) to the production of the public good, and C, the cost of providing the public good, is a random variable that has a continuous distribution F with density function f and support [a, b] E [0, cc). We assume that f > 0 and f is continuously differentiable in the interior of its support.

(A.l)

The values of s are restricted to 0 and 1 reflecting the assumed binary discreteness of the public good. The public good is produced if the aggregate contributions are sufficient to cover the cost of production. Individual j is endowed with wealth wj which he allocates between the private good yj and his contribution cj. We assume that the utility function of individual j is of the form: uj(s,

Yj) =

uj(s)

+

Yj,

(2)

and, for all j, uj( 1) > ~~(0). It is convenient to define: dj~U,(l)--j(O),

(3)

which can be interpreted as individual j’s monetary value of provision of the public good. To avoid corner solutions, we also assume that no individual would ever be willing to exchange all his income to obtain the public good. Specifically: dj
for all j.

(A4

360

Finally,

S. Nitzan and R.E. Romano, Private provision of a discrete public good

it is assumed

C Aj>a,

that (A.3)

to abstract from the trivial case where the socially efficient contribution level equals zero. (Sums are over all j unless otherwise indicated.) The players in our contribution game decide on the allocation of their resources before the realization of C, but with common knowledge of F and of the following refund policy. If the equilibrium aggregate contribution is less than the realization of C, then the public good is not produced and all contributors receive a full refund. If the aggregate contribution is equal to or greater than the realized C, then the public good is produced and any excess is not refunded to contributors. Rather, we assume it accrues as ‘profits’ to the provider of the public good. It is important for the eventual analysis of efficiency to recognize that excess contributions are not losses to society, but rather, a transfer to the producer of the public good. The strategy set of individual j is then [0, wj], and his payoff function is given by the expected utility E[Uj(s, Wj-Cj)]. A few comments about the model are in order. First, our no-refunds assumption when there is an excess of contributions is chosen only for simplicity. The results are qualitatively unchanged when there are refunds of various forms.3 Second, we assume full refunds when the equilibrium contribution is insufficient to cover the realization of cost. Whether refunding or not is more realistic in this event depends on the application (e.g. refunding better describes the above example of trying to attract a sports franchise but no refunding better describes the MCI case example), but the results hold a fortiori in the no refunding case.4 The intuition is that refunding removes an incentive to free ride which Palfrey and Rosenthal (1984) refer to as the ‘fear incentive’. Third, we do not permit cost-contingent strategies. That is, an individual cannot choose his contribution as a function of the realization of uncertain cost. If individuals could, then the uncertainty is of no consequence. We think that our characterization is more realistic in many cases, particularly since our results will often require only very little uncertainty. Fourth, note that each individual has complete information about the technology, including the density function of the random variable C, and about the wealth and preferences of the other individuals; and this information is assumed to be common knowledge. The contribution game is thus one of imperfect information (rather than incomplete information) because nature makes a random move which specifies the technological 3Probably the simplest case is where any excess is divided equally among contributors. It is straightforward to verify our results in this case. 4Actually, the analysis of the no refunding case is quite simple. Contact the authors for details. Also, see Waldman (1987) for an analysis of a special case of voluntary contribution to a discrete public good with uncertain cost, which has no refunding and identical players.

S. Nitzan and R.E. Romano, Private provision of a discrete public good

361

coefhcient C. With such a game the standard Nash equilibrium solution concept is appropriate. A more complex Bayesian-Nash equilibrium is not required. A final point regards some terminology we adopt. We will encounter three types of equilibria below; those where the probability the public good is produced equals zero, is between zero and one, and equals one. We refer to these equilibria respectively as nonsupporting, partially supporting, and totally supporting, the notion being the extent to which contributions support production of the public good. 3. Existence, uniqueness, and efficiency of equilibrium The first proposition regards the uncertainty case and is not new. It is a simplification of one of Bagnoli and Lipman’s results and is presented for purposes of comparison. In its statement and throughout this paper, by an ‘undominated equilibrium’ we mean a Nash equilibrium that is not Pareto dominated within the set of Nash equilibria. Proposition 1. Where f is degenerate so that C=a is known before contributions are chosen, the set of undominated equilibria are Pareto efficient equilibria. (Henceforth, by ‘efficient’ we mean ‘Pareto efficient’.) There are an infinite number of undominated equilibria. See the appendix.

Proof.

There may be other equilibria in this game, e.g. cj=O for all j, but such equilibria are inefticient and dominated.’ We will show that the set of undominated equilibria in the uncertainty case often contains only one element which is an inefficient contribution vector. Our focus on undominated equilibria is not without reservation. However, in our problem where there are dominated equilibria present, they will (with one noted exception) be associated with zero probability that the public good will be produced; and, hopefully, individuals can avoid playing such strategies. Turning now to the uncertainty problem, in a Nash equilibrium, the ith individual will maximize his expected utility given the refund policy and taking the contributions of others as fixed. He solves maxE[U,]=(Ui(l)-Ci+wi)F(~cj)+(uj(O)+Wi)[1-F(~Cj)],

(4)

Ci$O

where we need not employ the constraint can rewrite the problem: ‘Unless

J.P.E

E

di>a

for some i, there will be such dominated

that ci 5 wi because of (A.2). We and inefficient

equilibria.

362

S. Nitzan and R.E. Romano, Private provision of a discrete public good

where pi is the expected increase in utility from stochastic production public good. We assume that Oi is quasiconcave in ci for ci E [0, wJ and for all 1 cj.

of the

(A.4)

.ifi

It is shown in the appendix that a sufficient condition for (A.4) is that fO is nonincreasing in C. F(C)

(A.5)

We will sometimes adopt (AS) as an assumption which is actually quite general. It is shown in Barlow and Proschan (1965) that (A.5) defines the set of Polya Frequency Distributions of Order 2. This is a set of unimodal densities which includes (with appropriate restrictions on parameters) the uniform, exponential, beta, gamma, Weibull, normal, and truncated normal.‘j Given (A.l)-(A.4), any vector c* which satisfies either of the following systems is a Nash equilibrium in pure strategies: -F(CcT)+(di-Ci*)f(Ccj*)=O,

for c:>O,

(6.1)

-F(Ccj*)+dif(Ccf)~O,

for cz = 0,

(6.2)

andxcJ
(6.3)

or -1 +(di-Cr)f(b)>=O,

for cl > 0,

(7.1)

- 1 + Aif( b) 5 0,

for CT= 0,

(7.2)

andCcf=b.

(7.3)

The system (6.1)46.3) describes an equilibrium with F(xcj*) < 1. For those individuals with c: >O, aUi/aci must vanish. For those with cj+=O, aoi/aci

6Actually, Barlow and Proschan (1965) use concavity of In F to define this set of distributions, but it is very easy to verify this is equivalent to (AS) for continuous and differentiable distributions.

S. Nitzan and R.E. Romano, Private provision

ofa discrete

public good

363

can be negative. The system (6.1)<6.3) also subsumes the case of a nonsupporting equilibrium where F(x CT)= 0. The system (7.1)47.3) describes a totally supporting equilibrium where F(xcj*)= 1. rfi is nondifferentiable at ccj= b when f(b)>O, and this is why (7.1) differs from (6.1). When this system is satisfied the public good is always produced - no matter what the realization of C. Before describing the conditions under which the various types of equilibria arise, we show that there generally exists a supporting equilibrium. Proposition 2. Under (A.l)-(A.4) pure strategies with F(x CT) > 0.

there always

exists

a Nash

equilibrium

in

We show that there exists an equilibrium with F>O in an artificially restricted game and then show it to have properties that imply it is as an equilibrium in the game of interest. Let the modified Cartesian product of strategy sets be given by S= {c: c cjz a and, for all j, 05~~5 wjj. Clearly, S is a convex set. Moreover, (A.2) and (A.3) guarantee that S has an interior. Using (A.4) and, by immediate application of Friedman’s Theorem 2.4 (1986, p. 39), there exists an equilibrium in pure strategies in the restricted game with CGS. Now we show that any equilibrium vector c in the restricted game will not satisfy c cj=a. Assume that it does. Then this equilibrium is nonsupporting, i.e. F(x cj) = 0. Moreover, for at least one individual i, ci< di. This follows since, otherwise, cjz dj for all j; and thus F(x Cj)zF(x dj) >O [the last inequality by (A.3)], which contradicts F(x Cj)=O. We now show that the ith individual would have a better response in S. We have aiJi/aci= -F’(a) + (di-ci)f(a)=(di-ci)f(a). If f(a)>O, th en it follows immediately that there is a better response. If f(a) >O, then using a Taylor expansion, one can show that 13u,/&,>O for an arbitrarily small increase in ci.7 Hence, &‘(I cj) =0 yields a contradiction, which implies that any equilibrium in the restricted game must satisfy F>O. Since any equilibrium in the restricted game is not on the ‘lower’ boundary of S where xcj=a, it is clear that the equilibrium values must satisfy either the system (6.1)+6.3) or (7.1)-(7.3). The proof is now complete since F>O for any equilibrium in the restricted game, which is then also an equilibrium in the unrestricted game. Q.E.D. Proof:

Although there may also exist nonsupporting

equilibria, they are domi-

‘Doing a Taylor expansion on -F(a)+(d,-q)f(a) one obtains: -F(a+&)+ a )E, which is positive if f’(a) > 0. Sincef(a) = 0 and f is a density Cd,-(ci+e)]f(a+e)z(di-ci)f’( function, f’(a)zO. If the latter holds &ith equality, then one must look at the next higher order term to sign -F(a+e)+[d,-(c,+E)]f(a+&), but the fact that f(a)=0 and f is a density function ultimately implies that the expression is positive.

364

S. Nitzan and R.E. Romano, Private provision of a discrete public good

nated by any supporting equilibria and we have little more to say about them. The next proposition describes the conditions under which partially supporting and totally supporting equilibria arise. The following definition facilitates its statement. Let Ci G max (0, di - l/f(b)).

This is the maximum supporting equilibrium.

contribution

(8) of

the

ith

individual

in

a

totally

Proposition 3. Suppose (A. l)-(A.4) are satisfied. (a) cCjz b is necessary and sufficient for the existence of a totally supporting equilibrium. (b) cCj< b is necessary and sufficient for the existence of a partially supporting equilibrium. (C) Zf Ccj>b, then there are multiple undominated equilibria. If 1 Cj< b, then there is a unique undominated equilibrium. If, in addition, (A.5) is satisfied, then there is a unique equilibrium with F >O. Proof. (a) We prove sufficiency by construction. Let ~=(a~, . . . , .zN) denote any vector such that aj=O if Cj=O, ~~20 if cj>O, Cj-.sjzO, and C(Cj-sj)=b. That cCj>, b implies that such a vector exists. It is straightforward to verify that the vector with elements Cj-Ej satisfies (7.1)-(7.3) and is therefore a totally supporting equilibrium. We prove necessity by contradiction, Suppose c Fj < b. Then, in any vector which satisfies (7.3), CT >Ci for some i; and thus also CT >O. But, using (8), this implies that (7.1) is not satisfied for i. (b) We prove necessity by showing there is no equilibrium with 0 < F < 1 if c Cj 2 b. This is done by contradiction. Assume c Cj 2 b. An equilibrium with F < 1 must satisfy (6.1)-(6.3). The previous assumption and (6.3) imply cz O, so that neither (6.1) nor (6.2) can be satisfied for i. Sufficiency follows from Propositions 2 and 3(a). (c) Consider the first statement in Proposition 3(c) and refer to the sufficiency section of the proof of part (a) of this proposition. Given 1 Cj > b, it is clear that one can find multiple c-vectors and associated equilibria where, in comparing any two of the equilibria, some individual(s) contributes more and some less. Hence, within this set of equilibria, no equilibria are dominated. Moreover, part (b) of this proposition implies that there are no partially supporting equilibria so that this must constitute the set of undominated equilibria. Now consider the second statement in Proposition 3(c). There are no

S. Nitzan and R.E. Romano, Private provision of a discrete public good

365

totally supporting equilibria, so we must focus only on partially supporting equilibria. We show first that, if there are multiple partially supporting equilibria, then cc? must vary across them. This follows from the fact that, given cc?, CT is uniquely determined for all i, as can be seen by inspection of (6.1) and (6.2) taking note that xc? fixes F and f. Now we show that the partially supporting equilibrium with the highest cc; dominates. Consider two equilibria, denoted by c* and c**, where c cF* >c cJ. We show next that CT*2 CT for all j. Suppose to the contrary that there exists i such that c** Ai-c~. Using (6.1), this implies that ;(I &;!/(F?*) > d(c LT)/f(C 8) which using (6.1) and (6.2), implies that no individual his increased his coAt;ibutioi in going from the ‘*-equilibrium’ to the ‘**-equilibrium’. This contradicts cc?* >c CT, and so CT*zcj* for all j. Finally, then, every individual is strictly better off in the **-equilibrium since other individuals have collectively increased their contributions (and vi is monotonically increasing in cjzicj) and the individual in question chooses to increase (or possibly not change if CT=0) his contribution.8 Consider the last statement in Proposition 3(c). We have just shown that if there are multiple partially supporting equilibria, then cc? differs across them. Under (AS) this cannot be true, as is easily shown by contradiction. Suppose again there are two equilibria with xc?* >ccj*. (AS) implies that F(x cj**)/f(~ cj**)L F(x c~)/f(~ ~7). Th en, (6.1) and (6.2) imply CT*5 cj* for Q.E.D. all j. This contradicts 1 CT*> 1 cj*. Either there will be totally supporting equilibria or at least one partially supporting equilibrium, but not both. Under a fairly general condition [(A.S)], there will be a unique partially supporting equilibrium where no totally supporting equilibria exist. If (A.5) is unsatisfied, there can be multiple partially supporting equilibria, and this is the exception to the case that all dominated equilibria are nonsupporting. The case where totally supporting equilibria exist is similar to the certainty case as we discuss further below. However, partially supporting equilibria may be more common as suggested by the following proposition. Proposition 4. Suppose (A. l)-(A.4) are satisfied. (a) A necessary condition for the existence of a totally supporting equilibrium is that c Ajz b. (b) If Aj< l/f(b) for all j, then there does not exist a totally supporting equilibrium. ‘Taking any individual i, it must be that other individuals have collectively increased their contributions, i.e. C,+cf* >Cj+icj *. That cf*zcT for all j rules out ~jzic**<~jzic~. If this sum holds with equality, then CT*=@ by (A.4), which would then contradict ~jc:*>CjCT.

366

S. Nitzan

and R.E. Romano, Private provision of a discrete public good

(c) As uncertainty, measured by the range of variance, increases beyond some finite value of either measure, no totally supporting equilibria exist. Proof.

(a) From Proposition 3(a), this is easily verified using (8). (b) Using (8), the condition implies that Cj=O for all j; and, consequently, the result follows from Proposition 3(a). (c) Since a 20, b increases monotonically as the range is increased. The result then follows from part (a) of this proposition. Now consider using the variance (a’) as a measure of uncertainty. We have: c2=j(C-E[C])2f(C)dC<(b-a)2if(C)dC (I

(I

=b2+a2-2absb2+a2<2b2,

where the last two inequalities use the non-negativity of a. Then, using again part (a) of this proposition, for ~‘>2(xd~)~, no totally supporting equilibria exist.g Q.E.D. Proposition 4(a) indicates that if there is some probability that the public good will cost more than its aggregate value, then undominated equilibrium will be unique and partially supporting. Proposition 4(b) implies the same, under the relevant restriction, even if there is arbitrarily little uncertainty. This applies to any distribution where the density vanishes at the upper bound of its support, e.g. consider the exponential distribution with arbitrarily small variance (but adjusted for reasonableness so that a>O). Proposition 4(c) implies that the same is true whenever there is ‘enough’ uncertainty present. These results are important not only because of the uniqueness properties of equilibrium [recall Proposition 3(c)], but also because partially supporting equilibria are always inefficient, as we show next. Any Pareto efficient vector of contributions must maximize the sum of expected consumer utilities and expected profits of the producer of the public good. lo This sum, ‘expected welfare’, is given by: ‘An alternative proof shows that f(b) converges to zero as the variance grows (for a fixed mean) and then appeals to Proposition 4(b). This is notable since it does not require that cdj be finite, thus applying to an economy with an infinite number of consumers. Contact the authors for details. “This is so because the underlying utility functions and profit are linear in contributions in our model. Hence, utility is ‘transferable’. The private good serves as an indirect means of transferability. This means that the expected utility/profits frontier has a slope of minus one in all directions. Consequently, any Pareto ellicient combination of expected utilities/profits must maximize the unweighted sum of expected utilities and expected profit.

S. Nitzan and R.E. Romano, Private provision of a discrete public good

367

(9)

It is apparent by inspection of (9) that expected welfare depends on the aggregate contribution (1~~) and not the distribution of contributions.” The necessary and sufficient condition for Pareto efficiency is:

There

“This is because only a transfer.

the substitution

of one individual’s

contribution

for another’s

constitutes

368

S. Nitzan and R.E. Romano, Private provision of a discrete public good

the equilibrium contribution is below the efficient one by contradiction. Suppose 1 CT21 Aj. Then, for at least one individual i, c: 2 Ai. But then (6.1) cannot be satisfied. That is, for individual i, Cl? Ai cannot be a best response. Hence, c cy -CC Aj in equilibrium. The second statement in the proposition then follows, as does necessity in the first statement. Q.E.D. Inefficient provision arises in a partially supporting equilibrium as it does in the ‘standard continuous case’ of a voluntary contribution game [e.g. as in Bergstrom et al. (1986)]. In both cases, each individual’s private marginal benefit from contributing is declining and continuous in the vicinity of equilibrium, each individual who makes a positive contribution equates his private marginal benefit to the marginal contribution cost, and inefficiency results since the sum of the private marginal benefits exceeds the marginal contribution cost. In short, there is free riding. This is in contrast to the efficiency that results in any undominated equilibrium in the PalfreyRosenthal-Bagnoli-Lipman certain discrete problem. Although we have made a case for the prevalence of partially supporting equilibria in our problem, it is at least a theoretical possibility that multiple totally supporting and efficient equilibria will arise. It is interesting that here our uncertain discrete problem seems to correspond more closely to the certain discrete problem. For there to exist a totally supporting equilibrium in our problem, there must be a discontinuity in the private marginal benefit functions at 1 cj= b. Those who contribute positively are at a corner where there private marginal benefits are no less than the marginal contribution cost and drop to zero for a marginally increased contribution. Although the sum of private marginal benefits will exceed the marginal contribution cost, social optimality does not dictate an increased contribution since the social marginal benefit accordingly drops to zero. Hence, it is a discontinuity in the private marginal benefit functions, or, equivalently, a nondifferentiability of the payoff functions which can correct the free-rider problem. It is not necessary to have the discontinuity in the payoff functions which occurs in the certain discrete case to obtain efficiency.

4. Concluding remarks Recently, Palfrey and Rosenthal (1984) and Bagnoli and Lipman (1989) have drawn attention to the problem of the private provision of a discrete public good with certain cost. This contribution game has multiple undominated Nash equilibria, all of which are efficient. We show that uncertainty about cost often reverses both of these results. This is so if there is ‘very little’ uncertainty given some restrictions. The reversal of the results is always valid if there is sufficiently much uncertainty. The loss of efficiency is

S. Nitzan

and

R.E. Romano, Private provisionof a discrete public good

369

associated with the re-emergence of free riding as in the more standard nondiscrete voluntary contribution game. Underlying the results is that adding uncertainty to the discrete case changes payoffs’ from discontinuous to continuous and (almost everywhere) differentiable functions of contributions. Hence, it seems safe to conjecture that generalizations of the model which preserve this property will lead to analogous results. For example, one could introduce risk aversion by making utility a concave function of income. Another extension would be to introduce altruism by making utility depend also on one’s own contribution [see Palfrey and Rosenthal (1988)]. A more complicated extension would have incomplete information about players’ preferences, requiring a Bayesian-Nash approach. Appendix Proof of Proposition 1. Any vector c =(cl,. . , cN) satisfying 1 Cj= a and Ajz cj for all j is a Nash equilibrium. Any individual would experience

decreased utility by increasing his contribution. Alternatively, if the ith individual decreases his contribution, then the public good is not produced. With full refunds, the individual’s change in utility would equal at most ci- A,sO, and, consequently, there is no incentive to reduce one’s contribution. From (A.3), there are clearly an infinite number of such equilibria. Moreover, any such equilibrium is efficient since (A.3) implies that the public good ought to be produced, and contributions are at a minimum. Since these equilibria are efficient, they cannot be dominated. Q.E.D. Demonstration

that (A.5) is sufficient for (A.4).

Taking derivatives:

-

z=

(a.11

-F+(Ai-c,)f

I

and

2

g=f I

-f

+j7.(Ai-ci)

dCf(;f(x)l

I

Note first that there is an initial range CiE [0, a--_Cjzicj] over which oi=O if If there does not exist an interior local minimum for CiE [max(O, a-_Cizi~j), Wi], then (A.4) follows. This is shown to be true by demonstrating that where aoi/~ci vanishes in this range, d2 Oi/&$ ~0. From (a.2) and (A.5), to have a’O,/ac? 20 (given aui/aci=O) would require that f =0 and/or ci 2 A. Using (a.l), either of the latter imply a8Jaci ~0, a contradiction. Hence, we have quasiconcavity. Cj+iCj
J.PE.

F

370

S. Nitzan and R.E. Romano, Private provision of a discrete public good

References Bagnoli, M. and M. McKee, Can the private provision of public goods be efficient?: Some experimental evidence, Economic Inquiry, forthcoming. Bagnoli, M. and B.L. Lipman, 1989, Provision of public goods: Fully implementing the core through private contributions, Review of Economic Studies 56, 583-602. Barlow, R.A. and F. Proschan, 1965, Mathematical theory of reliability (J. Wiley & Sons, New York). Bergstrom, T., L. Blume and H. Varian, 1986, On the private provision of public goods, Journal of Public Economics 29, 2549. Chamberlin, J., 1974, Provision of collective goods as a function of group size, American Political Science Review 68, 707-716. Cornea R. and T. Sandler, 1986, The theory of externalities, public goods, and club goods (Cambridge Univeristy Press, New York). Friedman, James, 1986, Game theory and applications to economics (Oxford University Press, New York). Gradstein, M. and S. Nitzan, 1990, Binary participation and incremental provision of public goods, Social Choice and Welfare 7, 171-192. McGuire, M., 1976, Group size, group homogeneity and the aggregate provision of a pure public good under Cournot behavior, Public Choice 18, 107-126. Olson, M.,-1965, The logic of collective action (Harvard University Press, Cambridge, MA). Palfrev. T.R. and H. Rosenthal. 1984. Participation and the provision of discrete public -goods: A strategic analysis, Journal of Pubhc Economics 24, 171-i93. Private incentives effects of incomplete information and altruism, Journal of Public Economics Studies

54, 301-310.