A utilitarian approach to the provision and pricing of excludable public goods

Journal of Public Economics 89 (2005) 1981 – 2003 www.elsevier.com/locate/econbase

A utilitarian approach to the provision and pricing of excludable public goods Martin F. Hellwig Max Planck Institute for Research on Collective Goods, Kurt-Schumacher-Str. 10, D-53113 Bonn, Germany Received 8 June 2004; received in revised form 7 December 2004; accepted 16 December 2004 Available online 11 April 2005

Abstract This paper studies utilitarian welfare maximization in a large economy with an excludable public good where individual preferences are private information. If inequality aversion is large, optimal allocations involve the use of admission fees and exclusion to redistribute resources from people who benefit a lot from the public good to people who benefit little. If inequality aversion is close to zero, optimal admission fees are zero. Because of inequality aversion, information rents of people who benefit a lot from the public good receive less weight, so optimal provision levels for the public good are below first-best levels. D 2005 Elsevier B.V. All rights reserved. JEL classification: D61; D63; H21; H41 Keywords: Public-good provision; Admission fees for excludable public goods; Utilitarian welfare maximization

1. Introduction Ever since the seminal contribution of Samuelson (1954), the theory of public-goods provision has mainly focussed on goods that exhibit nonexcludability as well as nonrivalry in consumption. Less attention has been paid to goods that exhibit nonrivalry in consumption and yet allow for the possibility of individual exclusion. This neglect reflects the assessment that, in a first-best world, people should not be excluded from the

E-mail address: [email protected]. 0047-2727/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jpubeco.2004.12.006

1982

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

enjoyment of a good that exhibits nonrivalry and whose use involves no externalities from crowding or from mutual annoyance of participants. If it does not cost anything to provide an additional person with the opportunity to enjoy the public good, the ability to exclude individuals should not be resorted to. Samuelson (1958) actually used this argument to justify his focus on public goods exhibiting nonrivalry and nonexcludability at the same time. The present paper challenges this view. In a second-best setting involving private information about individual preferences, equity concerns can justify the active use of the ability to exclude individuals despite the efficiency losses that are thereby induced. If information about individual preferences is private, people with different preferences will end up with different payoffs. In particular, people who care a lot for the public good will end up being better off than people who do not care for the public good at all. Because people can always dissimulate their preferences, their contributions towards the provision of the public good cannot be made commensurate with the benefits they obtain. People who care a lot for the public good will therefore earn information rents. The distributive impact of these rents is reduced if admission to the public good is conditioned on the payment of a fee. By paying for admission, people who care a lot for the public good contribute more to its provision than people who do not care for it at all. For a given level of public-good provision, the information rents of people who care a lot for the public good are then lower, and the welfare of people who care little for the public good is higher. This equity gain may outweigh the efficiency loss from excluding people who care a little for the public good, but not enough to pay the admission fee.1 The paper studies this equity-efficiency tradeoff in a model of public-good provision by an inequality-averse utilitarian planner. The choice of an optimal admission rule is shown to depend on the planner’s degree of inequality aversion. If inequality aversion is low, an optimal incentive-compatible allocation involves completely open admissions and zero admission fees. If inequality aversion is high, an optimal incentive-compatible allocation involves positive admission fees and the exclusion of anybody who refuses to pay the fee. People who do not care for the public good at all then contribute less towards its provision. For extreme levels of inequality aversion, these people’s net contribution to the provision of the public good will actually be negative: an excess of admission fee revenues over provision costs can be redistributed in a lump-sum fashion, providing a net benefit to those people who do not care for the public good at all. The underlying logic of the analysis is well known from the theory of optimal income taxation initiated by Mirrlees (1971). If people differ in some unobservable characteristic, first-best incentive-compatible allocations can have undesirable distributional properties. A utilitarian planner may therefore prefer to forego first-best efficiency in order to improve on the distributional properties of the allocation. In the theory of optimal income taxation, people are assumed to differ in their earnings abilities. In this paper, they differ in their ability to benefit from the enjoyment of the public good. Though politically less relevant than differences in earning abilities or in wealth, these differences in preferences–and the differences in outcomes they induce–are an important

1

For a nonexcludable public good, this tradeoff has previously been discussed by Ledyard and Palfrey (1999).

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1983

aspect of public-good provision when there is incomplete information about individual preferences. In challenging the view that the ability to exclude individuals from the enjoyment of a public good should never be resorted to, this paper complements the recent work of Schmitz (1997) and Norman (2004); see also Moulin (1994) and Deb and Razzolini (1999a,b). In these papers, excludability is useful because, in bargaining under incomplete information with voluntary participation, the threat of individual exclusion makes participants more willing to contribute to the financing of the public good. Given that each agent can veto the arrangement and prevent the public goods from being provided at all, a version2 of the impossibility theorem of Myerson and Satterthwaite (1983) shows that first-best allocations cannot be implemented; second-best allocations involve admission fees and exclusion of anybody who fails to pay the fees. Indeed, in a large economy with independent private values where no individual is ever pivotal for the provision of the public good, such admission fees are the only way to obtain voluntary contributions towards the financing of the public good at all.3 The arguments are akin to the Ramsey–Boiteux argument concerning the desirability of raising prices above marginal costs in order to finance production with significant overhead costs when firms are subject to budget constraints.4 Like the Ramsey–Boiteux argument, the Schmitz–Norman argument on the desirability of admission fees is subject to the critique of Atkinson and Stiglitz (1976) that divergences of consumer prices from marginal costs are undesirable if one can use lump-sum taxes to provide the requisite funds. In Schmitz (1997) and Norman (2004), anything like lumpsum taxation is unavailable because agents must participate voluntarily, and any agent who has no use for the public good at all would veto any arrangement that requires him to make a lump-sum payment towards its provision.5 However, the voluntary-participation requirement is problematic if public-good provision is the subject of government activity rather than multilateral bargaining. The government has a power of coercion, which it can use to impose lump-sum taxes. In this case, the Atkinson–Stiglitz critique would seem to put us right back into a first-best world, where there is no point in using admission fees to finance public goods. From d’Aspremont and Ge´rard-Varet (1979), it is well known that, in the absence of interim participation constraints, first-best allocations can be implemented through Bayesian mechanisms even though each agent’s preferences are his own private information. 2

See Gu¨th and Hellwig (1986), and Mailath and Postlewaite (1990). Moulin (1994) and Deb and Razzolini (1999a,b) study the provision and financing of an excludable public good subject to interim participation constraints in a finite economy. They obtain voluntary contributions through the effects of pivotalness as well as exclusion threats. 4 The link between admission fees for excludable public goods and Ramsey–Boiteux pricing has been pointed out by Samuelson (1958, 1969) and Laffont (1982/1988); see also Dre`ze (1980). If the excludable-public-good provision problem is formulated in terms of services obtained by participants upon admission, one actually has a private-good provision problem with positive fixed costs and zero marginal costs, where the bquantityQ of the public good appears as a parameter determining the common bqualityQ of the private good, i.e. the services obtained by participants. 5 For explicit accounts of the relation between the mechanism design problem with interim participation constraints and the Ramsey–Boiteux problem with a government budget constraint, see Hellwig (2003a, 2004a). 3

1984

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

However, the first-best allocations that are thus obtained have unattractive distributive properties. Coercion to pay lump-sum taxes hits those people most who do not care for the public good at all. As mentioned above, under the given information assumptions, these are the people in the economy who are worst off. The inequality between agents who care and agents who do not care for the public good is therefore exacerbated if agents who do not care have to pay lump-sum taxes to contribute to financing the public good. Interim participation constraints, which are problematic on efficiency grounds, can actually be seen as a crude device for protecting those participants who are least well off against the imposition of a mechanism that worsens their position even further. Beginning with Mirrlees (1971), utilitarian analyses of equity-efficiency tradeoffs have usually focussed on distributional issues arising from unobservable differences in earning abilities.6 Thus, in their critique of the Ramsey–Boiteux approach, Atkinson and Stiglitz (1976) allow for differences in earning abilities and find that deviations from firstbest pricing rules for different consumption goods are unwarranted unless the demands for those goods vary systematically with the unobservable characteristic and hence with equilibrium labour supplies.7 Christiansen (1981) and Boadway and Keen (1993) consider optimal public-good provision, Blomquist and Christiansen (2001) optimal admission fees for excludable public goods. These papers deal with utilitarian distributive concerns that arise from heterogeneity in earning abilities, but, like Atkinson and Stiglitz (1976), they do not allow for other sources of heterogeneity. In particular, they do not consider the distributive implications of the heterogeneity of public-goods preferences which arises from the standard assumption that information about these preferences is incomplete. Empirically, differences in earning abilities or differences in wealth are the most obvious reasons for the differences in well-being that we observe. However, within the utilitarian model, earning ability is just one parameter, and there is no reason to assume that it is the only parameter which affects people’s well-being. If people differ in other parameters, the utilitarian approach must take these differences into account. Given the importance of the cardinal properties of utility representations of preferences in the utilitarian approach, one may feel uneasy about focussing on differences in preferences over private-good versus public-good consumption, but that would be an argument against the utilitarian approach altogether. There does not seem to be any reason why a reliance on the cardinal properties of utility representations of preferences should be any more suspect for preferences concerning private-good versus publicgood consumption than for the consumption-leisure choices that are treated in the income tax literature. As in Ledyard and Palfrey (1999), the cardinalization of public-goods preferences in this paper is based on the presumption that people who benefit more from having access to the public good are also better off than people who benefit less. This presumption would be inappropriate for something like hospital services, which are most needed by people in ill health who are intrinsically worse off than people who do not need hospital services. 6

An important exception is Ledyard and Palfrey (1999). In recent applications of this idea, Cremer et al. (2003) and Golosov et al. (2003) show that capital income taxation can be useful for improving the scope for redistribution through labour income taxation. 7

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1985

The presumption is, however, appropriate for something like a sports facility, an opera production, or a university education, which provide net benefits rather than compensation for disadvantages.8 In the following, Section 2 lays out the basic model and formulates the utilitarian welfare problem. The model involves one excludable public good and one private good in a large economy with private information about individual preferences. Under the additional assumption that, conditional on a person’s type, there is no randomization over admission to the public good, incentive compatible admission policies involve the use of admission fees so that people are admitted if and only if they pay the stipulated fee. Section 3 discusses the equity-efficiency tradeoff involved in the choice of these fees. Section 4 shows how optimal admission fees and optimal provision levels for the public good depend on the utilitarian planner’s inequality aversion. The paper concludes in Section 5 with a few remarks on the need for an integration of the argument presented here with other relevant arguments in policy applications.

2. The allocation problem I study a large-economy version of the model considered by Schmitz (1997) and Norman (2004). There is an atomless continuum of consumers with total mass equal to one. The economy has an aggregate production capacity Y, which can be used to provide an amount C of aggregate consumption of a private good and a level Q of a public good subject to the resource constraint C þ K ðQÞVY :

ð1Þ

The public good is excludable. For each individual in the economy, an allocation must determine how much of the private good he gets to consume and whether he is granted access to the public good or not. Consumers are assumed to be risk neutral. Given an expected consumption level c of the private good and a probability p of being granted access to the public good, which is provided at the level Q, a consumer obtains the payoff c þ phQ:

ð2Þ

The parameter h is taken to be the consumer’s private information. From the perspective of the other consumers, or of the system as a whole, h is the realization of a random variable h˜, which takes values in the unit interval and has a probability distribution F(d ) with a strictly positive, continuously differentiable density f(d ). The mean of h˜ is denoted as h¯ . The random variables h˜ pertaining to different consumers are assumed to be independent and identically distributed. I also assume that, by a large-numbers effect, with probability one, F(d ) is the cross-section distribution, and h¯ is the cross-section mean of the preference parameter in the population. In this setting, an allocation corresponds to a level Q of public-good provision and a pair p(d ), c(d ) of functions so that, for any ha[0, 1], p(h) is the probability that a 8

I am grateful to Ted Bergstrom for alerting me to the importance for my analysis of the distinction between these different categories of public goods.

1986

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

consumer with preference parameter h is granted access to the public good, and c(h) is his expected consumption of the private good. For ha[0, 1], vðhÞ : ¼ cðhÞ þ pðhÞhQ

ð3Þ

is the expected payoff of a consumer with preference parameter h under the allocation ( Q, p(d ), c(d )). An allocation ( Q, p(d ), c(d )) is said to be incentive compatible if vðhÞzcðhVÞ þ pðh VÞhQ for all h and hVa[0, 1]. An allocation is feasible, if it satisfies the constraint Z 1 cðhÞf ðhÞdh þ K ðQÞVY ;

ð4Þ

ð5Þ

0

R1 which is Eq. (1) with C ¼ 0 cðhÞf ðhÞdh. A utilitarian planner assesses allocations according to the welfare functional9 Z 1 W ðvðhÞÞf ðhÞdh:

ð6Þ

0

His problem is to maximize Eq. (6) over the set of feasible and incentive compatible allocations. I impose the following assumptions on the data of the model: A.I. The cost function K(d ) satisfies K(0) =0. Moreover, it is strictly increasing, strictly convex, and twice continuously differentiable, with K Vð0Þbmaxh hð1  F ðhÞÞ and limQ Yl K VðQÞNh¯ : A.II. The welfare function W(d ) in Eq. (6) is strictly increasing, strictly concave, and twice continuously differentiable. Assumption A.II expresses the notion that the planner is inequality averse. Following Atkinson (1973), I refer to the relative curvature q W (c) w (WW(c)/WV(c)) as a measure of inequality aversion. The boundary conditions in Assumption A.I ensure that the allocation problem has a solution and, moreover, this solution is nontrivial. The boundary condition for Q = 0, K V(0) b maxh h(1  F(h)) has the interpretation that a monopolist bent on maximizing profits from admission fees would want to have a strictly positive public-good provision

9

As written, the specification (6) presumes that the planner is concerned about each agent’s expected utility. An alternative specification would have the planner concerned about each agent’s ex post utility, e.g., with an integrand of the form (1p(h))W(c 0(h))+p(u)W(c 1(h)+hQ), where c 0(h) is the private-good consumption of a consumer who is excluded and c 1(h) is the private-good consumption of a consumer who is given access to the public good. Given the strict concavity of W(d ) though, under this alternative specification, the planner would find it optimal to set c 0(h)=v(h) and c 1(h)=v(h)hQ so that Eq. (6) is again the appropriate specification. Because consumers care only about the expected consumption c(h)=(1p(h))c 0(h)+p(h)c 1(h), there is no difficulty in arranging c 0(h) and c 1(h) so that, conditional on h, there is no further risk in consumers’ payoffs.

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1987

level. Lemma 3.1 below uses this assumption to show that the public-good provision levels chosen by averse planner are also strictly positive. Since R 1 the inequality R1 hð1  F ðhÞÞ ¼ h 0 dF ðgÞb h gdF ðgÞVh¯ , Assumption A.I also implies that KV(0) is less than h¯, so the level   Q * :¼ arg max Qh¯  K ðQÞ ; Q

ð7Þ

which would be chosen by a planner without concern for inequality, is also strictly positive. The boundary condition that limQYlKV( Q)N h¯ has the interpretation that Q* is finite. This property will ensure that public-good provision levels are always finite. If the preference parameters of the different consumers were publicly observable, the planner would not have to worry about incentive compatibility. Given the strict concavity of W(d ), he would choose an allocation satisfying p(h)= 1 and c(h) =c(0)  hQ for all h. Everybody would be granted access to the public good, and the payoff levels v(h) would all be equal to v(0) = c(0). Feasibility would imply c(0) = Y  K( Q) +h¯ Q. Q would therefore be chosen to maximize W( Y  K( Q)+ h¯ Q). This requires that KV( Q)= h¯; at the chosen Q, the marginal provision cost should just equal the expected marginal benefit per capita. However, the egalitarian first-best allocation is not incentive compatible. With p(h) =1 and c(h) =c(0)  hQ for all h, any consumer with h N 0 has an incentive to understate his preference for the public good in order to raise his consumption of the private good without having to reduce his enjoyment of the public good. If the preference parameter h is each consumer’s private information, the egalitarian first-best allocation cannot be implemented. Therefore the planner faces an equity-efficiency tradeoff. Efficiency considerations call for open admissions to the enjoyment of the public good, with p(h)= 1 for all h. But then incentive compatibility implies c(h)= c(0) and v(h)= Y  K( Q)+ hQ for all h. Agents with high h are strictly better off than agents with low h. Indeed, since K( Q) NhQ for h close to zero, the latter are made strictly worse of by the provision of the public good. Equity considerations call for some redistribution of the private good from agents with high h to agents with low h, providing some compensation for differences in the impact of the provision of the public good. Such redistribution requires a self-selection device that induces agents to reveal their information by the choices they take. The use of such a device induces an inefficiency. If an agent with high h is to accept a lower level of consumption of the private good, he must be compensated by a higher probability of being granted access to the public good. This is only possible if agents claiming a low h cannot be sure of being admitted, i.e. if p(h) b1 for h close to zero. Such exclusion of low-h agents is inefficient. The question is to what extent the utilitarian planner wants to impose it anyway in order to screen high-h agents from low-h agents and provide for some redistribution of the private good from the former to the latter.

3. Nonrandomized admissions, admission fees, and the equity-efficiency tradeoff I study the utilitarian allocation problem with the additional restriction that p(d ) takes only the values zero and one, i.e. that there is no randomization over whether someone is granted access to the public good or not. This restriction simplifies the analysis without

1988

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

distorting the conclusions. For a general analysis which allows for randomization, the reader is referred to Section 5 of Hellwig (2003b). The requirement that, conditional on people’s tastes, there is no randomization in access is natural if the planner is unable to control the identities of people claiming admission to the use of the public good and the allocation must satisfy the additional constraint of renegotiation proofness, i.e. of not providing incentives for trading admission tickets ex post. Renegotiation proofness requires that, with probability one, anybody who gets access to the public good has a higher valuation for it than anybody who does not. For some hˆ then, all agents with h Nhˆ get access with probability one and all agents with h b hˆ get access with probability zero.10 If p(d ) takes only the values zero and one, incentive compatible allocations have a simple structure: A person is granted access to the public good if and only if he or she pays an admission fee p. The fee is paid and access is gained if the benefit hQ that a consumer draws from the enjoyment of the public good exceeds p; the fee is not paid if hQ is less than p. An incentive compatible allocation ( Q, p(d ), c(d )) with Q N0 and p(h)a{0, 1} for all h thus takes the form cðhÞ ¼ c0 and pðhÞ ¼ 0; if hQbp;

ð8Þ

cðhÞ ¼ c0  p and pðhÞ ¼ 1; if hQNp

ð9Þ

for some c 0 and p z 0. For h = p/Q, the consumer is indifferent about paying p to enjoy the public good, and one may have either c( p/Q)= c 0 and p( p/Q) = 0 or c( p/Q)= c 0  p and p( p/ Q) = 1. An incentive compatible allocation with nonrandomized admissions is thus characterized by the public-good provision level Q, the base consumption c 0, and the admission fee p. The planner’s assessment of the allocation induced by Q, c 0, p is given as Z 1 W ðc0 þ maxðhQ  p; 0ÞÞf ðhÞdh; ð10Þ 0

and the feasibility condition (5) takes the form    p c0 VY  K ðQÞ þ p 1  F : Q

ð11Þ

The term p(1  F( p/Q)) in Eq. (11) represents the aggregate revenue from admission fees. This revenue provides scope to raise c 0 above Y  K( Q), the per capita amount that is available after deduction of the cost K( Q). The counterpart of c 0 exceeding Y  K( Q) by the amount p(1 F( p/Q)) is to be found in c 0  p falling short of Y  K( Q) by the amount pF( p/Q). The allocation problem is to choose Q, c 0, and p so as to maximize Eq. (10), subject to Eq. (11). In this problem, it is convenient to replace the admission fee p with the product hˆ Q, where hˆ is the value of the preference parameter at which a consumer is just

10

A formal treatment of renegotiation proofness in a more general model with m public goods is given in Hellwig (2004a). The underlying argument goes back to Hammond (1979) and Guesnerie (1995).

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1989

indifferent about paying the admission fee p and gaining access. Because the constraint (11) is obviously binding, one can also replace c 0 by      ð12Þ c0 Q; hˆ : ¼ Y  K ðQÞ þ hˆ Q 1  F hˆ : The problem then is to choose Q z 0 and hˆa[0, 1] so as to maximize Z 1         W * Q; hˆ : ¼ W Y  K ðQÞ þ hˆ Q 1  F hˆ þ Qmax h  hˆ ; 0 f ðhÞdh: 0

ð13Þ In the remainder of the paper, I study how the solutions to this maximization problem depend on the welfare function W(d ). Suppressing the role of the other exogenous data, I write ( Q(W), hˆ (W)) for a pair ( Q, hˆ ) which maximizes Eq. (13) for the given W. I begin with the observation that, under the given assumptions, the optimal public-good provision level Q(W) is bounded away from zero. Because KV(0) b maxh h(1 F(h)), there exists a pair ( Q 0, hˆ 0) such that  K( Q 0)+hˆ 0Q 0(1  F(hˆ 0)) N 0. By providing the public good at the level Q 0 and charging the admission fee hˆ 0Q 0, the mechanism designer earns a profit and can provide a base consumption c 0( Q 0, hˆ 0)N Y. Because the welfare provided by the optimal pair ( Q(W), hˆ (W)) is no less than the welfare provided by the pair ( Q 0, hˆ 0), and, by inspection of Eq. (13), the latter is no less than W(c 0( Q 0, hˆ 0)), it follows that      W * QðW Þ; hˆ ðW Þ zW c0 Q0 ; hˆ 0 : By inspection of Eq. (13), one also has         ðVW ðY  K ðQÞ þ QÞ W * Q; hˆ VW Y  K ðQÞ þ hˆ Q 1  F hˆ þ Q 1  hˆ

Þ

for all Q and hˆ . Upon combining inequalities, one infers that W( Y  K( Q(W)) + Q(W)) z W(c 0( Q 0, hˆ 0)), regardless of W, since c 0( Q 0, hˆ 0)N Y, the following lemma follows immediately. Lemma 3.1. Under Assumptions A.I and A.II, there exists eN0 such that, regardless of W, Q(W) Ne. Next, I argue that hˆ (W) is bounded away from one. More precisely, if h* is any maximizer of the product h(1 F(h)), then h* is less than one, and hˆ (W) is no greater than h*. The first claim is true because h(1  F(h)) N 0 for ha(0, 1) and 1(1  F(1)) = 0. The second claim follows from the observation that, for Q N 0 and hˆN h*, Eq. (13) yields W*( Q, hˆ )b W*( Q, h*): If the admission fee is lowered from hˆQ Nh*Q to h*Q, welfareˆ is increased because, without adverse effects on admission fee revenues and on c 0( Q, h ), the people who receive access to the public good have to pay less for it. Given that the argument applies to any maximizer of the product h(1  F(h)), this consideration yields: Lemma 3.2. Let h* :¼ min arg max hð1  F ðhÞÞ: h

Then, regardless of W, hˆ (W) V hT b1.

ð14Þ

1990

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

Given Lemmas 3.1 and 3.2, the first-order conditions for Q(W)and hˆ (W) can be written as Z 1 h     i W V  KVðQÞ þ hˆ 1  F hˆ þ max h  hˆ ; 0 f ðhÞdh ¼ 0 ð15Þ 0

and

     Z ˆ ˆ Q 1  F h  h f hˆ

1

0

W f ðhÞdh 

Z

0

hˆ



1

W Vf ðhÞdh ¼ 0;

ð16Þ

where in each integral, the derivative WV of the welfare function is evaluated at the point c 0( Q, hˆ ) +max(hQ  hˆ Q, 0).11 Condition (15) equates the marginal welfare benefits and costs of an increase in Q. Marginal benefits arise from the increase in the information rent hQ  hˆ Q of consumers with preference parameter h N hˆ and from the increase in the base consumption c 0( Q, hˆ ) which is due to the increase in the admission fee revenue hˆ Q(1 F(hˆ )). Condition (16) equates the marginal welfare benefits and costs of an increase in hˆ. A marginal welfare benefit arises if the increase in hˆ raises aggregate admission fee revenues and permits an increase in c 0( Q, hˆ ). A marginal welfare cost arises from the additional burden imposed on consumers with h Nhˆ, who pay the fee. Relying on Eqs. (15) and (16), I will compare Q(W) and hˆ (W) to the first-best solution ( Q*, 0) and to the choices of a monopolist bent on maximizing the excess hˆ Q(1  F(hˆ )) K( Q) of revenues over costs or, equivalently, a planner a` la Rawls (1971), who is bent on maximizing the welfare assessment W(c 0( Q, hˆ )) that is attached to the position of the worst-off individuals in the system and, hence, the base consumption c 0( Q, hˆ ) = Y  K( Q) +hˆ Q(1  F(hˆ )). The monopolist and the Rawlsian planner would choose hˆ = h*, where h* is any maximizer of the product h (1  F(h)), and they would choose the public-good provision level Q *(h*), where, for any hˆ ,   h    i Q* hˆ : ¼ arg max Qhˆ 1  F hˆ  K ðQ Þ ð17Þ Q maximizes the excess of revenues over costs when hˆ is given. From Eqs. (15) and (16), in combination with Lemmas 3.1 and 3.2, one obtains: Proposition 3.3. Under Assumption A.I and A.II,   Q* hˆ ðW Þ bQðW ÞbQ*; and 0Vhˆ ðW Þbh* :

ð18Þ ð19Þ

No matter what W(d ) may be, the public-good provision level Q(W) lies strictly ˆ between Q *(hˆ (W)) and Q*, and h (W) is strictly less than the smallest monopoly or Rawlsian level h *. Condition (18) follows directly from Eq. (15): With hˆ b 1, the information rent term in Eq. (15) satisfies Z 1 Z 1 Z 1     0b W Vmax h  hˆ ; 0 f ðhÞdhb W Vf ðhÞdh max h  hˆ ; 0 f ðhÞdh; ð20Þ 0 11

0

0

In principle, Eq. (16) should be a weak inequality, in combination with a complementary-slackness condition; however, for hˆ =0, the left-hand side of Eq. (16) is zero anyway.

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1991

it is less than the product on the right-hand side of Eq. (20). From Eq. (15) and the first inequality in Eq. (20), one obtains KV( Q) +hˆ (1 F(hˆ )) b0 and Q N KV1 (hˆ (1 F(hˆ ))) = Q * (hˆ ); from Eq. (15) and the second inequality in Eq. (20), one obtains    Z  KVðQÞ þ hˆ 1  F hˆ þ

hˆ

1



 h  hˆ f ðhÞdhN0;

R1 hence  KVðQÞ þ 0 hf ðhÞdhN0 and QbKV1 ðh¯Þ ¼ Q. Whereas the profit-maximizing monopolist or the Rawlsian planner would expand Q only to the point where, for given hˆ , the marginal revenue hˆ (1  F(hˆ )) just covers the marginal cost K V( Q), the non-Rawlsian mechanism designer takes account of the fact that the increase in Q also benefits the people with h N hˆ who pay to enjoy the public good. Therefore he is willing to raise Q above Q *(hˆ (W)). However, he does not raise it up to the first-best level Q*. Because WV(d ) is a decreasing function, in his assessment, the marginal increases in information rents of high-h consumers receive less weight in Eq. (15) than the marginal increases in information rents of low-h consumers; moreover, if hˆ (W) N0, consumers with h bhˆ (W) do not obtain any information rents at all. The marginal R 1 information rent term in REq. (15) is therefore less than the difference between hˆ 0 W VdF ðhÞ and 1 hˆ ð1  Fðhˆ ÞÞ 0 W VdF ðhÞ. The aggregate marginal welfare benefitR from an increase in Q 1 is assessed to be less than the first-best assessment between hˆ 0 W VdF ðhÞ, and Q(W) 12 must be less than Q*. Similarly, Eq. (19) is derived from the first-order condition (16). By Lemma 3.2, it suffices to show that one cannot have hˆ(W)= h *. By the definition of h *, one has Q(1  F(h *) h *f (h *)) =0. Beginning at h *, a small decrease in hˆ has no first-order effects on aggregate revenues from admission fees or on the base consumption c 0( Q, hˆ ). However, this decrease benefits the users of the public good. Therefore, the first-order condition (16) cannot be satisfied at hˆ = h *. The tradeoff between the marginal benefits and costs of an increase in hˆ can be interpreted as an equity-efficiency tradeoff. To show this, I rewrite condition (16) in the form    Z hˆ Q 1  F hˆ W Vðc0 ÞdF ðhÞ   0 hˆ Qf hˆ ¼ Z 1 W VdF ðhÞ 0  Z 1   QF hˆ W V c0 þ hQ  hˆ Q dF ðhÞ hˆ  : Z 1 W VdF ðhÞ

ð21Þ

0 12 This argument for underprovision of the public good relative to the first-best provision rule should be distinguished from the one given by Boadway and Keen (1993) for a model with heterogeneous earning abilities. In their analysis, with complete information about public-goods preferences conditional on earning abilities, deviations from the Samuelson–Lindahl rule for first-best public-good provision depend on the impact of publicgood provision on incentives to reveal earning abilities for what they are.

1992

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

The left-hand side of Eq. (21) measures the efficiency loss associated with ˆa marginal increase in the admission fee: If the fee is raised from p = hˆQ to p +dp =(hˆ +dh )Q, the set of participants who prefer to forego the enjoyment of the public good isˆ expanded to include all those whose preference parameter h lies between hˆ and hˆ + dh . The privategood equivalent of the surplus that is thereby lost is approximately hˆQf(hˆ)dhˆ: the set of consumers involved has approximately the mass f(h)dhˆ and each of these consumers loses an enjoyment that he considers to be worth approximately hˆQ units of the private good. This loss hˆQf(hˆ )dhˆ is a true deadweight loss because, in the absence of crowding or other external effects in the enjoyment of the public good, there is no real cost of giving these consumers access to the public good. The right-hand side of Eq. (21) provides a measure of the utilitarian redistribution gain that is derived from having all participants with h N hˆ reduce their consumption of the private good by dp = Qdhˆ units and using the proceeds to raise c 0 by Q(1 F(hˆ ))dhˆ units. For consumers with h b hˆ , this change entails a net increase of private-good consumption by Q(1  F(hˆ ))dhˆ, and for consumers with h Nhˆ, a net decrease by QF(hˆ )dhˆ. The first term on the right-hand side of Eq. (21) represents the welfare gain from the increase of private-good consumption of agents with h b hˆ, the second term represents the welfare loss from theR decrease of private-good consumption of agents 1 with h N hˆ. Through deflation by 0 W VdF ðhÞ, these terms are translated into an equivalent number of units of the private good. The difference between them provides a measure of the net welfare gains from this redistribution. To see that there are net welfare gains rather than losses, observe that the right-hand side of Eq. (21) can be rearranged as   Z 1h  i QF hˆ W Vðc0 Þ  W V c0 þ hQ  hˆ Q dF ðhÞ; ð22Þ R1 ˆ W VdF ð h Þ h 0 which is never negative. Indeed, with Q N 0 and hˆ b 1, the strict concavity of W(d ) implies that the integral on the right-hand side of Eq. (22) is positive, so the net welfare effect of redistributing private-good consumption from agents with h Nhˆ to agents with h b hˆ is positive if F (hˆ)N 0. The first-order condition for hˆ just balances this gain against the marginal efficiency loss derived from having more agents forego the enjoyment of the public good. The mathematical structure of this equity-efficiency tradeoff is the same as the mathematical structure of the tradeoff inherent in the standard first-order condition for optimal utilitarian income taxation, e.g., Eq. (27) in Mirrlees (1971). The different formalisms have different interpretations, but the underlying logic is the same.

4. Inequality aversion and optimal admission fees The preceding discussion of the equity-efficiency tradeoff has not actually shown that optimal admission fees are positive. Indeed at hˆ = 0, the first-order condition (16) is always satisfied, the left-hand sides and right-hand sides of Eqs. (21) and (22) are all equal to zero. The reason is that if one starts from p = hˆ Q = 0, then both the efficiency loss and the

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1993

redistribution gain from a small increase in uˆ are of the second order of smalls. The efficiency loss is of the second order of smalls because the few agents with ha[0, 0+ dhˆ ] suffer only a small loss if they forego the enjoyment of the public good. The redistribution gain is of the second order of smalls because the set of agents with ha[0, 0+ dhˆ ], who obtain a net increase of their private-good consumption, is small.13 For agents with h N 0 +dhˆ , the introduction of the small admission fee has hardly any effect. For them, the increase in the base consumption c 0 provides nearly full compensation for the increase in the fee. Even so, an admission fee equal to zero is not always optimal. If Q N 0 and Z 1 W Vðc0 ðQ; 0ÞÞN2 W Vðc0 ðQ; 0Þ þ hQÞdF ðhÞ; ð23Þ 0

a zero admission fee yields a local minimum of W* with respect to hˆ because the second derivative  Z 1 B2 W 4 ð Q; 0 Þ ¼ Qf ð 0 Þ W V ð c ð Q; 0 Þ Þ  2 W V ð c ð Q; 0 Þ þ hQ ÞdF ð h Þ ð24Þ 0 0 Bhˆ 2 0 is positive. To see why Eq. (23) might hold, go back to the equity-efficiency tradeoff in Eq. (21). At p = hˆQ = 0, the marginal efficiency loss on the left-hand side and the marginal redistribution gain on the right-hand side are both equal to zero. A marginal increase dp = Qdhˆ in the admission fee raises the marginal redistribution gain (22) by R1 0 ½W Vðc0 Þ  W Vðc0 þ hQÞ dF ðhÞ f ð0ÞQdhˆ ; R1 W VdF ð h Þ 0 which is strictly positive if Q N 0. An increase in the admission fee above zero thus raises the marginal redistribution gain that is to be expected from a further increase in the fee. Whereas a bfirstQ small increase in the admission fee above zero has no first-order effects on the distribution of private-good consumption at all because the people paying the fee are roughly the same as the people benefitting from the induced increase in c 0, a bfurtherQ small increase does induce a redistribution gain because the bfirstQ small increase in the fee above zero has created a clearly identified, nonnull set of agents who benefit from the redistribution involved in the bfurtherQ fee increase. If this increase in the marginal redistribution gain on the right-hand side is greater than the increase f(0)Qdhˆ in the marginal efficiency loss on the left-hand side of Eq. (21), then Eq. (23) holds, and a further increase in hˆ is desirable for the utilitarian planner. These considerations suggest that the desirability of a positive admission fee should depend on the curvature q W (c)= (WW(c)/WV(c))of the welfare function W(d ), i.e. the planner’s inequality aversion. Given that Q(W) is bounded away from zero, Eq. (23) is likely to hold if the curvature of W(d ) is large, and likely to fail if the curvature of W(d ) is small. This intuition is confirmed by the following two results.

13

In the theory of optimal income taxation, this consideration implies the desirability of zero marginal tax rates at the top and at the bottom of the earnings distribution, see Seade (1977).

1994

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

Proposition 4.1. Let {W k } be any sequence of increasing, concave, and twice continuously differentiable functions on Rþ such that lim kYl q W k (c) = 0 uniformly for ca[Y  K(Q*), Y  K(Q*) + Q*]. Then hˆ (W k )= 0 for any sufficiently large k and lim kYl Q(W k ) =Q*. Proposition 4.2. Let {W k } be any sequence of increasing, concave, and twice continuously differentiable functions on Rþ such that lim kYl q W k (c) = l uniformly in c. Then lim kYl hˆ (W k ) = h * and lim kYl Q(W k ) = Q *(h *). In particular hˆ (W) N 0 if q W (c) is uniformly large. ˆ If inequality aversion is small ,14 the pair ( Q (W), R 1 h (W)) is close to the pair ( Q*, 0) which maximizes the expected aggregate surplus h¯ hdF ðhÞQ  K ðQÞ. There is thus no discontinuity in the relation of the planner’s choice to the welfare function as inequality aversion goes to zero. Indeed, the optimal admission fee is not just converging to zero as inequality aversion goes to zero, but from some point onwards, optimal admission fees are actually equal to zero even though inequality aversion is still positive. In contrast, optimal admission fees are positive if inequality aversion is uniformly large. As inequality aversion goes out of bounds, the planner’s choices converge to the pair ( Q *(h *), h *) which would be chosen by a Rawlsian planner maximizing the base consumption c 0( Q, hˆ ). Proposition 4.2 thus confirms the well-known principle that utilitarian welfare maximization with a high degree of inequality aversion yields results similar to the Rawlsian approach of maximizing the payoff of the worst-off individuals in the economy.15 From the utilitarian perspective, public-good provision serves two purposes: First, it benefits the people who enjoy the public good. Second, admission fee revenues in excess of provision costs can be used to improve the position of low-h consumers. If overall inequality aversion is small, only the first concern matters: The public good is provided solely because it benefits users. If overall inequality aversion is large, the second concern also matters. This concern actually becomes paramount when overall inequality aversion goes out of bounds and the welfare weight of high-h consumers goes to zero. Revenues from optimal admission fees will usually differ from provision costs. If overall inequality aversion is small, revenues from admission fees are zero because the fees themselves are zero. In this case, following the logic of Atkinson and Stiglitz (1976), the public good is paid for by a lump-sum tax K( Q(W)) on all people in the economy, whether they draw utility from the public good or not. In contrast, if overall inequality aversion is large, revenues from admission fees exceed provision costs, providing a profit which is used to raise the base consumption of agents with low h. In either case, if overall inequality aversion is small and if overall inequality aversion is large, there is no room for a budget constraint a` la Ramsey–Boiteux. However, whether there is a surplus or a deficit from public-good provision with an admission fee depends on how strong a concern for redistribution the planner has. The limits in Propositions 4.1 and 4.2 satisfy Q * N Q *(h *) and 0 bh *. These inequalities reflect the fact that an increase in Q or a decrease in hˆ enhance the sensitivity of c 0( Q, 14 Uniformity of smallness or largeness of inequality aversion is not really needed; it helps avoid the need for complicated epsilon–delta arguments. 15 See, e.g., Arrow (1973) and Atkinson (1973).

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1995

hˆ )+ Qmax(h  hˆ , 0) with respect to h and hence the inequality16 inherent in the crosssection distribution of expected payoffs. Given this observation, one suspects that Q(W) might decrease and hˆ(W) might increase, or at least not decrease, if W changes in a way that raises inequality aversion. For welfare functions exhibiting constant inequality aversion, this conjecture can actually be proved. For arbitrary welfare functions, I have only been able to obtain the weaker result that at least one of the two monotonicity properties must hold. Proposition 4.3. If two welfare functions W 1 , W 2 are such that q W 2 (c) N q W 1 (c) for all c, then Q(W 2 ) bQ(W 1 ) or hˆ (W 2 ) N hˆ (W 1 ).

5. Concluding remarks As was pointed out in the introduction, the utilitarian analysis in this paper is based on a cardinalization of public-goods preferences under which people who benefit more from the enjoyment of the public good are better off than people who benefit less. This cardinalization is unsuitable for things like hospital services, which are most needed by people who are intrinsically worse off than people who do not need them. The cardinalization does, however, seem appropriate for a sports facility, an opera production, or a university education, the enjoyment of which provides a net benefit rather than a compensation for disadvantages. For such facilities, the paper argues that pricing should take account of distributive concerns in relations between the people who use them and the people who do not. Policy discussions of pricing for such facilities tend to be dominated by concerns about their properties as merit goods, about crowding and other externalities, and about public-sector budget constraints. Distributive concerns in these policy discussions tend to focus on correlations of tastes with earning abilities and incomes, e.g., on the question whether an opera or a university that is financed from general government funds provides benefits mainly to the rich, or whether they are a means of providing culture and education to the poor, who would otherwise be unable or unwilling to afford them.17 Without denying the importance of these concerns, this paper points out that distributive concerns related to benefits from the public goods in question should also be taken into account. The paper shows that these distributive concerns provide an argument for levying admission prices in excess of marginal admission costs (which in this paper were assumed to be zero). The relevance of these considerations is particularly obvious for higher education, where benefits translate directly into social status and

16

In the sense of a mean-utility-preserving spread; see Diamond and Stiglitz (1974). Thus Blomquist and Christiansen (2001) discuss utilitarian provision and pricing of excludable public goods when there is heterogeneity in earning abilities. Cremer and Laffont (2003) consider the implications of distributive concerns when people with different incomes also have different access costs for the public good in question. 17

1996

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

earnings (and are only partially reabsorbed through income taxation).18 For the other examples, the relevance of distributive concerns related to benefits is less obvious, but even so, they should be taken into account. This paper should not be read as saying that other concerns, from meritorious properties to budgetary concerns and correlations of tastes and earning abilities, do not matter. In principle, we need an integrated analysis which involves all the different concerns and provides a basis for assessing the relative importance of any one of them. Unfortunately, we do not as yet have a suitable framework for such an integrated analysis. The problem is partly conceptual, partly technical. At the conceptual level, an integrated assessment of the relative merits of budgetary concerns and of the different dimensions of distributive concerns in the provision and pricing of something like higher education would have to straddle the boundaries between the theory of public-goods provision, the theory of publicsector pricing under a government budget constraint, and the theory of optimal utilitarian income taxation. This would require us to come to terms with the relation between these different subfields into which public economics has traditionally been divided.19 At the technical level, we must overcome the difficulties posed by the multidimensional nature of incentive problems that arise when people differ in earning abilities as well as publicgoods preferences.20 As a first step towards resolving these difficulties, Hellwig (2004b) extends the utilitarian analysis of the present paper to allow for endogenous production and heterogeneity in labour productivities (earning abilities). Public-goods preferences and labour productivities (earning abilities) are private information. The difficulties of the multidimensional incentive problem are avoided by imposing the additional requirement that, as in the analysis of this paper, the allocation of private-good consumption and of admission tickets to public goods must be renegotiation proof and must not leave room for additional side-trading among the participants. Under this requirement, the multidimensional incentive problem is very much simplified, so that one is merely dealing with unidimensional incentive constraints, one for each of the hidden characteristics. The mechanism design problem can then be formulated in terms of a vector of admission fees for excludable public goods and a nonlinear income tax schedule.21 The conclusions of the present paper, in particular, the conclusions of Propositions 3.3, 4.1, and 4.2, continue to 18

Convincing empirical evidence on this point for the case of Germany is provided by Gru¨ske (1994). Most attempts to overcome the boundaries between the subfields are limited in scope by the assumption that heterogeneity in earning abilities is the only source of distributive concern. For prominent examples, see Atkinson and Stiglitz (1976), Christiansen (1981), Boadway and Keen (1993), or Blomquist and Christiansen (2001). The importance of sources of heterogeneity other than differences in earning abilities has been pointed out in assessments of the Atkinson–Stiglitz analysis by Laffont and Tirole (1993, pp. 194 ff.) and, more recently, by Cremer et al. (2001), who study a model with differences in endowments, as well as earning abilities. Bierbrauer (2005) introduces private information heterogeneity in public-goods preferences into the Boadway–Keen model of public-good provision and income taxation. 20 For an account of these difficulties of multidimensional mechanism design, see Rochet and Chone´ (1998). 21 With multiple public goods, the renegotiation proofness requirement is necessary, as well as sufficient, for this assertion. Hellwig (2004a) shows that, without renegotiation proofness, it is desirable to have mixed bundling, i.e. pricing schemes that involve discounts on combination tickets providing admission to several public goods at once. Fang and Norman (2003) show that even pure bundling can be preferable to separate provision at given prices. 19

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1997

hold in this more general framework if earning abilities and the taste parameters for public goods are affiliated.22 Income taxation meets those distributive concerns which relate to differences in earning abilities, admission fees those distributive concerns which relate to differences in public-goods preferences. Unless public-good preferences are perfectly correlated with earning abilities, differences in public-good preferences call for the use of admission fees for purposes of redistribution if inequality aversion is high.

Acknowledgements I am very grateful for helpful comments from Ted Bergstrom, Felix Bierbrauer, Thomas Gaube, Roger Guesnerie, Peter Norman, and two referees. I am also grateful to the Deutsche Forschungsgemeinschaft for research support through Sonderforschungsbereich 504 at the University of Mannheim.

Appendix A. Proofs The arguments for Lemmas 3.1 and 3.2 and for Proposition 3.3 are given in the text. Therefore no further proofs are given here. Proof of Proposition 4.1. For any A N0, if W(d ) is such that q W (c) V A for all c, one has dlnðW VðcÞÞ zA dc for all c. By a straightforward integration, one then obtains ln

W Vðc0 þ QðW ÞÞ z  AQðW Þ; W V ð c0 Þ

hence W Vðc0 þ QðW ÞÞzW Vðc0 ÞeAQðW Þ zW Vðc0 ÞeAQ4 :

ðA:1Þ

If A N0 is such that F hˆ     beAQ4 ; ˆ þ hˆ f hˆ hˆ a½0;1 F h sup

it follows that         F hˆ þ hˆ f hˆ W Vðc0 þ QðW ÞÞ þ W Vðc0 ÞF hˆ b0

ðA:2Þ

ðA:3Þ

22 As discussed by Milgrom and Weber (1982), affiliation involves a strong form of absence of negative correlation, even locally. Apart from cases of positive correlation, affiliation does encompass the case of stochastic independence.

1998

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

for all hˆ a(0,1]. Because WV(d ) is nonincreasing, it follows that      Z 1        F hˆ þ hˆ f hˆ W V c0 þ max h  hˆ ; 0 Q dF þ W Vðc0 ÞF hˆ b0 0

ðA:4Þ for all hˆ a(0,1], which is equivalent to Z      Z 1     1  F hˆ  hˆ f hˆ W V c0 þ max h  hˆ ; 0 Q dF  0

hˆ

1

W VdFb0: ðA:5Þ

Because Q(W) is bounded away from zero, it follows that no uˆ a(0,1] satisfies the firstorder condition (16) if q W (c)V A, where A satisfies Eq. (A.2). Therefore, one has hˆ (W) = 0 if q W (c) is sufficiently close to zero, uniformly in c. As for the second statement of the proposition, if hˆ (W)= 0, the first-order condition (15) for the public-good provision level takes the form Z

1

W V½  KVðQðW ÞÞ þ h dF ðhÞ ¼ 0:

ðA:6Þ

0

By the concavity of W, it follows that W Vðc0 ÞKVðQðW ÞÞzW Vðc0 þ QðW ÞÞh¯ : If q W (c)V A for all c, then by Eq. (A.1), it follows that KVðQðW ÞÞW Vðc0 Þzh¯ eAQ4 W Vðc0 Þ; and hence that

 QðW ÞzKV1 h¯ eAQ4 :

ðA:7Þ

Given that KV1(h¯ ) z Q(W), it follows that Q(W) is close to Q* =KV1(h¯ ) if A is close to zero. 5 Proof of Proposition 4.2. Fix A¯ N 0 and let W(d ) be such that q W (c)z A¯ for all c. I will show that, if A¯ is large, then c 0( Q(W), hˆ (W)) must be close to c*0 w max( Q, hˆ )c 0( Q, hˆ )= c 0( Q * (h *), h *). More precisely, for any e¯ N0, I will show that c*0 z c 0( Q(W), hˆ (W)) z c*0  e¯ if A¯ is sufficiently large. The first of these inequalities is trivial. The second of these inequalities is also trivial if e¯ N Q*.23 Suppose therefore that e¯ a(0, Q*]. To prove that the second inequality holds if A¯ is sufficiently large, I will show that for any pair 23 By inspection of Eq. (13), one has W*( Q(W), hˆ(W))V W(c 0( Q(W), hˆ(W))+Q(W)) and W*( Q *(h *), h *)zW(c 0( Q *(h *), h *)), so W*( Q(W), hˆ (W)) zW*( Q *(h *), h *) implies c 0 ( Q(W), hˆ(W))+Q(W) zc 0*, and the claim follows because Proposition 3.3 implies Q(W)bQ*.

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

1999

( Q, hˆ ) with c 0( Q, hˆ )b c*0  e¯ and Q V Q*, one has W*( Q, hˆ )b W(c*0 ) if A¯ is sufficiently large. By the concavity of W(d ), for such ( Q, hˆ ), one has     



  ˆ W c0 þ Q max h  hˆ ; 0  W c* 0 þ d V W V c* 0 þ d c0 Q; h þ hQ  c* 0 d

 V W V c*0 þ d ð  e¯ þ hQ*  dÞ for all d z 0 and all h. In particular,   



 W c0 þ Q max h  hˆ ; 0  W c* 0 VW V c* 0 ð  e¯ þ hQ*Þ for all h. Therefore Z e¯=Q4

 W *ðQ; hˆ Þ  W c* ðW ðc0 þ Q maxðh  hˆ ; 0ÞÞ  W ðc*0 ÞÞdFðhÞ V 0 0 Z 1 þ ðW ðc0 þ Q maxðh  hˆ ; 0ÞÞ  W ðc*0 þ dÞÞdFðhÞ e¯=Q4

þ W ðc0* þ dÞ  W ðc*0 Þ Z e¯=Q4 * ð  e¯ þ hQ*ÞdFðhÞ V W Vðc0 Þ 0

þ W Vðc* 0 þ dÞ

Z

1

ð  e¯ þ hQ*  dÞdFðhÞ e¯=Q4

þ ðW ðc0* þ dÞ  W ðc0*ÞÞ;

ðA:8Þ

for all d N 0. The first term on the right-hand side is negative because, for h b e¯ /Q*, one has e¯ +hQ*b 0. Depending on d, the second term on the right-hand side can be negative or positive, but I claim that, if d is small relative to e¯ and if A¯ is sufficiently large, this term is small relative to the first term. The third term on the right hand is positive, but this term is also small relative to the first term if d is small relative to ¯e . From Eq. (A.8), one obtains that Z  



 e¯=Q4 * ð  e¯ þ hQ*ÞdF ðhÞ W * Q; hˆ  W c* V W V c 0 0 0



 þ W V c* 0 þ d ðQ*  e¯ Þ þ W Vðc0* d V W Vðc0*Þ

Z

!

e¯=Q4

ð  e¯ þ hQ*ÞdFðhÞ þ e

A¯ d

ðQ*  e¯Þ þ d

0

R e¯=Q

ðA:9Þ

for all d z 0. If d ¼ 12 0 ðe¯  hQ*ÞdF ðhÞ and if A¯ is sufficiently large, the right-hand side of Eq. (A.9) is negative. Thus for e¯bQ*, as well e¯z Q*, c 0( Q, hˆ )bc*0  e¯ and Q V Q* imply W*( Q, hˆ ) bW(c*0 ) if A¯ is sufficiently large.

2000

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

For the given sequence {W k }with inequality aversion q W k (c) going out of bounds uniformly in c, it follows that limkYlc 0( Q(W k ), hˆ (W k ))=c*0 . Any limit point ( Q l, h l) of the sequence {( Q(W k ), hˆ (W k ))} must therefore be a maximizer of c 0(.,.). By inspection of Eq. (12), such a limit point satisfies h la arg max h (1F(h)) and Q l=Q *(h l). By Proposition 3.3, it follows that h * is the only maximizer of h(1F(h)) to which the sequence {hˆ(W k )} can converge. 5 Proof of Proposition 4.3. If the proposition is false, there exist W 1, W 2 satisfying q W 2(c)N q W 1(c) for all c such that for ( Q i , hˆ i ) = ( Q(Wi ), hˆ (Wi )), one has Q2 zQ1 and hˆ 2 Vhˆ 1 : For i = 1, 2, and any ha[0, 1], let     xi ðhÞ ¼ c0 Qi ; hˆ i þ Qi max h  hˆ i ; 0 ;

ðA:10Þ

ðA:11Þ

and note that, by Proposition 3.3, x i (h)a[ Y  K( Q*), Y  K( Q*)+ Q*] for all h. Note also that Bxi ðhÞ ¼ 0; if hbhˆ ðWi Þ; Bh and Bxi ðhÞ ¼ Qi ; if hNhˆ ðWi Þ; Bh so Eq. (A.10) implies Bx2 ðhÞ Bx1 ðhÞ z z0; Bh Bh

ðA:12Þ

where the first inequality is strict if ha(hˆ2, hˆ 1) or if h N hˆ1 and Q 2 N Q 1. Because x 2(d ) is optimal for W 2, one has Z 1 ½W2 ðx2 ðhÞÞ  W2 ðx1 ðhÞÞ dF ðhÞz0: 0

Therefore the concavity of W 2(d ) implies that Z 1 W2Vðx1 ðhÞÞ½x2 ðhÞ  x1 ðhÞ dF ðhÞz0:

ðA:13Þ

0

Because W 2V(x) is decreasing in x and, by Eq. (A.12), x 1(h) and x 2(h) x 1(h) are both nondecreasing in h, Eq. (A.13) implies Z 1 ½x2 ðhÞ  x1 ðhÞ dF ðhÞz0; ðA:14Þ 0

i.e. the mean of the random variable x 2(h˜) cannot be less than the mean of the random variable x 1(h˜).

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

By the definition of x 1(d ), one also has Z 1 Z 1 W1 ðx1 ðhÞÞdF ðhÞz W1 ðx2 ðhÞÞdF ðhÞ: 0

2001

ðA:15Þ

0

By Eqs. (A.14) and (A.15), there exists ka[0, 1] such that Z 1 Z 1 Wk ðx1 ðhÞÞdF ðhÞ ¼ Wk ðx2 ðhÞÞdF ðhÞ; 0

ðA:16Þ

0

where W k is given by W k (c) = (1 k)W 1(c)+ kc. One easily computes qWk ðcÞ ¼ 

ð1  kÞW1WðcÞ Vq ðcÞbqW2 ðcÞ ð1  kÞW1VðcÞ þ k W1

for all c. By Theorem 1, p. 128 of Pratt (1964), it follows that there exists a strictly increasing, strictly concave function u k so that W 2 = u k 8W k. From Eq. (A.12), in combination with Lemma 3.1 and Proposition 3.3, one also finds that, if one of the inequalities in Eq. (A.10) is strict, then the random variable W k (x 2(h˜)) is given by a meanpreserving spread of the random variable W k (x 1(h˜)).24 Given the strict concavity of u k , it follows that Z 1 Z 1 uk ðWk ðx1 ðhÞÞÞdF ðhÞN uk ðWk ðx2 ðhÞÞÞdF ðhÞ; 0

0

contrary to the definition of x 2(d ). The assumption that at least one of the inequalities in Eq. (A.10) is strict thus leads to a contradiction and must be false. Alternatively, suppose that Q 2 = Q 1 = Q and hˆ 2 = hˆ1 = hˆ, and consider the first-order condition (15) for Q 2 = Q. Using the fact that, again by Pratt’s theorem, W 2 = u 18W 1, this first-order condition can be written in the form Z 1 h i u1VðW1 ðx2 ðhÞÞÞW1Vðx2 ðhÞÞ  KVðQÞ þ hˆ ð1  Fðhˆ ÞÞ þ maxðh  hˆ ; 0Þ dFðhÞ ¼ 0: 0

ðA:17Þ By Proposition 3.3, hˆ b 1, so Eq. (A.17) implies that there exists h˘ , so that KV( Q) +hˆ (1F(hˆ )) + max(hhˆ , 0)T0 as hT h˘ . Given that u 1V(d ) is a strictly decreasing function, it follows that Z 1 u1VðW1 ðx2 ðhˇ ÞÞÞ W1Vðx2 ðhÞÞ½  KVðQÞ þ hˆ ð1  Fðhˆ ÞÞ þ maxðh  hˆ ; 0Þ dFðhÞb0: 0

ðA:18Þ ˆ ˆ ˆ Because Q(W 2)= Q(W 1)= Q and h (W 2)= h (W 1)=h imply x 1(h)=x 2(h) for all h, Eq. (A.18) is incompatible with the first-order condition (15) for Q 1=Q. The assumption that Q 2=Q 1 and hˆ 2=hˆ 1 thus also leads to a contradiction and must be false. 5

Equivalently, in the terminology of Diamond and Stiglitz (1974), x 2(h˜ ) is given by a mean-W k-utility preserving spread of x 1(h˜ ). 24

2002

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

References Arrow, K.J., 1973. Some ordinalist-utilitarian notes on Rawls’ Theory of Justice. Journal of Philosophy 70, 245 – 263. Atkinson, A.B., 1973. How progressive should income tax be? In: Parkin, M., Nobay, R. (Eds.), Essays on Modern Economics. Longman, London. Atkinson, A.B., Stiglitz, J.E., 1976. The design of tax structure: direct versus indirect taxation. Journal of Public Economics 6, 55 – 75. Bierbrauer, F., 2005. Optimal taxation and public good provision in a two-class economy, Preprint, Max Planck Institute for Research on Collective Goods, Bonn. Blomquist, S., Christiansen, V., 2001. The role of prices on excludable public goods, CESifo Working Paper No. 536, CESifo, Munich. Boadway, R., Keen, M., 1993. Public goods, self-selection, and optimal income taxation. International Economic Review 34, 463 – 478. Christiansen, V., 1981. Evaluation of public projects under optimal taxation. Review of Economic Studies 48, 447 – 457. Cremer, H., Laffont, J.-J., 2003. Public goods with costly access. Journal of Public Economics 87, 1985 – 2012. Cremer, H., Pestieau, P., Rochet, J.-C., 2001. Direct versus indirect taxation: the design of tax structure revisited. International Economic Review 42, 781 – 799. Cremer, H., Pestieau, P., Rochet, J.-C., 2003. Capital income taxation when inherited wealth is not observable. Journal of Public Economics 87, 2474 – 2490. d’Aspremont, C., Ge´rard-Varet, L.A., 1979. Incentives and incomplete information. Journal of Public Economics 11, 25 – 45. Deb, R., Razzolini, L., 1999a. Auction-like mechanisms for pricing excludable public goods. Journal of Economic Theory 88, 340 – 368. Deb, R., Razzolini, L., 1999b. Voluntary cost sharing for an excludable public good. Mathematical Social Sciences 37, 123 – 138. Diamond, P.A., Stiglitz, J.E., 1974. Increases in risk and risk aversion. Journal of Economic Theory 8, 337 – 360. Dre`ze, J., 1980. Public goods with exclusion. Journal of Public Economics 13, 5 – 24. Fang, H., Norman, P., 2003. An efficiency rationale for bundling of public goods. Discussion Paper, Yale University and University of Wisconsin. Golosov, M., Kocherlakota, N., Tsyvinski, A., 2003. Optimal indirect and capital taxation. Review of Economic Studies 70, 569 – 587. Gru¨ske, K.D., 1994. Verteilungseffekte der fffentlichen Hochschulnanzierung. In: Lu¨deke, R. (Ed.), Bildung, Bildungsfinanzierung und Einkommensverteilung, Schriften des Vereins fu¨r Socialpolitik NF vol. 221/II. Duncker & Humblot, Berlin, pp. 71 – 147. Guesnerie, R., 1995. A contribution to the pure theory of taxation. Econometric Society Monograph. Cambridge University Press, Cambridge, UK. Gu¨th, W., Hellwig, M.F., 1986. The private supply of a public good. Journal of Economics/Zeitschrift fu¨r Nationalo¨konomie Supplement 5, 121 – 159. Hammond, P., 1979. Straightforward individual incentive compatibility in large economies. Review of Economic Studies 46, 263 – 282. Hellwig, M.F., 2003a. Public-good provision with many participants. Review of Economic Studies 70, 589 – 614. Hellwig, M.F., 2003b. A utilitarian approach to the provision and pricing of excludable public goods. Discussion Paper No. 03-36, Sonderforschungsbereich 504, University of Mannheim, http://www.sfb504. uni-mannheim.de/publications/dp03-36.pdf. Hellwig, M.F., 2004a. The provision and pricing of excludable public goods: Ramsey–Boiteux pricing versus bundling. Discussion Paper 04-02, Sonderforschungsbereich 504, University of Mannheim, http:// www.sfb504.uni-mannheim.de/publications/dp04-02.pdf. Hellwig, M.F., 2004b. Optimal income taxation, public-goods provision and public-sector pricing: a contribution to the foundations of public economics. Preprint 04/14, Max Planck Institute for Research on Collective Goods, Bonn, http://www.mpp-rdg.mpg.de/pdf_dat/2004_14online.pdf.

M.F. Hellwig / Journal of Public Economics 89 (2005) 1981–2003

2003

Laffont, J.-J., 1982/1988. Cours de Theorie Microeconomique 1: Fondements de l’Economie Publique. Economica, Paris, Translated as: Fundamentals of Public Economics, MIT Press, Cambridge, Massachusetts. Laffont, J.-J., Tirole, J., 1993. A Theory of Incentives in Procurement and Regulation. MIT Press, Cambridge, MA. Ledyard, J., Palfrey, T., 1999. A characterization of interim efficiency with public goods. Econometrica 67, 433 – 448. Mailath, G., Postlewaite, A., 1990. Asymmetric information bargaining procedures with many agents. Review of Economic Studies 57, 351 – 367. Milgrom, P.R., Weber, R.J., 1982. A theory of auctions and competitive bidding. Econometrica 50, 1089 – 1122. Mirrlees, J., 1971. An exploration in the theory of optimum income taxation. Review of Economic Studies 38, 175 – 208. Moulin, H., 1994. Serial cost-sharing of excludable public goods. Review of Economic Studies 61, 305 – 325. Myerson, R.B., Satterthwaite, M.A., 1983. Efficient mechanisms for bilateral bargaining. Journal of Economic Theory 28, 265 – 281. Norman, P., 2004. Efficient mechanisms for public goods with use exclusions. Review of Economic Studies 70, 1163 – 1188. Pratt, J.W., 1964. Risk aversion in the small and in the large. Econometrica 32, 122 – 136. Rawls, J., 1971. A Theory of Justice. Harvard University Press, Cambridge, MA. Rochet, J.-C., Chone´, P., 1998. Ironing, sweeping, and multidimensional screening. Econometrica 66, 783 – 826. Samuelson, P.A., 1954. The pure theory of public expenditure. Review of Economics and Statistics 36, 387 – 389. Samuelson, P.A., 1958. Aspects of public expenditure theories. Review of Economics and Statistics 40, 332 – 338. Samuelson, P.A., 1969. In: Margolis, J., Guitton, H. (Eds.), Pure Theory of Public Expenditures and Taxation. Public Economics, Macmillan, London. Schmitz, P.W., 1997. Monopolistic provision of excludable public goods under private information. Public Finance/Finances Publiques 52, 89 – 101. Seade, J., 1977. On the shape of optimal tax schedules. Journal of Public Economics 7, 203 – 236.