Public and Private Provision of Public Goods in Democracy - Sfu

Public and Private Provision of Public Goods in Democracy Christoph L¨ ulfesmann∗ Simon Fraser University May 2010 Abstract This paper analyzes the provision of public goods in institutions with dual characteristics: the provision level (the ‘policy’) is chosen in democratic fashion by majority vote, but individual agents are free to top up their consumption levels before or after the collective decision is made. Important examples include policy making in federal states, charities, education, and health care. A key result establishes that regardless of the timing of private and public actions, and regardless of the degree of externalities, the results of the median voter theorem apply. Turning a one-tier public system into a dual system with ex-ante private contributions finds unanimous support in society, while adopting an ex-post contribution regime is at least majority preferred. Allowing for an endogenous timing of private contributions majority dominates both of those baseline institutions, and increases consumption of public goods. Finally, we show that with externalities, dual regimes with ex-post contributions are also majority preferred over a market-based private system.

JEL Classification: D02, D78, H11, H40, P16. Keywords: Public goods, Majority voting, private provision, dual provision, federalism, charities, health care. ∗

Department of Economics, Simon Fraser University, Vancouver B.C. email: [email protected].

Thanks to Antoine Loeper, Bard Harstad, David Baron, Richard Rogerson, Berthold Herrendorf, Anke Kessler, Steve Easton, and seminar participants at Arizona State, Bonn, Munich, Northwestern, Simon Fraser, Vienna, Cologne, Constance and Berlin for helpful comments and suggestions. Financial support by SSHRC is gratefully acknowledged.

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Introduction

Democratic policy decisions are often supplemented by private actions. Charities step in where tax dollars do not ensure a satisfactory level of aid. Countries with a taxfinanced public health care system often see individuals seek enhanced services from private doctors or hospitals. Various spending decisions or regulatory measures in federal states are taken at a central level of government, but member regions are allowed to tighten these policies at their own discretion. While some of these ‘private contributions’ have the characteristic of a private good whose benefits accrue only to the contributor, in many cases there are positive consumption externalities from one agent, or one federal region, to the other. A new or upgraded road or airport avails visitors from other countries, in the same way as helping the City’s poor may reduce crime rates, positively affecting more affluent citizens in the community. When a state government tightens federal regulation on vehicle or industry pollution, neighboring states are likely to benefit from the adoption of this local policy measure. The same is true when a state adopts mandatory vehicle liability insurance, which protects other drivers in case of an accident. The aim of the present paper is to explore situations of this type in a stylized setting. A group of agents takes a decision to provide goods or services in a democratic fashion, using majority voting to determine the level of spending. Each agent can also enhance the consumption of this good, by privately buying or contributing additional amounts. Public and private provision may generate consumption externalities across all agents in society. This framework follows a body of research (for a brief review, see below) that merges two well known baseline models: the democratic choice of policies under majority rule on the one hand, and voluntary private contributions to public goods on the other. While merging these two paradigms in a unified setting is of significant interest, overcoming the technical hurdles may be seen as challenging: since the voting and contribution decisions of all agents are interrelated and subject to strategic behavior, is is a priori unclear whether the median voter theorem as a convenient tool to characterize political outcomes remains applicable.1 1

Stiglitz (1974), Epple and Romano (1996) and Glomm-Ravikumar (1998) show that single-

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A main contribution of the paper is to show that these difficulties can be overcome in various versions of the baseline model analyzed here. We first explore a setting where private contributions are made simultaneously and non-cooperatively in a first stage, before a political decision on the uniform financing of the public good is taken in a second stage. In this regime with ex-ante voluntary private contributions (or for short ‘ex-ante regime’), individual preferences at the policy stage are single peaked regardless of initial contributions. The median voter theorem then applies and the individual with median preferences shapes the policy outcome. In contrast to the familiar majority voting scenario, though, the identity of the median voter is endogenous: since an individual’s private buying decision changes her subsequent preferences over policies relative to other individuals, the distinction of being a median voter depends on the vector of initial contributions. Nevertheless, we establish a ‘rank preservation’ property under which in equilibrium, the ‘natural’ median individual (the median voter under pure public provision without private contributions) always emerges as the pivotal decision maker. With this result, one can establish that only high-preference individuals contribute ex ante. Since these agents are well aware of the crowding out effect of their contributions, ex ante equilibrium contributions are not only smaller than those in a private system; they also fall short of the best responses to the subsequently chosen equilibrium policy. Nevertheless, the equilibrium policy turns out to be smaller and each agent’s public goods consumption to be larger than in a public system where topping up is infeasible. We then investigate the reverse case, which has been the focus of much of the existing literature. In this ‘ex-post regime’, democratic decisions occur before individual contributions can be made, which allows for a policy commitment vis-a-vis individual agents. Unfortunately, the analysis of this framework is severely hampered by the problem that in presence of externalities, individual preferences over policies are not single peaked. With any change in policy, the profile of subsequent private contributions and, potentially, the set of contributors changes. An individual may be a contributor for a small but not for a large policy, making the shape of his utility function difficult to predict. peakedness is generically violated in ’exit’ models where individuals can choose whether to consume a publicly provided good such as schooling, or to consume a private alternative instead.

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A key finding is that notwithstanding this problem, the results of the median voter theorem continue to hold. To prove this, we establish a separation property under which all individuals with preferences larger than the natural median prefer the policy that is globally preferred by the median individual over any smaller policy. Likewise, all lower-preference agents prefer the median voter’s bliss point over any larger policy. Hence, the policy preferred by the natural median cannot be defeated in majority voting, which makes her the democratic decision maker in society. This result then allows a precise characterization of equilibrium contributions and equilibrium policies. Equilibrium policies are found to be smaller than in the ex-ante regime, while private contributions are larger. An overall regime comparison is as follows. In both ex ante and ex post regime, only a subset of high-preference agents make private contributions, while individuals with lesser preferences including the median do not.2 Moreover, private contributions in the ex-post setting are larger than in the ex-ante setting, while equilibrium policies exhibit a reverse pattern. These findings can easily be understood in terms of the commitment capabilities assigned to the relevant agents. In the ex-post setting, the pivotal median voter knows that implementing a relatively small policy will trigger large private contributions, from which she benefits when externalities are involved. Conversely, in the ex-ante setting lower private contributions boost collective funding, which allows high-preference individuals to partially free ride on public provision. Not surprisingly, equilibrium policies are therefore larger in the ex-ante regime than the ex-post regime. Policies in each dual regime are also smaller than those in a one-tier public system: with normal preferences, the median voter finds it optimal to scale back public funding. Last not least and perhaps surprisingly, public goods consumption in dual systems does not always exceed the level achieved under one-tier public provision. While switching to the ex-ante system always raises public goods consumption, a switch to the ex-post system may even lower consumption levels of those services.3 2

Hence, a majority of the population makes no private contribution. With externalities, this majority includes even (some) agents with larger-than-median-preferences: in contrast to an agent’s spending increase in the political process, a larger private contribution is not matched by other agents. 3 The latter result holds because in the ex-post regime, the median voter may reduce (or even withhold) public funding in order to induce high preference agents to pay for public goods, even if the

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From a political economy perspective, we then address the crucial issue of the relative support for different regimes across society members. Regime changes usually require a well defined majority or super-majority of agents who prefer an alternative institution over the status quo. Our model allows to derive interesting and clear cut results. Perhaps most importantly, we find that a majority of citizens, comprising all individuals with low preferences for the public good, prefers the ex-post regime over both the ex-ante regime and pure public provision. At the same time, high-demand agents usually oppose a system switch towards an ex post system, in fear of exploitation by the majority that controls political decisions. In contrast and quite strikingly, there is unanimous support for institutional change from one-tier public provision to the ex-ante regime; in fact, all agents strictly support this transition which shows that adopting a dual system can be a Pareto improvement if designed in the right way: after the policy decision is taken, no additional private contributions can be allowed, while these contributions should be encouraged prior to the political decision.4 We then set up and analyze a more general version of the model. Specifically, we allow for endogenous timing so that agents are free to make private contributions before and/or after the policy is chosen. This institution appears as the most natural in environments where governments are unable to control the sequencing of private contributions. At first glance, one might think that economic outcomes must coincide with those for the ex-post regime; after all, the median voter is free to adopt the same equilibrium policy as in the ex-post model, thereby committing vis-a-vis high-demand citizens at the final stage. Perhaps surprisingly, this line of thinking turns out to be misleading. In fact, we show that the endogenous timing regime can strictly dominate the ex-post setting for a large majority of the population, including all high-demand individuals. The explanation is based on a reciprocal commitment argument. When exante contributions are feasible, high preference individuals will make them to alter the median voter’s strategic incentives at the subsequent policy stage. Specifically, these resulting provision level falls below the level in a pure public system. 4 This result has relevance for circumstances where the ex ante setting seems realistic, particularly, for situations where the policy decision (eg, at the federal level within a federation) finalizes the scale or design of the public project under consideration.

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ex-ante contributions offset the median agent’s incentive for strategic underprovision, causing her to embrace a policy level strictly larger than in the ex-post regime.5 We also find that the median voter and along with her, a large majority of citizens including all high-preference agents prefers endogenous timing over both the ex-ante and the ex-post dual regime. For these individuals, the general system is a win-win situation because it allows high preference agents to reduce their private contributions, while at the same time allowing the median voter to boost public funding without causing a crowding out effect. Hence, this most flexible regime emerges as the only stable institution under majority vote. We finally consider a scenario in which in contrast to our previous analysis, a private market based system represents the status quo. With externalities, such a system often dominates a one tier public system for a majority of voters, comprised of poor and rich individuals. However, we establish that a majority (including all individuals with preferences larger than the median) prefer the dual ex post regime over a private system.6 Since the same majority prefers the general system with endogenous contributions over the ex post system, the general system again emerges as stable instituion under majority vote, regardless of the intensity of externalities.

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Related Literature

We contribute to the growing literature on the allocative and political economy implications of a dual private-public provision of goods and services. Most existing papers confine attention to the ex-post system: first the political decision is made by majority rule, before individuals can privately contribute in a second stage. Epple and Romano (1996) pioneered the analysis of this setup. They explore a dual health care system, with a public-provision element being funded through linear income taxes, whose level 5

The equilibrium policy is then a best response to the (and only the) ex-ante private contributions.

In contrast, in the ex-post setting, the equilibrium policy choice is strictly lower than the best response to the equilibrium ex-post contributions: for strategic reasons, policy underprovision arises in order to stimulate subsequent private contributions. 6 We also show that the private system may majority dominate the ex ante dual system.

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is chosen in a democratic process. In this setting, the equilibrium level of public services is shown to exceed the provision level in both a pure public system, or in a market based private system. Moreover, a majority of citizens strictly prefer dual provision. These findings are confirmed and empirically tested in Fernandez and Rogerson (2003) who compare a variety of dual-provision systems of school funding. The democratic funding decision is taken at the provincial level, while private contributions are made by individual school districts.7 In both of these papers, the good under scrutiny choice is a private good which imposes no consumption externalities across agents in society. For this reason, public decisionmaking in the stage game is not implicated by strategic behavior on the voters’ parts, which guarantees the single peakedness of individual preferences and the validity of the median voter theorem. On the issue of institutional choice, this literature finds an ends-against-the middle pattern: a majority comprised of rich and poor individuals prefers the dual ex-post system over the one-tier public system, in contrast to the middle class that favors the larger public sector in the one-tier system.8 Moreover, the ex-post system is also shown to majority dominate private, market based provision. Accounting for the importance of externalities in many real life scenarios, Epple and Romano (2003) extend their analysis to the case of a pure public good. To deal with strategic voting, individuals are assumed to behave ‘myopically’: they disregard the effect of their first-stage voting behavior votes on second-stage private contributions. Such an equilibrium does not necessarily exist but if it does, the natural median voter is the pivotal individual. In a similar model with a focus on policymaking in federations, Hafer and Landa (2007) find that the lack of preference single peakedness in general renders the median voter theorem inapplicable. Using Cobb-Douglas preferences, they derive sufficient conditions for equilibrium existence, and provide a variety of additional characterization results. 7

Beyond analyzing institutional setups with private, public, or dual ex-post regime, their paper examines some other possible regimes, and provides a thorough empirical analysis. 8 In contrast, our paper (which assumes uniform taxes) shows that regardless of the degree of externalities, rich agents prefer the one-tier public system, while all other agents (a majority) prefers the dual ex-post system.

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Our work is closest to Cremer and Palfrey (2000, 2006) and Alesina et al. (2005). All these papers also assume that public policies are financed through uniform contributions.9 Cremer and Palfrey investigate ‘federal mandates’, imposed by a central government via majority vote among federal regions. A federal mandate is a regulation imposed by the highest tier of government on individual states, and compliance (e.g., environmental quality, food safety, etc.) requires states to incur costs. Moreover, while member states have to obey the mandated standard, they are free to impose stricter rules by state legislation, the ’private contributions’ element in our terminology. In Cremer and Palfrey (2000), policy choices and private contributions do not cause externalities across regions. Within this non-strategic context, the median voter theorem applies, and some high-demand states top up the mandate by imposing stricter local regulation. Moreover, a majority of the population in a majority of states strictly welcomes a switch from a one-tier public system to one with federal mandates. Cremer and Palfrey (2006) incorporate externalities but find that in this case, majority vote equilibria cannot be guaranteed to exist. As a remedy, they characterize ‘local’ policy equilibria, defined as the set of policies which is majority preferred to other policies in the vicinity.10 Alesina et al. (2005) use a similar framework, but with homogenous in-district voters. To the best of our knowledge, this is the only other paper to explore not only the ex-post regime, but also the ex-ante regime. While it finds that a majority of agents prefers the ex-ante system over public provision, we prove the stronger result that in fact, every agent strictly prefers the ex-ante regime. While some of our results are close to those in Alesina et al., they require the applicability of the median voter theorem which is assumed but not proven in their paper.11 9

All these papers are framed in the context of federal systems where uniform policy costs are very

plausible. However, uniform costs naturally occur in many other applications as well. Premiums for mandatory car liability insurance, for example, are generally unrelated to a driver’s income. Rather, they can be seen as a uniform tax that is imposed on every driver in the state. 10 There is one significant difference between the settings analyzed in Cremer and Palfrey, and our paper. While in Cremer and Palfrey (2006) each voter in each region participates in federal policy decisionmaking (in stage one), our setting can be interpreted as one where only the district medians cast their federal votes. Our results suggest that this seemingly minor modification restores the existence of global majority vote equilibrium. 11 In fact, Alesina et al. acknowledge that the theorem may not hold (See p. 614 of their paper).

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As discussed before, our paper extends existing results in several directions, most importantly, by incorporating externalities that often appear prevalent in reality. The key step here is to show that in dual institutional systems, the median voter theorem holds for an interesting class of models that admit varying degrees of externalities, and fully rational voters. With respect to empirical significance, we think that uniform cost sharing is often plausible. Contributions to a state funded health care system are made lump sum in Canada. In many regulatory settings including mandatory car liability insurance, insurance premia are unrelated to a driver’s income level.12 Federal mandates are usually uniformly imposed on all member regions, and compliance costs are not directly linked to a state’s wealth. In these and many other scenarios, uniform policy contributions appear as a good approximation of reality. The remainder of the paper proceeds as follows. Section 3 below describes the baseline model. Section 4 explores some benchmark scenarios. Section 5 provides a general analysis of the systems with ex-ante and ex-post contributions, and Section 6 highlights some special cases. Section 7 focuses on the endogenous timing of contributions, and Section 8 briefly compares dual regimes and a market system. Section 9 concludes.

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The Model

Consider an economy with N ≥ 3 agents, where N is odd. Individual i derives his utility U i (ci , Gi ) from the consumption of a private good ci , and a public good Gi which will be defined more precisely below. The utility function is quasiconcave, and both c and G are normal goods according to i’s preferences. Moreover, we impose the standard Inada conditions and normalize the prices of both public and private good to one. Agents may differ in their exogenous incomes, yi , as well as their preferences. All agents play a version of the following general game. In a stage 1, they can simultaneously and non-cooperatively make private ‘ex-ante’ contributions gˆi towards 12

While most countries (or states) in the developed world mandate minimum liability car insurance,

the required coverage amounts vary dramatically. For example, the mandatory third-party liability coverage is GBP 500,000 in the UK, but only US 30,000 per accident in California.

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the public good. These contributions become public information of all agents, and they can alternatively be viewed as individual i’s private purchase and consumption of the commodity, with spillover effects to other agents.13 In a subsequent stage 2, all individuals decide in a political decisionmaking process on a mandatory uniform lump-sum contribution g that is imposed on each agent. For concreteness, we follow the positive literature on public goods supply and assume that g is chosen by majority vote. Finally, in stage 3, individuals can simultaneously top up their public-goods consumption through additional ‘ex-post’ contributions, g˜i . Agent i’s consumption of the private good is then ci = yi − g − gˆi − g˜i . As we have already emphasized in the Introduction, private ex-ante and ex-post contributions allow us to explore the commitment effects of policy choices and of private consumption decisions, respectively. Regarding the characteristics of commodity G, we will allow for pure and impure forms of public goods or services. The public goods consumption of each individual is composed of a uniform amount GU which is provided to everyone through the political process, of the private contributions of the individual himself, and of the contributions of other individuals. In particular, GU = g(1 + β(N − 1)) where β ∈ (0, 1) indicates the degree of consumption externalities. For β = 0, there are no spillovers and GU becomes a private good with uniform consumption requirement. For β = 1, the commodity in question is a pure public good, while all interior specifications of β capture intermediate characteristics. The overall public good consumption of a citizen i is then Gi = GU + [ˆ gi + g˜i + β

∑

(ˆ gj + g˜j )],

j̸=i

where private contributions of individuals j ̸= i again cause spillovers at a rate β. Note that unless β = 1 where Gi = G for all i, larger private contributions translate into a higher overall consumption. Let g be a public policy and gˆ and g˜ be the vectors of ex-ante and ex-post private 13

In both interpretations, the ‘public good’ may be offered by suppliers in a competitive market, or it may be produced in home production where all individuals have access to the same technology.

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contributions, respectively.14 For subsequent reference it is useful to define ∆i (g, gˆ, g˜) =

UGi (ci , Gi ) Uci (ci , Gi )

as individual i’s marginal rate of substitution between the private and the public good. We impose Assumption 1. (Single crossing property) For any g and gˆ, g˜ with the property gˆi = gˆj , g˜i = g˜j for all i, j, the rank order of the vector ∆(g, gˆ, g˜) = {∆i }i=1,...,n is preserved. We say that an individual i = 1, ..., n with a higher index exhibits ‘larger preferences’ for the public good. The single crossing assumption allows us to order individuals according to their ‘natural’ rank in terms of preferences towards the public good: for any arbitrary public-good level generated by identical funding from each individual, the ranking of the marginal rates of substitution across individuals remains the same. Notice that the definition is flexible enough to accommodate not only taste, but also income differences. For example, consider two individuals with identical homogenous preferences but different incomes. For identical individual contributions to the public good, their private consumption differs, and the lower-income individual displays a smaller ∆(·) and is considered a lower-preference individual. Another important class of preferences consistent with Assumption 1 are quasilinear utilities, where agent preferences and incomes are heterogenous: U i = ci + αi H(Gi ), with αi > 0 being a preference parameter for the public good. In this representation, an individual with larger αi us characterized by a larger index i. In what follows, we will say that individual m is the ‘natural’ median individual according to the ordering described above. In what follows, we assume that g ∈ [0, g¯] where the maximum policy g¯ is smaller than the income of the poorest individual in society. Alternatively, we could assume that an agent’s policy contribution is min{yi , g}, and close the model by imposing a sufficiently harsh punishment in case 14

that the individual pays less than yi . This latter specification would not alter any qualitative results but significantly complicate the exposition.

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4

Benchmark cases

Before providing a general analysis, we briefly investigate the two simple benchmark cases of pure public provision, and of a situation where G is a private good. In the first of these frameworks, the consumption of the public good is determined exclusively through collective decision-making while in the second, no externality problem exists.

4.1

One-tier (Pure) Public System

Suppose public goods G are provided exclusively and uniformly through the political process, that is, no private consumption decisions can be made. Individual i prefers a policy outcome gipp = arg maxg U i (yi − g, g(1 + β(N − 1)), which is implicitly defined by the first-order condition ∆i (·) = UGi (ci , G)/Uci (ci , G) = 1/(1 + β(N − 1)) ≡ 1/z ≤ 1. Under our previous assumptions, the single-peakedness requirement on individual preferences is satisfied so that the median voter theorem applies. Moreover, since ∆i and thus gipp are increasing in the index i, the preferences of individual m with median bliss point represent the unique outcome of majority voting. We have pp is Proposition 1. Consider pure public provision. The unique equilibrium policy gm

determined by the preferences of the natural median voter m, and is implicitly given by ∆m =

pp pp 1 UGm (ym − gm 1 , gm (1 + β(N − 1)) = = . pp pp m Uc (ym − gm , gm (1 + β(N − 1)) 1 + β(N − 1) z

(1)

pp Under the Inada conditions, the selected policy gm is strictly positive. Moreover, and

in the special case where individuals differ only in their incomes, the median-income individual m becomes the pivotal voter in society.

4.2

Private goods

Now, suppose the publicly provided service is a pure private good, β = 0. As noted before, this case has received significant attention in the existing literature on dual institutions. Since consumption externalities are absent, voting is not subject to strategic behavior, a feature that considerably simplifies the analysis. 11

Suppose first that individuals make private consumption decisions and there is no collective provision of the ‘public’ good in question. In such a private market system, each individual buys and consumes the ‘public’ good at a level G∗i where his marginal rate of substitution ∆i equals the marginal costs. Since there are no externalities, this outcome is clearly Pareto efficient. Conversely, suppose Gi is provided through a one-tier public system. As found in Section 4.1, the median voter then determines pp the uniform provision and funding level gm , defined by ∆m (·) = 1. In this allocation,

any individual rather than m is hurt compared to a private system: lower-preference agents consume too much, and higher-preference agents consume too little. With this pretext, let us now find out the consequences of dual private-public provision regimes. To do so, we label a system in which private contributions are followed by public provision as the ’ex-ante regime’, and a dual system with reverse timing as the ’ex post regime. Our analysis yields Proposition 2. Suppose β = 0. In both the ex-ante and the ex-post regime, there exists pp a continuum of identical equilibria, characterized by equilibrium policies g ∗ ∈ [0, gm ].

In addition, a) Each high-preference agent with rank i ≥ m consumes her preferred amount of ‘public’ goods G∗i , defined by ∆i (G∗i ) = 1. b) For any equilibrium policy g ∗ satisfying g > g1pp (the bliss point of the lowestpreference individual), a subset of lowest-preference agents with ranks j < m is forced into overconsumption, ∆j (G∗j ) < 1. This subset increases in g ∗ , and it pp comprises all individuals j < m for g ∗ = gm .

pp ). Each dual systems Pareto-dominates pure public provision (unless g ∗ = gm

Proof: see the Appendix. In case of a publicly provided private good, dual public-private institutions display a continuum of equilibrium outcomes. Regardless of the timing (ex ante or ex post), 12

these equilibria are identical with respect to contribution vectors and equilibrium policies. In each equilibrium, individuals with preferences larger than the median replicate their consumption patterns in a private market system, by equating marginal rate of substitution and marginal rate of transformation. In contrast, low-demand agents consume more and pay more for the publicly provided good than desired. Not surprisingly, we can conclude that private markets Pareto dominate any system with collective provision if spillovers are absent.15 Since the upper bound of equilibrium policies in a dual system coincides with the (unique) equilibrium under one-tier public provision, it also follows that dual regimes Pareto-dominate the former regime (in almost every equilibrium) when externalities are absent.

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Agents with preferred consumption levels larger than g ∗ strictly prefer

a dual system because it allows them to consume their preferred levels of the ‘public’ good. But individuals with preferred consumption level less than g ∗ also benefit from a dual system, because it reduces overconsumption. Hence, every individual in society is strictly better off. Before concluding, it is worth emphasizing that in contrast to our setting with uniform taxation, unique equilibria arise in a setting with income-dependent funding (and β = 0).17 In particular, this result holds because with linear income taxes (Epple and Romano, 1996; Fernandez and Rogerson, 2003), each individual has strict preferences over tax rates. We will show that our general framework with externalities, which is analyzed next, restores the uniqueness of equilibrium for any β > 0. Of course, this is strictly true for equilibrium policies larger than g1pp . For smaller policies, the dual provision regimes and the market system of course have identical welfare implications. 16 This is not true in Cremer and Palfrey (2000) where (a) voters within districts are heterogenous, 15

(b) all voters in all districts decide on federal policies by majority vote (so the federal median decides), and (c) the ’topping up’ decision is taken by each regional median. Individuals in regions with median preferences below the federal median – and hence, a majority of the total population – remain unaffected by a transition towards the ex-post system. In contrast, a majority of agents in each region with median preferences above the median strictly gains from the transition, so that overall there are more winners than losers. 17 Cremer and Palfrey (2000) consider uniform taxation. They elegantly avoid multiplicity issue by assuming that no individual knows his place in the regional preference distribution. Voting in favor of the individually preferred consumption level is then dominant strategy in the federal voting stage.

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5

General Analysis

5.1

Dual System with Ex-ante contributions

Let us start the analysis with a system where only ex-ante private contributions gˆi are feasible. For example, the public good may be a new hospital, opera, or other public construction project whose technical specifications (design, size etc.) are irreversibly determined in the stage-2 political process.18 The ex-ante regime also captures situations in which individual agents - such as states in a federation, or charities - act as strategic leaders whose contributions initiate a project before political decisions on additional funding are made. In the context of charity work, for instance, the extraordinary capital endowment of the Bill and Melinda Gates foundation make this organization a strategic leader regarding health aid to developing countries. Similarly, in the context of the EU, large states such as Germany or France are sometimes said to preempt collective decisions at the Unon level in a variety of policy fields of common interest. Using subgame perfection, we first explore the stage-2 political equilibrium, before turning attention to the stage-1 private-contributions. Two crucial issues need to be addressed. First, it needs to be shown that individual stage-2 preferences are singlepeaked so that majority voting equilibrium exists and the preferences of the median voter prevail in political equilibrium. Second, upon establishing single-peakedness, we must find the stage-1 contributions equilibrium and ask whether or not the median voter induced by this contribution profile is the ’natural’ median m. Since individual policy preferences in stage 2 depend on the stage 1 decisions of all agents, the identity of the median voter is endogenous and the answer to this latter question not immediate. We start with a series of lemmas, whose proofs have been relegated to the appendix. Lemma 1. For any arbitrary stage-1 contribution profile gˆ = {ˆ gi }i=1,...,n , each agent i’s preferences in stage 2 over g are single-peaked, with bliss point giea . Hence, the choice 18

The political decision on project design then essentially determines the scale of the public good,

and additional private contributions would leave consumption levels of this good unaffected.

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ea of the individual M with stage-2 median preferences, gM (ˆ g ), prevails in majority vote.

To be more specific about the equilibrium outcome, we now investigate the privatecontributions game in stage 1. This yields Lemma 2. There exists an equilibrium in which M = m. Moreover, no low-demand individual j with index j ≤ m contributes in stage 1.

Lemma 3. The equilibrium as described in Lemma 2 is unique if β > 0.

The lemmas establish that the natural median individual m will be pivotal for the political decision in equilibrium, and that she herself will not make a private contribution. The key force behind this result is that no individual with larger natural preferences has an incentive to switch preference ranks with m. Voluntarily moving ‘to the left of the median’ means that an agent’s private contribution has been too large. This is true because for each agent, achieving larger consumption through private contribution is more costly than provision through the political process, in which all individuals collectively pay for the public good. But if no (high preference) individual has an incentive to switch ranks, m has no incentive to contribute herself: she is pivotal and therefore, can achieve her preferred public good consumption without wasting private resources. We can now summarize the equilibrium outcomes. Proposition 3. Consider the ex ante system where individuals can privately contribute in stage 1. In equilibrium, the ‘natural’ median-preference individual m becomes the ea median voter and chooses the equilibrium policy gm (ˆ g ). Moreover,

1) No individual j with natural preferences j ≤ m contributes in stage 1. These ea ea individuals consume uniform amounts of the public good, Gea j = Gm = gm [1 + ∑ β(N − 1)] + β i gˆiea .

2) Some high-preference individuals i > m may make a private contribution in stage 1. Equilibrium contributions are then rank-ordered, i.e., gˆN ≥ gˆN −1 ≥ ... ≥ gˆm+1 , with strict inequality for positive contributions. Accordingly, consumption levels 15

ea ea ea are Gea ˆiea (1 − β), with Gea i = Gj + g n ≥ Gn−1 ≥ ... ≥ Gm+1 (strict inequality

applies whenever β < 1 and gˆiea > 0 for at least one of two adjacent individuals). Proof: See Lemmas 1 to 3. The Appendix establishes monotonicity of contributions. Not surprisingly, higher-preference individuals make larger private contributions, and therefore consume larger quantities of the public good. In a next step, we now compare the economic outcomes in the dual-provision regime to those in a public system where no private contributions are allowed. This is done in Proposition 4. Consider the ex-ante system (superscript ea). In comparison to a pure public system (superscript pp), results are as follows: 1) equilibrium policies are characterized by g ea ≤ g pp , with strict inequality if β > 0, the private good is strictly normal, and if the set of contributors is non-empty. 2) the public-goods consumption of any non-contributor j (including the median) pp satisfies Gea j ≥ Gj , with strict inequality if β > 0, if the set of contributors is

non-empty, the public good is strictly normal. Of course, contributors consume even more unless β = 1. Proof: To establish 1), recall that m chooses public policy g in both regimes, and in a way that ∆m = 1/z. Let β > 0 and suppose the set of contributors in ea is nonempty. If consumption of the private good is strictly normal, the median voter’s private consumption cm in ea must be strictly larger than in pp, which requires a smaller g in the former institution. Part 2) follows immediately: with strictly normal public good preferences and because cm is larger in ea, public-goods consumption Gm will also be larger in this regime. 2 From a policy perspective, it is important to explore which individuals prefer the dual system over the pure public system, or vice versa. Our findings in this respect are surprisingly strong. Proposition 5. All individuals unanimously prefer the ex-ante system over the public system. This preference is strict if β > 0 and if the equilibrium set of private contributors is non-empty. 16

Proof: Consider an individual j who does not contribute in the dual system. Any such pp individual, consumes Grj = Grm in regime r ∈ {ea, pp}. By Proposition 4, Gea j ≥ Gj , ea pp gm ≤ gm (with strict inequality if β > 0 and if the set of contributors is non-empty). pp Hence, cea j ≥ cj so that individual j must prefer the ex-ante system. Consider now an

individual i who contributes gˆi > 0. If this agent had made no private contribution, he would have received a utility no smaller than in the public system: if he is the only contributor, a decision not to contribute would have triggered equilibrium policies and consumption levels identical to those in regime pp. Conversely, with other contributors, pp ea pp gˆi = 0 still induces Gea i ≥ Gi , and gm ≤ gm (with strict inequality if β > 0). Hence,

i prefers the ex-ante system, and strictly so if β > 0. By revealed preference, making a contribution must raise i’s utility even further. Hence, each equilibrium contributor i prefers ea over pp which completes the proof. 2 While resting on a simple logic, this result is quite intriguing. According to the Proposition, each agent in society prefers a public-private institution over pure public provision, and strictly so in the most plausible scenario of positive spillovers and a presence of private contributors. Hence, everybody welcomes a policy reform from a pure public to a dual system. We should emphasize that this finding remarkably differs from results in the existing body of literature on institutional comparisons. While some form of hybrid system is usually preferred by a simple majority, constitutional changes in reality require approval of some super majority of citizens. Proposition 5 shows that under the present assumptions on timing, even a Pareto improving institutional change is achieved.19 19

Notice that Alesina et al. (2005) investigate the same framework (with the restriction to quasilinear preferences), but arrive at a different result. While they argue that in general, only a majority of voters prefers the dual ex-ante system, we show that the much stronger statement of Proposition 4 can be made. Notice that in an extended setting with intra-regional preference heterogeneity (as analyzed in Cremer and Palfrey’s work), the result would have to be rephrased as saying that all regional median voters favor the dual system, while a minority of low-preference individuals in high-preference districts may oppose it.

17

5.2

Ex-post Contributions: Policy Commitment

This Section investigates the same setting as before, except that now the order of strategic moves in a dual public-private system is reversed. Suppose the policy g is determined in a first stage by democratic majority voting. In a subsequent second stage, each individual can top up the provision of the public good through a private contribution g˜i . As before, these ex-post contributions are made simultaneously and in a non-cooperative fashion. In this ex-post regime, prior political decisions shape the subsequent private behavior of individuals.20 Since this is seen as empirically most relevant, this form of a dual system has received most of the attention in the existing literature. As has been emphasized in this literature, in presence of externalities one cannot generally take the existence of a political equilibrium for granted. This is because the set of second-stage contributors, as well as the size of their respective contributions, depend on the policy g that was implemented in the first stage. Single-peakedness of preferences at the policy stage is then elusive, and the median voter theorem may not hold. A main result of the present paper is to show that even though the single-peakedness requirement may fail to hold, the result of the median voter theorem extends to the present setting. This is to say the only tax alternative which cannot be beaten in majority vote is the alternative preferred by the natural median individual, m. We also offer a precise characterization of this equilibrium, and provide a detailed comparison to pure public provision, and the ex-ante contributions regime. The central issue is the outcome of the policy choice stage. In the Appendix, we first establish that each individual j < m with preferences smaller than the natural median will prefer the policy which is preferred by m, over any larger public provision level (Lemma 4). Following our previous notation, we denote the policy preferred 20

In a federalism context, the strategic ‘leader’ role is imposed on the central government, and the ‘follower’ role on regions or lower-tier policy makers within a federation. In a regulatory context, states pass laws regarding, say, minimum car liability insurance; each citizen must obey these rules but can top up the level of coverage beyond the minimum if desired.

18

ep by m as gm , where ep stands for ex post contributions. Second, we show that any ep individual i > m with preferences larger than the natural median, prefers gm over any

smaller tax rate (Lemma 5). The next Proposition combines these results and shows that the median voter theorem must hold, the potential non-regularity of preferences notwithstanding. We have Proposition 6. In the ex-post regime, the preferred policy of the median individual m, ep gm , beats all other policy alternatives g in pairwise comparison.

Proof: By Lemma 4, m and all individuals j < m with smaller preferences for public ep ep goods than m prefer gm over any some larger policy g > gm . By Lemma 5, m and ep ep all individuals i > m prefer gm over any smaller policy g < gm . Combined, no other ep policy can beat gm under majority vote, which proves the result. 2

The result of Proposition 6 opens up the door for a substantive analysis of the ex-post regime. Our first goal is to characterize the ex-post private contribution levels. As in the ex-ante regime, it turns out that neither the median individual nor any individual with lower rank, will make a private contribution. This is shown in Proposition 7. Neither the median individual m, nor any smaller preference individual j < m will contribute in stage 2. Proof: see the Appendix. Proposition 7 is perhaps best understood by the following heuristic argument. Consider a policy level g sufficiently low so that m contributes in the stage-2 continuation equilibrium. The median voter knows that raising g further forces some low-preference individuals (who are non-contributors in stage 2) to provide additional resources for the supply of the public good. Hence, raising g transfers income from low-preference individuals to high-preference individuals. This allows the median voter m and all other contributors to reduce their private contributions not only by the amount of their increased policy payment but also by the amount of this transfer, while leaving their public goods consumption unaffected. As a result, m as a contributor can raise 19

her private consumption, and at the same time increase her utility level because naturally, non-contributors cannot reduce their private supplements. In this light, it does not surprise that the median voter will want to push collective payments g to the level where she ceases to contribute subsequently. The median voter m and all lower-preference individuals do not make a private contribution. What about higher-preference individuals? Similar to what we found for the ex-ante regime, a subset of highest-preference individuals may want to enhance their public goods consumption. The Proposition below compares the ex-post regime ep and the ex-ante regime regarding private contributions and equilibrium policies gm and ea gm , respectively. To simplify the exposition, we focus on a situation where the set of

contributors in both scenarios is non-empty.21 Proposition 8. With ex-post contributions, the equilibrium policy is strictly smaller ep ea than in the model with ex-ante contributions, gm < gm . Moreover,

a) each non-contributor (including every individual j ≤ m) prefers the ex-post regime ep over the ex-ante regime ea (and a fortiori, over the public regime pp); this is true even though ep may reduce their public goods consumption. b) Private contributions in ep are strictly larger, and the set of contributors may be larger than in ea. Contributors strictly prefer ea over ep, while their preference ranking between ep and pp is ambiguous. Proof: see the Appendix. The Proposition offers a variety of interesting results. Equilibrium policies in all three ea ep ≤ g pp , with strict inequalities as long as ≤ gm institutional regimes are ranked as gm

some agents privately contribute. In particular, the median agent m (who is pivotal in all three settings) chooses a lower collective contribution level in the ex-post regime, than in the ex-ante regime. The intuition here is the presence of a commitment effect: 21

If this set is empty in the ex-post regime, it will be empty in the ex-ante regime as well. Without private contributors, equilibrium policies can be identical, and private goods consumption can be larger in the ex-ante setting. See the example below.

20

by adopting a smaller policy in the ex-post regime, m can induce high-preference individuals to raise their private contributions. Quite naturally, all low-preference agents (including all agents j ≤ m) welcome this strategic commitment and hence, a majority of individuals prefers the ex-post setting over both alternative regimes.22 Conversely, high-preference contributing agents dislike the ex-post regime, which they perceive as a vehicle for funding relieve to low-preference agents. In fact, the collective funding cut in the ex-post regime can be so severe that for private contributors, even pure public provision may be preferable over the dual ex-post setting. Overall, these results imply that the ex-post regime is majority preferred (but not unanimously preferred) over both the ex-ante and the public-provision regime. Private contributions are clearly larger in a regime with ex-post contributions, for two reasons. First, with a smaller policy private contributions become more tempting for high demand individuals. Second, there is no strategic incentive for these agents to lower their contributions, in anticipation of a larger public response. Finally, regarding total public goods consumption, regime comparisons are generally ambiguous. One can show that if income effects are absent, equilibrium contributors consume most of the policy good in the ex-post regime, while non-contributors may consume least. For the special case of a pure public good, each agent consumes identical amounts in the expost regime and under one-tier public provision, and strictly more than in the ex-ante regime.23 22

This is due to revealed preferences: the median voter could always choose the same policy as in the ex-ante regime, thereby raising his utility (and the utilities of all lower-preference agents) because private contributions increase in the ex-post setting. While these contributions are best responses to a given policy in the ex-post setting, they are smaller than the best responses to the same policy level in the ex-ante setting. 23 Let there be a positive number of equilibrium contributors. The public goods consumption of any contributor h is characterized by ∆h (·) = 1 in the ex-post setting, but by ∆h (·) > 1 in the ex-ante setting where equilibrium contributions are less than the best responses to the equilibrium policy for strategic reasons. In absence of income effects, this implies a larger public goods consumption under public provision. However, with income effects, the larger equilibrium policy in the ex-ante regime expands h’s consumption set, and may cause h to consume more of the public good.

21

6

Extremes: Public vs. Private Goods

It is useful to compare the two limit case in which the public service is either a private or a purely public good. Consider first an ‘almost’ private good G for which externalities disappear in the limit, β → 0. In contrast to Proposition 2 which established multiple equilibria for β = 0, our results in Section 5.1 and 5.2 suggest that equilibrium policies are unique for any arbitrarily positive β > 0. Moreover, for β → 0 these equilibrium policies must be identical in all three regimes, for two reasons. First, we found that in each institutional regime, the natural median voter is pivotal and will never make a private contribution. Second, since externalities disappear in the limit, strategic voting motives become absent for the median individual or anybody else in society. Regardless of the institutional setup, this leads the median voter m to select a policy such that ∆m (cm , Gm ) = 1 holds in the limit. This policy is the equilibrium policy under pure pp public provision, gm : all individuals i > m privately contribute and consume the

desired amount of the public good, while all agents j < m are forced into an excessive consumption.24 Regime-independent equilibrium policies do not extend to a scenario with pure public goods, β = 1. In particular, private contributions can be very large in the an ex-post regime, but very small or even zero in the ex-ante regime. The opposite is true with regard to equilibrium policies, which can be zero in the ex-post regime and large in the ex-ante regime and public-provision regime. To make this point in the most simple way, consider quasi-linear preferences, U i = ci +αi H(Gi ), where αi > 0 is a preference parameter of agent i and H(·) some increasing pp and concave function. With public provision, the equilibrium policy gm > 0 yields

a public goods consumption Gpp as implicitly defined by H ′ (Gpp ) = 1/N αm , where N represents the number of agents and αm the median individual’s taste parameter. Assume in what follows that the size of N αm is ’close’ to the preference parameter of 24

Again, a simple majority of higher-preference agents are indifferent between both two dual systems and a private market system, and strictly prefer any of those institutions over a regime with pure public provision.

22

the highest-demand agent, αh .25 In the ex-post regime, the equilibrium policy g ep is then zero. The median voter m knows that with g = 0, h will privately contribute in a way that H ′ (Gep ) = 1/αh .26 But since αh and N αm are similar in size, the associated public goods level is close to Gpp at no cost to any agent other than the high demand individual h. In other words, the ability to commit to a policy makes it optimal for m (and a majority of citizens) to free ride on h’s contributions. For comparison, consider now the ex-ante setting. Here, h is aware that his private contributions will crowd out public provision. When N αm is close to αh , h will find it optimal to forgo any private contribution, thereby inducing a public pp policy gm and associated public goods consumption Gpp in the second stage. In this

latter regime, all individuals other than h are made worse off compared to the ex-ante system, and everybody in society achieves an equilibrium utility identical to the level under pure public provision.27 Overall, these arguments show that while public good consumption levels may be very similar or even identical across all regimes, different regimes can display extreme differences regarding the amount of private contributions, equilibrium policies, and the overall utility distribution when spillovers are important.

7

Endogenous Timing of Contributions

We now ask how these previous results play out in a more general framework where private contributions are feasible both at the ex-ante and the ex-post stage. At first glance, one might think that results will be the same as those in the ex-post system. Since policy decisions still precede private contributions, adding an ex-ante contributions stage may seem strategically irrelevant. Perhaps surprisingly, though, we will 25 26

For example, let N = 3, α1 = 1, α2 = αm = 2, and α3 = αh = 6. Standard results (Bergstrom et al., 1986) suggest that in absence of income effects, only the

highest-preference individual will privately contribute. 27 Suppose to the contrary that N αm and αh are dissimilar. With N αm sufficiently smaller than αh , equilibrium policies in both dual regimes are zero and h’s equilibrium contribution is defined by H ′ (G) = 1/αH , leading to a public goods consumption larger than under public provision. Conversely, pp for N αm sufficiently larger than αh , m implements a policy gm and h does not contribute, regardless of the timing of private contributions.

23

find this intuition to be misleading: endogenizing the timing of private contribution can generate equilibria that differ significantly from those in all previous settings. Importantly, we also find that an institution with endogenous timing of private contributions dominates both the ex ante and ex post system for a broad majority of members in society. While a full analysis of the general scenario must remain beyond the scope of this paper, interesting results are already obtained in a restricted setting. Specifically, we provide a characterization for the case of quasi-linear preferences, and a setting where β = 1 so that the good in question is a pure public good. Consider preferences of the form U i = ci + αi ln Gi with preference parameter αi increasing in i.28 Then, ci = yi − g − g˜i − gˆi indicates the private consumption of individual i with income yi , who provides funding g in the political process, makes a voluntary contribution of gˆi at an ex-ante stage, and another ex-post contribution g˜i after the policy g has been ∑ gi + gˆi ) is i’s consumption level of public implemented. Accordingly, Gi = gN + i (˜ services. We first analyze the ex-post decisions at stage 3. With a pure public good and quasiˆ = ∑ gˆi linearity, the only potential contributor is agent h with highest αh .29 Define G i

as the sum of individual ex-ante contributions. In stage 3, agent h maximizes ˆ U h (ˆ g , g, g˜h ) = yh − g − gˆh − g˜h + αh ln(gN + g˜h + G).

(2)

with respect to g˜h . The maximizer is easily computed as ˆ = max{αh − gN − G, ˆ 0}. g˜h∗ (g, G)

(3)

As expected, agent h’s ex-post contribution (if positive) decreases in policy g and in ˆ Moreover, for given policy g, the timing of h’s the sum of ex-ante contributions G. private contributions remains irrelevant as long as his stage-3 choice is interior. 28

The logarithmic structure is only imposed to generate closed-forms solutions, and for notational simplicity. The following results immediately extend to general quasi-linear preferences. 29 For arbitrary β, the identity of the highest-preference individual would be endogenous because it depends on the vector of ex-ante contributions. However, this problem does not arise for β = 1 where all individuals consume the same amount of the public good. Since any vector gˆ of private ex-ante contributions leaves the rank order of agents intact, only agent h with largest ‘natural’ preferences may possibly contribute at stage 3.

24

Consider now the policy decision at stage 2. For given first-period contribution vector gˆ, median individual m decides on g. Her goal is to maximize ˆ + g˜∗ (g, G)]. ˆ U m (g; gˆ) = ym − g − gˆm + αm ln [gN + G h

(4)

ˆ into account, this utility function is piecewise Taking individual h’s response g˜h∗ (g, G) defined. First, consider the range of policies for which h makes a positive ex-post contribution. Using (3), this range is defined as ˆ ˆ ≡ max{ αh − G , 0}. g ≤ g¯(G) N

(5)

For policies g from this interval, U m (·) monotonically decreases in g as any dollar of tax payments causes a total crowding out of h’s ex-post contributions.30 This eliminates any incentive for public funding, and the median voter’s local optimum in the considered range becomes g ∗ (g ≤ g¯) = 0. ˆ Here, d˜ Next, consider the opposite range of policies, g > g¯(G). gh∗ /dg = 0 and the first-order condition to (4) reads αm N ˆ gN + G

− 1 ≤ 0.

(7)

Using the lower policy bound g¯(·), an interior local policy optimum exists if and only if αm N > αh . This is because the median voter benefits from public funding only if her private return exceeds agent h’s propensity to finance the public good. Conversely, continuity of U m (·) in g in combination with the negative shape of U m (·) for any g < g¯ establishes that m’s globally preferred policy for αm N ≤ αh must be g = 0. Hence, we now concentrate on situations where αm N > αh which reflects an economic environment with many agents, or one where where the spread in tastes across society members is less pronounced. Condition (9) then describes an interior solution to m’s local choice problem for g > g¯(·), which is ˆ ˆ = max{αm − G , 0}. g + (G) N 30

(8)

The first-order derivative to (4) reads αm ˆ gN + g˜h + G

[N +

d˜ gh∗ ] − 1. dg

Since d˜ gh∗ /dg = −N , this derivative is negative everywhere.

25

(6)

To understand this, remember that m does not expect any ex-post contributions for policies in this range. Her preferred public funding level is then αm N (and her preferred policy is αm ), minus the sum of ex-ante contributions that have been made in stage 1. Overall, our previous arguments show that m’s preferred policy in the general system gs gs (subscript gs) is gm = 0 if αm N ≤ αh , and gm ∈ {0, g + } otherwise. Further investi-

gation is needed to find the global optimum in this latter case. To do so, we compare m’s stage-2 utility levels under both alternative policies, which are ˆ = ym − gˆm + αm ln max{αh , G}), ˆ U m (g = 0, G) and

ˆ ˆ = ym − gˆm + αm ln N αm − (αm − G ). U m (g = g + , G) N

Inspection reveals policy g + (if positive) to dominate if ˆ ˆ ≥ αm − G . αm ( ln N αm − ln max{αh , G}) N

(9)

ˆ ≤ αh (we will later verify this). Then, Remember that N αm > αh and assume G ˆ does not vary in G ˆ whereas U m (g + , G) ˆ is increasing in G. ˆ Note also that U m (0, G) ˆ converging to αh , g¯ converges to zero, while g + remains strictly positive. Since for G ˆ increases over the policy interval [¯ ˆ= U m (g, G) g , g + ], it follows that U m (g = g¯ = 0, G ˆ = αh ). We can conclude that g + must be m’s preferred policy αh ) < U m (g = g + , G ˆ exceeds some unique threshold G ˆ ∗ that satisfies (9) with equality, and with choice if G ˆ ∗ < αh .31 the property 0 ≤ G ˆ≥G ˆ ∗ , the median voter chooses g + > 0 in stage 2, and agent h makes no ex-post For G ˆ ∗ = 0 if the difference N αm −αh is sufficiently large. contribution in stage 3. Note that G In this latter case, m prefers a positive public policy g + even if no agent contributed ˆ G ˆ ∗ can constitute an equilibrium G. because each individual would benefit from lowering its ex ante contribution. The ˆ=G ˆ ∗ , the threshold remaining possibility are aggregate ex ante contributions of size G ˆ ≥ αh , no additional ex-post Formally, for any ex-ante contribution profile gˆ that satisfies G contributions will be made; any positive contribution made by an agent i < h’s then implies ∆i < 1 ˆ < αh , agent h’s ex-post contribution which cannot be optimal. Conversely, for contribution profiles G ˆ In anticipation of this, no agent i < h contributes ex ante, so that G ˆ = gˆh in equilibrium. is αh − G. 33

34

Accordingly, there is always underprovision relative to the social optimum, as characterized by the Samuelson rule. 35 ˆ < αh . Clearly, contribution profiles G ˆ > αh imply ∆i < 1 This is true for contribution profiles G for any individual i, which cannot be optimal.

27

level at which g + becomes the stage-2 equilibrium policy. To show the existence of ˆ ∗ .36 With this such an equilibrium, suppose only agent h contributes ex ante, gˆh = G contribution, he achieves a utility level ˆ ∗ ) = yh − G ˆ ∗ − g + (G ˆ ∗ ) + αh ln N αm . U h (g = g + , gˆh = G

(11)

ˆ ∗ because a larger threshold forces him to increase his Agent h’s utility decreases in G ˆ ∗ converge to αh and observe that ex-ante contribution to sustain g gs = g + > 0. Let G m

h

even in this least favorable situation, agent h strictly prefers (11) over (10) because his preferred stage-2 policy still exceeds g + .37 For public goods and quasi-linear preferences, the previous discussion allows us to fully characterizes equilibria of an institution with endogenous timing of private contributions. Defining ∆ = N αm − αh , we can state Proposition 9. Consider endogenous timing of private contributions, quasi-linear preferences, and a pure public good β = 1. 1) If N αm ≤ αh , the unique equilibrium policy g gs is zero and equilibrium outcomes coincide with those in the ex-post system. 2) If N αm > αh , the unique equilibrium policy is g gs = g + > 0, initial contributions ˆ = G ˆ ∗ ∈ [0, αh ), and no ex-post contributions are made. Moreover, satisfy G ˆ ∗ > 0, (a) public good consumption is strictly whenever ∆ is small enough that G larger than in regimes ea and ep; (b) the equilibrium policy is strictly smaller and satisfies g gs < g ep (≤ g ea ); (c) private contributions are strictly larger than in regime ea, and strictly smaller than in regime ep. 3) A majority of the population prefers regime gs over both ea and ep. This preferˆ ∗ > 0. ence is strict if N αm > αh and G ˆ ∗ is not to privately contribute at all. Note Clearly, any other individual’s best response to gˆh = G ˆ ∗ , but again G ˆ=G ˆ ∗ . In there may exist additional stage-1 contributions equilibria in which gˆh < G ˆ∗, any such equilibrium, only individuals of type i > m potentially contribute: by the definition of G 36

any individual i < m will at stage 2 prefer a policy g = 0 over policy g + . Hence, private contributions that prompt policy g + cannot be beneficial for those types. 37 Note that this is true even though the latter choice may imply a reduction in his consumption of private goods.

28

Proof: see the above discussion, and the Appendix. The Proposition conveys that for a broad range of parameter constellations, ex-post system and general system yield distinct outcomes in terms of equilibrium policies, contribution patterns, and consumption levels. Intuitively, the feasibility of ex-ante contributions allows high preference individuals to avoid being held up by the majority in a mutually beneficial way. Remember that in the ex-post system, the median voter chooses an artificially low equilibrium policy, and she does so in order to let high demanders finance the public good. Ex-ante contributions in the general system serve high-preference individuals as a commitment device not to make further contributions ex post. These contributions eliminate the median voter’s temptation for strategic underfunding, and induce her to adopt a larger policy than in the ex-post regime.38 Compared to the ex-post system, the overall outcome in the general system then features smaller private contributions, a larger policy, and larger total consumption levels of the public good. The welfare ranking of regimes is unambiguous. Note that the median voter would be free to choose policy g ep , and achieve (at least) the same utility than in the ex-post system. By not doing so and adopting a larger policy g gs instead, she and all higher-preference individuals must be better off by revealed preference, making the general system majority-preferred. The general system also majority-dominates the ex-ante regime: it restricts the commitment ability of high preference individuals to withhold private contributions. The general system exhibits less public funding and more public goods consumption, making this regime preferred over the ex ante regime for any individual other than the highest preference agent. This may have important policy implications. When citizens can choose the institutional structure by majority vote, the general system with endogenous timing emerges as the Condorcet winner: it is majority preferred over the pure public system and the ex-ante system (by all but the highest preference agents), and over the ex-post system (by all but the lowest preference agents). In an evolutionary sense, this institutional 38

As the Proposition shows, this is true when ∆ is positive but not too large. Remember that in the ex-post system, the equilibrium policy is smaller than m’s best response to the private equilibrium contributions. In contrast, it is a best response in the general system.

29

system can therefore be expected to prevail in the long run. Overall, our results suggest that equilibrium outcomes in the general system originate from a complex strategic interaction of the median voter, and of high-preference individual(s). In comparison to the ex post system, endogenous timing creates a win-win situations for these agents: highest preference agents reduce the hold-up power of the majority and benefit from higher taxes, and the median agent benefit from the larger consumption of public goods. In particular, she can avoid an inefficient downward adjustment of public policies for strategic reasons.

8

Private Provision as Status Quo Regime

The previous Sections compared public provision and dual public-private systems. In many situations of empirical importance, though, the status quo institution is one in which individuals privately provide those services. For example, many countries operate market based health care systems, some municipalities do not finance public transportation, and there are many policy areas of decentralized provision within federations. While private provision clearly Pareto dominates public provision in absence of externalities (β = 0), a welfare comparison for β > 0 is far less obvious. As is well know, only individuals with largest preferences deliver the public good in a private-contributions regime when β > 0. In equilibrium, individual contributions of any agent i are described by the system of first-order conditions ∆i (ci , Gi ) ≤ 1, which hold with equality for any contributor. Since contributors only pursue their own private interests, underprovision of public services is unavoidable. These characteristics suggest a basic tradeoff between public systems, and market based private provision. In a private system, low-preference individuals can free ride on the private supply of public goods, which they find more beneficial the larger the degree of externalities is. On the other hand, public provision may alleviate or mitigate the problem of underprovision that plagues a private system, which benefits everybody in society. These tradeoffs easily lead to cases where a private system majority dominates a one tier public system, or other situations where the opposite is true. The next Proposition 30

offers a welfare comparison of the private system, and dual public-private institutions. Proposition 10. Compare a private market system and a dual public-private regime. For any β > 0, 1) adopting the dual ex-post regime and a fortiori, the general system, is preferred by a majority of the population (strictly so if g ep > 0), including any i ≥ m. Conversely, a subset of lower-preference individuals may oppose. 2) adopting an ex-ante system may be opposed by a majority of individuals, including all low-preference individuals j ≤ m. Proof: see the Appendix. When private provision represents the status quo, adopting a dual system does not find unanimous support, regardless of the degree of externalities. Interestingly, this is because low-preference (or poor) individuals often prefer a private system over one which entails a public provision element, for two reasons. First, median voter policy forces small-preference individuals to pay more towards public goods consumption than privately desired. Second, and in presence of externalities, private provision allows these individuals to free ride on the contributions that are made by high-preference agents.39 Both arguments shift the preferences of low-preference individuals in the same direction: their loss on private resources in a dual system can be so severe that the enhanced public goods provision provides no sufficient compensation. Perhaps remarkably, we still find that (in contrast to pure public provision) the ex-post system unambiguously dominates the private market system for a majority of agents.40 39

The first point has been made in Cremer and Palfrey (2000), who analyze a setting without externalities. Note that the arguments against public provision remain valid in a scenario with income taxation: despite making smaller tax payments, poor individuals may view their tax contributions as excessive; with externalities, they may also prefer the free-riding opportunities in the private system. 40 This is not true in the Cremer-Palfrey setting (where β = 0): while agents in districts with highpreference medians are indifferent between the two regimes, a majority of agents in districts with low-preference medians suffers from the excessive amount of public provision. Hence, the number of losers in a dual system always exceeds the number of winners.

31

Moreover, the relative preferences of these agents are strict (whenever g ep > 0) for any arbitrary degree of externalities. The intuition is simple and again found in a revealed preference argument. If the median voter chooses a policy g = 0, she simply replicates the outcome of the private-provision setting. Conversely, if she chooses g > 0, and since all individuals i > m prefer even larger policies, all those individuals must be better off. This result does not extend to the ex-ante system which may be rejected by a majority. Because they move first, high preference individuals can reduce their contributions for strategic reasons. In the limit, they may not contribute at all, effectively turning the ex ante system into one with pure public provision. As a result, a large majority may favor the private system over the ex-ante system, in stark contrast to the case where public provision represents the status quo. We can conclude that for any β > 0, a switch from a private system towards a dual system that allows ex-post contributions is always majority preferred. Moreover, note that the majority in support of the ex post system over the private market system is the same majority that prefers the general system (with endogenous contributions) over the ex post system. Hence, the general system again emerges as the unique stable institution under majority vote. This finding is especially encouraging because as argued above, the transition from a private to a one tier public system may be rejected by a majority of the population. Our message here is that dual institutions can relax political feasibility constraints regarding the adoption of a public system; they can also respond to distributional concerns in society because high-preference agents continue to pay a larger share towards the funding of the public good.

9

Conclusions

This paper analyzes democratic choice in a framework where individuals can privately enhance their consumption of an impure public good that is provided through the political process. We distinguish different scenarios in which individuals can make their private provisions in a non-cooperative fashion, before and/or after the political out32

come is decided by majority vote. A variety of strong and interesting results can be established. First and foremost, we find that while the feasibility of private contributions in presence of externalities may destroy preference single peakedness, the outcome of the median voter theorem still applies: the median voter’s preferred policy beats any other policy alternative in pairwise comparison. We also establish that irrespective of the timing under consideration, private contributions are never rank-reversing in the sense that some individual other than the ‘natural’ median (the exogenous median agent in a one tier public system) can become the pivotal voter in equilibrium. We then use these insights to explore the characteristics of equilibria, and to provide a comparison of alternative regimes. The paper shows that as a general rule, the equilibrium provision of public services increases while public funding decreases when a dual institutional system is adopted. The only exception occurs in the ex-post system, where public service consumption possibly falls relative to the status quo of public provision. We also establish a clear cut ranking of equilibrium funding policies across regimes, from lowest (ex post system) over medium (general system and ex-ante system) to highest (pure public provision). An especially interesting issue is the welfare comparison of regimes. When the status quo is such that public goods are provided exclusively through the political process, adopting a system with additional private ex-ante contributions is beneficial for everybody in society. Unanimity is lost when moving to a system which allows only ex-post contributions; this latter regime is still preferred by majority but will in general be opposed by some high-preference individuals. Allowing a general system which allows private ex-post and ex-ante contributions finds the support of the majority of individuals. Specifically, all higher preference individuals prefer system over the ex-post regime; conversely, a different majority including all low-preference individuals prefers it over the ex-ante regime. These findings bear importance because in many empirically relevant situations, agents are free with respect to the timing of their individual contributions. Finally, we showed that if a market based private system is taken as the status quo, institutional comparisons change significantly. While a switch to the ex-post system is 33

still preferred by a majority, a strong supermajority may now oppose a transition to the ex-ante system. While this may suggest that support for a regime change is less likely when a private system represents the status quo, it remains true that under majority requirement, a dual system will be adopted whether or not the status quo institution is a public or private system. In either case, a system that allows for endogenous contributions always emerges as the stable equilibrium institution. Further research on these issues is clearly desirable to enhance our understanding of the evolution of institutions in setting with both private and public provision of public services.

34

Appendix – Proof of Propositions Proof of Lemma 1: Consider an arbitrary contribution profile gˆ. For any such profile, an individual i’s preferences over policies g are described by the utility function U i (yi − gˆi − g, g(1 + β(N − 1)) + β

∑

(ˆ gj + gˆi )).

j̸=i

Since preferences are quasiconcave by assumption, they are single-peaked in g and (if positive) the maximizer g i∗ for individual i is described by the first-order condition ∆i (ˆ gi , gˆ−i , giea ) ≡

UGi (·) = 1/(1 + β(N − 1)) = 1/z. Uci (·)

The individual i = M with ex-post median preferences then determines the policy outcome in pairwise majority vote.41 2 Proof of Lemma2: Consider the stage-2 median voter as induced by stage-1 contribution vector gˆ. Call this individual M (remember she is not necessarily the ‘natural’ median m). The proof is by construction. Suppose an equilibrium with M = m exists, and no j ≤ m contributes. Since the lowest-ranked (N + 1)/2 individuals do not contribute in stage 1, their ranking of policies g corresponds to the natural order of their ea preferences, and individual m by definition has the largest bliss point gm (ˆ g ) within this

group. Clearly, individual m then is the median individual M in stage 2 unless some ea agent i > m contributes gˆi in a way to switch ranks, giea (ˆ g ) < gm (ˆ g ).

We show that indeed no i > m will do so. ity the preferred second-stage policy g in gˆi . For gˆi = 0, this implies

giea

>

i∗

Note first that because of normal-

of any individual i is strictly decreasing

ea gm ,

irrespective of the contribution profile

gˆ−i = {ˆ g1 , ..., gˆi−1 , gˆi+1 , ..., gˆn } of all other individuals. Suppose one individual i > m ea ea (in order to generate m ̸= M there and giea ≤ gM contributes in a way that giea < gm

must be at least one such individual i). The stage-2 optimality condition for i then implies 1 ea ∆i (ˆ gi , gˆ−i , gM (ˆ g )) ≤ , z 41

(12)

For sufficiently large initial contributions gˆ, M may prefer the corner solution g = 0. In this case,

she shares her preferences with more than half of the population and remains pivotal in stage 2.

35

and holds with equality if M = i. We establish that for β > 0, this condition violates the optimality condition for i’s stage-1 behavior. For contributions gˆi inducing giea < ea gm , agent i’s stage-1 first-order condition reads ea ∆i (ˆ gi , gˆ−i , gM )=1+

ea 1 dgM i i [U (·) − U (·)z] . G Uci (·) c dˆ gi

(13)

The second term on the right-hand side reflects the effect of gˆi on the policy chosen ea by (and possibly, on the identity of) the median individual. Note that dgM /dˆ gi < 0

whenever β > 0: with normal preferences, a larger gˆi lowers the preferred stage-2 policy ea /dˆ gi > of any individual, and thus, of the induced median voter.42 In addition, dgM

−1 because otherwise, a raise in gˆi would yield less public goods consumption for M , violating preference normality. From the stage-2 optimality condition (2), the term in square brackets in (3) Uci (·) − UGi (·)z is zero if M = i, yielding an immediate contradiction since (2) and (3) become incompatible. Otherwise, when i < M according ea to induced stage-2 preferences, dgM /dˆ gi > −1 implies ∆i (·) > 1 − U i1(·) [Uci (·) − UGi (·)z].

Since ∆i (·)) =

i (·) UG , Uci (·)

c

this condition simplifies to 1 > z which is impossible. Taken

together, the assumed strategy cannot be optimal for agent i, implying individual i > m will never contribute in a way as to switch stage-2 preference ranks with m.43 To complete the argument, note that m = M if m does not contribute, and she will not contribute as she can always guarantee herself at least the same amount of both goods through the stage-2 choice of policies (i.e., m’s stage-1 optimization yields a corner ea solution). Finally, since gjea < gm for any j < m, none of these individuals will be a

contributor which proves the lemma. 2 Proof of Lemma 3: The proof of Lemma 2 implies that as long as gˆm = 0 in an equilibrium, the stage 2 equilibrium exhibits m = M . Specifically, for any individual i > m, ea g ) ≤ gM (ˆ g ) were shown to be incompatible with contributions with the property giea (ˆ

utility maximization. Suppose now there exists an additional equilibrium in which 42

This is true because raising gˆi can alter the identity of the median voter in only one direction: an individual k with a smaller stage-2 ∆k might become M . This indirect effect reinforces the direct effect of increasing gˆi . 43 For β = 0, the last Section has shown that each individual i > m will invest in such a way as to share stage-2 preferences with the median-income individual. Again, Lemma 2 continues to apply because m remains pivotal in stage 2. See our discussion below.

36

ea ea gˆm > 0. Two cases need to be distinguished. First, suppose that gm > gM in this

equilibrium. For this to be true, at least one individual i > m has contributed in a way ea ea that giea < gm (and giea ≤ gM ). But invoking the arguments of Lemma 2, contributions

with this property cannot be the best response to profile gˆ−i for any individual i > m. ea ea Second, suppose that gm < gM in the assumed equilibrium. We first establish that

regardless of the contribution profile gˆ−m , individual m can always generate stage-2 ea ea references gm ≥ gM : for gˆm = 0, m = M unless one individual i > m invests in a way ea that giea ≤ gm . By choosing gˆm = 0, M ’s stage-2 policy preferences in the assumed

equilibrium are therefore smaller than those of individual m. But contributing gˆm > 0 ea ea in a way that gm ≤ gM cannot be m’s best response to any contribution profile gˆ−m ea ea to apply, of other agents. In analogy to the arguments in Lemma 2, for gm ≤ gM ea (ˆ g )) ≤ 1/z(< 1). But m’s stage-2 references would have to satisfy ∆m (ˆ gm , gˆ−m , gM

contribution gˆm then violates m’s the stage-1 optimality condition (13), yielding a contradiction. In summary, there cannot exist equilibria in which individual m contributes gˆm > 0; as a consequence, M = m and the equilibrium characteristics given in Lemma 2 are unique. 2 Proof of Proposition 2 pp Consider the ex-post regime and a policy g ≤ gm . For any such stage-1 policy outcome,

each individual k privately contributes g˜k = max{gkpp − g, 0} in stage 2. Hence, agent k pp overconsumes relative to the privately efficient level G∗k if g > gm , while k realizes his pp preferred consumption G∗k otherwise. Consequently, any g ∈ [0, gm ] can be supported

as an equilibrium policy: each individual i ≥ m (a majority of agents) is indifferent among policies from this choice set, and none of them strictly prefers a policy larger pp pp than gm . Conversely, no g > gm is an equilibrium candidate because any such policy is pp and a majority that includes any individual j ≤ m. majority dominated by any g ≤ gm pp can Next, consider the regime with ex-ante contributions. We first show that no g > gm

become equilibrium policy in stage 2. First, in absence of private stage-1 contributions, any such policy is dominated by majority. Second, each individual’s preferred policy is single-peaked, decreasing in his own contribution, and (since β = 0) unaffected by the contributions of other agents. This proves the result. 37

pp On the other hand, any policy g ≤ gm can be supported as a majority voting equi-

librium. To verify this, consider stage-1 private contribution vectors in which each individual k either contributes gˆk = G∗k − g ∗ (if this expression is positive) for arbipp trary g ∗ ≤ gm , or contributes nothing otherwise. For any such contribution vector,

g ∗ is the majority preferred equilibrium policy in stage 2: every stage-1 contributor pp prefers g ∗ over any other policy, and contributors form the majority because g ∗ < gm

by construction. Next, verify that given the expectations on gˆ−k , each contributing agent’s best stage-1 response is indeed to provide gˆk = G∗k − g ∗ . Likewise, the best response of date-1 non-contributors in the alleged vector is not to contribute. This establishes existence of a continuum of equilibrium policies and shows that equilibrium characteristics in the ex-ante and the ex-post regime coincide. 2

38

Proof of Proposition 3 To establish the monotonicity properties stated in the Proposition, consider the private contributions equilibrium in stage 1. The utility maximizing ex-ante contribution of an individual i is ea ea ea ea giea +β (ˆ g−i , gˆi )[1+β(N −1))+ˆ (ˆ g−i , gˆi ), gM gˆiea = arg maxgˆi U i (yi −ˆ gi −gM

∑

gˆjea ). (14)

j̸=i

The set of contributors C must always include the highest-preference individuals. To see this, notice that the first-order derivative for individual i, dg(ˆ g) dg(ˆ g) dU i (·) = −Uci (·)[1 + ] + UGi [1 + (1 + β(N − 1))], dˆ gi dˆ gi dˆ gi is strictly increasing in index i: first, dg/dˆ gi is uniform across individuals, and second, the marginal rate of substitution ∆i (·) = UGi /Uci is increasing in i. Invoking normality, it is then impossible to have dU l (ˆ g−l , gˆl = 0)/dˆ gl ≤ 0 and at the same time U k (ˆ g−k , gˆk = 0)/dˆ gk > 0 for any two individuals k, l with rank order l > k.44 For any contributor i, the first-order condition to (14) holds with equality and can be rewritten as UGi (ci , Gi ) = Uci (ci , Gi ) 1+

1+ dg(ˆ g) [1 dgi

dg(ˆ g) dgi

+ β(N − 1)]

.

(15)

This system of first-order conditions defines the equilibrium contributions for any i ∈ C. To analyze these conditions, note first that for any β > 0, the marginal rate of substitution is not unity because any increase in private contributions triggers a smaller equilibrium public policy, dg(ˆ g )/dˆ gi < 0: with normal preferences, larger private contributions of some i > m cause the median voter m to raise her private consumption, with the consequence of a decrease in the public policy g. In anticipation of this negative response, each contributor contributes less, and contributions remain below the level which equates (taken the subsequent public provision into account) marginal 44

Hence, the set of contributors is non-empty iff the first-order derivative for individual n, −Ucn (·)[1 +

dg(ˆ g) dg(ˆ g) n ] + UG [1 + (1 + β(N − 1))], dˆ gn dˆ gn

is positive if evaluated at a contribution vector gˆ = 0.

39

rate of substitution and marginal costs of provision. Finally, the symmetry property ˆ i = dg/dˆ dg/dg gj suggests that for any two contributors i, j, equilibrium contributions must be strictly increasing in the index of natural preferences.45 Hence, public-goods consumption is increasing in the natural preference index, and strictly so for the set of contributors and unless β = 1. 2 Lemmas 4 and 5 ep Let gm indicate the policy preferred by m, where ep stands for ex post contributions.

Results evolve through a series of lemmas. ep ep Lemma 4: Each individual j < m prefers gm over any other policy g > gm .

Proof: Define • giT = g + g˜i (g) as the total contribution of individual i towards the supply of public goods; • G−i = Gi −giT = gβ(N −1)+β

∑ j̸=i

g˜j as the amount of i’s public goods consumption

funded through contributions of all other individuals. To prove the lemma, suppose to the contrary that individual j strictly prefers some ep larger policy, say gj∗ , over gm . We first show that if this is true, the total contributions ep of all other individuals under policy gj∗ exceed those under policy gm , that is, G−j (gj∗ ) > ep ep G−j (gm ). Suppose not. Then, j could under policy gm always replicate his public-goods ep ep consumption under policy gj∗ , by choosing g˜j (gm ) > 0 in a way that Gj (gm ) = Gj (gj∗ ).46

At the same time, his total public goods contribution gjT would be lower and his private ep consumption cj would be higher under policy gm , yielding a contradiction. Hence, if j’s ep preferences are strict, G−j (gj∗ ) > G−j (gm ), and j’s total consumption of public goods ep ) due to normality of under both policies regimes is characterized by Gj (gj∗ ) ≥ Gj (gm

preferences. 45

Suppose not, and the lower-preference individual j contributes more in equilibrium. Then, the first-order conditions (15) cannot simultaneously hold for individuals i and j. 46 Note this is always feasible; the required stage-2 contribution (if positive) would be g˜j = g˜j (gj∗ ) + ∗ ep ep ep (gj − gm ) − [G−j (gm ) − G−j (gj∗ )], and imply gjT (gm ) < gjT (gj∗ ). Note also that g˜j will in general ep ep not be j’s best response to policy gm and contributions G−j (gm ) of other individuals. However, g˜j defines j’s lower utility bound by revealed preference.

40

ep Consider now individual m who by definition prefers gm over gj∗ . Two cases need to be ep ep distinguished. First, G−m (gm ) > G−m (gj∗ ) may hold. If in addition Gm (gm ) ≥ Gm (gj∗ ) ep (this needs not be true, see below), j can strictly prefer gj∗ and m strictly prefer gm ep ep only if the conditions G−j (gj∗ ) > G−j (gm ) and G−m (gj∗ ) < G−m (gm ) are simultaneously T satisfied. Note that Gm (·) = Gj (·)+(1−β)[˜ gm (·)−˜ gj (·)] = Gj (·)+(1−β)[(gm (·)−gjT (·)].

Using this fact and the definition of G−i (·), a necessary condition for the validity of these conditions is T T ep ep gm (gj∗ ) − gm (gm ) > gjT (gj∗ ) − gjT (gm ). ep ep A second set of conditions is Gm (gm ) ≥ Gm (gj∗ ) and Gj (gj∗ ) ≥ Gj (gm ). Again using T the above definition of Gm (·), the validity of these two inequalities requires gm (gj∗ ) − T ep ep gm (gm ) ≤ gjT (gj∗ ) − gjT (gm ), yielding an immediate contradiction.47 ep ep Next, suppose again that G−m (gm ) > G−m (gj∗ ), but now let Gm (gm ) < Gm (gj∗ ). Notice

that this case can occur only if policy gj∗ forces m to make an excessive total contribution from his point of view, implying a stage-2 corner solution g˜m (gj∗ ) = 0. But then, agent j < m must also be a non-contributor (by definition, it is impossible to have T ∆j (·) > ∆m (·) at g˜j = g˜m = 0) and as a consequence, gm (gj∗ ) = gjT (gj∗ ), which implies T G−m (gj∗ ) = G−j (gj∗ ). Note now that under the rank-order assumption, gjT (g) ≤ gm (g) ep ep ep for any g. Hence, G−j (gm ) ≥ G−m (gm ). But since we consider the case G−m (gm )> ep G−m (gj∗ ), and since G−m (gj∗ ) = G−j (gj∗ ), this immediately implies G−j (gj∗ ) ≤ G−j (gm )

and hence, a contradiction. ep ep Next, suppose to the contrary that G−m (gj∗ ) ≥ G−m (gm ). Again, m’s preference for gm

can be consistent only if policy gj∗ is excessive from m’s point of view. Accordingly, her private contribution is g˜m (gj∗ ) = 0, and so is j’s private contribution under Assumption T ep 1. Hence, Gm (gj∗ ) = Gj (gj∗ ) and gm (gj∗ ) = gjT (gj∗ ). Consider now policy gm and suppose ep T ep ), so that both agents consume the same (gm ) = gm j’s total contribution satisfies gjT (gm

This is true for any β < 1. For β = 1 so that Gm (·) = Gj (·) ≡ G(·), satisfying conditions ep ep Gm (gm ) ≥ Gm (gj∗ ) and Gj (gj∗ ) ≥ Gj (gm ) requires public goods consumption under both policies to ∗ ep ep be identical, G(gj ) = G(gm ). To prefer gj∗ over gm , j’s total contribution must then be characterized 47

ep by gjT (gj∗ ) < gjT (gm ) (note that as an implication, j makes a positive stage-2 contribution under ep ep policy gm .). But this is inconsistent: because of preference normality, j would reduce gjT (gm ) to a ep level where Gj (gm ) < Gj (gj∗ ), yielding a contradiction.

41

ep ep ep amount Gm (gm ) = Gj (gm ). Since G−m (gj∗ ) ≥ G−m (gm ) by assumption, normality then ep (generically) implies Gm (gm ) < Gm (gj∗ ). Also, since ∆j (·) < ∆m (·) for identical total ep contributions of j and m (Assumption 1), j must prefer gm over gj∗ , given this is m’s ep T ep preference ranking.48 Moreover, by revealed preference, choosing gjT (gm ) ̸= gm (gm ) ep can only raise j ′ s utility under policy gm , reinforcing this preference. This yields a

contradiction and completes the proof. 2 ep ep Lemma 5: Any individual h > m prefers gm over any policy g < gm .

Proof: The proof mirrors the proof of Lemma 4, and is therefore omitted. Proof of Proposition 6 ep ep For any private contributor i, the second-stage first-order condition ∆i (gm , g˜ ) = 1

holds in equilibrium, which equalizes his marginal utilities from consuming public and ep private goods for given policy gm and ex-post contribution vector g˜ep . In contrast, ep ep ∆j (gm , g˜−j , g˜jep = 0) < 1 for any non-contributor j. By standard arguments, the set

of contributors in stage 2 is decreasing in g and at certain threshold levels g, the lowest-preference contributing individual becomes a non-contributor. The proof proceeds in several steps. We consider the range of first-period policies gm for which m contributes in the second stage, g˜m (gm ) > 0. We show that within this ∑ range g ∈ [g m , g¯m ], an increase in g raises the total funding GT ≡ N gm + i g˜i of all i ∈ N individuals. As an implication of this result, the total funding amount G−m m (g) of individuals other than m are shown to increase in g as well. Since U m (gm ) is increasing if and only if dG−m m /dgm > 0 (see Lemmas 4 and 5 above), the median voter m cannot ep be a stage-2 contributor in equilibrium. In other words, m’s globally optimal policy gm

is (weakly) larger than g¯m , and therefore larger than any upper-boundary threshold g¯j for individuals indexed j < m. This proves the result. 48

This is clearly true if both j and m have identical incomes. Suppose not and notice that total ep T contributions under policy gm must be characterized by gm (·) ≥ gjT (·) under Assumption 1. Since their ep individual contributions under the alternative policy gj∗ are identical, switching from gm to gj∗ causes j’s private consumption to decline more. Invoking the single-crossing assumption then establishes the result.

42

ˆ < N , and if Step 1: dGT (gm )/dgm > 0 whenever the number of contributors is N β < 1. ′ ′′ The proof is by contradiction. Suppose a raise in gm from gm to gm lowers total

funding GT (gm ) of the public good. Then, there must be at least one individual ′ k (necessarily, a contributor under policy gm ) who reduces his total contribution gkT ′′ under policy gm . We show that such a behavior violates normality. To see this, suppose ′′ first that k’s public goods consumption Gk is smaller under policy gm . But since his ′ ′′ , this than under policy gm private goods consumption must be larger under policy gm

yields a contradiction. Second, suppose other individuals compensate for the reduction in k’s total contribution in a way which allows k to (weakly) raise his public goods ′′ consumption Gk under policy gm . To see that this is impossible, notice that for any

unit reduction in gkT , the total provision of all other individuals, G−k k , must increase by at least 1/β > 1 to avoid a reduction in Gk . But then, total public goods provision ′′ ′ would be characterized by GT (gm ) > GT (gm ), a contradiction.

Step 2: dGT (gm )/dgm > 0 implies dG−m m (gm )/dgm > 0. ′′ −m ′ ′′ ′ Suppose not, and suppose instead G−m m (gm ) − Gm (gm ) = −ϵ for gm > gm , where

ϵ is a positive number. We show that m’s stage-2 contributions are characterized by ′′ ′ g˜m (gm )− g˜m (gm ) < βϵ. To see why, note that otherwise, m would raise her total public′′ ′ goods consumption in state gm relative to state gm . Her private consumption would

fall at the same time, which is inconsistent with the assumed normality of preferences. Since dGT (gm )/dgm > 0, the difference in total public goods supply between both ′′ ′ states would satisfy G(gm ) − G(gm ) ≤ −(1 − β)ϵ, which is negative for any β < 1 and

a contradiction to the result in Step 1. m ¯m . Step 3: dG−m m (gm )/dgm > 0 implies d U (gm )/dgm > 0 for any gm < g

For any gm < g¯m , m contributes a positive amount in stage 2. In the considered range gm ∈ [g m , g¯m ], m can always raise gm in such a way that her total contribution T = gm + g˜m constant. Since G−m gm m (gm ) increases in gm , this strategy would leave m’s

private consumption constant while raising her public goods consumption. Accordingly, m always raises her utility by raising gm to a level gm ≥ g¯m where she does not voluntary 43

contribute in continuation equilibrium. Finally, since arg maxgm U m (gm ) ≥ g¯m , neither ep m nor any individual j < m contributes g˜j > 0 in stage 2: this is because ∆m (gm , g˜m = ˜ −m (g ep )) ≤ 1 implies ∆m (g ep , g˜j = 0, G ˜ −j (g ep )) < 1 by normality of preferences. 2 0, G m

m

m

j

m

Proof of Proposition 7 We show that for a given policy level gm , private contributions in ep exceed those in ea. Notice that for each contributor i, the second stage optimality condition in ep reads ∆i (ci , Gi ) = 1 , as compared to ∆i (ci , Gi ) > 1 in ea. If in addition the equilibrium ep policies are characterized by gm ≤ g ea (this will be established below), the respective

Nash equilibria unambiguously satisfy g˜i > gˆi for each contributor i. In what follows, we show that the equilibrium policy level in ep is indeed smaller. To see this, note that in ea, m’s optimal policy choice satisfies the first-order condition ∆m (cm , Gm ) = 1/z. In the ex-post scenario, the corresponding condition reads ∆m (·) =

1 UG (·) = ∑ Uc (·) z+β j

d˜ gj dgm

.

By our previous arguments, this implies ∆m (cm , Gm ) > 1/z whenever the set of conep tributors is nonempty. By way of contradiction, suppose now that gm ≥ g ea , i.e., the

equilibrium policy in the ex-post setting is (weakly) larger. Since the median voter m does not privately contribute in either regime, her private consumption in ep is then (weakly) smaller. But in order to satisfy the respective first-order conditions for equilibrium policy choice in each regime, m’s public goods consumption Gm in the ex-post regime would have to be smaller as well. As a consequence, her utility in this regime would be smaller, which is impossible: by our above results, implementing a policy ea gm would raise m’s public-goods consumption relative to the ex-ante regime (for any

gm , private contributions are larger in the ex-post regime) while leaving her private ea ep . < gm consumption the same, a contradiction. Hence, gm

By revealed preference, our previous arguments imply that any non-contributor is betea , inducing the same private consumpter off in ep: m voter could have chosen policy gm

tion for non-contributors in both regimes. At the same time, public goods consumption in ep would have been larger because of larger private contributions (see above).49 49

ep ea ep Since gm < gm and since for commitment reasons, gm is smaller than the best response to the

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Conversely, contributors i are worse off in ep: with the lower equilibrium policy, achieving the same consumption Gi requires each of these these individuals to raise their total contributions in the ex-post regime. A contributor in ep may even prefer pp: the equilibrium policy in ep is not only smaller than in pp, but even smaller than the best response to equilibrium contributions in this regime (for commitment reasons). Hence, non-contributors contribute less towards a contributor’s public-goods consumption than in pp, which may outweigh i’s utility gain from being able to make private top up contributions. 2 Proof of Proposition 9 Most of these results have been established before. In all three regimes gs, ep, and ea, equilibrium outcomes depend on the sign and the size of ∆ = N αm − αh . Consider ∆ > 0. In this case, no individual will privately contribute in regime ea, and the ˆ ∗ > 0, ep is characterized equilibrium policy is g ea = αm . For ∆ not too large so that G by an equilibrium policy g ep = 0 and agent h privately contributes g˜h = αh . Conversely, in gs, the equilibrium policy is g gs = g + ∈ (0, αh ) and private contributions are gˆh = ˆ ∗ . This yields policies 0 = g ep < g gs = g + < g ea = αm and consumption levels G Ggs = Gea = αm > Gep = αh of the public good. Conversely, if ∆ is so large that ˆ ∗ = 0, equilibrium policies will instead be g ep = g ea = αm = g + (G ˆ = 0) = g gs , and G m m no individual privately contributes in either regime. Hence, Ggs = Gep = Gea in this case. Note that public goods consumption in gs is always at least weakly larger, and (unless ∆ is small) the equilibrium policy is weakly smaller than in the other regimes. For ∆ ≤ 0, g gs = g ep = 0 and Ggs = Gep = αh , with full financing of the public good provided by agent h. Regime ea replicates these results if ∆ = N αm − αh is very negative. In contrast, for negative but small ∆, g ea = αm , Gea = N αm and private contributions are zero. Hence, Ggs = Gep = αh > Gea = N αm and g gs = g ep = 0 < g ea = αm in this case. Note that in all these situations, public goods consumption in gs is at least weakly larger, and public policies are weakly smaller compared to the vector of subsequent private contributions, a comparison of public good consumption in both regimes is generally ambiguous. However, note that if β = 1 and preferences are quasilinear, public goods consumption in the ex-post regime will be larger.

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other regimes. We now compare gs with ea and ep with respect to equilibrium utilities. For ∆ < 0, outcomes in ep and gs coincide and so do utilities. For very negative ∆, results in ea are again the same. For ∆ negative but small so that h makes no private contribution in ea, every individual other than h strictly prefers gs (and ep) over ea because it allows those individuals to consume more of the public and the private good at the same time. Overall, gs and ep yield identical results while a (large) majority of agents prefers gs over ea. Finally, consider ∆ > 0. Since Ggs = Gea and g gs = g + < g ea , all agents except h prefer gs over ea. To compare gs and ep, consider first a situation ˆ ∗ > 0 and, accordingly, g ep = 0. In regime gs, in which ∆ is sufficiently small that G the median voter could adopt the same zero policy, and trigger an identical economic outcome. By choosing g + , she raises her utility by revealed preferences. But surely, this increases the utility (at least) for any agent with rank i > m as well, including ˆ ∗ = 0, the utility of each ex-ante contributor.50 Finally, for ∆ sufficiently large that G m implements g gs = g ep and private contributions are zero. In this case, individual utilities in both regimes are identical. 2 Proof of Proposition 10: To establish part 1), notice that for a policy g = 0, the outcome in ep is identical to the outcome in the private system. If the pivotal median voter m implements g ep > 0, she must be better off by revealed preference. In addition, every individual i > m prefers g ep over any smaller policy (Lemma 5), which validates the result. Finally, notice that for β → 0 where g ep → g pp , each individual j < m (and thus, a strong minority of citizens) opposes the regime change from private to ex-post system. To prove part 2), reconsider the quasi-linear example analyzed in Section 6. If G is a pure public good, the highest-preference h individual will contribute αh to its provision in a decentralized setting, while all other individuals free ride. In ea, the 50

To see this, suppose equilibrium contributor i had chosen not to contribute ex ante. He would

continue to prefer a policy larger than g + at stage 2, notwithstanding the fact that policy g = 0 gives him the same utility as in ep. By revealed preferences, contributing gˆi > 0 ex ante must further boost i’s wellbeing.

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highest-preference agent h will not privately contribute, but rely on public provision if ∆ = αm N − αh is positive (or not too negative). With equilibrium policy g ea = αm , ea and the private system then yield a similar public goods consumption if ∆ is sufficiently close to zero. Since funding is provided entirely by h in the private regime, and uniformly by all society members in the ex-ante regime, everybody (other than h) prefers private provision. 2

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Hafer, C. and D. Landa (2007), ”Public Goods in Federal Systems,” Quarterly Journal of Political Science, 2 (3), 253-275. Rubinchik-Pessach, A. (2005), ”Can Decentralization Be Beneficial?,” Journal of Public Economics, 89(7), 1231-1250. Stiglitz, J.E. (1974), ”The Demand For Education in Public and Private School Systems,” Journal of Public Economics, 3, 349-385.

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