# Voluntary Contribution Model of Public Goods

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_2952

## Abstract

This article surveys the literature on the model of voluntary contributions to public goods that has developed since the early 1980s. This literature draws explicitly on noncooperative game theory. We present a recent novel statement of the problem, based on ‘replacement functions’, which is both more powerful and more transparent than the traditional formulation that uses players’ best response functions. We survey existence, uniqueness and comparative static properties of the basic model, and also sketch some of the extensions of that model – impure public goods, weakest link and best shot – that have been proposed and applied to such problems as global public goods and the global commons. We also draw attention to recent attempts to dynamize the model.

### Keywords

Pure public goods Impure public goods Noncooperative games Neutrality Free riding Underprovision Weakest and weaker links Best and better shots Replacement functions Lotteries and public good provisionTwo classic papers by Samuelson (1954, 1955) played a major role in provoking interest in the problem of public good provision. However, they did not provide an explicit model of decentralized provision. His formal analysis focused on necessary conditions for their optimal provision. Elements of a positive model of decentralized provision – hereafter the standard model – were gradually developed during the following decades, and more complete formal analyses were provided by Cornes and Sandler (1985) and by Bergstrom et al. (1986).

## Introduction

*n*, of members. They have preferences over a private good and a public good. Player

*i*’s consumption of the private good is

*y*

_{i}, and the total provision of the public good is

*G*. Preferences, resource constraints, and the technology that converts individual contributions into the total available public good are summarized, respectively, by the following assumptions:

**Preferences**Player*i*’s preferences are represented by a utility function,*u*_{i}(*y*_{i},*G*), which is strictly increasing in both arguments and quasiconcave. Both goods are normal.**Resource constraint**\( {y}_i+{c}_i{g}_i\le {m}_i \), where player*i*’s unit cost as a contributor*c*_{i}, and money income*m*_{i}, are exogenously given.*g*_{i}is player*i*’s contribution to the public good.Technology of public good provision \( G={\varSigma}_{j=1}^n{g}_j \).

The model considers the Nash equilibrium of the static noncooperative game containing these elements when each player is choosing her best response, *ĝ*_{i}, to the choices made by all others, \( {G}_{-i}={\varSigma}_{j=1,j\ne i}^n{g}_j \).

This formulation slightly generalizes the standard model in that we allow unit costs to differ across contributors. This extension, initially explored by Ihori (1996), has interesting implications.

## A Graphical Treatment

Analyses typically derive a best response function for each player. This determines the player’s most preferred choice of contribution as a function of the choices made by all other players: \( {\widehat{g}}_i={b}_i\left({G}_{-i}\right) \), where *ĝ*_{i} is player *ī* ‘− *i*’s utility-maximizing response. A Nash noncooperative equilibrium is an allocation at which every player chooses her best response. Formally, it is a solution to the *n* equations provided by the individual best response functions in the *n* unknowns, *g*_{1}, *g*_{2},…, *g*_{n}. Questions about existence, uniqueness and other properties of equilibrium become questions about the existence, uniqueness and other properties of solutions to this set of equations. Such an approach, though naturally suggested by noncooperative game theory, is not the most helpful or transparent method of tackling these issues. We shall briefly sketch an alternative approach, suggested by Cornes and Hartley (2007a), which provides both a rigorous and powerful tool of analysis, and a simple and transparent geometric representation.

### Individual Behaviour

*i*’s preferences, constraints and choices. Suppose her income is

*m*

_{i}. If the sum of all other players’ contributions is \( {G}_{-i}^{\prime } \), player

*i*can devote all her money income to private goods consumption and enjoy the public good provided by others. This allocation is the point

*E*′. Each unit of private good consumption given up by

*i*augments total public good provision by the amount 1/

*c*

_{i}. Thus her budget constraint is the line

*E*′

*F*′. Her most preferred choice is the point of tangency,

*P*′. By varying

*G*

_{−i}parametrically, we can trace out the income expansion path

*II*, which summarizes the player’s behaviour. The locus

*II*is everywhere continuous, and slopes upwards at all points at which player

*i*is at an interior point, choosing strictly positive values of both

*y*

_{i}and

*g*

_{i}. If there is a finite value of

*G*

_{−i}at which \( {\widehat{y}}_i={m}_i \), at that point the locus become horizontal.

Note that, for any given value of *G*, the vertical distance between the income expansion path and the locus of the income constraint *mm* measures the implied value of expenditure by *i* on the public good, \( {c}_i{\widehat{g}}_i={m}_i-{\widehat{y}}_i \). If *c*_{i} = 1, this same distance measures the quantity of the public good. If *c*_{i} ≠ 1, then a simple scaling up or down of the vertical axis in panel (b) allows us to depict the quantity *g*_{i}. In any event, under our assumptions, to any given level of total public good provision *G* above a certain value there corresponds a unique level of contribution by player *i*, *ĝ*_{i}, that is consistent with that observed level, in the sense that *ĝ*_{i} is a best response to the quantity \( {G}_{-i}=G-{\widehat{g}}_i \). We write the implied functional relationship as \( {\widehat{g}}_i={r}_i(G) \) and call this player *i*’s replacement function. The figure suggests that every player has a replacement function that is continuous, everywhere non increasing, and strictly decreasing in *G* wherever the replacement value itself is positive.

*G*, player

*i*is a strictly positive contributor. Consider the consequence of an increase of Δ

*m*

_{i}in

*i*’s money income. At that given level of

*G*, Figure 1 panel (a) shows that her chosen allocation is unchanged. She consumes an unchanged quantity of the private good. Thus, her contribution to the public good changes by the amount

### Nash Equilibrium

*n-*player voluntary contribution game, it is an allocation at which

*R*(

*G*) is shown as the thick line in Fig. 2. It is simply the vertical sum of the individual graphs. A Nash equilibrium may be depicted graphically as a point where the graph of

*R*(

*G*) intersects the 45° ray through the origin in Fig. 2. This relationship describes a Nash equilibrium in the form of a single equation in a single unknown,

*G*, regardless of how many players there are, and how they differ with respect to preferences, unit costs and money incomes. Armed with the properties already sketched above of the individual replacement functions, scrutiny of this equation is sufficient to provide a complete positive analysis of the model. First, however, note the following simple points. First, the sum of two continuous functions is continuous. Second, the sum of two monotonic functions is itself monotonic.

## Properties of the Equilibrium

We now have all the ingredients for a rigorous analysis of the equilibrium properties of the model, which we now investigate.

### Existence of Equilibrium

Consider the player whose replacement graph reaches the 45° ray furthest from the origin in (*G*, *R*(*G*)) space. It is possible that, at that level of *G*, all other players are choosing to contribute zero. In this case, we have found an equilibrium, at which the chosen player is the sole contributor. Alternatively, there may be other players whose replacement values are positive. In this case, we have found a value of *G* at which \( R(G)\equiv {\sum}_{j=1}^n{r}_j(G)>G. \) Then monotonicity implies that, as *G* rises, the left-hand side of this inequality falls, while the right-hand side rises. Continuity implies that there must be a finite value of *G* at which the equilibrium condition holds. Either way, an equilibrium certainly exists. In Fig. 2, this is the point *G*^{N}, at which the sum of all players’ contribution levels that are individually consistent with *G*^{N} is also collectively consistent.

### Uniqueness of Equilibrium

Monotonicity implies that *R*(*G*) is everywhere nonincreasing. Clearly, *G* is a strictly increasing function of itself. Thus, there can only be one value of *G* at which *R*(*G*) = *G*.

### Presumptive Inefficiency of Equilibrium

In the basic model, in which a common unit cost is assumed across players, there is a general presumption that too little of the public good is provided at equilibrium, in the sense that Pareto-superior allocations can be obtained by increasing the level of public good provision. This may be confirmed by a simple envelope argument. Suppose that, at an equilibrium, players *j* and *k* are both positive contributors. Starting from the equilibrium, a small increase in player *j*’s contribution imposes a second-order cost on player *j*, but generates a first-order benefit for player *k*. Similarly, a small increase in player *k*’s contribution imposes a second-order cost for player *k* and a first-order utility gain for player *j*. Thus it is possible for both to be made better off if both raise their contributions slightly above their equilibrium levels. Furthermore, such a move will not hurt other players, and will generally benefit them. Thus it is Pareto-improving.

In the current model, in which unit costs are allowed to differ across players, this remains true. There is also, however, a second source of inefficiency. This arises from the fact that the ‘wrong’ people contribute at equilibrium. Consider an equilibrium at which both a high-cost and a low-cost contributor are making positive contributions. An initial transfer of income from the high to the low-cost player shifts the replacement function of the high-cost player down, and that of the low-cost player up, in the neighbourhood of the equilibrium value of *G*. But the latter shift is quantitatively greater, so that the aggregate replacement function shifts upwards. The equilibrium provision therefore must rise, and contemplation of Fig. 1a makes it clear that all players are better off in the new equilibrium.

Note that we talk of presumptive, not necessary, underprovision. This is for two reasons. First, as Cornes and Sandler (1996, p. 160) point out, if every player prefers to consume the private and public good in fixed proportions, so that their indifference curves are L-shaped, then the equilibrium is Pareto efficient. This possibility disappears if we allow some substitutability between the private and public goods. A second possibility, which certainly needs to be taken more seriously in policy discussions of public good provision than is sometimes done, is that the equilibrium involves zero total provision and that, even when provision is zero, the sum of all player’s marginal valuations is less than the minimum cost of producing an increment of the public good. In this case, the public good neither is, nor should be, provided.

### Neutrality

Suppose that two players – say *i* and *j* – have the same value for the cost parameter. Consider an equilibrium, *G*^{*}, at which both are strictly positive contributors. Now transfer an amount of income, Δ*m*, from one to the other. In the neighborhood of *G*^{*}, the recipient’s replacement graph shifts upwards by the amount Δ*m*. The donor’s graph shifts downwards by the amount Δ*m*. Thus, if both remain positive contributors at *G*^{*}, that value remains the sole equilibrium public good provision level. Nothing real has changed – equilibrium levels of private good consumption and of total public good provision, and therefore equilibrium utility levels, are unaffected by the income transfer. This is the famous neutrality property of the standard model, which assumes a common value of the cost parameter for all players. Often attributed to Warr (1983), it was foreshadowed in earlier work by Shibata (1971).

### Non-neutrality

The reasoning that led to the neutrality result allows us to understand easily the circumstances under which neutrality fails to hold. First, suppose that the source of the income transfer is initially choosing to contribute zero. Then, at the initial level of *G*, the reduction in her income cannot shift the relevant portion of her replacement function downwards – she is already contributing zero. The recipient’s replacement graph shifts upwards. Therefore the aggregate replacement graph shifts upwards, and the equilibrium provision of the public good must now be higher. Transfers between existing contributors and noncontributors will have real consequences, leading to changes in both the equilibrium total public good provision and also in individual equilibrium utility levels. It is even possible, as Cornes and Sandler (2000) point out, that transfers from each of several noncontributors to contributors leads to a Pareto-superior allocation.

Second, our discussion of the presumptive inefficiency of equilibrium has already shown that an income transfer from a high-unit-cost contributor to a low-unit-cost contributor will lead to a higher level of equilibrium provision and to a Pareto improvement.

### Implications of a Cost Change

Suppose that player *i* is initially a positive contributor, and that she enjoys an exogenous reduction in her unit cost. Consideration of Fig. 1 shows that the level of her preferred contribution that is associated with the initial equilibrium value of *G* must now be higher. In the absence of any other shocks, the equilibrium level of total provision must rise. Thus, every player except for *i* enjoys a higher equilibrium utility. However, player *i* herself may be either better or worse off – on the one hand, total provision is higher, but on the other hand she is now contributing a higher share of that total, since her fellow contributors have reduced their contributions.

### Limiting Behaviour as n Gets Large

The implications of adding players to the community are very easy to trace using our suggested approach. Suppose a fourth player joins the group of three depicted in Fig. 2. To identify the new equilibrium, we merely add the new player’s replacement graph to the existing ones. There are two possibilities. It is possible that, at the equilibrium of the three-player community the fourth player would choose to contribute zero. This will be the case if the extra player’s replacement graph hits the horizontal axis in Fig. 2 at a point to the left of *G*^{N}_{.} The equilibrium level of total provision, and the choices and utilities of the three initial players, are unchanged. The fourth player chooses to contribute nothing, and enjoys the existing level of public good, while allocating all of his money income to private good consumption. Alternatively, the replacement value of the new player is positive at the existing equilibrium. In this case, the graph of the aggregate replacement function shifts upwards in the neighborhood of the initial equilibrium. The new equilibrium involves a higher total provision level. Existing contributors will reduce their individual contributions, and all are advantaged by the addition of the extra player.

In the presence of a large number of potential contributors who may differ in terms of incomes, preferences or unit costs, the diagram strongly suggests the conclusion reached by Andreoni (1988) – namely, that when *n* is large, the proportion of players who make strictly positive equilibrium contributions may be vanishingly small. Almost all players choose zero contributions.

## Extensions

Early attempts to apply the voluntary contributions model – for example, to charitable giving, in which the aggregate *G* is the total quantity subscribed to some good cause – suggest that the very strong implications of the simple model – neutrality when unit costs are the same across contributors, and its implication that, when *n* is large, the number of strictly positive contributors will be a very small fraction of *n* – are difficult to square with empirical evidence. In addition, recent concerns with global and regional public goods have led to an interest in situations that naturally seem to involve public good technologies other than the summation one described above. We now briefly review some of the recent extensions and modifications of the model.

### Technology of Public Good Provision

Hirshleifer’s example of a best-shot public good involves defensive guns ringing a city, each trying to shoot down an approaching missile. What matters to the city’s inhabitants is the accuracy of the single most accurate shot. His example of a weakest-link involves a group of farmers, each owning a pie-shaped slice of land within a circular area surrounded by sea. Each is responsible for the maintenance of his part of the perimeter dyke. In the event of a storm that threatens to breach the dyke, it is the level of maintenance of the least well-maintained stretch of wall that determines the level of security enjoyed by all. Sandler (2004) suggests a wide range of situations involving regional or global public goods that are better captured by one or other of these formulations than by the standard summation formulation.

These formulations have very distinctive equilibrium properties. For example, consider a two-player model with the weakest link technology in which there is an equilibrium at which both contribute, say, ten units to the public good. Then any allocation at which each is contributing *x* units, where *x* lies between zero and ten, is also an equilibrium. After all, if the other player is contributing *x* units, it does not pay you to contribute any more than *x*, since the total provision is defined by the smaller individual contribution. This game there can have a continuum of equilibria. Hirshleifer himself suggested that the players may be expected to choose the Paretodominant equilibrium. However, experimental evidence suggests that players find it surprisingly hard to coordinate on the Pareto dominant equilibrium.

Cornes (1993) and Cornes and Hartley (2007b) consider the class of games in which the total level of a public good is generated by individual contributions according to a constant returns to scale CES production process: \( G={\left[{\sum}_{i=1}^n{g}_i^{\nu}\right]}^{\frac{1}{\nu }} \) The summation model is obtained by putting \( \nu =1:\nu \to +\infty \) generates the best shot, and \( \nu \to -\infty \) generates the weakest link. They show that, if −∞ < *ν* < 1, the resulting weaker link model has a unique equilibrium. It is only at the limit, when the isoquants associated with the production technology are L-shaped, that Hirshleifer’s problem of multiple equilibria arises. Moreover, if player *i* is contributing less than player *j* at an equilibrium – perhaps because her income is lower, or because she has less interest in the public good – then player *i* has a higher marginal product as a contributor. Hence, neutrality with respect to income transfers breaks down, and a transfer from player *j* to player *i* may lead to a higher equilibrium level of public good provision and may be Pareto improving.

Situations involving *ν* > 1 are better-shot games. Here, the production technology is inherently nonconvex, and again multiple equilibria may arise. For finite values of *ν*, an equilibrium may involve positive contributions by each of a team of positive contributors, while the rest make zero contributions. However, there may be many such equilibria, each involving a different team of contributors. In Hirshleifer’s best-shot case, if there are *n* players, there may be *n* equilibria, each of which involves a single ‘champion’, or ‘dragon-slayer’, who is the sole positive contributor, while all others make zero contributions. Again, achieving an equilibrium requires the players to resolve a tricky coordination problem.

### Preferences

*i*’s own contribution as an argument of her own utility function, in addition to the aggregate quantity

*G*:

*i*not only cares about the total amount given to the charitable giving,

*G*, but also experiences a ‘warm glow’ of satisfaction from her own contribution,

*g*

_{i}. If the standard resource constraint and public good technology are retained, this modification is sufficient to produce very rich comparative static possibilities: neutrality does not generally hold, and an increase in player

*i*’s money income alone may either increase or reduce the equilibrium level of

*G*. Finally, note that if the utility function is assumed to take the Cobb-Douglas form – \( {u}_i(.)={y}_i^{\alpha }{g}_i^{\beta }{G}^{\gamma } \) – then at any equilibrium every player will make a positive contribution to the public good. A proliferation of noncontributors as the number of players increases is no longer implied.

This extension significantly broadens the range of potential applications of the model. First, there is nothing to stop us from considering situations in which player *i* regards *G* as a public bad \( -\frac{\partial {u}_i(.)}{\partial G}<0 \). Thus the model may be interpreted as one involving congestion or pollution. Each may still be a positive contributor at equilibrium, the pollution or congestion being an incidental by-product that is jointly generated alongside the private good *g*_{i}. Kotchen (2006) and Ruebbelke (2002) have explored such models.

Morgan (2000) and Duncan (2002) have used a slight modification of this model to investigate the potential role that lotteries, or raffles, may play in raising the public good level above that implied by the voluntary contribution model. The basic idea is simple. The presence of the public good by itself involves a positive externality, and will tend to be underprovided. If individuals buy lottery tickets, each of which partially contributes to the public good and also gives its purchaser a probability of winning a money prize, a negative externality is thereby added – by buying a ticket, and increasing my chance of winning the prize, I inflict a negative externality on other ticket holders. There are two externalities, one beneficial and one harmful. The resulting equilibrium, at which these externalities tend to counteract each other, may involve a higher level of public good provision than if it were provided simply by individual contributions in the absence of the lottery.

### Dynamic Models

Up to this point, our discussion has remained firmly within the context of a one-shot static game. It is natural to wonder how the properties of equilibrium – in particular its presumptive inefficiency – are affected if we allow the contribution game to be played over many time periods. Schelling (1960, p. 45) suggested that such a setting may allow each player to make a small contribution, then wait to see whether others follow suit, before deciding whether to make a further small contribution. His suggestion has been analysed more formally by others, notably by Admati and Perry (1991) and Marx and Matthews (2000).

The last-named authors, whose analysis includes a useful discussion of the difference between their model and that of Admati and Perry, allow every player to choose a contribution level in each time period – any non-negative contribution, however large or small, is admissible. The properties of equilibria depend on (i) the degree of heterogeneity of players’ valuations of the public good, (ii) the rate at which future costs and benefits are discounted, and (iii) whether or not there is a significant step in the benefit function – for example, a bridge generates no benefits until it is completed, thus representing an extreme example of a benefit function with a discrete step. They provide good news and bad news. The good news is that, if contributions can be made in small increments over time, equilibria can be attained that are more efficient than the equilibrium associated with the one-shot game. They argue that, if players’ valuations are similar, and the rate of discount low, then nearly efficient equilibria exist. Furthermore, the presence of a significant benefit jump helps the prospects of successful completion of a project. An efficient equilibrium of the dynamic game may exist even in situations in which the only equilibrium of the static game involves zero contributions. The bad news is that, in common with many other dynamic games, there also exist other equilibria involving zero contributions.

Duffy et al. (2007) have investigated the properties of such dynamic models experimentally. They confirm that contributions do indeed tend to be higher in dynamic games of the kind proposed by Marx and Matthews, but their results cast doubt on the claimed importance of jumps in the benefit function.

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