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This paper considers a discrete public good subscription game under threshold uncertainty and private information on valuations and analyzes the effect of change in cost uncertainty on the private contribution equilibrium under a simultaneous institution. Comparative statics with respect to the changes in the cost distribution are derived. We find that if the cost distribution becomes more dispersed, in the sense of a mean-preserving spread, the expected total contributions to the public good will decrease. Our proposition provides a policy implication that if the suppliers are able to reduce the uncertainty of the cost distribution, the private contribution to the public good will increase.

This paper considers a discrete public good subscription game under threshold uncertainty and private information on valuations and analyzes the effect of change in cost uncertainty on the private contribution equilibrium under a simultaneous institution. Comparative statics with respect to the changes in the cost distribution are derived. We find that if the cost distribution becomes more dispersed, in the sense of a mean-preserving spread, the expected total contributions to the public good will decrease. Our proposition provides a policy implication that if the suppliers are able to reduce the uncertainty of the cost distribution, the private contribution to the public good will increase.

Discrete Public Good, Threshold Uncertainty, Private Information

This paper considers a model of provision of a discrete public good in a subscription game within an environment of threshold uncertainty and private information on public good valuations. The focus is on the comparative statics of a change in cost uncertainty on the private contribution equilibrium under a simultaneous institution. In a voluntary contributions mechanism, a discrete public good is provided if the total contributions are large enough to cover cost; otherwise, the public good is not provided. This kind of public good is also called a binary or a threshold public good. Typical examples of discrete public goods are roads, parks, community libraries, local radio programs, and school buildings. In a subscription game, the player contributions are refunded if the sum of the contributions does not cover the cost of the public good.

Palfrey and Rosenthal [

It is possible that the players do not face a certain threshold. For example, it might not be known how much money will be needed to build a community library or to complete a public good project. Realizing that the threshold uncertainty may affect the player’s strategic contribution behavior, Nitzan and Romano [

Papers by Menezes et al. [

Barbieri and Malueg [

Although McBride [

We show that if the cost distribution becomes more dispersed, in the sense of a mean-preserving spread, the expected contributions will decrease. Our results suggest that suppliers may increase contributions to the public good by reducing uncertainty over the cost distribution.

The rest of paper is organized as follows. Section 2 presents the model. Section 3 considers the comparative statics with respect to changes in cost distribution of the public good and characterizes the expected contribution. Section 4 is the conclusion.

Consider a subscription game with

A discrete public good can be provided if and only if the total contributions are equal to or larger than the cost of the public good,

The public good game we consider in this paper is the subscription game (Admati and Perry [

Given

Since the player does not know other players’ contributions when making the contribution decision, he needs to forecast the amounts other players will contribute,

The assumed objective for each player is to maximize expected payoff. If the public good is provided, then player

if the public good is not provided, then player i’s payoff is 0. Thus, player i’s expected payoff function can be written as:

Assume

Maximizing Equation (2) with respect to

Under the assumption of a common uniform distribution in [0, 1], the total expected contributions by other players,

From the best response function, Equation (4), we can find that

Using the definition of expected contribution,

With player values independently and uniformly distributed on [0, 1], the expected equilibrium contribution is determined by

We can solve (6) for K which yields is

Player

The cutoff point in equilibrium for each player to begin contributing a positive amount to the public good,

Since the lower bound of player’s value is 0, v_{p} has to be equal to or larger than 0. Thus, we can get a constraint that

Players may confront cost distributions with different levels of dispersion. If the cost distribution can be controlled or affected by supplier actions, the results in this paper may suggest to the producer of the public good the benefits of changing information related to the cost of the public good measured in terms of increased expected total contribution to the public good. In this section we consider the effect of changes in the cost distribution on equilibrium contributions.

Given each player’s value distribution is followed a common uniform distribution over [0, 1], assume the cost distribution becomes more uncertain in the sense of mean-preserving spread. For example, the new cost distribution is

Proposition.

A mean-preserving increase in the distribution of cost will decrease the total expected contributions.

From Equation (7), we know that

Since

Since

The proposition indicates that the players, on average, become less willing to contribute to the public good when the cost of the public good becomes more uncertain.

We have shown that the cutoff point in equilibrium is

increases, the cutoff point will increase2. It means that the player will begin contributing a positive amount to the public good at a higher value as the degree of cost uncertainty increases.

The change in

From Equation (11), we find that changing

We find that players start to contribute to the public goods at a higher cutpoint value and the contribution amounts at each possible value of public good weakly decrease, hence, the expected contribution to the public goods decreases in the degree of cost uncertainty,

Our proposition provides a policy implication that if the suppliers are able to reduce the uncertainty of the cost distribution, the private contribution to the public good will increase. The reduction in cost distribution uncertainty will encourage the players with low value to begin contributing to the public good and also increase contributions of inframarginal contributors.

In this subsection, we use a numerical example to illustrate that expected contribution,

In this example, we consider a subscription game with 5 players whose values are uniformly distributed in [0, 1]. Also, we assume that players do not know the cost of providing the public good but believe it is drawn from a uniform distribution with support [1, 5], i.e. the initial

If the valuation for the public good is complete information and identical for each player, McBride (2006) finds that the effects of increased cost uncertainty depend on the value of the public good. However, when we consider public good valuations as private information that expected contributions are monotonic and that a more dispersed cost distribution always decreases the expected contributions.

From a policy aspect, we suggest that supplier may increase the private contribution to the public good, if they can reduce the degree of uncertainty with respect to the cost distribution when both threshold uncertainty and private information on public good valuations exist.

We offer two directions for future research. Many real-world private contribution institutions are not simultaneous. Contributions are instead collected sequentially. For example, churches may announce organ fund campaign and report the updated contribution level frequently or local governments announce the seed donations to future contributors when they launch new public good projects. There is no published research that investigates how the sequential contribution would be affected by a change in the dispersion of the cost distribution or the value distribution in a subscription game under the threshold uncertainty and private information of valuation for the public good. A second research direction is to test the hypothesis from our theoretical model using experimental methods in a laboratory environment. These future studies may result in a more complete understanding of behavior in mechanisms of private contribution to public goods.

We have benefited from comments by Silvana Krasteva and participants at Ph.D. Student Presentation Seminar at Texas A&M University. We also thank anonymous referees for their comments. All errors are our own.