Private Information in the BBV Model of Public Goods

• DOI: http://doi.org/10.1111/jpet.12178
• Author: DAVID A. MALUEG,STEFANO BARBIERI
• Date: 01 December 2016
• Published date: 01 December 2016

PRIVATE INFORMATION IN THE BBV MODEL OF PUBLIC GOODS

STEFANO BARBIERI

Tulane University

DAVID A. MALUEG

University of California, Riverside

Abstract

To analyze the private provision of a public good in the presence of pri-

vate information, we explore the connections between two frameworks:

the binary public good model with threshold uncertainty and the stan-

dard continuous model `

alaBergstromet al. Linearity of best responses

in others’ contributions is key to matching the two frameworks. We

identify all utility functions that display this linearity, and we provide

conditions ensuring that the minimal properties that Bergstrom et al.

require for utilities are satisfied. Using techniques developed in the

threshold uncertainty framework, we show existence and uniqueness

of the Bayes-Nash equilibrium—thus generalizing existing results—

and we analyze its comparative statics properties. In particular, un-

der the reasonable assumption that agents’ income is stochastic and

private information, we complement the full-information crowding-

out and redistribution results of Bergstrom et al. If the government

taxes agents’ income proportionally and redistributes (expected) rev-

enues lump sum, equilibrium public good provision can increase or de-

crease, even if the set of contributors is unchanged. Similarly, we show

that crowding-out can be one-for-one, less than one-for-one, or more

than one-for-one. Finally, we extend our results to a multidimensional

framework in which agents’ unit costs of contributions are also private

information.

1. Introduction

The importance of “On the Private Provision of Public Goods” by Bergstrom, Blume,

and Varian (1986)—hereinafter referred to as BBV—to public economics cannot be

overstated. Their results on uniqueness, crowding-out, and income redistribution “are

now staples of public economics and are taught in most graduate courses around the

Stefano Barbieri, Department of Economics, TulaneUniversity, 206 Tilton Hall, New Orleans, LA 70118

(sbarbier@tulane.edu). David A. Malueg, Department of Economics, University of California, 3136

Sproul Hall, Riverside, CA 92521 (david.malueg@ucr.edu).

The authors thank two anonymous referees and Steven Slutsky for their generous comments on

earlier versions of this paper.

Received July 30, 2013; Accepted March 14, 2015.

C2016 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 18 (6), 2016, pp. 857–881.

857

858 Journal of Public Economic Theory

globe.”1While the full-information case is well understood, less is known about the vol-

untary provision of public goods in the BBV setup if one posits private information,

especially without appealing to mechanism design. As Martimort and Moreira (2010)

point out, much of the private-information literature assumes the existence of an unin-

formed mediator with full commitment power and then applies the tools of mechanism

design. While this assumption is certainly appropriate in some circumstances, Martimort

and Moreira argue that it is less so in others.2

Some interesting results appear in this literature. For instance, Gradstein, Nitzan,

and Slutsky (1994) provide examples that show how neutrality with respect to income

redistribution can easily fail when private information about willingness to pay is intro-

duced into the BBV model. However, while convincingly establishing that private infor-

mation has important consequences, the literature has not been able to marry generality

and tractability, as BBV and Cornes and Hartley (2007a) do for the full-information case.

On the one hand, for instance, the tractable private-information, public-good models of

Vesterlund (2003) and Andreoni (2006) consider a two- or three-point distribution of

uncertainty. Clearly, this limits the kind of information-related questions that these mod-

els can answer. On the other hand, Bag and Roy (2011) allow for quite general forms of

uncertainty but make no effort in the direction of establishing existence, uniqueness,

and comparative statics properties of equilibrium in their analysis of the simultaneous

game.3

If one abandons the continuous production function of BBV and focuses instead

on a binary public good (a setup such that the public good is either built or not; quan-

tity is not otherwise variable), while retaining continuous contributions, then progress

is possible. Earlier works on this subject were limited in focusing primarily on equilib-

rium existence with two agents, since multiple equilibria typically exist and complicated

strategies make deriving comparative statics results forbidding.4The recent contribu-

tion of Barbieri and Malueg (2010) adds threshold uncertainty to the discrete model

with the result that equilibrium is unique, readily characterized, and easily amenable

to comparative statics analysis and applications (see, e.g., Krasteva and Yildirim 2013;

Barbieri and Malueg 2014), even for asymmetric environments with many agents. How-

ever, Barbieri and Malueg (2010) operate in a binary public-good, subscription-game

framework.5While interesting, the applicability of this framework to real-world situa-

tions is not universal.

As Nitzan and Romano (1990) point out, binary public goods models with threshold

uncertainty are closely related to BBV’s setup because one can reinterpret the probabil-

ity of provision as a continuous production function. Then, a natural first question is:

1See Andreoni and Kanbur (2007, p. 1634). See also the results in Warr (1983) and Cornes and Sandler

(1985).

2Martimort and Moreira (2010) include examples such as health, environment, global warming, coun-

terterrorism, multilateral foreign aid, and lobbying.

3Their focus is on showing the dynamic game outperforms the static one. To do so, they derive an

upper bound on simultaneous-game contributions, rather than relying on a detailed characterization

of equilibrium.

4See, e.g., Alboth, Lerner, and Shalev (2001), Menezes, Monteiro, and Temimi (2001), Laussel and

Palfrey (2003), and Barbieri and Malueg (2008a, b).

5In their setup, the binary public good is built and paid for if and only if the sum of agents’ contribu-

tions surpasses a threshold. Otherwise, contributions are refunded. Refunds of insufficient contribu-

tion are what distinguish what Admati and Perry (1991) label the subscription game from the contribution

game, which has no refunds.