Submission Number:JPET-11-00213
Bayesian Equilibrium in a Public Good Economy
Shlomit Hon-snir Department of Economics, The Max Stern Academic Colloge of Emek Yezreel,ISRAEL
Benyamin Shitovitz Department of Economics, Haifa University,ISRAEL
Menahem Spiegel Department of Finance and Economics, Rutgers,NJ, USA
Abstract This paper studies the provision of a public good via voluntary contributions in an economy with uncertainty and differential information. Consumers differ in their private information regarding their future endowment as well as in their preferences. Each consumer selects her consumption exante, i.e., before knowing the state of nature. Contributions to the provision of the public good are determined ex-post, i.e., when the state of nature is realized. Assuming that some normality conditions hold, a Bayesian equilibrium exists. Further, equilibrium is unique, regardless of the number of consumers, when either (a) the information partitions of consumers can be ranked from the finnest to the coarsest, or (b) there are only two types of consumers.
Citation: Shlomit Hon-snir and Benyamin Shitovitz and Menahem Spiegel, (2011) ''Bayesian Equilibrium in a Public Good Economy'', Journal of Public Economic Theory, Vol. 12 No. 2 pp. 387-398. Contact: Shlomit Hon-snir -
[email protected], Benyamin Shitovitz -
[email protected], Menahem Spiegel -
[email protected]. Submitted: October 05, 2011. Published: October 11, 2011.
Bayesian Equilibrium in a Public Good Economy Shlomit Hon-Snir Department of Economics, The Max Stern Academic College of Emek Yezreel, Emek Yezreel 19300 ISRAEL E-mail:
[email protected] Benyamin Shitovitzy Department of Economics, University of Haifa, Mount Carmel, Haifa 31905 ISRAEL E-mail:
[email protected] Menahem Spiegel Department of Finance and Economics, Rutgers, NJ, USA E-mail:
[email protected] September 30, 2009
Comments received from Diego Moreno, Steve Slutzky, Myrna Wooders, two referees, and participants in GAMES 2008 in Chicago IL and in PET07 in Nashville TN are gratefully acknowledged. Financial support from RRC at RBS to M. Spiegel is gratefully acknowledged. y Corresponding author
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Bayesian Equilibrium in a Public Good Economy Shlomit Hon-Snir, Benyamin Shitovitz and Menahem Spiegel
Abstract This paper studies the provision of a public good via voluntary contributions in an economy with uncertainty and di¤erential information. Consumers di¤er in their private information regarding their future endowment as well as in their preferences. Each consumer selects her consumption exante, i.e., before knowing the state of nature. Contributions to the provision of the public good are determined ex-post, i.e., when the state of nature is realized. Assuming that some normality conditions hold, a Bayesian equilibrium exists. Further, equilibrium is unique, regardless of the number of consumers, when either (a) the information partitions of consumers can be ranked from the …nest to the coarsest, or (b) there are only two types of consumers. JEL codes: C72, D82, H41. Keywords: Di¤erential Information, Bayesian Equilibrium, Voluntary Contributions, Cardinal Normality, Public Good Economies.
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1
Introduction
This paper studies the provision of a public good via voluntary contributions in an economy with uncertainty and di¤erential information. A consumer’s private information about the state of nature is described by a partition. We assume that the set of possible states of nature is …nite, and that consumers have a common prior. In a typical pure public good economy each consumers maximizes her expected utility by allocating her endowment between consumption of the private good and her contribution to the public good –see, e.g., Bergstrom et al. (1986) and Bernheim (1986). In our model, a consumer makes this decision ex-ante, based on her information partition. Thus, a consumer determines her optimal consumption prior to the realization of the state of nature. Clearly, the nature of the uncertainty a¤ects the way consumers allocate their endowment between their consumption of the private good and their contribution to the public good, and therefore the total provision of the public good. In the literature several types of uncertainty are considered, and their implications on the provision of public goods are discussed. Gradstein (1992) assumes that consumers are uncertain about the contributions of other individuals. Under this uncertainty the time dynamics of the private provision of public good is derived. Gradstein et al. (1994) re-examines in the context of uncertainty Warr’s (1983) neutrality of the provision of a public good with respect to income distribution. In the model, uncertainty is about the consumers’income: each consumer knows her own endowment, but her information regarding the endowments of other consumers is incomplete. Keenan et al. (2006) examines the impact of increased uncertainty on the provision of the public good under a non-Nash response and symmetric equilibrium. Here again uncertainty is about the response of other contributors to a contribution to the public good. In this paper we consider a public good economy with di¤erential information regarding consumers income and preferences. We assume that both a consumer’s income and her utility function are state dependent, and are revealed only after the realization of the state of nature. The private information of each consumer is given by her information partition; that is, a consumer cannot distinguish between di¤erent states of nature that belong to the same element in her information partition. Under uncertainty, before the realization of the state, a consumer cannot 3
commit herself to make a speci…c contribution, as she does not know her (state-dependent) endowment. Therefore, in our model we assume that a consumer’s strategy, which is contingent on her information partition, speci…es a consumption of the private good. Thus, a consumer will commit herself ex-ante to a speci…c level of consumption of the private good, and the amount of her contribution for the provision of the public good will be determined ex-post, once her endowment is realized, as the residual; i.e., her contribution to the public good is the amount of the endowment not consumed. Hence a pro…le of consumers’strategies determines the ex-ante expected utility of each consumer. A Bayesian equilibrium is a strategy pro…le where each consumer maximizes her conditional expected utility –for de…nitions and analysis, see Tirole (1988), Allen and Yannelis (2001), Einy et al. (2002), Einy et al. (2003) and Chokler et al. (2006). In this non-cooperative environment, we are mainly concerned with the existence and the uniqueness of Bayesian equilibrium. Under full information, when utility functions are di¤erentiable and satisfy certain strict cardinal normality conditions in the interior of the consumption set, a typical public good economy admits a unique Nash equilibrium – see Shitovitz and Spiegel (2001). In our setting, these conditions imply that the expected utility of a consumer is continuous on the set of strategy pro…les and is concave with respect to the her own strategy, and therefore ensure existence of a Bayesian equilibrium. However, these conditions do not guarantee uniqueness without additional quali…cations on the information structure of the economy. Speci…cally, we show that under the above conditions a Bayesian equilibrium exists and is unique, regardless of the number of consumers, in the following two information settings: (a) When the information partitions of consumers can be ranked from the …nest to the coarsest. (b) When there are only two types of consumers (even if no consumer holds superior information). Information setting (a) refers to a public good economy where no restrictions are imposed on the size and/or the distribution of endowments in the di¤erent states of nature. Also, no additional restrictions are imposed on consumers’utility functions. Furthermore, information setting (a) includes the following interesting cases: (I) the “public information”case where all information partitions coincide, and (II) the case where some consumers have full information while others have null information. In information setting (b), while considering two types of consumers, no ranking restriction is imposed 4
on the two information partitions.1 The paper is organized as follows: Section 2 describes a public good economy with di¤erential information. In Section 3 some preliminary properties of Bayesian equilibrium are obtained. Section 4 is devoted to study uniqueness of Bayesian equilibrium in a public good economy under information setting (a), while in Section 5 uniqueness results are obtained under information setting (b). Concluding remarks are presented in Section 6.
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The Model
Consider an economy with two goods, a private good (x) and a pure public good (Y ), and n consumers. There is uncertainty about the state of nature. This uncertainty is described by a probability measure space ( ; F; ); where is a …nite set of states of nature, F is a set of events (which, w.l.o.g., coincides with the power set of ), and is a probability measure. The endowment of private good of consumer i 2 N := f1; :::; ng is state dependent, and is described by a function ei : ! 0: Moreover, Y (!) yi (!) = ei (!)
! ~ 2Ii (b !)
si (!) > 0 for every ! 2 Ii (b ! ): 9
Since si (:) is an interior solution on Ii (b ! ); it satis…es the …rst order condition for utility maximization; i.e., E(ui1 (si (b ! ); Y (:); :)
ui2 (si (b ! ); Y (:); :) j Ii (b ! )) = 0:
Replacing si (b ! ) by si (b ! ); and recalling that ui11 (:; :; :) ui12 (:; :; :) < 0 , we have ! ); Y (:); :) j Ii (b ! )) < 0: ! ); Y (:); :) ui2 (si (b E(ui1 (si (b Since for every ! 2 Ii (b ! ); Y (!) Y (!) and ui22 (:; :; :) ui12 (:; :; :) have E(ui1 (si (b ! ); Y (:); :) ui2 (si (b ! ); Y (:); :) j Ii (b ! )) < 0:
0; we
This contradicts that si (b ! ) > 0 maximizes consumer i’s conditional expected utility on Ii (b ! ). A direct consequence of Theorem 1 is that if s and s are two di¤erent Bayesian equilibria, then there exists ! 2 such that Y (!) 6= Y (!): (Contrariwise, if Y (!) = Y (!) for every ! 2 ; then Theorem 1 implies si = si .) Corollary 1 Let s and s be two di¤erent Bayesian equilibria, and let ! 1 be such that Y (! 1 ) < Y (! 1 ). Then there exist a consumer bi and a state of nature ! 2 2 Ibi (! 1 ); such that Y (! 2 ) > Y (! 2 ): Proof. Let ! 1 2
be such that
Y (!)
Y (!)
for every ! 2 Ii (! 1 ) and i 2 N: By Theorem 1 si (! 1 )
si (! 1 )
yi (! 1 )
yi (! 1 )
for every i 2 N: Therefore
for every i 2 N , which implies Y (! 1 )
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Y (! 1 ):
3.3
Equal Treatment of Bayesian Equilibria
De…nition 2: Consumer i and consumer j are of the same type if ui (x; Y; !) = uj (x; Y; !); Ii (!) = Ij (!) and ei (!) = ej (!) for all (x; Y ) 2 i; i.e., we assume that Fn ::: F2 F1 : This ordering of the information partitions together with the measurability of the strategy sets enable us to easily compare the quantity of the public good in di¤erent Bayesian equilibria. Theorem 2 A public good economy with di¤erential information such that Fn ::: F2 F1 admits a unique Bayesian equilibrium.
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Proof. Assume by way of contradiction that the are two di¤erent Bayesian equilibria, s and s . Denote by bi the consumer with the smallest index which satis…es that there exist ! 2 and ! 1 ; ! 2 2 I^{ (!) such that Y (! 1 ) < Y (! 1 ) and Y (! 2 ) > Y (! 2 ): The existence of consumer bi is guaranteed by Corollary 1. Since Y (! 1 ) =
n X
ei (! 1 )
i=1
n X
si (! 1 ) <
n X
si (! 2 ) >
i=1
n X
ei (! 1 )
i=1
n X
si (! 1 ) = Y (! 1 )
n X
si (! 2 ) = Y (! 2 );
i=1
and Y (! 2 ) =
n X
ei (! 2 )
i=1
we have
i=1
n X
(si (! 2 )
n X
ei (! 2 )
i=1
si (! 1 )) >
i=1
i=1
n X
(si (! 2 )
bi 1 X
(si (! 2 )
si (! 1 )):
i=1
::: F2 F1 , implies ! 2 2 Ij (! 1 ); sj (! 2 ) = Also ! 2 2 Ibi (! 1 ) and Fn sj (! 1 ) and sj (! 2 ) = sj (! 1 ) for j = bi; bi + 1; :::; n: Hence bi 1 X
(si (! 2 )
si (! 1 )) >
i=1
si (! 1 )):
i=1
Recall that Y (! 1 ) < Y (! 1 ), by the choice of bi; for consumer j = 1; :::; (bi 1) for every ! 2 Ij (! 1 ); Y (!) Y (!): By Theorem 1, it implies sj (! 1 ) sj (! 1 ): In a similar way we get sj (! 2 ) sj (! 2 ): That is, bi 1 X
(si (! 2 )
bi 1 X
si (! 2 ))
(si (! 1 )
si (! 1 ));
i=1
i=1
which contradicts the inequality obtained above. A direct application of Theorem 2 establishes uniqueness of Bayesian equilibrium when consumers have symmetric information (i.e., when information is public: F1 = F2 = ::: = Fn ): Another case of interest is that of an economy in which a consumer has either no information or full information (i.e., for each i 2 N either Fi = f ; g or Fi = F ). 12
5
Uniqueness when Consumers are of Two Types
Next we consider a pure public good economy with two types of consumers. For such economies we prove uniqueness of Bayesian equilibrium without any restrictions on the information structure. Let the economy consists of two types of consumers fA; Bg with nA and nB denoting the number of consumers of type A and B respectively. Recall that in a Bayesian equilibrium consumers of the same type follow the same strategy – Corollary 2. Therefore we denote by s = (sA ; sB ) the strategy pro…les that form a Bayesian equilibrium for such an economy. Theorem 3 A public good economy with di¤erential information which consists of two types of consumers admits a unique Bayesian equilibrium. = (sA ; sB ) are
Proof. Assume by contradiction that s = (sA ; sB ) and s two di¤erent Bayesian equilibria. Let x = max max nt jst (!) st (!)j t2fA;Bg !2
and denote by b t and ! b the consumer type and the state of nature for which x is obtained. Assume, without loss of generality, that sbt (b ! ) sbt (b ! ) > 0: Therefore nA [sA (!)
sA (!)] + nB [sB (!)
for every ! 2 Ibt (b ! ): Recall that,
sB (!)]
0
Y (!) = nA eA (!) + nB eB (!)
nA sA (!)
nB sB (!)
Y (!) = nA eA (!) + nB eB (!)
nA sA (!)
nB sB (!)
and for every ! 2 Ibt (b ! ): Therefore Y (!)
Y (!) = nA sA (!) + nB sB (!)
nA sA (!)
for every ! 2 Ibt (b ! ), which contradicts Theorem 1. 13
nB sB (!)
0;
6
Concluding Remarks
In this paper we study the Bayesian equilibria of a public good economy with di¤erential information and a …nite number of states of nature. Existence of a Bayesian equilibrium is assured by cardinal normality conditions.4 This is in contrast to Einy et al (2007), who show that Bayesian equilibrium in a Cournot oligopoly may not exist when prices are nonnegative. The proof of existence and uniqueness of Bayesian equilibrium in a public good economy with di¤erential information provides the essential foundation for the analysis of questions regarding the value of information, or the impact of information advantage (or disadvantage) on payo¤s.5 Our approach in this paper studies the private provision of a public good in a non-cooperative setting. We leave it to future research to study the core of a public good economy with di¤erential information. We conjecture that there may be a relation between the core and the set of Bayesian equilibrium allocations akin to that found by Shitovitz and Spiegel (2001). This line of research may provide more clear implications regarding the ine¢ ciency of the private provision of a public good in economies with di¤erential information.
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References 1. Bergstrom, T., L. Blume and H. Varian (1986) " On the Private Provision of Public Goods" Journal of Public Economics 29, 25-49. 2. Bernheim, B. D. (1986)" On the Voluntary and Involuntary Provision of Public Goods" American Economic Review 76 , 789-793. 3. Chokler, A., S. Hon-Snir, M. Kim, and B. Shitovitz (2006) " Information Disadvantage in Linear Cournot Duopolies with Di¤erentiated Products" International Journal of Industrial Organization 24, 785-793. 4. Einy, E., D. Moreno and B. Shitovitz (2002) " Information Advantage in Cournot Oligopoly" Journal of Economic Theory 106, 151-160. 4
Similar conditions were used in Shitovitz and Spiegel (2001) to prove the existence and uniqueness of equilibrium in a complete information public good economy. 5 Uniqueness of equilibrium is also essential in the determination of the value of information –see Einy et al. (2003) and Chokler et al. (2006).
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5. Einy, E., D. Moreno and B. Shitovitz (2003) " The Value of Public Information in a Cournot Duopoly" Games and Economic Behavior 44, 272-285. 6. Einy, E., O. Haimanko, D. Moreno and B. Shitovitz (2007) " On the Existence of Bayesian Cournot Equilibrium" Universidad Carlos III working paper 07-06. Games and Economic Behavior, forthcoming. 7. Gradstein, M. (1992) " Time Dynamics and Incomplete Information in the Private Provision of Public Goods", Journal of Political Economy 100, 581-597. 8. Gradstein, M., S. Nitzan and S. Slutzky (1994) " Neutrality and the private Provision of Public Goods with Incomplete Information" Economics Letters 46, 69-75. 9. Keenan, D., C. Iltae Kim and R. S. Warren J.R. (2006) " The Private Provision of Public Goods Under Uncertainty: A SymmetricEquilibrium Approach" Journal of Public Economic Theory 8, 863-873. 10. Milgrom, P. and J. Roberts (1990) " Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities" Econometrica 58, 1255-1277. 11. Shitovitz, B. and M. Spiegel (2001) " Stable Provision Vs. CournotNash Equilibrium in Pure Public Good Economies" Journal of Public Economic Theory 3, 219-224. 12. Tirole, J. (1988) The Theory of Industrial Organization, The MIT press: Cambridge Massachusetts. 13. Warr, P. (1983) " The Private Provision of a Public Good is Independent of the Distribution of Income" Economics Letters 13, 207-211.
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