The effect of group size on public good provision in a repeated game setting

Transcription

1 Journal of Publc Economcs 72 (1999) The effect of group sze on publc good provson n a repeated game settng Paul Pecorno* epartment of Economcs, Fnance and Legal Studes, Box , Unversty of Alabama, Tuscaloosa, AL , USA Receved 20 October 1997; receved n revsed form 20 June 1998 Abstract The ablty to cooperate n the provson of a pubc good s analyzed n a repeated game. Holdng the level of provson fxed, wth quas-lnear utlty we fnd that the crtcal value of the dscount parameter converges to 0 n the lmt. Thus, cooperaton s feasble n a large market. ext, we allow the level cooperaton to be adjusted optmally as the group sze ncreases, both for a specfc form of quas-lnear utlty and for obb ouglas utlty. In each case, we fnd that there are admssble values of the dscount parameter such that cooperaton may be mantaned n the lmt Elsever Scence S.A. All rghts reserved. Keywords: ooperaton; Publc goods; Repeated game JEL classfcaton: H41 1. Introducton It s well known that contrbutons to a publc good are subject to a free-rder problem. oes ths free-rder problem become worse as we ncrease the sze of the group whch benefts from provson of the publc good? What happens when the group sze becomes very large? There has been an extensve analyss of these questons n a statc settng, where there s only qualfed support for the * Tel.: ; fax: ; e-mal: ppecorn@cba.ua.edu / 99/ $ see front matter 1999 Elsever Scence S.A. All rghts reserved. PII: S (98)

2 122 P. Pecorno / Journal of Publc Economcs 72 (1999) proposton that the free-rder problem becomes worse as group sze ncreases (Sandler, 1992, pp ). In ths paper, the ablty to mantan cooperaton when group sze becomes large s analyzed n a dynamc settng n whch cooperaton may be mantaned through the use of a smple trgger strategy. When the dscount parameter d s suffcently large, a trgger strategy can mantan the cooperatve outcome n a repeated game settng. In such a game, a crtcal value of the dscount parameter d * may be defned such that f the actual dscount parameter les above t, cooperaton may be mantaned. If the actual dscount parameter les below d *, then cooperaton must break down. The effects of an ncrease n group sze n on the ablty to mantan cooperaton can then be analyzed through ts affect on d *. Ifd * rses wth n, then the ncrease n group sze s sad to make mantanng cooperaton more dffcult. If d * 1asn `, then cooperaton becomes mpossble n large markets. If d * converges to a number less than 1 as n rses wthout bound, then there are admssble values of the dscount parameter such that cooperaton may be mantaned even n large markets. To the extent that cooperaton can be mantaned n large markets, ths volates our ntuton about the ablty to mantan a cooperatve outcome n a large group. In ths paper, we wll focus on results for large markets for three specal cases of publc goods provson. Frst, we assume quas-lnear utlty, whle holdng the level of desred cooperaton fxed. We fnd that the crtcal value of the dscount parameter converges to 0 n the lmt as the group sze becomes large. Thus f ndvduals n a large economy place any weght at all on the future, cooperaton s feasble n the large market. ext we consder a case where the level of cooperaton s adjusted upwards to ts optmal value as the group sze ncreases. We do ths for a partcular specfcaton of the quas-lnear utlty functon and fnd that d * converges to a number less than 1 n the lmt. Thus, there are admssble values of the dscount parameter (.e., values less than 1) such that cooperaton may be mantaned n a large group. Fnally, we consder obb ouglas preferences, where the level of cooperaton s adjusted to the optmum as group sze vares. In contrast to the case wth quas-lnear utlty, obb ouglas preferences allow for ncome effects n the provson of publc goods. As a result, the noncooperatve level of provson s ncreasng n group sze. However, we agan fnd that there are admssble values of the dscount parameter such that cooperaton may be mantaned n the lmt. 2. Prevous lterature Olson (1965) s the classc reference on relatng publc goods provson to group sze. McGure (1974) and hamberln (1974) analyze the effects of group sze on provson n a statc settng and fnd that publc good provson s generally ncreasng n group sze. The level of contrbutons n the noncooperatve

3 P. Pecorno / Journal of Publc Economcs 72 (1999) equlbrum tends toward a fnte lmt as group sze approaches nfnty. The departure of the noncooperatve equlbrum from the group optmum wll grow n certan specal cases, but ths result does not hold n general (Sandler, 1992, pp ). Thus, n the statc settng there s only qualfed support for the proposton that the free-rder problem grows worse as group sze ncreases. McMllan (1979) consders the use of trgger strateges n the context of publc goods provson n a repeated game settng. See Sandler (1992) and ornes and Sandler (1996) for recent overvews of the publc goods lterature. The ablty of olgopolstc frms to cooperate has been extensvely studed n a repeated game settng. Fredman (1971) frst analyzes trgger strateges n the context of an nfntely repeated ournot olgopoly and fnds that cooperaton may be supported f the dscount factor s suffcently hgh. To mantan cooperaton n a large market, the crtcal value of the dscount parameter must converge to some number less than 1 as the number of frms n rses to nfnty. Lambson (1984), (1987) presents an analyss of large markets for both ournot and Bertrand olgopoly and fnds the restrctve condtons under whch cooperaton may be 1 mantaned n the lmt as n `. The restrctve nature of the condtons Lambson derves leaves us wth the presumpton that cooperaton must break down n large markets. Pecorno (1998a) has examned the effect of an ncrease n the number of dentcal frms n on the ablty of an ndustry to mantan a cooperatve level of tarff lobbyng. Hs two man fndngs are that an ncrease n n has an ambguous effect on the dffculty of mantanng cooperaton, and that n the lmt as n `, the crtcal value of the dscount parameter converges to a number less than 1. As a result, there are admssble values of the dscount parameter such that cooperaton may be mantaned wth an nfnte number of frms n the ndustry. Both of these results are at odds wth our ntuton on how ncreases n the number of frms affect the ablty to mantan cooperaton n tarff lobbyng. The publc goods analyss n ths paper dffers from the lobbyng analyss n Pecorno (1998a) n two ways. In Pecorno (1998a), the ndustry sze (measured by total ndustry captal) s held constant throughout the analyss. As a result, when the number of frms ncreases, frm sze, and therefore the ndvdual stake n the provson of the tarff, decreases. Thus, the noncooperatve level of provson s decreasng n group sze, and the optmal level of provson s nvarant to the number of frms. By contrast, here we have a pure publc good where an ncrease n group sze n no way dmnshes the benefts receved from the publc good by the orgnal members of the group. As a result, the optmal level of provson n ncreasng n group sze, and the noncooperatve level of provson s nvarant to group sze 1 Green (1980) constructs an example n whch cooperaton may be mantaned n large markets for ournot olgopoly. Lambson (1984) shows that ournot cooperaton n large markets requres very specal assumptons about demand and frm cost functons.

4 124 P. Pecorno / Journal of Publc Economcs 72 (1999) when there are quas-lnear preferences. One consequence of ths dfference s that Result 1 of ths paper s much stronger than the correspondng result n Pecorno 2 (1998a). Because they cover cases where the level of publc good provson rses optmally as the group sze grows, Results 2 and 3 of ths paper do not have drect analogs n Pecorno (1998a). Fnally, there are no ncome effects n the provson of a tarff (snce t s not a consumpton good). In essence, wth the tarff, one s always analyzng a quas-lnear case, n the sense that there are never any ncome effects. For more general preferences where the publc good s normal, the noncooperatve level of provson s ncreasng n group sze. In our analyss n Secton 6, we wll allow for ncome effects n the provson of the publc good by consderng obb ouglas preferences. Whle the standard ntuton s that t becomes harder to cooperate as group sze ncreases, the statc lterature on publc goods provson generally cannot address ths ssue. The reason s that there s no cooperaton n the one perod model. Rather the lterature has focused on the mportant queston of how the noncooperatve level of provson changes wth an ncrease n group sze. ooperaton cannot be mantaned n the sngle shot game because there are no future perods n whch to punsh behavor whch devates from a cooperatve soluton. In the repeated game, the dffculty of cooperaton wll be governed by how payoffs under defecton, cooperaton and noncooperaton evolve as the group grows larger. These n turn wll dctate the ncentve to cheat, the gan from cooperaton and the penalty for cheatng. As we shall see, these payoffs do not systematcally evolve so as to guarantee the breakdown of cooperaton n large markets. Before proceedng, t s worth makng a note on the termnology used n ths paper. All the games consdered n ths paper are noncooperatve n the sense that bndng contracts are not possble. However, I wll refer to a noncooperatve outcome as a shorthand for the sngle shot (or statc) ash equlbrum. I wll refer to the cooperatve outcome as a shorthand for an outcome of the game n whch some degree of cooperaton s supported as an equlbrum va a credble threat not to cooperate n future perods f anyone defects from cooperaton n the current perod. In the next secton, a generc repeated game s set out, allowng us to defne the crtcal value of the dscount parameter. In the ensung sectons, three specal cases are analyzed. In Secton 4, we examne the specal case of quas-lnear utlty, where the level of cooperaton s held constant as the group sze ncreases. In Secton 5, the level of cooperaton s mantaned at the optmum as group sze ncreases, but a specfc form of quas-lnear preferences s assumed. Wth quas-lnear preferences, there are no ncome effects on publc good provson, and so the level of contrbutons n the noncooperatve outcome s ndependent of group sze (ornes and Sandler, 1996, pp ). In Secton 6, we assume that 2 The correspondng result s Result 2.

5 P. Pecorno / Journal of Publc Economcs 72 (1999) preferences are obb ouglas n order to allow for ncome effects. For all of our specfcatons, we fnd that cooperaton may be mantaned n the lmt as group sze grows large. Secton 7 concludes the paper wth a dscusson of some the possble lmtatons of the model. 3. The crtcal value of the dscount parameter 3 All ndvduals are dentcal, and only symmetrc equlbra are analyzed. onsder the possblty of cooperaton n an nfntely repeated game, where cooperaton s supported through the use of a trgger strategy. In such an equlbrum, each person makes the cooperatve level of publc goods contrbuton n the current perod f all ndvduals cooperated n the prevous perod. If any person defected n the prevous perod, then everyone reverts to the sngle shot ash equlbrum forever. Assume that everyone observes, wthout error, the aggregate level of contrbutons made n the prevous perod. As a result, defectons are always detected. Let an ndvdual s payout be denoted p under the cooperatve outcome, and p under the noncooperatve outcome, respectvely. The payoff earned by a sngle ndvdual who defects from the cooperatve outcome s denoted p. If a person defects from the cooperatve outcome, she wll earn p for the current perod and p n all future perods. If ths payoff s greater than the payoff from contnued cooperaton, then she wll defect and cooperaton cannot be mantaned. Thus, a necessary condton for mantanng a cooperatve outcome under a trgger strategy wth nfnte ash reverson s ` ` t t p 1Od p #Od p, (1) t51 t50 where d denotes the dscount factor. efne d * to be the crtcal value of d such that (1) holds as a strct equalty. For all d $ d *, cooperaton can be supported, 4 whle for d, d *, cooperaton cannot be supported by our smple trgger strategy. Evaluate the summatons n (1) and solve for d * to get p 2 p d * 5 ]]]. (2) p 2 p 5 Snce, p $ p $ p, we have d * # 1. The effect of an ncrease n group sze n on the dffculty of mantanng 3 Pecorno (1998b) analyzes a tarff lobbyng game wth asymmetrc frms. 4 Even when d s suffcently hgh, the cooperatve outcome s clearly not unque (e.g., noncooperaton s always an equlbrum). 5 ote that the expresson n (2) s standard. See, for example, Shapro (1989) (p. 364).

6 126 P. Pecorno / Journal of Publc Economcs 72 (1999) cooperaton wll be measured through d *. If an ncrease n n ncreases d *, then t s sad to make cooperaton more dffcult. The major focus of ths paper are lmt results. If d * approaches 1 as the number of contrbutors rses wthout bound, then (absent the unrealstc case where there s no dscountng of future payoffs) cooperaton becomes mpossble as n grows large. If d * approaches a number less than 1 n the lmt as n grows large, then there are admssble values of the dscount parameter (.e., values less than 1) such that cooperaton can be mantaned n an nfntely large group. 4. A fxed level of cooperaton and quas-lnear preferences Each consumer s endowed wth ncome I. The constant prce of the prvate consumpton good s denoted P, and the prce of the publc good G s 6 normalzed to 1. Snce all consumers are dentcal, we wll conduct a representatve consumer analyss. Preferences are descrbed by U5 1 F(G), (3) where F9(G). 0 and F0(G), 0. Aggregate contrbutons to the publc good are denoted G wth the contrbuton by person denoted g. In the noncooperatve outcome, the consumer s problem s to maxmze utlty n (3) subject to the budget constrant P1 g 5 I, (4) where G 5 g 1 o j± j g. The frst order condtons to ths problem mply that F9(G) 5 1/P. (5) From (5), we can obtan the level of noncooperatve contrbutons G, where by the symmetry of the equlbrum, and g get g 5 g 5 G /n, (6) denotes the common level of ndvdual contrbutons. fferentate (5) to dg ]] 5 0, (7) dn 6 If we thnk of I as representng an endowment of a prmary commodty, then the constant prce of good mples that there s a constant rate of transformaton between commodty I and good (.e., we have constant returns to scale n producton of good ). In Andreon (1988), ths constant rate of transformaton s smply set to 1 so that P 5 1.

7 P. Pecorno / Journal of Publc Economcs 72 (1999) e. the level of contrbutons n the noncooperatve outcome s nvarant to group sze. From (3), (4) and (6), the ndvdual payoff n the noncooperatve outcome s p 5 (1/P )(I 2 [G /n]) 1 F(G ). (8) To acheve the optmal level of publc good provson for the group we can choose g to maxmze U 5 1 F(ng), (9) subject to the budget constrant n (4). Ths makes use of the fact that consumers are dentcal and the equlbrum s symmetrc. The frst order condton to ths problem mples nf9(g) 5 1/P. (10) O From (10), we can solve for the optmum level of contrbutons G. fferentate (10) to get O O dg F9(G ) ]] 52 ]]] O.0, (11) dn nf0(g ).e. the optmal level of contrbutons s ncreasng n group sze. onsder a constant level of cooperatve contrbutons G, where G, G, O O G. ote that whle G s constant by (7), and G s constant by assumpton, G s an ncreasng functon of n by (11). In the cooperatve outcome, each ndvdual contrbutes g 5 G /n. (12) From (3), (4) and (12), the payoff n the cooperatve outcome s p 5 (1/P )(I 2 [G /n]) 1 F(G ). (13) An ndvdual who defects from the cooperatve outcome wll cut her contrbu- ton to zero f F9h[(n 2 1)/n]G j, (1/P ), where [(n 2 1)/n]G represents contrbutons from all agents other than. Ths wll be the case when n s large, and snce we are focused on lmt results, we wll assume ths condton holds. When agent defects, she spends all her ncome on the prvate consumpton good and enjoys the payout S n 2 1 p 5 I/P 1 F]] G. (14) n Substtute (8), (13) and (14) nto (2) to get the crtcal value of the dscount parameter:

8 128 P. Pecorno / Journal of Publc Economcs 72 (1999) F 2 [F 2 (G /n)] d * 5]]]]]] F 2 [F 2 (G /n)] Fh[(n 2 1)/n]G j 2 [F(G ) 2 (G /n)] 5 ]]]]]]]]]]], (15) Fh[(n 2 1)/n]G j 2 [F(G ) 2 (G /n)] where the arguments of F have been ncluded after the second equal sgn. When an ndvdual defects from cooperaton, she gans her wthheld contrbuton G /n. Ths gan s partally offset by the reduced provson of the publc good under defecton,.e. F 5 Fh[(n 2 1)/n]G j, F(G ). In the lmt as n `, F F and the wthheld contrbuton G /n 0. Thus, as group sze becomes large, the gan from cheatng goes to zero, whle the beneft from cooperaton ( 5 p 2 p 7 5 F(G ) 2 F(G )) remans fnte. As a result, from (15) we can see that lm d * 5 0. (16) n ` ot only are there admssble values of the dscount parameter such that cooperaton may be mantaned n the repeated game, cooperaton remans an equlbrum n a large market f ndvduals place any weght at all on the future. Ths s summarzed as Result 1. Result 1. Wth quas-lnear preferences, f we hold the desred level of cooperaton fxed, d * approaches zero as group sze becomes large. If ndvduals put any weght on the future, cooperaton remans an equlbrum n an economy wth an nfnte number of agents. We can broaden the condtons n Result 1 and stll obtan the result that d * approaches 0 n the lmt. The gan from defecton s bounded above by G /n. As long as the cooperatve level of contrbutons rses less than proportonally wth n (e.g., wth the square root of n), the gan from defecton wll approach zero n a large market, as wll the crtcal value of the dscount parameter. Snce the sngle perod ncentve to defect s approachng 0 as n grows large, Result 1 s not senstve to the assumpton that the punshment perod s nfnte. Wth quas-lnear preferences, the level of contrbutons n the noncooperatve outcome G s ndependent of group sze. When G s held constant, ths assumpton ensures that G remans below the fxed level of G as n rses. Wth more general preferences (and assumng normalty), provson n the noncooperatve outcome s ncreasng n n. However, these contrbutons wll generally approach some fnte value as n ` (see Andreon, 1988; Sandler, 1992, p. 50). If 7 In Pecorno (1998a), the gans from defecton and the gans from cooperaton both go to zero as the number of frms rses wthout bound. Ths occurs because frm sze gets smaller as n rses. The crtcal value of the dscount parameter converges to a number less than 1, but t does not converge to zero.

9 P. Pecorno / Journal of Publc Economcs 72 (1999) a constant level of G s chosen to be above ths lmtng value of G, then Result 1 wll hold for more general preferences as well. Thus, wth a smple trgger strategy as an enforcement mechansm, there are a farly wde varety of cases n whch t becomes easy to mantan a cooperatve outcome as n `. 5. Mantanng an optmal level of provson The prevous secton establshes for quas-lnear preferences that f we hold the level of cooperaton constant (or let t grow less then proportonally wth n), n large markets t s not only possble to mantan cooperaton, but t s easy (snce d * 0). In ths secton, the goal s more ambtous, as we wll consder large markets where the level of publc good provson s mantaned at the optmal level as the group sze grows. However, we wll further restrct the quas-lnear utlty functon as follows: 12a G F(G) 5 ]], (17) 1 2 a where 0, a # 1. In the lmt as a 1, we have F(G) 5 ln(g). Usng Eqs. (5), (8), (10), (13), (14), (17), we may obtan the followng ndvdual payoffs: (12a)/a (12a)/a I (P ) (P ) p 5] 2]]] 1 ]]], (18a) P n (1 2 a) I a (12a)/a p 5] 1S]] (np ), (18b) P 1 2 a S S I 1 n a (12a)/a p 5] 1 ]] ]] (np ). (18c) P 1 2 a n Substtute (18a c) nto Eq. (2) to get (12a)/a 12a n h[(n 2 1)/n] 2 aj d * 5 ]]]]]]]]]]]] (12a)/a 12a. (19) n [(n 2 1)/n)] 2 [1 2 (1 2 a)/n] For a large market, we have lm d * a. (20) n ` Therefore, there are admssble values of the dscount parameter such that full cooperaton may be mantaned as the group sze rses towards nfnty. For a 5 1, F(G) 5 ln(g), and lm d * 5 0. Ths analyss s summarzed as Result 2. n ` Result 2. For the preferences descrbed by Eqs. (3) and (17), there are admssble values of the dscount parameter d (.e., values less than 1) such that the optmal

10 130 P. Pecorno / Journal of Publc Economcs 72 (1999) level of cooperaton may be mantaned wthn an nfntely large group. Ths optmal level of provson s tself ncreasng n the sze of the group. In Result 1, t s essentally guaranteed that cooperaton remans an equlbrum n the large economy, whle n Result 2 cooperaton remans an equlbrum only f d. 1 2 a. The dfferences n the results stem from the fact that, under Result 2, we are attemptng to mantan the optmal level of cooperaton as the group sze 12a 1/a rses. The ndvdual contrbuton n the cooperatve outcome g 5 (n P ). When we attempt to mantan the fully optmal level of provson, the ndvdual contrbuton not only remans fnte, but rses wth n (for the preferences descrbed by (3) and (17)). As a result, the gan from defecton s rsng n n. Offsettng ths n the lmt s the rsng beneft from cooperaton as group sze ncreases. 6. obb ouglas preferences onsder obb ouglas preferences over the prvate and publc goods: a 12a U 5 G, (21) where 0, a, 1. In the noncooperatve outcome, the consumer s problem s to maxmze (21) subject to (4). The soluton to ths problem for all n consumers mples S 1 2 a S]]]] (1 2 a) 1 an an I S]]]] S ] (1 2 a)n G 5 ]]]] I, (22a) (1 2 a) 1 an g 5 I, (22b) 5. (22c) (1 2 a) 1 an P Publc good provson s ncreasng n n, but approaches a fnte lmt as n grows large (lmn ` G 5 I(1 2 a)/a). Substtute (22) nto (21) to get the payout n the noncooperatve outcome: S S a a (12a) ni p 5 ] (1 2 a) ]]]]. (23) P (1 2 a) 1 an Agan, makng use of the symmetry of the equlbrum, for the optmal provson a 12a of the publc good, we can choose g to maxmze U 5 (ng), subject to the budget constrant n (4). The soluton to ths problem mples G 5 n(1 2 a)i, (24a) g 5 G /n 5 (1 2 a)i, (24b)

11 P. Pecorno / Journal of Publc Economcs 72 (1999) (a/p )I. (24c) Usng (21) and (24), the payoff n the cooperatve outcome s S a a 12a p 5 ] [(1 2 a)n] I. (25) P When ndvdual defects from the cooperatve outcome, she wthholds her contrbuton g 5 (1 2 a)i and spends all of her ncome on the prvate good. As a result, G 5 (n 2 1)(1 2 a)i, 5 I/P, and she enjoys the payout S a 1 12a p 5 ] [(1 2 a)(n 2 1)] I. (26) P Use (23), (25) and (26) n (2) to get 12a a 12a (1 2 a 1 an)[(n 2 1) 2 a n ] d * 5 ]]]]]]]]]] 12a a. (27) (1 2 a 1 an)(n 2 1) 2 a n For the large market we have a lm d * a. (28) n ` The contrbuton per person g 5 (1 2 a)i s constant, and ths suggests a constant upper bound on the ncentve to cheat. Ths s ncorrect, however, because the ncrease n the provson of the publc good as n rses mples a rsng margnal utlty of consumpton of the prvate good. Thus n utlty terms, the ncentve to cheat s ncreasng n n. Ths s offset n the lmt by the greater gans from cooperaton n larger markets. The analyss s summarzed n Result 3. Result 3. For the obb ouglas preferences n (21), there are admssble values of the dscount parameter d (.e., values less than 1) such that the optmal level of cooperaton may be mantaned wthn an nfntely large group. Ths optmal level of provson s tself ncreasng n the sze of the group. Based on the estmates of Reece and Zeschang (1985), Andreon (1988) argues that a reasonable value of a s , whch mples the coeffcent on publc goods n Eq. (21) s If we use ths estmate of a n Eq. (28) we get that the lmt as n ` of d * s Ths suggests that f agents place even a rather small weght on the future, the optmal level of cooperaton wll reman an equlbrum n the repeated game wth obb ouglas preferences.

12 132 P. Pecorno / Journal of Publc Economcs 72 (1999) oncluson The results of ths paper are drven by the effects on an ncrease n group sze on the payouts p, p, and p. Though the preferences analyzed n ths paper are not completely general, nothng about the effects of group sze on these payouts suggests that cooperaton must break down n large markets. In fact, under a farly wde range of crcumstances, Result 1 suggests that t becomes easy to mantan cooperaton n large markets. It s worth pontng out that, generally speakng, there are no monotoncty results for the effects of n on d * (the dervatve of d * wth respect to n s generally ndetermnate). Thus, not only s cooperaton possble n the lmt, but there s no presumpton that cooperaton gets harder (d * rses) as n rses. Ths, of course, s obvous for the case where d * approaches zero 8 n the lmt. There are two ways to nterpret the results of ths paper. The frst s to more or less accept them at face value and conclude that cooperaton s feasble n large markets. Andreon (1988) has convncngly argued that the statc verson of the model presented n ths paper (whch he refers to as an altrustc model of contrbutons) cannot explan the extent of prvate contrbutons to publc goods (n partcular, charty). He resolves ths problem by developng a model n whch ndvduals derve drect utlty from ther contrbuton to the publc good. The results here suggest that, n a repeated game settng, the altrustc model of publc good provson may be able to explan a hgh observed level of contrbutons. As a future extenson of the model, however, t would be useful to consder the types of asymmetres among agents that are consdered by Andreon. An alternatve s to mantan that cooperaton s more dffcult n larger groups, but that we need to look more deeply at the reasons why cooperaton must break down n the lmt. These reasons do not seem to le n the nature of the payouts n the noncooperatve, cooperatve and defecton outcomes. One canddate s nformatonal problems relatng to the unobservablty of preferences. Ths would tend to create a dffcult and well-known problem of assgnng contrbuton shares whch each party would have to meet to avod trggerng defecton. Informatonal problems mght also nteract wth uncertanty n a model along the lnes of Green and Porter (1984). The ablty to renegotate an agreement after a defecton has taken place may also hamper the mplementaton of trgger strateges n practce (see Bernhem and Ray (1989); Farrell and Maskn (1989); Mcutcheon (1997)). Suppose that the addtonal complcatons referred to above prevent cooperaton from ever beng acheved n a repeated game settng. Then we are left wth the statc results whch gve only lmted support for the proposton that an ncrease n group sze makes the free-rder problem worse. Provson s generally ncreasng n the noncooperatve outcome and t s not true n general that the gap between the 8 In ths case, we are not guaranteed that cooperaton becomes monotoncally easer ether.

13 P. Pecorno / Journal of Publc Economcs 72 (1999) noncooperatve and optmal provson s ncreasng n group sze, though t s true n certan specal cases (Sandler, 1992, pp ). If the complcatons referred to above allow cooperaton n small groups, but not large groups, then the proposton that the free-rder problem gets worse can at best stll only receve very qualfed support. If a small group can acheve cooperaton, an ncrease n the number of potental group members should not (for a pure publc good) undermne the ablty of the orgnal group to cooperate. The level of provson should never decrease wth group sze, though the gap between the actual level of provson and the global optmum may grow. Acknowledgements I would lke to thank Akram Temm and two anonymous referees for helpful comments on the paper. References Andreon, J., Prvately provded publc goods n a large economy: the lmts to altrusm. Journal of Publc Economcs 35, Bernhem, B.., Ray,., ollectve dynamc consstency n repeated games. Games and Economc Behavor 1, hamberln, J., Provson of publc goods as a functon of group sze. Amercan Poltcal Scence Revew 68, ornes, R., Sandler T., The Theory of Externaltes, Publc Goods, and lub Goods, 2nd ed. ambrdge Unversty Press, ambrdge. Farrell, J., Maskn, E., Renegotaton n repeated games. Games and Economc Behavor 11, Fredman, J.W., A non-cooperatve outcome for supergames. Revew of Economc Studes 38, Green, E.J., oncooperatve prce takng n large dynamc markets. Journal of Economc Theory 22, Green, E.J., Porter, R.H., oncooperatve colluson under mperfect prce nformaton. Econometrca 52, Lambson, V.E., Self-enforcng colluson n large dynamc markets. Journal of Economc Theory 34, Lambson, V.E., ynamc behavour n large markets for dfferentated products. Revew of Economc Studes 54, Mcutcheon, B., o meetngs n smoke flled rooms facltate colluson?. Journal of Poltcal Economy 105, McGure, M.., Group sze, group homogenety, and the aggregate provson of a pure publc good under ournot behavor. Publc hoce 18, McMllan, J., Indvdual ncentves n the supply of publc nputs. Journal of Publc Economcs 12, Olson, M., The Logc of ollectve Acton. Harvard Unversty Press, ambrdge, MA.

14 134 P. Pecorno / Journal of Publc Economcs 72 (1999) Pecorno, P., 1998a. Is there a free-rder problem n lobbyng? Endogenous tarffs, trgger strateges and the number of frms. Amercan Economc Revew 88, Pecorno, P., 1998b. Market structure, tarff lobbyng and the free-rder problem. Manuscrpt, Unversty of Alabama. Reece, W.S., Zeschang, K., onsstent estmaton of the mpact of tax deductblty on the level of chartable contrbutons. Econometrca 53, Sandler, T., ollectve Acton. Unversty of Mchgan Press, Ann Arbor, MI. Shapro,., Theores of olgopoly behavor. In: Schmalensee, R., Wllg, R.. (Eds.), Handbook of Industral Organzaton, Vol. 1. orth-holland, Amsterdam.