American Law & Economics Association Annual Meetings

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1 Amercan aw & Economcs Assocaton Annual Meetngs Year 2008 Paper 32 By-Product obbyng: Was Stgler Rght? Paul Pecorno Unversty of Alabama Ths workng paper ste s hosted by The Berkeley Electronc Press (bepress) and may not be commercally reproduced wthout the publsher s permsson. Copyrght c 2008 by the author.

2 December 2007 By-product obbyng: Was Stgler Rght? Paul Pecorno Department of Economcs, Fnance and egal Studes Box Unversty of Alabama Tuscaloosa, A E-Mal: ppecorn@cba.ua.edu Abstract Olson (965) argues that some large groups can overcome the free-rder problem through byproduct lobbyng. The by-product frm sells a prvate good to potental members of the nterest group and fnances lobbyng wth ts profts. George Stgler (974) argued that by-product lobbyng frms cannot survve competton wth for-proft frms, snce ths would compete away monopoly rents, leavng the frm unable to lobby. Pecorno (200) showed that by-product frms can survve n a monopolstcally compettve market structure f they use ther profts to provde a pure publc good. In ths paper, I show that by-product lobbyng wll fal f the publc good exhbts any degree of rvalry and the group sze s suffcently large. Ths suggests that Stgler s rght n the crcumstances under whch by-product lobbyng s most relevant. The results of ths paper have mportant mplcatons for the lterature on green goods and corporate socal responsblty. JE classfcaton: D7; H4 Keywords: By-product lobbyng; free-rder problem; Collectve acton Hosted by The Berkeley Electronc Press

3 . Introducton In The ogc of Collectve Acton, Mancur Olson argues that large groups are frequently unable to provde themselves wth publc goods. One mechansm whch may overcome the freerder problem nherent n publc good provson s by-product lobbyng. A by-product lobbyng frm s able to nduce potental members of an nterest group to jon ther lobbyng organzaton by sellng prvate goods to ts members and lobbyng the government wth the resultng profts. Consder the example of the Amercan Assocaton of Retred Persons (AARP). In return for ther dues, members receve a magazne and group dscounts. The AARP s then able to lobby the government wth the profts t makes on the sale of membershps. Closely related s the dea of corporate socal responsblty (Baron 200 and Besley and Ghatak 2007). A specfc example of corporate socal responsblty s green goods (Kotchen, 2006) under whch the profts from the sale of a prvate good are used to purchase envronmental amentes whch are a publc good. 2 George Stgler (974) crtczed the by-product theory by argung that competton from frms whch do not lobby would elmnate monopoly rents and leave the by-product frm unable to lobby. Pecorno (200) showed that by-product lobbyng frms could survve the free entry of for-proft frms n a monopolstcally compettve market structure. In partcular, the by-product frm s able to provde a hgher level of the publc good than would be obtaned through drect contrbutons n a sngle shot game. Importantly, n ths model the by-product frm uses ts profts to fnance provson of a pure publc good. Whle he does not use the term by-product lobbyng, t should be noted that the dscusson of ths phenomenon by Moore (96: 3-6) predates Olson. 2 Mathematcally, there s no dfference between a green good, a good provded by corporate socal responsblty or a good provded by by-product lobbyng. I am usng the term by-product lobbyng to relate the analyss to Olson s (965) dscusson of ths ssue.

4 2 In ths paper I consder by-product lobbyng where the publc good provded by the byproduct frm may exhbt a degree of rvalry. If the publc good exhbts even a small degree of rvalry, then n a suffcently large group the level of provson of the publc good wll approach zero. Ths s sgnfcant, because t suggests that by-product lobbyng breaks down under the very condtons for whch t s most relevant. These condtons are that the group be large and that the publc good dsplay some degree of rvalry. As mentoned above, by-product lobbyng s explctly posted as a way n whch large groups can overcome the free-rder problem. One reason why the free-rder problem s worse n large groups s that the stakes per person tend to be small (Olson 965: 46-52), whch mples that most publc goods do n fact exhbt a postve degree of rvalry. Thus Stgler s rght under precsely the condtons for whch by-product lobbyng s most relevant. The framework n ths paper s smlar to that provded n Pecorno (200), but there are some mportant dfferences. Pecorno (200) s a partal equlbrum analyss of monopolstc competton, whle the analyss here s a general equlbrum analyss based on the Krugman (980) model. In Pecorno (200) there s a pure publc good whch provdes the beneft G(S), where G s a concave functon of contrbutons S. In ths paper, the per person beneft from the publc good s G(S)/, where 0 and the populaton sze s denoted by. 3 If = 0, we have a pure publc good and f =, the good s fully rval. For ntermedate values of, the good n queston exhbts some degree of rvalry, but s less than fully rval. As demonstrated below, f > 0, then publc good provson by the by-product frm s monotoncally decreasng n the populaton sze. In addton, provson wll fall towards 0 as the group sze grows large. 3 In Secton 5 of the paper, I consder modelng rvalry n a more general way. Hosted by The Berkeley Electronc Press

5 3 Thus, Pecorno s (200) fndng that by-product lobbyng frms can successfully provde publc goods s contngent on the good n queston beng a pure publc good. By-product lobbyng fals under exactly the crcumstances for whch t s most relevant: group sze s large and the publc good exhbts a postve degree of rvalry. Ths s true, even when the degree of rvalry s small. Moreover, the ndvdual losses from the falure of byproduct lobbyng can, under some crcumstances, be qute large. 2. Other Related Works Posnett and Sandler (986) show that a charty n a perfectly compettve goods market can charge a premum for a prvate good whch reflects the prvate valuaton of chartable actvty, a publc good. In a perfectly compettve framework, a by-product lobbyng frm can generally only replcate the outcome assocated wth prvate contrbutons n a sngle shot noncooperatve game. 4 The by-product lobbyng mechansm n Pecorno (200) bears a resemblance to Morgan s (2000) lottery mechansm. In Pecorno (200), consumers have an ncreased wllngness to pay for a prvate good because they know that the proft on that good s used to fnance a publc good whch they value. Smlarly n Morgan, consumers are wllng to pay more for a lottery tcket, because they know that profts on the sale are used to fnance a publc good. 5 In the papers cted above there s no technologcal advantage (e.g., economes of scope) to havng a frm provde both a prvate good and a publc good. There are also no warm glow effects enjoyed by consumers who buy from a by-product frm. Thus the analyss here s dstnct from the related lterature on mpure publc goods n whch the technology of producton (or preferences,.e., the warm glow effect) leads a publc good and a prvate good to be jontly 4 On ths pont, also see Besley and Ghatak (2007).

6 4 suppled. 6 However, there s a lterature on by-product lobbyng whch ncorporates these assumptons. For example, Bagnol and Watts (2003) analyze by-product lobbyng assumng that consumers receve a warm glow beneft by makng a purchase from a socally responsble frm. They analyze a varety of market structures and whle publc goods are frequently underprovded, there are some stuatons n whch they are provded at an neffcently hgh level. By contrast, Kotchen (2006) analyzes green goods whle allowng a technologcal advantage for frms whch jontly provde the prvate good and publc good. The market for prvate goods s perfectly compettve. In contrast wth Bagnol and Watts, Kotchen also allows for drect contrbutons to the publc good as an alternatve to the jont purchase of the publc and prvate good. Among other results, he fnds an ambguous welfare effect arsng from the presence of by-product frms relatve to a stuaton where only drect prvate contrbutons to the publc good are avalable. In contrast to the papers above, Mayer (2002) takes the monopoly poston of the byproduct frm as gven and analyzes the behavor of ths frm. When prce dscrmnaton s not possble, the by-product frm faces a trade-off between two dstortons. If t prces at margnal cost, the frm can effcently provde prvate nputs to members of the ndustry assocaton, but t cannot lobby. If t charges a prce above margnal cost, t wll have funds wth whch to lobby, but there s an neffcent provson of the prvate nput. When prce dscrmnaton s possble, ths problem s allevated. In what follows, I smply assume that the by-product frm acts to maxmze ts own profts. In a suffcently large group, these profts go to zero as long as the publc good provded by the frm exhbts some degree of rvalry. If the by-product lobbyng frm cannot earn postve profts at the proft maxmzng prce, then clearly t cannot earn postve 5 Another related work s Engers and McManus (2007) who analyze charty auctons. Hosted by The Berkeley Electronc Press

7 5 profts at any lower prce that t mght set out of deference to the consumer nterests of the members of the nterest group. 3. The Model In the model whch follows, there s a monopolstcally compettve market structure for the prvate good, where one of frms n the ndustry lobbes wth ts profts to provde a publc good. The model of monopolstc competton s based on Krugman (980). 3.. Consumers Consumers are dentcal. Ther preferences reflect a love of varety for goods produced wthn the ndustry class and are gven by n U j = C θ j, () = where the ndex j dentfes the consumer, the ndex dentfes a varety wthn the commodty class and 0 < θ <. 7 The number of varetes n avalable to consumers s determned endogenously. I assume that the publc good n queston s an ntermedate nput nto producton whch enhances ndvdual productvty. 8 In partcular, ncreases n the level of the publc good ncrease an ndvdual s effectve labor supply above. The publc good can be thought of as nfrastructure, where ths nfrastructure may be subject to crowdng. 9 Thus, the per person beneft 6 For an overvew of the lterature whch addresses these ssues, see the work of Sandler (992) and Cornes and Sandler (996). Also see Andreon (989, 990). 7 In Secton 5 of the paper, I wll show that the results of the paper wll hold for more general preferences than those presented n (). 8 For a dscusson of publc goods as ntermedate nputs nto producton, see Sandler (992: 36-43). 9 The results whch follow wll go through even more strongly f the publc good s a consumpton good. Ths s dscussed n Secton 5.

8 6 of the nfrastructure may be decreasng n the total number of workers,. The total populaton sze s also assumed to equal. The benefts of the publc good are nonexcludable. The effectve labor supply of ndvdual j s Ej G( I ) = +, (2) where G( I ) reflects the output of the nfrastructure, s a coeffcent whch determnes the magntude of the crowdng effect arsng from ncreases n, and I s labor devoted to provdng nfrastructure. When = 0, the nfrastructure s a pure publc good as there s no crowdng. When =, the good s fully rval. The functon G has the usual propertes: G > 0 and G < 0. In addton, I assume that G ( I ) s fnte for I > 0 and for I suffcently large G ( I ) <. Ths last condton ensures that at an nteror equlbrum, there wll be a fnte level of provson for all values of. The consumer earns the wage w and has a budget constrant gven by n G( + I ) w P C = 0. (3) = Each frm produces a sngle varety of the good whch s unque. The producton technology s dentcal across goods and s gven by l = α + β, for X > 0 and X l = 0, for X = 0, (4) where X s output by frm, l s labor employed by frm, α s a fxed labor nput and β s the margnal labor nput requred to produce one more unt of the good. The parameter α leads to a Hosted by The Berkeley Electronc Press

9 7 fxed, but avodable cost, as ths cost s not ncurred f output s 0. The producton technology n (4) leads to ncreasng returns to scale. Each frm takes the wage rate w as gven. The technology n (4) leads to a fxed cost of αw and a constant margnal cost of βw. The goods = 2,... n, are produced by conventonal forproft frms. Good s produced by a by-product lobbyng frm. Ths frm uses the profts from the sale of good to hre labor at the wage w to produce the publc good. Thus we can express output of the publc good as G(Π /w), where Π denote profts of frm. Consumers recognze that when they make a purchase from frm, that ts profts wll rse by the dfference between prce and margnal cost whch equals P - βw. The by-product frm contrbutes ths margnal proft towards provson of the publc good and as a result, P /w - β unts of labor can be hred to produce the publc good. Ths ncreases the consumer s effectve labor supply by (P /w - β)g (Π /w)/ unts and the consumer s ncome by (P - wβ)g (Π /w)/. Consumers recognze ths beneft when makng a purchase from the byproduct frm and ths s reflected n the frst order condtons whch follow. Consumers act to maxmze ther utlty n () subject to both the budget constrant (3) and the relatonshp between ther purchase of commodty and the supply of the publc good. Equaton (5) s derved from the frst order condton wth respect to goods other than good, whle equaton (6) s derved from the frst order condton wth respect to good. θ λ j C j = P, = 2,.,n. (5) θ θ λ j G' ( Π / w) C j = P [ P βw]. (6) θ

10 8 Note that λ j s a the lagrangan multpler for person j and reflects the margnal utlty of ncome. If frm were an ordnary for-proft frm, the term n parentheses would smply be P, but because t s a by-product frm, we have the term P P βw][ G' ( Π / w) / ]. The dfference [ between prce and margnal cost s [ P βw], and the consumer recognzes that when he purchases good that contrbutons towards the publc good wll ncrease by ths amount. The second term, G'( Π / w) / ] s the margnal beneft of addtonal expendture on the publc [ good. Recall that ths arses from an ncrease n effectve labor supply. Snce all are dentcal, the values of consumpton and of the lagrangan multpler wll be dentcal across consumers. Thus n what follows, I wll wrte C to denote per person consumpton of commodty and λ to denote the common value of the margnal utlty of ncome. All frms have dentcal proft functons gven by Π = ( P βw) X αw. (7) Each frm chooses ts own prce to maxmze profts whle takng the prces of the other frms as gven. 0 Proft maxmzaton s subject to the frst order condton of the consumers gven n (5) and (6). As n Krugman (980), we assume that each frm takes λ, the margnal utlty of ncome as gven. The frst order condton for each frm s dπ dp = X dx ( P β w) = 0. (8) dp 0 I assume that the manager of the by-product frm s strongly commtted to provson of the publc good. By maxmzng profts she s maxmzng provson of the publc good. See the dscusson n Pecorno (200: ). Hosted by The Berkeley Electronc Press

11 9 The zero proft condton, whch apples for all the for-proft frms, may be expressed as P X α w βwx = 0, = 2,., n. (9) Snce the number of consumers s dentcal to the number of workers, we have X = C. (0) Usng (5), (8), (9) and (0), we can solve for prce, consumpton, and quantty produced for all goods other than the good produced by the by-product frm: βw P =, = 2,., n, (a) θ C αθ = β ( θ ), = 2,., n, (b) αθ X =, = 2,., n. (c) β ( θ ) The solutons n (a-c) are the same as those we would obtan n a model wthout by-product lobbyng. Usng equatons (6), (8), (9) and (0), we can solve for P, C, and X : G' ( Π / w) θ βw P = (2a) θ G' ( Π / w) C αθ = β ( θ ) (2b) αθ X = (2c) β ( θ ) As per usual, we can only solve for relatve prces. If both sdes of (a) are dvded by w, then we obtan the relatve prce of each good measured n unts of labor.

12 0 Usng equatons (7), (2a) and (2c), we can express profts for the by-product frm as G'( Π / w) Π = α G w (3) '( Π / w) Note that output and consumpton are the same as for goods produced by for-proft frms. Also, the prce set by the by-product frm exceeds the prce set by the for proft frm. Snce Π > 0, t exceeds the proft of the for proft frms, whch equals 0. These results match Result 2 parts (), () and () n Pecorno (200). Wll there be a postve level of profts such that (3) s satsfed? Suppose that G (0)/ <. The rght-hand sde of (3) s postve and the left-hand sde s 0 at Π = 0. As we rase Π above 0, the rght-hand sde monotoncally decreases, so there must be a unque postve value of Π such that the two sdes are equal. Smlarly, f G (0)/ >, the condtons on G guarantee that there s a fnte value of Π such that G / =. At ths pont, the left-hand sde of (3) s fnte and the rght-hand sde nfnte. As Π ncreases above ths value, the rght-hand sde wll monotoncally decrease mplyng the exstence of a unque value of Π such that (3) holds. Thus, we have a unque postve value of Π that solves (3). In addton, when profts are maxmzed the followng condton holds: G (Π Max /w)/ <. (4) Ths condton wll prove useful n Secton 4 when comparng the level of provson under byproduct lobbyng to that acheved under drect contrbutons. Hosted by The Berkeley Electronc Press

13 The demand for effectve labor unts must equal the effectve labor supply. The total labor demanded n the economy ncludes labor demanded to produce the prvate goods (ncludng good ) plus labor used to produce the publc good. et G denote the total labor demand for prvate good producton (ncludng good ) and I denote labor demanded to produce the publc good, both measured n effectve labor unts. Then we must have G + = = + G( Π / w) / ). (5) I E ( Snce I = Π / w, from (3) we have G'( I ) =Π = α G / w. (6) '( I I ) Substtute (6) nto (5) to solve for G. It s also true that n G = l. Usng (4), (c) and (2c), = we can solve for the equlbrum number of frms: θ G ( ) n=. (7) α From equaton (6), we can derve the effect of an ncrease n populaton on the quantty of labor hred to provde the publc good: di α G'( I ) = 0. (8) 2 d αg''( ) ( G'( )) I I

14 2 If = 0, d I / d= 0 so that the level of provson does not vary wth the populaton sze. Thus for a pure publc good, by-product lobbyng does not vary wth. 2 When > 0, d I / d< 0. Snce labor devoted to producton of the publc good falls, the per person provson of the publc good G( I )/ declnes even more rapdly wth when > 0. Also note that d I / d< 0 s equvalent to d ( Π / w) / d< 0. Ths leads to Result : Result : Under by-product lobbyng, f the publc good exhbts any degree of rvalry ( > 0), real frm profts Π /w and the per person provson of the publc good are both monotoncally decreasng n group sze. We can also use (6) to obtan a lmt result. Frst note that f = 0, Π /w does not vary wth so that the level of provson s both postve and constant n the lmt as grows large. If > 0 then as the populaton grows large,. Ths mples that Π /w 0 as. To see why, suppose ths were not true and that Π /w takes on a fnte postve value n the lmt. Snce G (Π /w) s fnte for Π /w > 0 ths mples that G (Π /w)/ 0 as. From (6), ths mples that Π /w 0 contradctng the assumpton that Π /w > 0. Thus, as, Π /w 0 whch mples that per person provson of the publc good G(Π /w)/ 0. Ths s summarzed as Result 2: Result 2: Under by-product lobbyng, f the publc good exhbts any degree of rvalry ( > 0), real frm profts Π /w and the per person provson of the publc good both approach 0 as group sze. 2 Ths matches Result 4 n Pecorno (200). Hosted by The Berkeley Electronc Press

15 3 Results and 2 ndcate that by-product lobbyng cannot overcome the fundamental problem of provdng publc goods n a large group. If the publc good s pure ( = 0), then by-product lobbyng can succeed n a large group, and as we wll see n the next secton, the level of provson under by-product lobbyng wll exceed that obtaned va drect contrbutons. However, by-product lobbyng s posted as a necessary response precsely because stakes per person tend to fall n a large group, and our results show that when > 0, by-product lobbyng performs poorly n a large group. Ths s true even when the good exhbts only a small degree of rvalry. It s obvous that f a publc good exhbts a degree of rvalry, the premum an ndvdual s wllng to pay a by-product frm for ther product wll decrease wth group sze. However, n a larger group there are more ndvduals, so t s not obvous that the profts of the by-product frm wll decrease monotoncally or approach 0 n a large group. However, that s exactly what Results and 2 mply, and ths s true even f the good n queston s close to beng a pure publc good (.e., > 0, but small). 4. A Comparson to the Drect and the Optmal evel of Contrbutons Wth the apparatus currently n place, we can easly examne what would happen f there were only drect contrbutons to the publc good and compare ths level of provson to that attaned under by-product lobbyng. Assume that all n frms are standard for proft frms and that each ndvdual may make a drect contrbuton s j 0 to the publc good. The consumer s budget constrant s now w n ( + G( S / w) / ) = s j + = PC j, (9)

16 4 where S = s j j=. Frms have access to the same technology as before. It s straghtforward to show that n ths model, P, C, and X are gven by equatons (a-c) for =,, n. If we assume an nteror soluton, then the frst order condton wth respect to provson of the publc good may be expressed as G' ( S / w) =. (20) If > 0, then we have d( S / w) d= / G''( S / w) < 0 ; drect contrbutons are monotoncally decreasng n group sze. If G (0) =, then as grows large, (9) can only hold f S/w approaches 0. (Recall that G (S/w) s fnte for S/w > 0). Alternatvely, suppose that G (0) s fnte. Then above some fnte value of there wll be a corner soluton wth S = 0 (because above some fnte, we have G (0)/ < ). Thus, when we have drect contrbutons to the publc good, we obtan analogs to Results and 2. We know from equaton (4) that under by-product lobbyng G (Π Max /w)/ <. By contrast, under drect contrbutons from (20) we have G '( S / w) / =. Ths mples (due to the curvature propertes of the G functon) that there s a greater level of publc good provson under by-product lobbyng than under drect contrbutons. Thus, when drect contrbutons and byproduct lobbyng are both allowed, we wll only observe by-product lobbyng. The reason s that proft maxmzaton by the by-product frm ensures that G (Π Max /w)/ <. Under ths condton, no ndvdual wll wsh to make a drect contrbuton. Ths s summarzed as Result 3: Result 3: () The level of provson of the publc good under by-product lobbyng exceeds that acheved under drect contrbutons. () When drect contrbutons and by- Hosted by The Berkeley Electronc Press

17 5 product lobbyng are allowed smultaneously, no drect contrbutons wll be made. Result 3 generalzes a result n Pecorno (200: 389) by allowng for a degree of rvalry n the provson of the publc good. Ths result ndcates that by-product lobbyng s superor to drect contrbutons as a method for provdng the publc good. However, Result 3 s strongly tempered by Results and 2: Provson s decreasng n group sze and approaches 0 n a suffcently large group f > 0.. It s straghtforward to show that an optmal provson of the publc good leads to the condton 3 G' ( I ) =. (2) From (2) we have d d O I o ( ) G'( I ) 0, (22) G''( ) = O I where the o superscrpt ndcates the optmal level of I and the strct nequalty holds f <. When =, we have a fully rval good and the optmal level of provson s a constant. For any lower degree of rvalry, the optmal level of provson ncreases n group sze. From Result, we know that the level of provson under by-product lobbyng s decreasng n group sze (or constant when = 0). Ths leads to Result 4: 3 Equaton (2) may be obtaned by optmzng socal welfare wth respect to the publc good whle takng the dstortons assocated wth monopolstc competton as gven. For-proft frms need to earn zero profts n equlbrum. Thus any attempt to mprove socal welfare va a reducton n prce of prvately provded goods would requre some compensatng subsdy to the frms. Such an nterventon seems unrealstc. Thus, t s reasonable to optmze wth respect to the publc good, whle takng the dstortons from monopolstc competton as gven.

18 6 Result 4: As the group sze ncreases, there s a growng dvergence between the optmal level of I and the level provded under by-product lobbyng. Ths s true for all values of between (and ncludng) 0 and. Whle Result 4 suggests for both by-product lobbyng and drect contrbutons that the loss from under provson of the publc good s greater n large groups, ths queston requres addtonal analyss. In partcular we need to analyze how, under an optmal level of provson, the per person provson of the publc good G( I )/ evolves. Usng (22) we have O G( I ) d d ( ) O 2 ( G' ( ) ) I = + O G( I ) G O G' '( ) I ''( O ) I. (23) Snce the numerator s of ndetermnate sgn, we are unable to say n general whether, under an optmal level of provson, the per person provson of the publc good s rsng or fallng. If = 0 (a pure publc good) the dervatve n (23) s postve, meanng that the optmal per person level of provson of the publc good ncreases wth an ncrease n. Under by-product lobbyng when = 0, the per person level of provson s constant so the ndvdual loss due to under provson s ncreasng n. Conversely, f = (a fully rval good), the dervatve n (23) s negatve mplyng that per person consumpton of the publc good decreases n group sze, even when t s optmally provded. Ths s summarzed as Result 5: Result 5: Under the optmal level of provson, the per person level of provson G( O I )/ ncreases n group sze for =0 and decreases n group sze for =. For ntermedate values of, the effect of ncreases n group sze on G( O I )/ s ndetermnate. Hosted by The Berkeley Electronc Press

19 7 Consder a smple parametrc example, where G( I ) = level of provson we have γ I. Usng (2), under the optmal γ γ γ γ G( O I ) = γ. (24) If γ > > 0, then the optmal per person level of provson grows wthout bound as ncreases, whle from Result 2 we know that the level of provson under by-product lobbyng goes to 0. In ths case, the ndvdual losses due to under provson ncrease monotoncally n and grow wthout bound. Conversely, f γ <, then G( O I )/ goes to 0 as group sze becomes large. As a result, the per person loss due to under provson must also go to 0 n a large group. Thus, the ndvdual consequences from the falure of by-product lobbyng are not very sgnfcant n ths case. 5. Dscusson In ths secton, I wll dscuss the robustness of the results as well as ther applcablty to other contexts. 5.. Modelng Rvalry In general, one can thnk of modelng the degree of rvalry as G( I )/F(), where F () 0. In ths paper, I have used the specfc form F() =. If F () > 0, then Results and 2 of the paper wll contnue to hold for any functon F such that lm F( ) =. If F () > 0, but lm F( ) = v, where v s fnte, then Result wll hold, but Result 2 wll not. In ths case we

20 8 have a pure publc good n the lmt. Thus, what s requred for Result 2 s that the good be rval n the lmt Preferences Over the Prvate Goods In ths subsecton, I show that the results of the model generalze beyond the specfc utlty functon n equaton (). In partcular, I wll show that the results contnue to hold f prvate goods enter the utlty functon n a symmetrc fashon. Ths ensures that the demand curves faced by all frms are dentcal. To see how ths works, frst note that Pecorno (200, Result 2) has shown that for a gven demand curve, a proft maxmzng by-product frm wll produce the same level of output as a for-proft frm. 4 Denote the common level of output obtaned under proft maxmzaton as X 0, and denote the prce charged by the for-proft frms as P 0. By the zero proft condton n (9) we have α w P0 + βw. (25) X = 0 et P be the prce charged by the by-product frm. When the by-product frm sells X 0 unts of the commodty, the wllngness to pay by consumers exceeds P 0, because the dfference between prce and margnal cost (= P - βw) s used to provde the publc good. The total beneft of byproduct lobbyng to the consumer s [P - βw]g (Π /w)/. Thus, we have G' ( Π / w) P = P0 + [ P βw]. (26) Profts for the by-product frm may be expressed as 4 The analyss n Secton 3 provdes an example where ths result holds. Hosted by The Berkeley Electronc Press

21 9 Π ) 0 = ( P βw X αw. (27) From (26) we obtan βwg'( Π / w) P0 P =. (28) G'( Π / w) Use (28) and (25) n (27) and smplfy to obtan G'( I ) Π = α G / w, (29) '( I ) whch s the same as the expresson for profts n (6). Thus, all the results of the paper contnue to hold as long as the prvate consumpton goods enter the utlty functon n a symmetrc fashon. If, for example, we used the preferences n Krugman (979), X 0 would be an ncreasng functon of. 5 What the analyss above has shown, however, s that the profts for the byproduct frm are (holdng constant) ndependent of the equlbrum level of X 0. Thus, when ncreases, Π decreases va the mechansms dentfed n Secton 3. The fact that X 0 ncreases has no addtonal effect on profts. 6 Combned wth the other assumptons of the model, the assumpton that goods enter the utlty functon n a symmetrc fashon s suffcent to ensure that the results presented n ths paper hold, but t s not necessary. I nterpreted equaton (25) as the zero proft condton for for- 5 Krugman (979) uses Dxt and Stgltz (977) preferences. 6 Ths echoes some dscusson n Pecorno (200: ).

22 20 proft frms, but I can also nterpret t to mean the followng: If the by-product frm behaved lke a for-proft frm, t would earn 0 profts. If ths s true, then regardless of how goods enter the utlty functon, we obtan equaton (29). Thus, the key condton for the results of the paper does not relate to preferences for prvate goods per se, but rather to the zero proft condton whch apples to for-proft frms Modelng the Publc Good as a Consumpton Good I have modeled the publc good as an ntermedate nput nto producton rather than as a consumpton good. One could model the publc good as a consumpton good wth a utlty functon of the form 7 U j = n = C θ j G( S / w) +. (30) Note that under ths utlty functon, there wll be a postve ncome effect on the demand for publc goods. Under the preferences n (30), the model provdes even stronger results because Results and 2 apply even for a pure publc good. The reason s that ncreases n lead to a wder array of prvate goods avalable for consumpton. Ths ncreases λ the margnal utlty of ncome. 8 Under by-product lobbyng, consumers contrbute to the publc good va ther wllngness to pay a premum for the good produced by the by-product frm. When λ ncreases, the opportunty cost of contrbutng to the publc good n ths fashon also ncreases and ths leads by-product lobbyng to become less effectve n a large group, even when t provdes a pure 7 In the nterest of space, I am not presentng the worked out results of ths verson of the model. However, I have worked ths out, and the dscusson below reflects the results of that formal analyss. 8 Ths can be verfed by solvng for λ. To do ths, substtute equatons (a) and (b) nto (5) and solve. Hosted by The Berkeley Electronc Press

23 2 publc good. 9 Ths problem does not occur when there s a publc ntermedate good. In ths case, ncreases n λ rase the margnal cost and the margnal beneft of contrbutng by equal amounts. Whle Result 2 would apply to a pure publc consumpton good under the specfcaton n (30), ths depends upon the property that the margnal utlty at a 0 level of consumpton s for the prvate good. As a result, λ ncreases wthout bound n. If the margnal utlty of consumpton of prvate goods at 0 s fnte, then λ s bounded, and Result apples to pure publc consumpton goods, but Result 2 does not. Further, f we thnk of by-product lobbyng n a context where the prvate goods consttute one ndustry class among many others, then t mght be reasonable to treat the margnal utlty of ncome as beng exogenous to that the sector of the economy. If λ s treated as a constant, then we obtan essentally dentcal results, regardless of whether or not the publc good s a consumpton good or an ntermedate nput nto producton. More partcularly, Results and 2 apply for > 0, but not for the case of a pure publc good consumpton good. These last two caveats suggest that the dfferences (wth respect to the man results of ths paper) between publc goods whch are ntermedate nputs nto producton and those whch are consumpton goods should not be overstated Applcaton to Other Models The results presented here are drectly relevant for Kotchen s (2006) green goods model. 20 In Kotchen, the by-product frm operates n a perfectly compettve market, but he allows for a possble technologcal advantage possessed by the by-product frm n provdng the publc good 9 Ths ssue s addressed n detal by Pecorno (2006) for a model wth drect contrbutons only. The same mechansm apples n the model wth by-product lobbyng.

24 22 (.e., economes of scope). As n ths paper, n Kotchen s model the prce consumers are wllng to pay to the by-product frm s a functon of how much they value the publc good. I have assumed that the by-product frm has access to the same publc good technology as ndvduals wshng to make drect contrbutons. However, the results of the paper wll contnue to hold f I allow for a technologcal advantage for the by-product frm. 2 Thus, f the green goods exhbt rvalry, Results and 2 of ths paper wll apply. Of course, some envronmental goods may be pure publc goods. When ths s true, the results of Kotchen stand unmodfed. What are the mplcatons of ths paper for the Bagnol and Watts (2003) model n whch consumers experence warm glow effects when they make a purchase from the by-product frm? Andreon (2007) has shown that warm glow effects are subject to congeston. Thus, the results of ths paper may be relevant for the Bagnol and Watts (2003) model, but t wll requre formal modelng to make more defntve statements. 6. Concluson Ths paper demonstrates that by-product lobbyng wll fal n a suffcently large group f the publc good n queston exhbts even a small degree of rvalry. 22 Thus, Stgler s (974) contenton that by-product lobbyng frms are unable to compete wth for-proft frms s correct under exactly the crcumstance for whch by-product lobbyng s most relevant: a large group sze and a publc good that exhbts some degree of rvalry. I also fnd that the level of the publc good provded va by-product lobbyng s monotoncally decreasng n the group sze whenever the publc good exhbts a postve degree of rvalry. 20 They are also relevant for Besley and Ghatak s (2007) model of corporate socal responsblty. If group sze s large, socally responsble frms wll be unable to provde publc goods whch exhbt any degree of rvalry. 2 Ths could be done by multplyng G( ) n equaton (2) by a parameter a >. 22 Techncally, we requre that the publc good reman rval n the lmt as group sze grows large. Hosted by The Berkeley Electronc Press

25 23 The work of Borcherdng and Deacon (972) and Bergstrom and Goodman (973) ndcates that publcally provded goods do n fact exhbt a hgh degree of rvalry. 23 Thus, the results of ths paper are qute relevant and suggest mportant lmtatons upon the doman for whch by-product lobbyng (or corporate socal responsblty) can succeed n provdng publc goods. Of course certan goods, partcularly certan envronmental goods, may truly be pure publc goods. If ths s so, then by-product lobbyng can succeed n provdng these goods, even for a large group. In addton, t has long been recognzed that by-product frms can succeed f they attan a poston of true monopoly power (Sandler, 992: 60). 23 Publcally provded goods such as polce and fre protecton clearly exhbt a hgh degree of rvalry.

26 24 References Andreon, J., 989. "Gvng wth Impure Altrusm: Applcatons to Charty and Rcardan Equvalence. Journal of Poltcal Economy 97, Andreon, J., 990. Impure Altrusm and Donatons to Publc Goods: A Theory of Warm Glow Gvng. Economc Journal 00: Andreon, J., Gvng Gfts to Groups: How Altrusm Depends on the Number of Recpents. Journal of Publc Economcs 9, Bagnol, M., Watts, S.G., Sellng to Socally Responsble Consumers: Competton, and the Prvate Provson of Publc Goods. Journal of Economcs and Management Strategy 2, Baron, D. P., 200. Prvate Poltcs, Corporate Socal Responsblty, and Integrated Strategy. Journal of Economcs and Management Strategy 0, Bergstrom, T. C. and Goodman, R. P., 973. Prvate Demand for Publc Goods. Amercan Economc Revew 63, Besley, T., Ghatak, M., Retalng Publc Goods: The Economcs of Corporate Socal Responsblty. Journal of Publc Economcs 9, Borcherdng, T. E. and Deacon, R. T., 972. The Demand for Servces of Non-Federal Governments. Amercan Economc Revew 62, Cornes, R., Sandler, T., 996. The Theory of Externaltes, Publc Goods, and Club Goods, 2nd ed. Cambrdge Unversty Press, Cambrdge. Dxt, A. K., Stgltz, J. E., 977. Monopolstc Competton and Optmum Product Dversty. Amercan Economc Revew 67, Engers, M., McManus, B., Charty Auctons. Internatonal Economc Revew 48, Hosted by The Berkeley Electronc Press

27 25 Kotchen, M. J., Green Markets and the Prvate Provson of Publc Goods. Journal of Poltcal Economy 4, Krugman, P. R., 979. Increasng Returns, Monopolstc Competton, and Internatonal Trade. Journal of Internatonal Economcs 9, Krugman, P. R., 980. Scale Economes, Product Dfferentaton, and the Pattern of Trade. Amercan Economc Revew 70, Mayer, W., On the Effcency of By-product obbyng. Publc Choce 2, Morgan, J., Fnancng Publc Goods by Means of otteres. Revew of Economc Studes 67, Moore, T. G., 96. The Purpose of censng. Journal of aw and Economcs 4, Olson, M., 965. The ogc of Collectve Acton. Harvard Unversty Press, Cambrdge, MA. Pecorno, P., 200. Can By-product obbyng Frms Compete? Journal of Publc Economcs 82, Pecorno, P., Monopolstc Competton, Group Sze and Publc Good Provson. Unversty of Alabama Workng Paper No Posnett, J., Sandler, T., 986. Jont Supply and the Fnance of Chartable Actvty. Publc Fnance Quarterly 4, Sandler, T., 992. Collectve Acton. Unversty of Mchgan Press, Ann Arbor. Stgler, G. J., 974. Free Rders and Collectve Acton: An Appendx to Theores of Economc Regulaton. Bell Journal of Economcs and Management Scence 5,