Special Interest Politics: Contribution Schedules versus Nash Bargaining

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3 Specal Interest Poltcs: Contrbuton Schedules versus Nash Barganng Achm Voss School of Economcs and Socal Scences, Unversty of Hamburg, Von-Melle-Park 5, Hamburg, Germany. Tel.: E-mal: Mark Schopf Correspondng author. Faculty of Busness Admnstraton and Economcs, Unversty of Paderborn, Warburger Str. 100, Paderborn, Germany. Tel.: E-mal: Abstract The artcle compares two models of lobby nfluence on polcy choce: The Grossman and Helpman (1994) contrbuton-schedule model and a negotaton between the lobbes and the government summarzed by a Nash-barganng functon. The lterature uses the models nterchangeably because they mply the same equlbrum polcy. We derve under whch condtons they lead to the same payments, equlbrum utltes, and total effcency. They concde under partcular assumptons about barganng power and dsagreement utlty. Keywords: Nash Barganng, Common-Agency Model, Lobbyng JEL: C71, C78, D72 1. Introducton The poltcal common-agency model of Grossman and Helpman (1994) s the workhorse model of a large lterature on the nfluence of specal-nterest groups. It assumes that the government would lke to maxmze welfare, but t s wllng to devate from ths am f t receves contrbuton payments as a compensaton. Its counterparts are the organzed specal-nterest groups or lobbes. Before the government chooses polcy, they smultaneously confront t wth contrbuton schedules defnng payments as functons of the polcy. The government then chooses polcy takng these schedules nto account. The basc contrbuton-schedules model assumes that all agents have constant margnal utlty of money. Equlbrum polcy then maxmzes a weghted sum of welfare and profts of all sectors wth lobbes. Ths would also be the outcome of an alternatve polcy process: Nash barganng between the government and the lobbes. Whle Grossman and Helpman

4 2. The Models (2001, 247) dsmss the possblty of barganng wth all lobbes because the polcymaker would not wsh to be seen as openly peddlng her nfluence, some authors use the multlateral Nash-barganng model, statng that the two models of poltcal nteracton are equvalent see for nstance Dharmapala (1999), Gawande et al. (2009), Goldberg and Magg (1999) and Magg and Rodríguez-Clare (2000). These papers focus on the choce of polcy, however, whle not analyzng the equlbrum contrbuton payments. The contrbuton of the present artcle s to compare the two models equlbrum propertes wth respect to payments, utltes and effcency. A clear understandng of the dfferences between multlateral barganng and the common-agency settng s relevant for three reasons. Frstly, to verfy the clam of a part of the lterature that they are equvalent. Secondly, there are stuatons that may be descrbed by both models, though not necessarly n domestc poltcs. For example, dfferent organzatons from ndustral countres may smultaneously try to nfluence the polcy n a developng country, a queston we address n Schopf and Voss (2016). Thrdly, the models have mplcatons for the nterest groups preferences for allowng more or less effcent polcy; determnng whether these mplcatons are the same n both models s mportant for analyzng the choce of nsttutons and consttutons. The followng Secton 2.1 ntroduces the agents, Sectons 2.2 and 2.3 analyze the polcy-choce mechansms, and Secton 2.4 dscusses dfferent assumptons for behavor n case of dsagreement. Secton 3 compares the models and concludes. 2. The Models 2.1. The Agents Lobbes nfluence polcy by payng contrbutons to a government. Denotng the polcy vector by p = {p k } k K, the utlty of lobby L s a lnear combnaton of ts gross utlty W (p) and a cost of payng contrbutons c : V = W (p) b c b 0. (2.1a) Smlarly, the government s utlty G depends on welfare W(p) and contrbuton payments c = {c } : G = W(p) + a c a 0. (2.1b) All W (p) and W(p) are assumed to be contnuous and sngle-peaked n p wth dfferent maxmzng polcy vectors. Achm Voss and Mark Schopf 2/12

5 2. The Models Ths settng s dentcal to that of Grossman and Helpman (1994) except for mnor notatonal adjustments that ease the exposton later on, and for the fact that both the cost of payng and the government s valuaton of recevng contrbutons may be lobby-specfc. A natural nterpretaton for W (p) s gross proft of sector (suggestng b 1, and b > 1 f there are addtonal costs of collectng contrbutons), and W(p) would be gross aggregate welfare ncludng that of sectors wthout a lobby, W(p) W (p) Contrbuton-Schedules Equlbrum Grossman and Helpman (1994) derve the subgame-perfect equlbrum of a two-stage game. In the frst stage, the lobbes smultaneously and non-cooperatvely offer contrbuton schedules to the government, defnng payments as a functon of the polcy: c = C (p). Afterwards, the government chooses polcy so as to maxmze ts utlty, gven the contrbuton schedules. Lettng a superscrpt o denote equlbrum, we have p = argmax p W(p) + a C (p). (2.2) Lobbes cannot offer negatve contrbutons. For postve contrbutons, attenton s restrcted to truthful contrbuton schedules, n whch a lobby s margnal payment cost equals ts margnal utlty gan due to the polcy. Ths determnes each contrbuton schedule up to a constant B 0: b C (p) = max [ 0, W (p) B ] for L. (2.3) (2.2) and (2.3) mply that the equlbrum polcy maxmzes a weghted sum of welfare and gross utltes: p = argmax p a W(p) + W (p). (2.4) b Lobby s polcy weght a b equals the rato of the margnal utlty of the government of recevng the lobby s money to the lobby s margnal payment cost. Fnally, each contrbuton schedule must mnmze the lobby s payment cost b c subject to the constrant that the government s better off by acceptng t nstead of rejectng t and recevng no contrbutons from the lobby. Thus, lobby sets the government ndfferent between choosng the Achm Voss and Mark Schopf 3/12

6 2. The Models optmal polcy wthout the lobby: p = argmax p and the equlbrum polcy wth lobby, p : W(p) + Cj (p), (2.5) \ W(p ) + Cj (p ) = W(p ) + Cj (p ) for L. (2.6) \ Rearrangng (2.6) and substtutng (2.3) yelds C (p ) = 1 a W(p ) + \ W j (p ) W(p ) \ W j (p ) > 0 for L. (2.7) In equlbrum, each lobby pays what the government and the other lobbes lose by accommodatng that lobby. Substtutng (2.7) for all lobbes nto (2.1) yelds the equlbrum utltes: V = W (p ) + b a G = W(p ) L W(p ) + W j (p ) W(p ) W j (p ) for L, (2.8a) + ( L 1 ) W(p ) + W j (p ) W(p ) W j (p ) L L 1 b. j \ (2.8b) The term n square brackets n (2.8a), whch represents the lobby s gan of offerng a contrbuton schedule, s postve by (2.4). Each lobby s equlbrum utlty s the utlty t would have wthout offerng contrbutons, plus a share of the addtonal jont surplus due to ts cooperaton. Smlarly, the government s equlbrum utlty s the utlty t would have on average f one lobby dd not pay any contrbutons, plus the jont loss of the government and the other L 1 lobbes on average due to the partcpaton of the resdual lobby. Ths jont loss must be offset by L 1 lobbes. If there were just one lobby, t would just compensate the government for the welfare loss Nash Barganng Soluton In ths secton, we drop the noton of a smultaneous offerng of contrbuton schedules. Instead, the government and all lobbes meet and bargan. The outcome s determned by an asymmetrc Nash barganng soluton, whch mplements the polcy and the profle of Achm Voss and Mark Schopf 4/12

7 2. The Models contrbuton payments that maxmze the Nash product N(p, c). Usng a superscrpt n to denote the outcome of barganng, we have γ N(p, c) = W(p) + a c G d (p n, c n ) argmax N(p, c), p,c [ ] W (p) b c V d γ, (2.9a) (2.9b) where γ denotes the barganng weght of lobby and γ that of the government. V d and G d are the respectve utlty values n case of dsagreement (see below). The frst-order condtons for maxmzng (2.9) are γ W(p n )/ p k W(p n ) + c n j G + d γa W(p n ) + c n j G d γ j W j (p n )/ p k W j (p n ) c n j Vj d γ b W (p n ) b c n V d N(p n, c n ) = 0 for k K, (2.10a) N(p n, c n ) = 0 for L. Wth N(p n, c n ) > 0, rearrangng (2.10b) and substtutng nto (2.10a) yelds W(p n ) p k + so that the barganed polcy can be wrtten as (2.10b) W j (p n ) = 0 for k K, (2.11) p k p n argmax p a W(p) + W (p), (2.12) b whch s dentcal to p from (2.4). Solvng (2.10b) as a system of equatons defnng c n for L yelds c n = 1 b γ + \ γ j γ + γ j + 1 a γ γ + γ j [ ] W (p ) V d G d + \ V d j W(p ) \ W j (p ) for L. (2.13) Thus, each lobby pays a share of what t gans due to cooperaton plus a share of what the government and the other lobbes lose due to ts cooperaton. If ts barganng power s low (γ 0), t contrbutes all ts gans, f ts barganng power s hgh (γ ), t just compensates the others. Substtutng (2.13) for all lobbes nto (2.1) yelds the equlbrum Achm Voss and Mark Schopf 5/12

8 2. The Models utltes: γ V n = V d + b a γ + W(p ) + γ j G n = G d γ + γ + W(p ) + γ j W j (p ) G d W j (p ) G d V d j V d j. for L, (2.14a) (2.14b) The term n square brackets n (2.14) represents the total gans of cooperaton. Thus, each lobby s and the government s equlbrum utlty are the respectve dsagreement utltes plus a share of the total gans of cooperaton, weghted by ther relatve barganng powers. The dsagreement utltes are determned by the polcy that would be chosen and the contrbutons that would be pad n that case: G d = W(p d ) + a c d and V d b c d for L. Thus, the total gans of cooperaton become whch s postve by (2.4). 1 W(p ) The Dsagreement Polcy n the Nash Barganng Soluton = W (p d ) W j (p ) W(p d ) W j (p d ), (2.15) By (2.12), the equlbrum polcy p s defned ndependently of the dsagreement stuaton. However, we need some assumpton about the polcy n case of dsagreement, p d, n order to derve the equlbrum utltes and payments. In contrast to the contrbutonschedules equlbrum where we have a polcy p wthout each respectve lobby barganng s a collectve agreement. Thus, we need to know the polcy that the government would choose f the barganng n total broke down. Ths choce depends on the commtment possbltes and the barganng opportuntes n case of dsagreement. The smplest case s that n whch there s no barganng after dsagreement. Then, no lobby can nfluence the polcy so that all dsagreement contrbutons c d are zero. If the government cannot commt to a dsagreement polcy p d ex ante, t just maxmzes welfare ex post: p d = argmax W(p). (2.16) p Else, f commtment s possble, t chooses p d so as to maxmze ts equlbrum utlty 1 The gans of cooperaton are ndependent of the contrbuton payments n case of dsagreement. Ths would not be true, however, f dsagreement mpled the formaton of addtonal lobbes because then (2.15) would become W(p ) + W j (p ) W(p d ) W j (p d ) + j / L c d j 0. Achm Voss and Mark Schopf 6/12

9 2. The Models (2.14b) for G d = W(p d ) and V d = W (p d ): p d = argmax p γ W(p) W j (p). (2.17) γ j (2.16) and (2.17) concde f γ = 0; n both cases, equlbrum utlty of the government s just dsagreement welfare. If γ > 0 and the government can commt to a dsagreement polcy, t ncreases the gans of cooperaton and thus ts own equlbrum utlty by reducng the dsagreement profts of the lobbes. In contrast to the mmedate polcy choce after a breakdown mpled by (2.16) or (2.17), dsagreement may allow subsequent barganng. Naturally, any potental coalton must nclude the government. In the context of Nash barganng, such a central role for one player n subsequent coaltons s allowed by the models of Compte and Jehel (2010) and Burguet and Camnal (2012). 2 Compte and Jehel (2010) ntroduce the coaltonal Nash barganng soluton: If the grand coalton breaks down, subsequent coaltons can form. A subsequent coalton s credble f each member s equal share exceeds the equal share n the grand coalton. A player s barganng poston n the grand coalton s determned by the number of subsequent coaltons he could take part n and ther common product. The coaltonal Nash barganng has a soluton f and only f the number of credble subsequent coaltons s smaller than the number of players. For nstance, t may hold that n case of dsagreement the L 1 coaltons contanng the government and all lobbes but one are credble. Then, the government receves more than the equal share of the surplus n equlbrum and all lobbes receve less. Burguet and Camnal (2012) ntroduce the R-soluton for Nash barganng of three players. Should the trlateral negotaton fal, all players bargan smultaneously n three blateral negotatons. If one blateral negotaton s surplus s greater than the sum of the other two, t takes place wth probablty one. Else, all three blateral negotatons take place wth a postve probablty. Wth one government and two lobbes, there are only two possble blateral negotatons and the one wth the hgher surplus takes place wth probablty one. If the other blateral negotaton s surplus s very small, the government and the lobby t actually bargans wth share ther surplus. Else, the government threatens to bargan wth the other lobby and thus ncreases ts share n the blateral negotaton. 2 See Okada (2010) for an n-person Nash barganng approach where there s no comparably central player. Achm Voss and Mark Schopf 7/12

10 3. Dscusson 3. Dscusson We now compare the contrbuton-schedules equlbrum and the Nash barganng soluton. From (2.4) and (2.12), the equlbrum polcy s dentcal: p = p n. Thus, the approaches concde f the contrbuton payments and, thus, the equlbrum utltes concde: V V V n = V n and G = G n. They dffer by: [ = b C (p ) c n ] = 1 ( ) ( γa V γa V d γ b G G d) γ b γ + γ j 1 ] G G n = [ C (p ) c n 1 1 = γ + γ j [ ( ) ( γ V j Vj d γj G G d)] for L, (3.1a) [ ( ) ( γ V j Vj d γj G G d)], (3.1b) where the frst parts of (3.1a) and (3.1b) follow from (2.1) and the second parts follow from substtutng (2.1) n (2.14). We characterze these dfferences n the followng Proposton: Proposton. The contrbuton-schedules equlbrum and the Nash barganng soluton concde f and only f γa ( V ) ( V d = γ b G G d) for all L. (3.2) Else, f the left-hand sde exceeds the rght-hand sde for lobby, ceters parbus, ts equlbrum utlty s greater n the contrbuton-schedules equlbrum than n the Nash barganng soluton, and vce versa. Proof. Substtutng V n γa ( V = V ) ( V d = γ b G G d) γ b + γ + γ j 0 = 1 and G n = G n (3.1) and rearrangng yelds: 1 [ ( ) ( γ V j Vj d γj G G d)] for all L, (3.3a) [ ( ) ( γ V j Vj d γj G G d)]. (3.3b) Achm Voss and Mark Schopf 8/12

11 3. Dscusson Substtutng (3.3b) n (3.3a) yelds (3.2). The remander of the Proposton follows from substtutng (3.2) n (3.1a) for j L\: V V n = 1 γ + γa \j γ j γ + γ j [ γa ( V ) ( V d γ b G G d)]. (3.4) The Proposton can be appled to a specal case. The contrbuton-schedules equlbrum and the Nash barganng soluton concde f the followng condtons are all fulflled: There s only one lobby ( L = 1), the government has no barganng power (γ = 0), and the dsagreement polcy s defned by (2.16) or (2.17) (whch concde for γ = 0). Wth γ = 0, the left-hand sde of (3.2) turns zero. By (2.16), G d then s maxmzed welfare. Wth only one lobby, (2.8b) mples G = W(p ), by (2.5), p s welfare-maxmzng as well. Thus, G = G d. In addton to our Proposton, we compare the two models concernng ther total effcency. Lobbyng s neffcent for two reasons. Frstly, the equlbrum polcy does not maxmze welfare. b Secondly, payng and recevng contrbutons causes socal costs f > a. 3 The equlbrum polcy s the same n both approaches. Thus, we only need to compare the socal costs due to payng and recevng contrbutons, (b a ) c. By (3.1), the dfference s (b a ) [ C (p ) c n ] = b a γa b γ b γ + γa ( V γ j As a specal case, assume that (3.2) s fulflled for j L\: (b a ) [ C (p ) c n ] = γ + γ j γ 1 γ j b a γa ) ( V d γ b G G d) [ ( ) ( γ V j Vj d γj G G d)]. (3.5) 1 b [ γa ( V ) ( V d γ b G G d)]. (3.6) Thus, gven that (3.2) s fulflled for j L\ and b a a for j L\, the contrbutonschedules equlbrum s more effcent than the Nash barganng soluton f the left-hand 3 If lobbyng provdes nformaton to the government that mproves polcy, t can also enhance welfare, see, e.g., Ball (1995) and Lagerlöf (1997). Ths can even be the case f polcy s dentcal wth and wthout lobbyng, see, e.g., Bhagwat (1980). Achm Voss and Mark Schopf 9/12

12 3. Dscusson sde of (3.2) exceeds ts rght-hand sde for lobby, and vce versa. On the one hand, lobby then pays less n the contrbuton-schedules equlbrum than n the Nash barganng soluton. On the other hand, G declnes so that the other lobbes must pay more to ensure that (3.2) remans fulflled for them. However, as long as b a a effcency ncreases because total payments declne. for j L\, total Fnally, note that our Proposton compares the equlbrum utltes for a gven set of avalable polces. However, the way that equlbrum polces are determned suggests that the lobbes would also care about the polcy nstruments avalable to the government. Grossman and Helpman (1994) suggest that a lobby would possbly prefer to restrct polcy choce to neffcent nstruments, because ths may ncrease the dfference between equlbrum utltes and the utltes n case the respectve lobby does not take part, whch reduces ts equlbrum contrbutons, see (2.8a). In the Nash-barganng model, the reasonng s smlar, but the lobby would prefer to restrct polcy choce so as to maxmze (2.14a). Even f the equlbrum utltes concde for a gven set of avalable polces, the preferred polcy nstruments may not. Thus, a clearer understandng of the approprate model of polcy settng s also crucal for understandng the consttutonal choce of allowed polcy nstruments. Our comparson also assumes a gven set of lobbes The equlbrum polcy and thus welfare s the same n both approaches for each set of lobbes. But the equlbrum utltes and the socal costs due to payng and recevng contrbutons can develop dfferently f, e.g., one addtonal lobby forms. 4 To sum up, the two models usually do not mply the same contrbuton payments and equlbrum utltes. They concde f the government has no barganng power (understood as the respectve parameter n the Nash-barganng functon) and there s only one lobby. Else, an addtonal assumpton s necessary for the Nash-barganng model: Whch polcy s chosen n case of dsagreement? By contrast, ths s endogenous for the contrbuton-schedules model. We demonstrate, for a gven assumpton about dsagreement utltes, that the two models only concde f each lobby has a certan relatve barganng power compared to that of the government. Fnally, we have compared the models concernng ther total effcency and ndcated that ther nsttutonal mplcatons can dffer. 4 For a gven set of lobbes, free entry nto an organzed sector leads to rent dsspaton. Ths may lead to a breakdown of the sector s lobby, for example f t cannot prevent free rdng of the entrants. See Grossman and Helpman (1996) and Baldwn and Robert-Ncoud (2007). Achm Voss and Mark Schopf 10/12

13 3. Dscusson Acknowledgements We would lke to thank Manuel Förster, Anke Gerber, B. Mchael Glroy and Nathale Schopf for helpful comments. References Baldwn, R.E., Robert-Ncoud, F., Entry and asymmetrc lobbyng: Why governments pck losers. Journal of the European Economc Assocaton 5, Ball, R., Interest groups, nfluence and welfare. Economcs and Poltcs 7, Bhagwat, J.N., Lobbyng and welfare. Journal of Publc Economcs 14, Burguet, R., Camnal, R., Smultaneous Nash barganng wth consstent belefs. Conference Paper 567. Latn Amercan Meetng of the Econometrc Socety. Compte, O., Jehel, P., The coaltonal Nash barganng soluton. Econometrca 78, Dharmapala, D., Comparng tax expendtures and drect subsdes: The role of legslatve commttee structure. Journal of Publc Economcs 72, Gawande, K., Krshna, P., Olarreaga, M., What governments maxmze and why: The vew from trade. Internatonal Organzaton 63, Goldberg, P.K., Magg, G., Protecton for sale: An emprcal nvestgaton. The Amercan Economc Revew 89, Grossman, G.M., Helpman, E., Protecton for sale. The Amercan Economc Revew 84, Grossman, G.M., Helpman, E., Rent dsspaton, free rdng, and trade polcy. European Economc Revew 40, Grossman, G.M., Helpman, E., Specal Interest Poltcs. The MIT Press, Cambrdge, Massachusetts. Lagerlöf, J., Lobbyng, nformaton, and prvate and socal welfare. European Journal of Poltcal Economy 13, Magg, G., Rodríguez-Clare, A., Import penetraton and the poltcs of trade protecton. Journal of Internatonal Economcs 51, Achm Voss and Mark Schopf 11/12

14 3. Dscusson Okada, A., The Nash barganng soluton n general n-person cooperatve games. Journal of Economc Theory 145, Schopf, M., Voss, A., Barganng over natural resources: Governments between envronmental organzatons and extracton frms. Mmeo. Achm Voss and Mark Schopf 12/12