Optimal Altruism in Public Good Provision

Transcription

1 ES Workng Paper Seres, September 2014 Optmal Altrusm n Publc Good Provson Robert W. Hahn, Smth School of Enterprse and the Envronment, Oxford Unversty Insttute for New Economc Thnkng (INET) Robert A. Rtz, Faculty of Economcs, Cambrdge Unversty Energy Polcy Research Group (EPRG) Abstract We present a model of altrustcally-mnded yet ratonal players contrbutng to a publc good. A key feature s the tenson between altrusm and crowdng-out effects. We present three man results: (1) More altrustc behavour often reduces socal welfare; (2) It s almost always optmal for a player to act more selfshly than her true preference; (3) A player s optmal altrustc commtment s often low or zero even wth strongly altrustc preferences. Applcatons to a range of publc good problems, ncludng clmate polcy, are dscussed. Our results hghlght that t wll generally be dffcult to nfer socal preferences from observed behavour. We would lke to thank Toke Adt, Davd Anthoff, Elzabeth Baldwn, John Feddersen, Reyer Gerlagh, Thomas Greve, Cameron Hepburn, Charles Mason, Grscha Perno, Rck van der Ploeg, John Quah, Robert Stavns, Paul Tetlock, Alexander Teytelboym, Rchard Tol, and Alstar Ulph for helpful comments and advce, semnar partcpants at EPRG, OxCarre, Cambrdge, and Toulouse for dscussons. The usual dsclamer apples.

2 1 Introducton There s a growng recognton that socal preferences may play an mportant role n explanng economc outcomes such as those arsng n problems of publc good provson. 1 We study the welfare mpact of unselfsh behavour by altrustcally-mnded yet ratonal players, and ask to what extent a preference for altrusm s optmally reflected n a player s contrbuton to a publc good. To our knowledge, ths s the frst attempt n the lterature to understand a noton of optmal altrusm. Our analyss s motvated n part by recent experence wth clmate polcy, whch many consder to be one of the bggest publc good problems of today (Stern 2008). Recent years have wtnessed a number of unlateral ntatves to combat clmate change at the local, natonal, and regonal levels. For example, the EU has a program to reduce greenhouse gas emssons by 20% (relatve to 1990 levels) by 2020 whle the UK ams to cut emssons by 80% by Such ntatves have taken place n the absence of a global agreement by countres to ontly reduce emssons, e.g., wth a global cap-and-trade scheme. Relatedly, there s an ncreasng use of the socal cost of carbon (SCC) n regulatory decson-makng. The SCC reflects the margnal beneft to the world from reducng CO 2 emssons rather than only to an ndvdual country or regon. Several European countres have appled the SCC (Watkss and Hope 2012), and the US has also developed a measure of the SCC (Greenstone, Kopts and Wolverton 2013) whch to date has been appled to selected energy and envronmental regulatons. At the same tme, many other countres do not ncorporate the SCC n polcymakng, and do not appear to have engaged n emssons abatement beyond busness-as-usual. There s some evdence that the domestc costs assocated wth unlateral polces exceed domestc benefts. For example, Tol s (2012) cost-beneft analyss of the European Unon s 20/20/20 polcy package fnds a beneft-cost rato < 1 across a range of scenaros. 3,4 In a smlar ven, the UK Department of Energy and Clmate Change s mpact assessment of the 2008 Clmate Change Act fnds the economc case for the UK contnung to act alone where global acton cannot be acheved would be weak (DECC 2009). It s dff cult to reconcle these unlateral ntatves wth standard economc theory, ncludng the theory of nternatonal envronmental agreements (Barrett 1994, 2005). Put smply, f unlateral acton by local, natonal, or regonal actors reduces ther own domestc welfare, then why are they dong t? But t seems possble that some of these clmate ntatves may be a reflecton of unselfsh or altrustc motves, n the sense of ncor- 1 See Sobel (2005) for an overvew of nterdependent preferences n economc analyss. 2 Smlar clmate-polcy ntatves, many on a relatvely small scale, also exst, for example, n Australa, Calforna, Chna, Japan, New Zealand and Norway, as well as at the cty level. 3 Ths polcy package targets a 20% cut n greenhouse gas emssons, a 20% share of renewable energy, and a 20% mprovement n energy eff cency by Smlar ssues have also emerged n the analyss of the recently proposed regulaton to reduce carbon doxde emssons from the US power sector (EPA 2014). The economc analyss conducted by the EPA suggests that the global benefts of ths regulaton exceed ts costs but, n some scenaros, domestc benefts accrung only to the US fall short of domestc costs (e.g., dependng on the extent to whch health-related benefts are taken nto account n addton to drect clmate benefts). 2

3 poratng benefts that accrue outsde the borders of the actng ursdcton. Ths paper seeks to understand the role that altrusm can play n such publc goods problems. 5 We begn wth a two-player model of (non-cooperatve) publc good provson wth the followng key features. A player s net beneft or natonal welfare Π k (k =, ) equals the beneft she derves from total contrbutons to the publc good (by both players) mnus the cost of her own contrbuton, whle global welfare W = Π + Π. 6 Preferences may depart from self-nterest: A player s true obectve functon S k = (1 θ k )Π k + θ k W places weght on both her own net benefts and global welfare, where θ k [0, 1] represents her degree of altrusm. 7 More altrustc behavour by player leads to an ncrease n ts own publc good contrbuton but nduces player to cut back ( crowdng out ). We refer to the rate at whch the other player s effort contracts as the leakage rate. The tenson between altrusm and leakage les at the core of our analyss. Our modellng approach s also consstent wth a characterstc shared by many (global) publc good problems: The absence of a world government means that solutons enforced by a central mechansm desgner play a lmted role. 8 Our analyss hghlghts three man fndngs. Frst, we obtan the seemngly paradoxcal result that more altrustc behavor by an ndvdual player often reduces socal welfare. For example, consder a small commtment to more altrustc behavor by player. Such a commtment rases the equlbrum net beneft enoyed by player but reduces s own net beneft. We show that welfare s more lkely to fall f player derves an above-average margnal beneft from contrbutons, and the leakage rate from her commtment s hgher. 9 Conversely, a necessary condton for more altrustc behavour to rase such a player s true obectve s that her degree of altrusm exceeds her leakage rate. Ths already shows that whether altrusm s prvately optmal and/or welfare-augmentng depends crucally on 5 Our analyss focuses on nternatonal altrusm between countres rather than ntergeneratonal altrusm between dfferent generatons of people n a sngle economy. We dffer from much of the lterature on altrusm n that we often thnk of our unt of analyss as a country rather than an ndvdual. Also, we do not wsh to clam that socal preferences are the only possble way of explanng unlateral clmate acton; n some cases, other explanatons, e.g., domestc poltcal economy, may be mportant. 6 Our results are robust to dfferent welfare defntons. Secton 5 provdes detals. 7 Our formulaton of altrusm has a contnuum of preferences, rangng from entrely selfsh to entrely altrustc preferences. On a hstorcal note, Edgeworth (1881) uses essentally the same formulaton, by wrtng S = Π + θ Π and callng θ the coeff cent of effectve sympathy. Some other formulatons of altrusm have condtonal elements. For example, the models of nequty averson due to Fehr and Schmdt (1999) and Bolton and Ockenfels (2000) feature utlty functons wth reference ponts whch determne the degree of perceved nequty n payoffs (and also affect players actons, e.g., dependng on whether they are ahead or behnd ). Lange and Vogt (2003) show that a preference for equty can generate cooperaton n nternatonal envronmental negotatons, whle Kosfeld, Okada and Redl (2009) argue that farness can play an mportant role n the formaton of nsttutons geared towards mprovng publc good provson. See also Rabn (1993) on farness n economc analyss. 8 Key contrbutons on voluntary publc-good provson nclude Bergstrom, Blume and Varan (1986) and Cornes and Sandler (1996). We work wth a smplfed model wth reduced-form beneft and cost functons, as s standard n much of the envronmental economcs lterature (e.g., Hoel 1991; Barrett 1994), whch captures the key feature that players contrbutons are strategc substtutes. 9 Hoel (1991) obtans a related result n an mportant early model of unlateral commtment n envronmental polcy that does not feature socal preferences. He shows that a small (exogenous) commtment to a hgher publc good contrbuton by a player, startng from a world n whch all players make entrely selfsh contrbutons, can reduce global welfare. Our work goes further by examnng a world n whch players can behave altrustcally to dfferent degrees, and dervng a noton of optmal (endogenous) altrusm. 3

4 the detals of the envronment; a player may thus wsh to fnd ways of makng publc contrbutons that departs from her true obectve. Second, we show that a player who genunely wants to maxmze global welfare almost always does best by beng at least somewhat selfsh. To see ths, suppose that player s true preference s entrely altrustc, θ = 1, whle player s altrustc only to some degree, θ < 1. Should make the contrbuton that maxmzes ts underlyng global-welfare obectve? No. Intutvely, a small decrease n ts own contrbuton only leads to a secondorder loss n global welfare (by the envelope theorem). But the resultng nduced ncrease n the other player s effort leads to a frst-order gan (whenever the other player s not already choosng the frst-best effort). 10 Ths s what we call reverse leakage a weaker commtment reduces free-rdng by other players, and ths can rase socal welfare. Thrd, we fnd that a player s optmal altrustc commtment s often low or zero even wth strongly altrustc preferences. In some cases, t s optmal for a player who cares about global welfare to act entrely selfshly, maxmsng only her own net beneft. We thus hghlght that cauton s requred n nferrng whether or not players are beng selfsh from ther observed behavour; selfsh behavour may be a welfare-maxmsng response to crowdng-out effects, especally wth heterogeneous players. We characterze optmal altrustc commtments usng the followng modellng devce: Player k has a strategc obectve functon Ω k = (1 λ k )Π k + λ k S k, where λ k 0 s her strategc preference. A player chooses a publc good contrbuton accordng to her true preference f λ k = 1, but whenever λ k < 1 (λ k > 1) acts more (less) selfshly than would be her true preference. We determne a player s optmal commtment λ k (θ, θ ) to ncorporate ts altrusm nto publc good contrbuton. In partcular, we always have λ k 1, almost always fnd λ k < 1 (for k =, ), and, n a range of cases, λ 0 and/or λ Only where all players have entrely altrustc preferences θ = θ = 1, s a full commtment λ k = 1 (for k =, ) optmal, n whch case the frst-best outcome obtans. We show that these results are very robust to a varety of dfferent model specfcatons. Ths ncludes the generalzaton to n 3 players where we explot the fact that players contrbutons are made n an aggregatve game (n the sense of Corchón 1994; see also Cornes and Sandler 2007); moderate degrees of cross-country cost spllovers (e.g., n renewable energy technologes such as solar or wnd); and dfferent representatons of altrusm n players obectve functons, ncludng the warm glow of Andreon (1989, 1990). Thus, our results apply wth both pure and mpure forms of altrustc preferences. One way of thnkng about how a player can commt to actons that depart from her true preference s n terms of the theory of strategc delegaton. For example, ctzens may delegate decson-makng on abatement targets to poltcans, and may wsh to appont poltcans whose clmate-polcy preference dffers from ther own (e.g., from those of the 10 These basc nsghts rely on crowdng-out effects but not on whether leakage rates are hgh or low. 11 To llustrate, two countres true preferences may be to apply the global SCC to 100% and 46% of proects respectvely, that s, (θ, θ ) = (1, 6 ). But f 13 (λ, λ ) = ( 1, 0), say, then optmal altrusm nvolves 2 usng the SCC only n 50% (= 1 1) of proects for country and not at all for country (= 0 6 ) (The detals underlyng ths numercal example are at the end of Secton 6.) 4

5 medan voter). Commtment can also be acheved by poltcal or regulatory nsttutons perhaps ndependent of government whch adopt partcular rules and practces. The classc reference on such delegaton s Schellng (1960), and the dea has been appled wdely to dfferent contexts such as barganng (Segendorff 1998), monetary polcy (Persson and Tabelln 1993), and the theory of the frm (Vckers 1985). It s farly wellknown that an ncentve to msrepresent preferences exsts n vrtually any game (Hefetz, Shannon and Spegel 2007) although ths, n tself, says lttle about how preferences wll be dstorted n a partcular game. We dffer from ths lterature n several respects. To begn wth, we consder a dfferent class of game, and examne a settng n whch agents are not drven by pure self-nterest; many of our themes thus have no analog n prevous models. 12 Moreover, our man applcaton to clmate polcy has at least two advantages compared to other delegaton applcatons. Frst, there s sgnfcant emprcal evdence that players efforts are strategc substtutes: A very large maorty of work on unlateral clmate polcy fnds that carbon leakage rates are postve, as n our model. 13 Second, clmate polcy s characterzed by somethng close to an nformatonal level playng feld between countres: The clmatechange debate s hghly publc and global (based, n part, on scentfc evdence) and countres abatement polces are commonly known (perhaps wth a few exceptons), as s whether or not they have adopted the SCC. 14 Our analyss also shows that altrusm can, at least n prncple, neutralze the strategc ncentve to dstort preferences whch s emphaszed by ths lterature. Our model always features strategc substtutes, so contrbutng less nduces a favourable response from the other player; but f both players are fully altrustc, they recognze that such preference dstorton no longer yelds any gan (as t nduces a move away from frst-best). So an ncentve to dstort play may exst n a standard game wth selfsh players but not n an otherwse dentcal game featurng socal preferences. Other applcatons. Questons of altrustc behavour arse n other envronmental problems. For example, there s an ongong debate about the motvatons behnd the Montreal Protocol to reduce chlorofluorocarbons (CFCs) whch deplete the ozone layer. Whle Barrett (1994) argues that the protocol was broadly consstent wth the outcome of a 12 Perhaps closest to us, though n a rather dfferent setup wthout altrusm, Roelfsema (2007) consders a model of mperfect competton wth strategc trade polces, n whch delegaton to a poltcan who cares more about the envronment than the medan voter can be optmal because ths nduces other countres to do the same. (Ths result reles on a partcular form of competton n product markets.) 13 Many emprcal estmates are derved from numercal smulatons of mult-sector, general equlbrum models whch focus on clmate ntatves by OECD countres that result n carbon leakage to non-oecd countres. These typcally fnd leakage n the range of 5 40%, wth many estmates below 20%. Leakage estmates for ndvdual sectors (such as the cement and steel ndustres n the EU Emssons Tradng Scheme) are frequently hgher, e.g., above 50%, but also rarely exceed 100%. See Babker (2005), Copeland and Taylor (2005), Rtz (2009), and the references cted theren. By contrast, n delegaton models on the theory of the frm, t s often dff cult to tell wth confdence whether competton between frms s n strategc substtutes (Cournot) or n strategc complements (Bertrand) and many results are known to depend crtcally on ths unobservable feature of the model. 14 In practce, there s sgnfcant uncertanty over the costs and benefts of CO 2 abatement; the key pont for us s that actons are easly observable and nformatonal asymmetres between countres are small. 5

6 non-cooperatve game, Sunsten (2007) notes that the US used a relatvely low dscount rate n evaluatng ts commtment whch mght be nterpreted as a form of altrusm. The key features of our model are shared by other problems of the commons. In fsheres polcy, for example, there s a strong tendency towards overexplotaton; ndvdual players have a suboptmal ncentve to lmt ther catch (Stavns 2011) and catch reductons are typcally strategc substtutes (Levrah and Mrman 1980), leadng to a leakage problem analogous to ours. Smlarly, n a classc paper, Olson and Zeckhauser (1966) suggest that small countres tend to free-rde on the defense nvestments of large countres, and observe that countres mltary expendtures are often strategc substtutes. It s more dff cult to pnpont altrusm emprcally n these applcatons, partly because there s no clear equvalent to the adopton of the SCC. However, t seems concevable that ndvdual European countres, say, also care about the welfare of the EU as a whole when t comes to polces affectng the envronment or defense. 15 Our results can also apply to problems from other domans that share publc good characterstcs. For example, suppose famly member pursues some useful actvty; famly member derves ndrect benefts from the actvty, and can help out at some cost. If altrustc, also cares about the benefts accrung to n choosng how much to help. But the more helps, the less does hmself the leakage problem. Whle s help always rases s prvate payoff, t need not rase overall welfare or s own altrustc obectve. Optmal altrusm typcally nvolves λ < 1, so s help falls short of her true preference. 16 To be concrete, a parent may want to help a chld wth ts homework on 4 out of 5 days a week (θ = 4 5, say), but realzes that, because of ncentve effects, λ = 1 2, say, s optmal and thus only helps twce a week. In practce, such a well-meanng but stern commtment may be acheved by puttng certan rules nto place, or the parent may engage a tutor (or sblng) twce a week and abstan from helpng drectly. 17 Plan for the paper. Secton 2 sets up our benchmark model. Secton 3 examnes the mpact of small altrustc commtments. Secton 4 analyzes n detal players optmal commtments, and Secton 5 shows that our man results are robust n a varety of drectons. Secton 6 ponts out some further propertes of our model, wth a focus on ts emprcal mplcatons. Fnally, Secton 7 dscusses recent clmate polcy ntatves n lght of our results, and offers some suggestons for future research. (The proofs are n Appendx A, and the detals of the robustness analyss are n Appendx B.) 15 A related applcaton s the problem faced by large chartes lke the Bll & Melnda Gates or Rockefeller foundatons. It seems clear that the broad obectve of such organzatons s to enhance some measure of global welfare. At the same tme, there are well-known concerns that ther contrbutons can crowd out others, such as local governments, the prvate sector, and smaller chartes. Ths corresponds qute drectly to the tenson between altrusm and leakage n our analyss. 16 The rotten kd theorem (Becker 1974) does not apply n our model. It states that, under certan condtons (Bergstrom 1989), an altrustc head who makes transfers to self-nterested household members nduces the eff cent outcome despte lmted altrusm n the famly overall. By contrast, our setup does not feature a desgn wth transfer payments (see also our concludng dscusson n Secton 7). (Recall also that the rotten kd theorem tself can fal n publc-good settngs, despte transfers.) 17 Our model assumes the tutor s optmally chosen and ncentvzed by the parent (strctly speakng, at zero cost) and has no specal sklls although ths s clearly not essental for the results. 6

7 2 A model of altrusm n publc good provson Setup of the model. Two players, and, contrbute to the provson of a publc good. Player k (k =, ) makes a contrbuton (e.g., shared nvestment, emssons reducton, or effort ) denoted by X k, and derves benefts B k (X + X ) whch depend on the aggregate effort by the two players. The margnal beneft satsfes B k ( ) > 0 and B k ( ) < 0. The cost functon C k (X k ) s player-specfc, wth margnal cost satsfyng C k ( ) > 0 and C k ( ) > 0. To guarantee an nteror soluton, assume C k (0) = C k (0) = 0 and B k (X k) C k (X k) < 0 for some X k <. Defne a player k s net beneft or natonal welfare as Π k = B k (X + X ) C k (X k ), and socal surplus or global welfare as W = Π + Π. In our model, each player s preferences may be at least partly altrustc. In partcular, player k s true obectve functon s gven by S k = (1 θ k )Π k + θ k W, (1) where the parameter θ k [0, 1] represents her true preference for altrusm. Player k s purely self-nterested f θ k = 0 (so S k = Π k ), and entrely altrustc f θ k = 1 (so S k = W ), n whch case her preference reflects the full global beneft of contrbutons, B + B. More generally, a hgher value of θ k represents a more altrustc preference that gves more weght to the other player s net beneft. For our applcaton to clmate polcy, we can nterpret θ k = 0 as an underlyng preference for the busness-as-usual (BAU) level of emssons, whle θ k = 1 corresponds to a desre to ncorporate the global socal cost of carbon (SCC) nto decson-makng. We next ntroduce a modellng devce n form of a strategc obectve functon: Ω k = (1 λ k )Π k + λ k S k. (2) A strategc obectve s a convex combnaton of a player s net beneft Π k and her true obectve S k, wth a relatve weght gven by the strategc preference λ k [0, θ 1 k ]. If λ k = 0, the strategc obectve s entrely selfsh, so Ω k = Π k (regardless of the underlyng true obectve S k ). If λ k = 1, the player s strategc obectve s dentcal to her true obectve, so Ω k = S k. We restrct attenton to λ k θ 1 k λ k θ k 1 to focus on the typcal stuaton where each player contrbutes too lttle to the publc good from a socal-welfare perspectve rather than too much. (Whenever θ k < 1, we do allow for the possblty that λ k > 1 so the strategc obectve could place more weght on altrusm than the true obectve although we wll see that, n equlbrum, ths does not occur.) Ths (λ, λ )-modellng devce allows us to analyze the welfare mpact of players followng through on ther altrustc preferences, and, buldng on ths, to understand the extent to whch players optmally engage n altrustc behavour. 18 The tmng of the model s as follows. At Date 0, each player s endowed wth a 18 In the lterature on nternatonal envronmental agreements, countres typcally make a bnary decson on onng an agreement ( n or out ). By contrast, countres here choose the ntensty of ther commtment. 7

8 beneft functon and a cost functon, B k ( ) and C k ( ), as well as wth a true obectve S k ( ) that reflects her degree of altrusm, θ k [0, 1]. Then, at Date 1, each player chooses her strategc preference λ k [0, θ 1 k ] to maxmze her true obectve S k. Fnally, at Date 2, each player or her agent chooses effort accordng to the strategc obectve functon Ω k. (For envronmental applcatons, a country s choce of X k s equvalent to choosng a domestc prce on emssons. 19 ) We focus on the subgame-perfect Nash equlbrum of the game, and follow the delegaton lterature n assumng that players strategc obectve functons, Ω and Ω, form credble commtments. 20 The plausblty of ths assumpton wll, of course, vary dependng on the applcaton n queston. As explaned n the ntroducton, we thnk that commtment value s reasonably lkely to obtan n the clmate-polcy context, gven somethng close to an nformatonal level playng feld between countres, as well as n other publc good problems. Key propertes of the model. We begn by establshng the key propertes of the model at Date 2. For player, say, the frst-order condton for ts contrbuton s Ω / X = (B C ) + λ θ B = 0. (3) The frst-best benchmark s nested where () both players have entrely unselfsh true preferences, θ = θ = 1, and () both players choose ther respectve effort levels accordngly, λ = λ = 1. In ths case, players at Date 2 make contrbuton decsons to max Xk W (where k =, ), thus each ncorporatng the full global beneft of ther actons. The frst-order condton also defnes player s best response to player s contrbuton, R (X ). The slope of ths functon s gven by R (X ) = ( ) B + λ θ B ( B + C λ θ B ) ( 1, 0). (4) A key property of the model s that players eff orts are strategc substtutes. Ths captures a crowdng out effect: If one player ncreases her effort, ths reduces the margnal beneft of effort for the other player, who therefore responds by cuttng back. In the context of clmate polcy, L [ R (X )] (0, 1) s the margnal rate of carbon leakage (IPCC, 2007) resultng from country s effort. Borrowng ths termnology: Lemma 1 The leakage rate due to player k s effort s gven by L k (0, 1). 19 To see ths, magne splttng country k s abatement decson at Date 2 nto two parts. At Date 2b, a representatve, prce-takng frm chooses emssons abatement X k to maxmze ts profts p k X k C k (X k ), where p k s the domestc emssons prce, such that p k = C k(x k ), n equlbrum. Ths defnes an upwardslopng abatement supply curve wth dx k /dp k = 1/C k (X k ) > 0. At Date 2a, polcymakers choose the domestc prce p k to maxmze the strategc obectve Ω k. Ths setup s exactly equvalent to the benchmark model snce choosng the domestc emssons prce s equvalent to choosng an abatement effort. 20 Ths s essentally equvalent to assumng that players contrbutons are publcly observable, whch, n turn, corresponds to knowng s λ θ when choosng her contrbuton polcy at Date 2 (as beneft and cost functons are commonly known). (We do not requre that then knows s true preference θ.) 8

9 Leakage rates quantfy the severty of the crowdng-out problem; they are postve but less than 100%. Ths s a common feature of publc good models across dfferent domans, ncludng envronmental problems, mltary protecton, fsheres, and chartable gvng. We next confrm the ntuton that more altrustc behavour by a player leads to an ncrease n her effort. (The result from Lemma 1 ensures that the equlbrum s unque, stable, and exhbts well-behaved comparatve statcs.) Lemma 2 If player k s true preference θ k > 0, her effort satsfes dx k /dλ k > 0. A hgher value of λ k nflates the margnal return to publc good contrbuton, whch, by stablty, also ncreases ts equlbrum level. So an ncrease λ, say, rases X (Lemma 2) and also rases X + X, but not by as much (Lemma 1). To complete our prelmnary dscusson, we show that a player wth an entrely selfsh true preference, θ k = 0, does not want to engage n a strategc commtment. Lemma 3 If player k s true preference θ k = 0, her optmal effort solves max Xk Π k. As a notatonal conventon, we refer to such an optmal commtment as λ k = The welfare mpact of small altrustc commtments To buld ntuton, we begn our analyss by consderng small commtments. Suppose that player s true preference θ > 0 s altrustc at least to some extent (whle θ 0), and that ntally both players act purely n ther self-nterest,.e., λ k = 0 for k =,. What s the mpact of a small commtment > 0 by player towards ncorporatng her true altrustc preference n her publc good contrbuton? Proposton 1 The mpact of a small unlateral commtment > 0 by player on her equlbrum true obectve ds = λ =λ =0 [ (θ B B L ) dx ] λ =λ =0 s (a) postve f the rato of margnal benefts satsfes B B and her true preference exceeds the leakage rate θ > L, and (b) negatve for a rato of margnal benefts B /B suff cently large or for a true preference θ suff cently small. Whether a small altrustc commtment s benefcal for a player depends crucally on the detals of the envronment. If she ether derves a relatvely large margnal beneft, or her true preference contans only a small degree of altrusm, t s never a good dea for someone to make such a commtment. However, two smple condtons whch are ontly 21 Ths conventon makes an altrustc player s strategc commtment drectly comparable wth a selfsh player; note that, f θ k > 0, then the contrbuton solves max Xk Π k f and only f λ k = 0. (Recall that we are restrctng attenton to cases where λ k 0.) 9

10 suff cent for ds > 0 are that the player has a relatvely low margnal beneft as well as a true preference that exceeds the rate of leakage. These results can be understood as follows. Wth a slght abuse of notaton, let dx > 0 denote the ncrease n s effort due] to ts small unlateral commtment > 0. (More formally, dx [(dx = /) λ =λ =0 > 0 by Lemma 2.) Due to the crowdng-out effect (Lemma 1), adusts ts effort by dx = ( L ) dx < 0 n response. By the envelope theorem, the drect effect of a small change n each player s effort on ts own net beneft s zero. The reason s that both players were ntally choosng ther respectve efforts selfshly to maxmze ther own net beneft, so any (small) change ther own contrbuton only has a second-order effect. However, the unlateral commtment by also has two strategc effects, one postve and one negatve. Frst, the ncrease n s effort yelds an ncrease n the benefts enoyed by the other player of B dx > 0. Second, the nduced reducton n s effort means that s beneft changes by B dx = ( B L ) dx < 0. Player s true obectve S = Π + θ Π places weght θ [0, 1] on the frst (postve) strategc effect and full weght on the second (negatve) strategc effect. The weghted sum of these effects, (θ B B L )dx, thus determnes the mpact of a small unlateral commtment on ts own true obectve functon and behaves accordng to Proposton 1. Intutvely, a unlateral commtment by ncreases the net beneft Π enoyed by but actng unselfshly hurts ts own net beneft Π. The commtment thus enhances ts own true obectve f (and only f) the former effect outweghs the latter. The postve effect wll be large f s margnal beneft s large, and receves large weght accordng to s degree of altrusm, θ. The negatve effect wll be small f there s lttle leakage, and f s own margnal beneft s small. We can also address when a small altrustc commtment mproves global welfare: Proposton 2 The mpact of a small unlateral commtment equlbrum global welfare dw = λ =λ =0 [ (B B L ) dx ] λ =λ =0 > 0 by player on s (a) postve f the rato of margnal benefts satsfes B B, and (b) negatve for a rato of margnal benefts B /B suff cently large. The logc underlyng Proposton 2 follows that of Proposton 1. Agan, the drect effects on each player s net beneft are both zero by the envelope theorem. The only dfference arses because, from a global-welfare perspectve, the combned effect of the two strategc effects depends on ther unweghted sum. So the ncrease n the benefts enoyed by the other player of (B )dx > 0 plus the nduced reducton n s beneft of [B ( L )]dx < 0 yeld an overall welfare mpact dw = (B B L )dx. The sgn of ths expresson, too, s ambguous. However, note that Proposton 2(a) mples that a small commtment must be global welfare-enhancng for at least one of the two players. 10

11 A small commtment, f t occurs, s of course more lkely to rase global welfare than s true obectve. For example, wth dentcal beneft functons, B (X + X ) = B (X + X ), s small commtment always rases equlbrum global welfare W (Proposton 2, snce L < 1) but ts own true obectve S may stll declne (Proposton 1, for θ < L ) Optmal altrustc commtments Ths secton develops our man results on optmal altrusm. Our analyss so far has already shown that a more altrustc commtment can rase or reduce socal welfare; ths already suggests that players may wsh to make publc good contrbutons n ways that depart from ther true obectves. We begn by dervng a generalzed formula for the welfare mpact of more altrustc behavour, and then use ths to establsh our man arguments. Frst, we show that optmal altrusm almost always means that players act more selfshly than would be ther true preference. Second, optmal commtments are often much lower than players true preferences and, n a range of cases, a socally-concerned player does best by actng entrely selfshly. Thus t wll generally be dff cult to emprcally nfer socal preferences from observed behavour. The general model. In the general verson of the model, each player chooses optmally how altrustcally to act so as to maxmze her true obectve S k ; ths yelds equlbrum values λ k (θ, θ ) for players strategc preferences (k =, ). Ths analyss s more complcated because our prevous argument, based on the envelope theorem, that the two drect effects of commtment are zero no longer apples (snce players no longer necessarly act selfshly at the outset ). By Lemma 2, however, a small ncrease > 0 n, say, s strategc preference (not necessarly startng from λ = 0), leads to an ncrease n ts own effort of dx > 0. By Lemma 1, adusts ts effort by dx = ( L ) dx < 0 n response. The two strategc effects of an addtonal commtment are also as before. Frst, the ncrease n s effort yelds an ncrease n the benefts enoyed by the other player of B dx > 0. Second, the nduced reducton n s effort means that s beneft changes by B ( L ) dx < 0. The drect effect of a small change dx n s effort on ts own net beneft Π, n general, s equal to (B C )dx. Usng s frst-order condton from (3), ths generalzed drect effect can also be wrtten as ( λ θ B )dx 0. Smlarly, the drect effect of a small (nduced) change dx on s net beneft Π s equal to (B C )dx. Agan, by ts frstorder condton, the generalzed drect effect equals ( λ θ B ) dx = (λ θ B L ) dx 0. The overall equlbrum mpact of an ncremental commtment by on ts true obectve 22 Smlarly, f margnal costs are dentcal n the ntal equlbrum C (X ) = C (X ), then a small commtment by always mproves global welfare W, but has an ambguous mpact on ts true obectve S. (In the ntal equlbrum, B k = C k (k =, ) as both are entrely selfsh.) 11

12 S = Π + θ Π takes nto account all of these effects, wth approprate weghts: ds = ( λ θ B )dx }{{} drect effect on player ( 0) + ( B L ) dx }{{} strategc effect on player ( < 0) ( [ λ θ B L ) dx }{{ } + θ }{{} true altrusm of player ( [0, 1]) drect effect on player ( 0) + ( B ) dx }{{} strategc effect on player ( > 0) ]. Ths decomposton shows that, n general, selfless acton reduces a player s own net beneft (dπ < 0) but helps the other player (dπ > 0). Wrtng t more compactly yelds:23 Lemma 4 The generalzed mpact of a small unlateral commtment > 0 by player on her equlbrum true obectve satsfes ds = [ (1 λ )θ B (1 λ θ θ )B L ] dx. Lemma 4 tells us the margnal equlbrum mpact of more altrustc behavour by player on her true obectve, takng nto account ts mpacts on both s own contrbuton effort and the ncentve effect on s contrbuton. By nspecton, t s clear that the mpact s ambguous n general. Man results. We can now establsh the key result that a full commtment wth λ = 1 s almost never optmal for player. Proposton 3 (a) If both players true preferences are entrely altrustc θ = θ = 1, then ther optmal commtments λ = λ = 1 acheve frst-best effort levels; (b) If at least one player has partally selfsh true preferences θ < 1 or θ < 1, then optmal commtments λ < 1 and λ < 1 and both efforts fall short of frst-best levels. Part (a) of the result shows that the frst-best outcome s sustanable n our model as long as both players want to be entrely unselfsh. The ntuton s that f both players care about global welfare, nether has an ncentve to unlaterally devate from a full commtment snce any such devaton, by constructon, causes global welfare to fall. Part (b) shows that ths optmstc concluson apples only where both players are entrely altrustc. Whenever at least one player places greater weght on domestc welfare n ts true obectve functon, both players optmal commtments fall short of a full com- 23 The formulae n Proposton 1 (λ = λ = 0) and Proposton 2 (λ = λ = 0 and θ = 1) can be obtaned as specal cases of Lemma 4. 12

13 mtment, λ < 1 and λ < 1. In such cases, gven the optmal strategc preference chosen at Date 1, player chooses Date 2 effort to max X Ω = Π + λ θ Π, wth λ θ < Thnk about the mpact of the last step towards a full commtment wth λ = 1. In ths case, the negatve drect effect on s suff cently negatve to entrely offset the weghted postve strategc effect on. The reason s that, wth a full commtment, already nternalzes the externalty of ts choce on (precsely to the extent t cares about her). Thus the mpact of the last step s determned solely by the two remanng effects, the strategc effect on plus the weghted drect effect on. Ths equals [ (1 λ θ θ )B L ] dx < 0, and s negatve snce θ < 1 or θ < 1 by assumpton (and also λ 1, n equlbrum). Therefore, the last step reduces the equlbrum value of s true obectve S. The same reasonng apples to the other player, so, n equlbrum, λ < 1 and λ < 1. It s optmal, for nstance, that each countres ctzens delegate decson-makng regardng publc good provson to poltcans whose preferences are closer to the natonal self-nterest. Perhaps the most strkng statement of ths latter result goes as follows: Suppose s entrely altrustc, so θ = 1, whle s unselfsh only to some degree wth θ < 1. Then part (b) says that the optmal commtment by satsfes λ < 1, so a full commtment s domnated by a weaker polcy. The optmal way for to maxmze global welfare W s to maxmze a strategc obectve Ω = (1 λ )Π + λ W that s partally skewed towards ts own natonal welfare. In other words, a player who genunely wants to maxmze global welfare does best by beng at least somewhat selfsh. Intutvely, why can do better than playng accordng to ts true, entrely altrustc preference? A small decrease n ts own effort leads only to a second-order loss n global welfare (by the envelope theorem). But the resultng nduced ncrease n the other player s effort creates a frst-order gan (whenever the other player s not already choosng the frstbest effort). So the reason why full commtment s almost never optmal s what we call reverse leakage a weaker commtment reduces free-rdng by the other player. 25 To further sharpen ths argument, we now turn to the opposte lmtng case: Our next result shows that, n a range of cases, the optmal commtment for one or both players s a zero commtment. More generally, we can show that optmal commtments are often low, despte players havng sgnfcantly altrustc preferences. Proposton 4 (a) If at least one player s true preference s not entrely altrustc, θ < 1 or θ < 1, and the rato of margnal benefts B /B s suff cently large, then player s optmal commtment λ = 0; (b) If players true preferences θ > 0 and θ > 0 but both suff cently small, then players optmal commtments λ = λ = Proposton 3 thus also rules out any values λ > 1 or λ > 1 as beng sub-optmal. The reason, loosely speakng, s that any such stronger commtment would drectly hurt s own net beneft by more than t can ever strategcally beneft. 25 A full commtment would become approxmately optmal for n lmtng cases where ts leakage rate tends to zero. Ths happens where players margnal benefts are approxmately constant (.e., B k 0 for k =, ), or where the other player s producton technology s hghly nflexble (.e., C ) so ts effort choce becomes almost non-strategc. 13

14 Part (a) of the result essentally gves a non-local verson of our earler fndngs, from Propostons 1 and 2, that a small commtment by an ndvdual player may not rase S, or ndeed W. In extreme cases, t s optmal for an entrely altrustc player (when θ = 1 but θ < 1) to choose her effort level n her own strct self-nterest (λ = 0). A further mplcaton s that a polcy of zero commtment may welfare-domnate one of full commtment. Suppose that has a completely altrustc true preference whle s entrely self-nterested, (θ, θ ) = (1, 0). By Lemma 3, we have that λ = 0 rrespectve of s polcy. But also, f B /B s suff cently large, then equlbrum global welfare W s hgher wth zero commtments (λ, λ ) = (0, 0) than wth (λ, λ ) = (l, 0) for any 0 l 1 (snce then dw / 0 for all λ [0, l]). In ths example, a global-welfare orented country does better by maxmzng natonal welfare than by maxmzng global welfare. The reason for part (b) s that a player who s only somewhat unselfsh places too lttle weght on the postve drect and strategc effects that accrue to the other player for the calculus to overcome the negatve mpact on ts own net benefts. Applyng ths logc to both players, optmal commtments are zero. Formally, the result requres that altrustc preferences are suff cently small, yet a key observaton s that ths s compatble wth large degrees of altrusm. To llustrate, let players have dentcal beneft functons, B ( ) = B ( ), wth altrusm parameters lower than leakage rates, θ k < L k (for k =, ). In ths settng, optmal commtments are zero, λ = λ = 0, even though socal preferences could be almost fully altrustc. 26 A smple corollary s that (suff cently small) ncreases n one or both players true levels of altrusm (θ and/or θ ), may, n equlbrum, have no mpact at all on the qualty of publc good provson snce they are endogenously offset by crowdng-out problems. Interor commtments. To complete ths part of our analyss, and move beyond the lmtng cases, we now provde a characterzaton of players optmal commtments n an nteror equlbrum, n whch (λ, λ ) (0, 1) 2 (and thus also θ > 0 and θ > 0). Proposton 5 In an nteror equlbrum wth (λ, λ ) (0, 1) 2, player s optmal commtment λ satsfes λ = where the equlbrum rates of leakage L = [ ] θ (1 L L ) (1 θ θ ) (B /B )L θ (1 θ θ L L ) (0, 1), [ ] 1 + λ θ (B /B [ ) ] (0, 1), 1 + (C / B ) + λ θ (B /B ) ( ) and player s equlbrum effort satsfes X = C 1 B + λ θ B > Of course, as long as leakage rates are even hgher; ths s always true, e.g., where players margnal costs are approxmately constant, C k 0. 14

15 Proposton 5 mplctly descrbes players optmal nteror commtments, leakage rates, and contrbuton efforts gven ther respectve beneft and cost functons as well as ther true preferences for altrustc behavour. In prncple, a numercal soluton for the sx unknowns can be obtaned by makng specfc assumptons on the functonal forms of B k ( ) and C k ( ). The basc nformatonal requrement s as follows: The rato of players margnal benefts, B /B, and slopes of margnal benefts, B the slopes of margnal cost to the slope of margnal benefts, C /B k / B k ; each player s rato of (wth all functons evaluated at equlbrum); and each player s true preference for unselfshness θ k (k =, ) The soluton can be smplfed under some commonly-made assumptons. Let player k s beneft functon B k ( ) = µ k B( ), where µ k > 0 s the weght placed on a global beneft functon B(X + X ). Ths has the advantage that the ratos B /B = B /B = µ /µ become nvarant to the detals of players contrbutons. Also assume that margnal costs and benefts are lnear, B k (X + X ) = [α k β k (X + X )] and C k (X k) = δ k X k, so that C k / B k = δ k/β k s constant, too. 27 Optmal commtments can then be determned more easly as the soluton to a system of four equatons and four unknowns (λ, λ, L, L ) (0, 1) 4, for gven underlyng true preferences (θ, θ ). Inferrng how altrustc players are. Suppose t s observed or otherwse estmated that player s publc good contrbuton appears to be entrely selfsh; ths corresponds to λ θ = 0 n our model. As the above analyss shows, t does not follow that ths player s underlyng true preference s completely selfsh. Lttle or no addtonal effort can be consstent even wth hghly altrustc true preferences smply because t may arse from λ = 0 rather than θ = 0. So cauton s requred n nferrng whether or not a player s beng selfsh from her observed behavour. More generally, how does players optmal altrusm compare wth true preferences? Proposton 6 (a) For true preferences 0 < θ = θ satsfy λ θ λ θ ; < 1, optmal commtments may (b) For true preferences 0 < θ < θ, optmal commtments may satsfy λ θ > λ θ ; (c) If true preferences 0 < θ < θ, as well as B B, and L L, optmal commtments n an nteror equlbrum satsfy (λ θ λ θ ) < (θ θ ). Part (a) observes that players wth dentcal true preferences toward altrusm may have dfferent degrees of optmal altrusm. Except n knfe-edge cases, ths wll always occur f they have dfferent beneft and/or cost functons. Part (b) notes that the general relatonshp between true and strategc preferences s even less clear-cut. A player who cares more about global welfare may, n equlbrum, be the player whose actons are closer to self-nterest. In short, players wth dentcal true degrees of altrusm may behave dff erently, and a more altrustc player may optmally behave less altrustcally than 27 Ths latter assumpton s essentally equvalent to the classc analyss of Wetzman (1974) on whether prce- or quantty-based regulaton s socally preferable. It can be seen as a second-order approxmaton to the unknown shapes of the underlyng cost and beneft functons (see also Barrett, 1994). 15

16 another player. Part (c) shows that, n an nteror equlbrum, there s a tendency for strategc consderatons to compress any cross-player dfferences n altrusm: The dfference n optmal degrees of altrusm s often less than that of true degrees of altrusm. Taken together, these fndngs pose obvous challenges for makng cross-country nferences on true degrees of altrusm based on countres observed choces. 5 Robustness of the man results The man results from the benchmark model are that the welfare mpact of an altrustc commtment s ambguous (Propostons 1 and 2), a full commtment s optmal only f both players have entrely unselfsh true preferences (Proposton 3), and, n some cases, a zero commtment may be optmal despte sgnfcantly altrustc preferences (Proposton 4). We have emphaszed these lmtng cases because, as explaned n detal n ths secton, we beleve that these nsghts are robust to a large varety of changes to the model s specfcaton. Dscusson. In the above, we have, for smplcty, wrtten each player s strategc obectve as a weghted average of the form Ω k = (1 λ k )Π k +λ k S k. But observe that our results on the ambguous mpact of a small altrustc commtment do not rely on Ω k at all. Player, say, rases her level of effort by a small amount dx what exactly nduces ths s rrelevant for the local results of Propostons 1 and 2. Moreover, our result that a full commtment s almost never optmal, due to reverse leakage, s based on small proftable (that s, S -ncreasng) devatons away from the case where λ k = 1. Agan, ths analyss does not depend mportantly on the functonal form of Ω k ( ). 28 (Of course, the precse values of (λ, λ ) n an nteror equlbrum are, n general, senstve to the formulaton of Ω k ( ), for nstance n Proposton 5.) Our results are also robust to dfferent defntons of global welfare. Our above defnton W = Π + Π s approprate for the clmate problem and corresponds to the usage of the SCC. In some applcatons, one mght nstead consder socal welfare to be W = S + S, where S = Π + θ Π (symmetrcally for ), whch drectly ncorporates players altrustc preferences. The only results potentally affected are Propostons 2 and 3. It s easy to see, usng W = (1 + θ )Π + (1 + θ )Π, that Proposton 2 certanly goes through as above f θ = θ. More generally, part (a) becomes that dw > 0 f B [(1 + θ )/(1 + θ )]B, whle part (b) remans unchanged. Moreover, Proposton 3 contnues to hold, notng only that the frst-best effort levels that maxmze W wll, n general, dffer from the benchmark model (agan, unless θ = θ ). We also assumed that a player s benefts B k (X + X ) depend on the unweghted sum of efforts, whch s an approprate assumpton for a range of applcatons. But observe that the underlyng ntuton does not depend crucally on the pure publc good property. 28 For nstance, we could wrte Ω k = (1 λ k )Π φ k k +λ ks 1 φ k k, wth the weght φ k (0, 1), or more generally Ω k = h k ((1 λ k )Π k, λ k S k ), where the functon h k (, ) s strctly ncreasng n each of ts arguments. Key s that maxmzng any of these alternatve strategc obectve bols down to maxmzng S k whenever λ k = 1. 16

17 The two mportant features of our setup, n addton to altrustc motves, are () that each player would, as such, lke the other player to contrbute more ( S / X > 0), and () the leakage problem that more effort by one player crowds out the other player (dx /dx < 0). Our basc nsghts also apply to many stuatons wth mpure publc goods, ncludng examples we dscussed n the ntroducton. 29 In the remander of ths secton, we show that our key results are also robust n several other drectons, n partcular, to the generalzaton to n 3 players, to moderate degrees of cross-country spllovers n costs, and to alternatve representatons of altrusm n the true obectve functon S k, ncludng the warm glow of Andreon (1989, 1990). (Appendx B provdes detaled proofs.) Generalzaton to n 3 players. The analyss quckly gets more complex as the number of players ncreases; each ndvdual player may have a dfferent beneft and cost functon, a dfferent true preference for altrusm towards other players, and her own leakage rate. Nevertheless, we can explot the fact that the model wth n 3 players remans an aggregatve game (Corchón, 1994) at Date 2 when players make contrbuton decsons. 30 The key s that an ncrease n s effort now nduces each of the n 1 other players to cut back; n other words, player-specfc leakage rates L [ R (X )] are postve. But the overall leakage rate L L (0, 1) remans less than 100% and so global contrbutons rse (correspondng to Lemma 2). 31 Consder a small commtment by player 1 (begnnng n a completely selfsh world wth {λ k } n k=1 = 0), and, to llustrate, suppose that n = 3. The ncrease n 1 s effort drectly rases the net benefts of 2 and 3. It also nduces 2 to contrbute less, whch hurts 1 but now also hurts 3. Smlarly, reduced effort by 3 hurts 1 and 2. So 1 s hurt twce due to leakage, and t now nduces postve and negatve effects on each of the other two players whch t cares about dependng on ts true preference for altrusm. In general, the number of effects to take nto account s of order n 2. The welfare mpact of a small commtment s, as before, ambguous. In partcular, we can show that our earler condtons from Proposton 1 generalze cleanly: ds > 0 holds whenever has a margnal beneft that s (weakly) below average B B, ts true preference exceeds the leakage rate θ > L, and the covarance between the n 1 other players margnal benefts and ther leakage rates s non-negatve cov(b, L ) The latter condton ensures that those players that cut effort back more strongly are also those 29 Our assumpton that players hold Nash conectures when choosng effort levels at Date 2 also does not seem crtcal for our results (that s, we could let conecture a non-zero response by when choosng X ). 30 In an aggregatve game, each player s payoff depends only on her own acton and a summary statstc of all other players actons (n our case, the unweghted sum of others efforts). 31 We note that many of our basc nsghts would also apply n settngs n whch some player-specfc leakage rates are zero (or even negatve), so long as the overall leakage rate remans suff cently hgh. 32 Formally, we defne ths covarance based the followng: ( ) ( ) (n 1) B (L L ) = (n 1) B (n 1) (L L) cov(b, L ). 17