Voluntary Contribution to Public Goods: Preferences and Wealth

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1 Theoretcal Economcs Letters, 06, 6, Publshed Onlne June 06 n ScRes. Voluntary Contrbuton to Publc Goods: Preferences and Wealth Oskar Nupa Department of Economcs, Unversdad de los Andes, Bogotá, Colomba Receved 8 Aprl 06; accepted 3 May 06; publshed 3 June 06 Copyrght 06 by author and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract Ths paper analyzes the role of wealth and ts dstrbuton n a model of voluntary-contrbuton to publc goods where ndvduals preferences for these goods depend on the level of ndvdual s wealth. We fnd that under such crcumstances a small redstrbuton of wealth among contrbutors s not necessary neutral. We then dscuss the condtons under whch a redstrbuton of wealth mght mprove effcency n the aggregate provson of publc goods. Unlkely what common sense mght suggest, we show that a redstrbuton of wealth can ncrease the prvate provson of publc goods regardless of whether ndvduals valuatons for these goods ncrease or decrease as ther level of wealth ncreases. Furthermore, we show that under some crcumstances, a regressve redstrbuton of wealth can ncrease the provson of publc goods. Keywords Publc Goods, Prvate Provson, Neutralty, Income Dstrbuton. Introducton We analyze n ths paper the role of wealth and ts dstrbuton n a model of voluntary-contrbuton to pure publc goods where ndvdual s preferences for these goods not only are determned by the respectve level of ndvdual s wealth, but also are affected by changes n ths wealth. Our postve analyss s motvated by the followng observaton. It s common n the lterature on publc goods to consder that ndvduals preferences (valuaton) for these goods mght be determned by the level of wealth (we dscuss ths lterature below). Nevertheless, when consderng ths possblty, ths lterature has nexplcably assumed that changes n ndvdual wealth do not affect ndvdual s preferences for the sad goods. From our pont of vew, ths s a clear theoretcal nconsstency that mght lead to msleadng analyss results. In ths study, we take nto account ths consderaton nto the standard mechansm of voluntary-contrbuton to publc goods and fnd some novel results. Be- How to cte ths paper: Nupa, O. (06) Voluntary Contrbuton to Publc Goods: Preferences and Wealth. Theoretcal Economcs Letters, 6,

2 O. Nupa sdes offerng a proper analyss of the prvate provson of publc goods under wealth dependent preferences, our analyss s of nterest because, as we dscuss below, there are several cases n whch ndvdual valuaton for publc goods can be affected by ther level of wealth. In our framework, pure publc goods are defned n the usual way: goods that are non-rval (consumpton of the good by one person does not reduce the consumpton avalable to another person) and non-excludable (once the good has been provded to one consumer, t s not possble to prevent other people from consumng t). The manner n whch we ntroduce our observaton (.e., that ndvdual s preferences for pure publc goods depend on the respectve level of ndvdual s wealth) nto the model of voluntary-contrbuton to publc goods s through the ndvdual margnal rate of substtuton (MRS) between the consumpton of a prvate good and the consumpton of a publc good. More precsely, we assume that the (absolute value of each) ndvdual s MRS s a functon of hs or her level of wealth. Three man and novel fndngs are obtaned from our analyss. Frst, a small redstrbuton of wealth among contrbutors s not necessary neutral. Second, unlkely what common sense mght suggest, a redstrbuton of wealth from a rcher contrbutor to a poorer contrbutor can ncrease the prvate provson of publc goods regardless of whether ndvduals valuaton for these goods ncrease or decrease as ther level of wealth ncreases. Thrd, under some crcumstances, a small redstrbuton of wealth from a poorer contrbutor to a rcher contrbutor can ncrease the provson of publc goods. We characterze the condtons under whch each of these possbltes occurs. By consderng a varety of crcumstances, several authors have analyzed the role of wealth and ts dstrbuton n a voluntary-contrbuton mechansm to publc goods. No one has analyzed (properly) ths under wealth dependent preferences. Furthermore, our results dffer from prevous results n the lterature under stuatons that can be apparently smlar to our case of study. For nstance, [] studed the effect of a redstrbuton of wealth on the fnal provson of publc goods when the margnal cost of provdng these goods vares across contrbutors. In ths context, they fnd that a transfer from a contrbutor wth a hgh margnal cost to a contrbutor wth a low margnal cost always ncreases equlbrum provson. The remander of ths paper s organzed as follows. In Secton, we motvate n more detal both our man assumpton and our analyss. Secton 3 presents a smple example to llustrate our man pont, and Secton 4 develops a general framework and presents the man results. Secton 5 concludes.. Wealth and Preferences for Publc Goods As sad above, t s common n the lterature on publc goods to consder that ndvduals preferences for these goods mght be determned by the level of wealth. There are several examples of ths. For nstance, the so-called ncome segregaton hypothess rests on the assumpton that rcher households value publc goods less than poorer households []-[4]. The opposte has been assumed by [5] n ther analyss of rent-seekng for pure publc goods (see pp. 54). Furthermore, lterature on prvate provson of publc goods has also consdered the possblty of dfferences n ndvdual s preferences caused by dfferences n ndvduals levels of wealth and, more generally, dfferences n ndvduals tastes (for nstances, see [] [6], pp ). 3 Nevertheless, although ths lterature has admtted that ndvduals preferences for publc goods mght be determned by the level of ndvduals wealth, ths has nexplcably assumed that changes n ndvdual wealth do not affect ndvdual s preferences for the sad goods. Does t make any sense? From our pont of vew, ths s a clear theoretcal nconsstency. If one consder that ndvduals preferences for certan types of publc goods are determned by ther level of wealth, then t must be true that changes n the level of ndvduals wealth must affect sad ndvduals preferences. Besdes ths theoretcal nconsstency n the lterature, there are several cases n whch ndvdual valuaton for A surprsng result n the lterature on the prvate provson of publc goods s that a small redstrbuton of wealth among those ndvduals makng a strctly postve contrbuton to the producton of a publc good does not have any effect on ts aggregate provson the so-called neutralty theorem [] []. It has been clamed that ths result holds true regardless of dfferences n ndvdual preferences. These crcumstances are: the presence of corner solutons []; mpure publc goods [3] [4]; large economes [5] [6]; non-nash behavor, such as the presence of nonzero conjectural varatons [7]; nformaton asymmetres [8]; nferor publc goods [9]; dfferences n the margnal cost of provdng publc goods [0] []. 3 Other studes have also assumed that ndvduals preferences can be affected by some of the usual exogenous varables n a decson model For nstance, many studes have analyzed the demand for prvate goods when ndvduals preferences are affected by market prces. Poneer studes n ths feld are [7] [8]. 433

3 O. Nupa publc goods can be affected by ther level of wealth. Frstly, consder envronmental qualty. Rcher ndvduals may value ther health and the health of ther chldren more than poorer ndvduals, because the opportunty cost of gettng sck or takng care of ther chldren s greater for them. Snce envronmental qualty mght mportantly affect ndvdual health, rcher ndvduals valuaton of envronmental qualty should also be greater than that of poorer ndvduals. Addtonally, changes n ndvdual wealth should affect ths valuaton. There s suffcent evdence demonstratng that ndvduals margnal wllngness to pay for envronmental goods ncreases as ndvdual wealth ncreases [9]-[3]. Although t s dffcult to know from these estmatons whether dfferences n the margnal wllngness to pay for such goods are due to dfferences n preferences, n the budget constrant or both, more recent emprcal analyses drectly suggest that ndvduals MRSs between prvate goods and envronmental qualty change as ndvduals level of wealth changes [4] [5]. In partcular, these studes fnd that rcher ndvduals are able to substtute a greater amount of prvate goods for a gven ncrement n envronmental qualty than poorer ndvduals. Another case n whch ndvduals valuatons of a publc good may also ncrease wth the level of wealth s publc safety. Snce rcher ndvduals are more attractve to crmnals than poorer ones, the former may value neghborhood publc safety more than the latter. Usng survey nformaton for Brazl s neghborhoods, [6] compute that a household s concern for publc safety ncreases as ts level of wealth ncreases. Although they relate ths result to a strong ncome effect, they also clam that t can be partly explaned by ncrements n the margnal utlty of publc safety as a household s wealth ncreases. One can also thnk of cases where the valuaton of publc goods may be negatvely correlated wth the level of wealth. For nstance, t may be that rcher ndvduals place less value on publc parks, nasmuch as they have access to prvate (excludble) spaces wth smlar characterstcs; poorer ndvduals, on the other hand, may value publc parks more, as they do not have access to such spaces. In ths case, an ndvdual s valuaton of the publc good ncreases as the ndvdual s wealth decreases. Unfortunately, as far as we know, no evdence to ths effect s currently avalable. 3. An Example: Log-Lnear Preferences To llustrate both how ndvduals preferences can be affected by the level of wealth, and how wealth and ts dstrbuton affect the fnal provson of publc goods n a voluntary-contrbuton mechansm, we begn our analyss wth a smple example n whch ndvduals have log-lnear preferences. Let us assume that there are two ndvduals ( =, ) who are nterested n producng a publc good. Each ndvdual has to decde how to allocate hs or her wealth ( w ) between the consumpton of a prvate good ( x ), and hs or her contrbuton ( g ) to the producton of a publc good. Thus, the ndvdual budget constrant of each ndvdual s gven as w px + g, where p s the relatve prce of the prvate good. The publc good s produced usng standard technology G = g+ g, where G s the total amount of the publc good produced. Each ndvdual decdes ndependently and smultaneously how much money to contrbute to the producton of the publc good. Wthout loss of generalty, through ths example, we assume that w > w. The new assumpton we ntroduce nto ths framework s the followng: preference ntenstes of ndvduals and for the publc good depend on ther respectve levels of wealth. Therefore, the log-lnear utlty functon for each ndvdual can be represented as u = α ln x + β( w) ln G, where, for smplcty, β s assumed to be a contnuous monotone functon of that ndvdual s level of wealth. Therefore, f β s an ncreasng functon of wealth, the ndvdual s valuaton of the publc good ncreases as hs level of wealth nβ. s a decreasng functon of wealth. Notce that, under these crcums- creases. The opposte happens f tances, the absolute value of the ndvdual s MRS, gven by β. > 0, MRS ncreases as ww ncreases. The opposte hap- vdual s respectve level of wealth. Hence, f pens f β < 0. MRS β. x =, s also a functon of the ndα G Each ndvdual smultaneously chooses x 0, and g 0 n order to maxmze hs utlty, subject to hs budget constrant and the publc good s producton technology, and takng the contrbuton of the other ndvd- ual as gven. In an nteror soluton, the requred Nash equlbrum condton for each s gven by MRS =. p Takng ths condton for each, and the respectve ndvduals budget constrants, ths s obtaned that, n equ- 434

4 O. Nupa lbrum, the aggregate provson of the publc good s gven by: W G = + α β w + α β w () where W = w w. + Remarkably, the prvate provson of G depends not only on the total wealth of the two ndvduals (W), but also on each ndvdual s respectve level of wealth, whch affects G through each nd- vdual s respectve preference ntensty for the publc good, β ( w ). Equaton () shows the dfference between properly and mproperly consderng the wealth dependent preferences assumpton n a voluntary-contrbuton mechansm. As sad before, n the standard analyss ths s common to assume that the valuaton of the rcher ndvdual s larger (or smaller) than the valuaton of the poorer ndvdual, but, that these valuatons are not affected by changes n ndvduals levels of wealth. In other words, n ths analyss β ( w ) s replacng by β, β ( w ) s replacng by β, and there s assumed that ether β > β or β < β. In ths case, the aggregate provson of the publc good n equlbrum s then gven by W G =, and ths only depends on the total amount of wealth (W), but not on ts dstrbuton between the two contrbutors. + α β + α β We can use Equaton () to study the effect of changes n an ndvdual s wealth on G. Dfferentatng G wth respect to w, we get: where z = α α + 0 β + >, and ( w ) β( w ) = + z ( w ) α ( w ) β( w ) ( θ G ) β θ =. The frst term n the parentheses n Equaton () β (multpled by z ) s always postve and corresponds to the ncrement n the ndvdual allocaton to publc good producton ( g ) due to an expanson n the respectve ndvdual s budget set. Actually, ths s the unque effect of w on G when ndvdual s preferences for G are not affected by changes n hs level of wealth. The second term n the parentheses n Equaton () emerges from consderng that ndvdual s preferences for G change as hs level of wealth changes. θ = 0 f ths consderaton s not taken nto account. Ths term s postve f and only f β > 0, and negatve f and only f β < 0. Interestngly, when the ndvdual s valuaton for the publc good decreases as hs respectve level of wealth ncreases, the fnal effect of an ncrement n w on G can be negatve. In ths case, the publc good s an nferor good. We can use Equaton () to study how a redstrbuton of wealth between ndvduals and affects the fnal provson of G. Let us consder a small progressve redstrbuton of wealth.e., a redstrbuton of wealth from ndvdual (the rcher ndvdual) to ndvdual (the poorer ndvdual), wth W remanng unchanged, and both ndvduals stll contrbutng a postve amount of resources to the publc good after redstrbuton. The fnal effect of ths redstrbuton on the total provson of publc good s gven by G =. Usng Equaton (), G transforms nto: G G = ( θ θ) (3) z G Snce 0 z >, the fnal effect of a progressve redstrbuton of wealth on the aggregate provson of G depends on the sgn of the expresson n parentheses n Equaton (3). There are two terms nvolved n θ. The frst β ( w ) term,, measures ndvdual s relatve change n hs preference ntensty for the publc good as w β w changes. The second term, α β w, measures s spendng on the prvate good relatve to total spendng n G. 4 () 4 In equlbrum, MRS ( w ) x β = =. From ths, t follows that α G p α β ( w ) px =. G 435

5 O. Nupa The larger ths term.e., the more mportant s ntal spendng on the prvate good the greater wll be the mpact of a change n s preference ntensty for the publc good on the aggregate provson of G. Thus, the effect of a redstrbuton of wealth on the fnal provson of G depends not only on how ndvduals preferences for a publc good change as ther respectve levels of wealth change, but also on how mportant ther respectve spendng on the prvate good s pror to the redstrbuton. 5 Once agan, f θ = 0, the wealth redstrbuton s totally neutral. α α To fx deas on how a redstrbuton of wealth can affect the level of G, let us assume that =. 6 β w β w Imposng ths assumpton n Equaton (3), t follows that: ( w ) ( w ) sgn( G ) sgn β β = β β The effect of a redstrbuton of wealth on G now depends exclusvely on the dfference between changes n the respectve ndvduals ntensty preferences for the publc good as w changes. The sgn of ths dfference depends on the assumptons mposed on β. For nstance, assumng that β > 0, and β 0, t follows from Equaton (4) that a progressve redstrbuton of wealth wll ncrease the aggregate provson of G. However, assumng that ndvdual s preferences for G ncrease as ther respectve levels of wealth ncrease s not enough to obtan ths result. If β > 0, but β > 0, a progressve redstrbuton of wealth mght decrease the aggregate provson of G. The same type of analyss can be done for the case where β < 0. The example n ths secton llustrates both how ndvduals preferences are affected by ndvduals levels of wealth when one consstently consder ths possblty, and how wealth and ts dstrbuton affect the fnal provson of publc goods n a voluntary-contrbuton mechansm. In the next secton, we generalze these fndngs. 4. A General Framework Consder the general model for the prvate provson of publc goods, wheren there s one publc good (G), one prvate good (x) and n consumers. Same as before, each consumer decdes how to allocate hs or her total exogenous wealth ( w > 0 ) between consumpton of the prvate good, and contrbutng to the provson of the publc good ( g ). The ndvdual budget constrant s gven as w px + g, where p s the relatve prce of the prvate good. Indvduals preferences are represented by the utlty functon u = u( x, G), whch s assumed to be strctly quas-concave, strctly ncreasng for both arguments, and twce-contnuously dfferentable. The n publc good s produced usng technology G = g =. We assume that ndvdual preferences are affected by respectve levels of ndvdual wealth. More precsely, u we assume that the absolute value of each ndvdual s MRS (.e., MRS = ) s not only a functon of u x the respectve level of x and G, but also a functon of wealth. Formally, we assume that: MRS = H x, G, w (5) ( ) If rcher ndvduals value G more than poorer ndvduals, then functon wealth ncreases. Conversely, than poorer ndvduals. We assume that H ncreases as ndvdual H decreases as ndvdual wealth ncreases f rcher ndvduals value G less H s a contnuous monotone functon of wealth. We concentrate on ths case because, as we mentoned n Secton, ths seems to be the more lkely stuaton for certan publc goods. However, our analyss can be extended to analyze the case where H s a non-monotone functon of wealth. Snce wthn our framework ndvdual wealth s totally exogenous, ndvdual preferences are stll determned only by exogenous parameters. 5 There s another way to understand the two terms n parentheses n Equaton (3). Actually, n the next secton, we relate these terms to changes n the (absolute value of) the ndvdual s MRS generated by changes n ndvdual s wealth and changes n prvate consumpton. 6 Ths assumpton mples several thngs. Frst of all, snce MRS MRS = our example, ths assumpton mples that MRS β ( w ) x α α =, t follows that α G β ( w ) β ( w ) (4) = f and only f. Actually, as noted n footnote 5, we use ths last nterpretaton n our analyss n the next secton. Notce also that n x = x. 436

6 O. Nupa Under the condtons descrbed above, each of n ndvduals smultaneously chose x 0, and g 0 n order to maxmze hs utlty, subject to hs budget constrant and the publc good s producton technology, and takng the contrbuton of the other ndvduals as gven. At an nteror soluton, the necessary Nash Equlbrum condton for each s gven by: 7 MRS H ( x, G, w ) = = (6) p Equaton (6) mplctly defnes each contrbutor s best response contrbuton to the publc good. For non-contrbutors, the best response s zero contrbuton to the publc good. 8 We denote by C the set of contrbutors, and by c n the number of contrbutors. Furthermore, wthout loss of generalty, we assume that those ndvduals =,,, c are contrbutors. Assumng that the condtons of the mplct functon theorem are satsfed, t follows from Equaton (6) that for each C : (,, ) x = x pgw (7) Pluggng Equaton (7) nto the ndvdual budget constrant and summng up over those s C, we get: Let W C C w = p x ( pgw,, ) + G (8) = w ; applyng the mplct functon theorem then, we obtan: C C G GW pw w w =,,,,, c (9) Equaton (9) represents aggregate prvate provson of the publc good n ths economy. Snce for a gven vector of ndvduals respectve wealth, ( w,, wn ), each consumer s maxmzaton problem s exactly the same as hs maxmzaton problem n the standard model of the prvate provson of publc goods, the exstence of an equlbrum n our framework s already a well-known result (see []). 4.. Changes n Wealth We begn by analyzng how changes n a contrbutor s level of wealth affect the prvate provson of G. We state ths result n Proposton. Proposton. Assume that MRS = H( x, G, w). For any contrbutor, > 0 f and only f H > ; and p H x < 0 f and only f H <. p H x Proof. Applyng mplct dfferentaton to Equaton (8), for any contrbutor, we get that, = p, Z H where Z = + p. It follows from Equaton (6) that =. Thus, the effect of an n- C H crease n a contrbutor level of wealth can be wrtten as: It also follows from Equaton (6) that 7 Notce that functon ndvduals for the term α. 8 An ndvdual s a contrbutor f H. = + p Z H. x (0) H =. Gven the propertes of the utlty functon, t fol- H H represents a partcular functon for each. For nstance, n our example n Secton 3, ths functon dffers across (, ) (, ) u w G u w G > p ; otherwse, ndvdual s a non-contrbutors. 437

7 O. Nupa lows that H > 0 and H < 0 for all contrbutor. Therefore, Z > 0. Furthermore, the sgn of depends on the sgn of the rght-hand term n the parentheses of Equaton 0. The result n Proposton follows from the sgn of ths term. QED Result n Proposton states how the aggregate provson of G changes as contrbutor s wealth changes. p H(.) Equaton (0) nvolves both the standard wealth effect, measured by (/Z), and a new effect gven by. Z H. Ths last effect accounts for the change n G due to the change n ndvdual s preferences as hs level of H wealth changes. Notce that s always postve f > 0. Nevertheless, can be negatve f both H < 0, and ths effect domnates the standard wealth effect (/Z). Proposton rases the possblty that the publc good n our framework can be an nferor good. When ths s the case, although equlbrum exstence s guaranteed (see []), unqueness s not (see [9] for some examples). In ths context, unqueness means that there s a unque Nash equlbrum (.e. a unque vector of contrbutons) wth a unque quantty of publc good and a unque set of contrbutors. The exstence of multple equlbra makes t dffcult to analyze how any redstrbuton of wealth affects the provson of publc good n our framework. For ths reason and smlar to [] and [] (see Theorem, pp. 9) n what follows, we concentrate on analyzng the effect of small redstrbutons of wealth (.e., those redstrbutons that only nvolve contrbutors and that do not affect the set of contrbutors) on the aggregate provson of G. 4.. Changes n the Dstrbuton of Wealth Let us now study how a small redstrbuton of wealth affects the aggregate provson of G n our framework. We state ths result n Proposton. Proposton. Let us consder contrbutors and wth w > w, and assume that MRS = H( x, G, w). a) A small redstrbuton of wealth, ether progressve (from to ) or regressve (from to ) s not neutral f H(.) H(.) and only f. H. H. b) If c) If H. H. >, a small progressve redstrbuton of wealth ncreases G. H. H. H. H. <, a small progressve redstrbuton of wealth decreases G. Furthermore, n H. H. ths case, a small regressve redstrbuton of wealth ncreases G. Proof. Consder contrbutors and wth w > w. The effect of a small progressve redstrbuton of wealth (from to ), keepng W unchanged, s gven by some smple algebrac manpulaton, ths equaton transforms nto: G = p H. H. Z H. x H. x = G. Replacng the dervatves and carryng out p If the redstrbuton s regressve, then the change n G s gven G. Snce 0 Z >, the results n Proposton follow from Equaton (). QED Proposton (a) states the condton under whch a small redstrbuton of wealth n not neutral. Clearly, ths H redstrbuton s neutral f, as t has been assumed n prevous lterature, = 0. Neutralty also emerges () 438

8 O. Nupa under the same mplct condtons that the standard model of the prvate provson of publc goods mposes on ndvduals behavor. As [] clam, n the standard framework, after a redstrbuton of wealth between contrbutors, each ndvdual consumes exactly the same amount of the prvate good that he dd pror to the redstrbuton. In other words, the standard model mplctly mposes the condton that when there s a redstrbuton of wealth, then = 0 for all the contrbutors. Actually, ths condton s unlkely to happen. Notce that, Equ- aton () can be also wrtten as G = p +. Therefore, f the condton that the standard model mposes on each ndvdual s behavor holds (.e. = 0 for all contrbutors), then t mmedately follows that a redstrbuton of wealth n our framework s also neutral. Moreover, our model shows that ths s not the only case where one can expect neutralty followng a redstrbuton of wealth. It can also happen f s equal for all the contrbutors nvolved n the redstrbuton. Results n Proposton (b) and (c) state the condtons under whch a progressve/regressve redstrbuton of wealth among contrbutors ncreases/decreases the aggregate provson of G. Ths result depends on the sgn of the rght-hand term n parentheses n Equaton (). For each ndvdual ( and ), ths expresson nvolves two H terms. The frst term s, whch captures the change n s preference ntensty for the publc good H (vs-a-vs the prvate good) as hs level of wealth changes. The other term,, captures the change n s preference ntensty for the publc good as hs prvate consumpton changes. Notce that, gven our (standard) assumptons on u, ths last term s always postve. The rato between these two terms then can be understood as the relatve change n the ndvdual s preference ntensty for publc goods (as wealth changes). 9 Therefore, results n Proposton (b) and (c) ndcate that f the relatve change n the ndvdual s preference ntensty for publc goods s larger for ndvdual (the poorer ndvdual) than for ndvdual (the rcher ndvdual), then a redstrbuton of wealth from to always ncreases the fnal provson of G. When the opposte happens, a redstrbuton of wealth from to always decreases the fnal provson of G. In order to ncreases G n ths case then, a regressve redstrbuton of wealth s requred. Thus, our results allow us to antcpate whch type of redstrbuton could be most useful for mprovng the effcency of the aggregate provson of G. Interestngly, to the extent that we assume that H result does not depend on the sgn of H s a monotone functon of an ndvdual s wealth, our. In other words, when an ndvdual s valuaton for G ncreases as hs or her respectve level of wealth ncreases, t s possble to observe ether an ncrement or a decrease n the fnal provson of G followng a progressve redstrbuton of wealth. The same thng happens when an ndvdual s valuaton for G decreases as hs or her respectve level of wealth ncreases. Notce that ths holds even f we H(.) H(.) assume that =, as we dd n the example n Secton 3. As Proposton ndcates, what matters n determnng the fnal effect of a redstrbuton of wealth on G s how the magntudes of the relatve change n the ndvdual s preference ntensty for publc goods compares across those ndvduals nvolved n the redstrbuton of wealth. 9 Notce that Equaton (3) n our example n Secton can be rewrtten as G α x =. Snce n ths case MRS p β ( w ) G = β w x β w x p αg αg z αg β( w) αg β w, where β x =, Equaton (3) can be drectly obtaned from Equaton (). Thus, as antcpated n Sec- α G ton 3, an analyss of Equaton (3) can be also done n terms of the changes n the absolute value of ndvduals MRSs as respectve ndvduals levels of wealth and prvate consumpton change. 439

9 O. Nupa As noted above, Equaton () can be also used to study the effect on G of a redstrbuton of wealth when H functon H s assumed to be a non-monotone functon of wealth. If ths s the case, the sgn for could be mportant for determnng how a redstrbuton of wealth affects G. 5. Conclusons In ths paper, we analyzed the role of wealth and ts dstrbuton n a model of voluntary-contrbuton to publc goods when ndvduals preferences for these goods are affected by the respectve level of ndvdual wealth. We fnd that, under these condtons, a small redstrbuton of wealth among contrbutor s not necessary neutral and that, unlkely what common sense mght suggest, redstrbutng wealth from a rcher contrbutor to a poorer contrbutor can ncrease the prvate provson of publc goods regardless of whether ndvduals valuatons for these goods ncrease or decrease as ther wealth ncreases. Furthermore, we show that under some crcumstances, a small regressve redstrbuton of wealth (from a poor ndvdual to a rch ndvdual) mght ncrease the provson of publc goods. Whch type of wealth redstrbuton mght be most useful n reducng the under-provson of publc goods n our framework depends on how the relatve change n ndvduals MRSs compares across ndvduals. Ths relatve change s defned as the rato between the change n the absolute value of the ndvdual s MRS as hs or her level of wealth changes, and the change n the absolute value of the ndvdual s MRS as hs or her prvate consumpton changes. If ths relatve change n MRSs s greater for poorer ndvduals than for rcher ndvduals, then a progressve redstrbuton of wealth wll always ncrease the fnal provson of G. When the opposte happens, a regressve dstrbuton of wealth s needed n order to ncrease the fnal provson of G. Our analyss calls the attenton on how to consder the assumpton of wealth dependent preferences for publc goods properly. Omttng that these preferences not only dffer across ndvduals wth dfferent levels of wealth but also are affected by changes n these levels mght lead to msleadng analyss results. Fnally, our results create space for publc nterventon for those publc goods where ndvduals valuatons are affected by ther respectve wealth. As mentoned n Secton, there are several cases n whch ndvdual valuaton for publc goods can be affected by ther level of wealth. Acknowledgements I thank Rchard Cornes, Jorge Garca, Perre Pesteau and Julo Robledo for very useful comments on a prevous verson of ths paper. References [] Warr, P. (983) The Prvate Provson of a Publc Good Is Independent on the Dstrbuton of Income. Economc Letters, 3, [] Bergstrom, T., Blume, L. and Varan, H. (986) On the Prvate Provson of Publc Goods. Journal of Publc Economcs, 9, [3] Cornes, R. and Sandler, T. (994) The Comparatve Statc Propertes of the Impure Publc Good Model. Journal of Publc Economcs, 54, [4] Andreon, J. (990) Altrusm and Donatons to Publc Goods: A Theory of Warm-Glow Gvng. The Economc Journal, 00, [5] Andreon, J. (988) Prvately Provded Publc Goods n a Large Economy: The Lmts of Altrusm. Journal of Publc Economcs, 35, [6] Fres, T., Goldng, E. and Romano, R. (99) Prvate Provson of Publc Goods and the Falure of the Neutralty Property n Large Fnte Economes. Internatonal Economy Revew, 3, [7] Sandler, T. and Posnett, J. (99) The Prvate Provson of Publc Goods: A Perspectve of Neutralty. Publc Fnance Revew, 9, [8] Gradsten, M., Ntzan, S. and Slutsky, S. (994) Neutralty and the Prvate Provson of Publc Goods wth Incomplete Informaton. Economcs Letters, 46, [9] Kerschbamer, R. and Puppe, C. (998) Voluntary Contrbutons When the publc Good Is Not Necessarly Normal. Journal of Economcs, 68,

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