Provision of public goods in a large economy

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1 Economcs Letters 61 (1998) Provson of publc goods n a large economy Mark Gradsten* Ben-Guron Unversty and the Unversty of Pennsylvana, Pennsylvana, USA Receved 13 Aprl 1998; accepted 25 June 1998 Abstract In ths paper, I consder a large economy wth a prvate and a publc good where the dstrbuton of costs (but not the ndvdual costs) assocated wth the provson of the publc good s known. A smple tax-subsdy polcy, whereby all contrbutors get a subsdy funded by a tax on non-contrbutors, s shown to attan the frst best allocaton provded that the socally optmal amount of the publc good s ndependent of the dstrbuton of ndvdual costs Publshed by Elsever Scence S.A. All rghts reserved. Keywords: Provson of publc goods JEL classfcaton: H4 1. Introducton It s well known that prvate provson of publc goods s neffcent, and that neffcency can only be aggravated n a large economy see Andreon, 1988; Fres et al., 1991 and Gradsten, 1992, for some recent studes of prvate provson of publc goods n a large economy. Government nterventon mght therefore be needed to restore the frst best allocaton. Ths s easly acheved when nformaton (on ndvdual tastes, producton technology, etc.) s complete, and the set of polcy nstruments s unlmted. For example, the government can attan the frst best by mposng Lndahl taxes, and such polcy benefts everyone. A more delcate stuaton emerges when nformaton s partal. For nstance, whle the dstrbuton of some characterstcs n the populaton may be known, each ndvdual s characterstcs may be not. In ths case, government polces cannot be condtoned on ndvdual characterstcs, although they may possbly depend on ndvdual actons. Furthermore, they must be coercve n the sense that some *Correspondng author. Tel.: ; fax: ; e-mal: grad@econ.sas.upenn.edu 1 The mechansm desgn lterature, whch seeks means of attanng effcent allocatons under ncomplete nformaton, has yelded many negatve results resultng n the mpossblty of voluntary and effcent provson of publc goods, thus mplyng necessty of coercve nterventon to attan effcency see Laffont and Maskn, 1979; Myerson and Satterhwate, 1983 and Rob, 1989, for some of the most drectly related lterature. Coercon can be avoded wth some specfc assumptons on the feasble set of economc envronments, see Gradsten, 1994, or under specfc allocaton of property rghts, see Neeman, / 98/ $ see front matter 1998 Publshed by Elsever Scence S.A. All rghts reserved. PII: S (98)159-1

2 23 M. Gradsten / Economcs Letters 61 (1998) ndvduals have to be forced to comply wth the polcy. For nstance, Lndahl taxes requre 1 nformaton on preferences and ncomes that ndvduals are not generally wllng to reveal. In ths paper, I consder a large economy where the cost assocated wth one s contrbuton to the publc good s prvately known, and the government acts knowng only the dstrbuton of these costs n the populaton. A smple tax-subsdy polcy s exhbted, where all contrbutors get a subsdy funded by a tax on non-contrbutors, whch attans the frst best allocaton provded that the socally optmal amount of the publc good s ndependent of the dstrbuton of ndvdual costs. Snce some ndvduals may oppose ths polcy, t turns out that ts mplementaton may nvolve coercon; and ts poltcal feasblty n a democratc regme hnges on the exstence of a suffcently large supportng coalton. These results are shown to hold under a varety of assumptons about the technology of publc goods provson, n partcular, they hold whether the publc good s contnuous or dscrete. 2. The basc model Consder an economy of ndvduals whose measure s the unt nterval. There are two goods one of whch (X) s purely publc and another (Y) prvate. Each ndvdual s endowed wth one ndvsble unt of the prvate good the value of whch s c. These values are assumed to be ndvdually specfc and prvately known; ther dstrbuton, G(c), however, s assumed to be common knowledge. The support of ths dstrbuton s assumed to be the unt nterval, and the dstrbuton functon s assumed to be contnuously dfferentable. Indvduals can partcpate n the provson of the publc good by contrbutng the prvate good. Lettng q denote the proporton of contrbutng ndvduals, X 5 H(q) s the amount of the publc good produced. H s the producton functon assumed to be monotonc wth H() 5 ; more detaled technology assumptons are lad down below. Let z denote ndvdual s decson varable, such that z 5 1 s the decson to contrbute to the provson of the publc good, and z 5 denotes the decson to refran from contrbuton. We assume that ndvdual tastes are dentcal and representable by an addtvely separable utlty functon, whch, after takng nto account the budget constrant can be wrtten as follows: V(X) 1 Y 5V(H(q)) cz (1) 2 where V s assumed to be monotonc and concave. Note that, wth our assumptons, c can also be nterpreted as the opportunty cost of a contrbuton to the publc good; we refer to t below as a contrbuton cost. We dstngush the followng producton technologes for the provson of the publc good Contnuous publc goods In ths case, the producton functon satsfes the standard concavty assumpton, H,. In the socal optmum, only ndvduals wth suffcently low costs contrbute to the provson of the publc 2 Ths paper consders standard pure publc goods, wthout the joy of gvng component (see Andreon, 1989, for a comprehensve study of the latter), although the man results could be extended to account for that possblty.

3 M. Gradsten / Economcs Letters 61 (1998) good. Lettng c denote the socally optmal threshold level such that only ndvduals wth a contrbuton cost below c make a contrbuton, ths threshold maxmzes the socal welfare functon (a constant whch represents the amount of the prvate good n the economy s omtted throughout) c 1 c E [V(H(G(c ))) 2 c]dg(c) 1E [V(H(G(c )))]dg(c) 5V(H(G(c )) 2E cdg(c) (2) c The socal welfare s gven by the dfference between the socal value of the publc good and the total contrbuton costs. Dfferentaton of (2) wth respect to c reveals the followng frst order condton: 2 c 1V9(H(G(c )))H9(G(c )) 5 (3) and our assumptons guarantee that the second order condton s satsfed. Assumng that V 9(H(1))H9(1),1 guarantees the exstence of a unque nternal soluton. In equlbrum, each ndvdual decdes whether or not to contrbute to the provson of the publc good on hs own. Snce n our large economy the level of the publc good s unaffected by a sngle ndvdual s contrbuton decson, t s clear that at equlbrum all ndvduals have an ncentve to free rde. Hence the equlbrum proporton of contrbutors to the publc good, and therefore the amount of the publc good, H(q*), equal. In contrast, the socally optmal amount, gven by (3), s strctly postve Dscrete publc goods One partcular type of publc goods, whch has receved a consderable attenton n the lterature, has a producton functon of the followng nature: the good s provded f and only f a certan mnmal requrement on the proporton of partcpants s met. Ths s the case for referendum votng and, more generally, for any publc choce procedure whch requres a quorum of partcpants. Publc goods wth ths feature have recently been studed n several theoretcal and emprcal papers see, for nstance, Dawes, 198; Ntzan and Romano, 199; Palfrey and Rosenthal, 1988, 1991; Van de Kragt et al., 1983; Schellng, 1978, contans examples of ths knd of a publc good. Let g denote the mnmal proporton of ndvduals requred for the provson of the above descrbed publc good. By arbtrarly choosng the scale, the producton functon can be descrbed as follows: H(q)5 f q,g, and H(q)51 f q$g. Let V(1)5v be the value assgned by the ndvduals to the publc good; then the frst best socal optmum requres that the publc good wll only be provded f where c v 2E cdg(c). (4) 21 c 5 G (g ) (5) n whch case ndvduals whose cost s less than c contrbute.

4 232 M. Gradsten / Economcs Letters 61 (1998) At equlbrum, on the other hand, the same consderatons as before ensure that the proporton of contrbutors s zero and therefore suboptmal. 3. Government nterventon Snce the government lacks suffcent nformaton on the ndvdual costs, t cannot smply pck the correct low cost contrbutors as requred by the socal welfare maxmzaton. Therefore, I consder a tax/subsdy scheme under whch each contrbutor s gven a subsdy of s, and each non-contrbutor s taxed by the amount of t. I also mpose the requrement of a balanced budget. Effcency requres that 3 the equlbrum of the game mples the socally optmal cutoff pont c. Note that the tax subsdy scheme preserves the socal optmalty condton because of the absence of ncome effects. Budget balance thus mples: sg(c ) 5 t[1 2 G(c )] (6) For c to consttute the equlbrum contrbuton threshold, the c ndvdual type must be ndfferent between makng a contrbuton and refranng from contrbutng. In other words, the followng should hold true: That s, V(H(G(c ))) 2 c 1 s 5V(H(G(c ))) 2 t (7) s 1 t 5 c (79) Combnng (6) and (79) we get t 5 cg(c ), s 5 c [1 2 G(c )] (8) where c s gven by (3). In Eq. (8) s determned the (unque) tax subsdy rates needed to mplement the frst best allocaton. Under ths polcy, the tax on non-contrbutors s the threshold contrbuton cost weghed by the fracton of the populaton whch contrbutes, and the subsdy to contrbutors s the threshold cost weghed by the fracton of non-contrbutors. Snce a hgher socally optmal threshold mples both a hgher threshold cost and a greater fracton of contrbutors, the tax ncreases as the socally optmal threshold ncreases. In contrast, the subsdy may go ether way. It defntely ncreases wth c for c close to and defntely falls wth c for c close to 1. Note that the optmal polcy depends not on personal characterstcs, nor on the producton functon, but only on ndvdual actons, and ts economc logc can be traced to Pgovan correctve subsdes. The ndvdual utltes under ths scheme are as follows: V(H(G(c ))) 2 c G(c ) for non 2 contrbutors (9) V(H(G(c ))) 2 c 1 c [1 2 G(c )] for contrbutors 3 Brto et al., 1991, study such polcy n a related context.

5 M. Gradsten / Economcs Letters 61 (1998) The equlbrum utlty level s a decreasng functon of the contrbuton cost; n partcular, snce the contrbuton cost of all contrbutors s less than the threshold socally optmal cost, at equlbrum the contrbutors (ndvduals wth low costs) are better off than the non-contrbutors. Furthermore, some ndvduals notably, the non-contrbutors who are requred to pay taxes may become worse off wth ths polcy. Turnng to the dscrete publc good case consdered at the end of the prevous secton, suppose that (4) s satsfed. Then at equlbrum the ndvdual type c as determned by (5) has to be ndfferent between contrbutng or not, and the budget balance condton (6) s to be respected. As n the case of the standard publc good, ths gves rse to (79). Combnng (5) and (79) we obtan t 5 G (g )g, s 5 G (g )(1 2 g ) (1) as a (unque) tax/ subsdy polcy mplementng the frst best allocaton. The ndvdual equlbrum utltes are 21 v 2 G (g )g for non 2 contrbutors 21 v 2 c 1 G (g )(1 2 g ) for contrbutors (11) Agan, the contrbutors are better off than the non-contrbutors. In addton, provded that v s small, some ndvduals (n partcular, non-contrbutors) may be worse off than they would be wthout government nterventon. Ths s the case, for nstance, f the cost dstrbuton s unform and 2 2 g /2,v,g. Then effcency requres that the publc good should be produced, but all noncontrbutors end up beng worse off. The adopton of ths polcy, therefore, hnges on the exstence of a wnnng coalton of low cost ndvduals who approve t. Ths, however, s not always ensured. Suppose that the publc choce s made by a smple majorty, such that the tax-subsdy polcy s adopted only f t s preferred to the prvate provson by at least 5% of the ndvduals. In ths example, all non-contrbutors vote aganst the polcy, and f g,1/ 2, ther vote s decsve. Hence, the suggested polcy s not always poltcally feasble. In a recent paper, Ledyard and Palfrey, 1994, present an alternatve, more sophstcated mechansm whch, n a related context, solves the problem of poltcal feasblty. Also note that the polcy we suggest does not acheve the frst best whenever the socally optmal amount of the publc good depends on the dstrbuton of ndvdual costs, the reason beng that the polcy tself affects the optmal amount of the publc good. It does, however, acheve ths goal whenever the socally optmal cutoff s not changed when ths polcy s ntroduced. Ths holds for any preferences under whch the optmal amount of the publc good s ndependent of the dstrbuton of the ndvdual characterstcs, whch as s proved n Bergstrom and Cornes, 1983, s satsfed for preferences gven by A(X)Y 1B (X), where and A and B are monotonc and concave functons, of whch (1) s a specal case. 4. Concludng remarks Ths paper presents a correctve polcy for the provson of publc goods when the ndvdual contrbuton costs are unknown, and the government polcy depends on ndvdual actons. The recent

6 234 M. Gradsten / Economcs Letters 61 (1998) lterature mples that a polcy yeldng voluntary and effcent provson of publc goods does not exst, hence some coercon has to be exercsed. Here t s shown that a smple tax subsdy polcy yeldng the frst best soluton can be devsed, provded that the socally optmal amount of the publc good s ndependent of the dstrbuton of ndvdual characterstcs. Snce ths polcy does not represent a Pareto mprovement relatve to prvate provson, ts poltcal feasblty hnges on the exstence of a suffcently large supportng coalton. References Andreon, J., Prvately provded publc goods n a large economy: The lmts of altrusm. Journal of Publc Economcs 35, Andreon, J., Gvng wth mpure altrusm: applcatons to charty and Rcardan equvalence. Journal of Poltcal Economy 97, Bergstrom, T.C., Cornes, R.C., Independence of allocatve effcency from dstrbuton n the theory of publc goods. Econometrca 51, Brto, D.L., Sheshnsk, E., Intrlgator, M.D., Externaltes and compulsory vaccnatons. Journal of Publc Economcs 45, Dawes, R.M., 198. Socal dlemmas. Annual Revew of Psychology 31, Fres, T.L., Goldng, E., Romano, R., Prvate provson of publc goods and the falure of the neutralty property n large fnte economes. Internatonal Economc Revew 32, Gradsten, M., Tme dynamcs and ncomplete nformaton n the prvate provson of publc goods. Journal of Poltcal Economy 1, Gradsten, M., Effcent provson of a dscrete publc good. Internatonal Economc Revew 35, Laffont, J.J., Maskn E., A dfferental approach to expected utlty maxmzng mechansms. In: Laffont, J.J. (ed.), Aggregaton and Revelaton of Preferences, North-Holland, Amsterdam, Ledyard, J.O., Palfrey, T.R., Votng and lottery drafts as effcent publc goods mechansms. Revew of Economc Studes 61, Myerson, R.B., Satterhwate, M.A., Effcent mechansms for blateral tradng. Journal of Economc Theory 28, Neeman, Z., Property rghts and effcency of publc-good mechansms under asymmetrc nformaton, mmeo, Ntzan, S., Romano, R., 199. Prvate provson of a dscrete publc good wth uncertan cost. Journal of Publc Economcs 42, Palfrey, T.R., Rosenthal, H., Prvate ncentves n socal dlemmas: The effects of ncomplete nformaton and altrusm. Journal of Publc Economcs 35, Palfrey, T.R., Rosenthal, H., Testng for effects of cheap-talk n a publc good game wth prvate nformaton. Games and Economc Behavor 3, Rob, R., Polluton clam settlements under prvate nformaton. Journal of Economc Theory 47, Schellng, T.C., Mcromotves and Macrobehavor, W.W. Norton, NY, Van de Kragt, A.J.C., Orbell, J.M., Dawes, R., The mnmal contrbutng set as a soluton to publc goods problem. Amercan Poltcal Scence Revew 77,