European Regonal Scence Assocaton 36th European Congress ETH Zurch, Swtzerland 6-3 August 996 Se-l Mun and Mn-xue Wang raduate School of nformaton Scences Tohoku Unversty Katahra Cho-me, Aoba-ku, Senda 98, JAPA ax:+8 63 9858 mun@skk.s.tohoku.ac.jp REE-RD, SCALE ECMY AD JT PRVS LCAL PUBLC D ABSTRACT: t s observed that each jursdcton does not necessarly have all types of local publc good. Therefore, some types of publc good may be used by people from other jursdctons. n ths stuaton, strategc local government may not provde ther own publc good and expect that ther resdents wll use the good n other jursdcton; we term ths nterjursdctonal free-rdng. Settng a two regon model, ths paper nvestgates ash equlbra of local governments wth respect to the provson of local publc good wth spll over effect. We dentfed the condtons under whch free-rdng occur as an equlbrum soluton. urthermore, we formulate the programmng problem to obtan socally optmal allocaton and compare wth ash equlbrum soluton. Dscusson about polcy nstruments are also provded.
ree Rdng, Scale Economy and Jont Provson of Local Publc ood by Se-l Mun and Mn-xue Wang. ntroducton t s observed that each jursdcton does not necessarly have all types of local publc good. Therefore, some types of publc goods may be used by people from other jursdctons, f the use of publc good s not excludable as postulated. n ths stuaton, strategc local governments may not provde ther own publc goods and expect that ther resdents wll use the goods n other jursdctons; we term ths nterjursdctonal free-rdng. The purpose of ths paper s to nvestgate the consequences of decentralzed decson makng by local governments ncludng the case of free-rdng, and to examne the effcency of such decentralzed system. The topc of ths paper s related to the lterature on local publc goods generatng spll-over effects(e.g. Wllams(966), Paully(97), ordon(983)). These authors ponted out that, under the presence of spll-over effects, decentralzed decson makng by local governments cannot acheve the effcent level of publc good, snce each local government does not take account of the beneft of non-resdents. They addressed varous ssues; how the equlbrum level of publc good devate from the optmal one, what the role of central government s, and so on. Recently, Wellsch(993) has demonstrated that ash equlbrum of competng local governments s socally effcent, f households are perfectly moble and local governments take account of the mgraton responses to ther polces. Most prevous works n ths feld, ncludng those mentoned above, are restrctve n the sense that they have focused only on nteror solutons n that each government necessarly provdes postve amount of publc good. To nvestgate the free-rdng problem, we should explctly deal wth corner solutons. Kuroda(989) and Tsukahara(995) are exceptons. Kuroda focused on the locaton of publc facltes as strategy of local government whle quantty of the publc goods are gven exogenously, and compared non-cooperatve and cooperatve solutons. Although he dentfed the case of corner soluton, t was elmnated from the man part of hs analyss; he consdered that the emergence of corner soluton should be avoded by central government. n contrast, Tsukahara shed lght on the corner soluton. He examned the case that two local governments jontly
provde one publc good, and derved the condton under whch local governments choose the jont provson nstead of ndependent provson. Snce the subject of our study s closely related to Tsukahara s work, we would dscuss t n more detal. rst, he assumed that resdents can use the publc facltes n other jursdctons only f ther government jons the coalton. Ths mples that free-rdng from other jursdcton s prohbted even f t s possble. Ths contradcts the basc character of publc good, non-excludablty. Second, n hs model, local governments are perfectly compettve n that each of them consders that ts behavor does not affect the behavor of others. Perfect competton s not compatble wth the fact that spll over effects are present among small number of neghborng jursdctons. Ths paper presents a dfferent approach to analyzng the smlar problem to that n Tsukahara s paper. We nvestgate not only jont provson as cooperatve soluton but also free-rdng resultng from non-cooperatve behavor of local government wth respect to the provson of local publc good wth spll over effect. We derve ash equlbra of local governments and dentfy the condtons under whch free-rdng occur as an equlbrum soluton. urthermore, we formulate the programmng problem to obtan socally optmal allocaton and compare wth ash equlbrum soluton. Dscusson about polcy nstruments are also provded.. The Model Suppose that the economy conssts of two regons(jursdctons). Total number of households n the economy s fxed at, and ther locaton dstrbuton, n( =, ) s exogenously gven. Thus followng relaton holds n+ n = () Each household supples one unt of labor force to the local frm. Each regon produces a sngle homogenous product, used as a numerare good. nputs for producton s labor; the producton functon s represented as Y = f ( n ), where f >, f < are assumed. ote that local populaton n s equvalent to the labor nput. All households have dentcal preferences and the utlty of a household n regon s a functon of ts consumpton x and servce level of local publc good, Z, denoted by ux (, Z). t s assumed that publc good n ether regon s accessble for all resdents n the economy, but ts servce level for resdents n other jursdcton s lower than that for resdents n the jursdcton where the good s provded. Ths reducton s, for example, attrbuted to the dstance or travel costs to vst facltes located n other jursdcton. We represent the
servce level of the publc good n regon as follows Z = max{, h }, j () j where s quantty of publc good provded n jursdcton, and h s a parameter rangng h. The above formulaton mples that households choose to use the publc good wth the hghest servce level. C ( ) = R S T + a, f >, f = The cost for producton of publc good n jursdcton s gven by where s the fxed cost and a s a constant representng the margnal cost. (3) 3. Decentralzed decson makng of local governments and ash equlbra 3. Decson makng of a local government The objectve of a local government s to maxmze the utlty of ther resdents. Wthout any nterventon of central government or nterregonal transfer, the followng resource constrant must be satsfed n each regon. Y nx C( ) = (4) The local government has two alternatve strateges: (A) to provde postve amount of by collectng tax from ther resdents, and (B) not to provde publc good and expect that ther resdents free rde on the publc good n another jursdcton. f the government takes the former strategy (A), t wll choose the quantty of publc good such that Max u Y C ( ), (5) n The frst order condton for optmzaton s n u u x KJ = C (6) Let be the soluton to the above equaton and V the utlty level correspondng to the soluton. n the other hand, f the government takes the free-rdng strategy (B), the resdents enjoy the utlty level gven by uy ( / n, h). The government wll choose the strategy that gves hgher j utlty. Decson makng of government depends on the strategy of another jursdcton j. Let j be the quantty of publc good n j satsfyng the followng relaton 3
uy ( / n, h ) = V (7) j When j s greater than j, the utlty for the free-rdng strategy (B) s greater than that for strategy (A). Thus the government wll not provde the publc good n the jursdcton. n the other hand, when j s smaller than j, the government wll provde ts own publc good accordng to the manner specfed n (5). The optmal polcy for the local government, as reactons to the strateges of another jursdcton j, can be summarzed as follows R S f > = T,, f j j j j (8) 3. ash equlbra ash equlbra concernng the provson of publc good by two local governments can be classfed to four cases. As llustrated n gure (a)-(d), the relatve poston of,, and determne whch case s emerged. below The condtons under whch each case s emerged are descrbed Case : nly jursdcton provdes postve quantty of publc good and resdents n jursdcton free-rde on the good n. Ths case emerges f > and < and the equlbrum soluton s (, ) = (, ). Case : Both jursdctons provde ther own publc good, thus nether spll-over nor free-rdng occurs. Ths case emerges f < and < (, ) = (, ). and the equlbrum soluton s Case 3: nly jursdcton provdes postve quantty of publc good and resdents n jursdcton free-rde on the good n. Ths case emerges f < and > and the equlbrum soluton s (, ) = (, ). Case 4: n ths case, as shown n the gure (d), the equlbrum soluton s not unque. Both governments seek to be free-rder. Ths case emerges f > and > and the equlbrum solutons are (, ) = (, ) and (, ) (, = ). 4
= h = h = h = h (a) Case (b) Case = h = h = h = h (c) Case 3 (d) Case 4 Equlbrum pont reacton of regon reacton of regon URE our cases of ash equlbrum 3.3 Propertes of ash equlbra To obtan the explct results, we specfy the form of utlty functon as ux (, Z) = log x+ ( ) log Z (9) where s a constant satsfyng < <. By usng the specfcaton above, we solve the equatons (6) and (7) to obtan = ( Y ) a 5 ()
= Yj j ah K J H Y K J j d,. () Then we derve explctly the condtons under whch each case emerges, as follows Case : Substtutng ()() nto the condtons, > and <, we rewrte them as h > h and h< h where h h Y Y b g ( Y ). () = H K J Y Y b g ( Y ). (3) = H K J Smlarly, condtons for the emergence of the remanng cases are: Case : h Case 3: h Case 4: h < h and h< h < h and h> h > h and h> h The values of h and h depend on varous parameters and we term them case boundary functons. To nvestgate the relaton between case emergence and populaton dstrbuton, we dfferentate (), (3) wth respect to n. h n h n = h = h L M L M l q f P< (4a) Y Q l( ) + q f + P> (4b) Y Y Y Q f ( ) Y ( )( Y ) Y f Y ( )( ) And we see from (),(3) that h = h when Y = Y,.e., n = n = /. urthermore, h (h ) as n, and h (h ) as n. Based on the above nformaton, we depct n gure the case boundary functons and show the range of h and n n whch each case emerges. rom the fgure, t s seen that free-rdng cases, such as cases,3 and 4, are lkely to emerge when the degree of spll-over, h s hgh and populaton dstrbuton s uneven. Hgher value of h mples that the declne n servce level due to dstance s not sgnfcant. n other words, two regons are not so dstant each other, or transportaton system s mproved. 6
Case 4 h Case 3 Case h h Case n n URE Parameters and cases of ash equlbrum Another mportant parameter affectng case dvson s the fxed cost for provson of the publc good. h Dfferentatng () wth respect to gves ( )( Y ) ( Y ) = h ( )( Y )( Y ) l q (5) Although the sgn of the RHS of (5) s ambguous n general, at least t s negatve when n > n. As shown n gure, h curve as the boundary between free-rdng case(case ) and ndependent provson case(case ) s meanngful only when n > n. Wthn ths range, ncrease n fxed cost shfts the curve downward. Smlar argument apples to the effect on h curve. Consequently, the lkelhood of free-rdng cases s ncreased as fxed cost becomes larger. nally we would make a remark on case 4, the case of multple equlbra. n ths case, we cannot predct whch soluton s realzed. But we can evaluate whch soluton s more socally preferable than the other. Some scholars clamed that players are lkely to choose the strategy such that preferable soluton s attaned when nformaton on focal pont s provded, or preplay communcaton s made. But ths s not always the case. The central government may play an mportant role n provdng the nformaton on focal pont, or coordnatng the pre-play communcaton, even f t does not ntervene by means of fscal polces. 7
4. Socal optmum 4. The problem formulaton n ths secton, we formulate and solve the problem as f the central government (or socal planner) can control everythng. We wll consder n the next secton the polcy nstruments n decentralzed settng. We choose the Benthamte socal welfare as the objectve functon to be maxmzed. As n the prevous secton, three cases are expected to emerge: () only regon provdes the publc good, () both regons provde, () only regon provdes. We wll solve the problem by two step; the frst step s to solve three problems, n each of whch one of three cases s supposed, the next s to choose the case that gves the hghest value of objectve functon. We start by descrbng the problem of each case n the frst step. Case : The problem under case s formulated as follows Max nu ( x, ) + nu ( x, h) (6) x, subject to Y + Y n x n x C( ) = (7) The frst order condtons for optmzaton are u = u (8a) x x n u u x n hu + = C (8b) u x (8b) states that the quantty of publc good n regon should be determned such that the sum of margnal beneft for all resdents n the economy s equal to margnal cost. n the decentralzed provson as descrbed by (6), the second term n LHS of (8b) s not appeared. Ths mples that decentralzed provson s not effcent because t gnores the beneft of non-resdents. Let x, x and be the soluton to the above problem, then the optmal value of the objectve functon n case s calculated as follows W = nu( x, ) + n u( x, h ) Case : Max nu ( x, ) + nu ( x, ) (9) x, subject to Y+ Y nx nx C ( ) C ( ) = () The frst order condtons for optmzaton are u = u (a) x x 8
n u u x = C, =, (b) And optmal value s W = nu( x, ) + n u( x, ) Case : Ths case s symmetry to Case. Thus we omt explanaton about ths case. Suppose that the optmal values, W, W and W are obtaned. Then the next task of the planner s to choose the optmal case. The planner should choose case f W > W and W > W ; case f W > W and W > W ; Case f W > W and W > W. 4. Propertes of optmal soluton The analyss n ths secton s based agan on the specfcaton of log-lnear utlty functon. By usng the explct solutons(see Appendx for detals), we derve the condtons under whch each case s chosen as optmal polcy. Case : The condtons, W > W and W > W become h> h and n > n, where h n n = H K J H K J + n n ( ) n Y Y Y + Y KJ. () Case : The condtons, W > W and W > W become h h < and h< h, where h n n = H K J H K J + n n ( ) n Y Y Y + Y KJ. (3) Case : The condtons, W > W and W > W become n > n and h> h. n cases and, quantty of publc good s determned takng account of the beneft of the resdents n both regons and. And n case (case ), regon (regon ) bears porton of the cost for provdng publc good, although the publc good s not located n ts terrtory. Therefore, we can regard cases and as jont provson. As n the analyss of ash equlbra, we nvestgate the shape of the case boundary functons, h and h, to see the relatons between parameters and the optmal polcy. Dfferentatng (), (3) wth respect to n, we have 9
h n h n = h = h ( f f ) n ( ) n ( Y + Y )( Y + Y ) R S T R S T ( f f ) n ( ) n ( Y + Y )( Y + Y ) + H K J + n H K J + U KJV W U KJV W n Y Y log (4a) Y + Y Y Y log (4b) Y + Y rom the assumpton of concave producton functon, f f s postve (negatve) when n < n (n > n). urthermore, n Y+ Y <, ( =, ), and <. Then we conclude that Y + Y h n <, f n > n and h n >, f n > n. Based on the above nformaton, we depct n gure 3 the case boundary curves and the area n whch each case s chosen as optmal polcy. t s seen n gure 3 that the jont provson cases are lkely to be optmal when h s hgh and populaton dstrbuton s uneven. Ths s smlar result to that n ash equlbrum. The decson rule concernng the choce between cases and s qute smple; the publc good should be provded by the regon wth larger populaton. As for the effect of fxed cost,, we obtan h = h Y+ Y ( ) n ( Y + Y )( Y + Y ) W < R S T Same result holds for h. U V These results mply that, as the fxed cost s larger, jont provson s (5) h Case Case n h o Case h o n URE 3 Parameters and cases of optmal solutons
more lkely to be optmal. 4. Comparson between ash equlbrum and optmum We frst examne whether the parameter range for the emergence of jont provson cases n optmum s larger than that of free-rdng cases n ash equlbrum. Ths comparson s meanngful because jont provson and free-rdng have common features n the sense that two regons have one publc good. gure 4 shows case boundary curves for equlbrum and optmum. n the fgure, free-rdng cases n ash equlbrum emerge n the area above h optmum emerges n the area above h and h curves, whle jont provson n and h curves. t s seen that the area of jont provson case s larger than that of free-rdng case, snce h and h curves are located above h and h curves unless the populaton dstrbuton s extremely uneven. n other words, the economy may produce two publc goods n ash equlbrum even though havng one publc good s optmal. ext, we examne whether the total quantty of publc good n the economy as a whole s larger n optmum than n equlbrum. The comparson was made for all possble combnatons of cases. After the drect comparson usng the explct solutons, () n ash equlbrum and (A)(A5)(A8) n optmal, we obtan + + where s the socally optmal quantty of publc good n jursdcton, and the good provded by jursdcton n ash equlbrum. the quantty of The above relaton mples that the total quantty of publc good n optmum s not smaller than that n ash equlbrum. Equalty holds only for the combnaton of case n equlbrum and case n optmum, although quantty n each jursdcton s dfferent. ote that the total quantty under case n equlbrum s smaller than that under case or n optmum; the sum of two publc good s smaller than one. Ths occurs n the range below h h curves(or lne A A A ) and above h h curves(or lne 3 B B B3) n gure 4. urthermore, n ths case, total expendture n ash equlbrum s larger than that n optmum ; ash equlbrum may cause excessve expendture for smaller publc good provson, snce fxed cost s double n case.
h A h B h o h o h B A A 3 B 3 n n ash equlbrum ptmum URE 4 Case boundares n equlbrum and optmum 5. scal polces toward effcent allocaton 5. rst best polcy n the last secton, t s shown that ash equlbrum of decentralzed decson makng cannot attan the socal optmal allocaton. ne polcy response s that the central government tself provdes the publc good by collectng tax from resdents. t s true that by choosng the tax optmally ths scheme attans the frst best allocaton. To verfy ths, we demonstrate below the problem and soluton for central provson of local publc good. Supposng case, the problem to be solved by the central government s formulated as Max T, T, nu Y T n u Y T, +, h (6) n KJ n KJ subject to T + T C( ) = (7) where T( =, ) s the total amount of tax leved n regon. As shown n (7), tax revenue s used to fnance the expendture for publc good provson. By manpulatng the frst order
condtons for the above problem, the dentcal expressons wth (8a)(8b) are easly derved. or log-lnear utlty, optmal tax n case s obtaned as T = ( p ) Y py + pc( ) j = ( p ) Y p ( Y ),, j =, ; j where p = n /. j (8) We can solve the problems n other cases by repeatng the same procedure. 5. Decentralzed decson makng wth tax-subsdy: a second best polcy f the central government tself provdes local publc good as supposed n the prevous secton, local governments may lose ther rason d'etre $. Advocatng the centralzaton s not our ntenton. n ths secton, we wll present an alternatve scheme n that central government collects tax and subsdze the regon provdng the publc good; on the sde of local government, t chooses the level of publc good gven the subsdy. Local governments are authorzed to provde publc goods accordng to ther own nterests; maxmzng the utlty of ther resdents. The central government optmally chooses the tax-subsdy, takng account of the above stated behavor of local governments. As we wll see later, ths scheme cannot attan the frst best allocaton. Thus we term ths the second best polcy. Suppose that case s chosen. The problem s formulated as Max n u Y T + ( ), +, h T, T, S n KJ n KJ (9) subject to T+ T S = (3) u Y T S C + ( ) = arg max, (3) n where S s the amount of subsdy gven to regon where publc good s provded. Consderng (3), the above problem s equvalent to the problem to determne the optmal transfer between regons and. or the log-lnear utlty functon, (3) s solved as = ( Y+ T ) (3) a Subsequently, we have KJ 3
x x = ( n Y T + ) (33) = ( n Y T ) (34) By usng the above relatons, the problem (9)-(3) s reduced to Max n T L M L M R S U V n Y T log ( + ) ( ) log ( Y T ) T W + + a R S U V RST RST + T W + n + n Y T h log ( ) ( ) log ( Y T ) a (35) Solutons to the above problem s n T Y n Y = K J ( ) (36) whch s dentcal to the frst best soluton obtaned by (8). Substtutng (36) nto (3)-(34), we obtan the quantty of publc good and consumpton under the second best polcy, as follows x n = a = K J + n n + KJ ( Y Y ) (37a) ( Y Y ) (37b) x Y Y = ( + ) (37c) t s seen that (37a)-(37c) do not concde wth the frst best solutons (A)(A). n other words, transfer gven by (36) cannot acheve the frst best soluton; resource allocaton s dstorted by decson makng of local government concernng the quantty of publc good. By comparng the second best solutons (37a)-(37c) wth the frst best soluton (A)(A) and ash soluton (), we obtan the followng relatons < < (38a) x < x < x (38b) x = x < x (38c) Under the second best polcy, publc good n regon s larger than that n ash equlbrum but stll undersuppled compared wth the frst best soluton. n case we have dfferent concluson; snce no spll-over effect exsts n ths case, the frst best polcy s chosen by local governments who consder the welfare of ther resdent only. Case 4 UVWQ P UVWQ P
s symmetry of case. Choosng the optmal case s the job of the central government; t performs that job by choosng the regon to be subsdzed. The procedure to fnd out the optmal case s same as n the prevous secton; dervng the case boundary functons and the condtons under whch each case s emerged. ootnotes. n case, by usng (A)(A), t s seen that Y n x > and Y n x C( ) < The above relatons mply that the resource s transferred from regon to. Thus we can say that resdents n regon bear the porton of cost for publc good n regon. Smlar argument apples to case.. Expendture for publc good provson n case of ash equlbrum s calculated by usng the soluton () as a + ( + ) = ( )( Y + Y ) + n the other hand, n case of optmal soluton, by usng (A), we have + a = ( )( Y + Y ) + t s obvous that the expendture n the former case s larger than that n the latter case. Appendx: Socally optmal solutons By usng the log-lnear utlty functon, we obtaned the soluton to the optmzaton problems and the optmal value functons of three cases as follows. Case : 5
x = x = ( Y + Y ) (A) = ( Y + Y ) (A) a W = log( Y+ Y ) + log + ( ) log + ( ) n log h (A3) a Case : x x Y Y = = ( + ) (A4) ( ) n = ( Y+ Y ), =, (A5) a n W Y Y n a n n = log( + ) + log + ( ) log + ( ){ log + log } (A6) Case : x = x = ( Y + Y ) (A7) = ( Y + Y ) (A8) a W = log( Y+ Y ) + log + ( ) log + ( ) n log h (A9) a References ordon, R. H., 983, An optmal taxaton approach to fscal federalsm, Quarterly Journal of Economcs 98, 567-586. Kuroda, T., 989, Locaton of publc facltes wth spllover effects: Varable locaton and parametrc scale, Journal of Regonal Scence 9, 575-594. Pauly, M. V., 97, ptmalty, publc goods, and local governments: A general theoretcal analyss, Journal of Poltcal Economy 78, 57-584. Tsukahara, K., 995, ndependent and jont provson of optonal publc servces, Regonal Scence and Urban Economcs 5, 4-45. Wellsch, D., 993, n the decentralzed provson of publc goods wth spll overs n the presence of household moblty, Regonal Scence and Urban Economcs 3, 667-679. 6
Wllams, A., 966, ptmal provson of publc goods n a system of local government, Journal of Poltcal Economy 74, 8-33. 7