Optimal Formulas for Subnational Tax Revenue Sharing

Transcription

1 Theoretcal Economcs Letters, 16, 6, Publshed Onlne August 16 n ScRes Optmal Formulas for Subnatonal Tax Revenue Sharng Raul Alberto Ponce Rodrguez, Ikuho Koch Departamento de Cencas Socales, Unversdad Autónoma de Cudad Juárez, Cudad Juárez, Méxco Receved 3 July 16; accepted August 16; publshed 5 August 16 Copyrght 16 by authors and Scentfc Research Publshng Inc Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY Abstract We develop an analyss of optmal formulas for subnatonal tax revenue sharng for two cases of nterest: when local publc goods (lpg s show spllovers and are nter-regonal perfect and mperfect substtutes Our analyss could be relevant to understand the determnants of feasble competng alternatves for the desgn of tax revenue sharng systems Our study shows that: 1 Inter-regonal spllovers should be taken nto account n the desgn of formulas for subnatonal revenue sharng when lpg s are perfect nter-regonal substtutes but should not be taken nto account to supply lpg s that are mperfect substtutes; The dstrbuton of preferences for lpg s and populaton should be consdered n formulas for revenue sharng when subnatonal governments provde both lpg s that are perfect and mperfect substtutes; 3 For local publc goods that are mperfect substtutes, the share of tax revenue s frst ncreasng and then decreasng wth ncreases n the populaton of the dstrct; 4 For perfect substtute-lpg s, the share of tax revenue n the dstrct s ncreasng and concave wth the dstrct s populaton; 5 The dstrbuton of ncome should not be consdered n the desgn of formulas for lpg s that are perfect and mperfect substtutes Other emprcally relevant comparatve analyses are consdered n the paper Keywords Local Publc Goods, State and Local Budget and Expendtures, Intergovernmental Transfers, Inter-Regonal Externaltes, Federalsm 1 Introducton Tax revenue sharng seeks to dstrbute revenue for central and subnatonal governments from a gven tax base Some of the advantages of tax revenue sharng recognzed n the lterature nclude: 1 Tax harmonzaton be- Correspondng author How to cte ths paper: Rodrguez, RAP and Koch, I (16 Optmal Formulas for Subnatonal Tax Revenue Sharng Theoretcal Economcs Letters, 6,

2 R A P Rodrguez, I Koch tween the central and subnatonal governments to reduce neffcency costs from tax competton; Rao [1] argues that tax revenue sharng leads to welfare gans by ensurng greater subnatonal fscal autonomy (by ensurng subnatonal governments a source of revenue, and 3 Revenue sharng s a tool to deal wth vertcal fscal mbalances Tax revenue sharng s commonly mplemented n many developed and developng countres through the use of formulas for the allocaton of tax revenue between the central and subnatonal governments Rao [1] classfes revenue-sharng systems n two types: n the frst type, the central and sub-natonal governments share a tax base and dfferent levels of governments can determne tax rates In the second type, one level of government could collect tax revenue that s shared among dfferent levels of governments 1 In spte of the emprcal relevance of ths polcy, there s lttle formal research on the optmal desgn of formulas for subnatonal revenue sharng Such analyss s worthwhle because knowng the propertes of dfferent forms of subnatonal revenue sharng systems would help us to dentfy potental advantages and shortcomngs of feasble polcy optons In ths paper we seek to contrbute to fll ths gap n the lterature and we analyze the optmal desgn of subnatonal tax revenue sharng To do so, we study a model wth a benevolent socal planner n whch subnatonal governments provde local publc goods wth nter-regonal spllovers The central government selects the natonwde tax rate on ncome to determne the optmal sze of tax revenue and ts shares to be allocated to subnatonal governments The man dstncton of our analyss wth respect the lterature s that we consder the optmal desgn of formulas for subnatonal revenue sharng when local governments provde publc goods wth spllovers n two cases of nterest: when local publc goods are perfect and mperfect nter-regonal substtutes 3,4 The man fndngs of our paper are the followng: 1 Inter-regonal spllovers should be taken nto account n the desgn of formulas for revenue sharng when lpg s are perfect nter-regonal substtutes but should not be taken nto account for mperfect substtutes; The nter-regonal dstrbuton of preferences for lpg s and populaton should be consdered n formulas for revenue sharng when subnatonal governments provde lpg s that are both perfect and mperfect substtutes; 3 For local publc goods that are mperfect substtutes, an ncrease n the populaton of the dstrct leads to an ambguous outcome n the share of tax revenue n the dstrct: for suffcently hgh (low ntal values of the share of tax revenue, an ncrease n the dstrct s populaton leads to a fall (ncrease n the share of tax revenue allocated to the dstrct; 4 For perfect substtute-lpg s, the share of tax revenue n the dstrct s ncreasng and concave wth the dstrct s populaton; 5 The dstrbuton of ncome should not be consdered n the desgn of formulas for lpg s that are perfect and mperfect substtutes Fnally, we develop other comparatve analyss on the optmal shares to be allocated among subnatonal governments The structure of the paper s as follows: n Secton, we develop optmal formulas for tax revenue sharng for an economy wth local publc goods that are mperfect nter-regonal substtutes Secton 3 contans the analyss for an economy wth perfect substtutes Secton 4 concludes Optmal Tax Revenue Sharng for Local Publc Goods that Are Imperfect Inter-Regonal Substtutes Consder an economy wth ndvduals lvng n dstrcts and In our economy, the central government s responsble for collectng tax revenue and allocatng (through the use of formulas the shares of tax revenue for all dstrcts For smplcty of the analyss, we assume that local governments use the resources allocated to them by the central government to provde local publc goods In dstrct,, there s a representatve ndvdual 1 It s common that, under ths type of revenue sharng, the central government s responsble for collectng tax revenue but subnatonal governments are n charge of the spendng choces and (or the admnstraton of publc programs There s a large lterature on the optmal desgn and effects of ntergovernmental transfers, see (among many others []-[5], and more recently [6] There has been also nterest on revenue assgnments, see [7] However, to the best of our knowledge, there has been lttle nterest on developng a formal theory of the optmal desgn of formulas for subnatonal revenue sharng 3 Spllovers of local publc goods lead to some degree of nter-regonal substtutablty among lpg s produced n dfferent regons Moreover, n real lfe, the moblty of households and frms makes lpg s more substtutable Hence, t s relevant to conduct an analyss of the dstncton between lpg s that are perfect nter-regonal substtutes (whch s the lmtng case and mperfect substtutes n the desgn of formulas for subnatonal tax revenue sharng However, for smplcty of the analyss, n ths paper we gnore nter-regonal moblty We leave such analyss for future research 4 In Secton, we provde a formal dstncton between local publc goods that are perfect and mperfect nter-regonal substtutes A real lfe example of a perfect substtute-local publc good wth nter-regonal spllovers could be productve local nfrastructure (such as local brdges roads etc To see ths, consder a frm (smlar examples can be found for the case of households that sells products n all dstrcts (regons If a brdge n some dstrct has a constant margnal rate of techncal substtuton wth respect a brdge provded n some other dstrct then these local publc goods show spllovers and are consdered as perfect nter-regonal substtutes For the case of mperfect substtutes, the margnal rate of techncal substtuton s not constant 89

3 R A P Rodrguez, I Koch wth an endowment g g are local publc goods suppled by dstrcts and Drect preferences are gven by µ = γ ln ( g + θ ln ( k g + α x where γ, θ, α ++ are postve constants related wth the ndvdual s ntensty of preferences over local publc goods g, g and the prvate good x 5 The parameter k [,1 measures the extent of spllovers, f k = there s no spllovers and f k were equal to one then the local publc good would be a pure natonal publc good The budget constrant of the ndvdual s gven by x = e ( 1 t where t s a proporton- al ncome tax mplemented by the central government In ths economy, there s nter-regonal heterogenety of preferences, that s γ γ, θ θ, α α and wthout loss of generalty we assume e > e We consder a benevolent socal planner rulng the central government that seeks to maxmze the naton s so cal welfare Ψ = ν ( B,, e + ν ( B,1 (, e where, are the populatons of dstrcts and The problem of polcy desgn for the central government s to maxmze Ψ by choosng the sze of the proportonal ncome tax t the budget B, and the formula for sharng revenue n dstrcts and,,1 ( to fnance local publc goods g, g The budget constrant of the central government s B= { e + e } t where B s the central government s budget The dstrbuton of tax revenue n the economy s determned by formulas for revenue sharng such that the budget constrants of subnatonal governments are gven by g = B and g = ( 1 B where [,1 ] s the share of the budget allocated to fnance the local publc good of dstrct For smplcty of the analyss, dstrcts and only supply, respectvely, g and g In ths secton, we analyze the case of local publc goods that are mperfect substtutes, whch are defned by a fnte elastcty of substtuton, whle for the case of local publc goods that are perfect substtutes (analyzed n Secton 3 the elastcty of substtuton s nfnte 6 Formally, the problem of tax revenue sharng for the central government when local publc goods are mperfect substtutes s: where ( B,, e e and hs ndrect preferences are gven by ( ν g, g, e where, {, } Ψ ν (,, ν (,1 (, B Max = B e + B e (1 : (,, ln ( ln ( ( 1 1 B st ν B e = γ B + θ k B + α e, e + e ν s the ndrect utlty functon of the representatve ndvdual of dstrct on feasble local publc goods (a smlar expresson s gven for ν ( B,1 (, e Proposton 1 The optmal budget B and formulas for dstrbuton of tax revenue,1 ( for local publc goods that are mperfect nter-regonal substtutes of dstrcts and are gven by: B = Wth ( ( ( γ + θ + ( γ + θ sα + s α e e s,1 : s = e e + e s + s = 1, and e e e e = ( 1 = γ + θ ( γ + θ + ( γ + θ γ + θ ( γ + θ + ( γ + θ 5 Our choce of the utlty functon s for smplcty of the analyss Moreover, ths type of utlty functon s common n the lterature see [8] and [9], among many others 6To dstngush local publc goods that are perfect vs mperfect nter-regonal substtutes, we use the elastcty of substtuton ( σ = d ln g g dg To see that our preference relaton for d ln dg ( MRS g g characterzes local publc goods that are mperfect nter-regonal substtutes, note that the elastcty of substtuton between dµ = ( MRS d ln ( g g g and g and g g ( (3 (4 (5 s postve but fnte snce ( d ln g g dg σ = where d ln dg ( MRS g g d d ln g g g g 1 1 g MRS = =, = = σ = > g >, g > In the case of perfect nter-rego- g g dg g dg g dg g g nal substtutes, the elastcty of substtuton s nfnte (see Secton 3 81

4 R A P Rodrguez, I Koch The frst order condtons for the government s problem are: Ψ γ θ γ θ α e α e + = + B + + = > B B B B B e + e Ψ γ θ θ γ = + = > ( 1 ( 1 Rearrange terms to obtan condtons (3, (4 and (5 Proposton The optmal budget B and formulas for dstrbuton of tax revenue,1 ( satsfy the followng propertes: 1 The mpled dstrbuton of local publc goods, g = B, g = ( 1 B s Pareto effcent s a constant: [,1] 3 The share of tax revenue allocated n dstrct,, also satsfes the followng: > γ a as and > γ + θ > ( b = = c e = e = d γ, that s s non decreasng wth γ, and γ e θ, that s s non decreasng wth θ, and θ See the Appendx Proposton says that the equlbrum dstrbuton of g, g s Pareto effcent snce, B takes nto account the natonwde dstrbuton of socal margnal costs and benefts of local publc goods For ths reason we can t fnd an alternatve feasble allocaton g, g n whch we can beneft at least one ndvdual wthout hurtng someone else n ths economy Some other nterestng results of our analyss are the followng: for a local government n dstrct supplyng an mperfect substtute-local publc good, the optmal formula for tax sharng,, s constant (see proposton and t depends only on the dstrbuton of parameters related wth the ntensty of preferences of ndvduals for local publc goods n dstrcts and, that s γ, θ, γ, θ, and the dstrbuton of the populaton n the economy (see condton 4 An ssue of nterest for the desgn of a polcy of tax revenue sharng s the relatonshp of wth the ds > γ trct s populaton Our analyss suggests that as Ths s relevant for polcy desgn > ( γ + θ snce emprcal evdence suggests the use of lnear formulas between and (see [1] However, our analyss suggests that the effect of an ncrease n the populaton of dstrct over the share of tax revenue to be γ allocated to dstrct s contngent to the level of For suffcently low values of (for ( γ + θ an ncrease n should lead to an ncrease n However, for suffcently hgh levels of (for γ > an ncrease n should lead to a fall n The explanaton of ths result s straghtforward: γ + θ ( Ψ γ θ θ γ from the optmalty condton = + = > where ( 1 ( 1 γ θ s the net socal margnal beneft for resdents of dstrct of ncreasng where γ ( 1 s the gross socal margnal beneft for resdents of dstrct of ncreasng (an ncrease n leads to a hgher level of the local publc good suppled by dstrct whch ncreases the wellbeng of resdents of that dstrct Moreover, a hgher 1 falls whch leads to a lower supply of g and the ben- means that ( (6 (7 811

5 R A P Rodrguez, I Koch efts for resdents of dstrct from consumng g fall by θ ( 1 Hence the net beneft of a margnal n- crease n s γ θ γ If then γ θ, ( 1 ( γ + θ > hence an ncrease n ( 1 leads to a net ncrease n the socal margnal benefts of resdents of dstrct whch leads to a hgher share of γ tax revenue allocated to ths dstrct (the opposte occurs f > γ + θ ( Other nterestng outcomes from the characterzaton of are: the formula for does not depend on the dstrbuton of spllovers k and k (see proposton 3b The explanaton of ths outcome s that the margnal socal benefts and costs of a change n do not depend on k and k (see condton 7 A smlar explanaton s gven to the outcome that the dstrbuton of ncome should not be consdered n the desgn of formulas for lpg s that are mperfect substtutes snce e = e = (see condton 3c Fnally, proposton 3 says that an ncrease n the ntensty of preferences of resdents n dstrct and of resdents of dstrct for the local publc good provded by dstrct (that s ncreases, respectvely, n γ and θ should ncrease the share of tax revenue allocated to dstrct Ths s the case because hgher values of γ and θ ncrease the naton wde s socal margnal benefts of the publc good provded by dstrct 3 Optmal Tax Revenue Sharng for Local Publc Goods that Are Perfect Inter-Regonal Substtutes In ths secton, we analyze the case for formulas for subnatonal tax revenue sharng when local publc goods are perfect nter-regonal substtutes whch means that the elastcty of substtuton between g and g s nfnte To dstngush our notaton from our prevous secton, we denote ˆB as the equlbrum level of the budget and ˆ as the allocaton formula of tax revenue for dstrct For ths case, ˆ ( ˆ ˆ,, ln ˆ ( ˆ ( 1 ˆ ˆ B ν B e = λ B+ k B + α e 1 s the ndrect utlty for a resdent n ds- e + e trct where λ, α are postve constants and g = ˆ Bˆ and g = ( 1 ˆ Bˆ 7 For ths economy, the problem of polcy desgn s 8 : Ψ ˆ, ˆ, ˆ,1 ˆ Max = ν B e + ν B, e (8 { B, } ( ( ( ˆ ˆ ( ˆ ˆ B,, ln ( B ( 1 ˆ B ˆ ˆ ˆ ˆ B ν e = λ + k + α e 1 (9 e + e Proposton 3 The optmal budget ˆB and formulas for dstrbuton of tax revenue ˆ,1 ( ˆ for local publc goods that are perfect nter-regonal substtutes n dstrcts and are gven by: ˆ λ + λ B = (1 sα + s α e e e e e Recall ( ( s,1 : s = e e + e s + s = 1, and 7 To see that lpg s are perfect nter-regonal substtutes, note that the drect preferences of the representatve ndvdual of dstrct are µ ( g, g, x = λ ln ( g + k g + α x where λ, α ++ Use the household s budget constrant and the government s budget constrant to characterze the ndrect preferences denoted by ( d ln ˆ, ˆ ( g g dg ν B, e Recall, σ = and snce d ln MRS dg MRS dg = = k and g g ( g g dg dµ = d ln MRS dg = σ = 8 The ndrect preferences of the representatve ndvdual of dstrct are gven by ˆ ( ˆ ( ˆ,1, ln( ( 1 ˆ ˆ ˆ ˆ B ν B e k = λ B+ B + α e 1 e + e e ( g g 81

6 And λ ( 1 λ ( 1 ( λ + λ ( 1k ( 1k k k k ˆ = ( 1 ˆ = λ ( 1k λ ( 1k k ( λ + λ ( 1k ( 1k The frst order condtons for the government s problem are: Ψ λ λ α e + α e = + = > Bˆ Bˆ Bˆ e + e ˆ B λ ( { ( } λ ( { ( } Ψ 1 k 1 k = = ˆ > ˆ ˆ ˆ 1 k + k 1 1k R A P Rodrguez, I Koch Rearrange terms to obtan condtons (1, (11 and (1 Proposton 4 The optmal budget ˆ,1 ˆ satsfy the followng propertes: 41 The mpled dstrbuton of local publc goods, ˆ ˆ gˆ = B, gˆ = ( 1 ˆ Bˆ s Pareto effcent 4 ˆ s a constant: ˆ [,1] 43 The share of tax revenue allocated n dstrct,, satsfes: a ˆ > and ˆ b ˆ ˆ > and > c ˆ ˆ and d ˆ ˆ e = e = e f ˆ λ > ˆ λ > f f ( λ + λ ( 1 k ˆ > > k ˆ ( 1 k (11 (1 (13 (14 ˆB and formulas for dstrbuton of subnatonal tax revenue ( wth wth λ > λ > See the Appendx Proposton 4 shows some comparatve statc results on the determnants of formulas for revenue sharng for local publc goods that are perfect nter-regonal substtutes In partcular, ˆ s non decreasng wth the populaton of the dstrct, however, ths relatonshp s not lnear snce ˆ In addton, the dstrbuton of tax revenue for dstrct s ncreasng on the spllovers of the publc good provded by dstrct k,, and decreasng n the spllovers of the local publc good suppled by dstrct k,, (see propostons 43b and 43c These last results are very ntutve: a hgher level of k ncreases the natonwde socal margnal benefts of g (by the effect of the spllovers of g over dstrct whch leads to an ncrease n the tax reve- nue allocated to dstrct A smlar effect occurs wth an ncrease n k whch makes more attractve to allocate more resources n dstrct and reduces ˆ The dstrbuton of ncome should not be consdered n the desgn of formulas for lpg s that are perfect substtutes snce e = e = (see condton 43c As n the prevous secton, the margnal socal benefts and costs of a change n do not depend on e and e (see condton 11 Proposton 4 also shows that ˆ has an ambguous relatonshp wth the dstrbuton of the ntensty of preferences for local publc goods To be specfc, ˆ λ > f ( λ + λ ( 1 k ˆ > and f 813

7 R A P Rodrguez, I Koch > k ˆ ( 1 k then ˆ λ > In terms of the practcal sgnfcance of ths paper, propostons two and four hghlghts the relevance of takng nto account the nter-regonal heterogenety of preferences and the dstrbuton of the country s populaton n the allocaton of tax revenue among sub-natonal governments, snce the heterogenety of preferences and populaton are the man determnants of formulas for local governments supplyng local publc goods that are perfect and mperfect nter-regonal substtutes In partcular, one common determnant of formulas for revenue sharng s the dstrct s populaton Our analyss shows that the relatonshp between the funds allocated n the dstrct and the dstrct s populaton s not necessarly lnear (a common practce n some countres, see [1], [1] and n fact t could also be negatve (an ncrease n the dstrct s populaton could lead to a lower share of the resources devoted to that dstrct, see proposton 3a Fnally, the least ntutve fndng of the paper s that the dstrbuton of ncome n the federaton should not determne the dstrbuton of tax revenue sharng among sub-natonal governments As mentoned before, n our paper ths fndng s explaned by the fact that both the margnal socal benefts and costs of allocatng funds to some dstrct n the federaton do not depend on the household s ncome Although ths result should be subject of further analyss to test whether ths fndng s general or closely related to the parametrc functons used n ths paper 4 Concludng Remarks In ths paper, we study the optmal desgn of formulas for subnatonal revenue sharng when local governments provde publc goods wth spllovers n two cases of nterest: when local publc goods are perfect and mperfect nter-regonal substtutes Even though, tax revenue sharng polcy s mplemented n many developed and developng countres there s lttle formal research on the desgn of formulas for subnatonal revenue sharng In ths paper we seek to contrbute to fll ths gap Our man contrbuton s to dstngush the determnants for optmal formulas for subnatonal revenue sharng when local governments provde local publc goods wth spllovers that are perfect and mperfect nter-regonal substtutes Ths analyss has the potental to provde relevant nformaton for polcy makers on the determnants and some propertes of formulas for subnatonal tax revenue sharng Our analyss suggests the followng: 1 Spllovers should be taken nto account n the desgn of formulas for revenue sharng when lpg s are perfect nter-regonal substtutes but should not be taken nto account for mperfect nter-regonal substtutes; The dstrbuton of preferences for lpg s and populaton should be consdered n formulas for revenue sharng when subnatonal governments provde both lpg s that are perfect and mperfect substtutes; 3 For local publc goods that are mperfect substtutes, an ncrease n the populaton of the dstrct leads to an ambguous outcome n the share of tax revenue n the dstrct: for suffcently hgh (low ntal values of the share of tax revenue, an ncrease n the dstrct s populaton leads to a fall (ncrease n the share of tax revenue allocated to the dstrct; 4 For perfect substtute-lpg s, the share of tax revenue n the dstrct s ncreasng and concave wth the dstrct s populaton; 5 The dstrbuton of ncome should not be consdered n the desgn of formulas for lpg s that are perfect and mperfect substtutes Other emprcally relevant comparatve analyss are consdered n the paper Although the paper provdes nsghts for the desgn of revenue sharng systems, t does not consder poltcal nsttutons (such as electoral competton, the nteracton between the executve and legslatve bodes, the role of specal nterest groups and other ssues rased by poltcal economy models that mght be central n shapng ncentves of polcy makers Analyss on revenue sharng systems should also ncorporate the role of moblty of households and frms, and adopt a systemc vew of the optmal composton of the tax structure and spendng of subnatonal governments and the central government Future research on ths topc should address these ssues Acknowledgements Raul Alberto Ponce Rodrguez thanks Consejo aconal de Cenca y Tecnología (COACyT for the fnancal support of the program estancas sabátcas en el extranjero 15 that helped to develop ths research References [1] Rao, MG (7 Resolvng Fscal Imbalances: Issues n Tax Sharng In: Boadway, RA and Shah, A, Eds, Intergovernmental Transfers, Prncples and Practce, The World Bank, Washngton DC,

8 R A P Rodrguez, I Koch [] Brd, RM, Ebel, RD and Wallsh, C (1995 Decentralzaton of the Socalst State: Intergovernmental Fnance n Transtonal Economes World Bank, Washngton DC [3] Boadway, R and Shah, A (7 Intergovernmental Fscal Transfers: Theory and Practce The World Bank, Washngton DC [4] Smart, M (1998 Taxaton and Deadweght Loss n a System of Intergovernmental Transfers The Canadan Journal of Economcs, 31, [5] Wldasn, DE (1984 The Welfare Effects of Intergovernmental Grants n an Economy wth Dstortonary Local Taxes: A Smple General Equlbrum Analyss Journal of Publc Economcs, 5, [6] Bracco, E, Lockwood, B, Porcell, F and Redoano, M (15 Intergovernmental Grants as Sgnals and the Algnment Effect: Theory and Evdence Journal of Publc Economcs, 13, [7] Ahmad, E and Broso, G (6 Handbook of Fscal Federalsm Edward Elwar Publshng, Massashussets [8] Besley, T and Coate, S (3 Centralzed versus Decentralzed Provson of Local Publc Goods: A Poltcal Economy Approach Journal of Publc Economcs, 87, [9] Lockwood, B (15 The Poltcal Economy of Decentralzaton In: Ahmad, E and Broso, G, Eds, Handbook of Multlevel Fnance, Edward Elgar Publshng, orthampton, [1] Peña-Ahumada, JA (11 Manual de Transferencas Federales para Muncpos Secretaría de Gobernacón Insttuto aconal para el Federalsmo y el Desarrollo Muncpal Méxco 815

9 R A P Rodrguez, I Koch Appendx ( B and formulas for dstrbuton of tax revenue,1 ( Proposton The optmal budget satsfy the followng propertes: 1 The mpled dstrbuton of local publc goods, g = B, g = ( 1 B s Pareto effcent s a constant: [,1] 3 The share of tax revenue allocated n dstrct,, also satsfes the followng: > γ a as and > γ + θ > b = = c e = e = d γ, that s s non decreasng wth γ, and γ e θ, that s s non decreasng wth θ, and θ 1 Effcency: local publc goods g, g are Pareto effcent Snce g, g ArgMax Ψ, g g g g : ν ( g, g, e > ν ( g, g, e ν ( g, g, e ν ( g, g, e s a constant: [,1] Ths follows trvally by the fact,, γ, γ, θ, θ are constants Mo reover, γ, γ, θ, θ ++ : lm 1 and lm { γ γ θ θ } { γ γ θ θ } 3 satsfes the followng: > γ a as and > γ + θ > From condton (4, = γ + θ ( ( γ + θ + ( γ + θ ( ( { γ + θ }( γ + θ ( ( γ = ( γ + θ + ( γ + θ γ + θ + γ + θ, the term ( ( γ ( γ + θ ( ( γ > Snce γ + θ + γ + θ > = Moreover from > γ + θ γ + θ + γ + θ >, t s smple to see that b = = From condton (4, = = c e = e = From condton (4, e = e = d s non decreasng on γ and γ γ + θ From (4, = 1 and γ γ + θ + γ + θ γ + θ + γ + θ ( ( ( ( ( snce ( ( γ + θ + ( γ + θ ( ( γ θ 1, γ γ + θ + γ + θ + = γ + θ γ γ = 1 e ( γ + θ + ( γ + θ s non decreasng on From (4, θ and θ + θ γ + θ + γ + θ γ + θ + γ + θ γ θ = 1 ( ( ( ( and 816

10 R A P Rodrguez, I Koch ( ( ( γ θ ( γ θ ( ( γ + θ = 1 θ γ + θ + γ + θ γ + θ θ θ = 1 ( γ + θ + ( γ + θ Proposton 4 The optmal budget satsfy the followng propertes: snce ˆB and formulas for dstrbuton of subnatonal tax revenue ˆ,1 ( ˆ 41 The mpled dstrbuton of local publc goods, ˆ ˆ gˆ B gˆ ( 1 ˆ 4 ˆ ˆ,1 43 The share of tax revenue allocated n dstrct a ˆ > and ˆ b ˆ > ˆ and > c ˆ ˆ and d ˆ ˆ e = e = e f s a constant: [ ] ˆ λ > ˆ λ > f f ( λ + λ ( 1 k ˆ > > k ˆ ( 1 k wth wth =, = Bˆ s Pareto effcent,, satsfes: λ > λ > 41 Effcency: local publc goods gˆ = ˆ Bˆ and gˆ = ( 1 ˆ Bˆ are Pareto effcent snce gˆ, gˆ ArgMax Ψ and feasble ˆ ˆ ( ( ˆ, :,,, ˆ, (,, ( ˆ, ˆ g g g g ν g g e > ν g g e ν g g e ν g g, e 4 ˆ s a constant: ˆ [,1] Snce,, λ, λ, k, k are constants Moreover,,, λ, λ ++ : lm 1 { λ λ } and lm { λ λ } 43 The share of tax revenue for dstrct, ˆ, also satsfes the followng: a ˆ > and ˆ From (11, ˆ λ ( 1 k ( λ ( 1 k λ ( 1 k k ( 1 k ( 1 k = 1 snce ( λ + λ ( 1k ( 1 k ( λ + λ ( 1k ( 1k ˆ [ ] ( λ ( 1k λ ( 1k k = ( k ( ( k ( ( λ + λ ( 1k ( 1k λ ( k ˆ ( ( ( ( ( { 1 1 k 1 k } λ λ 1 1,1 : ˆ, 1,1 1,1 ˆ 1 = > + k k ˆ It also holds that b ˆ ˆ > and > ˆ λ k λ ( 1k λ ( 1k k ote that = + λ + λ 1k 1 k λ + λ 1k 1k ( ( ( ( ( ( and 817

11 R A P Rodrguez, I Koch λ k ( λ + λ ( 1k ( 1k >, ( λ + λ ( 1k ( 1k ( 1k λ ( 1k λ ( 1k k ˆ ( λ + λ ( 1k ( 1k ( 1 k ˆ ˆ ˆ λ k = + > > k c ˆ and ˆ = ˆ ( 1 ˆ ( Defne ( ˆ 1 ˆ ˆ ˆ π π 1 : = + = + = = k k k k k ( 1 ˆ ( 1 ˆ ˆ ( 1 ˆ ( ˆ ˆ 1 > > = k k k d ˆ ˆ e = e = From condton (11, ˆ ˆ e = e = e ˆ λ > It s satsfed that and ( λ + λ ( 1 k wth ( 1 k ( ( ( f ˆ > λ > = From proposton 43b { λ ( 1 λ ( 1 } ( ( ( ( λ λ ( k k k = λ λ + λ 1k 1 k λ + λ 1k 1k 1 ˆ > as ˆ + = λ ( λ + λ ( 1 k ( λ + λ > 1 k Moreover, t s smple to show f ˆ λ > > k f ˆ ( 1 k λ > wth λ > From (11, ˆ k > > k = as ˆ λ ( λ + λ ( 1 k ( 1 k and λ Submt or recommend next manuscrpt to SCIRP and we wll provde best servce for you: Acceptng pre-submsson nqures through Emal, Facebook, LnkedIn, Twtter, etc A wde selecton of journals (nclusve of 9 subjects, more than journals Provdng 4-hour hgh-qualty servce User-frendly onlne submsson system Far and swft peer-revew system Effcent typesettng and proofreadng procedure Dsplay of the result of downloads and vsts, as well as the number of cted artcles Maxmum dssemnaton of your research work Submt your manuscrpt at: 818