Fiscal Externalities and Efficient Transfers in a Federation

International Tax and Public Finance, 7, 119–139, 2000.c© 2000 Kluwer Academic Publishers. Printed in The Netherlands.

Fiscal Externalities and Efficient Transfers in aFederation

MOTOHIRO SATO [email protected] University, Tokyo, Japan

Abstract

This paper investigates properties of the second best allocation in a fiscal federal system in which both federal taxand intergovernmental grants are involved and the taxation is distortionary. Also, optimal federal grants and taxpolicies in a decentralized fiscal system are examined. Our major findings are: (i) the second best does not requirethe equalization of marginal cost of public funds across regions in a conventional form; (ii) matching grants basedon either the local tax rates or tax revenues should be introduced to internalize the tax externality; and (iii) oncelump-sum and matching grants are optimized, federal tax policy becomes redundant so the optimal fiscal gap isindeterminate.

Keywords: tax externality, equalization of MCPFs, matching grants, optimal fiscal gap

JEL Code: H7

1. Introduction

There has been a growing emphasis on fiscal decentralization in many federations. Thecase for decentralizing expenditure responsibilities is widely accepted. Whether or nottaxation responsibility should be decentralized to the same extent is much more contro-versial, however. While a decentralized tax system can be supported from a public choiceperspective as a device for enhancing accountability, there are concerns of efficiency andequity, which have been emphasized mainly by normative public economists. To the ex-tent that lower jurisdictions have insufficient revenue-raising capabilities relative to theirexpenditure needs, the federal government inevitably must formulate a transfer policy toclose the fiscal gap. The purpose of this paper is to analyze optimal national tax and grantspolicy in a decentralized federation when governments are restricted to using distortionarytaxes.

There is a large literature on the efficiency of resource allocation in a federal systeminitiated by Buchanan and Goetz (1972) and Flatters, Henderson and Mieszkowski (1974).In a relatively simple framework, some key results have been obtained. Boadway andFlatters (1982) show that the existence of a fiscal externality arising from individual freemobility leads to an inefficient allocation of population among regions; they also showhow equalization grants can internalize this fiscal externality. On the other hand, Myers(1990) argues that federal grants are not necessary: migration efficiency can be achievedthrough voluntary inter-regional transfers when local authorities are allowed to use such aninstrument.

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Although these works have significant implications for policy making, they are still in theworld of the first best: head and rent taxes are assumed to be available so we do not haveto worry about the excessive cost associated with local or national tax policies. There islimited work on the properties of a fiscal federal system in the presence of distortionary tax.1

Recently, the relevancy of tax assignment for the welfare cost of local and national taxes hasbeen emphasized. In some federal nations including Canada and the United States, major taxbases, such as income, sales and/or payroll taxes are shared by federal and provincial/stategovernments. In this circumstance, a tax externality can arise, which will distort decisionmaking at each level of government. In a simple partial equilibrium framework, Dahlby(1994) shows that if a local government ignores the effect of its tax policy on the centralgovernment’s tax revenue, the MCPF (marginal cost of public fund) is underestimated andthere may be an excessive supply of local public goods.2 An analogous conclusion is derivedby Johnson (1988). This argument is extended to a general equilibrium model by Dahlby andWilson (1994). They establish that the second best allocation requires the social MCPFs tobe equalized not only among tax bases in each local jurisdiction, but also among regions. Theexplicit form of equalization grants to achieve this objective is also presented. In their model,however, immobility of households and capital is assumed as well as perfect cooperationamong governments. When the MCPF is underestimated by local authorities, they may raiselocal tax rates beyond the second best levels without binding agreement. Dahlby (1996)recommends the introduction of matching grants to internalize this tax externality. As longas the externality exerts a negative impact on the tax revenues of other governments, the rateof the grants reflecting the marginal external cost should be negative. His idea gives riseto the following question: how should the lump-sum (unconditional) and matching grantsbe combined? The case of mobile households is considered by Boadway and Keen (1996),who raise an important question: what is the optimal level of fiscal gap between local andcentral governments?3 They examine an economy consisting of homogeneous regions inwhich a labor income tax is co-occupied by local and central governments, and concludethat the existence of the tax externality may make the optimal fiscal gap negative. However,their model is restrictive in the sense that they consider only the symmetric equilibrium,and allow only lump-sum transfers from the central government.

This paper examines the second best policy at the federal level in the presence of bothheterogeneous regions and imperfect individual mobility. As cited above, so far, mostauthors have proceeded under the assumption of either immobility of households and/orhomogeneous regions. As a notion of imperfect mobility, we employ the home-attachmentmodel of Mansoorian and Myers (1993): by changing the degree of home-attachment,we can treat immobility and perfect mobility as polar cases. The basic framework of ourmodel is fairly standard. The economy consists of two regions and there are two levelsof government (local and federal governments). The regions differ in terms of productiontechnology and/or endowment of fixed factors. Individuals are homogeneous in all respectsbut the degree of home-attachment. As in Boadway and Keen (1996), we suppose that alabor tax is shared by the two governments in each region. In terms of game theory, wedescribe the central government as a fist mover and the local governments as followers.The local government provides a single local public good financed by local tax revenueand the transfer from the central government. The central government designs the inter-

FISCAL EXTERNALITIES AND EFFICIENT TRANSFERS IN A FEDERATION 121

regional transfer and federal tax schemes in addition to providing a national public good.The transfer scheme can involve both lump-sum and matching grants.

We begin by characterizing the second best outcome in the economy. We shall see thatin the presence of perfect or imperfect mobility of households, the heterogeneity of theregions casts a new light on its characteristics. Contrary to Dahlby and Wilson (1994), thesecond best does not generally require the equalization of the MCPFs between regions inconventional form. We will argue that either different criterion be applied for an efficientinter-regional resource and population allocation, or the MCPFs be redefined to incorporatedistortion associated with mobility of tax bases. In a decentralized system (non-cooperativesetting), we then examine the federal policy needed to replicate the second best allocation. Amatching grants scheme as a function of the local tax rate or tax revenue must be employedto internalize the tax externalities arising due to the misperception of the social MCPFsby the local authorities. A lump-sum transfer is also needed to equate the shadow pricesof raising public funds among the governments to ensure the second best provision of thenational public good and allocation of the population. Furthermore, it will be shown thatthe optimal fiscal gap is indeterminate, a consequence of the fact that federal tax policyis redundant as an instrument for the purpose of achieving the second best. This result issharply in contrast with Boadway and Keen (1996) in which only lump-sum transfer alongwith federal taxation is considered.

2. The Basic Structure of the Model

The federation consists of two regions denoted byk = A.B. The national population isnormalized to unity. Following Mansoorian and Myers (1993), we introduce imperfectmobility by supposing heterogeneous preferences with respect to home attachment.4 Thetype of households is denoted byν and is assumed to be distributed uniformly on the interval[0,1]. The utility function of typeν-household is:5

U (xA, hA)+ b(gA)+ B(G)+ a · (1− ν)if the household resides in region A, and

U (xB, hB)+ b(gB)+ B(G)+ a · νif he resides in region B, wherexk is a private good,hk is labor supplied,gk denotes alocal public good andG is a national public good. Preferences are strictly concave, strictlyincreasing inxk, gk andG, and strictly decreasing inhk. We assume that bothgk andG arepure in nature, but that the benefit of the local public good does not spill over across regions,while that of the national public good accrues to all households irrespective of where theyreside. The assumption of separability in the utility function implies thatgk andG do notaffect the leisure-consumption decisions of individuals. Moreover, individual residentialchoice will turn out to be independent of the national public good. A constant,a, designatesthe degree of home attachment. Thusa · (1− ν) (resp.a · ν) is the psychic benefit thattype ν-households obtain when residing inA (resp.B). The difference in the degree ofhome-attachment influences only inter-regional migration, not individual decision making

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within each region. Without loss of generality, we suppose that initially the householdswith ν < 1/2 (resp.> 1/2) reside in region A (resp. B). Fora = 0, there will be perfectmobility as is familiar in the literature. Complete immobility can be described as the otherextreme case (a→∞).

The decision making of the individuals can be divided into two stages: choice of laborsupply in each region and locational choice. The latter is done by comparing the utilitiesfor given local and federal policies, taking as given the size of population in each region.Each local government chooses the local tax rate and the amount of a local public goodsubject to local budget balance, and taking into account individual migration. The budgetbalance of other governments is, however, ignored: from a local viewpoint, the tax rate andthe public expenditures of other governments are taken as given.6 Following most of theliterature, we assume that the central government is a first mover and thus incorporates theeffects of its decision making on the Nash equilibrium. In other words, we can considerthe central government as a Stackelberg leader. Both federal taxes and intergovernmentalgrants, including a matching component, are available to the federal government. Theywill be used to manipulate the regional governments’ decisions and individual migrationin order to enhance efficiency. A key to understanding one of our main arguments, theindeterminacy of optimal fiscal gap, lies in the difference between the number of federalinstruments and the number of economic variables which the federal authority attemptsto manipulate. It will be seen that the former exceeds the latter, which implies that oneinstrument is redundant.

Let us begin with household optimization. Each household’s utility maximization inregionk (= A, B) is expressed by:

maxxk,hk

U (xk, hk)+ b(gk)+ B(G) subject to xk = (wk − τk)hk

wherewk is the wage rate andτk is the per unit tax on labor. The latter includes both thefederal tax,T and the local tax,tk, soτk = tk + T . In the present paper, the federal tax rateis assumed to be uniform across regions. Such a restriction is plausible when horizontalequity of individual treatment under federal tax system is required. Solving the aboveoptimization yields labor supply function,h(wk − τk). Throughout this paper, we assumeh′(wk − τk) > 0 for allwk − τk.

The production side of the economy is simple. Output, which can be used for privateconsumption, or local or national public good provision on a one to one basis, is producedby both labor and a fixed factor (land). The technology is represented by an increasing andstrictly concave production function,fk(nkhk), wherenk is the population in the region.Our analysis includes the case where both regions are heterogeneous with respect to eithertechnology or land. In equilibrium, the wage is equated to the marginal productivity oflabor so:

wk = f ′k(nkh(wk − τk)). (1)

Solving (1) forwk yields a regional market clearing wage function,wk(τk,nk).Rent is defined as a residual, or

rk(τk,nk) = f (nkh(w(τk,nk)− τk))− w(τk,nk)nkh(w(τk,nk)− τk). (2)


We assume that all the rent accrues to the public sector, and denote the proportion of the rentaccruing to the federal government and the local government byθ and 1− θ , respectively.This assumption implies either public ownership of the fixed factor or 100 percent tax onthe rent income. In the latter case,θ is the federal tax rate on the rent income. Throughoutthis paper,θ is assumed to be fixed.

The solution to the household problem yields the indirect utility (excluding the benefit ofhome-attachment) asvk(τk,nk)+ b(gk)+ B(G), which has the following properties:

dvk

dτk= −λkhk

Dk< 0 (3)

dvk

dnk= λk f ′′k h2

k

Dk< 0 (4)

whereλk is the marginal utility of income,hk = h(wk − τk) andDk = 1− nk f ′′k h′k > 0.Next, we turn to locational choice. Households differ in their attachment to a region, so

migration equilibrium for given policy instruments can be characterized by the marginalhousehold who is just indifferent between the two regions. The marginal household’s typeis equivalent to the population of region A due to the assumption of the uniform distributionof the types:

vA(τA,nA)+ b(gA)+ B(G)+ a(1− nA) = vB(τB,1− nA)+ b(gB)+ B(G)+ anA. (5)

Households withν < nA locate in region A and those withν > nA reside in region B.7

Local governments provide the local public goodgk, and finance it by the labor tax andthe transfer they receive in lump-sum and matching forms from the federal government.The local revenue constraint for region k is thus:

gk = Rk(tk, T,nk,mk, Sk) ≡ tknkh(w(τk,nk)− τk)+ (1− θ)rk(τk,nk)+mktk + Sk (6)

wheremk is the matching grant on the local tax rate, andSk is the lump-sum transfer.We assume that bothmk and Sk can be of either sign;mk < 0 implies that the federalgovernment taxes the local tax rate, whileSk < 0 designates the lump-sum tax on thelocal government.8 Throughout this paper, we assume thatmk and Sk can be regionallydifferentiated.

Following Burbidge and Myers (1994b) and Wellisch (1994), we employ the residents’utility excluding home-attachment,v(τk,nk)+ b(gk)+ B(G) (k = A, B), as the regionalobjective. This may correspond to the median voter objective (Mansoorian and Myers,1997; Wellisch, 1994).9 Each voter locating in one region will prefer a regional policywhich maximizes his own utility. Since all voters are identical except for locational pref-erences, and the latter is a parameter, the maximization of each voter’s utility is equivalentto maximizingv(τk,nk) + b(gk) + B(G). Therefore, there will be unanimous agreementfor the choice oftk andgk. Of course, the unanimity arises from the assumption that in-dividual preference is additively separable between home-attachment and consumption ofgoods.

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Local decision making incorporates the migration function, which can be obtained bysolving (5) with respect tonA and substituting (6). For future reference, we present itsderivatives with respect totk, Sk andT :

∂nk

∂tk= − 1

J

(∂vk

∂τk+ b′k

∂Rk

∂tk

)(7)

∂nk

∂Sk= − 1

Jb′k (8)

∂nA

∂T= − 1

J

{(∂vA

∂τA+ b′A

∂RA

∂T

)−(∂vB

∂τB+ b′B

∂RB

∂T

)}(9)

wherenB = 1− nA and

J =∑

k=A,B

(∂vk

∂nk+ b′k

∂Rk

∂nk

)− 2a. (10)

Stability requiresJ < 0.10 When this condition is satisfied, we can verify that∂nk/∂Sk > 0while ∂nA/∂T is ambiguous. The sign of the first equation is also ambiguous: while thedirect impact of a higher tax rate decreases the resident’s welfare, the expansion ofgk

following an increase in the tax revenue improves it. Obviously, iftk and thus the local taxrevenue are not high enough, the second term of (7) will dominate the first term: thus wehave∂nk/∂tk > 0, and vice versa. It can be shown, however, that in the regional optimum,these two effects offset one another.

Although most of the existing works include the migration function explicitly in thelocal optimization (Boadway, 1982; Myers, 1990), we find it more convenient to usenA

(or equivalentlynB = 1− nA) as a regional control variable by including the migrationequilibrium (5) in the regional optimization as a constraint.11 Adding nk artificially to theset of control variables does not increase the freedom of local government optimizationsince the migration constraint is also added. As cited above, we assume that the localgovernments take as given both public expenditure and tax rate of other governments.

For the local governmentk, tk andgk should be chosen to maximize per capita utility,υ(τk,nk)+ b(gk)+ B(G) subject to its own local budget constraints (6) and the migrationconstraint (5). The control variables aretk, gk, nA. Formally, the local optimizationmaximizes the following Lagrangian:

Lk(tk, gk,nk, µk, γkk )

= v(τk,nk)+ b(gk)+ B(G)

+ µk{v(τk,nk)+ b(gk)+ a(1− 2nk)− υ(τj ,nj )− b(gj )

}+ γ k

k {Rk(tk, T.nk,mk, Sk)− gk} (11)

wherek, j = A, B, k 6= j andnB = 1− nA. The multiplier associated with (6),γ kk ,

represents a regional shadow price of raising marginal tax revenue, which will not coincide


with the social value. For given federal policies,2 ≡ {T, Sk,mk(k = A, B)} andG, theNash equilibrium requires the decisions of the local governments to be consistent with eachother in the following sense: thetj andgj taken as given by another local authorityk (6= j )should be chosen byj . Given tk andgk (k = A, B) by (5), the value ofnA chosen in thelocal optimization is the same between the regions. Throughout this paper, uniqueness ofNash equilibrium for each federal policy is assumed. Due to the separability of the utilityfunction, however,G does not affect the structure of the Nash equilibrium. We can definethe values in the equilibrium as functions of2: say,tk(2) (k = A, B). As for the localpublic good, we can writegk(2) = Rk(tk(2), T,nk(2),mk, Sk). The federal policy exertsdirect and indirect impacts on the local public expenditure: the latter is done through thechange in the local tax rate and the induced migration. We can also write the welfare level inregionk in the resulting Nash equilibrium byVk(2,G, tj (2), gj (2)) (k, j = A, B, j 6= k).Recall that the policy parameters of the other local government are taken as given and thechanges in these parameters caused by changes in2 can affect the regionk’s welfare. Thisis why we includetj (2) andgj (2).

Central Government

Following Burbidge and Myers (1994b) and Wellisch (1994), the objective of the centralgovernment is assumed to be given by a weighted average of regional welfare levels:

δ{v(τA,nA)+ b(gA)+ B(G)} + (1− δ){v(τB,nB)+ b(gB)+ B(G)}, δ ∈ [0,1] (12)

As suggested by Mansoorian and Myers (1993) and Wellisch (1994), in the presence ofhome-attachment (a > 0), we can trace the second best frontier of regional utilities bychanging the welfare weight,δ. The next section contains a general characterization of thissecond-best frontier: it is independent of the value ofδ. However, it should be kept in mindthat the second best policy relies on the value ofδ.12 The task of the central government isto design federal policy to maximize social welfare subject to the budget constraint:

G = RF (tA, tB,nA, T, SA, SB,mA,mB)

≡ T∑

k=A,B

nkh(w(τk,nk)− τk)+ θ∑

k=A,B

rk(τk,nk)−∑

k=A,B

mktk −∑

k=A,B

Sk (13)

along with the dependency of the Nash equilibrium on2 andG. Formally, the Lagrangefunction is:

L F (2,G, γF ) = δVA(2,G, tB(2), gB(2))+ (1− δ)VB(2,G, tA(2), gA(2))

+ γF {RF (tA(θ), tB(2),nA(2),2)− G}. (14)

As mentioned above, the federal instruments are used to manipulate the incentives of thelocal authorities and internalize externalities such as the tax externalities arising in thefederal system.13

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3. The Second Best Allocation

In this section, we characterize the second best allocation. Dahlby and Wilson (1994)establish that the well-known Ramsey tax rule can be applied, given the immobility ofhouseholds: from a efficiency view point, the conventional MCPF should be equalized notonly among tax bases in each region, but also between regions and between local and federalgovernments. A similar conclusion is obtained by Boadway and Keen (1996) with perfectmobility under the restriction of a symmetric equilibrium. We will see, however, that whenthere is (perfect or imperfect) inter-regional movement and regions are not homogeneous,the equalization of the standard formula of MCPFs across regions does not necessarily hold.

At the outset, we should distinguish between the “conventional” MCPF and the “eco-nomic” one. The former applies to evaluate the marginal cost of providing a local publicgood, and according to the modified Samuelson condition, it should be equated with themarginal gain of the public good (Atkinson and Stiglitz, 1980). If we letε be the un-compensated elasticity of the taxed good (labor in this model) with respect to tax rate, theconventional MCPF is given by 1/(1+ ε).14 In an economy with a representative consumerand without mobility this MCPF is coincident with the value of the multiplier associatedwith the government’s revenue constraint in the optimum. This multiplier should be re-garded as the economic MCPF (see Atkinson and Stiglitz, 1980). Our argument in thissection is that, although we can still expect the equalization of the values of the multipliersassociated with local budget constraints in the second best, it does not imply the equalizationof conventional MCPFs across regions. As shown later, the conventional form of MCPFincorporates intra-regional distortion due to taxation, but does not include inter-regionalone. Henceforth we refer to the multipliers as shadow prices to avoid possible confusion.We use the term MCPF to express 1/(1+ ε) because this terminology is widely used. Itshould also be noted that, throughout this section, MCPF is defined in the social sense asreflecting all relevant costs associated with an increase in the labor tax rate.

The necessary conditions for the second best resource and population allocations arederived by maximizing the social welfare function (12) subject to the migration constraint(5) and the unified revenue constraint:

gA + gB + G =∑

k=A,B

{τknkh(w(τk,nk)− τk)+ rk(τk,nk)}. (15)

The control variables here areτk, gk (k = A, B), nA andG. Formally, the second bestoptimization problem is expressed for arbitrary value ofδ by the Lagrangian:

L = δ{v(τA,nA)+ b(gA)+ B(G)} + (1− δ){v(τB,nB)+ b(gB)+ B(G)}+ µ{v(τA,nA)+ b(gA)+ a · (1− 2nA)− v(τB,nB)− b(gB)}

+ γ{ ∑

k=A,B

{τknkh(w(τk,nk)− τk)+ rk(τk,nk)} −∑

k=A,B

gk − G

}.

We can establish the following first order conditions:15

−(δ + µ)λAhA + γnA(hA − τAh′A) = 0 (τA) (16)


−(1− δ − µ)λBhB + γnB(hB − τBh′B) = 0 (τB) (17)

(δ + µ)b′A − γ = 0 (gA) (18)

(1− δ − µ)b′B − γ = 0 (gB) (19)

(δ + µ)B′ + (1− δ − µ)B′ − γ = 0 (G) (20)

1

DA

{f ′′Ah2

A(λA(δ + µ)− γnA)+ γ τAhA}

− 1

DB

{f ′′Bh2

B(λB(1− δ − µ)− γnB)+ γ τBhB} = 2aµ (nA). (21)

The variables shown in parentheses are the instruments being optimized. Combining (16)and (17) with (18) and (19), respectively, yields the necessary conditions for the secondbest provision of local public goods:

nkb′(gk)

λk=(

1− τkh′khk

)−1

, k = A, B. (22)

This is a well known modified Samuelson condition where(1− τkh′k/hk)−1 (k = A, B)

is the conventional MCPF. Equation (22) requires that the provision of the local publicgood should be made so as to equate the marginal gain with the MCPF, which includes themarginal excess burden in addition to the resource cost. Solving (16) forδ+µ and (17) for1−δ−µ, respectively and inserting them into (20) gives the modified Samuelson conditionfor the national public good:∑

k=A,B

nkB′

λk

(1− τk

h′khk

)= 1. (23)

In the above, the marginal gain fromG in each region is weighted by reciprocal of the MCPFin each region. Only if the MCPFs are equalized across regions, will (23) reduce to a standardform of the Samuelson condition for the second best economy as in (22). Substituting (16)and (17) into (21) and making some manipulations establishes the necessary condition forthe second best population allocation:

−2anB

λB

(1− τB

h′BhB

)≤ τAhA − τBhB ≤ 2a

nA

λA

(1− τA

h′AhA

). (24)

This is analogous to the necessary condition in the economy with non-distortionary taxes,as derived by Wellisch (1994). It argues that the difference in the per-capita tax paymentacross regions is bounded by reciprocal of the conventional MCPFs weighted by 2ank/λk

(k = A, B). Alternatively, using the individual budget constraint, we can writeτkhk =wkhk−xk, which is the net social product (Boadway and Flatters, 1982). Thus the middle of(24) represents the difference in the net social product between two regions.16 The followingproposition summarizes the above discussion.

PROPOSITION1 The second best in the economy is characterized by Eq. (22)–(24).

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An alternative way of deriving the second best is to assume full policy coordinationamong governments. As examined by Dahlby and Wilson (1994), when policy coordinationis possible, the optimization is characterized by the maximization of the social welfarefunction subject to the set of revenue constraints. The lump-sum grants should be includedin the instruments being optimized. Since all externalities are incorporated, tax matchinggrants are not required. Letγk be the multiplier associated with local budget constraint (6)and letγF be the one for the federal constraint (13).17 From this alternative approach, weget an additional condition:

γA = γB = γF . (25)

COROLLARY 1 TO PROPOSITION1 In the second best, the shadow prices of taxation areequalized among governments.

Equation (25) is embedded in the integrated form of revenue constraint (15): note that theequalization of the shadow prices implies the unification of the budget constraints.

Let us turn to the issue of conventional MCPFs. In the case of immobility, which impliesµ = 0, solving (16) and (17) forγ yields:

γ = δλA

nA

(1− τAh′A

hA

)−1

= (1− δ)λB

nB

(1− τBh′B

hB

)−1

(26)

whereδλA/nA and(1− δ)λB/nB represent distributional concerns (Dahlby and Wilson,1994). Thus the last two terms may be called the MCPF with distributional weights, whichis a generalized form of the conventional MCPF. (26) reduces to the familiar formula, thatis, an equalization of the MCPFs across regions if and only if these weights are equal:(

1− τAh′AhA

)−1

=(

1− τBh′BhB

)−1

. (26′)

It is worth considering the relevancy of (26′) as a formula for second best inter-regionalresource allocation. Dahlby and Wilson (1994) derived (26′) by minimizing an aggregateexcess burden. Along with the assumption of homogeneous consumers, it is known that theminimization of excess burden is equivalent to maximization of a representative household’swelfare in the literature of optimal taxation (Auerbach, 1985). Therefore, the equalizationof the conventional MCPFs across tax bases is a relevant characteristic. The equivalencyof the two approach is not necessarily true in the context of a fiscal federal model, however,because consumers residing in different regions cannot be treated as a single agent ingeneral. (26′) should hold on a specific point of the second-best frontier, which correspondsto δλA/nA = (1− δ)λB/nB.18 Thus we can establish:

COROLLARY 2TOPROPOSITION1 If labor is immobile, the second best requires that MCPFswith distributional weights (rather than the conventional ones) be equated across regions.

In accordance with this corollary, the inter-regional transfers needed to realize the secondbest should be made from the region which would otherwise enjoy lower MCPF to the one


with higher MCPF initially. Below, we argue that the above corollary cannot be extendedto the case where household’s mobility is present.

In contrast, when mobility is perfect soa = 0, (24) reduces to:

τAhA = τBhB (27)

The residence-based tax payment is the same in both regions, which characterizes the effi-ciency of inter-regional population and resource allocation in the first best economy. Notethat (27) cannot be realized without inter-regional transfers in general and thus we caninterpret (27) as the formula for inter-regional transfers (equalization payments) as dis-cussed by Boadway and Flatters (1982). It is worth noting the contrast between (26) and(27): the former concerns the marginal cost of raising tax revenue, while the latter im-plies that the inter-regional transfer should equate per capita tax payments across regions.These two are consistent with each other in the case of symmetric regions as supposedin Boadway and Keen (1996). Otherwise, it is the equalization of per capita tax pay-ments that should be the criterion of the inter-regional transfers in the presence of perfectmobility.

In more general circumstances involving perfect mobility, using (16) and (17), the rela-tionship between the generalized MCPFs can be represented by:

δλA

nA

(1− τAh′A

hA

)−1

− (1− δ)λB

nB

(1− τBh′B

hB

)−1

= γ(

δ

δ + µ −1− δ

1− δ − µ). (28)

The right hand side cannot vanish unlessµ = 0. Along withδ = 1/2,µ = 0 applies in thesymmetric outcome assuming the homogeneity of the two regions. Otherwise,µ = 0 is nota general property of the second best solution. Equation (28) indicates that the equalizationof the generalized MCPFs is not an appropriate criterion of equalization payments wheninter-regional mobility is present and the regions are not identical. Rather, the equalizationformula should follow (24): the inter-regional resource transfer should be made so thatthe difference in the per-capita tax payment across regions is bounded by reciprocal of theconventional MCPFs weighted by 2ank/λk (k = A, B).

Our conclusion does not, however, eliminate all use of the conventional form of theMCPF with or without distributional weights. It is still valid for evaluating the marginalcost of expanding local public expenditure: we still have the familiar form of the modifiedSamuelson condition for local public goods. The presence of household’s mobility altersthe marginal gain ofgk and the marginal social cost of taxation, equally. This can be seenfrom comparing (16) with (18) for an instance: both conventional MCPF, 1− τAh′A/hA,and the marginal benefit,nAb′(gA)/λA, are weighted byδ + µ. (And the same is true forregion B.) This is not unusual in the literature. The presence of inter-regional mobilitydoes not change the criterion for efficient intra-regional resource allocation: in the firstbest world, the Samuelson condition holds for a local public good even when there isfree mobility (Boadway, 1982; Boadway and Flatters, 1982). However, mobility imposesadditional conditions for efficient population and resource allocation among regions, andthe inter-regional transfer must be designed to ensure this: the equalization of the MCPFsgiven in Eq. (26) is not compatible with this purpose.

130 SATO

4. Second Best Policy in a Decentralized Federation

The second best allocation achieves maximum social welfare when only distortionary tax-ation is available. The question is: can this be achieved in a decentralized framework? Inthe present model, it turns out that the central government can replicate the second best.19

To establish this argument, we begin with examining the characteristics of the Nash equi-librium for a given federal policy. Then we present the second best federal policy whichinvolves both tax policy and an intergovernmental transfer program.

For a given2 andG, solving the local government problem (11) for regionk yields:

λkhk(1+ µk) = nkhkγkk

(1− τk

h′khk

)+ γ k

k {nkh′kT − n2khkh′k f ′′k θ + Dkmk} (tk) (29)

b′k(1+ µk)− γ kk = 0 (gk) (30)

hk

Dk

{γ k

k tk + f ′′k hk{(1+ µk)λk − (1− θ)γ kk nk}

}− 2aµk = − hj

Djf ′′j hjµkλj (nk) (31)

where j 6= k. The variables being optimized are shown in parentheses. Recall thatnk isan artificial instrument; therefore (31) explains how population should be allocated froma regional viewpoint. Inserting (29) into (30) establishes the necessary condition for anoptimal provision ofgk from a regional viewpoint:

nkb′kλ

{(1− τk

h′khk

)+ nkh′k(T − nkhk f ′′k θ)+ Dkmk

nkhk

}= 1, k = A, B. (32)

The largest braces on the left-hand side corresponds to the reciprocal of the regional MCPF.The difference between the regional and social MCPFs in conventional form is representedby the second term in the bracket which can be rewritten as:

nkh′k(T − nkhk f ′′k θ)+ Dkmk = Dk

(−∂RF

∂tk

), k = A, B. (33)

In terms of Dahlby (1994, 1996),∂RF/∂tk represents a tax externality that the local gov-ernment imposes on the federal budget. Whenmk = 0, ∂RF/∂tk < 0 from (33) and, asargued by Dahlby (1994) and Boadway and Keen (1996), the MCPF is underestimated bythe local government. It may worth noting that the presence of the tax externality does notdepend on the degree of mobility.

Note that (32) can be restated as:

∂vk

∂τk+ b′k

∂Rk

∂tk= 0. (32′)

From (7), this implies∂nk/∂tk = 0: in the regional optimum, a marginal increase intk has noimpact on the migration. This property simplifies our analysis in this section substantially.For a given federal policy{2,G}, the Nash equilibrium can be obtained by solving thesystem of the equation (29)–(31) together with the migration constraint (5) and the revenueconstraints of the local governments (6) fortk, gk, nA, µk, γ k

k . This system consists of 9


equations for the same number of variables so we expect the system can be solved. As citedin section 2, the values in the equilibrium can be expressed as functions of2.

We now turn to the federal policy needed to replicate the second best outcome. De-note the values in the second best by asterisks. To replicate the second best,2 ≡{T, SA, SB,mA,mA} andG are required to satisfy the following relations:

τ ∗k = tk(2)+ T, g∗k = gk(2) = Rk(tk(2), T,nk(θ),mk, Sk) (k = A, B)

and

G∗ = RF (tA(2), tB(2),nA(2),2).

Note that migration equilibrium is ensured for each federal policy and onceτk = τ ∗k ,gk = g∗k andG = G∗ hold, nA is also at its second best value. Equivalently (and moreclearly),2,G should give a system of equations consisting of (13), (22), (23) and (24) withthe Nash equilibrium values of the local policy instruments, say,tk = tk(2). The system ofequations for2 should be solvable and, as a matter of fact, one federal instrument seemsto be redundant: there are six federal instruments variables for five equations. In the restof this section, we show that this conjecture is correct.

It is obvious that matching grants should be used to internalize the tax externality (Dahlby,1996). In other words,mk should be set so as to cause∂RF/∂tk to vanish in (33). Therefore,we can establish:

mk = nkhk

Dk

(nkh′k f ′′k θ −

h′khk

T

), k = A, B. (34)

With (34), (32) becomes coincident with the second best condition forgk, (22). The valueof mk depends onT as well asθ , but at least, it should be negative. This is because the taxexternality has a negative impact on the federal budget constraint.

The formula for revenue matching grants can be obtained from (34) (Dahlby, 1996). Letqk be the grant on the local tax revenueRk (k = A, B). The following relation should holdbetweenmk andqk:

mk = qk∂Rk

∂tk≡ qk

nkhk

Dk

{1− h′k

hktk − nkh′k f ′′k θ

}, k = A, B (35)

The right hand side represents the amount of a change in the matching grant paymentfollowing the change in the local tax rate. Equating (35) with (34) yields the second bestformula forqk:

qk

(1− h′k

hktk − nkh′k f ′′k θ

)= nkh′k f ′′k θ −

h′khk

T (36)

Let us know turn to the optimal formula for lump-sum grants. By solving the federaloptimization using the fact that∂RF/∂tk = 0 under (34), the necessary condition for optimallevel of Sk (k = A, B) can be obtained from:

γF = δγ AA − (1− δ)µBb′A + H

dnA

dSA= −δµAb′B + (1− δ)γ B

B + HdnA

dSB(37)

132 SATO

where

H = −δµAb′B∂RB

∂nA− (1− δ)µBb′A

∂RA

∂nA+ γF

∂RF

∂nA.

An additional increase ingk is worth γ kk from a regionk’s standpoint, while regionj

(6= k) values it at−µj b′k. In addition, the transfer followed by the increase ingk alsoinduces inter-regional migration, which alters the size of the tax base of all governmentsand, therefore, public expenditures. Such an induced migration exerts a first-order effect onsocial welfare.H summarizes the social value of the fiscal external effects associated withindividual mobility. To summarize, the middle and the right-hand sides of (37) representthe social (aggregate) net gain from exogenous revenue increase in regionk (= A, B). Wecan say that they are the shadow prices of the local tax revenues.γF is the shadow price ofthe federal tax revenue, and, therefore, we have the analogous expression to (25).

Making use of (37) along with (22), we can establish the necessary conditions for secondbest values ofG andnA, (23) and (24). We can provide an intuition for this result. Bothprovision ofG and residential movement necessarily follow vertical and horizontal reallo-cation of funds among the governments. Efficiency requires the marginal costs (the shadowprices) associated with such transfers to be equated among the governments, which canbe achieved by optimizing the lump-sum grants as represented in (37). Then the efficientallocation of funds (tax revenues) leads to (23) and (24).

The following proposition summarizes the above results.

PROPOSITION2 The federal government can replicate the second best by choosing thematching and lump-sum grants so as to satisfy (34) and (37), respectively.

We can define the second best transfer scheme involving both lump-sum and matchingcomponents as:

S̄k ≡ Sk +mktk or S̄k ≡ Sk + qk Rk, k = A, B

wheremk andSk (k = A, B) are the optimized values, andqk is the revenue matching grant,and is related to the second best value ofmk by (34).

In the present context, matching and lump-sum grants do different jobs. The formerinternalizes the tax externalities, which arise from the tax base sharing, while the latterunifies the revenue constraints of all governments by equalizing their shadow prices. Oncethe tax externalities are internalized, the formula forgk followed at the local level becomescoincident with the second-best one. Thus, the presence of the fiscal externality associatedwith residential movement does not distort the choice ofgk by the local governments evenin the case where they account for the individuals’ mobility: that is,nk is a function ofthe policy instruments from the local government’s view. There has been an argumentthat the lump-sum transfer should serve to internalize the fiscal externality (Boadway andFlatters, 1982). On the other hand, Myers (1990) suggests that a decentralized fiscalsystem can be inefficient because of lack of instruments of inter-regional transfers at thelocal level. In fact, the inefficiency of population allocation,nk, as well as provision ofG, will arise from the separate requirement of budget balance of all governments. Ingeneral, an allocation on the second best frontier does not ensure budget balance without


intergovernmental transfers. To fill the gap between revenues and expenditures of thegovernments in the second best, the transfers need to be made in lump-sum fashion. Inother terms, cross-subsidization is needed.20 As opposed to Myers (1990), however, theavailability of instruments of inter-regional transfers at the local level does not necessarilyresolve the problem. If the local authority does not take into account the budget balanceof the other governments, as assumed in the present paper, it will not have any incentiveto make transfers to the other governments: from the regional view, such a transfer justcontributes to the budget surplus of the others. Therefore, the second best can be achievedonly throughSk designed by the central government.

The main departure of the present model from Boadway and Keen (1996) is to introduceimperfect mobility and heterogeneous regions as well as the use of the tax matching grants.It is straightforward to see that the heterogeneity of regions requires regionally differentiatedtransfer scheme. A uniform formula for transfers, like the equalization system in Canadaand Australia, is not likely to achieve the efficient allocation. Although neither the degree ofmobility, a, nor the welfare weight,δ, influences the formula for the second best matchinggrants, the design of the lump-sum transfers is dependent on them (see (37)). It is interestingthat, irrespective of the values ofa andδ, givenT , the optimal transfer scheme should mixthe lump-sum and the matching components.21

It may be worth noting the difference between the formula of the grants derived in thepresent paper and the one of equalization in Canada. The latter also includes a matchingcomponent, but it is based on local tax bases, rather than local tax revenues or tax rates. Thisdifference will have a substantial implication for the tax externality issue. Smart (1998)argues that under the Canadian equalization formula, the regional MCPF is reduced furthersince the shrinkage of the regional tax base due to an increase in local tax rate is effectivelycompensated by equalization payments. In the extreme, the regionally perceived MCPFbecomes unity: this will be true for equalization receiving provinces whose tax rate is thesame as the national average. From the above discussion, it is straightforward to see thatmatching grants must be imposed on local tax revenue or tax rates in order to resolve thetax externalities. Insofar as the equalization payment is dependent on the tax base, theexternality is exacerbated.

What about the federal tax? The above proposition holds for any value ofT once thematching and lump-sum grants are optimized. This leads to the conclusion that the federaltax is irrelevant for achieving the second best. In fact, using the envelope theorem and (37),we can establish the following:

dSW

dT= H

{dnA

dT−

∑k=A,B

1

Dk

{nkh′kT − nkhk + (1− θ)n2

khkh′k f ′′k} dnA

dSk

}. (38)

Substituting (8) and (9) above, we can showdSW/dT = 0, that is, the increase inT hasno impact on the social welfare.22 An increase inT is followed by a reduction intk by anequal amount, which is achieved by adjustingmk and Sk so as to keep the aggregate taxrate,τ ∗, as well asg∗k andG∗ the same: accordingly,n∗A also remain the same. This resultis closely related to the issue of optimal fiscal gap examined by Boadway and Keen (1996).In the present context, the fiscal gap can be defined by the difference between the federal

134 SATO

tax revenue minus the public expenditure net of transfers:

Z = T∑

k=A,B

n∗kh(w(τ ∗k ,n∗k)− τ ∗k )+ θ

∑k=A,B

rk(τ∗k ,n

∗k)− G∗ (39)

where the values of the targeted second best are denoted by asterisks.Z is equivalent tothe total amount of the transfer from the central to the local governments and thusZ > 0(< 0) designates that the fiscal gap is positive (negative). Since the federal tax is irrelevant,the following is immediate:23

PROPOSITION3 Federal tax policy is redundant in the optimum and therefore the optimalfiscal gap is indeterminate.

The degree of decentralization of the tax system may be measured by the fiscal gap:higherZ implies a relatively centralized tax system and vice versa. The above propositionargues that when federal governments are equipped with sufficient instruments of inter-governmental grants, the existence of inefficiency in the federal system does not justifyeither centralization or decentralization of tax policies. Put differently, it must be the lackof instruments or other restrictions abstracted from in the present model that determinesthe optimal fiscal gap. The conclusion of Boadway and Keen (1996) in favor of a negativefiscal gap comes from the fact that matching grants are not available in their model.24

Proposition 3 does not necessarily deny any significance to the optimal fiscal gap issue.In the present model, we assume that the federal government can conduct regionally differ-entiated grants policy. In some circumstance, the federal government may be restricted touniform grants. That is,Sk and/or mk (k = A, B) may be constrained to be invariant acrossregions, which may be the case when there is asymmetric information between the centraland local governments about local characteristics such as preference for the public good:as is familiar from the optimal income taxation literature, the imperfectness of informationmakes it infeasible to conduct differentiated policy for different agents (Stiglitz, 1982).This kind of restriction decreases the number of federal instruments available, which canprevent the federal government from replicating the second best and/or make the coordina-tion between the inter-regional transfer and federal tax policies essential. If so, uniquenessof the optimal fiscal gap may result.25

It should also be mentioned that the irrelevancy of federal tax policy partly relies on thefollowing: if necessary, the federal government can make use of negative lump-sum grantsfor raising the revenue. The negative value ofSk does not cause an additional distortion inthe process of transferring the resource from the local to the central governments. In otherwords, the federal government has an option to tax on the local governments instead of thehouseholds. IfSk or S̄k is constrained to be non-negative,T must be sufficient to financethe grant payments, which will impose a lower-bound for the optimal fiscal gap.

Before closing this section, we should make a comment on the stability issue of thesecond best equilibrium in the presence of migration. As discussed by Stiglitz (1977) andBoadway and Flatters (1982), the stability of efficient equilibrium is not necessarily ensuredin fiscal federalism models. The well-known requirement for stability in an economywithout distortionary taxation is that the economy as a whole be over-populated in thatthe equilibrium size of each region gives a negative value for the net social product. This


condition may seem not to be compatible with a positive labor tax rate in the second best,which is the most interesting case: for the second best tax rates to be positive, the net socialproduct must be positive, which implies under-population of the economy. However, in thepresent model with two levels of government and imperfect mobility, we can show that thepositive aggregate tax rates (τk > 0) can result in the stable second best equilibrium.

The second best allocation,τ ∗k andn∗k (k = A, B), may be said to be locally stable if fora sufficiently small positive value ofε,

v(τ ∗k ,n∗k + ε)+ b(Rk(·))+ a · (1− (n∗k + ε))

< v(τ ∗j ,n∗j − ε)+ b(Rj (·))+ a · (n∗k + ε) (40)

wheren∗k+n∗j = 1. That is, local stability requires that the initial population allocation canbe restored after a small perturbation ofnk if τ ∗k andS∗k (k = A, B) left unchanged andgk

is adjusted so as to keep the balance of the budget. (40) can be restated as:

J =∑

k=A,B

hk

Dkb′k

{(τ ∗k − T)+ nkhk f ′′k

(θ − τ ∗k

h′khk

)}− 2a < 0

J should be negative when evaluated at the second best values. It is immediate that withsufficient degree of home-attachment,a, the bracket term can be negative as a whole: ascited by Wellisch (1994), home-attachment can improve the stability issue. Or ifθ issufficiently high and/or the elasticity of labor supply is low, the second term in the smallestbracket is likely to be negative and so is the sign ofJ.

Interestingly, federal tax policy can ensure the stability of the second best equilibrium:whenT is set high enough,J < 0 will result. This implies that, if necessary, federal taxpolicy should be conducted to stabilize the targeted second best equilibrium. Although theoptimal fiscal gap is still not unique, it should be bounded below sinceT should satisfy:

∑k=A,B

hk

Dkb′k

{τ ∗k + nkhk f ′′k

(θ − τ ∗k

h′khk

)}− 2a < T

∑k=A,B

hk

Dkb′k. (41)

The intuition behind the above argument is as follows. As shown in (4), an increase innk

reducesvk(τk,nk) because of decrease in the wage rate, while it expands the local tax base,Rk. The latter raises local expenditure and therefore improves welfare of the residents. Fora given second best rate ofτ ∗k , higherT implies lowertk. This in turn leads to a lower valueof ∂Rk/∂nk. Thus, for sufficiently highT , the former effect outweighs the latter in at leastone region (∂vk/∂nk + b′k∂Rk/∂nk < 0), which will let J < 0. To summarize:

COROLLARY TO PROPOSITION3 For the second best equilibrium to be stable, T should beset so as to satisfy (41), and this gives a lower bound to the optimal fiscal gap.

We can say the second best equilibrium can be stable even in the case ofτ ∗k > 0, that is,the economy is under-populated. The above corollary leads us to the conclusion that thecentralization of the tax system to some degree may be justified for the purpose of ensuringstability, rather than achieving the second best.

136 SATO

5. Conclusion

In this paper, we have attempted to generalize the existing works of Dalhby and Wilson(1994) and Boadway and Keen (1996) by introducing imperfect mobility and heterogeneousregions. In contrast to the familiar argument, the second best allocation does not requirethe equalization of the conventional form of the MCPFs across regions in the presence ofimperfect or perfect mobility. Regarding the implementation of the second best allocationin a federal setting, we have examined the case where local governments ignore the budgetconstraints of other governments. In the second best, matching grants internalize the taxexternalities, while lump-sum grants equalize the shadow prices of governments’ revenues.Under the behavioral assumption of the local governments considered in this paper, lump-sum grants must be combined with matching grants irrespective of the degree of home-attachment and social welfare weights. The optimal fiscal gap is indeterminate: in otherwords, we can establish an intergovernmental transfer scheme for any level of the federal taxrate to achieve a given second best allocation. The indeterminacy implies that the optimalfiscal gap, if it exists, comes from restrictions extraneous to our model. We have also shownthat when stability is an issue, there is a minimum level of federal tax needed to stabilizethe second best equilibrium, which may give a rationale for centralization of tax system tosome extent, even in the presence of matching grants.

We should mention a few restrictions in our framework of analysis. First, the introductionof imperfect mobility is done in a rather restrictive form. We have assumed a psychologicalcost of mobility (home attachment) introduced by Mansoorian and Myers (1993). However,there are other ways of formalizing imperfect mobility. One alternative will be to assumea resource cost related to the mobility as discussed by Boadway and Wildasin (1990).Different formulations of the imperfect mobility might bring different characteristics inthe second best. Second, we have abstracted from capital mobility. There are only afew attempts to incorporate both labor and capital mobility (Burbidge and Myers, 1994b;Wellisch and Wildasin, 1996). Introducing two sorts of mobility may give new insight onour analysis. Finally, in our model, the households are homogeneous except for locationalpreference. Recently, Boadwayet al (1997), Burbidge and Myers (1994a) and Wildasin(1994) examine income redistribution policy in a fiscal federal system. By extending ourmodel to include heterogeneous agents, we may find different formula for the second bestfederal policy.

Acknowledgment

The author is grateful to Robin Boadway, Dan Usher, participants in Public EconomicsWorkshop at Queen’s, four referees and the editor of this Journal for their helpful comments.This paper is supported by the Japan Society for the Promotion of Science.

Notes

1. A classical exception is Gordon (1983). Also see Wildasin (1983).


2. The MCPF is defined by the ratio of the change in household’s welfare (measured by the marginal utility ofincome) to the change in the tax revenue due to an additional increase in the tax payment. The well-knownformula is MCPF= 1/(1+ ε) whereε is an uncompensated elasticity of a taxed good with respect to taxrate. Dahlby and Wilson (1994) argue that this formula needs to be modified, however, if distributional issuesare considered. In the present paper, we use the term “MCPF” to designate the formula 1/(1 + ε) or itsgeneralization to include welfare weights.

3. The fiscal gap designates the imbalance of tax revenue and public expenditure among the different levelsof government. To fill the gap, there must be transfers from one level of government to another. In mostfederations including Canada, the fiscal gap is positive in that the federal government collects more revenuethan they need for their own expenditure and transfer funds to lower levels of government.

4. The home-attachment may reflect some aspect of culturally diverse regions such as Canada and the EU: dueto differences in language and/or religion, individuals may be reluctant to move.

5. Individuals differ only with respect to home attachment. Although it may seem to be ad-hoc, home attachmentis useful to model the imperfectness of mobility. See Mansoorian and Myers (1994) for the details of thisconcept.

6. This assumption about local government behaviour follows Boadway and Keen (1996) and Dahlby (1994,1996). There is another formulation, however. Boadway (1982), Myers (1990), Wellisch (1994) suppose thateach local jurisdiction takes into account regional resource constraints in other jurisdictions. In the presence ofdistortionary taxation, this assumption means that each local government incorporates the revenue constraintsof other governments. The author considered this alternative case, and the results are available upon request.

7. For simplicity, we assume that migration equilibrium is always interior and unique although it is well-knownthat there can be multiple equilibria (Atkinson and Stiglitz, 1980). Including such a multiplicity or possibilityof a corner solution would substantially complicate our analysis.

8. The matching grants correspond to the revenue grants examined by Dahlby (1996). Precisely, the latter is thegrant related to the local tax revenue,Rk, rather than the local tax rate,tk. Both tax rate and revenue matchinggrants are equivalent, however: we prefer the present formulation because it makes our analysis more tractable.

9. An alternative objective may be the total utility of residents. However, as cited by Mansoorian and Myers(1997), this formulation implies that each local authority has a preference for the population size, which maylead to inefficiency in the population allocation.

10. In fiscal federal models, it is well-known that the stability issue is closely related to the sign of aggregate taxrate,τk. We examine this problem in section 4.

11. As cited above, the migration function is obtained by solving (5) fornk. Therefore, these two approachesshould provide the same solution.

12. As an objective of the central government, it may be more plausible to include the home-attachment compo-

nents. Say, the objective attaches a weight,ων , on typeν-households where∫ 1

0ωνdν = 1 (Mansoorian and

Myers, 1997):∫ nA

0

ων{υ(τA,nA)+ b(gA)+ B(G)+ a · (1− ν)}dν

+∫ 1

nA

ων{υ(τB,nB)+ b(gB)+ B(G)+ a · ν}dν (12′)

Because of the presence of the migration constraint, (5), however, the above turns out to be equivalent to (12).Let δ =

∫ nA

0ωνdν. Given the values of other variables, the effect of marginal change innA on (12′) is:

δ∂vA

∂nA− (1− δ) ∂vB

∂nB+ ωnA{v(τA,nA)+ b(gA)+ B(G)+ a · (1− nA)}

−ωnA{v(τB,nB)+ b(gB)+ B(G)+ a · nA}

Due to (5), the final two terms offset each other. So the change inδ and thus 1− δ due to the change innA

does not affect the value of the objective. Therefore, whether or not home-attachment is included in the centralgovernment’s objective does not affect our argument.

13. It may be noteworthy that due to free-riding problem which would arise in the present context, the provisionof G cannot be decentralized. This provides a rationale that the federal government should be in place.

138 SATO

14. More formal definition is the following:MCPF= (dVk/dτk)/λkd R/dτk

whereR≡ RF +∑

j=A,BRj .

15. The details of calculations presented in this paper are available from the author upon request.16. The sign of the second best tax rate cannot be seen from (24) alone. Forτk > 0 to result, the net social product

must be positive, which implies that the economy is under-populated. Therefore, the second best is compatiblewith a positive tax rate only in the economy as such. Whether or not the economy is under-populated is closelyrelated to the stability issue and we will turn to it in section 4.

17. According to this alternative formulation, the Lagrange function can be written as:

L = δ{v(τA,nA)+ b(gA)+ B(G)} + (1− δ){v(τB,nB)+ b(gB)+ B(G)}+ µ{v(τA,nA)+ b(gA)+ a(1− 2nA)− v(τB,nB)− b(gB)}

+∑

k=A,B

γk{Rk(tk, T,nk,mk, Sk)− gk}

+ γF {RF (tA, tB,nA, T, SA, SB,mA,mB)− G}Optimizing with respect toSA andSB establishes (25) and the last three terms of the Lagrangian can reduceto (15) with the multiplier,γ = γF .

18. When the two regions are homogeneous and thus the second best allocation is symmetric, assigning equalwelfare weight on the two regions (i.e.,δ = 1/2) should be reasonable. Assuming the immobility of thehouseholds, (26) may reduce to (26′). In the case of heterogeneous regions, however, there is no rationale forletting δ = 1/2, so we cannot expectδλA/nA = (1− δ)λB/nB to hold in general.

19. Recall that the second best allocation is not unique in the presence of imperfect mobility. Thus the secondbest outcome targeted by the central government depends on the value ofδ.

20. Of course, because of the feasibility of the second best, the net transfers among governments sum to zero.21. This conclusion comes from the assumption that each local authority is not concerned with budget constraints

of the others. The author has considered an alternative setting in which the local governments incorporatethem: this sort of behaviour is assumed in Boadway (1982), Myers (1990) and Wellisch (1994) in the contextof the first best. We have found that (i) in the case of perfect mobility, only lump-sum grants are sufficient toachieve the second best, and (ii) ifδ = 1 (resp.δ = 0), the matching grant is not needed for regionA (resp.B). The details are available from the author on request.

22. In the second best equilibrium,dnA/dT can be written as

dnA

dT= − 1

J

{bA

DA

{T nAh′A − nAhA + (1− θ)n2

AhAh′A f ′′A}

− bB

DB

{T nBh′B − nBhB + (1− θ)n2

BhBh′B f ′′B}}

23. Since the second best value ofτk is given, the indeterminacy ofT implies that oftk, the local tax rate. For agivenτ ∗k , an increases inT is followed by the reduction intk by exactly the same amount.

24. The second best formula of the federal tax rate derived by Boadway and Keen (1996) is still valid in theheterogeneous region case when the regionally differentiated tax policy is allowed at the federal level (Lin,1995). LetTk be the federal tax rate applied to regionk (= A, B) and letTk = nkhk f ′′k θ . From (34), thiscorresponds to the case ofmk = 0. With (37), the second best can be realized and the fiscal gap is unique. Itmay be said that matching grants and regionally differentiated tax policy are substitute instruments.

25. Gilbert and Picard (1996) points out that when information is perfect, under a linear (matching) subsidy, anydegree of decentralization can do the same job as the unitary nation. Although they establish this in a differentcontext, their argument seems to be true here as well.

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Documents

Fiscal Externalities and Efficient Transfers in a Federation