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Tax competition and Leviathan-type politicians Aurélie Cassette,
Hubert Jayet, Sonia Paty*
MEDEE IFRESI-CNRS,
Faculté des Sciences économiques et sociales Bâtiment SH2
Université de Lille 1 59655 Villeneuve d’Ascq Cedex
Tél : 03-20-33-72-10
Abstract : In the literature, governments are traditionally assumed to be either benevolent or revenue-maximiser. The central purpose of this paper is to provide a framework within which these two contrasting views of tax competition can be articulated in order to be more realistic. We use a formulation in which policy-makers are potentially revenue-maximizers but also attach some value to the welfare of their citizens because they are concerned with their re-election. In this framework, our second objective is to determine the effect of interjurisdictional competition for capital on the performance of such policy-makers. We show that the supply of public goods is efficient in equilibrium with or without tax competition among governments. However, the result concerning the tax instruments depends on the assumptions concerning the ownership of capital. When the local stock of capital is owned by residents of competing governments, policy-makers are not likely to use the tax on capital while absentee ownership leads them to use it as tax exporting allows them to increase their rents and to make the decrease of return to capital borne by non-residents. Key-words : tax competition, Leviathan, capital taxation, strategic interactions JEL Code : H1; H2; H3
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1. Introduction In the now vast literature on tax competition as originated by Zodrow and Mieszkowski (1986) and surveyed by Wilson (1999), governments are assumed to be benevolent policy-makers who maximise their citizen’s welfare. They compete for investment through tax incentives and ignore the detrimental effect on other regions. The induced fiscal externality gives rise to too low tax rates and underprovision of public goods in equilibrium. The fiscal federalism literature has recognised the potential inefficiencies from non-cooperative tax-setting by lower-level jurisdictions when there is some mobility of tax base between them. However, this view is in sharp contrast to the Public Choice literature in which governments are intrinsically revenue-maximizers (Brennan and Buchanan, 1980; McLure, 1986; Drèze, 1993; Giersch, 1993). This approach concludes that competition among governments is beneficial because it tames Leviathan-type politicians. The basic argument is as follows. Each government is responsible for the provision of the institutional framework in which economic activities take place. If it provides high taxes but low-quality services, the decentralized federal system allows both capital and citizens to move out of jurisdictions with bad economic performance. This leads to a decline in income of the immobile factors. Voters will punish such government by electing another candidate. Since the government is interested in re-election, tax pressure is relieved on those who can move out of their jurisdiction and government is forced to improve its performance. Openness of the economy for mobile factors of production is an efficient way to limit the discretionary power of governments1. Neither of these approaches is realistic. Our research has two objectives. The central purpose of this paper is to provide a framework within which these two contrasting views of tax competition can be articulated in order to be more realistic. We use a formulation in which policy-makers are potentially revenue-maximizers but also attach some value to the welfare of their citizens because they are concerned with their re-election. In this framework, our second objective is to determine the effect of interjurisdictional competition for capital on the performance of such policy-makers. Our paper relates to a small literature that studies tax competition in models where benevolence and Leviathan coexist. Edwards and Keen (1996) address the following question: is international tax competition or interjurisdictional competition a good thing or a bad thing ? As re-election concerns are not modelled, the outcome of tax competition depends on the assessment of the relative strength of benevolence versus Leviathan. Starting from the non-cooperative equilibrium, they derive conditions under which tax coordination benefits the representative citizen when policy-makers are neither entirely benevolent nor wholly self-serving. Rauscher (1998) shows that interjurisdictional competition for mobile factors of production tames Leviathan governments. The governments raise the efficiency of the public sector under some assumptions concerning the kind of tax policy used for public goods supply. More precisely, tax competition may tame a Leviathan who finances public expenditure with user charges, but this result does not hold in the case of lump-sum taxes.
1 Moreover Apolte (2001) critically examines this hypothesis that institutional competition among governments may tame Leviathan. He shows that institutional competition between jurisdictions with free-choice of taxes induces Leviathan-type politicians to avoid taxation of mobile factors and to prefer taxation on immobile factor. In Nash equilibrium the amount of public goods provided is shown to be inefficiently low.
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Finally in some other models, competition to attract mobile households reduces wasteful behavior by rent-maximising policy-makers and increases resident utility. Gordon and Wilson (2001) use a model where residents initially set taxes while Leviathan governments decide on the distribution of public goods supply. Incumbents can be ousted from office if they spend more than the other officials. This is the so-called yardstick competition2 à la Besley and Case (1995). We propose a model where potentially Leviathan-type governments are both concerned with re-election and interjurisdictional tax competition for a mobile factor of production. Therefore incumbents are not wholly Leviathan as their re-election depends on the resident utility. Our paper is original in two points. First, like in most decentralized countries, we assume that officials can use two kinds of local tax instruments: a tax on the locally-invested mobile factor (capital) and a tax on the immobile factor (residents). Rent-seeking governments with free choice of tax instruments will not be able to escape competitive pressure by shifting taxes to the immobile factor because of their re-election concerns. Second, by contrast with the yardstick competition models (Besley and Case, 1995; Besley et Smart, 2001; Reulier et Rocaboy, 2003), the effect of re-election concerns on rent-seeking policy-makers decisions are considered without modelling the voting process. We also assume that there is no limit in the number of successive local mandates so that the Leviathan government is concerned with its electoral constraint in each period of the model. In such a framework we show that the supply of public goods is efficient in equilibrium with or without tax competition among governments. However, the result concerning the tax instruments depends on the assumptions concerning the ownership of capital. When the local stock of capital is owned by residents of competing governments, policy-makers are not likely to use the tax on capital while absentee ownership leads them to use it as tax exporting allows them to increase their rents and to make the decrease of return to capital borne by non-residents. The paper is organised as follows. In section 2 we set up a simple model without tax competition and derive the tax and spending decisions taken by the Leviathan of each jurisdiction. In section 3 we compare the sharing of the disposable budget between public and private consumption under benevolent and Leviathan governments. In section 4, we study the Leviathan policy-maker's decisions when capital is perfectly mobile. We then turn to the effect of strategic tax competition on its public choices (section 5). Finally, in section 6, we analyse the effect of yardstick competition on incumbents behaviour. 2. A rent-seeking Leviathan in a simple closed economy Let us start with a simple model of a Leviathan confronted to an electoral constraint, who has to decide the level of public good he provides and the level of rents he extracts. There is only one jurisdiction in the economy, inhabited by immobile homogenous households. Population size is normalized to unity and households’ preferences are represented by a strictly quasi- 2 In yardstick competition models, voters use the performance of the neighbouring policy-makers as a yardstick to evaluate the performance of their incumbent. If they disagree, they can punish him by not re-electing him. The effect of yardstick competition on residents’ welfare depends on the type of incumbents. Besley and Smart (2001) show that yardstick competition is most likely to be welfare improving for voters when it is more likely that politicians are benevolent and detrimental to welfare if it is more likely that policy makers are rent-seeking.
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concave, twice differentiable utility function, defined over consumption of private goods (C) and public goods (G).
U(C,G) with 0' >CU and 0' >GU His exogenous gross income (W) is used for private consumption, C, and a lump sum tax payment, θ. His private budget constraint is then:
W = C + θ (2.1) The tax revenue, raised by the planner from taxes on capital and on private income, can be used both to produce the public good, G, and to take personal rent, R. Every unit of public good is produced using one unit of private good as the sole input. The local planner is then subject to the following public budget constraint:
R + G = θ (2.2) Were the jurisdiction managed by a benevolent planner maximizing the citizen’s welfare (and then taking no rent, R = 0), he would supply an efficient amount of public goods satisfying the well known Samuelson’s rule, equalizing the marginal rate of substitution between public and private goods and their marginal rate of transformation: Max U(C,G) = U(W – θ, θ) ⇒ U’C = U’G ⇔ U’C / U’G = 1 But the policy-maker is of the Leviathan type and has an infinite lifespan (as a political party involved in rent-seeking activities): his utility depends on the maximization of flows of personal revenues (like rents which he will have succeeded to embezzle when collecting tax revenues). However the politician is constrained in his predation of local revenues by the existence of a local election at the end of each period. This election gives the immobile citizens the opportunity of expressing their dissatisfaction with respect to a bad management of the public funds, voting against the insider and throwing him out of office: The probability of re-electing the insider, π(u), is an increasing function of the residents’ utility generated by the planner’s policy, u. In contrast to many models in the literature of yardstick competition (Besley and Case, 1995; Bordignon Cerniglia and Revelli, 2002; Feld, Josselin and Rocaboy, 2002; Reulier and Rocaboy, 2003), there is no limit to the number of successive mandates for an elected official3. Consequently, Leviathan always faces the re-election constraint that prevents him from setting the highest level of rent at any period (he cannot use the entire tax revenue for personal profit). Choosing a tax policy, the planner must take account of the risk of losing future rents if he were not re-elected. If he is re-elected, he can again take rents. Otherwise, if he is not re-elected, he could not be elected any more later on and will receive an exogenous lump-sum revenue R at every period. If, at every period, the Leviathan’s current rent level is R, his intertemporal utility, V, is V = R + βπV + β(1 – π)V
where β is the Leviathan’s discount factor and ( )β−= 1RV is his intertemporal utility after losing the election. A straightforward calculation leads to
VRRV +−−
=βπ1
(2.3)
The tax and spending decisions are taken by the Leviathan of each jurisdiction so as to maximise his intertemporal utility, (2.3), subject to the public and private budget constraints, (2.1) and (2.2): 3 In many countries, e.g. France, there is no limit to the number of the successive local mandates.
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ϑ,max
G ( )( ) VGCU
RRV +−
−=
,1 βπ
st R = θ – G C = W – θ
By integrating these constraints in the objective function and differentiating relative to each of the control variables, we obtain the following first order conditions4:
( )[ ]GU UVVdGdV '.'.1
.11 πβπβ
−−−
−= (2.4)
( )[ ]CU UVVddV '.'.1
.11 πβπβθ
−+−−
= (2.5)
Or, equivalently, U’C / U’G = 1 and ( ) 1'.'. =− CU UVV πβ (2.6) Because he takes positive rents, the Leviathan chooses a combination of public and private consumption lower than that chosen by a benevolent planner. However this combination still satisfies the Samuelson’s rule and then is efficient. The level of rents chosen by the Leviathan is implicitly defined by (2.6), which equalizes the marginal benefit of increasing the current rent of one unit (equal to unity) and the current value of the expected loss in future rents implied by a lower probability of reelection: a one unit increase in rent decreases public and private consumption of one unit, hence of utility loss of U’C, a decrease in reelection probability of π’UU’C, and then a decrease in the current value of expected future rents of ( ) CU UVV '.'.πβ − . Figure 1 illustrates the difference between the benevolent and the Leviathan planner. Combining the public and private budget constraint, one gets
B = W – R = C + G The benevolent planner chooses a zero level of rent and then, the sum of the household’s public and private consumption equals W. The global budget constraint is the dotted line and the planner chooses the utility maximizing combination, corresponding to point B.
4 Details of the derivations are available in appendix 7.1
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Figure 1 : Private and public consumption
The Leviathan chooses a positive level of rent, R > 0. The sum of the household’s public and private consumption now equals W + E – R, and the global budget constraint is the plain line below the dotted line. Note that the expected value of the Leviathan’s flow of revenues, V, is determined by his current rent level, R, and his re-election probability, π. Thus, for every rent level, the Leviathan’s best choice is to maximize his re-election probability and then to maximize the resident’s utility: Along the plain line, the Leviathan choosing point L. Both the benevolent planner’s and the Leviathan’s choice are Pareto optimal, lying on the Engel curve corresponding to a unit price of the public good, L being closer to the origin. This result has been already found by Edwards et Keen (1996), Rauscher (1997) and Apolte (2001). A reformulation of the Leviathan’s choice As the Leviathan chooses an allocation on the Engel curve, thus without any loss of efficiency, we can reformulate the Leviathan’s choice as a choice on the Engel curve induced by a change in the rent level. Assume B to be the household’s disposable budget that can be allocated by the local planner between public and private consumption.
B = C + G = W – R The latter equality combining the public and private budget constraints. The tax price of the public good being unity, along the Engel curve, G solves the equation
( ) ( )GGBUGGBU GC ,, '' −=− . Let us write the solution to this equation as G = φ(B) = φ(W + E – R)
and C = B – φ(B) So that U(C,G) = U(B – φ(B), φ(B)) = ( )BU Then, the reelection probability may be written as a function of the household’s budget: π(U(C,G)) = π( ( )BU ) = ( )Bπ In order to determine the effect of increased rents on the resident’s welfare and the Leviathan’s reelection probability, we examine the effect of a change on the disposable budget of households:
BU ' = (1 – φ’(B))U’C + φ’(B)U’G = U’C = U’G
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because, along the Engel Curve, U’C = U’G. The consequence of this budget change on the Leviathan’s reelection probability is
π’UU’C = π’U BU ' = B'π Now, let us analyse the effect of an increase of the rents on the Leviathan’s welfare, which may now be written as
( ) VBRBWV +
−−−
=πβ1
So that ( )
( )[ ]BBB VVRBWVBV
dBdV '1
11'
111' 2 πβ
βππβ
βπβππ
π−−
−−=
−−−
+−
−=∂∂
+∂∂
=
And then the first order condition for an interior maximum is ( ) 1' =− BVV πβ
We find again (2.6). From the second order condition, we know that it is a maximum as soon as the re-election function π , is concave5. The planner will set R strictly between R and W. He cannot accept a rent level lower than R : were his private income higher than the income from governing (R < R ), he would not enter into politics. If R = R , then V = V and ( ) 0.11 >−= πβdRdV : the planner may therefore find it beneficial to set a level of rent higher than R . If R = W, then all the resources are captured by the Leviathan (B = 0) and we may assume that his probability of re-election will be null. We cannot exclude a maximum in this point. However it is reasonable to assume that a citizen stripped of his possession is highly sensitive to any increase (even small) of his budget. In this case +∞=B'π and 0<−∞=dRdV . So the rent will lie between R and W. 3. Leviathan-type policy-makers’ decisions when capital is perfectly
mobile: the “purely competitive” case Let us now introduce mobile capital. The private good is produced by perfectly competitive firms using two factors of production, capital and immobile labor, inelastically supplied by the local residents. Let f(K) be the technology used to produce the aggregate output with
KKK ff "0' >> . As before, this output can either be consumed or used as input into the provision of the local public good, the transformation rate being unity. Following the earliest literature on tax competition (Wilson, 1986 ; Zodrow, Mieskowski; 1986), we assume that capital is perfectly divisible and mobile and may then be used inside or outside the jurisdiction. It is paid its marginal product and then, inside the jurisdiction, its after-tax rate of return equals f’K – t, where t is tax on capital. Outside the jurisdiction, its rate of return, ρ, is a constant. Perfect mobility implying equality of rates of return, we find the well known capital-arbitrage condition.
f’K – t = ρ (3.1) which may be used to define the capital supply to the jurisdiction as a function of its net return K = K(ρ + t). Differentiating (3.1) we find
( ) ( )( ) 0"
1' <+
=+tKf
tKρ
ρ
5 See the appendix 7.2 for details on the second-order conditions
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The local resident receives the local return to the fixed factors. Furthermore, he owns a share α of the global exogenous existing stock of capital in the whole economy (inside and outside the jurisdiction), K . So, his income is W = f(K) – Kf’(K) + αρK (3.2) When the Leviathan has power to tax mobile capital, his optimal policy is determined by the following program (which solely change from the previous section by the addition of the capital supply function)
tG ,,max
ϑ ( )( ) VGCU
RRV +−
−=
,1 βπ (3.3)
St R = θ + tK – G C = W – θ f’K – t = ρ or, equivalently, K = K(ρ + t)
with W = f(K) – Kf’(K) + αρK Differentiating with respect to G and θ, we again find the first order conditions (2.4) and (2.5). Therefore, the Samuelson rule6 and (2.6) still apply, U’C / U’G = 1 and ( ) 1'.'. =− CU UVV πβ As before, even a Leviathan-type planner supplies public goods efficiently when he has to be re-elected because, doing so, for every level of rent, he maximises the consumers’ utility and then his re-election probability. As for the tax rate on capital, its optimal value satisfies the first order condition:
βπθ −+=
1'tK
ddVK
dtdV
We know that, at an optimal choice of ϑ , we have 0=ϑd
dV , and then
0.1'.<
−=
πβKt
dtdV
So at the optimum, t = 0 at least as long as a negative tax rate (a subsidy) is not allowed. The local Leviathan does not tax capital and finances his rents and the provision of public good out of the lump-sum tax only. At first sight, when he does not tax capital, the Leviathan has lower fiscal resources for providing the public good or withdrawing rents than with a positive tax. However, the quantity of mobile capital is maximised when it is not taxed. And the Leviathan is better off attracting more capital with a lower capital tax, generating higher local revenues which make him able to increase his fiscal resources through a higher lump-sum tax. This result is not surprising when the planners are benevolent and face no restriction on lump-sum taxation (Zodrow, Mieskowski, 1986); it has also been found by Apolte (2001) in the Leviathan-type case. The Leviathan provides an efficient amount of public good, and indirectly, by choosing t = 0, an optimal amount of capital stock. Once again, knowing that the planner efficiently shares the household’s budget between public and private consumption, we can reformulate the Leviathan’s problem as
tB ,max ( )
( ) VB
RBtKWV +−
−−+=
πβ1
6 Details of the first-order conditions in the appendix 7.3
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Where, taking account of (3.1), W + tK = f(K) – Kf’(K) + αρK +tK = f(K) – ρK + αρK (3.4) From (3.4), it is easy to understand why the Leviathan does not tax capital. Combining the public and private constraints in (3.3), one finds that W + tK = R + C + G = R + B. So, W + tK is the global amount of resources shared between the planner (who receives the rent, R) and the household (who receives B = C + G). Therefore, the best the Leviathan can do is to maximize this global amount of resources: the higher W + tK, the higher the rent he can obtain for a given household’s budget and then for a given re-election probability, the higher V. But, as one can see from (3.4), W + tK only depends upon the quantity of capital attracted by the jurisdiction K. And it is maximal when f’(K) = ρ, which implies t = 0. To summarize, the Leviathan chooses the level of the tax on mobile capital, t, so as to maximize the global amount of resources he will share with the consumer, W + tK. Then, he chooses the value of its current rent so as to maximise the expected value of his flow of revenues, V, taking account of the influence of the level of its rent on his re-election probability. Last, he uses the lump sum tax, θ, so as to efficiently share the household’s budget between public and private consumption. If the result differs from the benevolent planner’s choice, both reach Pareto-optimality. The impact of restrictions on the lump sum tax. In most models of fiscal competition, households are not taxed, which may be justified by the fact that local planners face constraints making them unable to levy taxes on households. We will now consider the slightly more general case where there is an upward bound to the lump-sum tax :
θθ ≤ When 0=ϑ , we are back to the standard case where the Leviathan is not allowed to tax households. Let λ be the Lagrange multiplier associated with this constraint. The first order conditions7 now imply that
11''
<−= µG
C
UU with µ = λ(1 – βπ)
and ( )( ) 0'
>++
−=tKtKt
ρρµ
When the constraint on lump-sum-taxation is not binding, λ = µ = 0 and we are back to the previous case: the Samuelson condition is met and there is no capital taxation. When the constraint is binding, as in the standard model with a benevolent planner, capital is taxed and the public good is underprovided since at equilibrium, its marginal utility is higher than the marginal utility of the private good. Let us also note that the modified Samuelson rule can be rewritten in :
µµ
µ −+=
−==
= 11
11
''
C
G
CteU UU
dGdC (3.5)
7 See the appendix 7.4
10
Because he cannot tax households at a high enough level, the local planner has to use capital taxation. However, taxing mobile capital instead of immobile households generates a distortion. The last term of (3.5), µ / (1 − µ), may be interpreted as the marginal cost of public funds generated by this distortion. So, we get a generalized Samuelson rule where the marginal cost of the public good is the sum of its marginal production cost and the marginal cost of public funds needed for financing production. 4. Strategic competition Let us now consider the case of strategic competition. There is a small number of jurisdictions, N. Let t = (tI,…, tN) be the vector of capital tax rates. Perfectly mobile capital equalizing posttax rates of returns among jurisdictions, the equilibrium stocks of capital, KI,…, KN, and the equilibrium after tax rate of return, ρ, solve the following system of equations:
( ) ii tKfi +=∀ ρ',
∑ =i
i KK
The solutions to these equations are Ki = Ki(t) and ρ = ρ(t), with, for all i and j ≠ i, ∂Ki / ∂ti < 0, ∂Ki / ∂tj > 0, and –1 < ∂ρ / ∂ti < 08. A higher tax rate in jurisdiction i generates a capital outflow from this jurisdiction toward its competitors; the posttax rate of return decreases. The capital stock located in jurisdiction i now depends upon the whole vector of capital tax rates generating strategic interaction. Nash equilibrium is characterized by N triplets ( )**,*, iii tG ϑ , i = 1,…,N such as, for any i, ( )**,*, iii tG ϑ solves the following maximization problem.
iii tG ,,max
ϑ ( )( ) iiii
iii V
GCURRV +
−−
=,1 βπ
St Ri = θi + tiKi(ti,t*–i)– Gi Ci = f(Ki(ti,t*–i)) – Ki(ti,t*–i)f’(Ki(ti,t*–i)) + αiρi(ti,t*–i) K – θ i
Where t = (ti,t–i), t–i being the vector of tax rates in jurisdiction competing with i. It is easy to prove that, once again, the Samuelson condition is met and then that the supply of public good is efficient. Solving the objective function of the local planner with respect to the capital tax rate and integrating the first order condition on θi, we find9:
( )
−−
∂∂
−=
iii
i
ii
ii
i
dtdKK
tKt
dtdV ρα
πβ .11 (4.1)
and
( )
−−
∂∂
−=
jii
j
ii
ij
i
dtdKK
tKt
dtdV ρα
πβ .11 (4.2)
It is interesting to look at the perfectly symmetric case where preferences, technology, and capital endowments do not change across jurisdictions. Of course, there is a symmetric Nash equilibrium, each jurisdiction receiving the same quantity of capital, NKKi = . Therefore, (4.1) and (4.2) may now be written as
8 See appendix 7.5 for more details 9 See appendix 7.6
11
( )( ) ( )
−+
−−
= KNNKf
tNNNdtdV i
ii
i απβ
1"
1.1
1 2
(4.3)
( ) ( ) 01".1
1 2
≥
−−
−−= NK
NKfNtN
dtdV
ij
i απβ
(4.4)
We know that ∂Ki / ∂ti and ∂ρ / ∂ti are both negative and that ∂Ki / ∂tj is positive When
KK ii α≤ , dVi / dti is non positive for every non negative ti, implying that the optimal capital tax rate is zero. Let us note that iKα is the quantity of capital held by the inhabitants of jurisdiction i. So, when the quantity of capital held by the jurisdiction is lower than the quantity of capital owned by its residents, a higher tax rate has two negative effects on the Leviathan’s welfare: first of all, the capital stock located in the jurisdiction decreases, hence lower tax receipts on which the Leviathan takes his rent; Secondly the net return of capital decreases, hence a lower households’ income and a lower probability of the Leviathan to be re-elected. As for the non strategic case, the Leviathan does not tax mobile capital, attracting the highest possible quantity, maximizing household’s revenue which is the basis of lump-sum taxation. He then avoids inefficient distortions in the provision of public good. However, a zero capital tax rate implies that
0.1
<−−
−=ji
ii
j
i
dtdKK
dtdV ρ
πβα
The jurisdiction receives a negative externality from the other jurisdiction. Note that, in the symmetric case, we can have αi = α = 1/N only: α > 1/N is impossible, as the sum of property shares cannot exceed unity. But, in (4.3), α = 1/N and ∂Vi / ∂ti = 0 together imply t = 0. There is no longer any externality and the Nash equilibrium is efficient. Let us now consider the opposite case, when KK ii α> : the jurisdiction now attracts more capital than the quantity its residents hold. Then, ∂Vi / ∂ti > 0 for ti = 0 and the Leviathan is better off taxing capital at a positive rate, determined by the nullity of the bracketed term in (4.1). What happens is that, now, the Leviathan is able to export taxes: capital taxes are, to some extent, paid by foreign owners. Moreover, ∂Vi / ∂tj > 0 and the local jurisdiction receives positive externalities from its competitors, as in the standard case of competition among Pigouvian planners. A higher capital tax rate in the other jurisdictions generates a capital inflow, allowing the local Leviathan to decrease its lump-sum tax or to provide more public good in order to raise his probability of re-election; or to take higher rents. The effect of a constrained lump-sum tax Let us now add a constraint on the households tax level, ii θθ ≤ . We find the same modified Samuelson condition we proved in the non-strategic context10,
11''
,
, <−= iGi
Ci
UU
µ with µi = λi(1 – βπi)
10 See the appendix 7.7
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where λi is the Lagrange multiplier associated with the constraint. Here, for simplicity, we only deal with the symmetric case. The optimal capital tax level is determined by the first order condition:
( )( )( ) ( )( )
−+−−+
−=
NKftNNKNNN
dtdV
i
i
"111
.11 2
αµµπβ
So that ( )( )( ) ( )
( )1"11
−−−+
−=NN
KNKfNNt αµµ
As 0.1 ≥− Nα , as soon as the constraint on lump-sum taxation is binding, t is positive and then mobile capital is taxed. Moreover,
( ) ( )( )[ ] 011.1
1>−−+
−−
= NNNKdtdV
j
i αµµπβ
.
Interjurisdictional externalities are positive with the consequence that mobile capital is undertaxed: local Leviathans would be better off making a coordinated increase in capital tax levels. 5. Concluding remarks Our central departure from previous models of tax competition with Leviathan planner is in the assumed objective function of the policy-maker in each jurisdiction which integrates his re-election function in order to put a constraint on the diversion of public funds. If the results on the efficiency of the public good provision and on the absence of capital taxation when the use of a lump-sum tax is allowed have already been found in the standard literature, some new results follow from the strategic context. These result follows from the assumptions on the ownership of capital. When residents own the whole local stock of capital, local planners will not use the tax on capital. But absentee ownership leads them to tax capital as the decrease of the net return of capital will not be borne by the local electorate. The model may be extended in several directions. For example, it would be convenient to add a marginal cost of public fund: when the Leviathan planner would increase the size of his budget compared to the benevolent planner, he would face an efficiency loss in the use of the disposable budget for households and the Samuelson condition would not hold any more even if there is no restriction on the use of the lump-sum tax. References Apolte T., 2001, “How tame will Leviathan become in institutional competition? Competition among governments in the provision of public goods”, Public Choice, 107, pp.359-381 Besley T., Case A., 1995, “Incumbent behavior: vote seeking, tax setting and yardstick competition”, American Economic Review, 85(1), pp.25-45 Besley T., Smart M., 2001, “Does Tax Competition Raise Voter Welfare?”, Working paper University of Toronto, Department of Economics, November Bordignon M., Cerniglia F., Revelli F., 2002, “In Search for Yardstick Competition : Property Tax Rates and Electoral Behavior in Italian Cities”, CESifo Working Paper 644, January
13
Brennan G., Buchanan J.M., 1980, The power to tax: analytical foundations of a fiscal constitution, New-York: Cambridge University press Drèze J., 1993, “Regions of Europe: A feasible status, to be discussed”, Economic Policy, 17, pp.206-307 Edwards J., Keen M., 1996, “Tax competition and Leviathan”, European Economic Review, 40, pp.113-134 Feld L.P., Josselin J.M., Rocaboy Y., 2002, « Le mimétisme fiscal : une application aux régions françaises », Economie et Prévision, pp. 44-49 Giersch H., 1993, Openness for prosperity, MIT press, Cambridge MA Gordon R., Wilson J.D., 2003, “Expenditure competition”, Journal of Public Economic Theory,5 , pp.399-417 McLure C., 1986, “Tax competition Is What’s Good for the Private Goose Also Good for the Public Gander”, National Tax Journal, 39, pp.341-48 Mintz J., Tulkens H., 1986, “Commodity tax competition between member states of a federation: equilibrium and efficiency”, Journal of Public Economics, 29, pp.133-172 Rauscher M., 1997, “Interjurisdictional Competition and the Efficiency of the Public Sector : The triumph of the Market over the State?”, CEPR Discussion Paper 1624, April Rauscher M., 1998, « Leviathan and Competition among jurisdictions : the case of Benefit Taxation », Journal of Urban Economics, 44, pp.59-67 Reulier E., Rocaboy Y., 2003, « Interactions stratégiques et compétition politique par comparaison, quelles conséquences sur le niveau d’imposition local ? », AFSE Lille, May Wildasin D.E., 1988, “Nash Equilibria in Models of Fiscal Competition”, Journal of Public Economics, 35, pp.229-240 Wilson J.D., 1986, "A Theory of Interregional Tax Competition," Journal of Urban Economics, 19, pp.296-315 Wilson J.D., 1999, “Theories of Tax Competition”, National Tax Journal, 52 (2), June, pp.269-304 Zodrow G.R., Mieskowski P., 1986, “Pigou, Tiebout, Property Taxation, and the Underprovision of Local Public Goods”, Journal of Urban Economics, 19, pp.356-370 6. Appendix
6.1. The closed economy case: First order conditions for a maximum Logarithmically differentiating (2.3), one gets
' '1 U G
V G R GU
V V R Rβ πβπ
∂ ∂ ∂ ∂= +
−− − and ' '
1 U C
V R CU
V V R R
θ θ β πβπ θ
∂ ∂ ∂ ∂ ∂= +− ∂− −
14
With R = W + θ – G ⇒ ∂R / ∂θ = – ∂R / ∂G = 1 C = W – θ ⇒ ∂C / ∂θ = – 1 And then
1' '
1 U G
V GU
V V R Rβ πβπ
∂ ∂= − +
−− − and 1
' '1 U C
VU
V V R R
θ β πβπ
∂ ∂= −
−− −
So that ( )( ) ( ) ( )
( )( ) ( ) ( )
11 1' ' 1 ' '1 1
11 1' ' 1 ' '1 1
U G U G
U C U C
V VV V V U V V UG R R
V VV V V U V V UR R
βπβ π β π
βπ βπ
βπβ π β π
θ βπ βπ
− −∂ = − − − = − − − ∂ − −− − −∂ = − − = − − ∂ − −−
6.2. The closed economy case: Second order conditions for a maximum
At the optimum, the first order condition is
( )[ ]BVVdBdV '1
11 πβπβ
−−−
−=
Differentiating with respect to B, one gets the second order condition:
( )( )[ ] ( )
( ) 0"'11
0"'1
'11 22
2
<
−+
−+
−=
<
−+
−+−−
−=
BBB
BBBB
VVdBdV
dBdV
VVdBdVVV
dBVd
πππβ
βπβ
πβ
πππβ
βπβπβ
πβ
But, when B meets the first order condition, dV / dB = 0 and then ( ) 0"
12
2
<−−
= BVV
dBVd π
πββ
As VV > , the ratio is strictly positive, hence 0" <BBπ . The probability of re-election must be a concave function of the disposable budget of the residents. When B increases, the probability of reeelection is growing at a decreasing rate.
6.3. Open economy: the competitive case Differentiating R = θ + tK – G and taking account of the fact that K = K(ρ + t), we get ∂R / ∂θ = –∂R / ∂G = 1 and ∂R / ∂t = K + tK’. Similarly, differentiating C = W – θ = f(K) – Kf’(K) + αρK – θ, we get ∂C / ∂θ = – 1 and ∂C / ∂t = – Kf”(K)K’ = – K. The partial derivatives ∂R / ∂θ = –∂R / ∂G = 1 and ∂C / ∂θ = – 1 being exactly the same as the derivatives calculated for the closed economy case, we get the same first order conditions,
( )[ ]
( )[ ] 0''11
1
0''11
1
=−−−
=∂∂
=−−−
−=∂∂
CU
GU
UVVV
UVVGV
πββπθ
πββπ
So that (2.6) still holds. As for the first order derivative with respect to t,
15
( )( )[ ]
( )( )[ ]βπθ
πββπ
πββπ
βπβπβπ
−+=+−−
−=
−−+−
=−−
+−
=
1''''1
11
'''1
1''11
12
tKddVKtKKUVV
KUVVtKKdtdCURR
dtdR
dtdV
CU
CUCU
and then, as ∂V / ∂θ = 0 when the Leviathan’s choice is an interior maximum,
βπ−=
1'tK
dtdV
6.4. Constrained lump sum taxation for the competitive case
The Lagrangian of the Leviathan’s maximization problem is
( ) ( )ϑϑλϑ −−= tGV ;;l Using the of V with respect to its three arguments calculated in the previous section, the first order conditions are
( )[ ] ( )
( )[ ] ( )
01
'
1''0''11
1
1''0''11
1
=−
+=
−=−⇒=−−−−
=−∂∂
=∂∂
=−⇒=−−−
−=∂∂
=∂∂
βπθ
µπβλπββπ
λθθ
πβπββπ
tKddVK
dtdV
UVVUVVV
UVVUVVGV
G
CUCU
GUGU
l
l
Where µ = λ(1 – βπ). Combining the first two conditions, one gets
11''
<−= µG
C
UU
Moreover, using the second condition, ∂V / ∂θ = λ, the third condition may be written as
0'
01
'1
'>−=⇒=
−+=
−+= µ
βπλ
βπθ KKttKKtK
ddVK
dtdV
6.5. Strategic case: Derivatives of the function of capital supply
Note ρ = ρ(t1,…, tN) and Ki = Ki(t1,…, tN) or equivalently ρ = ρ(ti,t−i) and Ki = Ki(ti,t−i), where t−i is the vector of tax rates chosen by the jurisdiction competing with i. ρ and Ki, i=1…N, solve the system of N + 1 equations.
( ) ii tKfi +=∀ ρ', and ∑ =i
i KK
Differentiating these equations.
( ) tddKdKf iii += ρ." and then ( )Kf
tdddKiKK
ii ''
+=
ρ
∑ =i
iKd 0
And then
( ) ( ) ( )10
" " "i i
i i ii i i
d d dt tdf f fK K Kρ ρ
+ = ⇔ − = ∑ ∑ ∑
hence
16
( )
( )
[ ]
1" 1, 01"
i
ijj
f Kt
f K
ρ∂ = − ∈ −∂ ∑
( ) 0"
1≤
∂∂+=
∂∂
Kft
tK
i
i
i
i ρ
And, for every ji≠ ,
( ) 0"
≤∂∂
=∂∂
Kft
tK
i
j
j
i ρ
In the symmetric case, where jurisdictions are all identical, they all choose the same tax rate,
ti = t and attract an equal amount of capital iKKN
= , so that
Nti
1−=
∂∂ρ
( ) ( )NKNfN
NKfN
tK
i
i
"1
"11 −
=−
=∂∂
( )NKNftK
j
i
"1
−=∂∂
6.6. Strategic case: First order conditions for an unconstrained maximum
Differentiating Ri = θi + tiKi – Gi, we get ∂Ri / ∂θi = –∂Ri / ∂Gi = 1, ∂Ri / ∂ti = Ki + ti(∂Ki / ∂ti) and ∂Ri / ∂tj = ti(∂Ki / ∂tj). Similarly, differentiating Ci = f(Ki) – Kif’(Ki) + αiρK – θi, we get ∂Ci / ∂θi = – 1 and
( ) ( )i
iiii
ii
ii
ii
iii
i
i
tKKK
tK
tK
tK
tKKfK
tC
∂∂
−−−=∂∂
+
∂∂
+−=∂∂
+∂∂
−=∂∂ ραραρρα 1"
( ) ( )j
iij
ij
iii
j
i
tKK
tK
tKKfK
tC
∂∂
−−=∂∂
+∂∂
−=∂∂ ραρα"
The partial derivatives ∂Ri / ∂θi = –∂Ri / ∂Gi = 1 and ∂Ci / ∂θi = – 1 being exactly the same as the derivatives calculated for the previous cases, we get the same first order conditions,
( )[ ]
( )[ ] 0''11
1
0''11
1
,,
,,
=−−−
=∂∂
=−−−
−=∂∂
CiUiiiii
i
GiUiiiii
i
UVVV
UVVGV
πββπθ
πββπ
As for the first order derivatives with respect to ti and tj,
( )
( ) ( )
∂∂
−+−−∂∂
+−
=
∂∂
−−
+∂∂
−=
∂∂
iiiiCiUiii
i
iii
i
i
iCiUi
i
i
i
i
ii
i
tKKKUVV
tKtK
tCURR
tR
tV
ραβπβπ
βπβπβπ
,,
,,2
''1
1
''11
1
17
( )
( ) ( )
, ,2
, ,
1 ' '1 1
1 ' '1
i i i iiU i C
j i j ji
ii i i i iiU i C
i j j
V R R R CU
t t t
Kt V V U K Kt t
βπβπ βπ
ρβπ αβπ
∂ ∂ − ∂= +
∂ − ∂ ∂− ∂ ∂ = − − − − ∂ ∂
And then, as ∂Vi / ∂θi = 0 for an interior maximum implies ( ) 1'' ,, =− CiUiii UVV πβ ,
( )
∂∂
−−∂∂
−=
∂∂
iii
i
ii
ii
i
tKK
tKt
tV ρα
βπ11
( )
−−
∂∂
−=
jii
j
ii
ij
i
dtdKK
tKt
dtdV ρα
πβ .11
In the symmetric case, replacing the derivatives by their values,
( )( )( ) ( )
−+
−−
=
−+
−−
=∂∂ KN
NKftNNN
NK
NK
NKNfNt
tV i
iii
ii
i απβ
αβπ
1"
1.1
11"
11
1 2
( ) ( ) ( ) 01".1
11"
1.1
1 2
≥
−−
−−=
−+−
−= NK
NKfNtN
NK
NK
NKNft
dtdV
iii
ij
i απβ
απβ
6.7. Strategic case: First order conditions for an constrained maximum
Our starting point is the same as in section 7.4. Differentiating the Lagrangian with respect to Gi and θi,
( ) ( )
( ) ( )
, , , ,
, , , ,
11 ' ' 0 ' ' 1
1
1 1 ' ' 0 ' ' 11
ii i i iiU i G iU i G
i i
ii i i i i i iiU i C iU i C
i i
VV V U V V U
GV
V V U V V U
β π β πβπ
λ β π λ β π µθ βπ
∂ = − − − = ⇒ − = ∂ −∂ − = − − − = ⇒ − = − ∂ −
Where λi is the Lagrange multiplier associated with the constraint and µi = λi (1 – βπi). Combining the first two conditions, one gets
11''
,
, <−= iGi
Ci
UU
µ
Then,
( )( ) ( )( )
−−−
∂∂
+−
=
−−−
∂∂
−+=
iii
i
iii
iiii
i
ii
ii
i
i
dtdKK
tKtK
dtdKK
tKtK
dtdV ραµµ
πβραµ
πβλ 1
.111
.11
And, after a similar calculation,
( )( )
−−−
∂∂
−=
jii
j
ii
ij
i
dtdKK
tKt
dtdV ραµ
πβ1
.11
In the symmetric case, after replacing the derivatives by their values,
( ) ( )( )
( )( )( ) ( )( )
−+−−+
−=
−−+
−+
−=
NKftNNKNNN
NNK
NKft
NN
NK
dtdV
i
i
"111
.11
11"
1.1
1
2
2
αµµπβ
αµµπβ
( ) ( )( )
−−+−
−= KN
NKfNtN
dtdV
j
i αµπβ
11".1
1 2
and, when t is optimal,
18
( )( )( )( ) ( )( ) ( ) ( )( )[ ] 011
.1111
111
.11 2
>−−+−
−=
−−+
−−−+
−= NNNKKN
NKNNN
dtdV
j
i αµµπβ
αµαµµπβ
