Remy Oddou
email: remy.oddou@free.fr

Welfarism and segregation in endogenous jurisdiction formation models

Keywords: JEL Classification: C78, D02, H73, R13 Jurisdictions, Segregation, Welfarism, Redistribution

This paper analyses how welfarism affects the segregative properties of endogenous jurisdiction formation, in a model where local jurisdictions produce a local public good and distribute an allowance to their households, both financed by a proportional tax based on the households' wealth. A jurisdiction is composed of all the households that live in the same place. Local wealth tax rates and the level of the allowance are determined to maximize a social welfare function. Households can "vote with their feet", which means that they can choose to move to the jurisdiction that offers the package "tax rate -amount of public good -allowance" that provides the highest utility level. The main result of this article is the proof that the maximin criterium is more segregative than the utilitarian one.

Introduction

Since the beginning of the XX th century, the share of local public spedings in the overall public spending has benn increasing in many countries. For instance, the total local public spendings in the US are almost equal to the federal spendings. As a consequence, the literature about local public economics has significantly risen. The first article dealing with local public goods is probably Tiebout's intuitions [START_REF] Tiebout | A pure theory of local expenditures[END_REF], in response to Samuelson's article [START_REF] Samuelson | The pure theory of public expenditure[END_REF].

Samuelson stated, among other significant results, that preferences over the public good are not observable, thus, a public good can not be financed through volontary contributions, or through contributions that depends on the marginal utility procured by the public good. Tiebout answered that preferences over public goods can be observable, as households can choose their jurisdiction based on a trade-off between local tax rates and amounts of public goods (vote with one's feet). Consequently, households whose marginal utility with respect to the public good is high will live in a jurisdiction where the tax rate and the amount of public good is high, while households whose preferences weakly depend on the public good will rather settle in jurisdictions wwith a low tax rate. Such a jurisdiction formation is therefore endogenous. Tiebout's article was mostly a non-formal intuition, but has laid the foundations of many formalized articles, either identifying the conditions ensuring the existence of an equilibrium, or examining the segregative properties of endogenous jurisdiction formation.

Westhoff [START_REF] Westhoff | Existence of equilibria in economies with a local public good[END_REF] considered a model based on Tiebout's intuitions, in which jurisdictions produce a local public good, financed by a local tax on wealth, and consumed by its inhabitants. Households' utility depends on the amount of public good available in their jurisdiction, and on the net-of-tax wealth. He found a sufficient condition for the existence of an equilibrium: the slopes of individuals' indifference curves in the "tax rate-amount of public good" space must be monotoc with respect to their private wealth. In addition, he proved that, under this condition, a stable jurisdiction structure will always be segregated.

Later, a necessary and sufficient condition has been identified by Gravel and Thoron [START_REF] Gravel | Does endogeneous formation of jurisdictions lead to wealth stratification[END_REF] to ensure the wealth-stratification of any stable jurisdiction structure, as per Greenberg and Weber's definition [START_REF] Greenberg | Strong tiebout equilibrium under restricted preferences domain[END_REF]. This condition is called the Gross Substitutability/Complementarity (GSC) condition. It is respected if and only if the Marshallian demand for the public good is always monotonic with respect to the private good price. Furthermore, the GSC condition is equivalent to the monotonicity of the favorite tax rate function with respect to the private wealth, for any level of prices and wealth.

Oddou [START_REF] Oddou | The effect of congestion and spillovers on the segregative properties of endogenous jurisdiction structure formation[END_REF] examined the robustness of this condition when the public good may suffer from congestion, and households may benefit from other jurisdictions' local public services. In such a framework, the GSC condition is affected neither by the existence of spillovers across jurisdictions, nor the congestion (at least if it is not too strong). Those articles prove the robustness of the GSC condition for several generalizations of Gravel and Thoron's model. However, as proven by Gravel and Oddou ([4]), two generalizations of the basic model mitigate the segregative properties : the existence of different kinds of public goods within each jurisdiction, and the presence of a housing market (if the preferences over the private good and the housing are not homothetic). These two elements make the GSC condition not sufficient to ensure segregation.

The closest model of this article is Biswas, Gravel and Oddou's article [START_REF] Biswas | The segregative properties of endogenous jurisdiction formation with a welfarist central governmment[END_REF]. They added a central government to Gravel and Thoron's model, whose objective is to implement a policy to pursue a welfarist objective. A central government policy is characterized by equalization payments between jurisdiction. Those equalization payments can be either vertical (the government taxes households and redistributes the revenues to jurisdictions), horizontal (the government redistributes local tax revenues between the jurisdictions), or mixed. Two types of social welfare functions are considered in turn: the maximin and the generalized utilitarism. Though a maximin government radically affects the results, they proved that the GSC condition remains necessary and sufficient to ensure the segregation of stable jurisdiction structure with a generalized utilitarian central government. However, their model only allows for redistribution among jurisdictions by the central government, not among households by the jurisdictions. This is the point that this article is generalizing upon.

In this article, we assume that jurisdictions can give their households an allowance in a model framework à la Gravel and Thoron. Jurisdictions choose their policy, i.e. the amount of allowances and their tax rates, in order to maximize a social welfare function. We require that the budgets of jurisdictions be balanced.

There is no cost of mobility and households have perfect information, so once all jurisdictions have chosen their policies, households move to the jurisdiction whose policy will satisfy their utility to the highest possible level. Equilibrium is reached as soon as there is no household left who can increase its utility by unilaterally leaving its jurisdiction for another one.

We prove that, in the presence of jurisdictional allowance, whose amount is determined by the local utilitarian government, the Gross Substitutability Complementarity Condition remains necessary, but is no longer sufficient to have all stable jurisdiction structure segregated.

The article is organized as follows: the next section introduces the formal model. Section 3 provides an example of how congestion and spillovers can modify a jurisdiction structure, section 4 states and proves the results. Finally, section 5 concludes.

The formal model

We consider an economy with a set of households I ⊂ N, the number of household in I is given by N . An economy is composed of three elements.

The first element is the preferences. All households share identical preferences, represented by a twice differentiable, increasing and concave utility function

U : R 2 ++ -→ R + (Z, x) -→ U (Z, x)
where 1. Z is the available amount of public good, 2. x is the amount of a composite private good.

We denote Z M (p Z , p x , R) and x M (p Z , p x , R) the Marshallian demands for the public good and the private good (respectively), when the public good price is p Z , the private good price, p x , and the revenue, R. We also define M RS(Z, x) as the Marginal Rate of Substitution of the public good to the private good.

We add two extra conditions : the regularity condition and the Inada condition.

Condition 1. If, for some public good price p Z , some income level R and some non degenerated interval I of positive private good price, one has

Z M (p Z , p x , R) = Z M (p Z , p x , R), ∀(p x , p x ) ∈ I 2 , then, one must have, ∀ px ∈ R + , Z M (p Z , p x , R) = Z M (p Z , px , R)
Condition 2. The preferences satisfies the Inada condition if and only if, ∀x > 0, lim

Z→0 U (Z, x) = ∞ and, ∀Z > 0, lim x→0 U (Z, x) = ∞
U representes the set of all functions satisfying the properties defined above.

The second element is the wealth distribution, given by ω :

I -→ R * + i -→ ω i -
household i is endowed with a wealth ω i ∈ R + -with ω being an increasing and bounded from above function. The third element is the set of conceivable locations. Households choose their place of residence among the finite set of locations, represented by L ⊂ N. The possibility for some locations to be empty is allowed. Households living at the same location form a jurisdiction. We denote J ⊆ L the set of jurisdictions. The subset of households living in j is denoted I j . As every household must live in one and only one jurisdiction, one has ∀(j, j ) ∈ J 2 , j∈J I j = I and j = j , I j ∩ I j = ∅.

Jurisdictions have two purposes: producing a public good, that will be consumed only by the households that compose it, and distributing an allowance to households. To finance those two functions, the jurisdictions raise a tax proportional to the households' wealth. The tax rate in jurisdiction j is denoted t j , the amount of allowance provided by jurisdiction j is denoted G j .

The amount of local public good produced by jurisdiction j is given by Z j = t j j -n j G j with :

• n j = card(I j ) being the number of households in j,

• j = i∈Ij ω i being the aggregated wealth in j.
The amount of the composite private good that household i living in jurisdiction j is given by

x ij = (1 -t j )ω i + G j
The preferences and the production of the public good having been introduced, we can now define the favorite tax rate function.

Definition 1. ∀( , n, G, ω i ) ∈ R 3 + × N, we define t * : R 3 + × N -→ [0; 1] ( , n, G, ω i ) -→ t * ( , n, G, ω i ) = argmax t∈[0;1] U (t -nG, (1 -t)ω i + G)
as the favorite tax rate function, eg the tax rate that maximizes the utility of a household endowed with private wealth ω i , in a jurisdiction with an aggregated wealth , a number of inhabitants equal to n, that grants an amount G of allowance.

Under the standard assumptions, ∀( , n, G, ω i ) ∈ mathbbR 3 + × N, the function U (t -nG; (1 -t)ω i + G) is single-peaked with respect to t, i.e. ∀ t < t * ( , n, G, ω i ) (resp. >), ∀t ∈] t; t * ( , n, G, ω i )[ (resp. ∀t ∈]t * ( , n, G, ω i ); t[), U ( t -nG, (1-t)ω i + G) < U (t -nG, (1-t)ω i +G) < U (t * ( , n, G, ω i ) -nG, (1-t * ( , n, G, ω i ))ω i +G).
Consequently, this "favorite tax rate function" always exists. For G = 0 (as in Gravel and Thoron's article), the monotonicity of the favorite tax rate function is equivalent to the GSC condition, but it is not the case otherwise. However, the next lemma will define the relation that exists between the favorite tax rate function and the Marshallian demand for the public good. Some conditions have to be respected for t * to exist. For instance, clearly, the function will not exist if < nG, because even if all the wealth was taxed, it would not be sufficient to finance the allowance.

The next lemma will define the relation between the favorite tax rate function and the Marshallian demand for the public good.

Lemma 1. For any preferences belonging to U, one has,

∀(ω i , , µ, G) ∈ R 4 + ,: t * ( , n, G, ω i ) ≡ Z M ( 1 , 1 ωi , 1 + G( 1 ωi + n )) + nG ) (1)
Proof. At the optimum, the MRS is equal to the price ratio. Hence, one has:

M RS = p Z p x ] (2) (3)
The first order condition (FOC) implies that:

U Z (t * -nG; (1 -t * )ω i + G) U x (t * -nG; (1 -t * )ω i + G) = ω i (4) (5)
Consequently, combining equations ( 2) and ( 4), we know that:

t * ( , n, G, ω i ) -µG = Z M (p Z , p x , R) (6)
(1

-t * ( , n, G, ω i )ω i + G = x M (p Z , p x , R) (7) (8)
when p Z = 1 , p x = 1 ωi and R = 1 + G( 1 ωi + n ). which leads to the result. We can now formally provide the definition of an economy. Definition 2. An economy is composed of 3 elements:

• Preferences represented by the utility function U ∈ U • A wealth distribution ω • A set of locations L ∈ N
We denote ∆ as the set of all economies that respect the assumptions presented above. Let us now define the notion of jurisdiction structure in the economy (U, ω, L).

Definition 3. A jurisdiction structure in the economy (U, ω, L) is a vector Ω = (J, {I j } j∈J ; ({t j } j∈J ); ({G j } j∈J ).
Literally, a jurisdiction structure is characterized by the set of jurisdictions, the partition of households among the different jurisdictions, and the vectors of the tax rates and amounts of allowance implemented in every jurisdiction.

Before presenting the notion of equilibrium, we must introduce the notion of welfarist local government. A welfarist local government chooses the policy that will maximize a certain social welfare function. As in [START_REF] Biswas | The segregative properties of endogenous jurisdiction formation with a welfarist central governmment[END_REF], we will only consider two kinds of objective for the local governments : the maximin and the utilitarism. First, we will present the objective of a maximin government. Definition 4. A local government of a jurisdiction j with a set of households I j and a aggregate wealth j pursues a maximin objective if and only if its policy

(t j , G j ) is such that, ∀(t, G) ∈ [0; 1] × R + , min i∈Ij U (t j j -n j G j ; (1 -t j )ω i + G j ) ≥ min i∈Ij U (t j - n j G; (1 -t)ω i + G).
In words, a maximin local government chooses the policy that provides the highest possible level of utility to the poorest household of the jurisdiction. Alternatively, a maximin government of the jurisdiction j chooses a policy (t max , G max ) that solves the maximization program max

(t,G)∈[0;1]×R+ inf i∈Ij U (t j -n j G, (1 -t)ω i + G) (9)
We can now provide a definition of a stable jurisdiction structure with maximin local governments. Definition 5. A jurisdiction structure Ω = (J, ({I j } j∈J ); ({t j } j∈J ); ({G j } j∈J ) is stable in the economy (U, ω, L) if and only if

1. ∀j, j ∈ J, ∀i ∈ I j , U (Z j , (1 -t j )ω i + G j ) ≥ U (Z j , (1 -t j )ω i + G j ), 2. ∀j ∈ J, Z j ≤ tj j -nj Gj p Z 3. ∀j ∈ J, ∀(t, G) ∈ [0; 1] × R + , min i∈Ij U (t j j -n j G j ; (1 -t j )ω i + G j ) ≥ min i∈Ij U (t j - n j G; (1 -t)ω i + G).
In words, a jurisdiction structure is stable if and only if :

1. No household can increase its utility by modifying its consumption bundle or by leaving its jurisdiction, 2. Every jurisdiction's budget is balanced, 3. Every jurisdiction applies a policy that maximizes the utility of its poorest inhabitant.

We can now switch to the definition of a utilitarian local government. Definition 6. A local government of the jurisdiction j with a set of households I j and an aggregate wealth equal to j is utilitarian if and only if it chooses the policy that maximizes the following program:

max (t,G)∈[0;1]×R+ i∈Ij U (t j -n j G, (1 -t)ω i + G) (10)
In words, a utilitarian local government chooses the bundle "tax rate-allowance" that maximizes the sum of household utilities. Thus, we can now define the notion of equilibrium with utilitarian local governments. Definition 7. A jurisdiction structure Ω = (J, ({I j } j∈J ); ({t j } j∈J ); ({G j } j∈J ) is stable in the economy (U, ω, L) if and only if

1. ∀j, j ∈ J, ∀i ∈ I j , U (Z j , (1 -t j )ω i + G j ) ≥ U (Z j , (1 -t j )ω i + G j ), 2. ∀j ∈ J, Z j ≤ tj j -nj Gj p Z 3. ∀j ∈ J, ∀(t, G) ∈ [0; 1] × R + , i∈Ij U (t j j -n j G j ; (1 -t j )ω i + G j ) ≥ i∈Ij U (t j - n j G; (1 -t)ω i + G).
Let us now formally express the definition of the segregation, which is the same definition as in [START_REF] Westhoff | Existence of equilibria in economies with a local public good[END_REF].

Definition 8. A jurisdiction structure Ω = (J, ({I j } j∈J ); ({t j } j∈J ); ({G j } j∈J ) in the economy (ω, U, L) is segregated if and only if ∀(ω h , ω i , ω k ) ∈ R 3 + such that ω h < ω i < ω k , (h, k) ∈ I j and i ∈ I j ⇒ Z j = Z j , G j = G j and t j = t j
In words, a jurisdiction structure is wealth-segregated if, except for groups of jurisdictions offering the same available amount of public good, the same amount of allowance and the same tax rate, the poorest household of a jurisdiction with a high per capita wealth is (weakly) richer than the richest household in a jurisdiction with a lower per capita wealth.

In the next section, we examine the robustness of the GSC condition to ensure segregation.

Results

This article proves that, with maximin local governemnts, the stable jurisdiction structure will always be segregated, but, with utilitarian local governments, the GSC condition is no longer sufficient to have all stable jurisdiction structures segregated. Let us first prove the first part of this statement.

With maximin local governments

First, we need to provide a solution to a maximin local government maximization program.

Lemma 2. Under a maximin local government, a solution of the program (9) is

t max = 1 and G max = argmax G∈[0; j n j ] ( j -n j G; G) 1 .
This solution is unique as soon as at least two strictly positive numbers of households in the jurisdiction j are endowed with different levels of wealth, while, if a

1 G max = (1-t * ( j ,n j ,G,ω)) j n j
with G = 0 and ω = j n j jurisdiction only contains households endowed with the same level of wealth ω i , then, ∀α ∈ [0; 1], t max = α + (1 -α)t * ( j , n j , 0, ω i ) and G max = αG max are solutions of the program [START_REF] Oddou | The effect of congestion and spillovers on the segregative properties of endogenous jurisdiction structure formation[END_REF].

Proof. The proof of this lemma is quite intuitive. Consider a jurisdiction j with a maximin government, whose poorest household is i. Obviously, the poorest household's private wealth is less than or equal to the average wealth of the jurisdiction, i.e. ω i ≤ j nj . This inequality becomes strict as soon as there is a non-null number of households strictly richer than households of type i in the jurisdiction.

Deriving U (t j -n j G, (1 -t)ω i + G) with respect to t and to G, one obtains:

j U (t j -n j G, (1 -t)ω i + G) ∂Z -ω i U (t j -n j G, (1 -t)ω i + G) ∂x (11)
-n j U (t j -n j G, (1 -t)ω i + G) ∂Z + U (t j -n j G, (1 -t)ω i + G) ∂x (12)
Let us suppose, first, that there exist at least one household in jurisdiction j that is richer than households i. Then, j nj > ω i , if, for some ( t, Ḡ)

∈ [0; 1] × R + , one has -n j U ( t j -n j Ḡ, (1 -t)ω i + Ḡ) ∂Z + U ( t j -n j Ḡ, (1 -t)ω i + Ḡ) ∂x = 0
then one has

U ( t j -n j Ḡ, (1 -t)ω i + Ḡ) ∂Z -ω i U ( t j -n j Ḡ, (1 -t)ω i + Ḡ) ∂x > 0
If one fixes t = 1, then, thanks to the Inada condition, we know that there exists

G max ∈]0; j nj [ such that -n j U ( j -nj G max ,G max ) ∂Z + U ( j -nj G max ,G max ) ∂x = 0.
As t is bounded from above to 1, and given that

j U ( j -n j G max , G max ) ∂Z -ω i U ( j -n j G max , G max ) ∂x > 0
we can conclude that t max = 1 and G max = argmax

G∈[0; j n j ] ( j -n j G; G)
is the unique solution of the program [START_REF] Oddou | The effect of congestion and spillovers on the segregative properties of endogenous jurisdiction structure formation[END_REF].

Let us now suppose that all households in jurisdiction j are endowed with the same wealth. The solution t max = 1 and G max = argmax G∈[0; j n j ] ( j -n j G; G) remains a valid solution, but is no longer unique.

One can observe that (11) = -ω i [START_REF] Westhoff | Existence of equilibria in economies with a local public good[END_REF]. So, if (11) = 0, then (12) = 0. By definition, t = t * ( j , n j , 0, ω i and G = 0 is another solution for the local government maximization program. Furthermore, ∀α ∈ [0; 1], t max = α + (1 -α)t * ( j , n j , 0, ω i ) and G max = αG max will generate the exact same amounts of public good and private good, thus there are solutions for the maximin local government maximization program.

Literally explained, a maximin local government can, by fixing the tax rate to 1 and redistributing a fixed allowance, equalize all households' wealth to the average wealth, which is the ultimate objective for a maximin government. Consequently, at equilibrium, every jurisdiction structure will be segregated, as stated by the following proposition.

Theorem 1. A stable jurisdiction structure with maximin local governements is always segregated.

Proof. Suppose that there exists a non segregated jurisdiction. Hence, there exists (at least) two jurisdictions 1 and 2, and (at least) three households (with a positive integer) with private wealth ω h < ω i < ω k such that:

U (t 1 1 -n 1 G 1 , (1 -t 1 )ω h + G 1 ) ≥ U (t 2 2 -n 2 G 2 , (1 -t 2 )ω h + G 2 ) (13) U (t 1 1 -n 1 G 1 , (1 -t 1 )ω i + G 1 ) ≤ U (t 2 2 -n 2 G 2 , (1 -t 2 )ω i + G 2 ) (14) U (t 1 1 -n 1 G 1 , (1 -t 1 )ω k + G 1 ) ≥ U (t 2 2 -n 2 G 2 , (1 -t 2 )ω k + G 2 ) (15) (16)
with at least one strict inequality. Thanks to lemma 2, we know that, at equilibrium, t 1 = 1. Hence, all households living in jurisdiction 1 have the same utility level, thus,

U (t 1 1 -n 1 G 1 , (1 -t 1 )ω i + G 1 ) = U (t 1 1 -n 1 G 1 , (1 -t 1 )ω k + G 1 ). In addition, as ω i < ω k , we know that U (t 2 2 -n 2 G 2 , (1 -t 2 )ω i + G 2 ) < U (t 2 2 -n 2 G 2 , (1 -t 2 )ω k + G 2 ).
Combining this inequality and equation (( 14) and ( 15), one obtains:

U (t 1 1 -n 1 G 1 , (1 -t 1 )ω k + G 1 ) ≥ U (t 2 2 -n 2 G 2 , (1 -t 2 )ω k + G 2 ) > U (t 2 2 -n 2 G 2 , (1 -t 2 )ω i + G 2 ) ≥ U (t 1 1 -n 1 G 1 , (1 -t 1 )ω k + G 1 ) which is impossible, as U (t 1 1 -n 1 G 1 , (1 -t 1 )ω i + G 1 ) = U (t 1 1 -n 1 G 1 , (1 - t 1 )ω k + G 1 ).
To be more specific, the only stable jurisdiction structure that can emerge are either the grand jurisdiction, or, if, at equilibrium, there is more than one jurisdiction, then all jurisdictions that contains at least two types of households with a non-null number implement the same policy and offer the same amount of public good. In this second situation, there can exist jurisdictions containing only households having the same wealth, who are richer than the richest household of any jurisdiction containing two different kinds of households. The following example present such a jurisdiction structure.

B

Let us consider households' preferences represented by the following utility function

U (Z, x) = ln(Z) + 4x -x 2 if x ≤ 2 ln(Z) + 4 otherwise
Such an utility function is continuous, twice differentiable, strictly increasing as long as x < 2 and concave with respect to every argument. Furthermore, such preferences violates the GSC condition. Consider an economy with three jurisdictions j 1 , j 2 and j 3 and four types of households a, b, c and d with private wealth ω a = 0.4, As long as x ≤ 2 and G = 0, the favorite tax rate function is given by

ω b = 2 - √ 2
t * ( , ω i ) = ω i -2 + (ω i -2) 2 + 2 2ω i
Here, the favorite tax rate function depends only on the private wealth, and not on the aggregate wealth, nor, obviously, on the number of households, as G = 0. Determining the preferred tax rate function if x > 2 will not be required, no household will be endowed with more than 2 units of private good.

One possible stable jurisdiction structure is where households of type a and b are placed in jurisdiction 1, households of type c, in jurisdiction 2, and households d, in jurisdiction 3. Hence, one will have 1 ≈ 2.7431 and n 1 = 5, 2 = 3 = 2 and n 2 ≈ 2 and n 3 = 1.

According to lemma 2, the local government of jurisdiction 1 will set t 1 = 1 and G 1 ≈ 0.2835. As jurisdictions 2 and 3 respectively contains households endowed with the same wealth, they can implement different policies, that would maximize the maximin social welfare function. Let us suppose that they both fix G = 0. Accordingly, the jurisdictions will set their tax rate to the favorite tax rate of their households when G = 0, i.e. t 2 = t * (2; 2; 0; 1) ≈ 0.36603 and t 3 = t * (2; 1; 0; 2) ≈ 0.35355. Therefore, households' utility level (rounded to three digits after the decimal point) in every jurisdiction would be: So, this jurisdiction structure is stable. We can make two remarks. First, if jurisdictions 2 or 3 had implemented another policy, for instance, by setting their tax rate to 1, and their allowance to G 2 = 0.6340 and G 3 = 1.293, households would have kept the same utility level in their own jurisdiction, but the jurisdiction structure would not have been stable anymore, as the new utility level would have been, in this case: The second remark is that, in such an economy, with such preferences, there exist two other stable jurisdiction structures. One, obviously, is the grand jurisdiction, containing all households, and the second one is where households of type a, b and c are placed in jurisdiction 1 and households d, in jurisdiction 3. Hence, one will have 1 = 4.7431 and n 1 = 7, 3 = 2 and n 3 = 1. Local government of jurisdiction 1 would set t 1 = 1 and G 1 ≈ 0.2061. Jurisdiction 3 would keep t 3 = t * (2; 1; 0; 2) ≈ 0.35355 and G = 0. The utility level would then be: j 1 j 3 a 1.976 0621 b 1.976 1.025 c 1.976 1.822 d 1.976 3.153 So this jurisdiction structure would be stable as well. As a consequence, as this ex-ample reinforces the proof of theorem 1, with maximin local governments, no stable jurisdiction structure can be segregated when local governments pursue a maximin objective.

We can now switch to the results with another welfarist objective: the utilitarism.

With utilitarian local governments

Let us start this subsection by defining the GSC condition.

Definition 9. The GCS condition holds if and only if one has either ∂Z

M (p Z ,px,R) ∂px ≤ 0∀(p Z , p x , R) (if Z is a gross complement to x) or ∂Z M (p Z ,px,R) ∂px ≥ 0∀(p Z , p x , R) (if Z is a gross substitute for x)
If G = 0, then the GSC condition and the monotonicity of the favorite tax rate function with respect to the private wealth are equivalent (the proof provided by Gravel and Thoron holds in this very case). However, if G > 0, the only relation between the Marshallian demand for the public good and the favorite tax rate function is that the favorite tax rate function will be decreasing with respect to the private wealth if the public good is a normal good and a gross substitute of the private good. Lemma 3. If the public good is a normal good and a gross substitute of the private good, then the favorite tax rate function is decreasing with respect to the private wealth.

Proof. Deriving equation (1) with respect to ω i , one has:

∂t * ( , n, G, ω i ) ∂ω i (17) = -1 p 2 x [ ∂Z M ( 1 , 1 ωi , 1 + G( 1 ωi + n ) ∂p x + G ∂Z M ( 1 , 1 ωi , 1 + G( 1 ωi + n ) ∂R ] (18)
As

∂Z M ( 1 , 1 ω i ,1+G( 1 ω i + n G ) ∂px
> 0 if the public good is a gross substitute of the private good, and

∂Z M ( 1 , 1 ω i ,1+G( 1 ω i + n G ) ∂R
> 0 if the public good is normal, then the favorite tax rate function will be decreasing if the public good is a gross substitute.

We can now formally state the main result of this article.

Theorem 2. For all economies belonging to ∆, the GSC condition is not sufficient to have any stable jurisdiction structure segregated. Proof. To prove this proposition, we consider, in turn, two different economies (with two different preferences), the public good being a gross complement to the private good with the first one, and a gross substitute for the second one, and, for each preference, we construct a stable and yet non-segregated jurisdiction structure.

Let us consider the following utility function : U (Z, x) = ln(Z) -1

x Such a utility function respects the assumptions presented above. The public good is a gross complement to the private good, as the Marshallian demand for the public good is given by:

Z M (p Z , p x , R) = 2R + p x -p 2 x + 4Rp x 2p Z
which is decreasing with respect to p x . Let us consider an economy with two jurisdictions j 1 and j 2 and three types of households a, b, c with private wealth ω a = 0.1, ω b = 1 and ω c = 20, with n a = 1, n b = 100, 000 and n c = 5, 000.

A stable jurisdiction structure would be households of type a and c living in one jurisdiction, denoted j 1 , with G 1 = 0.15171 and t 1 ≈ 0.7973665, hence one has Z 1 ≈ 78, 978, and households of type b, living in a second jurisdiction j 2 , with G 2 = 0 and t 2 ≈ 0.2680, hence one has Z 2 = 26, 795.

Consequently, one has the following utility level (rounded to four digits after the decimal point): Futhermore, one can check that the first order conditions, given by:

i∈Ij t -nG - ω i [(1 -t)ω i + G] 2 = 0 (20) i∈Ij n t -nG - 1 [(1 -t)ω i + G] 2 = 0 (21) (22)
are respected for both jurisdictions. Hence, the above jurisdiction structure is stable and yet, non-segregated.

Let us now consider the second economy, with a utility function for which the public good is a gross substitute of the private good, such as: U (Z, x) = ln(Z) + √ x This function also respects the assumptions developed in the previous section. The Marshallian demand for the public good is given by :

Z M (p Z , p x , R) = 2( p 2 x + Rp x -p x ) p Z
which is increasing with respect to p x We can construct a stable jurisdiction structure by placing 44 households of type a and 15 households of type c in jurisdiction j 1 , with G 1 = 0.7275 and t 1 = 0.4805, hence one has Z 1 ≈ 122.3695, and households of type b, living in a second jurisdiction j 2 , with G 2 = 0 and t 2 = 0.7320, hence one has Z 2 = 222.5280.

One will, therefore, have the following utility level (rounded to four digits after the decimal point): j 1 j 2 a 5.9237 5.9227 b 6.1361 6.1372 c 8.1413 7.7202 As previously, the first order conditions, given by:

i∈Ij t -nG - ω i 2 (1 -t)ω i + G = 0 (23) i∈Ij n t -nG - 1 2 (1 -t)ω i + G = 0 (24) (25)
are respected. Hence, the above jurisdiction structure is also stable and yet, non-segregated, which proves the proposition.

Conclusion

The main conclusion of this paper is that, when jurisdictions can distribute an allowance to its inhabitants, endogenous jurisdiction formation is more segregative with maximin local governments than with utilitarian local governments. The effect of this allowance is remarkable, as endogenous jurisdiction formation without allowances is, in terms of segregative properties, between the two welfarist approaches taken into account in this article.

Summed in one sentence, this article allows to state that struggling against wealth inequalities favors wealth segregation. Such intuitions were already expressed in [START_REF] Biswas | The segregative properties of endogenous jurisdiction formation with a welfarist central governmment[END_REF], with a welfarist central government, but they are reinforced with welfarist local governments.

However, one must note that our analysis is reduced to models à la Westhoff, and that we do not specify whether the GSC condition is necessary or not with utilitarian local governments. Also, we only consider two types of social welfare functions. Observing the segregative properties of other social welfare functions, such as the generalized utilitarism would be an interesting topic. Finally, we assume that all local governments share the same welfarist objective. For further researches, examining what would happen in a jurisdiction structure where both maximin and utilitarian local governments co-exist would be worthy of interest.

  , ω c = 1 and ω d = 2, and with number n a = 1, n b = 4, n c = 2 and n d = 1.

j 1 j 2 j 3 a 1 .

 1 336 1.822 3.153 b 1.336 1.822 3.153 c 1.336 1.822 3.153 d 1.336 1.822 3.153 So every household would have an incentive to move to jurisdiction 3.

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Remy Oddou
email: remy.oddou@free.fr

The effect of a local allowance on the endogenous formation of jurisdictions

Keywords: JEL Classification: C78, D02, H73, R13 Jurisdictions, Segregation, Allowance

This paper analyses the effect of allowances, whose amount depends on jurisdictions, on the segregative properties of endogenous formation of jurisdictions. Households choosing to live at the same place form a jurisdiction whose aim is to produce a local public good and to implement a redistribution policy, by granting every household an allowance whose amount is determined by the jurisdiction. In every jurisdiction, the production of the local public services and the allowance are financed with a local tax based upon the households' wealth. Local wealth tax rates and the level of the allowance are exogenously determined in every jurisdiction. Households are free to leave their jurisdiction for another jurisdiction that would provide them with their highest utility. We find that the existence of an allowance mitigates the segregative properties of endogenous jurisdiction formation, as the condition identified by Gravel and Thoron to ensure the segregation of any stable jurisdiction structure remains necessary, but is no longer sufficient.

Introduction

Local jurisdictions have a well-known role : the production of public goods. But, more and more, they choose to have a hand in wealth redistribution. Although, in most countries, local taxation is proportional, which prevents the implemention of a redistributive taxation policy, jurisdictions have other ways to implement a policy to reduce wealth inequality. They can, for instance, provide an allowance to households.

As highlighted by many economists (see, for instance, [START_REF] Madden | Problem of the century: racial stratification in the United States. E. Anderson and D. Massey[END_REF]), segregation by wealth seems to be an expanding phenomenon. Within an urban area, one can observe jurisdictions with mainly poor households, and jurisdictions with very few of them. Tiebout's 1956 article [START_REF] Tiebout | A pure theory of local expenditures[END_REF] provides a widely known explanation of this phenomenon: in his opinion, households choose their jurisdiction according to a trade-off between local tax rates and amounts of public services provided, which leads every jurisdiction to be homogeneous. Households that want to consume a large amount of public good will live in a jurisdiction where the tax rate and the amount of public good are high, while households who do not particularly enjoy the public good tend to stay in a jurisdiction where the tax rate is low. The formation of jurisdictions structure is therefore endogenous, due to the free mobility of households, that can "vote with their feet", that is to say leave their jurisdiction for another one that better fits their preferences. Numerous writings dealing with the formation of endogenous jurisdictions such as Tiebout's exists.

Westhoff [START_REF] Westhoff | Existence of equilibria in economies with a local public good[END_REF] must be mentioned as one of the first models based on Tiebout's article. In his model, households can consume a local public good, financed by a local tax on wealth, and only consumed by households living in the jurisdiction, and a composite private good. The main result of his article is that an equilibrium will exist if the slopes of individuals' indifference curves in the tax rate-amount of public good space to be ordered by their private wealth. Under this condition, at equilibrium, the jurisdiction structure will be segregated. It is important, though, to notice that this condition is sufficient, but not necessary. [START_REF] Gravel | Does endogeneous formation of jurisdictions lead to wealth stratification[END_REF] identified a necessary and sufficient condition to have every stable jurisdiction structure segregated, as defined by Westhoff. This condition, known as the Gross Substitutability/Complementarity (GSC) condition, requires that the public good must be, for all levels of prices and wealth, either always a complement or always a substitute of the private good. In their model, this condition is equivalent to having the favorite tax rate (i.e. the tax rate that maximizes the utility of a certain household, in a certain jurisdiction) being a monotonous function of the private wealth, for any level of prices and wealth.

Gravel and Thoron

Biswas, Gravel and Oddou [START_REF] Biswas | The segregative properties of endogenous jurisdiction formation with a welfarist central governmment[END_REF] integrated a welfarist central government that maximizes a generalized utilitarian social welfare function by implementing an equalization payment policy. Equalization payment can be either vertical (the government taxes households and redistributes the revenues to jurisdictions), horizontal (the government redistributes local tax revenues between the jurisdictions), or mixed. They found that the GSC condition remains necessary and sufficient. However, their model only allows for redistribution among jurisdictions, not among households.

Oddou [START_REF] Oddou | The effect of congestion and spillovers on the segregative properties of endogenous jurisdiction structure formation[END_REF] examined the robustness of this condition when the public good may suffer from congestion, and households may benefit from other jurisdictions' local public services. Within such a framework, the GSC condition is affected neither by the existence of spillovers across jurisdictions, nor the congestion (at least if it is not too strong). Those articles prove the robustness of the GSC condition to several generalizations of Gravel and Thoron's model. However, as proven by Gravel and Oddou ([5]), two generalizations of the basic model mitigate the segregative properties : theexistence of different kinds of public goods within each jurisdiction, and the presence of a housing market (if the preferences over the private good and the housing are not homothetic). These two elements make the GSC condition not sufficient to ensure segregation.

In this article, we start from Gravel and Thoron's model, and we assume that jurisdictions can give their households an allowance. The amount of the allowance is the same for every household, and is exogenously fixed by the jurisdiction, as is the tax rate and the amount of public good. One can suppose that it is determined by a vote, or according to a welfare function. the jurisdictions' budget must be balanced, so that the production cost of the public good added to the allowance must be equal to the tax revenue. This framework can be seen as a taxation that is progressive rather than proportional, because the share of the gross wealth that is taxed increases with respect to the wealth. In many countries, though, local taxes are more or less proportional to the income, because local taxes are based on housing values (at fixed rate), and the share of income that households devote to housing does not depend on the income (see [START_REF] Davis | Households expenditures, wages, rents. Review of Economics Dynamics[END_REF]). However, some countries have implemented or are implementing a progressive local taxation scheme.

Households are assumed to be freely mobile so once all jurisdictions have determined their tax rate and the amounts of public goods and allowance, households can leave their jurisdiction for another one that would increase their utility. Equilibrium is reached when no household has any incentive to unilaterall leavey its jurisdiction.

We prove that, in the presence of jurisdictional allowance, the monotonicity of the favorite tax rate function remains necessary to have every stable juridsiction structure segregated, but the GSC is no longer sufficient to ensure wealth-stratification. Furthermore, the GSC condition is not equivalent to the monotonicity of the favorite tax rate function.

If the public good is a gross complement of the private good, then, at equilibrium, the jurisdiction structure may look like the situation defined by Epple and Romano [START_REF] Epple | Ends against the middle: Determining public service provision when there are private alternatives[END_REF] as "the ends against the middles": on one hand, poor and rich households prefer living in jurisdictions with a high tax rate, the former because they can benefit from a high allowance, the latter because they can enjoy a high amount of public good, that is a complement to the private good. The intermediate households, on the other hand, prefer a jurisdiction with a low tax rate.

This paper aims at examining the segregative properties of the endogenous jurisdiction structure formation in such a framework. The article is organized as follows: the next section introduces the formal model, section 3 provides an example of how the existence of a local allowance affects the segregative properties of endogenous jurisdiction formation, section 4 states and proves the results and finally, section 5 concludes.

The formal model

We consider a model with a continuum of households on the interval I with Lebesgue measure λ, where, for any subset I 0 ⊂ I, the mass of household in I 0 is given by λ(I 0 ). Households' wealth distribution is given by a Lebesgue measurable function ω : I → R * + -household i is endowed with a wealth ω i ∈ R + -with ω being an increasing and bounded from above function.

All households share identical preferences, represented by a twice differentiable, increasing and concave utility function

U : R 2 ++ -→ R + (Z, x) -→ U (Z, x)
where 1. Z is the available amount of public good, 2. x is the amount of a composite private good.

We denote Z M (p Z , p x , R) and x M (p Z , p x , R) the Marshallian demands for the public good and the private good (respectively), when the public good price is p Z , the private good price, p x , and the revenue, R. We also define M RS(Z, x) as the Marginal Rate of Substitution of the public good for the private good.

We assume that both the public good and the private good are normal goods, which means that their Marshallian demands are increasing with respect to the revenue.

U represents the set of all functions satisfying the properties defined above.

We can now define the Marginal Rate of Substitution (MRS). as the ratio of the derivative of the utility function with respect to the public good over the derivative of the utility function with respect to the private good. Definition 1. The Marginal Rate of Substitution is the ratio of the derivative of the utility function with respect to the public good over the derivative of the utility function with respect to the private good:

M RS(Z, x) = U (Z,x) ∂Z U (Z,x) ∂x
Households choose their place of residence among the finite set of locations, represented by L ⊂ N. The possibility for some locations to be empty is allowed. Households living at the same location form a jurisdiction. We denote J ⊆ L the set of jurisdictions, and I j ⊆ I as the subsete of households that lives in jurisdiction j. As households must live in one, and only one, jurisdiction, one has j∈J I j = I and,

∀(j, j ) ∈ J 2 , I j ∩ I j = ∅.
Jurisdictions have two purposes: producing a public good, that will be consumed only by the households within the jurisdiction, and distributing an allowance to those households. To finance those two functions, the jurisdictions raise a tax proportional to the households' wealth. The taxe rate in jurisdiction j is denoted t j , the amount of allowance provided by jurisdiction j is denoted G j .

The amount of local public good produced by jurisdiction j is given by Z j = t j j -µ j G j with :

• µ j = λ(I j ) being the mass of households in j,

• j = i∈Ij ω i being the aggregated wealth in j.
The amount of the composite private good that household i living in jurisdiction j is given by

x ij = (1 -t j )ω i + G j Definition 2.
An economy is composed of 3 elements:

• Wealth distribution ω • Preferences represented by the utility function U ∈ U • A set of locations L ∈ N
We denote ∆ as the set of all conceivable economies. We must now define the notion of jurisdiction structure. Definition 3. A jurisdiction structure is a vector:

Ω = (J, {I j } j∈J ; ({t j } j∈J ); ({G j } j∈J ) .
In words, a jurisdiction structure is characterized by the set of jurisdiction that composes it, the partition of households among the different jurisdictions, and the policy (tax rate and amount fo the allowance) implemented by the different jurisdiction. Definition 4. A jurisdictions structure Ω = (J, ({I j } j∈J ); ({t j } j∈J ); ({G j } j∈J ) is stable in the economy (ω, U, (a, α), B, L) if and only if

1. ∀(j, j ) ∈ J 2 , ∀i ∈ I j , U (Z j , (1 -t j )ω i + G j ) ≥ U (Z j , (1 -t j )ω i + G j ), 2. ∀j ∈ J, Z j ≤ tj j -µj Gj p Z .
Literaly, a jurisdictions structure is stable if and only if :

1. No household can increase its utility by modifying its consumption bundle or by leaving its jurisdiction, 2. Every jurisdiction's budget is balanced.

Let us now express formally the definition of the segregation, which is the same definition as in [START_REF] Westhoff | Existence of equilibria in economies with a local public good[END_REF].

Definition 5. A jurisdictions structure Ω = (J, ({I j } j∈J ); ({t j } j∈J ); ({G j } j∈J ) in the economy (ω, U, L) is segregated if and only if ∀ω h , ω i , ω k ∈ R + such that ω h < ω i < ω k , (h, k) ∈ I j and i ∈ I j ⇒ Z j = Z j , G j = G j and t j = t j
In words, a jurisdictions structure is wealth-segregated if, except for groups of jurisdictions offering the same available amount of public good, the same amount of allowance and the same tax rate, the poorest household of a jurisdiction with a high per capita wealth is (weakly) richer than the richest household in a jurisdiction with a lower per capita wealth.

In the next section, we examine the robustness of the monotonicity of the favorite tax rate function and the GSC condition to ensure segregation.

Results

This article proves that the GSC condition is no longer equivalent to the monotonicity of the favorite tax rate function with respect to the private wealth, and that the monotonicity of the favorite tax rate function remains necessary to have all stable jurisdictions structures segregated, but is no longer sufficient, nor is the GSC condition. Let us define the GSC condition first. Definition 6. If the GCS condition holds, then, one has either

∂Z M (p Z ,px,R) ∂px ≤ 0∀(p Z , p x , R) (if Z is a gross complement for x) or ∂Z M (p Z ,px,R) ∂px ≥ 0∀(p Z , p x , R) (if Z is a gross substitute to x)
We now define the favorite tax rate function. Definition 7. ∀( , µ, G, ω i ) ∈ R 4 + , we define

t * : R 4 + -→ [0; 1] ( , µ, G, ω i ) -→ t * ( , µ, G, ω i ) = argmax t∈[0;1] U (t -µG), (1 -t)ω i + G)
as the favorite tax rate function, eg the tax rate that maximizes the utility of a household endowed with private wealth ω i , in a jurisdiction with an aggregate wealth , a mass of population equal to µ, that grants an amount G of allowance.

According to the assumptions made on the utility function, this "favorite tax rate function" always exists. For G = 0 (as in Gravel and Thoron's article), the monotonicity of the favorite tax rate function is equivalent to the GSC condition, but it is not the case otherwise. However, the next lemma will define the relation that exists between the favorite tax rate function and the Marshallian demand for the public good.

Some conditions have to be respected for t * to exists. For instance, clearly, the function will not exist if < µG, because even if all the wealth was taxed, it would not be sufficient to finance the allowance.

Let us define the following function, representing the utility with respect to the tax rate:

Φ : [0; 1] × R 4 + -→ R + (t; , µ, G, ω i ) -→ Φ(t; , µ, G, ω i ) = U (t -µG), (1 -t)ω i + G)
As the utility function is assumed to be concave, the function Φ(t; , µ, G, ω i ) is single-peaked with respect to t, i.e.

∀ t < t * ( , µ, G, ω i ) (resp. ¿), ∀t ∈] t; t * ( , µ, G, ω i )[ (resp. ∀t ∈]t * ( , µ, G, ω i ); t[, Φ( t; , µ, G, ω i ) < Φ(t; , µ, G, ω i ) < Φ(t * ( , µ, G, ω i ); , µ, G, ω i ).
The next lemma will define the relation between the favorite tax rate function and the Marshallian demand for the public good.

Lemma 1. For any preferences belonging to U, one has, ∀(ω i , , µ, G) ∈ R 4 + ,:

t * ( , µ, G, ω i ) ≡ Z M ( 1 , 1 ωi , 1 + G( 1 ωi + µ )) + µG ) (1)
Proof. At the optimum, the MRS is equal to the price ratio. Hence, one has:

M RS = p Z p x ] (2)
The first order condition (FOC) implies that:

U Z (t * -µG; (1 -t * )ω i + G) U x (t * -µG; (1 -t * )ω i + G) = ω i (3)
Consequently, combining 2 and 3, we know that:

t * ( , µ, G, ω i ) -µG = Z M (p Z , p x , R)(1 -t * ( , µ, G, ω i ))ω i + G = x M (p Z , p x , R) (4)
when p Z =1 , p x = 1 ωi and R = 1 + G( 1 ωi + µ ). which leads to the result. Thanks to this lemma, we can observe that, contrary to Gravel and Thoron's article, the GSC condition will not be equivalent to the monotonicity of the favorite tax rate function with respect to the private wealth 1 .

However, the previous lemma allows us to identify the following implication.

Lemma 2. If the public good is a gross substitute of the private good, then the favorite tax rate function decreases with respect to the private wealth.

Proof. Deriving 1 with respect to ω i , one has:

∂t * ( , µ, G, ω i ) ∂ω i = -1 p 2 x [ ∂Z M ( 1 , 1 ωi , 1 + G( 1 ωi + µ )) ∂p x + G ∂Z M ( 1 , 1 ωi , 1 + G( 1 ωi + µ )) ∂R ] (5)
As

∂Z M ( 1 , 1 ω i ,1+G( 1 ω i + µ G ) ∂px
> 0 if the public good is a gross substitute for the private good, and

∂Z M ( 1 , 1 ω i ,1+G( 1 ω i + µ G ) ∂R
> 0 because the public good is normal, then the favorite tax rate function will be decreasing if the public good is a gross substitute.

We can now formally state the main result of the article.

Theorem 1. For all economies belonging to ∆, the monotonicity of the favorite tax rate function with respect to the private wealth is necessary but not sufficient to have any stable jurisdiction structure segregated.

Let us start with the necessity part of the theorem. Proposition 1. For all economies belonging to ∆, the monotonicity of the favorite tax rate function with respect to the private welath is necessary to ensure the segregation of any stable jurisdiction structure.

Proof. To prove this proposition, we show that any violation of the condition allows to construct a stable and yet non segregated jurisdiction structure.

Let us suppose that the monotonicity of the favorite tax rate function with respect to the private wealth condition is violated, for some a non-degenerated interval of private wealth ]ω 1 ; ω 3 [, for some aggregate wealth ¯ , for some amount of allowance Ḡ and some mass of households μ. Let us suppose, with no loss of generality, that the favorite tax rate function is increasing on the interval ]ω 1 ; ω b [ and decreasing on the interval ]ω

b ; ω 3 [, ω b ∈]ω 1 ; ω 3 [.
Consequently, we know that there exists ω a < ω b and ω c > ω b such that :

t * ( ¯ , μ, Ḡ, ω a ) = t * ( ¯ , μ, Ḡ, ω c ) = t 1 < t 2 = t * ( ¯ , μ, Ḡ, ω b )
Let us denote Z j={1,2} = t j ¯ -μ Ḡ and x i={a,b,c} j={1,2} = (1 -t j )ω i + Ḡ. According to the definition of the favorite tax rate function, one has:

• U (Z 1 , x a 1 ) > U (Z 2 , x a 2 ) • U (Z 1 , x b 1 ) < U (Z 2 , x b 2 ) • U (Z 1 , x c 1 ) > U (Z 2 , x c
2 ) Let us consider 3 types of households, p, m and r 2 , such that households of type p have no wealth, households of type m are endowed with a wealth equal to ω c , and households of type r, to ωc(ωc-ωa) ω b -ωa . Clearly, one has ω p < ω m < ω r .

Let us place a mass µ p = of households p and a mass µ r =

(Z1+ x 1 1 )(ω b -ωa) ωc(ωc-ωa-(1-t1)(ω b -ωa))
of households r in a jurisdiction α with

t α = 1 -(1-t1)(ω b -ωa) ωc , G α = x 1
1 , and, thus,

α = (ωc-ωa)(Z1+ x 1 1 )(ω b -ωa) (ω b -ωa)(ωc-ωa-(1-t1)(ω b -ωa))
. Such a jurisdiction would produce an amount of public good equal to Z 1 , and would respectively provide households a, b and c with an amount of private good equal to (respectively) x a 1 , x b 1 and x c 1 .

Let us now place a mass This jurisdiction structure would therefore be stable and non segregated, which proves the proposition.

µ b = Z2 ωc-(1-t2)ω b of households b in a jurisdiction β with t β = 1-(1-t2)(ωi-ω h ) ω k , G β = x
Proposition 2. For all economies belonging to ∆, the GSC is not sufficient to ensure the segregation of any stable jurisdiction structure.

Proof. To prove this proposition, we pick two utility functions, the public good being a gross complement to the private good with the first utility function, and a gross substitute for the second one, and, for each preference, we construct a stable and yet non-segregated jurisdiction structure.

Let us consider the following utility function : U (Z, x) = ln(Z) -1

x Such a utility function is continuous, twice differentiable, strictly increasing and concave with respect to every argument. The public good is a gross complement to the private good, as the Marshallian demand for the public good is given by:

Z M (p Z , p x , R) = 2R + p x -p 2 x + 4Rp x 2p Z
which is decreasing with respect to p x . Consider an economy with two jurisdictions j 1 and j 2 and three types of households a, b, c with private wealth ω a = 1, ω b = 2 and ω c = 20, with n a = 62, n b = 34 and n c = 1.

A stable jurisdiction structure would be households of type a and c living in one jurisdiction, denoted j 1 , with G 1 = 0.4 and t 1 = 0.78, hence one has Z 1 = 38.76, and households of type b, living in a second jurisdiction j 2 , with G 2 = 0 and t 2 = 0.5, hence one has Z 2 = 34.

Consequently, one has the following utility level (rounded to two digits after the decimal point): Let us now consider the same economy, but with a utility function for which the public good is a gross substitute of the private good, such as:

U (Z, x) = ln(Z) + √ x
This function is also continuous, twice differentiable, strictly increasing and concave with respect to every argument. The Marshallian demand for the public good is givent by :

Z M (p Z , p x , R) = 2( p 2 x + Rp x -p x ) p Z which is increasing with respect to p x
We can construct a stable jurisdiction structure by placing households of type a and c in jurisdiction j 1 , with G 1 = 0.1 and t 1 = 0.48, hence one has Z 1 = 33.06, and households of type b, living in a second jurisdiction j 2 , with G 2 = 0 and t 2 = 0.5, hence one has Z 2 = 34. One will, therefore, have the following utility level (rounded to two digits after the decimal point): Hence, the above jurisdiction structure is also stable and yet, non-segregated, which proves the proposition.

Using this proposition, we can state that the monotonicity of the favorite tax rate function with respect to the private wealth is not sufficient either, because, thanks to lemma 2, we know that the favorite tax rate function will be increasing with respect to the private wealth if the public good is a gross substitute for the private good. Consequently, if the gross substitutability of the public good is not sufficient, then the monotonicity of the wealth is not sufficient either.

Conclusion

The conclusion of this paper is that the presence of an allowance mitigates the segregative properties of endogenous jurisdiction formation, as the GSC condition, that was proven to be robust to several generalizations of the model, is no longer sufficient to ensure the segregation of any stable jurisdiction structure, nor is the monotonicity of the favorite tax rate function with respect to the private wealth. Such a result is stricking for two reasons. First, it contradicts a widely shared belief that the redistribution of income favors segregation (see, for instance, [START_REF] Bierbrauer | Strategic nonlinear income tax competition with perfect labor mobility[END_REF]). Second, it proves that the GSC condition is not as robust as it seems, as it is not sufficient to ensure the wealth stratification of any stable jurisdiction structure.

It would be interesting, for further researches, to investigate whether the existence of a local allowance affects the conditions under which an equilibrium exists. Examining if the "poor chasing the rich" phenomenon, identified by Epple and Romano, exists when jurisdictions distribute an allowance would be an exciting challenge.

1 2 ,

 2 and, therefore, β = ω k Z2 ωc-(1-t2)ω b . Such a jurisdiction would produce an amount of public good equal to Z 2 , and would respectively provide to households a, b and c an amount of private good equal to (respectively) x a 2 , x b 2 ans x c 2 .

  45 3.43 Hence, the above jurisdiction structure is stable and yet, non-segregated.

If G = 0, as in Gravel and Thoron's article, the GSC condition is equivalent to the monotonicity of the favorite tax rate function with respect to the private wealth, but this results is no longer valid if G > 0.

p is for poor, m for middle and r for rich.

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