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The peer group effect and the optimality properties of head and income taxes by Francisco Martínez-Mora Documento de Trabajo 2011-07

April 2011

University of Leicester and FEDEA.

Jorge Juan, 46 28001 Madrid -España Tel.: +34 914 359 020 Fax: +34 915 779 575 infpub@fedea.es

Francisco MartÌnez-Mora

University of Leicester and FEDEA

Abstract

This paper studies a Tiebout model with two school districts, housing markets and peer e§ects to re-evaluate the optimality properties of the allocation of households to districts induced by head and income taxes. The main novel results reveal that head taxes are not superior to income taxes and that the indirect redistribution implied by income taxation is not necessarily at odds with location optimality or associated to welfare losses. Many combinations of head taxes di§erentiated by household type can sustain the optimal outcome as an equilibrium. While this may not be possible using di§erentiated income taxes, a combination of non-di§erentiated ones and di§erentiated head taxes levied on the residents of the rich district can lead to the optimal outcome and e§ect signiÖcant local redistribution. In turn, non-di§erentiated head taxes are suboptimal (unless optimality requires one of the districts to be type-homogeneous) and a combination of uniform income taxes and head taxes levied on the rich districtís population can do as well as them. Moreover, non-di§erentiated income taxes may generate smaller welfare losses than their lumpsum counterpart, a result which clashes with the beneÖt view of head taxes.

Key words: Tiebout, peer e§ects, head tax, income tax, optimality.

1 Introduction

In his seminal contribution, Tiebout (1956) suggested that head taxes would lead to an optimal sorting of households into homogenous jurisdictions, which would therefore select the optimal level of local public good provision unanimously. Tiebout did not formalise his argument, which thus became the Tiebout hypothesis. Bewley (1981) among others formalised the result, providing a set of conditions under which the Tiebout hypothesis holds. These included at least as many jurisdictions as household types, the use of local head taxes and a technology of production of the local public good exhibiting constant and identical marginal costs with respect to the population. In a recent contribution, Calabrese et al. (2010) extended the result to a less stylised framework with housing markets and a smaller number of exogenous jurisdictions than of household types. 1

Income taxes, in turn, generate welfare losses (Wildasin, 1986; Goodspeed, 1989, 1995). Since households with below average income face lower tax prices (that is, they receive greater amounts of spending per unit of tax paid), they do not internalise the true costs they impose on local Önances, and end up living in richer districts than they would otherwise do and beneÖtting from the implied redistribution. Moreover, to the extent that income taxes distort tax prices they also distort the political process.

The assumption that the costs of provision of the local public good are independent of the characteristics of the population is crucial to attain these conclusions. 2 Yet, such assumption is di¢ cult to justify, especially if the local public good considered is schooling. In that context, as parents, teachers, policy-makers and academics know well, the diversity of the student population and the importance of peer ináuences make the peer group e§ect an essential element of the problem. 3

The objective of this paper is to re-evaluate the optimality properties of the equilibrium district compositions induced by head and income taxes in the presence of peer e§ects (or, more generally, of non-anonymous crowding). The model represents a city divided into two jurisdictions (school districts) with exogenously given boundaries and housing supplies. Each district provides taxfunded public education to its residents and households choose where to live, a decision that subsumes the choice of school. 4 Households di§er continously in income, and are of one of two types that depend on the childís ability to beneÖt from education and on the cost of providing her with a unit of school quality. Peer e§ects are simply modeled as the outcome of these di§erential crowding costs: a school with a greater proportion of low cost pupils can provide the same level of school quality using less resources or provide greater quality with equal spending per pupil. In order to focus the analysis on locational optimality, the model assumes that local governments provide the (locally) optimal level of school quality. 5

1 The Tiebout hypothesis has also been formalised in the slightly di§erent framework of club theory. Club models do not incorporate a explicit representation of land markets and allow clubs to form endogenously. A joint review of the two literatures can be found in Scotchmer (2002). A succint review of the club literature on the Tiebout model is in Wooders (1999).
For example, de Bartolome (1990) revealed that the peer group e§ect could generate ine¢ ciencies even if local governments relied on head taxes to fund schools. In similar contexts, other authors (including Schwabb and Oates, 1991, Brueckner and Lee, 1989, and Conley and Wooders, 1998) have shown that di§erentiated taxes are required to achieve optimality.
3 Recent evidence documenting the impact of peer e§ects on student outcomes includes the work on Kenyan primary schools of Duáo et al. (forthcoming) and that of Lavy et al. (2009) on English secondary schools.
As the most prominent example of a local public good, the case of local schooling motivates the analysis. Results apply more generally to any local public good whose technology of production depends on the characteristics of the population that receives it.

The paper adopts a utilitarian normative framework, with the Social Welfare Function (SWF) being the unweighted sum of utility in the economy. In that setting, optimality requires households of the same type to be perfectly segregated by income across districts, with higher income ones living in the better school district. That requirement generates two necessary single-crossing conditions, one per household type. Optimality also demands some income mixing: relatively low income households of the low cost and high beneÖt type should reside in the good school district instead of higher income ones of the other type.

The results of the analysis critically depend on whether household cost-types are observable and taxes can be di§erentiated across them or not. In case they are, on the one hand, multiple head taxes that di§erentiate across household types can achieve optimality. 6 In a model with housing markets, therefore, these taxes need not cover the marginal cost of admitting a household into a district, as in a model without them (e.g. Schwabb and Oates, 1991). Instead, the extra taxes households of di§erent types pay in the rich district with respect to the poor one must provide them with the optimal relative location incentives (that is, they must generate di§erences in the cost of entry each type of household faces to live in the rich district that internalise the externalities emerging from the peer e§ect). On the other hand, although it is often possible to Önd combinations of di§erentiated income tax rates simultaneously satisfying the public sector budget constraints and providing cut-o§ households with the optimal location incentives, it may well be the case that none of them fulÖlls the two necessary single-crossing conditions optimality requires. Notwithstanding, it is always possible to reach optimality by combining anonymous income taxes with di§erentiated head taxes levied on the residents of the rich district. At most, the necessary single-crossing conditions will require the income tax rates to be equated across districts. Therefore, whilst the single-crossing conditions mark the limits to the compatibility between income taxes and optimality, they do not rule it out.

5 Most of the literature focuses on one of two e¢ciency questions: whether the emerging allocation of households to districts is e¢cient and whether the political process selects the e¢cient level of provision and taxation. A recent exception to this norm is Calabrese et al. (2010). In that paper, welfare losses stemming from voting distortions are small. Goodspeed (1989) notes that the main critic to the local use of income taxation concerns the distortions it introduces in the distribution of the population across districts.
6 Notice that these di§erentiated taxes depend on crowding types (which may be publicly observable) and not on tastes (which are not). See Conley and Wooders (1998).

Additional counter-intuitive results arise when local governments cannot tailor taxes to household types. In that case, it is not possible to unambiguously rank head and income taxes. Anonymous head taxes not only induce a suboptimal distribution of households across districts (unless optimality requires one of the districts to be type-homogeneous) but may also be more distortionary than an ability-to-pay tax. The intuition is the following: whereas anonymous head taxes do not a§ect households relative location incentives, income taxes generate di§erences in entry prices between the marginal (or cut-o§) household of each type living in the rich district (i.e. the lowest income residents of their type in that district). The reason is that, in general, these households have di§erent incomes in equilibrium. Under some circumstances, such di§erences will imperfectly internalise the location externalities emerging from the peer e§ect. 7 Furthermore, non-di§erentiated income taxes, combined with head taxes that are levied on (or with lump-sum transfers to) the rich districtís population, can match the outcome attained by head taxes. These results clash with the beneÖt view of local head taxes and imply that income taxes will not necessarily generate welfare losses with respect to them.

The rest of the paper is organised as follows. The next section presents the model. Section 3 derives the housing markets equilibrium condition when households of the same type are income-segregated across districts. The following section obtains the optimal allocation of households to districts. The analysis then turns to the comparison between head and income taxes. Section 5 studies the case where governments can use di§erentiated taxes, whereas section 6 does the same for the case in which taxes are anonymous. Finally, section 7 o§ers some concluding remarks and suggests questions for future research.

2 The model

A metropolitan area is divided into two communities (school districts) with Öxed boundaries, labeled as the urban area and the suburbs, and indexed with . Districts provide tax-funded, tuition-free public education of homogeneous quality to all their school-aged residents 8 . With no loss of generality, for the sake of deÖniteness and following the typical pattern of many European cities, the urban school corresponds to that of higher quality. 9

7 This possibility was conjectured by Goodspeed (1989, footnote 12).
8 Attendance to school is assumed compulsory. The probability of admission to a local school is equal to one if the household resides in the district and equal to zero

A population of households with mass normalised to 1 lives in the city. Every household has a school-aged child. Households di§er continuously according to income and discretely along two additional dimensions: the cost of providing the child with a given level of school quality and the beneÖt she receives from it. 10 To present results in the simplest possible setting, there are two cost-beneÖt types of households. Household types are indexed with measures the proportion of type 1 households in the population. The analysis presented below considers cases where crowding and beneÖt types are correlated as follows: 11

Assumption 1 Type 1 pupils impose smaller congestion costs onto schools and derive greater beneÖts from school quality than type 2 ones.

The interpretation is that type 1 households have higher ability o§springs and that the cost di§erence is the result of di§erential peer e§ects operating at the school level. Peer e§ects are thus modeled a-la-Lazear (2001) as the result of di§erential crowding e§ects: schools with a larger proportion of low cost pupils are able to provide the same level of school quality using less resources (e.g. with larger classes) or greater quality with equal spending per pupil. The technology of production of school quality is identical across districts and linear. It is described by the cost function:

\[C \left(n _ {j} ^ {1}, n _ {j} ^ {2}, e _ {j}\right) = \left(n _ {j} ^ {1} c _ {1} + n _ {j} ^ {2} c _ {2}\right) e _ {j}; \quad c _ {1} < c _ {2}\]

where stands for the mass of households of type i living in district and . Let denote the crowding cost di§erential

Household income is denoted with , and is distributed in the population according to ; 2. Income distribution functions are continuous, strictly increasing in all their support D and have densities 2. The total (and average) income is:

otherwise.
9 For simplicity sake, private schools are excluded from the analysis.
10 Most previous analyses in urban public Önance consider models in which households di§er along a single-dimension (e.g. Epple et al., 1984, 1993; BÈnabou, 1996; Nechyba 1999). There are however exceptions in which households di§er, additionally, along a taste parameter (Epple and Platt, 1998; Kessler and L¸lfessmann, 2005; Schmidheiny, 2006a, 2006b), a productivity one such as the o§spring ability (Epple and Romano, 2003), or both (Wildasin, 1986).
11 It
11 It is important to note that the results on the comparison between head and income taxes do not depend on the direction of correlation between cost and beneÖt types assumed. Only the circumstances under which income taxes may generate smaller welfare losses than head taxes change.

\[Y = \gamma \int_ {y} ^ {\overline {{y}}} y \phi_ {1} (y) d y + (1 - \gamma) \int_ {y} ^ {\overline {{y}}} y \phi_ {2} (y) d y,\tag{1}\]

whilst district jís income distribution functions are given by

The model of the housing market avoids well-known existence problems illustrated, for example, by Rose-Ackerman (1979) or Epple et al. (1984): districts have a Öxed (and identical) supply of homogeneous houses, denoted , and the total supply is equal to the mass of households that resides in the city: Absentee landlords lend these houses out to households in exchange of a rent. To avoid a source of indeterminacy, I normalise the suburbsírent away to zero. 13 Because housing is supplied inelastically, school quality and tax di§erentials will capitalise into housing prices in equilibrium. Equilibrium in the housing markets will therefore entail the existence of a rent premium in the good school district, 14

Preferences are deÖned over a private composite good (the numeraire), x, and the o§springís future human capital (or income), h. 15 The latter, in turn, depends on the quality of education received, e, and the availability of home inputs, y. A twice continuously di§erentiable utility function represents preferences over pairs . Following de Bartolome and Ross (2004), I adopt a quasi-linear speciÖcation of utility:

\[U _ {i} (x, e; y) = x + h _ {i} (e; y); \quad i = 1, 2,\tag{2}\]

where h is monotonically increasing in both arguments and strictly concave in e. Assumption 1 implies:

\[\partial h _ {1} (e, y) / \partial e > \partial h _ {2} (e, y) / \partial e, \forall (e, y).\]

The choice of a quasi-linear utility function not only simpliÖes the analysis but also completely separates e¢ciency and equity considerations: because the marginal utility of private consumption is constant and equal to one (i.e.

12 The assumption of equal district sizes does not a§ect any of the results presented below but avoids the need to determine whether it is optimal to have the larger district with the better school or viceversa (see Calabrese et al., 2010).
13 This implies that the opportunity cost of the residential use of land ñthe value of industrial or agricultural alternative usesñis normalised to zero.
14 The rent premium may be negative in equilibrium which would indicate the capitalisation of higher taxes in the urban area.
15 Because houses are homogeneous they are excluded from the preference relation.

households are risk neutral), the level of aggregate welfare attained by a particular allocation of households to districts is invariant to the distribution of private consumption and taxes in the population. Preferences satisfy:

Assumption 2 Education is a normal good.

Assumption 2 implies a positive income elasticity of demand for school quality, which agrees with the empirical evidence (Ross and Yinger, 1999). This assumption restricts quasi-linear preferences, requiring home and school inputs to be complements in the human capital production function. 16 The next assumption establishes the behaviour of local governments, focusing the analysis on locational optimality.

Assumption 3 Local governments provide the optimal level of school quality given the local population of pupils, and balance their budget.

Local governments fund their spending with head or proportional income taxes (indexed with that may be di§erentiated across types or not. In the latter case, which I refer to as anonymous taxation, a tax-bill function, denoted generically, derives from the corresponding local budget constraint. The tax-bill function must also meet the feasibility constraint that household tax payments must be smaller than household income. In the former case, the di§erentiated head tax bills or income tax rates will be required to fulÖll these two constraints too. The indirect utility function of a household of type i that has income y and lives in district is:

\[v _ {j} ^ {i} \left(e _ {j}, \tau_ {j} ^ {i k} (y), r _ {j}; y\right) = y - \tau_ {j} ^ {i k} (y) - r _ {j} + h _ {i} (e, y)\]

where represents their tax bill under the tax system k. A useful tool in the analysis that follows are the so-called bid-rent functions, which I deÖne next 17 :

DeÖnition 1 The bid-rent function provides the maximum amount of the numeraire a household of income y and type i is willing to pay as rent premium in the urban area, given the pairs of tax bills and school qualities . Bid rent functions are obtained by setting in the indi§erence condition:

\[v _ {u} ^ {i} \left(e _ {u}, \tau_ {u} ^ {i k} (y), r _ {i} ^ {k} (y), y\right) = v _ {s} ^ {i} \left(e _ {s}, \tau_ {s} ^ {i k} (y), r _ {s}, y\right).\tag{3}\]

The model is static. An equilibrium is an allocation of households to districts, a vector of local head tax bills or income tax rates and school qualities, and a value of the urban rent premium satisfying the following conditions:

16 This could be due, for example, to better labour market networking of better-o§ parents.
17 For a review of the literature on urban public Önance that extensively uses bidrent functions see Ross and Yinger (1999).

E1 Rational choices: no household can increase utility by moving into the other school district.

E2 Housing markets clear:

E3 Local governments balance their budget.

E4 School qualities are optimal given the local population of households.

3 The housing market constraint

In the utilitarian normative framework considered (explained in detail in the next section), the assumption that education is a normal good implies that, in an optimal allocation, households of the same type will be segregated across districts according to income, and that higher income ones will be allocated to the better school district. I call this property within-types income segregation.

DeÖnition 2 An allocation of households to districts satisÖes within-types income segregation (WTS) if, for any pair of households of the same type but of di§erent income who reside in di§erent districts, the higher income one lives in the urban area.

In an allocation satisfying the WTS property, households of the same type living in the same district belong to a single income interval and the intervals corresponding to each district do not overlap. The monotonicity of the income distribution functions then implies that, when households of a certain type i are present in the two districts, a unique cut-o§ income, denoted exists such that households of that type with income reside in the urban (suburban) district. A solution to the Social Planner Problem or an equilibrium may however have all households of a given type concentrated in one district. In particular, type 1 households may all concentrate in the urban district, where the good school is located, or type 2 ones may all concentrate in the suburbs. Clearly, the former case may only happen when , that is, if the proportion of type 1 households in the population is no larger than the size of the urban district. In that case, cut-o§ incomes are and , where is deÖned by:

\[1 / 2 - \gamma = (1 - \gamma) [ 1 - \Phi_ {2} (\overline {{y}} _ {2}) ].\]

The latter case may arise if and will have cut-o§ incomes and . Figure 1 represents examples of these three possibilities: panel (a) depicts an allocation with mixed districts, while panels (b) and (c) represent cases where the urban or the suburban district is type-homogenous. It is thus possible to express the mass of households of each type living in each district as a function of the corresponding cut-o§ income: and (where and . Moreover, the urban district housing market constraint (which implies the suburbs housing market constraint) can also be expressed in terms of the cut-o§ incomes:

\[H _ {u} = 1 / 2 = \gamma \left[ 1 - \Phi_ {1} (y _ {1}) \right] + (1 - \gamma) \left[ 1 - \Phi_ {2} (y _ {2}) \right]\tag{4}\]

Lemma 1 Consider allocations satisfying WTS. Cut-o§ incomes are linked through a continuously decreasing function z deÖned on a compact set ， where and , and and implicitly deÖned by: 18

\[1 / 2 - (1 - \gamma) [ 1 - \Phi_ {2} (y _ {2}) ] = \gamma [ 1 - \Phi_ {1} (z (y _ {2})) ]\tag{5}\]

Proof. The proof needs to establish that, for any value of , there exists a unique value of satisfying (4). Suppose then,

\[0 \leq (1 - \gamma) (1 - \Phi_ {2} (y)) \leq 1 / 2; \quad \forall y \in [ \underline {{y}}, \overline {{y}} ]\]

and the LHS of (5) belongs to the interval . Given that 0 and , continuity and strict monotonicity of the income distribution functions and the intermediate value theorem imply that, for any , there exists a unique such that the housing market constraint (4) holds. Suppose instead that then

\[0 \leq (1 - \gamma) [ 1 - \Phi_ {2} (y) ] \leq (1 - \gamma); \quad \forall y \in [ \underline {{y}}, \overline {{y}} ].\]

Hence, by continuity and strict monotonicity of there exists a unique income such that , and another such that . Moreover, given that , again the continuity and strict monotonicity of and and the intermediate value theorem imply that, for any , there exists a unique such that the housing market constraint (4) holds. Furthermore, these two properties guarantee that z is continuous and decreasing. Finally, apply the implicit function theorem to (4) in order to Önd the derivative:

\[\frac {d y _ {1}}{d y _ {2}} = - \frac {(1 - \gamma) \phi_ {2} (y _ {2})}{\gamma \phi_ {1} (z (y _ {2}))} < 0.\tag{6}\]

The analysis focuses on allocations satisfying WTS. These will be characterised simply with the type 2 cut-o§ income . Since the human capital production function displays decreasing marginal returns to school quality, Assumption 3 implies that every allocation of households to districts is linked to a unique pair of school qualities. For allocations satisfying WTS, then, optimal school qualities can be written as a function of . In the case of anonymous taxation, that is also true of the tax variables: given school qualities and the distribution of households across districts, and with some abuse of notation, it is possible to write:

18 For uneven district sizes, the domain of z depends on the size of the urban district relative to the proportion of each type of households in the population.

\[\tau_ {j} ^ {k} (y _ {2}) = \tau_ {j} ^ {k} \left(e _ {j} (y _ {2}), n _ {j} ^ {1} (y _ {2}), n _ {j} ^ {2} (y _ {2}), y _ {2}\right).\]

Therefore, under anonymous taxation, to every allocation satisfying WTS, y2, corresponds a vector of local policy variables:

\[\boldsymbol {\Pi} ^ {k} \left(y _ {2}\right) = \left[ e _ {u} \left(y _ {2}\right) e _ {s} \left(y _ {2}\right) \tau_ {u} ^ {k} \left(y _ {2}\right) \tau_ {s} ^ {k} \left(y _ {2}\right) \right], k = H, I\]

DeÖnition 3 Cut-o§ income bid rent functions, denoted K , provide for any type 2 cut-o§ income y2, the maximum rent premium households of each type with the corresponding cut-o§ income are willing to pay for a house in the urban area, when local policy variables are those in if local goverenments rely on anonymous taxation, or those chosen by the planner otherwise.

For instance, with anonymous taxation, cut-o§ income bid rent functions are implicitly deÖned by setting in the indi§erence conditions:

\[v _ {u} ^ {1} \left(e _ {u}, \tau_ {u} ^ {k}, \rho_ {1} ^ {k} (y _ {2}); z (y _ {2})\right) = v _ {s} ^ {1} \left(e _ {s}, \tau_ {s} ^ {k}, r _ {s}; z (y _ {2})\right)\tag{7}\]

\[v _ {u} ^ {1} \left(e _ {u}, \tau_ {u} ^ {k}, \rho_ {2} ^ {k} (y _ {2}); y _ {2}\right) = v _ {s} ^ {2} \left(e _ {s}, \tau_ {s} ^ {k}, r _ {s}; y _ {2}\right).\tag{8}\]

Importantly, note that because and are continuous, cut-o§ bid-rent functions are continuous too.

4 The optimal allocation

This section characterises the solution to the Social Planner Problem (SPP). I adopt a utilitarian approach and deÖne the Social Welfare Function (SWF) as the unweighted sum of utility in the economy. I consequently speak of optimality instead of e¢ciency. 19 The SWF includes the utility of the absentee landowners, which is assumed linear in the private composite good. The planner is allowed to use head taxes, which may di§er across household types, and can also make transfers to households and landlords, denoted and respectively. 20 The indirect utility function is thus:

19 The results in the paper emerge when it is optimal to have di§erentiated and segregated districts, as it is the case with the unweighted utilitarian SWF. Instead, one could assume a weighted SWF and consider sets of weights for which maximising the SWF requires districts providing di§erent levels of school quality and households segregating in the form described by WTS.

\[V _ {j} ^ {i} \left(e _ {j}, T _ {j} ^ {i}, r _ {j}, y, R _ {i} (y)\right) = y + R _ {i} (y) - T _ {j} ^ {i} - r _ {j} + h _ {i} (e _ {j}, y),\tag{9}\]

while the SWF is:

\[\begin{array}{r l} & {\int_ {\underline {{y}}} ^ {y _ {1}} y + R _ {1} (y) - T _ {s} ^ {1} - r _ {s} + h _ {1} (e _ {s}, y) \gamma \phi_ {1} (y) d y +} \\ & {\int_ {y _ {1}} ^ {\overline {{y}}} y + R _ {1} (y) - T _ {u} ^ {1} - r _ {u} + h _ {1} (e _ {u}, y) \gamma \phi_ {1} (y) d y +} \\ & {\int_ {\underline {{y}}} ^ {y _ {2}} y + R _ {2} (y) - T _ {s} ^ {2} - r _ {s} + h _ {2} (e _ {s}, y) (1 - \gamma) \phi_ {2} (y) d y +} \\ & {\int_ {y _ {2}} ^ {\overline {{y}}} y + R _ {2} (y) - T _ {u} ^ {2} - r _ {u} + h _ {2} (e _ {u}, y) (1 - \gamma) \phi_ {2} (y) d y +} \\ & {+ [ R _ {L} + r _ {s} H _ {s} + r _ {u} H _ {u} ].} \end{array}\tag{10}\]

The SWF is maximised with respect to subject to ten constraints, which include six nonnegativity constraints , the housing market constraint and two local governments budget constraints. 21 The housing market constraint has as its multiplier and, by Lemma 1, can be written as:

\[y _ {1} - z (y _ {2}) = 0.\tag{11}\]

The two local budget constraints have associated multipliers and are given by:

\[e _ {j} \left[ c _ {1} n _ {j} ^ {1} (y _ {1}) + c _ {2} n _ {j} ^ {2} (y _ {2}) \right] - T _ {j} ^ {1} n _ {j} ^ {1} (y _ {1}) + T _ {j} ^ {2} n _ {j} ^ {2} (y _ {2}) = 0; j = u, s.\tag{12}\]

The optimal demographic composition of districts is determined by the FOCs on the cut-o§ incomes . These yield the following Marginal Social Value functions:

\[M S V _ {i} (y) = h _ {i} (e _ {u}, y) - h _ {i} (e _ {s}, y) + c _ {i} [ e _ {s} - e _ {u} ]; i = 1, 2.\]

20 This is only for completeness; quasi-linearity of the utility function implies that such transfers do not a§ect aggregate welfare if this is given by the unweighted SWF considered.
21 The tenth constraint requires the transfersíbudget to balance :
y y γ R1 (y) 1(y)dy + (1 ) R2 (y) 2(y)dy + RL = 0 y y
R.

Marginal Social Value functions provide the marginal impact on social welfare of moving a household of type i with cut-o§ income from the suburbs to the urban area. 22 The following proposition characterises the solution to the SPP.

Proposition 1 A solution to the Social Plannerís Problem with school qualities exhibits WTS. Furthermore:

i) In an interior solution, cut-o§incomes satisfy: the housing market constraint and

ii) In a corner solution, y, y satisfy: , the housing market constraint and

iii) School qualities satisfy the Samuelsonian conditions:

\[\begin{array}{r} c _ {1} n _ {u} ^ {1} (y _ {1} ^ {*}) + c _ {2} n _ {u} ^ {2} (y _ {2} ^ {*}) = \gamma \int_ {y _ {1} ^ {*}} ^ {\overline {{y}}} \frac {\partial h _ {1} (e _ {u} ^ {*} , y)}{\partial e _ {u}} \phi_ {1} (y) d y \\ + (1 - \gamma) \int_ {y _ {2} ^ {*}} ^ {\overline {{y}}} \frac {\partial h _ {2} (e _ {u} ^ {*} , y)}{\partial e _ {u}} \phi_ {2} (y) d y \end{array}\tag{13}\]

\[\begin{array}{c} c _ {1} n _ {s} ^ {1} \left(y _ {1} ^ {*}\right) + c _ {2} n _ {s} ^ {2} \left(y _ {2} ^ {*}\right) = \gamma \int_ {\underline {{y}}} ^ {y _ {1} ^ {*}} \frac {\partial h _ {1} \left(e _ {s} ^ {*} , y\right)}{\partial e _ {s}} \phi_ {1} (y) d y \\ + (1 - \gamma) \int_ {\underline {{y}}} ^ {y _ {2} ^ {*}} \frac {\partial h _ {2} \left(e _ {s} ^ {*} , y\right)}{\partial e _ {s}} \phi_ {2} (y) d y. \end{array}\tag{14}\]

Proof. Deriving the MSV functions with respect to , it is straightforward to check that the normality of education implies that MSV functions are increasing in income, which entails that the solution satisÖes WTS. The FOCs on are:

\[- n _ {j} ^ {i} \left(y _ {i} ^ {*}\right) + \lambda_ {j} ^ {*} n _ {j} ^ {i} \left(y _ {i} ^ {*}\right) = 0; j = u, s, i = 1, 2,\tag{15}\]

which yield . Using that initial result and equation (6), the ones corresponding to the cut-o§ incomes simplify to:

\[\begin{array}{c} \left[ h _ {1} \left(e _ {u} ^ {*}, y _ {1} ^ {*}\right) - h _ {1} \left(e _ {s} ^ {*}, y _ {1} ^ {*}\right) - c _ {1} \left(e _ {u} ^ {*} - e _ {s} ^ {*}\right) \right] \gamma \phi_ {1} \left(y _ {1} ^ {*}\right) = - \lambda_ {h} ^ {*} \\ \left[ h _ {2} \left(e _ {u} ^ {*}, y _ {2} ^ {*}\right) - h _ {2} \left(e _ {s} ^ {*}, y _ {2} ^ {*}\right) - c _ {2} \left(e _ {u} ^ {*} - e _ {s} ^ {*}\right) \right] (1 - \gamma) \phi_ {2} \left(y _ {2} ^ {*}\right) = - \frac {\lambda_ {h} ^ {*} (1 - \gamma) \phi_ {2} \left(y _ {2} ^ {*}\right)}{\gamma \phi_ {1} \left(y _ {1} ^ {*}\right)} \end{array}\]

which together imply that must hold in an interior solution. Next, note that, since , Assumption 1 implies

22 Following Calabrese et al. (2010), I express MSV functions as the social value of moving a household from the suburbs to the urban area (i.e. the value of decreasing the cut-o§ income of a particular type) to facilitate comparability with the bid-rent functions and only for expositional purposes.

, where recall satisÖes . Since the MSV functions are rise e e eing in income, then, the optimal cut-o§ incomes must satisfy . If there is no such that e, a corner solution eemerges. In that case, since and functions are continuous, eand thus it is optimal to have the urban district populated exclusively by type 1 households if , or the suburbs exclusively by type 2 households if . Finally, the FOCs corresponding to the school quality variables and yield the usual Samuelsonian conditions (13) and (14).

An interior solution has households of the two types allocated to both districts, as in panel (a) of Figure 1; panels (b), where , and (c), where of the same Ögure correspond to corner solutions. The equality of the functions in an interior optimum implies the following optimality condition:

\[\left[ h _ {1} \left(e _ {u} ^ {*}, y _ {1} ^ {*}\right) - h _ {1} \left(e _ {s} ^ {*}, y _ {1} ^ {*}\right) \right] - \left[ h _ {2} \left(e _ {u} ^ {*}, y _ {2} ^ {*}\right) - h _ {2} \left(e _ {s} ^ {*}, y _ {2} ^ {*}\right) \right] = \Delta_ {c} \left[ e _ {s} ^ {*} - e _ {u} ^ {*} \right],\tag{16}\]

On the other hand, the Samuelsonian conditions that determine the optimal school qualities equate districtsí average marginal rates of substitution between school quality and numeraire consumption to their respective marginal cost of school quality

Remark 1 Income mixing is optimal. Households of type 1 with incomes between y and y are allocated to the urban area, while households of type 2 and identical income are assigned to the suburbs in the optimal allocation. The explanation is doublefold as type 1 pupils impose smaller congestion costs onto and derive greater beneÖts from the better quality school. 23

5 Decentralising the optimal allocation

In order to decentralise the optimal allocation, the planner must choose local tax variables that satisfy the local governmentsí budget constraints and ensure that, in the emerging equilibrium, households derive (weakly) higher utility in their socially optimal location. Bearing in mind that the optimal allocation satisÖes the WTS property, optimal tax combinations will need to make households with cut-o§ incomes indi§erent between the two residential alternatives. That requirement will generate a location-incentives constraint.

@h1 (e; y) =@e < @h2 (e; y) =@e
y1 < y2
23 Suppose assumption 1 does not hold but that, instead, preferences satisfy . In that case, though some income mixing will be optimal (except in special circumstances) which of the cut-o§ incomes should be smaller is ambiguous. The reason is that, while the lower costs of educating type 1 children tends to make optimal that y1 < y2, the greater beneÖt type 2 children derive from school quality has the opposite e§ect.

Two-single crossing conditions will then guarantee that the remaining households strictly prefer their socially optimal location. The analysis in this section restricts attention to interior solutions to the SPP for, as proposition 5 below demonstrates, corner solutions will be sustained as an equilibrium with anonymous head taxes.

5.1 Di§erentiated head taxes

With di§erentiated head taxes, the local budget constraints are:

\[E _ {j} = T _ {j} ^ {1} n _ {j} ^ {1} + T _ {j} ^ {2} n _ {j} ^ {2}, j = u, s,\tag{17}\]

where is districtís j total spending, whilst the indirect utility functions are:

\[v _ {j} ^ {i} (e _ {j}, T _ {j} ^ {i}, r _ {j}, y) = y - T _ {j} ^ {i} - r _ {j} + h _ {i} (e _ {j}, y)\tag{18}\]

Substituting (18) into the indi§erence condition (3) and normalising the rent of the suburbs to , one can derive the head-tax bid-rent functions (denoted

\[r _ {i} ^ {H} \left(y, e _ {u}, e _ {s}, T _ {s} ^ {i}, T _ {u} ^ {i}\right) = h _ {i} \left(e _ {u}, y\right) - h _ {i} \left(e _ {s}, y\right) + T _ {s} ^ {i} - T _ {u} ^ {i}, i = 1, 2.\tag{19}\]

Lemma 2 Suppose that , then the head-tax bid-rent functions are increasing in income, which implies that households induced preferences satisfy the following single-crossing conditions:

\[\begin{array}{r l} & v _ {i} ^ {s} (e _ {s}, T _ {s} ^ {i}, y _ {i}) = v _ {i} ^ {u} (e _ {u}, T _ {u} ^ {i}, r _ {u}; y _ {i}) \Rightarrow \\ & v _ {i} ^ {s} (e _ {s}, T _ {s} ^ {i}, y) < v _ {i} ^ {u} (e _ {u}, T _ {u} ^ {i}, r _ {u}; y); \forall y > y _ {i} \\ & v _ {i} ^ {s} (e _ {s}, T _ {s} ^ {i}, y) > v _ {i} ^ {u} (e _ {u}, T _ {u} ^ {i}, r _ {u}; y); \forall y < y _ {i}, i = 1, 2. \end{array}\tag{20}\]

Because a householdís tax burden does not depend on income, the normality of education ensures that head-tax bid-rent functions are rising in income. That property, in turn, implies that if households of type i and income are indi§erent between the two districts, then households of the same type and higher (lower) income strictly prefer the urban area (the suburbs).

Next, in order to derive the location-incentives constraint, let

\[\Delta_ {1} ^ {h} (y _ {2}) = h _ {1} \left(e _ {u} (y _ {2}), z (y _ {2})\right) - h _ {1} \left(e _ {s} (y _ {2}), z (y _ {2})\right)\]

and

\[\Delta_ {2} ^ {h} (y _ {2}) = h _ {2} (e _ {u} (y _ {2}), y _ {2}) - h _ {2} (e _ {s} (y _ {2}), y _ {2})\]

denote the gap in human capital each typeís cut-o§ households obtain from attending the urban school instead of the suburban one, when school qualities are determined optimally. The cut-o§ income bid-rent functions, derived from equations (7) and (8), can then be written as:

\[\rho_ {1} ^ {H} (y _ {2}) = \Delta_ {1} ^ {h} (y _ {2}) + T _ {s} ^ {1} - T _ {u} ^ {1}\tag{21}\]

\[\rho_ {2} ^ {H} (y _ {2}) = \Delta_ {2} ^ {h} (y _ {2}) + T _ {s} ^ {2} - T _ {u} ^ {2}\tag{22}\]

If an allocation is to be sustained as an interior head-tax equielibrium, households with cut-o§ income must be indi§erent between the two districts. That condition requires the equilibrium urban rent premium to satisfy , implying:

\[\Delta_ {1} ^ {h} \left(y _ {2} ^ {H}\right) - \Delta_ {2} ^ {h} \left(y _ {2} ^ {H}\right) = \left(T _ {s} ^ {1} - T _ {u} ^ {1}\right) - \left(T _ {s} ^ {2} - T _ {u} ^ {2}\right)\tag{23}\]

The location-incentives constraint is obtained by setting in the headtax equilibrium condition (23) and subtracting it from the optimality condition (16), which yields:

\[\Delta_ {c} \left[ e _ {s} \left(y _ {2} ^ {*}\right) - e _ {u} \left(y _ {2} ^ {*}\right) \right] = \left(T _ {u} ^ {1} - T _ {s} ^ {1}\right) - \left(T _ {u} ^ {2} - T _ {s} ^ {2}\right).\tag{24}\]

Combinations of head tax bills that fulÖll the previous equation ensure that, when school qualities are given by , the maximum bids households of each type with the optimal cut-o§ income are willing to o§er for a house in the urban area coincide; in other words, that there exists a level of the rent premium that makes them simultaneously indi§erent between the two districts.

DeÖnition 4 Let be the set of all combinations of di§erentiated head taxes satisfying the location-incentives constraint (24) and the local budget constraints (17) at the optimal allocation.

Clearly, the combination of head taxes covering the marginal cost of admitting a household of a given type in a particular district, , belongs to the set . In that case:

\[\left(T _ {s} ^ {i} - T _ {u} ^ {i}\right) = c _ {i} \left[ e _ {s} \left(y _ {2} ^ {*}\right) - e _ {u} \left(y _ {2} ^ {*}\right) \right],\]

and it is straightforward to check that the budget and the location-incentives constraints are satisÖed. Interestingly, inÖnitely many other combinations of di§erentiated head taxes lead to the optimal equilibrium as well.

Proposition 2 Consider an interior solution to the Social Planner Problem with optimal cut-o§ incomes , satisfying . For every combination of head-tax bills in there exists an optimal head-tax equilibrium exhibiting WTS with cut-o§ incomes and and rent premium . There are inÖnitely many such combinations.

Proof. First, notice that the elements of satisfy a system of three linearly independent equations in four unknowns, so that the system has one degree of freedom and inÖnitely many solutions. Proving existence requires checking that the four equilibrium conditions E1-E4 hold: E3 is satisÖed by the deÖnition of , E4 is fulÖlled by assumption 3, while proposition 1 implies that the housing market constraint E2 is satisÖed for . The rational choices condition E1, in turn, requires, Örst, the single-crossing conditions (20) to hold and, second, the cut-o§ bid-rent functions to be equal to each other and to the equilibrium housing rent premium at y: i.e. . The former was proved in Lemma 2, whereas the latter is again ensured by the deÖnition of , as its elements satisfy the location-incentives constraint (24).

Remark 2 In a model with housing markets, many combinations of di§erentiated head taxes can attain optimality. These need neither cover the marginal cost of entry of a household nor be greater for households with lower ability children. Instead, they must provide households with the correct relative location incentives. In other words, they must generate di§erences in the cost of entry each type of household faces to live in the rich district that internalise the externalities emerging from the peer e§ect. The result opens the door for di§erentiated head taxes to e§ect some redistribution across household types.

5.2 Income taxes

Under di§erentiated income taxation, the local budget constraint of district is:

\[e _ {j} \left(n _ {j} ^ {1} c _ {1} + n _ {j} ^ {2} c _ {2}\right) = t _ {j} ^ {1} n _ {j} ^ {1} Y _ {j} ^ {1} + t _ {j} ^ {2} n _ {j} ^ {2} Y _ {j} ^ {2}, j = u, s.\tag{25}\]

where stands for the local income tax rate district imposes on households of type i. Indirect utility functions are:

\[v _ {j} ^ {i} \left(e _ {j}, t _ {j} ^ {i}, r _ {j}; y\right) = y \left(1 - t _ {j} ^ {i}\right) - r _ {j} + h _ {i} \left(e _ {j}, y\right),\tag{26}\]

while the income-tax bid rent functions, obtained as before, are:

\[r _ {i} ^ {I} \left(y, e _ {u}, e _ {s}, t _ {s} ^ {i}, t _ {u} ^ {i}\right) = h _ {i} \left(e _ {u}, y\right) - h _ {i} \left(e _ {s}, y\right) + y \left(t _ {s} ^ {i} - t _ {u} ^ {i}\right); i = 1, 2.\tag{27}\]

There are two district-level variables whose impact on utility varies with income: school quality and the income tax rate. The former makes richer households willing to pay more than lower income ones for a house in the district o§ering higher quality of education. The latter makes them willing to pay more for a house located where their group income tax rate is lower. Therefore, the single-crossing conditions will be satisÖed if the richer district is able to fund its education spending with lower income tax rates than the poor district. More generally:

Lemma 3 Suppose . The income-tax bid-rent functions, , are increasing in income if and only if

\[\left[ \frac {\partial h _ {i} (e _ {u} , y)}{\partial y} - \frac {\partial h _ {i} (e _ {s} , y)}{\partial y} \right] > \left(t _ {u} ^ {i} - t _ {s} ^ {i}\right) \quad \forall y \in S, i = 1, 2.\tag{28}\]

If inequality (28) holds, households induced preferences satisfy:

\[\begin{array}{r l} & v _ {i} ^ {s} (e _ {s}, t _ {s} ^ {i}, y _ {i}) = v _ {i} ^ {u} (e _ {u}, t _ {u} ^ {i}, r _ {u}; y _ {i}) \Rightarrow \\ & v _ {i} ^ {s} (e _ {s}, t _ {s} ^ {i}, y) < v _ {i} ^ {u} (e _ {u}, t _ {u} ^ {i}, r _ {u}; y); \forall y > y _ {i} \\ & v _ {i} ^ {s} (e _ {s}, t _ {s} ^ {i}, y) > v _ {i} ^ {u} (e _ {u}, t _ {u} ^ {i}, r _ {u}; y); \forall y < y _ {i}, i = 1, 2. \end{array}\tag{29}\]

A su¢cient condition for these single-crossing conditions to hold is thus . If instead , they require the income elasticity of the demand for school quality to be large enough relative to the tax rate di§erential.

The cut-o§ bid-rent functions (denoted are again deduced from equations (7) and (8), yielding:

\[\rho_ {1} ^ {I} (y _ {2}) = \Delta_ {1} ^ {h} (y _ {2}) + z (y _ {2}) [ t _ {s} ^ {1} - t _ {u} ^ {1} ]\tag{30}\]

\[\rho_ {2} ^ {I} (y _ {2}) = \Delta_ {2} ^ {h} (y _ {2}) + y _ {2} [ t _ {s} ^ {2} - t _ {u} ^ {2} ]\tag{31}\]

If an allocation can be sustained as an interior income-tax equiliberium, the rent premium must satisfy , which requires:

\[\Delta_ {1} ^ {h} \left(y _ {2} ^ {I}\right) - \Delta_ {2} ^ {h} \left(y _ {2} ^ {I}\right) = y _ {2} ^ {I} \left[ t _ {s} ^ {2} - t _ {u} ^ {2} \right] - z \left(y _ {2} ^ {I}\right) \left[ t _ {s} ^ {1} - t _ {u} ^ {1} \right].\tag{32}\]

Therefore, the location-incentives constraint, obtained as before, is:

\[\Delta_ {c} \left[ e _ {s} \left(y _ {2} ^ {*}\right) - e _ {u} \left(y _ {2} ^ {*}\right) \right] = z \left(y _ {2} ^ {*}\right) \left[ t _ {u} ^ {1} - t _ {s} ^ {1} \right] - y _ {2} ^ {*} \left[ t _ {u} ^ {2} - t _ {s} ^ {2} \right].\tag{33}\]

Combinations of income tax rates that satisfy this equation make households of each type with the optimal cut-o§ income indi§erent between the two districts at the equilibrium rent premium when school qualities are given by

DeÖnition 5 Let be the set of feasible income tax rate combinations satisfying the location-incentives constraint (33) and the local budget constraints (25) at .

Lemma 4 is non-empty if and only if:

\[\Delta_ {c} \left[ e _ {u} \left(y _ {2} ^ {*}\right) - e _ {s} \left(y _ {2} ^ {*}\right) \right] \leq y _ {2} ^ {*} \bar {t} _ {u} ^ {2} + y _ {1} \bar {t} _ {s} ^ {1}\tag{34}\]

Proof. It is straightforward to check that (33) and the two local budget constraints (25) conform a system of three linearly independent equations in four unknowns which, therefore, has inÖnitely many solutions. That set must be restricted by eliminating the combinations of taxes that are not feasible. Let denote group tax base in district . If feasibility and the budget constraints require: , with if then and where Also, let and denote the local government budget constraints and substitute them into the location-incentives constraint:

\[y _ {1} ^ {*} t _ {u} ^ {1} - y _ {2} ^ {*} f \left(t _ {u} ^ {1}, y _ {2} ^ {*}\right) = \Delta_ {c} \left[ e _ {s} ^ {*} - e _ {u} ^ {*} \right] + y _ {1} ^ {*} t _ {s} ^ {1} - y _ {2} ^ {*} g \left(t _ {s} ^ {1}, y _ {2} ^ {*}\right).\tag{35}\]

Express each side of the equation as a function of . These functions are strictly increasing in and , respectively, as and . They reach a minimum at where and , and a maximum at , where and . Consequently, letting be the set of tax rates imposed by district on type 1 households, , that fulsill equation (35), i.e. such that , for any element of there exists a unique element in such that (35) holds. Finally, note that the sets , and thereby also , are empty sets if the images of and do not overlap. Otherwise, that is if (34) is satisÖed, and are non-empty.

Lemma 4 does not impose a restrictive condition: the RHS of (35) is the sum of the taxes a type 1 cut-o§ household would pay in district s if type 2 residents did not pay any and those a type 2 cut-o§ household would pay in district u if their type 1 neighbours did not pay any. That sum must be greater than the additional cost of educating a type 2 household (instead of a type 1) in the urban area (rather than in the suburbs). Nevertheless, the elements of may not sustain an optimal equilibrium.

Proposition 3 Consider an interior solution to the Social Planner Problem with optimal cut-o§ incomes and satisfying . A combination of local income tax rates in sustains the optimal allocation as an equilibrium with rent premium and cut-o§ incomes 2 if and only if the two pairs of tax rates ; satisfy Lemma 3.

Proof. The proof is analogous to the one of proposition 2 and is omitted for the sake of brevity.

Remark 3 The existence problem emerges because may be an empty set and, even if it is not, it may be that no element satisÖes the necessary single-crossing conditions. 24 Therefore, the optimal allocation may not be sustainable as a market equilibrium with di§erentiated income taxes.

Notwithstanding, the next proposition proves that it is possible to combine anonymous income taxes with another Öscal tool to correct for their distortionary location e§ects. If the anonymous income tax rates that balance the local goverenmentsíbudgets satisfy the single-crossing conditions, then the location externalities can be internalised with a self-funded lump-sum transfer scheme among households of di§erent types that live in the urban district. If they do not satisfy them, it must be the case that . Then, the proposal involves applying the suburbsí tax-rate to the urban area and imposing di§erentiated head taxes to correct for the location externalities and to fund the resulting deÖcit, 25

Proposition 4 Consider a solution to the Social Planner Problem with optimal cut-o§ incomes and satisfying . Suppose that local governments use anonymous income taxes to fund education.

1) If the vector of local policies I and household preferences satisfy lemma 3, then, the unique self-funded lump-sum transfers scheme from type 2 to type 1 urban residents satisfying

\[L _ {1} n _ {u} ^ {1} - L _ {2} n _ {u} ^ {2} = 0\tag{36}\]

\[- L _ {1} - L _ {2} = \Delta_ {c} \left[ e _ {s} ^ {*} - e _ {u} ^ {*} \right] - \left(y _ {1} ^ {*} - y _ {2} ^ {*}\right) \left(t _ {u} - t _ {s}\right)\tag{37}\]

sustains the optimal allocation as an income tax equilibrium.

2) In other case, setting the urban area tax rate at , the unique

24 Wildasin (1986) showed that a set of personalised income tax rates adjusted so that the tax payment of every household is equal to its marginal congestion cost in every district would achieve the normative objective. That solution, which is valid when all types are present in all districts in the optimal outcome, does not necessarily extend to the current setting where only two indi§erent types will exist in equilibrium. In general, the proposed taxes must also provide the optimal location incentives for types that concentrate in a subset of locations. This further restricts the set of income taxes that implement the optimal solution. In the current context in particular, the proposed taxes would also need to satisfy the relevant singlecrossing conditions.

25 Correcting for location externalities could be achieved in other ways as well. For example, with a transfer between households of a given type living in di§erent districts.

di§erentiated head-tax scheme imposed on urban residents satisfying

\[\hat {T} _ {1} n _ {u} ^ {1} + \hat {T} _ {2} n _ {u} ^ {2} = D _ {u}\tag{38}\]

\[\hat {T} _ {1} - \hat {T} _ {2} = \Delta_ {c} [ e _ {s} ^ {*} - e _ {u} ^ {*} ]\tag{39}\]

sustains the optimal allocation as an income tax equilibrium.

Proof. Clearly, both systems of equations have a unique solution. Equations (36) and (38) guarantee, in each case, that the scheme is either self-funded or covers the budget deÖcit arising in the urban district . Hence, both proposals ensure that the two local budget constraints are satisÖed. Under anonymous income taxation, the location incentives constraint (33) reduces to

\[\Delta_ {c} \left[ e _ {s} ^ {*} - e _ {u} ^ {*} \right] = \left(y _ {1} ^ {*} - y _ {2} ^ {*}\right) \left(t _ {u} \left(y _ {2} ^ {*}\right) - t _ {s} \left(y _ {2} ^ {*}\right)\right)\]

Therefore, without the proposed schemes, the optimal allocation cannot be sustained as an equilibrium. 1) In this case, the proposed scheme reduces the willingness to pay for a house in the urban area of every type 2 household by an amount equal to and increases that of type 1 households by . Equation (37) requires the sum of both to cover the di§erence between cut-o§ householdsíffoptimal" relative willingness to pay and the one induced by income taxes . Because lemma 3 holds by assumption, the scheme is thus able to sustain the optimal allocation as an equilibrium. 2) Here, , so that lemma 3 is satisÖed. Moreover, income taxes do not a§ect the location incentives of households: . Therefore, the proposed scheme needs to increase type 1 householdsíwillingness to pay for living in the urban area with respect to that of type 2 ones by , which is precisely what equation (39) imposes.

Remark 4 This result demonstrates that, while single-crossing conditions mark the limits to the compatibility between optimality, on one side, and income taxes and the implied redistribution, on the other, these are not incompatible with each other. As a matter of fact, the second proposal sets a lower bound for the amount of tax redistribution e§ected in the rich district, as the singlecrossing conditions could be satisÖed for some

The lump-sum transfers implied by the Örst proposal could be from type 2 to type 1 households or viceversa. The reason is that the latter have lower income so that the tax-price of entry to the urban area induced by anonymous income taxes may be too low for them relative to that required from type 2 cut-o§ ones. On the contrary, the di§erentiated head taxes that complement the uniform income tax scheme in the second proposal need to be greater for type 2 households.

6 The ambiguous comparison between anonymous head and income taxes

Suppose now that local governments observe the marginal cost of providing an additional unit of school quality to the district but cannot identify individual marginal congestion costs or cannot use that information to tax-discriminate across household types. This section proves that anonymous head and income taxes cannot be unambiguously ranked according to the distortions they generate and that income taxes can easily match the outcome of head taxes.

6.1 Head taxes

In this case, the budget constraints, indirect utility, bid-rent and cut-o§ bidrent functions are obtained by setting in (17), (18), (19), and in (21) and (22). The next proposition reveals that an interior head tax equilibrium is necessarily suboptimal, while a corner one is optimal.

Proposition 5 1) A head-tax equilibrium exists.

2) If the SPP has an interior solution, then there exists an interior head-tax equilibrium. Every interior head-tax equilibrium is suboptimal.

3) If the SPP has a corner solution and the sign of is positive (negative), then there exists a corner (interior) head-tax equilibrium. Every corner head-tax equilibrium is optimal.

Proof. 1) If , then, by lemma 2, the single-crossing conditions ensure the WTS property holds in that interval. Assumption 1 implies that . Then, if , the intermediate e evalue theorem and the continuity of the cut-o§ bid-rent functions ensure that there is a level of income such that eHence, such allocation satisÖes the rationality condition of equilibrium E1 for . Because cut-o§ bid-rent functions embed the equilibrium conditions E2 to E4, and the associated vector of local policies are an interior head-tax equilibrium with rent premium . If instead , the corner allocation of households to districts also satisÖes E1. In the case where , cut-o§ incomes satisfy and and, for the same reasons aforementioned, for every there exists a corner equilibrium with all type 2 households living in the suburbs. If , then and there is a corner equilibrium with every type 1 household living in the urban area and rent premium . Finally, note that if there exists such that , it is straighforward to check that such allocation of households to districts, y, the policy vector and the rent premium leave every household indi§erent between the two districts and, hence, constitute an equilibrium.

2) By proposition 1, in an interior solution to the SPP:

\[\Delta_ {1} ^ {h} (y _ {2} ^ {*}) - \Delta_ {2} ^ {h} (y _ {2} ^ {*}) = - \Delta_ {c} (e _ {u} ^ {*} - e _ {s} ^ {*}).\tag{40}\]

In an interior head-tax equilibrium, in turn, implies:

\[\Delta_ {1} ^ {h} \left(y _ {2} ^ {H}\right) - \Delta_ {2} ^ {h} \left(y _ {2} ^ {H}\right) = 0.\tag{41}\]

Because the RHS of (40) is negative: and . The latter implies that any interior head-tax equilibrium is suboptimal (i.e. that . The former, along with the fact that e, the continuity of the cut-o§ bid rent functions and the intermediate evalue theorem, entail the existence of a level of income such that efor which an interior head-tax equilibrium exists.

3) In a corner solution to the SPP:

\[\Delta_ {1} ^ {h} \left(\overline {{y}} _ {2}\right) - \Delta_ {2} ^ {h} \left(\overline {{y}} _ {2}\right) \geq - \Delta_ {c} \left(e _ {u} ^ {*} - e _ {s} ^ {*}\right).\tag{42}\]

Because the RHS of (??) is negative, its LHS may be negative or positive. In the former case, and, as proved above, there is a level of income such that for which an interior ehead-tax equilibrium exists. In the latter case, note that

\[\rho_ {1} ^ {H} (\overline {{y}} _ {2}) - \rho_ {2} ^ {H} (\overline {{y}} _ {2}) > 0 \Leftrightarrow \Delta_ {1} ^ {h} (\overline {{y}} _ {2}) - \Delta_ {2} ^ {h} (\overline {{y}} _ {2}) > 0\]

so that existence of a corner head-tax equilibrium implies that (??) holds and so the existence of a corner solution to the SPP, and viceversa.

Remark 5 Interior equilibria emerging with anonymous head taxes generate a suboptimal distribution of households across districts. Residential choices generate negative externalities because the homogenous tax-bill levied on the urban residents does not cover the marginal costs of admitting them into the district, being too low for high-cost (type 2) households and too high for low-cost (type 1) ones. Hence, too many high-cost households live in the good (urban) school district in equilibrium.

This subsection compares the distortions emerging in market equilibrium when local governments use non-di§erentiated head and income taxes. In the latter case, the local budget constraints and the indirect utility, bid rent and cut-o§ bid-rent functions are obtained by setting in (25), (26), (27), and in (30) and (31).

As in the case with di§erentiated taxes, the possibility that the single-crossing conditions might not be satisÖed implies that existence of an income-tax equilibrium satisfying WTS is not guaranteed. Following a similar argument as in the proof of the previous proposition, it can be shown that, if , then either there is a level of income for which either , or e. In both cases, equilibrium requirements E2-E4 hold but the implied allocation will only be an equilibrium if the singlecrossing conditions are also met. If an interior equilibrium exists, its cut-o§ incomes are derived from the equation of the two typesícut-o§ bid-rent functions, which yields:

\[\Delta_ {1} ^ {h} \left(y _ {2} ^ {I}\right) - \Delta_ {2} ^ {h} \left(y _ {2} ^ {I}\right) = \left(z \left(y _ {2} ^ {I}\right) - y _ {2} ^ {I}\right) \left(t _ {u} \left(y _ {2} ^ {I}\right) - t _ {s} \left(y _ {2} ^ {I}\right)\right).\tag{43}\]

The comparison between (43) and the optimality requirement (16) conÖrms that anonymous income taxes lead to a suboptimal allocation of households across districts. The next result shows that head taxes may induce greater welfare losses than income taxes. Figure 2 illustrates this possibility.

Proposition 6 Suppose an interior income-tax equilibrium exists with cut-o§ incomes and tax rates . Then, another head-tax equilibrium inducing larger locational distortions exists.

Proof. In an interior income-tax equilibrium, cut-o§ incomes satisfy (43). In turn, in an interior head-tax equilibrium, they fulÖll Because by assumption, the RHS of (43) is negative, implying . Using again the fact that , continuity of e ethe cut-o§ bid rent functions and the intermediate value theorem imply the existence of a level of income such that and for ewhich a head-tax equilibrium exists. Therefore,

Furthermore, anonymous income taxes can easily match the outcome achieved with anonymous head taxes.

Proposition 7 The results in proposition 5 apply to an income tax scheme that sets equal to and funds (gives back) the resulting urban budget deÖcit (surplus) through a uniform head tax levied on (with a uniform transfer paid back to) urban residents.

Proof. Because , the single-crossing conditions are satisÖed and the RHS of (43) is equal to zero so that:

\[\rho_ {1} ^ {I} (y) - \rho_ {2} ^ {I} (y) = \rho_ {1} ^ {H} (y) - \rho_ {2} ^ {H} (y).\]

■

Remark 6 Results in this section clash with the view of local head taxes as welfare-enhancing beneÖt taxes. When governments cannot observe the costparameters or use that information to tax-discriminate across households of di§erent types, head taxes may be more distortionary than an ability-to-pay tax such as the proportional income tax. The reason is that, if an income tax equilibrium has higher tax rates in the urban area the cut-o§ households of the high-cost type face a greater tax-price of entry into the urban area than the lower income cut-o§ households of the low-cost type. Thereby, the negative cost-externalities the former impose on the rest are (partially) internalised. 26

Remark 7 Moreover, income taxation can be combined with a uniform head tax levied on urban residents, or with a lump-sum transfer to them to match the outcome of head taxes. Therefore, the implied income redistribution does not generate more welfare losses than those resulting from the use of anonymous head taxes.

7 Concluding remarks

The analysis in this paper o§ers new insights into the relative normative merits of local head and income taxes. The main novel results reveal that head taxes are not superior to income taxes and that the indirect redistribution implied by income taxation is not necessarily at odds with location optimality or associated to welfare losses. In cases where local governments can observe the cost parameters of di§erent household types and tax-discriminate across them, both head and income taxes are able to sustain the optimal allocation in equilibrium. Remarkably, optimal di§erentiated head taxes need not cover marginal congestion costs, as the outcome of the location game depends on the relative willingness to pay for entering the good district. Because optimality requires the income segregation of households of the same type in the utilitarian normative framework considered, the necessary single-cossing conditions for a segregated equilibrium to exist mark the limits of the compatibility between income taxes and location optimality but do not rule it out. At most, it may be necessary to set anonymous and identical tax rates in the two districts and cover the budget deÖcit that results in the rich district by imposing di§erentiated head taxes on its residents. When di§erentiated taxes are not available, the two tax systems cannot be unambiguously ranked according to the welfare losses they generate as compared to the optimal outcome. Moreover, supplemented with head taxes levied on, or lump-sum transfers to, the residents of the rich district, income taxes could always match the outcome attained with head taxes.

26 It is important to stress that the essence of this result does not depend on the assumptions made over household types. In particular, an analogous result can be derived if households imposing smaller costs onto schools also derive smaller beneÖts from school quality.

It is important to check the robustness of these conclusions to alternative speciÖcations of the peer e§ect and of housing markets, that is to say, to relax the assumptions of linear crowding costs and of inelastic housing supplies. Two important questions for further research emerge. The Örst one concerns the comparison of income taxes to property taxes with and without zoning regulations, extending Calabrese et al. (2007) to include peer e§ects and income taxes. The second concerns the relative performance of alternative tax systems when local tax and spending policies are selected through an electoral process. It is worth mentioning to conclude that these results suggest as well that, in the presence of peer e§ects, the public sector could use exams and condition access to schools on the results to derive welfare gains.

Acknowledgements

I am very grateful for many useful comments to Subir Bose, Phillipe De Donder, Gianni De Fraja, ÕÒigo Iturbe, Miltos Makris, Clara PonsatÌ, Javier Rivas, to seminar attendants at the Institut díEconomia de Barcelona (IEB) and Universitat Rovira i Virgili and to the Department of Economics at Universitat Autonoma de Barcelona (UAB) where part of this research was completed. All remaining errors are my own.

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