Locational Stratification
Shin-Kun Peng, Ping Wang and Chia-Ming Yu∗
Our paper contributes to the literature on economic segregation by showing that heterogeneity in preferences for housing quality as a re-sult of the location-specific provision of public goods can generate locational stratification. We develop a general equilibrium frame-work of endogenous sorting in which agents are allowed to differ in their incomes, opportunity costs of commuting, and preferences for housing quality. Housing quality is endogenously determined by location-specific public infrastructure that is financed by property taxes. We characterize the configuration of the competitive spatial equilibrium. We find that complete integration arises only under a set of parameters of measure zero such that the ratio of the opportu-nity costs of commuting facing different types of agents is in a specific homothetic relation to the ratio of the marginal valuations of hous-ing quality. By contrast, locational stratification is generic. When the difference between agents’ commuting costs is sufficiently large (resp. small) compared to the difference in their respective preferences for housing quality, the equilibrium is segregated with all rich residing in the central city (resp. suburb) as observed in Asia and Europe (resp. U.S. metropolitan areas).
Keywords:economic segregation, endogenous sorting, housing quality, competitive spatial equilibrium
JEL classification:D50, H41, R20
∗Institute of Economics Academia Sinica and Department of Economics National
Tai-wan University, Department of Economics Washington University in St. Louis and Depart-ment of Economics National Tsing Hua University. We thank participants at the 2012 International Conference on Trade, Industrial and Regional Economics, the Second Asian Seminar in Regional Science, and the 59th North American Meetings of the Regional Sci-ence Association International for comments.
經濟論文叢刊(Taiwan Economic Review), 41:2 (2013), 195–217。
1 Introduction
Locational stratification has become a major concern of policymakers over the past 5 or 6 decades. In 2000, most of the 30 largest Metropolitan Sta-tistical Areas (MSAs) in the United States were highly segregated (see Peng and Wang (2005) using the dissimilarity index). As claimed in Glazer and Moynihan (1963), “The point about the melting pot...is that it did not happen.” Perlmann (1988) and Steinberg (1989) show that a multicultural society and pluralistic cities in the United States are evidence of the per-sistent existence of social and economic differences among ethnic groups. Jargowsky (1996) finds that in the 1980s there has been a large increase in economic segregation accompanied by a small decline in racial segrega-tion. In this paper, we argue that differences in housing quality as a result of location-specific provision of public goods can be a plausible force behind the locational segregation.
Since the pivotal contribution by Benabou (1996a), economists have highlighted human capital and non-human wealth as the key forces driv-ing locational segregation (e.g., see the income-sortdriv-ing model by de Bar-tolome and Ross (2003) and the skill-sorting model by Hanushek and Yil-maz (2007)). Also related to this literature, accessibility to capital mar-kets matters because of its effect on human capital investment (Benabou, 1996b). Beyond this conventional wisdom, Peng and Wang (2005) argue that preference heterogeneity for location-specific public goods can lead to locational stratification. More recently, Bayer et al. (2004) have empirically documented that preference heterogeneity for housing and neighborhood characteristics is crucial for stratified urban housing markets as observed. In this paper, we fully characterize the channels through which heterogeneous preferences for the housing quality can become a source of economic segre-gation. Our analysis will shed light on possible equilibrium outcomes that correspond to some of the outcomes observed in MSAs in the U.S. Asia, and Europe.
More specifically, we construct a general equilibrium framework of en-dogenous sorting with two locations/regions and two types of agents. One of the regions can be thought of as the central business district where all jobs are located. Agents differ in their incomes, opportunity costs of commut-ing, as well as preferences for housing quality. Each region has a fixed plot of land and each is administered by a local government which sets the location-specific property tax rate. Housing quality is endogenously determined by
location-specific infrastructure established by each local government, and fi-nanced by its property tax revenue. Via ‘sorting by foot’ in the spirit of Tiebout (1956), different types of agent may cluster together and economic stratification may arise as an outcome of the multi-class competitive spatial equilibrium (cf. see the original work by Hartwick et al. (1976) for the con-cept of such an equilibrium). We extend the insight on endogenous sorting to the study of economic segregation with housing quality depending upon the location-specific provision of public goods. In particular, we are able to establish necessary and sufficient conditions for the local economy to be-come segregated – when there are less rich than poor, locational segregation features agglomeration with all rich residing in the same location.
Since the pivotal work by Sweeney (1974), there have been numerous studies on housing quality and urban configuration, using static or dynamic frameworks (e.g., see Arnott et al. (1983); Arnott et al. (1999); Lin et al. (2004); Peng and Wang (2009), to name but a few). Nonetheless, a common feature of these previous studies is that households are homogeneous. As a result, one cannot analyze the relationship between housing quality and the agglomeration of high-income, high-skill labor. The heterogeneity of household plays a key role in our paper, driving the endogenous sorting outcomes in equilibrium.
The main findings of our paper are as follows. We show that the equilib-rium configuration crucially depends on the heterogeneity of preferences for housing quality versus that of the opportunity costs of commuting. When the difference between agents’ commuting costs is sufficiently large com-pared to the difference in their respective preferences for housing quality, the equilibrium is segregated with all rich residing in the central city. By contrast, when the difference in agents’ preferences for housing quality is sufficiently large, the equilibrium is segregated with all rich in the suburb. While the former captures the pattern observed in Asia and Europe, the lat-ter is often seen in the U.S. metropolitan areas. Complete integration arises only under a set of parameters of measure zero when the ratio of commuting costs between the two types is in a specific homothetic relation to the ratio of the marginal valuations of housing quality. Thus, locational stratification is a result of preference heterogeneity and a positive neighborhood external-ity in the form of housing qualexternal-ity financed by local property taxes. Where the rich reside, however, will crucially depend on their opportunity cost of commuting relative to their preferences for housing quality.
the basic structure of the model. Section 3 solves the short-run equilibrium for any fixed distribution of workers’ type. Section 4 solves the equilibria of Pattern I and Pattern II where high-skilled agents agglomerate in the CBD and suburb, respectively. Finally, some concluding remarks are offered in Section 5.
2 The Model
Consider an economy with two regions, j ∈ J ≡ {1, 2}, where region 1 is the CBD and the job place to which all workers commute. Both regions have the same land endowment. There are two types of agents, i ∈ M ≡ {H, L}, with total mass N > 0. Type-H agents (of mass H ) have a higher income and a stronger preference for housing quality than type-L agents (of mass L), where we restrict 0 < H < L and H + L = N. By denoting a type-i agent’s income and preference for housing quality as Yi and γi, respectively, i ∈ M, we thus have YH > YL > 0 and γH > γL > 0. Throughout this paper, the agents’ type is indexed by a superscript and their location is indexed by a subscript. The (endogenous) population for type-i agents living in j is denoted by ij, i ∈ M, j ∈ J . Following Thisse and Wildasin (1992) and Berliant et al. (2006), each agent is assumed to consume one unit of land. In the absence of vacant land, each region is occupied by residents of mass N/2. Therefore, the following population identities must hold true:
Hj+Lj = N
2, ∀j ∈ J. (1)
More specifically, the population identities can be summarized as follows. Population Region 1 Region 2 Total
Type-H H1 H2 H
Type-L L1 L2 L
Total N2 N2 N
Denote Ti > 0as the commuting cost between the CBD and the sub-urb for each type-i agent. Defining indicators I1 = 0 and I2 = 1, the commuting cost for each type-i agent residing in j becomes Ti ·I
j, where i ∈ M and j ∈ J . Since type-H agents may be thought to have a high valuation of commuting time, it is reasonable to assume that their commut-ing is more costly, i.e., TH > TL. Although every agent needs to travel to
the CBD to work, the local public goods are not the privilege of the CBD residents. Residents in each region are levied a property tax with a tax rate τj ≥ 0, j ∈ J . By using property tax revenue, each local government can establish location-specific infrastructure as well as LPGs that in turn improve the quality of houses in each region. In addition to the required sewer, road and utilities, such infrastructure includes ameliorating the living environ-ment, building parks and museums, and hosting festivities for the residents. Upon acquiring one unit of land in his/her region, the type-i agent in region j can enjoy the consumption of a numeraire good (cij) and housing quality (hi
j), i ∈ M, j ∈ J . To capture neighborhood externalities, we follow the Romer (1986) – Lucas (1988) convention, allowing the average consumption of housing quality in j ( ¯hj) to affect the utility facing each agent residing in j , where ¯hj = (HjhHj +LjhLj/N/2), which is higher when there are more type-H agents as neighbors. Specifically, we consider quasi-linear preference as proposed by Bergstrom and Cornes (1983): it is linear in consumption for the numeraire good and housing, and the utility sources from housing are given by γiU (hi
j, ¯hj) = γi(hij) αh¯1−α
j , (1/2) < α < 1, where it is assumed that own housing quality carries more weight than the average housing quality in the region j (i.e., α > (1/2)). The positive externality term ¯hj can be viewed as community housing quality in region j , j ∈ J . While agents can improve the quality of their own homes through individual investment in housing (hi
j), they are small so that they do not consider the influence of individual choices on the community quality. Notably, while Lin et al. (2004) consider a similar type of Romer-Lucas externalities via home production, our paper models such externalities from the preference side.
Denote pj as the price of each unit of housing quality in region j , j ∈ J. Each agent will exhaust her income for the consumption of the numeraire good, tax-included housing expenditure and commuting. The optimization problem for a type-i agent residing in region j can be written as: Vji = max c,h c + γ i U h, ¯hj , s.t. c + pj 1 + τj h + Ti·Ij =Yi. (2) Under quasi-linear preference, we can substitute the budget constraint (2) into the utility function to eliminate c and derive the first-order condition for h, as follows:
hij = " αγi pj 1 + τj #1−α1 ¯ hj, (3)
where there is no income effect on the individual demand for housing qual-ity. We can then plug (3) into (2) to obtain:
cji =Y i −Ti·Ij −pj 1 + τj −1−αα αγi 1 1−α ¯ hj, (4) i ∈ M, j ∈ J. While individual demand for housing quality is increasing in the average housing quality in the region, the individual consumption demand for the numeraire good is decreasing in it. Here, the consumption of the numeraire good absorbs the entirety of the income effect, captured by gross income net of the commuting cost.
3 Market Equilibrium
The aggregate demand for housing quality in region j can then be written as: HjhHj +LjhLj = α pj(1 + τj) 1−α1 jh¯j, (5) where j ≡Hj γH 1−α1 +L j γL 1−α1 , (6) representing the aggregate preference for housing quality in region j , j ∈ J . Most notably, the composition of residents in each region (type-H versus type-L) will affect both the average housing quality and aggregate preference for housing quality. To pin down the equilibrium price pj, we combine (5) and the definition of ¯hj to derive:
pj 1 + τj = α N
2 α−1
1−αj , ∀j ∈ J. (7) The equilibrium average housing quality in each region is determined by residents’ demand for housing quality and the tax revenue they contribute as a source of the regional supply of housing quality. Denoting Gj > 0 as the local public good provided by the local government in j (measured
in the same unit as the composite good), which is financed by property tax revenue, we have:
Gj =τj HjhHj +LjhLj pj, ∀j ∈ J. (8) Each local government’s development technology for housing quality is as-sumed to take a simple form: S(Gj) = g · (Gj)1/2, where g measures technological efficiency for local public good to enhance housing quality. That is, the more local public goods devoted to it, the higher the regional supply of housing quality will be. Since the local public good affects equilib-rium housing quality and hence the equilibequilib-rium level of utility achieved by households, our model resembles the setup by Greenhut and Mai (1980) in which public facility with endogenously determined location is consumed by households residing at the particular location. Market clearing for hous-ing quality requires:
HjhHj +LjhLj =S Gj , ∀j ∈ J, (9) which, together with (8) and the definition of ¯hj, yields,
Gj =τjpjS Gj , (10) ¯ hj =S Gj N 2 −1 . (11)
Substituting (11) and (7) into (3), (4) and (10), we obtain: hij = γi1−α1 −1 j S(Gj), cij =Yi −Ti ·Ij−α γi 1−α1 N 2 α−1 −αj S Gj , Gj =α τj 1 + τj N 2 α−1 1−αj S Gj , (12) where i ∈ M and j ∈ J . From (10) and the housing development technol-ogy, we have: τjpj = Gj S Gj = Gj 12 g , j ∈ J. (13) That is, the tax rate for each unit of housing quality can be written as τjpj from the residents’ viewpoint, or Gj/S(Gj) from the supply-side of local
government. In equilibrium, they should be equalized. (13) together with (7) yield the provision of the local public good Gj as well as the regional supply of housing quality S(Gj)(for j ∈ J ):
Gj = " αg τj 1 + τj N 2 α−1 1−αj #2 , (14) S Gj = αg2 τj 1 + τj N 2 α−1 1−αj . (15) Thus, both the provision of the local public goods and the regional supply of housing quality are increasing in the property tax rate (τj) and the aggregate preference for housing quality in region j (j). Moreover, the more efficient the production of housing quality is (a larger g), or the weaker the neigh-borhood externality is (a larger α), the larger the sizes of both the provision of the local public goods and the regional supply of housing quality.
We can further substitute (15) into (11) and (12) to obtain: ¯ hj =αg2 τj 1 + τj N 2 α−2 1−αj , (16) hij = γi1−α1 αg2 τj 1 + τj N 2 α−1 −αj , (17) cij =Yi −Ti ·Ij−α2g2 γi 1−α1 N 2 2α−2 1−2αj τj 1 + τj , (18) i ∈ M, j ∈ J . Intuitively, an increase in the aggregate preference for hous-ing quality leads to higher aggregate as well as average demand for houshous-ing quality (recall that the population in each region is fixed at N/2). As a re-sult of rising aggregate demand, the price must go up, thereby discouraging individual demand for housing quality so long as the positive neighborhood effect is not too strong. By substitution, individual demand for consump-tion must rise.
We are now prepared to characterize individual locational choices. De-note f (τj) ≡ (τj/1 + τj), kj ≡ (j)1−2α, j ∈ J , and ψi ≡ α(1 − α)g2(γi)(1/1−α)(N/2)2(α−1) > 0, i ∈ M. Substituting (16), (17) and (18) into the utility functions, we have:
Since α > 1/2, kj is decreasing with j, j ∈ J . That is, kj is an inverse index of the aggregate preference for housing quality in j , j ∈ J . When τj increases, a higher property tax is levied on the agents, but they can en-joy better housing quality (since there are more public good inputs used for improving the housing environment). Due to the positive neighborhood externalities in the Romer-Lucas convention (α < 1), the free-rider prob-lem arises, as one may be riding on positive spillovers from the neighbors in the absence of any public interventions. However, in our model econ-omy, a property tax is imposed and its revenue serves as the sole source for financing the local public good to enhance housing quality. In this case, the housing quality enjoyment effect dominates the property tax burden effect and, as a result, the indirect utility from living in region j is increasing in τj, for all τj ∈ R+, j ∈ J . Therefore, the property tax in this model is used as a centralized device in collecting revenue for the local public good provision, which forces individual investment in housing. As a result, it mitigates the free-rider problem and subsequently raises residents’ indirect utility. The reader is reminded, however, that this property need not hold for general utility functional forms with more general setups of location-specific supplies of housing quality or with more general methods of finance of the public good.
The equilibrium locations of type-i agents are determined by the differ-ence in the utility levels achieved by residing in the two regions,
V1i−V2i =Ti +ψi[k1f (τ1) − k2f (τ2)] . (20) That is, letting 8i(τ
1, τ2) ≡ ψi[k2f (τ2) − k1f (τ1)],
V1i T V2i if and only if Ti T 8i(τ1, τ2) , i ∈ M. 4 Equilibrium Characterization
Denote x1 ≡ H1/H (to be endogenously determined). The equilibrium distribution of agents can be summarized as follows:
H1=x1H, H2= (1 − x1) H, L1=
N
1 H L
2 L
Figure 1: Distribution of Agents’ Type in Pattern I
L2 = N
2 − (1 − x1) H.
Under the assumption of H < L, generically there are two equilibrium patterns: in Pattern I, all type-H agents reside in region 1 (x1 = 1); in Pattern II, all type-H agents reside in region 2 (x1 = 0). Of course, there is also a completely mixed equilibrium with some type-H agents in region 1 and some in region 2 (0 < x1 < 1). When there is a dispersion for both types of agents, (20) shows that TH/ψH =TL/ψLmust be satisfied. Therefore, only with a set of parameters of measure zero can such a mixed equilibrium arise.
It is noticed that j, and thus Gj, τjpj, and S(Gj)all depend on the distribution of agents, and therefore, a superscript I or I I is added for these variables for clarity when we characterize the two equilibrium patterns to which we now turn.
4.1 Pattern I: x1 = 1
In Pattern I, x1 = 1 indicates that all type-H agents reside in region 1. Thus, we have: H1 =H, H2 = 0, L1 = N 2 −H, L2 = N 2, as depicted in Figure 1.
In equilibrium, it must be the case that type-H agents achieve higher indirect utility by residing in region 1 and that type-L agents are indif-ferent to their residential locations. In Pattern I, the aggregate preference
variables become: I 1 = H (γH)1/(1−α)+((N/2) − H )(γL)1/(1−α) and I 2 = (N/2)(γL) 1/(1−α) , where I
1 > I2. For any given (τ1I, τ2I) ∈ R 2 +, type-H agents’ locational choice implies VH
1 ≥V2H, and thus:1 TH ≥8H τ1I, τ I 2; I 1, I 2 ≡ ψ HkI 2f τ I 2 − k I 1f τ I 1 , (21) while locational no-arbitrage facing type-L agents requires VL
1 =V2L; that is,
TL=8L τ1I, τ2I;I1, 2I ≡ ψLkI2f τ2I − k1If τ1I . (22) Let (τI ∗
1 , τ2I ∗) ≡ {(τ1I, τ2I) ∈ R2+|TL = 8L(τ1I, τ2I;I1, I2)}. For the existence of such a property tax pair (τI ∗
1 , τ2I ∗) ∈R2+to support the Pattern I equilibrium, it must be the case that kI
2f (τ I ∗ 2 ) > k I 1f (τ I ∗ 1 ), which implies:2 f τI ∗ 2 f τI ∗ 1 > kI 1 kI 2 = I 1 I 2 1−2α . (23)
That is, only when the ratio of the property tax rate in region 2 to that in region 1 is larger than the ratio of the aggregate preference for housing qual-ity among the two regions, can an equilibrium of Pattern I be constituted. Define k0i ≡Ti/ψi, i ∈ M. We can rewrite (21) and (22) conveniently as:
f τ2I ∗ ≤ k H 0 kI 2 +k I 1 kI 2 f τ1I ∗ , and (24) f τ2I ∗ = k L 0 kI2 +k I 1 k2I f τ1I ∗ , (25) where the latter equality can be expressed in terms of a transformed property tax ratio given as follows:
f τI ∗ 2 f τI ∗ 1 = kL 0 kI 2f τ1I ∗ + kI 1 kI2 . (26)
1Although the distribution pattern of H -type agents under Vi
1 = V2i can be arbitrary,
no i-type agent would have incentive to deviate from his/her equilibrium location given the equilibrium pattern of agents’ distribution.
2Recall that k
j ≡(j)1−2αis the inverse index for the aggregate preference for housing
Consider,
Condition I. (TH/TL) ≥ (γH/γL)(1/1−α).
Condition I requires that the ratio of agents’ commuting costs be sufficiently large compared to the ratio of their respective preferences for housing qual-ity. Intuitively, we note that the property tax augmented price of housing quality is higher in the region where type-H agents agglomerate (as shown in (7)) and the equilibrium is generically partially segregated (as shown in the beginning of this section). Thus, in order for type-H agents to agglomerate to the CBD, their valuation of commuting time must be sufficiently higher than that of L-type agents. Moreover, the higher the type-H agents’ relative preference for housing quality is, the greater their relative valuation of com-muting time will be required for supporting the Pattern I equilibrium. We can then establish:
Proposition 1. (Pattern I Equilibrium) Under Condition I, the market
equilibrium exists in which all type-H agents reside in region 1 whereas both regions are populated by type-L agents (Pattern I).
proof. Substituting (25) into the inequality (24) and manipulating, one
ob-tains: kH
0 ≥ k0L, which holds under Condition I. By plugging (26) into the inequality (23), we can easily see that this inequality holds because f (τI ∗
1 ) ≥ 0, k0L> 0, and kI2 > 0.
To illustrate the Pattern I equilibrium, we depict the relationship be-tween the equilibrium property tax rates (τI ∗
1 , τ2I ∗)in Figure 2. In the equi-libria of Pattern I, property tax rates fall on the line f (τI ∗
2 ) = (k0L/ k2I) + (k1I/ kI2)f (τ1I ∗). Since equilibrium property tax rates satisfy f (τ2I ∗) ≤ (kH0 / kI
2) + (k1I/ kI2)f (τ1I ∗)and f (τ2I ∗) > k1I/ k2I)f (τ1I ∗), for all τ1I ∗∈R+, there exists an equilibrium of Pattern I for all non-negative property tax rates.
Notice from (26) that when f (τI ∗
1 ) ≥ (kL0/ kI2 −k1I), τ2I ∗ ≤ τ1I ∗and that when f (τI ∗ 1 ) < (k L 0/ k I 2 −k I 1), τ I ∗ 2 > τ I ∗
1 . More specifically, since (kI
1/ kI2) < 1, whether (f (τ2I ∗)/f (τ1I ∗))may exceed one crucially depends on the first term on the right-hand side of (26), (kL
0/ k2If (τ1I ∗)). This term corresponds to an incentive adjustment effect, which is decreasing in τI ∗
1 . In-tuitively, the smaller τI ∗
1 is, the larger the ratio of τ2I ∗/τ1I ∗ that is needed to mitigate the free-rider problem so that type-L workers have no incentive to migrate from region 2 to region 1.
Using (14) and (26), we further compute the ratio of local public goods provided in the two regions:
45◦line f (τ1I ∗) f (τ2I ∗) kH0 k2I + kI1 kI2f (τ I ∗ 1 ) kL 0 kI 2 + kI1 kI 2 f (τ1I ∗) kI 1 kI2f (τ I ∗ 1 ) 0
Figure 2: The Relationship between Transformed Property Tax Rates in the Equilibrium of Pattern I GI ∗ 2 GI ∗1 = f (τ I ∗ 2 ) f (τ1I ∗) 2 I 2 I1 2(1−α) = kL 0 k2If (τ1I ∗) +k I 1 kI2 2 I 2 I1 2(1−α) . (27)
The second item on the right-hand side of (27), (I
2/ I1) < 1, indepen-dent of τI ∗
1 , represents an aggregate preference effect. Since region 1 resi-dents have a stronger aggregate preference for housing quality, they have higher willingness-to-pay for the local public goods. This effect tends to support (GI ∗
2 /GI ∗1 ) < 1. Nonetheless, as illustrated above, the first term, (k0L/ k2If (τ1I ∗))+(kI1/ k2I), could be greater or less than one. When f (τ1I ∗) ≥ (k0L/ k2I−kI1), the first term is less than one and as a result, we have (GI ∗
2 / GI ∗
1 ) < 1 unambiguously. Intuitively, when the incentive adjustment ef-fect is small, it is the aggregate preference efef-fect that dominates, so the willingness-to-pay becomes the crucial factor determining the provision of local public goods. In this case, although residing in the CBD results in the enjoyment of a higher housing quality and saves on the commuting cost, the residents must pay a higher property tax than those residing in the suburb.
In the case where f (τI ∗
1 ) < (k0L/ k2I −k1I), the incentive adjustment effect is sufficiently large and we must have τI ∗
2 > τ1I ∗ in order to miti-gate the free-rider problem for type-L workers to feel indifferent regarding residing in either region. That is, the case of a reversed taxation is present in the region with type-H residents that imposes a lower property tax rate. When f (τI ∗
1 ) ≥ (kL0/ k2I)((I1/ 2I)1−α−(k1I/ k2I))−1, even when the local governments set τI ∗
2 > τ1I ∗, the aggregate preference effect still dominates and, as a result, region 1 still provides more local public goods than region 2 (i.e., GI ∗
1 > GI ∗2 ). When f (τ1I ∗) < (k0L/ k2I)((1I/ I2)1−α−(k1I/ k2I))−1, in addition to reversed taxation, there exists a reversed provision of local public goods (i.e., GI ∗
1 < GI ∗2 ) as a consequence of a very strong incentive adjustment effect. In this case, not only does the region with type-H resi-dents impose a lower property tax rate, but it also provides fewer local public goods.
Since kL
0 is increasing in TLand decreasing in g, an increase in TLor a decrease in g shifts the type-L agents’ locational no-arbitrage condition (25) upward, which is depicted in Figure 2. As the result of a stronger incentive adjustment effect, the required property tax rate in region 2 rises for any given property tax rate in region 1. Similar results hold for a reduction in γH, as less efficient public good provisions and less willingness-to-pay by type-H agents in region 1 should have similar positive effects on (τI ∗
2 /τ1I ∗) (although in this case the type-L agents’ locational no-arbitrage condition is rotating counterclockwise, rather than shifting upward). In response to either a larger TL, a smaller g, or a lower γH, it is more likely that a reversion in the taxation and in the provision of local public goods may arise.
4.2 Pattern II: x1 = 0
We now turn to the Pattern II equilibrium with all type-H agents residing in region 2 (i.e., x1 = 0). Since H < L, we have:
H1 = 0, H2 =H, L1 = N 2, L2 = N 2 −H, as depicted in Figure 3.
In equilibrium, type-H agents achieve higher indirect utility by residing in region 2, whereas type-L agents are indifferent to residing in either re-gion. In this case, I I
1
H L
2 L
Figure 3: Distribution of Agents’ Type in Pattern II
((N/2) − H )(γL)1/(1−α), where I I
1 < I I2 . For any given (τ1I I, τ2I I) ∈ R2+, this equilibrium pattern requires that V1H ≤V2H and V1L=V2L, or,
TH ≤8H τ1I I, τ2I I;I I1 , I I2 ≡ ψHkI I 2 f τ I I 2 − k I I 1 f τ I I 1 , (28) TL=8L τ1I I, τ I I 2 ; I I 1 , I I 2 ≡ ψ LkI I 2 f τ I I 2 − k I I 1 f τ I I 1 . (29) Again, kI I
2 f (τ2I I ∗) > k1I I f (τ1I I ∗)is required which implies: f τ2I I ∗ f τ1I I ∗ > kI I1 kI I 2 . (30) Define (τI I ∗ 1 , τ2I I ∗) ≡ {(τ1I I, τ2I I) ∈ R2+|TL = 8L(τ1I I, τ2I I;I I1 , I I2 )}. We follow similar steps to those in the previous subsection, rewriting (28) and (29) as: f τ2I I ∗ ≥ kH 0 k2I I +k I I 1 k2I I f τ1I I ∗ , and (31) f τ2I I ∗ = k L 0 kI I 2 +k I I 1 kI I 2 f τ1I I ∗ , (32)
and further express the latter equality below: f τ2I I ∗ f τ1I I ∗ = k0L kI I2 f τ1I I ∗ + kI I1 kI I2 . (33) Consider now, Condition II. (TH/TL) ≤ (γH/γL)(1/1−α).
45◦line f (τ1I I ∗) f (τ2I I ∗) kL 0 kI I 2 +kI I1 kI I 2 f (τ I I ∗ 1 ) kH 0 kI I 2 +kI I1 kI I 2 f (τ I I ∗ 1 ) k1I I k2I If (τ I I ∗ 1 ) 0
Figure 4: The Relationship between Transformed Property Tax Rates in the Equilibrium of Pattern II
Condition II requires that the difference between agents’ commuting costs be sufficiently small compared to the difference in their respective prefer-ences for housing quality. We then have:
Proposition 2. (Pattern II Equilibrium) Under Condition II, the market
equilibrium exists in which all type-H agents reside in region 2 whereas both regions are populated by type-L agents (Pattern II).
proof. Plugging (32) into the inequality (31) yields: kH
0 ≤ k0L, which holds under Condition II. Substituting (33) into the inequality (30) shows that the inequality holds as long as f (τI I ∗
1 ) ≥ 0, k0L> 0, and k2I I > 0. The relationship between equilibrium property tax rates (τI I ∗
1 , τ2I I ∗)is depicted in Figure 4. In contrast to the Pattern I equilibrium, the Pattern II equilibrium always features τI I ∗
1 > τ I I ∗
2 and the type-L agents’ loca-tional no-arbitrage condition (32) always satisfies f (τI I ∗
2 ) ≥ (kH0 / k2I I) + (k1I I/ kI I2 )f (τ1I I ∗)and f (τ2I I ∗) > (k1I I/ kI I2 )f (τ1I I ∗). In this case, we can observe:
GI I ∗ 2 GI I ∗1 = f τ I I 2 f τI I 1 !2 I I 2 I I1 2(1−α) > 1, because (f (τI I ∗
2 )/f (τ1I I ∗)) > 1and (I I2 / I I1 ) > 1. That is, under Pat-tern II, both the incentive adjustment and the aggregate preference effects support a greater provision of local public goods in region 2 where type-H agents reside (i.e., GI I ∗
1 < GI I ∗2 ). That is, in region 2 where type-H agents reside, both the property tax rate and the provision of local public goods are always higher regardless of the level of τI I ∗
1 .
Under Pattern II, the effects of TLand g on the property tax rates are still transmitted through the incentive adjustment effect (via kL
0), so the results are the same as under Pattern I regardless of where type-H agents reside. On the contrary, the response to changes in γH crucially depends on type-H agents’ residential locations. With higher γH, type-H agents are more willing to pay for the local public good in region 2, thereby leading to a higher property tax rate in region 2 for a given property tax rate in region 1.
4.3 Equilibrium Configuration
We are prepared to summarize the equilibrium configuration by means of the following figure:
When the difference between agents’ commuting costs is sufficiently large compared to the difference in their respective preferences for hous-ing quality (i.e., Condition I is met), the market equilibrium is of Pattern I in which all type-H agents reside in region 1. On the contrary, when the difference between agents’ commuting costs is sufficiently small (i.e., Condi-tion II holds), the market equilibrium is of Pattern II with all type-H agents residing in region 2. Figure 5 displays the partition of the parameter space into the two equilibrium patterns. It is clear that the partition curve is above the 45◦line, indicating that Pattern II is more likely to arise than Pattern I.
In Pattern I, all high type agents agglomerate in the CBD, which cor-responds to the distribution patterns in Asian and European metropolitan areas. In Pattern II, all high type agents agglomerate in the suburb, which captures the distribution patterns observed in the U.S. metropolitan ar-eas. Therefore, under comparable measures of the valuation of commuting time, the commuting costs may be thought of as being similar in the above economies. Our model suggests that the observed distribution patterns in
γH γL
1−α1
Equilibria of Pattern I (Asia and Europe)
Equilibria of Pattern II (U.S.) 45◦line 1 γH γL TH TL
Figure 5: Equilibrium Configurations for Different Commuting Costs and Various Preferences for Housing Quality
different cities may arise from a stronger preference for housing quality in the U.S. compared to Asia and Europe.
Finally, when the ratio of commuting costs between the two types is in a specific homothetic relation to the ratio of the marginal valuations of hous-ing quality (precisely, (TH/TL) = (γH/γL)(1/1−α)), a mixed equilibrium arises where both the rich and the poor reside in both regions. A special case of such a mixed equilibrium is the fully integrated symmetric equilib-rium in which each region contains the same proportion of the rich (i.e., x1 =x2 = 1/2).
5 Conclusions
We have developed a general-equilibrium endogenous-sorting model to an-alyze the pattern of the multi-class competitive spatial equilibrium. We have established that the equilibrium configuration crucially depends on the het-erogeneity of preferences for housing quality versus that of the opportunity
costs of commuting. When the difference between agents’ commuting costs is sufficiently large compared to the difference in their respective preferences for housing quality, the equilibrium is segregated with all rich residing in the central city. When the reverse is true, the equilibrium is segregated with all rich in the suburb. While in the latter case the suburb with all the rich residing there features a higher property tax and more local public goods provision, in the former case there is no such clear-cut comparison. Un-der comparable measures of the valuation of commuting time, our results have suggested that the observed distribution patterns in many cities in the U.S. where more rich reside in suburbs may arise from a stronger preference for housing quality by the Americans compared to Asians and Europeans. Moreover, while locational stratification is a result of preference heterogene-ity and a positive neighborhood externalheterogene-ity in the form of housing qualheterogene-ity financed by local property taxes, where the rich reside will crucially depend on their opportunity costs of commuting relative to their preferences for housing quality.
There are several ways to extend the paper, among which we highlight but three. The first, and perhaps the most important, is to generalize the location-specific supply of housing quality. In particular, it would be in-teresting to allow individual resources to be devoted to enhancing housing quality `a la Peng and Wang (2009). This is likely to yield richer equilibrium configuration outcomes due to non-monotonic effects of property taxation. The second is to extend the model economy to a dynamic setting. In partic-ular, it would be interesting to allow the stock of housing quality to be de-termined by the existing capacity of local public infrastructure as well as the flow input of investment. This may be done in an overlapping-generations setup `a la Chen et al. (2009). An intriguing issue is whether locational strati-fication is more likely to be the stable equilibrium outcome. If it is, one may better understand why economic segregation is often observed and persists in many different societies.
Finally, a possible venue for future research toward a deeper understand-ing of sortunderstand-ing and segregation is to allow the number of types of agents to exceed the number of regions as analyzed by de Bartolome and Ross (2003) and Hanushek and Yilmaz (2007). Notably, when the number of types of agents equals the number of regions as in our benchmark economy, we have shown that one of the regions features generically homogeneous population in equilibrium. This result may not be what has been found in the em-pirical literature, as Hardman and Ioannides (2004) document, using the
1 2 H M L IA L 1 2 H M L IB L 1 2 H M L IIIA L 1 2 M H L IIIB L
Figure 6a: With Three Types of Agents and Two Regions, When H + M < L, There Are Four Equilibrium Patterns
1 2 H M L IIA M 1 2 M H M IIB L 1 2 H M L IIIA L 1 2 M H L IIIB L 1 2 H H L IVA M 1 2 H H M IVB L
Figure 6b: With Three Types of Agents and Two Regions, When H + M > L, There Are Six Equilibrium Patterns
American Housing Survey data, that the heterogeneity within a community typically exceeds the heterogeneity across communities. To rectify this prob-lem, one may consider the most stylized case where there are 3 types and 2 regions, with population Hj, Mj, Lj in region j , where H1 +H2 = H, M1 +M2 = M, L1+L2 = Land Hj +Mj +Lj = N/2, j ∈ J and where H < M < L. When H + M < L, the equilibrium features gener-ically Patterns IA, IB, IIIA, and IIIB (see Figure 6a). When H + M > L, the equilibrium features generically Patterns IIA, IIB, IIIA, IIIB, IVA, and IVB (see Figure 6b). Therefore, we have richer patterns of equilibria than in our benchmark model. In particular, while Patterns IA and IB are similar to
our benchmark Patterns I and II, in other equilibrium patterns all regions feature heterogeneous population as observed in reality.
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