ISSN (Print) 0473-453X

Discussion Paper No. 1105

ISSN (Online) 2435-0982

WHEN EDUCATION POLICYAND HOUSING POLICY INTERACT:CAN THEY CORRECTFOR THE EXTERNALITIES?

Yifan GongCharles Ka Yui Leung

October 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

When education policy and housing policy interact: Can they correct for theexternalities?
Yifan Gong, University of Western Ontario, Canada
Charles Ka Yui Leung, City University of Hong Kong, Hong Kong
This version: October, 2020.

Abstract:
We develop a simple spatial equilibrium model with the peer group effect and local public finance toanalyze the implications of housing policies such as public housing and housing voucher programs, andeducation policies such as school finance consolidation. The calibrated model can match several stylized
facts of the labor market and the housing market in the United States. Our counterfactual policy
analyses suggest that public housing and housing voucher programs have similar welfare implications on
the household level. However, within a household, the public housing program tends to benefit thechildren more than the parents, while the housing voucher program delivers the opposite result.
Combining the school finance consolidation policy with the public housing program could improve thewell-being of children from poor households without hurting other households' welfare. Some policies'
short-run welfare implications can deviate significantly from their long-run counterparts when all
choices are optimized.
JEL Classification: H00; I20; R00
Keywords: school finance consolidation, public housing, housing voucher, endogenous sorting
mechanism, short-run rigidity versus long-run flexibility.
Correspondence: Gong, Department of Economics, University of Western Ontario, Canada, (email)ygong48@uwo.ca ; Leung, Department of Economics and Finance, City University of Hong Kong,
Kowloon Tong, Hong Kong, (email) kycleung@cityu.edu.hk .

Acknowledgement: We are grateful to the comments by Mike Eriksen, Niikura Hiroaki, Anson
Ho, Alvin Murphy, Ed Olsen, Brian Peterson, Sinan Sarpca, Miki Seko, Kazuto Sumita, Paloma
Taltavull, Yuxi Yao, Kuzey Yilmaz, Norifumi Yukutake, especially an anonymous referee, Paul
Carrillo, as well as the seminar participants of the Bank of Canada, Kyoto University, Osaka
University, University of Hong Kong, AREUEA meeting, Asian Real Estate Society meeting,
European Real Estate Society meeting, Toyo University workshop. Part of this research is
conducted when Leung visits the Hoover Institution, whose hospitality is gratefully
acknowledged. Financial support from the City University of Hong Kong is gratefully
acknowledged. Leung’s visit to Osaka is supported by the Joint Usage/Research Center at ISER,
Osaka University and Grant-in-aid for Research Activity, Japan Society for the Promotion of
Science (15H05728, 20H05631), and the International Joint Research Promotion Program of
Osaka University. The usual disclaimer applies.

Title:When education policy and housing policy interact: can they correct for the externalities?
Abstract:
We develop a simple spatial equilibrium model with the peer group eﬀect and local public financeto analyze the implications of housing policies such as public housing and housing voucher programs,
and education policies such as school finance consolidation. The calibrated model can match several
stylized facts of the labor market and the housing market in the United States. Our counterfactual
policy analyses suggest that public housing and housing voucher programs have similar welfare
implications on the household level. However, within a household, the public housing program
tends to benefit the children more than the parents, while the housing voucher program delivers the
opposite result. Combining the school finance consolidation policy with the public housing program
could improve the well-being of children from poor households without hurting other households’
welfare. Some policies’ short-run welfare implications can deviate significantly from their long-run
counterparts when all choices are optimized.
JEL Classification: H00; I20; R00
Keywords: school finance consolidation, public housing, housing voucher, endogenous sorting
mechanism, short-run rigidity versus long-run flexibility.

11

Introduction

The United States practices "local public finance," where K12 education is substantially financed
by local property tax. The public school is then provided to heterogeneous agents within the samecommunity at zero price. Therefore, the housing market is linked to the education of children. The
competitions for housing in well-funded school districts, would drive up house prices, and driveout relatively low-income families.1 The peer group eﬀect literature informs us that quality ofeducation tends to increase with the proportion of students from high-income families (Sacerdote,
2011; Stinebrickner and Stinebrickner, 2006). As a result, oﬀsprings from disadvantaged families
could receive lower-quality education and are more likely to remain less privileged as adults.2
Can appropriately designed policies mitigate this intergenerational persistence in income phenomenon? To address this question, we develop a simple spatial general equilibrium model withthe peer group eﬀect and local public finance in this paper. We show that housing policies such as
public housing and housing voucher programs, and education policies such as school finance consolidation interact with one another and sometimes produce unexpected consequences. Thus, we
complement the literature on housing policies, which mainly relies on partial equilibrium analysis(Currie and Yelowitz, 2000; Olsen and Ludwig, 2013).3
The general equilibrium approach provides an avenue for understanding households’ decisions
and outcomes in a systematic fashion and allows us to analyze the feasibility of specific policies.
For example, housing market policies, such as public housing and housing voucher, often impose
additional tax burdens on non-participants, which might bring political resistance. While a partial
equilibrium setting may take the tax as given, our model can partially address this issue in two
ways. First, the property tax rate is determined through majority voting within the community.
Second, we attempt to find a Pareto-improving policy, i.e., to improve some members’ welfare andlet no other get hurt. Such a policy would be easier to receive support from voters. Notice thata Pareto-improving policy may be feasible in the present context because many forms of market
imperfections exist in the housing and education markets.
More specifically, our model is a variant of the Hanushek-Yilmaz (HY, henceforth) framework,
which incorporates all the externality and market imperfections discussed above.4 In an HY economy, households choose among the ex-ante identical communities for residence, given the school
1 According to Schwartz (2013, p.24~25), “... most children in the U.S., where you live determines where you go toschool... As of the 2008-2009 school year, 11 percent of children went to private schools, approximately 3 percent of
U.S. public school students attended charter schools, and another 5 percent attended magnet schools. Only 1 percent
of public school students enrolled in diﬀerent school district through interdistrict choice programs, even though 46
percent of school districts reported oﬀering such a program.”
The average level of housing costs in household budgets is around 20% in the United States, similar to the average
figure (18%) in the OECD countries.
2 Intergenerational persistence in income is high in the United States. One percent increase in parental income
is associated with a 0.5 percent increase in oﬀspring’s earnings (Solon, 1999; Black and Devereux, 2011). A childfrom an economically adverse family grows into a luxurious adult is unlikely (Chetty and Hendren, 2018; Currie andAlmond, 2011; Hanushek and Yilmaz, 2011).
3 Using a general equilibrium spatial model, Leung et al. (2012) show that housing vouchers generate higher
social welfare under some conditions than public housing. However, their paper does not explicitly model the intrahousehold allocation of resources. Hence, it is not particularly suitable for analyzing the intergenerational eﬀects ofhousing policies.
4 Papers adopting this framework includes Hanushek et al. (2011), HY (2007, 2013, 2015, 2020), and Leung et al.
(2012).

2

quality and property tax rate pair (Tiebout, 1956). Within a community, families also make thetradeoﬀ between commuting costs and spatially-diﬀerentiated rents in locational decisions (Alonso,1964). Thus, the HY framework links the housing market and education through local publicfinance, embedding the peer group eﬀect and spatial elements in a unifying framework.
Our contribution hinges on making two realistic modifications to the HY framework. Our firstmodification is to diﬀerentiate between the well-being of parents and oﬀspring explicitly. The
level of parental investment in oﬀspring well-being becomes a choice variable and can diﬀer acrosshouseholds. Also, parents decide the intra-household allocation of goods and residence space.
Hence, diﬀerent families can respond to policy changes diﬀerently, and we can assess both the intrahousehold and inter-household welfare changes across policy regimes. The second modification is
to model the peer group eﬀect as a function of the average quality of the peers (e.g., Blume andDurlauf, 2006), which, in turn, depends on their parents’ investment and per capita educational
spending of the local government. In contrast, previous Hanushek-Yilmaz papers assume peer groupeﬀects only depend on population composition, such as the fraction of educated parents within acommunity. To highlight the importance of this modification, we repeat our policy analysis underan alternative specification where the peer group eﬀect only depends on the population composition.
As we now note in a footnote and show in the appendix, our formulation could lead to quantitatively
diﬀerent policy implications from the case where the peer group eﬀect only depends on population
composition.
With both discrete choices (e.g., which community to reside) and externalities (e.g., peer groupeﬀect, public education budget) in the model, it may not be feasible to estimate the parameters
using a reduced-form approach directly. Instead, we follow the path of existing HY papers. Morespecifically, we calibrate our model to match several stylized facts of the U.S. economy circa 2010,enabling us to "identify" parameter values that might not be directly observed and ensure that ourmodel is consistent with those stylized facts. We perform counterfactuals based on the calibrated
model to analyze the implications of housing and education policies. Concerning the eﬀects ofhousing policies, we find that building public housing units for the poor can improve the well-beingof their oﬀspring and themselves at the cost of hurting the rest of the economy. The average welfare
also declines as location choices are restricted, and the incentives to work are distorted. However, we
also find that a policy that combines the public housing program with school finance consolidation
can be eﬀective (aiding the poor) and eﬃcient (Pareto-improving). Intuitively, this is the casebecause, in contrast to the results in HY (2007, 2013), we find that school finance consolidation
can improve welfare for all households, based on our calibrated model. We provide more discussion
and intuition in section 3.
Our policy experiments yield additional results. First, it is better to locate the public housing
units on the "edge of the city," i.e., the land that would not be occupied without the public housing
program. The intuition is that such an arrangement of public housing units would not reduce theamount of accessible area. Second, on the household level, housing voucher provides similar welfare
results as public housing policy does. However, their implications on the individual level diﬀer.
Adequately designed public housing policy can benefit the adult members in low-income families
and improve the well-being of their children. In contrast, while we still find an increase in the wellbeing of the less educated parents under the housing voucher program, their children’s well-being
is impaired. We also compare the short- and long-run welfare implications of public policies in thecontext of our model and find that there is a possibility that they would diﬀer. To our knowledge,
such comparison is relatively rare in the spatial general equilibrium literature and would hopefully
enrich our understanding.

3

The rest of the paper is organized as follows. We first present a modified version of the HYmodel and deliver the baseline results. The model-generated rent gradients are compared withthe empirical findings as a validity test of the model. We then analyze the welfare consequencesof introducing various housing policies and consolidating school finance, both in the short- andlong-run. We conclude in the final section and preserve the technical details in the appendix.

2

The Model

We first provide an informal overview of the model, where our subsequent policy analysis would
be based. Our model modifies the pioneering work in HY (2007), which combines the insights ofTiebout (1956) and Alonso (1964) in a multi-district spatial model. Our monocentric city is composed of two ex-ante identical jurisdictions, East (E ) and West (W ), and three types of households(low, medium, and high skill). The two jurisdictions are divided by a line that goes through theCentral Business District (CBD), where all households commute to work daily. For simplicity, we
assume that a family is formed via assortative matching, and parents make choices to maximize theutility of the whole household. The tradeoﬀ between the quantity and "quality" (or well-being) ofchildren diﬀers across household types. Household members consume housing, a consumption good,and leisure. Parents invest in their children directly (by purchasing a house and consumption goods)
and indirectly (through taxes that fund schools). School funding and peer group eﬀects jointly determine school quality, with the school funding itself determined by the property taxes collectedby the local government. While both property tax rate and peer group eﬀects are endogenous
and jurisdiction-specific, the determination mechanisms are diﬀerent. The former is determinedby majority voting, and the latter is determined by the average educational outcomes within theneighborhood. Figure 1 visualizes all these relationships.
(Figure 1 about here)

2.1

The General Set-up

Our formal model is a variant of the HY framework, with an elaborated form of parental altruism.5
We use the subscript  for parents and subscript  for oﬀspring. Parents’ well-being Ω depends on
the amount of goods  and lot sizes  that they consume and how much leisure time  ∈ [0 24]
they enjoy. Parents care about both the well-being of each oﬀspring Ω and the quantity  of theiroﬀspring, where  is the community the family chooses to live in (Becker, 1991; Hanushek, 1992).
The well-being of each oﬀspring Ω depends on both the public inputs, i.e., the education quality incommunity ,  and private inputs, i.e., the amount of consumption goods  and the residential
space  (Goux and Maurin, 2005; Gertler et al., 2004). We also assume that the marginal utility
derived from the number of oﬀspring declines as the fertility rate  increases. ( ) measures thedegree of altruism shown toward each child. Following HY (2007, 2013) and others, we assume
5 See also Becker and Barro (1988). Alternatively, we can assume that parents care about the utility of theirchildren. However, given the non-intergenerational nature of the model, the two formulations are observationallyequivalent.

4that households’ utility function is in the Cobb-Douglas form.6 Formally, the utility function of atypical household in district  ∈ { } is given by: (          ) = (Ω ) (Ω )  ( )




(1)



where Ω ≡ (     ), Ω ≡     , ( ) ≡ −
 , with the restrictions on parameters that +   +  = 1,  +  +   = 1,  +  = 1,   1. The parameters  and  capture the relative
importance of the parent part and the oﬀspring part, respectively.
We now describe the budget constraint faced by the household. Consider a household located miles away from the CBD. The two parents earn all the income for the whole family.  represents
the sum of their hourly wages. The parents allocate the hours in their days to work, leisure (hours), rearing oﬀspring (( ) hours), and commuting ( hours), where  is the time cost permile of their daily round-trip commute. For simplicity, we assume that ( ) =   where  is aconstant. Hence, the total income of this household is = [24 −  − ( ) − ]
The parents use their income to purchase consumption goods and pay for housing rents. We
normalize the price of the composite consumption goods as unity. We assume that the daily roundtrip commuting costs a total of  dollars. Formally, the total expenditure is = ( () +  () ())(1 +   ) () +  () +  () () + 
where  () is the amount of space consumed by the parents,  () is the number of oﬀspring in
the household,  () is the amount of space by each oﬀspring,   is the property tax rate,  ()is the daily unit rent in the location  Hence, ( () +  () ())(1 +   ) () is the after-tax
expenditure on space for the whole household. The total expenditure on consumption goods of thisfamily is  () +  () (). We allow all these variables to be functions of , reflecting that theconsumption of space and goods could depend on the location of residence. Our budget constraint
then equates the  to the . We reserve the details in the appendix.
2.1.1

Household Heterogeneity

In practice, households diﬀer in many dimensions. In this paper, we focus on the diﬀerences inincome and preference on the well-being of oﬀspring. We will show that these diﬀerences would
generate significant heterogeneity in equilibrium under local public finance practice. Table 1 showsthat, in data, more highly educated parents, on average, have higher earnings and lower fertility
rates. They spend more resources on fewer children, translating a diﬀerence in parents’ income
into a diﬀerence in oﬀspring well-being. Guided by these stylized facts, we classify the households
according to their adult members’ educational attainment and calibrate our model accordingly.
For simplicity, we assume that there are three types of households: "Not a High school graduate"(N.H.), "High School to associate degree" (H.A.), and "Bachelors degree or above" (B.A.). We
further assume that a family is formed with assortative matching. Hence, both of the two adults
6 There are several merits of assuming the Cobb-Douglas form utility function. Previous studies use that as well,and hence we can easily compare our results with them. The Cobb-Douglas form is tractable. The expenditure shareof each good is fixed, which seems to be consistent with the evidence (Davis and Ortalo-Magne, 2011).

5

(wife and husband) have attained the same educational attainment in a given household.7 Our
assumption that       is consistent with the U.S. data indicating that higher
educational attainment leads to higher wage income on average.
The second dimension of heterogeneity among parents is the quality-quantity tradeoﬀ of oﬀsprings (Becker, 1991; Hanushek, 1992). In our model, a higher value of  means that the parents
care more about the well-being of their oﬀspring than the number of oﬀsprings they have. To
capture the fact that adults who attain a higher level of education tend to bear fewer children, we
assume that       .8 With three types of agents and only two communities involved,
we have imperfect sorting, consistent with the empirical evidence (Davidoﬀ, 2005; Hardman and
Ioannides, 2004).
(Table 1 about here)

2.2

Basic Analysis of the Equilibrium

In this section, we define and characterize the equilibrium of our model. Our analytical characterizations hold for a broad and reasonable set of parameters and are broadly consistent with theempirical evidence. Hence, they provide some validity of our model. These characterizations also
assist our calibration in a later section.
2.2.1

Bid-rent Functions and Market Rent Curves

Like many spatial equilibrium models, all households bid for land on a featureless plane. Therefore,
we solve for the bid-rent function, which expresses a household’s willingness to pay for space witha given utility level . For a type  ∈ {   } household living in district  ∈ { }, themaximization problem is as follows:½¾ () −  −   −   −    (       ) =max(2)| () =      (1 +   )( +   )Solving this maximization problem, we obtain the following bid-rent function:  (       ) =⎧⎫1⎨     () ⎬ 

1
1 +  ⎩and the following bid-max lot size function:





⎭



(3)

7 In practice, marital sorting is not as extreme. According to Fernandez et al. (2005), the cross-country average ofassortative matching regarding spouse education level is about 06. For a review of the literature and new evidence
for assortative matching in marriage, see Bruze (2011), among others.
8 There are at least two ways to interpret the assumption that income is correlated with parents’ degree of
altruism. According to the warm-glow theory (Andreoni, 1990), a higher income leads to a higher degree of altruism.
Chowdhury and Jeon (2014) conduct a field experiment (dictator and recipient) and find support for the warm-glow
theory. The second interpretation is that less-educated parents may be less informed on how to “invest” in theiroﬀspring.
For a survey of the altruism literature, see Laferrere and Wolﬀ (2006), among others.

6 (       ) =  (       ) +   (       ) = ()1  (       )  1 +  (4)where  ,  ,  ,  are functions of parameters. (Interested readers may refer to the appendix
for details).
In the model, all of the lands are rented out via auctions. All three types of households andagricultural workers can bid for any location ( ).9 For each location, the right of usage goes to theagent who oﬀers the highest bid. Therefore, the equilibrium rent curve  () is the upper envelopeof the bid rent curves   (       ) of the three types of households and the agricultural rent  .
As the household moves away from the CBD, its bid rent declines due to the transportation cost.

It means that beyond a certain distance , the agricultural rent  dominates the bids oﬀered
by all of the households in the economy. Hence, no one resides there. Within the fringe distance, the spatial order of two adjacent types of households is determined by the relative steepnessof their bid rent curves at the intersection point. The one with the steeper curve resides closer tothe CBD. In other words, the condition for the equilibrium location of Household 1 being further
1 (·)
from the CBD than that of Household 2 is  (·)  1. Furthermore,2  +   (·)= −  (       )   ()Based on these observations, the following proposition becomes intuitive (all proofs are included in
the appendix):
Proposition 1 If    , then in each neighborhood, households with better-educated adults live
further from the CBD at the equilibrium.


There are opposing forces on the households’ location choice. As long as   1 (which is true
given    ),  (      ) is increasing in the distance to CBD . Thus, the income eﬀectof a higher wage creates more demand for lot size consumption and induces the household to live
farther away from the CBD. However, higher hourly wages also increase the opportunity cost ofcommuting time. This substitution eﬀect generates an incentive for the parents to live closer to the
CBD and therefore spend less time commuting. In our model, the income eﬀect always dominates.
Thus, a higher wage income and a stronger preference for the well-being of their oﬀspring drive
better-educated parents to reside farther away from the CBD. This prediction is consistent with along-lasting stylized fact in the United States that the nation’s poor are more likely to reside incentral areas of cities. In the year 1990, the majority (59%) of the poor poverty area residents lived
in central cities. 28% and 13% of them resided in outer-metropolitan areas and suburban areas,
respectively (Bureau of the Census, 1990).
2.2.2

Population and Fertility Decision

The total number of households for each type  is exogenously given at  in this model,  ∈{   }. However, the total population is endogenous as the fertility in each family is a
9 Following the urban economics literature, the agricultural workers are assumed to be self-sustained, except forthe participation of the land auction. They would not aﬀect any other aspect of the model economy.

7choice variable. Suppose that in equilibrium, the locations  miles away from the CBD in district are occupied by households of type , where  ∈ { }. Let () represent the amount ofland available per unit distance, at distance . Because the whole land is equally divided into
two districts, () = 12 2 =  in each district. The land market is cleared, which means that
within the fringe distance , () =  (       ) (       ), where  (       ) is theequilibrium number of households per unit distance in district  assuming that distance  is occupied
by type  household and  is the equilibrium utility of type  household. We introduce the function () to indicate the type of the residents at distance  of district . All of the households findlocations to reside, implying the following population constraint:
Z∞
Z∞





 (       )[ () = ] +  (       )[ () = ] =   ,
0

(5)

0

where [] is an indicator function that takes the value 1 when the condition in the bracket is satisfied
and 0 otherwise,  ∈ {   }. It is easy to verify that the household number distributionfunction in district  isX
 () = (       )[ () = ]
∈{ }The total population in this economy consists of adult and child populations, where the latteris endogenous. We denote  () to be the fertility choice of type  parents in district , located miles from the CBD. The solution of (2) suggests that () =  () = (),
 

which is independent of district .
1, then parents who care more about their oﬀspring’s well-being bear fewer
Proposition 2 If   2−children, other things being equal.
 () 0This proposition is rather intuitive. Hence, parents who care more about their oﬀspring leanmore heavily towards children’s well-being and bear fewer children. Together with our assumptionthat more altruistic adults have higher wage income, this proposition implies a negative incomefertility relationship, which is in line with the data.
Finally, the oﬀspring population located  miles from the CBD and in district  isX
 () = () (       )[ () = ]
∈{ }2.2.3

Property Taxes, Peer Group Eﬀect and School Quality

In the previous section, households take the property tax and school quality of each district asgiven in their location choice. This section shows how these subjects are determined. Recall that8

each of the two districts finances its school through the property taxes placed on the residential
land within that district. Because they do this independently, the education quality and property
tax rate packages (    ) may diﬀer between the two districts. As in the U.S., the publicly fundedschools in our model are only open to the residents in the same districts and do not charge tuition
fees. The local government of district  would have the following budget constraint:



Z

  =   ()()

(6)

0

X


where is the population of children from type  household in jurisdiction ,  =∈{ }is the total population of the children in jurisdiction ,  is expenditure per student in district and   is the property tax rate. Thus, equation (6) states that the total expenditure on studentsZ ()().
  needs to be financed by the local property tax collected within the district  
0In this model, the (local) education quality  has two determinants. First, a higher value of means that the local school can aﬀord better instructional facilities and instructors, and hence
provide better education quality  . Second, a higher value of the peer group eﬀect Π , which means
having more qualified peers, lifts a student’s educational achievement via several channels. For
instance, students may learn from their classmates during group works or even casual interactions.
Competing with well-educated peers in school may also induce a student’s motivation to study.
Following the HY (2007, 2013), we assume that the quality is the product of expenditure and peergroup eﬀect,10 =  Π 
(7)
We now turn to the determination of peer group eﬀect in the community, Π   ∈ { }. Some
previous studies assume that the peer group eﬀect is a function of the population composition (HY,2007, 2013). A higher proportion of skilled adults in the total population generates a higher positive
peer group eﬀect. Such formulation captures the ideas that (1) family has a significant impact on
student performance and (2) the abilities of parents and children are positively correlated. However,this formulation implicitly restricts the parental investment on oﬀspring to be identical acrosshouseholds, and hence the population composition would be suﬃcient to capture the peer groupeﬀect. This paper relaxes this assumption and allows parental investment to be an endogenous
decision. We also need a formulation that can apply to any finite number of household types.
Therefore, we assume that the peer group eﬀect is a function of the average quality of all students
in the community, as suggested by some previous studies (Blume and Durlauf, 2005; Liu et al.,2014; Sacerdote, 2011).11 This formulation captures the quality-dependent nature of peer groupeﬀects, while it remains tractable in a spatial general equilibrium model with many distortions.
Following HY (2013), we assume a similar functional form of Π (Ω ),Π (Ω ) = 1 + 2 exp(Ω ), 1  2  0
1 0 Notice that there is no fundamental unit for either school quality or peer group quality.

(8)
Therefore, we canrenormalize them so that (7) holds.
1 1 For simplicity, we assume that the "quality" of a child as a student is equal to her well-being, Ω , and use theterms "(student) quality" and "oﬀspring well-being" interchangeably.

9where Ω is the average quality of the students in the community, whose detailed specification isprovided in the appendix. Given our formulation, it is straightforward to show the following:
Proposition 3 In each neighborhood, better-educated adults produce higher quality students.
This proposition is intuitive. Parents with higher education levels have several advantages
in producing oﬀsprings who are higher quality students. They earn higher wage income, havefewer children, and spend a substantial proportion of that income on each child. As a result,the expenditure per child increases. Moreover, these households live farther away from the CBD,
where land rents are much cheaper. They can aﬀord larger lot sizes with the same budget, which
improve the well-being of their oﬀspring. This proposition also agrees with the perceived highintergenerational correlation between income and education in the United States.
To close the model, we now describe how the property tax rate   is determined. All of the adults(parents) in district  have the right to vote for their preferred tax rate.12 Hence, the preferredproperty tax rate of a particular household  is the tax rate which maximizes the utility, subject toall the constraints. Formally, it is the solution to the following maximization problem,

max () =subject to     ()¤ £(1 +   ) () =  Π and  =    

(9)where  is the total rent collected in community . Its detailed expression is presented in the
appendix. The solution takes the following simple form,  =  ( −  ) =  (  +   −  )

(10)Furthermore, the calibrated set of parameters ensures that   is positive for all three types ofhouseholds.
2.2.4

Stationary Equilibrium

We are ready to define the general equilibrium of this model economy. In the stationary equilibrium,
no household has an incentive to relocate after the electoral outcome is realized and observed. It
can be formally defined as follows:
Definition 1 An equilibrium is a set of utility levels {      }, market rent curves { ()  ()},school quality and property tax rate pairs {(    ) (    )}, household number/oﬀspring popu
lation distribution functions {( () 
 ()) ( ()  ())} and type functions { ()  ()}that show the equilibrium occupant of the location at distance  in district  producing the following
results.
• The households oﬀer their bids according to equation (3). The land is rented out throughauction. The household that oﬀers the highest bid wins a particular location if it is higher than the
agricultural rent. Otherwise, the land is left for agricultural use.
• Each household rents a certain amount of land according to equation (4). The land market
clears, and the population constraint (5) holds.
1 2 Following Nechyba (1997, 2003), parents are assumed to be “myopic” when voting and do not consider the
implications of their votes on the population composition, land prices and the peer group eﬀects in either communities.

10• Households of the same type attain the same utility level.
• Each jurisdiction finances its school through property taxes placed on residential land. The
property tax rate is determined by majority voting, according to equation (9). The local government
budget balances in all districts, as described in equation (6).
• School quality depends on both per-student spending and the peer group eﬀect, which is afunction of the average quality of the students in the school district, as shown in equation (8).
• All adults commute to the CBD for work and earn wage income according to their types.
Commuting presents both monetary and time costs.

2.3

Calibration

2.3.1

Parameter Set

We calibrate the stationary equilibrium of our model to match a large set of stylized facts from the
U.S. statistics circa 2010. We divide the parameters of our model into three categories, which are(1) budget constraint parameters  , ,  and , (2) preference parameters  ,   , ,  ,   , ,  , , and , and (3) macroeconomic environment parameters  ,   , 1 and 2 . Below we describe
the calibration of each category of parameters.
We start with the budget constraint parameters. Because we assume that the two parents
in one household attain the same level of education, the target wage income of the household
type  is the twice the wage income of a type  agent,  ∈ {   }. Hence, the averageannual earnings of type NH, HA, and BA households are about $51 432, $87 479, and $155 013,
respectively (U.S. Census Bureau, 2009). Based on the U.S. Department of Labor, the daily work
time is around 764 , and hence we set the hourly wages to be   = 20,  = 32 and = 55, accordingly.13 The monetary cost of commuting per mile in our city is about $055. Ina household in which two adults commute to work, the total round-trip pecuniary cost per mileis  = 2 × 2 × $055 = $22. Assuming the commuting speed in the city is 20 miles per hour, we
set  = 01. Zick and Byrant (1996) estimate that each parent in a wife-husband family with 
children spends an averageof about¡¢ 13607˜15110  on childcare every day. To mimic thisfact, we choose  = 12 13607+15110= 07179.
2
We then describe how we calibrate the preference parameters. Since we have imposed the restrictions that  +   +  = 1,  +  +   = 1 and  +  = 1, there are six free parameters
to be calibrated. We jointly choose values for these six parameters such that the baseline economy
approximates a list of "stylized facts" of the U.S. economy. In particular, we target the following
six moments: (1) share of total expenditure on children, (2) share of total expenditure on housing,(3) share of children’s expenditure on housing, (4) share of total "budget" on leisure, (5) preferredproperty tax rate,14 and (6) fertility rate. Table 2 summarizes these moments and their correspond1 3 We choose to match the national average hourly wages because, for each of the three types of households, theaverage hourly wage for urban residents is fairly close to the average hourly wage for the overall population. For NH
and HA households, this is the case because they do not enjoy large urban wage premium. For BA household, thisis the case because the vast majority (around 90%) of college graduates live in the urban area.
1 4 In the model, the property tax rate is the fraction of rents that are collected by the local government. In reality,
property taxes are typically based on the value of the house, which can be computed as rents divided by interest rate = 0025. Hence, in the numerical section, we report the property tax rate as the fraction of house value that iscollected by the local government instead. Mathematically, this property tax rate is equal to (  +  − ) .
 

11

 



ing expressions in our model. Each of these six moments’ model counterpart is often a functionof a subset of the preference parameters. The only exception is the fertility rate, which depends
on budget constraint parameters that are already chosen in previous steps. Therefore, for each of
the three types of households, the preference parameters can be determined as the solution to a
six-equation, six-unknown equation system. The calibrated preference parameters are reported inTable 3a.
(Table 2, 3a about here)
The last set of parameters to be determined are the macroeconomic environment parameters,i.e., agricultural rent  , number of household   and peer group eﬀect parameters 1 and 2 . We
fix the total number of households at 500 000. In the data sample that we have access to, about
10%, 55%, 35% of the mothers are of NH, HA, and BA types, respectively. We assume that thisratio also applies to the fathers and expect the proportion of college graduates to be slightly higher
in cities than in the national survey. Hence, we set the ratio of NH, HA, and BA type households
to be 10%, 50%, and 40%, respectively. Given the total number of households, agricultural rent determines the size of the city. The lower  is, the larger the city is. We set agricultural
rent  = $1 237 per acre per month to match the endogenous calibration targets for the fringe
distance, which is around 10˜15 miles. The peer group eﬀect parameters 1 and 2 determine thedemographic composition of the two communities. We normalize 2 = 1. We show in the appendix
that the larger 1 is, the stronger the sorting pattern is. We set 1 = 10 to match the endogenous
target that over 70% of the BA type households reside in and constitute the majority of the West.15
In the "Real data" column of Table 3b, we present some key equilibrium statistics about family,
the labor market, and the housing market in the data. Next to it is the "Baseline" column, which
reports the counterpart generated by our baseline model. Our model matches the data reasonably
well.
(Table 3b about here)
2.3.2

Baseline Equilibrium

This model has multiple equilibria. We focus on the asymmetric equilibrium, which is the stableone.16 It also permits us to discuss cross-district sorting related to the tradeoﬀ between the wellbeing of parents and oﬀspring. We summarize the baseline equilibrium outcomes in a series oftables and figures. Figure 2a shows that the market rents decline as the households move awayfrom the CBD. More prosperous families take advantage of the lower housing rents in remote areas
to purchase larger lots. Hence, a household’s lot size increases along with its distance from the
CBD, as shown in Figure 2b. In each district, the NH type agents, who have the lowest wage
income, live the closest to the CBD, followed by HA (middle-income) and BA (richest) types. This
spatial allocation of the population is a feature of the Alonso-Muth type model (Alonso, 1964;
1 5 We choose 70% as the target because it is approximately the lowest fraction to ensure that skilled (BA type)workers can constitute the majority of the West community and determine the property tax rate. In the appendix,
we perform robustness checks to examine whether increasing this target will lead to diﬀerent policy implications. We
find that, when we use a higher fraction as the target, SFC policy’s positive eﬀect becomes bigger. Consequently, ourmain finding is that combining public housing and SFC can lead to Pareto Improvement and help the poor become
even stronger.
1 6 The instability of symmetric equilibrium in spatial equilibrium models has been noticed by the literature (e.g.,
Fernandez and Rogerson, 1996).

12Muth, 1969), which describes a spatial structure similar to many US metropolitan cities. Figure 2cshows that population density decreases as residents move toward suburban areas. Two economic
decisions made by households drive this spatial pattern. A household’s lot size increases with its
distance from the CBD. Moreover, more aﬄuent families who tend to live further away from thecity center also tend to have fewer oﬀspring.
(Figure 2 about here)
The three types of households also diﬀer in other ways. Table 4a and the "Baseline" columnof Table 3b together indicate a few things. First, parents who attain a higher level of education
have higher incomes, achieve a higher level of well-being, and tend to have smaller families. Second,the oﬀspring with better-educated parents tend to be better. Third, the diﬀerences in parents’
well-being are much lower than the diﬀerences in oﬀspring well-being. The reason is simple. Bettereducated parents earn higher incomes, bear fewer oﬀspring, and spend a substantial proportion of
their incomes on their children’s consumption, consequently increasing the expenditure per child.
They also choose to live further away from the CBD, which allows them to provide more spacefor each child. All of these eﬀects work together and magnify the diﬀerence in oﬀspring well-beingΩ across households. However, the impact of higher education levels on parents’ well-being is
ambiguous. Devoting a larger share of expenditure on children leads to a smaller expenditure sharefor the parents. This intra-household allocation of resources harms parents’ well-being. According
to our benchmark calibration, the positive eﬀects of a higher total income and lower rents (weakly)
outweigh the adverse eﬀects.
(Table 4a about here)
The "Baseline" column of Table 4b indicates (partial) income-sorting or imperfect sorting acrossdistricts. Almost half of the households (46.93%) in the West community are BA type, while only
29.40% of families in the East are of type BA. The West has a smaller proportion of NH-type
households (7.05%) than the East (14.52%). Consequently, the West’s average annual income ishigher than that in the East ($116 192 vs. $102 106). Such spatial sorting has several implications.
First, the West has more high-quality students and a more substantial peer group eﬀect. Because
most of the households in the West comprise BA-type, its property tax rate is 1.4673%, higher thanthat in the East (1.3992%). A higher property tax rate and better peer group eﬀect make the Westa more desirable community that attracts more households than the East (about 60% of the total
population).
Given that the two districts are ex-ante identical, it is interesting to note that (1) the population
shares of the two communities are so diﬀerent (40% v.s. 60%) and (2) the equilibrium market rentin the West, the more populated neighborhood, is significantly higher ($41 076 vs. $37 703 in theEast). Because the schools are financed through property taxes, parents in the West eﬀectively
pay more instructional expenses. As a result, the school quality, which is the product of per-childeducation expenditure and peer group eﬀects, is much higher in the West than in the East.
(Table 4b about here)
Note that the schools in the West have a better quality that benefits the children in the community. However, such benefit comes at the expense of parents’ well-being because they need to
pay for higher housing rents. Table 4a also shows how the parents make the tradeoﬀ between thewell-being of their oﬀspring and themselves. For each household type, the average well-being of theparents (children) is higher (lower) in the East than in the West.
13

2.4

Model-implied Rent gradient

We have shown in Table 3b that our model can quantitatively match many targeted moments
related to the US economy. To further strengthen the credibility of our model, we check whether it
can simultaneously match some moments that are not used as calibration targets. Specifically, we
ask whether the rent gradient generated by our calibrated model is consistent with what is typicallyfound in the literature. To do so, we use the model to create some “artificial data” of house rent
in diﬀerent locations in the city, and then run a regression that resembles some existing empirical
works. We will then compare the model-generated rent gradient with the empirical counterpart.
Notice that the rent gradient in the model is not targeted in the calibration process.
To calculate the model-implied rent gradient, we draw a random sample that contains 2 800
observations for both the East and the West from their corresponding population in the model.17
For simplicity, we adopt a semi-logarithmic regression equation,log  =  0 +    +    +  

(11)

where  is the rent,  is the distance from CBD, and  are other control variables.
Notice that the location choice is endogenous, and hence, in the empirical literature, regression
models like (11) often include control variables  to mitigate the endogeneity issues. The control
variables  normally includes (1) variables that reflect the heterogeneity of the landlords/tenants;and (2) variables that represent specific housing unit characteristics. In our model, for simplification,
we assume that all housing units are identical except concerning lot-size and location. As thedependent variable is rent rate per square mile, by construction,  accounts for the eﬀect of thediﬀerence in lot-size. Therefore, we do not need to add other hedonic variables into  (Malpezzi,2003). Also, we assume that adults who purchase/rent the housing units diﬀer only concerning
their wage rates  and degree of altruism toward their children  . Notice further that  is notobservable, and it is perfectly correlated with wage  in our model. Therefore, it suﬃces to include
wage  in  . Thus, (11) can be rewritten as:log  =  0 +    +    +  

(12)

Table 5a summarizes the descriptive statistics of the two samples. The average rent and wage
income are higher in the West than in the East because the former attracts more skilled workers
than the latter. Other things being equal, the West is a more attractive community because it
provides better education to the younger generation. Consequently, more land is occupied in theWest. On average, families living in the West reside further away from the CBD than those in the
East.
(Table 5a, 5b about here)
We estimate equation (12) for each of the two groups separately. Table 5b shows the regression
results. All of the coeﬃcient estimates are highly significant. In the West, an additional dollar inoccupants’ hourly wage decreases market rents by about 011%, and the same change raises housing
rents by 008% in the East. Our focus is the coeﬃcient of distance from the CBD. The estimate of  falls around 009 indicating that rental rates are about 9% cheaper at locations 1 mile awayfrom the CBD. This estimate is comparable to the empirical finding of Eberts and Gronberg (1982).
It suggests that our model, as a first-order approximation of “reality,” is reasonably reliable.
1 7 We also estimate a large sample (over 100,000 observations) version. The results are almost identical to the small
sample version and are therefore omitted here.

14

3

Policy Analysis

Based on our calibrated model, we conduct a series of counterfactual experiments to analyze thewelfare implications of various education and housing policies. In this section, welfare can refer
to household utility, the well-being of parents, or oﬀspring, and we will clearly distinguish theirdiﬀerences whenever there is a chance of ambiguity. Our primary goal is to explore the possibilityof a Pareto-improving policy package. To facilitate the comparison with the previous literature,
the education policy that we study is school finance consolidation. For housing market policies,
we consider the provision of public housing units and housing voucher programs. To build ourintuitions in this highly complicated environment, we first study each regime separately. Then, we
consider some policy packages and their overall eﬀects.
To compare the well-being of parents, oﬀspring, and society’s average across diﬀerent policy
regimes, we turn to a widely used consumption-equivalent measure. More specifically, we search forthe discount factors/multipliers,  ,  ∈ { Ω   }, which must be imposed on the consumptionof parents and children in the new equilibrium to push their well-being back to their levels at thebaseline equilibrium. Therefore,   1 indicates that the households are worse oﬀ in the new
equilibrium because we need to multiply the amount of consumption of the household members tobring them back to the baseline utility level. Similarly,   1 indicates that the households are
better oﬀ in the new equilibrium. We report the value of 1 −  ,  ∈ { Ω   } in the summary
table so that the value is positive (negative) when welfare increases (declines). Here is our formal
definition.
Definition 2 For a particular group of households with average utility level, the well-being of parents and oﬀspring equal to  ,  , and Ω in the baseline equilibrium, the welfare measure ,  , and Ω satisfy the following equations:( (      )) =  ; ( (   )) =  ;(Ω (Ω   )) = Ω 
where  (    ),  (  ) and Ω (  ) are the average well-being of the household, parent
and oﬀspring in the new equilibrium, respectively.

3.1

School Finance Consolidation (An education policy may impact the
housing market)

School Finance Consolidation (SFC), or School District Consolidation, is an apparent post-war trend
in the US. The number of school districts that provide elementary and secondary education had
dropped from 117,108 in 1939~1940 to 13,862 in 2006-07 (National Center for Education Statistics).
In HY’s model setting, the central government moves all students to a single school, which it finances
through the property taxes collected from all of the lands in the economy. HY (2007) calibrate
their baseline equilibrium to match a representative United States city circa 1997. Based on theparameter set obtained from the calibration, they show that enforcing SFC hurts everyone in theeconomy. Their finding supports Fischel (2006) arguments, which describes consolidation policy as
an external distortion leading to welfare decline. In practice, although the government can equalize
the per-child educational spending, but may not be able to equalize the quality of the peer group, as
it depends on the agents’ choices made in equilibrium. To complement the literature, we assume15

that the two communities have the same per-child educational spending but can diﬀer in school
quality when considering SFC.
We summarize the new equilibrium in the “SFC” column of Table 3b and 4b and compare
this SFC equilibrium with our baseline equilibrium. After the consolidation of school districts, theproperty tax rate is voted by all the adult members in the economy. The same tax rate then applies
to both the West and the East communities. Because HA-type adults comprise the majority, theirpreferred property tax rate of 13992% will be the equilibrium property tax rate. It is slightly lower
than the level preferred by the BA. The drop in property tax rate decreases per-child educational
expenditure in the West, which induces BA- and HA- type households, who value their well-beingof oﬀspring, to move from the West to the East. Their movement increases (decreases) the average
income in the East (West), which boosts (lowers) the rent in the East (West). Because school
funding is derived from property taxes, which are proportional to housing rents, the per-childschool funding is higher (lower) in the East (West) than before. The movement also tends todecrease the gap in peer-group quality between the two communities.
In summary, the SFC narrows the school quality gap between the West and the East, which
substantially reduces cross-districts sorting at the equilibrium. As a result, the population is evenly
distributed in the two districts after the SFC policy is imposed. The thick solid lines in Figure 3ashows this pattern explicitly.
(Figure 3a about here)
Our welfare results are summarized in the "SFC" column of Table 6. Unlike in HY (2007), theSFC policy makes all types of households better oﬀ. On the one hand, SFC indeed restricts school
quality choices to one, potentially resulting in a more substantial average individual deviation from
optimal levels of school quality. On the other hand, the SFC policy substantially reduces the gap of
school qualities, rents, and population compositions between the two communities. Hence, underthe baseline equilibrium, the land in the West is "over-used," while the land in the East is "underused." In contrast, under the SFC policy, the land in both districts is evenly utilized, which leads
to more eﬃcient land use. Indeed, Table 3b shows that the population density is lower under the
SFC policy than in the baseline (5.43 v.s. 5.53). Whether the SFC policy is welfare-improving or
not depends on which of the two eﬀects dominates the other. It, in turn, depends on how extreme
the sorting pattern is in the baseline case since excessive sorting leads to ineﬃcient land use. The
sorting pattern is more extreme in our baseline case than the counterparts of the previous HY
papers.18 As a result, when we remove sorting, the positive eﬀect from the more eﬃcient use ofland dominates the negative impact of restrictions on choices. Our focus here is not to overturnthe conclusion of HY (2007, 2013) on SFC, but rather to highlight that SFC could potentially leadto Pareto Improvement in a variant of HY.
Furthermore, Table 6 shows that the adults in BA- and HA-type households enjoy better lives
while their oﬀspring suﬀer. In other words, SFC is Pareto improving at the household level, but not
the individual level. The reasons are simple. First, some of the BA- and HA-type families moveto the East where the housing rents are lower than the West, and hence, they can consume larger
lot sizes than they would if they stayed in the West. Second, cross-community sorting becomesmuch weaker. Recall that in our model, parents with a higher level of education tend to produce
children who would be better students. Consequently, children from BA- and HA-type households
are faced with lower peer group quality on average because they are pooled with children from
NH-type families. The converse applies to NH-type families.
1 8 For more robustness checks for alternative targets of community compositions, see the appendix.

16

(Table 6 about here)

3.2

Public Housing and Housing Vouchers (Housing market policies may
impact education)

In the previous section, we focus on SFC, an education policy that has implications on the housing
market. In this section, we consider two housing aid policies, including government-subsidized
public housing and housing vouchers, which would, in turn, aﬀect education quality. These housing
aid programs are designed to provide low-income groups with necessary residential spaces. In ourmodel, 10% of the households are of the NH-type. The income in these families is about 60%
and 33% of adults’ wages in the HA- and BA-type households, respectively. Hence, we assume
that public aid programs are only open to NH-type families. We further assume that all NHtype households can receive assistance.19 A central government finances the programs through theincome taxes paid by the adults. We assume that the NH-type adults, who would enjoy the housing
policy benefits, do not need to pay the income taxes and that all of the other adults are faced withuniform income tax rate . One may interpret this as a form of progressive taxation.20
We introduce public housing into the baseline model and study how it aﬀects economic outcomesand social welfare. Under this policy, each participating household enjoys a lot of size   andcontributes     to the program. The government receives contributions from the participants
and income taxes from the non-participants. Then, the government purchases land and builds
housing units for the program participants. Therefore, a particular program participant’s decision
problem is reduced tomax    ³´ ¡¢  (          ) =          1− subject to   () =   () +    () +     +  () +  () () +  () =   (), and the
For the sake of space, we show the exact formula for the new fertility decision, latest children’s well-being Ω  () in the appendix.
Recall that one of the main goals of public housing policies is to assist children from economically
adverse families and promote intergenerational mobility. Hence, we are interested in the eﬀects ofgovernment housing programs on the well-being of children from NH-type families. With the above
derivations, it is simple to prove the following propositions.

Lemma 1 At a given location, NH-type adults under housing program give birth to fewer oﬀspringthan they would at the baseline equilibrium if their contribution to the program is more prominent
than their expenditures on housing in the baseline situation
If      [(1 +   )  (        )  (       )] ,then ()  () 
1 9 In practice, more families are eligible for public housing programs than receiving assistance in the US. Leung etal. (2012) study the case when the public housing units are "under-supplied," and low-income families can get thoseunits through rationing.
2 0 Again, the uniform tax rate across HA- and BA-type households is imposed to simplify the analysis. The crucial
point is that NH-type families do not need to pay the income tax while receiving the benefits. The results would
carry to the environment when diﬀerent groups of agents all face diﬀerent tax rates.

17

Proposition 4 Given the same location and school quality, when NH-type households receive alarger lot size and pay more under the public housing program than in the baseline equilibrium, they
produce oﬀspring who would become higher-quality students.
If    and   [(1 +   )  (       )   (        )]  (        ) ,then Ω  ()  Ω  () 
The intuitions are straightforward. Facing the government-subsidized rent with public housing
units that are larger than they would otherwise rent from the market, the NH-type households
choose to have fewer children. Hence, as the number of children decreases, per-children spending
could increase, and children from those households could become better students. Moreover, thesecond inequality guarantees that the public housing policy does not reduce the consumption of lot
size. Consequently, the well-being of oﬀspring improves.
3.2.1

Public Housing Policy 1: Units Located at the Middle Ring of the City

In practice, public housing units are not evenly distributed within a city. In this paper, we consider
the case where public housing units are built in only one district, the East. The central government
must decide the locations of those public housing units within the neighborhood. We investigate
two alternatives. First, we consider a case similar to Leung et al. (2012), where public housing
units are located between  and  miles from the CBD. We assume that the land in this area
is rented from the market. Hence, the central government needs to calculate the rent she needs
to pay in a competitive rental market. The rental rates are determined at the auction with type  ∈ { } households, and agricultural workers. Since the analysis is analogous to thebaseline case with no public housing, we refer the interested readers to the appendix for details.
In addition to the rental costs, the central government must make a payment known as the
Payment in Lieu of Tax (PILOT) to compensate the local government in the East for some of theproperty tax revenue lost due to the public housing program. Here, we follow Leung et al. (2012)
to make the simplifying assumption that the PILOT is equal to the property taxes placed on the
public housing recipients’ contribution to the program,  =         (13)where   is the total number of NH-type households.
This program is financed by the residents’ contribution to public housing and the income
taxes paid by non-participating households. The former can be calculated by the simple formula =      . The central government adjust the parameters (,   ,   ,  ,and  ) of the public housing program so that the government budget constraint holds,  +   =  + (14)

where   is the sum of rents that the central government needs to pay to the private sector
to obtain the land for public housing,  is the total income tax revenue. We reserve the full
expressions of   and  in the appendix.
To compute the Public Housing Equilibrium, we impose some parameter values. The public
housing units are located only in the East and start from 4  away from the CBD. The familyspecific lot size is set to be 0001 square mile, which is about 25% larger than the average unit
18

within that band in the baseline equilibrium. We can then determine the outside boundary of
the public housing band, which is 6916 miles. The central government charges an income tax
rate  = 08% to balance its budget, which endogenously matches the calibration target for theparticipant contribution to the program of $2214 per month. The increase in utility that theparticipants obtain by joining the public housing program is close to the one derived from a 25%
consumption subsidy in the baseline equilibrium.
The thick dash-dotted lines in Figure 3a and the “PH1” column of Table 4b display powerfulsorting in equilibrium.21 Notice that the NH-type households, who are the public housing recipients,
all allocated to the East. They also care the least about the well-being of the oﬀspring. Thus, thepublic housing program leads to a sharp decline in the peer group eﬀect in the East. In response,
all of the BA-type and most of the HA-type households choose to live in the West. This intense
sorting makes the two districts significantly distinct. Both the property tax rates and the lot sizes
in the two communities are almost identical. However, with the strong sorting eﬀect, the West’s
market rent exceeds that in the East by a large margin, which leads to much higher expenditure
on education in the West. Combined with the peer group eﬀect, school quality in the West is alsomuch higher.
The "PH1" column of Table 6 presents the welfare eﬀects of this policy. At the household level,
the public housing residents (NH) are better oﬀ, and all of the other types are worse oﬀ. This result
is intuitive. Under the public housing policy, all of the household location choices deviate from theeﬃcient ones. As a result, the welfare of the whole economy declines. A careful inspection of theresults reveals that the improvement of NH-type households’ happiness level comes from an increasein parents’ well-being. These parents can enjoy large lot sizes without paying more. However, there
is no free lunch. The education quality in the East drops dramatically as all NH-type households
concentrate in one district. Consequently, public housing policy hurts the well-being of NH-typechildren.
On the other hand, the proportion of oﬀsprings from BA-type families increase in the West, andthrough the peer group eﬀect, BA-type children benefit from the outstanding quality of educationin the West. However, the public housing policy hurts adults from BA-type families the most for
two main reasons. First, the BA-type households all locate in the West and hence drive up the
housing rents significantly. Second, because they do not constitute the majority of their community,
the electoral outcome of the property tax rate is not their preferred outcome. Both parents andoﬀspring of HA-type households are worse oﬀ to a relatively mild extent.
3.2.2

Public Housing Policy 2: Units Located at the Edge of the City

In the previous section, we assumed that public housing units are located in areas that would
otherwise be occupied by non-participating households. Consequently, this type of public housing
policy decreases the amount of accessible land and probably results in higher market rents andless-eﬃcient land allocation. In this section, we consider an alternative public housing policy thatwould have minimal eﬀect on the already occupied land. More specifically, we assume the public
housing units are located at the edge of the city. We compare the new equilibrium with the one in
the previous section concerning resource allocation and welfare. Under this scenario, the households
who are not eligible for public housing would first compete with agricultural use on each piece ofland they desire. Then the government builds the public housing unit outside the fringe of the East
2 1 Although the sorting in our public housing equilibrium is qualitatively more substantial than that shown by
Leung et al. (2012), they are mostly similar in spirit.

19

district. In other words, the public housing units are located in areas that would not be occupied
by diﬀerent households, and thus, the government acquires the land at the agricultural rent   For
the sake of comparison, we maintain the assumption that   = 0001 square mile and  = 08%.
Each participating family must contribute $1994 to the program so that equation (14) holds. The
required contribution is smaller than that in the previous case, as the market value of the public
housing band is less than before. In the equilibrium, this band is located between 60871  and82997  away from the CBD. The thick dotted lines in Figure 3a show the housing rents andoccupants of all locations. For the public housing area, market rent is defined as the participants’contribution (per square mile).
A comparison between the "PH1" column with the "PH2" column of Table 4b highlights thediﬀerence between the two public housing policies. Although we still observe no BA-type householdin the East, the proportion of HA-type families residing in the East increases from 929% to 2133%.
The spatial sorting is weaker. At the same time, the BA-type households outnumber the HA-type
households in the West and determine the property tax rate to be 14673%. The corresponding“Community Comparison” column of Table 4b summarizes other essential statistics of this equilibrium. As more HA-type households move from the West to the East, the demand for land in
the East (West) becomes stronger (weaker), resulting in higher (lower) housing rents. Per-childeducation spending increases in both communities. In the East, this is a consequence of higher
rents. The community attracts more HA-type households and has a more substantial peer groupeﬀect than before. In the West, although housing rents decrease slightly, the property tax rate andtotal tax income increase. We observe better school quality in both communities in equilibrium.
Comparing the “PH2” with “PH1” columns in Table 6, we conclude that all of the family members from all of the household groups are better oﬀ after the government puts the public housing
units outside the fringe. It confirms the intuition that decreasing the amount of accessible land hasan adverse welfare eﬀect on the whole economy. In a sense, the government kills two birds with
one stone (i.e., implementing the second, at-the-edge type public housing policy, rather than thewithin-the-city type that is considered in the previous section). First, it can improve the welfareof the most impoverished families (NH) at a lower welfare cost imposed on other households in theeconomy. Second, it can reallocate resources between adults and children. Relative to the public
housing policy where units are located in the middle ring of the city, children are better oﬀ, andparents are worse oﬀ when the public housing units are located at the edge of the town.
Notice that our results are also broadly consistent with other empirical studies of public housing
unit residents. For instance, Olsen and Barton (1983) study the benefits and costs of public housing
based on New York City data, and find that “...the mean benefit of the program to these families issubstantial relative to their mean income but small compared with the cost to taxpayers.” Currie andYelowitz (2000), Jacob (2004), among others, also confirm that public housing per sec can improve
the younger generation of public housing residents.22
3.2.3

Housing Voucher

Instead of building public housing units and hence directly changing the housing consumption, thegovernment may distribute cash, also known as the housing voucher program (VC). Under thisscheme, the government collects income taxes from the skilled workers and redistributes them tothe poor in the form of a housing voucher, which can only be used to purchase housing services.
2 2 Public housing plays a more critical role in Asia. Among others, see Leung and Tang (2015), and the reference
therein.

20

Hence, program participants can still choose their desired locations, desired lot sizes, etc. We would
also examine the welfare implications of the housing voucher policy under the current setup, as we
did to other policies.
Here are the details of VC. For non-participating households, the utility maximization problem
is the same as that under public housing policy. For participating families, they are exempted fromthe income tax but instead receive housing vouchers from the government that amounts to  , which
is for housing consumption only. Hence, the program participant’s budget constraint becomes  () =  () +   () +  () +  () ()+ max{0 ( () +  () ())(1 +   ) () −  }

(15)

Since the utility function is monotonically increasing in lot size, all participating households spend
no less than the amount of their vouchers on housing. Whether they will have a higher expenditureon housing depends on their housing preference and income levels.
The rest of the program is simple. All of the lands are rented out through the same market
mechanism, as described in previous sections. The government chooses policy parameters (,  )correctly so that the income tax revenue  is equal to the total cost of financing the program,which is    .
The computation of the market equilibrium under the housing voucher program is similar toother cases. To be compatible with previous sections, we keep the income tax rate at 08%. In
equilibrium, it implies that each participating household receives the housing vouchers that amounts
to $2312 each day, and all participants spend more than this amount on housing units. As weobserve from the thick dashed lines in Figure 3a and the "VC" column of Table 4b, householdsorting is more substantial than that at the baseline and weaker than the one under public housing
policy (when all public housing units are placed outside of the fringe distance). Although there
are no BA-type households in the East under the VC program, we observe NH-type families thatamount to 269% of the total number of households in the West. Land demand and market rents
increase in the West, where all of the aﬄuent families live. They constitute the majority in theWest and determine the property tax rate to be 14673%, which is the same as the baseline case.
Having higher housing rents allows the West’s local government to collect more property taxes andprovide more funding to the schools.
Conversely, in the East, the average market rent declines from $37 703 to $35 172, indicatingthat less funding for schools is collected. The change in school funding amplifies the gap in school
quality between the two communities. Recall that skilled workers care more about their oﬀspringhence tend to produce children who would become higher-quality students. Under our model setting,schools with higher percentages of students from high-income families provide better peer groupquality. Therefore, the gap in school conditions is further enlarged because the West now attracts
all BA-type households. As a result, the educational qualities are 741 in the West and 123 in theEast. The counterparts in the baseline case are 564 and 280, respectively. More substantial sorting
occurs under the VC program, and an amplified gap in school quality is observed. NH-type parents
seek a better education for their oﬀspring and are willing to pay higher housing rents.
The "VC" column of Table 6 shows the welfare implications of the housing voucher program.
The overall welfare changes are very similar to those in the public housing case. The average utilityof program participants increases by about 185%, while other families’ overall welfare declines
slightly, as in Leung et al. (2012). The economy-wide average welfare decreases by only 0.19%.
Hence, the housing voucher program incurs less welfare loss than the public housing policy because
the former imposes fewer restrictions than the latter on household choices. Moreover, the increase21

in NH-type households’ welfare from the housing voucher program is less costly than that from thepublic housing program, which leads to a smaller welfare loss for HA- and BA-type families.
On the other hand, while the housing voucher program and public housing policy deliver similar
aggregate utility, their intra-household welfare implications are very diﬀerent. Recall that the
public housing policy improves the well-being of children from low-income families. Hence, the gaps
in children’s quality as students between diﬀerent households are reduced, potentially increasing
intergenerational social mobility. On the other hand, our welfare results show that VC policy
enlarges the gap between aﬄuent and low-income families. When the school quality of two districts
becomes drastically diﬀerent, skilled parents tend to cluster more heavily in the community withhigher rents and better schools. Unskilled parents do the opposite.

3.3

Policy Package: Combination of School Finance Consolidation andHousing Market Policies

The previous section, which studies each of the education and housing policies in isolation, delivers
the following lessons. First, when the baseline sorting pattern is extreme, imposing school financeconsolidation may increase the aggregate utility. Second, an appropriately designed public housing
program can be a handy tool concerning aiding children from low-income families. Third, housing
voucher tends to enlarge the quality gap between the children from high income and low-income
families. Based on these observations, this section addresses a natural question: is it possible to help
children in need while keeping other families at least as well oﬀ as before by combining educationand housing policies?
(Figure 3b about here)
We consider diﬀerent policy packages. One possibility involves combining school finance consolidation with a housing voucher program. Based on our model calibration, the housing voucher policy
alone does not perform well regarding increasing the living and educational qualities of children in
need. However, it is still possible that a housing voucher program with school finance consolidationdelivers better outcomes. The thick dashed lines in Figure 3b depicts the rent-distance relationshipof this policy package. The "SFC+VC" column of Table 6 summarizes the welfare changes causedby this combination. Comparing this column with the "VC" column, we find that adding a school
finance consolidation policy to the housing-voucher-only regime makes all types of households inthis economy better oﬀ. This finding may not be surprising, given that we observe similar results
when adding a school finance consolidation policy to the baseline economy. However, this policy
package is not a Pareto improvement over the baseline situation at the household level because nonparticipants are slightly worse oﬀ. Moreover, it does not help children from low-income families, as
their average well-being slightly decreases (−033%) comparing with the baseline case.
Hence, we consider another policy package, which combines the school finance consolidation
with public housing. It equalizes the per-child funding in the two districts and puts all public
housing units outside the fringe distance in the East. As the thick dotted lines in Figure 3b andthe "SFC+PH" column of Table 4b show, the cross-community sorting in this equilibrium is morepotent than the baseline case but much milder than the scenario of public housing policy only.
This observation confirms our conjecture: SFC makes the two districts less diﬀerent and weakens
the sorting magnitude. The "SFC+PH" column of Table 6 shows the welfare results of this policy
package. Comparing with the baseline equilibrium, all three types of households are better oﬀ.
We also observe a substantial improvement in the well-being of children from NH-type families.
22Hence, the government can bring welfare improvement to the economy and increase the well-beingof children from low-income families.

3.4

Short- vs. Long-run Analysis

Thus far, our policy analyses have followed the tradition of urban equilibrium models, which assumes
that markets are initially in equilibrium, upon which policy is imposed unexpectedly. Agents reoptimize their choices, such as their location and consumption. The markets instantly clear, andwe compare the welfare under the new equilibrium with the original one. With our static model,such analysis is interpreted as a long-run assessment of public policies. However, as Quigley andSwoboda (2010) and others recognize, some choices cannot be altered in the short-run.
Consequently, the long-run welfare implications for some policies can be dramatically diﬀerent
from their short-run counterparts. For instance, policy changes typically do not significantly alter
the housing supply in the short run, which is often assumed to be fixed. However, the long-run
supply of housing should arguably be flexible. Housing decays over time, and economic agents candecide to replace it. As such, it may be vital to re-examine our welfare results while considering
the short-run rigidity.23
To facilitate the comparison, we make minimal modifications to the current framework.24 Morespecifically, we diﬀerentiate variables according to their corresponding "flexibilities." In our model,
all of the choices are flexible in the long-run. In the short-run, some decisions are more flexible
than others. Table 7 summarizes the flexibility of diﬀerent choices. We consider fertility choices tobe the most inflexible. It is physically impossible to decrease the number of children that parents
have already had. It would take at least another year for parents to bear an additional child in any
given year.
Furthermore, residential choices cannot be easily adjusted. In many places, the term length ofa residential rental lease is typically one year or longer. Depending on the market condition, it may
take a similarly lengthy period to sell a house. On the other hand, households are free to decide
how many non-durable goods to consume, even in the short-run. We also assume that the amountof leisure can be easily adjusted.
The time allocation in our model is not entirely flexible. For instance, childcare time is a linear
function of the fertility rate and is therefore inflexible in the short-run, just like fertility choices.
A worker’s commute time is also rigid, as it is merely a multiple of the distance from the worker’s
home to the CBD. However, as some workers might adjust their work hours, the choice of leisuretime can be modified to a certain extent. Hence, we treat leisure time as a flexible choice.
(Table 7 about here)
The recognition that households are not allowed to move in the short-run restricts the set ofpolicies for which we can conduct short-run analysis. Since the public housing experiments force
program participants to relocate across districts, we perform only a "short- vs. long-run" analysis
for the school finance consolidation and housing voucher experiments and the experiment thatcombines the two.
2 3 In this section, the term “goods” is inclusive. All of the parents’ choices, including those related to fertility,
consumptions, space, and leisure, are considered “goods.”
2 4 Short- vs. long-run analysis is not a perfect substitute for the transition dynamics analysis, which is feasible only
when a fully dynamic general equilibrium (DGE) setting is available. In this paper, we focus on a static setup andleave the DGE for future research. For a review of the related literature, see Leung (2004), Leung and Ng (2019),
among others.

23

Table 8a summarizes some relevant statistics. Household’s consumption choices and working
hours in the short and long runs are similar. In our model, a household’s optimal expenditure on
consumption and leisure are constant fractions of their total potential income. Hence they do notdepend on their lot size choices. Thus, the fact that a household cannot adjust its residential areain the short run does not aﬀect its consumption and leisure decisions.
Nevertheless, the short- and long-run choices are not identical for at least two reasons. First,
a household’s location decision aﬀects its potential income and, consequently, its consumption andleisure choices. Therefore, as the economy has significantly diﬀerent household space distributionsin the short- and long-run, the average consumption and leisure time take diﬀerent values in thetwo situations. Second, a household’s decision on leisure and working hours are subject to time
constraints. Households allocate their time endowments to diﬀerent activities, including childcare,
commuting, and leisure and work enjoyment. Both childcare and commute time are fixed in the
short-run, as fertility and location choices are set in the short-run. Because agents are allowed to
re-optimize these decisions in the long-run, the leisure choice is aﬀected.
(Table 8a about here)
When the school finance consolidation is imposed, school quality almost does not change when
the time horizon moves from the short-run to the long-run. Recall that in this model, the school
quality of a community is simply the product of average education expenditure and average student
quality. Both of these components depend on the population share of highly educated households
in the city because their presence tends to drive up the rents, leading to more school funding. Also,
higher-quality students tend to be associated with higher-income parents. Under the school financeconsolidation regime, there is only one school district whose composition is always the population
distribution in both the short and long runs. Consequently, under SFC, school quality does not varyas much as under alternative policies when the economy moves from the short-run to the long-run.
Table 8b reports welfare comparisons. Households in the East are generally better oﬀ than thosein the West in the short run, especially when school finance consolidation policy is imposed. In
the long term, economic agents are mobile. Hence, the welfare of economic agents would dependon their types and not the residential location. Our setup enables us to conveniently decompose
the household’s total welfare into the well-being of parents and oﬀspring. The extra benefits that
residents in the East obtain come mainly in the form of higher well-being of the children. Recall
that the East has lower average rental rates and peer group quality. SFC policy enables children
from the East to go to better schools in the West, bringing significant welfare improvement tothose households. Simultaneously, because the agents are unable to move in the short run, families
in the East are eﬀectively given a "free-ride" on the better education provided in the West. In
other words, short-run rigidity prevents the housing market from functioning eﬃciently, and agents’
welfare becomes location-dependent.
(Table 8b about here)
Another important finding is that long-run equilibria are not always better than short-run
equilibria. According to Table 8b, the long-run average welfare of households is higher than the
short-run counterpart only when school finance consolidation policy is imposed. When the housing
voucher program alone is imposed, the economy’s general welfare increases in the short-run (relative
to the laissez-faire benchmark) but deteriorates in the long run. In particular, Group 2 agents (i.e.,
HA-type) is hurt in the short-run (relative to the laissez-faire benchmark) and hurt even more in24the long run. This finding is at odds with conventional wisdom. The intuition under the complete
market is as follows. Moving from the short-run to the long-run, households have more choices,and economic agents are usually better oﬀ. However, our agents live in a model with many formsof market imperfection. They include the peer group eﬀect (which is a form of externality) andthe non-convexity of consumption caused by the household’s location choices. Thus, HA-typeagents can be hurt more if the proportion of BA agents in their community drops dramatically inthe long run. Our numerical exercises with plausible parameter values suggest that it is indeed
the case. It is because when the housing voucher program is imposed, wealthy families cluster
more intensively in the West when given such an opportunity, greatly enlarging the diﬀerence incommunity composition.25

4

Concluding Remarks

This paper builds a simple spatial general equilibrium model that embeds fertility choice, location
choice, and work-leisure choice in a unifying framework. Our model distinguishes the welfare of thechildren from their parents. Our model distinguishes the welfare of the children from their parents.
It enables us to diﬀerentiate policies that benefit all at the household level from those that help allat the individual level. Our calibration confirms that the model can simultaneously match specificfamilies, labor markets, and housing market outcomes. While we do not include the rent gradientas a calibration target, our model-implied rent gradient also matches previous empirical estimates.
We analyze various educational and housing policies and their combinations. We demonstrate thatpublic housing policy can induce intense sorting among diﬀerent types of agents. The welfare result
depends crucially on whether the public housing units are built on land that would otherwise be
used. On the other hand, while the housing voucher program can increase the overall welfare of
low-income households, it is the parents who capture the welfare gains, and their oﬀsprings can behurt.
We also demonstrate that under some parameterization of the model, the policy package ofschool finance consolidation with appropriately located public housing units can help children from
low-income families without hurting other types’ welfare. We stress that our main intention is tohighlight an empirically plausible possibility, rather than producing results that can be directly
taken for policy recommendations. In practice, there are many issues to consider. For instance,
there is a lag of public housing construction, which means that by the time the public housing units
are available for end-users, the supply could be insuﬃcient or in excess. After the public housing
units are occupied, the demand conditions continue to vary, and hence both the quantity and theprice may fail to adjust to the market. Local governments might also have fiscal commitment issues.
As a result, many public housing projects in the U.S. are disappointing (Freeman, 2002; Malpezzi,
2020; Olsen et al., 2005; Wiltz, 2019).
On the other hand, we observe that public housing units are provided in Asia and Europe
under diﬀerent labels, while some form of school finance consolidation is implemented (Silver andDanielowski, 2019). Our analysis justifies this type of policy regime. We leave it to future research
to investigate whether the current level of government involvement in those Asian and European
countries are indeed at the socially optimal level.
We also compare the short-run (i.e., when some decisions are constrained), versus the long-run
2 5 The result here is consistent in spirit with studies of “second-best theory” in which the laissez-faire equilibriumis ineﬃcient.

25

(i.e., when all arrangements are flexible). To the best of our knowledge, other than the study by
Quigley and Swoboda (2010), such short- vs. long-run analysis has been relatively underexploredin the urban economics literature. For instance, we find that the housing voucher program canincrease the average welfare of the economy in the short-run but decreases it in the long run withplausible parametrization. This result also points to the possibility that some policies which canbring long-run gains to the economy may not be implemented due to their short-run adverse impact.
Besides, we demonstrate that in some situations, middle-income agents can lose more in thelong run than in the short run after a policy change. It is because their utility levels depend on the
proportion of high-income agents living in the same neighborhood. In the short term, agents stayin the original houses, and hence the welfare loss is simply the direct policy eﬀect. However, inthe long run, the high-income agents may move to another community, driving down the originalneighborhood’s school quality. The middle-income agents either stay and live with the depreciatedschool quality or migrate with the high-income agents and face possibly higher rents and taxes.
Future research should further explore such considerations.
There are critical dimensions to be further explored, as well. For instance, our model has a simple
commuting cost structure, while the reality may be more complicated (e.g., Leroy and Sonstelie,
1983, Yilmaz, 2019). Chetty and Hendron (2018b) find that long commute time is associated with
low intergenerational mobility. Thus, future research should re-examine the optimal location of
public housing units with a more realistic transportation system. Urban policy analysis in thoseenvironments remains a challenge to be met.

26

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30

Figure 1. Household Decision, Peer Group Effect and Public Expenditure EffectFigure 2. Rent, Lot-size, and Population Density at the Baseline Equilibrium
(a)

(b)

(c)

Figure 3a. Rent-Distance Curve under Alternative Policy Regimes

Figure 3b. Rent-Distance Curve under Policy Packages

Table 1 Educational Attainment, Annual Income and Fertility Rate
Level of
Education

Male
Income(in $)

Femaleincome(in $)

Income of
PseudoHousehold(in $)
46,192

FemaleFertility
Rate

Average
Income($) acrossGroupsAverageFer. Rate
acrossGroupsLess than
26,604
19,588
9th grade
2.521
51,432
2.521
9th to 12th
33,194
23,47856,672
gradeHigh school
43,14032,22775,367
1.954
graduate
1.918
87,479Some
52,58036,553
89,133
1.892
college
Associate
55,63142,307
97,9381.869
degreeBachelors
1.682
degree92,81562,198155,013155,013
1.652
Graduate or
1.597
professional
degreeTotal
62,44544,857
107,302
1.888
N/A
N/A
Note: Income data is in current dollar and is from U.S. Census Bureau (2009). Fertility data
is from U.S. Census Bureau (2010). Due to data availability, we calculate the across groupaverage income by taking simple average. The average fertility rate is accurately calculated.

Table 2 Statistics and Expressions
Statistics
Share of Total
Expenditure on Children

31% - 47%

Share of Total
Expenditure on Housing

20%

Share of Children’s
Expenditure on Housing

31%

Share of Total‘Budget’ on Leisure65% - 70%

Property Tax Rate
1.22% - 1.47%(of house value)
Fertility Rate
2.521 - 1.652

Expression( o   o ) koi( p   p ) k ip  ( o   o ) koi

 o koi   p k pi
( p   p )k pi  ( o   o )koi
o
 o  ok ip1  koi  koi
 p k ip   o koi   koi
kni
Yi (r)
wi ckTi

Table 3a Symbols for Variables and Parameter Values(Panel 1)
Variable Symbolslqnog ( no )  gno C ( no )  cnorInterpretation
lot size forparents (offspring)consumption goods
for parents (offspring)
Parent leisure time
Educational quality of the public school
number of offspring
degree of altruism
toward each childtime cost of bearing no offspring
distance from the CBD

(Panel 2)
Parameter
Symbols

Short Description
per mile pecuniary cost

Parameter
Value
2.2

per mile commuting time cost

0.1

time cost of bearing eachoffspring(Intercept) parameter in peergroup effect function

0.7179(slope) parameter in peer groupeffect function
Wage rate of the BA (workers
with bachelor degrees)

1

10

55

32

Wage rate of the HA (workers
who graduated from high schoolor obtained associate degrees)
Wage rate of the NH (workers
who did not graduate from highschool)Population of BA

200,000

Population of HA

250,000

Population of NH

50,000Agricultural rent per acreper month

1,237

20

SourceSimilar to Hanushek andYilmaz (2007)
Similar to Hanushek andYilmaz (2007)
Zick and Byrant (1996)
Target: 70% of the populationof the West is BA (bachelor
degree holders)
Normalization
Our calculations based on the
U.S. Census Bureau andDepartment of Labor
Our calculations based on the
U.S. Census Bureau andDepartment of Labor
Our calculations based on the
U.S. Census Bureau andDepartment of Labor
Our calculations based on the
U.S. Census Bureau
Our calculations based on the
U.S. Census Bureau
Our calculations based on the
U.S. Census Bureau
To match a fringe distancearound 10~15 miles, similar
to Hanushek and Yilmaz(2007)

(Panel 2, continued)
Parameter
Symbols

Short Description
Utility weight of offspring for BA
Utility weight of offspring for HA
Utility weight of offspring for NH
Utility weight of parent for BA,
1
Utility weight of parent for HA
1
Utility weight of parent for NH
1
Utility weight of log size in offspring
direct utility
Utility weight of consumption goods inoffspring direct utility
Utility weight of public school quality
in offspring direct utility
1
Utility weight of log size in parent
direct utility
Utility weight of consumption goods inparent direct utility
Utility weight of leisure inparent direct utility
1(slope) parameter of the parent
altruism towards children(scale) parameter of the parent altruism
towards children

Parameter
Value
0.176
0.166
0.141
0.824
0.834
0.859
0.25880.5761

Target
To jointly match six targets,
including:(1) share of total
expenditure on children,(2) share of total
expenditure on housing,(3) share of children’s
expenditure on housing,(4) share of total "budget"on leisure,(5) preferred property tax
rate,(6) fertility rate.

0.1651

0.04
0.18
0.78

0.8
1

Normalization

Table 3b Statistics and Calibration Results
TargetReal data

AnnualIncome($)
Time Spenton Working
per Day(hour)

Group 1
Group 2
Group 3
Group 1
Group 2
Group 351,43287,479155,013

Fertility
Rate

Group 1
Group 2
Group 3
Group 1
Group 2
Group 3

2.521
1.918
1.652
1.3607~1.5110Group 1
Group 2
Group 331%~
47%

Group 1
Group 2
Group 3Around20%Child-careTime Cost
per Day(hour)
ProportionofExpenditureonChildren
Proportionof Total
Expenditureon Housing

7.64

Baseline
SFC
PH1
PH2
Labor Market-related variables
51,233
51,21946,311
46,83788,28888,28887,58387,522
153,394 153,412 152,030 152,075
7.02
7.02
6.34
6.42
7.56
7.56
7.56
7.55
7.64
7.64
7.63
7.64

VC

SFC+PHSFC+VC45,366
87,615
152,108
6.21
7.56
7.64

48,68687,492
152,225
6.67
7.55
7.64

45,23487,607
152,232
6.20
7.56
7.64

Family-related variables
2.566
2.567
2.604
1.913
1.913
1.915
1.624
1.626
1.604
1.8421
1.84271.86961.37361.37361.3745
1.16571.16701.1511

2.568
1.919
1.608
1.84371.3777
1.1546

2.683
1.912
1.612
1.92611.37281.1573

2.464
1.921
1.625
1.76911.3793
1.1664

2.691
1.913
1.626
1.93201.3732
1.167035.33%
43.03%
44.77%

38.39%

35.29%

38.39%

23.10%

4.93%

23.10%

38.39%

35.44%

Housing Market-related variables
23.10%
5.82%
5.18%
23.70%
23.92%

Group 1
Share ofChildren’s Group 2
Expenditure Group 3
on Housing
Population per Acre
Group 1
Preferred
Property
Group 2
Tax Rate
Group 3

31%

31%

31%

8.45%

7.55%

31%

7.20%

31%

5.87

5.24

5.48

31%

4.63
About
1.40%

5.53

5.43

5.82

5.69
1.22%
1.40%
1.47%

Key 1: Group 1: Not a high school graduate; Group 2: High school graduate to Associate degree; Group 3: Bachelor’s degree or above.
Key 2: SFC: School finance consolidation regime; PH1: Public Housing policy regime (Public housing units locate within the fringe distance); PH2: Public Housing policy regime (Public housing units locate outside thefringe distance); VC: Housing Voucher; SFC+PH: School finance consolidation and Public Housing policy (Public housing units locate outside the fringe distance) are imposed simultaneously; SFC + VC: School financeconsolidation and Housing Voucher policy are imposed simultaneouslyTable 4a Cross-community Welfare Comparison at the Baseline equilibrium
Total Welfare
Parent Direct Utility
Offspring Quality(Utility level)
West
East
West
East
Group 110.4342
12.3364
12.4337
0.98570.9441
Group 2
11.4934
13.4727
13.59102.36472.2709Group 313.6434
14.9952
15.21095.07054.6966
Average
12.2474
14.0249
3.0554
Key 1: Group 1: Not a high school graduate; Group 2: High school graduate to Associate degree; Group 3: Bachelor’s degree or above.

Table 4b Equilibrium Outcome Summary
Variables

Baseline

Number Group
1ofHousehold Groupin the
2
West
Group
3
Number Group
1ofHousehold Groupin the
2
East
Group
3
(W) School Quality/
Property Tax Rate
(E) School Quality/
Property Tax Rate(W) Average Rent ($)(E) Average Rent ($)(W) Annual Income($)(E) Annual Income ($)(W) Annual EduSpending ($)(E) Annual EduSpending ($)

4.26%

SFC
PH1
PH2
Household Distribution
5%
0%
0%

VC

SFC+PH SFC+VC

2.69%

0%

5%

27.83%

25%

45.35%

39.34%

32.72%

27.66%

25%

28.38%

20%

40%

40%

40%

30.62%

20%

5.74%

5%

10%

10%

7.31%

10%

5%

22.17%

25%

4.65%

10.66%

17.28%

22.34%

25%

11.62%

20%

0%

0%

0%

9.38%

20%

564/
1.47%
280/
1.40%
41,07637,703
116,192

Community Comparison
406/
618/
774/
741/
1.40%
1.40%
1.47%
1.47%
406/
34/
64/
123/
1.40%
1.40%
1.40%
1.40%
39,515 44,709 43,861 43,309
39,515 28,488 29,316 35,172
110,631 117,810 120,090 120,269

632/
1.40%
264/
1.40%
41,212
34,405
121,491

386/
1.40%
386/
1.40%
39,51339,513
109,219

102,106 110,631
5,321
4,815

59,255
5,362

67,752
5,668

75,187
5,627

92,726
4,578

109,219
4,757

4,308

957

1,651

2,931

4,578

4,757

4,815

Key 1: Group 1: Not a high school graduate; Group 2: High school graduate to Associatedegree; Group 3: Bachelor’s degree or above.
Key 2: SFC: School finance consolidation regime; PH1: Public Housing policy regime (Public
housing units locate within the fringe distance); PH2: Public Housing policy regime (Public
housing units locate outside the fringe distance); VC: Housing Voucher; SFC+PH: School
finance consolidation and Public Housing policy (Public housing units locate outside thefringe distance) are imposed simultaneously; SFC + VC: School finance consolidation andHousing Voucher policy are imposed simultaneously.

Table 5a Summary Statistics of the Model-Generated Data
Variables
Ri

Mean
West
East
52,605
46,692

MinimumWestEast
26,042
26,050MaximumWestEast102,408 82,834

Standard Deviation
West
East
20,56015,812

Di

7.28575.79090.00310.001514.5629

11.6291

4.19373.3888wi

41.843637.1336

20

20

55

55

13.3761

13.3541

Table 5b Regression Results
log Ri   0   D Di   w wi  ui
Point Estimate(Standard Deviation)
West
East
11.4857
11.24970(0.0018)(0.0016)-0.0880-0.1005
D(0.0002)(0.0003)-0.00110.0008w(0.0001)(0.0001)
Sample Size
2,800
2,800
2
0.99620.9951R
F-statistics
370,278
286,228

Table 6 Equilibrium Welfare Comparison
Household Type

SFC +
PH
Welfare Comparison (Consumption-Equivalent Measure %)Panel A: Household Level
Group 1+0.02 +19.98 +20.22 +18.59+17.53

Group 3

BenchmarkGroup 2

Baseline

Average

SFC

PH1

PH2

VC

SFC +
VC

+18.93

+0.02

-6.30

-2.69

-2.14

+0.45

-1.62

+0.01

-5.55

-2.26

-2.06

+0.28

-1.62

+0.02

-3.46

-0.35

-0.19

+1.92

+0.28

-0.44

+31.13+25.81+25.79+7.52+22.89

+0.15

-5.59

-2.08

-1.58

+0.87

-0.91

+1.33

-10.33

-9.02

-7.83

-0.79

+0.36

+0.61

-3.45

-1.98

-1.27

+0.78

+2.05

+0.75

-10.54

+8.12

-8.13

+39.92

-0.33

-0.24

-6.96

-3.63

-2.66

-1.02

-2.74

-2.28

+3.45

+8.58

+7.14

+1.74

-4.93

-1.30

-1.78

+3.62

+1.85

+3.58

-4.63

Panel B: Parent Direct Utility (PDU) Only
Group 1

Group 3

BenchmarkGroup 2Average

Panel C: Offspring Quality (OQ) Only
Group 1

Group 3
Average

BenchmarkGroup 2

Key 1: Group 1: Not a high school graduate; Group 2: High school graduate to Associatedegree; Group 3: Bachelor’s degree or above.
Key 2: SFC: School finance consolidation regime; PH1: Public Housing policy regime (Public
housing units locate within the fringe distance); PH2: Public Housing policy regime (Public
housing units locate outside the fringe distance); VC: Housing Voucher; SFC+PH: School
finance consolidation and Public Housing policy (Public housing units locate outside thefringe distance) are imposed simultaneously; SFC + VC: School finance consolidation andHousing Voucher policy are imposed simultaneously.

Table 7 Flexibility of Various Choices
Type of Goods
Fertility Rate
Lot Size
Rental Rate
Residential Location
Non-durable Good
LeisureProperty Tax Rate
School Quality

Short-runInflexible
Inflexible
Inflexible
Inflexible
Flexible
Flexible
Flexible
FlexibleLong-runFlexible
Flexible
Flexible
Flexible
Flexible
Flexible
Flexible
FlexibleTable 8a Short-run VS Long-run (Statistics)
Statistics

Baseline

Group 1
Group 2
Group 3
Group 1
AnnualConsumption Group 2($)
Group 3
Hours
Group 1
Worked perGroup 2
Day
Group 3
Voucher Received per
Day ($)
Property Tax
West
Rate
East
Annual EduWest
Spending ($)
East

51,23388,288
153,394
38,55464,227110,175
7.02
7.56
7.64
N.A.

SR
51,17088,149153,090
38,57664,279110,293
7.01
7.55
7.62
N.A.

LR
51,21988,288
153,41238,56764,224110,299
7.02
7.56
7.64
N.A.

SR
44,99088,428
153,645
40,752
63,660109,197
6.16
7.57
7.65
23.1249

LR
45,366
87,615
152,108
40,311
63,678
108,505
6.21
7.56
7.64
23.1223

SFC + VC
SR
LR
44,92745,23488,28987,607
153,339 152,232
40,77340,434
63,712
63,696109,315 109,415
6.15
6.20
7.56
7.56
7.64
7.64
23.1249 23.1326

1.47%
1.40%
5,321

1.40%
1.40%
4,761

1.40%
1.40%
4,815

1.47%
1.40%
5,321

1.47%
1.40%
5,627

1.40%
1.40%
4,761

1.40%
1.40%
4,757

4,308

4,761

4,815

4,308

2,931

4,761

4,757

AnnualIncome($)

Peer Quality

SFC

VC

West
38.67
30.18
30.75
38.02
48.08
29.87
29.62
East
23.73
30.18
30.75
23.61
15.11
29.87
29.62
School
West
564
394
406
554
741
390
386
Quality
East
280
394
406
279
123
390
386
Key 1: Group 1: Not a high school graduate; Group 2: High school graduate to Associatedegree; Group 3: Bachelor’s degree or above.
Key 2: SFC: School finance consolidation regime; VC: Housing Voucher; SFC + VC: School
finance consolidation and Housing Voucher policy are imposed simultaneously.
Key 3: SR: Short-run; LR: Long-run.
Key 4: N.A.: Not Applicable.

Table 8b Short-run VS Long-run (Welfare)Household Type

SFC
VC
SFC + VC
SR
LR
SR
LR
SR
LR
Welfare Comparison (Consumption-Equivalent Measure %)Panel A: Household Level
Group 1W
-3.07
+0.02 +19.01 +18.59+16.55+18.93
E

+3.30

+0.02

+19.16+18.59+21.79+18.93

W

-3.53

+0.02

-1.30

-2.14

-4.80

-1.62

+3.72

+0.02

-1.17

-2.14

+2.54

-1.62

-3.70

+0.01

-1.33

-2.06

-5.00

-1.62

+3.88

+0.01

-1.18

N.A.

+2.69

-1.62

-0.70

+0.02

+0.62

-0.19

-0.04

+0.28

W

+0.71

+2.05

+25.55+21.61+26.06+24.80

E

0

-2.32

+25.63+26.58+25.63+21.45

+0.78

+2.28

-1.25

-3.76

-0.26

+1.25

0

-2.58

-1.26

+3.68

-1.26

-3.66

+0.80

+3.59

-1.29

-5.26

-0.27

+2.64

0

-4.38

-1.27

N.A.

-1.27

-5.40

+0.48

+0.61

+1.52

-1.27

+2.00

+2.05

W

-10.69

-3.56

+4.92

+1.08

-5.04

-4.68

E

+9.28

+3.91

+5.26

-10.23+13.93+2.87-10.68

-3.37

-1.38

+0.16

-11.99

-5.95

+9.28

+3.64

-1.04

-11.02

+8.20

+1.24

-10.67

-6.22

-1.38

+3.56

-11.99

-8.97

+9.28

+7.00

-1.04

N.A.

+8.20

+4.60

-2.96

-1.30

-0.97

+1.85

-3.88

-4.63

E
Group 3W

BenchmarkGroup 2

BaselineE
Average
Panel B: Parent Direct Utility Only
Group 1W
E

Group 3W

BenchmarkGroup 2

E
Average
Panel C: Offspring Quality Only
Group 1W
E

Group 3W
E

Average

BenchmarkGroup 2

Key 1: Group 1: Not a high school graduate; Group 2: High school graduate to Associate degree; Group 3: Bachelor’s
degree or above.
Key 2: SFC: School finance consolidation regime; VC: Housing Voucher; SFC + VC: School finance consolidationand Housing Voucher policy are imposed simultaneously.
Key 3: SR: Short-run; LR: Long-run.
Key 4: W: West; E: East.
Key 5: N.A.: No Group 3 household reside in the East under VC policy in the LR.

Appendix
Alternative Calibration Targets
In the paper's main text, we choose parameters in the peer effect function: 70% of BAtype households reside in the West and determine the property tax rate there. To justify
this choice, in this appendix, we consider alternative targets (55%, 75%, and 80%) andexamine whether our main results are robust to the choice of target. We recalibrate themodel for each alternative target and redo the analysis for SFC and SFC + PH. The
results are summarized in Table A1 and A2.
We find that 70% is approximately the lowest fraction to ensure that skilled (BA type)workers can constitute the majority of the West community and determine the property
tax rate. When we use a higher fraction as the target, the positive effect of SFC policy
becomes bigger. Consequently, our main finding is that combining public housing
programs and the school finance consolidation policy can lead to Pareto Improvement
and help the poor become even more potent. If we further reduce this fraction (from 70%
to 55%), the BA-type household no longer constitutes the majority of the West, and thesorting pattern becomes much weaker. In this case, the overall welfare effect of the SFC
policy becomes negative, consistent with the findings in previous HY papers. However,even in this case, we still find that combining SFC and PH can lead to Pareto
improvement, suggesting that our main finding is robust to this.

Table A1 Equilibrium Welfare Comparison (SFC): Different Targets
Household Type

70%
75%
80%
55%
(used inthe paper)
Welfare Comparison (Consumption-Equivalent
Measure %)Panel A: Household Level
Group 1+0.02
+0.05 +0.08 -0.04
Group 2

+0.02

+0.06

+0.09

-0.03

Group 3

+0.01

+0.05

+0.08

-0.03

Average

+0.02

+0.05

+0.06

-0.06

Panel B: Parent Direct Utility (PDU) Only
Group 1

-0.44

-0.69

-1.02

-0.04

Group 2

+0.15

+0.18

+0.22

-0.00

Group 3

+1.33

+1.92

+2.67

+0.07

Average

+0.61

+0.86

+1.15

-0.01

Panel C: Offspring Quality (OQ) Only
Group 1

+0.75

+1.25

+1.86

-0.02

Group 2

-0.24

-0.22

-0.21

-0.08

Group 3

-2.28

-3.29

-4.58

-0.21

Average

-1.30

-1.81

-2.54

-0.23

Key 1: SFC: School finance consolidation regime.
Key 2: Group 1: Not a high school graduate; Group 2: High school graduate to an
Associate degree; Group 3: Bachelor’s degree or above.
Key 3: The fractions shown in the first row are the fraction of Group 3 households inthe West in the baseline case.

Table A2 Equilibrium Welfare Comparison (SFC + PH): Different Targets
Household Type

70%
75%
80%
55%
(used inthepaper)
Welfare Comparison (Consumption-Equivalent Measure %)Panel A: Household Level
Group 1+17.53 +17.34 +17.9 +17.42
0Group 2
+0.45
+0.52
+0.56
+0.42
Group 3

+0.28

+0.35

+0.38

+0.25

Average

+1.92

+1.99

+2.04

+1.88

Panel B: Parent Direct Utility (PDU) Only
Group 1

+7.52

+7.04

+7.55

+7.70

Group 2

+0.87

+0.91

+0.94

+0.72

Group 3

-0.79

-0.24

+0.50

-2.01

Average

+0.78

+1.00

+1.35

+0.20

+39.63

Panel C: Offspring Quality (OQ) Only
Group 1+39.92+40.12

Group 2

-1.02

-0.92

+40.4
5
-0.91

Group 3

+1.74

+0.97

-0.22

+3.69

Average

+3.58

+3.26

+2.56

+4.66

-0.82

Key 1: SFC+PH: School finance consolidation and Public Housing policy (Public
housing units located outside the fringe distance) are imposed simultaneously.
Key 2: Group 1: Not a high school graduate; Group 2: High school graduate to an
Associate degree; Group 3: Bachelor’s degree or above.
Key 3: The fractions shown in the first row are the fraction of Group 3 households inthe West in the baseline case.

HY-type Peer Group Effect Function
In the paper's main text, we assume that the peer group effect in a district is determinedby the average quality of students residing in that district, which differs from Hanushek
and Yilmaz (2007, 2013). They assume that the peer group effect is determined by thefraction of college-educated households in the district. Our formulation implies that thepeer group effect depends on parental investment in children and government spending
on education, all equilibrium objects. Since the equilibrium cross-district allocation ofthese objects changes when counterfactual policies are imposed, our formulation canpotentially lead to different policy implications from the original HY formulations.
To quantify the importance of our formulation, we redo all of the counterfactual policy
analysis considered in the paper, adopting a peer group effect function similar in spirit
to the original HY formulations. Expressly, in each of the counterfactual, when
computing the peer group effect in a district, we assume that the average quality ofstudents for each household type is the same as its counterpart in the baseline
equilibrium. Since the overall average quality of students in a district is just a weighted(by population structure in the neighborhood) average of average qualities of children
for each type of household, the peer group effect in a community only depends on the
district's population composition under this alternative formulation. We re-solve for theequilibrium using this formulation. The results are summarized in Table A3.
Comparing Table A3 to Table 6 in the paper, we find both qualitative and quantitative
differences. Perhaps most notably, we find substantially larger positive effects of public
housing and housing voucher programs on students' quality from NH (Not a Highschool graduate) type households under this alternative HY-type formulation of peergroup effects. For example, as shown in the VC column of Panel C in Table 6, students'
average quality from NH type households drops by 8.13% after the housing voucher
program is implemented under our peer group effects. In contrast, the VC column ofPanel C in Table A3 shows that the housing voucher program leads to a 0.93% increasein the average quality of students from NH type households under the alternative HYtype formulation. Similar patterns can be found by comparing the PH1 and PH2columns of the two tables.
The reason behind this difference is as follows. As shown in the last two columns(Annual Edu-Spending) of Table 4b, the three policies (PH1, PH2, and VC) lead tomore extreme sorting of households and lower the per capita educational spending theEast district, where all or most NH type households reside. As a result, the average
quality of students from NH type households decreases in these policy regimes. Under
our formulation, this decrease in average student quality has an additional negative
effect through the peer group effect. The peer group effect is weaker in the East district,resulting in lower school quality for NH type households who primarily reside in theEast. Under the alternative HY-type formulation, this additional negative effect isassumed away.

Table A3 Equilibrium Welfare Comparison (Alternative Peer Group Effects)
Household TypeBaseline

SFC

PH1

PH2

VC

BenchmarkSFC +PH
Welfare Comparison (Consumption-Equivalent Measure %)Panel A: Household Level
Group 1
+0.1 +20.19 +20.18 +19.28 +20.14
7Group 2
+0.1
-6.36
-1.78
-1.12
-1.82
9
Group 3
+0.2
-5.64
-1.70
-1.10
-1.74
0
Average
+0.2
-3.51
+0.30
0.77
+0.27
0
Panel B: Parent Direct Utility (PDU) Only
Group 1

SFC +VC

+19.28
-1.12
-1.10
0.77

+29.58+21.87+22.89+21.77+22.89+0.1
5Group 3
+1.3
3
Average+0.6
1
Panel C: Offspring Quality (OQ) Only

-5.48

-0.10

-0.91

+0.03

-0.91

-10.24

-7.77

+0.36

-7.65

+0.36

-3.58

-1.01

+2.05

-0.92

+2.05

-4.04

+18.59+0.93+18.72+0.93

Group 2

Benchmark-0.44Group 1

Group 3
Average

BenchmarkGroup 2

+1.2
0
+0.2
2
-1.82

-7.41

-4.79

-1.46

-5.10

-1.46

+3.10

+7.83

-3.61

+7.68

-3.61

-0.84

-1.82

+3.53

-3.32

+3.33

-3.32

Key 1: Group 1: Not a high school graduate; Group 2: High school graduate to an
Associate degree; Group 3: Bachelor's degree or above.
Key 2: SFC: School finance consolidation regime; PH1: Public Housing policy regime(Public housing units located within the fringe distance); PH2: Public Housing policy
regime (Public housing units located outside the fringe distance); VC: Housing
Voucher; SFC+PH: School finance consolidation and Public Housing policy (Public
housing units located outside the fringe distance) are imposed simultaneously; SFC +VC: School finance consolidation and Housing Voucher policy are imposed
simultaneously.