ISSN (Print) 0473-453X

Discussion Paper No. 1214

ISSN (Online) 2435-0982

DOES SPACE MATTER?
THE CASE OF THE HOUSING
EXPENDITURE CAP

Yifan GongCharles Ka Yui Leung

September 2023

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

Does Space Matter? The Case of the Housing
Expenditure Cap
Yifan Gong, University of Nebraska-Lincoln
and Charles Ka Yui Leung, City University of Hong Kong
This version: September, 2023
Abstract
In our evaluation of the housing expenditure share cap, a macroprudential
policy, we discover the importance of modeling space. The spatialconsiderations allow households to sort into segmented housing markets basedon income. Our model generates the observed negative relationship between
housing expenditure share and income. More importantly, the cap policy causesa more considerable reduction in housing costs for low-income families than for
high-income families in a spatial model. Depending on the assumption ofhouseholds' preference, this mechanism leads to a minor increase or even a
modest decrease in welfare inequality in a spatial model than in a spacelessmodel.
JEL Classification Numbers: D04, R20, R30
Keywords: housing expenditure share, monocentric model of a city, spatialsorting, welfare inequality

This version is a substantial revision of an earlier version. The authors thank Xiaoying Deng,
William Larson, Meng Li, Chandler Lutz, Steve Malpezzi, Dan McMillen, Haoning Sun, Shihan
Xie, and seminar participants of the National Taiwan University, AREUEA Conference, AsRES
meeting, Asian Meeting of the Econometric Society, CEANA meeting, City University
Affordability Workshop, Jinan‐SMU‐ABFER workshop, and especially the anonymousreferees and the co‐editor David Krisztian Nagy for helpful comments. Part of the research is
conducted when Leung visits the Hoover Institution, whose hospitality is gratefully
acknowledged. The authors are also grateful for the financial support from the City
University of Hong Kong, University of Western Ontario, and University of Nebraska‐Lincoln.
The usual disclaimer applies.

1

Introduction

This paper argues that spatial considerations are essential for modeling housing markets
and evaluating housing-related policies. We illustrate this point through a cap policy on
the ratio of housing expenditures to disposable income, referred to as the housing expenditure share henceforth. We document some stylized facts regarding housing expenditure
shares across different income classes. We can generate them through a homothetic CES
utility function in a canonical monocentric city model or a “spaceless” model with nonhomothetic preference. Yet, subject to the same expenditure cap policy, these two models
have divergent welfare inequality predictions.
Our research is partly motivated by the booming housing research in recent years.
After the 2008 Global Financial Crisis, many became aware that an increase in house
prices could translate into an “excessive” amount of credit allocated to the real estate
sector and increase the risk of having a financial crisis.1 In response, many countries have
implemented macroprudential policies, which help control the growth of house prices andhousing-related credit.2 According to IMF (2018), “...141 countries reported a total of
1,313 macroprudential measures, for an average of measures per country of 9.3” (p.6).
The scale and diversity of all these macroprudential measures are enormous. It is almost
impossible to assess all different macroprudential policies in a unifying framework.3 To
complement the literature on mortgage debt and risk-taking, this paper focuses on a policy that caps the housing expenditure share, a common concern of many policymakers.4
This type of cap policy has been explicitly and implicitly adopted in many contexts.
For instance, in some affordable housing programs, the U.S. Department of Housing and
Urban Development (HUD) imposes that rent “does not exceed 30 percent of the adjusted
income of a family whose annual income equals 65 percent of the median income for the1
2

For a review of the literature, see Leung and Ng (2019), Leung (2023), among others.
See, e.g., Akinci and Olmstead-Rumsey (2018), Carreras, Davis, and Piggott (2018), Claessens,

Ghosh, and Mihet (2014), and Gambacorta and Murcia (2020).
3
Empirical works suggest that the relationship between macroprudential policies and inequality may
be non-monotonic and depend on several factors (Biljanovska et al., 2023; Carpantier, Olivera, and Van
Kerm, 2018; Casiraghi et al., 2018; Guerello, 2018). Favilukis, Mabille, and Van Nieuwerburgh (2019)combine the life-cycle dynamics with the spatial choice between two communities. They calibrate theirmodel to match specific features of New York City.
4
For instance, OECD (2019) states that “...Another common measure for housing affordability usedhere is the ‘housing cost overburden rate’, which measures the proportion of households or populationthat spend more than 40% of their disposable income on housing cost (in line with Eurostat methodology).” USA President Joe Biden also said that “every American in every zip code should have access tohousing that is affordable - taking up no more than 30% of income so they have money left over to meetother needs.” See Biden (2020) for more details.

2area, as determined by HUD.”5 Among macroprudential policies implemented in many
countries, caps are imposed on the debt servicing ratio in various types of stress tests(IMF, 2014).6 In Appendix B, we show through an extended version of the model that
this is equivalent to imposing a cap on the housing expenditure share in our static model.
However, the empirical evidence for the implications of such caps for housing markets
and welfare is mixed. According to Biljanovska et al. (2023), many studies based on
aggregate data generate statistically insignificant results, and studies based on microdata tend to suggest more significant effects of macroprudential policies. One potential
explanation is that some researchers assume away the spatial dimension of the macroprudential policies, i.e., there is typically only one housing market in the model (e.g.,
Alpanda, Cateau, and Meh, 2018).7 In reality, the housing markets are segmented even
within the same city. This paper shows that models differ in spatial considerations andpreference choices can deliver very different policy implications.
To substantiate our claim, we proceed in several steps. We start by establishing threestylized facts in the United States that guide our modeling exercise and are relevant to
the welfare implication of the cap policy. The facts are (1) the approximate constancy inthe working hours across income groups, (2) the negative correlation between the share of
housing-related expenditure and income, and (3) the positive association between commute time and distance with income. Simultaneously satisfying these three facts imposes
stringent restrictions on the equilibrium model. For instance, a spaceless model needs thenonhomothetic preference to produce the observed negative relationship between housing
expenditure share and income. In addition, because there is only one housing market, thisspaceless model is silent on the observed spatial correlation between household income
and residential locations. In contrast, we show that a canonical monocentric city model
can simultaneously account for all three stylized facts even with homothetic preference(CES) (Brueckner, 1987; Mills, 1972; Muth, 1969).
Welfare implications from government policies such as an expenditure cap are dra5

According to the California Department of Housing and Community Development, “State statu-

tory limits are based on federal limits set and periodically revised by the U.S. Department of Housing
and Urban Development (HUD) for the Section 8 Housing Choice Voucher Program. The comparable
federal limit, more widely used, is 30 percent of gross income, with variations.” For more details, see
https://www.hcd.ca.gov/grants-and-funding/income-limits.
6
For instance, the Hong Kong Monetary Authority reports that a household could borrow a mortgage
loan of up to 157 times the households monthly income. After the stress-testing requirement came into
effect on 15 September 2010, the maximum amount that the household could borrow was less than 80% of
the situation before the HKMA’s recent prudential measures were introduced (HKMA, 2010). Thus, themacroprudential measures effectively put a ceiling on the income share of housing-related expenditures
for households without additional financial resources.
7
There are exceptions. For instance, Acharya et al., 2022 show that caps on loan-to-value and loanto-income ratios lead banks to reallocate mortgage credit from low- to high-income households and fromurban to rural communities and change the relative house prices among different places as a result. This
paper generates similar results in a structural model.

3matically different in a spatial model than in a model without space. If preferences are
homothetic, a welfare cap reduces welfare inequality in a spatial model but increases it
in a model without space. This is because, under the homothetic preference, the expenditure share on housing is independent of the income level. In a spaceless model,
lower-income households are forced to compete with higher-income agents in the unified housing market. As a result, lower-income families spend a larger fraction of theirincome on housing. Under the expenditure cap, lower-income households are forced tospend less on accommodation, reducing their welfare. At the same time, the housing
expenditure cap is non-binding for the higher-income agents, and hence, the impact on
their welfare is minimal. In a spatial model, each “location” in a monocentric city is ageographically segregated housing market, and agents self-select into different submarkets. Under the cap, each submarket’s “market price” adjusts differently. Low-income
families enjoy disproportionately more significant reductions in endogenous equilibrium
rent than high-income families. In our baseline model, the welfare gain from these rent
reductions entirely offsets the welfare loss due to being constrained by the cap.
On the other hand, if preferences are sufficiently non-homothetic, welfare inequality
increases in both models but less so in the model with space. This is because agentsendowed with non-homothetic preferences allocate different portions of their incomes tovarious goods. Under our calibration, lower-income agents spend a larger percentageof their income on housing and, hence, are more likely to be constrained by the cap
policy. In the non-homothetic spaceless model, the higher-income agents would be ”benefited” by the expenditure cap as the lower-income competitors are restricted to spending
on housing. With non-homothetic preference, the higher-income agents would spend asmaller portion of their income on housing, reducing the housing rent, which partly offset
the adverse effect of the cap policy on the lower-income households. In the homothetic
monocentric city model, economic agents are self-selected into different ”submarkets” tobegin with. Hence, lower-income households cannot benefit from the demand reduction
effect of the higher-income agents.
We explore the relevance of this key mechanism and the robustness of our welfare
results in monocentric cities with a variety of alternative model settings. In Appendix
B, we consider 1) a multi-period setting and 2) a setting with imperfect spatial sorting
while keeping the utility function homothetic (CES). We show that our main conclusionthat a housing expenditure cap has negligible effects on welfare inequality remains underthese settings. In Section 5, we investigate the role of homothetic preference for ourwelfare results by considering a non-homothetic monocentric city model.8 We find that,while the cap policy still aggravates welfare inequality in this model, the magnitude ofthis effect is smaller than that in the non-homothetic spaceless model because of the8

After all, while many existing models adopt homothetic preference (Epple and Romer, 1991;

Nechyba, 2000, 2003), there are also examples where non-homothetic preferences are employed (Albouy,
Ehrlich, and Liu, 2016; Wang and Xie, 2022).

4general equilibrium effect. Thus, we conclude that the key mechanism discussed above isalways present with its importance depending on households’ preferences. We discuss theimplications of this finding for future research in the conclusion (Section 6) and provide
a comprehensive summary of the value of modeling space in Section 5.3.
This paper builds on and complements several strands of the literature. On top of
the few we briefly discussed above, the first literature that inspires this paper is theone on housing affordability and the evaluation of related government policies (BenShahar, Gabriel, and Oliner, 2020; Gabriel and Painter, 2020; Quigley and Raphael,2004). They typically employ a rich micro-data set and adopt a reduced-form estimation
approach. The second literature concerns the design of macroprudential policies, such as
Buch, Vogel, and Weigert (2018), Gadanecz and Jayaram (2017), etc. Third, this paper isrelated to the literature on how loan-to-value ratio policy (LTV), which is a different kindof macroprudential policy, would affect the housing market, such as Aastveity, Juelsrud,and Wold (2020), Armstrong, Skilling, and Yao (2019), Bekkum et al. (2019), Laufer andTzur-Ilan (2021), Tzur-Ilan (2019). Fourth, this paper is related to the literature thatemphasizes the housing market heterogeneity at the city-level or regional-level (Beraja
et al., 2019; Fratantoni and Schuh, 2003; Leung and Teo, 2011; Piazzesi, Schneider, andStroebel, 2020; Sun and Tsang, 2018).
Our paper complements the literature in different ways. First, we build a structuralmodel and hence complement the literature based on the reduced-form approach.9 Second, we abstract from the macroeconomic risk of the housing market and the dynamic
considerations that the previous literature has studied. Instead, our baseline model focuses on a static environment where ex-ante heterogeneous agents would choose among
infinitely many locations within a city. Thus, rather than investigating the effects ofthe cap policy on financial and housing market stability, this paper examines the importance of spatial considerations for understanding the cap policy’s implications on welfare
inequality.

2

Empirical Facts

This section documents several crucial empirical findings that motivate the mechanism
we explore in this paper. The first stylized fact is that work hours are roughly constantconcerning income; the hours worked on average are around 8 hours for all income groups.
Table 2 shows the evidence for 2017.
Second, we document that the housing expenditure share is decreasing with income.
Table 1 shows the average annual before-tax income, total expenditure, expenditure on
housing, and the fraction of total expenditure spent on housing for each before-tax income
9

There is an emerging view that the reduced form approach and structural model approach arecomplementary to each other (Heckman, 2010; Todd and Wolpin, 2023). A merit of the “structural”
approach is facilitating counterfactual analysis (Wolpin, 2013).

5

decile of the population in 2016.10
A comparison of the first two rows of Table 1 shows that the pre-tax income and thepost-tax (and transfer) expenditure can differ significantly, especially for lower-income
households. This observation is also consistent with recent research that pre-tax and posttax in the U.S. can be very different (Auten and Splinter, 2019; Splinter, 2020). Hence,
we measure the housing expenditure share (ratio of housing expenditures to disposable
income) by the fraction of total expenditure spent on housing throughout the paper
because it considers the effect of progressive taxes and transfers on disposable income.
This is conceptually valid because total expenditure equals disposable income in thestatic models we will present in later sections. As shown in the fifth row of Table 1, thehousing expenditure share decreases from 0.41 to 0.30 in a roughly monotonic manner.
Table 1: Average Total Expenditures and Housing Expenditures by Income Decile
Item(1) Income before taxes(2) Averageannual expenditures(3) Average annual
expenditures on housing(4) Average annual
expenditures on transportation(3)/(2)Item(1) Income before taxes(2) Averageannual expenditures(3) Average annual
expenditures on housing(4) Average annual
expenditures on transportation(3)/(2)LowestSecondThird

FourthFifth

10 percent

10 percent

10 percent

10 percent

10 percent$6,502$16,229$24,432$33,499$43,931$23,588$26,675$34,221$39,308$43,975$9,567$10,961$12,829$14,271$15,511$1,037$1,218$1,530$1,848$2,1390.40560.41090.37490.36310.3527

Sixth

SeventhEighth

Ninth

Tenth

10 percent

10 percent

10 percent

10 percent

10 percent$57,192$73,568$94,739$127,268$269,644$51,351$59,395$70,411$87,432$136,873$17,119$19,285$22,085$26,719$40,547$2,450$2,776$3,220$3,802$5,3050.33340.32470.31370.3056

0.2962

Source: Consumer Expenditure Survey, 2016; U.S. Bureau of Labor Statistics.

The third critical empirical finding is that more highly paid workers live farther awayfrom their workplace. Using the 2010 American Community Survey (ACS) data, we first
10

In Table 1, we use pre-tax total income. Table 2 is based on a different dataset; hence, we use annual

wage, which is the labor income. In the United States, mortgage applications are often based on labor
income and the stable part of the capital income. Leung and Tang, 2023 employ labor income only.

6

Table 2: Hours Worked on an Average Day by Weekly Earnings
Income Range
Hours$0 - $590$591 - $920

$921 - $1,440$1,441 and higher

7.87

8.19

8.28

8.12

Source: American Time Use Survey, 2017; U.S. Bureau of Labor Statistics.

compute households’ average commute time and distance in each annual wage and salary
income distribution decile. We find that both commute time and distance are monotonically increasing with income. Recognizing that the differences in average commute timeand distance across income groups might be partially driven by income-based geographic
sorting, we also use a fixed-effects model to estimate the effect of income on commutetime and distance.11 Using decennial Panel data from the Census and the ACS from 1980
- 2010, we find that a 1,000 (2017) dollars increase in annual wage and salary income
is associated with a 0.0359-minute increase in commute time, with this coefficient being
significant at a 1% level. In 2017, the average commute speed is 23.42 miles per hour.12
Therefore, in 2017, a 1,000-dollar increase in annual wage and salary income is associated with a 0.0140-mile increase in commute distance on average.13 Another supporting
evidence is that, as shown in the fourth row of Table 1, the average annual expenditures
on transpiration increase with pre-tax income.
This paper develops a canonical monocentric model with an extended CES utility
function to reproduce these empirical observations. More specifically, the utility function
combines numeraire consumption goods, housing, and leisure to produce the final utility. While leaving the details for later sections, we provide some basic intuitions here.
First, we show that wealthy families would reside in relatively remote areas with low
housing/land prices. Intuitively, this is because the pecuniary component of commute
cost is “less expensive” for more affluent households in a relative sense. Standard consumer theory predicts that the fraction of total expenditure spent on one commodity
would increase with the price of this commodity if the elasticity of substitution between
this commodity and other commodities is smaller than 1. Therefore, combining the features that (1) the unit price of housing is lower for more affluent households and (2) theelasticity of substitution between housing and numeraire consumption goods is greater11

We do not control for household characteristics such as age and family size in the regression. Con-

sistent with our treatment of the association between income and housing expenditure share, we aretrying to capture the “total” association between income and commute time, including the part thatarises through these household characteristics.
12
See National Household Travel Survey 2017, https://www.bts.gov/topics/national-household-travelsurvey.
13
While the magnitude of this association might seem small, it is close to what we find in our baselinemodel. In our baseline model, a 1,000-dollar increase in annual wage and salary income is associatedwith a 0.0175-mile increase in commute distance. In Appendix B, we consider an alternative model
setting under which we can perfectly match this association by adding unobserved preference shocks tohouseholds’ location decisions. We find quantitatively similar results in this alternative model.

7than 1, we generate the pattern that the fraction of housing expenditure is decreasing inincome. By the same token, as the elasticity of substitution between leisure and othercommodities is unity, the work hours are roughly constant concerning income.

3

A Monocentric City Model: Baseline

We present a stylized static model that describes the resource allocation within a city. An
absentee landlord owns all the land and allocates land through auction. The householdthat offers the highest bid acquires the land for each location if this highest bid exceeds
an exogenous agricultural rent Pa .14 Households then enjoy the housing service fromtheir acquired land and commute to a location r miles away from their home to work.
This paper does not explicitly model housing development and uses the terms land andhousing interchangeably. After paying for the land, households spend the rest of theirincome on non-durable consumption goods and commutes.
In terms of spatial configuration, in this section, we consider a monocentric city modelin which the city is built around a central business district (CBD), i.e., all households
commute to the city center for work. There are infinitely many housing markets in thismodel. These markets are defined by the distance r to the CBD and have potentiallydifferent equilibrium rents. We first analytically show that this monocentric city model’s
spatial feature helps us generate the observed negative relationship between income andthe housing expenditure share without non-homothetic preference. We then calibrate thebaseline model to match the U.S. economy circa 2017.
3.1

Household’s Problem

This subsection defines and solves a household’s utility maximization problem. The
solutions to the maximization problem have several relevant implications for the empiricalfindings in Section 2.
3.1.1

Utility Function, Budget Constraint, and Household Heterogeneity

A household’s utility is determined by its consumption on numeraire good c, housing lot
size h, and leisure l. Formally, we assume that its preference is homothetic and its utility
is given by:α

U (c, h, l) = l1−α [θc1−ρ + (1 − θ)h1−ρ ] 1−ρ .

(1)

Our specification extends the Cobb-Douglas utility function typically used in previous
work on computable spatial equilibrium models (e.g. Hanushek and Yilmaz, 2007, 2013).
We can interpret it as a two-step aggregator. First, housing h and consumption goods
c are combined in a general CES manner with an elasticity of substitution of ρ1 . Then,
their aggregate enters with leisure l in a Cobb-Douglas way to produce the final utility(Krusell et al., 2000; Ogaki and Reinhart, 1998).
14

Alternatively, we could consider a city with exogenous boundaries, and the agricultural rent would
be endogenously determined in equilibrium. Our main results are robust to this alternative specification.

8

We now describe the budget constraint faced by the household. Consider a householdwith an hourly wage w, located r miles away from the CBD.15 The household allocatesthe 24 × 7 = 168 hours in a week to work, leisure (l hours), and commuting (br hours),
where b is the time cost per mile of weekly round-trip commute. Hence, the total incomeof this household is given by:
Income = [168 − l − br]w.

(2)

The household pays for the consumption goods and housing rents with income. We
normalize the price of the composite consumption goods to 1 and denote the rent for
one unit of housing at location r as P (r). To capture the pecuniary cost of commuting,
we assume that the weekly round-trip commute costs ar dollars. Formally, the total
expenditure is given by:
Expenditure = c + P (r)h + ar.

(3)

As usual, total income should equal total expenditure at the equilibrium. Denoting
Y (r) ≡ (24 − br)w − ar, the budget constraint implies the following:
Y (r) ≡ (168 − br)w − ar = wl + c + P (r)h.

(4)

Households in this model differ only in their hourly wages w. Limited by data availability and computational tractability, we categorize households into ten equal-sizedgroups based on hourly wages.16 We refer to households whose hourly wages belong
to the ith decile as Type i households and add subscript i to all relevant variables todifferentiate among types. To ease the notation, we suppress the subscripts in most partsof this paper, and the readers should keep in mind that income levels differ across types
of households.
3.1.2

Optimal Consumption Allocation and Indirect Utility Function

Taking equilibrium market rent P ∗ (r) as given, a household living r miles from the CBDmaximizes the utility function (Equation 1) subject to the budget constraint (Equation
4):
V (P ∗ (r), r) = max U (c, h, l) s.t. Y (r) = wl + c + P ∗ (r)h,(5)(c,h,l)where Y (r) ≡ (168 − br)w − ar.
15

A household can only live in one location. Hence, the consumption set is not convex. The First

Welfare Theorem does not apply here. See Rogerson (1988) and Shell and Wright (1993) for morediscussion.
16
The income distribution of the U.S. is well-approximated by a log-normal distribution. Hence, eachgroup’s income range is different to generate equal-sized groups.

9

Solving this maximization problem yields the following optimal choices of consumption, housing, and leisure:ακ(P ∗ (r))Y (r),P ∗ (r)h(P ∗ (r), r) =c(P ∗ (r), r) = α[1 − κ(P ∗ (r))]Y (r),ar(1 − α)Y (r)= (1 − α)(168 − br − ),wwl(P ∗ (r), r) =1

where κ(x; θ, ρ) =

( 1−θ
)ρ x
θ
1

ρ−1
ρ

1+( 1−θ
)ρ x
θ

ρ−1
ρ

(6)
(7)
(8)

.

Finally, substituting the optimal choices shown in Equation (6)-(8) in the utility
function shown in Equation (1) yields the indirect utility function V (P ∗ (r), r):αρ1V (P ∗ (r), r) = [1 − κ(P ∗ (r); θ, ρ)] ρ−1 KY (r)( )1−α ,
w

(9)

α

where K = αα (1 − α)1−α θ 1−ρ .
3.1.3

Work Hours and Housing Expenditure Share

In our model, the weekly work hours are obtained by subtracting leisure time and time
cost of commute, br, from 168 hours. Further, Equation (8) shows that the optimal leisuretime is approximately a constant, 168α, if the time and monetary cost of commute, arand br, are small. Hence, we conclude that all types of households have roughly the same
work hours (Stylized Fact 1) when the commute cost is low. In the calibration section
later, we show ar and br are indeed small relative to income.
The κ(P ∗ (r); θ, ρ) function defined in the previous subsection provides a measureof housing expenditure share in the model. Equation (6) and (7) together imply that∗

P (r)h∗κ(P ∗ (r); θ, ρ) = c+P∗ (r)h . Noting that the total expenditure is equal to c + P (r)h + ar,κ(P ∗ (r); θ, ρ) measures the ratio of housing expenditures to total after-commute-cost
expenditures and is hereafter referred to as the ACC housing expenditure share. Since
ar is relatively small compared to the total expenditure, κ(P ∗ (r); θ, ρ) is quite close to
the housing expenditure share.
Note that κ(P ∗ (r); θ, ρ) does not depend on hourly wages. Since all households havethe same preferences (θ and ρ), we simply write the κ(P ∗ (r); θ, ρ) function by suppressing
the θ and ρ arguments. In Appendix A, we show that κ(P ∗ (r)) is increasing in equilibrium
rent P ∗ (r) if the elasticity of substitution between consumption and housing is smaller
than one.
> 0.
Proposition 1. If ρ > 1, then ∂κ(x)∂xAs we will explain in the following subsection, this proposition is vital for generating
the negative relationship between income and housing expenditure share.

10

3.2

Basic Analysis of the Equilibrium

This subsection defines and characterizes the equilibrium of our model. We will show that
our monocentric city model is broadly consistent with the empirical evidence documented
in Section 2. Also, these characterizations assist our calibration in a later section.
3.2.1

Bid-rent Functions and Market Rent Curves

As with many spatial equilibrium models, all households bid for land on a featurelessplane. The common practice is to solve the bid-rent function, which expresses a household’s willingness to pay for the equilibrium utility level u∗i . For a Type i ∈ {1, 2, ..., 10}
household, the maximization problem can be mathematically expressed as follows:ψi (u∗i , r) = max(c,h,l)Yi (r) − c − wi l∗
|U (c, h, l) = ui ,h(10)where Yi (r) ≡ (168 − br)wi − ar.
Technically speaking, this bid-rent maximization problem is the dual problem to theutility maximization problem defined in Equation (5). Hence, we can obtain the following
bid-rent function by inverting the indirect utility function Vi (P ∗ (r), r):ψi (u∗i , r) =(Yi (r)Ku∗i wi1−α

 ρ−1
αρ

ρ
) ρ−1

−1

(

1
θ ρ−1
) ,
1−θ

(11)

α

where K = αα (1 − α)1−α θ 1−ρ .
In the model, all of the lands are rented out via auctions. The ten types of households
and agricultural workers can bid for any location indexed by its distance from the CBD,
r.17 For each location, the right of usage goes to the agent who offers the highest bid.
Therefore, the equilibrium rent curve P ∗ (r) is the upper envelope of the bid rent curves
ψi (u∗i , r) of the ten types of households and the agricultural rent Pa . As the family moves
away from the CBD, its bid rent declines and eventually hits 0. It means that beyonda certain distance Rf? , the agricultural rent Pa dominates the bids offered by all of thehouseholds in the economy. Hence, no one resides there. We introduce the function t? (r)to indicate the type of the residents at distance r. Formally, within the fringe distanceRf? , t? (r) is given by:t? (r) = arg max ψi (u∗i , r).
i

The equilibrium rent P ∗ (r) is then given by:X
P ∗ (r) = max(
ψi (u∗i , r)I(t? (r) = i), Pa ),

(12)

(13)

i∈{1,2,...,10}

where I(·) is an indicator function that takes the value 1 when the condition in thebracket is satisfied and 0 otherwise.
17

Following the urban economics literature, the agricultural workers are assumed to be self-sustained,

except for the participation of the land auction. They would not affect any other aspect of the model
economy.

11

To understand how different agents are distributed spatially, we consider the spatial
order of two adjacent types of households. It is determined by their bid rent curves’
relative steepness at the intersection point. The one with the steeper curve resides closer
to the CBD. In other words, the condition for the equilibrium location of Household 1
1 (·)/∂r< 1. In Appendix A,
being farther from the CBD than that of Household 2 is ∂ψ∂ψ2 (·)/∂r

we show that ∂ψ∂ri (·) is always negative and that | ∂ψ∂ri (·) | is larger for households with lower
hourly wage wi . Hence, the following proposition holds.
Proposition 2. Households with higher hourly wage wi live farther from the CBD at theequilibrium.
The equilibrium rent P ∗ (r) decreases in the distance r. Combining Proposition 2 and
1, we conclude that households with higher hourly wages spend a smaller fraction of their
total after-commute-cost expenditures on housing services.
Proposition 3. Consider two arbitrary households. We denote their types as i and j,
respectively. Let ri∗ and rj∗ denote their distance from the CBD at the equilibrium. If
ρ > 1 and wi > wj , then κ(P (ri∗ )) < κ(P (rj∗ )) at the equilibrium.
3.2.2

Population Density and Land Market Clearance

Proposition 3 shows how agents are allocated across the city given the equilibrium rent
schedule. Now we explain how the equilibrium rent schedule is determined. Following theliterature, the total number of households for each type i, i ∈ {1, 2, ..., 10}, is exogenously
given at Ni in this model. Suppose that Type i households in equilibrium occupy thelocations r miles from the CBD, and L(r) represents the amount of land available perunit distance. Our model has L(r) = 2πr.18
The clearing of the land market means that within the fringe distance Rf? , the following
equation holds, L(r) = hi (P ∗ (r), r)mi (P ∗ (r), r), where mi (P ∗ (r), r) is the equilibrium
number of households per unit distance, assuming that distance r is occupied by Typei household and u?i is the equilibrium utility of Type i household. All households findresidence locations, implying the following population constraint.
Z∞mi (P ∗ (r), r)I(t? (r) = i)dr = N i , for i ∈ {1, 2, ..., 10}.

(14)

0

It is easy to verify that the household number distribution function m(r) is given by:m(r) =X

mi (P ∗ (r), r)I[t? (r) = i].

(15)i∈{1,2,...,10}
18

In reality, one unit of land might correspond to more housing units at locations closer to the city’scenter. One way to incorporate this possibility is to let L(r) = 2πrD(r), while D(r) is decreasing in
r. It is important to note that our main welfare result (Proposition 4) remains in this case because its
validity does not depend on the choice of L(r).

123.2.3

Stationary Equilibrium

We now define the general equilibrium of this model economy. In the stationary equilibrium, no household has an incentive to relocate. It can be formally defined as follows:
Definition 1. An equilibrium is a set of utility levels {u∗1 , u∗2 , ..., u∗10 }, market rent curveP ∗ (r), and type function t∗ (r) that show the equilibrium occupant of the location at
distance r that satisfy the following conditions.
• Households offer their bids according to Equation (11). The land is rented out
through an auction. The household that offers the highest bid wins a particular location
if the bid is higher than the agricultural rent. Otherwise, the land is used for agriculture.
Type function t∗ (r) records the auction’s winner.
• Market rent P ∗ (r) is determined as the upper envelope of bids from all types ofhouseholds and the agricultural rent according to Equation (13).
• All Type i households attain the same utility level, i.e., u∗i = Vi (P ∗ (r), r) for all r
such that t∗ (r) = i.
• Each household rents a certain amount of land according to Equation (6). The landmarket clears, and the population constraint (Equation 14) holds.
3.3

Calibration

3.3.1

Parameter Set

This section shows that our baseline model can be calibrated to match the stylized factsof the U.S. circa 2017 documented in Section 2. The parameters of our model can bedivided into three categories, which are (1) macroeconomic environment parameters Pa
and N i , (2) budget constraint parameters wi , a, and b, and (3) preference parametersα, θ, and ρ. Below we describe the calibration of each category of parameters. Table 3
summarizes our calibration exercise.
We start with the macroeconomic environment parameters, i.e., agricultural rent
Pa and the number of households N i . In 2017, the average population and area of
cities with more than 100,000 people were 303,322 and 94.08 square miles.19 In ourmodel, all households commute to the CBD for work. In the U.S., the employment-topopulation rate is consistently around 60%. Hence, we fix the total number of households
at 303, 322 × 0.6 = 181, 993. Each income group constitutes 10% of the total population
implying that N i = 18, 199 for all i. Given the total number of households (and otherparameters), agricultural rent Pa determines the city’s size. The lower Pa is, the larger
the city is. We set agricultural rent to Pa = $505, 340 per square mile per week to matchpthe endogenous calibration targets for the fringe distance Rf∗ , which is 94.08/π miles.
We then describe the calibration of the budget constraint parameters. Note that ourmodel is static and does not feature income tax. Hence, instead of calibrating hourly
wage wi to match annual before-tax income (reported in the first row of Table 1), we
19

Authors’ calculation based on the Census data.

13

calibrate wi to match total yearly expenditures (reported in the second row of Table1). Recall that Table 2 shows that the hours worked on average are around 8 hours for
all income groups. Hence, under a reasonable assumption that a typical worker works
five days each week, the weekly and annual work times are around 5 × 8 = 40 hours
and 40 × 52 = 2080 hours, respectively. Therefore, we obtain the hourly wages for each
income group by dividing the average total annual expenditures reported in the second
row of Table 1 by 2080.
The weekly per mile pecuniary cost a and time cost b are chosen to match averagetransportation expenditure and commute time. The average annual spending on transportation for each income group has been reported in Table 1. Dividing those numbersby 52 (weeks) yields the calibration targets for the group-specific averages of ar. In the
U.S., the average one-way commute time is 26.1 minutes, according to 2017 Census data.
We assume each household does a round-trip commute to the CBD for five days weekly.
Hence, the calibration target for the average br is 26.1 × 2 × 5/60 = 4.35 hours. The
calibrated values of a and b are 17.46 and 1.56, respectively.
The last set of parameters to be determined are the preference parameters. Equation
24−br−nl, where n represents the weekly hours(8) shows that α = 1 − 168−br−
ar = 1 −168−br− arww

of work. As documented above, n is roughly 40 hours and does not vary much acrosshouseholds. The average commute time br is 4.35 hours. The average ar
can be obtainedw
this way. We first calculate the ratio of the group-specific averages of ar to group-specificweekly wage wi . We then take the average over income groups. The result is 1.81 hours.
168−4.35−40= 0.236.
Hence, we set α to be 1 − 168−4.35−1.81

Finally, we calibrate parameters θ and ρ to match the observed pattern of the housing
expenditure share. Intuitively speaking, ρ determines the speed at which this fraction
decreases with total income; for any fixed ρ, θ pins down the overall level of this fraction.
We search for the set of θ and ρ that minimizes the sum of the squared differences between
model-implied shares for each income group and their data counterparts. We find thatθ = 0.0068 and ρ = 2.85.
3.3.2

Baseline Equilibrium

Table 4 summarizes the baseline equilibrium outcomes. Panel A of Table 4 shows thatthe calibrated model can closely match data on total expenditures, housing expenditures,
housing expenditure share, work hours, commute time, and pecuniary commute cost.
Figure 1 shows that our model can match the observed negative relationship between
income, or equivalently total expenditures, and the housing expenditure share. The reddashed line is obtained by plotting the fifth row of Panel A of Table 4 against the firstrow of Panel A of Table 4. The housing expenditure share is roughly monotonicallydecreasing in total weekly spending. The solid blue line, obtained by plotting the sixth
row of Panel A against Panel A’s second row, represents the model-implied counterpartof the red line. Figure 1shows these two lines are very close.

14

15

Parameter

Population Size of Type i

Weekly Commute Cost per Mile (hours)abInverse of the Elasticity of Substitution between Numerare Goods and Housing

Importance of Numerare Goods relative to Housing“Free” Consumption of Numerare Goods (dollars)θ (CES)ρ (CES)βC (Non-homothetic)c̄ (Non-homothetic)and Hourly Wage Rate wi

Relationship between Housing Expenditure Share

Weekly Hours of Work

Average Commute Time

Average Transportation Expenditure

Total Expenditures by Income Type

Average Working Population of U.S. Cities

Average Size of U.S. Cities

Targeted Moment

N.A.

N.A.

2.85

0.00680.236

505,340

Value: CES Spatial

1.56

17.46

240

0.7191

N.A.

N.A.

N.A.

Varies by Income Type18,199703,070

644,910Non-homothetic Spaceless Non-homothetic Spatial

w9 = 42.03, w10 = 65.80.

Note: Here we report type-specific hourly wages: w1 = 11.34, w2 = 12.82, w3 = 16.45, w4 = 18.90, w5 = 21.14, w6 = 24.69, w7 = 28.56, w8 = 33.85,

Importance of Consumption relative to LeisureImportance of Numerare Goods relative to Housingα (CES)

Preference Parameters

Hourly Wage Rate (dollars)

Weekly Commute Cost per Mile (dollars)wiBudget Constraint Parameters

Weekly Agricultural Rent (dollars per square mile)

Pa

Interpretation

Ni

Macroeconomic Environment Parameters

Table 3: Calibration Targets and ResultsPanel B of Table 4 presents the additional results. Notably, the second row of Panel
B shows that more affluent households reside farther away from the CBD, where the rent
is lower. More affluent households can afford housing with larger lot sizes with higher
disposable income, as shown in the first row of Panel B. Finally, the last row of Panel B
reports the average value of κ for each income group. Comparing them to the sixth row
of Panel A, we find that κ is systematically larger than but quite close to the housing
expenditure share. This is consistent with our interpretation of κ in Section 3.1.2.
Figure 1: Total Annual Expenditure and the Fraction Spent on Housing

4

A Monocentric City Model: Housing Expenditure Share Cap

In this section, we examine the welfare implications of imposing a cap κ̄ on the after∗

P (r)hcommute-cost (ACC) housing expenditure share, c+P∗ (r)h , in the context of the mono-

centric city model described in Section 3. In Appendix B, we show that our results
are robust to adding dynamics and unobserved preference shock to the baseline static
deterministic model.

16

17

Second

Third

$519

$456

Model

$664

$186

Model

$200

$211
$247

$247

0.4090Model

0.38590.410940.36

0.33790.3334

$335

$329

$991

$988

2.61

3.30

$11

Model

1.62681.9479miles )

2

2

2.6137

$29

$29

Model

κ

Model

0.4364$1,145,300

0.4096$1,028,100
0.3960$946,900

Average Weekly Rent (unit: per miles )

Model

−4

$21

$23

Average Lot-size (unit: 10

$20

Data

Average Weekly Pecuniary Commute Cost
$44

$41

$52

$47

0.3838$877,900

3.12540.3726$817,100

3.6306

0.3619$760,900

4.4020Panel B: Additional Outcomes

$37

$36

4.61

3.95

1.90

Model

1.01

around 4.35 hours

Data

Average Weekly Commute Time

40.15

40.31

40.38

0.34810.3527

$297

$298

$852

$846

Model

40.46

0.35970.3631

10 percent

Sixtharound 40 hours

40.17

0.37250.3749

$274

$274

$763

$756

10 percent

Fifth
Panel A: Model-fit

10 percent

FourthData

Average Weekly Work Hours

0.4056

Data

Fraction of Expenditures Spent on Housing

$184

Data

Average Weekly Housing Expenditures

$513

$454

Data

$658

10 percent 10 percent 10 percent

LowestAverage Weekly Expenditures

Item

0.3512$707,100

5.2905

$59

$53

5.30

39.98

0.32770.3247$374$371$1,142$1,14210 percent

SeventhTable 4: Summary of Baseline Equilibrium Outcomes

0.3402$654,400

6.5291

$68

$62

6.04

39.74

0.3176

0.3137$427$425$1,345$1,35410 percent

Eighth

0.3286$600,600

8.4751

$77

$73

6.85

39.42

0.30720.3056$509$514$1,657$1,681

10 percent

Ninth0.3157$538,400

14.0459

$89

$102

7.92

38.81

0.29620.2962$756$780$2,554$2,63210 percent

Tenth

4.1

Optimal Consumption and Indirect Utility Function

P ∗(r)h, we modify the utility maximization problem
When a cap of κ̄ is imposed on c+P∗(r)hdefined in Equation (5) accordingly to obtain the indirect utility function Ṽi (P ∗ (r), r):Ṽi (P ∗ (r), r) = max U (c, h, l)(c,h,l)P ∗ (r)h≤ κ̄,s.t. Yi (r) =wi l + c + P (r)h, andc + P ∗ (r)h∗(16)

where Yi (r) ≡ (168 − br)wi − ar.
There are two possibilities. First, the cap is not binding. Hence, the utility maximization problem is identical to the one defined in Equation (5). As shown in Section 3.1.2,
1∗

P (r)hin this case, the optimal value of c+P
is equal to κ(P ∗ (r)) =
∗(r)h

( 1−θ
) ρ P ∗ (r)
θ
1

ρ−1
ρ

1+( 1−θ) ρ P ∗ (r)
θ

ρ−1
ρ

, whichis monotonically increasing in the market rent P ∗ (r). Hence, the cap binds when P ∗ (r)
ρ

1

κ
θ
is sufficiently high. It is easy to verify that the cutoff value for P ∗ (r) is ( 1−κ) ρ−1 ( 1−θ) ρ−1 .

The second possibility is that the cap binds. The choices of consumption, housing,and leisure are given by:h̃i (P ∗ (r), r) =ακ̄Yi (r),P ∗ (r)(17)c̃i (P ∗ (r), r) = α(1 − κ̄)Yi (r),(18)

˜li (P ∗ (r), r) = (1 − α)Yi (r) = (1 − α)(168 − br − ar ).
wi
wi

(19)

Comparing these choices to their counterparts in the unconstrained case (Equation
6-8), we find that under a binding housing expenditure cap, households spend less money
on housing and more money on consumption, and enjoy the same amount of leisure time.
Moreover, we can combine results for both the constrained and the unconstrained casesin the following expression:Ṽi (P ∗ (r), r) = [(1 − κ̃)1−ρ +α
1 − θ κ̃1−ρ 1−ρ
1] KYi (r)( )1−α ,∗1−ρθ P (r)wi(20)

where κ̃ = min(κ(P ∗ (r)), κ̄) represents the ACC housing expenditure share chosen by thehousehold at the equilibrium.20 We will see later whether the cap binds directly affectthe welfare results.
4.2

Bid-rent Function and Spatial Order under Housing Expenditure Cap

This section studies how agents are distributed in the monocentric city under the housing
expenditure cap. Recall that the spatial order of two adjacent types of households isdetermined by the relative steepness of the bid-rent function ψ̃i (u∗i , r) at the intersection,and the household with a steeper bid-rent function lives closer to the CBD.
20

We obtain the indirect utility function for this constrained case by substituting the choices shownin Equation (17)-(19) in the utility function shown in Equation (1).

18

When the cap is non-binding, this bid-rent function is identical to the one shown inEquation (11). Hence, as discussed in Section 3.2.1, among the unconstrained households,
those with higher hourly wages wi live farther away from the CBD.
When the cap binds, we obtain the bid-rent function ψ̃i (u∗i , r) by inverting the indirect
utility function Ṽi (P ∗ (r), r) shown in Equation (20) with κ̃ taking the value of the cap κ̄:ψ̃i (u∗i , r) = κ̄(Yi (r)Ku∗i wi1−α

 ρ−1
α

1
) ρ−1

− (1 − κ̄)1−ρ

(

1
θ ρ−1
) .
1−θ

(21)

As with the unconstrained case, we can show that 1) ∂ ψ̃∂ri (·) is always negative, and 2)| ∂ ψ̃∂ri (·) | is larger for households with lower hourly wage wi in the constrained case as well.
Hence, those with higher hourly wages wi also live farther away from the CBD among
the constrained households.
To sum up, we conclude that Proposition 2, which states that households with higher
hourly wage wi live farther from the CBD at the equilibrium, remains valid under the
housing expenditure cap.21
4.3

Welfare Analysis

This subsection analyzes and quantifies the welfare implications of imposing a cap on the
ACC housing expenditure share, especially on how the cap would affect the welfare/utility
inequality. Since utility is unit-free, direct cross-sectional comparisons of individual utility
do not convey meaningful information. Hence, we adopt a widely used consumptionequivalent measure (CE) to quantify the cap’s effect on the utility level for each typeof household (Cooley and Hansen, 1989). We focus on the cross-sectional relationshipbetween CE and income level. For example, if low-income households tend to have smaller
gains or more losses in CE under the cap, we would conclude that the cap policy leads toa more considerable welfare inequality. Formally, the consumption-equivalent measure ξ
is defined as follows:
Definition 2. Consider a household whose numeraire goods consumption, housing consumption, and leisure time in the baseline equilibrium are cbase , hbase , and lbase , respectively. Further, denote ucap as the utility level this household obtains under a cap on
the ACC housing expenditure share. The consumption-equivalent measure ξ for thishousehold satisfies the following equations:
U ((1 + ξ)cbase , (1 + ξ)hbase , lbase ) = ucap .
Definition 2 states that ξ measures the faction of numeraire goods and housing consumption that a household needs to obtain in the baseline to achieve the same level of
21

i (·)
Noting that ∂ ψ̃∂ris negative for all households, we also conclude that the equilibrium rent P ∗ (r) is

decreasing in the distance from the CBD r. Hence, Proposition 1 ( ∂κ(x)∂x > 0) implies that the housing
expenditure share, κ̃ = min(κ(P ∗ (r)), κ̄), is weakly monotonically decreasing in the distance r and in
the hourly wage wi at the equilibrium.

19

utility in the economy with the cap policy. A positive ξ represents a utility gain, while anegative ξ represents a utility loss.
It is easy to verify that, for any given cap κ̄, households of the same type have thesame ξ. We then denote the consumption-equivalent measure for type i household as ξi .
The Appendix shows that when a commute’s monetary and time costs are negligible, ξiis roughly the same for all households. Formally, the following proposition holds.
Proposition 4. For any κ̄ ∈ (0, 1] and any Type i and Type j households, we havelim

a−>0+ ,b−>0+

ξi − ξj = 0.

Proposition 4 implies that, in contrast to the single housing market, the welfare costs
incurred by different types of households are similar under the policy cap inflicted in a
monocentric city model when commuting costs are negligible. The intuition is as follows.
A monocentric city model contains infinitely many housing submarkets indexed by the
distance r. Because the equilibrium exhibits perfect sorting by income, constrained
low-income households only need to compete with other deprived low-income families
interested in the same submarket, resulting in a lower equilibrium rent for these locations.
Hence, the utility loss due to constrained housing expenditure share is compensated by
the utility gain from lower market rent for the constrained households.
Next, we quantify the effect of the cap on equilibrium utility based on our calibrated
model. We consider five different values for κ̄, 20%, 25%, 30%, 35%, and 40%. For eachvalue, we solve for the equilibrium utility and the consumption-equivalent measure ξi for
all types of households.
Panel A of Table 5 reports the results. As shown in Section 3.3, commute costs arequite small in our calibrated model. Hence, consistent with Proposition 4, we find very
similar ξi across household types for all κ̄ that we consider. If anything, we find that ξi
is generally smaller or more negative for relatively higher-income households.22 Both thetheoretical and quantitative results suggest that restricting the fraction of total income ahousehold can spend on housing consumption could lead to a smaller welfare inequality
in a monocentric city environment.

5Spaceless Model and Non-homothetic PreferenceThis section will further explore the interaction of non-homothetic preference and theassumption of space. In particular, we will show that adding a simple Stone-Geary typenon-homothetic utility function to a spaceless model drastically alters the implications
for welfare inequality (Geary, 1950; Stone, 1954). In comparison, we also consider amonocentric city model with non-homothetic preference and discuss the implications ofthese results for the value of explicitly modeling space.
22

In many cases, we see ξi is small but positive for all types of households. Notice that this does not

represent an economy-wide Pareto improvement. Equilibrium market rents in these cases are lower thanin the baseline, meaning the absentee landlord bears the cost.

20

21

Cap κ̄

-3.84
-7.64

-6.71

-11.81

-0.13

-0.91

-2.33

-4.78

-8.86

25%

20%

40%

35%

30%

25%

20%

10 percent

Fourth10 percent

Fifth
10 percent

Sixth
10 percent

Seventh

-3.14

1.40

1.53

0.48

0.04

-3.24

1.33

1.47

0.44

0.04

-3.33

1.27

1.43

0.41

0.04

-3.46

1.18

1.37

0.38

0.03

-3.58

1.10

1.31

0.35

0.03

-7.26

-3.39

-1.14

-0.12

0

-6.59

-2.94

-0.88

-0.04

0

-5.78

-2.40

-0.59

0

0

-5.16

-2.00

-0.40

0

0

-1.64

-0.49

0.02

-5.68

-2.40

-0.68

0

0.05

-4.89

-1.84

-0.34

0.13

0.04

-4.36

-1.49

-0.15

0.17

0.04

-3.79

-1.11

0.04

0.17

0.03

-3.38

-0.86

0.14

0.14

0.03

Panel C: Monocentric City Model: Non-Homothetic Preference

-8.24

-4.08

-1.55

-0.27

0

Panel B: Spaceless Model: Non-Homothetic Preference-10.56

-5.77

-2.65

-0.83

-3.31

-0.06

-2.97

30%

-2.91

20%

1.52

-1.22

1.56

25%

1.62

35%

1.65

30%

0.54

-0.19

0.56

35%

0.06

10 percent

Third
Panel A: Monocentric City Model: CES Preference

10 percent

Second40%

0.07

10 percent

Lowest

40%

Item

-3.01

-0.64

0.21

0.11

0.02

-4.55

-1.63

-0.24

0

0

-3.73

1.00

1.25

0.33

0.03

10 percent

Eighth

-2.68

-0.47

0.24

0.08

0.01

-3.94

-1.27

-0.12

0

0

-3.90

0.89

1.18

0.31

0.03

10 percent

Ninth

Table 5: Consumption Equivalent Measure ξi (%) for Alternative Housing Expenditure Share Cap κ̄

-2.28

-0.30

0.21

0.05

0.01

-3.10

-0.81

-0.01

0

0

-4.21

0.70

1.08

0.28

0.02

10 percent

Tenth

5.1

Spaceless Non-homothetic Model

Recall that the work hours are roughly constant across income groups. Hence, we consider
a simpler version where all households work the same amount of time (40 hours perweek), and a household’s utility depends only on its consumption of numeraire good cand housing lot size h. Formally, we assume that its preference is non-homothetic andits utility is given by:
U (c, h) = (c + c̄)βc h1−βc ,(22)where c̄ is “free” consumption of numeraire goods, which, as shown below, can help
generate a decreasing (with income) housing expenditure share.
In this spaceless model, all households reside at a location r∗ miles away from theirworkplace, the CBD. The (weekly) budget constraint faced by a household with an hourly
wage of w is given by
40w = c + P ∗ (r∗ )h + ar∗ ,(23)

where, as before, P ∗ (r∗ ) is the equilibrium market rent and ar∗ is the pecuniary cost ofcommuting.
Solving the household’s problem yields the following equations for the optimal housing
expenditure and the indirect utility function:P ∗ (r∗ )h(P ∗ (r∗ ), r∗ ) = (1 − βc )(40w − ar∗ + c̄),V (P ∗ (r∗ ), r∗ ) = βcβc (1 − βc )1−βc (40w − ar∗ + c̄)[P ∗ (r∗ )]βc −1 .

(24)
(25)

The optimal after-commute-cost (ACC) housing expenditure share in this model isthen given by:c̄
P ∗ (r∗ )h(P ∗ (r∗ ), r∗ )= (1 − βc )(1 +
).
(26)
∗
40w − ar40w − ar
It is trivial to see that this housing expenditure share is decreasing in hourly wage w,
which is consistent with our second stylized fact.
Next we examine the implications of imposing a cap of κ̄ on the ACC housing expen∗

∗

∗

∗

∗

)h(P (r ),r )diture share, P (r 40w−ar. Notice that because our model features an open city, the∗equilibrium rent equals the exogenous agricultural rent Pa with or without the housing
expenditure cap. Hence, if the cap is not binding, the utility maximization problem faced
by a particular household is identical to the baseline case. Consider a household with anhourly wage of wi . If the optimal ACC housing expenditure share is lower than κ̄, thecap is not binding for this household. In this case, this household will attain the samelevel of utility as in the baseline. On the other hand, if this share is greater than κ̄, thishousehold cannot allocate consumption expenditures optimally. As a result, it will attainutility that is strictly lower than the baseline level.
Because high-income (higher w) households have a lower optimal housing expenditure
share, they are less likely to be constrained by the housing expenditure share cap κ̄compared to low-income households. Therefore, high-income households may not be22affected for a moderate κ̄, while low-income families may be constrained and attain lower
utility (than in the baseline). Since high-income households have relatively higher utility
than low-income households in the baseline (V (P ∗ (r∗ ), r∗ ) is increasing in w), this policy
cap on the ACC housing expenditure share would increase utility/welfare inequality.
We also quantitatively examine the welfare implications of the cap policy in thisspaceless model. We calibrate the model to match the same targeted moments as in
Section 3.3.1. The only three parameters calibrated to have different values from thosefor the monocentric city model are the agricultural rent Pa and preference parametersβc and c̄. Pa is the minimum market rent in the monocentric city model and the uniquemarket rent in the spaceless model. Hence, to keep the city’s size constant, Pa in thespaceless model needs to be larger than its counterpart in the monocentric city model.
The calibrated weekly agricultural rent Pa is $703,070 per mile2 . The preference parameters, βc and c̄, are chosen to match the relationship between housing expenditure shareand total weekly expenditure/income (Figure 1) as closely as possible. Their calibrated
values are βc = 0.7191 and c̄ = $240, respectively.23
Panel B of Table 5 reports the welfare results from this spaceless model. Consistent with the discussion above, a household experiences a drop in welfare (negative CE
measure) only when its optimal ACC housing expenditure share is higher than κ̄. Since
low-income families have a higher optimal ACC housing expenditure share, they are moreheavily (negatively) influenced by the cap. As a result, the consumption-equivalent measure ξi tends to be much more negative for low-income households. For example, a 20%
cap imposes 3.10% welfare loss (in terms of ξi ) for families in the top decile of the income
distribution and 11.81% welfare loss for households in the bottom decile of the income
distribution.
5.2

Spatial Non-homothetic Model

The previous sections have shown the welfare implications of the housing expenditure
cap in a monocentric city model with CES preference and a spaceless model the nonhomothetic preference (Panels A and B of Table 5). In this section, we will show inparallel the welfare implications of having non-homothetic preference in a monocentriccity model (Panel C of Table 5). Expressly, we assume a household’s problem is that
described in Section 5.1 while keeping all the other model settings the same as thosedescribed in Section 3.
Appendix C.2 shows that this spatial non-homothetic model can generate the same
spatial order as the spatial CES model - wealthier households live farther away fromthe CBD. We focus on quantitatively evaluating the welfare implications of imposing
a housing expenditure share cap, κ̄. Most calibrated parameters of the spatial nonhomothetic model are the same as those of the spaceless non-homothetic model. The
only exception is the weekly agricultural rent Pa , calibrated to be 644,910 per mile2 .
23

See Appendix C.1 for the fit of the spaceless model.

23

We again measure welfare changes using the consumption-equivalent (ξi ) defined in
Definition 2. Panel C of Table 5 reports the welfare results. The general pattern isqualitatively similar to those of the spaceless model (Panel B). When a cap is imposed on
the housing expenditure share, poorer households suffer a larger welfare drop than more
affluent households. For example, a 20% cap imposes 2.28% welfare loss for households
in the top decile of the income distribution and 8.86% welfare loss for households in thebottom decile of the income distribution, which results in a larger welfare inequality.
However, the quantitative differences are worth noting. Again using a 20% cap as
an example, the welfare drop in the spatial model is smaller than that in the spacelessmodel (3.10% for the top decile and 11.81% for the bottom decile). We then turn ourattention to the effect of the cap on welfare inequality. One measure of welfare inequalitychange is the difference between the top income decile’s and bottom income decile’s CE,
ξ10 − ξ1 . This difference’s positive (negative) value indicates an increase (decrease) in
welfare inequality. For a 20% cap, this difference equals 6.58% (8.86% - 2.28%) in thespatial model and 8.71% (11.81% - 3.10%) in the spaceless model. In Figure 2, we plot
this measure of welfare inequality change for a wide range of κ̄ and for all three models
we have considered. For all levels of κ̄, Figure 2 shows that while, unlike the case in thespatial CES model, we still see aggravated welfare inequality due to the cap policy in thespatial non-homothetic model, the magnitude of this effect is smaller compared to thatin the spaceless non-homothetic model.
The quantitative difference between the welfare results in the spatial non-homothetic
model and the spaceless non-homothetic model stems from the general equilibrium priceadjustment and income-based spatial sorting discussed in Section 4.3. The welfare inequality still increases in the spatial non-homothetic model because the general equilibrium channel is not strong enough when households have (non-homothetic) Stone-Geary
preference. In this case, market rent only affects welfare through its influence on housing
affordability. In contrast, in the CES model, market rent influences housing affordability
and determines the optimal housing expenditure share. This second additional effectstrengthens the general equilibrium channel, allowing it to entirely offset the aggravating
impact of the cap on welfare inequality in the spatial CES model.
5.3

Discussion: The Value of Modeling SpaceIn examining the effect of a housing expenditure share cap, the value of modeling spaceis three-fold. First, it enables our model to match the empirical relationship between
commute distance and household income. In contrast, by definition, a spaceless model issilent on this empirical finding.
Second, because of the general equilibrium channel discussed above, keeping household preference unchanged, the aggravating effect of the cap on welfare inequality isalways more potent in a model without space than in a model with space. This is thecase even when the preference is non-homothetic (Stone-Geary).

24

Figure 2: Changes in Welfare Inequality and Housing Expenditure Share Cap

25

Third, perhaps more importantly, by explicitly modeling space, we can generate thenegative relationship between income and housing expenditure share using homothetic
preference (CES). This presents an important unidentification issue because both thespatial CES model and the spatial Stone-Geary model can match the three stylized facts
documented in Section 2 but have drastically different policy predictions. Intuitively,
these two models represent two extremes. The optimal housing expenditure share isdetermined entirely by equilibrium market rents in the former and entirely by hourly
wages in the latter. In reality, this share likely depends on both. That means the true
effect of the cap policy on welfare inequality is somewhere between the blue solid line
and the black dashed line in Figure 2. To point-identify this effect, one needs first to
bring in more data to recover households’ preferences precisely.

6

Conclusion

This paper reinforces the intuitive and essential point that spatial consideration mattersin economic analysis. We frame the statement in the context of the housing expenditure
share cap. Variants of this policy have been explicitly and implicitly adopted as somecomponents of macroprudential policies worldwide. We document three cross-sectional,
stylized facts: the housing expenditure share decreases with income, the apparent constancy of work hours across income groups, and the positive relationship between income
and the commute distance from the CBD. While non-homothetic preference is needed
for a spaceless model to match the first two facts, a simple monocentric city model canmatch all three facts with both homothetic and non-homothetic preferences.
Moreover, because all households face the same housing price, the spaceless model
would predict that the cap policy leads to a disproportionately high welfare loss for
low-income families constrained by the cap, resulting in substantially larger welfare inequality. The monocentric city model has a different prediction. Because of income-based
spatial sorting and geographically differentiated housing price adjustments, constrained
low-income families experience a disproportionally more considerable drop in endogenous
equilibrium rents than unconstrained high-income families. Depending on households’
preferences, the welfare gain from this mechanism can partially (non-homothetic preference) or entirely (homothetic preference) offset the welfare loss due to the cap policy,
resulting in a smaller increase (non-homothetic preference) or even a mild reduction(homothetic preference) in welfare inequality after the cap is imposed.
Our results suggest that policymakers should incorporate spatial considerations in
the design of macroprudential policies. Moreover, our finding that the exact welfare
results depend on households’ preferences highlights the importance for future research
to precisely estimate a flexible utility function that allows the housing expenditure shareto depend on both household income and housing rents/prices.
We have considered several extensions in this paper, including introducing idiosyncratic location preference shock and extending to a more realistic, dynamic setting. Fu26

ture research can explore other extensions, including collateral constraints, dynamic location choice, endogenous housing supply, endogenous tenure choice, multi-community
within a city, and portfolio choices.24 These new features can facilitate a more comprehensive and in-depth analysis of macroprudential policies.

24

See Beraja et al. (2019), Combes, Duranton, and Gobillon (2021), Hanushek and Yilmaz (2007,2013), Leung and Teo (2011), Piazzesi, Schneider, and Stroebel (2020), and Yao (2023), among others.

27

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32

AppendicesA

Proofs

Proposition 1. If ρ > 1, then ∂κ(x)
> 0.
∂x
1

Proof. Recall that κ(x; θ, ρ) =respect to x, we have:

( 1−θ
)ρ x
θ
1

ρ−1
ρ

)ρ x
1+( 1−θθρ−1ρ

. Taking the derivative of this function with

1

ρ−1

ρ−1

1

[1 + ( 1−θ) ρ x ρ ]( 1−θ) ρ x ρ −1 ρ−1∂κ(x; θ, ρ)
θ
θ
ρ
=
ρ−1
1
∂x
[1 + ( 1−θ ) ρ x ρ ]2
θ

ρ−1
ρ

1

−

( 1−θ
)ρ x
θ

1

)ρ x
( 1−θ
θ

ρ−1
−1 ρ−1
ρ

1

ρ−1
ρ

1

−1

[1 + ( 1−θ
)ρ x
θ

ρ

]2

)ρ x ρ
( 1−θ
ρ−1
θ
.
=
ρ [1 + ( 1−θ ) ρ1 x ρ−1
ρ ]2

(27)

θ

This derivative has the same sign as ρ−1. Hence, if ρ > 1, then ρ−1> 0, which impliesρρthat ∂κ(x;θ,ρ)> 0.
∂x

Proposition 2. Households with higher hourly wage wi live farther from the CBD at theequilibrium.
Proof. Recall that the bid-rent function for Type i household, ψi (u∗i , r), is given by:
ψi (u∗i , r) =(Yi (r)Ku∗i wi1−α

 ρ−1
αρ

ρ
) ρ−1

−1

(

1
θ ρ−1
) ,
1−θ

(28)

where Yi (r) ≡ (168 − br)wi − ar.
The first order derivative of this function with respect to the distance r is given by:

i

 ρ−1
αρ

)−1


 ρ−1
∂ψi (·)ρYi (r)K
ρ − 1 Yi (r)K αρ Yi0 (r)∗= ψi (ui , r)
−1
∂r
ρ−1
αρ u∗i wi1−α
Yi (r)
u∗i wi1−α
ρ−1
h
i αρ
Yi (r)K∗1−α∗ψi (ui , r)−(a + bwi )
ui wi
=
.
ρ−1
h
i
α
(168 − br)wi − arαρYi (r)K−1
u∗ w1−α
(

(29)

i

h

i ρ−1

αρ
The fact that ψi (u∗i , r) > 0 implies that uY∗iw(r)K− 1 > 0. Hence, we conclude
1−αi ithat ∂ψ∂ri (·) < 0 from Equation (29). It means all bid-rent functions are decreasing in the

distance from the CBD, r.
∗
Let rB,idenote the distance from the CBD at which Type i households and Typei + 1 households intersect in the baseline equilibrium. Note that, at the equilibrium, we
33∗∗have ψi (u∗i , rB,i
) = ψi+1 (u∗i+1 , rB,i) (same rental rate for both types of households at theintersection). Together with Equation (28), it implies that:∗
ψi (u∗i , rB,i)h Y (r∗ )K i ρ−1
αρ
i

B,i
u∗i wi1−αh Y (r∗ )K i ρ−1
αρ

α

i

B,i
u∗i wi1−α

=

α

−1

ρ−1
∗
αρ
i+1 (rB,i )K1−α∗
ui+1 wi+1
ρ−1∗ )K
Yi+1 (rB,i
αρ

hY

∗
ψi+1 (u∗i+1 , rB,i
)

h

i

i

1−α
u∗i+1 wi+1

.

(30)

−1

Denoting the value of this term as Cons∗ψ,i , we have:∗∂ψi (u∗i , rB,i)a + bwi| = Cons∗ψ,i,∂r(168 − br)wi − ar∗)∂ψi+1 (u∗i+1 , rB,ia + bwi+1| = Cons∗ψ,i.
|∂r(168 − br)wi+1 − ar

|

(31)

Then, the difference between the two slopes is given by:|∂ψi (·)∂ψi+1 (·)
a + bwia + bwi+1|−|| = Cons∗ψ,i [
−
]
∂r
∂r
(168 − br)wi − ar (168 − br)wi+1 − arCons∗ψ,i {(a + bwi )[(168 − br)wi+1 − ar] − (a + bwi+1 )[(168 − br)wi − ar]}=[(168 − br)wi − ar][(168 − br)wi+1 − ar]
168Cons∗ψ,i a(wi+1 − wi )=[(168 − br)wi − ar][(168 − br)wi+1 − ar]
> 0,

(32)

where the last line follows from the fact that Type i households has lower hourly wages
than Type i + 1 households (wi < wi+1 ).
Thus, we have shown that | ∂ ψ̃∂ri (·) | is larger for households with lower hourly wage wi ,which implies that households with higher hourly wage wi lives farther from the CBD at
the equilibrium.
Proposition 3. Consider two arbitrary households. We denote their types as i and j,
respectively. Let ri∗ and rj∗ denote their distance from the CBD at the equilibrium. If
ρ > 1 and wi > wj , then κ(P (ri∗ )) < κ(P (rj∗ )) at the equilibrium.
Proof. If wi > wj , then Proposition 2 implies that ri∗ > rj∗ . Because the equilibrium rent
function P (r) is decreasing in the distance r, we have P (ri∗ ) < P (rj∗ ). Hence, Proposition
1 implies that κ(P (ri∗ )) < κ(P (rj∗ )).
Proposition 4. For any κ̄ ∈ (0, 1] and any Type i and Type j households, we havelim

a−>0+ ,b−>0+

ξi − ξj = 0.

Proof. Consider two equilibria, the baseline equilibrium and the equilibrium with the∗∗cap κ̄. Let rB,iand rκ̄,idenote the distances from the CBD at which Type i households

and Type i + 1 households intersect in the baseline equilibrium and in the equilibriumwith the cap, respectively. Let PB∗ (r) and Pκ̄∗ (r) denote the rent functions in the baseline34equilibrium and in the equilibrium with the cap, respectively. Further, noting that κ̃ =min(κ(P ∗ (r)), κ̄) only depends on the equilibrium rent P ∗ (r) (and the cap κ̄), we let κ̃∗B,iand κ̃∗κ̄,i denote the value of κ̃ at the distances from the CBD at which Type i households
and Type i + 1 households intersect in the baseline equilibrium and in the equilibriumwith the cap, respectively.25∗
The utility that Type i household obtains in the baseline, ṼB,i, is given by:∗ṼB,i= [(1 − κ̃∗B,i )1−ρ +∗1−ρ
α
1 − θ (κ̃B,i )1∗)( )1−α .
] 1−ρ KYi (rB,i∗∗1−ρθ PB (rB,i )
wi

(33)

α

, where K = αα (1 − α)1−α θ 1−ρ .
When this household is given ξ fraction more consumption (both c and h), it is trivial∗
to see that the utility ṼB,i(ξ) is given by:
α

∗
∗
ṼB,i
(ξ) = (1 + ξ) 1−ρ ṼB,i(ξ)α= (1 + ξ) 1−ρ [(1 − κ̃∗B,i )1−ρ +α
1 − θ (κ̃∗B,i )1−ρ 1−ρ
1∗] KYi (rB,i)( )1−α .
∗ ∗1−ρθ PB (rB,i )
wi

(34)

∗
, is
The utility that Type i household obtains in the equilibrium with the cap, Ṽκ̄,igiven by:1−θ∗Ṽκ̄,i= [(1 − κ̃∗κ̄,i )1−ρ +θ(κ̃∗κ̄,i )1−ρ α
1 1−α
1−ρ KY (r ∗ )().
i κ̄,i∗ )1−ρ ]Pκ̄∗ (rκ̄,i
wi

(35)

.
The consumption-equivalent measure for Type i household, ξi , is defined as the solu∗∗. Solving this equation, we have:(ξi ) = Ṽκ̄,ition to the equation ṼB,i∗1−ρ
1−θ (κ̃κ̄,i )
∗ 1−ρ
∗
)
+
(1
−
κ̃
∗
∗
κ̄,i
Yi (rκ̄,i ) 1−ρθ Pκ̄ (rκ̄,i )1−ραξi = []− 1.
∗(κ̃∗B,i )1−ρYi (rB,i)(1 − κ̃∗B,i )1−ρ + 1−θ
∗∗1−ρθ PB (rB,i )(36)

Following the same steps, we can show that the consumption-equivalent measure forType i + 1 household, ξi+1 , is given by:∗1−ρ
1−θ (κ̃κ̄,i )
∗ 1−ρ
∗
(1
−
κ̃
)
+
∗
∗
κ̄,i
Yi+1 (rκ̄,i ) 1−ρθ Pκ̄ (rκ̄,i )1−ραξi+1 = []− 1.
∗(κ̃∗B,i )1−ρYi+1 (rB,i)(1 − κ̃∗B,i )1−ρ + 1−θ
∗∗1−ρθ PB (rB,i )(37)

The key here is to notice that Type i and Type i + 1 households share the same rent andhousing expenditure share at the intersection of the two types.
Thus, we conclude that:ξi+1 − ξi = {[where Cons∗i ≡25∗∗
Yi+1 (rκ̄,i) 1−ρ
Yi (rκ̄,i
) 1−ρ
α − [
]
] α }Cons∗i ,∗∗Yi+1 (rB,i )Yi (rB,i )

1−ρ
(κ̃∗
κ̄,i )∗ (r ∗ )1−ρPκ̄κ̄,i(κ̃∗ )1−ρ
B,i
1−θ
∗
1−ρ
(1−κ̃B,i )+ θP ∗ (r ∗ )1−ρB B,i

(1−κ̃∗κ̄,i )1−ρ + 1−θθ.

In the baseline equilibrium, κ̄ = 1.

35

(38)

Recall that Yi (r) ≡ (168 − br)wi − ar. Hence, we have:∗∗∗)wi − arκ̄,i)(168 − brκ̄,iYi (rκ̄,i=∗∗∗Yi (rB,i)(168 − brB,i)wi − arB,i
∗
∗
∗
∗
∗
∗
(168 − brB,i)wi − arB,i))wi + a(rB,i− rκ̄,ib(rB,i− rκ̄,i=+∗∗∗∗(168 − brB,i )wi − arB,i(168 − brB,i )wi − arB,i∗∗(a + wi b)(rB,i− rκ̄,i)=1+∗∗(168 − brB,i )wi − arB,i

−−−−
−−−→
1.
+
+

(39)

a→0 , b→0

This proves that ξi+1 − ξi −−−−−−−→
0 for all i = 1, 2, ..., 9. As a result, we show that++a→0 , b→0

ξj − ξi −−−−−−−→
0 for all i = 1, 2, ..., 10 and j = 1, 2, ..., 10.
++a→0 , b→0

B

Extensions

Our main analysis is based on a static model. While it is compatible with the previous
literature (e.g, Hanushek and Yilmaz, 2022), there is room for further improvement. For
one thing, a static model does not distinguish between the for-sale and rental markets.
In practice, buyers typically do not pay in full but instead, make a mortgage loan.
Therefore, many macroprudential measures target the debt-to-income ratio (IMF, 2014,2018). In addition, our model features perfect sorting of income. In reality, while robust
income-based sorting exists, such a sorting pattern is not perfect, i.e., households withdifferent income levels reside in the same neighborhood (Davidoff, 2005; Gaignea et al.,2022). Therefore, we consider two extensions of the baseline model in this section. First,
we extend our model to a two-period setting, which enables us to consider the for-salemarket and mortgage loans. Hence, we can explicitly investigate the effect of a cap on
the mortgage debt-to-income ratio, a commonly adopted macroprudential policy. We
introduce unobserved preference shock to the baseline model in the second extension.
This feature allows us to examine how our conclusions would change under imperfectincome sorting. Our main finding that the cap policy does not lead to immense welfare
inequality remains in both extensions.
B.1

A Two-Period ModelIn this section, we extend our model to a two-period setting and then analyze the welfare
effect of the cap on the debt-to-income ratio.
B.1.1

Model

A household’s lifecycle is divided into two periods, with period t corresponding to qt
weeks. We normalize a household’s lifespan to one week so that q1 + q2 = 1. To facilitate
the comparison with the earlier sections, we assume that the periodic utility function ineach period is the same as the one in the baseline static model (Equation 1) and thathouseholds do not discount future utility. Households only make residential decisions
36

(location and housing lot size) in the first period. The total utility in this two-period
model is given by:
U (c1 , c2 , h, l1 , l2 ) =Xα+ (1 − θ)h1−ρ ] 1−ρ ,qt lt1−α [θc1−ρ
t

(40)

t=1,2

where h denotes the housing lot size and, for t = 1, 2, ct denotes the consumption per
week on numeraire goods and lt denotes leisure time per week.
As before, in period t, the total income of a household living r miles away from the
CBD with an hourly wage w is qt (168−lt −br)w, where b is the time cost per mile of weeklyround-trip commute. In the first period, the household pays for the consumption goods
and pecuniary commute cost (q1 (c1 +ar)) and purchases a housing unit by combining theirperiod-1 income and D amount of mortgage debt. In the second period, the householdpays back the mortgage and spends the remaining period-2 income on consumption goods
and commute (q2 (c2 + ar)). Since households do not discount future utility, we set the
mortgage interest rate to zero, as alternative values for the interest rate would not change
our substantive conclusions. Finally, we assume that the maximum amount of mortgage
debt D̄ that a household can take is proportional to the household’s period-2 aftercommute-cost (ACC) income, i.e., D̄ = κ̄D q2 [(168 − l2 − br)w − ar].26 We refer to κ̄D as
the debt-to-income ratio. To summarize, the household’s budget constraints are given as
follows:
q1 (c1 + ar) + P (r)h = q1 [168 − l1 − br]w + D,
q2 (c2 + ar) + D = q2 [168 − l2 − br]w,
D ≤ κ̄D q2 [(168 − l2 − br)w − ar].

(41)

Other aspects of the model, including the land allocation mechanism and market
clearing conditions, remain the same as the baseline model.
B.1.2

Model Solution and Welfare Analysis

As before, this dynamic model can be solved using a bid-rent function approach. We
first note that, when the borrowing does not reach the limit, a household (optimally)chooses to have the same consumption and leisure time per week in the two periods, i.e.,c1 = c2 and l1 = l2 . In this case, the utility function and the budget constraints of the twoperiod model become identical to the static model. As a result, the equilibrium allocationand housing consumption of households in the two-period model are the same as thosein the static model. The optimal (ACC) housing expenditure share given equilibrium∗∗)h, is still given by κ(P ∗ (r∗ ); θ, ρ), where κ(x; θ, ρ) =housing price P ∗ (r∗ ), q1 c1 +qP2 c(r∗ ∗2 +P (r )h
1

( 1−θ
)ρ x
θ
1

ρ−1
ρ

1+( 1−θ
)ρ x
θ
26

ρ−1
ρ

. This equivalence result implies that we can assign the calibrated parameter

Alternatively, we can assume that D̄ is proportional to the household’s period-1 ACC income, i.e.,D̄ = κ̄D q1 [(168 − l1 − br)w − ar]. This does not change our qualitative results.

37

values from the baseline static model to the two-period model. The only parameter
unspecified in the static model is the length of the first period, q1 (and q2 = 1 − q1 ).
While this parameter does not influence the equilibrium outcomes when all households
are unconstrained, it can affect our counterfactual welfare analysis. Assuming that thefirst period corresponds to ages 20-40 and the second period corresponds to ages 40-80,
we let q1 = 1/3 and q2 = 2/3. Alternative parameterization of q1 and q2 does not lead to
substantially different results.
When the borrowing constraint is binding, it restricts the amount of debt the household can take to finance their home purchase, effectively imposing a cap on the housing
expenditure share. One can show that a household is borrowing constrained if and only
if the unconstrained optimal housing expenditure share κ(P ∗ (r∗ ); θ, ρ) is larger than thecap on the debt-to-income ratio, κ̄D . Thus, if no one is constrained by the housing
expenditure share cap in the static model (κ̄), then no one will be constrained by adebt-to-income ratio cap (κ̄D ) of the same value in the two-period model. However,
it is essential to note that this result does not imply full equivalence regarding welfare
implications between the housing expenditure share cap and the debt-to-income ratio
cap. To examine the welfare implications of the debt-to-income ratio cap, we still needto numerically solve the two-period model with the cap and compute the consumption
equivalence measures, ξ.
Recall that ξ is defined as the faction of numeraire consumption goods (c1 and c2 )and housing consumption (h) that a household needs to obtain in the economy without
debt limits to achieve the same level of utility in the economy with a debt limit. A
higher value of ξ is associated with more utility gain in percentage terms. We solve
for the equilibrium utility and the consumption-equivalent measure ξi for all types ofhouseholds for κ̄D = 20%, 25%, 30%, 35%, and 40%. Panel A of Table 6 reports theresults. Comparing with the results in the baseline static model (Panel A of Table 5),
we find that the welfare impact of the cap on the debt-to-income ratio is much smaller
than that of the cap on the housing expenditure share. The intuition behind this result is
simple. Even in the extreme case where the household cannot borrow against the future(i.e., κ̄D = 0), the household can still have a considerable amount of housing expendituresusing their income from the first period. Despite these differences in the magnitude of
the welfare impact, our qualitative conclusions about welfare inequality based on thestatic model remain valid in the two-period model. For all five levels of caps that we
considered, ξi is generally smaller for relatively higher-income households. Thus, a capon the debt-to-income ratio does not magnify the welfare inequality in a monocentriccity model.
B.2

A Model with Unobserved Preference Shock

Notice that our principal result regarding the welfare consequences of the housing expenditure cap relates to income sorting in the housing market. Recall that low-income

38

39

10 percent

Third
10 percent

Fourth10 percent

Fifth
10 percent

Sixth
10 percent

Seventh

0.03

0.03

20%

0.03

0.03

0.01

0.01

0

0.03

0.03

0.01

0.01

0

0.02

0.03

0.01

0.01

0

0.02

0.03

0.01

0.01

0

0.02

0.03

0.01

0.01

0

0.02

0.02

0.00

0.01

0

10 percent

Eighth

0.04

0.32

0.92

1.69

-3.13

40%

35%

30%

25%

20%

-3.17

1.66

0.89

0.31

0.03

-3.27

1.59

0.83

0.26

0.02

-3.34

1.55

0.79

0.24

0.02

-3.39

1.51

0.76

0.22

0.02

-3.48

1.45

0.72

0.20

0.02

-3.57

1.40

0.68

0.19

0.02

-3.66

1.34

0.64

0.18

0.02

Ninth

-3.78

1.27

0.60

0.17

0.01

0.02

0.02

0.00

0.01

0

10 percent

Panel B: Model with Unobserved Preference Shock, Housing Expenditure Share Cap κ̄

0.03

0.04

0.02

0.02

30%

25%

0.01

0.01

35%

0

Panel A: Two-Period Model, Debt-to-Income Ratio Cap κ̄D

10 percent

Second0.00

10 percent

Lowest40%

Cap κ̄D /κ̄

Item

-3.99

1.16

0.54

0.15

0.01

0.02

0.02

0.00

0.01

0

10 percent

Tenth

Table 6: Consumption Equivalent Measure ξi (%) for Alternative Debt-to-Income Ratio Cap κ̄D and Housing Expenditure Share Cap κ̄households self-select into neighborhoods near the CBD in our static model. Thus, under
the housing expenditure cap, they tend to compete with other low-income households,
who are also constrained. Therefore, the equilibrium rent decreases and low-income
households would not suffer more than their high-income counterparts. While we believe
this is a reasonable first-order approximation, there are issues to be considered. Theoretically, the conditions needed for perfect income sorting may be strong and not be satisfied
in all situations (Gravel and Oddoub, 2014). In practice, sorting is often imperfect, andhouseholds with different income levels live near each other. In addition, amenities may
also affect the geographic sorting (Gaignea et al., 2022). Hence, it would be worthwhile
to re-examine our main welfare conclusion under imperfect sorting. To proceed, we firstbuild a model where imperfect sorting arises in equilibrium, which is the focus of thenext section. We then calibrate the model to match the data and perform counterfactual
policy analysis.
B.2.1

Model

Income sorting across neighborhoods is imperfect (Davidoff, 2005). One possibility is
the amenities differ across neighborhoods (Carlino and Saiz, 2019).27 In this section, we
extend our baseline model to allow the utility that a household derives from a location
to depend on an additive unobserved location-specific preference shock. Formally, whena housing expenditure share cap of κ̄ is imposed, the total utility ṼiT associated with alocation r miles away from the CBD is given by:
ṼiT (P ∗ (r), r) = Ṽi (P ∗ (r), r) + σir ,(42)

where Ṽi (P ∗ (r), r) is the indirect utility that household i obtains given location r andequilibrium rent P ∗ (r) (Equation 16), ir has an i.i.d. Type-I extreme value distribution,and σ measures the magnitude of the magnitude of the unobserved preference shock.28
As is standard in the literature, we assume a household chooses the location associatedwith the highest total utility ṼiT to reside. This decision is made based on both aconstant component Ṽi (P ∗ (r), r) and an unobserved random component σir . As a result,the equilibrium location decision for a particular type of household is described by ahousehold density distribution over all possible locations. In our model, each location
corresponds to one dot on the two-dimensional plane within the fringe distance Rf∗ , whichis the distance at which the market rent P ∗ (r) equals the agricultural rent Pa .
Then, under the assumption of Type-I extreme value distribution for ir , the equilibrium density of Type i household at all locations r miles away from the CBD is given
27

There are alternative explanations. For instance, De Bartolome and Ross (2003) explains the im-perfect income sorting by fiscal competition between the inner city and suburb governments. Here weabstract from the fiscal competition and leave such possibility to future research.
28
Note that, when κ̄ = 1, Ṽi (P ∗ (r), r) equals the unconstrained indirect utility V (P ∗ (r), r) given by
Equation (5).

40

by:
e

mi (P ∗ (r), r) = R R∗
f0Ṽi (P ∗ (r),r)
σ

e

L(r)

Ṽi (P ∗ (r),r)
σ

N i,

(43)

L(r)drwhere L(r) = 2πr is the amount of locations available per unit distance, at a distance r,
and N i is the total number of Type i households.
To close the model, we impose the market clearing condition at all locations at theequilibrium. It means that for all distance r ≤ Rf∗ , the following equation is satisfied.
10X

mi (P ∗ (r), r)h̃i (P ∗ (r), r) = L(r),

(44)

i

where h̃i (P ∗ (r), r) (Equation 17) is the house size demanded by Type i household at
distance r given equilibrium rent P ∗ (r).
B.2.2

Re-calibration and Welfare Analysis

Compared to the baseline model, this extended model has one additional parameter,σ, which captures the magnitude of the unobserved preference shock. Intuitively, thisparameter determines the degree of income-based sorting at the equilibrium. In one
extreme case where σ → ∞, we have mi (P ∗ (r), r) → R Rf∗L(r)0

N i , implying that, for any

L(r)drType i, households at all income levels are uniformly distributed across all locations. As
a result, we do not observe income-based sorting at the equilibrium. In the other extreme
case where σ → 0, all equilibrium objects converge to their values in the case without
the unobserved preference shock, and we observe perfect income-based sorting. Hence,
it is essential to find a plausible value of σ for our policy analysis.
In the current context, the stronger the income-based sorting is, the more the average
household income co-move with the commute distance. Hence, following the discussion
above, we calibrate the value of σ to match our empirical finding reported in Section 2 thata 1,000 dollars increase in annual wage income is associated with a 0.0140-mile increase incommute distance.29 More specifically, for each possible value of σ, we randomly sample
households from the equilibrium and regress a household’s annual gross income on thehousehold’s distance from the CBD.30 We find that σ = 0.4524 is the value that produces
a regression coefficient of 0.0140.31 As a comparison, in the baseline model where theunobserved preference shock is not present, the regression coefficient is 0.0175.
29

We also need to re-calibrate the preference parameters to match the type-specific housing expenditure

shares in the data, θ, and ρ. This is because the relationship between the housing expenditure shareand income changes with the degree of income-based sorting.
30
Note that, for each type of household, the wage rate wi is calibrated to match annual expendituresinstead of annual gross income. We multiply the model-implied annual expenditures by the ratio of
annual pre-tax income (Item 1 in Table 1) to annual expenditures (Item 2 in Table 1) to obtain modelimplied annual gross income. Hence, we ensure the consistency between the model and the data for eachtype of household.
31
The re-calibrated values of ρ and θ are 5.47 and 3.1139 × 10−6 , respectively.

41

To visualize how unobserved preference shocks influence the degree of spatial sorting,we plot the relationship between the average wage rate and the distance from the CBD
for both the baseline model and the current extended model in Figure 3. This figure
shows that, as one moves away from the CBD, the average wage rate increases slower inthe extended model, which has the unobserved preference shocks, than in the baselinemodel. The difference is particularly salient at locations closer to the city’s boundary.
Figure 3: Average Hourly Wage and Distance from the CBD (Mile)We then conduct welfare analysis based on the re-calibrated model. Note that the
average utility of all Type i households is the same as the expected utility of a Type ihousehold, which is given by:
Z Rf∗

ṼiE (P ∗ (r), r) = E[ṼiT (P ∗ (r), r)] = σγ + σ log[eṼi (P ∗ (r),r)σ

L(r)dr],(45)0where γ = 0.5772 is the Euler’s constant and L(r) = 2πr is the amount of locations
available per unit distance, at a distance r.
As before, we define the consumption equivalence measure, ξi , as the faction of numeraire goods and housing consumption that Type i household needs to obtain in theeconomy without the cap policy to achieve the same level of average utility ṼiE (P ∗ (r), r)
in the economy with the cap policy.

We solve for the equilibrium utility and theconsumption-equivalent measure ξi for all types of households for κ̄ = 20%, 25%, 30%,42

35%, and 40%. The results reported in Panel B of Table 6 are quantitatively similar towhat we found earlier based on the baseline model (Panel A of Table 5). For all five levelsof caps that we considered, ξi is generally less positive or more negative for relatively
higher-income households. Thus, the housing expenditure ratio cap does not exacerbate
welfare inequality under realistic income sorting.

C

Non-homothetic Model

C.1

Fit of the Spaceless Model

Figure 4: Total Annual Expenditure and the Fraction Spent on Housing (Non-homothetic
Model)

C.2

Spatial Order in the Monocentric City Model

The bid-rent function for Type i household in the spatial non-homothetic model can beobtained by inverting the indirect utility function shown in Equation (25):1
β βc (1 − βc )1−βc (40w − ar + c̄) 1−βψi (u∗i , r) = [ c
] c.
∗
ui

43

(46)

The first order derivative of this function with respect to the distance r is given by:∂ψi (·)
1
β βc (1 − βc )1−βc (40wi − ar + c̄) −1 βcβc (1 − βc )1−βc (−a))= ψi (u∗i , r)[ c]∂r
1 − βc
u∗i
u∗i
−a
(47)
= ψi (u∗i , r)
(1 − βc )(40wi − ar + c̄)It is easy to see that ∂ψ∂ri (·) < 0 from Equation (47). It means all bid-rent functions∗
are decreasing in the distance from the CBD, r. Let rB,idenote the distance from the

CBD at which Type i households and Type i + 1 households intersect in the baseline∗∗
equilibrium. Note that, at the equilibrium, we have ψi (u∗i , rB,i
) = ψi+1 (u∗i+1 , rB,i) (samerental rate for both types of households at the intersection). It implies that:|a∂ψi+1 (·)
a∂ψi (·)|−|| = ψi (u∗i , r)[
−
]
∗
∂r
∂r
(1 − βc )(40wi − ar + c̄) (1 − βc )(40wi+1 − ar∗ + c̄)ψi (u∗i , r)a[(40wi+1 − ar∗ + c̄) − (40wi − ar∗ + c̄)]=(1 − βc )(40wi − ar∗ + c̄)(40wi+1 − ar∗ + c̄)
40aψi (u∗i , r)(wi+1 − wi )=(1 − βc )(40wi − ar∗ + c̄)(40wi+1 − ar∗ + c̄)
> 0,

(48)

where the last line follows from the fact that Type i households has lower hourly wages
than Type i + 1 households (wi < wi+1 ).
Thus, we have shown that | ∂ ψ̃∂ri (·) | is larger for households with lower hourly wage wi ,which implies that households with higher hourly wage wi lives farther from the CBD at
the equilibrium in the spatial non-homothetic model.

44