NBER WORKING PAPER SERIES

AFFORDABLE HOUSING AND CITY WELFARE

Jack Favilukis
Pierre Mabille
Stijn Van Nieuwerburgh

Working Paper 25906

http://www.nber.org/papers/w25906

NATIONAL BUREAU OF ECONOMIC RESEARCH

1050 Massachusetts Avenue
Cambridge, MA 02138
May 2019, Revised June 2021

We thank Hae-Kang Lee for superb research assistance. We thank John Humphries, Timothy
McQuade, Adam Guren, Charles Nathanson (discussants), and Morris Davis for valuable
comments. We have benefited from participants at the 2018 Society for Economic Dynamics
conference in Mexico City, the Urban Economics Association conference at Columbia
University, the 2019 American Economic Association in Atlanta, the Macro Finance Society
meeting at Kellogg, the Conference on Low-Income Housing Supply and Housing Affordability
in Madrid, and seminar participants at the University of Miami, Columbia GSB, Atlanta Fed and
Emory University, NYU Stern, University of Connecticut, University of Luxembourg, John’s
Hopkins University, PUC in Santiago, Boston College, London Business School, and the San
Francisco Federal Reserve Bank. The views expressed herein are those of the authors and do not
necessarily reflect the views of the National Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been
peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies
official NBER publications.

© 2019 by Jack Favilukis, Pierre Mabille, and Stijn Van Nieuwerburgh. All rights reserved. Short
sections of text, not to exceed two paragraphs, may be quoted without explicit permission
provided that full credit, including © notice, is given to the source.
Affordable Housing and City Welfare
Jack Favilukis, Pierre Mabille, and Stijn Van Nieuwerburgh
NBER Working Paper No. 25906
May 2019, Revised June 2021
JEL No. E21,E6,G11,G12,G18,H2,H76,R1,R13,R21,R28,R3,R31,R41,R51

ABSTRACT

Housing affordability is the main policy challenge for most large cities in the world. Zoning
changes, rent control, housing vouchers, and tax credits are the main levers employed by policy
makers. How effective are they at combatting the affordability crisis? We build a dynamic
stochastic spatial equilibrium model to evaluate the effect of these policies on the well-being of
its citizens. The model endogenizes house prices, rents, construction, labor supply, output,
income and wealth inequality, the location decisions of households within the city as well as
inter-city migration. Its main novel features are risk, risk aversion, and incomplete risk-sharing.
We calibrate the model to the New York MSA. Housing affordability policies carry substantial
insurance value but affect aggregate housing and labor supply and cause misallocation in labor
and housing markets. Housing affordability policies that enhance access to this insurance
especially for the neediest households create substantial net welfare gains.

Jack Favilukis Stijn Van Nieuwerburgh

Sauder School of Business Columbia University
University of British Columbia Graduate School of Business
Henry Angus Building 2053 Main Mall Uris Hall, office 809
Office 867 3022 Broadway
Vancouver, BC V6T 1Z2 New York, NY 10027
Canada and NBER
jack.favilukis@sauder.ubc.ca svnieuwe@gsb.columbia.edu

Pierre Mabille
INSEAD
Boulevard de Constance
77300 Fontainebleau
France
pierre.mabille@insead.edu
1 Introduction
The increasing appeal of major urban centers has brought on an unprecedented housing
affordability crisis. Ever more urban households are burdened by rents or mortgage pay-
ments that take up a large fraction of their paycheck and/or by long commutes. The share
of cost-burdened renters in the United States has doubled from 24% in the 1960s to 48%
in 2016. Over this period, median home value rose 112%, far outpacing the 50% increase
in the median owner income (Joint Center for Housing Studies of Harvard University,
2018). Hsieh and Moretti (2019) argue that our most productive cities are smaller than
they should be because of lack of affordable housing options, underscoring the impor-
tance of the issue. The Covid-19 pandemic lent new urgency to the affordability crisis
with 10 million American renters behind on $60 billion in rent as of February 2021, and
with above-10% house price appreciation in 2020.
Policy makers throughout the world are under increasing pressure to improve afford-
ability.1 They employ policy tools ranging from rent regulation, upzoning, inclusionary
zoning, housing vouchers, to developer tax credits. While there is much work, both em-
pirical and theoretical, on housing affordability, what is missing is a general equilibrium
model that quantifies the aggregate and distributional impact of such policies on individ-
ual and city-wide welfare. This paper provides such a model. It endogenizes prices and
quantities of owned and rented housing, the spatial distribution of housing and house-
holds, commuting patterns, incentives to work, income and wealth inequality within and
across neighborhoods, output, and in- and out-migration to other metropolitan areas. We
calibrate the model and use it as a laboratory to conduct various housing policy experi-
ments, allowing us to study the various policy instruments and compare their effective-
ness. This new framework is well suited for studying the impact of place-based policies.
We find that the expansion of a range of housing affordability policies improves wel-
fare. In an incomplete markets model with risk and risk aversion, affordability policies
play a quantitatively important role as an insurance device. The housing stability they
provide disproportionately benefits low-income households. These insurance benefits
trade off against the aggregate and spatial distortions in housing and labor markets that
accompany such policies. Our results highlight the importance of general equilibrium ef-
1 Fifteen cities in California have rent regulation. A November 2018 California state ballot initiative

proposed to overturn the 1995 Costa-Hawkins Act, clearing the way for more rent control. Oregon imposed
statewide rent control in 2019. New York State passed the most sweeping expansion of rent regulation laws
in a generation in June 2019. So did the District of Columbia. Bill de Blasio, the mayor of New York City,
was elected on a platform to preserve or add 200,000 affordable housing units. Many local policymakers
are trying to overturn preemption laws, on the books in in 36 U.S. States, that prevent local governments
from adopting rent regulation laws.

1
fects, which often reverse partial equilibrium logic, and of how the affordability policies
are financed.
Our model consists of two metropolitan areas. The first metro is the one we focus on
and whose housing policies we study. We think of this metro as a “gateway” city with a
housing affordability problem. This metro consists of two zones, the urban core (zone 1)
and the suburbs (zone 2). Working-age households who live in zone 2 commute to zone 1
for work. Commuting entails both an opportunity cost of time and a financial cost. Zones
have different sizes, captured by limits on the housing stock, provide different amenity
benefits, and households who live in the urban core enjoy higher productivity, captur-
ing agglomeration benefits. The second metro serves as an alternative, naturally more
affordable, location for residents of the first metro. We study migration between the two
metros, which is subject to moving costs. The spatial aspects of the model are important
since affordable housing policies affect the spatial allocation of labor and housing, both
within and across metros.
The economy is populated by overlapping generations of risk averse households who
face idiosyncratic labor productivity and mortality risk. They make dynamic decisions
on location, non-housing and housing consumption, labor supply, tenure status (own
or rent), savings in bonds, primary housing, investment property, and mortgage debt.
Since households cannot perfectly hedge labor income and longevity risk, markets are
incomplete. Progressive tax-and-transfer and social security systems capture important
insurance mechanisms beside affordable housing policies. The model generates a rich
cross-sectional distribution over age, labor income, tenure status, housing wealth, and
financial wealth. This richness is paramount to understanding both the distributional
and the aggregate implications of housing affordability policies.
On the firm side, each metro produces tradable goods and residential housing, sub-
ject to decreasing returns to scale. As an area approaches its housing limit, construction
becomes increasingly expensive, and the housing supply elasticity falls. Wages, house
prices, and market rents in each metro are determined in equilibrium.
We calibrate the first metro to the New York metropolitan statistical area (NY MSA),
designating Manhattan as zone 1 and the remaining 24 counties of the NY MSA as zone
2. Our calibration targets key features of the data, including the relative housing stock
and population of zone 1 and zone 2, the income distribution in the New York MSA, ob-
served commuting times and costs, the housing supply elasticity, current zoning laws,
the current size and scope of the rent stabilization system, and the current federal, state,
and local tax-and-transfer system. The baseline model generates realistic income, wealth,
and home ownership patterns over the life-cycle for various percentiles of the income dis-

2
tribution. It matches both income and wealth inequality. The model also matches house
price and rent levels for the MSA. Finally, it generates a housing affordability problem,
with high price-income and rent-income ratios, and over half of renters that are cost-
burdened.
We think of the second metro as the rest of the U.S., it is an outside option for current
New Yorkers, and a source for migration into New York. It is calibrated to the average
of the next 74 largest MSAs in the U.S. It only has one zone, no rent stabilization, and a
lower income level for top-productivity households, but is otherwise similar to the NY
MSA. We calibrate moving costs to match in-migration rates and out-migration rates by
age and income for the NY MSA.
We model rent regulation in NY as rent stabilization (RS). The government mandates a
slower rate of growth for rents on RS housing units than for market rentals. In a stationary
economy, this translates into a rent level discount relative to the market rent, which grows
with the length of stay in the RS unit (tenure). Developers in the model must make a fixed
share of the housing stock rent stabilized. RS units are allocated by lottery to capture the
random nature of the allocation process in reality. RS is subject to a housing size constraint
but has no tenant income qualification and suffers from low turnover. This results in
substantial misallocation of RS housing in the model, mimicking that in the data.
We define access to insurance as the likelihood that a household in the bottom half of
the income distribution that experiences a negative labor productivity shock gains access
to a RS housing unit. We define the stability of insurance as the likelihood that a household
in the bottom half of the income distribution that is already in a RS unit can remain there
the next period. The value of insurance depends on the size of the RS unit, how deeply the
rent is discounted, and on household risk aversion.
The RS system creates multiple inefficiencies which trade off with the insurance ben-
efits. The first one is that the housing stock is misallocated. Because there is no income
qualification, some RS units go to higher-income households by sheer luck, taking away
affordable units from the needy. Given the maximum size constraint on RS units, these
households often under-consume housing. This under-consumption may get worse as
households age because of rising labor income profiles and growing discounts. Other,
lower-income RS tenants over-consume housing because of the RS discount. Households
choose to live in a zone where they otherwise would not because they won the RS lottery
in that zone. A lower-productivity household, with a lower opportunity cost of commut-
ing, may be taking the place of a higher-productivity household in the urban core. RS
may therefore trigger spatial misallocation of labor as well as housing.
The second distortion is on the supply of housing, which in the model encompasses

3
the maintenance of the existing housing stock. The rent regulation mandate results in
lower average prices for new housing development and a lower equilibrium housing
supply. Reductions in housing supply result in higher equilibrium rents and prices, all
else equal, worsening the affordability problem for market renters and potential home
owners. Since the RS mandate varies across zones, so does the distortion. For example,
in Manhattan, half of rentals are RS, a much larger share than in zone 2.
We study housing policy reform in a sequence of experiments. We ask whether they
improve welfare. Most policies benefit some households while hurting others. Since
the reforms in the NY MSA affect the attractiveness to live there, they change both out-
and in-migration decisions. Generally, the mobility margin dampens the welfare effects
since households have another margin of adjustment, namely to “vote with their feet.”
Migration complicates welfare analysis since the set of households that live in the NY
MSA is different before and after the reform. As our main aggregate welfare criterion, we
focus on a fixed set of households who live in NY in the period before the reform, and
compute the change in their value function in the first period after the reform regardless
of whether they still live in NY. We take cross-sectional averages of value functions before
and after, implicitly giving more weight to the needy (high marginal-utility households).
In a first set of experiments, we study policies aimed at reducing the misallocation
caused by RS. These policies hold the share of the housing stock that is RS constant. We
introduce income qualification in various ways. Among these ways, a policy that income-
tests every RS tenant every period, but provides housing stability for existing RS tenants,
delivers substantial welfare gains ( 0.66%). RS housing becomes much better allocated,
improving access to insurance without compromising stability of insurance. Four forces
limit the welfare gains. First, income qualification reduces incentives to work. Second,
by replacing long-term tenants by new tenants, the policy reduces the average subsidy
RS housing provides since new tenants start with much smaller rent discounts. Third, the
policy results in a larger NY population increasing the competition for affordable housing
units. Fourth, because of a constraint on the maximum size (quality) of RS units, the least-
needy households do not choose RS even in the absence of an income qualification.
A second set of policy experiments changes the scope of the affordable housing man-
date. Surprisingly, a 50% increase in the share of square footage set aside for RS housing
units increases welfare by a large 0.91%. With more RS housing units, access to insurance
increases without hurting the stability of insurance. The benefits this brings to lower-
income households outweigh the costs that arise from weaker incentives to construct and
maintain housing, higher rents in market units, and more spatial misallocation of labor
and housing. We explore both smaller and larger fractions of affordable housing units

4
and find that welfare increases monotonically with the scope of the affordable housing
mandate.
A third set of policy experiments operate on the spatial aspects of affordable housing.
First, we relocate all RS housing from the urban core to the suburbs, which increases wel-
fare by 0.25%. Gentrification of the core ensues, with fewer but higher-income residents,
larger apartments, and more home ownership. Increased socio-economic segregation,
with low-income households missing out on the agglomeration benefits of living in the
core, and a better spatial allocation of labor, with more high-productivity households in
the core, are two sides of the same coin. In one variant we provide subsidized transit for
RS tenants in addition at a cost of $800 million. The distortionary taxes needed to pay for
the transit subsidy more than offset the higher benefits to the recipients in the model with
migration.
The next policy experiment increases the maximum amount of housing that can be
built in zone 1, for example through a relaxation of land use or height restrictions. This
“upzoning” policy increases the equilibrium population share and the housing stock of
zone 1. Rents fall, which benefits both market and RS renters. It generates a modest
welfare gain of 0.11%. The policy involves less redistribution, creating benefits for all
age, productivity, income, and wealth groups, at least in the long-run. But it generates
only modest improvement in the plight of low-income households.
Housing vouchers are transfers provided to low-income households for housing ex-
penditures. We find that a $800 million expansion of the voucher system produces no
average welfare gain despite large benefits to low-income, high-marginal utility house-
holds. A powerful interaction between taxation and migration bedevils this program. We
assume, consistent with reality, that a voucher expansion must be financed via distor-
tionary income taxes. Since higher taxes prompt an outflow of high-productivity house-
holds, the tax rate must be even higher to finance the same dollar expansion in housing
vouchers. This prompts further out-migration, etc. The labor supply distortions are much
larger than in a model without migration. The same migration response triggers a drop
in the housing stock and higher rents. The housing voucher program triggers an inter-
esting spatial response. In equilibrium low-income households are not more likely to
“move to opportunity,” but end up living in the same areas they were before. In fact,
the urban core gentrifies, housing a larger share of top-productivity and higher-income
households, with fewer renters and higher rents. Vouchers “remove from opportunity”
some medium-productivity households who end up farther from their jobs or migrating
to a different MSA.
Finally, a $800 million developer tax credit policy also generates no gain, and little ad-

5
ditional housing. It suffers from similar taxation-induced distortions on labor supply and
housing demand, underscoring the importance of considering how housing affordability
policies are paid for in equilibrium.

Related Literature Our work is at the intersection of the macro-finance and urban eco-
nomics literatures. A large literature in finance solves partial-equilibrium models of port-
folio choice between housing (extensive and intensive margin), financial assets, and mort-
gages.2 Recent work in macro-finance has solved such models in general equilibrium,
adding aggregate risk, endogenizing house prices and sometimes also interest rates.3 Like
the former literature, our model features a life-cycle and a rich portfolio choice problem
and captures key quantitative features of observed wealth accumulation and home own-
ership over the life-cycle. Like the latter literature, house prices, rents, and wages are
determined in equilibrium. We abstract from aggregate risk which is not central to the
question at hand. Our contribution to the macro-finance literature is to add a spatial
dimension to the model and to evaluate a rich set of housing policies.
A voluminous literature in urban economics studies the spatial location of households
in urban areas in spatial equilibrium models. This literature studies the trade-off between
the commuting costs and housing expenditures.4 Guerrieri, Hartley, and Hurst (2013)
study house price dynamics in a city and focus on neighborhood consumption external-
ities, in part based on empirical evidence in Rossi-Hansberg, Sarte, and Owens (2010).
Couture, Gaubert, Handbury, and Hurst (2018) uses a similar device to explain the return
of rich households to the urban core over the past decades, reversing an earlier wave of
suburban flight. Our model also features such luxury amenities in the city center. Urban
models tend to be static, households tend to be risk neutral or have quasi-linear pref-
erences, and landlords are absentee (outside the model). The lack of risk, investment
demand for housing by local residents, and wealth effects makes it hard to connect these
spatial models to the macro-finance literature. When there is no risk, there is no insurance
role of affordability policies.5
2 Examples are Campbell and Cocco (2003), Cocco (2005), Yao and Zhang (2004), and Berger, Guerrieri,
Lorenzoni, and Vavra (2017). Davis and Van Nieuwerburgh (2015) summarize this literature.
3 E.g., Landvoigt, Piazzesi, and Schneider (2015), Favilukis, Ludvigson, and Van Nieuwerburgh (2017),

Guren and McQuade (2019), and Kaplan, Mitman, and Violante (2019). Imrohoroglu, Matoba, and Tuzel
(2016) study the effect of the 1978 passage of Proposition 13 which lowered property taxes in California.
4 Brueckner (1987) summarizes the Muth-Mills monocentric city model. Rappaport (2014) introduces

leisure as a source of utility and argues that the monocentric model remains empirically relevant. Rosen
(1979) and Roback (1982) introduce spatial equilibrium. Recent work on spatial sorting includes Van
Nieuwerburgh and Weill (2010), Behrens, Duranton, and Robert-Nicoud (2014) and Eeckhout, Pinheiro,
and Schmidheiny (2014).
5 Hizmo (2015) and Ortalo-Magné and Prat (2016) bridge some of the gap between these two literatures

6
 Because it is a heterogeneous-agent, incomplete-markets model, agents’ choices and
equilibrium prices depend on the entire wealth distribution. Because of the spatial di-
mension, households’ location is an additional state variable that needs to be kept track
of. We use state-of-the-art methods to solve the model. We extend the approach of Fav-
ilukis et al. (2017), which itself extends Gomes and Michaelides (2008) and Krusell and
Smith (1998) before that. The solution approach accommodates aggregate risk, though
we abstract from it in this model.
The resulting model is a new laboratory that can be used to study how place-based
policies affect the spatial distribution of people, labor supply, house prices, output, and
inequality. Favilukis and Van Nieuwerburgh (2021) use a related framework to study the
effect of out-of-town investors on residential property prices. They do not study housing
affordability policies and do not consider the inter-city migration decision, which adds
both substantial complexity to the model solution and richness to the analysis. The idea
that the migration margin provides insurance against adverse income shocks appears
prominently in recent work by Bilal and Rossi-Hansberg (2021).
Our model connects to an empirical literature that studies the effect of rent control and
zoning policies on rents, house prices, and housing supply. Autor, Palmer, and Pathak
(2014, 2017) find that the elimination of the rent control mandate on prices in Cambridge
increased the value of decontrolled units and neighboring properties in the following
decade, by allowing constrained owners to raise rents and increasing the amenity value
of those neighborhoods through housing market externalities. The price increase spurred
new construction, increasing the rental stock. Diamond, McQuade, and Qian (2019) show
that the expansion of rent control in San Francisco led to a reduction in the supply of
available housing, paradoxically contributing to rising rents and the gentrification of the
area. While beneficial to tenants, it resulted in an aggregate welfare loss. We also find
a lower housing stock and higher rents from an expansion of rent regulation, but an ag-
gregate welfare gain for the entire MSA in spatial equilibrium. Davis, Gregory, Hartley,
and Tan (2017) study the effect of housing vouchers on location choice and children’s
schooling outcomes in a rich model of the Los Angeles housing market, while Davis,
Gregory, and Hartley (2018) study Low Income Housing Tax Credits (LIHTC) and their
effect on demographic composition, rent, and children’s adult earnings. Diamond and
McQuade (2019) find that LIHTC buildings in high- (low-)income neighborhoods have
negative (positive) effects on neighboring property prices. Earlier work by Baum-Snow
by studying a problem where households are exposed to local labor income risk, make a once-and-for-all
location choice, and then make an optimal financial portfolio choice. Their models are complementary to
ours in that they solve a richer portfolio choice problem in closed-form, but don’t have preferences that
admit wealth effects nor allow for consumption and location choice each period.

7
and Marion (2009) focuses on the effects of LIHTCs on low income neighborhoods and
Freedman and Owens (2011) focuses on crime. Luque, Ikromov, and Noseworthy (2019)
summarize financing methods for low-income housing development.
We sidestep the question whether housing policy is the optimal policy to redistribute
and provide insurance. In the spirit of Diamond and Saez (2011), we evaluate policy
reforms that are extensions of existing policies, limited in complexity and potentially po-
litically feasible. Housing policies are omnipresent, making it paramount to understand
their macro-economic and distributional effects. One reason for their prevalence may
simply be that housing policies are the most accessible levers for local policy makers to
influence their citizen’s welfare; they may have limited control over tax-and-transfer poli-
cies. While the public finance literature has generally argued for the superiority of cash
transfers (following Atkinson and Stiglitz, 1976), it has also identified several rationales
for in-kind transfers. Currie and Gahvari (2008) discuss paternalism, interdependent pref-
erences, imperfect information on the part of the government, self-targeting, and mitigat-
ing income tax distortions.6 We contrast housing policies to two cash transfer programs,
similarly financed with distortionary taxation. A highly progressive cash transfer scheme
generates the largest net benefit in our model, while a cash transfer system that follows
the housing voucher design in all but its in-kind nature does not result in a welfare gain
and is dominated by that of other housing policies.
The rest of the paper is organized as follows. Section 2 sets up the model. Section 3 de-
scribes the calibration to the New York metropolitan area. Section 4 discusses the bench-
mark model’s implications for quantities and prices, the distribution of households, and
housing affordability. Section 5 studies the main counter-factual policy experiments. Sec-
tion 6 concludes. Appendix A provide detail on the data, Appendix B on the calibration,
and Appendix C studies the affordable housing policies in a model without inter-city mi-
gration, helping to isolate the role of migration. Appendix D contains sensitivity analysis.

2 Model
The model consists of two metropolitan areas (MSA), a “gateway” MSA, whose housing
affordability problem we study, and an outside MSA. The gateway MSA has an urban
core (zone 1) and a suburban area (zone 2). Zone 1 is the central business district where
all employment takes place. Households living in zone 2 face a commuting cost. While
6 In-kind
transfers are very important in developed and developing countries alike. Health/food, edu-
cation, and housing are the main ones. Currie and Gahvari (2008) conclude it is “more the norm than the
exception for governments to conduct redistribution in-kind.”

8
clearly an abstraction of the more complex production and commuting patterns in large
cities, the monocentric city assumption captures the essence of commuting patterns (Rap-
paport, 2014) and is the simplest way to introduce a spatial aspect in the model. House-
holds face a moving cost to move into or out of the gateway MSA; the population of the
gateway MSA is endogenous and will respond to housing policy changes.

2.1 Households
Preferences The economy consists of overlapping generations of risk averse house-
holds. There is a continuum of households of a given age a. The total population in
the economy is fixed.
Each household maximizes a utility function u over consumption goods c, housing
h, and labor supply n. Utility depends on location ` and age a, allowing the model to
capture commuting time and amenity differences across locations.
The period utility function is a CES aggregator of c and h and leisure l:
h i 1− γ
χ`t ,a C(ct , ht , lt )
U (ct , ht , nt , `t , a) = , (1)
1−γ
h η i η1
η
C(ct , ht , lt ) = (1 − αn ) ((1 − αh )cet + αh het ) e + αn lt
ht ≥ h (2)
(
1 − φT` − lt ≥ n if a < 65
nta = (3)
0 if a ≥ 65



 1 if in Outside MSA
 χ NY

if ` = 2
χ`t ,a = (4)
 χ NY χ1
 if ` = 1 and a < 65

 χ NY χ1 χ R

if ` = 1 and a ≥ 65

The coefficient of relative risk aversion is γ. The parameter e controls the intra-temporal
elasticity of substitution between housing and non-housing consumption.
Equation (2) imposes a minimum house size requirement (h), capturing the notion that
a minimum amount of shelter is necessary for a household. The city’s building code often
contains such minimum size restrictions.
Total non-sleeping time in equation (3) is normalized to 1 and allocated to work (nt ),
leisure (lt ), and commuting time φT` . Since we will match income data that exclude the
unemployed, we impose a minimum constraint on the number of hours worked (n) for

9
working-age households. This restriction will also help us match the correlation between
income and wealth. There is an exogenous retirement age of 65. Retirees supply no labor.
The taste-shifter χ`,a captures the relative amenity value of the various locations. They
are allowed to depend on age as follows. The amenity value of the Outside MSA is nor-
malized to 1. The value of living in the gateway MSA is governed by χ NY . The additional
amenity value of zone 1 relative to zone 2 is given by χ1 . For retirees living in zone 1,
there is an amenity shifter χ R which will help the model match the fraction of retirees
living in zone 1.
There are two types of households in terms of the time discount factor. One group
of households have a high degree of patience β H while the rest have a low degree of
patience β L . This preference heterogeneity helps the model match observed patterns of
wealth inequality and wealth accumulation over the life cycle.

Endowments A household’s labor income ylab

t depends on the number of hours worked
n, the wage per hour worked W, a deterministic component G a which captures the hump-
shaped pattern in average labor income over the life-cycle, and an idiosyncratic labor
productivity z, which is stochastic and persistent.
To capture the effect of living in the urban core on current and future income, we
assume that households working in the city center experience higher productivity. This
productivity shifter A1 > A2 = 1 will help the model match the income differential
between zone 1 and zone 2.7
After retirement, households earn a retirement income which is the product of an ag-
gregate component Ψ and an idiosyncratic component ψ a,z . The idiosyncratic component
has cross-sectional mean of one, and is determined by productivity during the last year of
work. Labor income is taxed linearly at rate τ SS to finance retirement income. Other taxes
and transfers are captured by the function T (·) which maps total pre-tax income into a
net tax (negative if transfer). Net tax revenue goes to finance a public good which does
not enter in household utility.
Households face mortality risk which depends on age, p a . Although there is no inten-
tional bequest motive, households who die leave accidental bequests. We assume that the
number of agents who die with positive wealth leave a bequest to the same number of
agents alive of ages 21 to 65. These recipient agents are randomly chosen, with one restric-
tion. Patient agents (β H ) only leave bequests to other patient agents and impatient agents
7 Thisproductivity shifter is a reduced-form way of capturing production agglomeration effects in the
urban core. It includes network effects, better access to good schools, etc. Note that because all employment
takes place in the urban core, traditional production agglomeration effects are already maximized.

10
(β L ) only leave bequests to other impatient agents. One interpretation is that attitudes
towards saving are passed on from parents to children. Conditional on receiving a be-
bt+1 is a draw from the relevant distribution, which differs
quest, the size of the bequest b
for β H and β L types. Because housing wealth is part of the bequest, the size of the be-
quest is stochastic. Agents know the distribution of bequests, conditional on β type. This
structure captures several features of real-world bequests: many households receive no
bequest, bequests typically arrive later in life and at different points in time for different
households, households anticipate bequest sizes to some degree, and there is substantial
heterogeneity among bequest sizes for those who receive a bequest.

Affordable Housing We model rent stabilization (RS) in the gateway MSA to capture
key features in reality. A fraction η ` of rental housing units in zone ` are rent stabilized.
The rent per square foot is a fraction κ1 (d) < 1 of the free-market rent. The discount is
increasing in the length of tenure d.8 We model the assignment of RS units to renters as a
lottery. Every household in the model enters the affordable housing lottery every period.
A household that wins the lottery in a zone can choose to turn down the affordable unit,
and rent or own in the location of its choice on the free market.9 There is a maximum
RS housing size. The model allows for an income qualification requirement whereby the
income of a RS tenant must be below a fraction κ2 of area median income (AMI). In the
baseline model, as well as in reality, there is no income qualification requirement for RS
units, so that κ2 = ∞. We explore policies below that lower κ2 .
Households that lived in a RS housing unit in a given zone in the previous period
have an exogenously set, high probability of winning the RS lottery in the current period,
p RS,exog .10 This parameter determines the average length of tenancy in the RS system. For
households that were not previously in RS, the probability of winning the lottery for each
zone is endogenously determined to equate the residual demand (once accounting for
RS stayers) and the supply of RS units in each zone. Households form beliefs about this
probability. This belief must be consistent with rational expectations, and is updated as
part of the equilibrium determination. The presence of the RS housing program distorts
labor supply, location choice, housing demand, and housing supply, as discussed further
8 In reality, a government entity sets the rent growth increases at rates that are below the growth rate
of market rents. Since our model is stationary, a growing discount with tenure in a stationary model is
equivalent to a slower growth rate of RS rents. Appendix A.4 shows the discount in the data as a function
of tenancy for the New York MSA. Our calibration below adopts this schedule.
9 There is a single lottery for all affordable housing units. A certain lottery number range gives access

to affordable housing in zone 1, while a second range gives access to housing in zone 2. Households with
lottery numbers outside these ranges lose the housing lottery.
10 For these households, the probability of winning the RS lottery in the other zone is set to zero.

11
below.

Migration Households who lived in the gateway MSA in the previous period optimally
decide to either remain or to migrate out to the outside MSA by comparing the value
functions associated with each choice:
n o
V = max V NY , m( a, z)V ∗

where m(·) is a moving cost that depends on age a and on productivity z. Since the value
function is negative m( a, z) > 1 denotes a moving cost and m( a, z) < 1 a moving benefit.
Migration into the gateway MSA depends on the value in remaining in the outside
MSA relative to the value of being in the gateway MSA and paying a moving cost:
n o
max m∗ ( a, z)V NY , V ∗

The moving-in cost function is allowed to differ from the moving-out cost, as explained
in the calibration section below.

Location and Tenure Choice within Gateway MSA Denote by p RS,` the probability of
winning the RS lottery and being offered a RS unit in zone `. The household chooses
whether to accept the RS option with value VRS,` , or to turn it down and go to the private
housing market with value Vf ree . The value function, conditional on being in the gateway
MSA, V NY is:
 
V NY = p RS,1 max VRS,1 , Vf ree + p RS,2 max VRS,2 , Vf ree + 1 − p RS,1 − p RS,2 Vf ree .
 

A household that loses the lottery or wins it but turns it down, freely chooses in which
location ` ∈ {1, 2} to live and whether to be an owner (O) or a renter (R).

Vf ree = max {VO,1 , VR,1 , VO,2 , VR,2 } .

The Bellman equations for VRC,` , VR,` and VO,` are defined below.

Tenure Choice in Outside MSA In the outside MSA, there is only one zone (zone 1)
and no RS system. The value function, conditional on being in the outside MSA, is:

V ∗ = max {VO∗ , VR∗ } .

12
State Variables Let St be the vector which includes the wage Wt , the housing price Pt` ,
the market rent R`t and previous housing stock Ht`−1 for each zone `. There is a similar
state variable for the outside location St∗ except that there is only one zone in the outside
MSA. The household forms beliefs about (St , St∗ ). The household’s individual state vari-
ables are: net worth at the start of the period xt , idiosyncratic productivity level zt , age
a, and housing status in the previous period dt . The housing status is equal to 0 if the
household was a market renter or owner in the gateway MSA and takes non-zero values
to record both where and how long the household has been in the RS system (since the
discount depends on the length of tenancy). We suppress the dependence on β-type in
the problem formulation below, but note here that there is one set of Bellman equations
for each β-type.

Market Renter Problem In the gateway MSA, a renter household on the free rental
market in location ` chooses non-durable consumption ct , housing consumption ht , and
working hours nt to solve:

VR,` ( xt , zt , a, dt ) = max U (ct , ht , nt , `t ) + (1 − p a ) βEt [V ( xt+1 , zt+1 , a + 1, 0)]

ct ,ht ,nt ,bt+1
s.t.
t + Ψt ψ + πt + xt − T yt
ct + R`t ht + Qbt+1 + φF` = 1 − τ SS ylab z tot ,

ylab ` a
t = Wt nt A G zt , 
1
ytot lab
t = yt + Q − 1 xt + πt ,
bt+1 ≥ 0,
x t + 1 = bt +1 + b
and equations (1), (2), (3), (4).

The renter’s savings in the risk-free bond, Qbt+1 , are obtained from the budget constraint.
Pre-tax labor income ylab
t is the product of wages W per efficiency unit of labor, the num-
ber of hours n, and the productivity per hour A` G a z. The latter has location-, age-, and
individual-specific components. Total pre-tax income, ytot , is comprised of labor income
and financial income. Financial income is the sum of interest income on bonds and a
share of firm profits πt , defined below. Net tax (taxes owed minus government transfers
received) as a function of total pre-tax income is given by the function T (ytot
t ). It captures
all insurance provided through the tax code. Additionally, a Social Security tax τ SS is
applied to labor income. Next period’s financial wealth xt+1 consists of savings bt+1 plus
any accidental bequests bbt+1 . Housing demand and labor supply choices are subject to
minimum constraints discussed above. In addition to a time cost, residents of zone 2 face
a financial cost of commuting φF` .

13
 In the outside location, the problem of a market renter is the same, with value function
VR∗ and last term Et [V ∗ ( xt+1 , zt+1 , a + 1)].

RS Renter Problem In the gateway MSA, a renter household in the RS system in loca-
tion ` chooses non-durable consumption ct , housing consumption ht , and working hours
nt to solve:

VRS,` ( xt , zt , a, dt ) = max U (ct , ht , nt , `t ) + (1 − p a ) βEt [V ( xt+1 , zt+1 , a + 1, `)]

ct ,ht ,nt ,bt+1
s.t.
` = 1 − τ SS ylab + Ψ ψz + π + x − T ytot ,
ct + κ1 (dt ) R`t ht + Qbt+1 + φF,t



t t t t t
bt+1 ≥ 0,
x t + 1 = bt + 1 + b
ylab
t ≤ κ2 Y t if dt = 0,
ht ≤ κ3` ,
and equations (1), (2), (3), (4).

The per square foot rent of a RS unit is a fraction κ1 (dt ) of the market rent R`t , which
depends on length of tenancy. In versions of the model where RS has income qualifi-
cation, labor income must not exceed a fraction κ2 of area median income (AMI), Y t =
Median[ylab,i
t ], the median across all residents in the MSA. There is no income qualifica-
tion requirement in the benchmark model (κ2 = ∞). The last inequality imposes that the
maximum size for a RS unit must not exceed a threshold κ3` . Length of tenancy in the RS
system is updated through the state variable dt+1 .

Owner’s Problem In the gateway MSA, an owner in location ` chooses non-durable

consumption ct , housing consumption ht , working hours nt , and investment property b
ht
to solve:

VO,` ( xt , zt , a, dt ) = max U (ct , ht , nt , `t , a) + (1 − p a ) βEt [V ( xt+1 , zt+1 , a + 1, 0)]

ct ,ht ,b
ht ,nt ,bt+1
s.t.
ct + Pt` ht + Qbt+1 + κ4` Pt` b ` = 1 − τ SS ylab + Ψ ψz + π + x + κ ` R` b tot ,



ht + φF,t t t t t 4 t ht − T yt
bt+1 + Pt`+1 ht (1 − δ` − τ P,` ) + κ4` Pt`+1b
x t + 1 = bt +1 + b ht (1 − δ` − τ P,` ),
− Qt bt+1 ≤ Pt` θ ht + κ4` b ht − κ4` R`t bht − (ytot
t − c t ),
ht ≥ 0,
b
κ4` = 1 − η ` + η ` κ1 ` ,
and equations (1), (2), (3), (4).

14
 Local home owners are the landlords to the local renters. This is a departure from the
typical assumption of absentee landlords in urban economics.11 Our landlords are risk-
averse households inside the model. For simplicity, we assume that renters cannot buy
investment property and that owners can only buy investment property in the zone of
their primary residence. Landlords earn rental income κ ` R` b
ht on their investment units b
4 t ht .
Per the affordable housing mandate, investment property is a bundle of η ` square feet of
RS units and 1 − η ` square feet of free-market units. The effective rent earned per square
foot of investment property is κ4` R`t . It depends on the RS discount κ1 ` , which depends
on both the discount by tenancy κ1 (d) and the fraction of RS renters at each tenancy in
zone `. Since the average rent is a multiple κ4 ≤ 1 of the market rent, the average price of
rental property must be the same multiple of the market price, κ4` Pt` . Because prices and
rents scale by the same constant, the return on investing in rental property is the same
as that on owner-occupied housing. As a result, landlords are not directly affected by RS
regulation. However, the lower average price for rental property (κ4 < 1) has important
effects on housing supply/development, as discussed below.
The physical rate of depreciation for housing units is δ` . The term P` hδ` is a financial
costs, i.e., a maintenance cost. As shown in equation (10) below, the physical depreciation
can be offset by residential investment undertaken by the construction sector.12
Property taxes on the housing owned in period t are paid in year t + 1; the tax rate
is τ P,` . Property tax revenue finances local government spending which does not confer
utility to the households.13
Housing serves as a collateral asset for debt. For simplicity, mortgages are negative
short-term safe assets. In practice, mortgage rates are higher than bond rates but mort-
gage interest is also tax deducible. We assume these two effects cancel out. Households
can borrow a fraction θ of the market value of their housing.14 We exclude current-period
11 The majority of rentals in the urban core are multi-family units owned by local owner-operators. For
example, According to 2015 Real Capital Analytics data, 81% of the Manhattan multifamily housing stock
is owned by owner-operator-developers which tend to be overwhelmingly local. Non-financial firms, some
of which are also local, own 3%. The remaining 16% is owned by financial firms, private equity funds, or
publicly listed REITs, with at least some local investors. The majority of rentals outside the urban core are
single-family rentals. About 99% of those are owned by small, local owners. A substantial minority are
multi-family units, with again a non-trivial local ownership share.
12 The model can accommodate a higher rate of depreciation for renter-occupied properties, possibly to

reflect the higher rate of depreciation for RS housing units. We are not aware of empirical evidence that
shows that RS housing results have higher depreciation rates than market rental units. In contrast, rent
controlled and public housing units are often associated with severe under-maintenance. Nevertheless, as
a robustness check, Appendix D considers an exercise where RS housing depreciates at a higher rate.
13 This is equivalent to a model where public goods enter in the utility function, but in a separable way

from private consumption. A model where the public good enters non-separably in the utility function
would require taking a stance on the elasticity of substitution between private and public consumption.
14 It is easy to introduce a different LTV ratio for primary residences and investment property. The em-

15
rental income and savings from the pledgable collateral. In light of the fact that one pe-
riod is four years in the calibration, we do not want to include four years worth of (future)
rental income and savings for fear of making the borrowing constraint too loose.15
For the outside location, the ownership problem is the same. We denote the value
function VO∗ , and the last term on the right-hand side of the Bellman equation is
Et [V ∗ ( xt+1 , zt+1 , a + 1)].

2.2 Firms
Goods Producers There are a large number n f of identical, competitive firms located in
the urban core (zone 1), all of which produce the numéraire consumption good.16 This
good is traded nationally; its price is unaffected by events in the city and normalized
to 1. The firms have decreasing returns to scale and choose efficiency units of labor to
maximize profit each period:

Πc,t = max Nc,tc − Wt Nc,t

ρ
(5)
Nc,t

Developers and Affordable Housing Mandate In each location ` there is a large num-
ber n f of identical, competitive construction firms (developers) which produce new hous-
ing units and sell them locally. All developers are headquartered in the urban core, re-
gardless of where their construction activity takes place.
The cost of the affordable housing mandate is born by developers. Affordable housing
regulation stipulates that for every 1 − η ` square feet of market rental units built in zone
`, η ` square feet of RS units must be built. Developers receive an average price per square
foot for rental property of κ4` Pt` , while they receive a price per square foot of Pt` for owner-
occupied units.17 Given a home ownership rate in zone ` of hot` , developers receive an
`
average price per square foot Pt :
 
`
Pt = hot` + (1 − hot` )κ4` Pt` . (6)

pirically relevant case is θres ≥ θinv . We abstract from this for simplicity.
15 This assumptions helps the model match the home ownership rate. However, the affordable housing

policies would have similar effects without it.

16 We assume that the number of firms is proportional to the number of households in each MSA when

solving the model. With this assumption, our numerical solution is invariant to the total number of house-
holds in the economy. Due to decreasing return to scale, the numerical solution would depend on the
number of households otherwise.
17 Recall that κ ` = 1 − η ` + η ` κ ` , where κ ` =
4 1 1 ∑d ω ` (d)κ1` (d) and ω ` (d) is the share of RS square feet in
a zone that goes to RS tenants in that zone with tenure d, such that ∑d ω ` (d) = 1.

16
The cost of construction of owner-occupied and rental property in a given location is the
same. After completion of construction but prior to sale, some of the newly constructed
housing units are designated as rental units and the remainder as ownership units. The
renter-occupancy designation triggers affordable housing regulation. It results in a lower
rent and price than for owner-occupied units. Developers would like to sell ownership
units rather than rental units, but the home ownership rate is determined in equilibrium.
`
Developers are price takers in the market for space, and face an average sale price of Pt .
A special case of the model is the case without rent stabilization: κ4` = 1 either because
`
η ` = 0 or κ1 = 1. In that case, Pt = Pt` . Without RS, the higher sale price for housing
increases incentives to develop more housing.

Zoning Given the existing housing stock in location `, Ht`−1 , and average sale price of
`
Pt , construction firms have decreasing returns to scale and choose labor to maximize
profit each period: !
` Ht`−1
Π`h,t =
ρ
max Pt 1− N`,th − Wt N`,t (7)
N`,t H`
The production function of housing has two nonlinearities. First, as for consumption
good firms, there are decreasing returns to scale because ρh < 1.
Second, construction is limited by zoning laws and space constraints. The maximal
amount of square footage zoned for residential use in zone ` is given by H ` . We interpret
H ` as the total land area available for residential use multiplied by the maximum possi-
ble number of floors that could be built on this land. This term captures the idea that,
the more housing is already built in a zone, the more expensive it is to build additional
housing. For example, additional construction may have to take the form of taller struc-
tures, buildings on less suitable terrain, or irregular infill lots. Therefore, producing twice
as much housing requires more than twice as much labor. Laxer zoning policy, modeled
as a larger H ` , makes development cheaper, and all else equal, will expand the supply of
housing.
When H ` is sufficiently high, the model’s solution becomes independent of H ` , and
the supply of housing is governed solely by ρh . When H ` is sufficiently low, the housing
supply elasticity depends on both H ` and ρh .18
18 Inthis sense, the model captures that construction firms must pay more for land when land is scarce or
difficult to build on due to regulatory constraints. This scarcity is reflected in equilibrium house prices.

17
Profits Per capita profits from tradeable and construction sectors are:

Πt = Πc,t + Π1h,t + Π2h,t .

These profits represent a competitive compensation to capital and pure profit. Equiva-
lently, the production function in both sectors contains a term K ρk , with K is normalized
to 1. Aggregate profit is ((1 − ρc − ρk )/(1 − ρc )) Πt . We assume that these profits go to
local residents; the πt term in the household budget constraint. Profits received depend
on household age and productivity.

2.3 Equilibrium
Given parameters, a competitive equilibrium is a price vector (Wt , Pt` , R`t ) and an alloca-
tion, namely aggregate residential demand by market renters HtR,` , RS renters HtRS,` , and
owners H O,` , aggregate investment demand by owners H
t
b ` , aggregate housing supply,
t
aggregate labor demand by goods and housing producing firms (Nc,t , N`,t ), and aggre-
gate labor supply Nt in each MSA, as well as a population share in each MSA, such that
households and firms optimize and markets clear in each MSA.19
The following conditions characterize the equilibrium. First, given wages and aver-
age prices given by (6), firms optimize their labor demand, resulting in the first-order
conditions:
  1
1− ρ h

Ht`−1 `
 1 − H` Pt ρh 
  1
ρc 1− ρ c
Nc,t = and N`,t =  . (8)
Wt  Wt 

Second, labor demand equals labor supply in efficiency units:

!
nf Nc,t + ∑ N`,t = Nt = ∑ nit zit G a A` . (9)
` i,a,`

Third, the housing market clears in each location `:

!
Ht`−1
(1 − δ` ) Ht`−1 + n f N`,th = HtO,` + H
b t` .
ρ
1− (10)
H`
is one price and allocation vector for each MSA, e.g., Wt for the gateway MSA and Wt∗ for the
19 There

outside MSA, etc. To ease notational burden we did not separately list all the variables for the outside MSA.

18
The left-hand-side is the supply of housing which consists of the non-depreciated hous-
ing stock and new residential construction. The right-hand-side is the demand for those
housing units by owner-occupiers and landlords. Fourth, the supply of rental units in
each location ` must equal the demand, from market tenants and RS tenants, respectively:

b t` (1 − η ` ) = H R,` ,
H b t` η ` = H RS,`
H (11)
t t

Fifth, total pension payments equal to total Social Security taxes collected:

Ψt Nret = τ SS Nt Wt , (12)

where Nret is the total number of retirees. Sixth, total profits collected equal total profits
distributed to households. Seventh, the population of the gateway MSA is endogenously
determined by the value functions of living in each MSA and the moving costs.20 Eighth,
the value of all bequests received is equal to the wealth of all agents who die. Ninth,
the aggregate state (St , St∗ ) evolves according to rational expectations. We focus on the
model’s steady state where all aggregate quantities and prices are constant.

2.4 Welfare Effects of Affordability Policies

We compute the welfare effect of an affordability policy using the following procedure.
Denote agent i’s value function under benchmark policy θb as Vi,t ( x (b), z, a, S(b); θb ). Con-
sider an alternative policy θc , which goes into effect in the gateway MSA in period t +
1, with value function Vi,t+1 ( x (c), z, a, S(c); θc ). Prices and hence asset valuations and
wealth may be different under this new policy, hence the dependence of x and S on the
policy. Because of endogenous migration, the set of households that is present in the
gateway MSA before (at t) and after the reform (at t + 1) may be different. Our main
welfare measure averages over a fixed group of households that were present in the gate-
way MSA prior to the reform, the set gt with cardinality G, and tracks them at time t + 1
regardless of their mobility decisions. Finally, we express the welfare change in consump-
tion equivalent units rather than utils. To summarize, our main welfare measure is:

1
1
!
G ∑i ∈ gt Vi,t+1 ( x ( c ), z, a, S ( c ); θc )
(1−γ)(1−αn )
Wg = 1
− 1. (13)
G ∑i ∈ gt Vi,t ( x ( b ), z, a, S ( b ); θb )
20 We assume that the population of the outside MSA is large and not affected by out-migration from the

gateway MSA.

19
This welfare criterion is utilitarian in that it weighs each household in the group equally.
But because of the curvature of the value function, lower-income households implicitly
receive a larger weight. We also use (13) to compute welfare for subgroups of gt , for ex-
ample by labor productivity type, by income quartile, or net worth quartile. This welfare
change is calculated in the first period after the reform, the first period of the transition
towards a new steady state. Of course, the value function is forward-looking and incor-
porates the expected risk-adjusted present discounted values, but state variables have not
settled down to their new steady state levels yet.21

3 Calibration
We calibrate the model to match important features of the New York MSA. The outside
MSA is calibrated to the average of the 75 largest U.S. MSAs except for New York. Data
sources are described in Appendix A. Table 1 summarizes the chosen model parameters.
The parameters are the same in the two MSAs unless explicitly mentioned. Some param-
eters are set exogenously, while others are chosen to match a moment in the data.22

Geography The New York MSA consists of 25 counties located in New York (12), New
Jersey (12), and Pennsylvania (1). We assume that Manhattan (New York County) repre-
sents zone 1 and the other 24 counties make up zone 2. The zones differ in size, measured
by the maximum buildable residential square footage permitted by existing zoning rules,
H ` . Appendix A describes detailed calculations on the relative size of Manhattan and the
rest of the metro area, which imply that H 1 = 0.0238 × H 2 . We then choose H 2 such that
the ratio of households living in zone 1 to households living in zone 2 is 12%, the fraction
observed in the NY data. Since the model has no vacancy, we equate the number of NY
households in the model with the number of occupied housing units in the NY data.
In the outside MSA, H ∗ is chosen to match the average population-weighted housing
supply elasticity of 1.55 among the largest 75 U.S. MSAs outside the New York MSA using
data from Saiz (2010).

Production and Construction We assume that the return to scale ρc = 0.66. This value
implies a labor share of 66% of output, consistent with the data.
21 For the no-migration model in Appendix C, we also report a welfare measure that uses the steady-state

value function under the alternative policy, Vi,∞ ( x (c), z, a, S(c); θc ).

22 As in Andrews, Gentzkow, and Shapiro (2017), one parameter affects multiple moments but often has

a disproportionate effect on one moment. With that caveat, we associate parameters with individual mo-
ments.

20
 For the housing sector, we also set ρh = 0.66 in order to match the housing supply
elasticity, given the other parameters. The long-run housing supply elasticity in the model
is derived in Appendix B.3. Saiz (2010) reports a housing supply elasticity for the New
York metro area of 0.76. The model delivers 0.69. The housing supply elasticity is much
lower in zone 1 (0.08) than in zone 2 (0.71), because in zone 1 the housing stock is much
closer to H (12% from the constraint) than in zone 2 (70% from the constraint). Since
the housing stock of the metro area is concentrated in zone 2, the city-wide elasticity is
dominated by that in zone 2.

Demographics The model is calibrated so that one model period is equivalent to 4 years.
Households enter the model at age 21, work until age 64, and retire with a pension at age
65. Survival probabilities p a are calibrated to mortality data from the Census Bureau.

Recall that pre-tax labor income for household i of age a is ylab i a ` i

Labor Income t = Wt nt G A zt ,
where the household takes wages as given and chooses labor supply nit . The choice of
hours is subject to a minimum hours constraint, which is set to 0.5 times average hours
worked. This constraint rules out a choice of a positive but very small number of hours,
which we do not see in the data given the indivisibility of jobs. It also rules out unem-
ployment since our earnings data are for the (part-time and full-time) employed. This
constraint binds for only 11.15% of workers in equilibrium.
Efficiency units of labor A` G a zit consist of a deterministic component that depends on
the location of the household (A` ), a deterministic component that depends on age (G a ),
and a stochastic component zi that captures idiosyncratic income risk.
The agglomeration parameter that governs the extra productivity a household derives
from living in the urban core A1 = 1.09757 > A2 = 1 is chosen to match the 1.66 ratio of
average income in zone 1 to zone 2 in NY. Since the outside region has only one zone, it
does not have this parameter.
The G a function is chosen to match the mean of labor earnings by age. We use data
from ten waves of the Survey of Consumer Finances (1983-2010) to estimate G a .
The idiosyncratic productivity process z is chosen to match earnings inequality and
persistence in household earnings. We discretize z as a 4-state Markov chain. Appendix
B.1 explains how we choose the productivity grid points and the transition probabilities
between states. In a nutshell, the model matches the pre-tax household income distribu-
tion for the NY metro and the outside metro. Income data are from the IPUMS Census
data set. The model matches the persistence of labor income of 0.9. It matches how the
variance of earnings rises with age in the SCF, and it delivers the observed correlation

21
between income and wealth in the SCF. The first productivity bin contains the lowest
25%-productivity households. The second productivity level contains the next 50% of
households. Bin 3 is the next 12.5%, and bin 4 contains the 12.5% most productive house-
holds. The income calibration is an iterative process since both labor supply and MSA
location are endogenous choices that depend on all other parameters and features of the
model.

Taxation Since our model is an incomplete markets model, housing affordability poli-
cies can act as an insurance device and help to “complete the market.” Therefore, a re-
alistic calibration of the redistribution provided through the tax code is important. We
follow Heathcote, Storesletten, and Violante (2017) and choose an income tax schedule
that captures the observed progressivity of the U.S. tax code in a parsimonious way. Net
taxes are given by the function T (·):

T (ytot ) = ytot − λ(ytot )1−τ

The parameter τ governs the progressivity of the tax and transfer system. We set τ = 0.17
to match the average income-weighted marginal tax rate of 34% for the U.S. It is close to
the value of 0.18 estimated by Heathcote et al. (2017). We set λ to match federal, state, and
local government spending to aggregate income, which ranges between 15-20%.23 This
delivers λ = 0.75. Appendix B.2 shows the resulting tax rate and after-tax income as a
function of before-tax income. This tax-and-transfer system includes a baseline level of
government transfer spending on housing vouchers.

Retirement Income Social Security taxes are proportional to labor earnings and set to
τ SS = 0.10, a realistic value. Retirement income is increasing in the household’s last pro-
ductivity level prior to retirement, but is capped for higher income levels. We use actual
Social Security rules to estimate each productivity group’s pension relative to the aver-
age pension. The resulting pension income states are ψz = [0.44, 1.25, 1.51, 1.51], where
z reflects the last productivity level prior to retirement. They are multiplied by average
retirement income Ψ, which is endogenously determined in equation (12) to balance the
social security budget. Average retirement income Ψ is $33,189, which corresponds to
27% of average earnings.
23 Forexample, depending on what share of NY state and NJ state spending goes to the NY metro area,
we get a different number in this range.

22
 Table 1: Calibration

Description Parameter Value Target

Panel A: Geography and Income
Agglomeration zone 1 NY A1 1.095 Relative income NY vs. outside MSA
Productivity states NY Z NY [0.191 0.901 1.923 4.187] Income level & inequality - App. B
Productivity states outside Z∗ [0.191 0.741 1.507 3.148] Income level & inequality - App. B
Transition prob. ( p11 , p22 , p33 , p44 , p H ) (0.93,0.92,0.28,0.64,0.02) Pop. shares of income groups, income persistence - App. B
Tax progressivity τ 0.17 Heathcothe et al. (16)
Tax burden λ 0.75 Govt. spending to GDP
Average pension τ SS 0.10 Avg. contribution rate to social security
Pension distr. ψz [0.44 1.25 1.51 1.51] U.S. pension progressivity rules
Available space NY ( H1 , H2 ) (0.05,2.07) Max. residential buildable area, pop. share Manhattan
Available space outside H∗ 7.00 Housing supply elasticity avg. MSA, Saiz (10)
Time-Commuting cost φT2 0.037 Average commuting time NY
Financial-Commuting cost φ2F 0.01 Average commuting cost NY
Panel B: Production
Return to scale consumption sector ρc 0.66 Labor income share of 2/3
Return to scale housing sector ρh 0.66 Housing supply elasticity for NY, Saiz (10)
Panel C: Preferences
Risk aversion γ 5 Standard value

23
Leisure weight αn 0.63 Average work week
Housing consumption weight αh 0.41 Avg. rent/avg. income of 23%
CES parameter consumption/housing e -0.50 Elasticity of substitution consumption/housing of 2/3
Minimum hours worked n 0.15 Min. hours worked 50% of available time
Time Preference (4yr) (βH , βL ) (1.204,0.925) Avg. wealth/income and wealth Gini in SCF
Extra utility NY χ NY 1.036 Avg. wealth/income in NY equal to outside
Extra utility zone 1 χ1 1.036 Rent per sf ratio of zones in NY data
Extra utility zone 1 retirees χR 1.071 Fraction of retirees ratio of zones in NY data
Panel D: Finance, Housing, Construction
Bond Price (4yr) Q 0.89 Price/rent ratio in NY metro
Property tax rate NY (4yr) (τ P,1 , τ P,2 ) (0.029,0.053) Prop. tax rate zone 1 and 2 of NY MSA -Brookings
Property tax rate outside (4yr) τ P,∗ 0.063 Prop. tax rate 74 largest MSAs - Brookings
Depreciation rate NY (4yr) ( δ1 , δ2 ) (0.058,0.096) Res. depr. BEA, rel. depr by zone
Depreciation rate outside (4yr) δ∗ 0.095 Res. depreciation BEA
Maximum LTV θ 0.90 Avg. mortgage downpayment
Minimum housing size h 0.20 Obs. minimum housing size
Panel E: Affordable Housing
Fraction rent stabilization (η 1 , η 2 ) (56.37%,29.72%) Frac. of HHs in RS of 37.3% and 12.0%
Prob. to stay in RS p RS,exog 0.83 Fraction of RS HHss in same unit for ≥ 20 years
Rental discount κ1 ( d ) 7%–45% Observed rental discount - App. A.4
Income threshold for RS κ2 ∞ No income testing in NY
Maximum RS house size κ3 0.50 Avg. size of RS and market rentals equal
Commuting Cost We choose the time cost to match the time spent commuting for the
average New York metro area resident. This time cost is the average of all commutes,
including those within Manhattan. We normalize commuting time for zone 1 residents to
zero: φT2 > φT1 = 0. For φT2 , we target the additional commuting time of zone 2 residents.
The additional commuting time amounts to 25 minutes per trip for 10 commuting trips
per week.24 The 4.2 hours represent 3.7% of the 112 hours of weekly non-sleeping time.
Hence, we set φT2 = 0.037.
As we did for the time cost, we normalize the financial cost of commuting for residents
of zone 1 to zero: φ1F = 0. The financial cost of commuting φ2F is set to 1.8% of average
labor earnings, or $2213 per household per year. This is a reasonable estimate for the
commuting cost in excess of the commuting cost within Manhattan.25
We assume that retirees have time and financial commuting costs that are 1/3 of those
of workers. This captures that retirees make fewer trips, travel at off-peak hours, and
receive transportation discounts.

Preferences The functional form for the utility function is given in equation (1). We set
risk aversion γ = 5, a standard value in the macro-finance literature.
The observed average workweek is 42.8 hours or 38.2% of available non-sleeping time.
Since there are 1.64 workers on average per household, household time spent working is
38.2%× 1.64/2=31.3%. We set αn to match household time spent working. The model
generates 29.3% of time worked.
We set the labor supply elasticity parameter η = 1. This generates an (endogenous) av-
erage Frisch elasticity of 1.08 when estimated from macro and 1.42 when estimated from
micro data.26 This is in line with estimates based on macro data and on the intensive mar-
gin of labor supply in micro data. This is an important object because the misallocation
coming from workers’ persistent location and labor supply decisions depends on how
sensitive labor supply is to wage changes.
We set αh in order to match the ratio of average market rent to metro-wide average
24 The 25 minute additional commute results from a 15 minute commute within Manhattan and a 40
minute commute from zone 2 to zone 1. With 10.5% of the population living in Manhattan, the average
commuting time is 37.4 minutes per trip or 6.2 hours a week. This is exactly the observed average for the
New York metro from Census data.
25 In NYC, an unlimited subway pass costs around $1,400 per year per person. Rail passes from the

suburbs cost around $2400 per year per person, depending on the railway station of departure. If zone 1
residents need a subway pass while zone 2 residents need a rail pass, the cost difference is about $1000
per person. With 1.64 workers per household, the cost difference is $1640 per household. The cost of
commuting by car is at least as high as the cost of rail once the costs of owning, insuring, parking, and
fueling the car and tolls for roads, bridges, and tunnels are factored in.
26 The [25%,75%] of the distribution of Frisch elasticities across agents is [0.86,1.92] in the model.

24
income. The model generates 23.9%. This value is close to the 24% value calculated from
decennial Census data for a cross-section of MSAs by Davis and Ortalo-Magne (2011).
We set the intratemporal elasticity of substitution between housing and non-housing
consumption equal to 2/3 (e = −0.5), a value in the middle of the (wide) range of esti-
mates in the literature.
We set β H = 1.204 (1.047 per year) and β L = 0.925 (0.981 per year). A 25% share
of agents has β H , the rest has β L . This delivers an average β of 0.99,27 chosen to match
the average wealth-income ratio which is 5.69 in the 1998-2010 SCF data. The model
generates 6.00. The dispersion in betas delivers a wealth Gini coefficient of 0.74, close to
the observed wealth Gini coefficient of 0.80 for the U.S.
Three parameters govern the amenity value of housing in (4). The taste-shifter for NY
relative to the outside MSA, χ NY = 1.0335, is chosen to keep the ratio of net worth to av-
erage earnings equal between the NY metro and the outside metro. Living in Manhattan
relative to the rest of the NY metro gives a utility boost χ1 = 1.036, chosen to match the
2.78 ratio of rents in zone 1 to zone 2 in the NY metro. Being a retiree in Manhattan gives
an additional utility boost of χ R = 1.071, chosen to match the 0.91 ratio of retirees in zone
1 to zone 2 in the NY MSA. Retirees have lower time and financial costs of commuting,
giving them a comparative advantage to living in zone 2. A retiree preference for living
in Manhattan is needed to offset the commuting effect.

Housing The price of the one-period (4-year) bond Q = 0.89 targets the average house
price to rent ratio for the New York MSA, which is 17.79. The model delivers 16.75. Under
the logic of the user cost model, the price-to-rent ratio depends on the interest rate, the
depreciation rate, and the property tax rate.
The property tax rate in Manhattan is τ P,1 = 0.029 or 0.73% per year, and that in zone
2 is τ P,2 = 0.053 or 1.33% per year. The property tax rate in the outside MSA τ P,∗ is 1.60%
per year. These match the observed tax rates averaged over 2007-2011 according to the
Brookings Institution.28
The housing depreciation rate in Manhattan is δ1 = 0.058 or 1.45% per year, and that
in zone 2 is δ2 = 0.096 or 2.41% per year. This delivers a metro-wide average depreciation
rate of 2.39% per year. For the outside MSA, we set depreciation to 2.45%, equal to the
average depreciation rate for privately-held residential property in the BEA Fixed Asset
tables for the period 1972-2016. The annual depreciation wedge of 1.0% between NY
27 Notethat because of mortality, the effective time discount factor is (1 − p( a)) β.
28 Thezone-2 property tax rate is computed as the weighted average across the 24 counties, weighted
by the number of housing units. The outside property tax rate is computed as the population-weighted
average of the property tax rates of the largest 74 MSAs outside NY using the same data source.

25
zones 1 and 2 is chosen to match the relative fraction of buildings that were built before
1939.29
Given its higher property tax and depreciation rates, the outside metro has a lower
price-rent ratio of 15.7.
We set the maximum loan-to-value ratio (LTV) at θres = 0.9, implying a 10% down
payment requirement. The observed mean combined LTV ratio at origination for U.S.
mortgages in the U.S. is 87.3% as of October 2016 according to the Urban Institute and
has consistently been above 80% since the start of the data in 2001.
Finally, we impose a minimum housing size of 506 square feet. This is 31% of the
average housing unit size of 1644 square feet in NY and 26% of the average house size
of the 1980 square feet in the outside MSA. This is a realistic value for New York given
the model is solved at the household level (with 1.64 members on average). While the
average NY house size is a normalization constant, set to match the data, the outside MSA
house size is endogenously determined as is the house size distribution in NY. More on
the model’s house size implications below.

Affordable Housing Rent regulation plays a major role in the New York housing mar-
ket, as discussed above. In this paper, we focus on rent stabilization which is by far the
most prevalent affordable housing program in New York. We find that 37.3% of zone-1
households and 12.0% of zone-2 households live in RS units. Appendix A.4 contains a
detailed description of data and definitions. We set the share of square feet of rental hous-
ing devoted to RS units, η 1 = 56.37% and η 2 = 29.72%, to match the share of households in
the entire population that are in RS units in each zone. This fraction is endogenous since
housing size and ownership are choice variables.
We find that the rent discount on RS units depends strongly on tenure. The average
discount is 7% for households who have lived in the unit for four years or less, and grows
to 45% for tenure of 12 years or longer. The growing discount reflects smaller annual
rent increases on RS than on market rental units, which cumulate as long as households
remain in their RS unit. We use the observed discount-tenure schedules, computed in
Appendix A.4, in the calibration of κ1 (d).
RS housing units are available to anyone; there is no income qualification (κ2 = ∞).
We assume that households who were in RS in the previous period have a probability of
83.4% to qualify for RS in the same zone this period. The value is chosen to match the
29 Datafrom the 5-year American Community Survey from 2017 give the distribution of housing units
by year built for each of the 25 counties in the New York MSA. In Manhattan, 42.8% of units are built
before 1939. The housing-weighted average among the 24 counties of zone 2 is 26.6%. Assuming geometric
depreciation, matching this fraction requires a 1.0% per year depreciation wedge.

26
fraction of RS tenants who have lived in a RS unit for 20 years or more. That number in
the data is 23.1%.30 It is 26.1% in the model.
The maximum RS size κ3` is set such that the average size of market and RS rentals is
equal in that zone.

Moving Costs To compute migration rates, we sort households in each of six age groups
(26-34, 35-44, 45-54, 55-64, 65 and up) into four income groups (bottom 25%, middle 50%,
next 12.5%, top 12.5%) and compute the out- and in-migration rates for each group. The
migration data are described in Appendix A.5. While migration decisions are endoge-
nous and depend on the full structure of the model, moving costs are crucial to hit the
migration targets. The moving cost functions m( a, z) for moving out of New York takes
the following form:

m( a, z) = m0 + m1 ( a) + m2 (z) + σm u; u ∼ U (0, 1).

The coefficient m0 is chosen to match the average of the out-migration and in-migration
rates of 2.5%. The coefficients m1 ( a) and m2 (z) are chosen to best match the out-migration
profile by age-by-income groups. The volatility parameter governs the sensitivity of
the moving rate to the moving shock u. This parameter helps the model to better fit
the rent/income ratio in New York. The functional form for the moving cost function
m∗ ( a, z) for moving into New York is the same, but the parameters are chosen to match
the in-migration rates instead.31 The parameter σm is restricted to be the same in the
in-migration as for out-migration moving cost function.
Figure 1 shows that the model closely matches the migration rates out of (top panel)
and into (bottom panel) the New York MSA by age and income. Migration is declining
in age and U-shaped in income. Among those 65 and older, out-migration exceeds in-
migration.

Profit Share Two-thirds of output in the data goes to labor, 27% to investment, and 7%
goes to profits. Firms in the model also make profits. Since the model has no capital,
we scale these profits so that they represent 7% of output in the model. These profits
are distributed to local residents according to a profit distribution that depends on age
30 SeeTable H of the NYU Furman Institutes’ 2014 “Profile of Rent-Stabilized Units and Tenants in NYC.”
31 Sincethe model is stationary, the average out- and in-migration rates must be the same. Therefore, we
set m0 to the average of these two numbers. The constant m0∗ is chosen to keep the population of the NY
metro at an arbitrary constant of 2000. This helps make the model comparable to a model without migration
with 2000 agents in New York. Note that when we conduct policy experiments, we do not recalibrate this
parameter. Thus, the population of NY can rise or fall relative to the 2000 number in the baseline model.

27
 Figure 1: Inter-MSA Migration Rates by Age and Income

Out-migration
By Age By Income
5.5 5.5
Data Data
5 Model Model
5

Outmigration Rate (annual %)

Outmigration Rate (annual %)

4.5
4.5

4
4
3.5
3.5
3
3
2.5

2.5
2

1.5 2

1 1.5
<26 26-34 35-44 45-54 55-64 >64 1 2 3 4
Age Income Group

In-migration
By Age By Income
5.5 5
Data Data
5 Model Model
4.5
4.5
Inmigration Rate (annual %)

Inmigration Rate (annual %)

4 4

3.5
3.5
3
3
2.5

2 2.5

1.5
2
1

0.5 1.5
<26 26-34 35-44 45-54 55-64 >64 1 2 3 4
Age Income Group

Note: The top panel reports annual out-migration rates out of New York into the Outside MSA. The bottom
panel plots in-migration rates into the New York MSA from Outside. Migration in the data is computed
based on IRS tax returns from New York State.

and income. The latter is calibrated to how the share of private business income to total
household income in the Survey of Consumer Finances depends on age and income.32

4 Baseline Model Results

We start by discussing the implications of the baseline model for the spatial distribution
of population, housing, income, and wealth. We also discuss house prices and rents for
the city as a whole and for the two zones. Then we look at the model’s implications
for income, wealth, and home ownership over the life-cycle. Table 2 shows some key
32 The
business income shares from the SCF by age and income are rescaled such that, given the popula-
tion distribution in the simulation of the model, total profit redistributed is equal to total profit generated.
The redistribution differs slightly between NY and Outside because the age-income distribution is slightly
different, and therefore also the normalization constant.

28
moments; moments in boldface are not directly targeted by the calibration.

4.1 Demographics, Income, and Wealth

Demographics The first three rows of Table 2 show that the model matches basic demo-
graphic moments. In the model, we get 19.5%. The average NY resident above age 21 is
47.6 years old in the data and 46.5 years old in the model. In both model and data, zone
1 skews younger than zone 2. People age 65 and over comprise 19.1% of the NY popu-
lation age 21 and over in the data; in the model this share is 19.9%. Migration decisions
and mortality rates combine to produce these results.

Table 2: New York Metro Data Targets and Model Fit

Data Model
metro ratio zone 1/zone 2 metro ratio zone 1/zone 2
1 Households (thousands) 7124.9 0.12 7124.9 0.12
2 Avg. hh age, cond. age > 20 47.6 0.95 46.5 0.86
3 People over 65 as % over 20 19.1 0.91 19.5 0.90
4 Avg. house size (sqft) 1644 0.59 1644 0.63
5 Avg. pre-tax lab income ($) 124091 1.66 124165 1.69
6 Home ownership rate (%) 51.5 0.42 58.4 0.57
7 Median mkt price per unit ($) 510051 3.11 496649 2.25
8 Median mkt price per sqft ($) 353 5.24 280 3.55
9 Median mkt rent per unit (monthly $) 2390 1.65 2471 1.76
10 Median mkt rent per sqft (monthly $) 1.65 2.78 1.39 2.77
11 Median mkt price/median mkt rent (annual) 17.79 1.89 16.75 1.28
12 Mkt price/avg. income (annual) 3.99 1.86 4.00 1.33
13 Avg. rent/avg. income (%) 24.0 1.00 23.9 1.04
14 Avg. rent/income ratio for renters (%) 42.1 0.81 33.9 1.21
15 Rent burdened (%) 53.9 0.79 54.7 1.32
16 % RS of all housing units 14.63 3.11 14.20 3.28

Note: Columns 2-3 report the values for the data of the variables listed in the first column. Data sources
and construction are described in detail in Appendix A. Column 3 reports the ratio of the zone 1 value to
the zone 2 value in the data. Columns 4 and 5 are the corresponding moments in the model. Moments in
boldface are not directly targeted by the calibration.

Mobility Within the New York MSA The model implies realistic moving rates from
zone 1 to zone 2 and vice versa in the New York MSA, even though there are no moving
costs within the MSA. Intra-MSA mobility rates are not targeted by the calibration. Figure
2 shows that mobility in he model is highest for the young (ages 21–32). For these groups,
the annual mobility rate is around 4%. The overall mobility rate across neighborhoods in
the model is about 2% annually. These intra-MSA mobility rates are consistent with the
2.1% county-to-county migration rates in the New York MSA described in Appendix A.5.

29
Given the attractive rents, the model generates lower mobility rates for RS tenants at all
ages. The same is true in the data.

Figure 2: Intra-MSA Moving Rates by Age in Model

All Homeowners
0.06 0.06

0.04 0.04

0.02 0.02

0 0
21-24
25-28
29-32
33-36
37-40
41-44
45-48
49-52
53-56
57-60
61-64
65-68
69-72
73-76
77-80
81-84
85-88
89-92
93-96

21-24
25-28
29-32
33-36
37-40
41-44
45-48
49-52
53-56
57-60
61-64
65-68
69-72
73-76
77-80
81-84
85-88
89-92
93-96
Age Age
Renters RS Renters
0.06 0.06

0.04 0.04

0.02 0.02

0 0
21-24
25-28
29-32
33-36
37-40
41-44
45-48
49-52
53-56
57-60
61-64
65-68
69-72
73-76
77-80
81-84
85-88
89-92
93-96

21-24
25-28
29-32
33-36
37-40
41-44
45-48
49-52
53-56
57-60
61-64
65-68
69-72
73-76
77-80
81-84
85-88
89-92
93-96
Age Age

Note: Mobility rates are measured as the annual probability to move across zones.

Housing Units In the data, the typical housing unit is much smaller in Manhattan than
in the rest of the metro area. We back out the typical house size (in square feet) in each
county as the ratio of the median house value and the median house value per square
foot, using 2015 year-end values from Zillow. We obtain an average housing unit size of
1,021 sf in Manhattan and 1,718 sf in zone 2; their ratio is 0.59. In the model, households
freely choose their housing size subject to a minimum house size constraint. The model
generates a similar ratio of house size in zone 1 to zone 2 of 0.63.
The top row of Figure 3 shows the distribution of house sizes. The model (left panel)
matches the data (right panel) reasonably well, even though these moments are not tar-
geted by the calibration. The size distribution of owner-occupied housing is shifted to the
right from the size distribution of renter-occupied housing units in both model and data.
The average house size in the outside MSA is 27% larger than in the NY MSA in the
model, or 2089 versus 1644 square feet.

30
 Figure 3: House Size Distribution New York
Model Data
All All
0.5 0.5

0 0
49

49
9

9
0

0
9

9
99

99
00

00
49

99

49

99

99

49

99

49

99

99
<7

<7
0-

0-
>4

>4
-1

-1

-2

-2

-3

-1

-1

-2

-2

-3
75

75
00

00

00

00

00

00

00

00

00

00
10

15

20

25

30

10

15

20

25

30
Sqft Sqft
Homeowners Homeowners
0.5 0.5

0 0
9

9
9

9
00

0
9

9
4

4
99

99

00
49

99

49

99

99

49

99

49

99

99
<7

<7
0
0-

0-
>4

>4
-1

-1

-2

-2

-3

-1

-1

-2

-2

-3
75

75
00

00

00

00

00

00

00

00

00

00
10

15

20

25

30

10

15

20

25

30
Sqft Sqft
Renters Renters
0.5 0.5

0 0
9

9
9

9
0

0
9

9
4

4
99

99
00

00
49

99

49

99

99

49

99

49

99

99
<7

<7
0-

0-
>4

>4
-1

-1

-2

-2

-3

-1

-1

-2

-2

-3
75

75
00

00

00

00

00

00

00

00

00

00
10

15

20

25

30

10

15

20

25

30
Sqft Sqft

Note: Left panel: model. Right panel: data. Data source: American Housing Survey for the New York
MSA, U.S. Census Bureau, 2015.

Income Average income in the NY MSA and the ratio of income in zone 1 to zone 2 are
matched by virtue of the calibration (row 5 of Table 2). The model also matches the ratio
of average income in the New York MSA to the Outside MSA of 1.26. It is informative
to explore the underlying sorting by productivity type. Zone 1 contains workers that
are on average 44% more productive than in zone 2. Only 21.6% of working-age, top-
productivity households live in zone 1.
Figure 4 plots how households with different productivity sort across space and across
tenure status. The vertical axes measures the total square footage devoted to the various
types of housing in each zone. Values reported on the top of the bars correspond to the
percentage of households in each category. These percentages add up to 100% across the six
housing categories in NY. Colors correspond to productivity levels: increasing in shade
from yellow (low, z = 1) to red (high, z = 4) for working-age households, and green
for retirees. The graph shows that retirees and top-productivity households consume a
disproportionate share of zone-1 housing. The graph also illustrates enormous housing
inequality. The bottom 25% of households by productivity (yellow) consume a small

31
share of the housing stock.

Figure 4: Geographic Distribution of Households by Productivity.

Note: The colors indicate productivity levels. For working-age households: red indicates a top 12.5%
productivity household, brown a household in the next 12.5% of the productivity distribution, okra: a
household in the middle 50% of productivity, and yellow a household in the bottom 25%. Retired house-
holds of all productivity levels are indicated by green. The vertical axes measures the total square footage
devoted to the various types of housing in each zone. Numbers reported atop each of the six vertical bars
are the percentage of households; they sum to 100% across the six housing status categories in zone 1 and
zone 2, and they sum to 100% across the two housing status categories in the outside MSA.

The top panel of Figure 5 shows household labor income over the life-cycle, measured
as pre-tax earnings during the working phase and as social security income in retirement.
We plot average income for the bottom 25%, for the middle (25-75%), and for the top 25%
of the distribution, as well as the overall average income. The model’s earnings Gini of
0.54 is close to the 0.47 value in the 2015 NY metro data. Earnings inequality in New York
in the model is lower within zone 1 (Gini of 0.49) than within zone 2 (Gini of 0.54).

Wealth The model makes predictions for average wealth, the distribution of wealth
across households, as well as how that wealth is spatially distributed. Average wealth to
average total income (ytot ) in New York is 6.00. Wealth inequality is high, with a wealth
Gini coefficient of 0.74 in New York and 0.78 in the Outside MSA. They are close to the
data by virtue of the calibration.
The middle panel of Figure 5 shows household wealth over the life cycle for the same
income groups as in the top panel. These moments are not targeted. The graph shows
that the model generates substantial wealth accumulation as well as a large amount of
wealth inequality between income groups. Wealth inequality grows with age during the
working phase.

32
 Figure 5: Income, Wealth, and Home Ownership Over the Life-Cycle

Labor Income
400 Mean
Bottom 25% income
Middle 50% income
200 Top 25% income

0
20 30 40 50 60 70 80 90

3000
Net Worth

2000
1000
0
20 30 40 50 60 70 80 90

1
Homeownership

0.5

0
20 30 40 50 60 70 80 90
Note: The figure plots the model-implied income distribution (top row), wealth distribution (middle row),
and home ownership distribution (bottom row) for he New York metropolitan area. The different lines in
each panel refer to the different income groups.

4.2 Home Ownership, House Prices, and Rents

Next, we discuss the model’s predictions for home ownership, house prices, and rents.
The model manages to drive a large wedge between house prices, rents, and home own-
ership rates between zones 1 and 2 for realistic commuting costs.

33
Home Ownership The model generates a home ownership rate of 58.4% in New York,
fairly close to the 51.5% in the New York data. The home ownership rate in the Outside
MSA is 65.2%, close to the nationwide average over the past 50 years.
The bottom panel of Figure 5 plots the home ownership rate over the life-cycle. It
displays a hump-shape over the life-cycle with variation across income groups. High-
income households become home owners at a younger age than low-income households,
achieve a higher ownership rate, and remain home owners for longer during retirement.
These patterns are broadly consistent with the data.
Row 6 of Table 2 shows that the ratio of the home ownership rate in Manhattan to
zone 2 is 0.42 in the data. The model also generates a much lower home ownership rate
in zone 1 than in zone 2, with a ratio of 0.56.

House Prices and Rents Table 2 shows the median price per housing unit (row 7), the
median price per square foot (row 8), the median rent per unit (row 9), and the median
rent per square foot (row 10).
The model closely matches the price and rent levels in the NY metro. While these
moments are not directly targeted, they follow fairly directly from the calibration which
targets the NY price-rent ratio and the aggregate housing expenditure share. The median
house value in the NY metro area is $510,051 in the data compared to $496649 in the
model.33 The data indicate a monthly rent on a typical market-rate unit of $2,390 per
month in the metro area; the model predicts $2471.
The model understates the ratio of house prices and price-rent ratios in zone 1 to zone
2, mostly because it understates the ratio of price per square foot. The price-rent ratio
in the model is well approximated by the user cost formula (1 − Q × (1 − δ − τ P ))−1 .
Differences in the price-rent ratio between zones must arise from the wedges between
depreciation and property tax rates. These wedges are too small to generate the observed
gap in the price-rent ratio across zones.34 Since the model matches the rent differences
33 To ensure consistency with the empirical procedure, we calculate the median house size in each zone in
the model from both owner- and renter-occupied units but excluding RS units. Call these h̄` . We form the
median price per unit as the product of the market price times the typical unit size P` h̄` . The market rent is
R` h̄` . The price-rent ratio is simply P` h̄` /R` h̄` = P` /R` . To form metro-wide averages, we use the number
housing units in each zone as weights, just like in the data.
34 Several factors outside of the model may help bridge the gap. First, houses in Manhattan may be less

risky than in zone 2 which would increase the price-rent ratio wedge in a richer model with meaningful
risk premia. Second, owner-occupied housing in Manhattan may be of higher quality than in zone 2 in
ways not fully captured by the lower depreciation rate in zone 1 than in zone 2. Third, price-inelastic
out-of-town investors may well be pushing up relative prices since they are disproportionately active in
Manhattan (Favilukis and Van Nieuwerburgh, 2021). Fourth, the higher price of a Manhattan apartment
may partly stem from its value as a shared/part-time rental via platforms such as AirBnB. Fifth allowing

34
between zones well and rents are more directly linked to housing affordability, this is not
a crucial miss for the model.

4.3 Housing Affordability

Price-Income and Rent-Income Row 12 of Table 2 reports the ratio of the median value
of owner-occupied housing to average earnings in each zone. Average earnings are pre-
tax and refer to all working-age residents in a zone, both owners and renters. The median
home price to the average income is an often-used metric of housing affordability. In the
NY metro data, the median owner-occupied house costs 3.99 times average income. Price-
income is 6.7 in Manhattan compared to 3.6 outside Manhattan, a ratio of 1.86. The model
generates a price-income ratio of 4.00 for the MSA, very close to the data. It generates a
ratio across zones of 1.33, understating the ratio for the reason noted above.
Row 13 reports average rent paid by market renters divided by average income of
all residents in a zone; 24% in the data. This moment was the target for the housing
preference parameter αh . To get at the household-level rent burden, we compute two
additional moments reported in rows 14-15 of Table 2, using PUMS-level data from the
American Community Survey. The first statistic computes household-level rent to income
ratio for renters with positive income, caps the ratio at 101%, and takes the average across
households. For this calculation, income is earnings for working-age households and so-
cial security income for retirees. The observed average share of income spent on rent by
renters is 42.1% in the metro area. The model generates an average rent-income ratio for
renters of 33.9%, which is lower than in the data.35 The second statistic computes the frac-
tion of renters with positive income whose rent is over 30% of income. These households
are known as rent-burdened. In the data, 53.9% of households are rent-burdened; in the
model this fraction is 54.7%. The model generates a large “housing affordability crisis,”
with more than half of renters spending more than 30% of their income on rent.

Rent Stabilization By virtue of the calibration, the model generates the right share of
RS households in the population in each zone (row 16 of Table 2).
Figure 6 zooms in on the allocation of RS housing units by age and income. It plots
the fraction of households that are in RS for the bottom 25%, middle 50% and top 25% of
for a zone-specific sensitivity of the housing production function to the distance from the building limit
may help.
35 Within the class of homothetic preferences over housing and non-housing consumption it is difficult to

generate large deviations in the housing expenditure ratio without preference heterogeneity in αh . Never-
theless, the model generates a rent-to-income ratio for renters that is 10% points above the average rent-to-
average income ratio of 24%.

35
 Figure 6: Prevalence of Rent Stabilization

Model Data
0.3 0.3
Mean
Bottom 25% income
0.25 0.25

Share of RS households
Middle 50% income
Share of RS households

Top 25% income

0.2 0.2

0.15 0.15

0.1 0.1

0.05 0.05

0 0
40 60 80 40 60 80
Age Age

Note: The figure plots the share of households in rent stabilized rental housing units out of all housing
units. Age is on the horizontal axis. At each age, we split households into the bottom-25% of income,
the middle 50%, and the top-25%. The results for the model are plotted on the left. The results from the
data are plotted on the right. Since RS status by age and income is only available from the New York City
Housing and Vacancy Survey, the data only pertains to the five counties of New York City rather than to
the full MSA. For the purposes of this graph only, we include rent-controlled units in the numerator of the
RS share. The shares are rescaled to deliver the overall RS share in the entire MSA.

the income distribution at each age. The model is on the left, the data on the right. In both
model and data, high-income households are less likely to be in RS. At young ages, both
low and middle-income households are about equally likely to be in RS in both the model
and the data. The difference in the RS share between high and low-income households is
larger in the model than in the data. In the model, the prevalence of RS is increasing in
age. In the data, it is declining at young ages and flat thereafter. In sum, there appears
to be tremendous misallocation of RS housing units in the data. Even though the model
offers RS housing units to households randomly and without income qualification, and
matches the observed RS rent discounts as well as the persistence in RS tenancy, it still
generates less misallocation than in the data.
Affordable housing acts as an insurance device in our incomplete markets model. We
calculate the probability of getting a RS unit in the current period for a household that
was not in a RS unit in the previous period and that suffered a negative productivity
shock from the second to the first or from the third to the second productivity level. This
probability measures access to the insurance that RS provides for middle- and low-income
households who fall on hard times. If it is difficult for such a household to get into the RS
system, then the value of that insurance is low. The access to insurance metric is 5.4% in

36
the metro area. This breaks down into 0.3% for zone 1 and 5.2% for zone 2 RS housing.
Including low-income households that already were in RS, the likelihood of getting RS
housing is 16.5%.
We also define the stability of insurance as the probability of staying in a RS unit for
a household that was in a RS unit in the previous period and that currently is in the bot-
tom quartile of the income distribution. This probability is 79.9% in the baseline model.
Risk averse households prefer a stable housing situation, i.e., low volatility of changes in
the marginal utility of housing. In a complete market, households can perfectly smooth
consumption and marginal utility ratios are constant over time; their volatility is zero.
Our benchmark model displays severe incompleteness with volatilities of 0.60 for both
the marginal utility growth of non-housing consumption and housing consumption.36

5 Affordability Policies
Having developed a quantitatively plausible dynamic stochastic spatial equilibrium model
of the New York housing market, we now turn to policy counterfactuals. The first set of
policies aims to improve the targeting of the rent stabilization system, the second set of
policies changes the scope of the RS system, while the third set of policies affect the spa-
tial aspects of housing. We highlight some differences with the same policy experiments
conducted in a closed-economy model of the metropolitan area, where there is no migra-
tion to or from other metro areas. The “no migration” results are discussed in detail in
Appendix C.

5.1 Improving the Targeting of RS

As shown above, RS in the benchmark model suffers from misallocation. The first set of
policies we consider aim to improve the allocation of a given amount (square footage) of
affordable housing by better targeting it on the most needy households.

Introducing Income Qualification Requirement The first experiment, reported in col-

umn (1) of Table 3, introduces an income qualification requirement for RS housing. This
requirement is only imposed when households enter RS, not on existing tenants. Specifi-
36 The volatilities of marginal utility growth ignore the risk of being born as a low productivity household.

The housing policies we study below play a role in insuring this risk.

37
cally, we lower κ2 from ∞ to 60% of AMI in zone 1 and to 50% of AMI in zone 2.37 As in
the benchmark model, incumbents can remain in RS housing with exogenous probability
far exceeding the endogenous entry probability for everyone else (p RS,exog >> p RS ).
The policy is successful at allocating affordable housing units to low-income house-
holds. There is a 58.39% increase in the fraction of Q1-income households in RS (row 4)
relative to the benchmark model. This number exceeds the increase in the fraction of all
households in RS of 11.35% (row 3). Because the households in RS are lower-income, they
choose smaller apartment units (rows 6 & 7), and the RS system can accommodate more
households in the same square feet of affordable housing space.
RS becomes a better insurance device. Row 27 reports that lowering the income qual-
ification threshold greatly improves access to insurance for lower-income households who
have fallen on hard times (49.53%). Row 28 reports that income qualification leaves the
stability of insurance nearly unaffected (0.86%). Rows 29 and 30 report the time-series
standard deviation of marginal utility growth of non-housing and housing consumption,
averaged across households. The policy lowers both but mostly the volatility of housing
consumption (-2.68%). By offering households more housing stability, the policy brings
the economy closer to complete markets.
Income targeting increases the share of households in the urban core (3.76%, row 10).
Average income in zone 1 falls (-3.41%, row 20) since many of the new residents in zone
1 are lower-income households living in RS units. Many of those are of working age (the
share of retirees in zone 1 falls; -5.51%, row 11).
Overall, this policy generates a modest welfare gain averaged across households that
live in NY in the period prior to the reform (W g = 0.39%, row 31). Figure 7 shows how
welfare changes are distributed across age, productivity, income, and wealth groups. The
policy benefits lower-productivity, low-income, and low-wealth households at the ex-
pense of the other groups.
Several factors work to offset the benefits from better targeting. First and foremost,
a fraction 1 − p RS,exog of existing RS tenants lose the right to stay, and are now subject
to (i) the luck of the draw, and (ii) income qualification. Even if they qualify and regain
access, their RS tenure clock restarts. They now receive a smaller rent discount (since
the rent discount κ1 (d) is rising in tenure d). In other words, some–possibly less needy–
37 For the purposes of income qualification, household income is defined as total pre-tax household in-
come, including labor income, pension income, and financial income. AMI is the area median income
among all households in the entire metropolitan area, including retirees. 60% of AMI is a common income
threshold in affordable housing policy. Setting the income cutoff to 50% of AMI in both zones would lead
to excess supply of RS housing in zone 1 and lack of market clearing. Intuitively, RS housing in zone 1 is
still relatively expensive for low-income households.

38
long-tenured tenants with large rent discounts are replaced by–more needy–tenants with
small rent discounts. The welfare losses from the former offset the welfare gains from
the latter group. Second, income qualification leads some to reduce labor supply in order
to qualify. This effect is masked because of net in-migration (row 32) and more clearly
visible in the no migration model. Third, by exempting most existing tenants from in-
come qualification, there is little churn in the RS program and limited scope for income
qualification to reduce misallocation. Fourth, changes in migration result in lower output
(-0.42%, row 25), which hurts welfare. Fifth, the maximum house size cutoff for RS units
already reduces misallocation by making RS unattractive for high-income households.38
The policy affects inter-city migration in two ways. First, it results in a larger popu-
lation for the NY MSA ( 2.17%). This increases aggregate labor supply (row 23). Second,
the composition of the population changes. There are fewer low-productivity and more
higher-productivity households moving out than in the benchmark model; see left panel
of Figure 8. Total hours worked in efficiency units declines (row 24), despite the large
increase in total hours. Efficiency gains in the housing market from better targeting result
in efficiency losses in the labor market.

Re-applying Each Period In the second policy experiment, we introduce income quali-
fication at 60% of AMI in each zone but force each RS tenant to go through income testing
each period (four years).39 By setting the parameter p RS,exog = p RS , the endogenously
determined probability of winning the RS lottery p RS is now the same for everyone and
increases substantially. The removes the persistent misallocation of tenants to RS units.
The results in column (2) of Table 3 show that access to insurance improves dramatically
(120.61%). The fraction of low-income households in RS grows strongly (59.82%), show-
ing the improved targeting of the RS system. The overall fraction of households in RS
also increases (19.23%), as poorer households choose much smaller RS unit sizes.
However, the policy experiment results in an aggregate welfare loss ( -0.22%). First,
the policy dramatically lowers the stability of RS insurance (-79.66%, row 28). Housing
consumption becomes more volatile (+4.79%, row 30).
Second, the higher churn of RS residents lowers the average tenure and thereby the
average rent discount that RS tenants enjoy. The policy makes the average RS housing
38 In unreported results, we find that introducing an income cutoff leads to larger welfare gains if there is
no RS size cutoff.
39 Since RS housing becomes less attractive in this experiment, the RS housing market in zone 2 does not

clear at a 50% of AMI income cutoff. Hence, for this experiment, we set the income limit at 60% of AMI in
both zones. As in the previous experiments, these cutoffs reflect the point at which RS housing is in excess
supply.

39
 Table 3: Better Targeting of Rent Stabilization–Main Moments

(1) (2) (3) (4) (5)

Benchm. Inc Qual New Inc Qual All Inc Qual Stay 0.50 RS discount RS size
1 Avg(rent/inc.) renters in Z1 (%) 39.5 -0.49% 12.12% 14.00% -4.16% -0.48%
2 Avg(rent/inc.) renters in Z2 (%) 32.8 5.54% 11.92% 11.24% 4.90% -0.07%
3 Frac. of HHs in RS (%) 14.20 11.35% 19.23% 29.23% 7.05% 7.69%
4 Frac. in RS of those in inc. Q1 (%) 16.48 58.39% 59.82% 105.06% 19.36% 20.63%
5 Frac. rent-burdened (%) 54.7 12.10% 18.95% 22.28% 0.06% 2.02%
6 Avg. size of RS unit in Z1 (sf) 869 -16.89% -43.31% -48.65% -11.59% -5.77%
7 Avg. size of RS unit in Z2 (sf) 862 -17.22% -22.27% -38.04% -8.35% -10.37%
8 Avg. size of a Z1 mkt unit (sf) 1156 -0.21% -2.33% -2.52% 1.25% 0.48%
9 Avg. size of a Z2 mkt unit (sf) 1824 0.42% 1.54% -2.61% 3.47% -0.12%
10 Frac. of pop. living in Z1 (%) 10.5 3.76% 14.91% 10.61% 5.61% 0.48%
11 Frac. of retirees living in Z1 (%) 17.5 -5.51% 92.44% 114.80% 8.56% 3.33%
12 Housing stock in Z1 – -0.42% 0.46% 0.37% 0.40% -0.01%
13 Housing stock in Z2 – 0.23% 0.83% 1.17% 0.28% -0.13%
14 Rent/sf Z1 ($) 3.55 2.07% 1.57% 6.94% -1.61% 1.08%
15 Rent/sf Z2 ($) 1.28 2.25% 1.55% 7.48% -1.99% 1.13%
16 Price/sf Z1 ($) 884 2.16% 1.62% 6.82% -1.64% 1.10%
17 Price/sf Z2 ($) 249 2.33% 1.72% 7.65% -2.01% 1.15%
18 Homeownership rate in Z1 (%) 34.8 -0.90% -7.77% -6.26% -4.96% 0.52%
19 Homeownership rate in Z2 (%) 61.1 0.03% 0.35% -2.62% 0.68% -1.50%
20 Avg. inc. Z1 working-age HHs ($) 167840 -3.41% -20.34% -17.02% -2.61% 0.35%
21 Avg. inc. Z2 working-age HHs ($) 99755 0.10% 0.46% -0.64% 0.18% -0.22%
22 Frac. of top-prod. HHs in Z1 (%) 21.6 1.35% -0.61% 4.73% -2.43% 1.11%
23 Total hours worked – 1.88% 1.27% 6.69% -1.28% 0.89%
24 Total hours worked in effic. units – -0.67% -1.12% -0.99% -0.06% -0.17%
25 Total output – -0.42% -0.69% -0.63% -0.06% -0.11%
26 Total commuting time – 1.28% 2.12% 9.18% -1.97% 0.94%
26 Developer profits – 0.25% 2.52% 3.10% 0.88% -0.06%
27 Access to RS insurance (%) 4.1 49.53% 120.61% 94.73% 7.06% 12.43%
28 Stability of RS insurance (%) 79.9 0.86% -79.66% 2.03% -0.01% 0.31%
29 Std. MU growth, nondurables 0.60 -1.25% 3.94% 2.80% -0.51% 0.40%
30 Std. MU growth, housing 0.60 -2.68% 4.79% 3.51% 0.20% 0.42%
31 Aggr. welfare change (NY pop) – 0.39% -0.22% 0.66% -0.43% 0.18%
32 NY population 7124.9 2.17% 3.73% 9.90% -1.68% 1.09%
33 Aggr. welfare change (no migr.) – 0.04% -0.78% 0.32% -0.96% 0.16%

Notes: Column “Benchmark” reports values of the moments for the baseline model.

unit less affordable.

Third, the policy results in a larger NY MSA population ( 3.73%) since the likelihood
of obtaining RS is higher now that the preference for incumbents is gone. As in the first
policy experiment, the additional immigrants tend to be of lower average productivity
than residents in the previous period. This lowers aggregate labor supply in efficiency
units, and aggregate output falls by -0.69%. Figure 8 shows lower out-migration rates for
low-income households, as well as high net in-migration for households in the second
productivity group (the middle 50% of the productivity distribution).
The (endogenously) lower average rent discount in the RS system reduces distortions
for developers. It results in higher developer profits (row 26) and higher housing supply.

40
But since housing demand also rises–due to less misallocation in the RS system,–both
housing stock and rents go up in equilibrium in both zones.

Figure 7: Better Targeting of Rent Stabilization–Welfare Heterogeneity

Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

3 1.5
Inc Qual New
2 Inc Qual All
1 Inc Qual Stay
0.50 RS discount
1 RS size
0.5
0
0
-1

-0.5
-2

-3 -1
40 60 80 1 2 3 4
Age Productivity Level
Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

1.5 1

1
0.5

0.5
0
0

-0.5
-0.5

-1 -1
1 2 3 4 1 2 3 4
Income Quartile Net Worth Quartile
Notes: The baseline model has the following parameters: η 1 = 56.37, η 2 = 29.72, κ1 = 7%, κ2 = 1000.00, κ3 = 0.50. Policy experiments,
each panel: Top left panel: by age. Top right panel: by productivity level. Bottom left panel: by income quartile. Bottom right panel:
by net worth quartile. The welfare changes are measured as consumption equivalent variations for an average household in each
group.

Restoring the Preference for Insiders In the third experiment, we impose income qual-
ification in each period on all tenants (60% of AMI in zone 1 and 50% of AMI in zone 2),
but we allow existing tenants who qualify to remain in RS with high probability; p RS,exog
is set to its baseline value. Compared to the baseline model, the experiment in Column (3)
of Table 3 imposes an income cutoff on all RS residents in every period. Compared to the
experiment in Column (1), the income cutoff now applies also to existing RS tenants. This
experiment generates a welfare gain of 0.66%. It provides the fairness of income qual-
ification while avoiding excessive churn in the RS system. The targeting of RS units to
low income households improves further (105.06%, row 4). Access to insurance is much
higher than in the baseline (94.73%), but now comes without loss in the stability of that
insurance (2.03%). The welfare gains are larger than in column (1) since now income qual-
ification is applied to existing tenants, reducing the misallocation that builds up over time
as tenant income grows. The reform replaces high-income insiders with low-income out-

41
siders. Like in the previous experiment, the fraction of all households in RS increases. In
sum, the combination policy with income targeting and preference for insiders produces
non-trivial welfare gains, yet requires no expansion of the scope of the RS program nor
additional taxes.
Figure 7 shows that most of the benefits from this experiment flow to 50-80 year-olds.
The policy redistributes from high-income and net-worth to low-income and low-net
worth households.
Rent-income ratios among renters increase in zone 1 (14.00%, row 1) and in zone 2
(11.24%, row 2). These changes reflect the new socio-economic make-up of the two zones.
The fraction of rent-burdened households increases (22.28% , row 5). This suggests that
rent-income ratios and rent burden, the most common metrics of housing affordability,
must be interpreted carefully as they reflect equilibrium rents and the income of the peo-
ple who have sorted into each area in spatial equilibrium.
The spatial allocation of labor productivity worsens. The policy results in a much
larger share of retirees, a smaller share of top-productivity households, and lower average
income in zone 1. There is more “income mixing” in the urban core.
The policy attracts significant net in-migration ( 9.90%). While the chances of gaining
access to RS conditional on receiving a negative income shock are not as high as in the
previous experiment, low-income households have a higher unconditional probability of
ending up in a RS unit. Also, the value of the RS insurance is now higher due to the pref-
erence for insiders. For both reasons, there is much less out-migration of poor households
(Q1). Out-migration offers a way for some of those who are adversely affected to escape
the adverse consequences of the policy. Indeed, Figure 8 shows more out-migration in
productivity group 3. The migration option ends up raising the welfare gains of this pol-
icy for the NY population (row 33 versus 31). The policy triggers less in-migration, which
can be understood from the much higher cost of living in NY after the reform.

Varying the RS Discount Column (4) of Table 3 reports on an experiment that changes
the size of the rent discount for RS units relative to the market rent. We multiply the
entire discount schedule κ1 (d) by a factor of 0.5. Reducing the discount by half results
in a large welfare loss ( -0.43%). Figure 7 shows that a smaller discount hurts older and
low-productivity households the most. The age effect arises because the discounts are
increasing in tenure and the effects on rent are therefore largest in absolute magnitude for
the old.
There are interesting equilibrium effects on targeting. Making RS less generous makes
it less attractive, which reduces competition for it in the absence of income qualification,

42
 Figure 8: Better Targeting of Rent Stabilization–Migration
20 40

Out-migration Relative to Benchmark (%)

In-migration Relative to Benchmark (%)

10 30

0 20

-10 10

-20 0

-30 -10
Inc Qual New
Inc Qual All
-40 Inc Qual Stay -20
0.50 RS discount
RS size
-50 -30
1 2 3 4 1 2 3 4
Productivity Level Productivity Level
Notes: The left (right) panel reports the percentage difference in out-migration (in-migration) rates between the experiment and the
benchmark model for each productivity group.

and results in more bottom-quartile households ending up in it (19.36%). Access to insur-

ance improves, while the stability of that insurance does not deteriorate.
The policy reduces distortions to development, traditionally emphasized in the rent
regulation literature. The reform results in higher developer profits, a larger housing
stock, and lower rents. Finally, labor supply and output are nearly unchanged despite a
reduction in the NY population. Out-migration is high for second-quartile productivity
households, relative to the benchmark economy, since many of these households had
valuable RS housing prior to the reform but lost that benefit. Lower rents and house
prices attract more high-productivity households to the NY MSA. In sum, the policy that
reduces the subsidies to RS housing has several benefits, but its average welfare effect is
negative.

Maximum Size of Rent Stabilized Unit Because it is subsidized and not income-tested,
some higher-income residents in RS may over-consume housing. Column (5) of Table 3
studies a reduction in the maximum size of a RS housing unit by 10% compared to the
baseline. The average size of RS units falls by less than 10% in zone 1 and by slightly more
than 10% in zone 2. The average welfare effect is a gain of 0.18%, and is concentrated
on low-income households. Making RS units smaller is a reasonably effective way of
targeting RS units to low-income households, simply because higher-income households
do not want to live in small housing units. Because units are now smaller but the same

43
total square feet are devoted to RS, more households qualify. For both reasons, access to
insurance improves. Since low-income households benefit, they are less likely to move
out of the metro area.

5.2 Varying the Scope of the Rent Stabilization Mandate

The second policy instrument we study is the scope of the affordable housing mandate.
We symmetrically change η ` in each zone, the fraction of rental square footage that must
be set aside for affordable units, varying it from 0.25× to 1.5× its benchmark value. Table
4 reports the results. Average welfare rises monotonically in the scope of the RS mandate.
Higher welfare is associated with higher fractions of households and of bottom-income
quintile households in RS. Both access to RS insurance and the stability of that insurance
increase in the share of rentals that are rent stabilized. Figure 9 plots average welfare as a
function of η ` and confirms the increasing pattern. At 1.5×, all households who want RS
have access to it, so that further increases would lead to excess supply and lack of market
clearing in the RS segment.

Figure 9: Varying the Scope of the RS Mandate–Aggregate Welfare

1

0.8
Welfare Relative to Benchmark (%)

0.6

0.4

0.2

-0.2

-0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7
RS Share of Rented Sqft
Notes: The baseline model has the following parameters: η 1 = 56.37, η 2 = 29.72, κ1
= 7%, κ2 = 1000.00, κ3 = 0.50. The share of
RS rented sf is calculated as a (total sf-)weighted average of η 1 , η 2 . The larger dots on
the right-hand side of the graph represent the
maximum RS share above which markets do not clear. The welfare changes are measured as consumption equivalent variations for
an average household.

Increasing RS is not without cost. It induces increasing distortions to housing and

labor markets, both in terms of supply and spatial misallocation. We discuss these in
turn.

44
 An increase in the RS mandate weakens developers’ incentives to build since it lowers
the average sale price they earn on new housing units and hence their profits. The over-
all housing stock decreases. This is consistent with the traditional view on rent regula-
tion and with the empirical literature, which finds that increased incentives of landlords
to renovate their properties and of developers to invest in new construction generate a
modest housing boom in decontrolled areas (Autor et al., 2014; Diamond et al., 2019).
The lower housing stock results in higher market rents (12.03% in zone 1 and 3.21%
in zone 2 in column 5). Developers “pass through” the increased housing affordability
targets into the market rent. The fraction of rent-burdened renters increases by 3.97%
metro-wide. The increased rent burden reflects the fact that low- and middle-income
households who do not win the RS lottery face higher rents. Since many more households
now enjoy the lower rents that come with RS, the rent burden metric must be interpreted
cautiously.
A proportional increase in RS in both zones shifts the population towards the urban
core (14.09% in column 5). The home ownership rate in zone 1 is much lower as there are
now many more RS tenants in zone 1. Zone 1 residents choose smaller housing units both
in the market and RS rental segments, increasing density in the urban core. The different
population mix in zone 1 is reflected in a much lower average income, much fewer top-
productivity households, and more retirees. In other words, the spatial allocation of labor
productivity deteriorates. The reallocation of the housing stock towards affordable units
pushes some middle- and upper-middle-income households out of the urban core. The
socio-economic make-up of the urban core is more diverse. This pattern is consistent with
the empirical evidence in Autor et al. (2014), who show that richer households moved into
units previously occupied by poorer RS tenants after a reduction in control in Cambridge,
MA.
There is a substantial reduction in individual labor supply by those in RS who face
lower rent. The resulting reduction in aggregate labor supply can be seen clearly in the
model without migration but is fully offset by the increase in labor supply due to the
rising NY population in the model with migration. Effective labor supply and output are
unchanged.
In summary, the model generates the well-known distortions in labor supply and
housing associated with expansions of RS. But these costs are outweighed by the benefits
to the RS recipients. RS policy completes markets by reducing the volatility of marginal
utility growth of housing and non-housing consumption (rows 29 and 30).
Figure 10 studies the heterogeneity in welfare changes. Expanding rent stabilization
benefits younger and especially low-productivity and low-income households the most.

45
 Table 4: Varying the Scope of the RS Mandate–Main Moments

(1) (2) (3) (4) (5)

Benchm. 0.25× 0.50× 0.75× 1.25× 1.50×
1 Avg(rent/inc.) renters in Z1 (%) 39.5 -1.91% -0.93% 0.49% 1.85% -0.58%
2 Avg(rent/inc.) renters in Z2 (%) 32.8 4.74% 3.24% 1.58% -1.36% -2.58%
3 Frac. of HHs in RS (%) 14.20 -160.70% -86.22% -37.97% 32.40% 64.47%
4 Frac. in RC of those in inc. Q1 (%) 16.48 -155.46% -81.30% -35.17% 32.13% 58.72%
5 Frac. rent-burdened (%) 54.7 -1.32% -0.88% -0.89% 2.04% 3.97%
6 Avg. size of RC unit in Z1 (sf) 869 1.45% 1.09% 0.40% -0.68% -3.82%
7 Avg. size of RC unit in Z2 (sf) 862 0.21% 0.28% 0.19% -0.70% -1.27%
8 Avg. size of a Z1 mkt unit (sf) 1156 -3.63% -2.81% -1.25% 2.62% -6.54%
9 Avg. size of a Z2 mkt unit (sf) 1824 -1.18% -0.95% -0.67% 1.05% 5.50%
10 Frac. of pop. living in Z1 (%) 10.5 -0.35% -0.54% -0.64% 2.14% 14.09%
11 Frac. of retirees living in Z1 (%) 17.5 -39.20% -29.92% -15.81% 16.87% 22.88%
12 Housing stock in Z1 – 0.65% 0.49% 0.46% 0.21% 0.36%
13 Housing stock in Z2 – 0.39% 0.25% 0.04% -0.29% -0.25%
14 Rent/sf Z1 ($) 3.55 -5.81% -4.50% -2.34% 3.10% 12.03%
15 Rent/sf Z2 ($) 1.28 -4.01% -3.09% -1.40% 1.85% 3.21%
16 Price/sf Z1 ($) 884 -5.69% -4.37% -2.28% 3.04% 11.28%
17 Price/sf Z2 ($) 249 -4.00% -3.05% -1.42% 1.88% 3.17%
18 Homeownership rate Z1 (%) 34.8 43.06% 35.61% 23.31% -30.06% -116.32%
19 Homeownership rate Z2 (%) 61.1 9.80% 6.84% 3.65% -5.11% -10.75%
20 Avg. inc. Z1 working-age HHs ($) 167840 17.76% 14.49% 9.56% -13.87% -42.47%
21 Avg. inc. Z2 working-age HHs ($) 99755 -3.54% -2.77% -1.62% 2.40% 6.69%
22 Frac. of top-prod HHs in Z1 (%) 21.6 22.31% 18.54% 12.53% -17.68% -82.31%
23 Total hours worked – -2.72% -1.91% -1.11% 1.12% 1.58%
24 Total hours worked in effic. units – 0.08% 0.12% -0.01% -0.16% 0.02%
25 Total output – 0.03% 0.06% -0.02% -0.09% 0.03%
26 Total commuting time – -3.69% -2.67% -1.42% 1.29% 0.49%
26 Developer profits – 1.29% 0.90% 0.51% -0.68% -0.55%
27 Access to RC insurance (%) 4.1 -171.61% -89.22% -40.93% 37.33% 68.83%
28 Stability of RC insurance (%) 79.9 -10.75% -1.03% -0.16% 0.44% 0.21%
29 Std. MU growth, nondurables 0.60 2.14% 1.03% 0.92% -0.46% -1.21%
30 Std. MU growth, housing 0.60 1.38% 0.84% 0.95% -1.18% -4.14%
31 Aggr. welfare change (NY pop) – -0.40% -0.37% -0.31% 0.40% 0.91%
32 NY population – -3.46% -2.54% -1.28% 1.29% 1.89%
33 Aggr. welfare change (no migr.) – -0.98% -0.90% -0.68% 0.53% 1.36%

Notes: Column “Benchmark” reports values of the moments for the baseline model. The other columns report log differences between
the moments in the policy experiment and in the baseline. The different columns scale down or up the share of square feet that
developers must set aside for RS housing.

Interestingly, it also benefits high-income and high-wealth households. The reason is that
RS now directly benefits several higher-income households given that there is no income
or wealth qualification. Also, the increase in house prices benefits home owners who tend
to be the wealthy. This benefit is partially offset by lower values for investment homes
(due to the growing gap between primary and investment housing κ4 ), and by lower
developer profits (which disproportionately flow to the high-wealth households). Arnott
(1995) argues that developers exert market power and that rent regulation is a way to
limit this market power. Our model has a flavor of this in that developers make profit,
which mostly flows to wealthy households, and more rent regulation reduces that profit.

46
Welfare Relative to Benchmark (%) Figure 10: Varying the scope of RS – Welfare Heterogeneity

Welfare Relative to Benchmark (%)

1.5 1.5
0.25 RS share
0.50 RS share
0.75 RS share
1 1.25 RS share
1 1.50 RS share

0.5
0.5
0

0
-0.5

-1 -0.5
40 60 80 1 2 3 4
Age Productivity Level
Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

1
1

0.5
0.5

0
0

-0.5 -0.5
1 2 3 4 1 2 3 4
Income Quartile Net Worth Quartile
Notes: The baseline model has the following parameters: η 1 = 56.37, η 2 = 29.72, κ1 = 7%, κ2 = 1000.00, κ3 = 0.50. Policy experiments,
each panel: Top left panel: by age. Top right panel: by productivity level. Bottom left panel: by income quartile. Bottom right panel:
by net worth quartile. The welfare changes are measured as consumption equivalent variations for an average household in each
group.

Higher prices and rents in turn dissuade in-migration, as can be seen in the right
panel of Figure 11. The NY population rises solely because of reduced out-migration for
bottom-half productivity households. Migration provides an extra margin of adjustment
and dampens the welfare effects from either reducing or increasing the scope of the RS
mandate (compare rows 31 and 33 of Table 4).

5.3 Geographic Location of Affordable Housing

Columns (1) and (2) of Table 5 conduct two policy experiments that shift all RS housing
from zone 1 to zone 2. The experiment in Column (2) is coupled with subsidized public
transit for RS tenants. The aggregate transit subsidy is $800 million and the subsidy is the
same for all recipients at about $800 per year. It is akin to a discounted metro/rail pass for
RS tenants. The subsidy is paid for by higher income taxes on all NY residents, engineered
through a lower λ. By construction, the experiment keeps the number of households in
RS constant.
In both experiments, the urban core gentrifies with more high-income and top-productivity

47
 Figure 11: Varying the scope of RS – Migration
30 25

Out-migration Relative to Benchmark (%)

In-migration Relative to Benchmark (%)

25 20

20
15
15
10
10
5
5
0
0
0.25 RS share -5
-5 0.50 RS share
0.75 RS share
-10 1.25 RS share -10
1.50 RS share
-15 -15
1 2 3 4 1 2 3 4
Productivity Level Productivity Level
Notes: The left (right) panel reports the percentage difference in out-migration (in-migration) rates between the experiment and the
benchmark model for each productivity group.

households and a much higher home ownership rate. There are also a lot fewer retirees
in zone 1. Moving affordable housing to the suburbs improves the spatial allocation of
labor.
The reform eliminates developer distortions in zone 1, which results in a higher hous-
ing stock in zone 1 and lower market rents. However, rents are still much higher in zone
1 than in zone 2. The urban core actually loses population share and sees reduced density.
Developer distortions increase in zone 2, lowering the housing stock. Market rents
rise due to higher aggregate demand for rentals in zone 2. Rent-income ratios fall in both
zones, and the share of rent-burdened households falls.
There is an aggregate welfare gain of 0.25% in Column (1) and 0.18% in column (2). A
consequence of the reform is that the targeting of RS improves; the fraction of Q1-income
households in RS increases substantially. Figure 12 shows that this policy benefits low-
income households the most. Their benefit is even larger with subsidized transit. There
is a welfare loss for high-income residents who shoulder most of the cost of the transit
subsidy, which explains why the welfare gain is not larger in the experiment with than
the one without transit subsidy.
The welfare effects are dampened relative to the no-migration model (row 33), where
the benefit of the policy with transit subsidy far exceeds that without. Tax rates must
rise much more in the model with migration than in the model without to raise the $800
tax revenue for the transit subsidy. The policy increases the population of NY (row 32).

48
Since the policy benefits lower income households, there is reduced out-migration for this
group. Figure 13 also shows higher in-migration for agents in the second productivity
group and lower in-migration rates for households in the top-25% of the productivity
distribution.

Table 5: Spatial Housing Policies–Main Moments

(1) (2) (3) (4) (5) (6) (7)

Benchm. All RS All RS Zoning Vouchers LIHTC Cash Cash
in Z2 in Z2 + transit transfer V transfer P
1 Avg(rent/inc.) renters in Z1 (%) 39.5 -4.74% -5.13% 0.28% -0.36% 0.34% -0.41% -1.39%
2 Avg(rent/inc.) renters in Z2 (%) 32.8 -0.87% -0.67% -0.09% 0.63% -0.26% 0.79% 4.10%
3 Fraction of HHs in RS (%) 14.20 – – 0.14% 4.44% 0.93% 3.21% 5.11%
4 Frac. in RS of those in inc. Q1 (%) 16.48 22.43% 21.23% 0.16% 6.18% 2.21% 4.52% 5.98%
5 Frac. rent-burdened (%) 54.7 -3.96% -0.92% -0.18% 3.88% 0.18% 4.08% 4.12%
6 Avg. size of RS unit in Z1 (sf) 869 – – 0.19% -0.17% -0.01% -0.26% -0.29%
7 Avg. size of RS unit in Z2 (sf) 862 -0.13% 0.09% 0.02% -1.76% -0.27% -1.60% -1.86%
8 Avg. size of a Z1 mkt unit (sf) 1156 -3.30% -5.17% 0.40% -1.13% 0.08% -1.35% -0.77%
9 Avg. size of a Z2 mkt unit (sf) 1824 -2.05% -4.70% -1.04% -4.61% 0.08% -5.14% -4.99%
10 Frac. of pop. living in Z1 (%) 10.5 -8.43% -7.67% 6.95% -1.95% 0.46% -1.95% -3.01%
11 Frac. of retirees living in Z1 (%) 17.5 -34.48% -21.96% 4.14% 3.13% 1.30% 2.91% 0.65%
12 Housing stock in Z1 – 0.62% 0.19% 8.80% -0.50% 0.58% -0.44% -0.20%
13 Housing stock in Z2 – -0.65% -2.41% -0.62% -2.69% 0.12% -2.74% -1.86%
14 Rent/sf Z1 ($) 3.55 -1.55% -1.32% -0.29% 1.79% -0.36% 2.01% 2.38%
15 Rent/sf Z2 ($) 1.28 1.45% 1.45% 0.24% 1.80% -0.45% 1.97% 2.44%
16 Price/sf Z1 ($) 884 -1.38% -1.06% -0.27% 1.72% -0.37% 1.89% 2.41%
17 Price/sf Z2 ($) 249 1.48% 1.60% 0.23% 1.69% -0.43% 1.87% 2.45%
18 Homeownership rate in Z1 (%) 34.8 54.26% 54.10% 8.30% 4.26% -0.08% 4.07% 5.09%
19 Homeownership rate in Z2 (%) 61.1 -6.30% -9.43% -0.64% -6.13% -0.45% -4.98% -8.03%
20 Avg. inc. Z1 working-age HHs ($) 167840 29.20% 28.37% 2.70% 4.60% -0.11% 4.54% 5.83%
21 Avg. inc. Z2 working-age HHs ($) 99755 -6.39% -7.75% -1.31% -2.09% -0.20% -2.37% -2.26%
22 Frac. of top-prod HHs in Z1 (%) 21.6 32.35% 33.58% 10.10% 5.85% 0.56% 6.26% 6.07%
23 Total hours worked – 1.69% 0.62% 0.76% 0.56% -0.11% 0.53% 1.56%
24 Total hours worked in effic. units – -0.06% -2.77% 0.28% -3.48% -0.07% -3.68% -2.69%
25 Total output – -0.08% -1.83% 0.16% -2.24% -0.06% -2.37% -1.73%
26 Total commuting time – 2.36% 1.81% 0.05% 1.10% -0.05% 1.25% 2.28%
26 Developer profits – -0.24% -2.74% 0.21% -3.37% 0.98% -3.69% -2.78%
27 Access to RS insurance (%) 4.1 13.55% 13.54% 1.24% 2.42% 1.14% 3.80% 8.28%
28 Stability of RS insurance (%) 79.9 0.90% 1.23% -0.10% 0.62% 0.21% 0.55% -0.06%
29 Std. MU growth, nondurables 0.60 3.42% 6.45% 0.78% 2.73% 0.71% 3.72% 0.41%
30 Std. MU growth, housing 0.60 1.42% 3.75% 1.13% 1.45% 0.61% 2.08% -1.09%
31 Aggr. welfare change (NY pop) – 0.25% 0.18% 0.11% -0.00% 0.01% -0.02% 3.90%
32 NY population 7124.9 2.66% 3.32% 1.18% 2.12% 0.06% 2.37% 3.06%
33 Aggr. welfare change (no migr.) – 0.37% 0.96% 0.40% 0.53% 0.05% 0.56% 4.66%

Notes: Column “Benchmark” reports values of the moments for the baseline model.

5.4 Upzoning the Urban Core

The next experiment studies a zoning change that allows for more housing in the city
center in an effort to capture the agglomeration benefits associated with higher density in

49
the urban core (A1 > 1). We think of this policy as relaxing height (increasing allowable
1
floor-area ratios) or other land use restrictions. We increase H by 10%. The equilibrium
housing stock in Manhattan increases by 8.80%, as shown in Column (3) of Table 5. Since
a fixed fraction of square feet must still be set aside for RS units, the expansion in the
housing stock also creates more affordable units in zone 1. This is akin to a mandatory
inclusionary housing policy.
Because of the increased housing supply, rents (-0.29%) and prices (-0.27%) in zone
1 fall, making housing in the urban core more affordable. The reform brings more top-
productivity households to zone 1 (10.10%). The average income and home ownership
rate rise in the core. In zone 2, the housing stock falls (-0.62%) and rents rise (0.24%) as
developers shift their activity towards zone 1.
Upzoning is welfare increasing with a modest average benefit of 0.11%. As can be
seen in Figure 12, the upzoning policy brings positive benefits to all age, productivity,
income, and wealth groups, unlike most of the other policies which disproportionately
benefit the poor. Real-world resistance to upzoning (“NIMBYism”) can be understood
in the model by observing the lower equilibrium house prices in zone 1. They represent
capital losses to existing homeowners.
Zoning reform increases the NY population ( 1.18%). Figure 13 shows a reduction in
out-migration rates that is more pronounced for agents in the 25th–87.5th percentiles of
the productivity distribution.

5.5 Housing Vouchers

An important pillar of U.S. affordable housing policy is the Section 8 voucher program,
housing assistance provided by the federal government to low-income households. We
consider a policy that spends $800 million on vouchers in the New York metro, the same
amount as was spent on the free transit experiment above and the tax credit experiment
below. In the model and in the data, vouchers are allocated by lottery to households who
make less than 50% of AMI. The voucher amount is set to $8,300, the observed amount in
the data.40 Households who win the voucher lottery and accept must spend the voucher
amount plus 20% of household income on housing. They can choose to turn down the
voucher if they do not wish to abide by this housing constraint.41 The experiment pays for
40 Data compiled from the Housing and Urban Development department show that the housing author-
ities responsible for the 25 counties in the New York MSA disbursed $2.06 billion in 246,000 Section 8
vouchers in the year 2013 (latest available). This amounts to an average of $8,300 per year per voucher.
41 In the data, the housing expenditure constraint is the voucher amount plus 30% of household income.

Lower-income households may choose a lower voucher amount to alleviate the constraint. To prevent the

50
 Figure 12: Spatial Housing Policies–Welfare Heterogeneity
2 0.8
Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

All RS in Z2
0.6 All RS in Z2+transit
1.5 Zoning
0.4 Vouchers
LIHTC
1 0.2 Cash transfer V

0
0.5 -0.2

-0.4
0
-0.6

-0.5 -0.8
40 60 80 1 2 3 4
Age Productivity Level

0.6 0.4
Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

0.4
0.2
0.2

0 0

-0.2 -0.2
-0.4
-0.4
-0.6

-0.8 -0.6
1 2 3 4 1 2 3 4
Income Quartile Net Worth Quartile
Notes: The baseline model has the following parameters: η 1 = 56.37, η 2 = 29.72, κ1 = 7%, κ2 = 1000.00, κ3 = 0.50. Policy experiments,
each panel: Top left panel: by age. Top right panel: by productivity level. Bottom left panel: by income quartile. Bottom right panel:
by net worth quartile. The welfare changes are measured as consumption equivalent variations for an average household in each
group.

96,385 vouchers, or about 1.35% of NY households in the baseline model. The program is
paid for by higher labor income taxes, engineered by a decline in λ. The voucher lottery
is independent from the RS lottery in the model. As in the real world, the voucher can be
used to pay for rent-stabilized housing.
Column (4) of Table 5 shows a zero welfare change for the voucher program ( -0.00%).
Figure 12 shows that older, low-productivity, low-income, and low-wealth households
gain substantially from the policy at the expense of middle-class and rich households.
Several effects work to offset the large welfare benefits to the poor. Chief among them
are labor supply distortions. First, since vouchers are income-tested, they affect the labor
supply of the recipients. Second, in the model as in data, vouchers are paid for with dis-
tortionary labor income taxes, reducing labor supply among non-recipients as well. Total
hours worked in efficiency units fall sharply (-3.48%), as does output (-2.24%), despite
NY MSA population growth. Lower out-migration of low-productivity households and
higher out-migration of high-productivity households reduces average productivity in
constraint from being unrealistically tight in our model where the voucher amount is fixed, we reduce the
own contribution to 20% of household income.

51
NY, resulting in a lower income tax base, and a higher equilibrium tax increase needed
to pay for the voucher program. The same Laffer-curve phenomenon was at work in the
subsidized transit experiment. In the no-migration economy, the tax changes needed to
pay for the vouchers are not nearly as pronounced. Output does not fall and the wel-
fare gain from the voucher policy is substantially larger ( 0.53%) than in the migration
economy.
Further limiting the welfare benefit of housing vouchers is the fact that the voucher
amount is not contingent on income, conditional on qualifying (income below 50% of
AMI). Because many households qualify, the chances of winning the voucher lottery are
slim, reducing its insurance benefit. Finally, households must re-apply for the voucher
each period, resulting in housing instability.
The voucher expansion, which is location-neutral in its design, has interesting spatial
equilibrium effects. The policy leads to a reduction in the population share of zone 1 (-
1.95%) and an increase in commuting (1.10%). It increases the average income of zone
1 (4.60%), the fraction of top-productivity households who live there (5.85%), and the
homeownership rate. In other words, the urban core gentrifies. In equilibrium, low-
income households are not more likely to live in zone 1, where they can take advantage of
the agglomeration effects on current and future labor income (A1 > 1). This is consistent
with the empirical evidence in Collinson and Ganong (2018) that vouchers do not “move”
lower-income households “to opportunity.” Rather, lower-income households end up liv-
ing in the neighborhoods were they were already living prior to the voucher expansion. In
fact, some middle-income households leave the urban core, whose size shrinks, and even
the metro area. The voucher program “removes them from opportunity.” This experiment
underscores the importance of studying housing vouchers in general equilibrium.
Finally, a change to one housing affordability program may affect the benefits from
other programs. Expanding vouchers results in more (low-income) households in RS
housing, each occupying a smaller unit on average.

5.6 Housing Tax Credits

The last housing policy we study is a version of the low-income housing tax credit (LI-
HTC) program. As explained in detail in Appendix B.4, the program subsidizes construc-
tion costs associated with affordable housing development. For comparability with the
previous policies, the policy is sized to also cost $800 million. The tax credit raises the
`
average price P developers earn by 4.07% in zone 1 and 1.22% in zone 2, thereby stim-
ulating new construction. The envisioned increase in the housing stock materializes in

52
 Figure 13: Spatial Housing Policies–Migration
15 25

Out-migration Relative to Benchmark (%)

In-migration Relative to Benchmark (%)

10 20

5 15

0 10

-5 5

-10 0
All RS in Z2
-15 All RS in Z2+transit -5
Zoning
Vouchers
-20 LIHTC -10
Cash transfer V
-25 -15
1 2 3 4 1 2 3 4
Productivity Level Productivity Level
Notes: The left (right) panel reports the percentage difference in out-migration (in-migration) rates between the experiment and the
benchmark model for each productivity group.

equilibrium, and is stronger in zone 1 than in zone 2. Equilibrium rents fall in both zones,
which helps with housing affordability. However, the effects are small. Tax credits in
difficult-to-develop gateway cities, like New York, create too few additional affordable
housing units to make a meaningful dent in the welfare of low-income households. In
addition, there is the aforementioned welfare offset from the distortions resulting from
the tax increases needed to pay for the tax credits. All told, we find that the tax credit
program does not generate a welfare gain ( 0.01%). The same is true for the LIHTC ex-
periment in the no-migration economy. This experiment underscores the importance of
targeted policies and how housing affordability programs are financed.

5.7 Comparison to Cash Transfers

Finally, it is useful to contrast the housing policies to cash transfer program. Columns (6)
and (7) of Table 12 study two cash transfer policies. Both of them raise the income tax
rate (by lowering λ) and collect the same additional $800 million in tax revenues as in the
housing voucher, tax credit, and subsidized transit experiments. Both of them redistribute
that extra tax revenue to households who earn less than 50% of AMI.
The policy in column (6) is identical in its benefit allocation to the housing voucher
experiment of column (4). It simply removes the minimum housing expenditure con-
straint, thereby replacing the in-kind transfer with a cash transfer. The aggregate welfare
effect of the “Cash transfer V” experiment (V stands for voucher) is -0.02%, close to the

53
welfare gain in the housing voucher experiment. This goes to show that the housing con-
straint does not distort consumption allocations very much in the housing voucher exper-
iment. Figures 12 and 13 confirm that the distributional and migration consequences of
the voucher-like cash transfer program are also similar to the housing voucher program.
We note that several of the housing policy reforms discussed above produce strictly larger
welfare benefits than this cash transfer program.
The policy in column (7) aims to improve on the design of the cash transfer by chang-
ing the benefit allocation. Specifically, each household with income below 50% of AMI
receives a recurring cash transfer for the maximum of zero and X-30% of pre-tax house-
hold income. The parameter X is set such that the program costs exactly the targeted
amount in the aggregate. When X = $5, 530, the program costs $800 million in equi-
librium. Note that the transfer is larger the lower is the household’s income; the cash
transfer is progressive (hence the label P). There is no lottery. The policy distributes dol-
lars from the lowest income level upwards, until the money runs out. While the average
benefit amount is much lower (about $1,1000 per year) than the cash transfer V ($8,300
per year), the cash transfer P policy is better targeted, both at the extensive margin (7%
of the population are recipients versus 1.35%) and intensive margins (progressive nature
of the benefit). While the benefit fluctuates over time as household income changes, it is
more stable than the cash transfer V. This policy creates a large aggregate welfare gain of
3.90%. Hence, it is possible to conceive of cash transfer policies that dominate the housing
policies we study. Achieving this optimal targeting may be difficult in practice.
Enacting meaningful tax-and-transfer reform may be outside of the remit of local pol-
icy makers. (Currie and Gahvari, 2008) offer several explanations for the widespread
adoption of in-kind transfers, including in the realm of housing. Whether they are the
constrained optimal policy or just an important practical alternative to tax policy, hous-
ing policies are important to analyze given their prevalence.
Finally, cash transfer policies interact in powerful ways with the housing market through
taxation and migration. As already discussed in the context of housing vouchers, cash
transfers also result in a lower long-run housing stock (-0.20% in zone 1 and -1.86% in
zone 2 in column 7), despite the population growth, and in much higher rents (2.38% and
2.44%) and house prices (2.41% and 2.45%). The higher cost of living reduces in-migration
and accelerates out-migration of high-productivity households. The latter reap the capital
gain. Transfer policies have (unintended) spatial consequences, resulting in gentrification
of the urban core (more high-income residents, less density, and more homeownership).

54
6 Conclusion
In a world with rising urbanization rates, the high cost of housing has surfaced as a daunt-
ing challenge. Existing affordable housing policy tools affect the supply of housing, how
the housing stock is used (owned, rented, affordable), and how it is distributed in space.
Households of different tenure status, age, income, and wealth are differentially affected
by changes in policy. This paper develops a novel dynamic stochastic spatial equilibrium
model with wealth effects and rich household heterogeneity that allows us to quantify
the welfare implications of the main housing affordability policy tools.
The model is calibrated to the New York metropolitan area, but allows for migration
to another metro area. It matches patterns of average earnings, wealth accumulation,
and home ownership over the life-cycle, delivers realistic house prices, rents, and wages,
as well as large spatial differences in income and rents between the urban core and the
periphery. The calibration captures the key features of New York’s affordable housing
system as well as restrictions on residential land use.
We use the model to evaluate changes to the rent stabilization system, zoning policy,
an expansion of the housing voucher system, and tax credits for the development of af-
fordable housing. These policies have quantitatively important aggregate, distributional,
and spatial implications. General equilibrium effects are sometimes at odds with partial
equilibrium logic.
Consistent with conventional wisdom, increasing the housing stock in the urban core
by relaxing zoning regulations is welfare improving. Contrary to conventional wisdom,
increasing the scope of rent stabilization and housing voucher systems are also welfare
improving. The main reason is that housing affordability policies generate important in-
surance benefits which trade off against the larger housing and labor market distortions.
Increasing the housing safety net for the poorest households creates welfare gains for so-
ciety. How the affordability policies are financed has first-order effects on welfare gains.
Finally, the insurance view of affordability points towards advantages from better target-
ing of RS housing towards the neediest households.
These results underscore the need for rich models of household heterogeneity to un-
derstand both the aggregate and the distributional implications of place-based policies.
Future work could use this framework to analyze investment in transit infrastructure, the
effects of working from home or driverless cars on commuting costs, or the effects of local
tax changes on migration. Applying this framework to study other cities with different
institutional features is another useful direction for future inquiry.

55
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58
A Data Appendix
A.1 The New York Metro Area
U.S. Office of Management and Budget publishes the list and delineations of Metropolitan Statis-
tical Areas (MSAs) on the Census website (https://www.census.gov/population/metro/data/
metrodef.html). The current delineation is as of July 2015. New York-Newark-Jersey City, NY-
NJ-PA MSA (NYC MSA) is the most populous MSA among the 382 MSAs in the nation.
NYC MSA consists of 25 counties, spanning three states around New York City. The complete
list of counties with state and zone information is presented in Table 6. As previously defined, only
New York County (Manhattan borough) is categorized as zone 1; the other 24 counties are cate-
gorized as zone 2. For informational purposes, the five counties of New York City are appended
with parenthesized borough names used in New York City.

Table 6: Counties in the New York MSA

County State Zone

New York (Manhattan) NY Zone 1
Bergen NJ Zone 2
Bronx (Bronx) NY Zone 2
Dutchess NY Zone 2
Essex NJ Zone 2
Hudson NJ Zone 2
Hunterdon NJ Zone 2
Kings (Brooklyn) NY Zone 2
Middlesex NJ Zone 2
Monmouth NJ Zone 2
Morris NJ Zone 2
Nassau NY Zone 2
Ocean NJ Zone 2
Orange NY Zone 2
Passaic NJ Zone 2
Pike PA Zone 2
Putnam NY Zone 2
Queens (Queens) NY Zone 2
Richmond (Staten Island) NY Zone 2
Rockland NY Zone 2
Somerset NJ Zone 2
Suffolk NY Zone 2
Sussex NJ Zone 2
Union NJ Zone 2
Westchester NY Zone 2

A.2 Population, Housing Stock, and Land Area

The main source for population, housing stock and land area is US Census Bureau American
FactFinder (http://factfiner.census.gov). American FactFinder provides comprehensive

59
survey data on a wide range of demographic and housing topics. Using the Advanced Search op-
tion on the webpage, topics such as population and housing can be queried alongside geographic
filters. We select the DP02 table (selected social characteristics) for population estimates, the DP04
table (selected housing characteristics) for housing estimates, and the GCT-PH1 table (population,
housing units, area and density) for land area information. Adding 25 counties separately in the
geographic filter, all queried information is retrieved at the county level. We then aggregate the 24
columns as a single zone 2 column.
Since the ACS (American Community Survey) surveys are conducted regularly, the survey
year must be additionally specified. We use the 2015 1-year ACS dataset as it contains the most
up-to-date numbers available. For Pike County, PA, the 2015 ACS data is not available and we
use the 2014 5-year ACS number instead. Given that Pike County accounts only for 0.3% of zone
2 population, the effect of using lagged numbers for Pike County is minimal.
The ratio of the land mass of zone 1 (Manhattan) to the land mass of zone 2 (the other 24
counties of the NY MSA) is 0.0028. However, that ratio is not the appropriate measure of the
relative maximum availability of housing in each of the zones since Manhattan zoning allows for
taller buildings, smaller lot sizes, etc.
Data on the maximum buildable residential area are graciously computed and shared by
Chamna Yoon from Baruch College. He combines the maximum allowed floor area ratio (FAR) to
each parcel to construct the maximum residential area for each of the five counties (boroughs) that
make up New York City. Manhattan has a maximum residential area of 1,812,692,477 square feet.
1
This is our measure for H . The other four boroughs of NYC combine for a maximum buildable
residential area of 4,870,924,726 square feet. Using the land area of each of the boroughs (expressed
in square feet), we can calculate the ratio of maximum buildable residential area (sf) to the land
area (sf). For Manhattan, this number is 2.85. For the other four boroughs of NYC it is 0.62. For
Staten Island, the most suburban of the boroughs, it is 0.32. We assume that the Staten Island ratio
is representative of the 20 counties in the New York MSA that lie outside NYC since these are
more suburban. Applying this ratio to their land area of 222,808,633,344 square feet, this delivers
a maximum buildable residential square feet for those 20 counties of 71,305,449,967 square feet.
Combining that with the four NYC counties in zone 2, we get a maximum buildable residential
2 1 2
area for zone 2 of 76,176,377,693 square feet. This is H . The ratio H /H is 0.0238. We argue that
this ratio better reflects the relative scarcity of space in Manhattan than the corresponding land
mass ratio.

A.3 House Prices, Rental Prices, and Home Ownership

Housing prices and rental prices data come from Zillow (http://www.zillow.com/research/
data) indices. Zillow publishes Zillow Home Value Index (ZHVI) and Zillow Rent Index (ZRI)
monthly. The main advantage of using Zillow indices compared to other indices is that it over-
comes sales-composition bias by constantly estimating hypothetical market prices, controlling for
hedonics. Zillow uses a machine-learning algorithm that ensures that the ZHVI and ZRI pertain
to the same, typical, constant-quality unit, in a particular geography. We use 2015 year-end data
to be consistent with the ACS dataset. There are a few missing counties in ZHVI and ZRI. For the
five counties with missing ZHVI index price, we use the median listing prices from the Zillow
website instead. For the two counties with missing ZRI index price, we estimate the rents using
the price/rent ratio of comparable counties. Zillow excludes non-arms’ length transactions and
rent-regulated rentals. To aggregate across the 24 counties in zone 2, we calculate the median price
as the weighted average of the median prices in each county, where the weights are the shares of

60
 Table 7: New York City Housing Units by Type

Type Number of Units Share

owner occupied 31.6%
owner occupied condo, coop, conventional 950,404 29.8%
Mitchell lama or Article 4 coop 57,064 1.8%
market rental 936,660 29.4%
rent stabilized (RS) 966,442 30.3%
rent controlled (RC) 8.8%
public housing 186,175 5.8%
rent controlled 21,751 0.7%
Mitchell lama or Article 4 Rental 43,529 1.4%
HUD or other regulated (Loft Board, Municipal Loan Program) 26,845 0.8%
In rem housing 1,661 0.1%

housing units. Similarly, for the median rent of zone 2, we average median rents of the 24 counties
using housing unit shares as weights.
Home ownership data is directly from American FactFinder. In table DP04 (selected housing
characteristics), the Total housing units number is divided by Occupied housing units and Vacant
housing units. Occupied housing units are further classified into Owner-occupied and Renter-occupied
housing units, which enables us to calculate the home ownership ratio.

A.4 Rent Regulation

The main source for rent regulation data is US Census Bureau New York City Housing and Va-
cancy Survey (NYCHVS; http://www.census.gov/housing/nychvs). NYCHVS is conducted ev-
ery three years to comply with New York state and New York City’s rent regulation laws. We
use the 2017 survey data table, which is the most recent survey data. This survey also corrects a
misclassification in the type of housing units in earlier surveys. Table 7 below lists the non-vacant
housing units by type. In New York City, owner-occupied units make up 31.6% of the housing
units, market rentals make up 29.4%, rent stabilized units (RS) make up 30.3%. We group the re-
maining housing units (8.8%), under the auspices of seven different affordable housing programs,
into a category we call rent controlled (RC) units. Most of those are public housing units; only
0.7% of the housing stock are actual rent controlled units.
Rent stabilization generally applies to apartments in buildings with six or more units con-
structed before 1974. Rent stabilized units are restricted in terms of their annual rent increases,
which is set by the Rent Guidelines Board. The vast majority of units built after 1974 that are rent
stabilized are so voluntarily. They receive tax abatement in return for subjecting their property to
rent stabilization for a defined period of time. The rent on RS units does not depend on tenant
income level, apartment size, how many people live there, or any other needs-based factors. In
the past, landlords could increase rents after a tenant left based on individual apartment improve-
ments made to the unit or major capital improvements made to the building. That ability became
severely curtailed with the passage of new affordable housing legislation in the state of New York
in July 2019. Our data do not include the period after July 2019.

61
A.4.1 Measuring the Discount in RS Housing Units
To measure the extent of the rental subsidy in RS and RC rental units, we perform the following
exercise. We first estimate a hedonic model for market rental units, where we regress the out-of-
pocket rent (which equals the contract rent for this group) on a list of unit characteristics. We omit
all apartments whose rent is top-coded; they have rents above $5,995 per month. Figure 14 plots
the actual rent against the predicted rent by this hedonic model; the R2 from this regression is
63.9%. The figure notes list the characteristics included in the regression.

Figure 14: Fitting Market Rent for Market Rental Units

Predicted vs Actual Rents for Market Units
6000
4000
2000
0

0 2000 4000 6000

Contract rent

Fitted values Contract rent

Notes: Horizontal axis: contract market rent (in dollars per month). Vertical axis: predicted market rent from a linear regression
of market rent on a set of characteristics: sub-borough area, categorical variables for number of bedrooms, number of bathrooms,
year built, number of units in the building, number of stories in the building, elevator, complete kitchen, working kitchen, complete
plumbing, peeling paint, deep-wear floors, cockroaches, heating breakdowns, maintenance deficiencies, any holes in walls, any holes
in floors, air conditioning, cable/internet, condition of the building, missing brick in walls, years living in the unit, household interest
income, and householder race. Data are from the 2017 New York City Housing and Vacancy Survey.

In the next step, we use the hedonic coefficients estimated from market rentals and multiply
them by the characteristics of the RC and RS units to obtain an “imputed market rent.” We set
a unit’s actual contract rent as the floor for the imputed market rent. We define the “discount”
as 1 - (out-of-pocket rent/imputed market rent). Figure 15 plots a histogram for the discount on
RS units (left panel) and RC units (right panel). The median discount for RS units is 26.2% with
interquartile range of [0.5,53.5%]. For RC units, the median discount is 77.7% with interquartile
range [61.4,86.4%].
Finally, we analyze the distribution of the discount, conditional on how long the tenant has
lived in the unit. We sort tenants in four groups: 0-4 years in the unit, 4-8 years, 8-12 years, and
12 or more years. The box plot on the left of Figure 16 is for RS tenants while the plot on the right
is for RC tenants. For RS units the discount is strongly increasing in length of tenancy, while for
RC units it is not. For RS, the discount rises from 7% for tenancies between 0 and 4 years, 20%
for 4-8years, 27% for 8-12 years, and 45% for 12 or more years. In unreported results, we find
that considering additional breakdowns beyond 16 years are not statistically different from one
another. This is the discount schedule we use in the calibration.

A.4.2 Measuring Share of RS Units

The NYHVS provides the number of all renter-occupied units and the number of RS units for
New York City, as reported in Table 7, for each of the five counties in New York City. The share

62
 Figure 15: Rent Discount
Rent Stabilized Rent Controlled

8
20

6
15
Percent

Percent
4
10

2
5
0

0
0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1
Rent Discount Rent Discount

Notes: Histogram of rent discounts for rent stabilized units (left panel) and rent controlled units (right panel). The discount for a
housing unit is defined as 1 - out-of-pocket rent/imputed market rent. The implied market rent uses the characteristics of the units
and the hedonic coefficients estimated from the sample of market rentals. The floor for the imputed market rent is the contract rent
for the unit. Data are from the 2017 New York City Housing and Vacancy Survey.

of all housing units that are rent-stabilized in Manhattan (zone 1) is 37.29% in 2017. The share of
renter-occupied units that are RS in Manhattan is 49.48%. To measure the number and share of RS
units for zone 2 of the New York metro, we need data on total renter-occupied and RS units for the
remaining twenty counties outside New York City. For the other 20 counties outside of New York
City, we measure the total stock of renter-occupied units from the 2018 American Community
Survey (5-year survey). We obtain the number of rent stabilized units from Affordable Housing
Online (http://affordablehousingonline.com) at the county level. We add up the total number
of RS units and the total number of occupied units across the 24 counties of zone 2. The resulting
RS share out of all housing units in zone 2 is 12.01%. The share of RS units out of renter-occupied
units in zone 2 is 26.65%.

A.4.3 Rent-Stabilization by Age and Income

We calculate the prevalence of rent stabilization for renter households of various ages and income
levels. We perform this analysis using microdata available from the NYHVS, i.e., for the five
counties in New York City. For the purposes of this graph, we compute define the RS share as the
ratio of RS plus RC units to the sum of RS plus RC plus market rental units. This is plotted in the
right panel of Figure 6 in the paper.

A.5 Migration Data

A.5.1 Migration Out and Into New York MSA
The best available data to measure migration by age and income out-of and into the New York
MSA comes from the Internal Revenue Service SOI Tax Stats Migration data 2011-2012 (https:
//www.irs.gov/statistics/soi-tax-stats-migration-data-2011-2012). The file computes
migration rates by comparing address of residency in two successive tax returns in 2011 and 2012.
The data universe is the universe of individuals who file Tax Form 1040. Tax returns are matched
on the taxpayer identification number of the primary, secondary, and dependent filers. Since only
the state-level migration data contain information on age and income, we use the New York State

63
 Figure 16: Rent Discount Conditional on Tenancy
Rent Stabilized Rent Controlled
1

1
.8

.8
.73 .74 .74 .76
.6

.6
Rent Discount

Rent Discount
.45
.4

.4
.27

.2
.2

.2
.07
0

0
0 4 8 12+ 0 4 8 12+
excludes outside values excludes outside values

Notes: Histogram of rent discounts for rent stabilized units, conditional on tenancy (left panel) and rent controlled units, conditional
on tenancy (right panel). The discount for a housing unit is defined as 1 - out-of-pocket rent/imputed market rent. Tenancy is defined
as the number of years a tenant has lived in the unit. We define four tenancy groups: 0-4 years, 4-8 years, 8-12 years, and more than
12 years. Data are from the 2017 New York City Housing and Vacancy Survey.

file as a proxy for the New York MSA. The in-migration rate is defined as the ratio of the number
of inflow tax returns to the sum of same-state tax returns plus non-migrant tax returns plus inflow
tax returns. The out-migration rate is defined as the ratio of the number of outflow tax returns to
the sum of same-state tax returns plus non-migrant tax returns plus inflow tax returns.

A.5.2 Migration Within New York MSA

We use county-to-county migration data for 2006-2010 and 2010-2014 from the 5-year American
Community Survey for the 25 counties in the New York metropolitan area. For Manhattan (zone
1), we compute the frequency of moves to one of the 24 counties in zone 2. For each of the 24
counties in zone 2, we compute the likelihood of moving to Manhattan. We aggregate these mo-
bility rates by computing the population-weighted average across the 25 counties. We compute
the annual probability of moving by 4-year age groups, and also separately for home owners and
renters. The resulting mobility rates are plotted in Figure 2.

B Calibration Appendix
B.1 Earnings Calibration
Before-tax earnings for household i of age a is given by:
lab,`
yi,a = Wt nit G a zi A`

where G a is a function of age and zi is the idiosyncratic component of productivity. Since en-
dogenous labor supply decisions depend on all other parameters and state variables of the model,
exactly matching earnings in model and data is a non-trivial task.

Age Profile We determine G a as follows. For each wave of the Survey of Consumer Finance
(SCF, every 3 years form 1983-2010), we compute average earnings in each 4-year age bucket

64
 (above age 21), and divide it by the average income of all households (above age 21). This gives
us an average relative income at each age. We then average this relative age-income across all 10
SCF waves.

Income Dispersion by Age We also use SCF data to determine how the dispersion of income
changes with age. We choose four grid points for income, corresponding to fixed percentiles (0-25,
25-75, 75-87.5, 87.5-100). To compute the idiosyncratic income z a,i of each group i ∈ {1, 2, 3, 4} at a
particular age a, relative to the average income of all households of that age we do the following:

Step 1: For each positive-earnings household, we compute which earnings group it belongs to
among the households of the same age.

Step 2: For each 4-year age bucket, we compute average earnings of all earners in a group.

Step 3: We normalize each group’s income by the average income in each age group, to get each
group’s relative income.

Step 4: Steps 1-3 above are done separately for each wave of the SCF. We compute an equal-weighted
average across all 10 waves to get an average relative income for each age and income group.
This gives us four 11x1 vectors z a,i since there are 11 4-year age groups between ages 21 (en-
try into job market) and 65 (retirement). Note that the average z across all households of a
particular age group is always one: E[z a,j | a] = 1.

Step 5: We regress each vector, on a linear trend to get a linearly fitted value for each group’s relative
income at each age. The reason we perform Step 5, rather stopping at Step 4 is that the
relative income at age 4 exhibits some small non-monotonicities that are likely caused by
statistical noise (sampling and measurement error). Step 5 smoothes this out.

The steps above apply to both the New York and Outside MSA income profiles.

Income Dispersion by Productivity - New York The four productivity grid points at the
average age are chosen to match the NY metro income distribution. The data is the 2018 U.S.
Census Bureau’s IPUMS. For all countries in the New York metro, we obtain total pre-tax household
income of all household members older than 15. The data are not top-coded. We compute average
income of the bottom 25%, the middle 50%, the next 12.5%, and the top 12.5%. The resulting
averages in each bin are reported in Panel B of Table 8. We divide these numbers by $90,234,
which is the average in the Outside region defined below, to arrive at the productivity grid zi ∈
Z NY = [0.1907, 0.9012, 1.9227, 4.1867]. The average income in the NY metro area is $113,883,
which is 26.2% higher than the average income Outside. Hence, the mean productivity in NY is
1.262, while mean productivity is normalized to 1.00 Outside.

Income Dispersion by Productivity - Outside MSA For the outside MSA, we use the Of-
fice of Management and Budget MSA definitions and include 261 MSAs, excluding NYC. House-
hold income is measured from the same 2018 IPUMS data set we use for New York. Aver-
age pre-tax household income by group is reported in Panel C of Table 8. We divide by av-
erage income of $90,234, so that the grid averages to 1. This results in states: zi ∈ ZOutside =
[0.1906, 0.7410, 1.5067, 3.1480].

65
Income Risk The 4 × 4 transition probability matrix for zi is age-invariant, but is allowed to
depend on β type. Specifically, the expected duration of the highest productivity state is higher
for the more patient agents. The transition probability matrix for z is P for β L agents. We impose
the following restrictions:
 
p11 1 − p11 0 0
 (1 − p22 )/2 p22 (1 − p22 )/2 0
P=


 0 (1 − p33 )/2 p33 (1 − p33 )/2 
0 0 1 − p44 p44

For β H types, the transition probability matrix is the same, except for the last two entries which
are 1 − p44 − p H and p44 + p H , where p H < 1 − p44 . We pin down the five parameters

( p11 , p22 , p33 , p44 , p H ) = (0.93, 0.92, 0.28, 0.64, .02)

to match the following five moments. We match the population shares in each of the four income
groups, which are fixed at 25%, 50%, 12.5%, and 12.5%. Given that population shares sum to one,
that delivers three moments. We match the persistence of individual labor income to a value of
0.9, based on evidence form the PSID in Storesletten, Telmer, and Yaron (2006). Finally, we choose
p H to match the fraction of high-wealth households in the top 10% of the income distribution. The
transition probability matrix is identical in the two MSAs.
Table 8 summarizes the results. Average earnings are reported annually. Earnings autocorre-
lation and volatility are reported for 4 years.

Table 8: Labor Earnings Calibration

All Group 1 Group 2 Group 3 Group 4

Panel A: Common Parameters
Earnings autocorr. 0.77
Earnings vol. 0.125
Corr. (income,wealth) 0.23
Pop. shares Model 25% 50% 12.5% 12.5%
Pop. shares Data 25% 50% 12.5% 12.5%
Panel B: New York Metropolitan Area
Avg. earnings Model 15,257 71,247 158,526 375,374
Avg. earnings Data 17,207 81,333 173,526 377,848
Panel C: Outside Metropolitan Area
Avg. earnings Model 16,699 65,500 129,118 292,959
Avg. earnings Data 17,200 66,872 135,972 284,089

66
B.2 Progressive Taxation
1
Following Heathcote et al. (2017), households with income ytot < y0tot = λ τ receive transfers
T (ytot ) < 0, and those with ytot ≥ y0tot pay taxes T (ytot ) ≥ 0. We set τ = 0.17 and λ = 0.75, as dis-
cussed in the calibration section. As a result of our calibration, 39% of households are subsidized
by the progressive tax system, and 35% receive a subsidy after subtracting Social Security taxes.
Figure 17 describes the progressive taxation system. At low total income values, some households
receive a subsidy, which progressively decreases. At higher incomes, taxes increase faster than
income. This is reflected in households’ after-tax income, shown in Figure 18.

Figure 17: Progressive Taxes

Progressive Tax and Social Security Tax

4
10 104
15 15
Progressive Tax

10 10

5 5

0 0

0 1 2 3 4 0 1 2 3 4
5 5
Total Before-Tax Income
10
Total Before-Tax Income
10

Notes: Horizontal axis: total income (in dollars, annual), measured as the sum of labor earnings, pensions, and financial income.
Vertical axis: taxes minus transfers excluding Social Security taxes (in dollars, annual; left panel), total taxes minus transfers including
Social Security taxes (in dollars, annual; left panel). The dashed line plots the zero-tax case.

Figure 18: After-tax Total Income

105 105
4 4
After Progressive Tax Income

3.5 3.5
After Total Tax Income

3 3

2.5 2.5

2 2

1.5 1.5

1 1

0.5 0.5

0 0
0 1 2 3 4 0 1 2 3 4
105 5
Total Before-Tax Income Total Before-Tax Income
10

Notes: Horizontal axis: total income (in dollars, annual). Vertical axis: post-tax income excluding Social Security taxes (in dollars,
annual; left panel), post-tax income including Social Security taxes (in dollars, annual; left panel). The dashed line is the 45 degree
line.

67
B.3 Housing Supply Elasticity Calibration
We compute the long-run housing supply elasticity. It measures what happens to the housing
quantity and housing investment in response to a 1% permanent increase in house prices. Define
housing investment for a given zone, dropping the location superscript since the treatment is
parallel for both zones, as:  
h Ht−1 ρ
Yt = 1 − Nt h .
H
Note that Ht+1 = (1 − δ) Ht + Yth , so that in steady state, Y h = δH. Rewriting the steady state
housing investment equation in terms of equilibrium quantities using (8) delivers:
  1−1ρ ρh −ρh
1 H h 1− ρ h
ρh
H= 1− ρh P 1− ρ h W 1− ρ h
δ H
Rewrite in logs, using lowercase letters to denote logs:

1 ρh ρh
h = − log(δ) + log(1 − exp(h − h)) + p− w
1 − ρh 1 − ρh 1 − ρh

Rearrange and substitute for p in terms of the market price p = log(ho + (1 − ho )κ4 ) + p:

1 − ρh 1
p= h− log(1 − exp(h − h)) + k
ρh ρh

where
1 − ρh
k≡ log(δ) + w − log(ho + (1 − ho )κ4 )
ρh
Now take the partial derivative of p w.r.t. h:

∂p 1 − ρh 1 exp(h − h) ∂k
= + +
∂h ρh ρh 1 − exp(h − h) ∂h

Invert this expression delivers the housing supply elasticity:

∂h ρ
= h h i (14)
∂p exp(h−h) (1−κ 4 )
1 − ρh + 1−exp(h−h) ∂h − ρ h ho +(1−ho )κ
+ ρh ∂w ∂ho
∂h
4

∂h ≈ 0) and either the RC distortions are

If (i) the elasticity of wages to housing supply is small ( ∂w
small (κ4 ≈ 1) or the home ownership rate is inelastic to the housing supply ( ∂ho
∂h ≈ 0), or (ii) if the
two terms in square brackets are positive but approximately cancel each other out, then the last
two terms are small. In that case, the housing supply elasticity simplifies to:

∂h ρh
≈
∂p exp(h−h)
1 − ρh + 1−exp(h−h)

Since, in equilibrium, Y h = δH, ∂yh /∂p = ∂h/∂p.

Note that h − h measures how far the housing stock is from the constraint, in percentage terms.
exp(h−h)
As H approaches H, the term 1−exp(h−h)
approaches +∞ and the elasticity approaches zero. This

68
is approximately the case in zone 1 for our calibration. If H is far below H, that term is close to
ρ
zero and the housing supply elasticity is close to 1−hρh . That is approximately the case for zone 2
in our calibration. Since zone 2 is by far the largest component of the New York metro housing
stock, zone 2 dominates the overall housing supply elasticity we calibrate to.
In the calibration, we use equation (14) to measure the housing supply elasticity and set ∂w∂h =
0.05 based on evidence from Favilukis and Van Nieuwerburgh (2021), who study a model with
aggregate shocks to housing demand driven by out-of-town home buyers.

B.4 Tax Credits

Tax credits directly incentivize the development of affordable housing units by giving developers
subsidies to offset the cost of affordable housing construction.

B.4.1 Institutional Background

Developers who receive tax credits for affordable housing development can sell them to other
profitable firms; they fetch prices above 90 cents on the dollar. In other words, they are (nearly)
equivalent to cash subsidies. The main program, the federal Low Income Housing Tax Credit
(LIHTC) subsidizes 30% of the construction cost associated with affordable housing units. This
is known as the 4% program. A 4% subsidy of construction costs is given over a 10-year period,
and is worth 30% of construction costs in present-value terms. There is a second program, the
9% subsidy for 10 years, which is worth 70% in present-value terms, which is aimed at more
deeply affordable housing units. We focus on the 4% LIHTC program. Total spending on LIHTC
is $9 billion annually nationwide; about $50 million in the New York MSA. Additional programs,
like Tax Incremental Financing, are ran by municipal governments. LIHTCs are often used to
subsidize mixed market rate-affordable housing projects. In practice, it is up to the states to decide
how to spend their federal LIHTC allocations depending on which areas and which points of the
income distribution they want to target. In places like Manhattan with high costs of land and high
construction costs, the only way to break even on an affordable housing development through tax
credits, given the rules of the LIHTC, is to build a mixed property with market-rate and affordable
units. Such areas are known as Difficult to Develop Areas. They are our focus. Davis et al. (2017)
study instead low-income tracts with a poverty rate above 25%, the so-called QRT areas.

B.4.2 Modeling Tax Credits

Our experiment considers an expansion of the LIHTC in the New York MSA. In the model, de-
velopers in a given zone earn a price per sf built equal to the market price times a discount; recall
equation (6), repeated here for convenience:
 
`
Pt = hot` + (1 − hot` )κ4` Pt` .

The discount depends on the fraction of units that are owned (hot` ) and the rent discount due to
RS housing κ4` . We now assume that developers receive a subsidy to help offset the rent discount
due to RS:
κ4` = 1 − η ` + η ` κ1 ` (1 + LI HTC ),
where
κ1 ` = ∑ ω` (d)κ1` (d)
d

69
and ω ` (d) is the share of RS square feet in a zone that goes to RS tenants in that zone with tenure
d, such that ∑d ω ` (d) = 1.
In the benchmark model, the parameter LI HTC = 0. In the LIHTC experiment, we subsidize
30% of the construction costs for 50% of the development. Thus we choose LI HTC such that the
total value of LIHTC subsidies, aggregated first across all firms within each zone and then across
zones, is equal to 50% × 30% = 15% of the construction costs associated with the construction of
RS housing. We compute the construction costs of RS housing as follows. Since the only input is
labor, we take the total wage bill in each zone (aggregating across firms) and multiply it by the
share of RS sf to compute the construction costs associated with RC housing in that zone; then
we sum across zones and multiply by 15%. That gives us the total value of LIHTC subsidies.
Matching the construction costs of RC housing requires a value for the parameter LI HTC of 0.14.
We assume it is identical across zones. When accounting for ownership rates and shares of RC
`
sf in each zone, the tax credits increases the average price Pt earned by developers by 4.07% in
zone 1 and 1.22% in zone 2. We then change the value of λ in the tax-and-transfer function to
generate enough additional tax revenue to exactly pay for the aggregate tax credit outlay. The
policy is budget neutral, like the previous experiments. The size of the program, the extra tax
dollars raised (and spent), is calibrated to be the same as for the voucher program and equals
$800m. Hence, these two programs are directly comparable. Developers continue to build market
rate and affordable units in proportions 1 − η ` and η ` .

C No Migration Model
This appendix discusses the results for a model where there is no migration in or out of the
metropolitan area. The model consists of just the gateway MSA with its zone 1 and zone 2, but no
outside location. We begin by discussing the calibration, to the extent that it differs from the base-
line model with migration. Then, we show baseline results for the no migration model. Finally,
we revisit the same policy experiments we studied in the migration model.

C.1 Calibration
We set H 2 = 1.69 such that the ratio of households living in zone 1 to households living in zone
2 is 12%, the fraction observed in the NY data. The ratio H 1 /H 2 is the same as in the benchmark
model. The no migration model delivers 0.63. The housing supply elasticity is much lower in
zone 1 (0.07) than in zone 2 (0.65), because in zone 1 the housing stock is much closer to H (10%
from the constraint) than in zone 2 (66% from the constraint).
The minimum hours constraint is the same, but now binds for 10.05% of workers in the no
migration equilibrium.
The agglomeration parameter that governs the extra productivity a household derives from
living in the urban core A1 = 1.047 > A2 = 1 is chosen to help the model match the 1.66 ratio of
average income in zone 1 to zone 2 in NY.
The financial cost of commuting φ2F is set to 1.9% of average labor earnings, or $2393 per house-
hold per year.
The values for β H and β L are the same as in the benchmark. The no-migration model implies
an average wealth-income ratio of 6.25, compared to 5.69 in the 1998-2010 SCF data. The model’s
wealth Gini coefficient is 0.77, close to the observed wealth Gini coefficient of 0.80 for the U.S. Both
numbers are slightly higher than in the migration model.

70
 Two parameters govern the amenity value of housing in (4); since there is no Outside region,
χ NY = 1.000. Living in Manhattan relative to the rest of the NY metro gives a utility boost χ1 =
1.063. And being a retiree in Manhattan gives an additional utility boost of χ R = 1.050. We choose
these parameters to match the 2.78 ratio of rents in zone 1 to zone 2, and the 0.91 ratio of retirees
in zone 1 to zone 2. In the model, these ratios 2.78, and 0.92, respectively.
We impose a minimum housing size of 544 square feet. This is 33% of the average housing
unit size of 1644 square feet in NY.
We set the share of square feet of rental housing devoted to RS units, η 1 = 56.92% and η 2 =
31.63%, to match the observed share of households in the entire population that are in RS units
in each zone, namely 37.3% in zone 1 and 12.0% in zone 2. Households who were in RS in the
previous period have a probability of 83.7% to qualify for RS in the same zone this period. As in
the benchmark model, this value is chosen to match the fraction of RS tenants who have lived in
an RS unit for 20 years or more, which is 23.1% in the data and 25.7% in the model.

C.2 Baseline Model Results for No Migration Model

Table 9 shows that the no migration model matches demographic, income, house size, home own-
ership, rent, and house price moments well.

Table 9: New York Metro Data Targets and Model Fit, No Migration Model

Data No Migration Model

metro ratio zone 1/zone 2 metro ratio zone 1/zone 2
1 Households (thousands) 7124.9 0.12 7124.9 0.12
2 Avg. hh age, cond. age > 20 47.6 0.95 47.2 0.86
3 People over 65 as % over 20 19.1 0.91 21.4 0.92
4 Avg. house size (sqft) 1644 0.59 1644 0.55
5 Avg. pre-tax lab income ($) 124091 1.66 124203 1.66
6 Home ownership rate (%) 51.5 0.42 59.3 0.55
7 Median mkt price per unit ($) 510051 3.11 560308 2.07
8 Median mkt price per sqft ($) 353 5.24 311 3.57
9 Median mkt rent per unit (monthly $) 2390 1.65 2792 1.61
10 Median mkt rent per sqft (monthly $) 1.65 2.78 1.55 2.78
11 Median mkt price/median mkt rent (annual) 17.79 1.89 16.72 1.28
12 Mkt price/avg. income (annual) 3.99 1.86 4.52 1.24
13 Avg. rent/avg. income (%) 23.0 1.00 27.0 0.97
14 Avg. rent/income ratio for renters (%) 42.1 0.81 34.9 0.90
15 Rent burdened (%) 53.9 0.79 58.5 0.80
16 % RS of all housing units 14.63 3.11 14.60 3.20

Note: Columns 2-3 report the values for the data of the variables listed in the first column. Data sources
and construction are described in detail in Appendix B. Column 3 reports the ratio of the zone 1 value to
the zone 2 value in the data. Column 4 and 5 are for the no migration model. Column 5 reports the same
ratio in the model.

Mobility Intra-MSA mobility rates are shown in Figure 19. The overall mobility rate across
zones in the model is about 3% annually, consistent with the 2.1% county-to-county migration
rates in the New York MSA.

71
 Figure 19: Intra-MSA Moving Rates by Age, No Migration Model
All Homeowners
0.06 0.06

0.04 0.04

0.02 0.02

0 0
21-24
25-28
29-32
33-36
37-40
41-44
45-48
49-52
53-56
57-60
61-64
65-68
69-72
73-76
77-80
81-84
85-88
89-92
93-96

21-24
25-28
29-32
33-36
37-40
41-44
45-48
49-52
53-56
57-60
61-64
65-68
69-72
73-76
77-80
81-84
85-88
89-92
93-96
Age Age
Renters RS Renters
0.06 0.06

0.04 0.04

0.02 0.02

0 0
21-24
25-28
29-32
33-36
37-40
41-44
45-48
49-52
53-56
57-60
61-64
65-68
69-72
73-76
77-80
81-84
85-88
89-92
93-96

21-24
25-28
29-32
33-36
37-40
41-44
45-48
49-52
53-56
57-60
61-64
65-68
69-72
73-76
77-80
81-84
85-88
89-92
93-96
Age Age

Note: Mobility rates are measured as the annual probability to move across zones.

House Sizes Figure 20 shows the distribution of house sizes. The model (left panel) matches
the data (right panel) quite well, even though these moments are not targeted by the calibration.
The size distribution of owner-occupied housing is shifted to the right from the size distribution
of renter-occupied housing units in both model and data.

Income The New York productivity distribution is substantially different in the two zones.
Zone 1 contains workers that are on average 40% more productive than in zone 2. Productive
working-age households have a high opportunity cost of time and prefer to live close to work
given the time cost of commuting. Mitigating the high opportunity cost of time is the high cost
of living in Manhattan. Indeed, some high-productivity workers may still be early in the life-
cycle when earnings are lower and accumulated wealth smaller. Only 17.2% of working-age,
top-productivity households live in zone 1.
Figure 21 plots how households of different productivity types sort across space. Zone-1 hous-
ing consists mainly of retirees that are home owners and RS renters and top-productivity house-
holds that are owners and market renters. There are also some middle-income households in
zone 1, mostly market and RS renters. The bottom 25% of households by productivity (yellow)
consume a small share of the housing stock.
The model generates a large amount of income inequality at every age. The model’s earnings
Gini of 0.52 is close to the 0.47 value in the 2015 NY metro data. Earnings inequality is lower
within zone 1 (Gini of 0.44) than within zone 2 (Gini of 0.53) in the model.

Home Ownership The model generates a home ownership rate of 59.3%, slightly overstating
the 51.5% in the New York data. Row 6 of Table 9 shows that the ratio of the home ownership

72
 Figure 20: House Size Distribution New York, No Migration Model
Model Data
All All
0.5 0.5

0 0
49

49
00 9

00 9
0

0
00 9

00 9

00 9

00 9

00 9

00 9

00 9

00 9

9
10 99

10 99
00

00
15 49

20 99

25 49

30 99

99

15 49

20 99

25 49

30 99

99
<7

<7
0-

0-
>4

>4
-1

-1

-2

-2

-3

-1

-1

-2

-2

-3
75

75
Sqft Sqft
Homeowners Homeowners
0.5 0.5

0 0
49

49
00 9

00 9
0

0
00 9

00 9

00 9

00 9

00 9

00 9

00 9

00 9

9
10 99

10 99
00

00
15 49

20 99

25 49

30 99

99

15 49

20 99

25 49

30 99

99
<7

<7
0-

0-
>4

>4
-1

-1

-2

-2

-3

-1

-1

-2

-2

-3
75

75
Sqft Sqft
Renters Renters
0.5 0.5

0 0
49

49
9

9
0

0
9

9
99

99
00

00
49

99

49

99

99

49

99

49

99

99
<7

<7
0-

0-
>4

>4
-1

-1

-2

-2

-3

-1

-1

-2

-2

-3
75

75
00

00

00

00

00

00

00

00

00

00
10

15

20

25

30

10

15

20

25

30
Sqft Sqft

Note: Left panel: model. Right panel: data. Data source: American Housing Survey for the New York
MSA, U.S. Census Bureau, 2015.

rate in Manhattan to zone 2 is 0.42 in the data. The model also generates a much lower home
ownership rate in zone 1 than in zone 2, with a ratio of 0.55.

House Prices and Rents The model produces price and rent levels that are close to but some-
what higher than in the data.
The model generates a ratio of house prices in zone 1 to zone 2 of 2.07, the product of a house
size ratio of 0.55 and a price per sf ratio of 3.57. That ratio is 3.11 in the data, and is the product
of a ratio of house sizes of 0.59 and a ratio of prices per sf of 5.24. The relative price-rent ratio in
zone 1/zone2 is 1.28 compared to 1.89 in the data.

Price-Income and Rent-Income Row 12 of Table 9 reports the ratio of the median value of
owner-occupied housing to average earnings in each zone. The model generates a price-income
ratio of 4.52, and a ratio across zones of 1.24.
Row 13 reports average rent paid by market renters divided by average income of all residents
in a zone; while rows 14 and 15 report two moments related to household-level rent burden. The
model generates an average rent-income ratio for renters of 34.9%, which is lower than in the
data.The model generates a large “housing affordability crisis,” with 58.5% of renters spending
more than 30% of their income on rent. The affordability crisis is worse in zone 2 than in zone 1,
both in the model and the data.

Rent Stabilization Figure 22 zooms in on the allocation of RS housing units by age and income.
It plots the fraction of households that are in RS for the bottom 25%, middle 50% and top 25% of the

73
Figure 21: Geographic Distribution of Households by Productivity, No Migration Model.
Zone 1 Zone 2
25 56.9
250

20
200

15
150

10
100

3.4

5 3.6 3.6 50
23.9

8.7
0 0
Owner Renter RS Owner Renter RS

Note: The colors indicate productivity levels. For working-age households: red indicates a top 12.5%
productivity household, brown a household in the next 12.5% of the productivity distribution, okra: a
household in the middle 50% of productivity, and yellow a household in the bottom 25%. Retired house-
holds of all productivity levels are indicated by green. The vertical axes measures the total square footage
devoted to the various types of housing in each zone. Numbers reported atop each of the six vertical bars
are the percentage of households in each of the six tenure categories; they sum to 100%.

income distribution at each age. The model is on the left, the data on the right. The no migration
model generates less misallocation than in the data.

C.3 Affordability Policies

We now consider the same policy experiments as we performed in the main text but in a model
without inter-city migration. We report an additional aggregate welfare measure, which is com-
puted the same way as our main welfare measure except that it compares the value function in
the new steady state (rather than the value function in the first period of the transition) to the
value function in the old steady state. Hence, this measure shows how welfare changes in the
long-run after all state variables (e.g., the housing stock and the wealth distribution) have ad-
justed to their new long-run stochastic steady state. This long-run welfare measure cannot be
meaningfully computed for the migration model since the population composition changes after
a policy change, making the measure difficult to interpret in that model. This is not the case in the
no-migration model since the NY metro population is fixed.

C.3.1 Improving the Targeting of RS

The first set of policies aims to improve the targeting of the RS system. Table 10 contains the
results.

Introducing Income Qualification Requirement Column (1) introduces income qualifica-

tion for RS at 60% of AMI in zone 1 and at 50% of AMI in zone 2. The requirement is enforced
only at the time of entry; existing RS tenants are exempt. This experiment creates essentially no
change in aggregate welfare ( 0.04% in row 31) in the transition and a modest welfare gain in the
long-run ( 0.32%, row 32). The policy is successful at allocating more affordable housing units to
low-income households. There is a 26.08% increase in the fraction of Q1-income households in

74
 Figure 22: Prevalence of Rent Stabilization, No Migration Model

Model Data
0.3 0.3
Mean
Bottom 25% income
0.25 0.25

Share of RS households
Middle 50% income
Share of RS households

Top 25% income

0.2 0.2

0.15 0.15

0.1 0.1

0.05 0.05

0 0
40 60 80 40 60 80
Age Age

Note: The figure plots the share of households in rent stabilized rental housing units out of all housing
units. Age is on the horizontal axis. At each age, we split households into the bottom-25% of income,
the middle 50%, and the top-25%. The results for the model are plotted on the left. The results from the
data are plotted on the right. Since RS status by age and income is only available from the New York City
Housing and Vacancy Survey, the data only pertains to the five counties of New York City rather than to
the full MSA. For the purposes of this graph only, we include rent-controlled units in the numerator of the
RS share. The shares are rescaled to deliver the overall RS share in the entire MSA.

RS (row 4), which exceeds the overall increase in the fraction of households in RS of 1.59% (row
3). Because the households in RS choose smaller apartment units (rows 6 & 7), the RS system ac-
commodates more households in the same square feet of affordable housing space. RS becomes a
better insurance device. Income qualification improves access to insurance for lower-income house-
holds who have fallen on hard times (27.16%, row 27). Row 28 reports that the stability of insurance
is nearly unaffected (0.46%) since income testing is only at the point of first entry and existing
tenants can stay with high probability p RS,exog . Row 30 reports that the policy lowers the volatility
of the marginal utility growth of housing consumption (-1.94%), offering the average household
more housing stability.
It may be surprising that income qualification is not more beneficial in this model. The first
reason is that the maximum size of RS units–such that the average RS unit has the same size as the
average market rental—already discourages many higher-income and higher-wealth households
from choosing a RS unit. The second reason is that the income qualification does not create much
change in the RS system since existing tenants are exempt. The third reason is that, since the
benefit of staying in RS grows with tenure, long-tenure RS tenants have a growing incentive to
stay even if their income levels would suffice to rent a (potentially larger) market rental. Some of
the long-time tenants do lose access (with probability 1-p RS,exog ). When they do, they lose a large
RS discount. They are replaced by a more needy tenant but one who receives a smaller discount
due to the reset of the tenure clock. The welfare losses from the former group offset the welfare
benefits from the latter group.
Figure 23 shows that the oldest retirees, and the bottom productivity, income, and wealth
groups benefit, while all others lose.
This policy does not change developer distortions relative to the baseline model, and there-

75
 Table 10: Policies That Better Target RS—No Migration Model

(1) (2) (3) (4) (5)

Benchm. Inc Qual New Inc Qual All Inc Qual Stay 0.50 RS discount RS size
1 Avg(rent/inc.) renters in Z1 (%) 32.0 3.29% 31.01% 33.03% -0.24% -1.83%
2 Avg(rent/inc.) renters in Z2 (%) 35.5 3.13% 9.25% 6.49% 6.28% 1.37%
3 Fraction of hhs in RS (%) 14.60 1.59% 6.51% 9.77% 1.58% 4.56%
4 Frac. in RS of those in inc. Q1 (%) 22.97 26.08% 31.23% 61.28% 18.78% 21.09%
5 Frac. rent-burdened (%) 58.5 4.45% 10.07% 10.63% 2.67% -1.92%
6 Avg. size of RS unit in Z1 (sf) 719 -8.05% -21.50% -21.50% -3.40% -7.23%
7 Avg. size of RS unit in Z2 (sf) 719 -6.19% -11.47% -15.16% -8.27% -9.06%
8 Avg. size of a Z1 mkt unit (sf) 1079 -0.81% -2.28% -0.43% 0.34% 0.10%
9 Avg. size of a Z2 mkt unit (sf) 1861 1.08% 2.89% 3.36% 1.23% 1.26%
10 Frac. of pop. living in Z1 (%) 10.5 3.66% 9.37% 8.54% 0.75% 2.39%
11 Frac. of retirees living in Z1 (%) 19.6 -6.22% 83.87% 81.75% 3.38% 1.70%
12 Housing stock in Z1 – 0.11% 0.41% 0.36% 0.32% -0.14%
13 Housing stock in Z2 – 0.34% 0.75% 0.78% 0.47% 0.13%
14 Rent/sf Z1 ($) 3.98 0.37% 0.07% 0.46% -0.39% -0.05%
15 Rent/sf Z2 ($) 1.43 0.45% 0.04% 0.58% -0.45% -0.03%
16 Price/sf Z1 ($) 993 0.38% 0.05% 0.45% -0.41% -0.05%
17 Price/sf Z2 ($) 278 0.46% 0.03% 0.64% -0.46% -0.03%
18 Homeownership rate in Z1 (%) 33.9 -1.57% 0.66% -0.93% 4.47% -1.78%
19 Homeownership rate in Z2 (%) 62.2 3.24% 3.61% 3.84% 2.28% 1.59%
20 Avg. inc. Z1 working-age HHs ($) 163506 -3.57% -15.64% -16.18% 0.68% -1.85%
21 Avg. inc. Z2 working-age HHs ($) 99463 0.42% 2.31% 2.45% -0.21% 0.18%
22 Frac. of top-prod. HHs in Z1 (%) 17.2 -1.18% -2.17% -2.29% 0.07% -0.02%
23 Total hours worked – -0.15% 0.30% 0.10% 0.31% -0.03%
24 Total hours worked in effic. units – 0.02% 0.27% 0.20% 0.31% 0.08%
25 Total output – -0.00% 0.11% 0.06% 0.16% 0.03%
26 Total commuting time – -0.55% 1.41% 1.41% -0.02% -0.25%
26 Developer profits – 0.75% 1.97% 2.07% 1.49% 0.25%
27 Access to RS insurance (%) 5.9 27.16% 112.23% 65.89% 18.70% 23.07%
28 Stability of RS insurance (%) 80.8 0.46% -82.57% 0.87% 0.17% 0.32%
29 Std. MU growth, nondurables 0.60 0.23% -0.94% -0.66% -0.37% -0.01%
30 Std. MU growth, housing 0.59 -1.94% 2.38% 0.86% 0.83% 0.75%
31 Aggr. welfare change (NY pop) – 0.04% -0.78% 0.32% -0.96% 0.16%
32 Aggr. welfare change (Long-run) – 0.32% -0.55% 0.73% -0.57% 0.37%

Notes: Column “Benchmark” reports values of the moments for the baseline model.

fore has only minor implications for the housing stock, market rents, house prices, and home
ownership rates in both zones (rows 12-19). Because there are fewer middle- and high-income
households in RS units after the introduction of income qualification, there are fewer households
who are choosing sub-optimally small apartments.
Rent-income ratios among renters increase in zone 1 (3.29%, row 1) and in zone 2 (3.13%, row
2). These changes reflect the new socio-economic make-up of the two zones. There are more low-
income households in zone 1 because of the policy change, so that the average income of zone
1 falls (-3.57%, row 20). The opposite is true in zone 2 (0.42%, row 21). This suggests that rent-
income ratios, the most common metric of housing affordability, must be interpreted carefully as
they reflect both equilibrium rents and the income of the people who have sorted into each zone
in spatial equilibrium. Even the fraction of rent-burdened households rises (4.45%, row 5); it does
not accurately reflect the improved access to affordable housing for lower-income households.

76
 Figure 23: Policies That Better Target RS—No Migration Model
Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

2 1
Inc Qual New
Inc Qual All
1 0.5 Inc Qual Stay
0.50 RS discount
RS size
0 0

-1 -0.5

-2 -1

-3 -1.5
40 60 80 1 2 3 4
Age Productivity Level
Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

1 0.5

0.5
0
0

-0.5
-0.5

-1

-1
-1.5
1 2 3 4 1 2 3 4
Income Quartile Net Worth Quartile
Notes: The baseline model has the following parameters: η 1 = 56.37, η 2 = 29.72, κ1 = 7%, κ2 = 1000.00, κ3 = 0.50. Policy experiments,
each panel: Top left panel: by age. Top right panel: by productivity level. Bottom left panel: by income quartile. Bottom right panel:
by net worth quartile. The welfare changes are measured as consumption equivalent variations for an average household in each
group.

Re-applying Each Period In the second policy experiment, we force every household to go
through the RS lottery each period (four years). We enforce an income limit of 60% of AMI in both
zones on every RS tenant each period. (The RS market in zone 2 does not clear if we set the income
cutoff at 50% of AMI in zone 2 in this experiment.) By setting the parameter p RS,exog = p RS , the
endogenously determined probability of winning the RS lottery increases substantially. Relative
to the baseline model, this policy removes the preference for insiders and introduces income qual-
ification. The results in column (2) show that access to insurance improves dramatically (112.23%).
The fraction of low-income households in RS grows strongly (31.23%), showing the improved tar-
geting of the RS system. The overall fraction of households in RS also increases (6.51%), as poorer
households choose smaller RS unit size.
However, the policy experiment results in a substantial welfare loss of -0.78% per New Yorker
(and -0.55% in the long-run). The reason is twofold. First, the policy dramatically lowers the
stability of this insurance (-82.57%, row 28). Housing consumption becomes more unstable over
time (2.38%, row 30). Second, the higher churn of RS residents lowers the average tenure and
thereby the average rent discount that RS tenants enjoy, since the rent discount κ1 (d) is rising
in tenure d. The policy effectively makes the average RS housing unit less affordable. In sum,
the welfare gains from better targeting are more than offset by the welfare losses from excessive
housing churn.
The (endogenously) lower average rent discount in the RS system reduces distortions for de-
velopers. It results in higher developer profits (row 26) and a larger equilibrium housing supply
in both zones (rows 12 and 13). With more RS tenants living in smaller housing units, the fraction

77
of the population that lives in the urban core grows (9.37%). Again, rent-income ratios and rent
burden suggest worsened housing affordability, but actually pick up composition effects due to
spatial resorting.

Restoring the Preference For Insiders In the third experiment, we introduce income quali-
fication for RS at 60% of AMI in zone 1 and at 50% of AMI in zone 2 but restore the preference for
RS insiders: p RS,exog is set to its baseline value. Compared to the baseline model, the experiment
in Column (3) of Table 10 imposes an income cutoff on all RS residents in every period. Com-
pared to the experiment in Column (1), the income cutoff applies also to existing RS residents, not
only to new entrants. This experiment strikes us as the most realistic way of introducing income
qualification.
This experiment generates a welfare gain of 0.32% (and 0.73% in the long run). It combines the
fairness of income qualification while avoiding excessive churn in the RS system by giving prefer-
ence to income-qualifying insiders. The targeting of RS units to low income households increases
substantially (61.28%) as does the access to insurance metric (65.89%). These gains are much larger
than in column (1) since now income qualification is applied to existing tenants, reducing the mis-
allocation that builds up over time as tenant income grows. Making RS less persistent helps to
reduce the misallocation since it replaces high-income insiders with low-income outsiders. Like
in the previous experiment, the fraction of all households in RS increases (9.77%). In sum, the com-
bination policy with strict income targeting for all RS tenants and stability for insiders produces
non-trivial welfare gains, yet requires no expansion of the scope of the RS program nor additional
taxes.
The policy results in a larger share of retirees living in zone 1, lowering zone-1 average income,
thereby creating more income mixing. There is more commuting, even though a larger share of
the population lies in the urban core. A smaller share of top-productivity households live in zone
1. Figure 23 shows that the benefits from this experiment flow to the middle-aged and retirees.
There are substantial welfare gains for low-income households and welfare losses for all other
income quartiles.

Varying the RS Discount Column (4) of Table 10 reports on an experiment that changes the
size of the rent discount for RS units relative to the market rent. We multiply the entire discount
schedule, which is a function of tenure, by 0.5. Reducing the discount by half results in a large
welfare loss ( -0.96%).
There are interesting equilibrium effects on access to insurance. Making RS less generous
makes it less attractive, which reduces competition for it in the absence of income qualification,
and results in more bottom-quartile households ending up in it (18.78%). By this metric, the tar-
geting is almost as good as it is in the income qualification experiment of column (1). Access to
insurance improves, while the stability of that insurance does not deteriorate. The policy reduces
distortions to development, resulting in a higher equilibrium developer profits, a larger housing
stock, and lower rents. Finally, labor supply and output rise. All of these factors offset the negative
welfare effects from lower rent discounts which hurt low-income households the most.
Indeed, Figure 23 shows that a smaller discount hurts older and low-productivity households
the most. The age effect arises because the discounts are increasing in tenure and the effects on
rent are therefore largest in absolute magnitude for the old.

Maximum Size of RS Unit Column (5) of Table 10 studies a reduction in the maximum size
of a Rent Stabilized housing unit by 10% compared to the baseline. The actual average size of

78
RS units falls by slightly less than 10%. The aggregate welfare effect is a gain of 0.16% (0.37% in
the long-run). Making RS units smaller is an effective way of targeting RS units to low-income
households. Higher-income households do not want to live in small RS units. Because units are
now smaller but the same total square feet are devoted to RS, more households qualify. For both
reasons, access to insurance improves. The policy benefits low-income households. Overall, the
policy in column (5) is quite similar to the income qualification policy in column (1).

C.3.2 Expanding the Affordable Housing Mandate

The second set of policy experiments we study change the scope of the affordable housing man-
date. We symmetrically vary η ` in each zone, the share of all rental square footage that must be set
aside for affordable units. Table 11 shows the results, varying the share of RS housing from 0.25
to 1.75 times the value in the benchmark economy. In the migration model, we only computed
up to 1.5x the benchmark RS share. In the no-migration model, we can compute the 1.75x case as
well before running into the issue of excess supply of RS housing in equilibrium (and hence lack
of market clearing in the RS market).
As we increase the share of square feet of affordable housing η ` , the fraction of households
in RS also increases (row 3), as does the share of low-income households in RS (row 4). Access
to insurance and the stability of that insurance rise monotonically with the share of RS rentals.
Housing stability improves (row 30). Row 31 shows large welfare losses from strong cuts in the
scope of the RS system and large welfare gains from strong expansions of the RS mandate.
Expanding the RS mandate increases distortions. First and foremost, these distortions affect
developers. Developer profits fall, which in turn affects household wealth adversely. A larger
share of rental housing that is RS lowers the size of the equilibrium housing stock and increases
equilibrium rents and prices. It increases the fraction of households that are rent burdened. With
higher house prices and more RS rental households, the home ownership rate naturally falls in
the RS share. A second distortion comes in the form of labor market distortions. Labor sup-
ply and effective labor supply both decrease in the scope of the RS mandate. As a result output
falls. The spatial allocation of labor worsens as the RS share grows, with a smaller fraction of
top-productivity households and a larger fraction of retirees that live in zone 1. These are the
distortions from RS emphasized by economists. We find that they are quantitatively modest, even
though RS engenders much misallocation (no income qualification, preference for insiders).
How does welfare change with the scope of RS? Figure 24 shows that aggregate welfare, ex-
pressed as a percentage change relative to the benchmark, increases with the scope of the af-
fordable housing mandate. The insurance benefits from expanding the RS system continue to
outweigh the costs until all households that want RS obtain such a unit (large circle).
Figure 25 illustrates the redistributive effects of a more expansive RS system. A more ubiqui-
tous RS system benefits the young and the low-productivity/low-income households more. How-
ever, for large increases in RS, even the Q4-income and Q4-wealth households benefit. This is be-
cause an increasing fraction of them now also access RS; recall that there is no income qualification
for RS in this experiment. For example, the share of top income-quartile households in RS is three
times higher in the “1.75x” case as in the benchmark, and two times higher for the other income
groups. This highlights the value of RS insurance; the option value of needing RS in the event of
a future adverse income shock is valuable even to fairly rich households.
It is worth noting that our welfare criterion is calculated in the first period of the transition.
In the long-run, the average welfare gains are very similar (row 32 of Table 11). However, we
have confirmed that the first-period gains for high-income and high-wealth households are larger
than the long-run gains. The reason is that the housing stock has not adjusted much yet in the

79
 Table 11: Varying the Scope of the RS Mandate—No Migration Model

(1) (2) (3) (4) (5) (6)

Benchm. 0.25x 0.50x 0.75x 1.25x 1.50x 1.75x
1 Avg(rent/inc.) renters in Z1 (%) 32.0 -19.12% -11.81% -5.41% 4.85% 10.54% 17.54%
2 Avg(rent/inc.) renters in Z2 (%) 35.5 7.27% 5.30% 2.97% -2.97% -5.25% -7.11%
3 Fraction of hhs in RS (%) 14.60 -160.66% -85.72% -37.55% 30.33% 56.20% 84.41%
4 Frac. in RS of those in inc. Q1 (%) 22.97 -162.10% -87.97% -39.62% 33.14% 58.88% 82.01%
5 Frac. rent-burdened (%) 58.5 -5.40% -3.50% -1.65% 1.79% 4.04% 9.08%
6 Avg. size of RS unit in Z1 (sf) 719 -0.22% -0.01% 0.04% 0.05% 0.04% -0.30%
7 Avg. size of RS unit in Z2 (sf) 719 0.77% 0.68% 0.31% -0.48% -0.64% -1.09%
8 Avg. size of a Z1 mkt unit (sf) 1079 -9.12% -7.51% -6.06% 10.60% 17.73% 70.80%
9 Avg. size of a Z2 mkt unit (sf) 1861 -5.64% -4.13% -2.20% 2.70% 6.54% 12.08%
10 Frac. of pop. living in Z1 (%) 10.5 -1.90% -0.98% 0.80% -2.16% 0.51% 9.46%
11 Frac. of retirees living in Z1 (%) 19.6 -44.50% -29.34% -15.68% 14.17% 23.91% 48.62%
12 Housing stock in Z1 – 0.38% 0.32% 0.24% -0.22% -0.45% -1.08%
13 Housing stock in Z2 – 0.76% 0.55% 0.28% -0.27% -0.60% -0.84%
14 Rent/sf Z1 ($) 3.98 -1.37% -1.01% -0.44% 0.34% 1.31% 4.49%
15 Rent/sf Z2 ($) 1.43 -0.64% -0.44% -0.22% 0.33% 0.71% 1.49%
16 Price/sf Z1 ($) 993 -1.33% -0.98% -0.43% 0.33% 1.28% 4.24%
17 Price/sf Z2 ($) 278 -0.65% -0.44% -0.23% 0.33% 0.70% 1.48%
18 Homeownership rate in Z1 (%) 33.9 28.54% 19.21% 7.07% -2.01% -17.33% -133.25%
19 Homeownership rate in Z2 (%) 62.2 9.16% 7.04% 4.23% -4.71% -9.17% -13.13%
20 Avg. inc. Z1 working-age HHs ($) 163506 14.21% 9.70% 2.73% 0.12% -8.75% -55.94%
21 Avg. inc. Z2 working-age HHs ($) 99463 -2.82% -1.92% -0.62% 0.15% 1.60% 8.15%
22 Frac. of top-prod. HHs in Z1 (%) 17.2 21.56% 13.45% 3.64% -0.72% -17.21% -169.76%
23 Total hours worked – 0.60% 0.43% 0.24% -0.29% -0.58% -0.95%
24 Total hours worked in effic. units – 0.58% 0.44% 0.24% -0.11% -0.36% -0.87%
25 Total output – 0.30% 0.21% 0.10% -0.11% -0.26% -0.54%
26 Total commuting time – -0.41% -0.33% -0.36% 0.52% 0.43% 0.06%
26 Developer profits – 2.15% 1.60% 0.89% -0.76% -1.67% -2.88%
27 Access to RS insurance (%) 5.9 -179.47% -97.41% -45.72% 41.25% 77.31% 114.48%
28 Stability of RS insurance (%) 80.8 -1.05% -0.85% -0.39% 0.49% 0.91% 1.36%
29 Std. MU growth, nondurables 0.60 -0.30% -0.27% -0.16% 0.12% 0.23% 0.41%
30 Std. MU growth, housing 0.59 1.52% 1.38% 1.21% -2.42% -1.81% -2.43%
31 Aggr. welfare change (NY pop) – -0.98% -0.90% -0.68% 0.53% 1.36% 2.20%
32 Aggr. welfare change (Long-run) – -1.39% -1.06% -0.59% 0.63% 1.42% 2.44%

Notes: Column “Benchmark” reports values of the moments for the baseline model.

first period of the transition and neither have rents risen much yet. As a results, profits from
development, which go disproportionately to the rich, have not suffered much yet.

C.3.3 Geographic Location of Affordable Housing

Columns (1) and (2) of Table 12 conduct two policy experiments that shift all RS housing from
zone 1 to zone 2. In the experiment of Column (2), the experiment is coupled with subsidized
transportation for RS tenants. The total transit subsidy amount is $800 million (about 0.05% of
metro-area GDP) and the subsidy is the same for all recipients. The subsidy is paid for by higher
income taxes, engineered through a lower λ. By construction, the experiment keeps the number
of households in RS constant (row 3 shows that this is essentially the case).
There is an aggregate welfare gain of 0.37% in Column (1) and 0.96% in the experiment with
subsidized transit. The reason for the larger gain is that financial transportation costs are im-
portant for low-income households. Subsidizing this cost makes RS housing substantially more

80
 Figure 24: Aggregate Welfare for Varying RS Share—No Migration Model.
2.5

Welfare Relative to Benchmark (%)

1.5

0.5

-0.5

Risk aversion=5
Benchmark
-1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
RS Share of Rented Sqft
Notes: The baseline model has the following parameters: η 1 = 56.92, η 2 = 31.63, κ1
= 7%, κ2 = 1000.00, κ3 = 0.50. The share of
RS rented sf is calculated as a (total sf-)weighted average of η 1 , η 2 . The larger dots on
the right-hand side of the graph represent the
maximum RS share above which markets do not clear. The welfare changes are measured as consumption equivalent variations for
an average household.

Figure 25: Varying the Scope of the RS Mandate—No Migration Model.

Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

4 3
0.25 RS share
0.50 RS share
3 0.75 RS share
2
1.25 RS share
2 1.50 RS share
1.75 RS share
1
1
0
0

-1
-1

-2 -2
40 60 80 1 2 3 4
Age Productivity Level
Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

3 3

2 2

1 1

0 0

-1 -1

-2 -2
1 2 3 4 1 2 3 4
Income Quartile Net Worth Quartile
Notes: The baseline model has the following parameters: η 1 = 56.92, η 2 = 31.63, κ1 = 7%, κ2 = 1000.00, κ3 = 0.50. Policy experiments,
each panel: Top left panel: by age. Top right panel: by productivity level. Bottom left panel: by income quartile. Bottom right panel:
by net worth quartile. The welfare changes are measured as consumption equivalent variations for an average household in each
group.

81
attractive to those low-income households. The policies result in greater share of low-income
households in RS, better access to RS insurance, more housing stability, and slightly lower volatil-
ity of marginal utility growth of non-housing consumption.
In both experiments, the urban core gentrifies with more high-income and top-productivity
households and a much higher home ownership rate in zone 1. There are also a lot fewer re-
tirees in zone 1. As anticipated, moving affordable housing to the suburbs improves the spatial
allocation of labor. This can be seen in the growing wedge between labor supply and labor sup-
ply in efficiency units. It shows that high-productivity households supply relatively more labor
compared to the average worker, relative to the benchmark model. The increases in aggregate
labor supply is smaller in the case with subsidized transit since the subsidy dissuades labor sup-
ply among the recipients and because the transit is paid for with distortionary income taxation.
Aggregate output actually falls in column (2); the distortionary labor income effects dominate the
productivity gains from a better spatial allocation of labor.
The reform eliminates developer distortions in zone 1, which results in a higher housing stock
in zone 1 and lower market rents. Developer distortions increase in zone 2, lowering the housing
stock in zone 2. Market rents also fall in zone 2 which is due to the lower aggregate demand
for rentals in zone 2 after higher-income renters leave for zone 1. Rent-income ratios fall in both
zones, and the share of rent-burdened households fall by the most in any of our experiments.
Figure 26 shows that this policy benefits households across the age distribution, but more so
young and old households. It benefits low-productivity, low-income, and low-wealth households
the most. This is consistent with more of the RS going to low-income households in this experi-
ment.

C.3.4 Upzoning the Urban Core

The next experiment studies a zoning change that allows for more housing in the urban core. We
1
increase H by 10%. The equilibrium housing stock in Manhattan increases by 9.13%, as shown
in Column 3 of Table 12. Since a fixed fraction of rental square feet must still be set aside for
RS units, the expansion in the housing stock also creates more affordable units in zone 1. This
is akin to mandatory inclusionary housing policies. This policy is welfare increasing with an
aggregate benefit of 0.40% (and 0.33% in the long-run). Rents and prices in zone 1 fall (-0.74% for
rents). The population share of Manhattan rises by 9.89%. Because of the population reallocation,
average income in Manhattan decreases (-3.73%). Average rent-to-income among renters increases
by 0.11% in zone 1, despite the rent decline. More middle-income households can now afford
Manhattan and the additional affordable housing that is associated with the new construction
also brings in lower-income households. Hence upzoning increases the socio-economic diversity
of the urban core. With more working-age households in zone 1, aggregate commuting time falls
substantially (-1.22%). In equilibrium, most of the aggregate time saved commuting goes towards
leisure, boosting utility. Output is nearly unaffected.
The housing stock in zone 2 falls in the long-run (-0.53%) as developers shift their resources
towards zone 1 where the population has swelled. Rents in zone 2 also fall (-0.78%) because of
the weaker demand for housing in zone 2 due to the population pivot to Manhattan. Average
rent-to-income among renters increases by 0.67% in zone 2. The rising rent-income ratios in both
zones obscure the fact that more households can now afford to live close to work. Again, it mostly
reflects compositional changes in the income profile of each neighborhood.
In contrast to the preceding policies, upzoning is much less redistributive in nature. As can be
seen in Figure 26, the upzoning policy brings modest but similar benefits to all age, productivity,

82
 Table 12: Spatial Housing Policies—No Migration Model

(1) (2) (3) (4) (5) (6) (7)

Benchm. All RS All RS Zoning Vouchers LIHTC Cash Cash
in Z2 in Z2 + transit transfer V transfer P
1 Avg(rent/inc.) renters in Z1 (%) 32.0 -29.50% -29.26% 0.11% 0.24% -0.18% 0.30% 0.02%
2 Avg(rent/inc.) renters in Z2 (%) 35.5 -1.90% -2.38% 0.67% 0.27% -0.15% 0.42% 2.43%
3 Fraction of hhs in RS (%) 14.60 -0.10% 0.76% 0.76% 0.28% 0.19% -1.52% 0.79%
4 Frac. in RS of those in inc. Q1 (%) 22.97 21.40% 16.76% 0.51% -0.64% 1.07% -2.52% 0.98%
5 Frac. rent-burdened (%) 58.5 -7.51% -8.64% 0.96% -0.06% 0.20% -0.10% 0.70%
6 Avg. size of RS unit in Z1 (sf) 719 – – 0.11% -0.03% 0.06% -0.02% -0.00%
7 Avg. size of RS unit in Z2 (sf) 719 0.70% 2.09% -0.40% 0.21% -0.11% 0.32% 0.08%
8 Avg. size of a Z1 mkt unit (sf) 1079 -11.78% -11.62% -1.19% 0.14% 0.01% -0.25% -1.13%
9 Avg. size of a Z2 mkt unit (sf) 1861 2.40% 2.17% 0.62% -0.35% 0.30% -0.40% 0.15%
10 Frac. of pop. living in Z1 (%) 10.5 -1.29% -1.40% 9.89% 0.09% 0.14% 0.35% 0.74%
11 Frac. of retirees living in Z1 (%) 19.6 -68.09% -68.01% 1.31% -0.87% 0.20% -1.12% -0.78%
12 Housing stock in Z1 – 0.37% 0.41% 9.13% 0.15% 0.21% 0.23% 0.00%
13 Housing stock in Z2 – -0.49% -0.69% -0.53% -0.28% 0.24% -0.16% -0.02%
14 Rent/sf Z1 ($) 3.98 -1.41% -1.47% -0.74% 0.39% -0.34% 0.30% 0.04%
15 Rent/sf Z2 ($) 1.43 -0.26% -0.36% -0.78% 0.47% -0.42% 0.36% -0.02%
16 Price/sf Z1 ($) 993 -1.36% -1.36% -0.73% 0.33% -0.31% 0.24% 0.08%
17 Price/sf Z2 ($) 278 -0.28% -0.32% -0.78% 0.41% -0.38% 0.29% 0.03%
18 Homeownership rate in Z1 (%) 33.9 51.95% 51.16% -3.52% 0.40% -0.20% 0.74% -1.78%
19 Homeownership rate in Z2 (%) 62.2 -3.43% -4.55% 0.95% 0.26% 0.03% 1.79% -0.59%
20 Avg. inc. Z1 working-age HHs ($) 163506 18.71% 18.69% -3.73% 0.55% -0.23% 0.32% -1.21%
21 Avg. inc. Z2 working-age HHs ($) 99463 -3.89% -3.87% -0.02% -0.03% 0.03% -0.01% 0.18%
22 Frac. of top-prod. HHs in Z1 (%) 17.2 28.01% 27.94% 2.01% 1.18% -0.13% 1.03% -2.59%
23 Total hours worked – -0.18% -0.48% -0.02% -0.23% -0.01% -0.36% -0.79%
24 Total hours worked in effic. units – 0.22% 0.09% 0.11% 0.05% 0.01% 0.03% -0.03%
25 Total output – 0.04% -0.04% 0.03% 0.03% -0.01% 0.01% -0.02%
26 Total commuting time – -0.73% -0.71% -1.22% -0.00% -0.01% -0.04% -0.10%
26 Developer profits – -0.12% -0.50% -0.04% 0.27% 0.97% 0.11% -0.03%
27 Access to RS insurance (%) 5.9 13.12% 8.07% 2.28% -0.22% 1.67% -2.62% 1.19%
28 Stability of RS insurance (%) 80.8 0.43% 0.31% 0.04% 0.06% 0.02% 0.01% 0.03%
29 Std. MU growth, nondurables 0.60 -0.00% -0.15% -0.19% 0.87% -0.09% 0.86% -5.43%
30 Std. MU growth, housing 0.59 -0.57% -0.57% 1.46% 0.35% 0.24% 0.51% -2.24%
31 Aggr. welfare change (NY pop) – 0.37% 0.96% 0.40% 0.53% 0.05% 0.56% 4.66%
32 Aggr. welfare change (long-run) – 0.33% 0.90% 0.33% 0.48% 0.19% 0.66% 5.25%

Notes: Column “Benchmark” reports values of the moments for the baseline model.

income, and wealth groups. The real-world resistance to upzoning, usually driven by home own-
ers, can be understood from the lower equilibrium house prices which represent capital losses to
existing home owners (rows 16 and 17).

C.3.5 Housing Vouchers

We now study the effects of increasing the size of the housing voucher program. Voucher recip-
ients must earn less than 50% of AMI, receive $8300 per year in in-kind housing subsidies, and
must spend at least the voucher amount plus 20% of household income on housing expenditures.
There is a voucher lottery to allocate the vouchers. The $800 million voucher program is paid for
by higher income taxes, engineered through a lower λ.
Column (4) of Table 12 shows that aggregate welfare increases substantially when the housing
voucher program is expanded ( 0.53% in the transition and 0.48% in the long-run). Figure 26

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Welfare Relative to Benchmark (%) Figure 26: Spatial Housing Policies—No Migration Model

Welfare Relative to Benchmark (%)

2 1.5
All RS in Z2
All RS in Z2+transit
Zoning
1.5 1
Vouchers
LIHTC
Cash transfer V
1 0.5

0.5 0

0 -0.5
40 60 80 1 2 3 4
Age Productivity Level
Welfare Relative to Benchmark (%)

Welfare Relative to Benchmark (%)

1.5 1.5

1 1

0.5 0.5

0 0
1 2 3 4 1 2 3 4
Income Quartile Net Worth Quartile
Notes: The baseline model has the following parameters: η 1 = 56.37, η 2 = 29.72, κ1 = 7%, κ2 = 1000.00, κ3 = 0.50. Policy experiments,
each panel: Top left panel: by age. Top right panel: by productivity level. Bottom left panel: by income quartile. Bottom right panel:
by net worth quartile. The welfare changes are measured as consumption equivalent variations for an average household in each
group.

shows that low-productivity, low-income, and low-wealth households gain much from the policy,
while the wealthy and high-income households lose.
The welfare gain in the voucher experiment occurs despite severe distortions. Chief among
them is tax-induced labor supply distortions. In the model, as in the real world, vouchers must
be paid for with distortionary labor income taxes. Total hours worked fall (-0.23%). However,
hours in efficiency units (0.05%) and total output (0.03%) rise slightly by virtue of a more efficient
spatial allocation of labor. These distortions are much smaller than in the migration model since
high-productivity households who are hurt by the reform cannot leave for the outside MSA. As
a result, the tax increase that is needed to raise the $800 million in equilibrium to pay for the
vouchers is lower in the no-migration model.

C.3.6 LIHTC
Column (5) of Table 12 studies a policy that subsidizes construction costs associated with afford-
able housing development, modeled after the federal Low Income Housing Tax Credit (LIHTC)
`
program. The tax credit increases the average price P developers earn, thereby stimulating new
construction. The policy is sized to have the same cost of the voucher expansion and subsidized
transportation experiments in columns (2) and (4), namely $800 million. It generates a trivial wel-
fare gain of 0.05% in transition and a small welfare gain 0.19% in the long-run. The envisioned
increase in the housing stock materializes in equilibrium, but is blunted by a reduction in housing
demand. The latter arises from the distortionary taxation required to pay for the tax credits. Tax

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credits in difficult-to-develop gateway cities, like New York, create too few additional affordable
housing units to make a meaningful dent in the welfare of low-income households. This experi-
ment underscores the importance of general equilibrium effects, of targeted policies, and of how
affordability programs are financed. The welfare benefit per dollar of taxpayer money spent is
much lower for developer tax credits as for a voucher expansion.

C.3.7 Cash Transfers

Columns (6) and (7) of Table 12 study two policies that raise taxes by lowering λ, collecting an
additional $800 million in tax revenues, and then redistributes that extra revenue to households
earning less than 50% of AMI. For the policy in column (6), labeled cash transfer V, the benefit
allocation is the same as it was for the housing voucher program in column (4). For the policy in
column (7), labeled cash transfer P, recipients receive a recurring “check” for the maximum of zero
and X-30% of pre-tax household income. The parameter X = $5, 980 per household is set such
that the program costs exactly $800 million in the aggregate. About 7% of households receive a
positive cash transfer. The average transfer is $900.
Cash transfer V creates an aggregate welfare gain of 0.56%, only slightly higher than the
housing voucher program with the extra housing expenditure restriction. The policy is highly
redistributive, as can be seen from Figure 26.
Cash transfer P creates an aggregate welfare gain of 4.66%. This policy does not affect rents
or prices much nor does it have spatial effects. It mostly lowers the volatility of marginal utility
growth of non-housing and housing consumption, resulting in a large welfare gain. It serves
as a useful benchmark for gauging the welfare benefits of the housing policies. This policy is
perfectly targeted on the low-income households, and the size of the transfers grows the poorer
the household is. Like the other policies that increase taxes, the policy has adverse effects on
aggregate labor supply, but only modest effects on effective labor supply and output. This policy
does not create more housing nor does it alleviate rent burden. It has some interaction with RS
policy in that it results in a small increase in low-income households in RS units.

D Higher Depreciation of Rent-Stabilized Housing Units

Table 13 studies how sensitive the welfare changes from the RS expansion policy (50% increase in
the sf share devoted to RS) are to the depreciation rate on RS housing. In the benchmark policy
exercise, we assumed that the depreciation rate of market and RS rental housing was the same.
In this robustness check, we assume instead that the depreciation rate on RS housing is 0.5%
per year higher. The welfare gain is 0.52%, which compares to 0.91% in the main text. We
repeat this exercise for the no migration model. The welfare gain from RS expansion with higher
RS depreciation is 0.57%, compared to the baseline gain of 1.36%. In both cases, higher RS
depreciation moderates the welfare benefits from an expansion of rent regulation but does not
eliminate them.

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Table 13: Varying the Scope of the RS Mandate with Higher RS Depreciation–Migration
Model

(1)
Benchm. 1.50×
1 Avg(rent/inc.) renters in Z1 (%) 42.9 -7.28%
2 Avg(rent/inc.) renters in Z2 (%) 33.0 -0.49%
3 Frac. of HHs in RS (%) 12.70 72.01%
4 Frac. in RC of those in inc. Q1 (%) 15.56 58.00%
5 Frac. rent-burdened (%) 54.6 6.28%
6 Avg. size of RC unit in Z1 (sf) 847 -6.88%
7 Avg. size of RC unit in Z2 (sf) 851 -1.25%
8 Avg. size of a Z1 mkt unit (sf) 1142 -11.54%
9 Avg. size of a Z2 mkt unit (sf) 1816 9.92%
10 Frac. of pop. living in Z1 (%) 10.5 22.71%
11 Frac. of retirees living in Z1 (%) 15.0 25.31%
12 Housing stock in Z1 – -0.98%
13 Housing stock in Z2 – 0.18%
14 Rent/sf Z1 ($) 3.66 16.25%
15 Rent/sf Z2 ($) 1.30 3.21%
16 Price/sf Z1 ($) 863 4.47%
17 Price/sf Z2 ($) 248 0.46%
18 Homeownership rate Z1 (%) 49.8 -159.15%
19 Homeownership rate Z2 (%) 63.3 -7.07%
20 Avg. inc. Z1 working-age HHs ($) 186249 -53.29%
21 Avg. inc. Z2 working-age HHs ($) 98078 9.17%
22 Frac. of top-prod HHs in Z1 (%) 24.2 -88.08%
23 Total hours worked – -1.29%
24 Total hours worked in effic. units – -0.91%
25 Total output – -0.51%
26 Total commuting time – -3.40%
26 Developer profits – 1.84%
27 Access to RC insurance (%) 3.6 73.92%
28 Stability of RC insurance (%) 80.0 -0.31%
29 Std. MU growth, nondurables 0.60 -2.36%
30 Std. MU growth, housing 0.60 -5.36%
31 Aggr. welfare change (NY pop) – 0.52%
32 NY population – -1.32%
33 Aggr. welfare change (no migr.) – 0.57%

Notes: Column “Benchmark” reports values of the moments for the baseline model with higher depreciation of RS units.

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