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In this paper, we investigate to what extent shocks in housing and …nancial mar- kets account for wage and employment variations in a frictional labor market. To explain these interactions, we use a model of job search with accumulation of wealth as liquid funds and residential real estate, in which house prices are randomly persistent. First, we show that reservation wages and unemployment are increasing in total wealth. And, second, we show that reservation wages and unemployment are also responsive to the composition of wealth. Speci…- cally, when house prices are expected to rise, holding a larger share of wealth as residential real estate tends to increase reservation wages, which deteriorates employment transitions and increases unemployment. We estimate our model structurally using National Longitudinal Survey of Youth data from 1978 to 2005, and we …nd that more relaxed house …nancing conditions, in particular lower down payment requirements, decrease employment rates by 5 percentage points in the short run and by 2 percentage points in the long run. We also …nd that worse labor market conditions immediately increase homeownership rates by up to 5 percent points, whereas in the long run homeownership decreases by 8 percentage points.
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WORKING PAPER NO. 15-27
INTERACTIONS BETWEEN JOB SEARCH AND HOUSING
DECISIONS: A STRUCTURAL ESTIMATION
Sílvio Rendon
Federal Reserve Bank of Philadelphia
Núria Quella-Isla
Barnard College, Columbia University
July 2015
Interactions Between Job Search and Housing
Decisions: A Structural Estimation
Sílvio Rendon Núria Quella-Isla
Federal Reserve Bank of Philadelphia Barnard College, Columbia University
July 2015
Abstract
In this paper, we investigate to what extent shocks in housing and …nancial mar-
kets account for wage and employment variations in a frictional labor market.
To explain these interactions, we use a model of job search with accumulation
of wealth as liquid funds and residential real estate, in which house prices are
randomly persistent. First, we show that reservation wages and unemployment
are increasing in total wealth. And, second, we show that reservation wages
and unemployment are also responsive to the composition of wealth. Speci…-
cally, when house prices are expected to rise, holding a larger share of wealth
as residential real estate tends to increase reservation wages, which deteriorates
employment transitions and increases unemployment. We estimate our model
structurally using National Longitudinal Survey of Youth data from 1978 to
2005, and we …nd that more relaxed house …nancing conditions, in particular
lower down payment requirements, decrease employment rates by 5 percentage
points in the short run and by 2 percentage points in the long run. We also …nd
that worse labor market conditions immediately increase homeownership rates
by up to 5 percent points, whereas in the long run homeownership decreases
by 8 percentage points.
Keywords:
job search, housing, savings, structural estimation
JEL
Classi…cations:
J64, E21, E24, R21
Our e-mails: silvio.rendon@phil.frb.org, nquella@barnard.edu. We thank participants at the
Midwest Macroeconomics Conference 2010 in East Lansing, the SED Meetings in Ghent 2011, the
NYU Alumni Conference in 2013, the NASM Conference in Minneapolis in 2014, the SED Meetings
in Toronto in 2014, the Eastern Economic Association Annual Conference in 2015 in New York City,
and seminar participants at Fordham University, Barnard College, Columbia University (SIPA),
Yeshiva University, Queens College, and the Federal Reserve Bank of Philadelphia. Special thanks
to Morris Davis for his useful comments. The views expressed in this paper are those of the authors
and not necessarily those of the Federal Reserve Bank of Philadelphia, the Federal Reserve System,
or the Office of the Comptroller of the Currency. This paper is available free of charge at
www.philadelphiafed.org/research-and-data/publications/working-papers/.
2
1 Introduction
The e¤ects of deteriorating labor market conditions on wealth accumulation have been
su¢ ciently studied and are well understood. The same cannot be said for the impact
of changes in home …nancing conditions and the composition of wealth on job search
dynamics, which are relatively less understood. This paper aims to contribute to the
understanding of interactions between the housing and labor markets when there are
frictions in both, and individuals also face …nancial constraints. To this purpose, we
develop a model of job search, extended to allow for accumulation of wealth as liquid
funds and residential real estate, where individuals decide whether to own or rent a
house, and where house prices are stochastic. This allows us to evaluate the dynamics
of homeownership, wealth, unemployment, employment transitions, and wages under
varying home lending conditions.
In our economy, the underlying mechanism by which easier access to home credit
causes higher unemployment rates is a change in reservation wages. While the labor
literature has already established that wealth matters in the process of job search,
we go on to a¢ rm that, in the presence of frictions in the housing market, it is
also the composition of wealth that matters. Thus, house price ‡uctuations have an
ect on the labor market via changes in wealth. We show that when house prices
are low, and are therefore expected to increase, reservation wages increase in the
share of residential wealth. This, in turn, deteriorates employment transitions and
decreases employment. And so, we are able to explain the robust empirical …nding
that homeownership is positively associated with unemployment.
We perform the structural estimation of our model using data from the National
Longitudinal Survey of Youth (NLSY) and show that it accurately replicates the
main observed trends of homeownership rates, residential and liquid wealth, as well
as wages, employment status, and employment transitions. With the recovered struc-
tural parameters, we evaluate the e¤ect of unexpected counterfactual regime varia-
tions in the housing market on the labor market and the reciprocal e¤ect of regime
variations in the labor market on the housing market. We …nd that more relaxed
home lending conditions increase workers’reservation wages, making them more se-
lective in their job search. This, in turn, causes their employment transitions to
deteriorate: Job …nding and job-to-job transition rates decline, while job loss rates
increase, so that the overall employment rate decreases by 5 percentage points in the
short run and 2 percentage points in the long run. We also …nd that worsening labor
3
market conditions decrease homeownership rates by 8 percentage points in the long
run, though they brie‡y increase by 5 percentage points in the very short run.
In the last decade, the economic literature has paid increased attention to housing
as the household’s main investment and, consequently, to its important macroeco-
nomic e¤ects. Several dynamic models consider housing in their analysis of business
cycles and taxation, notably Gervais (2002); Sanchez (2007); Yang (2009); Díaz and
Luengo-Prado (2010); Chambers, Garriga, and Schlagenhauf (2011); Fisher and Ger-
vais (2011); Bajari et al. (2013); Iacoviello and Pavan (2013); Sommer, Sullivan, and
Verbrugge (2013); Head, Lloyd-Ellis, and Sun (2014); and Hedlund (2014a, 2014b).
However, in all these models the individual’s income process is assumed to be exoge-
nous, so that the housing market has no direct ect on the labor market. Rather,
it is deteriorating labor market conditions (and lending to those who lose their jobs)
that have a negative impact on the housing market.1
A very recent and growing strand of literature does establish a connection between
housing and job search via the “lock in ect, whereby declining home prices put
homeowners with home loans “under water.”
2These homeowners then have less home
equity to use as a down payment for a new mortgage loan in a new geographical
location. They, therefore, are forced to reject job o¤ers that would require them to
move. See, for example, Laufer (2008), Sterk (2010), Head and Lloyd-Ellis (2012),
Rupert and Wasmer (2012), Davis et al. (2013), Karahan and Rhee (2013), and Nenov
(2013). This mechanism appears to provide strong support for the positive association
between homeownership and unemployment rates …rst shown by Oswald3(1997), then
later by Bover, Muellbauer, and Murphy (1989), Chan (2001), Engelhardt (2003),
Mian and Su… (2011), Ferreira, Gyourko, and Tracy (2010, 2012), and Blanch‡ower
and Oswald (2013). Thus, the lock in e¤ect temporarily gained traction as one of
the main explanatory mechanisms of the Great Recession. However, more recently,
1Haurin, Hendershott, and Wachter (1997) …nd ownership to be quite sensitive to potential
earnings, the cost of owning relative to renting, and especially borrowing constraints. Chambers,
Garriga, and Schlagenhauf (2009) show in a quantitative exercise that innovations resulting in a
lowering of the downpayment requirement can help explain the rise— and fall— in homeownership
since 1994.
2Homeowners are “under water”when they owe more on their mortgage than their home is worth
on the market.
3Oswald’s nding was based upon analysis of time series and cross-section data for Organisation of
Economic Co-operation and Development (OECD) countries and for regions within selected OECD
countries. Green and Hendershott (2001) replicated Oswald’s results for the U.S. states and by
age group within states. Flatau, Forbes, Hendershott and Wood (2003) provided a summary of
the empirical results on the relationship between unemployment and homeownership, with some
empirical tests supporting this nding and some refuting it.
4
Aaronson and Davis (2011), Mumford and Schultz (2013), and Valletta (2013) …nd
evidence that homeowners are not less likely to move from their current location than
renters, which challenges the existence of a housing lock in e¤ect.
This article’s approach stems from the literature on how wealth accumulation
in‡uences job search.4Our explanation for this interaction, however, is not based
on geographical mobility, but rather, it is based on the e¤ect of homeownership and
wealth on reservation wages. We show that individuals are more selective in their job
search the larger the share of residential wealth in their total wealth and the more
relaxed their credit conditions.5Thus, our mechanism also provides an explanation
for the positive relation between homeownership and unemployment rates found in
the earlier empirical studies. But, contrary to the lock in mechanism, we …nd a
positive association between house prices and unemployment. When house prices
are low, agents expect prices to rise and their wealth to increase, so they are more
selective in their job search and unemployment increases, the more so the larger the
percentage of wealth held in the form of residential real estate. This does not happen
because agents with relatively more residential wealth are locked in and have thus
less geographical mobility, but rather, because they are more selective in their job
search.
Beyond a¤ecting labor supply decisions, the housing crisis deteriorated labor
demand by putting …nancial institutions in distress, which tightened liquidity con-
straints for employers who, in turn, reduced labor demand.6We address this declining
labor demand by simulating displacements of the wage o¤er distribution and increases
in the layo¤ rate.
To estimate our model, we input its policy rules in a simulated method of moments
(SMM). The NLSY provides us with a detailed work history of individuals in the U.S.
from 1978 until 2010, including their employment transitions, wages, and several types
of wealth, including residential wealth. The model …ts reasonably well with the data
on the evolution of homeownership rates, the composition of wealth as residential real
4The job search model in this paper is set up along the lines of Mortensen (1977) and Burdett and
Mortensen (1998) and includes wealth accumulation as in Danforth (1979), Acemoglu and Shimer
(1999), Costain (1999), Rendon (2006), Lentz (2009), and Lise (2013).
5A mechanism similar to ours is provided by Herkenho¤ and Ohanian (2013) who emphasize the
ect of foreclosure delays on unemployment with a detailed chararecterization of the reservation
wage. Delayed foreclosures alleviate liquidity contraints in housing with the e¤ect of increasing
reservation wages and unemployment.
6According to our model, falling house prices imply lower reservation wages, which should have
set, at least partly, the increase in unemployment due to a falling labor demand. Testing for this
mechanism during the Great Recession is, however, beyond the purpose of this paper.
5
estate and liquid funds, as well as wages, employment, and employment transitions.
Once the behavioral parameters are recovered, we evaluate the dynamics of em-
ployment, employment transitions, wealth accumulation, and homeownership under
two counterfactual scenarios: (i) more relaxed conditions for housing credit and (ii)
deterioration of labor market demand. We accomplish these scenarios by modifying
the underlying parameters that we estimated previously. The …rst regime change aims
to assess the evolution of lending conditions for housing in the early 2000s, whereas
the second regime change endeavors to evaluate the decline in hiring by …rms once
the crisis broke up in the late 2000s.
The …rst regime change reveals that more lenient home loan conditions, in par-
ticular a reduction of 15 percentage points in down payment requirements, decrease
employment substantially, by up to 2 percentage points. The second regime change
shows that falling labor market demand, in particular a reduction in the wage o¤er
of 20 percentage points, decreases homeownership rates by up to 8 percentage points.
Furthermore, we also evaluate the impact of …scal policies that extend unemployment
bene…ts,7and we …nd that there is an additional impact of 3 percentage points on
the employment rate. On the other hand, this policy is able to reduce the drop in
homeownership rates by 2 percentage points.
The remainder of the article is organized as follows. The next section explains
the model and characterizes the optimal solution. Section 3 describes the data used
in the estimation and documents their basic trends. Section 4 details the estima-
tion procedure, namely the criterion function of the SMM estimation and the target
statistics. Section 5 presents the results of the estimation, the behavioral parame-
ters and an assessment of how well the model …ts the data. Section 6 performs the
policy experiments mentioned previously. The main conclusions of this article are
summarized in Section 7. A sensitivity analysis, proofs, details on the solution of the
model, and further explanation of the data used in the estimation are contained in
the appendices.
2 Model
In our model, individuals maximize their expected lifetime utility by choosing their
home size and home tenure status (owning versus renting the house in which they
7This has been discussed, among others, by Rothstein (2011), Hagedorn et al. (2013), and
Mitman and Rabinovich (2014).
6
live), the level of consumption of nondurables, and acceptable wage o¤ers.
Agents …nish their schooling and immediately enter the labor market in quarter 1.
That is, agents are active, employed, or unemployed, during t2 f1; :::; T gquarters,
then retire and live for t2 fT+ 1; :::; TFgadditional quarters. There are no bequests,
so agents die in period TFwithout assets.
Speci…cally, in quarter 1, agents are age 18, then stay in the labor market for a
total of 47 years until age 65, when they retire; that is, agents are active for T= 188
quarters. Once agents retire, they live an additional 16 years (or 64 quarters) until
the age of 81; that is, agents live for a maximum of TF= 252 quarters.
In each period t, individuals derive utility from consumption of nondurables, Ct,
and from consuming the services of a house of size Ht2H; H, which they can
own or rent, so that their period-by-period utility function is U(C; H). Renters can
adjust the level of housing services they consume without cost as long as they remain
renters. But …rst-time buyers and owners who change the size of their house must
bear a price-dependent adjustment cost of the form
c(Ht1; Ht; pt) = pt(HtHt1) + abptmax (HtHt1;0) asptmin (HtHt1;0) ;
where ptis the price per house unit in period t, so that house value is ptHt1;ab
is equivalent to a fee per unit of increase in house size, and asis a similar fee per
unit of decrease in house size.8When there is no variation in house size, c() = 0.
There is no house depreciation and no house maintenance spending. The house price
follows a Markov process P(pt+1jpt), parameterized as an AR(1) process: ln pt+1
Nln pt; 2
p, where pt2p; p;0< p < p < +1.
Rental payments are rhHt, where rhis the rent per house unit. Renting does not
give the same utility as owning; so a renter of a house of size Htonly enjoys a fraction
gof that house size: gHt.
Homeowners can rent part of their house and live in the remaining space, denoted
by st2[0;1]. If they do, they receive an extra income equal to (1 st)rhHt, but
they only derive utility from the part of the house that is not rented stHt.
Agents can buy or sell a house throughout their lifetime, but they can only ex-
perience employment transitions during their active period, that is, when tT.
Unemployed agents receive transfers b, which include nonlabor income (such as fam-
ily transfers), plus unemployment compensation net of out-of-pocket search costs.
8This cost is equivalent to requiring the house be sold each time there is a variation in house size,
as in Flavin and Nakagawa (2008).
7
Retired agents receive a pension bRfor t>T. These transfers allow agents to rent
at least the smallest house, so that they are never homeless. This constitutes the
no-homelessness condition:
b; bRrhH:
All agents enter active life as unemployed and with an initial stock of liquid wealth
A0. Agents have a subjective discount factor 2(0;1), can save at a rate of return
r, and can borrow up to a limit Bt+1 at that same rate. This borrowing limit is
determined by the liquidation value of a house owned in property that is used as
collateral:9
Bt+1 (Ht) kt(1 d)p(1 as)Ht;
where dis the down payment rate; pis the lowest possible price per house unit and
constitutes a “natural borrowing limit”for banks, that will not lend more than what
they can recover with probability one; asis the adjustment fee when a house is sold,
and it must be deducted from the house value, as it will not be available to the owner in
the event of a sale. And the time adjustment factor kt=(1+r)T(1+r)t
(1+r)T1is a stylized way
of capturing a house mortgage loan in a …nite horizon environment without an extra
state variable.10 Because this constraint becomes tighter every subsequent period,
the indebted homeowner will want to pay this home loan. Thus, this borrowing limit
ensures that home loans are always repaid and there is no bankruptcy.
Unemployed agents receive a wage o¤er xwith probability u
t, drawn from a wage
er distribution F(), (x2(w; w);0< w < w < 1). They become employed if
they receive and accept a wage o¤er; otherwise, they remain unemployed. Employed
agents are laid o¤ with probability t, and with probability t, they receive a wage
er drawn from the same distribution F(). They change employer when they receive
an o¤er and accept it. They continue to work for the same employer when they are
not laid o¤, when they receive an o¤er that they do not accept or when they do not
9Banks do not lend to retirees, not on account of their pension, nor on account of their home, if
they own it.
10 The idea here is that if at period 0 the agent borrows (1 d)p(1 as)Ht, then from pe-
riod 0until period Tthe agent must make a …xed payment mper period, as if it were a
mortgage, so that (1 d)p(1 as)Ht=PT
s=0
m
(1+r)s. Consequently, the lower bound on liq-
uid wealth at period tis Bt=PT
s=t
m
(1+r)st, that is, Bt=(1 d)ktp(1 as)Ht, where
kt=PT
s=t
1
(1+r)st
PT
s=0 1
(1+r)s=1(1
1+r)Tt
1(1
1+r)T=(1+r)T(1+r)t
(1+r)T1. A fraction sof future secure income could also
be included in the borrowing limit: Bt+1 =sPT
j=t
brhH
(1+r)jkt(1 d)p(1 as)Ht. We previously
conducted estimations of this speci…cation and found swas very close to zero.
8
receive an o¤er at all and the current job is preferable to unemployment. Employed
individuals can always quit to become unemployed. Arrival rates t, layo¤ rates
t, and wages wt(!)are age-speci…c. The evolution by age of these labor market
parameters captures the accumulation of human capital over time.
The present discounted utility Vr
tfor an individual who decides to rent, has liquid
wealth holdings At, income y, for a house of size Ht1and unit house price pt, is
Vr
t(At; y; Ht1; pt) = max
At+1Bt+1 ;Hr
tUAt+w(y)c(Ht1;0; pt)rhHr
tAt+1
1 + r; gH r
t
+EtVt+1 (At+1; y; 0; pt)g:
As explained previously, a renter only derives utility from a fraction of the total rented
house, gHr
t. A renter who owned a house in the past period must have sold it, so
c(Ht1;0; pt). When the individual is indebted, liquid assets are negative.
If the agent decides to own a house, the present discounted utility Vo
tfor the same
state variables is
Vo
t(At; y; Ht1; pt) = max
At+1Bt+1 ;Ht;stUAt+w(y)c(Ht1; Ht; pt) + (1 st)rhHtAt+1
1 + r; stHt
+EtVt+1 (At+1; y; Ht; pt)g:
This owner may decide to rent a fraction of his property and earn additional income
(1 st)rhHt, which implies that the remaining fraction of the house yields some
satisfaction, stHt.
Then, the value function is the maximum between the value of renting and the
value of owning:
Vt(At; y; Ht1; pt) = max [Vr
t(At; y; Ht1; pt); V o
t(At; y; Ht1; pt)] ;
When the agent is retired, t=T+ 1; :::; TF,y= 0,w(0) = bR, and the agent only
expects ‡uctuations in future house prices:
EtVt+1(At+1 ;0; Ht; pt) = ZVt+1 (At+1;0; Ht; pt+1 )dP (pt+1jpt):
When the agent is active, t= 1; :::; T . If the agent is unemployed, then y= 0,
w(0) = b, and
9
EtVt+1(At+1 ;0; Ht; pt) =
t+1ZZ max [Vt+1(At+1; x; Ht; pt+1); Vt+1 (At+1;0; Ht; pt+1)] dF (x)dP (pt+1 jpt)
+(1 t+1)ZVt+1 (At+1;0; Ht; pt+1 )dP (pt+1jpt):
And if the agent is employed, y=!, and
EtVt+1(At+1 ; !; Ht; pt) = f(1 t+1 )
t+1ZZ max Ve
t+1(At+1 ;max (x; !);0; pt+1 ); V u
t+1(At+1 ;0; pt+1)dF (x)dP (pt+1 jpt)
+ (1 t+1 )Zmax Ve
t+1(At+1 ; !; 0; pt+1 ); V u
t+1(At+1 ;0; pt+1))dP (pt+1 jpt)
+t+1 t+1ZZ max Ve
t+1(At+1; x; 0; pt+1); V u
t+1(At+1 ;0; pt+1)dF (x)dP (pt+1 jpt)
+(1 t+1)ZVu
t+1(At+1;0; pt+1)dP (pt+1 jpt):
In the absence of bequest motives, ATF+1 = 0 and HTF= 0. Active agents solve a
dynamic problem with a …nite horizon Tand a “salvage value,”which is the present
discounted utility at retirement: Vt(At; !; Ht1; pt) = Vt(At;0; Ht1; pt), at t=T+ 1.
At all times, the solution to this problem is contained in the liquid wealth accumula-
tion rule and the housing decision rule: At+1(At; !; Ht1; pt)and Ht(At; !; Ht1; pt),
respectively. That is, wealth accumulation and house consumption depend on liquid
wealth, employment status, wages if employed, and on homeownership status and
house prices. See Appendix A1 for details on the solution. When agents are active,
there exists a reservation wage that indicates the lowest acceptable wage o¤er:
!
t(At; Ht1; pt) f!jVt(At;0; Ht1; pt) = Vt(At; !; t1Ht1; pt)g:
The reservation wage depends on holdings of liquid wealth, residential wealth, and
house prices. Thus, this model creates an explicit connection between wealth accu-
mulation, homeownership, house prices, and job transitions.
Proposition 1 If adjustment fees are zero (ab=as= 0), lifetime value functions
will be una¤ected by the liquidity composition of wealth and will only be determined
10
by the amount of total wealth Zt=At+ptHt1:
Vt(At; !; Ht1; pt) = VtZt; !; (1 )Zt
pt
; pt;
for all 2[0;1],At; !; Ht1, and pt.
Proof: In Appendix A2.
If there are no adjustment fees, objective functions are una¤ected by the compo-
sition of wealth in liquid funds and illiquid property, and owning a house of size Ht1
has the same e¤ect on lifetime utility as holding its value ptHt1in liquid funds. In
other words, in the absence of adjustment fees, the composition of wealth is irrelevant
for an agent wishing to maximize lifetime utility; only the total amount of wealth is
relevant. This property is re‡ected in the reservation wage.
Corollary 1 As value functions are determined only by total wealth, so are reserva-
tion wages. If adjustment fees are zero (ab=as= 0), the reservation wage will be the
same whether the individual owns a house of size Ht1or whether she holds ptHt1in
liquid funds:
!(At; Ht1; pt) = !Zt;(1 )Zt
pt
; pt;
for all 2[0;1],At; !; Ht1, and pt.
If the housing market is frictionless and there are no adjustment costs, an individ-
ual’s reservation wage depends only on the total size of his wealth, and the liquidity
structure of this individual’s wealth is irrelevant. The “neutrality” of reservation
wages to a workers asset liquidity can be viewed as equivalent to the neutrality of a
rm’s investment to its asset structure in the Modigliani-Miller Theorem. If unem-
ployment allows workers to reject low wage o¤ers and attain better job matches, then
it is bene…cial for the whole economy, similarly to …rms’investment and increased
production.
Because there are no closed-form solutions to this model, we assume speci…c func-
tional forms. Our utility function is a Cobb-Douglas function embedded into a con-
stant relative risk aversion (CRRA) utility function:
U(C; H) = (CH1)11
1;
11
where is the coe¢ cient of relative risk aversion, and represents the share of
consumption of nondurables.
The base wage o¤er distribution is a truncated lognormal F(x):ln !N(; 2j!; !); 0 <
! < ! < 1:Age-dependent arrival and layo¤ rates are logistic: qk
t=exp(0
q+qt)
1+exp(0
q+qt),
where q=f; ; g:Finally, the wage growth function has the form wt(!) = !exp (1t+2t2):
Approximation to the policy rules and value functions is done numerically. We
allow wealth and wages to be continuous while we discretize house size and house
prices. Accordingly, we use the Euler equation and an interpolation algorithm to
solve for wealth next period and a numerical maximization to solve for housing. We
integrate the value functions for wages exploiting an interpolation technique while we
integrate over prices by using a weighted summation. The dynamic problem is solved
backward, starting with retirement and ending in period one. See Appendix A3 for
a detailed explanation of the numerical solution to the model.
[Figure 1 here]
The policy rules are illustrated in Figure 1, which shows the reservation wage and
the value of the house owned as a function of liquid wealth.11 Both are increasing in
wealth, that is, wealthier workers are more selective in their job search and buy larger
houses. The house value increases monotonically in liquid wealth until it reaches its
largest possible level. The reservation wage is lowest when the agent is most indebted,
grows rapidly as debt levels decrease, and it stabilizes when the agent buys the largest
possible house and saves.
[Figure 2 here]
Figure 2 shows the reservation wage and house size as a function of liquid wealth
for two di¤erent current house values and two di¤erent price levels. In Figures 2a and
2b, we can see that for all levels of liquid wealth, both owning a more valuable house
and facing a higher price per house unit imply higher reservation wages. Figure 2c
shows that increasing the value of the house currently owned by the individual (from
$67,000 to $121,000) implies owning a larger house also in the next period for any
11 We use the estimated parameters in Table 4 to characterize the model in these policy rules and
for the sensitivity analysis that follows. The benchmark house value is $67,000 at the lowest price,
0.825. We then increase the current house value to $121,000, and the price to 1.212 (see the price
vector on page 19).
12
level of liquid wealth. And, for any given level of liquid wealth Figure 2d shows a
higher house price implies owning a smaller house in the next period, in accordance
with the law of demand. After this price increase, a house initially worth $67,000
increases in value to $80,000, as indicated by the horizontal dotted lines. The agent
then switches ownership to a smaller house and increases his liquid wealth.
[Figure 3 here]
Figures 3 and 4 show the e¤ect of homeownership on the reservation wage when
total wealth is held constant at four di¤erent levels. In Figure 3, the price level is
low, whereas in Figure 4 the price level is held high. As per Proposition 1, when
adjustment fees are zero, the reservation wage is unected by the composition of
wealth. In our simulation adjustment fees are positive, so the composition of wealth
does a¤ect the reservation wage at all levels of total wealth. Note that when the value
of the house owned by the individual surpasses the total wealth level, the individual
is in debt, the more so the higher the value of the house. In Figure 3, we can see
that for individuals with the lowest level of total wealth ($48,000), the reservation
wage is unresponsive to low house values. However, when house values are high,
the individual is in debt and cannot a¤ord to be selective in his job search, as a
relatively low reservation wage indicates. Thus, the reservation wage of low-wealth
individuals is decreasing in house value only for higher levels of residential wealth.12
For a medium-to-low level of total wealth ($62,000), reservation wages are clearly
increasing in residential values. The agent expects current home prices to rise and,
therefore his total future wealth to increase, the more so the larger the proportion
of total wealth held as real estate. Accordingly, his reservation wage goes up. For
medium-to-high and high levels of total wealth ($76,000 and $99,000, respectively),
the reservation wage pro…le is unresponsive to residential value.
[Figure 4 here]
Figure 4 shows how the reservation wage responds to high house prices when
total wealth is held constant at four di¤erent levels. When house prices are high, the
reservation wage is markedly unresponsive to residential wealth for high and low levels
12 The agent with the lowest level of total wealth can only own a home worth more than $48,000
if he borrows. Buying a house worth $148,000 means this individual has a debt of $100,000, which
eventually will make him less selective in accepting job o¤ers.
13
of total wealth ($71,000 and $145,000, respectively). Only for medium levels of total
wealth is the reservation wage decreasing slightly in residential wealth. When house
prices are high, individuals expect them to fall; consequently, replacing residential
wealth with liquid wealth makes the agent slightly less selective in a job search.
In sum, this model exhibits an explicit mechanism whereby the housing market and
the individual’s …nances crucially a¤ect the job search. In this frictional labor market,
an individual’s amount of total wealth, the composition of this wealth, and price
uctuations in the housing market are very important determinants of an individual’s
wages and employment status.
Most analyses on the connection between the …nancial and real estate sectors focus
on …rms and on how a …rm’s collateral constrains how much the …rm can borrow.
When the Modigliani-Miller assumptions are not ful…lled, a …rm’s ability to expand
its scale of production depends on how much capital is available to the …rm. Hereby,
we show an additional connection centered on the worker and on job search. The
amount a worker can borrow is constrained by the value of the collateral available to
him. The single most important collateral of an individual is his house, so that his
ability to reject low-wage o¤ers in his job search throughout his active life will be
determined by the value of his house. Therefore, more collateral and easier housing
credit conditions allow an individual to be more selective in the job search and obtain
higher wages, but they also lengthen the duration of unemployment spells and, thus,
increase overall unemployment rates.
We also perform a sensitivity analysis of all the parameters of the model, which
we detail in Appendix A4. This analysis establishes the connection between para-
meter variations and the response of several observed statistics. We are particularly
attentive to “cross-ects,” that is, e¤ects of housing market parameters on labor
market observables, and of labor market parameters on housing market observables.
Both types of e¤ects are fairly important in the model and in our analysis of possible
regime changes. In the next sections, we examine how our model stands up to the
actual data.
3 Data
We use data from the National Longitudinal Survey of Labor Market Experience -
Youth Cohort (NLSY), a national strati…ed sample of 12,686 individuals who were be-
tween 14 and 21 years of age in January 1979 and who have been interviewed annually
14
from 1979 until 2010. This data set contains information on personal characteristics
of the individual, household composition, educational status and attainment, mili-
tary experience, labor market activity and transitions, detailed week-by-week work
history, income and several forms of wealth, including residential property.
Total wealth is the net market value of the sum of residential wealth and liquid
wealth. Residential wealth is the market value of the house owned by the individual
if it were sold at the time of the interview. Liquid wealth consists of business assets,
nancial assets, vehicles, and other assets (such as jewelry or furniture), all net of
debts, minus all debts on residential property. Annual data on residential wealth
and the various forms of liquid wealth are available from year 1985 onward; this
information is assigned to the calendar quarter in which the interview took place.
Out of the total number of respondents, we select only those individuals who are
most likely to conform to our theoretical model: white males born after December 31,
1960, without military experience, who …nished high school and attended college for a
maximum of one year,13 and for whom wealth and housing data are available. We are
left with a sample of 268 individuals. However, our theoretical model does not include
out-of-the-labor force as an employment status, nor does it consider search intensity.
Therefore, to avoid further reducing the sample, we also classify as unemployed those
individuals who are not working and not looking for a job, those who work less than
20 hours a week, and those for whom information on wages is missing.14 For each
individual in the sample, we have up to 132 quarterly observations on wealth, housing,
wages, employment status, and employer. Appendix A5 provides further explanations
on these variables.
[Table 1 here]
Table 1 presents summary statistics for the variables used in the model. On
average 37% of the individuals in the sample are homeowners, with the mean house
value around $60,000 of 1982-1984. However, the standard deviation on this value
is also around $60,000. These individuals hold, on average, around 12,000 constant
13 A sample exclusively composed of high school graduates would be too small. Therefore, our
sample also includes individuals who have attended college for a maximum of one year.
14 It is for this reason that our “unemployment”rates are higher than those reported in employment
surveys. To avoid confusion, we will use the term “nonemployment rate”to denote the percentage
of people who are not working (either because they are not searching or because they do not …nd
employment), are underemployed, and for whom we do not have enough observations on wages. We
will use “employment rate”for the reciprocal concept.
15
dollars worth of liquid assets. About 33% of the individuals have negative liquid
wealth, that is, they are in debt for this amount. However, total wealth (net liquid
wealth plus the market value of the house) amounts to about $40,000, and only 8%
of individuals exhibit negative total wealth. Clearly, owning a home is the reason for
these debts. As for labor market statistics, around 72% of individuals are employed,
while the average wage is $4,726 per quarter, or 8.26 in log wages.
[Table 2 here]
Table 2 shows summary statistics by homeownership and employment status. The
average house value is about $59,500, and 43% of individuals own a house that is worth
less than $40,000. Only 21% of homeowners own houses that are worth $80,000 or
more. Renters hold slightly more liquid wealth than owners: $12,300 and $11,300,
respectively. About 60% of homeowners and 12% of renters are in debt, that is, they
have negative liquid wealth. However, the totality of renters wealth is liquid, whereas
on average, only about 16% of the total wealth of homeowners is held in liquid assets.
Speci…cally, homeowners’total wealth is around $71,000, and only 3% of them have
a negative total wealth position. Out of all renters, 48% are not employed, compared
with only 22% of homeowners. Average wage for renters is $3,400 (7.92 in log wages),
whereas it is $5,100 (8.35 in log wages) for homeowners.
If we look at statistics by employment status, we can see that employed individuals
are more than twice as likely to be homeowners: 47% of the employed are owners
compared with only 21% of the nonemployed. Moreover, the average value of the
houses owned is also higher for the employed: $63,000 compared with $47,000 for the
nonemployed. And 58% of the owners who are not employed own a house that is
worth less than $40,000; this percentage is 38% for employed individuals. Similarly,
liquid and total wealth are both greater for the employed. While the employed have
around $15,000 worth in liquid wealth and around $49,000 of total wealth, these values
for individuals who are not employed are, respectively, around $6,700 and $19,000.
However, among the employed the proportion of those with negative liquid wealth is
higher: 38% compared with 25% among those who are not employed. Because the
employed are more likely to be homeowners, they also are more likely to have home
loans.
[Table 3 here]
16
Table 3 shows the main variables of our model 4, 12, and 20 years after the
individual left high school. Homeownership increases from 5% in year four to 66%
in year 20, while the value of the house owned increases from around $31,000 to
$64,000 for these same years. Similarly, liquid wealth also increases for both renters
and homeowners. Log wages increase 60% from year four to year 20, from 7.9 to
8.5. The dispersion of log wages is relatively stable over time at 0.5. Meanwhile,
the nonemployment rate declines from 43% in year four to 10% in year 20. This is
the result of increasing rates for job taking and job-to-job transitions and decreasing
rates for job separations.
These trends are informative of the evolution of the main variables of the model
and of their interconnections. They suggest that better labor market outcomes are
associated with homeownership, which involves borrowing and, hence, negative hold-
ings of liquid wealth. Accordingly, homeowners typically have more total wealth
but less liquid wealth than renters. Our next step is an estimation of the behavioral
parameters of the model.
4 Estimation
The estimation strategy aims to recover the behavioral parameters of the theoretical
model. The estimation procedure is an SMM in which the parameter estimates of
the theoretical model are the minimizers of this function. We build a set of simulated
data and use it to compute some selected moments that are then matched to the
actual moments. For each individual in the sample, we generate 100 simulations.
Because wealth and housing are observed only since 1985, there are several periods
for which we can only observe employment and wage data but not wealth and housing
data. For these periods, data for simulations are assumed to be the same as the
actual data. From the quarter that we …rst observe wealth onward, we use the policy
rules that solve the dynamic programming problem and random numbers for the
stochastic components (job o¤ers, layo¤s, wage o¤ers, and house price ‡uctuations) to
generate simulated career paths. That is, simulated data are based on observed liquid
wealth and observed employment status and wages, assuming no homeownership and
a medium housing unit price. At each iteration of the parameter computation, we
construct a measure of distance between the observed and the simulated moments.
Since for many quarters there are very few actual observations, we compute biannual
periods both in the actual and the simulated moments.
17
In the estimation, we only use data for the …rst 26 years of labor market experience,
which approximately contain observations from 1978 until 2005 for most people. That
is, we are trimming the sample to cover the period just before the Great Recession.
We …x the down payment rate at 20% to reinforce identi…cation of other para-
meters, in particular the coefficient of risk aversion. The rate of discount …xed at
0.9872 and the interest rate at 0.012272 are the quarterly values that match the
annual values of 0.95 and 0.05, respectively.
The parameters to estimate are then  = fab,as,,p,rh,,,g,b,,,,
,,1,2,wg. The moments used in this estimation are the following:
1. percentage of owners by year after graduation,
2. value of the house by year after graduation,
3. liquid wealth holdings by year after graduation and homeownership status,
4. log-wage means and standard deviations by year after graduation,
5. employment rates and employment transitions by year after graduation,
6. liquid wealth variation when buying and selling a house,
7. residential wealth variation when buying and selling a house,
8. percentage of people ever buying and ever selling a house, and
9. duration of the period to buy a house for the …rst time and average house tenure
period.
The SMM procedure relates a parameter set to a weighted measure of distance
between sample and simulated moments:
S() = m0W1m;
where m= (mamp)is the distance between each sample and simulated moment
and Wis a weight matrix. The estimated behavioral parameters are thus b
 =
arg min S(). We minimize this function by means of the Powell algorithm, as in
Press et al. (1992), which uses direction set methods to …nd the minimum and relies
on function evaluations, not gradient methods.
18
5 Results
The estimates and their corresponding asymptotic standard errors are reported in
Table 4. As we show, these estimates reproduce the observed trends in the evolu-
tion of homeownership status, house value, liquid wealth by homeownership, wages,
nonemployment rate, and employment transitions.
The share of nondurable consumption is identi…ed by the relative evolution of ob-
served liquid wealth with respect to residential wealth. The coe¢ cient of risk aversion
is pinned down by the speed of wealth accumulation and homeownership over time.
The subjective value of a rented house is pinned down by variations in homeownership
that happen without large variations in wealth, while the rent parameter is identi…ed
by variations in homeownership that happen together with relatively large variations
in wealth. The adjustment fee parameters are identi…ed mainly by variations in liquid
and residential wealth when buying and selling a house, as well as the time before
buying a house for the …rst time. Since the available data set does not contain data
on house prices, these prices are treated as an unobserved random variable required
to estimate the model.15 Labor market parameters are identi…ed mainly by observed
wages and transitions per quarter, as it is well established in prior estimations.16
[Table 4 here]
Adjustment fees both for buying and selling are around 9% of the variation in
house size: 9.15% for buying and 9.50% for selling. The stochastic process for the
housing price has an autocorrelation coe¢ cient of 0.87, which reveals a moderate
persistence over time. The volatility for this process is 0.11, which is also moderate.
Given that we work with seven house prices, these parameters generate the following
15 This has to be solved numerically because the model does not admit a closed-form solution.
Similarly, the analysis on the identi…cation of the model’s parameters by observables is also numer-
ical. The sensitivity analysis in Appendix A4 discusses the e¤ects of each parameter on the main
observables.
16 The identi…cation of behavioral labor market parameters from data on wages and employment
transitions is discussed by Flinn and Heckmann (1982a, 1982b) and Wolpin (1992). The identi…cation
in models of wealth accumulation and job search is discussed by Blundell, Magnac, and Meghir (1997)
and Rendon (2006).
19
price vector and transition matrix:
p=
2
6
6
6
6
6
6
6
6
6
6
6
4
0:825
0:880
0:938
1:000
1:066
1:137
1:212
3
7
7
7
7
7
7
7
7
7
7
7
5
; P (pt+1jpt) =
2
6
6
6
6
6
6
6
6
6
6
6
4
0:526 0:215 0:149 0:074 0:027 0:007 0:001
0:330 0:227 0:209 0:139 0:066 0:023 0:007
0:171 0:186 0:228 0:202 0:129 0:059 0:025
0:073 0:118 0:194 0:229 0:194 0:118 0:073
0:025 0:059 0:129 0:202 0:228 0:186 0:171
0:007 0:023 0:066 0:139 0:209 0:227 0:330
0:001 0:007 0:027 0:074 0:149 0:215 0:526
3
7
7
7
7
7
7
7
7
7
7
7
5
:
Annual rent is around 2.034% of the average house value, which is in line with
observed rent-to-value ratios. The coe¢ cient of risk aversion is 0.73, which is lower
than in prior estimations. This is not unrealistic considering in our model a house is
both a consumption good and residential wealth and that, moreover, it determines
the individual’s borrowing constraint. Models in which these aspects are absent do
require higher coe¢ cients of risk aversion to account for observed savings. The share
of nondurable consumption is 83.8%, in line with descriptive data. The subjective
valuation of a rented house is 72%, similar to values used in other models.
The estimated amount of net transfers while not employed is about $432 per
quarter. At the beginning of active life, when the agent has no work experience,
the probability of receiving an er is 28%. A growth parameter of 0.004576271
implies that 20 quarters (6 years) after graduation, this probability becomes 30%, and
it is 32% and 37% 40 quarters (10 years) and 80 quarters (20 years) after graduation,
respectively. The probability of receiving an er while employed with no experience
is 1.28%. A parameter of 0.000884202 implies that 20 quarters after graduation
this arrival rate becomes 1.31% and is 1.33% and 1.38% after 40 and 80 quarters,
respectively. The base lay rate is 13.9%. A parameter of -0.009323368 implies
that at quarters 20, 40, and 80 after graduation this rate becomes 11.8%, 10.0%, and
7.1%, respectively. The estimated base mean of the underlying distribution of log
wages is 6.75, and the corresponding variance is 0.99. The values of parameters 1 and
2 imply that, at quarter 20 after graduation, the base mean log-wage er becomes
6.80, an increase in mean wages of 6.10%; and at quarters 40 and 80 after graduation,
the base log-wage values are, respectively, 6.83 and 6.89, implying increases in mean
wages of 7.8% rst and then 13%. These parameters imply that wages peak at quarter
135, when the individual is around 51, out of the sample used in the estimation. As
seen in Table 3, the implied increase of wages from year 4 to year 20 is around 60%,
20
while these parameters imply an increase of wages of only 10% for the same period.
These parameters exhibit, thus, a slow variation of arrival rates and of the wage
er distribution, which suggests that they are not the main drivers of the observed
variations in wages and employment transitions. Rather, the increase in reservation
wages is a product of the life-cycle accumulation of wealth, both in the form of liquid
assets and residential real estate.
Asymptotic standard errors are calculated using the outer-product gradient es-
timator; they are, in general, small. To assess whether these parameter estimates
capture the essential features of the observed data, we compare the observed and
the predicted trajectories of homeownership, house value, wealth by homeownership
status, wages, employment status, and employment transitions.
[Table 5 here]
[Figures 5 and 6 here]
Table 5 provides a summary of the actual and predicted distribution of all variables
for years 4, 12, 20, and 28 after graduation.17 It also shows goodness of …t tests: 2
for discrete variables and R2for continuous variables.18 In addition, Figures 5 and
6 present a graphical comparison of actual and predicted variables by year after
graduation. Figures 5a and 5b show that the predicted path of the homeownership
rate and the house value are relatively close to the actual path, which can be conrmed
by looking at the 2and R2statistics in Table 5. Figures 5c and 5d compare the actual
and predicted liquid wealth of renters and owners. Both show some overprediction in
the later years. In spite of some noise in the liquid wealth data, the model reproduces
quite well the observed trend in wealth accumulation, as the R2statistics in Table 5
con…rm.
Figures 6a and 6b show that the model reproduces well the wage distribution, es-
pecially in the later periods, as is conrmed by their respective R2statistics in Table
17 This last year is not used in the estimation, but it is reported as an out-of-sample prediction,
as a means to perform a cross-validation of the model.
18 For discrete variables 2= T
t=1
(nt^nt)2
^nt, where ntis the actual number of observations at
time t,^ntis the model predicted counterpart, and Tis the number of periods. This statistic has an
asymptotic 2distribution with T1degrees of freedom. For continuous variables R2=Pb
Y2
Pb
Y2+Pe2,
where b
Yis the predicted continuous variable and e=Yobs b
Yis the predicted error. Squaring and
summing across observations, we obtain PY2
obs =Pb
Y2+ 2 Pb
Y e +Pe2, and it is not necessarily
true that Pb
Y e = 0, as in the linear regression framework.
21
5. Figure 6c shows that the model captures well the overall trend of the nonemploy-
ment rate in spite of some underprediction in the early years and some overprediction
in the later years. Similarly, in Figures 6e and 6f, we can see the model replicates
fairly well the overall trend of actual job loss rates and job-to-job transitions.
[Table 6 here]
Finally, Table 6 shows actual and predicted variations of liquid and residential
wealth when buying a house for the …rst time and then selling it, as well as the per-
centage of agents involved in these transactions and the duration of time for buying
and selling. In the data, wealth is only observed every four quarters, that is, annu-
ally, and only for some years, whereas the model always exhibits quarterly variations.
This precludes an exact comparison between observables and their predictions. How-
ever, these moments are important for identifying some parameters, particularly the
adjustment fees.
As shown in Table 6, variations of liquid wealth are larger in the model, but
residential wealth variations are larger in the data. The model also exhibits more
turnover in the housing market: More people buy and more people sell their houses,
so that they are homeowners for eight and a half years, one and a half years less than
in the data. The model, however, replicates well the duration to sell a house, at about
ve years.
In short, both graphically and formally, the model is fairly successful in replicating
the main features of the data.
6 Regime Changes
After recovering the underlying parameters of the model and assessing their success in
replicating the data, we perform two regime changes: relaxing housing credit condi-
tions and deteriorating labor market demand. We assess these changes in two derent
ways, when agents start their career with the new regime and when agents start their
careers with the benchmark regime and the new regime is introduced without antici-
pation in derent periods. In Table 7, we report the e¤ects on several observables on
three selected years, when agents start their careers with the new regime. In Figures
7 and 8, we report the e¤ects on just one relevant outcome variable, when these same
regime changes are introduced unanticipatedly in di¤erent years once agents have
entered the labor market.
22
[Table 7]
In the …rst experiment we decrease the down payment required to buy a house, d,
by 15 percentage points, and decrease the persistence of the housing price process, ,
by 10 percentage points. We are interested in evaluating how these regime changes,
one at a time and combined, in‡uence the labor market.
A reduction in the down payment increases homeownership rates and the wealth
of renters, who now have an incentive to save more, and decreases liquid wealth held
by homeowners, who can now buy larger houses. This variation also increases both
wages and nonemployment rates, while it deteriorates employment transitions, which
is consistent with an increase in the reservation wage. Nonemployment rates increase
1.2 percentage points 20 years after graduation. That is, easier access to housing
credit generates higher nonemployment rates.
A fall in the persistence of the housing price process means currently low prices
are more likely to increase and currently high prices are more likely to fall. This
fall increases the rate of homeownership, while it decreases wealth of any type for
both renters and owners. In the labor market, this fall is re‡ected in an increase in
wages and the nonemployment rate, which is consistent with increasing reservation
wages. Employment transitions also worsen in that transitions from nonemployment
to employment fall, while transitions from employment to nonemployment increase.
The combined e¤ect of these regime changes is shown in Table 7, where both kinds
of wealth for renters and owners decrease, while homeownership increases. In terms
of the labor market, this change generates an increase of nonemployment of up to 2
percentage points, while wages increase by a maximum of 4 percentage points.
[Figure 7 here]
In Figure 7, we can see the immediate jump in nonemployment once housing mar-
ket conditions are loosened. This jump is higher (5%) for relatively older individuals
and lower for individuals who have recently entered the labor market. Over time,
this increase fades, but then it increases again, reaching almost 2 percentage points,
which is the common long-run variation for the years in which the regime change is
introduced.
The second experiment consists of deteriorating labor demand; that is, decreasing
the base mean wage o¤er and increasing lay rates and nonemployment transfers.
A decrease in the mean wage o¤er reduces homeownership rates and the wealth of
23
both renters and owners, and it also pushes wages down. However, nonemployment
rates decline. It is the decrease in reservation wages that makes it possible for wages
and nonemployment to both fall. Accordingly, job …nding rates increase while job
separations and job-to-job transitions do not change much.
An increase in the layo¤ rate decreases homeownership by 5 percentage points,
while it decreases liquid wealth for renters and residential wealth for owners. The lay-
rate is crucial in determining savings for precautionary reasons. Thus, its increase
deteriorates labor market conditions, but it also boosts savings for homeowners. As a
result of this change, wages fall and nonemployment rates increase, while job …ndings
and job separations both rise, and job-to-job transitions fall.
When the previous regime changes happen simultaneously, there is a larger fall in
homeownership of up to 8 percentage points. Liquid wealth holdings of both renters
and owners fall by larger amounts than in any single change (except in initial years
for owners), which suggests that, because of this labor market deterioration, only
the initially wealthy buy houses. Over time, fewer and fewer of the initially wealthy
buy houses while their liquid wealth falls further. This combined change implies an
increase in the nonemployment rate and a large fall in wages, particularly in the later
years (23 percentage points). There is also lower job …nding and fewer job-to-job
transitions but more job separations.
We also consider the additional impact of an increase in nonemployment transfers,
which captures the government’s attempt to counteract deterioration of labor market
conditions, as in Rothstein (2011) and Hagedorn et al. (2013). Under this combined
regime, the decline in homeownership rates is also large (6 percentage points) but
below what it would be without an increase in nonemployment transfers, 8 percent-
age points. Wealth of renters also falls by less, but the liquid holdings of owners in
the later years decrease by a bit more. Wages fall by 6 percentage points, substan-
tially less than the 12 percentage point decrease without transfers. Nonemployment
increases by larger amounts, up to 6 percentage points instead of the previous 3 per-
cent increase, mostly due to slower transitions from nonemployment to employment.
In sum, a government intervention that increases nonemployment transfers is e¤ec-
tive in alleviating the substantial fall in homeownership rates and liquid wealth that
follow the decrease in labor demand. This policy also impedes wages to fall further
while it generates larger nonemployment rates.
[Figure 8 here]
24
In Figure 8, we can see the e¤ect of labor market deterioration on homeownership.
After an initial jump of 5 percentage points, homeownership systematically falls by
almost 8 percentage points, especially for individuals who are hit by this shock early
in their careers and are, hence, exposed longer to this regime change.
Thus, housing and labor markets are closely connected, and their respective shocks
ect one another. These reciprocal e¤ects are understandably larger the longer the
exposure to the speci…c regime change. For both kinds of shocks, there are relatively
large short-term ects that work in opposite directions to the long-term e¤ects, which
are important.
7 Conclusions
In this paper, we have developed a framework that allows us to analyze interac-
tions between the housing, …nancial, and labor markets when there are frictions.
We propose an extended model of job search that allows for savings and for choice
of homeownership status and house size under random house prices. Contrary to
most other dynamic models of housing where the income process is exogenous, in our
search-theoretic framework, workers decide to accept or reject wage o¤ers. Individu-
als have, therefore, some control over their income, which produces a feedback e¤ect
between collateralized …nancing and house price ‡uctuations on the one side and be-
tween wages and unemployment on the other. In this environment, reservation wages,
and thereby unemployment, are not only increasing in total wealth but, under costly
housing adjustment, are also responsive to the composition of this wealth as liquid
funds and residential real state. In particular, when house prices are low and expected
to increase, holding a larger share of residential wealth tends to increase reservation
wages, which deteriorates employment transitions and increases unemployment. In
this manner, we are able to explain how higher homeownership rates are associated
with higher nonemployment rates without relying on the mechanism of homeowners
geographical lock in.
Moreover, our …nding that wealth composition and access to credit a¤ect the
individual’s reservation wage and, hence, unemployment is consequential in that it
suggests an additional mechanism through which …nance a¤ects real economic activ-
ity. Liquidity constraints and internal …nance are thus not only relevant for the …rm
but also for the worker.
We have …t this model to data from the NLSY and recovered the behavioral
25
parameters, which are realistic for both the housing and the labor market. The
model exhibits a fairly good capability to replicate the main observables, namely,
the evolution of homeownership rates, house value, liquid wealth, wages, employment
status, and employment transitions over time.
We have shown that more relaxed conditions for home loans decrease employment
rates by around 2 percentage points. We have also evaluated the e¤ects of deteri-
orating labor demand on the housing market and found that less job creation by
employers, captured by lower arrival rates, higher layo¤ rates and lower wage o¤ers,
reduces homeownership by 8 percentage points.
Future research can extend our framework to explore other important dimensions
of the connection between housing markets and job search. In our model, lenders
do not share risk with borrowers, so that they recover their loans with probability
one. One could explore how the model’s predictions change when a borrower has
the option to default on his loan.19 One could also model the mortgage structure,
allow for foreclosures in case of default, consider underwater mortgages, or relax the
no-homelessness condition and have some agents become homeless.
As for the labor market, an obvious concern is to allow for an equilibrium frame-
work with matching. This would enable the model to assess …rms’reactions to relevant
regime changes, such as those discussed in this article. For instance, if workers are
more selective in their search, …rms will have a harder time …nding a worker. Once
rms realize this, the value of a vacancy will decrease: Firms will open fewer vacan-
cies and the unemployment rate will increase, potentially more so than in a partial
analysis. On the other hand, an equilibrium framework will also enrich the analysis
on the e¤ect of …nance on the real activity, as liquidity constraints will act on both
sides of the frictional labor market. Firms’investment and vacancy creation as well as
workers’acceptable wages will be limited by assets used as collateral on loans, in one
case by the amount of a …rm’s capital and in the other by the size of an individual’s
home.
19 A possibility is to declare bankruptcy, which will prevent the individual from borrowing at all
in the future.
26
Appendix
A1. Solution of the Model
As in the main text, yt=bRif t>T, and if tT; yt=bwhen the agent is unemployed,
and yt=wt(!)when the agent is employed. Then, the solution for renting, Ht= 0 is:
Vt(At; y; Ht1; pt) = max
At+1;H r
tUAt+ytc(Ht1;0; pt)rhHr
tAt+1
1 + r; gH r
t+EVt+1 (At+1; y; 0; pt);
where U(Ct; gH r
t) = C
t(gH r
t)111
1:
Then, we have the following FOC for At+1:
C
t(gH r
t)11
Ct
=(1 + r)EVA(At+1 ; y; 0; pt);
which is inverted in the following way:
Ct= (1 + r)EVA(At+1 ; y; 0; pt)
(gH r
t)(1)(1)!1
(1)1
:(1)
If the borrowing constraint is binding, At+1 =Bt+1 (0), and thus, the Euler equation
does not hold, we have the following equation for consumption:
Ct=At+bc(Ht1;0; pt)rhHr
tBt+1 (0)
1 + r:(2)
The corresponding FOC for Hr
tis
C
t(gH r
t)11
Ct
rh+(1 )
Hr
t= 0;
which implies
Ctrh=(1)
Hr
tand can be expressed in the following way:
Hr
t=1
rhCt:(3)
From (1) and (3), we obtain
Hri
t=1
rh2
6
4(1 + r)EVA(At+1 ; y; 0; pt)
g(1)
rh(1)(1)3
7
5
1
:
If the borrowing constraint (2) is binding, then
Hri
t=1
rhAt+yt+pt(1 as)Ht1Bt+1 (0)
1 + r;
27
which holds if the rented house size lies between the house size bounds. Since there is a
minimum and a maximum house size, Hr
t2H;H, in general this expression has to be
bounded:
Hr
t= min max Hri
tH; H:
In any case, conditional on At+1, once Hr
tconsumption Ctis simply
Ct=At+bc(Ht1;0; pt)rhHr
tAt+1
1 + r:
Solution for owning, Ht>0:
V(At; y; Ht1; pt) = max
At+1;Ht;stUAt+ytc(Ht1; Ht; pt) + rh(1 st)HtAt+1
1 + r; stHt
+EVt+1 (At+1; y; Ht; pt)g;
where U(Ct; stHt) = C
t(stHt)111
1:
Optimal house size is determined by discrete choice of the house size that maximizes the
value function. Consumption (or wealth next period) and the fraction of the owned house
that is rented are determined by the …rst-order conditions for At+1 and stbelow.
For At+1:
C
t(stHt)11
C=(1 + r)EVA(At+1 ; y; Ht; pt);
which is inverted in the following way:
Ct= (1 + r)EVA(At+1 ; y; Ht; pt)
(stHt)(1)(1)!1
(1)1
:(4)
If the borrowing constraint is binding At+1 =Bt+1 (0), and the Euler equation does not
hold, we have the following:
Ct=At+ytc(Ht1; Ht; pt) + rh(1 st)HtBt+1 (0)
1 + r:(5)
For st:
s(1)(1)
tC
tH1
t1
Ct
rhHt+(1 )
st= 0;(6)
which can be expressed as follows:
si
t=Ct
Ht
(1 )
rh
:
28
From (4) and (6) we obtain
Ct=2
6
4(1 + r)EVA(At+1 ; y; Ht; pt)
(1)
rh(1)(1)3
7
5
1
:
And when the borrowing constraint binds, (5) and (6), consumption is given by
Ct=At+bc(Ht1; Ht; pt) + rhHtBt+1 (0)
1 + r:
These expressions, however, apply only when st<1. If Ctis high enough, then st= 1 and
(4) and (5) directly provide an expression for consumption. A general expression for stis
then
st= min Ct
Ht
(1 )
rh
;1:
There is no scenario under which st= 0, as Ct>0and H > 0.
A2. Proof of Proposition 1
Proof of Proposition 1 If ab=as= 0, then Atc(Ht1; Ht; pt) = At+ptHt1ptHt=
ZtptHt.
At time TF,VTFZt;0;(1 )Zt
pt; pt= max
Hr
t
U(Zt+bRrhHr
t), so that only total
wealth matters, that is, VTFZt;0;(1 )Zt
pt; pt=VTF(Zt;0;0; pt)for any 2[0;1].
Then we proceed backward and de…ne the value function for renting
Vr
tZt;0;(1 )Zt
pt
; pt= max
At+10;H r
tUZt+bRrhHr
tAt+1
1 + r; gH r
t
+ZVt+1 (Zt+1;0;0; pt+1)dP (pt+1jpt);
and for owning a house
Vo
tZt;0;(1 )Zt
pt
; pt= max
At+10;Ht;stUZt+bRptHt+ (1 st)rhHtAt+1
1 + r; stHt
+ZVt+1 (Zt+1;0;0; pt+1)dP (pt+1jpt):
In none of these functions, the value of , the share of liquid wealth in total wealth, makes a
di¤erence. At every time, the composition of wealth is irrelevant, that is, VtZt;0;(1 )Zt
pt; pt=
Vt(Zt;0;0; pt)for any 2[0;1]. The process is repeated backward until the …rst period of
retirement. Then the process is repeated backward throughout the agent’s active life, only
paying attention to the particular employment status and its associated expected value func-
tion, until the …rst period of active life is reached.
29
A3. Numerical Solution of the Model
Continuous and discrete variables
As mentioned in the main body of the paper, the model is solved by means of gridpoints.
Liquid wealth and wages are continuous variables, only discretized to support the compu-
tation of any value on their domains, whereas house sizes and house prices are discretized.
Table A1 gives further details of this discretization.
Table A1. Discretization of Variables
Liquid Wealth Base Wages House Size House Price
Original variable A ! H p
Discretized variable A(i; h1; t)!(j)H(h)p(l)
Gridpoints i= 1; :::; NAj= 1; :::; N!h= 1; :::; Nhl= 1; :::; Np
Gridpoint location Left Left Middle Middle
Number of gridpoints NA= 51 N!= 31 Nh= 7 Np= 7
Number of intervals NA1N!1NhNp
Lower bound A=B(h1; t)!= max (1;500; rhH(1)) H= 20;000 ln p=p
p12
Upper bound A=p(Np)H(Nh) (1 as)!= 10;000 H= 180;000 ln p=p
p12
Gridsize A=AA
NA1w=ln !ln !
N!1H=HH
Nhp=ln pln p
Np
The lower bound on liquid wealth is set so that an agent can borrow up to some
fraction of the lowest possible value of the house owned, net of value of rent of the
smallest possible house. The fraction is determined by the down payment coe¢ cient
and the remaining active lifetime. Accordingly, the wealth array depends on the size
of the house owned and on time. A retired agent cannot borrow.
B(h1; t) = (1 d)ktp(1) H(h1) (1 as) + rhH(1) , if tTF;
B(h1; t) = 0, if t > TF:
The lowest possible base wage allows an individual to rent the smallest possible
house (no homelessness condition).
Wage as a function of base wage and time w(!; t)becomes w(j; t) = !(j) exp (1t+2t2).
Wage and house price distributions
Wages: For each wage interval j= 1; Nw1, we compute three truncated moments
of the log-normal base wage distribution (see Jawitz, 2004):
F(!j+1)F(!j) =
ln !j+1
!ln !j
!
ln !
!ln !
!;
E[!j!j!!j+1] = exp +2
!
2ln !j+1
!!ln !
!!
ln !
!ln !
!;
E!2j!j!!j+1= exp 2+ 22
!ln !j+1
!2!ln !
!2!
ln !
!ln !
!:
30
House prices: Following Tauchen (1986), the discrete conditional probability for
a price p(i)is
pl0jl=8
>
>
>
<
>
>
>
:
ln p(l0)+p=2ln p(l)
pfor l0= 1;
ln p(l0)+p=2ln p(l)
pln p(j)p=2ln p(i)
p1< l0< Np;
1ln p(l0)p=2ln p(l)
pl0=Np:
:
Value function, policy rules, and expected value function
These are approximated by
Vt(At; !; Ht1; pt) = V[i; j; h1; l; t]
A0(At; !; Ht1; pt) = A0(i; h1; t)
H(At; !; Ht1; pt) = H(h) (i; h1; t)
EV e
t(At; !; Ht1; pt) = EV [i0; j; h; l]:
Only value functions and policy rules are stored. Expected value functions are over-
written at each iteration.
Numerical solution
The numerical solution starts at period t=TF= 252 and proceeds backward.
In reaching period T= 188, the agent becomes active, then the solution keeps going
backward until reaching period t= 1. Table A2 illustrates how current income di¤ers
in both stages:
Table A2. Current Income
Agent Retired Active
Unemployed Employed
t=TF; :::; T + 1 T; :::; 1
j= 0 0 1; :::; N!
y(j; t) = bRb w(j; t)
The following steps are done for each i,j,h1, and l:
1. Initialization. In the last period, t=TF, the agent is retired and does not
bequest any liquid or illiquid wealth, so ATF+1 = 0 and HTF= 0. The agent
just rents to maximize his current utility.
De…ne the discretized value function for each i,h1,l:
V[i; j; h1; l; t] = U(A(i) + y(j; t)c(H(h1);0; p (l)) rhHr(h); gHr(h)) ;
Hr(h) = min max 1
rh
(A(i) + y(j; t) + pt(1 as)H(h1)) ; H; H:
To improve legibility, arguments h1and tare omitted in A(i; h1; t), that
becomes A(i).
31
2. Integration. Go backward one period, to t1. The function previously com-
puted, V[i; 0; h1; l; t], becomes V[i0;0; h; l0; t + 1].
For the retired agent,20 and for each i0,j,h, and l, de…ne
EV [i0; j; h; l] =
Nl
X
l0=1
V[i0; j; h; l0; t + 1] P(l0jl):
3. Di¤erentiation. Compute the derivative of this object over liquid wealth using
a cubic interpolation.
EVA[i0; j; h; l] = EV [i0+ 2; j; h; l] + 4EV [i0+ 1; j; h; l]3EV [i0; j; h; l]
A(i0+ 2) A(i0); if i0= 1;
=EV [i0+ 1; j; h; l]EV [i01; j; h; l]
A(i0+ 1) A(i01) ; if NA> i0>1;
=3EV [i0; j; h; l]4EV [i01; j; h; l] + EV [i02; j; h; l]
A(i0)A(i02) ; if i0=NA:
To improve legibility, arguments hand t+ 1 are omitted in A(i0; h; t + 1), that
becomes A(i0).
4. Policy rule inversion. We use the endogenous gridpoints methods as in Carroll
(2006). For each i0and h, and renting of the house owned Is(= 0, if the owner
does not rent; = 1, if the owner does rent), optimal consumption C(i0; h; IS)is
found.
Renting, h= 0:
Hr
t(hr) = 1
rh2
6
4(1 + r)EVA[i0; j; 0; l]
g(1)
rh(1)(1)3
7
5
1
;
C(i0;0;1) = (1 + r)EVA[i0; j; 0; l]
(gHr
t(h))(1)(1)!1
(1)1
:
Owning, h > 0:
No renting, Is= 1, that is, if s= 1, then
C(i0; h; 1) = (1 + r)EVA[i0; j; h; l]
H (h)(1)(1)!1
(1)1
:
20To improve legibility of this pseudo-code, the integration of the value function for the active
agent is explained in step 2.
32
Renting, I2= 2, that is, if s < 1, then
C(i0; h; 2) = 2
6
4(1 + r)EVA[i0; j; h; l]
(1)
rh(1)(1)3
7
5
1
:
5. Smoothing. Conditional on hand Is= 1;2, regress C(i0; h; Is)on A(i0). When-
ever there are nonmonotonicities (see below) in C(i0; h; Is)over A(i0), use pre-
dicted consumption instead of actual consumption:
b
C(i0; h; Is) = b
b0+b
b1A(i0) + b
b2[A(i0)]2:
6. Inverse solution. Find liquid wealth at time tas a function of i0and h, denoted
by e
A, for each l,j, and h1,j, and l:
Renting, h= 0:
Hr(h) = 1
rhb
C(i0;0;1) ;
e
A(i0;0) = b
C(i0;0;1) y(j; t)c(h1;0; l)rhHr(h)A(i0)
1 + r:
Owning, h > 0:
De…ne
s= min "1
rhb
C(i0; h; 2)
H(h);1#:
If s= 1, then e
A(i0; h) = b
C(i0;0;1) bRc(h1; h; l)A(i0)
1+r;
if s < 1, then e
A(i0; h) = b
C(i0;0;2)y(j; t)c(h1; h; l)+ (1 s)rhH(h)A(i0)
1+r:
In both cases, store s(i0; h) = s:
7. Conditional solution. Reposition current liquid wealth e
Ato …nd the solution
conditional on housing h.
Interior solution. For each i, locate i0such that e
A(i0; h)< A(i0; h)<e
A(i0+1; h),
then compute the linear interpolations
A(i; h1; h) = aA(i0; h) + (1 a)A(i0+ 1; h);
s=as(i0; h) + (1 a)s(i0+ 1; h);
EV =aE V (i0; h) + (1 a)EV (i0+ 1; h);
where a=A(i;h)e
A(i0;h)
e
A(i0+;h)e
A(i0;h), and we drop price and time arguments to improve
legibility.
33
Corner solutions. If A(i; h)<e
A(i0; h), then i= 1; if A(i; h)>e
A(NA; h), then
i=NA:
A(i; h1; h) = A(i; h)
s=s(i; h)
EV =E V (i; h):
Then,
C(i; h1; h) = A(i; h) + y(j; t)c(h1; h; l) + (1 s)rhH(h)A(i; h1; h)
1 + r;
e
V(i; h1; h) = U(C(i; h1; h); sH(h)) + EV :
8. Solution. Find the optimal choice of housing.
Choose h2[0; Nh]that maximizes the value function:
V(i; h1; t) = max
he
V(i; h1; h);
h(i; h1; t) = arg max
he
V(i; h1; h);
A0(i; h1; t) = A(i; h1; h(i; h1; t)) ;
C(i; h1; t) = C(i; h1; h(i; h1; t)) :
Choice variables with static solutions, such as Hrand s, are not stored, as i)
they can always be recovered from hand C, and ii) they do not have observable
counterparts in the available data set.
9. Go back to step 2. and repeat the process until reaching period t= 1.
2’ Integration. Active agent.
For all three consecutive wage gridpoints, the value function while employed
(for simplicity expressed dropping all arguments except wages) is interpolated
using a quadratic function:
Vj(w) = aj+bj!+cj!2:
For j= 2; :::; Nw1, these three points are j1,j, and j+ 1. For j= 1, these
three points are j,j+ 1, and j+ 2, that is, the calculated coe¢ cients are the
same as those of j= 2.21 Then cj=Vj+1Vj1d
(!j+1!j1)2,bj=d
!j+1!j12!j1cj, and
aj=Vj1bj!j1!2
j1cj, where d=VjVj1x2(Vj+1Vj1)
xx2and x=!j!j1
!j+1!j1.22
21 Notice that i) wage intervals are not of equal size, because the discretization of this variable is
done over log wages, and ii) this interpolation is done over wage levels; it cannot be done in the
unit interval as other interpolations, because the purpose is not just to interpolate some value but
to integrate a function.
22 To determine these coe¢ cients, we …rst interpolate Vover x:V(x) = Vj1+dx +ex2,where
34
With this quadratic function the expected value function for each wage interval
is computed exploiting the previously de…ned truncated moments:
E[Vj(!)j!j!!j+1] = Zwj+1
wjaj+bj!+cj!2dF (w)
=aj[F(!j+1)F(!j)] + bjE[!j!j!!j+1 ]
+cjE!2j!j!!j+1:
The computation of the reservation wage is also facilitated by this interpolation.
It proceeds then in two steps:
1. Find jsuch that V(!j)V(0) V(!j+1).
2. Then …nd !=fwjV(0) = Vj(!) = aj+bj!+cj!2g, that is, !=bj+pb2
j4(ajV(0))cj
2cj:
We compute the expected value for an interval conditional on wages exceeding
the reservation wage, E[Vj(!)j!!!j+1]in a similar manner. Then we
compute the expected value conditional on wages being above any current wage
or the reservation wage: