Simultaneous Search in the Labor and Marriage Markets with
Endogenous Schooling Decisions∗
Luca Flabbi
UNC-Chapel Hill
CPC, IZA
Christopher Flinn
New York University
CCA
October 26, 2015
Very Preliminary and Incomplete
Abstract
Labor and marriage market decisions are joint decisions. If individuals are engaged
in a stable relationship, labor market decisions are taken at the household level. If
individuals are not yet engaged in a stable relationship, marriage market decisions
and opportunities are strongly influenced by the individual current and expected labor
market position. The literature recognizes the joint nature of these decision processes
but usually focuses only on one side of the decisions tree. Our analysis, instead,
develops and estimates a model designed to determine the joint equilibrium distribution
of labor market outcomes and marriage market statuses, together with an endogenous
schooling decision. The model is estimated by the Method of Simulated Moments using
labor market information from the Current Population Survey and marriage market
information from the American Community Survey.
Using the estimates of the model, we perform several comparative statics exercises.
On top of being of interest in themselves, they provide an empirical assessment of the
importance of taking into account the joint nature of the decision process in the two
markets. We also plan to use the parameter estimates to perform a series of policy
experiments comparing a labor income tax system based on individual taxation with
a system based on joint taxation.
JEL Codes: D13,J12,J64
∗
Sergio Ocampo and Mauricio Salazar Saenz provided excellent research assistantship. Partial funding
from the Inter-American Development Bank (IDB) ESW grant RG-K1415 is gratefully acknowledged. The
views expressed in this paper are those of the authors and should not be attributed to the IDB.
1
1
Introduction
Labor and marriage market decisions are strongly interrelated. If individuals are engaged in
a stable relationship, labor market decisions are among the most important choices affecting
overall household welfare, and therefore are not taken in isolation but at the household level.
If individuals are not yet engaged in a stable relationship, marriage market decisions and
opportunities are strongly influenced by the individual’s current and expected labor market
position. The literature often recognizes the joint nature of these decision processes, but
usually focuses only on one side of the decisions tree.
With respect to the impact of the marriage market on labor market decisions, there
now exist a number of estimated models of household search1 that are able to examine the
impact of one spouse’s labor market status on the other spouse’s current and future labor
market choices. However, these contributions ignore the process leading to the formation
of the household and frequently lack intrahousehold behavior, since they typically impose
the existence of a household utility function.2
With respect to the impact of the labor market on marriage market decisions, there
exists a huge literature in economics, demography and population studies focusing on
how an individual’s current labor market state and future labor market performance may
impact outcomes in the marriage market. However, most of these contributions ignore
dynamic considerations and all of them consider labor market characteristics as a fixed
individual-level characteristic. In other words, these contributions ignore the fact that the
individual’s current labor market state may endogenously change as a result of marriage
market decisions.3
Our analysis utilizes an innovative methodological framework designed to determine
the joint equilibrium distribution of labor market outcomes and marriage market statuses.
1
See Dey and Flinn (2008); Gemici (2011); and Flabbi and Mabli (2012). Guler, Guvenen and Violante
(2012) do not estimate the household search model but provide an exhaustive theoretical discussion.
2
The notable exception is Gemici (2011) which includes some intrahousehold behavior and endogenous
marriage choices. However, intrahousehold behavior is mainly limited to the impact on the geographical
location decisions of the household and the marriage decisions are derived without imposing equilibrium
conditions on the marriage market.
3
Recent examples include DelBoca and Flinn (2014) which looks at married people but recognize that
marital sorting is influenced by labor market variables. However, these labor market variables are state
variables that cannot change as a result of the marriage. This is also the case in Chade and Ventura
(2005) which allows for endogenous household formation and dissolution (but not for wage dispersion.)
There also exists a rich literature on marriage sorting processes which exploits equilibrium conditions in the
marriage market (Pollack (1990); Dagsvik (2000); Choo and Siow (2006); and Choo (2015).) This setting
studies sorting over individual level characteristics which in principle may include labor market outcomes
but they are instead limited to demographic characteristics. Finally, the search and matching literature
with two-sided heterogeneity has been applied to study sorting in the marriage market (see for example
Shimer and Smith (2000)). Wong (2003) is an application estimating the model with the inclusion of labor
market variables (wages). However, wages are just one component of a single index describing individual
heterogeneity and they are permanent components, not allowed to change as a result of marriage market
dynamic.
2
Informed by the importance of human capital investments in both the marriage and labor
market, we also introduce an endogenous schooling choice which takes place before entering
both markets. Exploiting a number of simplifying assumptions, we are able to estimate the
model using cross-sectional wage distributions, unemployment durations, and transitions
across marriage market statuses taken from readily available and nationally representative
data sources. As a result, we can empirically assess the importance of taking into account
the joint nature of the decision process in the two markets and the cost of ignoring it when
using models for policy evaluation purposes.
We assume individuals begin adult life by making a schooling decision. After schooling is completed, individuals enter the marriage and labor market, jointly searching (in
continuous time) in both. Each market is characterized by frictions and by match-specific
shocks. Household interaction is assumed to be non-cooperative with each spouse making
job choices solely along the extensive margin, i.e., they decide to accept or reject an offer
if receiving one themselves and they decide to stay or quit their current job when their
spouse’s employment state changes. We solve the potential equilibrium multiplicity that
may arise in such a dual searcher setting by assuming that the spouse who is receiving the
offer is the first-mover, while the other spouse acts as a follower.
A few recent contributions introduce elements of interactions between the marriage
and labor markets. Jacquemet and Robin (2013) model endogenous household formation,
allowing for wage dispersion and a labor supply decision. However, only the labor supply
decision is endogenous and allowed to change with marriage market outcomes, while wages
are treated as a permanent individual-specific component. Greenwood, Guner, Kocharkov
and Santos (2015) develop a similar setting, adding, like us, an endogenous education
decision. However, the interaction with the labor market is even more limited than in
Jacquemet and Robin (2013), since only women are allowed to endogenously adjust labor
supply. Finally, Chiappori, Dias and Meghir (2015) allow for the same three choices we
consider in our model: schooling, marriage, and jobs. However, the interactions between
the labor and marriage market choices is more limited than in our case, since the marriage
decision is irrevocable and taken before entering the labor market. On the other hand, labor
market dynamics are allowed to possess a life-cycle pattern. These patterns are essentially
ruled out within our stationary setting.
After developing the model, we estimate it by the Method of Simulated Moments
(MSM) using labor market information from the Current Population Survey (CPS) and
marriage market information from the American Community Survey (ACS), under the
assumption that a monthly CPS sample is a point sample from the steady state distribution. With no on-the-job search in the labor market and no on-the-marriage search in the
marriage market, identifying the transition rate parameters associated with the labor and
marriage markets is reasonably straightforward, as are the wage offer distributions under
parametric (log normal) assumptions. The identification of the distribution of marriage
match values is more problematic, but it is solved by exploiting the rich combination of
“types”of couples arising in equilibrium. Types are described by combinations of school3
ing levels, labor market states, and wage levels when an individual is employed. Finally,
the distribution of schooling costs is the one on which we observe the least amount of
information and as a result we assume the distribution to be known up to one parameter.
Using the estimates of the model, we perform several comparative statics exercises.
In addition to being of direct interest in themselves, these exercises provide an empirical
assessment of the importance of taking into account the joint nature of the decision process
in the two markets. First, we explore the impact of changes in the labor market structure on
the joint equilibrium distribution of both markets. We analyze two changes: reducing labor
market frictions and eliminating gender differences in the wage offer distributions. Second,
we plan to estimate lifetime returns to schooling, showing the importance of both markets
in determining the final equilibrium outcomes. Some exercises will show how ignoring one
of the two markets, or ignoring the interaction between them, may impact the estimated
returns. Finally, we plan to assess the impact of changes in the marriage market structure,
such as recent demographic changes, on the joint equilibrium distribution of both markets
and on overall welfare.
As a final contribution, we plan to use the parameter estimates to perform a series of
policy experiments comparing a labor income tax system based on individual taxation with
a system based on joint taxation. Specifically, we will set a tax revenue objective, impose
alternative taxation schemes, and obtain the tax schedule necessary to satisfy the tax
revenue target under the two regimes. We will then obtain the joint equilibrium distribution
of schooling levels, labor market outcomes, and marriage market statuses under the two
regimes and compare their associate lifetime welfare levels.
The paper is organized as follows. Section 2 presents the model and section 3 the data.
Section 4 describes identification and econometric issues. Section 5 contains the preliminary
estimation results and section 6 the preliminary experiments results. No policy section is
included at the moment. Section 7 concludes.
2
Model
2.1
Environment
Individuals begin adult life by making a schooling decision, which involves a comparison
of the values of entering the labor and marriage markets with schooling type s ∈ {L, H},
denoting a low and high schooling level. The benefit of acquiring schooling is the access
to a schooling-specific labor market which is completely segregated from the labor market
of the other schooling level. The access to a schooling-specific labor market also has an
impact on marriage market opportunities. The cost of acquiring the high schooling level is
heterogeneous in the population and it is gender-specific. We denote gender with g ∈ {f, m}
and the cost of schooling with q ∼ Q(q|g).
After schooling is completed4 , individuals enter the marriage market and the labor
4
The timing implies that we ignore any time cost involved in the acquisition of schooling and any marriage
4
market. Each market is characterized by frictions and by match-specific shocks. In the
labor market, we denote the Poisson rate of arrival of job offers with λ (g, s) . A job offer
is fully characterized by a wage w ∼ F (w|g, s) . If a job is accepted and a match realized,
it is terminated by an exogenous shock η (g, s) . A job-match may also be endogenously
terminated as a result of changes in the marital status or a as a result of changes in the
labor market status of the other spouse. There is no on-the-job search in the current
version of the model.
Marriage markets are also characterized by frictions and offers arrive following a Poisson
process. Any member of one gender, no matter the education group, may meet and marry
a member of the other gender, no matter the education group. However, the probability of
meeting between and within schooling groups are allowed to be different and to be governed
by a CRS matching function Γ [S (g, s) , S (g 0 , s0 )] where S (g, s) denotes the measure of
single of gender g and schooling level s in the economy and the superscript 0 denotes the
other spouse variables. As a result, the Poisson arrival rate of a marriage opportunity to
an individual of gender g and schooling s with an individual of the opposite gender5 and
schooling s0 is:
Γ [S (g, s) , S (g 0 , s0 )]
λM g, s, s0 =
(1)
S (g, s)
A marriage offer is characterized by the gender, education level and labor market status
of each of the two spouses and by a match-specific utility of marriage denoted by θ ∼
G (θ). One individual of each gender is necessary to create a married couple. Only single
individuals are allowed to meet in the marriage market. Marriages can be terminated only
by an exogenous process with rate ηM .
Each single individual of gender g and educational level s has a utility function given
by:
u(l, c|g, s) = α (g) ln(l) + (1 − α (g)) ln(c)
(2)
where:
c = wh (g, s) + y
T
= l + h (g, s)
Consumption c is equal to the sum of labor income wh (g, s) and nonlabor income y. Time
T is allocated between leisure l and work but there is not intensive margin decision on
labor supply: h (g, s) is the gender and school-specific amount of hours required by each
job contract and individuals cannot choose it. An unemployed agent sets l = T , where T
is the upper bound on time available for leisure and work at any moment in time.
market or labor market activities happening during the schooling completion process. The schooling decision
is fixed after entering the labor and marriage markets.
5
Driven by data availability, we impose that marriage only happens between individuals of opposite sex.
5
Each married individual of gender g and educational level s married with an individual
of gender g 0 and educational level s0 has utility function given by:
u(l, c, θ|g, s, s0 ) = α (g) ln(l) + (1 − α (g)) ln(c) + θ
(3)
where:
c = wh (g, s) + w0 h g 0 , s0 + y + y 0
T
= l + h (g, s)
Consumption c is a public good and it is equal to the sum of both spouses labor and
nonlabor incomes. θ is another public good and by the utility of staying together. Time
T is allocated between the private good, leisure l and work, under the constraint mention
above that h (g, s) is not a choice variable but a job requirement. For the case in which a
spouse s is unemployed, then w = 0, and the same is true for spouse s0 .
Household interaction is assumed to be non-cooperative. Each spouse decides to accept
or reject a job offer and whether to keep or quit their current job. The spouse who is
receiving the offer is the first-mover, while the other spouse acts as a follower. This timing
convention solves the equilibrium multiplicity that may arise in the dual-searcher setting
we are analyzing. It may be justified by assuming that the spouse receiving the offer may
restrain from informing the other agent about the job offer unless it is optimal to do so.6
We assume that agents live forever and they they face a common instantaneous discount
rate, ρ.
2.2
Value Functions
The value function for an agent of gender g and schooling s, employed at wage w and
married to an agent with schooling s0 employed at wage w0 and enjoying a marriage flow
value of θ is:
ρ + η (g, s) + η g 0 , s0 + ηM V w, w0 , θ|g, s, s0 = u(l, c, θ|g, s, s0 )
(4)
0
0 0
0
+η (g, s) V 0, wR w , 0|g , s , s , θ|g, s, s
+η g 0 , s0 max V w, 0, θ|g, s, s0 , V 0, 0, θ|g, s, s0
+ηM max {V (w|g, s) , V (0|g, s)}
The value function is conditioned on the gender of the agent, their schooling, and the
schooling of the spouse, and it is function of wage of the agent (a wage of zero denotes
unemployment), the wage of the spouse, and marriage match-value. Given the state,
three shocks may hit the agent: an exogenous termination of her job at a rate η (g, s); an
6
When two single employed agent meet and decide to marry, it is not clear who should be considered
and who should be considered the follower conditioning on the criteria just described. In this case, we
randomize assuring a 50% chance to each gender.
6
exogenous termination of her spouse’s job at a rate η (g 0 , s0 ); and exogenous termination of
the marriage at a rate ηM . Notice that each spouse reacts optimally to a shock to the other
spouse’s labor market status, following the household interaction process described above.
We denote the optimal reaction (quitting or not quitting the current job) with the function
wR . For example wR (w0 , 0, θ|g 0 , s0 , s) in equation (4) assumes either value w0 if the spouse
keeps the current job after the agent has been exogenously terminated or value 0 if the
spouse quits the current job as a result of the agent’s change in labor market status. The
agent also reacts optimally to exogenous divorce, deciding whether to stay in the current
job or to quit in order to look for a better job offer. Notice we have introduced the notation
for value functions of single agents: they are just a function of the agent’s labor market
state.
The value function for an agent of gender g and schooling s, unemployed and married
to an agent with schooling s0 , working at job w0 and enjoying a marriage flow value of θ is:
ρ + λ (g, s) + η g 0 , s0 + ηM V 0, w0 , θ|g, s, s0 = u(T, c, θ|g, s, s0 )
(5)
Z
+λ (g, s) max V w, wR w0 , w, θ|g 0 , s0 , s , θ|g, s, s0 , V 0, w0 , θ|g, s, s0 dF (w|g, s)
+η g 0 , s0 V 0, 0, θ|g, s, s0
+ηM V (0|g, s)
Given the state, three shocks may hit the agent: a job offer at a rate λ (g, s), an exogenous
termination of her spouse’s job at a rate η (g 0 , s0 ), and an exogenous termination of the
marriage at a rate ηM . When the spouse’s job is terminated, no endogenous reaction follows
since agents are not allowed to divorce as a result of a change in labor market status. When
the agent receives a job offer, she will decide to accept or reject the job by maximizing over
the alternative value functions, anticipating the spouse’s optimal reaction. As before, we
denote the spouse’s optimal reaction with wR . If the marriage is terminated, no additional
actions are available.
The value function for an agent of gender g and schooling s, employed at wage w and
married to an agent with schooling s0 , searching for a job, and enjoying a marriage flow
value of θ is:
ρ + η (g, s) + λ g 0 , s0 + ηM V w, 0, θ|g, s, s0 = u(l, c, θ|g, s, s0 )
(6)
0
+η (g, s) V 0, 0, θ|g, s, s
Z
+λ g 0 , s0
V wR w, w0 , θ|g, s, s0 , w0 , θ|g, s, s0 dF w0 |g 0 , s0
A0 (w|g,s,s0 )
+ηM max {V (w|g, s) , V (0|g, s)}
where:
A0 w|g, s, s0 = w0 : V w0 , wR w, w0 , θ|g, s, s0 , θ|g, s, s0 > V 0, w, θ|g 0 , s0 , s
7
Given the state, three shocks may hit the agent: exogenous job termination at a rate
η (g, s), a job offer to her spouse at a rate λ (g 0 , s0 ), and exogenous termination of the
marriage at a rate ηM . When the job is terminated, no endogenous reaction follows since
agents are not allowed to divorce as a result of a change in labor market status. When the
spouse receives a job offer, she will decide whether to accept the job. If he accepts, the
agent in question will react optimally, as denoted by the reaction function wR . But not all
the job offers are acceptable to the spouse and in general the acceptance region depends on
the agent’s labor market state and type. We denote by A0 (w) the support of the job offers
distribution which is acceptable to the spouse given the agent’s job (w, h) . If the marriage
is terminated, the agent decides whether to continue employment at their current job or
to quit into unemployment so as to look for a better job.
The value function for an unemployed agent of gender g and schooling s, married to an
unemployed agent with schooling s0 , and enjoying a marriage flow value of θ is:
ρ + λ (g, s) + λ g 0 , s0 + ηM V 0, 0, θ|g, s, s0 = u(T, c, θ|g, s, s0 )
(7)
Z
+λ (g, s) max V w, 0, θ|g, s, s0 , V 0, 0, θ|g, s, s0 dF (w|g, s)
Z
0 0
+λ g , s
V 0, w0 , θ|g, s, s0 dF w0 |g 0 , s0
A0 (0|g,s,s0 )
+ηM V (0|g, s)
Given the state, three shocks may hit the agent: an exogenous job offer at a rate λ (g, s), a
job offer to the spouse at a rate λ (g 0 , s0 ), and exogenous termination of the marriage at a
rate ηM . When the agent receives the offer, she is the first mover and decides whether to
accept the offer. When the spouse receives the offer, the spouse decides whether to accept
the offer. In both cases, the other spouse has no ability to respond. If the marriage is
terminated, the agent is back to the single unemployed state.
The value function for an unemployed single agent of gender g and schooling s is:
[ρ + λ (g, s) + λM (g, s, L) + λM (g, s, H)] V (0|g, s) = u(T, c|g, s)
Z
+λ (g, s) max {V (w|g, s) , V (0|g, s)} dF (w|g, s)
X
+
λM g, s, s0 ×
s0 ∈{L,H}

U (g 0 , s0 )
max {V (0, 0, θ|g, s, s0 ) , V (0|g, s)} dG (θ) +
0 0
B(0,0|g


R
R ,s ,s)


0
0
max {V (0, wR (w0 , 0, θ|g 0 , s0 , s) , θ|g, s, s0 ) , V (0|g, s)} 
 E (g , s )


0
0
0
0
0
C(g ,s ) B(w ,0|g ,s ,s)
0
0
0
0
0
0
dG (θ) dF (w |g , s , w ∈ C (g , s ))

R
8
(8)
where:
B w, w0 |g, s, s0
≡
θ : V wR w, w0 , θ|g, s, s0 , wR w0 , w, θ|g 0 , s0 , s , θ|g, s, s0 > V (w|g, s)
C (g, s) ≡ {w : V (w|g, s) > V (0|g, s)}
Given the state, two shocks may hit the agent: a job offer at a rate λ (g, s) and a
marriage offer at a rate λM (g, s, L) if coming from a low schooling-level individual or at
a rate λM (g, s, H) if coming from a high schooling-level individual. Given the schooling
level, the potential spouse can be unemployed or employed: we denote the endogenous
measures of these two sets in equilibrium with U (g 0 , s0 ) and E (g 0 , s0 ) . The set of employed
singles is drawn only from the accepted wage distribution of this type, which has support
C (g 0 , s0 ). The set B (w, w0 |g, s, s0 ) takes into account the fact that marriage is consensual
and the current agent can decide about marrying a potential spouse only if the potential
spouse also agrees to marry.
The value function for a single agent of gender g and schooling s, employed at wage w
is:
[ρ + η (g, s) + λM (g, s, L) + λM (g, s, H)] V (w|g, s) = u(l, c|g, s)
(9)
+η (g, s) V (0|g, s)
X
+
λM g, s, s0 ×
s0 ∈{L,H}






U (g 0 , s0 )
max {V (wR (w, 0, θ|g, s, s0 ) , 0, θ|g, s, s0 ) , V (w|g, s)} dG (θ) +
0 0
B(0,w|g

R
R ,s ,s)

0
0
E (g , s )
max {V (wR (w, w0 , θ|g, s, s0 ) , wR (w0 , w, θ|g 0 , s0 , s) , θ|g, s, s0 ) , V (w|g, s)} 

C(g 0 ,s0 ) B(w0 ,w|g 0 ,s0 ,s)
0
0
0
0
0
0
dG (θ) dF (w |g , s , w ∈ C (g , s ))
R
Given the state, two shocks may hit the agent: an exogenous job termination at a rate
η (g, s) and a marriage offer at a rate λM (g, s, L) if coming from a low schooling-level
individual or at a rate λM (g, s, H) if coming from a high schooling-level individual. As
before, meeting in the marriage market may be with singles of any schooling level who can
be employed or unemployed. And as before, each agent reacts optimally with respect to her
labor market decision when considering marriage. However, now there is the possibility
that two single employed agents meet in the marriage market, generating an ambiguity
about which one of the two is the leader in the game. In the notation, above we are
assuming that the agent for whom we are writing the value function is the leader. In
simulation and estimation, we randomize which one of the two agents is the leader when
meetings of two employed singles occur.
9
2.3
2.3.1
Equilibrium
Definition
The optimal decision rules have a reservation value property. In the labor market the
reservation value is defined over the wage w; in the marriage market over the marriage
match value θ; and the schooling reservation value is defined with respect to the schooling
cost q.
First, we look at labor market decisions. When single, decision rules are identical
to a standard single agent search model and the reservation wage above which offers are
accepted is defined as:
w∗ (g, s) : V (w∗ |g, s) = V (0|g, s)
(10)
When married, labor market decision rules also depend on the labor market status of the
spouse, due to the nonlinearity of the utility function.7 As a result, the reservation wage
of an individual with schooling s, married to a spouse with schooling s0 and labor market
status characterized by w08 , in a marriage generating a match value θ is given by:
w∗ w0 , θ|g, s, s0 :
(11)
∗
0
∗
0 0
0
0
V w , wR w , w , θ|g , s , s , θ|g, s = V 0, w , θ|g, s, s
Next, we consider marriage market decisions. Again, given the assumptions on the
utility function, the two markets are interdependent and the marriage market decision
depends on the labor market status of both the agent and the potential spouse. The
reservation match value for the marriage to occur (from this agent’s perspective) is:
(12)
θ∗ w, w0 |g, s, s0 :
∗
0
∗ 0 0
0 ∗
0
V wR w, w , θ |g, s, s , wR w , w, θ |g , s , s , θ |g, s = V (w|g, s)
Finally, we look at schooling decisions. Schooling decisions have a different timing than
labor market and marriage market decisions because they are taken before entering the two
markets. Once a schooling decision is made, it cannot be changed and the individual enters
simultanously the marriage and labor market as a single unemployed agent of schooling
level s. As a result, the reservation cost of acquiring the high level of schooling H is given
by:
q ∗ (g) :
(13)
∗
V (0|g, H) − q = V (0|g, L)
where individuals with cost less than or equal to q ∗ (g) acquire schooling level H.
We can now propose the following:
7
See Dey and Flinn (2008). A systematic treatment of the issue is also provided by Guler, Guvenen and
Violante (2012), while Flabbi and Mabli (2012) exploit the result in estimation.
8
Recall that w0 > 0 defines employement and w0 = 0 defines unemployment.
10
Definition 1 Given g ∈ {f, m}, s ∈ {L, H}, s0 ∈ {L, H} and:


α (g)
λM (g, s, s0 )

 λ (g, s)
ρ
ηM
η (g, s) ,
, Q(q|g)
,


h(g, s)
G(θ)
F (w|g, s)
an equilibrium is a set of values
V ., ., θ|g, s, s0 , V (.|g, s)
that solves equations (4)-(9) under the optimal decisions rules characterized by equations
(10)-(13).
2.3.2
Computational Method
A closed form solution for the value function characterized in Definition 1 is not available
and therefore we use simulation methods to solve for an equilibrium at given parameter
values.
The model is solved by evaluating the value functions in a discretized grid of wages and
marriage match-specific values, given the set of parameter values and the model’s steady
state equilibrium conditions. The model’s steady state equilibrium conditions are particularly challenging in our context since the equilibrium distribution of singles is endogenous
and it is necessary to compute the value functions. Notice that we have to keep track not
only of the proportion of singles but also of their equilibrium distribution over labor market states (including over accepted wages while employed) and schooling levels since the
labor market state and schooling level of single agents has an impact on marriage market
decisions.
The procedure works as follow. Given a set of parameters and a guess of the relevant
steady state equilibrium distribution, a first set of value functions is found by solving the
fixed point problem. The fixed point problem is over a quite a high-dimensional vector of
values since we have to jointly iterate over value functions in the marriage market and the
labor market for each value of the discretized wage and value of marriage grids, and for
each gender and education level. The fixed point over the value functions for given steady
state equilibrium distribution constitutes the “inner loop” of our simulation procedure.
Given the value functions, we can obtain an updated value of the steady state equilibrium
distribution that can be compared with the starting distribution and, in case of lack of
convergence, can be used to find a new set of value functions. The computation of the
steady state equilibrium distribution for given value function constitutes the “outer loop”
of our simulation procedure. The process is iterated until convergence is reached using
usual tolerance criteria.
Given this general structure, additional details of the simulation procedure need to be
solved. First, we have to decide whether to jointly simulate both sides of the marriage
11
market or whether to directly utilize the steady state distributions. In order to reduce the
computational burden, we chose the second alternative. As a result, when a marriage offer
is received by a given individual of gender g, a potential spouse is drawn from the steady
state distribution of single agents of gender g 0 .
Second, our leader-follower approach cannot solve the issue of two single employed
agents meeting in the marriage market. In this particular case, we choose the randomization
implemented by some previous papers in this literature: when the agent is single-employed
and another single-employed agent is drawn as a potential spouse, the leader of the marriage
game is chosen randomly assigning a 0.5 probablity to each agent to be the leader.
Third, the simulation is done agent by agent, storing each agent’s state. The state of
an agent is determined by her schooling level, wage and marriage status. If the agent is
married, the spouse’s schooling level and wage and the couple’s match-specific marriage
utility are also stored. Mirroring the data we will use in estimation, we store each agent’s
state every three months.
Finally, the steady state used in the “outer loop” is computed using the final 30% of the
total number of periods simulated. A total of 5,000 agents of each type (gender-schooling)
are simulated for 540 months.
2.3.3
Discussion
A graphical representation of the equilibrium outcomes both in the marriage and in the
labor market is reported in Figures 1 through 3.
We first focus on labor market decisions. When an agent is single and unemployed,
or in a couple with both spouses unemployed, the employment decision is characterized
by the usual reservation wage policy rule. When an unemployed agent is married to an
employed agent, the employment decision is more complex since it depends on the wage
and schooling level of the spouse. In this situation, when the unemployed spouse receives
an offer, three outcomes are possible: the agent can reject the job offer, the agent can
accept his job offer and the spouse quit her current job, or the agent can accept his job
offer and the spouse keep her current job. Figure 1 shows the solution to this game: the
wife is employed at one of the wages reported on the x-axis and the husband is receiving a
job at one of the wages reported on the y-axis. Each panel represents one of the possible
schooling combinations. The darkest area at the bottom of each panel shows the region in
which the offer is rejected; the most lightly-shaded region is where both agents keep their
jobs; the remaining middle gray area to the left of each panel shows the region where the
agent accepts the offer and the spouse quits her job. As expected, this area corresponds to a
combination of relatively low spouse’s wages and relatively high wage offers. Similarly, the
darkest area corresponds to combinations of low wage offers and high spouse wages. Only
when the husband receives a relatively high wage offer and the wife is already employed at
a job paying a relatively high wage, will they both be employed in equilibrium.
Next, we look at marriage market decisions. When two unemployed agents meet, the
12
marriage decision depends only on the match draw θ and, for a given schooling level
combination, there will exist a unique reservation θ∗ above which the agents decide to
marry. The marriage decision when two employed agents meet is more complicated because
each θ draw is defining an acceptance region over the space of the couples’ accepted wages.
Figure 2 represents such region for a given θ value. The equilibrium is characterized by
four regions: the darkest area (with wages below the reservation values and therefore
never active) indicates non-marriage; the lightest area indicates marriage with both agents
keeping their current job; the second darkest (next to the vertical axis) indicates marriage
and quitting of the current job by the man; and the second lightest (next to the horizontal
axis) indicates marriage and quitting by the woman. Note that, similarly to what happened
in Figure 1, quitting is induced when one wage is relatively low when compared to the other.
As a result, we observe married couples with both agents employed only when they both
have relatively high wages.
Figure 3 displays the joint distributions of wages for a married couple where both
spouses are employed, conditioning on the four schooling levels combinations. The joint
distributions show the assortative mating in wage levels implied by the equilibrium behavior
we have shown in Figure 1 and 2. Figure 3 also shows how both markets, combined
with the spouses’ schooling levels, have a genuinely joint impact on the four equilibrium
distributions.
3
Data
We use the Current Population Survey (CPS) to extract moments referring to: proportion
across labor market states, unemployment durations, means and standard deviations of
accepted wages, and correlations between spouses accepted wages. We use the American
Community Survey (ACS) to extract moments referring to proportions of the population
in the various marriage market states and transitions between marriage market states over
time. All the moments are computed by gender and schooling level.
We impose the following restrictions on the sample:
• Age: 25-49;
• Education:
– High level of schooling: College completion or more;
– Low level of schooling: Associate degree, some college, HS completed or less;
• Race: White;
• Year: 2007.
13
The states in the two markets are defined as follows. Married individuals are individuals
who declare that they are currently married. We classify all the individuals who are
not married as single, including those cohabiting. Employed individuals are individuals
currently working. All the other individuals are considered searching in the labor market,
even if they declare to be out of the labor force.
4
Econometric Issues
The identification of the model requires a set of additional functional form assumptions. As
shown by Flinn and Heckman (1982), we need to assume recoverable wage offer distributions
if we want to identify them from accepted wages information. We assume the wage offers
distributions to be lognormal with gender- and schooling-specific parameters µ (g, s) and
σ (g, s).
The marriage match-specific value θ is unobservable but the equilibrium shows it has
an impact on the “type” of marriage that is realized,9 where type is defined by the labor
market state and schooling level of the spouses. We assume a normal distribution with
parameters µθ and σθ .
The cost of acquiring the high schooling level with respect to the low schooling level
has only an impact on the schooling level acquired before entering the marriage and labor market. This simple threshold-crossing impact forces us to assume a one-parameter
distribution. We assume a negative exponential but with gender-specific parameters τ (g).
Finally, we need to impose a functional form for the contact rate functions. Since we
can observe gender and schooling specific transitions, we can identify marriage market
meeting rates that are gender-specific and schooling-specific in both spouses’ schooling. As
a result we can identify a two-parameter matching function which we will assume has the
following frequently used Cobb-Douglas specification:
0
0
Γ S (g, s) , S g 0 , s0 = β s, s0 S (g, s)ν(s,s ) S (g, s)(1−ν(s,s ))
(14)
The estimation procedure involves three main steps. First, we fix the following parameters:
Θ1 = {ρ, T, h (g, s)}g∈{f,m},s∈{L,H}
(15)
The discount rate is fixed to 5% a year, the time endowment to 80, and the job hours
requirements to the mean of each specific gender-schooling group.
Second, we use the Method of Simulated Moments (MSM) to estimate the following
set of parameters:


0)
λ
(g,
s)
λ
(g,
s,
s


M




η (g, s)
ηM
Θ2 =
,
, α (g)
(16)
µ (g, s)
µθ






σ (g, s)
σθ
g∈{f,m},s∈{L,H},s0 ∈{L,H}
9
See in particular Figure 2.
14
where the first column refers to labor market parameters, the second to marriage market
parameters, and the third to the utility function parameter.
Third, we recover the following set of parameters:
Θ3 = τ (g) , β s, s0 , ν s, s0 g∈{f,m},s∈{L,H},s0 ∈{L,H}
by solving identities (1) and by inverting the identity equating the observed proportion in
the high schooling-level group with the equilibrium proportion implied by the model.
5
Preliminary Estimation Results
Preliminary point estimates from the estimation procedure are presented in Table 2. The
first row reports the estimate of the preference parameter α: the weight on leisure in the
log-linear utility function we assume (see equations 2 and 3.) We estimate a value quite
similar to previous existing work but, contrary to previous literature, we find it to be
slightly higher for men than women.
The second group of parameters describes the labor market structure for each gender
and schooling level. The mobility parameters are estimated to be similar to previous
empirical work using a comparable setting. The location and scale parameters of the
lognormal wage offers distribution imply a gender differential in average wage offers of
about 6% in the low schooling group and of about 4% in the high schooling group. The
returns to acquiring a high level of schooling in terms of mean wages offers is about 39%
for women and about 37% for men. Both results are derived by computing the average
wage offers conditional on gender and schooling at our estimated parameters. Means and
variances of wage offers for the four groups are reported in the top two rows of Table 3.
The third group of parameters describes the marriage market structure for each gender
and schooling level. The arrival rate of marriage offers λM follows the expected ranking,
with offers more likely to arrive from individuals with the same schooling level. However,
they are estimated to be extremely low, predicting that the average duration for receiving
a marriage offer is about 26 years. We are currently working to improve the performance
of the estimated model along this dimension. The location and scale parameters of the
marriage match value distribution predict an expected value of about 0.307 (see Table 3.)
Given that θ is additively separable in the flow utility where leisure and consumption enter
in logs and since the estimated variance is fairly low, we can infer that the marriage match
value plays a significant but not major role in the marriage decision for the average couple.
Tables 4 to 6 present some measures of model fit. Table 4 reports a major labor market
outcome: accepted wages. The fit is reasonable on the first moments for all groups but it
is not on the standard deviations of the high schooling group, both in the male and female
samples. We are currently working to improve the performance of the estimated model
along this dimension. Table 5 reports the joint steady state proportions over labor market
state, marriage market status, schooling level and gender. The average fit is good but
15
there is one particular feature we systematically over-estimate: the proportion of couples
with one spouse working and the other spouse unemployed. This is another dimension
we are currently working on to improve the fit. We suspect it may require some changes
in our behavioral model. Finally, Table 6 presents fits over transitions between marriage
and labor market states. We fit some moments quite well while others less so, even if we
always manage to produce estimates of the right order of magnitude (a weak argument,
we realize).
6
Preliminary Experimental Results
In this preliminary version, we present just two comparative statics exercises. The first,
reported in Table 7, looks at the impact of decreasing labor market frictions in the market
for the high level of schooling. Since the value of participation in such a market is increasing (less frictions mean better labor market opportunities) we expect the proportion
of individuals acquiring a college degree to increase. This is exactly what we observe in
the two bottom rows of Table 7. When the arrival rate of offers double, the proportion of
women acquiring the high level of schooling increases by about 18%, and the proportion of
men increases by about 20%. We also observe a small increase in the proportion of married people. It results from the fact that more individuals have a high level of schooling
(a desirable characteristics in the marriage market, too) and that those who do also have
better labor market opportunities (because of the lower level of frictions). Even if limited in magnitude, this type of effect shows how even a minor change in the labor market
structure gets transferred to marriage market outcomes.
The second comparative statics exercise is reported in Table 8. The experiment consists
in setting the female labor market structure (arrival and termination rates; wage offers at
each schooling level) equal to the estimated male labor market structure. The motivation
rests on the gender differential literature and it indicates how gender asymmetries in labor
market structure impact gender differences in marriage rates and schooling rates. A first
result concerns schooling decisions: women acquire less education than at baseline. This
is due to the fact that low schooling labor market parameters for men are relatively better
than those for women. A second result relates to labor market outcomes: the gender gap
on them essentially disappears. Still, this is informative because it indicates that gender
asymmetries in the marriage market do not negatively impact women’s labor market performance. Finally, the third result is directly informative about the joint decision process in
the two markets. Despite a large change in women’s labor market parameters, the marriage
rate barely changes, moving from 52.45% to 52.43%.
16
7
Conclusion
The paper presents a tractable framework to analyze simultaneous search in the labor and
marriage markets in the presence of endogenous schooling decisions. After developing the
model, we propose an identification and estimation strategy of its structural parameters.
We implement it using data from the Current Population Survey (CPS) to describe the
labor market dynamic and from the American Community Survey (ACS) to describe the
marriage market dynamic. Preliminary results generate reasonable point estimates and a
good fit along numerous dimensions. However, we still consider them preliminary because
they are not able to reproduce two data features we judge crucially important: variance of
accepted wages for the high schooling groups and the proportion of married couples where
one spouse works and the other is unemployed.
We also provide a set of preliminary comparative statics exercises showing the magnitude of the interactions between the two markets. We plan to implement a larger set of
comparative statics exercises in order to provide an empirical assessment of the importance
of taking into account the joint nature of the decision process in the two markets. Finally,
we plan to use the parameter estimates to perform a series of policy experiments comparing a labor income tax system based on individual taxation with a system based on joint
taxation.
17
References
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Economic Dynamics 8, 565–599.
Chiappori, P.-A., Dias, M. C. and Meghir, C. (2015), The Marriage Market, Labor Supply and Education Choice, Working Paper 21004, National Bureau of Economic
Research.
Choo, E. (2015), ‘Dynamic marriage matching: an empirical framework.’, Econometrica
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Choo, E. and Siow, A. (2006), ‘Who marries whom and why’, Journal of Political Economy
114(1).
Dagsvik, J. (2000), ‘Aggregation in matching markets.’, International Economic Review
41(27- 57).
DelBoca, D. and Flinn, C. (2014), ‘Household behavior and the marriage market’, Journal
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of Econometrics 145, 43–63.
Flabbi, L. and Mabli, J. (2012), ‘Household Search or Individual Search: Does it Matter?
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18
Table 1: Descriptive Statistics CPS Sample
Gender:
Education:
Low
Female
High
Tot
Low
Male
High
Tot
32.8
31.2
26.7
4.4
63.9
16.2
19.9
7.1
12.8
36.1
48.9
51.1
33.8
17.2
100.0
37.5
32.4
25.6
6.8
69.9
13.6
16.5
4.2
12.3
30.1
51.1
48.9
29.8
19.1
100.0
3.3
60.6
63.9
0.8
35.3
36.1
4.1
95.9
100.0
3.7
66.2
69.9
0.7
29.4
30.1
4.4
95.6
100.0
13.4
5.8
23.2
11.2
16.2
7.4
26.3
13.5
4.2
5.5
4.3
5.2
3.9
4.9
3.2
4.2
31,565
17,828
36,056
15,521
Marriage Market:
Single
Married
with Low
with High
Total
Labor Market:
Unemployed
Employed
Total
Wages:
Mean
SD
U Durations:
Mean
SD
N. Observations
19
49,393
51,577
Table 2: MSM Estimated Parameters
Gender (g):
Schooling (s):
α (g)
Female
Low
High
Male
Low
High
0.167
0.174
λ (g, s)
η (g, s)
µ (g, s)
σ (g, s)
0.320
0.022
2.445
0.423
0.370
0.015
2.980
0.309
0.352
0.020
2.483
0.468
0.446
0.019
3.009
0.337
λM (g, s, L0 )
λM (g, s, H 0 )
ηM
µθ
σθ
0.0031
0.0020
0.0021 0.0028
0.0031 0.0020
0.009
0.307
0.465
0.0020
0.0039
20
Table 3: Exogenous Heterogeneity implied by MSM Estimates
Gender (g):
Schooling (s):
Female
Low High
Male
Low High
Wage Offers:
E (w|g, s)
V (w|g, s)
12.61
31.14
13.36
43.66
20.64
42.61
Marriage Match Values:
E (θ)
V (θ)
Cost of Schooling:
E (q|s)
233.9
21
21.45
55.18
0.307
0.217
299.4
Table 4: Model Fit: Accepted Wages
Gender (g):
Schooling (s):
Female
Low High
Male
Low High
15.18
16.50
23.55
25.69
16.52
18.45
24.26
27.25
5.26
5.19
6.22
6.18
6.63
6.75
6.81
7.02
13.40
14.12
23.16
23.87
16.18
19.12
26.34
30.12
5.83
6.01
11.16
11.25
7.41
7.91
13.51
13.82
Estimated:
Mean
Single
Married
SD
Single
Married
Sample:
Mean
Single
Married
SD
Single
Married
22
Table 5: Model Fit: Steady State Proportions (%)
Gender (g):
Schooling (s):
Low
Female
High Total
Low
Male
High Total
Estimated:
Single:
U
E
30.1
3.0
27.1
16.9
0.9
16.0
46.9
3.9
43.1
34.3
2.8
31.5
13.8
0.7
13.1
48.1
3.5
44.7
Married:
UU
UE
EU
EE
29.3
0.4
5.6
2.9
20.6
23.7
0.2
1.8
5.2
16.6
53.1
0.5
7.3
8.0
37.2
31.0
0.3
5.9
3.2
21.5
20.9
0.2
1.9
3.9
14.8
51.9
0.5
7.9
7.2
36.3
Total
59.4
40.6
100.0
65.3
34.7
100.0
Single:
U
E
32.8
2.2
30.6
16.2
0.4
15.7
48.9
2.6
46.3
37.5
2.7
34.8
13.6
0.5
13.1
51.1
3.1
48.0
Married:
UU
UE
EU
EE
31.2
0.2
1.0
0.9
29.1
19.9
0.0
0.4
0.3
19.3
51.1
0.2
1.4
1.1
48.4
32.4
0.2
0.9
1.0
30.4
16.5
0.0
0.2
0.3
15.9
48.9
0.2
1.1
1.3
46.4
Total
63.9
36.1
100.0
69.9
30.1
100.0
Sample:
23
Table 6: Model Fit: Transitions (%)
Single
SL
SH
MLL
Married
MLH MHL
Total
MHH
Estimated:
Males:
SL
SH
ML
MH
91.1
0.0
6.8
0.0
0.0
85.9
0.0
4.5
5.5
0.0
55.2
0.1
3.3
0.0
37.9
0.0
0.0
6.9
0.0
46.6
0.0
7.2
0.0
48.8
100.0
100.0
100.0
100.0
Females:
SL
SH
ML
MH
90.3
0.0
5.7
0.0
0.0
87.3
0.0
4.3
6.3
0.0
60.7
0.1
3.4
0.0
33.6
0.1
0.0
6.6
0.1
51.8
0.0
6.1
0.1
43.7
100.0
100.0
100.0
100.0
Males:
SL
SH
ML
MH
95.3
0.0
3.8
0.0
0.0
92.3
0.0
2.1
3.3
0.0
73.7
0.0
1.3
0.0
22.5
0.0
0.0
1.6
0.0
24.4
0.0
6.1
0.0
73.4
100.0
100.0
100.0
100.0
Females:
SL
SH
ML
MH
95.0
0.0
4.3
0.0
0.0
91.9
0.0
2.5
4.2
0.0
81.2
0.0
0.8
0.0
14.5
0.0
0.0
2.9
0.0
35.3
0.0
5.3
0.0
62.2
100.0
100.0
100.0
100.0
Sample:
24
Table 7: Experiments: Reducing Labor Market Frictions for Schooling Level H
Baseline
Increase in λ (g, H)
10%
50% 100%
Labor and Marriage Mkts. Proportions:
Single
U
E
47.55
3.66
43.89
47.45
3.50
43.95
47.34
3.61
43.73
47.26
3.51
43.75
Married
UU
UE
EU
EE
52.45
0.49
7.62
7.62
36.72
52.55
0.53
7.53
7.53
36.96
52.66
0.50
7.76
7.76
36.63
52.74
0.45
7.89
7.89
36.50
37.27
31.31
40.99
34.75
43.91
37.55
Proportion H Schooling:
Female
Male
36.10
30.10
25
Table 8: Eliminating Gender Differences in Labor Market Structure
Baseline
Female Male
Experiment
Female Male
Schooling:
Proportion H
36.10
30.10
32.26
30.03
24.25
27.25
16.52
18.45
25.42
27.47
16.39
18.10
24.26
27.46
16.41
18.89
Labor and Marriage Markets:
Wages:
Single E (w|g, H, E)
Married E (w|g, H, E)
Single E (w|g, L, E)
Married E (w|g, L, E)
23.55
25.69
15.18
16.50
Proportions:
Single
U
E
47.55
3.66
43.89
47.57
3.55
44.02
Married
UU
UE
EU
EE
52.45
0.49
7.62
7.62
36.72
52.43
0.44
8.36
8.36
35.28
Note: The experiment consists in setting the female labor market structure (arrival and termination
rates and wage offers) equal to the estimated male labor market structure.
26
Figure 1: Job Offer to Married UE couples - Husband leads
Emp. Decision MUE−LL − T 1
Emp. Decision MUE−LH − T 1
100
2
80
80
60
60
Wage
Wage
100
40
3
20
0
40
3
20
1
0
2
20
40
60
Wife Wage
80
0
100
1
0
20
Emp. Decision MUE−HL − T 1
2
80
80
60
60
40
3
20
0
20
40
60
Wife Wage
80
100
2
40
3
20
1
0
80
Emp. Decision MUE−HH − T 1
100
Wage
Wage
100
40
60
Wife Wage
0
100
27
1
0
20
40
60
Wife Wage
80
100
Figure 2: Marriage Offer between two Employed - Husband leads
M Lead LS−HS − T 3
100
80
80
Woman Wage
Woman Wage
M Lead LS−LS − T 3
100
60
40
20
0
60
40
20
0
20
40
60
Man Wage
80
0
100
0
20
100
80
80
60
40
20
0
80
100
80
100
M Lead HS−HS − T 3
100
Woman Wage
Woman Wage
M Lead HS−LS − T 3
40
60
Man Wage
60
40
20
0
20
40
60
Man Wage
80
0
100
28
0
20
40
60
Man Wage
Figure 3: Joint Accepted Wages distribution - Married couple
MEE LH
MEE LL
6000
1000
4000
500
2000
0
0
0
0
0
50
wW
100
100
0
50
50
wW
wM
50
100
MEE HL
1000
2000
500
0
0
0
0
wW
0
100
0
50
50
100
wM
MEE HH
4000
50
100
wW
wM
29
50
100
100
wM