1. society
2. social institution
3. marriage

Marriage Patterns under Uncertainty
Pavel Jelnov
April 2, 2015
Abstract
I analyze two marriage models with heterogeneous individuals, whose true type is unobserved. The
question is whether the last-50-years change in the marriage pattern is driven by technical changes, such
as a more ecient dating technology and slowing down of the female biological clock, or by changes in
the heterogeneity. Both models indicate that the second possibility is more likely to be correct. The rst
model, which emphasizes endogenous divorce, is simulated for a wide set of parameter values, and the
second model, which incorporates pre-marital relationship, is calibrated, comparing the 1970 and 1995
marriage age distributions.
1 Introduction
Since the 1960s, the marriage pattern in the US dramatically changed.
The age of marriage of men and
women increased, the mean spousal age gap narrowed, and the variance of ages at rst marriage sharply rose.
Figure 1 demonstrates the dierences between the 1970 and 1990 American marriage patterns.
It shows
the distribution of rst marriages by age of the spouses, using the Vital Statistics marriage records. In this
paper, I address this changed pattern, using two alternative marriage models with heterogeneous individuals
and female biological clock.
The question that is analyzed is whether the last 50 years change in the marriage pattern is driven by
technical changes, including the dating technology and the female biological clock, or by changes in the
heterogeneity.
Both models indicate that the second possibility is more likely to be correct.
1
Specically,
Figure 1: The actual new marriages distribution - 1970 (left) and 1990 (right)
the rst model addresses the uncertainty regarding the true type of the partner using endogenous divorce,
while the second model incorporates pre-marital relationship. The rst model is simulated for a wide set of
parameters' values, and the second model, which is relatively simple, is structurally calibrated using the 1970
and 1995 Vital Statistics data.
In the rst model, time is continuous, and single men and women randomly meet. The partner's true
type, which can be low or high, is unobserved before marriage. While meeting the potential spouse, only a
signal of his or her true type is observed. This signal updates a prior which is the proportion of the high type
among the singles of each age. However, after the marriage the true type is known. Divorce is endogenous and
unilateral, but the possibility of divorce arrives as a Poisson shock, and the frequency of the shock measures
for the easiness of divorce. When the Poisson shock arrives, each of the partners can unilaterally decide to
divorce. The key issue is that time is costly, especially for women, and the arrival of the shock may be too
late. Thus, when the shock is seldom, many initially bad marriages turn to be good at the time when
divorce is possible.
The simulated results of the model indicate that slowing down of the biological clock
can not explain the recent increase in the age of marriage, but the liberalization of divorce, interacted with
2
increasing inequality between the two types, can do it.
In the second model, only men can be of low or high type, and the type may be observed after one period
of pre-marital relationship. The prior of the mate's type is the proportion of high types among single men
of each age. The three parameters of the model (proportion of the high type in the population, biological
clock of females relatively to this of males, and the frequency of meetings) are structurally calibrated and the
model is found to t pretty well the empirical marriage patterns of 1970 and 1995. The calibrated parameters
imply that the change in the biological clock does not drive the changes in the marriage pattern between 1970
and 1995, but increasing heterogeneity does. Specically, the calibrated proportion of the high type in the
population decreased during this period from 0.92 to 0.56. It implies increased variance and Gini coecient
of the marriage payo. An extension of the model allows for a longer period of pre-marital relationship, when
each period a signal of the man's true type arrives, updating the woman's prior. Each period she decides
whether to marry, to stay in a relationship, or to split and return to the singles' population. The analytical
result shows that the length of pre-marital relationship weakly decreases in the age of the female partner but
follows a hump-shape in the age of the male partner.
The correlation between income inequality and the age of rst marriage was documented in many papers (Keeley (1977, 1979); Gould and Passerman (2002); Loughran (2002); Coughlin and Drewianka (2011);
Danziger and Neuman (1999); Bergstrom and Schoeni (1996); Blau et al. (2000); Fortin and Lemieux (2000);
Mu and Xie (2011); Bloom and Bennett (1990)). I show the impact of uncertainty interacted with increased
inequality.
Two branches of literature analyze the role of uncertainty in the marriage market.
My paper
relates to the literature that emphasizes the unobservable true type of the potential marriage partner. This
literature builds on Bergstrom and Bagnoli (1993), who show that the age of marriage increases when men are
more unequal and only old men's true type is observed. Another important paper is Weiss and Willis (1997),
who show the impact of surprising partner's income outcomes on probability of divorce. In my rst model, I
assume, similarly to Bergstrom and Bagnoli (1993), that uncertainty, measured by the variance of the signal,
decreases with the partner's age.
Dierently from their paper, both men and women are heterogeneous,
and there exists possibility of divorce. Technically, the model belongs to the group of recent age-of-marriage
search models, such as Sautmann (2011). The second branch of literature emphasizes quality of match rather
3
than the type of each of the partners. This literature builds on Jovanovic (1979) model. Recently, Marinescu
(2012) presented a divorce model where the qua lity of match is a random walk. Divorce not as a result of
learning or shocks on the quality match, but as a result of on-the-job search of married women is modeled
in Rotz (2011). Divorce as a result of Rubinstein bargaining game between the spouses is modeled in Hyee
(2011). Brian et al (2006) structurally estimate a model of cohabitation, marriage, and divorce with quality
of match. The second model in my paper incorporates pre-marital relationship, but, dierently from Brian
et al (2006), in a much simpler framework, empirically comparing two distinct points in time: 1970 versus
1995, and using a very large data set: the US Vital Statistics marriage records.
The paper is organized as follows. Section 2 present the rst model and discusses its results. Section 3
presents the second model, and estimates it using the moments of the 1970 and 1995 marriage age distributions. Section 4 concludes.
2 The rst model (endogenous divorce)
2.1
Setup
Continuum of heterosexual individuals of two genders exist in environment with continuous time. The single
individuals do not obtain utility and seek marriage. Each individual has three observable properties: sex, age
and type. The age
a
¯
a
is in a continuous range
¯],
[a, a
where
a
is the age of entry into the marriage market, and
is the maximal age that an individual gives and receives utility in marriage. It means that never-married,
widowed and divorced individuals drop out of the marriage market at the age of
females and males, respectively. The individual's congenital type
is
f
or
m
for
Further down,
is the sex opposite to i. Thus,
(i, j, a) is used to indicate each individual in a group sharing
{(i, j, a), (i0 , j 0 , a0 )}.
The proportion of the high type in the population is
δf (a)
j ∈ {l, h}.
i
capturing her characteristics and I indicate her mate with
the same characteristics. Couples are indicated with
factor, are
is low or high,
The sex
(i, j, a),
I indicate each individual with a triple
(i0 , j 0 , a0 ) where i0
j
a
¯.
for women and
δm (a)
for men.
p.
The death rates, determining the discounting
Let us indicate the measure of population of sex
4
i
at
age
a
with
xi (a).
xi (a) = xi (a)e
´a
−
a
The total population of sex
δi (τ )dτ
where
xi (a) =
´a
¯
a
with a Poisson rate
ρ.
i
of marriageable ages
1
´t
δi (τ )dτ
− a
e
.
[a, a
¯]
is normalized to be 1.
Then,
Individuals meet randomly the opposite sex mates
dt
Then they marry each other if and only if the expected utility of each of the two
partners from the marriage is above their values as singles. The model has endogenous divorce. The divorce
is possible only with arrival of a Poisson shock that hits the married couple with rate
γ
γ.
Thus, the parameter
measures the easiness of divorce. The divorce is unilateral and occurs if and only if at least one of the sides
is better o if single than in the existing marriage. In the case of divorce, the two individuals return to the
marriage market. Similarly, widows and widowers return as well to the marriage market.
2.2
Utility
The utility from marriage depends only on the partner and there is no match-specic payo. The utility is
a ow of non-transferable payos of the form
Vij (a) ≡ ϕi (a)ψ(a)λj
when the partner is
The term
ψ(a)
(recall that
j
(i, j, a).The
term
ϕi (a)
(1)
captures the biological clock, that is age (a) and sex (i) specic.
is an additional payo component that depends on age. The partner's type premium is
is the type, low or high).
the high type premium to be
λh = λ.
Let us normalize the low type premium to be
When the
(i, j, a)
λl = 1 ,
λj
and name
(i0 , j 0 , a0 )
partner, her total
dt + Dijj 0 (a, a0 )
(2)
individual marries a
expected utility is
0
a
¯−max(a,a
)
ˆ
0
Vi0 j 0 (a0 + t)e−
Ψijj 0 (a, a ) =
´t
0
Qijj 0 (a+t,a0 +t)dt
0
where
Dijj 0 (a, a0 )
is the expected utility from possibility of future dissolution of the marriage due to the
death of the partner or divorce.
Qijj 0 (a, a0 )
of the partners or divorce so that
e−
´t
0
is the rate of destruction of a marriage due to the death of one
Qijj 0 (a+t,a0 +t)dt
is the survival hazard of the marriage.
5
2.3
Number of singles
Let us symbolize the population of
(i, j, a) singles with uij (a).
The dynamics of
uij (a) are determined by the
dierential equation
u˙ ij (a) = −Gij (a)uij (a) + (γbij (a) + πij (a))(xij (a) − uij (a))
(3)
The right hand side of (3) is the sum of the outow and the inow out and into the population of singles.
The outow is due to marriage or death and occurs with rate
partner with rate
πij (a)
or divorce with rate
γbij (a).
Gij (a).
The inow is due to the death of the
The solution of (3) is (4). The rst term of (4) is the
measure of the never-married and the second term is the measure of singles who are ever-married.
uij (a) = uij (a)e
−
´a
a
ˆ
Gij (t)dt
a
(γbij (t) + πij (t))e−
+
´a
t
Gij (τ )dτ
dt
(4)
a
The population of the never-married
(i, j, a)
decreases with rate
a
¯
X ˆ
Gij (a) = δi (a) + ρ
ui0 j 0 (τ )Mijj 0 (a, τ )dτ
j 0 =l,h a
because of death or marriage. The married become singles because of death of the partner with rate
πij (a) =
a
¯
X ˆ
δi0 (a0 )yijj 0 (a, a0 )da0
(5)
j 0 =l,h a
where
yijj 0 (a, a0 )
The term
rate
γ
bij (a)
where
0
is the measure of the{(i, j, a), (i
is the measure of
bij (a) =
X ´ a¯
a
(i, j, a)
, j 0 , a0 )}
marriages, or because of divorce with rate
γbij (a).
who are involved in bad marriages that may be dissolved with
bijj 0 (a, a0 )da0 . Mijj 0 (a, a0 )
is the index that determines whether
(i, j, a)
and
j 0 =l,h
(i0 , j 0 , a0 )
mutually agree to marry. The initial condition of (4), promising that at the age
are single, is
uij ≡ uij (a).
Given that
population, the initial conditions are
p
a
all individuals
is the exogenous proportion of the high type individuals in the
uih = pxi (a)
and
6
uil = (1 − p)xi (a).
2.4
Value as a single
A single individual
(i, j, a)
has a value
vij (a).
This value is determined by the dierential equation
v˙ ij (a) = Gij (a)vij (a) − hij (a)
where
hij (a)
is the expected utility from possibility of marriage at age
hij (a) =
a
¯
X ˆ
j 0 =l,h
a:
Ψijj 0 (a, τ )ui0 j 0 (τ )Mjj 0 (a, τ )dτ
a
The solution is
ˆ
a
¯
hij (t)e−
vij (a) =
´t
a
Gij (τ )dτ
dt
(6)
a
The initial condition for integration is
vij (¯
a) = 0.
This initial condition means that at the age of
a
there
is no value for being single, as the individual's participation in the marriage market is over.
A marriage between a man and a woman who are
if
(m, j, a)
Ej 0 (Ψmjj 0 (a, a0 )) > vmj (a) and Ej (Ψf j 0 j (a0 , a)) > vf j 0 (a0 ).
2.5
and
(f, j 0 , a0 ),
respectively, occurs if and only
The probability that it happens is
Mmjj 0 (a, a0 ).
Divorce
A married couple receives opportunities to divorce with rate
γ.
This parameter implies how easy the divorce
is. The shocks are not unique, so that each couple receives repeating opportunities to divorce. The marriage
may be dissolved endogenously, if one of the partners or both have a higher outside option relatively to their
value within the current marriage. There is no on the job search, so the outside option is singlehood. The
bad marriages are dissolved if the divorce opportunity arrives.
If divorced, an individual returns to the
marriage market. Note that a marriage may be bad today but good tomorrow, because the partners become
older and their outside option decreases.
First, let us dene the measure of bad
{(i, j, a), (i0 , j 0 , a0 )}
7
marriages as
bijj 0 (a, a0 ).
Bad marriages are
the ones that are dissolved if the couple has such an opportunity. The rate of unilateral endogenous divorces
from these marriages is
γbijj 0 (a, a0 ).
The measure of such bad marriages is
bijj 0 (a, a0 ) = ζijj 0 (a, a0 )yijj 0 (a, a0 )
where
yijj 0 (a, a0 )
vmj (a)}I{Ψf j 0 j (a0 , a) > vf j 0 (τ )}
2.6
{(i, j, a), (i0 , j 0 , a0 )}
is the measure of
marriages and
is an indicator stating whether
(7)
ζijj 0 (a, a0 ) = 1 − I{Ψmjj 0 (a, a0 ) >
{(i, j, a), (i0 , j 0 , a0 )}
1
is a bad marriage .
Expected utility from the marriage dissolution
The previous section presents the probability of divorce. Another reason for returning to the marriage market
is the death of the partner. The probability that this happens depends on the partner's age. Thus, in order
to calculate the rate of becoming a widow or a widower (5), one needs to know the distribution of marriages
by the spouses ages. As dened above, the measure of
assume without loss of generality that
a0 − a = k > 0.
{(i, j, a), (i0 , j 0 , a0 )}
Then, the measure of
marriages is
yijj 0 (a, a0 ).
{(i, j, a), (i0 , j 0 , a0 )}
Let us
marriages is
determined by dynamics
∂yijj 0 (a, a0 )
= zij (a, a0 ) − Qijj 0 (a, a0 )yijj 0 (a, a0 )
∂a
where
and
zij (a, a0 ) = ρMijj 0 (a, a0 )uij (a)ui0 j 0 (a0 )
Qijj 0 (a, a0 ) = δi (a) + δi0 (a0 ) + γζijj 0 (a, a0 )
yijj 0 (a, a0 ) = 0
for all
a0 ,
is the rate of entrance into the
ˆa
zij (t, t + k)e−
yijj 0 (a, a ) =
marriages
is the rate of destruction of marriages. Under initial condition
that means that at the minimal age
0
{(i, j, a), (i0 , j 0 , a0 )}
´ a−t
0
a
everyone is single, the solution is
Qijj 0 (t+τ,t+k+τ )dτ
dt
(8)
a
Equation (8) implies that the measure of
1 If
{(i, j, a), (i0 , j 0 , a0 )}
couples is the number of marriages with the
the divorce is not unilateral, but requires the agreement of both sides, then
ζijj 0 (a, a0 ) = (1 − I{Ψmjj 0 (a, a0 ) > vmj (a)})(1 − I{Ψf j 0 j (a0 , a) > vf j 0 (a0 )})
8
age gap
k
that once formatted and survived until the spouses are aged
a
and
a0
respectively. Finally, the
expected utility from possibility of future dissolution of the marriage due to death of the partner or divorce
that appears in (2), is
0
a
¯−max(a,a
)
ˆ
Dijj 0 (a, a0 ) =
(δi0 (a0 + t) + γζijj 0 (a + t, a0 + t))vij (a + t)e−
´t
0
Qijj 0 (a+τ,a0 +τ )dτ
dt
0
2.7
Uncertainty
The dierence between this model and the majority of the existing marriage models is that the mutual
agreement to marry,
Mijj 0 (a, a0 ),
is the probability that
(i, j, a)
is not a matrix of zeros and ones, but a matrix of probabilities.
and
(i0 , j 0 , a0 )
Mijj 0 (a, a0 )
mutually agree to marry. The agents signal about their type,
and their mates calculate the probability that an agent is of the high type conditional on the received signal.
Then, the value of marriage is evaluated according to this probability and compared to the value of the outside
option. Thus, for each individual who meets a mate, there is a reservation value of the signal, depending on
the mate's age and sex, that above it the individual agrees to marry her mate. I assume that the variance
of the signal decreases with age, which is an assumption corresponding to the Bergstrom and Bagnoli (1993)
model. Age makes the true type of the individual clearer. Dierently from Bergstrom and Bagnoli (1993),
not only the men's true type is uncovered with age, but also the women's one.
Vi0 j 0 (a0 ) ≡ ϕi0 (a0 )ψ(a)λj 0
where
(i0 , j 0 , a0 )
Recall utility structure (1):
are the partner's characteristics. In fact, the partner's type
j0
is
unobserved at the moment of meeting. One must marry her to uncover her type. What is observed is a signal
w = λj 0 ε where ε is i.i.d.
σ 2 (a) −→ 0.
and is a realization of a log normal distribution such that
ln(ε) ∼ N (0, σ 2 (a)) where
It means that the variance of the noise of the signal decreases with age. Thus, the uncertainty
about the individual's type is a function of his age. The type of an individual gets more transparent as he
gets older.
The reason to assume positively skewed log normal distribution is that individuals put eort to make
good impression during dating.
Of course, sometimes high type individuals are not able to express their
personality during a date and in their case
ε < 1.
9
While meeting a potential mate, an individual (of sex i, type
j
and age
a) observes her age a0
and signal
w.
Her type is unobserved. However, he can calculate a Bayesian expectation of her type from the distribution
of
wj 0 (a0 ).
The individual is indierent between marriage with a mate of age
a0
and remaining single if his
value as single is equal to the expected utility from marriage:
vij (a) = Ej 0 (Ψijj 0 (a, a0 ))
What is this expected utility?
(9)
It is calculated according to the conditional probabilities of the two types
given the observed partner's age and signal (the signal distribution depends on age):
Ej 0 (Ψijj 0 (a, a0 )) = P (low|w, a0 )Ψijl (a, a0 ) + P (high|w, a0 )Ψijh (a, a0 )
The conditional probabilities
P (low|w, a0 )
and
P (j 0 |w, a0 ) =
where
fj (w|a0 )
P (high|w, a0 )
(10)
can be calculated using Bayes rule:
fj 0 (w|a0 )Pi0 j 0 (a0 )
fl (w|a0 )Pi0 l (a0 ) + fh (w|a0 )Pi0 h (a0 )
is the probability density of the signal
w
of type
j
at the age
a0 ,
and
Pi0 j 0 (a0 )
is the a-priori
(i0 , j 0 , a0 ):
probability measure of meeting a single
Pi0 j 0 (a0 ) =
ui0 j 0 (a0 )
ui0 l (a0 ) + ui0 h (a0 )
Then the indierence condition (9) can be rewritten as
Kij (a, a0 ) ≡
Pi0 h (a0 ) Ψijh (a, a0 ) − vij (a)
fl (w|a0 )
=
Pi0 l (a0 ) vij (a) − Ψijl (a, a0 )
fh (w|a0 )
(11)
the likelihood ratio of the signal probability density of low-type to high-type individual. The signal
solves (9) is the partner's signal that will make a mate
single. Note that this equation makes sense only if
side of (11) is negative. If
Ψijl (a, a0 ) > vij (a)
w
that
(i, j, a) indierent between marrying her and remaining
Ψijl (a, a0 ) < vij (a) < Ψijh (a, a0 ).
Otherwise the left hand
the individual will agree to marry the partner no matter what
10
her signal is. If
Ψijh (a, a0 ) < vij (a)
the low type premium
dene
θ = ln(λ).
λl
he will not marry her no matter what her signal is. Let us normalize
to be 1, the high type premium
Using the assumption that
λh
to be
λ
2 and let us solve (11) for
wl ∼ lognormal(0, σ 2 (a))
and
w.
Let us
wh ∼ lognormal(θ, σ 2 (a)),
the
likelihood ratio is
θ2
fl (w|a0 )
− θ
= e 2σ2 (a) w σ2 (a0 )
0
fh (w|a )
Thus, the solution of (11), the reservation signal of
sex and age
a0 ,
(i, j, a) while meeting a potential mate of the opposite
is
θ
wj∗ (a, a0 ) = e 2 Kij (a, a0 )−
σ 2 (a)
θ
(12)
Because signals have a log-normal distribution, we can calculate the probability of the mutual acceptance,
Mijj 0 (a, a0 ):
0
Mijj 0 (a, a0 ) = (1 − Fja0 (wj∗ (a, a0 ))(1 − Fja (wj∗0 (a0 , a))
where
(13)
Fja (w) is a cumulative distribution function of log-normal distribution that associates with a normal dis-
tribution with parameters
µj (a) and σ 2 (a).
Equation (13) means that a probability that a
{(i, j, a), (i0 , j 0 , a0 )}
couple marries is a probability that the signals of both of them are above the reservation signals of each other.
Let us eliminate index
i
and call
aggregation of four functions
M (a, a0 )
Mjj 0 (a, a0 )
Mjj 0 ,
the mutual agreement probability of
Mjj 0 ,
a
to women of age
a0
is a measure of marriages of
λ, and there is proportion p of high type in
1 + p(λ − 1) and the variance is p(1 − p)(λ − 1)2 . Thus, the heterogeneity
maximum at p = 0.5.
the value of low-type individual is 1 and the value of high-type individual is
λ
M (a, a0 )
and, formally,
the population, then the mean value of the population is
(variance) increases in
M (a, a0 )
weighted by measures of single of each sex, type and age. The weights dene
the probabilities of random meetings for each such combination. Thus,
men of age
The
one for each combination of man's and woman's type, into one function
that describes the distribution of marriages in the population, is the result of the model.
is the average of the four
2 If
{(m, j, a), (f, j 0 , a0 )}.
and is parabolic in
p
with
11
X X
M (a, a0 ) =
j=l,h
Mjj 0 (a, a0 )umj (a)uf j 0 (a0 )
(14)
j 0 =l,h
The measure of rst marriages is calculated similarly, but with never-married population.
Note that rst
marriage of a bride does not mean rst marriage of her groom and vice versa. If we are interested in rst
marriages of women, it is
Mff (a, a0 ) =
X X
Mjj 0 (a, a0 )umj (a)uf j e
−
´ a0
a
Gf j 0 (t)dt
j=l,h j 0 =l,h
and, similarly, the measure of rst marriages of men is
f
Mm
(a, a0 ) =
X X
j=l,h
2.8
´a
a
Gmj (t)dt
uf j 0 (a0 )
j 0 =l,h
The result of the model is a marriage pattern, captured by the density function
{(i, j, a), (i0 , j 0 , a0 )}
couples.
the partner's biological clock,
ψ(a)
which is 1 for the low type and
α
−
Numerical results
Parameters
of the
Mjj 0 (a, a0 )umj e
λ
Recall that the payo function is
Vij (a) = ϕi (a)ψ(a)λj
is additional age-dependent component and
λj
The term
where
ϕi (a)
is
is the high type premium
for the high type. I assume that the female biological clock ticks with rate
and men do not experience biological clock (but, unfortunately, die). This implies that
ϕw (a) = e−α(a−a) .
yijj 0 (a, a0 )
ψ(a)
ψ(a) = e0.05a−0.0005a
is set to be the Mincer wage
2
ϕm (a) = 1
and
. In addition, I use the
3 to calculate the men's and women's death rates by age, δ (a) and δ (a).
m
f
US Actuarial Life Table for 2007
The marriage market is set to be between ages of 17 and 70 (a
Uncertainty is assumed to decrease with age, means
= 17, a
¯ = 70).
σ 2 (a) −→ 0.
Specically, I assume
σ 2 (a) = σ 2 e−β(a−a)
The two parameters are
3 Downloaded
σ and β .
The former is the amount of uncertainty about young individual's type. The
from the Social Security website.
12
later captures how fast this uncertainty decreases when the individual becomes older. This parametrization
echoes Bergstrom and Bagnoli (1993). Dierently from their paper where uncertainty exist only for men's
true type, in my model the uncertainty exists for both men and women.
To conclude, there are seven parameters in the model
ρ
- the rate of random meetings
γ
- the rate of divorce opportunities
α
- the rate of the biological clock ticking
λ
- the high type premium
p
- the proportion of the high type in the population
σ
- uncertainty (the lognormal distribution parameter)
β
- the rate of decrease of uncertainty with age
The model was numerically solved for a wide range of the parameters values, keeping constant the congenital heterogeneity on 50-50, means
p = 0.5.
The marriage pattern
The further is the solution of the model for the following values of the parameters: two random meetings a
year (ρ
(λ
= 2),
= eθ = 2),
no divorce (γ
= 0),
fast women aging (α
= 0.05)4 ,
the high type are 50% of the population (p
decreases fast (σ
the high type is twice as good as low type
= 0.5),
the uncertainty about young is high but
= 1, β = 0.05)5 .
Figure 2 shows the probability of mutual agreement to marry by ages and types. Observe that in this
example women only little trade age for type.
The reason is that women's aging is very fast.
However,
the high type men marry young women when they are old enough so their high type is observed: in the
two bottom panels the mutual agreement at middle ages moves out of the diagonal. There is no symmetry
between the genders because of women's aging: low type women have to pay more age to marry high type
men than another way around.
Observe that the movement of the mutual agreement out of the diagonal
4 It means that women lose about 40% of their value until age 27.
5 This value of β means that about 40% of σ 2 decreases until age of
13
27.
Figure 2: Probability of mutual agreement to marry by age and type
Parameters:
ρ = 2, p = 0.5, α = 0.05, β = 0.05, γ = 0, σ = 1, λ = 2
is not the same for all ages.
6
Very young and very old are less picky than middle ages .
In addition, the
high uncertainty about the young's type plays a role in adding more mutual agreement to negatively selected
young couples.
The left panel of Figure 3 shows the proportion of positive assortative marriage by type
(proportion of high with high and low with low out of all marriages) out of
bride/groom ages combination. The right panel shows the same but for
ow
stock
of new marriages for each
of all marriages.
Comparative statics
Table 1 shows the results. The Table presents the linear regression estimation, where the mean age at rst
marriage in years is regressed on the values of the parameters and the interactions between them. Columns
6I
can see it with my friends who are single after 30.
14
Figure 3: Proportion of positive assortative marriage
Parameters:
ρ = 2, p = 0.5, α = 0.05, β = 0.05, γ = 0, σ = 1, λ = 2
(1)-(3) show the eects of the parameters on the male age at rst marriage, and columns (4)-(6) show the
eects on the female one. The main result, observed in the Table, is that the age of marriage of men and
women dierently responds to the changes in the parameters values.
Inequality, measured by the high type premium
is not signicant for the female one.
λ,
positively aects the male age of rst marriage, but
This is possibly implied by the fact that heterogeneity is the main
source of variation in the male payo in marriage, while the main source of variation in the female payo in
marriage is age. Regarding uncertainty, the lognormal distribution parameter
σ,
that determines the noise of
the signal, is found not to be a signicant contributor to the age of rst marriage. However, the parameter
β
that counts for how fast does the uncertainty regarding the mate disappears with his age, is an important
contributor to the age of marriage. In accordance with the Bergstrom and Bagnoli (1993) results, the age of
marriage decreases when the uncertainty evaporate as individuals are relatively young. The surprising result
is that slowing down of the female biological clock actually decreases the male age of marriage (a positive
coecient of the
ticks slower).
α
parameter implies that men marry younger when
α
is smaller, means, the biological clock
The intuition is that when older women become attractive because of the slowing down of
15
the biological clock, men need a shorter search for a mate. However, this eect narrows when the divorce
is liberalized (a negative coecient of the
α·γ
interaction). One of the roles of divorce is that a man can
dissolve a marriage with a woman who gets older and remarry a younger one. Thus, the eect of the changes
in the biological clock is less dramatic when divorce is easy.
For women (columns (4)-(6)), the picture is dierent from men.
increases as the biological clock slows down (increasing
α
The age of rst marriage of women
means a faster ticking of the female biological
clock, so the negative coecient means that women marry later when the biological clock ticks slower).
Similarly, the liberalization of divorce (a higher
in the rate of the random meetings,
ρ,
γ)
increases the female age at rst marriage. An increase
decreases the female age of marriage.
Moreover, the increase in
the age of marriage as a result of the divorce liberalization is smaller when the rate of meetings increases
(observe the negative, and statistically signicant, coecient of the
ρ·γ
interaction). This result implies that
the importance of the divorce liberalization depends on the easiness of meeting potential partners after the
divorce.
The dierent response of the male and female age at rst marriage to the divorce liberalization and the
biological clock slowing down is graphically illustrated in Figure 4. The Figure shows the mean age of men
and women as a function of the rate of meetings
clock (α
= 0.01
versus
α = 0.05)
ρ,
and compares dierent regimes: slow and fast biological
interacted with conservative and liberal divorce (γ
= 0.1
versus
γ = 1).
The simulations keep the inequality and uncertainty constant. The results show that the age of marriage of
both men and women decreases in the rate of meetings
ρ.
However, the female biological clock has a dierent
impact on men and women: as it slows down, women marry older and men younger, closing the age gap.
While women marry older as a result slowing down of the biological clock and as a result liberalization of
divorce, men present an interaction: they marry younger in a new regime with a slow female biological
clock and a liberal divorce, relatively to an old regime with a fast female biological clock and a conservative
divorce. This interaction is the surprising result implying that changes in the biological clock can not explain
the shift toward a later marriage of both men and women observed in data of the last decades.
In addition, the comparative statics show that the proportion of marriages which are positive assortative
by type (low with low or high with high) increases if meetings are rare (low
16
ρ),
divorce is dicult (low
γ)
Figure 4: Age at rst marriage without uncertainty
Parameters:
p = 0.5, λ = 2.
Aging: slow
and women's aging is fast (high
α).
α = 0.01,
fast
α = 0.05.
Divorce: easy
γ = 1,
dicult
γ = 0.1.
Adding uncertainty does not have a large impact on the age at rst
marriage, but decreases the proportion of marriages which are positive assortative by type, especially if the
aging is fast and divorce is dicult.
3 The second model (pre-marital relationship)
3.1
Introduction
The second model deals with a similar question, but emphasizes the role of increasing pre-marital cohabitation (or other forms of pre-marital relationship) in the changing marriage pattern. This simple model is
structurally calibrated using the Vital Statistics marriage records of 1970 and 1995. Similarly to the theoretical results in the rst model, the empirically calibrated parameters of the second one imply that the slowing
down of the female biological clock is not the force driving the change in the marriage pattern between 1970
and 1995. Instead, the increasing inequality is the dominant factor.
Increasing rates of cohabitation are recorded in all Western countries.
17
In the US, the VI Cycle of the
Table 1: Comparative statics
dep. var. mean male age at rst marriage
dep. var. mean female age at rst marriage
(1)
(2)
(3)
(4)
(5)
(6)
β
-1.848
-2.229*
-2.229*
-1.048*
-1.067**
-1.067**
(1.344)
(1.115)
(1.115)
(0.570)
(0.508)
(0.508)
h
0.781***
0.740***
0.740***
0.044
0.063
0.063
(0.245)
(0.204)
(0.204)
(0.104)
(0.093)
(0.093)
σ
α
ρ
γ
-0.096
0.099
0.099
-0.325
-0.238
-0.238
(0.476)
(0.397)
(0.397)
(0.202)
(0.181)
(0.181)
75.624***
81.419***
72.739**
-11.663***
-22.533
-22.131
(8.602)
(29.402)
(29.395)
(3.645)
(13.387)
(13.384)
-0.631***
-0.785
-0.814
-0.361***
-0.369
-0.367
(0.147)
(0.576)
(0.577)
(0.062)
(0.262)
(0.263)
5.093***
0.242
5.983*
5.115
0.936***
5.053***
(0.619)
(3.085)
(3.098)
(0.262)
(1.405)
(1.411)
-86.803***
omitted
4.017
omitted
α·γ
(20.872)
α·ρ
ρ·γ
9.325
3.795
3.661
(10.377)
(10.504)
(4.725)
(4.782)
-1.235
-0.945
-1.475***
-1.488***
(1.081)
(1.097)
(0.492)
(0.500)
α·ρ·γ
constant
N
(9.503)
6.432
-28.934***
1.339
(6.957)
(3.168)
21.933***
21.708***
21.795***
22.330***
22.280***
22.276***
(1.138)
(1.907)
(1.909)
(0.482)
(0.868)
(0.869)
48
48
48
48
48
48
Standard errors are given in parentheses.
18
National Survey of Family Growth (2002) reported that out of the 20-24 year old women, 23.1% are married
and 15.7% cohabit. Out of the 25-29 year old, 51.6% are married and 12.9% cohabit. Out of the 30-34 yer
old, 61.8% are married and 7.9% cohabit.
Moreover, the rate of cohabitation decreases with the level of
education: the rate of cohabiting women is three times larger among those who did not obtain a high school
diploma relatively to women with a bachelor or higher academic degree. All the dierence is explained by
a higher proportion of married among the educated relatively to the non-educated. This later observation
implies that a positive signal regarding the partner's quality, education in this case, promotes marriage,
while lack of positive signals promotes cohabitation. In the model, I develop this idea by assuming that one
period of pre-marital relationship allows to observe the male partner's true type.
3.2
Setup
The model operates in discrete time. There exist masses of men and women. Women dier by age
dier by age
m
and type that can be low or high. In population, proportion
q
f.
Men
of men are of the high type.
The man's type is his private information.
The man's utility in the rst marriage with a woman
Tf
f
is 1 if
f ≤ Tf
and 0 otherwise. Thus, the parameter
measures the female biological clock, and, dierently from the rst model, it is discrete. Women require
from their partners both quality and age. Woman's utility in her rst marriage with a man
m is 1
if
and he is of the high type, and 0 otherwise. Men have a slower than women biological clocks, that is
m ≤ Tm
Tf ≤ Tm .
I assume that marriage is costly in the sense that individuals do not marry for a zero utility.
There is no divorce and a remarriage market in the model, but this is equivalent to an assumption that
there is remarriage market, but it is separated from the rst marriage market and the utility in the remarriage
does not depend on the partner's age (for example, because all births are given in the rst marriage so the
biological clock does not matter for the following marriages).
3.3
Pre-marital relationship and search
One period of pre-marital relationship is sucient for a woman to uncover her boyfrined's true type.
it turns out that he is of a low type, she resolves the relationship; otherwise, they marry.
19
If
In order the
pre-marital relationship to be informative, I assume that the low type men obtain some small utility from a
successful pick-up, encouraging them to enter pre-marital relationship even if they know it will not lead to
marriage.
The dating market operates through indirect search. Probability of meeting between a single
j∈ {low, high}
f
and
m
of type
is
p(mj , f ) = ujm uf
where
ujm , uf
3.4
Pure strategies equilibrium
are measures (out of 1) of vacant men of age
m,
type
j,
and of women of age
The equilibrium is dened, similarly to the rst model, by vectors of masses of singles
f.
ulm , uhm uf ,
matrices
of masses of couples currently in pre-marital relationship or married, and three matrices determining the
agents' strategies:
L, H ,
and
W.
(the age can be between 1 and
The matrices are of the size
A).
men, respectively, and the matrix
female ages
Similarly,
m
and
The matrices
W
L(m, f ) = H(m, f ) = of f er
(b)
W (m, f ) = pre − marital
(c) There is exist
(d)
T∗
for all
for all
such that
determine the strategies of the low and high type
and determines the marriage oers of the low-type men.
W (m, f ) ∈ {accept, reject, pre − marital}.
m ≤ Tm
m < Tm
and
and
W (m, f ) = reject
f ≤ Tf .
f < Tf .
when
m = Tm
and
f ≤ T ∗ and W (m, f ) = accept
when
f > T ∗.
W (m, f ) = accept
Proof:
is the maximal possible age
There exists an equilibrium in pure strategies, in which:
(a)
and
H
A
Women can immediately accept the marriage oer, reject the mate, or
enter a one-period pre-marital relationship:
m = Tm
and
where
determines the strategy of women. for each combination of male and
f , L(m, f ) ∈ {reject, of f er}
H(m, f ) ∈ {reject, of f er}.
Proposition 1 :
L
A×A
(a), (b), (d)
H
when
f = Tf .
oer to all women.
L
play the same as
and there is no pre-marital relationship. But then
play the same, women, whose age is below
Tf ,
L
H
- otherwise, their true type is known,
deviate to get immediately married. Because
require pre-marital relationship. Women who are
20
L
Tf
and
H
accept
any oer because the expected utility is positive and this is their last period in the marriage market. (c) The
value of further search depends on the woman's age. It is zero for a woman who is
a man who is
Tm
Tf .
Thus, if she meets
(pre-marital relationship is not possible), she marries him if the value of further search is
above the expected utility from marrying him, which is
uhTm .
QED.
Because the utility function is binary, the value of woman's further search is the probability to marry, one
day in the future, a high-type man. Therefore, the value of the threshold age
f = T∗
satises the indierence
condition that her probability to marry some time in the future a high-type man is equal to the a-priori
probability that a single man of age
3.5
Tm
is of the high type.
Calibration
Data and parameters
I use the US Vital Statistics marriage records to structurally estimate the model's parameters. This data set
covers the period of 1968 to 1995. I am looking at two distinct points in time, separated by 25 years: 1970
and 1995. I focus on rst marriages of white individuals whose ages lie between 17 and 65. The maximal age
of a man to bring utility to a woman in a rst marriage is, therefore, assumed to be 65.
To calculate the model's moments, I assume, consistently with data, that the sex ratio at birth is dened
by nature to be 1.05 males to females. The parameters to be calibrated are
Tm
- the number of periods a man can survive in the rst marriage market.
Tf
Tm - the female biological clock.
q
- the rate of high type men at birth.
Therefore, the length of a search period, in years, is calibrated. It is
67−17
Tm
=
50
Tm . It captures the intensity
of search, or the dating technology. This term replaces, in my model, the Poisson rate of meetings, which
often presents in the search literature.
21
Table 2: The actual and tted moments of the marriage age distribution
1970
1995
data
tted
data
tted
E(m)
22.4
22.3
25.5
23.9
E(f )
20.6
20.8
23.7
24.1
520.3
525.5
675.2
686.0
2
434.5
440.9
586.9
596.5
· f)
471.3
464.8
622.8
596.7
2
E(m )
E(f )
E(m
Table 3: The calibrated parameters
1970
Tm
- the number of periods of men in the rst marriage market
q
Tf
Tm - the female biological clock
- the high type proportion at birth
1995
100
94
0.261
0.254
0.915
0.564
Results
The estimation procedure is GMM. It is looking for parameters values that minimize the weighted squared
dierence between the rst marriage age distribution moments in data and in the model. I am considering
the rst moments of the distribution. Table 2 presents the actual and tted values of the moments. It is
observed that the model, with its three calibrated parameters, captures very well the ve moments of the
actual marriage age pattern in both 1970 and 1995.
Table 3 presents the calibrated parameters. The standard errors are calculated using bootstrap procedure
with 50 iterations. The results show that both the intensity of search and the female biological clock changed
very little, if at all, between 1970 and 1995. The intensity of dating is once in half a year. It implies the limited
impact of technological improvements in both dating and fecundity, on the marriage market. However, the
heterogeneity parameter
q
counts for most of the change in the marriage pattern. The decrease of
q
from
0.915 to 0.564 means that while in 1970 a vast majority of men was quantied as high type, in 1995 almost
half of men are of the low type. The variance of male quality distribution, which is measured by
rose, therefore, from 0.078 in 1970 to 0.246 in 1995. The Gini coecient, which is
0.436.
22
1 − q,
q(1 − q),
rose from 0.085 to
3.6
Endogenous length of pre-marital relationship
The assumption that pre-marital relationship lasts for a xed term of one period is consistent with data,
showing that the mode of the separation rate of cohabiting couples is after two years of cohabitation. However,
the length of pre-marital relationship can be endogenous, as the following extension shows. A simplied idea
of this model was simulated by zoologists (Wachmeister and Enquist (1999)), investigating the courtship
periods of monogamous species.
In this extension, each period the low type men bring to their partners utility 1 with probability p and
utility 0 with probability 1-p. The high type men always bring one unit of utility. In this case,
W (m, f ) ∈ {reject, accept, 1, 2, ...}
where
W (m, f ) is the length of the pre-marital relationship,
required by the woman. It is
accept if she agrees
to marry immediately. If she receives utility 0, she resolves the relationship, because in this case she certainly
knows that her mate is of the low type. If she receives 1, she updates the probability that her partner is of
the high type. The posterior probability, after
t
periods of pre-marital relationship, is
qm,t =
where
qm is
qm
qm + (1 − qm )pt
the proportion of the high type among single men of age
required length of pre-marital relationship decreases in
f,
(15)
m
who oer to her. It follows that the
because the female outside option diminishes with
her age. However, it follows from (15) that the length of pre-marital relationship increases in
the a-priori probability for a high type decreases in
m,
m,
because
as more and more high type men leave the marriage
market and never return. On the other hand, it can not increase monotonically given the limited horizon in
the marriage market. For this reason, we would expect a hump-shaped length of pre-marital relationship as
a function of the male age: it is short for young and old men, and long for the middle ages.
The dierence in the length of pre-marital relationship gradient with respect to the male and female
age leads to interesting theoretical results. For example, assuming that the partners' utility depends on the
23
spousal age gap such that there is some spousal age gap that maximizes the utility of the spouses, may explain
the empirically prevalent two-years pre-marital relationship of many couples, who either marry or split after
this period of time. This constant length may be the optimal pre-marital relationship length regardless of
the partners' ages, because it oppositely depends on the male and female age. Thus, couples of dierent age
but a constant age gap may endogenously agree on pre-marital relationship of the same length.
4 Conclusions
This paper presents two simple marriage market frameworks with uncertainty regarding the true type of a
partner. The rst model shows the eect on the age of marriage of interactions between the key marriage
market parameters: inequality, the search frictions, easiness of divorce, uncertainty, and a biological clock.
When both men and women are heterogeneous but women have a faster than men biological clock, the
increased inequality positively aects the male age of rst marriage. Men marry younger as female biological
clock slows down, because they compete on a larger pool of good women. However, this eect is overturned
if the search technology improves and divorce gets easy, because women become more selective. Women marry
older as the biological clock slows down, but this eect is statistically insignicant when interactions between
the parameters are included. It turns out that the female age of marriage increases when divorce is easy,
because in this case the women's age is less important as marriage does not include a long term commitment.
But interaction between easy divorce and improving search technology decreases the age of female marriage
because the price of making a mistake is smaller. Consistently with Bergstrom and Bagnoli (1993), the faster
uncertainty evaporates with age, the younger marry both men and women. The second framework, that in
extremely simple way compares the two-dimensional distributions of the age of rst marriage in 1970 versus
1995, shows that the change in the moments of the distribution is explained by increasing inequality, and not
by slowing down of the female biological clock or improving search technology. The main conclusion from
my study is that mechanical changes, such as a better health care or a better search technology, are not the
main parameters that determine the demographic patterns. However, changes in the distribution of marital
payos and in the ability of individuals to face uncertainty, are a strong demographic engine.
24
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26