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Courtship as a Waiting Game

Author(s): Theodore C. Bergstrom and Mark Bagnoli

Source:

Journal of Political Economy,

Vol. 101, No. 1 (Feb., 1993), pp. 185-202

Published by: The University of Chicago Press

Stable URL: http://www.jstor.org/stable/2138679

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Courtship as a Waiting Game

Theodore C. Bergstrom

University of Michigan

Mark Bagnoli

Indiana University

In most times and places, women on average marry older men. We

propose a partial explanation for this difference and for why it is

diminishing. In a society in which the economic roles of males are

more varied than the roles of females, the relative desirability of

females as marriage partners may become evident at an earlier age

than is the case for males. We study an equilibrium model in which

the males who regard their prospects as unusually good choose to

wait until their economic success is revealed before choosing a bride.

In equilibrium, the most desirable young females choose successful

older males. Young males who believe that time will not treat them

kindly will offer to marry at a young age. Although they are aware

that young males available for marriage are no bargain, the less

desirable young females will be offered no better option than the

lottery presented by marrying a young male. We show the existence

of equilibrium for models of this type and explore the properties of

equilibrium.

In most times and places, men, on average, are older than their wives.

A recent United Nations (1990) study reports the average age of

marriage for each sex for more than 90 countries over the time in-

We are grateful for encouragement, assistance, and helpful comments from Ken

Binmore, Arthur Goldberger, David Lam, Robert Schoeni, and Hal Varian. Berg-

strom's participation was partially supported by the National Institute for Child Health

and Development (grant RO1-HD19624).

[Journal of Political Economy, 1993, vol. 101, no. 1]

? 1993 by The University of Chicago. All rights reserved. 0022-3808/93/0101-0001$01.50

185

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186 JOURNAL OF POLITICAL ECONOMY

terval between 1950 and 1985.1 In every country and in every time

period reported, the mean age at marriage of males exceeded that

of females. The smallest difference in mean ages was 1 year (Ireland)

and the largest difference was 10.9 years (Mali). In 1985 in the United

States, the difference was 1.9 years, in western Europe about 2.5

years, and in southern and eastern Europe about 3.5 years. In Japan

the difference was 3.7 years, in India nearly 5 years, and in the Mid-

dle East about 4 years. In the Caribbean the age gap is about 5 years,

in Central America about 4 years, and in South America between 2

and 3 years. In African countries, this gap ranges between 5 and 10

years. In most countries, the age difference between the sexes at

marriage has diminished substantially between 1950 and 1985, but

nowhere has it disappeared altogether.2

This paper proposes a partial explanation for the difference in age

at marriage of males and females, for why this difference is diminish-

ing over time, and for why it tends to be greater in traditional societies

than in modern societies. We suggest that this difference is a result

of the different economic roles of males and females and a corre-

sponding difference between the sexes in the rate at which evidence

accumulates about one's "quality" as a possible marriage partner.

In societies in which male roles as economic providers are relatively

varied and specialized, information about an individual male's eco-

nomic capabilities may be revealed only gradually after he has spent

time in the work force. In contrast, for a female whose anticipated

tasks will be childbearing, child care, and traditional household labor,

it may be that once she has reached physical maturity, the passage of

time adds little information about her capabilities for these tasks.

We propose a model in which males who expect to prosper will

delay marriage until the evidence of their success allows them to

attract more desirable females. The most desirable females, on the

other hand, have little to gain by postponing marriage since the rele-

vant information about their quality is available at an earlier age. In

the long-run stationary equilibrium of this model, males with poor

prospects marry at an early age, whereas those who expect success

will marry later in life. All females marry relatively early in life. The

more desirable females marry successful older males and the less

desirable females marry the young males who do not expect to

prosper.

1 The average computed is the "singulate mean age at marriage." This statistic esti-

mates the average number of years spent in the single state by those who marry before

age 50 and is computed from census statistics on the proportion of the population that

have never married in each age group. See Hajnal (1953) for details.

2 In 72 of the 91 countries listed in App. B, the age gap decreased and in 14 countries

the gap increased. Exceptions to this pattern are Japan, Germany, and several countries

in southern and eastern Europe.

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COURTSHIP 187

This model predicts that males who marry young will tend to have

lower earnings in later life. There is evidence, at least for the United

States, that this is the case. According to the U.S. Census Bureau

(1980), 35 percent of males aged 45-54 who married before age

20 had annual incomes below $10,000. For those who first married

between ages 21 and 29, only 17.5 percent had incomes below

$10,000. The median income of persons who married before 18 was

$14,500, the median income of those who married between 18 and

20 was $16,800, and the median income of those who married be-

tween 22 and 29 was $19,000.3

The formal model presented in this paper is starkly oversimplified.

We confine the analysis to two possible ages of marriage for each sex.

We assume much more dramatic differences between the sexes than

is justified by reality. Furthermore, the model lacks a realistic treat-

ment of search costs. While our model lacks sophistication in many

directions, it is unusual in its explicit treatment of the dynamics of

an assignment equilibrium taking place in real time. Though we make

no claim that this model is detailed enough to "explain" the observed

distributions of age at marriage and income after marriage, we be-

lieve that we have identified an important influence on marriage pat-

terns and have taken a useful first step in untangling the logic of a

dynamic marital "lemons" model. We hope that this model will be

useful as a building block for more realistic and detailed theories.

I. Preferences, Information, and the Distribution

of Quality

We consider a population of constant size, in which people are born,

marry, and die. In every year, equal numbers of males and females

reach maturity. People can choose to marry in either the first or the

second year of maturity. Those who marry in the first year are said

to marry at age 1 and those who marry in the second year of maturity

are said to marry at age 2. Marriages are monogamous, and there is

no divorce or remarriage.

Some people are more desirable marriage partners than others,

but it is assumed that members of each sex agree in their rankings

of the opposite sex. It is further assumed that all members of each

sex have identical von Neumann-Morgenstern utility functions over

lotteries in which their marriage partners are randomly selected from

the opposite sex. Let xi be the von Neumann-Morgenstern utility

3 Bergstrom and Schoeni (1992) examined U.S. census data in detail and showed

that average male incomes are an increasing function of age at first marriage until

approximately age 30. But it is interesting to note that males who marry for the first

time after age 30 earn less than those who marry in their midtwenties.

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188 JOURNAL OF POLITICAL ECONOMY

that males assign to the prospect of marrying female i and let yi be

the von Neumann-Morgenstern utility that females assign to the

prospect of marrying male i. We shall call xi or yi the "quality" of

individual i. The quality of females is distributed over an interval [Lg,

Ug] with a cumulative distribution function Fg(x), and the quality of

males is distributed over an interval [Lb, Ub] with a cumulative distri-

bution function Fb(y). Other things being equal, everyone would pre-

fer marrying at age 1 to marrying at age 2. The utility cost of delaying

marriage from age 1 to age 2 is Cb for males and cg for females.

Marrying even the least desirable member of the opposite sex is pre-

ferred to the prospect of remaining single.4

The quality of each female is known to all persons when she reaches

age 1. The quality of a male does not become public information

until he reaches age 2. At age 1, each male knows what his own quality

will be at age 2.5 To the females, his prospects are indistinguishable

from those of his contemporaries, except insofar as his choice of

when to marry acts as a signal.

II. Marriage Market Equilibrium

We model the marriage market as a game of incomplete information.

Players have only two available strategies: to marry at age 1 or to

marry at age 2. The quality of males of age 2 and of females of all

ages is common knowledge. The quality of a male of age 1 is known

only to himself. Members of each generation make simultaneous

choices about when to marry without observing the choices made by

their contemporaries. Thus each individual believes that his or her

choice of strategy will not alter the choices made by contemporaries.

In equilibrium, although knowledge of the quality of specific age 1

males is private information, the distribution of quality among young

males who choose to marry and among young males who choose to

wait will be common knowledge.

The payoffs to each strategy are determined by a matching rule

applied to the set of people whose choose to marry in any time period.

Females who marry in any period are matched to males of corre-

sponding expected quality who choose to marry in the same period. If

the quality of all persons in the marriage market were public informa-

tion, then this matching would be entirely straightforward. The most

desirable male would be matched to the most desirable female, the

4 This assumption involves no loss of generality since the population referred to

consists only of those persons for whom being married is better than being single.

5This model extends without formal alteration to the case in which young males are

not certain about how well they will turn out but have some private information about

their prospects. Then yi is interpreted as i's expectation of his quality in period 2.

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COURTSHIP 189

second most desirable male to the second most desirable female, and

so on until the supply of persons of at least one sex is exhausted. If

the number of available persons of one sex exceeds that of the other,

then some people from the lower tail of the quality distribution will

be left unmatched. Unmatched persons of age 1 may reappear in the

marriage market in the next period.

The actual matching rule is complicated by the fact that males of

age 1 are indistinguishable to females and hence have equal expected

quality. Applying the principle of matching by corresponding rank

leads to the following assignment. At time period t, the best unmar-

ried male of age 2 will be matched to the best female who chooses

to marry at time t, the second-best unmarried male will marry the

second-best unmarried female, and so on until the supply of males

whose quality exceeds the average of available age 1 males is ex-

hausted.

The assignment of partners for the remainder of the population

follows directly from the principal of matching by corresponding

rank and from the fact that females cannot distinguish between males

who choose to marry at age 1.

Let Nm(t) be the number of age 1 males who choose to marry at

time t. Let Nf(t) be the number of females who choose to marry at

time t and who are not matched to a male who is better than a random

draw from the available age 1 males. There are three possible cases.

1. If Nm(t) = Nf (t), then each of the males who choose to marry at

age 1 will be randomly assigned a partner from the set of Nf (t) fe-

males who want to marry in this period and are not already taken by

an age 2 male.

2. If Nm(t) < Nf (t), then the best Nm(t) of the Nf (t) available females

will be randomly matched to the males of age 1. The remaining Nf (t)

- Nm(t) females will be matched in order of corresponding quality

with any remaining males of age 2 who have lower quality than the

average available male of age 1. Females left over at the end of this

process will be left unmatched. Those who are of age 1 may reenter

the marriage market in the next period at age 2.

3. If Nm(t) > Nf (t), then a random draw of Nf (t) males from the

set of available males of age 1 will be paired with the Nf (t) females

who are available and have not been matched with a male of higher

quality. Males who chose to marry at age 1 but did not receive part-

ners in the random assignment will be able to reenter the marriage

market in the next period at age 2.

With these matching rules, the assignment of partners within the

set of people who choose to marry in any given year has the core or

stable marriage assignment property (Gale and Shapley 1962; Shapley

and Shubik 1972). That is to say, no two people of opposite sexes who

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190 JOURNAL OF POLITICAL ECONOMY

marry in the same year would both get higher expected utility from

marrying each other than they do from their actual choices.6

Equilibrium must determine when each person chooses to marry

as well as how the people who choose to marry at a given time are

matched up. In equilibrium, each person's choice of whether to marry

at age 1 or age 2 must maximize his or her expected utility, given the

choices of all other individuals.7 Optimal strategies for individuals

depend nontrivially on the actions of others because these choices

determine the quality distribution in the marriage pool in each year

and thus determine the payoffs from marrying at age 1 or age 2. In

equilibrium, no person who marries at age 1 would have a higher

expected payoff from waiting to marry at age 2, and no person who

marries at age 2 would have a higher expected payoff by marrying

at age 1.

By restricting strategic choices to a decision of whether to marry at

age 1 or age 2, we have arbitrarily excluded strategies that may be

preferred by some males. Since males who choose to marry young

are matched randomly to females who are willing to marry young

males and since the quality of young females is common knowledge,

one might expect some males to try a strategy of the following form:

Go into the marriage market while young. If you are lucky enough

to draw one of the better females who is willing to marry a young

man, marry her. If you draw a female from the lower end of the

distribution, don't marry but wait until you are older. Indeed there

is nothing in this model to prevent this strategy from being preferred

by some males to accepting a random draw. A thorough treatment

of strategies of this type must await a model with a more detailed

search theory and with more than two possible ages of marriage.

III. Long-Run Stationary Equilibrium

Since we have assumed that the number of persons born in each year

is constant and that quality distributions and preferences are the same

in each generation, we can expect to find a long-run stationary equi-

6 Because we have assumed that persons of each sex agree in their rankings of the

opposite sex, the only assignment in the core from an ex ante standpoint is the assign-

ment that matches persons in order of expected quality. For further references and a

masterful survey of the general problem of stable marriage assignments, see Roth and

Sotomayor (1990).

7This is a Bayesian equilibrium of an agreeably simple nature. In this model, we do

not have to wonder what inferences are to be drawn about a player's type if he or she

deviates from equilibrium behavior. By the time that a deviation is observed by another

player, the deviator will have reached age 2. The type of every age 2 person is common

knowledge, so there is no mystery about how to regard someone who has in the past

deviated from equilibrium strategies.

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COURTSHIP 191

librium in which each generation behaves in exactly the same way as

all preceding generations.

Analysis of equilibrium is much simplified by the fact that in any

equilibrium, whether it is stationary or not, the following proposition

is true.

PROPOSITION 1. At any time t, the set of males who choose to wait

until age 2 to marry will be an "upper tail" of the quality distribution,

that is, a set of the form {lyI ' -y} for some yt E [Lb, Ub].

Proof. Consider two males, born at the same time, of quality y' and

y (y' > y). If these males both marry at age 1, then they face the same

lottery and their expected payoff will be the same. If they wait until

age 2 to marry, then the male of quality y' will be matched to a female

whose quality is at least as great as the quality of the female matched

to the male of quality y. From this it follows that if it is worthwhile

for a male of quality y to wait until age 2 to marry, any male of higher

quality than y will find it worthwhile to wait. Q.E.D.

It turns out that in long-run equilibrium, all females marry at age

1. There is a threshold level of quality, y*, such that in each time

period, males of higher quality than y* marry at age 2 and males of

lower quality marry at age 1. The highest-quality male from a genera-

tion will marry at age 2 the highest-quality female from the next

younger generation. The second-highest-quality male will marry at

age 2 the second-highest-quality female of age 1, and so on until the

threshold quality y* is reached. Males of quality lower than y* will

choose to marry at age 1 and will receive a random assignment from

the set of age 1 females who did not have sufficiently high quality to

be matched with the available males of age 2.

Let us define a function g such that x = g(y) means that a female

of quality x has the same ordinal rank among females that a male of

quality y has among males. Thus g(z) is the (unique) solution to the

equation Fb(z) = Fg(g(z)). We further define Ob(Y) to be the quality of

the "average male who is no better than a male of quality y" and

define a similar notation I'g(y) for females. Formally,

rY zdFb (Z)

11b (Y) b Fb (f )

and

~g (Y) =JY zdFg(z)

Lg Fg (y)

In long-run equilibrium, a male of threshold quality y* will be just

indifferent between marrying at age 1 and marrying at age 2. If he

marries at age 2, he will be matched to a female of quality g(y*) and

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192 JOURNAL OF POLITICAL ECONOMY

his utility will be g(y*) - Cb, where Cb is the utility cost of waiting. If

he marries at age 1, he will be indistinguishable from the other males

who marry at age 1 and will be randomly assigned to one of the

females who cannot marry an older male of quality y > y*. The aver-

age quality of females in this pool is ,ug(g(y*)). Therefore, his ex-

pected utility if he marries at age 1 is Vtg(g(y*)) and he will be indiffer-

ent between marrying at age 1 and marrying at age 2 if

llg(g(Y*)) = g(Y*) -Cb (1)

Long-run stationary equilibrium is fully characterized by the follow-

ing result.

PROPOSITION 2. If y* E [Lb, Ub] satisfies equation (1), then there is

a long-run stationary equilibrium such that, in every generation, each

male of quality y - y* marries at age 2 a female of age 1 whose quality

is g(y), and each male of quality y < y* marries at age 1 a female

randomly selected from the set of females in his own generation of

quality x < g(y*). Conversely, every long-run stationary equilibrium

is of this type.

Proof. The assertion that the proposed arrangement is an equilib-

rium will be demonstrated if we can show that no individual can gain

by deviating from the proposed equilibrium strategy. Consider a male

of quality y > y*. If he chooses to marry at age 2, he will be matched

to a female of quality g(y) and his payoff will be g(y) - Cb. If he

chooses to marry at age 1, he will have an expected payoff of

,ug(g(y*)).8 Since g is an increasing function of y, it follows from equa-

tion (1) that he cannot gain by marrying at age 1 rather than at age 2.

Consider a male of quality y < y*. If he marries at age 1, he will

have a random draw from the set of females of quality less than g(y*)

and his expected payoff will be ,ug(g(y*)). If he waits until age 2 to

marry, then his quality will be common knowledge. All the males

from his own generation of quality y - y* will be in the marriage pool

at this time and will be matched to all the age 1 females of quality x

- g(y*). Therefore, his payoff from marrying at age 2 will be smaller

than g(y*) - Cb. From equation (1), it follows that he cannot gain by

marrying at age 2 rather than at age 1.

Consider any female. If she deviates from the strategy of marrying

at age 1, the expected quality of her partner will be no higher than

8 Since we have assumed that people in the same generation choose their age of

marriage simultaneously, his choice to marry at age 1 will not change the set of females

who choose to marry at age 1, nor will it change the set of unmarried age 2 males.

Therefore, the pool of females who are available to marry age 1 males does not change

in response to his decision to marry at age 1. It follows that the expected payoff from

marrying at age 1 remains Lu(g(y*)) whether or not he chooses to marry at age 1.

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COURTSHIP 193

the expected quality she can get at age 1.9 Since waiting is costly, she

would not gain from choosing to marry at age 2 rather than at age 1.

We have shown that if y* satisfies equation (1), no person can gain

by deviating from the proposed equilibrium strategies. All that re-

mains is to show that every long-run stationary equilibrium is of the

type described in this proposition. From proposition 1, it follows that

in any equilibrium, the set of males divides into an upper-quality

interval who marry at age 2 and a lower-quality interval who marry

at age 1. If equilibrium is to be stationary, then the threshold quality

at which these groups divide must be some constant y*. If the pool

of available males is the same in every period, then (since waiting is

costly) it can never be worthwhile for females to choose to marry at

age 2 rather than at age 1. Therefore, in equilibrium all females must

marry at age 1. Finally, it is straightforward to verify that males better

than y* will choose marriage at age 2 and males worse than y* will

choose marriage at age 1 only if equation (1) is satisfied. Q.E.D.

IV. - Existence and Uniqueness of Long-Run

Equilibrium

The questions of existence and uniqueness of long-run equilibrium

reduce to the question of whether equation (1) has a solution and

whether that solution is unique. Let us define the difference between

a male or female's own quality, z, and the quality of the average male

or female who is no better than z. Let 8b(Z) = Z - 4b(Z) and bg(z) =

z - Ilg(z). Then equation (1) is equivalent to

bg(g(Y*)) = Cb. (2)

The following two assumptions will be sufficient for the existence

and uniqueness, respectively, of a solution to equations (1) and (2).

ASSUMPTION 1. The distribution function for the quality of each

sex is continuous, and the difference between the quality of the most

desirable female and the average quality of females exceeds the cost,

Cb, to a male of waiting to marry at age 2.

ASSUMPTION 2. The function bg(x) (which is the difference between

x and the average quality of females worse than x) is a monotone

increasing function of x.

9 There is a slight complication. If she decided to delay marriage, then when her

age is 1, the number of males in the marriage market would exceed the number of

females by one. Therefore, a randomly selected male who chose to marry at age 1

would not find a mate. He would reappear in the marriage market in the next year.

But the addition of a randomly selected male from the set of males of quality y < y*

will not improve the expected quality assignment for any female who waits until the

next period to marry.

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194 JOURNAL OF POLITICAL ECONOMY

PROPOSITION 3. If assumption 1 holds, then there exists at least one

long-run stationary state equilibrium in which y* solves equation (1).

Proof. By assumption 1, 6(Ug) > Cb. From the definition of the func-

tion be ), it follows that b(Lg) = 0 < Cb. The function 6(x) inherits

continuity from the distribution function for x. Therefore, from the

intermediate value theorem, there must be at least one solution, x*,

to the equation 8(x*) = Cb. Let y* = g-'(x*). Then b(g(y*)) = Cb.

Therefore, there exists a solution to equations (1) and (2). From prop-

osition 2, it follows that there exists a long-run stationary equilibrium.

Q.E.D.

PROPOSITION 4. If assumption 2 holds, then any long-run stationary

equilibrium is unique.

Proof. From assumption 2 and the monotonicity of g, bg(g(y)) - Cb

must be a monotonic increasing function of y, and hence there can

be only one y* for which b(g(y*)) = Cb. From proposition 2 it follows

that every long-run stationary equilibrium must satisfy this equation.

Q.E.D.

An example.-Suppose that the quality of females is uniformly dis-

tributed on an interval [0, a] and the quality of males is uniformly

distributed on the interval [0, b]. Then the function that maps males

to females of corresponding quality rank is g(y) = (alb)y. For the

uniform distribution, the average quality of females worse than x is

just x/2. Thus we have ILg(x) = x/2 and bg(x) = x - ILg(x) = x/2. We

see that bg(x) is an increasing function of x, so that assumption 3 is

satisfied. In fact we can readily solve for the unique equilibrium. The

equilibrium condition bg(g(y*)) = Cb will be satisfied if (al2b)y* = Cb

or, equivalently, if y* = 2bcbla. Therefore, if 0 < 2Cb < a, there

will exist a unique solution for y* in the interval (0, b). In long-run

equilibrium, all males of quality y < y* = 2bcbla will choose to marry

at age 1. Males who marry at age 1 will get a random draw from the

population of females of age 1 whose quality is lower than g(y*) =

2Cb. The expected payoff of a draw from this pool will then be Cb. If

a male of quality y* marries at age 2, he will be paired with a female

of quality g(y*) = 2Cb, but he has to bear the cost of waiting until age

2. His utility payoff from waiting is 2Cb - Cb = Cb, which is the same

as the payoff from marrying at age 1. Females of quality x > 2Cb will

marry age 2 males of quality bxla. Females of quality x < 2Cb will get

a random draw from the population of males who choose to marry

at age 1.

We have shown that the monotonicity of the function bg(z) is suffi-

cient for the uniqueness of equilibrium, and as our example shows,

if the quality of females is uniformly distributed, then bg(x) is strictly

monotone increasing. It would be useful to know more generally

what probability distributions have this property. As it happens, the

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COURTSHIP 195

class of distributions that have this property is quite large, and many

of its members can be identified by an easily checked sufficient con-

dition.

It turns out that a necessary and sufficient condition for bg(x) to be

an increasing function of x is that the log of the integral of the cumu-

lative density function be a concave function.'l This fact is not as

useful as one might hope because it is rarely possible to find a closed-

form expression for the cumulative density function, let alone its

integral. Therefore, it is not easy to verify whether a random variable

has this property. But we are rescued by the remarkable fact that

"log concavity begets log concavity" (under integration). This result

seems to have been discovered by Prekopa (1973) and has since ap-

peared in several places in the literature on operations research, sta-

tistics, and economics (see, e.g., Flinn and Heckman 1983; Gold-

berger 1983; Caplin and Nalebuff 1988; Dierker 1989). This result,

which is proved in Appendix A, follows.

LEMMA 1. If f(x) is a differentiable, log concave'1 function on the

rea,1 interval [a, b], then the function F (x) = aff(t)dt is also log concave

on [a, b], and so in turn will be the function G(x) = fa F(t)dt.

In a recent study, Bagnoli and Bergstrom (1989) examine the log

concavity of density functions, cumulative density functions, and their

integrals for numerous common probability distributions. All the fol-

lowing probability distributions have log concave densities and hence

monotone increasing 8g(x) functions: uniform, normal, logistic, ex-

treme value, chi-squared, chi, exponential, and Laplace. Therefore,

according to their theorem 2, if the distribution of female quality

belongs to any of these families, equilibrium will be unique. The

following probability distributions have log concave density functions

for some but not all parameter values: Weibull, power function,

gamma, and beta.

Log concavity of the density function is a sufficient but not a neces-

sary condition for bg(x) to be monotone increasing. Bagnoli and Berg-

strom (1989) show that although the lognormal distribution and the

Pareto distribution do not have log concave density functions, they

do have log concave cumulative density functions and monotone in-

creasing 8(x).12

10 To see this, integrate the expression for bg(x) by parts.

11 A function f is said to be log concave if log f is a concave function.

12 In fact, we are aware of no probability distributions famous enough to be "named"

for which b(x) is not monotone increasing. However, as Bagnoli and Bergstrom showed,

the probability distributions defined as "mirror images" of the Pareto distribution and

the lognormal distribution have, respectively, monotone increasing and nonmonotonic

a(x).

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196 JOURNAL OF POLITICAL ECONOMY

V. The Trajectory to Long-Run Equilibrium

If the population starts out in long-run stationary equilibrium, it will

remain there. But if initially the population is not in long-run station-

ary equilibrium, it will not immediately jump to a stationary equilib-

rium."3 Somehow the system must move gradually toward equilib-

rium. During the process of adjustment to long-run equilibrium,

some people are going to have to be left without partners.

When the system does not start out in long-run equilibrium, the

dynamics are complicated by the fact that, in some time periods,

females will choose to delay their date of marriage because the sup-

plies of available males may be more favorable to them in the second

period of their lives than in the first. A complete general characteriza-

tion of the behavior of the system outside of long-run equilibrium

appears to be very difficult. Here we settle for a pair of general

results, one for each sex, and an example.

Proposition 1, which we proved earlier, states that along any equi-

librium path the set of males who choose to wait until age 2 to marry

is-an upper tail of the distribution of males. The behavior of females

is much more complicated and is not fully described here, but we do

have one rather interesting general result.

PROPOSITION 5. In equilibrium, no female will ever marry a male

of age 2 whose quality rank is higher than her own.

Proof. If in period t some female marries a male of higher quality

rank than her own, then there will have to be some young female of

higher quality who does not marry a male of quality rank as high as

her own in period t. This second female must, therefore, have volun-

tarily postponed marriage. But if she is willing to postpone her mar-

riage, she must get a male whose quality exceeds that of her quality

match by at least Cb. This means that a third female of yet higher

quality must be displaced one generation later. The process would

have to continue, with females of ever higher quality in later genera-

tions being displaced. Eventually there would be no male sufficiently

good to compensate the best displaced female for waiting until age

2. Q.E.D.

We conclude with an example that is simple enough that we can

work out an exact solution for the pattern of marriage that starts

out from a position off of the long-run equilibrium path and moves

gradually toward long-run equilibrium.

13 This problem is echoed by occasional informal arguments to the effect that per-

haps the reason that women marry older men is that somehow people got started

doing things this way and now it cannot be stopped because there are so many unmar-

ried older men around who compete the women away from younger men.

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COURTSHIP 197

An example.-For each sex, "quality" is uniformly distributed on the

interval [0, 1]. Equal numbers of males and females are born in each

period. In the initial period, there are no unmarried persons of age

2 available from either sex. The utility cost of marrying at age 2

rather than at age 1 is c < 1/2 for members of either sex. A person's

desirability to members of the opposite sex neither increases nor de-

creases between ages 1 and 2.

In long-run equilibrium, males of quality lower than 2c marry at

age 1, males of quality higher than 2c marry at age 2, and all females

marry at age 1. But this population will not go all the way to long-run

equilibrium in a single step. If it did so, then no age 1 males of quality

y > 2c would marry, and since there are no males of age 2 available,

any female who marries in the first period would have to accept a

random young male whose expected quality would be c. But by wait-

ing until the next period when some high-quality age 2 males become

available, females of the highest quality could get spouses of nearly

quality 1. Since, by assumption, 1 - 2c > 0, it must be that 1 -c >

c, so some of the best females will be better off waiting to marry at

age 2.

For this example, the pattern of ages at marriage converges to

long-run equilibrium in a simple but rather surprising way. The pro-

portion of males who choose to marry at age 2 goes immediately to

the equilibrium level and stays there. But females divide into four

groups. In each period after the first, some females of age 1 at the top

of the quality distribution marry males of age 2. Some of intermediate

quality wait until age 2 to marry, at which time they marry males of

age 2. The females of age 1 just below these marry males of age 1.

Finally, at the bottom of the quality distribution of females are those

who are left without partners.

Let X' denote the set of females born in year t who at age 1 marry

males of age 2, let X2 be the set of females born in year t who at age

2 marry males of age 2, let X3 be the set of females born in year t

who at age 2 marry males of age 1, let X4 be the set of females born

in year t who at age 1 marry males of age 1, and let X5 be the set of

females born in year t who are left without mates. If initially there

are no unmarried persons of age 2, each of these sets is an interval.

These intervals take the following form: X1 = (x1, 1), X2 = (x2, x/1),

X3 is empty, X4 = (xt4, x2), and X5 = (0, X 4). Specifically, it turns out

that

t= 2c + (1 -2c)(1/2),

t = 2c + (1 -2c)(1/2)t

x4 = (1 -c)(/2)t.

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198 JOURNAL OF POLITICAL ECONOMY

This means that xl starts out at 1 in the first period. In the second

period, xi moves halfway from 1 to the equilibrium value 2c and in

each subsequent period again moves halfway from its previous loca-

tion to 2c. Notice also that, for all t, x2 = xi I and that the length of

the interval X2 of females who marry at age 2 is halved in every

period and is being squeezed asymptotically to 2c. The interval set

X' of females who are left unmatched is being halved in every period.

In the limit, the behavior of females approaches the long-run equilib-

rium in which all females of quality x > 2c belong to Xl and all females

of quality x < 2c belong to X4.

VI. Remarks and Possible Extensions

Becker (1974) suggests a reason to expect that high-wage males might

marry earlier rather than later. He argues that high-wage males have

more to gain from marriage than low-wage males because they will

enjoy greater returns to specialization (by marrying low-wage females

who will specialize in doing household work). Since there is more to

be gained from being married, they will spend less time searching

and hence marry earlier. Keeley (1977) investigates this relation em-

pirically using a sample of households from the 1967 Survey of Eco-

nomic Opportunity. Although he finds a positive relation between

age at first marriage and income if one does not include years of

schooling as an explanatory variable, he finds a negative relation be-

tween age at first marriage and income when years of schooling is

included. Using 1982 census data, Bergstrom and Schoeni (1992)

find that controlling for education does reduce the positive relation

between age at first marriage and income in later life, but even in

this case, expected male income in later life increases with age at first

marriage up to an age in the midtwenties and then decreases for

older ages.

As a test of our model, it seems inappropriate to "control for educa-

tion" in exploring the relation between age at first marriage and eco-

nomic success. To do so seems to beg the question of why it is that

people who get more years of education tend to marry later. It is

hard to see why the benefits from marriage are likely to be smaller

for those who are attending universities than for persons of the same

age who are working for wages.'4 One of the most convincing ways

that a young man can demonstrate to potential mates that he is able

and diligent is to finish a college degree.

14 Those who have observed fraternities at large universities will find it hard to

believe that this environment is as well suited to scholarship as married life.

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COURTSHIP 199

Of course we would not be so narrow-minded to claim that our

model is a full explanation of when people marry or that the consider-

ations suggested by Becker and Keeley can be neglected. To confront

the data more convincingly, one would like to have a much more

elaborate model than we have presented. The model should be en-

riched to incorporate search costs, to allow more varied roles for

females, to allow the gradual accretion of evidence about members

of each sex as time passes, and to take into account the role of nonhu-

man wealth. There is much interesting work to be done.

Appendix A

Proof of Lemma 1 (Log Concavity Begets Log Concavity)15

By elementary calculus, F(x) will be log concave if 0 2 F '(x)/F(x) = f' (x)F(x)

- f(x)2. Iff is log concave, then also by elementary calculus it must be that,

for x t, f '(x)lf(x) 2 f '(t)lf(t). Therefore, for all x E [a, b],

f (x) f(t)dt f (t)dt

But

Ixf((t) dt f '(t)dt = f(x) - f(a).

Therefore,

f'( F(x) ?x f(x) -f(a) ? f(x),

and hence 0 2 f'(x)F(x) - f(x)2. Therefore, F(x) is log concave. Since F(x) is

log concave, the same argument can be applied to show that G(x) inherits log

concavity from F(x). Q.E.D.

15 The idea for this proof is borrowed from Dierker's (1989) proof of the same

proposition.

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Appendix B

TABLE BI

MEAN AGE AT FIRST MARRIAGE

DIFFERENCE

MALES FEMALES BETWEEN

THE SEXES

Average Average

Mean Age Annual Mean Age Annual In Age Average

at First Change, at First Change, at First Annual

COUNTRY Marriage 1950-85 Marriage 1950-85 Marriage Change

Asia and the Middle East

Brunei 26.1 .01 25.0 .20 1.1 -.19

Hong Kong 29.2 .02 26.6 .19 2.6 -.17

Indonesia 24.8 .07 21.1 .13 3.7 -.06

Japan 29.5 .08 25.8 .04 3.7 .04

Korea 27.8 .11 24.7 .14 3.1 -.03

Malaysia 26.6 .07 23.5 .15 3.1 -.08

Nepal 21.5 .07 17.9 .07 3.6 .0

Philippines 25.3 .01 22.4 .01 2.9 .0

Singapore 28.4 .10 26.2 .26 2.2 - .16

Thailand 24.7 .01 22.7 .05 2.0 -.04

Bangladesh 23.9 - .01 16.7 .04 7.2 -.05

India 23.4 .09 18.7 .11 4.7 -.02

Pakistan 24.9 .09 19.8 .10 5.1 -.01

Sri Lanka 27.9 .03 24.4 .12 3.5 -.09

Algeria 25.3 - .02 21.0 .03 4.3 - .05

Cyprus 26.3 .11 24.2 .09 2.1 .02

Egypt 26.9 .05 21.4 .08 5.5 - .03

Iraq 25.2 -.06 20.8 .01 4.4 -.07

Iran 24.2 -.07 19.7 .12 4.5 -.19

Israel 26.1 .02 23.5 .10 2.6 -.08

Jordan 26.8 .10 22.8 .12 4.0 -.02

Kuwait 25.2 .01 22.4 .18 2.8 -.17

Morocco 27.2 .09 22.3 .17 4.9 - .08

Syria 25.7 .02 21.5 .09 4.2 - .07

Tunisia 27.8 .02 24.3 .18 3.5 -.16

Turkey 23.6 .06 20.7 .07 2.9 - .01

North America, Oceania, and Europe

United States 25.2 .05 23.3 .08 1.9 -.03

Canada 25.2 .0 23.1 .02 2.1 - .02

Australia 25.7 .01 23.5 .09 2.2 -.08

New Zealand 24.9 -.03 22.8 .02 2.1 -.05

Denmark 28.4 .06 25.6 .13 2.8 - .07

Finland 27.1 .04 24.6 .06 2.5 -.02

Norway 26.3 -.05 24.0 .03 2.3 -.08

Sweden 30.0 .10 27.6 .19 2.4 -.09

Ireland 24.4 -.23 23.4 -.11 1.0 -.12

England 25.4 - .02 23.1 .03 2.3 - .05

Austria 27.0 -.02 23.5 -.03 3.5 .01

Belgium 24.8 -.05 22.4 -.03 2.4 -.02

France 26.4 .0 24.5 .05 1.9 -.05

West Germany 27.9 .01 23.6 -.03 4.3 .04

Luxembourg 26.2 -.08 23.1 -.05 3.1 -.03

Netherlands 26.2 -.04 23.2 -.05 3.0 .01

Switzerland 27.9 -.01 25.0 .01 2.9 -.02

Greece 27.6 - .07 22.5 - .11 5.1 .04

Italy 27.1 -.05 23.2 -.05 3.9 .0

Portugal 24.7 - .08 22.1 - .08 2.6 .0

Spain 26.0 -.10 23.1 -.11 2.9 -.01

200

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TABLE B 1 (Continued)

DIFFERENCE

MALES FEMALES BETWEEN

THE SEXES

Average Average

Mean Age Annual Mean Age Annual In Age Average

at First Change, at First Change, at First Annual

COUNTRY Marriage 1950-85 Marriage 1950-85 Marriage Change

Bulgaria 24.5 .03 20.8 -.01 3.7 .04

Czechoslovakia 24.7 - .08 21.7 - .04 3.0 - .04

East Germany 25.4 -.01 21.7 -.07 3.7 .06

Hungary 24.8 -.06 21.0 -.05 3.8 -.01

Poland 25.9 .02 22.8 .04 3.1 - .02

Romania 24.9 .04 21.1 .08 3.8 - .04

USSR 24.2 .0 21.8 .07 2.4 - .07

Yugoslavia 26.1 .07 22.2 .0 3.9 .07

Caribbean, Central America, and South America

Cuba 23.5 - .09 19.9 - .08 3.6 - .01

Dominican Republic 26.1 .02 19.7 .05 6.4 -.03

Haiti 27.3 - .04 23.8 .06 3.5 - .10

Trinidad 27.9 .05 22.3 .12 5.6 - .07

Costa Rica 25.1 -.03 22.2 .01 2.9 -.04

El Salvador 24.7 -.03 19.4 -.01 5.3 -.02

Guatemala 23.5 -.02 20.5 .06 3.0 -.08

Honduras 24.4 -.05 20.0 .16 4.4 -.21

Mexico 24.1 - .02 20.6 - .03 3.5 .01

Nicaragua 24.6 -.08 20.2 .01 4.4 -.09

Panama 25.0 .01 21.3 .10 3.7 -.09

Argentina 25.3 - .07 22.9 - .01 2.4 - .06

Bolivia 24.5 .0 22.1 - .02 2.4 .02

Brazil 25.3 -.09 22.6 -.04 2.7 -.05

Chile 25.7 -.04 23.6 .0 2.1 - .04

Colombia 25.9 -.04 22.6 .03 3.3 -.07

Ecuador 24.3 - .04 21.1 .0 3.2 - .04

Paraguay 26.0 - .02 21.8 .03 4.2 - .05

Peru 25.7 .01 22.7 .05 3.0 -.04

Uruguay 25.4 - .13 22.4 - .03 3.0 - .10

Venezuela 24.8 - .05 21.2 .10 2.6 - .15

Africa

Benin 24.9 .0 18.3 .07 6.6 -.07

Central African Republic 23.3 .04 18.4 .07 4.9 -.03

Congo 27.0 .13 21.9 .18 5.1 -.05

Ghana 26.9 .06 19.4 .15 7.5 - .09

Kenya 25.5 .08 20.3 .11 5.2 -.03

Mali 27.3 .05 16.4 .01 10.9 .04

Liberia 26.6 .03 19.4 .12 7.2 -.09

Mauritius 27.5 .06 23.8 .15 4.7 -.09

Mozambique 22.7 - .04 17.6 - .06 5.1 .02

Reunion 28.1 .03 25.8 .10 2.3 - .07

Senegal 28.3 .02 18.3 .05 10.0 -.03

South Africa 27.8 .02 25.7 .10 2.1 -.08

Togo 26.5 .07 17.6 -.07 8.9 .0

Tanzania 24.9 .07 19.1 .11 5.8 -.04

Zambia 25.1 .06 19.4 .11 5.7 -.05

SOURCE.-United Nations (1990), tables 7, 18, 34, and 44.

NOTE.-The time trends for some countries are not the same as the rest: Bangladesh, 1974-81; Iraq and

Romania, 1966-77; Iran, 1957-77; Bulgaria, 1956-75; and USSR, 1979-85. The time intervals for Africa vary

from country to country depending on the available data.

201

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202 JOURNAL OF POLITICAL ECONOMY

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