The Value of Commitment: Marriage Choice in the Presence of Costly Divorce

The Value of Commitment: Marriage Choice in the
Presence of Costly Divorce
Anna Yurko
Very preliminary
October 2008
Abstract
Two individuals may choose to enter a coresidential relationship, which may
or may not include joint property ownership and raising children. In addition to
that, individuals may decide to formalize this contractual relationship by hold-
ing a marriage ceremony. The literature on coresidential relationships typically
assumes that the formal marriage contract o?ers additional tangible bene?ts
to the couple. However, this assumption is not obvious, and these potential
bene?ts (or costs) may vary greatly across societies and time. What remains
invariant, however, is the typically higher costs of terminating a relationship
that has been “sealed” by marriage.
The goal of this paper is to develop a simple dynamic model that can ex-
plain the existence of marriage contract. The model assumes that the only
di?erence between marriage and non-marital cohabitation is the higher con-
tract termination cost associated with marriage, and that the agents are free
to enter either type of relationship contract. The quality of the match evolves
randomly and independently over time, and every period each of the paired
agents makes his or her decision on whether to stay in the given relationship
or terminate it and seek another one based on the current match quality and
expectations about the future. The relationship survives only if both agents
choose to not terminate it. The unilateral decision to end a relationship by one
agent may impose a negative externality on her partner, if he still would prefer
to maintain it. The main ?nding is that when break-up costs are su?ciently
high, choosing a marriage contract that provides even higher termination cost
may reduce the expected break-up externality and result in greater welfare.
Keywords: family economics, marriage, cohabitation, externalities
JEL classi?cation: D62, D91, J12

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1
Introduction
Over the past half a century, the developed world has observed a dramatic decline in
legal marriage rates, combined with an even more dramatic increase in divorce rates.1
At the same time, an alternative relationship form to legal marriage, cohabitation, has
become vastly more popular. Exact cohabitation rates are more di?cult to measure;
however, survey numbers show a tenfold increase for the US from 1970 to 2006.2 These
trends suggest a decline in relative bene?ts from legal marriage versus cohabitation.
The twentieth century has seen the body of laws governing family relationships
in the majority of developed countries shifting its focus from protecting marriage
to protecting rights of individuals.3 This shift re?ects the socio-economic and tech-
nological changes that have made it possible to construct more ?exible relationship
contracts.4 Currently, Sweden leads the way with its most neutral legal approach to
family forms, and the highest cohabitation rate of 28% of all couples in 2005. For
comparison, the average cohabitation rate for the US and Europe is around 9%.5 As
cohabiting couples get increasingly similar treatment in the eyes of the law, and the
legal bene?ts to being married get smaller, will the institution of marriage disappear?
This paper’s answer is negative. As long as it is more di?cult to terminate the legal
marriage than to break up a cohabiting union, marriage remains the most committed
relationship option, and people who value commitment prefer it even in the absence
of additional bene?ts.
The goal of this paper is to provide an explanation and a modeling framework for
1According to OECD (2005) report, the average number of marriages per 1,000 residents in the
twenty seven OECD countries has experienced a 36% decline, from around 8 in 1970 to slightly
above 5 in 2001. The average number of divorces per 1,000 residents for this group of countries
has increased by 90% over the same time period, from slightly above 1 to more than 2. The exact
numbers vary across countries, but the overall trends remain roughly similar.
2Olson and Olson-Sigg (2007) report this statistic using the US Census Bureau data.
3See, for example, Weitzman, Lenore J. (1974).
4Smith (2004) argues that prior to the invention of e?ective contraception methods and reliable
paternity testing technology, the primary function of legal marriage was to mitigate the hazards of
sexual opportunism, by endowing the man with control rights over his wife’s sexuality. His paper
also gives a good and brief overview of the marriage literature.
5These numbers are from USA Today (Jul 18, 2005) and UNECE Gender Statistics.
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the choice of legal marriage over cohabitation in the absence of additional bene?ts
from marriage. Previous literature on cohabitation versus marriage has assumed that
marriage provides additional bene?ts to the agent, in order to induce her to choose this
option (Brien, Lillard, and Stern (2006)). This assumption is not obvious, however.
But if marriage does not o?er additional bene?ts, why would couples choose it over
cohabitation? The analysis in this paper suggests that in the presence of substantial
break-up costs, either emotional, social, or ?nancial, the higher termination costs
associated with marriage can actually result in greater lifetime utility for the agent
contemplating marriage versus cohabitation.6 Marriage is the most committed type
of relationship because it is the costliest one to dissolve, and the commitment is
valuable.
The intuition is as follows. Any relationship survives if and only if both partners
chose to maintain it. If one person enjoys the relationship and prefers to stay in it,
she would only be able to do so if her partner also prefers not to terminate it. If he
is no longer happy in the relationship and chooses to end it, his decision imposes a
negative externality on her. When the break-up is painful and costly, this externality
may be quite large. Higher costs of terminating a legal marriage may induce her
partner to change his mind and preserve the relationship, eliminating the break-up
costs they both would have had to pay otherwise. If break-up costs are su?ciently
large, and as long as there is a chance that he may eventually become happy again
in this relationship, additional termination costs from the marriage option result in
higher welfare even when agents end up staying in less than satisfactory relationships
for prolonged periods of time.
This paper does not aspire to explain all aspects, costs and bene?ts included,
6The adoption of no-fault divorce laws by the majority of developed countries has reduced the
costs of terminating a legal marriage. Friedberg (1998) shows that the adaption of the no-fault
divorce law has contributed to the increase in the divorce rates across states for the US. However, the
costs of terminating a marriage are still perceived to be signi?cantly higher than those of breaking-up
a cohabiting relationship. See Crouch et al. (2005) for a summary of divorce procedures in di?erent
states in the US and countries in Europe, including the length of time it takes for a couple to obtain
the no-fault divorce with and without partner’s consent.
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of di?erent family structures. However, it may provide an additional insight into
the establishment of the marriage institution. If many individuals would prefer a
relationship with higher termination costs, the society is bound to come up with a
way of providing this option. The incentive to do so is even stronger if there is a
general consensus that break-up imposes costs not only on the adult parties to the
relationship contract, but also on their children.
The next section presents a very simple model without the marriage option. The
goal is to use the simplest setting to study how break-up costs a?ect decisions and
values of agents when they are single and when they are in a relationship. Section
III adds the marriage option to the model, and ?nds the lowest divorce cost for each
value of the break-up cost, such that a single agent would weakly prefer to marry the
next good partner she meets to cohabiting with him. The model is also solved for
the divorce cost that would give a single agent the highest expected lifetime utility.
Section IV concludes with plans for future work.
2
Basic Model and Analysis
This section introduces the basic framework for the analysis. The model presented
here does not even have a marriage option in it. The purpose is to study how the
break-up cost a?ects the decisions and values of agents when they are single or in a
relationship.
2.1
Model
The agents are identical (no di?erences for men and women) and in?nitely lived. In
every period, each agent can be Single or Paired. Denote the value of being single by
S, and the value of being paired by R.
There are two sources of uncertainty in the model:
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1. Single agents get matched in every period. With probability µg they draw a
good match, and with probability (1 ? µg) they get a bad one. A good match gives
utility ug to the agent and her match partner in the current period. A bad match
gives utility ub to both of them, where ub < ug. Note that the agents may choose to
remain single, which gives them utility 0 in the current period (normalization).
2. Paired agents experience a shock to the quality of their match in every period.
When they choose to get matched, they start out with the same match quality, which
is the utility from their match. However, after this initial period their match qualities
evolve independently according to transition probability matrix
?
?
?gg ?gb
? = ?
? .
?bg ?bb
Timing
For Single Agents:
• Arrive in period t.
• Draw a match quality shock uq, where ’q’ stands for ’g’ or ’b’.
• Choose whether to stay single, or be paired.
• ”Consume” the option of choice.
The current period utility from being single is normalized to 0, the utility from
being paired is equal to match quality uq.
For Paired Agents:
• Arrive in period t.
• Shocks to match satisfaction are realized for both partners.
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• Simultaneously make decisions to stay together or break up. Possible outcomes:
1. Both partners are satis?ed with the match and decide to stay paired.
2. Only one of the partners is satis?ed, the other would be better o? splitting.
They split.
3. Both want to split.
”Mixing” is also possible. For example, one partner is in a good state, the other one
is in a bad one. The ?rst partner wants to stay together, the second one is indi?erent
and would split with probability 0.5. That means they would stay together with
probability 0.5.
• ”Consume” the option of choice.
Value functions
For single agents, the decision vector ? =
?
solves
g
?b
S (q) = max ? ES, uq + ? Eq ,q /qR q , q
,
?
?
where q is the agent’s current match quality, q? is her partner’s, and primes
denote next period values. If the agent chooses to stay single, ?q = 0. It is equal to
1 otherwise.
Denote by ? =
?
the decision vector for paired agents in
gg
?gb ?bg ?bb
every state (a state is current match qualities for the agent and her partner). In
every state (q, q?), ?qq is the probability (decision) of staying together.
?
Then, for paired agents,
R (q, q?) = max ?qq ?q
/q,q R q , q
+ 1 ? ?
?
?
qq
q
?
?
?q
uq + ? Eq ,q? ?
?
?q
[?c + ? ES] ,
qq?
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where c is the break-up cost.
Equilibrium
The agents solve their decision problems independently, taking as given the part-
ner’s action. In a symmetric equilibrium, the agents make the same state-dependent
decisions.
2.2
Results
Here the decisions and value function have been calculated for the following parameter
values:
ug = 5
ub = ?2
?gg = ?bb = 0.98
µg = 0.05
? = 0.99
Note that the shocks to match quality are quite persistent. The break-up cost is
allowed to vary from 0 to 150.
Decisions
For single agents the decisions turn out to be independent of break-up cost for
these parameter values. The agent always chooses to get paired when the initial
match quality is good (?g = 1), and remain single if the match quality draw is bad
(?b = 0).
For paired agents the decisions are plotted in Figure 1 below. When the break-up
cost is 85 or below, the agents wish to stay together if their own match quality is good,
and chose to break up otherwise (?gg = ?gb = 1, ?bg = ?bb = 0). Thus, the match
survives only if both agents are in a good state. Higher break-up cost makes the agent
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in a bad state not so positive about breaking up when her partner is in a good state
(?gg = ?gb = 1, 0 < ?bg < 1, ?bb = 0). Increasing c makes the agent in (bad,good)
state less likely to break up, and the match more likely to survive. The agents break
up for sure only if both of them are unhappy with the match. Finally, when cost of
break-up is 105 or higher, the agents never break up (?gg = ?gb = ?bg = ?bb = 1).
Figure 1: Decisions of paired agents
Value functions
How do these decisions a?ect value functions?
Figure 2 plots the value functions for a pared agent. For paired agent in states
(bad,good) and (bad,bad) the values are always equal, and lower than the values in
other states.
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Figure 2: Value functions of paired agent
If cost of break-up is 85 or below, for agent in state (good,bad) the value is the
same as that in states (bad,good) and (bad,bad), since the relationship would always
end in break-up in all of these three states. When cost of break-up is higher, the
agent in (good,bad) is more likely to get to keep the match, and her value increases.
That causes the values in other states to increase as well. Finally, when c ? 105, no
one ever breaks up and bears the cost of it.
Figure 3 plots the value functions of single agent in each state. It re?ects the
changes in R as function of the cost.
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Figure 3: Value functions of single agent
Thus, if the cost or terminating a relationship is above a certain value, increasing
this cost may be bene?cial for agents in all states. And that could be the role of
marriage contract. This ?nding is robust as long as ?bb = 1, that is, as long as there
is a chance for a bad relationship to become a good one again.
3
Marriage versus Cohabitation
3.1
Model
This section adds the option of marriage to the previous setting, by allowing the single
agent to decide between one of the three options: stay single and wait for a better
match, enter a cohabiting relationship with the given partner, or get married.7 The
7Cohabiting partners may also be allowed to change their relationship type to marriage. In fact,
in the real world, many marriages start as cohabitations. In the current model, however, agents
that have cohabited do not have any additional incentives to marry relative to the newly matched
agents. Thus, it is su?cient to only o?er this option to the single agents.
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