An Empirical Dynamic Marriage Matching Game

Jinhui Liang

November 5, 2018

Abstract

In order to better understand modern marriage life cycle with highturnover, we develop and estimate a dynamic two-sided matching gamewith transferable utility in a large marriage market. Forward lookingagents are facing household formation and dissolution decisions each pe-riod. Extending the Choo and Siow (2006) static matching framework,we derive a dynamic marriage matching function based on their currentmarriage gain and expected future utility. Using Hotz and Miller (1993)conditional choice probabilities estimators with expectation maximizationalgorithm, we are able to estimate marriage and divorce decisions with un-observed match quality. We apply this model to the marriage market inU.S from 1968 to 2015. We find that couples follows positive assorta-tive matching in education and ages. Strong education gradient existsas higher educated couples enjoy more marital output. We also examinethe role of match-specific capital such as match duration and childrenin marriage stability. We find slightly negative effect on marital outputfrom duration and positive effect from children which indicates childrencontribute positively toward marriage stability.

1 Introduction

Marriage and family life has always been the object of study for social scientists.Understanding of individuals decisions about marriage and divorce is vital sincethese decisions have great impacts on individuals life cycle. Marriage and divorcerates with the number of children in a household can no longer draw a pictureof modern marriages. Modern marriages are characterized by high turnoveras Andrew Cherlin named his book about the state of marriage and family inUS The Marriage-Go-Around : most studies suggest that 40 to 50 percent ofmarriage would be disrupted eventually (Cherlin 2010). About 80 percent ofdivorcees go on to remarriage. At the same time, marriage and fertility aredecoupling: the percentage of children born outside of marriage increase from4 percent in 1950 to 39.7 percent in 2007 (U.S. National Center for HealthStatistics, 2005, 2009b).

The institution of family is far from static, so does the research that modelthe marriage decisions. Gary Becker laid the foundation of a static transferable

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utility model of marriage market (1973,1974). Becker introduces the positiveassortativeness of marriage matching: high type males tend to marry high typefemale given the types are strategic complements. Followed by the seminalpaper of Choo and Siow (2006), they develop a tractable method to estimate astatic transferable utility model of a two-sided market under general equilibriumframework. From the information of who married to whom, relative importanceof agent characteristic can be revealed. If we observe more older men pairingwith younger women, then these pairings must be preferable. Or there maybe an abundant supply of older men and/or of younger women. Choo (2015)expands the marriage market into dynamic matching. However, it operates at arestrictive assumption that married people cannot endogenously divorce at theirwill. Bruze et.al (2014) incorporates divorce decisions into two-sided matchingmarket but do not follow general equilibrium conditions. Fox (2008) is anotherattempt of extending the static matching models into dynamic.

In this paper, we develop and estimate a dynamic two-sided matching modelwith transferable utility that is consistent with the data on matching and di-vorce patterns. The innovation lies in the incorporation of endogenous divorcedecisions in two-sided matching model. This model is a dynamic extension ofChoo and Siow (2006) that allows agents to switch endogenously given theircharacteristics and market transfers. The modeling choice of two-sided deci-sions and using a general equilibrium framework is essential given the fluidityof marriages in US: the relative availability of potential match in the marketwould impact marital decisions. Extending static two-sided matching modelsto dynamic setting enable us to build in forward looking behaviors in agentswhen matching patterns over time is available. But more importantly, buildinga dynamic matching model helps us understand the high turnover in modernmarriage: if marriage is out of date already, why most people still get married?If not, why so much marriages end in divorce? Researchers contribute maritalinstability to uncertainty and evolution of information (eg., Becker et.al, 1977;Weiss and Willis, 1997). Jovanovic (1979) constructs a model of job separationwith uncertainty and firm-specific human capital. Marriages can be viewed asan ”experience” good too. In our paper, couples lack information about theirmatch quality before they are married. And after they learn their match qual-ity in the beginning of their marriage, the decision to divorce each period isbased on their individual characteristics as well as the match-specific capitalthey accumulate. Estimation of the model would tell us what drives people’smate selection and what matters to the stability of marriages.

We face great computational challenge when comes to estimation of themodel. As Fox (2008) points out in the paper: the challenges of an empiricaldynamic matching model lies in two sources of large dimension. Firstly, thepossible matches in a static matching goes up very quickly with the growingpopulation. Secondly, continuation value in the dynamic setting would raisewith the number of states. In addition, the incorporation of unobserved matchquality in marriage adds more to the problem. In order to address the cursesof dimensionality, we employ the Hotz-Miller conditional choice probabilitiesmethod. Hotz and Miller (1993) found a one-to-one mapping between the nor-

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malized choice-specific value and conditional choice probabilities in the dynamicdiscrete choice problem. Given the similar structure between two-sided match-ing and discrete choice problem, we discover that many of estimation strategiesin dynamic discrete choice can be utilized to solve a dynamic two-sided matchingproblem. Using nonparametric estimation of conditional choice and transitionprobabilities, match-specific value function can be written as the combinationof current period utility and future utility adjustment. Complication rises whenthe transfer in these two-sided matching is often unobserved. Therefore insteadof minimizing a criterion function based on the estimated condition choice prob-abilities in the discrete choice model, we have to minimize the criterion func-tion subject to equilibrium constraints in transferable utility matching models.For the unobserved match quality, expectation maximization (EM) algorithmis needed. Arcidiacono and Miller (2011) combines EM algorithm with CCP toestimate a dynamic discrete choice model with unobserved heterogeneity. An-other innovation of this paper is to apply these methods in two-sided matchingcontext.

In our dynamic two-sided matching model, forward looking unmarried indi-viduals are deciding whom to match (including not to match) every period withperfect information in a large market. Individuals are weighting choices basedon not only their current period utility, which is determined by their character-istics and transfer, but also their future utility given these choices. For example,given certain matches were expected to end in a shorten period, individuals mayplace less weight on these matches if their possible choices are worsen in the fu-ture. The matching market clears every period and transfers between matchedindividuals are determined by the equilibrium constraints. We also derive anexplicit dynamic marriage matching function that is comparable to the staticone in Choo and Siow (2006). We show that the future utility adjustment thatis unique to dynamic structure is consist of normalized conditional probabilitiesthat the agents were to choosing unmatched next period. Married individualsbehave similarly as their choices are restricted to stay in current marriage or todivorce. The choices are driven by their characteristics and match-specific cap-ital, together with their match quality they learn at beginning of the marriage.If the marriage is disrupted, they revert back to unmarried and start anothermarriage-go-around.

We apply our dynamic model to the PSID marriage history data from 1968to 2015. We examine the marriage patterns based on the characteristics ofagents such as age and education. In addition, we incorporate the duration ofa match and the number of children as potential match-specific capital. We areable to recover their dynamic joint surplus and marital output from matchingand divorce patterns.

Our results are mostly consistent with other literatures. We find that posi-tive assortative matching by age and by education is still valid under dynamicsetting. Individuals prefer spouse with similar age and education. We also findstrong educational gradient in matching: higher educated couples enjoy a highermarital output due to the stability of their match and their better position inmatching market. We then examine how possible match-specific capital such as

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children and match duration would affect marriage stability. We find that thecurrent marital output decreases with duration. At the same time, the presentof children positively impact on the stability of marriage. The strength of ourapproach is allowing us utilizing all marital decisions of men and women. Un-der the reveal preference framework, we are able to recover the utility directlyrelated to each choice.

The remainder of this paper is organized as follows. We will discuss relatedmarriage and matching literature in section 2. In section 3 we will introduce ourdata. Then in section 4, we present our theoretical dynamic two-sided matchingmodel. We will detail estimation strategy in section 5 and presents the resultsin section 6. We end with a discussion of possible learning in section 7. Section8 is the conclusion.

2 Literature Review

The changing American family life in the past few decades are documented byeconomists and demographers. The traditional linear progression from beingsingle to married to have children is no longer the norm of a American marriagelife cycle. Teachman, Tedrow and Crowder (2000) recognizes the diversity ofAmercian families with respect to racial, ethnic and social class. Demographerslike Andrew Cherlin are paying more attentions to the deviations from marriagessuch as cohabitation, divorce and remarriage. Andrew Cherlin (2010) pointsout these deviations are blurring the boundaries of family units and familydemographers need to rethink the types of models they use. Stevenson andWolfers (2007) calls for a combination of empirical work and general equilibriumtheorizing at the end of their survey.

Developing and estimate an economic model of marriage and divorce that isconsistent with data is always the objective of economists. Aiyagari, Greenwoodand Guner (2000) developed a general equilibrium search model of marriage anddivorce to analyze family structure. They calibrate a two-period dynamic modelto evaluate policies such as child support. Brien, Lillard and Stern (2006) buildsan model of cohabitation, marriage and divorce with agents learning their matchquality in the relationships.

Choo and Siow (2006) pioneered nonparametric identification of preferencesin a static two-sided matching market with transferable utility. Choo (2016)extends the static matching market to a dynamic matching market with theassumption that divorces are exogenous. Building on Choo and Siow frame-work, Bruze et.al (2015) formulate a dynamic model of marriage, divorce andremarriage. In this dynamic model, people are driven into marriage by totalmarital surplus that are part identified and part unobserved. Meanwhile, cou-ples consider staying or leaving their marriage in order to get a better match inthe marriage market.

This paper builds on this strand of literature. We try to uncover the drivingforces behind marriage formations and dissolves. It is essential to include agentsmarital choices over their life cycle and also relevant state variables. The main

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challenge of these literatures are the computation burden caused by both thelarge dimension of possible matches and the dynamic nature of marital choices.The Choo and Siow framework reduces the computation burden by groupingagents into types. While this solves the problem at the matching part, extendingthe framework into matching part is not obvious. In Bruze et.al (2015), theybring the systematic matching surplus which is identified in the matching marketinto the utility after married. But at the same time, the unobserved type specifictaste shock is not found in marriage. They used a individual specific taste shockinstead. The comprise is due to the nature of marriage: while you share similarpreferences within your type, each match evolves differently. Married coupleseach have their individual two people market and the transfer in their personalmarket is affected by idiosyncrasies in each match.

No much work has been done on how match-specific capital affects marriagestability. Bruze et.al (2015) finds that divorce costs have a mild U-shape withrespect to marriage duration. As couples marry longer, we would expect someaccumulation of match specific capital such as children and therefore divorcecosts should be higher the longer they married. It doesn’t seem to be the caseand no literature has explored the mechanism behind it.

3 Data

We collect marriage record data between 1968 and 2015 from Panel Study ofIncome Dynamics (PSID). PSID collects an array of household data includingeducation,income,marriage and childbearing in United States. PSID helpfullycompiles 1985-2015 Marriage History File to access information collected in 1985through 2015.

Marriage-eligible age individuals in responding PSID families are asked abouttheir retrospective marriage history. Each set of records contains cumulativedata about the timing and circumstances of his or her marriage. If one hasnever married, one record is assigned to the individual. The marriage historyfile also links to individual file and the 1985-2015 Childbirth and Adoption His-tory File.

We follow all individuals in the marriage history file who are born after1948. 37998 out of 46843 records are marriage records and the rest has neverbeen married. 21 percent of marriage records end in divorces over the samplingperiod.

The delay into first marriage is one of the most striking change in familyunions. We focus on males and females between age 16 and 60 in the samplingperiod. Figure 1 presents the marriage frequency distribution with respect toage, for male and female. The median age of marriage for male is 25 while forfemale is 24 over the sampling period.

Figure 2 presents the distribution of marriage frequency with respect to agedifference of the couple which is defined as the husband’s age minus the wife’sage. It shows strong positive assortative matching in terms of age: 80 percentof the marriages have age difference of less or equal than 5 years old. The

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husband on average is two years older than the wife. Agents can be pairedwith each other with no age difference restriction in some papers such as Chooand Siow (2006) and Choo (2015) while only the most popular age pairingsare in the sample in Bruze et.al (2015). We strike a balance by restricting theage difference between husband and wife to be smaller than 14 years old whichcovers most of the marriages

We divide the agents into two education group given their highest gradescompleted: agents who receive less or equal than 12 years of education are inthe low education group while others are in the high education group. Table1 shows the sample distribution of completed education. Table 2 presents thedistribution of education matching. Positive assortativeness is also strong interms of education. 70 percent of the couples belong to the same educationgroup. 18.6 percent of the marriages is high education wife paring with loweducation husband while only 11.2 percent of the marriages with low educationwife matched with high education husband.Chiappori, Salanie and Weiss (2017)finds that the proportion of marriage in which husband is more educated fallsdramatically. Education as a characteristic in matching is fixed through time.Agents are believed to match with their expected completed grades even at theperiods when the education is not completed. Table 3 summarizes the proba-bilities of divorce in education groups. Couples who share the same educationlevel have lower chance of divorces comparing to others.

The duration of match is collected from the marriage record. In Figure3, weplot the frequency of divorces against the duration of marriage. Divorces arerare in the first year or less. Then divorces increase drastically in the secondyear and sustain at a high level until seventh year then finally divorces dropgradually afterward. We suspect a long lasting match may induce a higher perperiod utility through the accumulation of match-specific capital that woulddestroy upon ending of the match.

We collect the information on children using PSID 1985-2015 Childbirth andAdoption History File. We cross the childbirth history file with the marriagehistory file to identify childbirth in each marriage. We focus on children whosebirth parents are the husband and wife in the family. 31.9 percent of the mar-riages do not have any children. 19 percent of the marriages have one and 20.9percent of the marriages have two children. We also try to capture the match-specific capital by using fertility decisions in the marriage. The perception ismore children, especially young children, in the family would deter divorces. Atthe same time, couples are less likely to have children when they expect divorcesin a few years.

PSID contains information about cohabitation. Cohabitation has emergedas a precursor or a substitute to formal marriage in recent decades. This raisesnew questions regarding the changing definition of household and its decouplingwith marriage. We don’t plan to tackle these questions in the paper. We intendto build a general repeated matching game and we see cohabitation as anotherthreshold of matching. We decide not to burden our model with cohabitation.

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4 Model

4.1 General Motivation

In this paper we construct a dynamic model of match formation and dissolutionin a large two-sided market with transfer between agents. The general approachis combining two-sided matching model with transfer pioneered by Choo andSiow (2006) with dynamic discrete choice model to understand the driving forcesof marriages and divorces. The strand of literature we build on features fric-tionless matching with unobserved heterogeneity which contrasts with searchmodel. In the absence of frictions, agents have perfect information of their po-tential partners even if they are in a large market. The main reason for takingthe frictionless approach is that we don’t observe the transfer and bargainingin the marriage data we use. Modeling frictions with only matching patternswould not be feasible.

Under this framework, individuals enter the marriage market when they areyoung and exit the market when they reach old ages. They start as singlesand each period they choose whom to marry in a large market by maximizingtheir discounted present value of utility. Transfer between potential couples aredetermined by their willingness of marrying each other and are solved by marketclearing conditions. Married individuals learn the match quality at the firstperiod of their marriage. The match quality is not observed by the economistsand it does not change overtime Married individuals decide if they stay in thecurrent match or divorce. Transfer between the husband and wife is made eachperiod so that their willingness to divorce align with each other.

Some of the modeling assumptions are made to relieve computational bur-den. Single men and women are classified into a number of types according totheir characteristics such as age and education. The crucial assumption is thatindividuals are matched by types and those who share the same type are equiv-alent to each other in the matching market. Besides observable characteristics,each type of individuals have unobserved taste shock toward their spouse’s type.We assume that the unobserved taste shock takes the form of type I extremevalue distribution conditional on observable characteristics. Single individualsmatch on observables and unobservables.

Married individuals are not restricted by their types. The unobserved typespecific taste shock is replaced by unobserved taste shock toward their currentspouse and toward divorce (single). Moreover, married couples receive a level ofmatch quality after they married. The couple’s characteristic change over timedue to their decision to stay in the marriage and possible investment in match-specific capital such as children. Unobserved taste shock along with matchquality and observable characteristics precipitate possible divorce decisions overthe duration of their marriage.

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4.2 Utility

We begin by describing the two-sided matching market. Every period, singlewomen and men choose whom they marry (including matching with oneself). Apairing ai denotes a male a is matched with a female i, i ∈ {0, 1, · · · , i, · · · , I},i = 0 denotes the decision to stay single. The states of male a before the match isdenoted sa0 which contains his previous match 0 (since everyone is single at thematching market) and his characteristics sa0 = {xma } where the m superscriptdenotes male. We assume all characteristics are discrete and finite. The currentperiod utility of pairing ai for male a is

(1)um(ai, sa0) + wai + εmai

with um(ai, sa0) denoting the systematic marriage gain for male a and wai de-noting the transfer from female to male. wai enters utility additive separably.It can be positive or negative and is solved by market clearing conditions. εmai isa vector of type-specific taste shock toward all female types, {εma0, εma1, . . . , εmaI}.We make a simplifying assumption that the type-specific taste shocks are inde-pendent and identically distributed following type I extreme value distribution.Thus the driving force of marrying female i for male a is divided into the deter-ministic utility flow um(ai, sa0) + wai and unobserved taste shifter εmai.

Similarly, the current period utility of pairing ai for female i is

(2)uf (ai, s0i)− wai + εfai

where s0i = {xfi } denotes the state of female i, uf (ai, s0i) denoting the system-atic marriage gain for female i. Since we can only identified the difference ofutility between choices, we normalize systematic marriage gain um(a0, sa0) anduf (0i, s0i) to zero if they decide to stay single. Transfers wa0 and w0i are zeroif they are not matched. Note that there is no costs associated with forming amatch. And the marriage gain um(ai, sa0), transfer wai and unobserved tasteshock εmai are all based on the types of agents, meaning the utility are the samefor newly form match in which the husbands and wives share the same type.

The interchangeability of spouse within the same type is a striking featureof the Choo and Siow (2006) two-sided matching framework. While it allows anexplicit matching function and it reduces the curse of dimensions, the extensionof the framework into decisions within the marriage is not obvious. The transferand unobserved shock in marriage is no longer type-specific given they are facingthe same spouse repeatedly. And also married men and women are not able torematch until they end their current match, therefore their decisions to stayor to leave the current marriage is confined to each individual market. Thusthe transfer cannot be solved using equilibrium conditions as before. Thereforewe introduce unobserved match quality with unobserved taste shock into thecurrent period utility for married couples.

If married, a male a and a female i generates marital output together, thecurrent period utility for the male a choosing to stay in the marriage is

(3)um(ai, sai, q) + τ + ηmai

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where τ is the transfer from wife to husband that is specific to this marriage.q is match quality of this marriage that is revealed to the agents after theyget married. um(ai, sai, q) is systematic marital output of male a. ηmai is theunobserved taste shock for male a when choosing to stay in the match thisperiod and we also impose the type I extreme value distribution on the shock.Similarly, the current period utility for female i staying in the marriage is

(4)uf (ai, sai, q)− τ + ηfai

where uf (ai, sai, q) is the systematic marital output of female i and ηfai denotesthe taste shock for female i. τ can be positive or negative like wai. However, τ nolonger type-specific. It is endogenously determined by equilibrium conditions.

If the married couple ai chooses to divorce, they revert their state to single.The current period utility of male a choosing to divorce from paring ai is

(5)um(a0, sa0) + ηma0

where um(a0, sa0) is already normalized to zero. Note that no penalty or explicitcost is dictated when divorced. We also have the current period utility fordivorced female i

(6)uf (0i, s0i) + ηf0i

Notice the difference between new divorcee and an unmarried individual choos-ing to stay single this period lies in the taste shock εmai and ηma0.

4.3 Timing and Beliefs

Agents start at the marriage market as single when they turn age 16. Everyperiod unmarried agents make matching decisions in the large matching market.At the beginning of each period, state variables are known to all agents. Thenagents receive taste shocks and they make matching decisions simultaneouslyby maximizing expected utility, while at the same time the equilibrium transferand corresponding beliefs are determined. Unmarried agents don’t know thevalue of match quality q before they made the decision. As for the steadystate beliefs, it is essential to assume agents believe their actions would notaffect expected transfer or the distribution of potential choices. In addition, wesimplify our model by assuming stationarity. The state transition distributionis hθ1(s

′

ai|sa0, s0i, ai) with prime superscript denoting the state next period.Unmarried agents exit the matching market when they get married or whenthey reach age 60.

Married agents decide whether to stay in the marriage or to divorce everyperiod. Agents learn their match quality q at the first period of marriage whichis constant through this match. State variables are known to both agents. Andas they receive taste shocks at the beginning of each period, transfer betweenthe spouse are determined. As the spouse with higher willingness to stay in themarriage would make positive transfer to the other party until they are equallyhappy. If no such transfer is possible, divorce happens. They revert back tosingle as they divorce at the same time they lose the match-specific capitalaccrued during the marriage.

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4.4 Agent Characteristics

The model include components that are believed to be important to agents’decisions. The unmarried individuals state variables xma and xfi include agents’age and education at each period.They are deterministic and finite. One weak-ness of employing a general equilibrium two-sided matching model is that wehave to limit the number of characteristics to reduce the dimensions. We trackagents from age 16 to 60. And we classify agents who complete high schoolor less as low education group and those who complete more than high schoolas high education group. Combining age and education, there are 88 types ofagents on each side of the market. Without restriction, the possible match typeswould be 7744. Given agents usually form matches with similar age spouse, wewill restrict the age differences of the potential match to be less than 14 whichare majority of the matches.

As for the married individuals, their state variables include age, education,the number of children they have, the duration of the match and unobservedmatch quality. Children, as an important part of the family structure, have sub-stantial impacts on marriages and divorces. Intuitively, unmarried individualsare more likely to select into marriage if they are more willing to have children.And common perception suggests married couples are less likely to divorce ifthey have children. Duration of the match is included because of potential pos-itive increase in marital output with duration. Jovanovic (1979) shows thatthere is negative duration dependence associated with exiting a job after a crit-ical period of time. This is caused by the present of unobserved heterogeneity.Therefore, a persistent unobserved match quality is also included. Children,duration and match quality contribute to the accumulation of match-specificcapital. The match-specific capital will be lost if they choose to divorce. Inthis sense, we are estimating the parents’ utility toward children but rather thecontribution of children toward their parents’ marriage.

Modeling the fertility decisions on top of this model would be too difficult.Instead, we allow the probability of having children in the marriage conditioningon their characteristics.

4.5 The Matching Decisions

With the introduction of model environment, we are now equipped to exploreindividual’s problem in the marriage life cycle. We will illustrate agents’ utilitymaximization problem starting with unmarried individuals. Unmarried individ-uals make matching decisions by maximizing their expected present discountedutility given their beliefs and state variables. Individuals solve their optimizationproblem by backwards recursion. As a finite-horizon dynamic discrete choiceproblem, we introduce the Bellman equations. We define a continuation value

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of male a at sa0 as

V m(sa0) =

∫maxi

[um(ai, sa0) + wai + εmai

+ β

∫V m(s

′

ai)hθ1(s′

ai|sa0, s0i, ai)ds′

ai]hm(εa)dεa

(7)

The continuation value function V m(sa0) represents the expected utility ofmale a on the matching market, which is the sum of current period utility andthe discounted expected value of choosing i from potential match {0, 1, . . . , I}.β denotes the discount factor between 0 and 1. The expectation over the dis-counted future value term is taken over the continuation value of next periodgiven future state variables.hm denotes the joint density of male unobservedtaste shock. Female i has similar continuation value function at state s0i

V f (s0i) =

∫maxa

[uf (ai, s0i)− wai + εfai

+ β

∫V f (s

′

0i)hθ1(s′

0i|sa0, s0i, ai)ds′

ai]hf (εi)dεi

(8)

where hf denotes the joint density of female unobserved taste shock.With the assumption that the unobserved taste shocks follow independent

and identically distributed type I extreme value distribution, the choice of aspouse can be modeled using a multilogit model. We define the match-specificvalue function vm(ai, sa0) for male a as

vm(ai, sa0) ≡ um(ai, sa0) + wai + β

∫V m(s

′

ai)hθ1(s′

ai|sa0, s0i, ai)ds′

ai (9)

Then we have the probability of male a choosing female i given his state sa0

Prm(ai|sa0) =exp(vm(ai, sa0))∑I

k=1 exp(vm(ak, sa0)) + 1

(10)

where vm(ai, sa0) is the normalized match-specific value function given by

vm(ai, sa0) ≡ vm(ai, sa0)− vm(a0, sa0) (11)

Similarly, the choice probability of women i choosing spouse a is

Prf (ai|s0i) =exp(vf (ai, s0i))∑A

l=1 exp(vf (li, s0i)) + 1

(12)

where the normalized match specific value function for female i vf (ai, s0i) islikewise defined.

Male a chooses his spouse unilaterally given the state variables and thetransfer, so as female i. Recall that we assume all characteristics to be discreteand finite, thus the states {s0i} and {sa0} are finite. Given the finite number

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of types at both side of the market and the nature of one-to-one matching, wemust have Nm(ai, sa0) = Nf (ai, s0i) unless the match is with themselves (staysingle). Nm(ai, sa0) denotes the number of type a single male who marry witha type i female and Nf (ai, s0i) denotes the number of type i single female whomarry with a type a male. The number of husbands must equal the number ofwives. Combining with (10) and (12), we have the equilibrium constraints:

Nf (s0i)Prf (ai|s0i) = Nm(sa0)Prm(ai|sa0) (13)

for all types a ∈ {1, 2, . . . , A} and i ∈ {1, 2, . . . , I}. Nf (s0i) and Nm(sa0)denotes the number of female i with state s0i and male a with state sa0 respec-tively.

Transfer wai between partners are used to clear the market. Given tasteshock, the choice probability of choosing partners are functions of the transfer.And now we are equipped to define the equilibrium in our model

Definition 1. An equilibrium is a function wai(sa0, s0i) that fulfills the follow-ing conditions 1. Equilibrium constraints (13) for all potential matches,for alla ∈ {Nm(sa0)/0} and i ∈ {Nf (s0i)/0} are binding

2. Bellman equations for male (7) and female (8) are satisfied

In Fox (2008), he explains that given the number of unknowns equal thenumber of equations, it is possible that such equilibrium exists for any param-eterization.

4.6 The Dynamic Marriage Matching Function

The formulation of our dynamic two-sided matching model lies closely to Fox(2008) but with the modification of aggregate states. The curse of dimensionsremains the major challenge in computing these models. One contribution ofthis paper is utilizing Conditional choice probability (CCP) estimators which isoriginally developed to solve single-agent dynamic discrete choice. Proposed byHotz and Miller (1993) and later improved by Aguirregabiria and Mira (2002),CCP estimators exploit the mapping from the value functions to the probabili-ties of individuals choices given state variables. And the expected value of futureutilities given optimal choice can be expressed as a function of flow payoffs andthe conditional choice probabilities for any future choices.

In our dynamic logit setting, the continuation value function (7) can berewritten as a function of CCP and match-specific value function of stayingsingle:

V m(sa0) = −ln[Prm(a0|sa0)] + vm(a0, sa0) + c (14)

where c is Euler’s constant. This closed form representation of the value func-tion offers us enormous advantage in estimation since we no longer require todo numerical integration over unobserved taste shocks for future value term.It requires type I extreme value distribution assumptions of taste shocks andconditional independence of the state transitions.

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This enable us to formulate the normalized one-period ahead match specificvalue function for male a

vm(ai, sa0) =um(ai, sa0) + wai

− β∫lnP rm(a0|s

′

ai)hθ1ds′

ai

+ β

∫lnP rm(a0|s

′

a0)hθ1ds′

a0

(15)

where we denote the CCP as the estimated CCP P rm(a0|s′ai) since we areestimating them in first stage nonparametric estimation to further reduce com-putation burden. And we have the similar match-specific value function forfemale i

vf (ai, s0i) =uf (ai, s0i)− wai

− β∫lnP rf (0i|s

′

ai)hθ1ds′

ai

+ β

∫lnP rf (0i|s

′

0i)hθ1ds′

0i

(16)

We now make a simplifying assumption about transition rule of the states. Ina general case, the parameters that govern transition of states can be estimatedby nonparametric regression methods. Here in our example, the state variablesonly include individual’s age and education. Thus there is no parameters toestimate in the transition of states since the evolution of ages is deterministic andwe assume individuals have perfect information about their eventual educationlevel.

This simplification enable us to develop the dynamic matching function.Under the conditional logit setting, we have the dynamic log-odds ratio for thefemale as

lnPrf (ai|s0i)Prf (0i|s0i)

= vf (ai, s0i) (17)

and after some manipulations, we have the dynamic matching function

lnNm(ai, sa0) + lnNf (ai, s0i)− lnNm(a0, sa0)− lnNf (0i, s0i) =

um(ai, sa0) + uf (ai, s0i)− β(lnP r

f(0i

′ |s′ai)

P rf(0i′ |s′0i)

+ lnP r

m(a

′0|s′ai)

P rm

(a′0|s′a0))

(18)

The right hand side represents the dynamic joint surplus of the match whilethe left hand side is the ratios of number of the match to the number of un-matched ones. This function is directly comparable to the static matching func-tion of Choo and Siow (2006). The noticeable difference between the two liesin the composition of the dynamic joint surplus: in addition to the joint mar-riage gain um(ai, sa0) + uf (ai, s0i), there is an adjustment term for the future

−β(lnP r

f(0i

′|s

′ai)

P rf(0i′ |s′0i)

+lnP r

m(a

′0|s

′ai)

P rm(a′0|s′a0)

) in the dynamic joint surplus. The adjustment

term accounts for the expected utility loss due to the probability of breaking up

13

next period. If the normalized probabilities of agents choosing to divorce arehigher, then the dynamic joint surplus would be lower. Subsequently, the ratioof agents choosing this match would be lower. The intuition is quite simple:agents place less value on relationship that doesn’t last, at the same time, theyplace less value on relationships in general if they can easily find new matchesin the future.

4.7 The Divorce Decisions

Now we change gear to a structurally simpler side of the model, the divorcedecision. Married individuals are confined to two choices each period: to staymarried to the same spouse or to get a divorce. As we illustrated before, marriedindividuals bargain with their spouse to determine the transfer τ . If the transferis not enough to make both side satisfied, the match ends.

Married individuals learn the value of match quality q at the first period ofmarriage, which is unobserved to the economists. Husbands and wives decideif they stay in the marriage or get a divorce each period given match quality,marital output and unobserved taste shock.

The match-specific value function for married couple ai when male a chooseto stay in the marriage is defined in the same way as before

vm(ai, sai) ≡ um(ai, sai) + τ

+β

∫V m(s

′

ai)hθ1(s′

ai|sai, ai)ds′

ai

−β∫V m(s

′

a0)hθ1(s′

a0|sai, a0)ds′

a0

(19)

Notice that the state sai now contains match quality q. Similarly, we have matchspecific value function for female i when she choose to stay

vf (ai, sai) ≡ uf (ai, sai)− τ

+β

∫V f (s

′

ai)hθ1(s′

ai|sai, ai)ds′

ai

−β∫V f (s

′

0i)hθ1(s′

0i|sai, 0i)ds′

0i

(20)

Given the distribution assumption of taste shock, we derive the probability ofstaying in the match ai

Prm(ai|sai) =exp(vm(ai, sai))

exp(vm(ai, sai)) + 1(21)

for male a.

Prf (ai|sai) =exp(vf (ai, sai))

exp(vf (ai, sai)) + 1(22)

for female i.

14

The equilibrium condition entails the willingness to divorce for both spousesmust be equal. Otherwise the spouse has a higher surplus in marriage wouldmake positive transfer τ to the other side in order to save the marriage. Theequilibrium condition is

Prf (ai|s0i) = Prm(ai|sa0) (23)

for every marriage. Which in turn means

vf (ai, sai) = vm(ai, sai) (24)

In our model with perfectly transferable utility within marriage, Becker-Coasetheorem is valid. You are only as happy as your spouse. And divorce would onlyhappen when the joint utility is smaller than the sum of their reserved utilities.Give this flexibility, divorce is efficient under transferable utility.

5 Estimation

The estimation of model is done in two parts: the estimation of joint marriagegain in the matching market and the estimation of joint marital output withinthe marriage. These two parts are linked through the conditional choice prob-ability next period. The existence of unobserved match quality q within themarriage requires an expectation-maximization approach (EM) in the estima-tion within marriage which is detailed in Arcidiacono and Miller (2011). Onthe other hand, the estimation of dynamic matching function stems from theestimation of static matching function in Choo and Siow (2006).

5.1 The estimation of dynamic matching function

We derive the dynamic marriage matching function in (18). The estimation ofjoinnt marriage gain requires a first stage estimation of conditional choice prob-abilities. We employ two different methods to estimate CCPs. For the CCPs

that conditional on next period being single, P rm

(a′0|s′a0) and P r

f(0i

′ |s′0i),frequency estimator is used

(25)P rf(0i

′|s

′

0i) =

∑kl I(0i

′, s

′

kl = s′

ai)∑kl I(s

′kl = s

′ai)

for the simplicity and unbiasedness given that we are in a large matching market.Smoothing methods such as Nadaraya-Watson kernel estimator is needed.

We approximate the CCPs that conditional on next period being marriedusing flexible logit model since the state space increase drastically when wemodel the utilities within marriage. Employing first stage estimation of transi-tion functions is essential given the computation burden. Since the first stageestimation in our dynamic matching model presents little difference comparingto the standard dynamic discrete choice. We assume away the estimation oftransition functions in our example.

15

Since only the joint marriage gain is identified, we denote the joint surplusu(ai, sa0, s0i) ≡ um(ai, sa0) + uf (ai, s0i). The first step of estimation is param-eterizing for the joint surplus, denote as uθ(ai, sa0, s0i). From (15) and (16), wehave the dynamic joint surplus as

vfθ (ai, s0i) + vmθ (ai, sa0) = uθ(ai, sa0, s0i)− β(lnP r

f(0i

′ |s′ai)

P rf(0i′ |s′0i)

+ lnP r

m(a

′0|s′ai)

P rm

(a′0|s′a0))

(26)And we can rewrite equilibrium constraints (13) as

vfθ (ai, s0i)− vmθ (ai, sa0) = lnNm(sa0)− lnNf (s0i)

−ln(

I∑k=1

exp(vmθ (ak, sa0)) + 1) + ln(

A∑l=1

exp(vfθ (li, s0i)) + 1)(27)

Combining (26) and (27), we have a system of nonlinear equations in match-

specific value functions vfθ (ai, s0i) and vmθ (ai, sa0). Suppose we have A types offemale and I types of male, we will have 2A(A + 1)I(I + 1) of unknowns and2A(A+ 1)I(I + 1) of equations.

While we manage to reduce a part of computational burden by circumventingthe dynamic programming problem using two-stage CCP estimators, we do haveto solve for the system of nonlinear equations every time we try a new θ. Nextwe construct the partial log-likelihood in order to use the MPEC approach bySu and Judd (2012).

The partial log-likelihood function is

L(θ) =

a=A∑a=0

i=I∑i=0

logPrfθ (ai|s0i)

+

a=A∑a=0

i=I∑i=0

logPrmθ (ai|sa0)

(28)

where Prfθ (ai|s0i) is given by (12) and Prmθ (ai|sa0) is given by (10).θ is estimated by constrained maximization of partial log-likelihood (28)

subject to (26) and (27).

5.2 The estimation of divorce cost: EM algorithm

From the equilibrium condition we established in (23) and (24), we constructthe joint marital output of a married couple ai by combining (19) and (20)

2vθ(ai, sai) = u(sai)− β(lnP r

f(0i

′ |s′ai)

P rf(0i′ |s′0i)

+ lnP r

m(a

′0|s′ai)

P rm

(a′0|s′a0)) (29)

where v(ai, sai) ≡ vm(ai, sai) = vf (ai, sai) and u(sai) ≡ um(sai) + uf (sai).

16

The conditional choice probability of staying in current match Prm(ai, sai)and Prf (ai, sai) is equal after the transfer within marriage. Therefore, we de-note Pr(ai, sai) ≡ Prm(ai, sai) = Prf (ai, sai)

Pr(ai, sai) =exp(v(ai, sai))

exp(v(ai, sai)) + 1(30)

We choose to parameterize joint marital output u(sai) as

u(sai) = q + α2agem + α3age

f + α4edum + α5edu

f

+α6dur + α7dur2 + α8kid+ α9kid

2(31)

where the joint marital output are determined by their current age, education,match duration and the number of children they have.

Complication arises with the match quality q in match-specific value func-tion. Married agents learn the value of q at the beginning of each marriage andthey treat it as another state variable such as education and age. In estimation,we do not observe match quality. Therefore we need the EM algorithm.

We use π to denote the probability distribution over the unobserved statesconditional on observed states. In the expectation step, we update π usingconditional probabilities of being in each unobserved state. The maximizationstep follows as if we observed the unobserved state variables.

At the t+ 1 iteration, we evaluate the likelihood function at parameters weget from previous iteration θt, πt

Lt =

n=N∑n=1

I(ai, sai)logPrθt(ai, sai) (32)

We derive the joint likelihood function given the data and the unobserved stateq1 in a similar fashion

Lt(q = q1) =

n=N∑n=1

I(ai, sai)logPrθt(ai, sai, q = q1) (33)

Then we calculate the probability of marriage ai being in unobserved stateq1 using

π(q = q1)t =Lt(q = q1)

Lt(34)

We also need to update our CCPs using flexible logit at each iteration. Att+ 1 iteration, the conditional probabilities of being in unobserved state π(q|x)are used as weight in flexible logit.

t+ 1 iteration ends with the maximization of weighted likelihood

n=N∑n=1

∑(q=q1)

qI(ai, sai)logPrθt+1(ai, sai) (35)

17

5.3 Identification

The model parameters are identified through individuals decisions in marriagemarket. Given the similarity between the dynamic matching function (18) andthe static matching function in Choo and Siow (2006), the identification resultsare comparable. The future adjustment term in the dynamic matching functioncan be estimated using nonparametric regression methods without solving thedynamic programming problem. The joint marriage gain um(ai, sa0)+uf (ai, s0i)is identified. However, individual surplus um(ai, sa0) or uf (ai, s0i) cannot beidentified. Transfer wai cannot be identified either. These results are in linewith identification of static transferable utility models in the literature.

Given the divorce patterns, joint marital output u(sai) is identified. Wecan also identify the distribution of the unobserved match quality q, whichwe assume it is a normal distribution. Parameters in the joint marital outputparameterization (31) capture the effect of each state variable.

6 Results

The results are presented in four parts. First we will present the quality of fitcomparing the marriage rates and divorce rates from the data with those fromthe model. Second we demonstrate evidences for positive assortative matchingin education and in age. Third we show the change of flow utilities conditionalon the duration of the match. Lastly we discuss the role of children in marriage.

6.1 Quality of Fit

In Figure 4 we show that our estimated model fits closely to the data in term ofmarriage rates. We are showing the constructed and empirical marriage rates ofthe couples in which husbands are two years older than the wives. We can seethe marriage rates are peaking between the age of 20 and 28 with a second peakat age 36 for these couples. In Figure 5, we compare the divorce hazard ratesin the data with hazard rates constructed by the model. We plot the divorcehazard rates against duration. As for the fitting, we are overestimate the divorcehazard rates at the first two years of marriage and slightly underestimate themafterward.

6.2 Positive Assortative Matching

Since Gary Becker proposes the sorting of valuable characteristics would leadto positive assortative matching (PAM) in marriage market, testing for PAMis pervasive in the matching literature. In this section, we will show that PAMstill exists when we extend into dynamic two-sided matching.

In Figure 6, we plot the marriage gain for both new couples and existingcouples against the age difference, defined as the husband’s age minus wife’sage. Marriage gain of the new couples is the highest where the husband isslightly older than the wife (Age difference is at 2). And the gain decreases as

18

the age differences increase. As for the marriage gain of the existing couples,no such positive assortativeness patterns exist. Though couples with large agedifference rarely select into marriage, once they do, the age difference does notseem to affect the marriage stability. Notice the marriage gain is negative dueto the fact that we normalized the current utility of choosing single as zero.Therefore if the number of individuals choosing single exceeds those choosingmarriage, the marriage gain would be negative (smaller than zero).

In Table 4 and Table 5, we show the current period marriage gain and the fu-ture utility by education groups for the new couples. We construct the weightedaverage of all marriage gains in education groups with weights being the esti-mated conditional probabilities. In both Table 4 and Table 5, the highest utilityare from the positive assortative matches on the diagonal when the couples areat the same education level. Couples with higher education are enjoying highermarriage utility mostly due to their expected future utility, meaning they areslightly more likely to get married because of their expectation of marriage sta-bility. These results are widely confirmed by previous literatures. In addition,the education mismatch couples are having less utility comparing with coupleswho both have low education. This is particular interesting given that the fe-males in US gain more education than males in recent decades. Chiappori et.al(2009) give an explanation that women are choosing more education not only fortheir human capital but also for a better chance to match with men with highereducation. However, the equilibrium of marriage market predicts that excesssupply of high education women would have to choose men with less education(if they decide to get married). And our results show these pairings are havingless utility comparing to other groups.

In Table 6,7 and 8, the positive assortativeness are mostly preserved forthe existing couples. Table 8 shows that couples with mismatch education arehaving less dynamic marriage output than those with same education level.And couples with higher education, even those mismatch ones, are enjoyingmore dynamic marriage output than the lower education ones. Table 6 and 7dissect the dynamic marriage output into the current period utility and futureutility. The interesting results are the source of education gradient of marriages:though the higher education couples enjoy less utility in current period, theyhave a much better expected future utility.

6.3 Marital Output for Existing Couples

Table 9 presents the estimation of joint marital output (31). Note that allestimates are significant at 5% level. The first column shows shows the modelwithout unobserved match quality q and the second column shows the modelwith q. For the model with heterogeneity q, we assume q is drawn from a normaldistribution. The EM algorithm gives us the mean of the normal distribution tobe -0.4406 and the standard error to be 0.9316. The introduction of unobservedmatch quality also change most the estimates. We find that q is systematicallydifferent across education with lower education group has a higher match quality,see Table 10.

19

Though it is counterintuitive to have a negative estimates for education in thepresent of unobserved match quality. Note that higher education couples enjoyless current period marriage output comparing to those with lower educationas we discuss Table 6. This is partly due to the lower education couples have ahigher match quality. The education of wife has more negative impact towardthe marital output comparing to the husband. This is possibly due to greateremployment opportunities for highly educated women. Given the husband isalways expected to be employed while women have to take responsibilities forboth work and family, it is reasonable to expect the education level of the wifehas a larger impact on current marriage output.

Next we plot the current period marital output over time with each educa-tion group in Figure 7. The ranking of each group remain unchanged comparingwith Table 6. As time pass, the two education groups with low educated hus-band converge together and remain the highest. We can see an upward trendof marital output across education groups. Recall that the divorce rates arefluctuating in a high level after the second year of marriage in Figure 5. Itshows little negative duration dependence comparing with other literature. Theduration coefficient is negative while the duration square coefficient is positive.It implies the effect of duration on marital output is negative for the first 20years of marriage. The negative duration coefficient offsets by the present ofper period random taste shock. Thus the divorce rates fluctuate rather thanincreasing.

This result goes against the hypothesis that couples accumulate match-specific capital as their marriage endures. We have not found any evidenceto support marriage output would rise with duration at the first 20 years ofmarriage. The literature are mixed in this regard: Bruze et.al (2015) have asimilar U-shape but the utility is the lowest at 5 or 6 years of marriage (they useDenmark marriage data); Brien et.al(2006) has a positive duration coefficient.

Another possible mechanism of accumulating match-specific capital is throughchildbearing. It is a common perception that having children, especially youngchildren, would increase marriage stability. We can see that in Figure 8 wherewe plot the divorce rates for couples who have children with those who don’t.The divorce rates for those who don’t have children are mostly higher. Wetreat fertility decisions as exogenous in the model but we do allow the arrivalof children are conditioning on couple’s characteristics. It is well documentedthat more and more children is born out of wedlock after 1970 in US. Numberof children that are involved in divorces is growing. In our dataset, 1/5 of thechildren are born before the parents get married.

The result shows that the present of children has a positive impact on thestability of marriage. Though the children coefficient in Table 9 is negative,the children square coefficient is positive. We can see the result more directlyin Table 11. We list the number of children in the family and their respectivecurrent utility, future utility and the dynamic output (which is the sum of cur-rent martial output and future utility). Although the current marital outputof having children are lower than not having children. The difference in futureutility offset the negative for couples who have children. And in all, the dynamic

20

marriage output is the highest for those who have more children. Having chil-dren may have negative impact on couples current period utility due to budgetand time constraint required for taking care of them. It brings enough futureutility through increasing expected marriage stability. We interpret the resultas evidence in children being a match-specific capital.

7 Extension:Learning match quality in marriage

In this section, we revise our assumption that married couples learn about theirmatch quality q in the first period of marriage. Instead, married couple receivea signal every period if they decide to stay in the marriage. The population ofmatch quality q is normal with mean q and variance σ. Couples have beliefsabout the their match quality when they get married. And each period they stayin the marriage, couples learn about the true value of q through accumulationof experience.

Married individuals have prior belief that is normally distributed, qt ∼N(pt,Φt). Each period the couple choose to stay, they receive a signal:

δt = q + ξt (36)

where ξ ∼ N(0, σ2ξ ).

Then the couple updates the distribution of his beliefs subject to the Bayesianupdating equations:

pt+1 = Φt+1(Φ−1t pt +δtσ2ξ

) (37)

Φt+1 = [Φ−1t +1

σ2ξ

]−1 (38)

In our baseline model, married couple observe the match quality when theyget married and they treat it as another state variable. Now they treat theirprior beliefs p and Φ. The match-specific value function is

vθ(ai, sai, pt) = uθ(ai, sa0, s0i) + γE[q] + 2∆(sai)

−∫mt+1

{β(lnP r

f(0i

′ |s′ai)

P rf(0i′ |s′0i)

+ lnP r

m(a

′0|s′ai)

P rm

(a′0|s′a0))}dF (mt+1)

(39)

Matsumoto (2014) illustrates how to incorporate a learning framework withdynamic discrete choice. The expected future value term requires integratingover all future beliefs thus it requires integrating over all possible realizationsof signal. Agents calculate the variance of future beliefs conditional on choices.Realization of signal does not matter since the transitions of prior beliefs areconditional on choice. Therefore, it is still possible to write the future valueterm as a function of CCPs.

21

8 Conclusion

In this paper, we build and estimate a dynamic two-sided matching model withendogenous match formation and dissolution. Then we use it to study marriagemarket in US from 1968 to 2015. We first construct the theoretical dynamicmarriage model and explore forward looking individuals decision makings undertwo-sided market. We derive the dynamic marriage matching function which isan extension of the static matching function in Choo and Siow (2006). Withthe introduction of dynamic, the joint surplus of the match consists of twoparts: the current period joint surplus and the expected future utility. And theexpected future utility term has a intuitive explanation: individuals place lessvalue on unstable relationships and they don’t value relationships all togetherif they have a strong position in future marriage market.

Then we fit the model to PSID marriage history data in US from 1968 to2015. We select a few key variables such as age, education, number of childrenand duration of the match and evaluate their relative importance in marriageand divorce over agents life time. We also allows for unobserved match qualityin the match to ensure rich heterogeneity. The results confirm the positiveassortativeness in matching in terms of education and age. Moreover, familybehaviors diverge among education groups. While the lower educated couplesenjoy higher current period marriage output, higher educated couples makes upthe differences by having higher expected future utility. The source of maritaleducation premium is the stability of the matches and their better positions infuture marriage market.

We explore the contribution of possible match-specific capital such as chil-dren and match duration toward marriage stability. We find that current mar-riage output decreases with duration for a long period of time. At the same timechildren contributes to the marriage stability by increasing the future expectedutility of the match.

It is worth noting that we do not model the fertility decision in this paper.And as parents are increasingly having children outside of marriage, fertilitydecisions are clearly intertwined with decisions to select into and out of mar-riage. Another short coming is the limited variables we select in the two-sidedmatching. And also we do not allow for cohabitation as a choice. Most of theshort coming are due to computational constraints which presents interestingfuture research opportunities.

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24

20 30 40 50 60

0400

800

1200

Age

Fre

quency

FemaleMale

Figure 1: Marriage Frequency with respect to Age

−15 −10 −5 0 5 10 15

0500

1500

2500

Age Difference

Fre

quency

Figure 2: Marriage Frequency with respect to Age Difference

0 5 10 15 20 25

50

150

250

Duration of Marriage

Fre

quency o

f D

ivorc

es

Figure 3: Divorce Frequency with respect to Marriage Duration

25

Table 1: Sample Distribution of Completed Education

Completed education Male FemaleHigh School or less 0.542 0.570

More than High School 0.458 0.430No. of Observations 27332 26729

Table 2: Distribution of Education Matching

Husband’s Education

Wife’s education Low HighLow 0.339 0.112High 0.186 0.363

Table 3: Probability of Divorce with respect to Education

Husband’s Education

Wife’s education Low HighLow 0.203 0.222High 0.228 0.153

20 25 30 35 40 45

0.0

02

0.0

04

0.0

06

Age of husband

Marr

iage

Rate

DataModel

Figure 4: Marriage rates in the model and from the data

26

0 5 10 15 20 25

0.0

05

0.0

10

0.0

15

0.0

20

0.0

25

0.0

30

Duration

Div

orc

e R

ate

Data

Model

Figure 5: Divorce rates in the model and from the data

−15 −10 −5 0 5 10 15

−23

−22

−21

−20

−19

Age Difference

Marr

iage G

ain

−3.4

−3.2

−3.0

−2.8

New CouplesExisting Couples

Figure 6: Positive Assortative Matching in Age

27

Table 4: Marriage Gain for New Couples

Husband’s Education

Wife’s education Low HighLow -18.00 -19.38High -18.63 -17.92

Table 5: Future Utility for New Couples

Husband’s Education

Wife’s education Low HighLow 6.58 6.31High 6.04 7.27

Table 6: Current Period Marriage Output for Existing Couples

Husband’s Education

Wife’s education Low HighLow -2.556 -3.244High -3.216 -3.995

Table 7: Future Utility for Existing Couples

Husband’s Education

Wife’s education Low HighLow 6.585 7.923High 7.624 9.134

Table 8: Dynamic Marriage Output for Existing Couples

Husband’s Education

Wife’s education Low HighLow 4.028 4.679High 4.408 5.139

28

Table 9: Parameter Estimates of Marital Output for Existing Couples

Parameter Description Without heterogeineity With heterogeineity

q match quality 1.464** (0.1733) -0.4406 (0.9316)α2 age of husband 0.0709** (0.0095) 0.0921** (0.0098)α3 age of wife -0.061** (0.010) -0.0021** (0.0103)α4 education of husband 0.4727** (0.0853) -0.1160** (0.0825)α5 education of wife -0.6336** (0.0845) -0.3236** (0.0776)α6 duration -0.1320** (0.0128) -0.1757** (0.0126)α7 duration square 0.0052** (0.0003) 0.0044** (0.0003)α8 number of children 0.3175** (0.0903) -0.1585** (0.0902)α9 number of children square -0.1006** (0.0302) 0.0841** (0.0304)

Notes:1. Standard errors are presented in parentheses. Double asterisk are signifi-

cant at 5% level2. The match quality q with heterogeneity is a normal distribution with the

mean of -0.4406 and standard error of 0.9316.

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Table 10: Unobserved Match Quality for Existing Couples

Husband’s Education

Wife’s education Low HighLow -0.386 -0.451High -0.454 -0.475

30

0 5 10 15

−4.0

−3.5

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

Duration

Mari

tal O

utp

ut

Wife Low Husband LowWife Low Husband HighWife High Husband LowWife High Husband High

Figure 7: Marital Output by Education Groups

0 5 10 15 20 25

0.0

05

0.0

10

0.0

15

0.0

20

0.0

25

0.0

30

Duration

Div

orc

e R

ate

No Kid

Have Kid

Figure 8: Divorce Rates Conditional on Number of Children

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Table 11: Divorce Cost, Future Utility, Match Quality by the Size of Family

Children Current Marital Output Future Utility Dynamic Marriage Output

0 -3.171 7.633 4.4621 -3.429 8.013 4.5852 -3.482 8.264 4.7823 -3.318 8.322 5.005

32