Costly Divorce and Marriage Rates

Transcription

1 MPRA Munich Personal RePEc Archive Costly Divorce an Marriage Rates Anna Yurko NRU - Higher School of Economics 1. February 2012 Online at MPRA Paper No , poste 3. April :31 UTC

2 Costly Divorce an Marriage Rates Anna Yurko February, 2012 Abstract This paper evelops a moel of choice between marriage an cohabitation to stuy the effect of ivorce costs on marriage ecision. The paire agents are heterogeneous, the utility is non-transferable, an break up an ivorce ecisions are moele explicitly as unilateral, that is, it takes the ecision of only one partner to terminate a relationship. This framework is empirically relevant, since unilateral ivorce is legal in many countries, an multiple empirical stuies of the effect of changes in ivorce laws on ivorce rates emonstrate that Coase theorem oes not hol partners cannot bargain efficiently). The moel seeks to reconcile the conflicting empirical evience on the relationship between marriage rates an ivorce costs. Keywors: family economics, marriage, cohabitation, ivorce, externalities JEL classification: D62, D91, J12

3 1 Introuction Marriage rates vary significantly across countries. For instance, in 2009 Slovenia ha the lowest marriage rate among the OECD countries, 3.17 versus 7.31 for the US. 1 Comparisons with countries outsie of the OECD reveal even larger ifferences in marriage rates aroun the worl. Marriage rates in the evelope countries have also ecline over time, an most of the ebate aroun the institution of marriage has focuse on these trens. Since 1970, the average marriage rate for the twenty seven OECD countries has fallen by almost 40%. 2 Economists ten to seek the explanation for social phenomena from the point of the cost / benefit analysis. The ata inicate that there are vast ifferences among countries in social, economic, an legal relative costs an benefits of marriage, an that the relative benefits of marriage must have ecline over the past few ecaes. This paper focuses on the effect of ivorce costs on marriage ecisions an the resulting relationship between ivorce costs an marriage rates. Since cohabitation is often a precursor or even a substitute for legal marriage, agents in the moel can choose to cohabit or marry legally. 3 Empirical evience inicates that higher ivorce costs can result in either higher or lower marriage rates an the moel seeks to reconcile this evience. Rasul [9] presents evience that the aoption of unilateral ivorce in the US has contribute to the ecline in marriage rates. Matouschek an Rasul [7] emonstrate that propensity to ivorce is lower for couples marrie after the introuction of unilateral ivorce laws in the US. Their paper an the work of Rasul [10] also evelop theoretical moels of marriage that explain these finings: marriage contract acts as a commitment evice an lower exiting costs unermine its ability to serve this 1 Marriage rate is efine as the number of marriages performe in a given year ivie by total population an multiplie by Source: OECD [8]. The exact numbers vary across countries, but the overall trens remain roughly similar. 3 Stevensen an Wolfers [11] iscuss recent trens in cohabitation an marriage. 1

4 purpose. Aoption of unilateral ivorce laws can be interprete as a reuction in costs of obtaining ivorce. Figure 1 epicts the relationship between marriage rates across countries with unilateral ivorce laws an a ifferent measure of ivorce costs: the length of the manatory separation perio before the ivorce is legalize. 4 Countries with longer separation requirements are interprete to have higher ivorce costs. Figure 1 shows that marriage rates are higher in countries with lower ivorce costs. Figure 1: Marriage rates an no-consent ivorce laws BY Marriage Rate, average RU UA FI SE EE CZ ES NL HU SI LT DK IS AU HR NO CA VE MX NZ BE LV DE AT IT PT LU UY AR CL CH IE GR UK FR Number of years to no consent ivorce Data, 36 countries Fitte line slope 0.3, signif. 5%) 4 The sample inclues countries with unilateral ivorce legislation for which the author was able to obtain information on legal grouns for ivorce. There are 36 countries in the sample: Argentina AR), Australia AU), Austria AT), Belarus BY), Belgium BE), Canaa CA), Chile CL), Croatia HR), Czech Republic CZ), Denmark DK), Estonia EE), Finlan FI), France FR), Germany DE), Greece GR), Hungary HU), Icelan IS), Irelan IE), Italy IT), Latvia LV), Lithuania LT), Luxembourg LU), Mexico MX), Netherlans NL), New Zealan NZ), Norway NO), Portugal PT), Russian Feeration RU), Slovenia SI), Spain ES), Sween SE), Switzerlan CH), Ukraine UA), Unite Kingom UK), Uruguay UY), Venezuela VE). Note that the US is not on the sample since the ivorce law iffers by state. 2

5 The moel in this paper seeks to reconcile this evience. It combines two incentives to marry: 1) exogenous benefits to marriage, social, financial, or legal, an 2) commitment value of marriage, which is an enogenous outcome of higher separation costs associate with ivorce. Single agents are matche in pairs an are free to enter either type of relationship contract or to remain single. The ecisions are base on the observe initial match quality an expectations of the future match qualities, an the utility is non-transferable. For every cohabiting or marrie agent future realizations of the aitional match quality shock are ranom. The agents are heterogeneous: paire agents may receive ifferent realizations of this match quality signal. Paire agents observe their iniviual realization of the match quality shock, an each agent makes his or her ecision on whether to stay in the current relationship or terminate it. The match survives only if both partners choose to preserve it. The assumption of non-transferable relationship utility is very important. Since Becker, Lanes an Michael [1], most of the theoretical literature on marriage an ivorce typically assumes that the Coase theorem applies to marital bargaining, so spouses can always reach ivorce agreements by the reistribution of welfare. If this assumption is vali, changes in consent versus no-consent ivorce laws shoul have no effect on inciences of ivorce an marriage. Empirical stuies, however, emonstrate that the change to no-fault unilateral ivorce laws has cause a small, 5 but statistically significant increase in the ivorce rates in the US an Europe. More recent moels of marriage assume that Coasian bargaining is not possible in marriage: for example, Fella et al [3], Rasul [10], an Guha [6]. This paper follows in their footsteps, emonstrating that the inability of partners to efficiently compensate each other when separation is esirable by only one of them plays a crucial role in etermining the relationship between ivorce costs an the ecision to marry. The non-transferability of utility only matters when the agents are heterogeneous. Otherwise, they make the same ecisions an no transfers between partners are ef- 5 See Frieberg [4] an Wolfers [12] for the US an Gonzalez an Viitanen [5] for Europe. 3

6 ficient. To illustrate the effect of the heterogeneity assumption on the moel s preictions when the utility is non-transferable this paper consiers three cases: 1) the paire agents are homogeneous, that is, they receive the same future realizations of the match quality shock; 2) the agents are heterogeneous an their utility epens only on their own realization of the relationship quality signal, but not their partner s; 3) the agents are heterogeneous an the utility of each partner epens not only on his/her realize match quality, but also on that of the other partner. The first case introuces the framework an shows that when the agents are homogeneous or, alternatively, the utility is perfectly transferable, the moel cannot generate a positive relationship between ivorce costs an marriage rates. The secon case contains the main result of the paper, an the thir case is a robustness check. The analysis in the paper cases two an three) emonstrates that when the agents are heterogeneous an the utility is non-transferable, higher costs of ivorce may increase the value of marriage an make it preferre to cohabitation even in the absence of aitional benefits to marriage. The intuition is as follows. Any relationship survives if an only if both partners chose to maintain it. If one person enjoys the relationship an prefers to stay in it, she woul only be able to o so if her partner also prefers not to terminate it. If he is no longer happy in the relationship an chooses to en it, his ecision imposes a negative externality on her. Higher costs of terminating legal marriage may inuce her partner to choose to preserve the relationship, eliminating the negative externality. Thus, commitment is valuable an ivorce costs make marriage a more committe relationship form. If marriage oes not carry any aitional benefits relative to cohabitation, the moel generates a positive relationship between ivorce costs an marriage rates, consistent with the evience on the ecline in marriage rates after the aoption of unilateral ivorce laws from Rasul [9]. What about the negative relationship between marriage rates an the ifficulty of obtaining ivorce in Figure 1? If marriage provies aitional utility relative to cohabitation, however small, the 4

7 preicte relationship between the costs of ivorce an marriage rates becomes U- shape: the marriage rate eclines for lower values of ivorce costs an increases for the costs of ivorce above some threshol value. When the ivorce costs are relatively low, the exogenous marriage benefit serves as the main incentive for marriage, an couples that marry for this incentive are iscourage with higher ivorce costs. Once the ivorce costs are high relative to the marriage benefit, the commitment effect ominates the marriage benefit incentive, an the aitional couples choose marriage for the lower likelihoo of separation. Thus, the moel is capable of reconciling empirical evience when both incentives to marry are present. Rasul [10] was the first to emonstrate the commitment value of greater ifficulty of obtaining ivorce in the mutual consent ivorce regime with non-transferable utility an heterogeneous agents. He stuies how the move from mutual consent ivorce to unilateral ivorce affects the marriage market outcomes with a moel of choice between marriage an singlehoo. The moel can explain the observe ecline in marriage rates after the aoption of unilateral ivorce laws. Theoretical moels of greater commitment in marriage relative to cohabitation have been evelope by Wyick [13] an Matouschek an Rasul [7]. They have homogeneous agents an use a repeate prisoner s ilemma game setting to show that marriage can foster cooperation better than cohabitation ue to its higher termination costs. The implication is that partners in marriage are more likely to act cooperatively towars each other an behave themselves than the cohabiters, so marriage generates enogenous benefits relative to cohabitation. Brien, Lillar, an Stern [2] an Matouschek an Rasul [7] have moels of choice between cohabitation an marriage with marriage proviing an exogenous benefit to both spouses. These moels preict a negative relationship between ivorce costs an marriage rates. The rest of the paper is organize as follows. Section 2 escribes the moel. The solution of the moel with homogeneous agents an the results are given in Section 5

8 3. Section 4 solves the moel with heterogeneous agents an inepenent utilities an presents the main intuition an the results of the paper. Section 5 explores the implications of assuming that paire agents have interepenent utilities. Section 6 conclues. 2 Moel Setup This section presents the basic framework for analyzing the relationship between ivorce costs, marriage benefits, an marriage market outcomes. Men an women in the moel are symmetric. The moel has two perios. In the beginning of perio one iniviuals are matche in pairs an raw the same couple specific match quality shock q from uniform istribution on [q L, q H ], where q L < 0 < q H. The match quality etermines the relationship utility each agent receives in perio one if he / she were to enter a househol sharing relationship with the current match partner. Assume that the relationship utility in perio one is equal to the realization of match quality q. The utility from remaining single is normalize to zero. Upon observing q, each agent ecies whether to remain single or enter into cohabitation or marriage with the current match partner. Since the initial match quality is the same for both match partners, they make ientical ecisions. Single agents remain single in perio two an receive the total utility of zero. Cohabiting an marrie agents receive an aitional iniviual relationship quality shock x from the same istribution. Upon observing the quality shocks, each agent unilaterally ecies whether to preserve the relationship or to exit it. If the relationship is preserve, each agent in a couple receives relationship utility of r q, x, x ) in perio two, where x is the realization of the agent s own aitional match quality shock an x is that of his/her partner. If the agents separate, each receives the utility of zero in perio two. The relationship survives only if both agents make the 6

9 ecision to maintain it. For what values of q o single agents ecie in perio one to remain single, cohabit, or marry? Denote the expecte value of remaining single when the observe initial match quality is q by S q), the expecte value of cohabiting by R q), an that of getting marrie by W q). There is no iscounting between perios. The expecte value of cohabitation an marriage epen on the ecisions of agents in perio two. Each agent observes the aitional shocks to match quality in perio two an compares the value from staying together to that of splitting. Assume that for cohabiting agents the cost of breaking up is zero, an that for marrie agents the cost of obtaining ivorce is > 0. Then, each cohabiting agent will choose to stay in the relationship as long as the resulting relationship utility in perio two excees zero, an each marrie agent will ecie to stay marrie if the total utility from being marrie excees. Assume also that marrie agents may receive an aitional exogenous utility bonus M 0 in every perio of their marriage. Next we explore three cases: 1. The agents are homogeneous: both partners receive the same aitional match quality shock x. Thus, they make ientical ecisions an break-up or ivorce occurs only if beneficial to both. This case is stuie in Section The agents are heterogeneous: the aitional match quality shocks are inepenent raws from the same istribution. Also, for each agent in a couple the utility in perio two epens only on the realization of own aitional match quality shock x, rq,x,x ) x = 0. Thus, match partners can make ifferent ecisions. The relationship is preserve only if both prefer not to terminate it. If one agent ecies to en the relationship while the other woul prefer to maintain it, this ecision imposes a negative externality on the pro-relationship partner. For simplicity, assume that the relationship utility is perio two is r q, x, x ) = q x. This case is explore in Section 4. 7

10 3. The agents are heterogeneous with inepenent realizations of the aitional relationship quality shock in perio two an rq,x,x ) x 0. That is, the relationship utilities are interepenent. Specifically, assume u = αx + 1 α) x an v = αx + 1 α) x, where 0.5 < α < 1. Then, in perio two if the couple stays intact the relationship utility of the first agent is r q, x, x ) = q u, an that of the secon agent is r q, x, x) = q v. Section 5 analyzes this case. In orer to solve the moel for each of the three cases it remains to specify the istribution for the aitional match quality shock x. Assume that it is istribute exponentially with mean 1/. The cumulative istribution function is 1 e x, x 0 F x x) = 0, x < 0. 1) 3 Homogeneous agents Here we assume that the paire agents are homogeneous, that is, the realize value of the aitional relationship quality shock is the same for both agents in a cohabiting or marrie union, x = x. The relationship utility for paire agents in perio two if the agents stay together is r q, x, x) = q x. The purpose of the analysis in this section is to emonstrate that in the absence of heterogeneity an when the utility is non-transferable, higher ivorce costs cannot increase the relative value of marriage. That is, for any positive value of the marriage benefit, higher ivorce costs result in lower marriage rates. We begin by formulating the problem of cohabiting agent, an then procee to that of marrie agent. In perio two a cohabiting agent woul prefer to stay in the relationship if the relationship utility r q, x, x ) = q x 0. Since both agents in a couple receive the same realization of the match quality shock, the probability of them staying together is F x q). 8

11 The expecte value of cohabiting with the initial relationship quality q is R q) = q + q 0 q x) F xx). With the exponential cumulative istribution function R q) = 2q 1 e q, q 0 q, q < 0. 2) A marrie agent with the initial relationship quality q woul choose to stay marrie in perio two if the total utility r q, x, x ) + M = q + M x. Thus, the expecte value of marriage is W q) = q + M) + q+m+ 0 q + M x) F x x) + [1 F x q + M + )] ) or W q) = 2 q + M) 1 e q+m+), q M q + M), q < M. 3) To solve the moel we nee to fin the values of q for which matche agents choose marriage in perio one for given an M. Denote the lowest value of q above which matche agents prefer to marry by q. Recall the moel s assumptions of > 0 an M 0. Proposition below establishes the following results: Proposition 1 Let q be the initial relationship quality such that i) q solves W q) = max {0, R q)}, where W q) an R q) are given by equations 3) an 2) respectively; ii) For all q q, W q) max {0, R q)}. Then, a) For M = 0, q oes not exist. That is, with no aitional exogenous benefits marriage is never preferre to cohabitation or singlehoo; b) For any M > 0, q is increasing in. Proof. 9

12 Consier two cases: 1) R q) 0 or, equivalently, q 0 an 2) M q < 0. Note that when q < M marriage is never chosen, so q cannot belong to this range of q values. 1) q 0 From W q ) = R q ) obtain q = 1 [ ln 1 e M+) ) ln 2M) ]. This value is 0 when M,, an are such that 2M 1 e M+). For part a) of the proposition, note that the solution exists only if M > 0. The solution also satisfies conition ii) of the proposition when M is strictly positive. To obtain b), ifferentiate q with respect to : q = e M+). 1 e M+) The erivative is positive for all strictly positive an finite values of M,, an. 2) q [ M, 0) The cut-off value q solves W q) = 0, where W q) = 2 q + M) 1 e q+m+) from equation 3). Unfortunately, no close-form solution can be obtaine. Analysis of the function W q) yiels the following: i) W q) is strictly convex; ii) It has a unique minimum that is below zero; iii) lim q W q) = lim q W q) = ; Thus, equation W q) = 0 has two roots. Denote these roots by q an q, with q > q. W q) 0 for q, q ] an q [q, ). Since W M ) = 2 < 0, q < M < q. Thus, the unique caniate solution for q is q. It is the solution if also q < 0, which is true for all M,, an such that 2M > 1 e M+). For a), note that this conition is never satisfie when M = 0. 10

13 To show b), implicitly ifferentiate equation W q ) = 0 with respect to : q = e q+m+) 2 e q+m+) > 0 for any q [ M, 0). Marriage rate can be obtaine as the fraction of matches with initial relationship quality of at least q. If F q enotes the cumulative istribution function for q, then MR = 1 F q q ). The main result of this section follows straightforwarly: Corollary 2 Divorce Costs an Marriage Rates: the Case of Homogeneous Agents) Uner the assumptions of the moel with homogeneous agents, a) If M = 0, marriage rate is also zero for any value of ivorce cost ; b) For any M > 0, marriage rate is a ecreasing function of. Figure 2 illustrates the relationship between marriage rates an ivorce costs for ifferent values of the exogenous marriage benefit M when = 0.5 an q is uniformly istribute on [q L, q H ], with q L = 5 an q H = Figure 2 A) epicts the lowest value of q above which matche agents prefer to marry q) as a function of the ivorce cost for M = 0, M = 0.1, M = 0.5, M = 1, M = 2. Figure 2 B) shows the respecte marriage rates, that is, the fraction of agents that choose marriage in perio one. Observe that without exogenous benefits to marriage M = 0), marriage is never optimal. For any M > 0, agents with higher values of the initial match quality prefer to marry, an more agents marry for higher values of M. For any value of M marriage rates ecline in the cost of ivorce. Figure 3 A) shows the net ivorce rate for each value of M as function of ivorce cost, where net ivorce rate is a fraction of marrie agents that ivorce in perio two. The ivorce rates are lower with higher cost of ivorce. 6 Note that higher values of reuce the likelihoo of ba relationship utility shocks in perio two. The likelihoo of staying together in perio two is higher for both marrie an cohabiting couples, so the ivorce costs play a relatively smaller role in the couple s ecision of what type of union to form. 11

14 Figure 2: Homogeneous agents: Cut-off values of q an Marriage rate 6 A) Cut off value of q 4 q B) Marriage rate MR M=0 M=0.1 M=0.5 M=1 M= Average relationship welfare is epicte in Figure 3 B). It is compute without the exogenous benefit M, since it is obvious that aing a positive constant to utility increases it, an we are intereste in comparing the welfare from the matches for various levels of M an. Observe that higher exogenous marriage benefits reuce relationship welfare, since matches of lower quality result in marriages. The average welfare also eclines with, since the ivorce costs are incurre with positive probability in any marriage. For any value of M, the welfare maximizing ivorce cost is zero. 4 Heterogeneous agents: Inepenent utilities In this section the agents are heterogeneous: in perio two each paire agent raws an aitional match quality shock from the same istribution, an the raws are inepenent. The relationship utility of each agent in perio two epens only on the agent s own realization of the match quality shock x an is inepenent of the 12

15 Figure 3: Homogeneous agents: Divorce rates an Average relationship welfare DR A) Divorce rate M=0 opt small) M=0.1 opt small) M=0.5 opt small) M=1 opt small) M=2 small) opt B) Relationship welfare no M) Wel value receive by his/her partner x, specifically, r q, x, x ) = q x. Upon observing own quality shock, each agent unilaterally ecies whether to stay in the relationship or to exit it. The relationship is preserve only if both partners ecie to maintain it. Thus, exiting ecision by one agent may impose a negative externality on his/her partner if the partner woul prefer to stay in the relationship. The purpose of this section is to emonstrate that when the utility is non-transferable an the agents are heterogeneous, positive ivorce costs may help eliminate the externality an increase the value of marriage. Thus, for some values of marriage benefits an ivorce costs marriage rate can increase in the cost of ivorce. Paire agents in perio two raw inepenent realizations of the aitional relationship quality shock x from the same istribution F x x) from 1). They remain paire only if both partners prefer to stay together. Thus, the expecte value of cohabiting with initial match quality shock q is R q) = q+f x q) q 0 q x) F xx), an that of being marrie is W q) = q + M) + F x q + M + ) q+m+ 0 q + M x) F x x) + [ 1 Fx q + M + ) 2] ). 13

16 With the exponential cumulative istribution function q + 1 e q) [ ] q 1 e q, q 0 R q) = q, q < 0. 4) an q + M + 1 e q+m+)) [ ] q + M + 1 e q+m+), q M W q) = q + M, q < M 5) Solving the moel involves fining the values of q such that matche agents choose marriage in perio one for given an M. Figure 4 shows the cut-off values of the initial relationship quality q above which matche agents prefer to marry as a function of the ivorce cost for several values of M when = Figure 4: Heterogeneous agents with inepenent utilities: Cut-off values of q q M=0 M=0.1 M=0.5 M=1 M=

17 First, note that for M = 0, the cut-off value of q is lower for higher values of ivorce costs, that is, when the ivorce cost is larger it inuces marriage for lower match quality couples. The analysis in the previous section with homogeneous agents emonstrate that in the absence of aitional exogenous benefits, marriage is never preferre to cohabitation. This is no longer the case when the agents are heterogeneous with inepenent aitional relationship quality shocks: marriage is preferre by matche agents with higher initial match quality an the fraction of couples choosing marriage increases with higher ivorce cost. This is the commitment effect of ivorce costs. For high values of the marriage benefit M here it is for M > 1) the relationship between the cut-off value of q an the cost of ivorce is reverse. Agents with lower initial match qualities choose marriage, an their main incentive is obtaining the exogenous marriage benefit. These couples are iscourage by higher ivorce costs. The commitment effect of higher ivorce costs is ominate when the exogenous marriage benefit is sufficiently large. For intermeiate values of the marriage benefit here for M 0, 1]) the relationship between the cut-off values of q an the cost of ivorce is bell-shape. For small ivorce costs, couples with matches of lower quality marry to obtain the exogenous benefit M. Since they ivorce with positive probability, higher ivorce costs reuce the relative value of marriage, resulting in higher cut-off values of q for marriage. When is sufficiently large, the commitment effect ominates. Higher ivorce costs help eliminate the break-up externality an inuce marriage for lower match quality couples. The following proposition formally establishes these results: Proposition 3 Let q be the initial relationship quality such that i) q solves W q) = max {0, R q)}, where W q) an R q) are given by equations 5) an 4) respectively; ii) For all q q, W q) max {0, R q)}. 15

18 Then, A) For M = 0, 1) q 0 an 2) q is ecreasing in ; B) M > 0, such that for any M > M, q is increasing in ; C) For M 0, M], q can be increasing or ecreasing in. Proof. Let D t) = 1 e t) [ t 1 ] e t = t 1 ) + e t ) 2 t e t 6) for any real t 0. Let q) = W q) max {0, R q)}. From 4), 5), an 6) q) = M ) + D q + M + ) D q), q 0 q + M ) + D q + M + ), q [ M, 0] q + M ), q M. 7) Then, q is the value of initial relationship quality q such that 1) q ) = 0 an 2) for all q q, q) > 0. Note that for any q M, q) < 0. Thus, in what follows, this case is not consiere. Claim A): For M = 0, 1) q 0 an 2) q is ecreasing in, that is, its erivative with respect to is negative: q ) < 0. For any real t 0, let v 0 t) = e t 2 t e t), 8) Then, from 6) an 8), 16

19 D t) = t 1 ) + 1 v 0 t) 9) The properties of function v 0 t) are explore in the Appenix. properties to prove parts 1) an 2) of Claim A): We use these 1) Show that q 0. Suppose not, i.e., q [, 0). Then, from 9) an 7), q) = ) 2q v 0 q + )) < 0 q < 0. This is because v 0 t) 1 for any t 0 See Appenix). Thus, q < 0 oes not exist. 2) Consier q 0 an show that q ) < 0. For M 0, q solves q) = M ) + D q + M + ) D q) = 0. Equivalently, q) = 2M + 1 [v 0 q + M + )) v 0 q)] = 0. When M = 0, q) = 0 is equivalent to v 0 q + )) = v 0 q). From the properties of v 0 t) See Figure 11 A) in the Appenix), conclue that q t 0, t 1 ) an q ) < 0, where t 0 solves v 0 t) = 0 an t 1 solves v 0 t) = 0. Next we show claim C) an use the result to establish claim B). Claim C): For any fixe value of M > 0, q ) can be positive, negative, or zero. Let q ) be the root of equation q, ) = 0 for any fixe M 0. That is, q ), ) = 0. Fully ifferentiate this equation with respect to to fin the erivative q ): where q an enote the erivatives of q) with respect to q an, respectfully. q ) = ) q ), ), q q ), By ifferentiating 7) for q M, obtain 17

20 D q + M + ) D q), q 0 q = 1 + D q + M + ), q [ M, 0] an = D q + M + ) 1, q M. Differentiate 6) to obtain D t) = 1 + e t t 3 + 2e t), t 0. Let Then, v 1 t) = e t t 3 + 2e t). 10) Using 11) obtain D t) = 1 + v 1 t). 11) v 1 q + M + )) v 1 q), q 0 q = 2 + v 1 q + M + )), q [ M, 0] 12) an = v 1 q + M + )), q M. 13) The properties of function v 1 t) are explore in the Appenix. To etermine the signs of q an, consier the two cases: 1) q [ M, 0) an 2) q 0: 1) q [ M, 0) q = 2 + v 1 q + M + )) > 0 since v1 t) 1 t 0 See Figure 11 B) in the Appenix). 18

21 = v 1 q + M + )), thus, is where t 1 solves v 1 t) = 0. 2) q 0 First, show that q < t 1. 0, q [ M, t 1 M ] 0, q [ t 1 M, 0 ), As previously establishe, q 0, q solves q) = 2M+ 1 [v 0 q + M + )) v 0 q)] = 0. Equivalently, 2M = v 0 q ) v0 q + M + )). 14) Suppose q t 1. For all t t 1, v 0 t) is an increasing function, so v 0 q + M + )) > v 0 q ) q t 1. Thus, equation 14 cannot hol an q < t 1. Next, etermine the signs of q an for all q 0. Analysis for remains as before. For q [ 0, t 1 ), q = v 1 q + M + )) v1 q ) > 0. Combining the two cases, obtain q ) is 0, q [ M, t 1 M ] 0, q [ t 1 M, t 1 ). Claim B): M > 0, such that for any fixe M > M, q ) > 0. Let q M) be the root of equation q, M) = 0 for any fixe > 0. As before, can obtain the erivative of q M) with respect to M by fully ifferentiating this equation 19

22 with respect to M: q M M) = ) M q M), M ), q q M), M The erivative of q) with respect to M is M = 1 + D q + M + ) = 2 + v 1 q + M + )) > 0, q M. Since q > 0 q [ ) M, t 1, conclue that qm M) < 0. M ) = 2 an q) is continuous an strictly increasing with respect to M. As M gets larger, the root of q) = 0 shifts to the left, closer to M ). Thus, there exists M such that q M) = t 1 M < 0 an for M > M, q [ M, t 1 M ) an q ) > 0. Marriage rate is the fraction of matches with initial relationship quality of at least q. As before, let F q enote the cumulative istribution function for q, then MR = 1 F q q ). The following corollary establishes the main result of this section: Corollary 4 Divorce Costs an Marriage Rates: the Case of Heterogeneous Agents) Uner the assumptions of the moel with heterogeneous agents, a) If M = 0, marriage rate is increasing in ivorce cost ; b) M > 0, such that for any M > M, marriage rate is ecreasing in ; c) For M 0, M], marriage rate can be ecreasing or increasing in. Figure 5 epicts the relationship between marriage rate an ivorce costs for = 0.5, q L = 5, an q H = 10. This simple moel suggest the following explanation of the empirical evience. Rasul [9] fins that marriage rates in the US ecline after reuction in ivorce costs. 20

23 Figure 5: Heterogeneous agents with inepenent utilities, continuous case: Marriage rate M=0 M=0.1 M=0.5 M=1 M=2 MR This evience is consistent with the commitment moel of marriage. Here the commitment effect ominates when the exogenous benefits to marriage are low relative to the cost of ivorce. The relationship between marriage rates an the measure of ifficulty of obtaining ivorce across countries is, however, of the opposite sign. The exogenous benefits part of the marriage ecision story appears to be more relevant. Figure 6 A) shows the net ivorce rate as a ecreasing function of the ivorce cost. Figure 6 B) epicts the average relationship welfare for ifferent values of M. As before, it is lower for higher values of M since matches of lower quality result in marriage. The total welfare maximizing ivorce cost epens on the value of M. For small values of the exogenous marriage benefit, M = 0 an M = 0.1, the welfare is increasing in, so the optimal ivorce cost is equal to the highest value in the given range, = 10. Higher ivorce costs help eliminate the negative break-up externality, increasing the expecte value of marriage an welfare. For higher values of M the commitment effect of higher ivorce costs is weaker, thus, the welfare-maximizing 21

24 Figure 6: Heterogeneous agents with inepenent utilities, continuous case: Divorce rates an Average relationship welfare A) Divorce rate DR M=0 opt large) M=0.1 opt large) M=0.5 opt small) M=1 opt small) M=2 small) opt B) Relationship welfare no M) 0.53 Wel ivorce cost is small, just above zero. We conclue that when the agents are heterogeneous, the relationship utility is non-transferable, an the cost of ivorce is high relative to the marriage benefit, ecreasing the ivorce costs can result in lower marriage rates. When the ivorce costs are relatively low, the marriage rate is a ecreasing function of the ivorce costs for any positive marriage benefit. The assumption of heterogeneous agents with inepenent utilities, just like the assumption of homogeneous agents, may be too strong. In the next section we relax this assumption an solve the moel for the case of heterogeneous agents with interepenent utilities. 22

25 5 Heterogeneous agents: Interepenent utilities Consier any two agents comprising a cohabiting or marrie couple in perio two. Refer to these agents as agent 1 an agent 2. The aitional match quality shock of agent 1 in perio two is x, an that of agent 2 is x. As before, both x an x are rawn inepenently from the same exponential istribution with mean 1/. Let u = αx + 1 α) x an v = αx + 1 α) x, where 0.5 < α < 1. The relationship utility of agent 1 in perio two is r q, x, x ) = q u an that of agent 2 is r q, x, x) = q v. Thus, the aitional utility obtaine by each agent in perio two is a linear combination of the agent s own match quality shock an that of his or her partner, with a larger in absolute value) weight assigne to the own relationship quality shock. Then, the expecte value of cohabiting for agent 1 similar for agent 2) with the initial relationship quality q is R q) = q + F u,v q, q) q 0 q u) f uu) F u q) u, an that of being marrie is W q) = q + M) + F u,v q + M +, q + M + ) q+m+ 0 q + M u) [1 F u,v q + M +, q + M + )] ), f uu) F uq+m+) u + where F u,v u, v) is the joint cumulative istribution function of u an v, f u u) is the marginal ensity function of u, an F u u) is the marginal cumulative istribution function of u. To fin the joint istribution of u an v let the joint probability ensity function of inepenent x an x be ) e x e ) x, if x 0, x 0 f x,x x, x ) =. 0, otherwise Define A = {x, x ) : f x,x x, x ) 0} an B = {u, v) : u = αx + 1 α) x, v = αx + 1 α) x, x, x ) A}. Then, the joint probability ensity function of u an v is 23

26 f u,v u, v) = 2 2α 1 e u+v), if u, v) B 0, otherwise 15) an the marginal probability ensity function an the cumulative istribution function for u are, respectively, ) e uα e u 1 α, if u 0 2α 1 f u u) = 0, otherwise an α1 e α u ) 1 α) 1 e u ) 1 α, if u 0 F u u) = 2α 1. 0, otherwise Unfortunately, analytical solution cannot be obtaine in this case. The moel is solve numerically for = 0.5 an three values of the utility interepenency parameter α: 0.6, 0.75, an Figure 7 presents the cut-off values of the initial relationship quality q above which the agents prefer to marry for each value of α an ifferent values of the marriage benefit M: M = 0, M = 0.1, M = 0.5, M = 1, an M = 2. Figure 8 compares the marriage rates for ifferent values of α an M. 7 The results are robust to changes in the parameter values. As previously mentione, higher values of reuce the likelihoo of averse relationship utility shocks in the secon perio. With higher values of the ivorce costs play a relatively smaller role in the paire agents choice of marriage versus cohabitation, since they are less likely to separate in either case. 24

27 Figure 7: Heterogeneous agents an interepenent utilities: Cut-off values of q A) alpha= B) alpha= C) alpha=0.9 M=0 M=0.1 M=0.5 M=1 M=2 q Figure 8: Heterogeneous agents an interepenent utilities: Marriage rate 0.8 A) alpha= B) alpha= C) alpha= MR M=0 M=0.1 M=0.5 M=1 M=2 Notice that even for a large egree of utility interepenency α = 0.6) the cut-off value of q is ecreasing an the marriage rate is increasing in the cost of ivorce when M = 0. The U-shape relationship between the marriage rate an the cost of ivorce is also present for small values of the marriage benefit an becomes stronger as the utilities become less interepenent M = 0.1 an M = 0.5 on Figure 8). 25

28 Marriage rates are higher when the utilities of paire agents are less interepenent, since the value of eliminating the negative break-up externality is higher, inucing couples with lower values of the initial relationship quality to marry. Figure 9 shows that the net ivorce rate is a ecreasing function of the ivorce cost for any value of the exogenous marriage benefit M. Figure 10 epicts the mean relationship welfare of agents for ifferent values of M an α an the welfare-maximizing cost of ivorce. As before, the graph shows the welfare without the marriage benefit, so for any value of α the relationship welfare is lower for higher values of M since couples with lower initial match quality choose marriage. The optimal ivorce cost, however, is chosen so as to maximize the total welfare, taking into account the marriage benefit M. Observe that for any α the highest relationship welfare is achieve when M is small an is large. Figure 9: Heterogeneous agents an interepenent utilities: Divorce rate A) alpha=0.6 B) alpha=0.75 C) alpha= M=0 M=0.1 M=0.5 M=1 M= DR

29 Figure 10: Heterogeneous agents an interepenent utilities: Average welfare A) alpha= B) alpha= C) alpha= Welfare M=0 opt large) M=0.1 small) opt M=0.5 opt small) M=1 opt small) M=2 opt small) M=0 opt large) M=0.1 small) opt M=0.5 opt small) M=1 opt small) M=2 opt small) M=0 opt large) M=0.1 opt large) M=0.5 opt small) M=1 opt small) M=2 opt small) The analysis in this section emonstrates that even a small egree of heterogeneity low values of α) affects the ecision to marry an alters the relationship between the cost of ivorce an marriage rates when the utility is non-transferable. Explaining the empirical evience on this relationship oes not require the assumption of heterogeneous agents with inepenent utilities; the assumption of some egree of inepenency an utility non-transferability is sufficient as long as the aitional benefit to marriage is not too large. 6 Conclusion References [1] Gary Becker, Elisabeth Lanes, an Robert Michael. An economic analysis of marital instability. Journal of Political Economy, 85: , [2] Michael Brien, Lee Lillar, an Steven Stern. Cohabitation, marriage, an ivorce in a moel of match quality. International Economic Review, 472): , [3] Giulio Fella, Paola Manzini, an Marco Mariotti. Does ivorce law matter? Journal of the European Economic Association, 24): ,

30 [4] Leora Frieberg. Di unilateral ivorce raise ivorce rates? evience from panel ata. American Economic Review, 883): , [5] Liberta Gonzalez an Tarja Viitanen. The effect of ivorce laws on ivorce rates in europe. European Economic Review, 53: , [6] Brishti Guha. Sex ratios, ivorce laws an the marriage market. Paper no. 28, SMU Economics an Statistics Working Paper Series, [7] Niko Matouschek an Imran Rasul. The economics of the marriage contract: Theories an evience. Journal of Law an Economics, 51:59 110, [8] Society at a Glance. OECD Social Inicators, 2005, 2006, [9] Imran Rasul. The impact of ivorce laws on marriage. Unpublishe manuscript, University of Chicago, Grauate School of Business, [10] Imran Rasul. Marriage markets an ivorce laws. Journal of Law an Economics, 221):30 69, [11] Betsey Stevenson an Justin Wolfers. Marriage an ivorce: Changes an their riving forces. Journal of Economic Perspectives, 212):27 52, [12] Justin Wolfers. Di unilateral ivorce laws raise ivorce rates? a reconciliation an new results. American Economic Review, 965): , [13] Bruce Wyick. Granma was right: Why cohabitation unermines relational satisfaction, but is increasing anyway. KYKLOS, 604): , Appenix Recall functions v 0 t) = e t 2 t e t ) an v 1 t) = e t t 3 + 2e t ) from equations 8) an 10), respectively. Functions v 0 t) an v 1 t) are use in establishing the results in Proposition 3, so it is useful to consier their properties. Figure 11 plots these functions. Observe the following: 28

31 Figure 11: Functions v 0 t) an v 1 t) A) Function v 0 1 v t 0 ~ t 1 ~ B) Function v 1 0 v t 1 t ~ t v 0 0) = 1; v 0 t 0 ) = 0, v 0 t) > 0 for t < t 0, an v 0 t) < 0 for t > t 0, t As t, v 0 t) 0. Function v 0 t) has a unique minimum at t 1, where t 1 solves v 0 t) = v 1 t) = 0, an t , v 0 t 1 ) v 1 0) = 1; v 1 t) < 0 for t < t 1, an v 1 t) > 0 for t > t 1. As t, v 1 t) +0.Function v 1 t) has a unique maximum at t , v 1 t 2 )