A good man is hard to find

Journal of Economic Behavior & Organization Vol. 47 (2002) 27–53

A good man is hard to find Marriage as an institution Russell D. Murphy, Jr.∗ Department of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Received 7 July 1999; received in revised form 10 November 2000; accepted 18 November 2000

Abstract This paper presents a model of marriage as an institution that changes the incentives of a mating game between men and women. Unlike other models of the family, decisions to invest in children are not contractible ex ante, but must be sub-game perfect given that intimacy and pregnancy are sequential. Marriage and divorce, which are publicly observable, create costs for exiting a match; informal relationships do not. Providing an institution which makes a match observable, marriage, improves incentives for men to invest costly unobservable effort in their children. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: J12; D19 Keywords: Marriage; Marital dissolution; Family structure; Household Behavior

1. Introduction Even today, ordinary Americans are likely to make a clear distinction between couples who are married and those who are not; 30 years ago, they undoubtedly did. Economists, it seems, do not. The theory of the family largely ignores the distinction, and to the extent that it differentiates between marriages and less formal arrangements, it assumes that marriage provides a contract for child rearing services (Becker, 1991). In fact, the availability of contracts makes marriage less understandable; with perfect contracts, why marry? I argue that the social distinctions between marriages and informal relationships are better understood by recognizing that marriage is not a contract. Marriage is a public declaration of commitment which creates a cost of leaving the match. This exit penalty helps to mitigate a problem presented by children: if a couple is intimate, the woman may bear a child and incur a cost. If so, she prefers that her partner invest in ∗ Tel.: +1-540-231-4537. E-mail address: [email protected] (R.D. Murphy, Jr.).

0167-2681/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 1 ) 0 0 1 6 7 - 6

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raising the child. A man may view intimacy as a benefit and child rearing as a cost; but, he faces the decision to invest only after the couple has been intimate. This sequencing suggests that his investment decision must be sub-game perfect. A public declaration of commitment — marriage — allows prospective partners to discriminate against men less likely to invest in children by “tagging” those whose previous marriages failed. This informational penalty increases the likelihood that choosing to invest in children will be sub-game perfect. This declarative view of marriage accounts well for several stylized facts about marital status and children and provides three main lessons. First, marriage improves incentives for men to invest in their children when investment is non-contractible; good outcomes for children and intact marriages will be positively correlated. Second, marriage is not a guarantee of investment. Some marriages end in divorce because no investment occurs and others last — even when fathers do not invest in their children. And third, a society may want to impose additional, exogenous penalties for divorce and illegitimacy to increase the average investment in children. However, social penalties are crude instruments and their imposition creates a tradeoff — some mothers and children benefit from increased investment by fathers, but others suffer from being “trapped” in bad marriages. In a related paper (Murphy, 1999), I explore some counter-intuitive implications of this policy tradeoff and show that the apparent (or measured) benefit to children of having intact families is inversely related to the “strength” of family values. There are several reasons for trying to make sense of the social distinction between formal and informal matches. Although couples have traditionally showed a strong preference for marriage, behavior has changed dramatically over the past few decades (Table 1). Fewer women marry and those that do have fewer children. More women today are unmarried and they have more children while unmarried than their did predecessors. Marriages that do form are more likely to end in divorce. Evidence suggests that children of intact marriages are more likely to remain in school, more likely to enter the labor force, and less likely to Table 1 Marital behavior (women aged 15–44) Married women Of all 1950 1960 1970 1980 1990 a

0.706 0.712 0.644 0.571 0.546

womena

Unmarried women Birthsb

Birth

3490.4 4033.6 3332.7 2946.5 2992.8

141.0 156.6 121.1 97.0 93.2

ratec

Birthsd

Birth

141.6 224.3 398.7 665.7 1165.4

14.1 21.6 26.4 29.4 43.8

ratee

Divorce Of all 0.040 0.053 0.107 0.184 0.280

birthsf

Rateg 10.3 9.2 14.9 22.6 20.9

Women 15–44; (National Center for Health Statistics, Public Health Service, 1995), Tables 1-76, 1-77. Births in 1000; (National Center for Health Statistics, Public Health Service, 1995), Tables 1-1, 1-76. c Births per 1000 married women 15–44. 1940: (National Center for Health Statistics, Public Health Service, 1995), Tables 1-1, 1-76, 1-77; 1950-1990: (U.S. Dept. of Health and Human Services, 1995). d Births in 1000; (U.S. Dept. of Health and Human Services, 1995), Table I-1. e Births per 1000 unmarried women 15–44; (U.S. Dept. of Health and Human Services, 1995), Table I-2. f Proportion of all births accounted for by unmarried women; (U.S. Dept. of Health and Human Services, 1995), Table I-3. g Per 1000 married women > 15; (National Center for Health Statistics, Public Health Service, 1996), Table 2-1 and (U.S. Bureau of the Census, 1996). b

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experience teen births. 1 A framework with which to evaluate policies which change the relative costs of forming and dissolving different types of matches would be useful; policies such as paternity rights for unmarried men, income tax deductions for parents with children, and expanded child support enforcement are frequently proposed. Finally, one might want to understand why societies distinguish between match forms voluntarily chosen by rational individuals (e.g. why should I care if my neighbor has a child out-of-wedlock?); the distinctions are intriguing from the perspective of individual self interest and rational choice. In this paper, I ignore many of the issues of the existing literature on the family: household division of labor, intra-household bargaining, and labor supply. Instead, I focus on the difficulty of ensuring investment of effort in children when effort is non-contractible. To address this, I borrow two themes from the industrial organization literature. First is that of commitment. I observe that although a man might be willing to promise to help care for his child if his partner becomes pregnant, the promise must be rationally fulfilled after the couple has already been intimate. The man’s expected cost of investing in a child might be outweighed by his expected gain from intimacy, but the gain has already been realized once the child is born. A promise is not credible because he cannot commit to make the investment ex post. The second issue is the boundary between the firm and the market. Why have most couples “organized” so that child rearing services are provided within a marriage, instead of being traded in a market 2 or within alternative family structures? One explanation for the boundary between firm and market is the difficulty or ease of contracting for various goods and services (Williamson, 1985); firms keep internal those relationships over which performance is hard to specify and hard to verify. Here, I argue that a similar difficulty in contracting for investment in children provides an inducement for couples to make use of marriage. 3 Interestingly, even though current popular debate tends to assign viewpoints to either the “right” (families within marriage, difficult divorce, distinguishing legitimate from illegitimate births) or “left” (cohabitation or marriage, convenient divorce, no legitimacy distinction), the extreme cases of no divorce and completely costless divorce both reduce incentives for providing for children. A threat of divorce 4 helps to ensure that marriage provides appropriate incentives. Divorce availability and incentives are intertwined because a choice of the bad action by a man is (partially) revealed when his partner leaves. No divorce and completely costless divorce both remove the threat by which (some) men are induced to take the good action. Furthermore, I show that additional “taxes” on out-of-wedlock child bearing and on divorce may increase children’s welfare (on average), but will make some children strictly worse off. This paper highlights the important sense in which marital status contingent payoffs affect parents’ decisions on multiple margins. 1 This association is reduced, but persists after accounting for factors such as income; see, for instance McLanahan and Sandefur (1994) and Mayer (1997). 2 Becker (1991) discusses this point but primarily concludes that marriage is a contract and largely ignores the distinction between a marriage and an informal relationship. I argue that marriage is not a contract and furthermore, if a contract was available, marriage would be unnecessary. 3 The industrial organization literature, in analyzing commitment problems, primarily considers inefficiencies which arise from transactions which do not occur. I ignore the inefficiency arising from couples not consummating matches. Instead, I focus on the level of aggregate investment in children. 4 No divorce here means no breakup is possible. One might want to distinguish no breakup from no remarriage (separation possible, but remarriage is not).

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The paper is organized as follows: Section 2 briefly reviews related literature; Section 3 presents a model of divorce (without “marriage”). Section 4 models marriage by allowing an ex ante choice of whether matches that end, end in divorce. That is, women choose whether a match will be formal (prior match status is observable in the event of a breakup) or informal (prior match status is unobservable). Section 5 considers the effects of penalizing divorce and out-of-wedlock child bearing with social sanctions, or “taxes;” Section 6 concludes.

2. Literature Previous work in this area falls into two groups; several papers suggest theoretical explanations for changing marital status and fertility patterns and a large empirical literature which searches for relationships between dependent variables such as marital status, fertility, and children’s outcomes and potential explanatory factors such as demographic characteristics and transfer payments. Neither set of papers provides a rational agent-based explanation for distinguishing marriages from informal relationships. I build upon an observation of Becker (1991) who suggests a contracting explanation: Since married women have been specialized to childbearing and other domestic activities, they have demanded long-term “contracts” from their husbands to protect them from abandonment and other adversities. Virtually all societies have developed long-term protection for married women; one can even say that “marriage” is defined by a long-term commitment between a man and a woman. 5 However, this question of contracting and commitment is not fully developed. The remainder of Becker (1991) makes little distinction between households and marriages. The existence of and compliance with appropriate contracts is presumed. 6 Missing from the models is parental or inter-temporal conflict. 7 Also missing is a choice between marital and out-of-wedlock child-bearing. But the distinction between marrying and not marrying has been very important socially. And, if contracts are available, what role does marriage play? Marriages are public declarations; most involve almost no specified performance requirements. 8 Instead, participants know a good marriage when they experience it. And, if they do not experience one, they often leave. Marriage involves something hard to identify ex ante, hard to specify, and hard to verify. Another assumption in Becker’s models is that parents are altruistic. However, an assumption that parents jointly maximize family utility at some initial date and then carry out 5

Becker (1991, pp. 30–31). See Becker (1991, p. 137) on families maximizing a utility function including expenditures per child, pp. 155–156 on parental utility as a function of children’s utility, and p. 179 on children’s income as a function of parents’ (plural) income. 7 By inter-temporal conflict, I mean the potential time-consistency problem faced by a man who might be willing to invest ex ante in an expected value sense, but who, ex post, would prefer to avoid investing: a conflict between his ex ante and ex post selves. 8 Becker reports that historically, several cultures — including Jewish, Islamic, and late Roman republic — have had explicit marriage contracts with clearly specified penalties for breach. But, to be parochial, these examples are notable, in part, because they are dissimilar to our current culture. 6

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planned actions as time unfolds 9 is not entirely appropriate here. For instance, child support awards are often quite difficult to set and enforce. Although altruism is frequently assumed in the economics of the family (and elsewhere), its empirical grounding seems limited (see Altonji et al. (1992) and Behrman et al. (1995)). Relaxing the altruism assumption is appropriate and perhaps necessary to understand many of the incentives surrounding marriage. Willis (1994) models a world in which men and women choose between various strategies to maximize their reproductive success. Men may choose to pursue serial partners in an environment in which women are relatively both numerous and wealthy. Unfortunately, Willis’s distinction between households and marriages is not entirely compelling. Men and women either form monogamous households (marriage, to Willis) or they sleep around. According to Akerlof et al. (1996) the increase in the illegitimacy ratio 10 in the US is the consequence of a decrease in “shotgun weddings;” this, in turn, results from the wider availability of abortion after Roe v. Wade. They use two models to make this point. The first considers the incentives for women to solicit a promise of marriage before agreeing to be sexually intimate. When abortion is unavailable, an equilibrium may prevail in which women solicit and men make promises of marriage. When abortion is available, women with high costs of single motherhood no longer need to seek a promise of marriage, since at a low cost they can obtain an abortion if they become pregnant. Men may no longer be willing to promise to marry since they can more easily find another woman who will be intimate without requiring a promise. In the second model, men feel guilty about burdening their partner with an out-of-wedlock birth and have to decide on marriage if the woman chooses to not have an abortion. Akerlof et al. (1996) provide an interesting exploration of some effects of introducing a technological change such as abortion, but their story is incomplete as an explanation of marriage and marital behavior. First it is not clear what role marriage plays. By assumption, women receive positive payoffs after marrying and men negative ones. Presumably, these arise from transfers made from men to women after marriage. But the first model assumes that promises to marry are binding yet implies that marriage is needed to elicit the transfers. If promises are binding, then why marry? What does marriage do that other promises would not do? In the second model, the assumption of male guilt is convenient, necessary, and hard to justify. Also troubling is the timing of the marriage decision. In both models, the decision to marry occurs only after pregnancy, in keeping with the focus on shotgun marriages. However, the notion that marriage occurs primarily after conception is empirically questionable. Even in the late 1980s, 59% of women aged 15–34 were married at the time their first birth was conceived. 11 In the early 1960s, the figure was 72%. The bulk of marriages, both

9 See also the literature on intra-household resource allocation which considers the possibility of conflicting interests within a household (Manser and Brown, 1980; Browning et al., 1994). 10 The proportion of all births accounted for by unmarried women. 11 More precisely, birth occurred at least 8 months after marriage. See Bachu (1991), Tables E and G; these figures were calculated by taking the reported post-maritally conceived births (which are actually conceived post-first marriage) and subtracting births occurring between first and second marriages. This is a small adjustment (a high of 2.4% for 1980–1984 and a low of 1.0% for 1960–1964) and is conservative since some of those births may have been conceived during the first marriage.

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before and after Roe v. Wade, occurred before conception. 12 In short, the decline in shotgun marriages seems likely to have contributed to the increase in the illegitimacy ratio, but the Akerlof et al. (1996) models, as explanations of the margin between marital and non-marital child-bearing, seem incomplete. The empirical literature is fairly large, takes the existence of marriage as given, and focuses on correlates of marital behavior across time, space, income, and race. A popular explanation for the growth of illegitimacy (among blacks) is offered by Wilson (1987). He suggests that significant declines in the number of eligible (employed) young men relative to the number of young women have made marriage unattractive or infeasible for women. However, as Wilson acknowledges, declines in the numbers of eligible black men are not similarly observed among white men, yet whites have had proportionately greater increases in illegitimacy. Less aggregated evidence also seems to argue against this “marriageable-men index” model (Wood, 1995). So, what characteristics would be desirable in a model of marriage? At the most basic level, we want a model in which men and women choose a match type. In equilibrium, some choose to marry while others do not, some who do not marry have children while others do not, and some marriages survive while others do not. The payoffs that induce men and women to choose marriage should arise endogenously. The model should account for the stylized facts and contradictory incentives surrounding marriage. Marriage has traditionally been preferred, but the preference is diminishing. Children, on average, are better off in married couple families, yet some benefit from divorce. This was one of the justifications for liberalizing divorce laws: if divorce was less costly, fewer children (and adults) would be “trapped” in bad marriages. Women may also face a tension between their own interests and those of their children. A good match for a woman (in emotional, physical or financial terms) may involve a man who neglects their children; alternatively, a good father may not be the most compelling companion. Finally, the model should incorporate some sense of the “declarative” nature of marriage: even a simple civil ceremony requires a public avowal of one’s current and future interest in a partner. Yet, with a few cultural exceptions, marriage ceremonies involve few specifications of performance or of penalties for breach.

3. Divorce Imagine a couple that has been dating. They know how well they get along and must decide whether to further pursue their relationship (i.e. be sexually intimate). If they do, they realize some benefit beyond what they receive from dating, but the woman may also become pregnant and have a child. She would find the child costly to raise, but would do so nonetheless. She would like her partner to help. Unfortunately, while some men are “loving” 12 These figures condition on having children and some marriages do not involve children, so they understate the extent to which marriages occur outside of the (Akerlof et al., 1996) shotgun marriage regime. Since 12.2% of ever-married women aged 35–39 in 1990 had had no children, this is not a trivial understatement; a significant fraction of women marry but reach age 35 without having a child. Some of these women may be in (or were in) marriages which were intentionally childless; neither this paper nor (Akerlof et al., 1996) have much to say about their marriages. Others, however, were unintentionally childless and their marriages are inexplicable in the (Akerlof et al., 1996) models.

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Table 2 Notation: primitives m θi wj p µ β

Match specific payoff; m has a continuous pdf f (m) Cost of investing effort in a child for a parent of sex i Mother’s payoff from her child’s welfare given that she stays in the match and the father chooses (invest effort or not) j ∈ {E, N } Probability of pregnancy Ex ante probability that a man likes children (is an L-type) Discount factor

fathers, others are “opportunistic.” The woman in our representative couple does not know, before the birth, her partner’s type. This is the general problem that I analyze below. This is the point at which the marriage decision will be considered: when the man and woman are relatively familiar with each other, but before the woman is pregnant. 13 For analytical convenience, I assume that prior to this acceptance decision, the couple has not been intimate (or has practiced perfectly effective contraception). If the match is accepted, the couple will be intimate (or contraception will no longer be perfect). The model economy has two periods and many agents; each agent is either male or female. At the beginning of each period, men and women without partners (in period 1, all of them) are matched in couples. Each individual observes a match-specific utility payoff m (personal compatibility, mutual physical attraction, etc.) which is the same for both partners, but is unobservable outside of the match. The match payoff m is drawn from a continuous probability density function f (m) > 0 and is independent of the payoffs that partners would receive in other matches. If the match is accepted by both parties, each receives m; children entail additional payoffs. With probability 0 < p < 1 a woman becomes pregnant in an accepted match and gives birth to a child. Children’s outcomes, which are modeled only indirectly here, are dependent on the investment of effort by parents. Effort is observable within the match, but not outside of it. Women always choose to invest effort and incur the cost θW > 0, while men may choose to invest effort or not. Not investing effort is costless. There are two types of men; some men find it easy to invest in children while others find it costly. With probability 0 < µ < 1, a man likes children and is designated an L-type man. Men who like children are no better or worse than men who do not, but they enjoy investing L < 0. 14 With probability 1 − µ , a man effort in their children; they face an effort cost θM does not “like” children. His investment in children will be opportunistic; designate such a man as an O-type. These men do not necessarily dislike children; the difference between them and their counterparts is that they find investing effort costly and they only do so if O > 0 (see the benefits outweigh the costs. They (the O-type men) face an effort cost of θM Table 2 for notation). Types are unobservable. For a woman, a match has two important dimensions: its match quality, m, and the cost k , k ∈ {L, O}. The two characteristics are independent; a match type of her partner, θM 13

This is unlike Akerlof et al. (1996) in which marriage only occurs after pregnancy. Some L-type men might actually dislike children, but because of religious, moral, or social concerns, would always invest effort in their children. In that case, I take their fixed decisions as preferences, and classify them as “liking” children. 14

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characterized by a high m may be with either a man who likes children (an L-type) or with one who does not (an O-type). After pregnancy status is realized and effort choices have been made, men and women can choose to break off their matches before the man’s investment outcome is realized. That is, the payoff from the investment is realized over time, but effort choices are observed quickly (within the household). Women always receive custody of children. In period 1 there are populations of size 1 of men, women, and matches (although not all matches will be consummated). Women care about their children’s welfare, which is increasing in their fathers’ effort. The additional payoff to a mother whose partner chooses to invest effort is wE > 0 and her additional payoff when he invests no effort is wN < 0. Matches that break up, do so before wN or wE are realized; a mother’s payoff from the child’s welfare is then normalized to 0. At the end of the period, children are grown, leave their homes, and join the general population. No one dies. If men and women remain with their current partners for the next period, their match payoff will again be m; if they leave their partners (or their partners leave them), they get a new match and a new draw of m. Individuals discount second period payoffs by β ∈ (0, 1). The sequence of decisions and payoffs within one period is shown in Fig. 1. The extensive form of the one period game shown in Fig. 1 begins with (relatively uninteresting) moves: by Nature (N — choosing the match payoff m), the man and the woman (M, W — each choosing to accept or reject), and Nature again (choosing the pregnancy outcome). Then, if the woman becomes pregnant, the man chooses either to invest effort or not. After his effort choice, both he and his partner choose to stay in this match or return to the pool. The game, as drawn in Fig. 1, is quite similar to alternatives with slight variations in setup. As shown, the man makes his effort choice, then chooses to stay or go, and finally, his partner responds by also choosing to stay or go. I can show that, for likely parameter

Fig. 1. Decisions and one period payoffs (woman, man [of type k ∈ {L, O}]).

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values, the analysis would be unaffected by switching the man’s and woman’s “stay/go” decisions (i.e. having the woman choose first). The intuition for this is that the critical decision for the man is his effort choice; the critical decision for the woman is to stay or go. His choice depends on his cost of effort and the match quality. Her decision depends on his effort choice and the match quality. His decision to stay or go is of little consequence and is included largely for symmetry. Letting him decide simultaneously with his effort choice or after observing his partner’s decision has little bearing on the major decisions but simplifies the analysis. Alternatively, one might prefer to prune the game tree by removing the man’s “invest effort/go” path since it is never chosen. I leave it in however, since his action space includes {effort, no effort} × {stay, go}. It seems more appropriate to prune this path endogenously rather than by fiat. Neither the presence nor absence of this choice, however, affects the analysis since he has incurred the effort cost prior to his stay/go decision. Men and women newly matched in period 2 might, in period 1, have been: • children (and are now adults) • in matches that failed • in rejected matches (never consummated) Those that were in period 1 matches that failed are identified as divorced; all others available for new matches in period 2 are single. There is no choice of marriage (yet). All men are single at the start of period 1, while in period 2 proportion α are. The proportion of L-type (likes children) single men in period 2 is µS and the proportion of L-type divorced men in period 2 is µD . These characteristics, α, µS , and µD , of the distribution of men available in period 2 are determined endogenously. Note that the analyses presented here are not for all possible parameter values. Instead, I focus on plausible and interesting cases. 3.1. Equilibrium overview Strategies for women consist of, for each period, rules governing rejection, staying when pregnant, and staying when not pregnant. The rejection rule consists of a simple cutoff Rt for each period such that she rejects matches for which m < Rt . The stay when pregnant rule will be contingent on the match quality m and the man’s effort choice (which she observes). The stay when not pregnant rule will be contingent only on m. The optimal stay when pregnant rule can be described by (m, m) where for m < m, she never stays (even if he invests effort) and for m ≥ m, she always stays (even if he invests no effort). For m ∈ [m, m) she stays contingent on her partner’s choice of investing effort or not. To stay when she is not pregnant, she requires m ≥ N . Her optimal strategy will then be described by {RW,t , mt , mt , Nt } for t ∈ {1, 2}. For women, I focus attention on the first period cutoff rules m1 and m1 , and also on RW,1 (Table 3 recaps the notation for derived expressions). Strategies for men consist of, for each period, rules for rejection, investing effort, and staying when their partners are not pregnant. The rejection rule will be a cutoff rule, contingent on type. The rules for invest effort and stay when not pregnant will also be cutoffs, describing (again, contingent on type) the minimum levels of m required to induce a man first, to invest effort and stay (when she is pregnant) and second, to stay (when she is not

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Table 3 Notation: endogenous expressions Ri,t m m Ni,t Ej α µ

The smallest value of m such that an individual of sex i prefers accepting m = R rather than waiting one period for a new match The value of m such that a woman prefers to seek a new match after a period in which she got pregnant and her mate chose to invest effort The level of m high enough so that a woman will never leave The value of m such that an individual of sex i in period t would prefer to seek a new match after the woman did not get pregnant The level of m required such that a man of type j prefers to invest effort and stay Probability of being matched, in period 2, with a single man Probability that a man is a L-type, conditional on his being available in period 2

pregnant). His optimal strategy will be described by {RM,t , Et , Nt } for t ∈ {1, 2}. For men, I concentrate on the invest effort rule E and particularly on the O-type’s effort rule E O . Assume that RM,t < RW,t , 15 or that the match acceptance decision is more important for women. Conversely, the effort decision (E) is important only for men (by assumption, all women invest effort). My focus is on Ri,2 , m1 , E1O , and m1 , so I will ordinarily drop the time subscript when referring to decision rules. 3.2. Decision rules The game ends after period 2, so the optimal decisions in that period are straightforward; women, moving last, stay in a match only if their partner invests effort. Men receive the match payoff m before choosing to invest effort or not; therefore only L-type men invest effort. Payoffs for women from accepted matches with L-type men are m − p(θW − wE ). For women matched with O-type men, the payoffs are m − pθW . Faced with a new potential partner of match quality m who is an L-type with probability µ, a woman would have an expected payoff of m − pθW + pµwE . The worst match quality, then, that a woman would be willing to accept in period 2 is R(µ): R(µ) ≡ pθW − pµwE

(1)

Men in period 2 are willing to accept any match such that m ≥ 0. Assume that θW − wE > 0 (children are costly, even with a father who invests effort); the woman’s rejection rule determines which matches are accepted. Then, for a woman in period 1, the expected value of a new match with a man who is an L-type with probability µ is:
∞ v(µ) = (m − pθW + pµwE ) dF (m) (2) R(µ)

And, if the probability of being matched with a single man is α, the expected value (to a woman) of a new match in period 2 is αv(µS ) + (1 − α)v(µD ). 15 For period 2, this is a conclusion, not an assumption; for period 1, it is true if the reputation cost of divorce to men is less than the cost of children to women, which is plausible — and likely in the context developed here.

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There are three potential behavior rules for women who have accepted period 1 matches: always remain with their partners, never remain with their partners, and conditionally remain with their partners. Always remaining with a partner is only optimal if the intrinsic match quality is sufficient to offset the cost she incurs from his neglect of their child. She always stays if m ≥ m: m = αv(µS ) + (1 − α)v(µD ) −

wN + R(0.0) β

(3)

When match quality reaches m, a woman’s divorce threat is no longer credible. Since the cost to her of his low effort invested in their child is wN < 0, m increases when low effort becomes more costly to tolerate. If her partner invests effort but m is low, then she is willing to remain only if m is not less than m (i.e. she divorces even a known good father if m < m) where: m = αv(µS ) + (1 − α)v(µD ) −

wE + R(1.0) β

(4)

Since wE > 0, m decreases (she stays in lower quality matches) as a man’s effort investment in her child is more valuable. The term R(1.0) reflects the fact that, at the margin, she leaves a known loving father. Since wN < 0 < wE , m < m. For m ∈ [m, m) a mother is willing to stay with her partner if and only if he chooses to invest effort in their child. But will he? A loving father always chooses to invest effort. An opportunistic father, however, only does so to avoid losing his current match. That is, there is no reason to invest if his partner will never leave (m ≥ m), nor is there a reason to invest if his partner will always leave (m < m). If his partner’s threat to leave is credible (m ≤ m < m), then a O-type man will invest effort if m ≥ E O :
∞ θO O (5) m dF (m) + M E = β R(µD ) There are additional decision rules for behavior within accepted matches when the woman does not become pregnant; 16 the not pregnant rules are not particularly interesting, so I largely ignore them.

16 If the woman is not pregnant then individuals prefer to remain in their current match if m ≥ N where the optimal N varies for women, O-type men, and L-type men:

NW = αv(µS )+(1−α)v(µD )+R(µ ) O = NM

L = NM




∞

R(µD )




∞

R(µD )

m dF (m)

L m dF (m)+F (R(µD ))pθM

Assume that the woman’s cutoff when not pregnant is higher than the men’s cutoffs; N = NW .

(6) (7)

(8)

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Fig. 2. Best response regions.

3.3. Characteristics of equilibrium Given the decision rules, I can characterize couples on the basis of the match qualities (m) drawn from the distribution f (m) (with cumulative distribution F (m)). Fig. 2 shows (qualitatively) the various cutoffs and the choices made in equilibrium. Knowing these equilibrium choices, I can determine the salient characteristics of the second period distribution of men: the probability that a divorced man is an L-type (µD ), the probability that a single man is an L-type (µS ), and the probability that a woman is matched with a single man (α). The unconditional probability that a man likes children is µ . Define the following population groups of men available in the period 2 matching market: • • • •

matches rejected in period 1: φR = F (pθW ) children in period 1 (new): φN = p[1 − F (pθW )]/2 matches that always break up: φA = [1 − p][F (N ) − F (pθW )] + p[F (m) − F (pθW )] matches in which men invested no (zero) effort: φ0 = p(1 − µ )[F (E O ) − F (m)]

The divorced group is φD = φA + φ0 while the single group is φS = φR + φN . The only divorced men who like children (L-types) are those in the group whose matches always break up (i.e. in φA ). Therefore, a divorced man is less likely to be an L-type than is a single man (the φ0 group has no L-types): µD =

µ φ A < µ φA + φ 0

(9)

However, knowing that a man is single adds nothing to our priors about his cost type: µS =

µ (φR + φN ) = µ φR + φ N

(10)

The probability that a woman is matched with a single man in period 2 is α = φS /(φS +φD ). Assume that m ≤ E O < m (i.e. the man’s investment decision is not trivial). Then 0 < α < 1. Marital status allows women to identify and hence discriminate against men who are less likely to be L-types. Therefore, men who choose not to invest in their children face a cost.

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Theorem 1. A divorce tag creates an endogenous penalty for men whose partners divorce them. (All proofs can be found in Appendix A) Theorem 1 is simply a consequence of women, at the margin, choosing to avoid men who are less likely to invest in their children. 3.4. Implications Of the matches in period 1: • some are rejected • some that are accepted break up because they are less attractive than a new match in period 2 • some break up because the woman becomes pregnant and the man does not invest effort • some form and remain intact because the man invests effort • some involve payoffs to the woman that are high enough that she remains in the match even though her partner invests no effort The choices that agents make determine the distribution of men in period 2. The characteristics of the distribution of interest are α, µD , and µS . Because µD < µS , women rationally discriminate against divorced men in period 2 (i.e. R(µD ) > R(µS )). This discrimination reduces the value of divorcing and looking for a new match; this induces more opportunistic men to invest effort in their children. More of them invest effort than would if they were not identified as divorced in period 2 (i.e. if unattached men could not be distinguished from each other). Theorem 2. A divorce tag increases m, increases m, and reduces E O . Relative to a world in which period 1 match status is unobservable in period 2, the divorced/single distinction in period 2 increases both m (women are choosier about potentially permanent partners) and m (women’s divorce threats are more credible); the distinction also reduces E O (opportunistic men are more likely to invest effort). Both the reduction in E O and the increase in m raise the number of children whose fathers invest effort within the set of matches that always remain intact (m ≥ E O ). At one margin, more men are willing to invest effort because their future prospects are worse in the event of a breakup (E O ↓); at the other, more women are able to credibly threaten divorce because their prospects are better in the event of divorce (m ↑). Women’s prospects in period 2 improve because they can better distinguish O-type and L-type men. However, the effects of observability (divorce) are not unambiguously positive. Fraction µ of the matches between the smaller (divorce unobservable) and larger (divorce observable) values of the marriage threshold m would have involved fathers investing effort and mothers choosing to stay. With divorce, because these mothers’ future prospects are better, they leave even when their partners invest effort. Theorem 3. Relative to the pool of available period 2 men in an environment in which marital status is unobservable, the pool of divorced men is worse than average, and the pool of single men is better than average.

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Table 4 Changes in children’s welfare (period 1) Change in welfare

Range of children affected

Proportion affected

Effect

− + +

mN < m < mD EDO < m < ENO mN < m < mD

µ 1 − µ 1 − µ

Formerly invest effort and stay, now go Formerly no effort, now invest effort and stay Formerly no effort and stay, now invest effort

When divorce is observable, the available men in period 2 are split, largely on the basis of their investment decisions, into two pools. Let xD and xN indicate agents’ decision rules in the divorce and no divorce regimes, respectively. Children’s welfare (in period 1) improves in two ranges and worsens in another (see Table 4). Define (in an abuse of notation) F (x) as the fraction of matches affected by decision rule x’s shift due to the change in regime: F (x) ≡ F (xD ) − F (xN )

(11)

Theorem 4. The average welfare of children is higher when divorce is observable if: µ F (m) − F (E O ) < 1 − µ F (m)

(12)

The inequality comes directly from Table 4: who benefits from the regime change and who does not? This condition is more likely to be true when either the proportion of loving fathers in the population (µ ) is small or when F (m) is small. The blunt sword of reputation (the cost of divorce) is less useful in a society of saints than in a more selfish land. What have we learned? If men face this commitment problem, they choose to invest costly effort for two reasons. Some (those between E O and m) invest to avoid losing a good match. Others (those between EDO and ENO ), when faced with an exit penalty, invest to avoid the penalty. Observability, the public identification of those whose matches failed, generates such a penalty. It seems natural to suppose that marriage and divorce provide this sort of public identification of match status. Marrying typically involves a public declaration such as: I, N, take you, N, for my lawful wedded wife; to have and to hold, from this day forward, for better, for worse, for richer, for poorer, until death do us part. Yet under few circumstances does a marriage involve more detailed specification of “do”s and “don’t”s. Marriages are unlikely to specify penalties for breach. Public observability, and not a contract, is likely to be the key element that distinguishes a marriage from an informal match. This observability generates a penalty for men trying to exit a match and therefore improves incentives for them to invest effort in their children. However, even if we conclude that marriage offers observability and hence improves incentives, not everyone marries. And, match acceptance (sleeping together) is ordinarily

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unobservable; observable match breakups are clearly not de rigueur. To the extent that observability is normative, it has to arise from individuals choosing to marry. How, then, should we think about this choice?

4. Marriage Consider a small variation of the previous section’s model in which after a match is made and m is drawn (and observed), the woman has a new first move: she chooses whether the match will be observable or unobservable (a formal or an informal match). If she chooses a formal match (marriage) then splits are observable in the sense of Section 3; otherwise they are unobservable. Men whose matches fail in period 1 are identified in period 2 as divorced if they were in formal matches and as single otherwise (women are identified also, but for now this does not matter). This is a natural conception of marriage since many of the characteristics distinguishing marriages from informal matches are social. Introducing even this small degree of additional complexity makes solving for the full set of explicit strategy rules cumbersome. Instead of solving for the explicit equilibrium strategies, I characterize the qualitative effects of this choice on the key decision rules. Assume that a woman chooses marriage if she strictly prefers it; if indifferent, she chooses marriage unless all men (both O and L-types) would strictly prefer no marriage. She assumes that her choice has no effect on the probability that a given (period 2) unattached man was in a period 1 marriage. If a woman chooses marriage, then we know that, for that match, E O is lower. She strictly prefers marriage over the range between the two values of E O (observable divorce versus no observable divorce): m ∈ [EDO , ENO ). Otherwise, she is indifferent; by assumption she chooses marriage when indifferent except when m < m (then every man strictly prefers an informal union because he would be identified as divorced in period 2 after the match breaks up — which it will with certainty). There are no costs to marrying. By assumption, she believes that her choice has no effect on the probability that an unattached man in period 2 is divorced, so she does not believe that her choice will affect m (the credible divorce threat point). Nonetheless, in equilibrium, if, with positive probability, women choose marriage and men are divorced, m will be higher than it would be without marriage (that is if status were unobservable) since the expected value of a second period match for women will be larger. Single men in period 2 now include those who were in period 1 informal relationships, new adults, and those whose (period 1) matches were rejected: φS = [F (m) − F (pθW )] + 21 p[1 − F (pθW )] + F (pθW )

(13)

There are two groups of divorced men. Some had marriages which failed because they did not invest effort: φD,0 = p(1 − µ )[F (E O ) − F (m)]

(14)

Others had marriages which failed when their wives did not have a child: φD,NC = (1 − p)[F (N) − F (m)] Then φD = φD,0 + φD,NC .

(15)

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There are three groups of married men. Those with very high match qualities are in marriages that never fail: φM,V = 1 − F (m)

(16)

An “untested” group had match qualities high enough to succeed (and low enough to fail) but the men were not forced to make an investment decision because their wives did not have children: φM,U = (1 − p)[F (E O ) − F (N)]

(17)

The remainder all chose (or would choose) to invest effort: φM,E = pµ [F (E O ) − F (m)] + F (m) − F (E O )

(18)

Then φM = φM,V + φM,U + φM,E . A man in the divorced group is a loving father with probability: µD =

µ φD,NC < µ φD,0 + φD,NC

(19)

Single men are no worse than the unconditional average: µS =

µ φ S = µ φS

(20)

And, as before, both single and divorced men are present in the pool of period 2 available men: φS α= (21) φS + φ D Again, since f (m) > 0, 0 < α < 1. In equilibrium, there are rejected matches, informal matches (and children born out of wedlock), and marriages. Some marriages survive and some end in divorce. Marriage does not guarantee investment of effort in children, but does increase the likelihood of it. Investment in children (in period 1) takes place as shown in Table 5. Consider also the likelihood of investment in children conditional on observed match status: pr(wE |parents married) =

φM,E + µ φM,V >0 φM,E + φM,V

(22)

pr(wE |parents divorced) = 0

(23)

pr(wE |parents unmarried) = 0

(24)

This simple variation on the basic model suggests that when observability is limited (and child rearing is non-contractible), a choice of marriage helps induce more investment in children. Nonetheless, not all women marry and not all marriages survive. Even in intact marriages, children are not guaranteed a good outcome, since when m ≥ m, some men invest no effort but their wives do not divorce them.

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Table 5 Fathers’ choices and children’s outcomes Observed match form

Unobserved match quality

Effort outcome

Proportion of men

Proportion of children

Informal

m ∈ [pθW , m)

E 0 N

µ – 1 − µ

0 1 0

Divorce

m ∈ [m, E O )

E 0 N

0 – 1

0 1 0

Marriage

m ∈ [m, m)

E 0 N E 0 N

1 – 0 µ – 1 − µ

1 0 0 µ 0 1 − µ

m ∈ [m, ∞)

A notable feature of social life in America, at least through the recent past, was that not only did women (and men) face a decision of whether or not to marry, but they also faced large social penalties for the “wrong” decisions. These penalties are exogenous to the model, but a simple extension shows how they can have both positive and negative effects on children’s welfare.

5. Making divorce difficult The model suggests that children fortunate enough to be born in surviving marriages receive, on average, higher investment than do children whose parents never marry or marry and then divorce. Noting this, a society might attempt to improve the lot of its children not only by developing an institution of marriage, but also by “strengthening” it. By penalizing divorce and illegitimacy, a society might try to reduce the incidence of “broken” homes and so increase the average welfare of children. 5.1. Exogenous stigma Suppose that society has some means (e.g. social penalties or taxes, broadly defined) of penalizing individuals. 17 By assumption, only limited information is available on which to base the penalties. Effort devoted to raising children is unobserved, but marriage, divorce, and child birth are observable (as is whether or not a woman is married at the time of birth). Assume that women who bear children in informal unions face a “tax” of I : births out-of-wedlock are considered illegitimate (only mothers face the tax because paternity is 17 The discussion that follows avoids important questions of how social penalties might be imposed, how a community ensures compliance in imposing the penalties (Kandori, 1992), and what form government or community imposed penalties might take. I also ignore the problem of what the optimal levels of such penalties would be.

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unobserved); men and women who divorce (end a formal marriage) face a “tax” of D. Note that both partners face equivalent divorce taxes in keeping with the notion that the actions within the marriage are unobservable outside of it; “blame”, then, is unassignable. The divorce tax shifts m down; divorce threats are less credible: D wN + R(0.0) − (25) m = αv(µS ) + (1 − α)v(µD ) − β β The O-type man’s effort cutoff also shifts down:
∞ θO D m dF (m) + M − EO = D β β R(µ )

(26)

Men are more likely to invest effort at one margin (E O ↓), but less likely at the other (m ↓). The penalties I and D might be set such that women and L-type men choose to remain single rather than to accept matches and then divorce. Illegitimacy could be sufficiently penalized that informal unions are unacceptable to women. Women would have a lower bound on the matches they will accept (RI ); when m is at least this large, they propose marriage. If RI ≥ E O , no marriages end in divorce (those that would, the ones to O-type men when m is not high “enough”, never form). Note that if the penalties are imposed to benefit children (for instance, to increase the proportion of children whose fathers invest effort) then I and D are not independent policy instruments. Suppose they were and a benevolent dictator decided to impose very large illegitimacy penalties. Now women would have a strict preference for formal matches. But, as long as RM,1 < RW,1 (the man’s divorce penalty is not too large relative to the woman’s), all of what were previously informal matches still form. Because I affects only her incentives regarding the form of the match and not those affecting its creation or destruction, her other decision cutoffs are unaffected (his are unaffected as well). Her concern when evaluating the informal versus formal threshold m is with the value of her partner’s effort investment, wE , (and the chances of finding another man who might invest effort next period) and with m (relative to its distribution). Similar concerns drive the determination of her credible divorce threshold m and the O-type man’s effort threshold E O . Matches that would have resulted in illegitimate births still form; now they are marriages, but marriages that immediately dissolve. Children are unaffected. They would receive effective investments of 0 (in the range m < m) with no illegitimacy penalty; they do with a large I as well. If one were trying to increase investment in children, an illegitimacy penalty by itself would be inappropriate. The effects of a divorce penalty, D, are slightly different from those of I . Changes in D do change the decision cutoffs. Suppose I = 0. For women, the prospect of a divorce penalty of D reduces her lower and upper cutoffs (m, m) by (D/β). If an O-type man also faces the penalty D (in addition to his endogenous penalty), then his effort cutoff (E O ) also shifts down by (D/β). But, in addition to changing the cutoffs m, m, and E O , the prospect of D also changes the woman’s incentives to choose a formal versus an informal match. As D gets “large”, m, E O , and m become “small”. However, for D large enough, women will refuse to marry unless they are sure that they will not face divorce: only if m ≥ E O . Then, all matches for which pθW ≤ m < E O will be informal. If m is large enough, some informal matches will survive when men invest effort. In this case (D large) not only does the tax fail to ameliorate the single-parent household problem, it worsens the situation for

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children in some married-couple households (if women always marry when m ≥ E O ). As D increases, m falls and more women are “stuck” in matches in which their opportunistic partners invest no effort now, but would have invested effort if m was larger. Instead, a benevolent dictator will have to adjust I and D simultaneously. To reduce the man’s effort threshold E O , the dictator increases the divorce penalty D-which induces effort from more O-type men. He raises the rejection threshold for informal matches by increasing the illegitimacy penalty I and for low quality formal matches by increasing D. Both rejection thresholds (formal and informal) would have to be at least as large as E O for the society to have no illegitimacy and no divorce. Increasing these penalties involves a tradeoff because the penalties lower the divorce threat point m and so reduce investment in (some) children when match quality is no longer below the credible divorce cutoff (the new m). 5.2. Children’s welfare? Imposing these penalties may or may not increase the average welfare of children. Define aggregate (children’s) welfare, γ , as the fraction of children who benefit from fathers’ investment of effort (i.e. their parents remain together and their fathers invest effort). This includes fraction µ of matches m ≥ m, all of matches m ∈ [E O , m), and fraction µ of matches m ∈ [m, E O ). γ =

µ [1 − F (m)] + [F (m) − F (E O )] + µ [F (E O ) − F (m)] 1 − F (pθW )

(27)

Under plausible conditions, increasing the penalties men and women face for divorce and illegitimacy leads to higher welfare for children. Theorem 5. Children’s welfare is increasing in greater divorce and illegitimacy penalties if f (m) is Gaussian, m > E O , and f (m) < f (E O ). These are sufficient conditions for greater penalties to increase aggregate children’s welfare. However, if f (m) is, for instance uniform, increasing penalties reduces average welfare γ . Then, the number of matches between E O and m is fixed; increasing D simply reduces both cutoffs which increases the number of matches in the region above m. To increase welfare (γ ) when m has a uniform distribution, one would want to increase m as much as possible (make divorce as easy as possible). Even  ∞if f (m) is normal, but the divorce threat point m is already very low (e.g. E O < m < −∞ m dF (m)), then aggregate welfare need not be increasing in D. In this case, increasing D (and reducing m) might increase the fraction of women “stuck” in matches (m ≥ m) enough to reduce average children’s welfare γ . A society might want to impose exogenous penalties for divorce and illegitimacy to increase the average welfare of children. However, the penalties are not independent; increasing children’s welfare requires simultaneously setting a divorce and an illegitimacy penalty. Furthermore, there is a tradeoff between the increase in effort enjoyed by some women and children and the reduction in welfare suffered by others who can no longer credibly threaten divorce.

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6. Conclusions The social institution of marriage has a plausibly rational basis; society should differentiate between couples in informal relationships and those marrying formally. The model presented here shows that differentiating between formal and informal matches improves the incentives for men to invest unobservable effort in their children. The benefits arise from publicly identifying men who are likely to have high costs of investing and hence are less likely to do so. The incentives flow from a credible divorce threat by women; without a credible threat (by women) to leave or a penalty conditional on leaving (for men), men have no incentive to invest beyond their direct benefit or cost. Clearly, the model is quite stylized: • • • • • • • •

it has only two periods, individuals can have only one partner at a time, women always keep children and choose high effort, neither contraception nor abortion are available and couples cannot choose to remain childless, there are only two effort choices: invest or not. But, despite its simplifications, this model of marital behavior has several desirable features: the model deals explicitly with the time consistency characteristics of the match acceptance decision for men and women, marriage, divorce, and illegitimacy arise in equilibrium from model primitives and agent maximizing behavior, a formal match, a marriage, provides incentives for parents to invest in children, even though not all marriages generate high investment; some women have no credible divorce threat and hence they (and their children) are “stuck” in marriages that are, at least in one respect, “bad”.

The model provides both policy and methodological lessons. Many government programs, such as those that would grant paternity rights to unmarried fathers, expand child support enforcement for unmarried mothers, or streamline the divorce process change the relative costs of choosing different match types. If the role of marriage is, in part, to induce the investment of unobservable effort in children, these policies may have unintended consequences that are not easily compensated for. Marriage has an advantage over a centrally administered program in that it improves the likelihood that the parents of children provide effort, in a decentralized manner; it uses parents (only mothers here) to monitor their partners (fathers here). The model also has implications for empirical work on children and families. Regressing child outcomes on parents’ marital status and other explanatory variables may provide misleading results. The coefficient estimated for a married dummy variable in an equation measuring children’s outcomes (e.g. education) is not a consistent estimate of “the effect of marriage on children.” Instead, it is an estimate of the likelihood of the outcome conditional on parents getting and remaining married. Consider the differences between the 1950s and the 1990s; conditioning on getting and remaining married is very different in 1995 than it was in 1955. Marriage changes incentives for parents, but does not guar-

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antee a child’s education, a job, or any other outcome. A related paper (Murphy, 1999) shows that the “divorce penalty” faced by children of unmarried parents should have increased over the past decades, even as divorce and single parent families became less stigmatized and more common. Estimates of the dropout likelihoods of 17-year-old using samples from the US Census are consistent with this prediction. Several extensions seem worth pursuing. When considering how to test the model, one is immediately confronted with the fact that, unlike the agents in the model, individuals vary significantly in their endowments. Children’s outcomes, as well as match acceptance decisions, probably reflect these endowments. Extending the model to incorporate systematic endowment differences should allow for sharper testable implications (for instance, general equilibrium predictions concerning wages of those in formal versus informal matches, disentangling the effects of wages and household status). Second, allowing a choice of “pregnancy technology” would allow for consideration of issues related to the declining costs of contraception and abortion over the past few decades (e.g. Roe v. Wade). In addition, additional symmetry — e.g. allowing women to choose to invest in children — may be interesting to explore.

Acknowledgements I thank Peter Doeringer, Nancy Folbre, Michael Manove, Jeff Miron, Sharun Mukand, Andrew Weiss, Alwyn Young, an anonymous referee, and seminar participants at Boston University, the University of Massachusetts at Amherst, and the Virginia Polytechnic Institute and State University for helpful comments and discussions. This paper is based on Chapter 2 of my dissertation; I am quite grateful for the valuable discussions with and suggestions from my advisors Glenn Loury, Kevin Lang, and Eli Berman. With apologies to Flannery O’Connor.

Appendix A Theorem 1. A divorce tag creates an endogenous penalty for men whose partners divorce them. Proof. The value of a period 2 match of quality m to a man is:  L if θ = θ L < 0  m − pθM M M VM (m) = m O >0 if θM = θM The expected value of a new draw for a single man is: ∞ L  R(µS ) m dF (m) − [1 − F (R(µS ))]pθM vM,S ≡   ∞ R(µS ) m dF (m)

(A.1)

L <0 if θM = θM O >0 if θM = θM

(A.2)

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and for a divorced man: ∞ L  R(µD ) m dF (m) − [1 − F (R(µD ))]pθM vM,D ≡   ∞ D m dF (m) R(µ )

L <0 if θM = θM O >0 if θM = θM

(A.3)

Since µD < µS , R(µD ) > R(µS ). Since θW − wE > 0, R(µS ) > 0 and vM,D < vM,S .
Definition 1. The woman’s lower decision bound with no (observable) divorce is denoted by mN : mN ≡ v(µ) −

wE + R(1.0) β

(A.4)

Definition 2. The woman’s lower decision bound with (observable) divorce is denoted mD : mD ≡ αv(µS ) + (1 − α)v(µD ) −

wE + R(1.0) β

(A.5)

Lemma 1. (∂mD /∂α) > 0. Proof. ∂mD = ∂α




∞ R(µS )


(m − pθW + pµS wE ) dF (m) −

∞ R(µD )

(m − pθW +pµD wE ) dF (m) (A.6)

and m − pθW + pµS wE > m − pθW + pµD wE

(A.7)

For m ≥ R(µD ), the first term in ∂mD /∂α is strictly larger than the second. For m < R(µD ), the second term is 0 while m − pθW + pµS wE ≥ 0. Since f (m) > 0, (∂mD /∂α) > 0.
Lemma 2. For α ∈ [0, 1], (dµD /dα) < 0. Proof. µD = µ

φA φA + φ 0

(A.8)

and α=

φR + φN φR + φ N + φ A + φ 0

(A.9)

If α = 0, then φR + φN = 0. Let φR increase: φA + φ 0 dα = >0 dφR (φR + φN + φA + φ0 )2

(A.10)

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and dµD φ0 = −µ <0 dφR (φA + φ0 )2

(A.11)

But φR increases as α does; therefore (dµD /dα) < 0.




Lemma 3. (dmD /dα) > 0. Proof. From Lemma 1, (∂mD /∂α) > 0. The last two terms in mD are fixed. From Eq. (10) µS = µ so that R(µS ) = pθW − pµ wE . Therefore, there are no indirect effects of α in the first term. The second term varies directly with α and indirectly through changes in µD and, through µD , in R(µD ). But by Lemma 2, (dµD /dα) < 0. As α increases, the expected value of a given match to a divorced man: m − pθW + pµD wE

(A.12)

declines for all m while the cutoff level of m required to accept a divorced man: R(µD ) = pθW − pµD wE

∞

(A.13) pµD w

increases. Both of these effects reduce R(µD ) (m − pθW + E ) dF (m). The second term in mD gets smaller directly as α increases through (1 − α) and also indirectly
through µD . Lemma 4. For α = 0, mD = mN . Proof. Let α = 0. Then
∞ wE (m − pθW + pµD wE ) dF (m) − mD = + pθW − pwE β R(µD )

(A.14)

Since α = 0, φR + φN = 0. Then, with no observability, an available man is an L-type with probability φR + φ N + φ A φA µ = µ = µ (A.15) φR + φ N + φ A + φ 0 φA + φ 0 or µ = µD . But then R(µD ) = pθW − pµwE and
∞ wE + R(1.0) (m − pθW + pµwE ) dF (m) − mD = mN = β R(µ)




(A.16)

Definition 3. The woman’s upper decision bound with no (observable) divorce is denoted by mN : wN + R(0.0) (A.17) mN ≡ v(µ) − β Definition 4. The woman’s upper decision bound with (observable) divorce is denoted mD : wN mD ≡ αv(µS ) + (1 − α)v(µD ) − + R(0.0) (A.18) β

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Lemma 5. (∂mD /∂α) > 0. Proof. ∂mD = ∂α







∞ R(µS )

(m − pθW + pµS wE ) dF (m) −

∞ R(µD )

(m − pθW +pµD wE ) dF (m) (A.19)

From Eq. (9) µD = µ (φA /(φA + φ0 )) < µ and from Eq. (10), µS = µ , so for all m m − pθW + pµS wE > m − pθW + pµD wE

(A.20)

Also, for R(µS ) < m < R(µD ), the contribution of the second term in Eq. (A.19) is 0 while the contribution of the first term is positive.
Lemma 6. (dmD /dα) > 0. Proof. Since µS = µ , there are no indirect effects of α in the first term of Eq. (A.19). However, (dµD /dα) < 0 (Lemma 2) so the indirect effect on payoffs when matched with a divorced man is negative: d (m − pθW + pµD wE ) < 0 dα

(A.21)

and on the cutoff R(µD ) is positive: d (pθW − pµD wE ) > 0 dα ∞ Both effects make R(µD ) (m − pθW + pµD wE ) dF (m) smaller.

(A.22)


Lemma 7. For α = 0, mD = mN . Proof. Let α = 0
∞ wN mD = (m − pθW + pµD wE ) dF (m) − + R(0.0) D β R(µ ) mD = v(µ) −

wN + R(0.0) β

and, when α = 0, µD = µ (Lemma 4). Theorem 2. A divorce tag increases m and m, and reduces E O . Proof. In turn: 1. m increases α > 0; by Lemma 4, mD = mN if α = 0 and by Lemma 3, (dmD /dα) > 0. 2. m increases: α > 0; by Lemma 7, mD = mN if α = 0 and by Lemma 6, (dmD /dα) > 0. 3. E O falls:

(A.23) (A.24)


R.D. Murphy, Jr. / J. of Economic Behavior & Org. 47 (2002) 27–53

51

By Theorem 6, the expected value of a new draw is lower for a man identified as divorced. If at some m, when divorce is unobservable, an opportunistic man is indifferent between investing and not investing, then by Theorem 6 he will strictly prefer to invest when divorce is observable.
Theorem 3. Relative to the pool of available period 2 men in an environment in which marital status is unobservable, the pool of divorced men is worse than average and the pool of single men is better than average. Proof. When divorce is unobservable, α = 0; µ(divorce unobservable) = µD (Lemma 4) and, (dµD /dα) < 0. Therefore, µD < µ < µS .
Theorem 4. The average welfare of children is higher when divorce is observable if: µ F (m) − F (E O ) <  1−µ F (m)

(A.25)

Proof. Children’s welfare improves for a group of size: (1 − µ )(F (mD ) − F (mN ) − (F (EDO ) − F (ENO )))

(A.26)

(F (E O ) < 0 by Theorem 7) and declines for a group of size: µ (F (mD ) − F (mN ))




(A.27)

Theorem 5. Children’s welfare is increasing in greater divorce and illegitimacy penalties if f (m) is Gaussian, m > E O , and f (m) < f (E O ). Proof. Let D and I be high enough that R1 ≥ m. γ =

µ [1 − F (m)] + [F (m) − F (E O )] + µ [F (E O ) − F (R1 )] 1 − F (R1 )

(A.28)

dγ 1 = × ([1 − F (R1 )] dD (1 − F (R1 ))2   dF (m) dF (m) dF (E O ) dF (E O ) dF (R1 ) + − + µ − µ × −µ dD dD dD dD dD  − µ (1 − F (m)) + F (m) − F (E O ) + µ (F (E O ) − F (R1 )) 

dF (R1 ) × − dD



(A.29)

52

R.D. Murphy, Jr. / J. of Economic Behavior & Org. 47 (2002) 27–53

Then (dγ /dD) > 0 if

dF (m) dF (E O ) − dD dD dF (R1 ) dF (R ) 1 > (1 − F (R1 ))µ − (1 − µ )(F (m) − F (E O )) dD dD dF (R1 )  −µ (1 − F (R1 )) dD

(1 − F (R1 ))(1 − µ )

or if:





dF (m) dF (E O ) (1 − F (R1 )) − dD dD



> −(F (m) − F (E O ))

dF (R1 ) dD

(A.30)

(A.31)

Or, (dγ /dD) > 0 if (dF (m)/dm)(dm/dD) − (dF (E O )/dE O )(dE O /dD) F (m) − F (E O ) (dF (R1 )/dR1 )(dR1 /dD) >− 1 − F (R1 )

(A.32)

But, (dm/dD) = (dE O /dD) = −(1/β) and (dF (x)/dx) = f (x). Therefore, the necessary condition is −

1 f (m) − f (E O ) f (R1 )(dR1 /dD) >− O β F (m) − F (E ) 1 − F (R1 )

(A.33)

Because m > E O and f (m) < f (E O ), the left-hand side is positive. (dR1 /dD) > 0 since R1 increases as D (and I ) increases.


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