On Population Structure and Marriage Dynamics

Transcription

1 On Population Structure and Marriage Dynaics Eugenio P. Giolito Universidad Carlos III de Madrid and IZA February 15, 2010 Abstract I develop an equilibriu, two-sided search odel of arriage with endogenous population growth to study the interaction between fertility, the age structure of the population and the age at first arriage of en and woen. I show that, given an increase of the desired nuber of children, age at arriage is affected through two different channels. First, as population growth increases, the age structure of the population produces a "thicker arket" for young people, inducing early arriages. The second channel coes fro differential fecundity: if the desired nuber of children is not feasible for older woen, young woen becoe relatively less choosy than young en. In equilibriu, woen are ore likely to arry older en and single en outnuber single woen. In the second part of the paper I extend the odel to a finite nuber of periods and, using fertility data, show that two echaniss described above ay have acted as persistence echaniss after a fertility shock like the baby boo in the U.S. Specifically, I show that deographic dynaics ay account for up to a third of the increase in en s age of arriage since JEL Classification: J12, D83. Keywords: arriage, population structure, sex ratio Departent of Econoics. Universidad Carlos III de Madrid. C/ Madrid, 126, Getafe (Madrid, Spain. E-ail address: egiolito@eco.uc3.es. I thank Javier Diaz-Gienez, Nezih Guner, John Knowles, Seth Sanders, Yora Weiss, participants at the conference "Incoe Distribution and the Faily" (Kiel, and at seinars at Duke University, University of Arhus and Universidad Carlos III de Madrid for their helpful coents. I gratefully acknowledge the financial support of the Spanish Ministerio de Educación y Ciencia (Grant BEC and of the European Coission (MRTN-CT Errors are ine. 1

2 1 Introduction In this paper, I investigate the relationship between fertility, the age structure of the population and the tiing of arriage of en and woen. Looking at the evolution of those variables in the U.S., we observe several facts that suggest the existence of such a relationship. First, the edian age of arriage of en and woen appears to be negatively correlated with the total fertility rate (TFR, although with a delay. 1 Observe in Panel (a of Figure A.5 that fertility rates were sharply falling fro the early sixties to the id seventies, while the age of arriage of en and woen starts to unabiguously increase only when fertility rates becoe stable. The relationship between arriage tiing and fertility can be reconciled if we take a closer look at the deographic iplications of the U.S. "Baby Boo". Notice in Panel (b of Figure A.5, which shows the evolution of the share of population aged s-old on those aged 15-44, together with the TFR, that this easure of population structure iics fertility rate with a lag of about 20 s. Specifically, this share starts increasing in the early sixties, when the early "booers" becae 15 -olds and peaks in the id-seventies, when they turned 30, to then decrease until the id-nineties. Panel (c shows the coparison between this easure of population structure and the edian age of arriage of en and woen. Notice that both easures of arriage tiing start increasing approxiately at the sae tie the population starts to becoe older again, around The second fact that links fertility and arriage tiing is related to the age gap at arriage between en and woen. As shown in Panel (d of Figure A.5, the difference between the average age at first arriage en and woen shows a positive correlation with the total fertility rate. Observe that the age gap increases in the forties and starts to decrease just by the end of the baby boo, around In this paper, I will argue that the tiing of arriage of en and woen is affected by two endogenous sources of frictions. The first one is the age structure of the population, which is a function of past fertility and produces a "thicker arket externality" that induces en and woen to arry earlier in the case where the population becoes younger. The second source is the ratio of single en and single woen, which produces asyetries in the arriage opportunities of en and woen. As I will show in this paper, assuing that there are always equal total quantities of en and woen in the econoy, the age gap and the sex ibalances are jointly deterined by the behavior of the agents. Observe in Panel (d of Figure A.5 the positive correlation between the sex ratio and the age gap in the tie series. Finally, notice in Panel (f that the evolution in 1 Throghout this paper I use the edian age at first arriage as a easure of arriage tiing, because population structure is a key factor in this paper. The U.S. Census Bureau publishes ly the estiated edian age at first arriage since 1947, using an indirect ethod based on the proportion of people who were ever arried within the single age cohort. This ethodology prevents the age of arriage fro being affected echanically by population structure. Since the ean age of arriage is conditional on arriage, this easure is subject to those echanical effects. However, I use eans of age at first arriage in order to calculate the age gap. 2 In contrast to the previous arriage literature, which assues that the age difference is constant, Ni Brolcháin (2001, has shown that the age gap changes according to the size of the cohorts. 2

3 the sex ratio does not appear to follow changes in the relative total quantity of en and woen, at least since In Section 2 I present the basic, steady state odel. Specifically, I develop an overlapping generations general equilibriu odel with two-sided search in which agents only differ in their age and are able to arry once in their lifeties. I assue that they live as adults for two periods and that they obtain utility fro the (idiosyncratic quality of their arriage and fro the joy of having children. I also assue that both en and woen desire to have a given nuber of children, which I keep as exogenous. Given desired fertility, population growth, and therefore the age structure of the population, are endogenous. In this odel, as people get older they becoe less attractive to their prospective younger ates. In the case of both sexes, this is because younger spouses know that they will lose the utility of being arried relatively early in life when their older spouse dies. In the case of woen, the disutility of aging increases because they lose their fecundity relatively early. Given the above assuptions, I show that, given an increase of the desired nuber of children, age at arriage is affected through two different channels. First, as population growth increases, the age structure of the population is altered, which reduces the frictions that en and woen face in order to eet other younger counterparts, and therefore increases their likelihood to arry younger. The second channel coes fro differential fecundity: if the biological constraints for older woen are binding, young en becoe ore reluctant to arry older woen. Consequently, young woen anticipate their shorter fecundity horizon and reduce their reservation atch quality for both younger and older en. Therefore, woen increase their likelihood of arrying older en, who have a greater willingness to arry than younger en. In the steady state equilibriu, as young woen are ore likely to arry older en, young en are (relatively ore likely to wait and single en will outnuber single woen. The results above have dynaic iplications, which I study in Section 3, where I extend the odel to a full life-cycle odel with people living a finite nubers of ly periods. In order to study arriage dynaics, I proceed in two steps. First, I assue that the econoy in its steady state and calibrate the odel paraeters to U.S.1930 deographics, to obtain an initial state Then, since desired fertility is exogenous in this odel, I take those paraeters fro the U.S. tie series data and assue that desired fertility reaches a new steady state value in Fro that onwards, arriage tiing is exclusively affected by a persistence echanis that is basically caused by the change of the age structure of the population due to the "baby boo". Therefore, until the age structure of the population reaches its new steady state level (around the 2050 in the odel, people tend to arry younger than in the steady state because they face relatively higher atching rates of young ates.the calibration exercise decline in fertility in the period ay account for ore than a 35% of the increase in the age of arriage occurred during the period However, the slow convergence to the steady state is particularly observed in the case of en. Differently, woen reach their steady state age of arriage uch sooner, slightly after fertility rates stabilize. 3 However, total and single sex ratios appear to be correlated between 1900 and

4 In order to understand the how uch influence the deographic structure ay have had in en s arriage slow convergence, I perfor two counterfactual experients. In the first experient, I assue than fertility rates converge slowly fro the 1940 levels to the replaceent rate, as if the "baby boo" never occurred. In the second experient, I assue that even there is even there is an increase in desired fertility fro the 1945 to 1965, the inflow of people to the econoy is constant, so I a isolating the persistence echanis. What I find in either case that en would have reached their steady state edian at first arriage in the id eighties, 40 s before than in the benchark odel. In the case of woen, the effects are in both experients negligible. Review of the literature. The earliest work devoted to the explanation of the age gap in econoics is based on gender roles of traditional societies. Bergstro and Bagnoli (1993 present a odel with incoplete inforation where woen are valued as arriage partners for their ability to bear children and to anage a household, and en are valued for their ability to ake oney. Woen s ability is observable earlier in life than en s ability, which is private knowledge. Therefore, en who expect to be successful delay arriage until they are able to give a signal that allows the to attract ore desirable woen. Siow (1998 introduced the issue of the shorter fecundity period of woen in the econoics literature. In a two-period odel with uncertainty over huan capital accuulation, and where utility coes exclusively fro having children, old and young en (all fertile copete for young, fertile woen. Consequently, soe old en, those who successfully obtain a higher wage, are able to arry, displacing soe of the young en in the copetition over scarce fertile woen. In this paper, different fro the works entioned, search for a partner takes tie and there is no role for perceptions about future earnings but, as in Siow (1998, woen have a shorter fecundity period than en. 4 However, Siow also explores how differential fecundity affects gender roles, but the analysis is focused on the steady state with no population growth. Here, as stated above, I explore the role of population structure in arriage tiing and the transitional dynaics of the odel. 5 The observed relationship between sex ibalances and the age gap akes search odels particularly useful to analyze this proble. 6 The present work is also related to previous studies on arriage tiing, for exaple, 4 In this paper, differential fecundity produces the age gap through a different echanis than the one in Siow (1998. In Siow s paper, older en copete over scarce young woen and provide the with a higher copensation (through their higher wages. Here, young woen becoe less choosy and therefore ore likely to arry older en. Therefore, differential fecundity is a suffi cient condition for the existence of the age gap. 5 Another related paper is Tertilt (2005, who also allows for endogenous population growth to study the effects of polygyny in certain African countries. Also in the developent literature, Edlund (1999 discusses possible links between son preference in certain Asian countries and arriage patterns (for exaple spousal age gap, as here. 6 Seitz (2002 studies of the ipact of ibalances of the total sex ratio (like ortality or inigration on differences in the arriage tiing between whites and blacks. Differently fro this paper, she does not allow for arriage across cohorts, and therefore the sex ratio is only affected by exogenous shocks on the total population of en and woen. 4

5 Caucutt, Guner and Knowles (2002. They study the roles played by wage inequality, huan capital accuulation, and returns to experience for the tiing of arriage and fertility. They show that highly productive woen arry and have children later in life than their less productive colleagues. With a richer structure than this odel, they abstract fro cross-age arriages and fro population growth. Section 2 of the paper is a copletely revised version of Giolito (2004, which is, to the extent of y knowledge, the first attept to odel the age of the individuals, both as a state variable and as a source of agents heterogeneity, within the search literature studying equilibriu arriage behavior with ex-ante heterogeneous agents and non-transferrable utility (NTU. 7 While here I preserve the ain characteristics of that odel, the previous version focused exclusively on the steady state with no population growth. 8 More recently, Díaz-Giénez and Giolito (2008 have developed a full life-cycle setting that accounts for differences in fecundity, earnings and ortality profiles between en and woen. With a richer structure than in this paper, but focusing only in the steady state and abstracting fro population growth, they find that differential fecundity is all that atter to account for age differences at first arriage, and that the distribution of age differences along the life cycle, which shows a pattern which is increasing with age for en, and reducing (in absolute value for woen is also ore consistent with a partially rando atching search odel that with a odel where people exclusively eet within their own age group. Recently, Coles and Francesconi (2007 have analyzed a two-sided search odel where en and woen differ in their earnings (which evolve stochastically and in their fitness, which decays deterinistically with age. They show that when arket opportunities are siilar for both sexes, then the equilibriu exhibits toy-boy arriages (old woen with young en and toy-girl arriages (old en with young woen. When labor arket opportunities are better for en than for woen, toy-boy arriages disappear. The ain difference with this work is that Coles and Francesconi abstract fro population growth and also assue that atching rates are independent of the arriage patterns across ages. 9 Therefore, they abstract fro the relationship between the sex ratio and the age gap, which is crucial in this odel. 7 Following Burdett and Coles (1997 and Sith (2006, there is an extensive literature in NTU equilibriu search odels of arriage. See Burdett and Coles (1999 for a survey. More recent exaples include Bloch and Ryder (2000, Cornelius (2003, Burdett et al. (2004, Wong (2003.and Coles and Francesconi ( More recently, Díaz-Giénez and Giolito (2008 have developed a full life-cycle setting that accounts for differences in fecundity, earnings and ortality profiles between en and woen. With a richer structure than in this paper, but focusing only in the steady state and abstracting fro population growth, they calibrate the odel to U.S. data and show that gender differences in fecundity are all iportant in accounting for life-cycle arriage profiles, and that differences in earnings atter little. 9 They assue that the population of unarried en is always equal to the population of unarried woen. 5

6 2 A two-period odel of arriage 2.1 Deographics The econoy is inhabited by a continuu of en and woen who live for three periods, one as children and two as adults: young (age 1 and old (age 2. Adult en and woen derive utility fro being arried and fro having children, do not discount tie, and their ain econoic activity is searching for a ate. In the steady state, the population grows every period at a constant rate n, which is endogenous. Therefore, in any period t, the econoy is populated by N t young and N t 1 old agents, where N t = N t 1 (1 + n = N 0 (1 + n t. Since half of the agents are ale and half are feale, I assue for siplicity that N 0 = 2. The population dynaics in this econoy deterines the steady state age structure of the population. The fraction of young people in this econoy is given by q t = N t N t + N t 1 = N 0 (1 + n t N 0 (1 + n t 1 (2 + n = 1 + n 2 + n, (1 which is increasing in the population growth rate. Each period t, equal easures s w 1,t of young single woen and s 1,t of young single en enter the econoy, where s 1,t = s w 1,t = (1 + n t. Consequently, every period there is a easure, St w = (1 + n t + s w 2,t, of single woen in the econoy, of which (1 + n t are young and s w 2,t are old. Siilarly, there is a easure, St = (1 + n t + s 2,t, of single en, of which (1 + n t are young and s 2,t are old. Since the easures of single old en and woen are endogenous in this odel, the total easures of singles, and therefore the ratio of single en to single woen (henceforth, the sex ratio, φ t = S t are also endogenous. St w Since en and woen arry in pairs, the total nuber of brides, w 1,t young plus w 2,t old, ust be equal to the total of groos, h 1.t young plus h 2,t old. Since in the steady state the sex ratio is constant, φ t = φ, the fact that every period an equal nuber of young en and woen enter the arket requires the nuber of en who exit the arriage arket either because they arry, or because they die single to also be equal to the nuber of woen who leave the arket. Consequently, w 1,t + w 2,t + u t = h 1,t + h 2,t + u w t (2 where u t and u w t denote the nuber of en and woen who die unarried, respectively. Notice that condition (2 and the fact that people arry in pairs iply that u t = u w t. 2.2 Matching technology I assue that singles of both ages eet randoly at ost once every period and decide whether or not to arry. Once arried, both en and woen reain so until the end of their lives and people whose spouse dies are not allowed to search for a new spouse For a odel of arriage with endogenous separations, see Brien, Lillard and Stern (2002 and Cornelius (2003. Moreover, Chiappori and Weiss (2006 exaine equilibriu divorce incentives where there is a thick arket externality in the rearriage arket. 6

7 Whenever a an and a woan eet, either one can propose arriage to the other. If they both accept, they arry. Otherwise they both stay single. The total nuber of eetings between singles is represented, in the spirit of Pissarides (1990, by the following constant returns to scale atching function, η t = ρ (S t θ (S w t 1 θ (3 where θ (0, 1 and ρ (0, 1 2 ].11 As is standard in the search and atching literature, the probabilities of being atched depend on the sex ratio φ. Therefore, a single an eets a single woan with probability λ = η t S t = ρ (φ θ 1 (4 Siilarly, a single woan eets a single an with probability λ w = ρ (φ θ. Since the sex ratio is fully endogenous in this econoy, here the atching function also captures the way in which the aggregate effects of the arriage decision feed back into the individual decision proble. The probability that an individual eets either a young or and old agent of the opposite sex depends on the age structure of the single population. For exaple, the probability that a single an eets a single young woan is λ p w 1 and the probability of eeting an old single woen is given by λ p w 2 = λ (1 p w 1, where p w a = sw a,t is the fraction of single St w woen of age a, which is constant in the steady state. Siilarly, fro a single woen point of view, the probabilities of eeting a young and an old an are given by λ w p 1 and λ w (1 p 1, respectively Payoffs Table 1 define the utility that people derive fro their arital status. I assue that arried people derive utility both fro the atch quality and fro the nuber of children they are able to have with a given spouse, and the utility of being single is noralized to zero. I assue that en and woen desire to have k [ 1, k] children during their 2 lifetie, where each "child" is a pair of twins, one girl and one boy. I also assue that woen bear all their children in the sae period they arry. Finally, en and woen differ in their fecundity horizons. On one hand, while young woen are able to bear k children, old woen are only capable of bearing at ost one child. On the other hand, en of age 1 and of age 2 are able to have k children. The utility that a woan derives fro being arried to a specific an depends also on the an s type, assued to be a rando variable with cuulative distribution G (y, support in [0, y], and ean µ y. Siilarly, I assue that the utility that a an derives 11 Given that the sex ratio can be unbalanced in equilibriu, I assue that ρ is sall enough to ensure that the atching probabilities are not higher than one. The interval assued for ρ ensures the uniqueness of a syetric equilibriu in a two-period econoy. 12 Given that the atching technology is CRS and that s 1,t = s w 1,t, it is straightforward that λ w p 1 = η S w t (1+n t S t = λ.p w 1 7

8 fro being arried to a specific woan depends on the woan s type, which is a rando variable with cuulative distribution G w (x, support in [0, x], and ean µ x. I further assue that both cuulative distributions are continuous and differentiable, and that their probability density functions are g (y and g w (x, respectively. Since I assue that singles are hoogeneous in their quality, the distributions G and G w are identical across singles, and the realizations that people draw at each eeting are independent. 13 For exaple, if a young woan arries a young an, their utilities are 2ky and 2kx, because the arriage will last for two periods. If a young woan arries an old an, they will also receive ky and kx but only for one period. On the other hand, if a young an arries an old woan, their utilities are only in{k, 1}x and in{k, 1}y for one period, respectively. Since old woen are less fecund and only able to bear one child, they receive the sae utility regardless of the age of their spouse. The rationale for a functional for where the nuber of desired children and the type of the spouse are copleents can be interpreted as if the joy of a having a faily" is a function of the atch quality. In other words, people enjoy having children ore with people they care about ore. Young Spouse Old Spouse Woen Marry young 2ky ky Marry old in{k, 1}y in{k, 1}y Never arry 0 0 Men Marry young 2kx in{k, 1}x Marry old kx in{k, 1}x Never arry 0 0 Table 1: Table Payoffs for en and woen Furtherore, since in the odel econoy people who do not arry iss out on the joys of both copanionship and childbearing, I noralize to zero the utility of being single. In order to ensure existence of an equilibriu, I assue that the axiu value of k is such that the utility of arrying an average spouse and having k children with the is not higher than the joy of finding a perfect atch and having only one child. That is, kµ x x and kµ y y, which iplies { x k = in, y }. (5 µ x µ y 2.4 The decision proble of singles Since people atch randoly in our econoy, the single en s proble is to choose reservation values for young and old single woen. Let Ra,b denote the steady state 13 The assuption about the two separate distributions for en and woen, as in Burdett and Coles (1997 is ade exclusively for analytical tractability and does not affect the results. 8

9 reservation value that single en of age a set for single woen of age b, and, siilarly, let Rb,a w be the reservation values that age b single woen set for age a single en. Since I assue that people eet only once per period, and that they derive zero utility fro being single, the reservation values of old single en for both young and old single woen is zero, trivially. That is, R2,1 = R2,2 = 0. Siilarly, the reservation values of old single woen for both old and young single en are also zero, that is, R2,1 w = R2,2 w = Expected utility of arriage for old single people Denote as γ j a,b the steady state probability that a single agent of age a and gender j {, w} arries soeone of age b. This probability depends first, on the probability of eeting soeone of the opposite sex and age b, λ j p j b. Given the eeting probability, the probability of arriage depends on utual acceptance. That is, for exaple for en, γ a,b = λ p w b [1 G (R w b,a][1 G w ( R a,b ] (6 Let us observe the case of old single people. Since I assue that old people who do not arry obtain zero utility, they accept any arriage offer. For exaple, if an old single an (woan eets a young single woan (an they will arry provided the youngest agent accepts. Therefore, γ 2,1 = λ p w 1 [1 G (R w 1,2] and γ w 2,1 = λ w p 1 [1 G w ( R 1,2 ] (7 On the other hand, if an old single an eets an old single woan, they arry with certainty. Without loss of generality, I assue that everybody eventually arries at soe point in life. Consequently, those old en and woen that either were unatched or rejected by younger counterparts will face a frictionless technology that autoatically atches the aong theselves. 15 Therefore γ 2,2 = [1 γ 2,1] and γ w 2,2 = [1 γ w 2,1]. Given the payoffs described in Table 1, the expected utility of arriage for old single en is U 2 = γ 2,1kµ x + ( 1 γ 2,1 in{k, 1}µx (8 = λ p w 1 [1 G (R w 1,2]kµ x + ( 1 λ p w 1 [1 G (R w 1,2] in{k, 1}µ x For old single woen, since they will enjoy the sae utility regardless of who they arry, we have, U w 2 = γ w 2,1 in{k, 1}µ y + ( 1 γ w 2,1 in{k, 1}µy (9 = in{k, 1}µ y. 14 This rules out the trivial equilibriu where all woen reject all arriages because they know that all en will reject the and vice versa. 15 This assuption is for convenience, and will be relaxed in the next section, when I study the transitional dynaics of the odel. Recall that by condition (2, steady state iplies that an equal nuber of old en and woen would reain unatched after period 2. 9

10 2.4.2 Expected Utility of Marriage for Young Single People Since young single en and woen have a chance to arry in the future, they choose their reservation values taking into account the next period s prospects. For exaple, provided that a single young an eets a single young woan (with probability λ p w 1 = λ w p 1, they will arry if both of the find each other utually acceptable. A single young woan will accept a single young an with probability [1 G (R1,1], w and he will accept her with probability [1 G w (R1,1]. Therefore, the probability that a young an arries a young woan (or vice-versa is γ 1,1 = γ w 1,1 = λ p w 1 [1 G w ( R 1,1 ][1 G (R w 1,2] (10 On the other hand, if young an or woan eets an old single agent of the opposite sex (with probability λ p w 2 or λ w p 2, respectively, the acceptance of the young agent will be suffi cient for arriage to occur. Consequently, and γ 1,2 = λ p w 2 [1 G w ( R 1,2 ] (11 γ w 1,2 = λ w p 2 [1 G ( R w 1,2 ], (12 for en and woen, respectively. The expected utility of arriage for young single en, U 1, is then, U 1 = γ 1,12kE [ x x R 1,1] + γ 1,2 in{k, 1}E [ x x R 1,2], (13 and the one for woen, U w 1, U w 1 = γ w 1,12kE [ y y R w 1,1] + γ w 1,2 ke [ y y R w 1, The young single people s optiization proble Young single people of gender j {, w} choose the reservation values for young and old single agents of the opposite sex, R j 1,1 and R1,2, j which axiize their expected lifetie utility. In order to do this, they solve the following proble: V j 1 = ax R j 1,1,Rj 1,2 [ U j 1 + ( 1 Γ j 1 ] U j 2 ] (14 (15 subject to Γ j 1 = γ j 1,1 + γ j 1,2 (16 where Γ j 1 stands for the steady state probability that a single individual of gender j arries at age 1. The solutions to this proble require a young single agent to be indifferent between arrying when young or resapling again the following period. If a young single agent eets an old individual of the opposite sex he/she ust copare the utility of a one period arriage with the value of arrying when old. In the case of a young an who eets an 10

11 old woen, this type of arriage becoes less valuable if their desired nuber of children is greater than the one that an old woan is able to conceive (k > 1. Therefore, in{k, 1}R1,2 = U2 (17 R1,2 = U2 in{k, 1} The case of a young woan who eets an old an is different. Since en of all ages are able to have k children, she ust copare the utility of arrying young with the value of waiting until the next period and facing the liited fecundity which will be binding in the case where k > 1. Then we have hence kr w 1,2 = U w 2 = in{k, 1}µ y, R1,2 w = in{k, 1}µ y. (18 k Now we focus on the reservation values that young singles set for people of their sae cohort. Given that a young single an eets a young single woan (or vice-versa, they have to copare between the value of waiting and the utility of a two-period arriage with k children. Therefore, 2kR j 1,1 = U2, j or R j 1,1 = U j 2 (19 2k for j {, w}. To obtain the reservation values chosen by the young single en, substituting expression (8 into Equations (17 and (19, respectively, we get and R1,1 = µ { x in{k, 1} + λ p w 1 [k in{k, 1}] [ ( ]} 1 G R w 2k 1,2, (20 R 1,2 = µ x { 1 + λ p w 1 [k in{k, 1}] [ 1 G ( R w 1,2 ]} = 2k in{k, 1} R 1,1. Notice that the young single en s reservation values depend positively on the average atch quality of single woen, µ x, and on the difference between their desired nuber of children and the nuber that old woen are capable to bear (k in{k, 1}. When the desired nuber of children is not biologically binding for old woen (k 1, we have that R 1,1 = µ x 2 and R 1,2 = µ x. On the other hand, when k > 1, we have that R 1,1 = µ x 2k (21 { 1 + λ p w 1 [k 1] [ 1 G ( R w 1,2 ]}, (22 11

12 and R 1,2 = µ x { 1 + λ p w 1 [k 1] [ 1 G ( R w 1,2 ]}, (23 which are increasing on the probability of eeting a single young woan, λ p w 1, and decreasing on R1,2, w i.e., the reservation value of young single woen for old single en. Naturally, when R1,2 w decreases, that is, when it becoes ore likely that young single woen accept the proposals of old single en, young en are ore willing to wait and therefore becoe choosier. Finally, notice that while R1,1 is decreasing in the desired nuber of children, k, R1,2 is increasing in k. That is, as the desired nuber of children increases, young en becoe choosier with respect to old woen and less choosy with young woen. On the other hand, reservation values chosen by young single woen are siply, R w 1,1 = in{k, 1} µ 2k y, (24 and R1,2 w in{k, 1} = µ k y. (25 Notice that, in this case, when fecundity is binding (k > 1, the reservation values that young woen set for all en becoe decreasing functions of their desired nuber of children Steady state deographics The nuber of single en and woen This section presents the endogenous nuber of single people in the steady state econoy as a function of the probabilities of arriage at age 1, Γ j i for j {, w} and the rate of population growth, n. Since the population of old single en and woen is coposed of the unarried young singles of the previous period, and given that the population of young singles is s 1,t = s w 1,t = (1 + n t, we have that s j 2,t = s j 1,t 1 ( 1 Γ j 1 = (1 + n t 1 ( 1 Γ j 1, (26 Therefore, the total nuber of singles is S j t = s j 1,t + s j 2,t = (1 + n t 1 ( 2 + n Γ j 1 (27 The age distributions of singles that is the fraction of young single people of gender j {, w}, is p j 1 = sj 1,t (1 + n S j = t 2 + n Γ j, ( Notice that the reservation values of en and woen are all continuous in the value of the paraeter k, which allows us to find an equilibriu for each value of k. 12

13 Finally, the sex ratio is φ = S t S w t = 2 + n Γ 1. ( n Γ w 1 Note that in the steady state the sex ratio and the fractions of young single people are constant. Moreover, the fractions of young singles depend positively on the population growth rate Fertility and population growth To obtain the rate of population growth, recall that a woan that arries at age 1 is capable of having k pairs of twins, and that a woan who does so at age 2 is able to bear only one pair of twins. Given those assuptions, we have that the nuber of young en and woen who enter the arket each period are s 1,t = s w 1,t = ks w 1,t 1Γ w 1 + in{k, 1}s w 2,t 1. (30 Given that s w 1,t = (1 + n t, substituting (26 in (30,and anipulating we have (1 + n 2 = k (1 + n Γ w 1 + in{k, 1} (1 Γ w 1. (31 Solving the quadratic equation (31 and taking the positive root we have that the rate of population growth in this econoy is n = kγw (kγ w in{k, 1} (1 Γ w 1. (32 2 Notice that woen s fecundity is noralized to obtain n = 0 when k = 1. The population growth reaches its iniu when k = 0.5 and Γ w 1 = 1 (n = 0.5 and its axiu when k = k and Γ w 1 = 1 ( n = k Steady State Equilibriu Marriage arket clearing Since en and woen arry in pairs, the total nuber of brides ust be equal to the total of groos in each type of arriage. Therefore, for en of age a and woen of age b, for a, b {1, 2}, we have s a,tγ a,b = s w b,tγ w b,a. (33 This condition is trivially satisfied when a = b = 1, since s 1,t = s w 1,t = (1 + n t and, by (10, γ 1,1 = γ w 1,1. In the case of people arrying across cohorts we have, s 1,tγ 1,2 = s w 2,tγ w 2,1 (34 s 1,tλ 2 [1 G w ( R 1,2 ] = s w 2,t λ w p 1 [1 G w ( R 1,2 ] s η t 1,t St s w 2,t St w = s w 2,t η t S w t s 1,t S t. 13

14 Siilarly, we have that. s 2,tγ 2,1 = s w 1,tγ w 1,2 (35 s 2,tλ p w 1 [1 G ( R w 1,2 ] = s w 1,t λ w p 1 [1 G ( R w 1,2 ] s η t 2,t St s w 1,t St w = s w 1,t η t S w t s 2,t S t Definition Given a desired nuber of children, k, a steady state equilibriu for this econoy is a vector of reservation values for young single en, R1 = (R1,1, R1,2, a vector of reservation values for young single woen, R1 w = (R1,1, w R1,2, w a probability of arriage for young single en, Γ 1 = γ 1,1 + γ 1,2, a probability of arriage for young single woen, Γ w 1 = γ w 1,1 + γ w 1,2, and a population growth rate n, such that: (i Given R w 1, Γ 1, Γ w 1 and n, R 1 solves the young single en s decision proble described in expression (15, and is deterined by equations (20 and (21. (ii Given R 1, Γ 1, Γ w 1 and n, R w 1 solves the young single woen s decision proble described in expression (15, and is deterined by equations (24 and (25. (iii The young en s probability of arriage, Γ 1, deterined by (10 and (11, is consistent with the reservation values chosen both by the young single en and the young single woen, the probability of arriage of young single woen and the rate of population growth. (iv The young woen s probability of arriage, Γ w 1, deterined by (10 and (12, is consistent with the reservation values chosen both by the young single en and woen, the probability of arriage of young single en and the rate of population growth. (v The rate of population growth, n, deterined by (32, is consistent with the reservation values and arriage probabilities of single young en and woen. 17 (vi Marriage arket clears, by (34 and ( Existence Here I show the existence of a steady state equilibriu for this econoy. Notice that, by expression (20, R 1,1 is uniquely deterined by R 1,2. In addition, the reservation values of woen are copletely deterined by k and µ y (equations (24 and (25. Therefore, I have to show the existence of an interior solution of a syste of four equations in four unknowns: R 1,2, Γ 1, Γ w 1 and n. 17 Notice that, strictly speaking, the reservation values for old single en, R2 = (R2,1, R2,2, and old single woen, R2 w = (R2,1, w R2,2, w are also part of the steady state equilibriu of this econoy. However, as shown in Section 2.4, they are trivially zero, and I have oitted the fro the definition of the equilibriu for the sake of siplicity. 14

15 Theore 1 (Existence A steady state equilibriu exists for this econoy, i.e. The following result rules out the existence of corner solutions and establishes that in equilibriu arriages across cohorts occur with positive probability. Corollary 2 The equilibriu is interior, and R 1,1, R 1,2 (0, x, Γ 1 (0, 1, Γ w 1 (0, 1, n ( 0.5, k 1. Furtherore, in equilibriu all type of arriages occur with positive probability, Proof. See Appendix A.1 γ 1,1 = γ w 1,1 > 0 γ 1,2, γ w 2,1 > 0 γ w 1,2, γ 2,1 > Equilibriu characterization In this section I characterize the equilibriu of the econoy, assuing that the distribution of types is identical for en and woen, that is G = G w = G, denoting its ean by µ. The analysis is divided into two parts. Next, I analyze the case where old woen s biological restriction is not binding (k 1. This case is called "syetric" because the decision probles faced by the single en and the single woen are identical. Then, in Section I analyze the case of differential fecundity, that is, when the desired fertility is higher than the nuber of children that old woen are capable of bearing Syetric case Consider the case were people desire to have k [0.5, 1] children. 18 Consequently, differential fecundity plays no role because the decision probles faced by the single en and the single woen are identical. Moreover, in this case the reservation values of single en and single woen are identical, and so are the nubers of single en and single woen in the econoy. Notice that in this case the reservation values of en and woen are independent of their desired fertility. 18 Note that the results depend exclusively on syetry and not on the negative population growth rates iplied by the assuption of k 1. This assuption is ade erely for expositional reasons and will be relaxed in the next section. 15

16 Assuption 1 G = G w = G with ean µ and density g. Assuption 2 The probability density function g is log-concave. Under Assuption 1, by (21, (20, (24 and (25 we trivially have and R 1,1 = R w 1,1 = µ 2, (36 R 1,2 = R w 1,2 = µ. (37 Proposition 1 Under Assuption 1, if people only want to have a nuber of children that is biologically feasible for old woen, k [ 1, 1], there is always a steady state equilibriu 2 where en and woen arry at age 1 with equal probability, Γ 1 = Γ w 1 = Γ s 1, and where population is not increasing over tie, n 0. This iplies that the nuber of single en is equal to the nuber of single woen, St = St w, and therefore the sex ratio is equal to one, φ = 1. Furtherore, the econoy is ostly populated by old agents, q 1. 2 Proof. See Appendix A.2 Proposition 2 Under Assuptions 1 and 2,there is a unique steady state equilibriu for k [ 1 2, 1]. Proof. See Appendix A.3 Given that en and woen face identical probles the existence of an unique equilibriu where en and woen arry at the sae age sees trivial. However, the interesting feature of the syetric case coes fro coparative statics. As we show below, when en and woen are identical, an increase in the desired nuber of children leads to en and woen arrying earlier as population growth increases, despite the constant reservation rules derived above. Proposition 3 Under Assuptions 1 and 2, if k [ 1, 1, given an increase in the desired 2 nuber of children: n 1. Population growth rate increases: > 0. This iplies that the fraction of young k q people in the econoy increases: > 0. k 2. Men and woen arry at age 1 with higher probability: Γs 1 k > 0. Proof. See Appendix A.4 The result above coes fro the interaction between young people s preferences for arrying within their cohort and the effect of desired fertility on the age structure of the population. When k increases, population growth increases and therefore the age structure of the econoy is affected, with a larger share of young people q = 1+n. Therefore young 2+n people eet people of the sae cohort with higher probability. Since young agents obtain higher utility fro arrying other young people (and therefore set lower reservation values for the, they tend to arry younger as age the structure of population is also becoing younger. Notice that this relationship is consistent with cross section data of U.S. states for the 2000 (see Figure??, Panels (a and (b for en and woen, respectively. 16

17 2.7.2 Differential fecundity Now consider the case where people desire to have k (1, k] children. If k > 1, old woen are not able to bear their desired nuber of children. As shown in Section 2.4.1, if en wait until age 2 they still can arry a young woan and have k children. Recall fro (8 that the expected utility that a an obtains fro arrying at age 2 is U2 = γ 2,1kµ + ( 1 γ2,1 µ. On the other hand, if woen wait until age 2, they will get the sae utility regardless of who they arry, U2 w = µ, (9. Since in equilibriu γ 2,1 > 0 (fro Theore 1, this iplies that differential fecundity increases the value of waiting for en. Therefore, en becoe relatively choosier than woen. Proposition 4 Under Assuption 1, if people want to have ore children than those that old woen are able to bear k (1, k], there is always a steady state equilibriu where young en are choosier than young woen, that is R 1,1 > R w 1,1 and R 1,2 > R w 1,2. Proof. When k > 1 we have fro (24 and (25 that R w 1,1 = µ 2k and Rw 1,2 = µ k. Substituting R w 1,2 in (20 and (21 we have, R 1,1 = µ 2k { [ ( µ ]} 1 + λ p w 1 [k 1] 1 G > R w k 1,1, and { [ ( µ ]} R1,2 = µ 1 + λ p w 1 [k 1] 1 G > R w k 1,2, since λ p w 1 > 0. Fro siple observation of (24 and (25 it is clear that woen s reservation values both for young and old en old en are decreasing in k. Fro Proposition 4 we know that if differential fecundity plays a role, en are choosier than woen. The previous result does not iply that as desired fertility increases en s reservation values will necessarily increase. In fact, as I show below, if desired fertility increases en also reduce their reservation values for young woen, but to a lesser extent than woen do. However, young en will increase their reservation values for older woen, which is the ain driving force behind the difference in the tiing of arriage. Proposition 5 Under Assuptions 1 and 2, then in the neighborhood of the unique steady state equilibriu where k = 1, given a sall increase in the desired nuber of children 1. Young en becoe choosier with old woen and less choosy with young woen: R12 > 0 and R 11 < 0. In the latter case, the decrease in the reservation value is k k lower than the decrease in young woen s reservation value: R 11 < Rw Woen arry young with higher probability: > 0. In addition, the difference k between the probability of arriage at age 1 of woen and en, Γ w 1 Γ 1, is positive and increasing in k. Γ w 1 k k 17

18 3. The ratio of single en to single woen, φ, is greater than 1 and increasing in k. 4. Siilarly to Proposition 3, population growth increases, therefore the fraction of young people in the econoy, q increases. Proof. See Appendix A.5 Proposition 5 establishes that when people desire to have ore children than the nuber that a woan who arries at age 2 is able to bear, differential fecundity plays a role and single en becoe choosier than single woen. Consequently, single woen tend to arry younger than en, and the age gap is an increasing function of the desired nuber of children. Moreover, it establishes that, as the desired nuber of children increases, single en increasingly outnuber single woen. The echanis behind Proposition 5 is the following. If people in the econoy want to have a nuber of children that is feasible for woen that arry at age 2, the biological differences between en and woen play no role in the arriage arket. However, biological gender asyetries do appear if society want to have ore children than those that woen of age 2 are able to bear (k > 1. Therefore, young en are less willing than young woen to accept an older spouse and the value of waiting until age 2 increases for en relative to woen. Notice that en also decrease their reservation values for young woen (albeit in a lesser extent because if they wait they ay not be able to arry a young woan when old. The increase in the sex ratio is the necessary counterpart of the increase in the age gap. As young woen are ore likely to arry old en, in equilibriu ore young en have to wait until period 2 in order to arry. Therefore, it ust be the case that single en outnuber single woen. This result has soe iplications in light of the recent epirical literature on arriage tiing. It is a standard practice in this literature to use the single sex ratio as a easure of relative availability of en and woen, ignoring the endogeneity proble caused by the relationship between the sex ratio and the age gap. 19 Notice that an while and increase in k akes woen unabiguously arry younger, it has abiguous effects on en s tiing of arriage. In the case of en there are two countervailing effects: because of differential fecundity, they tend to arry older as k increases, but because of the change in population structure they would arry younger (Proposition 3. These two effects will be analyzed in ore detail in the next section when we study the odel dynaics. As we will see later, the transitional dynaics of the odel is particularly affected by the level of the sex ratio, φ t, which will act as a persistence factor in deterining the age gap, and by the evolution of the structure of the population, q t, which will affect the arriage tiing of both en and woen. 3 An extended odel In this section I relax the assuption of a two-period life horizon in the odel developed in Section 2 The objective of this extension is to analyze in ore detail the effects of the 19 See, for exaple, Fitzgerald (1991, McLaughlin et al. (1993, and Parrado and Centeno (

19 results found above on the dynaics of arriage behavior, and specifically to copare the odel results with the evolution of the age of arriage in the U.S. starting. fro the period known as the "baby boo". Since in this stylized odel desired fertility is considered exogenous, the "baby boo" will be odeled as a shock on desired fertility. However, our focus of interest will not be this period itself, but the deographic consequences of it and their potential influence in the arriage behavior of the generations born during and after the "baby boo". In this section, I study a odel econoy populated by a continuu of en and a continuu of woen. Men and woen live for at ost J s, which we denote with subindex j = 15, 16,..., J. Men and woen in this odel econoy differ in their fecundity, and in their longevity. And they derive utility fro being arried and having children. Fro now on I relax the assuption of the constant population growth rate aintained thorough Section 2. For the sake of brevity, in this section I describe only the ain assuptions of this odel econoy. A detailed description of the T-period odel econoy is contained in Appendix??. 3.1 Mortality In our odel econoy each period every person faces an exogenous probability of dying until the following period. We denote these probabilities by δ i j,t, for i {, w}and we assue that they are gender, age and tie dependent. Since people live at ost for J periods, δ J,t = δ w J,t = 1 for any period t = 1, 2,..... These assuptions also iply that the probability that a person of gender i who is a -old in period t survives until age b is b 1 ( Ψ i t(a, b = 1 δ i,j,t+j a j=a ( Matching technology Costly double sided search for spouses is a distinguishing feature of our odel econoies. The probabilities of being atched depend on an exogenous paraeter that easures the search frictions, and on the ratio of available singles. Let s i j,t denote the nuber of j -old singles of gender i in period t and let St i = J j=15 si j,t denote the total nuber of singles of gender i. Then the probability that a single an eets a single b -old woan is [ ( ] S λ w b,t = ρ in t s w b,t, 1 (39 where paraeter 0 < ρ 1 easures the search frictions. 20 In this odel econoy the decision proble of single en and woen are exactly identical. For notational convenience I describe the proble and variables that pertain to 20 Notice, that this atching function is different fro the one in Section 2. In the first part of the paper a differentiable function was required, but keeping this function in the calibration exercise would ean one ore free paraeter to calibrate. This change does not affect the results at all. S t S w t 19

20 single en only. To obtain the corresponding variables for single woen siply substitute the s for w s and the b s for a s. 3.3 Fecundity This expected nuber of children depends on an exogenous, tie varying paraeter that desired fertility k t and biological constraints based in en s and woen s age. (deterined at the tie of arriage and biological constraints. Even though that this is not an easy atter, I will ake the following assuptions that are roughly consistent with the literature in the subject. 21, 22. Define as c w t (a, b the nuber of children that a woan of age a expects to have if she arries at tie t a an of age b. Then, k t if a 30 and b 55 in [k t, 3] if 30 < a 35 and b 55 c w in [k t (a, b = t, 2.5] if 35 < a 38 and b 55 (40 in [k t, 1] if 41 < a 44 and b 55 in [k t, 0.5] if 44 < a 49 and b 55 0 if a > 49 or b > 55. In a siilar way is defined c t (a, b. Therefore, I assue that en are fecund until age 55 and woen arried after than age 30 have not enough tie to have ore than three children (this restriction will be only binding during the peak of the baby boo Payoffs fro arriage The period utility of arriage that a single en a obtains fro arrying in period t with soeone of age b and a given arriage quality is zc t (a, b, where z is an independent and identically distributed realization fro G (z, the sae for en and woen (Assuption 1. The period utility of single people is constant endowent of A.. The expected values of arriage. The value that an a -old groo expects to obtain 21 According to Wood and Weinstein (1988, woen s probability of any conception changes rapidly after age 40, as a result of large changes in the ovarian function. Between ages 25 and 40, tthis probability of woen is rearkably constant. This finding suggests that any reduction in the physiological capacity to bear children between ages 25 and 40 is attributable to an elevation in intra-uterine ortality rather than to a decline in the ability to conceive. But, even accounting for intra-uterine loss, the pattern of effective fecundability reains fairly flat between ages 20 and According to Hassan and Killick (2003, the effect of en s age on fecundity reains uncertain. Experients that study the effect of age on ale fecundity have been criticized on ethodological grounds for using age at conception and for not taking into account confounding factors such as the age of the feale or coital frequency. Hassan and Killick study ale infecundity by coparing the tie to pregnancy easured fro the onset of the attepts to achieve pregnancy for en of different age groups. They found that ale aging leads to a significant increase in the tie to pregnancy, especially after ages 45 to Here "2.5 children" should be interpreted as "two children and a 50% probability of a third child". 20