A Theory of OutofWedlock Childbearing: Lewis Team … Theory of Out of Wedlock Childbearing: Lewis...

35
A Theory of OutofWedlock Childbearing: Lewis Team Notes 1 Presented by Marrium Khan Lauren Russell March 30, 2015 1 This picture is taken from http://www.slate.com/articles/news_and_politics/the_best_policy/2008/03/_and_baby_makes_two.html

Transcript of A Theory of OutofWedlock Childbearing: Lewis Team … Theory of Out of Wedlock Childbearing: Lewis...

A Theory of Out­of­Wedlock Childbearing: Lewis Team Notes

1

Presented by Marrium Khan Lauren Russell

March 30, 2015

1 This picture is taken from http://www.slate.com/articles/news_and_politics/the_best_policy/2008/03/_and_baby_makes_two.html

Table of Contents

1. Introduction

2. Single Mother, Unidentified Father

3. Establishment of Paternity and the Gains to Marriage

4. Out­of­Wedlock Fatherhood

5. Marriage Markets and Childbearing within Marriage

6. Marriage Markets and Out­of­Wedlock Childbearing

7. Discussion and Conclusions

8. Extensions

9. Empirical Evidence

11. Bibliography

12. Math Appendix A

13. Math Appendix B

14. Math Appendix C

1. Introduction In his paper A Theory of Out­of­Wedlock Childbearing, Robert Willis presents a theoretical framework to explain the steady growth in the percentage of childbirths to unmarried couples that occurred between the 1960s and today. In his model, he considers men and women as separate decision makers who are both seeking to optimize their individual utility, and the child is considered a collective good. In their 1985 paper titled Children as Collective Goods and Divorce Settlements, Weiss and Willis argue that, when parents live separately, they reach an inefficient, noncooperative equilibrium where both parents give too few resources to their children because they are unable to monitor the other’s spending activity. Willis then takes that model and builds upon it to explain why some men might find single fatherhood preferable despite this inefficiency. Some men might prefer to have children out of wedlock at a low or zero cost to them instead of having children within marriage where they must share the cost with their wife. This will occur if a man is able to find a sufficient number of women willing to have his children with little or no support from him. This happens if women’s incomes are high absolutely and relatively to the incomes of men and if women are more numerous than men. Therefore, there will be a fraction of low­income men who choose to father many children out­of­wedlock by more than one woman from the lower end of the income distribution if the women will assume all financial responsibility for the children. In this case, expenditures on children end up being lower than they would be if the parents were married. As we will show later in our notes, this has substantial impact on minority communities (mainly blacks and hispanics) because of the distinctly higher occurrence of single motherhood. This paper has 269 citations and has been used as a basis for understanding the impact of out­of­wedlock childbearing on overall child welfare. Variable Description

cf Mother’s consumption

n Number of children mother bears

q Quality per child

yf Mother’s full income

π Marginal cost of child quality relative to adult consumption

kf Mother’s non­labor income

wf Mother’s wage

τ Mother’s total supply of time

us Utility of single mother

qs Single mother’s demand function for child quality

If Difference between mother’s utility of having one child and not having a child

cm Father’s consumption

ym Father’s full income

u Arbitrary utility assigned to single mother

vs Utility of single father

αf Mother’s altruism

αm Father’s altruism

T Income transfer by father to mother

T * Optimal income transfer by father

P Number of partners father has

dij d=1 if father lives with ith child of jth household.

ε Elasticity of (n)α

θ Elasticity of transfers with respect to partners

η Elasticity of B(P) with respect to P

P︿ Reservation number of partners

Sm Supply of men to out­of­wedlock fatherhood

R Population ratio female to male

OWF Demographic availability function

2. Single Mother, Unidentified Father A woman considering motherhood has a utility function

(c , n, q)u f and a budget constraint

.nqyf = cf + π

Variable Explanation

cf mother’s consumption

n number of children mother bears

q quality per child

yf mother’s full income

π marginal cost of child quality relative to adult consumption

From the above budget constraint, we notice that, given an income as the number of children,,yf n, or the desired quality per child, q, increase, the mother’s non­child­related expenditure, ,cf must decrease. This is because a larger portion of the mother’s income must now be allocated to child rearing. If labor income is a part of and children are relatively time­insensitive, then mother’s incomeyf and marginal cost of child services ( ) are positive functions of her wage. A woman’s income π function can thus be written as:

,τyf = kf +wf

Variable Explanation

kf mother’s non­labor income

wf mother’s wage

τ mother’s total supply of time

where is a woman’s nonlabor income such as unearned income or income from welfarekf programs for single mothers. Since , then if a woman works more hours, she will spend(w )π = π f less time with her child and this will increase the marginal cost of child services.

One of the assumptions of this model is that a woman can either choose to remain childless or have a child. Using this assumption, her utility function when she decides to remain childless will be:

(c , 0, 0) (y , 0, 0) (y )u f = u f ≡ ϕ f When a woman decides not to have a child then , which means implying . n = 0 nq π = 0 yf = cf So mother’s utility in a childless state is dependent on her income as she is spending it all on private consumption. On the other hand, her utility function with one child is:

(c , 1, q) (y q (y , ), 1, q) (y , π)u f = u f − π sf π ≡ ψ f

Here represents woman’s demand function for child quality that is dependent on her( qs , )yf π total income and marginal cost of child services. Given she has one child and allocates her income optimally between her own consumption and child expenditures, the MRS between q and will be equal to .cf π

RSM f = ucuq = π

It is important to remember here that MRSf denotes the rate at which the mother is willing to decrease her own private consumption to receive one more unit of child quality while remaining at the same level of overall welfare. A woman’s decision to become a mother can be understood by:

.(y , π) (y )I f = ψ f −ϕ f > 0 If the utility she gets from having a child is greater than the utility of remaining childless, she will choose to become a mother. Here, the fertility demand is assumed to be normal so there is a positive relationship between income and childbearing. This means that with higher income the quality of child will also increase. Hence, normality holds when . AI /ΔyΔ f f = ψy −ϕy > 0 woman’s marginal utility of consumption is higher when she has a child as compared to a childless state. Therefore, a woman would value consumption more as a mother because you have to consume less in general in order to provide for child care. An alternative explanation of this is that the woman’s marginal utility of private consumption is diminishing and independent of parenthood (because positive expenditures on the child implies higher consumption level in a childless state). If, however, then the woman will decide to remain childless unless her income increasesI f < 0 past a critical value ( ), holding constant. An increase in will reduce utility in a state with ŷ π π children but it will have no effect on utility from childless state. So a woman will choose to become a mother if and critical value increases as increases .(π)yf > ŷ π Δŷ/Δπ ) ( > 0

3. Establishment of Paternity and Gains to Marriage The utility function for males is represented by v = v (cm, n, q).

Variable Explanation

cm man’s consumption on all goods (excluding any child­related expenses)

n number of children

q quality per child

This utility function assumes that a man’s utility from children does not depend on interacting with them or cohabiting with the mother of his children. (Willis makes this assumption in order to simplify the analysis.) Therefore, a male without any identified children has the utility function v = v (ym, 0, 0). Here, ym replaces cm because all of his income is spent on consumption not related to childcare. This can be seen in the following equation representing his budget constraint:

ym = cm + nq π For a man with no identified children, is equal to zero. Thus, ym = cm. Therefore, when anq π single woman decides to have a child (implying that her income is greater than ŷ( ) and the) π father does not recognize or plan to financially support the child, she incurs all of the costs and spends q=qs(yf, ) on the child. π

Willis then moves to consider the case where a father recognizes that he has a child and observes the level of child quality. It is important to note that the father’s utility is increased by establishing paternity (i.e. recognizing that he’s the father) if he:

1) has a positive preference for children 2

2) and, if the mother’s expenditure on the child is sufficiently high. This would be the case of a single father whose child is being financially supported solely by its mother. The utility for this type of father is denoted vs and is represented by:

vs = v(ym, 1, qs(yf, )) > v(ym, 0, 0) π Here, it is important to note that the utility of a single father where only the mother financially supports the child is greater than the utility of a man with no identified children because the single father has a positive preference for children.

2 Father might have a minimum standard for child quality below which he would prefer not having a child. If, however, mother’s standard of child quality is at least as high as the father’s, then she can voluntarily have the child. This constraint will not be binding for the father.

However, because the father has a positive preference for children, this situation is not pareto optimal. That is, a pairing other than (us, vs) can be chosen so that both the mother and father are better off. This new pairing can be reached by increasing the level of child expenditure to a level greater than qs(yf, ), assuming that both parents share the expense. In π order to prove this, it is first important to note that having a positive preference for children implies that the father will always value increases in q. Therefore, MRSm = vq / vc > 0, meaning that the father is willing to decrease his own consumption in order to increase q for his child. Since MRSf = , it follows that π

MRSf + MRSm > π if the the father is identified, but only the mother financially supports the child. This inequality violates the Samuelson condition for efficiency in the provision of a collective good. This 3

condition implies that , where MRT is the marginal rate of transformation ofRS RT∑n

i=1M i = M

parents income into child quality. However, if the mother’s utility is maximized subject to the joint budget constraint of

both parties, y = yf + ym + nq π

and the constraint vs, which ensures that father is not made worse off by sharing the expense, v ≥ the mother’s utility will exceed , which is the result of the utility maximization for a single u mother with a non­contributing father That is, when both parents share in the cost of child­rearing, the mother is better off than when she has to fully support the child on her own. The resulting shared resource allocation now satisfies the Samuelson condition

MRSf + MRSm = . π The gains to efficient allocation can be represented by the indifference curve diagram below

3 The Samuelson conditions, according to Samuelson (1954), holds in case of efficient allocation of goods. In an economy with n consumers, the summation of marginal rate of substitution of all the individuals will give the marginal rate of transformation of the economy between a public good and an arbitrarily selected private good.

Important Observations from Graph

1. The father’s indifference curves are derived by substituting the couple’s joint resource constraint into his utility function.

v = v(cm, n, q) = v(ym + yf ­ cf ­ , 1, q)q π

2. The set of efficient points satisfying the Samuelson condition is given by the contract curve generated by the locus of tangency points between the mother’s and father’s indifference curves.

3. The segment ab (the highlighted portion of the contract curve) indicates the Pareto­improving allocations that increase the welfare of one or both parents relative to the allocation at point s.

4. V increases as cf is the decreases. U increases as cf increases.

5. With pooled resources, the couple can attain both higher individual utility and a higher level of child quality.

Therefore, if an efficient level of child care is chosen, the couple’s demand function for child quality can be written as

q = q (yf + ym, , ).e π u

Although we are looking at a collective demand function, it has properties of unitary model. This model assumes that in a normal case, the income effect will be positive and the price effect will be negative.

Properties of Demand Function

1. A single decision maker controls total family resources; this implies that in a normal case there will be a positive income effect and a negative price effect.

2. The demand for q may depend on the distribution of welfare between the father and mother in collective models. This is reflected in which denotes the level of welfareu received by the mother. If mothers are more altruistic than fathers than and q may beu positively related.

3. When the father captures the gains to coordination of parental resources, the level of q is lower than when the mother captures all the gains (i.e. ).qa < qb

4. The level of q is always higher when parental resources are allocated efficiently regardless of the distribution of welfare. qs < qa < qb This is because the marginal cost of child quality faced by each parent is less that π when parents share the cost of child.

The effect of distribution of welfare on q can be determined by totally differentiating the following equations:

(1) MRSf + MRSm = π (2) y = yf + ym + nq π (3) (c , q) u f = u

Solving this system of equations for the effect of an increase in mother’s utility on quality of child, we get (see math appendix for full derivation):

.)( )dudq = ( dCf

dMRSf − dCmdMRSm 1

D

The value of D is positive; hence, an increase in mother’s welfare will increase child quality along the contract curve if > .dCf

dMRSfdCm

dMRSm These results can be best be summarized in the table below.

Condition Implication Explanation

If MRSf = MRSm givenΔ Δ cf = cmΔ Δ

= 0dudq Increasing mother’s utility will have no impact on child quality when you equally increase mom’s consumption and decrease dad’s consumption.

Mother and father are equally altruistic.

If MRSf > MRSm givenΔ Δ cf = cmΔ Δ

> 0dudq Increasing mother’s utility will positively increase child quality when you equally increase mom’s consumption and decrease dad’s consumption.

Mother is more altruistic than father.

If MRSf < MRSm givenΔ Δ cf = cmΔ Δ

< 0dudq Increasing mother’s utility will negatively increase child quality when you equally increase mom’s consumption and decrease dad’s consumption.

Mother is less altruistic than father.

That is, in order for a child to be better off by increasing the mother’s welfare/happiness, a one dollar increase in the mom’s private consumption raises her marginal value of child quality by more than a one­dollar decrease in the father’s private consumption decreases his marginal valuation of child quality. In this paper, Willis assumes that, on average, mothers are more

altruistic than father. Therefore, in this paper, > 0.dudq

Special Case Suppose that both parents have altruistic utility functions:

n(c ) ln qu = l f + αf n(c ) ln q v = l m + αm

Further assuming that mothers are more altruistic than fathers (i.e. ), we can derive :αf > αm 4

dCfdMRSf − dCm

dMRSm = qα − αf m

This implies that and that the contract curve is positively sloped.dudq > 0

To avoid complication due to the correlation between spending on a collective good and income distribution, utility is assumed to be transferable . This means that the demand for collective 5

goods is independent of distribution of welfare. According to Bergstrom and Cornes (1983) for this to hold, a necessary and sufficient condition is that all individuals must have the following utility function:

(q)c (q) ui = A i + Bi According to the author, incorporating these results into the model have no effect on the demand for q if and , where A(q) is a common arbitrary function(q)c (q)u = A f + Bf (q)c (q) v = A m + Bm for both parents and differs between both parents. Given these preferences, there will exist (q) Bi a vertical contract curve and the collective demand function will be the same as unitary demand function.

Important Implications

Given transferability condition, a man and woman who allocate their resources efficiently will choose to have a child if their joint income exceeds the same critical value established in section one .(π)y = ym + yf > ŷ

Cooperation between mother and father is important for childbearing in “traditional” households where men provide the major share of parental income.

Since father also gains from enhanced well­being of the child, pooling of parental resources is beneficial for both parents and for the children as well. An assumption for this to hold true is that married couples can achieve efficient allocation of resources that is not feasible for separated couples.

Now assume unmarried couples play a noncooperative Stackelberg game where the mother has child’s custody and the father can only influence expenditures on child by transferring income to the mother. The mother’s behaviour is identical to that of a single mother with unidentified father except for the income transfer.

4 Derivation shown in Math Appendix B. 5 Transferability of utility is when the pooled income of both parents are under a cooperative game (M. Browing, Chiappori, & Lechene, 2006). In this case one player can loosely transfer a portion of their utility to the other player.

If T denotes the amount of his income that the father gives to the mother, then the mother’s demand function for child quality is qs (yf + T, ) and MRSf = . π π The father will then choose T so that his utility is maximized. That is, he will maximize

v(ym ­ T, qs(yf + T, )) subject to T 0. π ≥ The resulting first order condition implies

MRS m

≤ 1π (dq/dy)

where (dqs/dy) is the mother’s marginal propensity to spend on children (i.e. the proportion of π her increased income that she will spend on children).

Important Implication

The marginal cost to the father of inducing the mother to increase expenditure on the child by one dollar is equal to the reciprocal of the mother’s marginal propensity to spend on children, which is greater than one dollar.

For example, if the father transfers $1 to the mother and she spends only $0.25 more on the child and $0.75 more on her own consumption. That is, if T = 1 and = 0.25, then MRSm = $4.00.dy

dq Therefore, in order to increase the mother’s expenditure on the child $1.00, the father will have to transfer $4.00 to the mother. Because of this, the level of child quality in the noncooperative equilibrium will be smaller than it is within marriage. This is shown on the graph below.

Important Observations from the Graph

The mother’s income expansion path is represented by sy. This line represents the result to the single mother’s utility maximization problem subject to a budget constraint that increases due to T.

At point s, the father transfers no money to the mother and, as a result, she supports the child wholly on her own income. Therefore, she chooses qs.

Father maximizes his utility at the tangency between sy and his indifference curve at c by making a transfer of Tc to the mother. She then chooses qc.

If the mother and father married and pooled their resources, they they would choose qm.

Note, qs < qc < qm. The child will always receive the highest level of quality when the parents are married even if the father transfers his entire income to the mother.

This model demonstrates that when parents marry the father is incentivized to spend more on the child. These reasons are summarized in the table below:

Single Married

Father Marginal cost to him of increasing expenditure on children is greater (perhaps much greater) than a dollar.

“Induced to substitute away from his own consumption toward child quality because the marginal cost to him of increasing expenditures on children is less than a dollar when he shares child costs with his wife.”

Mother Faces MRSf = . π Reduced shadow price of q faced by mother from when π she is a single mother to MRSf = ­ MRSm when she π is married.

IMPORTANT: Marriage leads both parents to substitute away from their own consumption towards child quality. Therefore the joint income required for a couple to desire to have children is lower within marriage than it is for the same couple to desire a child out of wedlock. 4. Out­of­Wedlock Fatherhood As we have seen in the previous section, out­of­wedlock fatherhood leads to a non­cooperative equilibrium that increases marginal cost of child quality. It may also decrease the ‘cost of fatherhood’ to a lower level or even to zero. Through the various parts of this section the author shows that a man prefers nonmarital fatherhood under the following conditions:

a. lower absolute level of male income, b. higher female incomes relative to male, c. and, higher absolute female income.

Impact of Out­of­Wedlock Fatherhood on Cost of Fatherhood Suppose that the amount of transfer made by the father of an out­of­wedlock child is determined by the noncooperative Stackelberg game. Then, the optimal amount of the transfer is determined by the following function:

(y , y , π)T * = T m f It is important to note that the amount of money that the father is willing to transfer to the mother increases when his income increases and decreases when mother’s income increases:

and dymdT* > 0 dyf

dT* < 0

If initially the husband’s and wife’s income corresponds to point s in figure 2, then the optimal transfer occurs at point c where father’s indifference curve is tangent to mother’s income­expansion path. So the optimal transfer T* corresponds to mother’s budget of yf + T * and father’s budget of .T ym − * Impact of an Increase in Mother’s Income on Cost of Fatherhood Now, let us suppose that the mother’s income rises toYf + T* independent of the father (e.g. she receives a raise at work). This is represented by the blue dashed line in the graph below.

Let us also assume that the father’s income falls by T*. The mother is now able to afford qc without the assistance of the father. Let us also assume that the father’s income decreases by T*. Therefore, if he transfers no money to the mother, he is able to remain on the same indifference curve, and the child still receives qc. In this case, the mother bears all financial responsibility for the child, so the cost of fatherhood is zero. If the mother’s income continues to increase, the father will continue to transfer nothing; however, his overall utility will increase because the mother will choose increasingly higher levels of child quality. On the other hand, if the mother’s income is held constant at the dashed blue line below, an increase in the

father’s income would cause his demand for child quality to increase, and he will choose to make a transfer of T>0 to mother and the child will receive q > qc. Advantages of Out­of­Wedlock Fatherhood In this section, we now move to consider the advantages of out­of­wedlock childbearing for the father when:

a. he has children by more than 1 woman b. the women assume if not all then the most of the financial responsibility for

children c. women are relatively more numerous than men.

Assumptions

Women have identical preferences and income.

Male’s utility function is additively separable.

Each of the children of the ith partner receives the same level of expenditure so that . ni qij = qi

The father co­resides with either all or none of the children from the ith partner so that .dij = di

The father places higher weight on the quality of children with whom he lives.

Variable Explanation

P number of partners male has children with

dij binary variable for whether the father lives with the child or not

ni number of children with the ith partner

Explanation of Equation Components

is the utility attained by the father from his private consumption(c ) V m

is the weight a man gives to the utility attained by his children from all partners(P ) β

is the degree to which father cares about the quality of child jth child that is dependent(n , )α i dij on the number of children he has from the ith partner and whether he resides with the ith partner.

is the total weight father places on n number of children that he has(n , ) α(n , )∑ni

j=1α i dij = ni i dij

from partner i.

is the demand for child quality for ith child from jth partner.(q )f ij

Fatherhood with Single Partner With a single partner we have and , and we assume that each child receives the(1) β = 1 di = 1 same level of expenditure. The fathers utility in this case will be .Holding(c ) (n)nf(q) v = v m + α the number of children constant, the marginal utility of additional expenditure on child quality is positive but decreasing. Following Becker and Barro (1988) model, the degree of altruism towards each child is assumed to decrease as number of children increases. Even though , α′ < 0 total altruism increases (i.e. , where is the elasticity of ). Thus if[α(n)n]/dn 1 )n d = ( + ε > 0 ε (n) α

is more elastic the father will be more altruistic. These assumptions are important for(n) α empirical implication of models of marital fertility with altruistic parents so the quality per child, q, has a positive income elasticity. One reason the degree of altruism per child decreases as number of children grows is that parent’s interaction with each child reduces with more children. Similarly, having two children with one partner would give the father higher utility as opposed to having two children with different partners. In addition to this, increasing the number of partners means the time spent with each partner decreases which increases the uncertainty of whether the children born to a given partner are his own. These disadvantages of fathering children with multiple partners are reflected in the function , that is a decreasing utility function of the number of partner one(P ) β has (i.e. ).(P ) β′ < 0 Deciding Whether to Have Children Out­of­Wedlock

Now assume a married man cannot father and provide for children out of his marriage without breaking his marriage. Thus he must choose whether to marry one woman or have children out­of­wedlock. It is also assumed that a man who marries and has children with only

one woman obtains a utility level , that is greater than any level of utility that he can obtain vm from fathering the same number of children by the same woman out­of­wedlock.

If the males chooses to have children out of wedlock by P different women, the number of children and the level of child quality are determined by each woman. That is, she will choose her optimal number of children and the optimal level of quality for each child subject to her budget constraint. If she receives any money from the father, this amount will be included in her budget constraint. Therefore, the mother’s demand for children can be represented by

ni = ns(yfi + Ti, ) π and her demand function for child quality can be represented by

qi = qs(yfi + Ti, ). π Here, the subscript i specifies the mother. Willis assumes that Ti, ni, and qi are determined in P separate Stackelberg games between the mother and the father (this is discussed in Section 3). Next, assume that the woman’s optimal choice of number of children and child quality is denoted by (ni*, qi*). She chooses this bundle if the man makes his optimal transfer of T* to her. If we assume that each woman has the same preferences and income, then the man’s transfer function is given by T* = T(ym, yf, , P). π

Important Observations Explanation

> 0dymdT* As the father’s income increases,he transfers more money to the

mother.

< 0dyfdT* As the mother’s income increases, he transfers less money to the

mother.

T* = 0 Occurs when the mother’s income is substantially high relative to the father’s income.

< 0dPdT* As the number of partners increases, the father transfers less

money to the mother if T* > 0 . This is due to that fact that additional partners reduce the income he has available for his own consumption and < 0, his interest in his children’s welfareβ′ relative to his own consumption tends to decrease.

The relationship between an unmarried male’s utility and the number of partners is given by

Differentiating this equation with respect to P, gives the effect of an additional partner on the males utility. This is represented by

The variables in this equation are summarized in the table below.

Variable Value Explanation

θ < 0 θ Assume: < 1 θ| |

elasticity of transfers with respect to the number of partners.

< 1 implies that a male’s private consumption θ| | decreases as the number of partners increases because the increases in transfers caused by adding an additional partner is not offset by decreases in transfers to the other partners

η < 0 η Assume: < 1 η| |

elasticity of with respect to the number of partners.(P ) β

ε < 0 ε Assume: < 1 ε| |

elasticity of with respect to the number of children.(n) α

δ > 0δ elasticity of f(q).

sT share of transfers in the mother’s total budget sT = .T*

y + T*

Eny Eny > 0 mother’s income elasticity of demand for the number of children.

Eqy Eqy > 0 mother’s income elasticity of demand for the quality of children.

Given the equation and assumptions above, the impact of an additional partner on a father’s utility depends on the balance between the negative impact on his private consumption and the positive effect of added utility from additional children. A graphical representation of male utility as determined by his number of partners is depicted below.

If for him T* = 0, then his utility is a monotonically increasing function of the number partners (represented by VsA). That is, if the father gives no financial support to his children, then he will be “better off” each time he has a child out­of­wedlock with a different woman. However, if his T* > 0, then at some point the disutility from transfers dominates and additional partners will decrease utility (represented by VsB).

Males are presented with three options for fatherhood:

1. be single and childless 2. get married and have all of his children within that marriage 3. be single and have all children out­of­wedlock

If the joint income of a man and a woman is above the income threshold to have children, then the male will choose either option 2 or option 3 (pending he as has a positive preference for children). In the graph above, it can be seen that the utility of males in marriage is higher than that of males who decide to have children out­of­wedlock with one woman (P = 1). However for men who have T* = 0, there exists some number of partners where a man is indifferent between having children within marriage and having children out­of­wedlock. This number of partners is denoted , and Willis refers to this at the “reservation number of partners.”P

︿

Condition Explanation

P < P︿ Better for man to be married than be single and have kids with P partners

even if T*=0

P > P︿ Better for man to be single and have kids with P partners and T*=0 transfer

Therefore, a man will avoid marriage if he has at least partners available to him. Note, forP

︿

men who T* > 0, may not exist. That is, they may never be “better off” fathering childrenP︿

out­of­wedlock. However, there is also a psychological reason why some men may not feel “better off” having children out­of­wedlock. This could occur because, as the number of partners and children increases, the father’s interest in (utility derived) from children can rapidly decrease.

If a woman has an income lower than the threshold income where should we agree to have a child, the man must make a positive transfer in order to increase her income to at least the threshold amount if she is to bear his child. Therefore, the price of having, ni, children by the ith partner for a single father can be given by

Here, f ( i , i) is the minimum income that a woman would require in order to decided toy︿ π η have child and yf is her income. Notice that as yfi increases, 0. That is, when a woman’s T

︿→

income is sufficiently high without transfers from a man, she would be willing to have his child and take on all of the financial burden. As mentioned above, the value of will determine if T

︿P︿

exists (see chart below).

Condition Explanation

= 0 T︿

existsP︿

> 0 T︿

may or may not exist but has a higher probability of existing with the lower P︿

P︿

T︿

If exists, then it is given by the function = P(ym, yf, ). In order find , you must equateP

︿P︿

π P︿

the utility of an unwed father with that of a married one.

The value of P that solves this equation will represent the number of partners that a man would need to be indifferent between marriage and unwed fatherhood.

Assuming that all partners have the same income and women do not obtain transfers higher than the supply price, the effect of male income on can be found by differentiating theP

︿

above equation with respect to ym. The resulting equation is

In order to understand this equation, first, note the following inequalities.

Married Single Explanation

cmm < cm

s Male private consumption for men is higher outside of marriage.

Vymm > Vym

s The male marginal utility of income is higher in marriage.

The numerator of this equation is always positive, and the denominator is positive when Vs intersects Vm from below (when P < , men get higher utility from marriage). As Ym increasesP

︿

increases. The more money a man makes, the number of partners at which Vm = VsP︿

increases. Therefore, the more money a man makes, the more preferable marriage becomes. The impact of an increase in the woman’s income, depends on the value of T*. If T* > 0, then the impact can be determined by the differentiating the same equation from above with respect to yf. This is given by

If a woman’s income increases, then the father will decrease T* by that amount. While the sign of this expression is ambiguous, it is more likely to be negative the larger the number of partners. If T* = 0, then the impact of an increase in the woman’s income is given by

. The sign of this equation is also ambiguous, but is more likely to negative the larger the number of partners.

The resulting signs of these three derivatives suggest the following implications:

When men have higher incomes, they prefer marriage to being single fathers. When women have higher incomes, men increasingly prefer being single fathers because

they are able to contribute less or no money to the child’s expenses. When women have relatively higher incomes than men, the number of reservation

partners is reduced and men increasingly prefer being single fathers. 5. Marriage Markets and Childbearing within Marriage Decision of marriage and childbearing no longer involves bargaining between a man and a woman. Thus, men compete with other men and women compete with other women in forming sexual and reproductive alliances. This can be understood by Beckerian model:

Given equal number of men and women with identical preferences and income levels

Assumptions Implication

If joint incomes are sufficiently high to justify having a child

All persons will choose to marry, have children and spend efficiently on each child.

If joint incomes are too low All partners are assumed to single and childless.

The marriage market does not determine how the gains of marriage are distributed between husband and wife. Bargaining helps determine their respective level of private consumption. When incomes vary among members of each sex, standard Beckerian marriage market outcomes follow what is called a traditional market equilibrium where:

If males outnumber females or if females have relatively low income as compared to men

Assumptions Implication

Existence of household collective goods (in this case children)

Leads to positive assortative mating where highest income man marries highest income woman and so on until no unmarried woman remains or joint income of highest ranked man and woman fall short of critical y.︿

Men and women who fail to marry remain childless.

The distribution of welfare between husbands and wives within marriage will depend on the numerical balance of genders.

If both men and women have identical preferences and incomes

Assumptions Implication

If men are in excess supply Competition amongst men implies that married men will receive the same utility as single men and women capture all gains from marriage.

If women are in excess supply Married and single women will receive the same utility and men will capture all gains from marriage.

The distribution of welfare within a marriage between a man and a woman is bounded by the utility each would receive in the next best marriage.

If income varies

Assumptions Implication

As the number of participants grow Competition in the marriage market increases to determine the equilibrium price that gives the parametric utility to men and women conditional to their income.

All couples for whom marriage and childbearing is optimal will receive higher utility from marriage.

6. Marriage Markets and Out­of­Wedlock Childbearing The marriage market equilibrium changes dramatically when men are relatively scarce and women have higher incomes compared to men. Their behavior can be seen by where they lie in terms of income distribution.

Income Distribution Behavior

Upper portion of the income distribution Men and women choose to have children within

marriage

Lower portion of income distribution Women bear children out­of­wedlock with one or more men, who choose to father children by more than one women rather than marry.

Assumption

All members of each sex have identical preferences. However, men and women may not have identical preferences.

Individuals differ from others of the same sex by income level.

Each woman may have at most one child so that a female’s fertility choice is whether or not to have a child.

The aggregate supply curve of men to out­of­wedlock fatherhood for any given number of partners per man, P, as a fraction of number of men, , for whom , is given by:Sm P ≤ P

︿

,(P ) (P (y , , )Sm = Sm ≤ P︿

= Sm ≥ P m yf π where is assumed to be monotonically increasing function of male earnings such(y , , )P

︿= P m yf π

as the curve in the figure below. If the male population is ordered by income in ascendingSSm m order when all females have identical incomes, will be an increasing function of . ThisP

︿ym

means that men with higher incomes require more number of partners in order to prefer out­of­wedlock fatherhood. Thus, shows the value of for males such that there isSm P

︿

one­to­one correspondence between and along . The shape of the curve comes fromym P︿

SSm m cummulative distribution of male earnings transformed by .(y , , )P m yf π

If there is dispersion in female income then we will consider assortative mating where marriage is ruled out and mates play a Stackelberg game to determine the shape of Males have anS .Sm m incentive to marry highest­income female because they have lower supply price of child quality. If females face competition, they will be indifferent to male income. However, high income males will pay a premium above the supply price to mate with high income females and maintain positive assortative mating by attracting a preferred female. Hence, even if female income varies, positive assortative mating will still hold in a nonmarital setting and shape of will beSSm m related to cummulative distribution of male earnings. The availability of sex partners to a single man depends on the overall number of men and women minus the men and women who are married:

P = N −MfN −Mm

If there are more females than males, then the number of partners per male increases as a fraction of men who can marry increases. This is reflected in the “demographic availability” function:

WFO = 1 − MNm =

R−1P−1

Here OWF is the fraction of men who father out­of­wedlock and P>R>1 holds when females are in excess. Equilibrium holds when:

(P ) WF (P ) Sm ≥ P︿

= O This is shown in figure 4, where demographic availability curve intersects At theS .Sm m intersection, the marginal man requires at least two partners in order to choose out­of­wedlock fatherhood as oppose to marriage. If female income increases relative to male income the curve will shift to the left and menSSm m will require less partners to reject marriage.

7. Discussion and Conclusions

Important Implications

Out­of­wedlock childbearing can be explained in an equilibrium outcome of marriage market behavior given that men may free­ride and women are able to raise children at low economic cost themselves.

Out­of­Wedlock childbearing will exist when women are in excess of men, they have sufficient income to support their families, and relative gains to marriage are small because male income is low.

“Winners” in an out­of­wedlock equilibrium: 1. This equilibrium benefits adults at the cost of child. Children born out­of­wedlock will

have lesser resources available to them than those born to married couples. 2. Men would benefit at the expense of those women who would normally marry and would

receive higher gains from marriage through their spouse’s resources. 3. Some low­income women who would normally remain childless in a traditional

equilibrium.

8. Extensions Willis proposes to extend this paper by shifting to a dynamic framework where having children out­of­wedlock has an adverse impact on a woman’s future marriage prospects. Therefore, a woman’s perception that she will have fewer marriage prospects that will be able to contribute financially will incentivize her to have a child as a single mother and bear all financial responsibility. This increase in the number of women willing to be single mothers will, in turn, increase the attractiveness of single fatherhood to men. Willis also proposes to look at the consequences that this model has on marital search as an extension. Marriage typically tends to follow assortative mating where spouses are matched by education and income. Out­of­wedlock childbearing reduces the incentive for assortative mating because the father plays a smaller role in the household’s economy. In the paper, The Spread of Single Parent Families in the United States since 1960, the authors go beyond Willis’ model of out­of­wedlock childbearing to further explore the impacts of households led by single mothers on white and black communities. They also note the role that single motherhood can play in the persistence of poverty in low income areas. Additionally, their claims fully support Willis’ model in that they explain the increase of out­of­wedlock childbearing as the result of the increased importance of economic calculations when making marriage and fertility decisions instead of sexual attraction.

9. Empirical Evidence (Interesting Charts and Graphs)

10. Bibliography

1. C. Becker, Lecture notes on “The CES function, demand, comparative statistics, and integrability,” Lecture 4, Econ 205 Microeconomics Theroy, Duke University (2009)

2. C. Becker, Lecture notes on “Money Matrics and Comparative Statisticcs,” Lecture 5, Econ 205 Microeconomics theory, Duke University (2009)

3. D. T. Ellewood, & C. Jencks, (2004). The Spread of Single­Parent Families in United States since 1960. KSG Working Paper No. RWP04­008.

4. M. Browning, P. Chiappori, & V. Lechene, (2006). Collective and Unitary Models: A Clarification. Review of Economics of Household.

5. P. A. Samuelson, (1954). The Pure Theory of Public Expenditure. The Review of Economics and Statistics, Vol. 36, No. 4

Math Appendix A The father’s preferences are represented in space by indifference curves that are derivedc , q)( f by substituting the couple’s joint resource constraint into his utility function. That is,

.(c , n, q) (y q, 1, q)v = v m = v m + yf − cf − π The slope of his indifference curves are given by in space. In order to find the slope ofdq

dcf c , )( f q his indifference curves, we start with the Samuelson condition:

MRSf + MRSm = π Note, ­MRSf = MRSm ­ because MRSf is always negative because the mother’s indifference π curves are downward sloping in space.c , )( f q Therefore,

Uc

Uq = V cV q − π

.dqdU 1

dCdU = V c

V q − π

dqdC = V c

V q − π RSdq

dC = M m − π The father’s indifference curves are positively or negatively sloped, depending on whether his marginal valuation of child quality is greater or less than the marginal cost of child quality; they have a zero slope when the two are equal. Math Appendix B By totally differentiating the following equations:

1. MRSf + MRSm = π 2. y = yf + ym + = cf + cm + nq π nq π 3. (c , q)u f = u

We get the following system of equations: 1’. )dc )dc dn )dq du dy π( Uc

Uqc −U2c

U Uq ccf + ( V c

V qc − V 2c

V Vq ccm + 0 + ( Uc

Uqq −U2c

U Uq cq + V cV qq − V 2

c

V Vq cq = 0 + 0 + d

2’. qdn ndq du y qdπdcf + dcm + π + π = 0 + d − n 3’. dc dc dn dq u dy dπUcf f + 0 m + 0 +Uq = d + 0 + 0 Let the following matrix be A:

)( Uc

Uqc −U2c

U Uq cc )( V cV qc − V 2

c

V Vq cc 0 )( Uc

Uqq −U2c

U Uq cq + V cV qq − V 2

c

V Vq cq

1 1 q π n π

Ucf 0 0 Uq

where matrix A is multiplied by the following 4x1 matrix:

dcf

dcm

n d

q d

We can also write the right hand side of the equation in form of matrices to show how we arrive to the solution of Thus, let the following be matrix B:.du

dq

0 0 1

0 1 π − n

1 0 0

which can be written as a multiple of the following 3x1 matrix:

u d

y d

π d

Using the cramer rule we can arrive at the solution by taking the inverse of matrix A and solve for these system of equations to get:

Bdudq = A−1

Hence, by solving this system we can write as:dudq

)( )( dCfdMRSf − dCm

dMRSm 1D

Where D is a positive determinant.

Math Appendix C Let and , then we can show that if mothers are moren(c ) ln(q)u = l f + αf n(c ) ln(q) v = l m + αm altruistic than fathers will be positive.du

dq Given , from the utility function u we get:RSM f = Uq

Ucf

RSM f = 1/cf

α /qf = qα cf f

Similarly using utility function v and MRSm we will get

RSM m = 1/cmα /qm = q

α cm m By derivating the MRS function with respect to consumption we can show that:

dCfdMRSf − dCm

dMRSm = qαf − q

αm = qα − αf m

Since this term will be positive which shows that in equation 12.αf > αm dudq > 0