Dec 11, 2001 - equilibrium model of nonmarital cohabitation, marriage, and divorce that is consistent with current ... For example, marriage taxes and bonus have been implemented by allowing for ...... âSeparate Spheres Bargain- ing and the ...
Cohabitation, Marriage, and Divorce in Equilibrium Michael J. Brien Arthur Andersen LLP
Shannon N. Seitz Queen’s University
Steven Stern University of Virginia December 11, 2001 Abstract The objective of this research is to develop and estimate an economic equilibrium model of nonmarital cohabitation, marriage, and divorce that is consistent with current data on the formation and dissolution of relationships. We use a matching model with endogenous relationship-speci…c capital accumulation.
1
Introduction
There is much discussion about the role of government in in‡uencing marriage markets. The most obvious place where government in‡uences marriage markets is in the rules it promulgates with respect to ending marriages. Most of the rules are meant to discourage divorce by increasing the cost of divorcing. Over time, society has reduced those costs, for example, by instituting no fault divorce provisions in law.1 But, even in no fault divorce states, the cost of divorce, and even the incremental cost associated with government rules ]. Besides increasing divorce costs, many sociis signi…cant. [ eties provide signi…cant subsidies to, or extract signi…cant taxes from, married couples. For example, marriage taxes and bonus have been implemented by allowing for di¤erent brackets and deductions for married couples versus single individuals in the tax code.2 The e¤ect of marriage taxes have been stud-
Quantify
1 A recent literature examines the e¤ect of no fault divorce laws on divorce rates. See, for example, Stevenson and Wolfers (2000) and Friedberg (1998). 2 There are also instances throughout history where taxes were imposed directly on individuals choosing to remain single. The Globe and Mail (December 6, 1926) reports “In Rome, Premier Mussolini’s cabinet approved a special tax on bachelors between the ages of 25 and 65. The measure was based ‘on the principle that it is a man’s duty to marry and rear children, and that the Government must intervene to provide judicial punishment for failure on the part of citizens to ful…ll their moral obligations”’.
1
ied extensively in the literature (Chade and Ventura (2001a, 2001b); Alm and Whittington (1995); Sjoquist and Walker (1995)). There remains the question of why society should have an interest in reducing divorce rates or subsidizing marriage. There are a number of possibilities put forth in the literature. The most obvious reason to discourage divorce is that, in families with children, there is signi…cant evidence that children are hurt by divorce (see Gruber (2000) for a discussion of this literature). To the degree that parents do not internalize the cost of their divorce on to their children, there is an externality that government might be able to help parents internalize by increasing divorce costs. There are a number of problems with this argument. First, it is not so obvious that divorce hurts children once one takes into account the selection e¤ect. The relevant choices for children in families with unhappy parents is to have the parents divorce and construct a reasonable custody arrangement or to have the parents stay together unhappy. The choice set does not include living in a family with happy parents who want to be with each other. [ ]. Second, it is di¢cult to measure to what degree potentially divorcing parents internalize the cost to their children of a divorce. If parents internalize the whole cost, then there is no role for government. Third, there are signi…cant divorce costs even in families without children. Obviously, any argument about the cost of divorce to children does not apply here. Others have suggested that marriage is an institution that increases welfare, ). Thus, society might have a role in encouraging especially for men ( it (or discouraging divorce). The problems with this argument is that a) the empirical evidence in support of this argument is very weak in that it is not successful in controlling for the endogeneity or marriage; b) similar evidence suggests marriage is not such a great deal for women; and c) there is no reason to believe that, if such e¤ects did exist, potential couples would not internalize them and make e¢cient decisions about marriage. One could argue that marriage, at least partially, serves an insurance role ). Each member of the couple can insure the other against bad shocks, ( at least to the degree they are uncorrelated. Society might have a role in discouraging divorce to help strengthen the enforcement mechanisms of such insurance arrangements. The argument is that, without signi…cant divorce costs, the member of the couple transferring resources would have an incentive to renege, especially if the transfer was large or the negative shock was permanent. While such an enforcement mechanism might increase the value of the insurance component of marriage, there would be nothing that prevented the couple from writing a marriage contract that provided the optimal enforcement mechanism. Furthermore, an optimal enforcement mechanism would specify a division of assets that recognized the insurance component of marriage rather than impose costs on separation. Finally, it is not clear that discouraging divorce would prevent reneging (see, for example, Lundberg and Pollack, 1993). Some research has focused on the endogenous nature of investment in the
Discussion of literature on this
point
cites
cites
2
cite
relationship ( ). The idea is that the couple might invest more in the relationship if the members of the couple know that the relationship is less likely to end. If society increases the cost of divorce, then couples will divorce less frequently and invest more in their marriages, thus improving their marriages (and decreasing the probability of marriage even further). Thus, it might be worthwhile for government to discourage divorce to increase the amount of investment in relationships, especially if each member of the couple does not internalize the increased value of lower divorce probabilities to their spouse. We show in this paper that such an argument does not work even when there is signi…cant endogenous investment in relationships because, in an environment with Nash bargaining, each member of the couple does internalize the total cost (of reduced investment) of divorce probabilities. Finally, some literature (Drewianka, 1999) suggests that there are externalities on the size of the marriage market when a couple divorces. If a couple divorces, then one extra man and one extra woman are available to other single women and men respectively to marry. This improves the expected outcome of participating in the marriage market for the remaining single women and men as it becomes easier to make a contacts with other singles. Yet the divorcing couple does not internalize the bene…t of their divorce to other single women and men. It may therefore be the case that this externality results in an ine¢cient number of matches in equilibrium. In fact, there is another e¤ect as well. A divorcing man entering the marriage market is not the average single man. He may be an unusually good “catch” given that he was once a good enough match for someone to marry. If this is the case, then there is an extra bene…t to single women associated with his divorce. Alternatively, he may be an unusually bad “catch” given that he divorced. If this is the case, then there is an extra cost to single women associated with his divorce. This paper models these equilibrium e¤ects on the distribution of single people and then estimates their size. We anticipate that the results will show that government e¤orts to discourage divorce are not welfare improving.
2
Model Speci…cation
Every period single people meet other single people. The match speci…c component of the match at the beginning of the match is µ 1 . Let the characteristics of a potential partner be X k for k = h (male or husband); w (female or wife). Note that X k does not vary over time. We do this to greatly simplify equilibrium computation later and without much loss in generality. In Brien, Lillard and Stern (2001) (BLS), the only time varying exogenous variables in X were education and a dummy for whether one was in school. If we assume that one knows his/her path of X’s over time, then we can capture both education and an “in school” dummy by comparing …nal education to age. It is assumed that the µ1 between male h (husband) and female w (wife) at time t is ¢ ¡ µ1 » iidN 0; ¾ 2µ : 3
Let dt be the duration of the relationship. It is assumed that µ evolves over time according to µd ¡ µ 1 = ½ (µd¡1 ¡ µ1 ) + ´ t with 0 · ½ · 1,
(1)
¢ ¡ ´t » iidN 0; ¾2´
as long as the relationship remains intact. Note that, if j½j < 1, then µd is stationary around µ1 , while, if ½ = 1, then µd is a random walk starting at µ1 . Each period t, a potential couple can choose to not have a relationship (mt = 1), cohabit (mt = 2), or be married (mt = 3). Couples in a relationship must choose how much to invest in the relationship it . The cost of investing is q (it ) with q 0 (it ) > 0, q 00 (it ) > 0. Investment it allows the couple to accumulate relationship-speci…c capital Ht (RSC) according to ½ H1 if dt = 1 Ht = : (1 ¡ ±) Ht¡1 + it if dt > 1 We also allow for a nondivisible investments, children. Let Ct = (c1t ; c2t ) where c1t is the number of children (truncated at ¿ c1 ) and c2t is the age of the youngest child (truncated at ¿ c2 ). Let bt = 1 i¤ the couple chooses to have a birth at t. Then c1t c2t
= max [c1t¡1 + bt ; ¿ c1 ] ; ½ max [c2t + 1; ¿ c2 ] if bt = 0 = : 0 if bt = 1
The utility ‡ow to the male or female is ¡ ¢ ukm t; X h ; X w ; Ct + µdt + rk Ht ¡ skmt¡1 1 (m = 1; mt¡1 > 1) ¡ q (it )
(2)
for k = h; w where rk is the rate of return on RSC and skmt¡1 is a separation cost depending on the type ¡ ¢ of relationship one is separating from. The …rst two ‡ow ¢including the match terms, ukm t; X h ; X w ; Ct + µdt , are the direct ¡ utility k h w speci…c component µdt . We assume that um t; X ; X ; Ct and its derivatives are bounded from above and below for all values of its arguments. Also, we assume that ¡ ¢ ¡ ¢ ¡ ¢ @uk2 X h ; X w ; Ct @uk1 X h ; X w ; Ct @uk3 X h ; X w ; Ct ¸ ¸ ; (3) U > @c1t @c1t @c1t ¡ ¢ ¡ ¢ ¡ ¢ @uk3 X h ; X w ; Ct @uk2 X h ; X w ; Ct @uk1 X h ; X w ; Ct ¡U < · · uk2 and sk3 > sk2 : (4) k
k
k
k
¢ ¡ The state variables are St = ¹mh ; ¹mw ; X h ; X w ; µ 1 ; µdt ; Ht¡1 ; Ct¡1 ; mt¡1 , and the choice variables, other than mt , are Pt = (it ; bt ) where Pt 2 ªt with restrictions, it ¸ 0, it = 0 if mt = 1, bt 2 f0; 1g, and mt 2 f1; 3g if mt¡1 = 3. The …rst restriction (it ¸ 0) says that the couple can not invest a negative amount in their relationship. The next (it = 0 if mt = 1) says one can not invest in a relationship if one is not in a relationship. The next (bt 2 f0; 1g) says that births are discrete and rules out multiple birth pregnancies. The last (mt 2 f1; 3g if mt¡1 = 3) assumes that a couple can not move from a marital relationship to a cohabiting relationship (as in BLS). We assume a Nash bargaining solution with equal bargaining power (though not necessarily equal threat points). Then the value to k (h or w) of choosing mt and Pt , given St , is Vtk (mt ; Pt ; St ) = Wtk (mt ; Pt ; St )
(5)
if mt = 1, and it is Vtk (mt ; Pt ; St ) = Wtk (1; Pt ; St ) +
¤ 1 X £ l Wt (mt ; Pt ; St ) ¡ Wtl (1; Pt ; St ) 2 l=h;w
(6)
if mt > 1 where ¡ ¢ ¡ ¢ Wtk (Pt ; St ) = ukm t; X h ; X w ; Ct + rk Ht + µdt 1 (mt > 1)
(7)
¡skmt¡1 1 (mt
= 1; mt¡1 > 1) · ¸ 1 k max Vt+1 (Pt+1 ; St+1 ) j Pt ; St ¡ q (it ) + ¯E Pt+1 2ªt+1 2
for t < T ¤ . This is the set of value functions for a Nash bargaining solution with equal bargaining power. De…ne ¸ ¸ · · i¤t (m) = arg max max Vth (m; i; b; St ) = arg max max Vtw (m; i; b; St ) ; b b i i h h i i b¤t (m) = arg max max Vth (m; i; b; St ) = arg max max Vtw (m; i; b; St ) ; b
Vtk¤ (mt ; St )
=
Vtk
i
b
i
(mt ; i¤t (mt ) ; b¤t (mt ) ; St ) :
At T , it is assumed that each agent has no choices and must remain where they
5
are until Te when he/she dies. The value function at T is therefore VTk
(mT ; PT ; S
T¤
) =
Te X
t=T
¯ (t¡T )
X
k=h;w
¡ ¢ ukmT t; X h ; X w ; Ct +
(8)
³ ´ rk (1 ¡ ±)(t¡T ) HT + E (µdt j µdT ) 1 (mT > 1) :
3
Properties of Model
3.1
Choices about Relationship Type
First we show that the value functions exist and are well behaved. In particular, they are bounded and have bounded derivatives. These properties are necessary for any of the results that follow. Theorem 1 (Existence) a) Vtk (mt ; it ; bt ; St ) is …nite for all …nite values of its arguments; b) Vtk (mt ; it ; bt ; St ) is bounded from above for all …nite values of µ1 , µdt , ¹mi , ¹mj , and Ht ; c) derivatives of Vtk (mt ; it ; bt ; St ) with respect to each of its arguments is bounded from above and below. Proof. At T , VTk (mT ; iT ; bT ; ST ) satis…es conditions (a) through (c) given the bounding assumptions made about ukm and equation (1). Now assume there exists at time t¤ : Vtk (mt ; it ; bt ; St ) satis…es conditions (a) through (c) for k = h; w for all t > t¤ . Then equations (5) through (7) and the bounding assumptions about ukm imply that Vtk¤ (mt¤ ; it¤ ; bt¤ ; St¤ ) satis…es condition (a). The only arguments of Vtk (¢) that are unbounded are it , Ct , and the set of variables in condition (b). As it¤ ! 1, Vtk¤ (mt¤ ; it¤ ; bt¤ ; St¤ ) ! ¡1 because of the assumptions made about q (it ). As Ct¤ ! 1, Vt¤ (mt¤ ; it¤ ; bt¤ ; St¤ ) ! ¡1 because of the assumptions made about derivatives of ukm with respect to Ct . Thus Vtk¤ (mt¤ ; it¤ ; bt¤ ; St¤ ) satis…es condition (b). Condition (c) follows as well. The potential couple must make a decision about whether to form a relationship. The next theorem determines when it is bene…cial to form a match. Theorem 2 The potential couple (h; w) maximizes X Vtk (m; it ; bt ; St ) : Yt (m; it ; bt ; St ) =
(9)
k=h;w
Proof. The Nash bargaining rule in equations (5) and (6) implies the result.
Our goal is to show the existence of reservation values of µ associated with the relationship choice. As in BLS, …rst we show that there are single crossing points for the value function with respect to µ. This is done by showing that @ k @µ Vt (m; it ; bt ; St ) can be ranked with respect to relationship type. 6
@ @ @ Vtk (m; it ; bt ; St ) = @µ Yt (m; it ; bt ; St ) = 0 for m = 1, and @µ Vtk (m; it ; bt ; St ) ¸ Theorem 3 @µ @ 1 and @µ Yt (m; it ; bt ; St ) ¸ 2 for m > 1. @ Proof. >From equations (5), (7), and (1), it is clear that @µ Vtk (m; it ; bt ; St ) = 0 for m = 1. From equation (8), when mT > 1, e T
X (t¡T ) @ @ k VT (mT ; PT ; ST ) = 2 ¯ E (µdt j µdT ) > 2 @µ @µdT t=T
@ Yt (m; it ; bt ; St ) = 0 for m = 1, if ½ ¸ 0 in equation (1). This implies that @µ @ and @µ Yt (m; it ; bt ; St ) ¸ 2 for m > 1. Now assume that there exists t¤ : @ k ¤ Then, at t¤ , @µ Vt (m; it ; bt ; St ) ¸ 2 for m > 1 for k = h; w for all t > t . equations (6) and (7) imply that · ¸ @ @ Yt¤ (mt¤ ; Pt¤ ; St¤ ) = 1 + ¯E max Yt¤ +1 (mt¤ +1 ; Pt¤ +1 ; St¤ +1 ) j Pt¤ ; St¤ : Pt¤ +1 2ªt¤ +1 @µ @µ (10) @ Yt¤ +1 (mt¤ +1 ; Pt¤ +1 ; St¤ +1 ) ¸ 0 for all mt¤ +1 , Pt¤ +1 ; St¤ +1 , the second Since @µ @ Yt¤ (mt¤ ; Pt¤ ; St¤ ) ¸ 2. The term in equation (10) is nonnegative. Thus @µ @ k results for @µ Vt (m; it ; bt ; St ) follow from the Nash bargaining rule.
We need a slightly di¤erent result when one of the choice variables, b, is discrete. First we prove a general result similar to proofs in Milgrom and Segal (2001) and Blundell, Chiapori, Magnac, and Meghir (2001), and then we apply it to our problem where a couple must not only choose what type of relationship to have but how much to invest (continuously and discretely) in their relationship conditional on the relationship type. Lemma 4 Consider a function # (x; y; z) where x and y can vary continuously and z can vary only discretely. Let #¤ (x) = max # (x; y; z) ; y;z ¸ · z ¤ (x) = arg max max # (x; y; z) y
z
Then @#¤ (x) =@x = @# (x; y; z) =@x. Proof. At points where @z ¤ (x) =@x = 0, @#¤ (x) =@x = @# (x; y; z) =@x by the Envelope Theorem. At points where limx"x¤ z ¤ (x) 6= limx#x¤ z ¤ (x), it must = 0 at x¤ . Otherwise, without loss of generality, be the case that ¢#(x;y;z) ¢z > 0. Then consider the case where limx"x¤ z ¤ (x) 6= limx#x¤ z ¤ (x) and ¢#(x;y;z) ¢z it must be the case that z ¤ (x¤ ¡ ") = z ¤ (x¤ ) for some small ", thus violating the condition that limx"x¤ z ¤ (x) 6= limx#x¤ z ¤ (x) (because z ¤ (x) is discrete in
7
x). Thus, at points where limx"x¤ z ¤ (x) 6= limx#x¤ z ¤ (x), @z ¤ (x) @x
= =
@# (x; y; z) ¢z ¤ (x) ¢# (x; y; z) + @x ¢x ¢z @# (x; y; z) : @x
@ @ @ Vtk¤ (1; St ) = @µ Yt¤ (1; St ) = 0 for m = 1, and @µ Vtk¤ (m; St ) ¸ Theorem 5 @µ @ ¤ 1 and @µ Yt (m; St ) ¸ 2 for m > 1. Proof. This is a direct application of Lemma (4) and Theorem (3).
De…ne Yt¤ (m; St ) =
X
Vtk¤ (m; St ) :
k
The next theorem shows the existence of critical values of µ that determine whether one should begin a relationship or remain single. The result follows from the fact the one can sign the relative derivatives of Yt¤ (m; St ) over di¤erent relationship types, m. Theorem 6 Given Ct¡1 , 9µ¤t (1; m) for m > 1 : Yt¤ (m; St ) > Yt¤ (1; St ) 8µ1 > µ¤t (1; m) and Yt¤ (m; St ) < Yt¤ (1; St ) 8µ 1 < µ ¤t (1; m) : @ @ Vtk¤ (1; St ) = 0 for m = 1, and @µ Vtk¤ (m; St ) ¸ 2 Proof. By Theorem (5), @µ ¤ for m > 1. Thus there must exist a single crossing point µt (1; m) that satis…es the theorem. The next theorem shows the existence of critical values of µ that determine whether one should remain in a relationship or separate and become single. ¡ ¢ Theorem 7 Given S ¤ = ¹mh ; ¹mw ; X h ; X w ; µ 1 ; Ht¡1 ; Ct¡1 ; m , 9µ¤t (m; 1) for m > 1 : Yt¤ (m; St ) > Yt¤ (1; St ) 8µ d > µ¤t (m; 1) and Yt¤ (m; St ) < Yt¤ (1; St ) 8µd < µ¤t (m; 1) : Proof. The proof is the same as in Theorem (6). The next theorem shows that there are some values of the match value µ where the couple would separate if cohabiting but remain in the relationship if married. The result follows given the assumption that separation costs are higher in marriage than in cohabitation.
8
Theorem 8 Given S ¤ , µ¤t (3; 1) < µ¤t (2; 1) : Proof. Consider Dt (m) = Yt¤ (m; St ) ¡ Yt¤ (1; St )
for m > 1 and for mt¡1 = m evaluated at µ¤t (2; 1). By de…nition, Dt (2) = 0. Then Dt (3) ¸ Yt (3; i¤t (2) ; b¤t (2) ; St ) ¡ Yt¤ (1; St ) = [Yt (3; i¤t (2) ; b¤t (2) ; St ) ¡ Yt (2; i¤t (2) ; b¤t (2) ; St )] + X¡ ¢ sk3 ¡ sk2 [Yt (2; i¤t (2) ; b¤t (2) ; St ) ¡ Yt¤ (1; St )] + k
=
Yt (3; i¤t (2) ; b¤t (2) ; St )
¡ Yt (2; i¤t (2) ; b¤t (2) ; St ) +
because
(11)
X¡ ¢ sk3 ¡ sk2 k
Yt (2; i¤t (2) ; b¤t (2) ; St ) ¡ Yt¤ (1; St ) = Dt (2) = 0: Note that DT ¤ (3) =
T ¤¤ X
¯
(t¡T ¤ )
t=T ¤
"
# X¡ ¢ X¡ k ¢ k k k u3 ¡ u2 + s3 ¡ s2 > 0 k
k
µ¤T ¤
(2; 1) > µ ¤T ¤ (3; 1). Now assume by equation (4). Since @Vtk¤ (3; St ) =@µ > 0, ¤ there exists a t¤ : Dt (3) > 0 and µt (2; 1) > µ¤t (3; 1) for all t > t¤ . Then equation (9) can be substituted into equation (11) and evaluated at t¤ to get X¡ ¢ X¡ k ¢ uk3 ¡ uk2 + s3 ¡ sk2 Dt¤ (3) ¸ (12) k
+¯
X k
E
·
k
max
Pt+1 2ªt+1
Yt+1 (Pt+1 ; St+1 ; mt = 3)
¡ max Yt+1 (Pt+1 ; St+1 ; mt = 2) j Pt ; St Pt+1 2ªt+1 X¡ ¢ X¡ k ¢ uk3 ¡ uk2 + s3 ¡ sk2 > k
+¯ Pr [µdt¤ ¸
k
µ¤t¤
(2; 1)]
¸
X¡ ¢ uk3 ¡ uk2 : k
The strong inequality in the last line of equation (12) follows from the fact that the missing term is equal to Z µ¤t¤ (2;1) ¤ £ [Yt¤¤ (3; St ) ¡ Yt¤¤ (2; St )] Pr µ dt¤ j µdt¤ ¡1 dµdt¤ : ¯ (3;1) µ¤ t¤
Note that the integrand in the equation is positive for µ¤t¤ (3; 1) · µ dt¤ · µ ¤t¤ (2; 1). Since all of the terms in the last line of equation (12) are positive, Dt¤ (3) > 0 which implies that µ¤t¤ (3; 1) < µ ¤t¤ (2; 1). 9
@ @ maxit ;bt Yt (3; it ; bt ; St ) > @µ maxit ;bt Yt (2; it ; bt ; St ) : Theorem 9 Given S ¤ , @µ ¤ Proof. Given S , equation (9) implies that ¸ · @ max Yt (3; i; b; St ) ¡ max Yt (2; i; b; St ) i;b @µ i;b ¸ Z µ¤t (2;1) · ¤ £ = ¯ max Yt (3; i; b; St ) ¡ max Yt (2; i; b; St ) Pr µ dt j µdt¡1 dµdt > 0: µ¤ t (3;1)
i;b
i;b
The next theorem shows that there is a critical match value µ determining whether a couple deciding to form a relationship chooses cohabitation or marriage. The result again follows from the single crossing property of the value functions for cohabitation and marriage. ¤ ¤ Theorem 10 Given S ¤ , 9µ¤¤ t (m) for m < 3 : Yt (3; St ) > Yt (2; St ) 8µd > ¤¤ ¤¤ ¤ ¤ µt (m) and Yt (3; St ) < Yt (2; St ) 8µ d < µt (m) : Proof. This follows directly from Theorem (9) and the single crossing principle.
The next two theorems show that both marriage and cohabitation will exist. Marriage exists because, as µ rises, eventually the value of marriage is greater than the value of cohabitation (because the marriage value function rises faster in µ than the cohabitation value function). Cohabitation exists as long as, for some people, the separation cost from marriage is high enough relative to the separation cost from cohabitation. Theorem 11 9St ; sm : Yt¤ (3; St ) > Yt¤ (2; St ) : Proof. This follows directly from Theorem (9).
¤ ¤ Theorem 12 9St ; sm ; um t) : P: Y¡t k(3; Stk)¢< Yt (2; Sk¤ k¤ u = 0, V ¡ u Proof. Since when 3 2 t P(3; k ¡ Skt ) < kV¢t (2; S¤t ), by the continuity principle, there exists an " > 0 : when k u3 ¡ u2 > 0, Yt (3; St ) < Yt¤ (2; St ).
3.2
Choices about RSC and Children
De…ne ¡¿ ;t = Pr
"
¿ [
l=0
¤ Yt+l
(1; St+l ) >
max
mt+l >1;mt+l 2ªt+l
¤ Yt+l
(mt+l ; St+l ) j St
#
to be the probability of a separation by t + ¿ given a relationship at t and a set of state variables St . The next three theorems show that the probability of a separation in the future declines with the match value µ. 10
Theorem 13 @µ@d ¡0;t < 0: t Proof. This follows from Theorem (5).
Theorem 14 @µ@d ¡1;t < 0: t Proof. This follows from equation (1) which implies that £ ¤ @ Pr µdt+1 < x j µdt 1: Theorem 17 @H t Proof. Note that
@ Yt (mt ; it ; bt ; St ) (13) @Ht · ¸ X @ = rk + ¯ E max Yt+1 (Pt+1 ; St+1 ) j Pt ; St Pt+1 2ªt+1 @Ht k "Z ¤ µ t+1 (mt ;1) X ¢ ¡ @ k ¤ = r +¯ Yt+1 (1; St+1 ) Pr µdt+1 j µ dt dµdt+1 @Ht ¡1 k Z µ¤¤ t+1 (2) ¢ ¡ ¤ +1 (mt = 2) Yt+1 (2; St+1 ) Pr µdt+1 j µ dt dµdt+1 µ¤ t+1 (2;1)
+1 (mt = 2)
Z
µ¤¤ t+1 (2)
µ¤ t+1 (2;1)
+1 (mt = 3)
Z
1
µ¤ t+1 (3;1)
¢ ¡ ¤ Yt+1 (2; St+1 ) Pr µdt+1 j µ dt dµdt+1 ¤ Yt+1 (3; St+1 ) Pr
11
# ¢ ¡ µdt+1 j µ dt dµdt+1 :
Since the derivative of each integral in equation (13) is positive (and we do @ ¤ i (mt+1 ) by the Envelope Theorem), the result not have to worry about @H t t+1 follows.
Theorem 18 i¤t (m) is …nite. Proof. This follows directly from the argument in Theorem (1) that as it ! 1, Vtk (mt ; it ; bt ; St ) ! ¡1. The next three theorems show that investment in RSC increases with the match value µ. The basic idea is that, as µ increases, the probability of future separation decreases, thus increasing the pro…tability of investing in RSC. Theorem 19 If ½ > 0, then Proof. Note that
@2 @µdt @it Yt (mt ; it ; bt ; St )
> 0 for mt > 1:
@ Yt (mt ; it ; bt ; St ) (14) @it · ¸ X @ = rk ¡ q 0 (it ) + ¯ E max Yt+1 (Pt+1 ; St+1 ) j Pt ; St Pt+1 2ªt+1 @it k=h;w "Z ¤ µ t+1 (mt ;1) X ¢ ¡ @ ¤ rk ¡ q 0 (it ) + ¯ Yt+1 (1; St+1 ) Pr µdt+1 j µdt dµdt+1 = @it ¡1 k=h;w Z µ¤¤ t+1 (2) ¢ ¡ ¤ +1 (mt = 2) Yt+1 (2; St+1 ) Pr µdt+1 j µdt dµ dt+1 +1 (mt = 2)
µ¤ t+1 (2;1) 1
Z
µ ¤¤ t+1 (2)
+1 (mt = 3)
Z
1
¢ ¡ ¤ Yt+1 (3; St+1 ) Pr µ dt+1 j µdt dµdt+1
µ¤ t+1 (3;1)
# ¢ ¡ ¤ Yt+1 (3; St+1 ) Pr µdt+1 j µdt dµ dt+1 :
The derivative of equation (14) with respect to µdt is positive because each of the integrals in equation (14) have a positive derivative with respect to µdt because of Theorem (17), equation (1) and ½ > 0.
Theorem 20
@2 max Yt (mt ; it ; bt ; St ) @µdt @it m >1;m t t 2ªt ;bt
>0 :
Proof. This follows directly from Theorem (19) and Lemma (4).
12
Theorem 21 (Endogenous Investment)
@ @µdt
0:
¸ · arg max maxYt (mt ; it ; bt ; St ) > bt
it
Proof. This follows directly from Theorem (20) and the fact that ¸ · arg max maxYt (mt ; it ; bt ; St ) bt
it
maximizes maxYt (mt ; it ; bt ; St ) over it (i.e., that ¸ · bt at arg max maxYt (mt ; it ; bt ; St ) . it
@2 maxYt (mt ; it ; bt ; St ) @i2t b
< 0
t
bt
We can show a similar result with respect to endogenous childbirth because childbirth is really only another form of investment in RSC. The only complication is that the investment is indivisible. Theorem 22 (Endogenous child birth) @µ@d [Yt (mt ; it ; 1; St ) ¡ Yt (mt ; it ; 0; St )] ¸ t 0 for mt > 1, ¸ · @ max Yt (mt ; it ; 0; St ) ¸ 0; max Yt (mt ; it ; 1; St ) ¡ mt >1;mt 2ªt @µdt mt >1;mt 2ªt and ¸ · arg max maxYt (mt ; it ; bt ; St ) it
bt
is nondecreasing in µdt . Proof. De…ne ¢ ¡ Stc = ¹mh ; ¹mw ; X h ; X w ; µ1 ; µ dt ; Ht¡1 ; mt¡1 ;
i.e., (Stc ; Ct¡1 ) = St . Also de…ne
¢Yt (mt ; it ; bt ; St ) = Yt (mt ; it ; bt ; Stc ; c1t¡2 + 1; 1) ¡Yt (mt ; it ; bt ; Stc ; c1t¡2 ; c2t¡2 + 1) : Then, given equation (3), ¢Yt (3; it ; St ) ¸ ¢Yt (2; it ; St ) ¸ ¢Yt (1; it ; St ) : Using equation (9), @ @ ¢Yt (mt ; it ; St ) = ¯ E @µ dt @µ dt ¡
·
max
¡ ¢ c Yt+1 Pt+1 ; St+1 ; c1t¡1 + 1; 1 (15) ¸ ¡ ¢ c Yt+1 Pt+1 ; St+1 ; c1t¡1 ; c2t¡1 + 1 j Pt ; St : max
Pt+1 2ªt+1
Pt+1 2ªt+1
13
At T ¤ , @ ¢YT ¤ (mT ¤ ; iT ¤ ; ST ¤ ) @µdT ¤ ¤¤
T X X £ ¡ ¢ ¤ @ ukmT ¤ t; X h ; X w ; c1T ¤ ¡1 + 1; 1 ¯ (t¡T ) @µdT ¤ t=T ¤ k=h;w ¡ ¢¤ k h w ¡umT ¤ t; X ; X ; c1T ¤ ¡1 ; c2T ¤ ¡1 + t
=
which is equal to 0. Therefore, · @ 0 = YT ¤ (mT ¤ ; iT ¤ ; 1; ST ¤ ) ¡ max @µ dT ¤ mT ¤ >1;mT ¤ 2ªT ¤ ¸ max YT ¤ (mT ¤ ; iT ¤ ; 0; ST ¤ ) mT ¤ >1;mT ¤ 2ªT ¤
¸ · by Lemma (4). Therefore, arg max maxYT ¤ (mT ¤ ; iT ¤ ; bT ¤ ; ST ¤ ) does not de¤
pend on µdT ¤ . At T ¡ 1,
=
bT ¤
iT ¤
@ ¢YT ¤ ¡1 (mT ¤ ¡1 ; iT ¤ ¡1 ; ST ¤ ¡1 ) @µdT ¤ ¡1 · @ ¯E max YT ¤ (PT ¤ ; STc ¤ ; c1T ¤ ¡2 + 1; 1) PT ¤ 2ªT ¤ @µdT ¤ ¡1 ¡
max
PT ¤ 2ªT ¤
YT ¤
(PT ¤ ; STc ¤ ; c1T ¤ ¡2 ; c2T ¤ ¡2
(16)
¸
+ 1) j PT ¤ ¡1 ; ST ¤ ¡1 :
Because of the additive separativeness in equation (2) between Ct and µ dt , @ max YT ¤ (PT ¤ ; STc ¤ ; c1T ¤ ¡2 + 1; 1) @µdT ¤ ¡1 PT ¤ 2ªT ¤ @ ¡ max YT ¤ (PT ¤ ; STc ¤ ; c1T ¤ ¡2 ; c2T ¤ ¡2 + 1) @µdT ¤ ¡1 PT ¤ 2ªT ¤ = 0:
(17)
But Pr [mT ¤ = 2 j ST ¤ ] = 0 (because separation is no longer an option after T ¤ ), and Pr [mT ¤ = 3 j ST ¤ ] @ >0 @µdT ¤ ¡1 Pr [mT ¤ = 1 j ST ¤ ] from Theorem (13). Thus, there are values of µdT ¤ where the optimal mT ¤ does not change with small changes in µ dT ¤ (and then equation (17) applies), and there are values of µ dT ¤ where the optimal mT ¤ changes from 1 to 3 with small
14
changes in µ dT ¤ . At such points, equation (3) implies that · @ max YT ¤ (PT ¤ ; STc ¤ ; c1T ¤ ¡2 + 1; 1) @µdT ¤ PT ¤ 2ªT ¤ ¡ ¸ 0:
max
PT ¤ 2ªT ¤
¸ YT ¤ (PT ¤ ; STc ¤ ; c1T ¤ ¡2 ; c2T ¤ ¡2 + 1)
Thus, given equation (1), equation (16) is positive. The condition that ¸ · arg max max YT ¤ ¡1 (mT ¤ ¡1 ; iT ¤ ¡1 ; bT ¤ ¡1 ; ST ¤ ¡1 ) bT ¤ ¡1
iT ¤ ¡1
is nondecreasing in µdT ¤ ¡1 follows immediately. Now assume that there exists t¤ : @ [Yt (mt ; it ; 1; St ) ¡ Yt (mt ; it ; 0; St )] ¸ 0 @µdt for mt > 1, · @ max Yt (mt ; it ; 1; St ) ¡ @µdt mt >1;mt 2ªt
max
mt >1;mt 2ªt
¸ Yt (mt ; it ; 0; St ) ¸ 0;
and arg max bt
¸ · maxYt (mt ; it ; bt ; St ) it
is nondecreasing in µdt for all t > t¤ . Then, the expected value in equation (15)
15
can be written as "Z ¤ µ t¤ +1 (mt¤ ;1) ¡1
¡ ¢ Yt¤ +1 1; 0; b¤t¤ +1 (1) ; Stc¤ +1 ; c1t¤ ¡1 + 1; 1 ¢
(18)
Pr (µdt¤ +1 j µ dt¤ ) dµdt¤ +1 Z µ¤¤ t¤ +1 (mt¤ ) ¡ ¢ +1 (mt¤ = 2) Yt¤ +1 2; i¤t¤ +1 (2) ; b¤t¤ +1 (2) ; Stc¤ +1 ; c1t¤ ¡1 + 1; 1 ¢ (mt¤ ;1) µ¤ t¤ +1
Pr (µdt¤ +1 j µ dt¤ ) dµdt¤ +1 Z 1 ¡ ¢ +1 (mt¤ = 2) Yt¤ +1 3; i¤t¤ +1 (3) ; b¤t¤ +1 (3) ; Stc¤ +1 ; c1t¤ ¡1 + 1; 1 ¢ (mt¤ ) µ ¤¤ t¤ +1
Pr (µdt¤ +1 j µ dt¤ ) dµdt¤ +1 Z 1 ¡ ¢ ¤ +1 (mt = 3) Yt¤ +1 3; i¤t¤ +1 (3) ; b¤t¤ +1 (3) ; Stc¤ +1 ; c1t¤ ¡1 + 1; 1 ¢ (mt¤ ;1) µ¤ t¤ +1
Pr (µdt¤ +1 j µ dt¤ ) dµdt¤ +1 ] "Z ¤ µt¤ +1 (mt¤ ;1) ¡ ¢ Yt¤ +1 1; 0; b¤t¤ +1 (1) ; Stc¤ +1 ; c1t¤ ¡1 ; c2t¤ ¡1 + 1 ¢ ¡ ¡1
Pr (µdt¤ +1 j µ dt¤ ) dµdt¤ +1 Z µ¤¤ t¤ +1 (mt¤ ) ¡ ¢ ¤ +1 (mt = 2) Yt¤ +1 2; i¤t¤ +1 (2) ; b¤t¤ +1 (2) ; Stc¤ +1 ; c1t¤ ¡1 ; c2t¤ ¡1 + 1 ¢ (mt¤ ;1) µ¤ t¤ +1
Pr (µdt¤ +1 j µ dt¤ ) dµdt¤ +1 Z 1 ¡ ¢ +1 (mt¤ = 2) Yt¤ +1 3; i¤t¤ +1 (2) ; b¤t¤ +1 (3) ; Stc¤ +1 ; c1t¤ ¡1 ; c2t¤ ¡1 + 1 ¢ (mt¤ ) µ ¤¤ t¤ +1
Pr (µdt¤ +1 j µ dt¤ ) dµdt¤ +1 Z 1 ¡ ¢ +1 (mt¤ = 3) Yt¤ +1 3; i¤t¤ +1 (3) ; b¤t¤ +1 (3) ; Stc¤ +1 ; c1t¤ ¡1 ; c2t¤ ¡1 + 1 ¢ (mt¤ ;1) µ¤ t¤ +1
Pr (µdt¤ +1 j µ dt¤ ) dµdt¤ +1 ] :
The derivative of the integrand in each term in equation (18) with respect to µdt¤ is zero because of the additive separateness of equation (2) with respect to Ct and µdt . But (using the same argument for T ¤ ¡ 1) there are values of µdt¤ where small increases in µ dt¤ change the optimal mt¤ from 1 to 2 or 3 and values of µ dt¤ where small increases in µ dt¤ change the optimal mt¤ from 2 to 3. At these values, equation (3) implies a positive derivative. Integrating over Pr (µdt¤ +1 j µ dt¤ ) then implies that @ [Yt¤ (mt¤ ; it¤ ; 1; St¤ ) ¡ Yt¤ (mt¤ ; it¤ ; 0; St¤ )] ¸ 0 @µ dt¤
16
for mt¤ > 1. Lemma (4) implies that ¸ · @ Yt¤ (mt¤ ; it¤ ; 1; St¤ ) ¡ max Yt¤ (mt¤ ; it¤ ; 0; St¤ ) ¸ 0: max mt¤ >1;mt¤ 2ªt¤ @µ dt¤ mt¤ >1;mt¤ 2ªt¤ The condition that
¸ · arg max maxYt¤ (mt¤ ; it¤ ; bt¤ ; St¤ ) bt¤
it¤
is nondecreasing in µdt¤ follows immediately. Remark 23 We could avoid all of the problems associated with bt being discrete by changing the model so that bt is a random variable whose realization depends upon a continuous “e¤ort” variable that the couple chooses. This would change the analysis of birth behavior so that it is just like it behavior.
3.3
Value of Divorce
First we consider the value of being able to divorce for a couple, ignoring equilibrium e¤ects. The …rst theorem tells us that it is never in the interest of the couple to restrict the option to divorce when there is no endogenous investment. The intuition here is the same as the argument in Becker; i.e., couples divorce i¤ the joint value of separating is greater than the joint value of remaining together. Theorem 24 (Divorce can’t be bad for a couple without endogenous investment) @ Yt (mt ; it ; bt ; St ) < 0 8l > 1 8t < If it and bt are not choice variables, then @s l ¤ T ¡ 2: Proof. At T ¤ , one no longer has the option to separate, so @ YT ¤ (mT ¤ ; iT ¤ ; bT ¤ ; ST ¤ ) = 08l > 1: @sl At T ¤ ¡ 1,
@ YT ¤ ¡1 (mT ¤ ¡1 ; iT ¤ ¡1 ; bT ¤ ¡1 ; ST ¤ ¡1 ) @sl = ¡1 (mT ¤ ¡1 = 1; mT ¤ ¡2 = l) · ¸ @ +¯ E max YT ¤ (mT ¤ ; iT ¤ ; bT ¤ ; ST ¤ ) j PT ¤ ¡1 ; ST ¤ ¡1 mT ¤ @sl = ¡1 (mT ¤ ¡1 = 1; mT ¤ ¡2 = l) < 0
if mT ¤ ¡1 = 1 and mT ¤ ¡2 = l. At T ¤ ¡ 2,
@ YT ¤ ¡2 (mT ¤ ¡2 ; iT ¤ ¡2 ; bT ¤ ¡2 ; ST ¤ ¡2 ) (19) @sl = ¡1 (mT ¤ ¡2 = 1; mT ¤ ¡3 = l) · ¸ @ ¤ ¤ ¤ ¤ ¤ ¤ ¤ +¯ E max YT ¡1 (mT ¡1 ; iT ¡1 ; bT ¡1 ; ST ¡1 ) j PT ¡2 ; ST ¡2 : mT ¤ ¡1 @sl 17
Equation (19) is negative if mT ¤ ¡3 = l and zero if mT ¤ ¡3 6= l because @ max YT ¤ ¡1 (mT ¤ ¡1 ; iT ¤ ¡1 ; bT ¤ ¡1 ; ST ¤ ¡1 ) · 0 @sl mT ¤ ¡1 for all mT ¤ ¡1 and mT ¤ ¡2 . At T ¤ ¡ 3, @ YT ¤ ¡3 (mT ¤ ¡3 ; iT ¤ ¡3 ; bT ¤ ¡3 ; ST ¤ ¡3 ) @sl = ¡1 (mT ¤ ¡3 = 1; mT ¤ ¡4 = l) · ¸ @ +¯ E max YT ¤ ¡2 (mT ¤ ¡2 ; iT ¤ ¡2 ; bT ¤ ¡2 ; ST ¤ ¡2 ) j PT ¤ ¡3 ; ST ¤ ¡3 mT ¤ ¡2 @sl which is negative always because @ max YT ¤ ¡2 (mT ¤ ¡2 ; iT ¤ ¡2 ; bT ¤ ¡2 ; ST ¤ ¡2 ) · 0 @sl mT ¤ ¡2 for all mT ¤ ¡2 , @ max YT ¤ ¡2 (mT ¤ ¡2 ; iT ¤ ¡2 ; bT ¤ ¡2 ; ST ¤ ¡2 ) < 0 @sl mT ¤ ¡2 if mT ¤ ¡2 = l, and Pr [mT ¤ ¡2 = l] > 0: Now assume there exists t¤ : Then
@ @sl Yt (mt ; it ; bt ; St )
< 0 8l > 1 8t¤ < t < T ¤ ¡ 2.
@ Yt¤ (mt¤ ; it¤ ; bt¤ ; St¤ ) @sl = ¡1 (mt¤ = 1; mt¤ ¡1 = l) · ¸ @ ¤ ¤ ¤ ¤ ¤ ¤ ¤ +¯ E max Yt +1 (mt +1 ; it +1 ; bt +1 ; St +1 ) j Pt ; St mt¤ +1 @sl < 0 by the same argument that applied for T ¤ ¡ 3. By similar arguments, we can show that having the option to cohabit rather than marry is always of value to a couple. Again the idea is that, in some cases, cohabitation will be a better option than marriage (to avoid large and relatively likely divorce costs), and it is only in those cases that a couple will choose to cohabit rather than marry. Theorem 25 (Cohabitation must be good without endogenous investment) If it and bt are not choice variables, then @¹@ Yt (mt ; it ; bt ; St ) > 0 and @¹@ Yt (mt ; it ; bt ; St ) > 2i 2j 0 8t < T ¤ ¡ 1: 18
Proof. At T ¤ , @¹@ YT ¤ (mT ¤ ; iT ¤ ; bT ¤ ; ST ¤ ) > 0 and @¹@ YT ¤ (mT ¤ ; iT ¤ ; bT ¤ ; ST ¤ ) > 2i 2j 0 if mT ¤ = 2, and they are both zero if mT ¤ 6= 2 (from equation (8). At T ¤ ¡1, if mT ¤ ¡1 = 3, then it is infeasible to have mT ¤ = 2; thus @ @ YT ¤ ¡1 (3; iT ¤ ¡1 ; bT ¤ ¡1 ; ST ¤ ¡1 ) = YT ¤ ¡1 (3; iT ¤ ¡1 ; bT ¤ ¡1 ; ST ¤ ¡1 ) = 0: @¹2i @¹2j If mT ¤ ¡1 6= 3, then Pr (mT ¤ = 2) > 0, so @ YT ¤ ¡1 (3; iT ¤ ¡1 ; bT ¤ ¡1 ; ST ¤ ¡1 ) > 0; @¹2i @ YT ¤ ¡1 (3; iT ¤ ¡1 ; bT ¤ ¡1 ; ST ¤ ¡1 ) > 0: @¹2j Now assume there exists a t¤ : @¹@ Yt (mt ; it ; bt ; St ) > 0 and 2i 0 8t¤ < t < T ¤ ¡ 1. Then by the same argument,
@ @¹2j Yt (mt ; it ; bt ; St )
@ Yt¤ (3; it¤ ; bt¤ ; St¤ ) > 0; @¹2i @ Yt¤ (3; it¤ ; bt¤ ; St¤ ) > 0: @¹2j
Conjecture 26 (Discouraging divorce is bad even with endogenous investment) @ Yt (mt ; it ; bt ; St ) < 0 for l > 1: Even when it or bt are choice variables, @s l Example 27 Consider the (leading) case where ¡ ¢ ukm t; X h ; X w ; Ct = rk
=
skm
=
1 um ; 2 1 r; 2 1 sm : 2
Also abstract away from children, so that Pt = (mt ; it ). This implies that St = (µ 1 ; µdt ; Ht¡1 ; mt¡1 ). Finally, assume that um is small enough so that
19
>
maxi Yt (3; i; St ) > maxi Yt (2; i; St ) always. Then we can write Yt (3; it ; St ) = u3 + rHt + µdt ¡ q (it ) Z µ¤t+1 (3;1) +¯ Yt+1 (1; 0; St+1 ) g (µ dt +1 j µdt ) dµdt +1 ¡1 Z 1 +¯ max Yt+1 (3; it+1 ; St+1 ) g (µdt +1 j µdt ) dµdt +1 ; it+1 µ¤ t+1 (3;1)
Z
µ¤ t+1 (1;3)
Yt+1 (1; 0; St+1 ) g (µdt +1 ) dµdt +1 Yt (1; 0; St ) = u1 ¡ s3 1 (mt¡1 = 3) + ¯ ¡1 Z 1 +¯ max Yt+1 (3; it+1 ; St+1 ) g (µdt +1 ) dµdt +1 : it+1 µ¤ t+1 (1;3)
At T ¤ , ¤¤
YT ¤ (3; iT ¤ ; ST ¤ ) =
T X
¯ (t¡T
¤
t=T ¤ ¤¤
YT ¤ (1; 0; ST ¤ ) =
T X
)
h
t¡T ¤
u3 + r (1 ¡ ±)
i HT ¤ + E (µdt j µ dT ¤ ) ;
¤
¯ (t¡T ) u1 :
t=T ¤
Note that @ @ YT ¤ (3; iT ¤ ; ST ¤ ) = YT ¤ (1; 0; ST ¤ ) = 0: @s3 @s3 At T ¤ ¡ 1, YT ¤ ¡1 (3; iT ¤ ¡1 ; ST ¤ ¡1 ) = u3 + rHT ¤ ¡1 + µ dT ¤ ¡1 ¡ q (iT ¤ ¡1 ) Z µ¤T ¤ (3;1) ¢ ¡ YT ¤ (1; 0; ST ¤ ) g µ dT ¤ j µdT ¤ ¡1 dµdT ¤ +¯ ¡1 Z 1 ¢ ¡ +¯ max YT ¤ (3; iT ¤ ; ST ¤ ) g µdT ¤ j µdT ¤ ¡1 dµdT ¤ ; (3;1) iT ¤ µ¤ T¤
YT ¤ ¡1 (1; 0; ST ¤ ¡1 )
Z
µ¤ T ¤ (1;3)
YT ¤ (1; 0; ST ¤ ) g (µdT ¤ ) dµdT ¤ = u1 ¡ s3 1 (mT ¤ ¡2 = 3) + ¯ ¡1 Z 1 ¯ max YT ¤ (3; iT ¤ ; ST ¤ ) g (µ dT ¤ ) dµ dT ¤ : (1;3) iT ¤ µ¤ T¤
20
Note that @ YT ¤ ¡1 (3; iT ¤ ¡1 ; ST ¤ ¡1 ) @s3 ¤ £ @iT ¤ ¡1 @YT ¤ ¡1 (3; iT ¤ ¡1 ; ST ¤ ¡1 ) ¡ ¯ Pr µdT ¤ · µ ¤T ¤ (3; 1) j µ dT ¤ ¡1 = @s3 @iT ¤ ¡1 @µ ¤T ¤ (3; 1) +¯ [YT ¤ (3; iT ¤ ; ST ¤ ) ¡ YT ¤ (1; 0; ST ¤ )] @s ¤ £ 3 = ¡¯ Pr µdT ¤ · µ ¤T ¤ (3; 1) j µdT ¤ ¡1 < 0
by Envelope Theorem and the fact that
YT ¤ (3; iT ¤ ; ST ¤ ) = YT ¤ (1; 0; ST ¤ ) at µ ¤T ¤ (3; 1); @ YT ¤ ¡1 (1; 0; ST ¤ ¡1 ) = ¡1 (mT ¤ ¡2 = 3) · 0: @s3 Induction follows straightforwardly.
Conjecture 28 (Discouraging being single and encouraging marriage may be good or bad with endogenous investment) Assume it or bt are choice variables. Consider a balanced budget tax program that transferred money from single people to married people (e.g., marriage tax credit). Let ¿ 3 be the size of the transfer, measured in utils, to married people. If people are risk neutral, then @¿@ 3 Y1 (1; 0; bt ; St ) > 0. If people are risk averse enough, then @ @¿ 3 Y1 (1; 0; bt ; St ) < 0. Example 29 Case 1: Assume people are risk neutral, and cohabitation is not an option. Let ¿ 1 be the penalty single people must pay to …nance ¿ 3 . Then, ¿ 1 and ¿ 3 must satisfy ¿3
T ¤¤ X
t
¯ Pr (mt = 3) = ¿ 1
t=1
T ¤¤ X
¯ t Pr (mt = 1) :
(20)
t=1
What are the …rst order e¤ects (if any)? e¤ects?
If not, what are the second order
Example 30 Case 2: Assume people are risk averse, and cohabitation is not an option. Then equation (20) becomes ¿3
T ¤¤ X
¯ t Pr (mt = 3) = b¿ 1
t=1
T ¤¤ X t=1
21
¯ t Pr (mt = 1)
where b=¡
@ @¿ 1 @ @¿ 1
PT ¤¤ t=1 PT ¤¤ t=1
¯ t Yt (Pt ; St ) Pr (mt = 1) ¯ t Yt (Pt ; St ) Pr (mt = 3)
>1
is a ratio of weighted averages of marginal utilities in the two states. greater than 1 as long as ¤¤
T X
It is
¤¤
t
¯ Yt (Pt ; St ) Pr (mt = 3) >
t=1
T X
¯ t Yt (Pt ; St ) Pr (mt = 1) :
t=1
It is obvious that there is a value of b large enough so that the loss of utility when single will dominate the gain when married.
4 4.1
Equilibrium Equilibrium Densities
Assume the utility function in equation (2) is ¢ ¡ ½ ¡ ¢ ¹mi + v ¡X h ; X w ; Ct ¢ k h w um t; X ; X ; Ct = ¹mj + v X h ; X w ; Ct
if k = h : if k = w
Assume ¹1i = 0 (identifying restriction). De…ne ¹i = (¹2i ; ¹3i ) and ¹i » iidGk1 (¢) with density g1k (¢) and E¹i = ®1 at t = 1. ¡Let nk be the proportion of people ¢ of type k, k = h; w, nh + nw = 1. Let ¨kt mt ; µdt j mt¡1 ; ¹; µ dt ¡1 ; µ1 ; X k be the probability that a type k person in period t who was in state mt¡1 in period t ¡ 1 with ¹i = ¹ and other characteristics X k is in state mt : ¨st (mt ; µ dt j ¹; X) · ¸ = 1 mt = arg max maxYt (m; it ; bt ; St ) Pr (µ 1 ) m
if mt¡1
it ;bt
= 1 ¨rt (mt ; µdt j mt¡1 ; ¹; µ dt ¡1 ; µ1 ; X) · ¸ = 1 mt = arg max maxYt (m; it ; bt ; St ) Pr [µdt j µdt ¡1 ; µ 1 ] m
(21)
it ;bt
if mt¡1 > 1 ¢ ¡ with X = X h ; X w . This is just the transition probability from (mt¡1 ; µdt ¡1 ) to (mt ; µ dt ) conditional on (¹; µ 1 ; X). Note: we assume that each person knows his/her own X k and can observe X =k of any potential partner once the partner is sampled. 22
The last issue to be addressed is the rate at which single people sample other single people and married people. Let ¸m be the rate at which people of relationship status m are sampled. Conceptually, consider a world where all matches occur in singles bars. At one extreme, if only single people go to singles bars, then ¸1 > ¸2 = ¸3 = 0. At the other extreme, if relationship status has no e¤ect on propensity to go to a singles bar, then ¸1 = ¸2 = ¸3 > 0. Assume everyone is single in period 1: m1 = 1. De…ne ·Z Z ¸ Z Z ¡ h h¢ h h s¤ h w w w w w gt = min gt ¹ ; X d¹ dX ; gt (¹ ; X ) d¹ dX Xh
¹h
Xw
¹w
and °t =
¸1 gts¤
+
P
m>1 ¸m
R
¸1 gts¤ gs¤ : ¢¢¢ g2r (¹; µ 1 ; µd ; X; m) dµ d dµ1 d¹dX t R
Note that, if ¸2 = ¸3 = 0, then
° t = gts¤ : ¢ ¡ Then, the joint density of ¹; X k single people in period 2 is Z Z ¡ k k¢ ¡ k k¢ sk sk = g1 ¹ ; X ¡ ° 1 ¨s2 (1; µ1 j ¹; X) ¢ g2 ¹ ; X X =k
¹=k
¢ =k
¹=k ; X ¢ ¡ ¢ =k ¹=k ; X =k d¹=k dX =k g X =k ¹=k 1 ¡ ¢ g1sk ¹k ; X k R R d¹=k dX =k k (¹k ; X k ) d¹k dX k g k k 1 X ¹ R
for k = h; w where
s=k ¡
R
=k =
g1
w h
if k = h ; if k = w
and the joint density of (¹; µ1 ; µ1 ; X) people in relationships (m > 1) in period 2 is g2r (¹; µ1 ; µ 1 ; X; m) = 2° 1 ¨s2 (m; µ 1 j ¹; X) ¢ ¡ ¢ g1sh ¹h ; X h R R ¢ g sh (¹h ; X h ) d¹h dX h X h ¹h 1 R
Xw
g1sw (¹w ; X w ) : sw w w w w ¹w g1 (¹ ; X ) d¹ dX
R
23
¡ ¢ In general, the joint density of ¹k ; X k single people in period t is ¡ ¢ ¡ k k¢ sk = gt¡1 ¹ ;X gtsk ¹k ; X k Z Z Z Z + X =k
¹=k
µ1
X
µd¡1 l>1
(22)
¨rt (1; µd j l; ¹; µd¡1 ; µ1 ; X) ¢
r (¹; µ1 ; µ d¡1 ; X; l) dµ d¡1 dµ1 d¹=k dX =k gt¡1
1 ¡ ° t¡1 2
Z
X =k
Z
¹=k
Z X µ1 l>1
¨st (l; µ1 j ¹; X) ¢
¡ ¢ g1sk ¹k ; X k ¢ g sk (¹k ; X k ) d¹k dX k X k ¹k 1 ¢ s=k ¡ =k ¹ ; X =k g1 dµ 1 d¹=k dX =k ; R R ¢ s=k ¡ =k =k d¹=k dX =k ¹ g ; X X =k ¹=k 1 R
R
and the joint density of (¹; µ1 ; µ d ; X; m) people in relationships in period t is Z X gtr (¹; µ1 ; µd ; X; m) = ¨rt (m; µd j l; ¹; µ d¡1 ; µ1 ; X) ¢ (23) µ d¡1 l>1 r gt¡1 (¹; µ1 ; µ d¡1 ; X; l) dµd¡1 +2° t¡1 ¨st (m; µ1 j ¹; X) 1 (µ1 = µd ) ¢ ¡ ¢ gsh ¹h ; X h gsw (¹w ; X w ) RR sw t¡1 RR sh t¡1 : gt¡1 (¹w ; X w ) d¹w dX w gt¡1 (¹h ; X h ) d¹h dX h
Equations (22) and (23) can be solved iteratively. The value function in equation (9) depends on the density of ¢ St+1 given St . ¡ Remember that St+1 = ¹mi ; ¹mj ; X h ; X w ; µ1 ; µdt ; Ht ; Ct ; mt whose density ¡ k k¢ ¹ ; X . Equilibrium requires that we use for single people depends¡ on gtsk ¢ equilibrium values of gtsk ¹; X k and gtr (¹; µ1 ; µ d ; X; m) in evaluating equation (9).
4.2
A “Simple” Example
Assume that a) neither X k variables nor t a¤ect utility, b) separation costs do not depend on k, c) there are¡ no births and no investment, and d) there is ¢ ; µ ; µ ; m , the joint value function no cohabitation. Then St = ¹hm ; ¹w 1 d t¡1 t m becomes " # X k 2#m + ¹m + 2µdt 1 (m > 1) ¡ 2smt¡1 1 (m = 1; mt¡1 >(24) 1) Yt (m; St ) = k
·
+¯ESt+1 max Yt+1 (mt+1 ; St+1 ) j St mt+1
24
¸
where #m is a potential government subsidy (tax) to state m; the transition probabilities become · ¸ s ¨t (mt ; µdt j ¹) = 1 mt = arg maxYt (m; ; St ) Pr (µ 1 ) (25) m
if mt¡1
= 1
if mt¡1
> 1;
· ¸ ¨rt (mt ; µ dt j mt¡1 ; ¹; µdt ¡1 ; µ 1 ) = 1 mt = arg maxYt (m; St ) Pr [µdt j µdt ¡1 ; µ 1 ] m
gts¤
= min
°t =
·Z
gtsh
¹h
¸1 gts¤ + ¸3
R
¡ h¢ h ¹ d¹ ;
Z
¹w
gtsw
w
w
(¹ ) d¹
¸
;
¸1 gts¤ g s¤ ; ¢¢¢ gtr (¹; µ 1 ; µd ; 3) dµ d dµ1 d¹ t R
and the equilibrium densities become Z ¡ k¢ ¡ k¢ sk sk g2 ¹ = g1 ¹ ¡ ° 1
¹=k
¨s2 (1; µ1 j ¹) ¢
(26)
¡ =k ¢ ¡ ¢ ¹ g1sk ¹k R d¹=k ; R k s=k ¡ =k ¢ k k =k k g1 (¹ ) d¹ ¹ d¹ g ¹ =k 1 ¹ ¡ ¢ g1sh ¹h g1sw (¹w ) r s R ; g2 (¹; µ 1 ; µ1 ; 3) = 2° 1 ¨2 (3; µ 1 j ¹) R sh h h g (¹ ) d¹ ¹w g1sw (¹w ) d¹w ¹h 1 Z Z Z Z ¡ k¢ ¡ k¢ 1 sk sk = gt¡1 ¹ + ¨r (1; µd j 3; ¹; µd¡1 ; µ 1 ) ¢ gt ¹ 2 ¹=k µ1 µd¡1 µd t s=k
g1
r (¹; µ 1 ; µd¡1 ; 3) dµd dµd¡1 dµ1 d¹=k gt¡1 Z Z ¡° t¡1 ¨st (3; µ1 j ¹) ¢ ¹=k
gtr
(¹; µ1 ; µ d ; 3) =
¡
¢
µ1
g1sk ¹k R sk k k ¹k g1 (¹ ) d¹
Z
µd¡1
¡ =k ¢ ¹ dµ 1 d¹=k ; R s=k ¡ =k ¢ =k ¹ d¹ g ¹=k 1 s=k
g1
r ¨rt (3; µd j 3; ¹; µd¡1 ; µ 1 ) gt¡1 (¹; µ1 ; µ d¡1 ; 3) dµd¡1
+2° t¡1 ¨st (3; µ1 j ¹) 1 (µ1 = µd ) ¢ ¡ h¢ sh ¹ gt¡1 g sw (¹w ) R swt¡1 R sh : h h gt¡1 (¹w ) d¹w gt¡1 (¹ ) d¹
25
Note that XZ k
=
" X Z
¡ ¢ gtsk ¹k d¹k + sk gt¡1
k
¡ k¢ 1 ¹ + 2
Z
¢¢¢
Z
¹=k
Z
gtr (¹; µ1 ; µd ; 3) dµd dµ1 d¹
Z Z µ1
µ d¡1
Z
µd
¨rt (1; µd j 3; ¹; µ d¡1 ; µ 1 ) ¢
r (¹; µ1 ; µd¡1 ; 3) dµ d dµd¡1 dµ1 d¹=k gt¡1 3 ¡ k¢ Z Z s=k ¡ =k ¢ sk ¹ ¹ g g 1 1 ¡° t¡1 ¨st (3; µ 1 j ¹) R dµ1 d¹=k 5 sk (¹k ) d¹k R s=k ¡ =k ¢ =k =k g k ¹ µ1 ¹ d¹ 1 ¹ ¹=k g1 Z Z Z r + ¢¢¢ ¨rt (3; µd j 3; ¹; µd¡1 ; µ1 ) gt¡1 (¹; µ1 ; µd¡1 ; 3) dµd¡1 µ d¡1
¡ h¢ sh ¹ gt¡1 g sw (¹w ) R swt¡1 j ¹) 1 (µ 1 = µd ) R sh h h gt¡1 (¹w ) d¹w gt¡1 (¹ ) d¹ Z Z Z X ¡ k¢ k sk r gt¡1 ¹ d¹ + ¢ ¢ ¢ gt¡1 = (¹; µ 1 ; µd¡1 ; 3) dµ d¡1 dµ1 d¹: +2° t¡1 ¨st (3; µ1
k
Also note that · ¸ ESt+1 max Yt+1 (mt+1 ; St+1 ) j St mt+1 Z = max Yt+1 (mt+1 ; St+1 ) Pr (St+1 j St ) dSt+1 mt+1 Z ¤ £ = max Yt+1 (mt+1 ; St+1 ) Pr µdt+1 j µdt ; µ1 dµ dt+1 mt+1
if mt
if mt
> 1;
¢ s=k ¡ gt¡1 ¹=k max Yt+1 (mt+1 ; St+1 ) R s=k ¡ ¢ = ° t¡1 Pr (µ1 ) dµ1 d¹=k mt+1 =k =k gt¡1 ¹ d¹ ¢ ¡ + 1 ¡ ° t¡1 Yt+1 (1; St+1 ) = 1: Z Z
It becomes clear from equation (26) that solving for equilibrium must involve a numerical process because solving for the value function in equation (24) involves solving backwards recursively while solving for the probabilities in equations (26) requires solving forwards recursively. © ªw Theorem 31 Given an initial density g1k k=h , an equilibrium set of functionªT © als z = ¨st ; ¨rt ; gtsh ; gtsw ; gtr ; Yt t=2 exists that solve equations (24), (??), and (26). T Proof. De…ne z¨ = f¨st ; ¨rt gt=2 with z¨ 2 ¥¨ where ¥¨ is the space of all continuous functions. Given a candidate choice of z¨ , equation (26) implies
26
zg = zY = De…ne
© sh sw r ªT gt ; gt ; gt t=2;k=h;w .
fYt gTt=2
Given the implied zg , equation (24) implies
with zY . Given zg and zY , equation (26) implies a new z¨ .
¤ (z¨ ) = z¨ (zY (zg (z¨ )) ; zg (z¨ )) : ¸ · By construction (0 · 1 mt = arg maxYt (m; St ) j ¹; µ1 ; µd · 1 and Pr [µdt j ¹; µdt ¡1 ; µ1 ] m
and Pr (µ1 ) are densities), ¤ (z¨ ) maps ¥¨ into a (possibly improper) subset of ¥¨ . ¤ (z¨ ) is continuous because z¨ , zY , and zg are continuous.
we still need to show that
is compact.
Note:
¥¨ Assuming ¥¨ is compact, Schauder’s …xed point theorem implies that there is a …xed point z¤¨ where ¤ (z¤¨ ) = z¤¨ . We can still do some incremental policy analysis. Consider an equilibrium where equations (24), (??), and (26) hold. Now consider a tax credit of the form discussed earlier. In particular, increase #3 and decrease #1 so that budget balance is maintained: ¾ Z ½ Z Z T X £ ¡ ¢ ¤ ¯t gtr (¹; µ 1 ; µd ; 3) dµ 1 dµd d¹ = 0: #1 gtsh ¹h + gtsw (¹w ) + #3 µ1
t=1
4.3
µd
E¢ciency in a “Simpler” Example
Several externalities may arise in our marriage market. For one, when forming a match, individuals do not take into account the e¤ect they have on the remaining singles waiting to match. As as result, it may be the case that the competitive equilibrium in the marriage market may be ine¢cient in the absence of marriage taxes or subsidies. We illustrate this point by showing that the reservation match qualities that arise in equilibrium di¤er from those chosen by a social planner. Assume further that a) ¹k does not a¤ect utility and b) there are no separation costs. For simplicity, we also assume µ does not change over time within any marriage so that we can focus on the externality in‡uencing marriage rates. In this case, Wtk (mt ; St ) = ukmt + µ1 1(mt > 1) · ¸ k +¯ESt+1 max Vt+1 (mt+1 ; St+1 )jSt : mt+1
and Yt (mt ; St ) = uhmt + uw mt + 2µ1 1(mt > 1) · ¸ k (mt+1 ; St+1 )jSt : +¯ESt+1 max Yt+1 mt+1
27
4.3.1
Competitive Equilibrium
Agents must solve for reservation match qualities for entering a match in each period. The reservation quality must be such that the potential couple is indifferent between marriage and remaining single, i.e. Yt (3; µ¤t ; 1) = Yt (1; 1).
(27)
In the last period, YT (3; µ ¤T ; 1) = YT (1; 1) or ¤ h w uh3T + uw 3T + 2µT = u1T + u1T :
Therefore the reservation match quality in the terminal period is h w uh1T + uw 1T ¡ u3T ¡ u3T : 2
(28)
YT ¡1 (3; µ ¤T ¡1 ; 1) = YT ¡1 (1; 1).
(29)
µ¤T = In period T ¡ 1,
¸ · ¤ k uh3T ¡1 + uw + 2µ + ¯E Y (m ; S )jS = 3 max ST T T T ¡1 T ¡1 3T ¡1 T mT ¸ · k = uh1T ¡1 + uw 1T ¡1 + ¯EST max YT (mT ; ST )jST ¡1 = 1 ; mT
µ¤T ¡1
=
h w uh1T ¡1 + uw ¯µ¤T (1 ¡ ± 3 ) 1T ¡1 ¡ u3T ¡1 ¡ u3T ¡1 + 2 [1 + ¯(1 ¡ ± 3 )] [1 + ¯(1 ¡ ± 3 )] ³R ´ 1 ¤ ¯° T ¡1 µ¤ [µ ¡ µT ] g(µ)dµ T + [1 + ¯(1 ¡ ± 3 )]
(30)
In the remaining periods of an agent’s life, reservation match quality must satisfy: Yt (3; µ¤t ; 1) = Yt (1; 1).
uh3t
+ uw 3t
·
+ 2µ¤t
k (mt+1 ; St+1 jSt+1 max Yt+1 m
+ ¯ESt+1 t+1 ¸ · h w k = u1t + u1t + ¯ESt+1 max Yt+1 (mt+1 ; St+1 )jSt = 1 mt+1
28
(31) ¸ =3
¢ ¯° 2 R 1 µg(µ)dµ ¡ h h w t u1t + uw µ¤ 1t ¡ u3t ¡ u3t t+1 + = 2 [1 + ¯(1 ¡ ± 3 )] 2 [1 + ¯(1 ¡ ± 3 )] R1 £ ¤ k ¯ 2 ° t µ¤ ESt+2 maxmt+2 Yt+2 (mt+2 ; St+2 jSt+1 = 3 g(µ)dµ
µ¤t +
t+1
2
¡
¯ °t
R1
µ¤ t+1
2 [1 + ¯(1 ¡ ± 3 )] £ ¤ k ESt+2 maxmt+2 Yt+2 (mt+2 ; St+2 jSt+1 = 1 g(µ)dµ
2 [1 + ¯(1 ¡ ± 3 )] £ ¤ k (mt+2 ; St+2 jSt+1 = 1 ¯ 2 (1 ¡ ± 3 ) ESt+2 maxmt+2 Yt+2 + 2 [1 + ¯(1 ¡ ± 3 )] £ ¤ 2 k ¯ (1 ¡ ± 3 ) ESt+2 maxmt+2 Yt+2 (mt+2 ; St+2 jSt+1 = 3 ¡ 2 [1 + ¯(1 ¡ ± 3 )] ´ " #³ h h w Z 1 u1t+1 + uw ¡ u ¡ u 1t+1 3t+1 3t+1 +¯ 1 ¡ ± 3 ¡ ° t g(µ)dµ ¤ 2 [1 + ¯(1 ¡ ± )] 3 µ t+1 Since ¸ · ¤ k uh3t+1 + uw + 2µ + ¯E Y (m ; S )jS = 3 max S t+2 t+2 t+2 t+2 3t+1 t+1 mt+2 t+2 ¸ · k = uh1t+1 + uw + ¯E Y (m ; S )jS = 1 ; max S t+2 t+2 t+1 t+2 1t+1 t+2 mt+2
the reservation match quality can be re-written as ¢ ¯° 2 R 1 µg(µ)dµ ¡ h h w t u1t + uw µ¤ 1t ¡ u3t ¡ u3t t+1 ¤ + µt = 2 [1 + ¯(1 ¡ ± 3 )] 2 [1 + ¯(1 ¡ ± 3 )] i R1 h ¤ h w g(µ)dµ ¡ u ¡ u + 2µ ¯° t µ¤ uh3t+1 + uw t+1 3 1 1 t+1 t+1 t+1 t+1 ¡ 2 [1 + ¯(1 ¡ ± 3 )] i h ¤ h w ¯ (1 ¡ ± 3 ) uh3t+1 + uw 3t+1 ¡ u1t+1 ¡ u1t+1 + 2µt+1 + 2 [1 + ¯(1 ¡ ± 3 )] ´ " #³ h h w Z 1 u1t+1 + uw 1t+1 ¡ u3t+1 ¡ u3t+1 +¯ 1 ¡ ± 3 ¡ ° t g(µ)dµ 2 [1 + ¯(1 ¡ ± 3 )] µ¤ t+1 =
h w uh1t + uw ¯ (1 ¡ ± 3 ) µ ¤t+1 1t ¡ u3t ¡ u3t + 2 [1 + ¯(1 ¡ ± 3 )] [1 + ¯(1 ¡ ± 3 )] ¤ R1 £ ¯° t µ¤ µ ¡ µ ¤t+1 g(µ)dµ t+1 : [1 + ¯(1 ¡ ± 3 )]
for t = 1; 2; :::T ¡ 2 . 29
4.3.2
Social Planner’s Problem
¤¤ ¤¤ The social planner chooses reservation match qualities (µ¤¤ 1 ; µ 2 ; :::µT ) to maximize:
1 ¦ = g1sh V1h (1; S1 ) + g1sw V1w (1; S1 ) + g1m Y1 (3; µ1 ) 2
(32)
µ1 = E [µ 1 jµ1 ¸ µ¤¤ 1 ]
(33)
where
subject to the equilibrium conditions gtsh + gtsw + gtm
= 1
g1sh
= g0sh ¡
g1sw
= g0sw ¡
g1m
= 2
Z
1 µ¤¤ 1
(34)
Z
1
µ¤¤ Z 11 µ ¤¤ 1
g (µ) d (µ) g0s¤
(35)
g (µ) d (µ) g0s¤
(36)
g (µ) d (µ) g0s¤
(37)
Di¤erentiating ¦ with respect to µ T yields: @¦ @µ T
=
@g1sh h @V h (1; S1 ) sh @g1sw w @V w (1; S1 ) sw V1 (1; S1 ) + 1 g1 + V1 (1; S1 ) + 1 g1 @µT @µT @µ T @µT m 1 @g1 + Y1 (3; µ1 ) 2 @µT £ ¤ 1 @ Y1 (3; µ1 ) m + g1 2 @µ T
and µ¤¤ T =
¢ ¡ h h w u1T + uw 1T ¡ u3T ¡ u3T = µ¤T : 2
The reservation match quality chosen by the social planner is equal to the reservation match quality in the competitive equilibrium because there is no externality on future marriage markets in the terminal period.
30
Di¤erentiating ¦ with respect to µ T ¡1 yields: h i uh1T ¡1 + uw ¡ uh3T ¡1 ¡ uw 1 3 T ¡1 T ¡1 ¯(1 ¡ ± 3 )µ ¤T + = µ ¤¤ T ¡1 2 [1 + ¯(1 ¡ ± 3 )] [1 + ¯(1 ¡ ± 3 )] R1 ¯° T ¡1 µ¤ (µ ¡ µ¤T ) g(µ)dµ T + [1 + ¯(1 ¡ ± 3 )] i h³ ´¡ R1 ¢ @° ¡1 R 1 ¤ sh sw m + ± ¯ @µTT ¡1 (µ ¡ µ ) g(µ)dµ 1 ¡ ° g(µ)dµ g + g 2g ¤ ¤ 3 T T ¡2 µ T ¡1 1 1 1 µT ¢ ¡ sh + sw g(µ2 )° T ¡2 [1 + ¯(1 ¡ ± 3 )] g1 + g1 i h³ ´¡ ¢ R1 @° T ¡1 R 1 ¯ @µT ¡1 µ¤¤ (µ ¡ µ¤T ) g(µ)dµ 1 ¡ ° T ¡2 µ¤ g(µ)dµ g1sh + g1sw + ± 3 2g1m T T ¡1 ¢ ¡ = µ ¤T ¡1 + g(µT ¡1 )° T ¡2 [1 + ¯(1 ¡ ± 3 )] g1sh + g1sw
The reservation match quality chosen by the social planner in period T ¡ 1 is equal to the reservation match quality in the competitive equilibrium plus an additional term describing the e¤ect of a change in the reservation match quality on the matching function. The latter is the externality agents do not take into account when they decide to match. Notice · ¸ 2 ¸1 gts¤ @ ¸1 gs¤ +¸3 g3 t t @° t = ¤¤ @µ t @µt £ ¤ £ ¤ ¤¤ ¤¤ s¤ s¤ s¤ 3 s¤ s¤ s¤2 2¸1 gt g (µ ¤¤ t ) gt¡1 ¸1 gt + ¸3 gt ¡ ¸1 g (µt ) gt¡1 ¡ 2¸3 g (µt ) gt¡1 ¸1 gt = 2 [¸1 gts¤ + ¸3 gt3 ] 2
=
¤¤ ¤¤ s¤ s¤ s¤ 3 s¤ s¤ ¸21 gts¤2 g (µ¤¤ t ) gt¡1 + 2¸1 gt g (µt ) gt¡1 ¸3 gt + 2¸3 g (µt ) gt¡1 ¸1 gt 2
[¸1 gts¤ + ¸3 gt3 ]
> 0 as long as the arrival rates are positive. Therefore, in our simple example, the reservation match quality chosen by the social planner is higher in period T ¡ 1 than the reservation match quality in the competitive equilibrium. As a result, too many matches are formed in the competitive equilibrium.
Note: we still need to solve for the remaining T-2 reservation match qualities in the planner’s problem
4.4
Solving for Equilibrium
Solving for equilibrium is an iterative process. 1. Make an initial guess of ¨st (mt ; µ dt j ¹; X) and ¨rt (mt ; µdt j mt¡1 ; ¹; µ dt ¡1 ; µ1 ; X) for all arguments of ¨s and ¨rt . A good guess would be the empirical transition probabilities. Set ¶ = 0, and call your initial guesses s r ¶ ¨t (mt ; µdt j ¹; X) and ¶ ¨t (mt ; µdt j mt¡1 ; ¹; µ dt ¡1 ; µ1 ; X) : 31
2. Use ¶ ¨st¡(mt ; µ d¢t j ¹; X) and ¶ ¨rt (mt ; µdt j mt¡1 ; ¹; µdt ¡1 ; µ1 ; X) to solve for ¶ gtsk ¹k ; X k and gtr (¹; µ 1 ; µd ; X; m) for all arguments of g using equations (22) and (23) repeatedly. ¡ ¢ 3. Solve for ¶ Yt (Pt ; St ) in equation (9) using ¶ gtsk ¹k ; X k and ¶ gtr (¹; µ1 ; µ d ; X; m).
4. Use ¶ Yt (Pt ; St ) in equation (21) to solve for ¶ ¨st (mt ; µdt j ¹; X) and ¶ ¨rt (mt ; µdt j mt¡1 ; ¹; µ dt ¡1 ; µ1 ; X). £ ¤ 5. Check for convergence of ¶ ¨st ;¶ ¨rt ;¶ Yt ;¶ gtsk ;¶ gtr . If not, increment ¶ and return to step (2). ¡ ¢ One could probably hold gtsk ¹k ; X k and gtr (¹; µ 1 ; µd ; X; m) …xed at their empirical density for a while to avoid having to iterate over steps (2) through (5). This is¡ especially true in an econometrics exercise when the deviation ¢ between gtsk ¹k ; X k and gtr (¹; µ1 ; µ d ; X; m) and their empirical counterparts are in the statistical objective function.
4.5
Policy Analysis
Given the algorithm in the previous section, we can now measure direct and equilibrium e¤ects of policy or environmental changes. ¡ ¢ We can measure direct changes in behavior by treating gtsk ¹; µ1 ; µd ; m; X k and gtr (¹; µ1 ; µd ; X; m) as …xed and evaluating how the policy or environmental change a¤ects Yt (Pt ; St ) and ¨st (mt ; µdt j ¹; X) and ¨rt (mt ; µ dt j mt¡1 ; ¹; µdt ¡1 ; µ1 ; X). We can measure equilibrium e¤ects by using the algorithm above.
4.6
Assortative Mating
Once one speci…es a functional form for the utility function in equation (2), the model has very strong results with respect to assortative mating. The degree of assortative mating depends upon interaction terms in the utility function, on the methods people use to search for relationship partners, and the Nash bargaining assumption. Assortative mating depends upon the utility function in obvious ways. For example if utility is much higher when the race (religion) dummies in X h and X w are the same, then people of the same race (religion) will be more likely to have relationships with each other and less likely to separate. Also, if @2 @2 uk3 > uk (= 0) ; @Educi @Educj @Educi @Educj 1 then people will sort themselves by education. Assortative mating depends on the search methods used because they may limit (or skew) the sample from which one draws potential partners. For example, if people generally …nd mates at church (or synagogue) then, independent of preferences re‡ected in ukm (¢), people of common religion are more likely to marry. Of course the search mechanism may be endogenous; e.g., if having a
32
common religion is of value, then it is e¢cient to look for potential mates where there are many people of the same religion. These two e¤ects on assortative mating have very di¤erent implications. For example, the preference argument implies that blacks take longer to get married because they want to marry within their race but the pool of blacks with other characteristics (e.g., earning power) is not so good. On the other hand, the sampling argument implies that blacks take longer to marry because they sample too many whites? We can possibly identify these two e¤ects with data in two di¤erent ways. First, while the sampling argument a¤ects the marriage rate (e.g., it reduces the marriage rate for blacks), its e¤ect on the divorce rate is usually the other way (e.g., it unambiguously reduces the divorce rate because it will be di¢cult to …nd another mate). On the other hand, the preference argument has the same qualitative e¤ect on the marriage rate and an ambiguous e¤ect on the divorce rate (e.g., blacks are more likely to divorce because their spouse has not such great X’s, but they are less likely because the distribution of other X’s is not so good). Second, if the sampling argument is the dominant e¤ect, then it implies that people of ethnic groups (e.g., race, religion) of similar size will have similar marriage rates at least after controlling for variation in X’s. Thus, we might be able to measure the sampling e¤ect by comparing the marriage rate for Jews vs. blacks. If, as is my guess, Jews get married much faster even after controlling for variation in X’s, providing some support for the sampling argument. In my own personal experience, though I have a strong preference for marrying another Jew, the marriage market at my synagogue is so thin, it is not a viable market. The Nash bargaining assumption has assortative mating implications in that the side-payment implications may cause negative assortative mating. For example, assume that there is no preference for marrying someone of the same race, and that whites enjoy marriage more than blacks. In particular, assume that uk3 ¡ uk1 = ° 0 + ° 1 Blackk + µ with ° 1 < 0, Blackk = 1 i¤ partner k is black, and 2 (° 0 + µ) > 2 (° 0 + µ) + ° 1 > 0 > 2 (° 0 + ° 1 + µ) : Then if both partners are white they will marry (2 (° 0 + µ) > 0), if both are black they will not (0 > 2 (° 0 + ° 1 + µ)), and mixed partners will marry (2 (° 0 + µ) + ° 1 > 0). In the case of the mixed marriage, the white partner essentially provides the black partner with a necessary (because ° 0 + ° 1 +µ < 0) side-payment to make the marriage worthwhile. Thus, the availability of sidepayments “causes” negative assortative mating by race. This is essentially a selection e¤ect. The same type of selection e¤ect could exist even in the presence of a preference to marry someone of the same race. It suggests a negative bias associated with any estimate of assortative mating where there is also the potential for variation in preferences for marriage by the characteristic and the potential for side payments. 33
5
Data
We have not yet decided on a data set. But, we know there are a number of choices where we can construct histories of men and women similar to those we constructed in BLS. We will also need a data set that can provide us with a consistent estimate of the initial joint density of single people and their observed characteristics. For example, Census data would provide such an estimate.
6
Estimation
Estimation proceeds iteratively. Start o¤ by making initial guesses of the parameters. Solve for the equilibrium densities of married and single people. Then update parameter guesses using methods similar to those used in BLS holding the equilibrium density of married and single people …xed. Somewhat infrequently, update the equilibrium density of married and single people so that it is consistent with the decisions predicted by the model at the parameter values guessed. We anticipate that such a procedure will be far more computationally e¢cient than updating the equilibrium density of married and single people at each guess of the parameters because changes in the equilibrium density of married and single people will be second order relative to changes in the value functions of individual agents. There is an issue associated with the size of the state space and whether we will have the memory available to store the matrix of value functions for a single representative person (as described in BLS) in core. We may have to compromise on what variables we allow to remain in the state space and what values they take on. However, relative to BLS, we are saving space by not allowing for learning, and available computer memory has increased signi…cantly since we estimated the model in BLS.
7
References
References [1] Alm, James and Leslie Whittington (1995). “Does the Income Tax A¤ect Marital Decisions?” The National Tax Journal 48(4), 565-572. [2] Blundell, Richard, Andre Chiapori, Thierry Magnac, and Costas Meghir (2001). “Collective Labor Supply: Heterogeneity and Nonparticipation.” Unpublished manuscript, University College London. [3] Brien, Michael, Lee Lillard, and Steven Stern (2001). “Cohabitation, Marriage, and Divorce in a Model of Match Quality.” Unpublished manuscript, University of Virginia.
34
[4] Chade, Hector and Gustavo Ventura (2001a). “On Income Taxation and Marital Decisions.” Unpublished manuscript, University of Western Ontario. [5] Chade, Hector and Gustavo Ventura (2001b). “Taxes and Marriage: A Two-Sided Search Analysis.” International Economic Review. Forthcoming. [6] Drewianka, Scott (1999). “A Search-Based Theory of Social Interactions in Family Structure Decisions.” Unpublished manuscript, University of Chicago. [7] Friedberg, Leora (1998). “Did Unilateral Divorce Raise Divorce Rates? Evidence From Panel Data.” American Economic Review, 83(3). [8] Gruber, Jonathan (2000). “Is Making Divorce Easier Bad For Children? The Long Run Implications of Unilateral Divorce.” NBER Working Paper 7968. [9] Lundberg, Shelley and Robert Pollack (1993). “Separate Spheres Bargaining and the Marriage Market.” Journal of Political Economy. 101(6): 9881010. [10] Milgrom, Paul and Ilya Segal (2001). “Envelope Theorems for Arbitrary Choice Sets.” Econometrica. Forthcoming. [11] Sjoquist, David. and Mary Beth Walker (1995). “The Marriage Tax and the Rate and Timing of Marriage.” The National Tax Journal 48(4), 547-558. [12] Stevenson, Betsey and Justin Wolfers (2000). “‘Till Death Do Us Part: Effects of Divorce Laws on Suicide, Domestic Violence and Spousal Murder.” Unpublished Manuscript, Harvard University.
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