Marriage as a commitment device

Abstract

Non-cooperative couples are inefficient. Cooperation raises the utility of both parents, and of each child, but does not guarantee efficiency. In the presence of credit rationing, a cooperative equilibrium may not exist outside marriage, because the main earner cannot credibly promise to compensate the main childcarer at some future date, and may not be able or willing to do so at front. By allowing the main childcarer to credibly threaten divorce if the main earner does not deliver the promised compensation when the time comes, marriage makes that promise credible, and thus increases the probability that a cooperative equilibrium will exist. In a separate-property jurisdiction, a reduction in the cost or difficulty of obtaining a divorce increases married women’s participation in the labour market. In a community-property one, it has no such effect.

Appendix

1.1 Efficiency

The first-order conditions yield

$$ \frac{u^{\prime }\left( a_{i1}\right) }{u^{\prime }\left( a_{i2}\right) }=r+ \frac{\rho }{\mu }, $$

(35)

$$ \frac{v_{t}}{v_{c}}=-\left( y_{f1}^{\prime }+\frac{y_{f2}^{\prime }}{r+\frac{ \rho }{\mu }}\right) \frac{\partial L_{f}}{\partial t}-\left( y_{m1}^{\prime }+\frac{y_{f2}^{\prime }}{r+\frac{\rho }{\mu }}\right) \frac{\partial L_{m}}{ \partial t} $$

(36)

and

$$ \frac{v}{v_{c}}=c-\left( y_{f1}^{\prime }+\frac{y_{f2}^{\prime }}{r+\frac{ \rho }{\mu }}\right) \frac{\partial L_{f}}{\partial n}-\left( y_{m1}^{\prime }+\frac{\mu y_{f2}^{\prime }}{r+\frac{\rho }{\mu }}\right) \frac{\partial L_{m}}{\partial n}, $$

(37)

where μ is the Lagrange-multiplier of (10), necessarily positive, and ρ that of (11), positive if the couple is credit constrained, zero otherwise.

In view of (3, 4), and assuming that n is positive (or there would be no division of labour, and no gain from the union),

$$ \frac{\partial L_{f}}{\partial t}<0, \quad \frac{\partial L_{f}}{\partial n} <0\hbox { and }\frac{\partial L_{m}}{\partial t}=\frac{\partial L_{m}}{ \partial n}=0 $$

(38)

if $\left( h_{f},h_{m}\right)$ satisfies (13), and the division of labour is consequently (14),

$$ \frac{\partial L_{f}}{\partial t}=0,\quad \frac{\partial L_{f}}{\partial n} <0, \quad\frac{\partial L_{m}}{\partial t}<0\hbox { and }\frac{\partial L_{m}}{\partial n}<0 $$

(39)

if $\left( h_{f},h_{m}\right)$ violates (13), and the division of labour is then (15).

1.2 Cournot-Nash equilibrium

For i’s (i = f, m) first-order conditions on the choice of $\left( a_{i1},a_{i2},s_{i},c_{i},t_{i}\right) ,$

$$ u^{\prime }\left( a_{i1}\right) =\beta v_{c}, $$

(40)

$$ \frac{u^{\prime }\left( a_{i1}\right) }{u^{\prime }\left( a_{i2}\right) }=r+ \frac{\rho _{i}}{\mu _{i}} $$

(41)

and

$$ \frac{v_{t}}{v_{c}}=\left( y_{i1}^{\prime }+\frac{y_{i2}^{\prime }}{r+\frac{ \rho _{i}}{\mu _{i}}}\right) n, $$

(42)

where μ _i is the Lagrange-multiplier of i’s period-2 budget constraint, and ρ _i that of i’s credit constraint (i = f, m). Additionally, for f’s first-order condition on the choice of n, 

$$ \frac{v}{v_{c}}=c_{f}+\left( y_{f1}^{\prime }+\frac{y_{f2}^{\prime }}{r+ \frac{\rho _{f}}{\mu _{f}}}\right) \left( t_{0}+t_{f}\right). $$

(43)

Using (5, 6) and (40–42), we find that, at the CN equilibrium, a _f1 = a _m1, a _f2 = a _m2, μ _f  = μ _m , ρ _f  = ρ _m and U _f  = U _m . As f could always choose n = 0, and thus L _f  = 1, the common utility level will be at least equal to U ^S. In view of (3–6) and (43), we also find that

$$ \begin{aligned} &\left( 1+2\alpha \left[ 1-\left( t_{0}+t_{f}\right) n\right] +\frac{\mu \alpha \left( 1+\alpha \right) }{r\mu +\rho }\right) h_{f} \\ =&\left( 1+2\alpha \left( 1-nt_{m}\right) +\frac{\mu \alpha \left( 1+\alpha \right) }{r\mu +\rho }\right) h_{m}. \\ \end{aligned} $$

where μ is now used to denote the common equilibrium value of μ _f and μ _m , and ρ that of ρ _f and ρ _m , at the CN equilibrium. This tells us that, if h _f  = h _m and, consequently in view of (7), b _f  = b _m , f and m will split everything down the middle. Otherwise, the monetary cost of the children will be divided equally between them, but the parent with the larger human capital endowment will supply more child care, and less market work, than the one with the larger money endowment. In other words, the parties will specialize against their comparative advantages.¹² The opportunity-cost of the children will not be minimized in either case.

The equilibrium will be inefficient, even if ρ = 0, because the MRS of t for c is equated to each parent’s, rather than to the couple’s, marginal opportunity-cost of providing attention, and that of n for v to the full cost of an extra child for f, rather than for the couple. As the full cost for the couple is inefficiently large, however, we cannot be sure that f’s share of this cost will be smaller than the efficient total. Therefore, we cannot say in general whether n will be too large or too small.

1.3 Nash-bargaining equilibrium without marriage

From the first-order conditions, we find that, at each point of the UPF,

$$ \frac{u^{\prime }\left( a_{i1}\right) }{u^{\prime }\left( a_{i2}\right) }=r+ \frac{\rho _{i}}{\mu _{i}}, \quad i=f,m, $$

(44)

$$ \frac{v_{t}}{v_{c}}=-\left( y_{f1}^{\prime }+\frac{y_{f2}^{\prime }}{r+\frac{ \rho _{f}}{\mu _{f}}}\right) \frac{\partial L_{f}}{\partial t}-\left( y_{m1}^{\prime }+\frac{y_{m2}^{\prime }}{r+\frac{\rho _{m}}{\mu _{m}}} \right) \frac{\partial L_{m}}{\partial t} $$

(45)

and

$$ \frac{v}{v_{c}}=c-\left( y_{f1}^{\prime }+\frac{y_{f2}^{\prime }}{r+\frac{ \rho _{f}}{\mu _{f}}}\right) \frac{\partial L_{f}}{\partial n}-\left( y_{m1}^{\prime }+\frac{y_{m2}^{\prime }}{r+\frac{\rho _{m}}{\mu _{m}}} \right) \frac{\partial L_{m}}{\partial n}, $$

(46)

where μ _i is again the Lagrange-multiplier of i’s period-2 budget constraint, and ρ _i that of i’s credit constraint (i = f, m). As cooperative parents specialize according to their personal comparative advantages, the signs of $\frac{\partial L_{i}}{\partial t}$ and $\frac{\partial L_{i}}{\partial n}$ are those shown in (38) if the initial endowments satisfy (??), or those shown in (39) if they do not. If ρ _f  = ρ _m  = 0, (44–46) reduce to (35–37), and the allocation is then efficient at every point of the UPF. Otherwise, f’s intertemporal trade-off, $r+\frac{\rho _{f}}{\mu _{f}},$ will be different from m’s, $r+\frac{\rho _{m}}{\mu _{m}}.$ If that is the case, at some point or everywhere along the UPF, the allocation will be inefficient despite the fact that cooperative parents specialize according to their comparative advantages.

Let j denote the main childcarer, and k the main earner (j, k = f, m). As x ₁ becomes larger (more positive if k = m, more negative if k = f), U _j will rise relative to U _k , but k’s credit ration will become tighter relative to j’s. In the case where

$$ 0<\rho _{j}\leq \rho _{k}\quad\hbox{for}\,x_{1}=0, $$

(47)

the allocation will then become more inefficient. As the opportunity-cost of U _j in terms of U _k increases faster than it would if k could either borrow, or postpone the payment, the UPF will be steeper than the efficiency locus, and lie inside it, everywhere in the $\left( U_{j},U_{k}\right)$ plane. The case where

$$ 0<\rho _{k}\leq \rho _{j}\quad \hbox{for}\, x_{1}=0 $$

(48)

is more complicated. Up to the point where ρ _k  = ρ _j , any increase in the size of x ₁ will make the allocation less inefficient. The opportunity-cost of U _j in terms of U _k will then rise more slowly than it would if k were more tightly credit constrained than j, and the UPF will consequently be flatter than the efficiency locus, but still lie inside it. From that point onwards, we are back to the previous case. In view of (7) and (13) , $\left( b_{k}-b_{j}\right)$ will be negative if k = f, but can have any sign if k = m. In the absence of any information about $\left( b_{f},h_{f}\right)$ and $\left( b_{m},h_{m}\right)$, other than that they satisfy (7), and given that $\left(y_{k1}-y_{j1}\right)$ is positive anyway, the likelihood of (47) is then higher if we observe k = f, than if we observe k = m.

1.4 Nash-bargaining equilibrium with separate-property marriage

From the first-order conditions, we find that, at each point of the married UPF,

$$ \frac{u^{\prime }\left( a_{i1}\right) }{u^{\prime }\left( a_{i2}\right) }=r+ \frac{\rho _{i}}{\mu _{i}-\xi _{i}},\quad i=f,m, $$

(49)

$$ \frac{v_{t}}{v_{c}}=-\left( y_{f1}^{\prime }+\frac{y_{f2}^{\prime }}{r+\frac{ \rho _{f}}{\mu _{f}-\xi _{f}}}\right) \frac{\partial L_{f}}{\partial t} -\left( y_{m1}^{\prime }+\frac{y_{m2}^{\prime }}{r+\frac{\rho _{m}}{\mu _{m}-\xi _{m}}}\right) \frac{\partial L_{m}}{\partial t} $$

(50)

and

$$ \frac{v}{v_{c}}=c-\left( y_{f1}^{\prime }+\frac{y_{f2}^{\prime }}{r+\frac{ \rho _{f}}{\mu _{f}-\xi _{f}}}\right) \frac{\partial L_{f}}{\partial n} -\left( y_{m1}^{\prime }+\frac{y_{m2}^{\prime }}{r+\frac{\rho _{m}}{\mu _{m}-\xi _{m}}}\right) \frac{\partial L_{m}}{\partial n}, $$

(51)

where ξ _i denotes the Lagrange-multiplier of i’s divorce-threat constraint (i = f, m), and the other variables are defined as in the last section.

If neither divorce-threat constraint is binding (ξ _f  = ξ _m  = 0), these conditions reduce to (35–37), and the married UPF coincides with the unmarried one. Not so if either of these constraints is binding at some point. Given that only j’s divorce-threat constraint can be binding, k’s intertemporal trade-off remains $\left( r+\frac{\rho _{k}}{\mu _{k}}\right) ,$ but j’s becomes $\left( r+\frac{\rho _{j}}{\mu _{j}-\xi _{j}}\right)$. In view of (28), ξ _j  ≥ 0 as U _j  ≤ U _k . If (47) is true, the difference between k’s and j ’s intertemporal trade-offs is initially smaller, and the allocation less inefficient, than it would be without marriage. As x ₁ becomes larger, however, ξ _j decreases. Therefore, the married UPF is steeper than the unmarried one for all U _j  ≤ U _k , and lies outside it for all U _j  < U _k . By contrast, if (48) is true, the difference between the two trade-offs will be initially larger, and the allocation more inefficient, than it would be without marriage. As x ₁ becomes larger, ξ _j will again decrease, but the married UPF will now be flatter than the unmarried one for all U _j  ≤ U _k . In either case, the married UPF will coincide with the unmarried one for all U _j  ≥ U _k .

1.5 NB equilibrium with community-property marriage

For the first-order conditions, at each point of the married UPF,

$$ \frac{u^{\prime }\left( a_{i1}\right) }{u^{\prime }\left( a_{i2}\right) }= \frac{\mu r+\rho }{\mu -\xi _{i}}, \quad i=f,m, $$

(52)

$$ \frac{v_{t}}{v_{c}}=-\left( y_{f1}^{\prime }+\frac{\mu -\xi _{f}}{\mu +\rho } y_{f2}^{\prime }\right) \frac{\partial L_{f}}{\partial t}-\left( y_{m1}^{\prime }+\frac{\mu -\xi _{f}}{\mu +\rho }y_{m2}^{\prime }\right) \frac{\partial L_{m}}{\partial t} $$

(53)

and

$$ \frac{v}{v_{c}}=c-\left( y_{f1}^{\prime }+\frac{\mu -\xi _{f}}{\mu +\rho } y_{f2}^{\prime }\right) \frac{\partial L_{f}}{\partial n}-\left( y_{m1}^{\prime }+\frac{\mu -\xi _{f}}{\mu +\rho }y_{m2}^{\prime }\right) \frac{\partial L_{m}}{\partial n}, $$

(54)

where μ denotes the Lagrange-multiplier of the couple’s joint period-2 budget constraint, ρ that of their joint credit constraint, and ξ _i that of i’s divorce-threat constraint, i = f, m. As only one of ξ _f and ξ _m can be positive, and irrespective of whether the credit constraint is or is not binding, an allocation will then be efficient if and only if neither divorce-threat constraint is binding.