Relational contracts for household formation, fertility choice and separation

Abstract

This paper applies the theory of relational contracts to a model in which a couple decides upon fertility and subsequently on continuation of the relationship. We formalize the idea that within-household-cooperation can be supported by selfinterest. Since the costs of raising children—a household public good—are unequally distributed between partners, a conflict between individually optimal and efficient decisions exists. Side-payments can support cooperation but are not legally enforceable and thus have to be part of an equilibrium. This requires stable relationships and credible punishment threats.Within this framework, we analyze the effects of separation costs and post-separation alimony payments on couples’ fertility decisions. We derive the predictions that higher separation costs and higher alimony payments facilitate cooperation and hence increase fertility. We present empirical evidence based on a recent German reform that reduced rights to post-divorce alimony payments. We find that this reform reduced in-wedlock fertility.

Appendices

Appendix A: Omitted proofs

Lemma 2

Assume players can sign a contract at the beginning of period t=1, with court-enforceable discretionary transfers and player 1 getting a share α∈[0, 1] of the resulting net surplus. Such a contract will yield the efficient outcome, i.e. n ^∗ =n ^FB , and the couple separates if and only if $v>\hat {v}$.

Proof

We have to show that transfers exist that make both partners agree to the contract (with the alternative being the non-cooperative outcome) and that make both partners agree to the efficient outcome. There, we assume that p ₁ is solely used to redistribute gains from a higher fertility, and p ₂ is solely used to eventually keep the relationship together, i.e., p ₁(n) and p ₂(v ₁,v ₂). This assumption can be made without loss of generality, since efficient as well as non-cooperative fertility levels are independent of all second-period events.

First, we analyze period 1, i.e. the couple’s fertility choice. There, assume the following contract: The couple agrees on a fertility level n ^∗ = n ^FB. If n ^FB is actually chosen, the payment $p_{1}^{*}>0$ (where the actual value is determined below) is made from primary to second earner. For any other realized fertility level, no payment is made. This implies that if a deviation occurs, the second earner will only agree on the non-cooperative level $\underline {n}$.

To compute the value of p ₁, we calculate the net surplus of having n ^FB, compared to $\underline {n}$:

$$\begin{array}{@{}rcl@{}} &&\left( \varphi_{2}(n^{FB})+\varphi_{1}(n^{FB})-\varphi_{2}(\underline{n})-\varphi_{1}(\underline{n})\right)\left( 1+\delta\right)+w_{21}\left( c(\underline{n})-c(n^{FB})\right)\\ &&+\delta\left( w_{22}(n^{FB})-w_{22}(\underline{n})\right). \end{array} $$

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Note that no other fertility level than n ^FB could yield a larger net surplus.

Assuming that player 1 gets a share α of this net surplus, his total utility abstracting from relationship utility R, outside utilities and associated redistributive transfers (which are unaffected by n) is

$$\begin{array}{@{}rcl@{}} \hat{U}_{1}&=&w_{1}(1+\delta)+\varphi_{1}(\underline{n})\left( 1+\delta\right)\\ &&+\alpha[\left( \varphi_{2}(n^{FB})+\varphi_{1}(n^{FB})-\varphi_{2}(\underline{n})-\varphi_{1}(\underline{n})\right)\left( 1+\delta\right)\\ &&+w_{21}\left( c(\underline{n})-c(n^{FB})\right)+\delta\left( w_{22}(n^{FB})-w_{22}(\underline{n})\right)]. \end{array} $$

Equivalently, we have for player 2:

$$\begin{array}{@{}rcl@{}} \hat{U}_{2}&=&w_{21}\left( 1-c(\underline{n})\right)+\varphi_{2}(\underline{n})+\delta\left( w_{22}(\underline{n})+\varphi_{2}(\underline{n})\right)\\ &&+\left( 1-\alpha\right)[\left( \varphi_{2}(n^{FB})+\varphi_{1}(n^{FB})-\varphi_{2}(\underline{n})-\varphi_{1}(\underline{n})\right)\left( 1+\delta\right)\\ &&+w_{21}\left( c(\underline{n})-c(n^{FB})\right)+\delta\left( w_{22}(n^{FB})-w_{22}(\underline{n})\right)]. \end{array} $$

The transfer p ₁ then has to make up for the difference between a player’s utility with realized fertility n ^FB (still abstracting from R, outside utilities and associated redistributive transfers) and their assigned utility $\hat {U}_{i}$.

Hence, $w_{1}(1+\delta )+\varphi _{1}(n^{*})\left (1+\delta \right )-p_{1}^{*}=\hat {U}_{1}$, or alternatively $w_{21}\left [1-c(n^{*})\right ]+\varphi _{2}(n^{*})+\delta \left (w_{22}(n^{*})+\varphi _{2}(n^{*})\right )+p_{1}^{*}=\hat {U}_{2}$. There, note that both approaches naturally yield the same value $p_{1}^{*}$.

Since the net surplus is positive and the alternative would be the non-cooperative fertility level $\underline {n}$, the transfer $p_{1}^{*}$ will make both partners agree on n ^FB and also willing to sign such a contract. Hence, if a transfer contingent on realized fertility is exogenously enforced, a contractual solution can yield first-best fertility.

Concerning the second period, the following transfer $p_{2}^{*}(v_{1,}v_{2})$ makes both partners sign the contract, and then agree to stay together if and only if this efficient: If v ₁ + v ₂≤2R, the transfer $p_{2}^{*}(v_{1},v_{2})$ is paid if both agree to stay together, otherwise no payment is made. The transfer is determined such that $R-p_{2}^{*}(v_{1},v_{2})=v_{1}+\alpha \left (2R-v_{1}-v_{2}\right )$, or alternatively $R+p_{2}^{*}(v_{1},v_{2})=v_{2}+(1-\alpha )\left (2R-v_{1}-v_{2}\right )$, where $p_{2}^{*}(v_{1},v_{2})$ might be positive or negative. If v ₁ + v ₂>2R, no transfer is determined by the contract. □

Corollary 1

Without loss of generality, the transfer p ₁ can be positive.

Proof

Assume the HRC involves a negative p ₁ and the share α. Replacing p ₁ by $\tilde {p}_{1}=0$ and α by $\tilde {\alpha }=\frac {-p_{1}+\delta \alpha \overline {s}_{2}(n^{*})}{\delta \overline {s}_{2}(n^{*})}$ leaves all constraints and profits as expected at the end of period t=1 unaffected. Since $-p_{1}\leq \delta (1-\alpha )\overline {s}_{2}(n^{*})$, we still have $\tilde {\alpha }\in [0,1]$. □

Corollary 2

(IC1) can be omitted.

Proof

This follows from $\underline {n}_{1}>\underline {n}_{2}$, $\underline {n}=\min \left \{ \underline {n}_{1},\underline {n}_{2}\right \}$, $n^{*}\ge \underline {n}$ and the concavity of $u_{10}^{0}+\delta \overline {u}_{12}^{d}$. □

Corollary 3

A fertility level n ^∗ can be enforced if and only if it satisfies the (IC-DE) constraint

$$ \varphi_{2}(n^{*})-\varphi_{2}(\underline{n})-w_{21}\left( c(n^{*})-c(\underline{n})\right)+\delta\overline{s}_{2}(n^{*})+\delta\left( \overline{u}_{22}^{d}(n^{\ast})-\overline{u}_{22}^{d}(\underline{n})\right)\geq0. $$

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Proof

Necessity follows from adding (IC2) and (DE1). For sufficiency, assume that Eq. 32 is satisfied. Set $p_{1}^{+}\equiv \alpha \delta \overline {s}_{2}(n^{*})$ and plug it into (IC-DE), which becomes $\varphi _{2}(n^{*})-\varphi _{2}(\underline {n})-w_{21}\left (c(n^{*})-c(\underline {n})\right )+p_{1}^{+}+\delta (1-\alpha )\overline {s}_{2}(n^{*})+\delta \left (\overline {u}_{22}^{d}(n^{\ast })-\overline {u}_{22}^{d}(\underline {n})\right )\geq 0$. Thus, (IC2) is satisfied. Furthermore, (DE1) holds by by construction of $p_{1}^{+}$. □

Proposition 1

There exists a threshold value of the relationship capital, $\overline {R}^{0}\ge 0$ , such that (IC-DE) binds for $R<\overline {R}^{0}$ and does not bind for first-best fertility otherwise.

Proof

The (IC-DE) constraint equals

$$\varphi_{2}(n^{*})-\varphi_{2}(\underline{n})-w_{21}\left( c(n^{*})-c(\underline{n})\right)+\delta\overline{s}_{2}(n^{*})+\delta\left( \overline{u}_{22}^{d}(n^{\ast})-\overline{u}_{22}^{d}(\underline{n})\right)\geq0, $$

where

$$\overline{s}_{2}(n^{\ast})=G(\hat{v})\left[u_{12}^{\ast}(n^{\ast})+u_{22}^{\ast}(n^{\ast})-\left( \overline{u}_{12}^{d}\left( n^{\ast}\mid v\le\hat{v}\right)+\overline{u}_{22}^{d}\left( n^{\ast}\mid v\le\hat{v}\right)\right)\right]. $$

Hence, we can rewrite the constraint as

$$\begin{array}{@{}rcl@{}} &&\varphi_{2}(n^{*})-\varphi_{2}(\underline{n})-w_{21}\left( c(n^{*})-c(\underline{n})\right)+\delta G(\hat{v})\left[2R-\frac{1}{G(\hat{v})}\int\limits_{0}^{\hat{v}}\tilde{v}g(\tilde{v})d\tilde{v}\right]\\ &&+\delta\left[w_{22}(n^{*})+\varphi_{2}(n^{*})\right]-\delta\left[w_{22}(\underline{n})+\varphi_{2}(\underline{n})+ {1}_{m=1}\left\{ \phi\cdot\left( w_{1}-w_{22}(\underline{n})\right)\right\} \right]\geq0. \end{array} $$

Note that $2R=\hat {v}$ and $G(\hat {v})=\int \limits _{0}^{\hat {v}}g(\tilde {v})d\tilde {v}$. Hence, $G(\hat {v})\left [2R-\frac {1}{G(\hat {v})}\int \limits _{0}^{\hat {v}}\tilde {v}g(\tilde {v})d\tilde {v}\right ]=\hat {v}\int \limits _{0}^{\hat {v}}g(\tilde {v})d\tilde {v}-\int \limits _{0}^{\hat {v}}\tilde {v}g(\tilde {v})d\tilde {v}=\int \limits _{0}^{\hat {v}}\left (\hat {v}-\tilde {v}\right )g(\tilde {v})d\tilde {v}$.

Then, (IC-DE) equals

$$\begin{array}{@{}rcl@{}} w_{21}\left( c(\underline{n})-c(n^{\ast})\right)+\varphi_{2}(n^{\ast})-\varphi_{2}(\underline{n})\\ +\delta\left( w_{22}(n^{\ast})+\varphi_{2}(n^{\ast})-w_{22}(\underline{n})-\varphi_{2}(\underline{n})\right)\\ +\delta\int\limits_{0}^{\hat{v}}g(\tilde{v})(\hat{v}-\tilde{v})d\tilde{v}\ge0, \end{array} $$

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Note that $\varphi _{2}(\underline {n})-w_{21}c(\underline {n})+\delta \left (w_{22}(\underline {n})+\varphi _{2}(\underline {n})\right )-\left [\varphi _{2}(n^{\ast })-w_{21}c(n^{\ast })+\delta \left (w_{22}(n^{\ast })+\varphi _{2}(n^{\ast })\right )\right ]\ge 0$ because of $\underline {n}=\underline {n}_{2}\le n^{*}$ (if this expression were negative, $\underline {n}_{2}$ would not be the second earner’s preferred fertility level absent transfers). Hence, the left hand side of Eq. 33 without the term $\delta \underset {0}{\overset {\hat {v}}{\int }}g(\tilde {v})(\hat {v}-\tilde {v})d\tilde {v}(\ge 0)$ is negative.

Furthermore, since $\hat {v}=2R$, the term $\int \limits _{0}^{\hat {v}}g(\tilde {v})(\hat {v}-\tilde {v})d\tilde {v}$ - and thereby the left hand side of (IC-DE) - increases in R. Hence, there exists a value of R(n ^#)≥0 for any given fertility level $n^{\#}\ge \underline {n}$ such that (IC-DE) is satisfied for n ^#. This proves the existence of $\overline {R}^{0}$ (for efficient fertility), as well as the claim that (IC-DE) binds for values of R below $\overline {R}^{0}$ and does not bind for values above. It follows that $\frac {d\overline {R}^{1}}{dk_{i}}\le 0$. □

Proposition 2

For general values of k _i and ϕ, and k ₁ +k ₂ sufficiently small, there exists a threshold value of the relationship capital, $\overline {R}^{1}\ge 0$ , such that (IC-DE) binds for $R<\overline {R}^{1}$ and does not bind—for first-best fertility as described by Eq. 12 —otherwise. Furthermore, $\overline {R}^{1}$ is decreasing in k ₁ +k ₂ and ϕ. If k ₁ +k ₂ is sufficiently large, (IC-DE) does not bind even for R=0.

Proof

Now, the (IC-DE) constraint can be written as

$$w_{21}\left( c(\underline{n})-c(n^{\ast})\right)+\varphi_{2}(n^{\ast})-\varphi_{2}(\underline{n}) $$

$$+\delta\left( w_{22}(n^{\ast})+\varphi_{2}(n^{\ast})-w_{22}(\underline{n})-\varphi_{2}(\underline{n})\right) $$

$$ +\delta\int\limits_{0}^{\hat{v}}g(\tilde{v})(\hat{v}-\tilde{v})d\tilde{v}+\delta\phi\left( w_{22}(\underline{n})-w_{22}(n^{\ast})\right)\ge0, $$

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Still, because of $\underline {n}=\underline {n}_{2}\le n^{*}$, the term $\varphi _{2}(\underline {n})-w_{21}c(\underline {n})+\delta \left (w_{22}(\underline {n})+\varphi _{2}(\underline {n})+\phi w_{22}(\underline {n})\right )$

$-\left [\varphi _{2}(n^{\ast })-w_{21}c(n^{\ast })+\delta \left (w_{22}(n^{\ast })+\varphi _{2}(n^{\ast })+\phi w_{22}(n^{\ast })\right )\right ]\ge 0$ . Hence, the left hand side of Eq. 34 without the term $\delta \underset {0}{\overset {\hat {v}}{\int }}g(\tilde {v})(\hat {v}-\tilde {v})d\tilde {v}(\ge 0)$ is negative.

Furthermore, since $\hat {v}=2R+k_{1}+k_{2}$, the term $\int \limits _{0}^{\hat {v}}g(\tilde {v})(\hat {v}-\tilde {v})d\tilde {v}$—and thereby the left hand side of (IC-DE)—increases in R. However, (IC-DE) might already be satisfied for first-best fertility even if R=0, namely when k ₁ + k ₂ is sufficiently large. The reason is that $\hat {v}$ and hence $\int \limits _{0}^{\hat {v}}g(\tilde {v})(\hat {v}-\tilde {v})d\tilde {v}$ increases in k _i . Now assume that k ₁ + k ₂ is small enough that (IC-DE) is not satisfied for first-best fertility given R=0. The existence of $\overline {R}^{1}\ge 0$ then follows, as well as $\frac {d\overline {R}^{1}}{dk_{i}}\le 0$.

Now, assume that $\overline {R}^{1}\ge 0$ exists. Then, $\frac {d\overline {R}^{1}}{d\phi }=-\frac {\left [(1+\delta )\varphi _{2}^{\prime }(\underline {n})-w_{21}c^{\prime }(\underline {n})+\delta \left (w_{22}^{\prime }(\underline {n})+\phi w_{22}^{\prime }(\underline {n})\right )\right ]\frac {d\underline {n}}{d\phi }+\delta \left (w_{22}(\underline {n})-w_{22}(n^{\ast })\right )}{2\delta G(\hat {v})}$, where however $\left [(1+\delta )\varphi _{2}^{\prime }(\underline {n})-w_{21}c^{\prime }(\underline {n})+\delta \left (w_{22}^{\prime }(\underline {n})+\phi w_{22}^{\prime }(\underline {n})\right )\right ]=0$ is the first-order condition determining $\underline {n}$. Because of $w_{22}(\underline {n})-w_{22}(n^{\ast })\ge 0$, $\frac {d\overline {R}^{1}}{d\phi }\le 0$ follows. □

Corollary 4

Assume $R<\overline {R}^{1}$ , i.e. (IC-DE) binds. Then, equilibrium fertility n ^∗ is increasing in k _i and ϕ.

Proof

The binding (IC-DE) equals

$$w_{21}\left( c(\underline{n})-c(n^{\ast})\right)+\varphi_{2}(n^{\ast})-\varphi_{2}(\underline{n}) $$

$$+\delta\left( w_{22}(n^{\ast})+\varphi_{2}(n^{\ast})-w_{22}(\underline{n})-\varphi_{2}(\underline{n})\right) $$

$$ +\delta\int\limits_{0}^{\hat{v}}g(\tilde{v})(\hat{v}-\tilde{v})d\tilde{v}+\delta\phi\left( w_{22}(\underline{n})-w_{22}(n^{\ast})\right)=0. $$

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First of all, note that the first partial derivative of the left hand side of (IC-DE) with respect to n ^∗, denoted ∂(I C−D E)/∂ n, is $-w_{21}c^{\prime }(n^{\ast })+\delta w_{22}^{\prime }(n^{\ast })\left (1-\phi \right )+\varphi _{2}^{\prime }(n^{\ast })$ and has to be negative for the (IC-DE) constraint to bind. Otherwise, higher fertility would relax the constraint, contradicting that it binds and fertility is too low at the same time. Since $\frac {d\int \limits _{0}^{\hat {v}}g(\tilde {v})(\hat {v}-\tilde {v})d\tilde {v}}{d\hat {v}}=G(\hat {v})$ and $\hat {v}=2R+k_{1}+k_{2}$, we have - as $\underline {n}\leq n^{\ast }$:

$$\begin{array}{@{}rcl@{}} \frac{dn^{\ast}}{dk_{i}} & = & -\frac{\delta G(\hat{v})}{\partial(IC-DE)/\partial n}>0 \end{array} $$

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$$\begin{array}{@{}rcl@{}} \text{ }\frac{dn^{\ast}}{d\phi} & = & -\delta\frac{\overline{w}_{2}(\underline{n})-\overline{w}_{2}(n^{\ast})}{\partial(IC-DE)/\partial n}>0 \end{array} $$

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□

Appendix B: German reform of alimony law

In Germany, a major reform of the alimony law became effective on January 1st, 2008. It was implemented to adapt the existing law to changes in society, visible in increasing divorce rates, increasing shares of children born out-of-wedlock, increases in single parenthood and couples having additional children with a new partner after a divorce. The main goals of the reform were to increase child welfare, to increase partner’s responsibility in supporting their own living after a divorce and to simplify the family law.²³

The German government published a draft of the reform in alimony law in 2006. At least since then it was thus known that changes in alimony law could be expected, possibly leading to anticipatory changes in behavior. The key aspect of the reform for the current paper, however, was not included in the first draft of the law but was only included during the year 2007 after a ruling of the German constitutional court. The latter ruled in February 2007 that previously very unequal alimony rights to single parents, depending on whether the children were born in- or out-of-wedlock, were unconstitutional. The first version of the reform did not address these differences. Following the publication of the constitutional court’s ruling in May 2007,²⁴ the legislative procedure for the reform was thus interrupted and the reform adapted to include an equalization of rights to single parents. The changes were signed into law at the end of November 2007.

The key aspect of the reform for the current paper was this equalization of alimony payments for single parents. Before 2008 in case of a divorce, the partner who cared for the children was entitled to alimony payments to cover her living expenses—in addition to alimony payments (or child support) to cover the children’s expenses—at least until the youngest child in the household had turned 8. In contrast, single parents who were not previously married to the other parent were only entitled to alimony payments for their own living expenses until the youngest child had turned 3. If the youngest child was older than 3, the single parent was expected to earn her own living. Whether children were born in- or out-of-wedlock did not matter for the child support. The reform in the alimony law set the general cutoff age to 3 also for partners who were previously married, although allowing for longer alimony payments if necessary, for example in the absence of adequate child care options.