United but (un)equal: human capital, probability of divorce, and the marriage contract

Abstract

This paper studies how the risk of divorce affects the human capital decisions of a young couple. We consider a setting where complete specialization is optimal with no divorce risk. Couples can self-insure through savings which offers some protection to the uneducated spouse, but at the expense of a distortion. Alternatively, for large divorce probabilities, symmetry in education, where both spouses receive an equal amount of education, may be optimal. This eliminates the risk associated with the lack of education, but reduces the efficiency of education choices. We show that the symmetric allocation will become more attractive as the probability of divorce increases, if risk aversion is high and/or labor supply elasticity is low. However, it is only a “second-best” solution as insurance protection is achieved at the expense of an efficiency loss. Finally, we study how the (economic) use of marriage is affected by the possibility of divorce.

Appendix: General utility

Throughout this paper, we assume quasi-linear preference (with no income effect) and a utility function given by u(c _i −v(ℓ _i )). This specification implies that labor supply is always increasing in wage, and one might be tempted to consider that this is crucial for our results. However, this is not the case. We use this specification for simplicity and to be able to reduce the problem to a single dimension. But this is simply a matter of exposition. Proposition 1 continues to be valid for general utility functions. This is obvious for items (iii) and (iv), but needs to be established for (i) and (ii).

The result that maximum wage differentiation is optimal for π = 0 with general utility functions follows directly from Cremer et al. (2011). The theoretical argument effectively formalizes the intuition explained in Section 3.3 above. To make this paper self-contained, we briefly sketch this main argument. Assume the general utility function u(c _i ,ℓ _i ), then for equal wages and π = 0 we will have c _i =w _i ℓ _i =ℓ _i and s = 0 (i.e., spouses consume their own incomes, which are equal and saving is zero). Welfare is then given as follows,

$$ \mathcal{W}^{E}=4u\left(\ell^{E},\ell^{E}\right) , $$

(A1)

where

$$ \ell^{E}=\arg \max_{\ell }\;u\left(\ell ,\ell \right) . $$

(A2)

With unequal wages, w _i = 2 and w _−i = 0, the spouse with w _i = 0 does not work and earns no income but he/she receives a compensation from the spouse with higher human capital, as optimally marginal utilities are equalized. Denoting this transfer T, the couple’s welfare is given as follows,

$$\begin{array}{@{}rcl@{}} \mathcal{W}^{MD} &=&\max\limits_{\ell_{i},\ell_{-i},T}\;2u\left(w_{i}\ell_{i}-T,\ell_{i}\right) +2u\left(w_{-i}\ell_{-i}+T,\ell_{-i}\right) , \\ &=&2u(2\ell_{i}^{MD}-T,\ell_{i}^{MD})+2u(0+T,0), \end{array} $$

where we have ℓ _−i = 0. Observe that as long as π = 0 we have s = 0, and the two periods are perfectly symmetrical. Now assume the individual with higher human capital simply gives half of his/her income to his/her spouse ($T=\ell _{i}^{MD}$) so that consumption levels are equalized (which is generally not the optimal level). Additionally, set $\ell _{i}^{MD}=\ell ^{E}$, so that the individual with w _i = 2 works the same number of hours with equal wages (which is also generally not optimal). With these two assumptions, we have

$$\mathcal{W}^{MD}=2u\left(\ell^{E},\ell^{E}\right) +2u\left(\ell^{E},0\right) >\mathcal{W}^{E}. $$

In other words, under wage differentiation the couple can achieve the same consumption levels as under wage equalization by having only a single individual work (the same amount as under wage equalization). Intuitively, this is simply a generalization of the argument discussed in Section 3.3 and represented in Fig. 1. This establishes that item (i) of Proposition 1 continues to be valid with more general utility function.

Introducing a positive π, a couple’s welfare with general utility functions is redefined as follows:

$$\begin{array}{@{}rcl@{}} \mathcal{W}=& u\left(c_{m}^{1c},\ell_{m}^{1c}\right) +u\left(c_{f}^{1c},\ell_{f}^{1c}\right) +\pi \left[ u\left(c_{m}^{2s},\ell_{m}^{2s}\right) +u\left(c_{f}^{2s},\ell_{f}^{2s}\right) \right] \notag \\ & +(1-\pi )\left[ u\left(c_{m}^{2c},\ell_{m}^{2c}\right) +u\left(c_{f}^{2c},\ell_{f}^{2c}\right) \right] . \end{array} $$

(A3)

Budget constraints are unchanged and continue to be given by Eqs. (3)–(6).²⁰ Under equal wages, maximizing (A3) subject to Eqs. (3)–(6) yields $c_{i}^{tj}=\ell _{i}^{tj}=\ell ^{E}$ defined by Eq. (A2) so that welfare is given by $\mathcal {W}^{E}=4u\left (\ell ^{E},\ell ^{E}\right )$ which does not depend on π. Under unequal wages (w _i = 2 and w _−i = 0), differentiating welfare with respect to π, while using the envelope theorem yields

$$\frac{\partial \mathcal{W}}{\partial \pi }=\left[ u\left(c_{m}^{2s},\ell_{m}^{2s}\right) +u\left(c_{f}^{2s},\ell_{f}^{2s}\right) \right] -\left[ u\left(c_{m}^{2c},\ell_{m}^{2c}\right) +u\left(c_{f}^{2c},\ell_{f}^{2c}\right) \right] \leq 0. $$

To establish the inequality, observe that the second term in brackets is the utility of the married couple, while the first term is the utility of the divorced spouses. Since savings and productivities are the same in both cases, the utility of the married couple is always at least as large as that of the divorced spouses. This is because the (c,ℓ) bundles chosen by the divorced spouses are feasible for the married couple, while the opposite is not true. Consequently, item (ii) of the proposition stating that wage differentiation becomes less attractive as π increases, remains valid with general utility. To summarize, none of these results requires an increasing labor supply function.