## NOTES

1 The exception is the state of Mississippi where there is ‘common property law’. This means that marital assets are divided according to who has the legal title.

2 Assets are observable and contractible in their model, as they are in ours. This means that the standard individual Euler equation does not hold, it needs to be augmented with terms which reflect the impact of wealth on the participation constraints.

3 In the Appendix, we show that our analysis can be extended to the case of a home good whose consumption is public to the household.

4 It is trivial to show that in *t* = 1 and assuming
$\lambda _1 \approx \frac{1}{2}$
the analogous participation constraints are slack. To see why, note that the spouses are identical in period 1 and benefit from being together since ξ > 0 and because of risk sharing against period 2 shocks.

5 For the class of utility functions we will utilize it is straightforward to show monotonicity with respect to λ_{2}. The conditions basically say that given productivity there is a positive surplus in the marriage and a sharing rule consistent with participation.

6 Obviously if the weights sum to one or not is immaterial for the optimal program. Moreover, the fact that we omit ξ from the objective is also without loss of generality (ξ matters only for the participation constraints).

7 As discussed previously, in the two period model it is straightforward to show analytically that the utility of individual *g* is monotonic in the relative weight. This means that the values of *v*
_{2, g
} are consistent with the updating scheme in (3) (when a participation constraint binds the relative weight is uniquely given and so on). In the more general multiperiod setup this can also be illustrated based on the properties of utility and the fact that a low weight implies lower consumption.

9 It is worth emphasizing that the property that the couple can for any ε_{
m
}, ε_{
f
} pair replicate the ‘bachelor’ allocation is not a general feature of the dynamic model of intrahousehold bargaining. It holds in the two period model precisely because in *t* = 2 the household does not have to make an optimal savings decision. If we assumed a third period, ε_{
m
} > ε_{
f
} and ξ = 0, then the male spouse would be better off outside the marriage since period 2 savings are split equally between the spouses in period 3. Such features are present in the dynamic model of Section 3 where we have to assume ξ > 0 in order to sustain an equilibrium without divorce.

10 It is worth noting that general equilibrium effects, that are left out from (5), typically operate in the opposite direction. To see this, note that in the short run, with the economy’s capital being fixed, a fall in labor taxes will affect wages and interest rates through hours worked. Since hours will increase in response to the fall in distortionary taxation, the wage rate *w* will decrease and the interest rate *r* will rise (under a Cobb–Douglas production technology). Consequently, the above expressions have to be multiplied by
$1+\frac{ d w}{ d (1-\tau _N ) } \frac{(1-\tau _N)}{w}$
. Moreover, an additional term that reflects the effect of higher financial income on the constraint set has to be included. This term is given by
$\frac{d \; r (1-\tau _K) a_1 }{ d\; (1-\tau _N)}>0$
times the partial derivative of λ^{
L
}
_{2} with respect to *A*_{c}
.

We can show that
$\frac{d \lambda _2^L}{\ d \; A_c } <0$
. In words, when labor taxes fall, the rise in the interest rate will increase the importance of financial income to the household’s budget and relax the constraints.

Finally, notice that the magnitude of the term
$\frac{ d w}{ d (1-\tau _N ) } \frac{(1-\tau _N)}{w}$
has to be implausible to invalidate Proposition 1. For example, assume the technology is *K*
^{ζ}
*N*
^{1 − ζ} this gives *w* = (1 − ζ)*K*
^{ζ}
*N*
^{− ζ}. Moreover, note that we can write the term
$\frac{ d w}{ d (1-\tau _N ) } \frac{(1-\tau _N)}{w}$
as
$\frac{ d w}{ d N } \frac{ d N}{ d (1-\tau _N ) } \frac{(1-\tau _N)}{w}$
. Let *e*_{N}
be the elasticity of *N* with respect to (1 − τ_{
N
}), it follows that
$\frac{ d w}{ d N } \frac{ d N}{ d (1-\tau _N ) } \frac{(1-\tau _N)}{w} = -\zeta e_N$
. If we assume that ζ (the capital share in value added) is one third, the elasticity has to be above three to reverse the sign of the term
$1+\frac{ d w}{ d (1-\tau _N ) } \frac{(1-\tau _N)}{w}$
. This seems to be implausibly high given the empirical estimates of the labor supply elasticity.

11 We model single households explicitly and include them in the model for two reasons: First, so as to not bias the responses of prices and quantities from the reform; as we will later show the response of labor supply, consumption and capital accumulation to the change in policy will differ between single and couple households; in general equilibrium we should include both household types. Second, singles are important for welfare comparisons, as they are the well-known benchmark in the literature.

12 Under our assumption of wealth pooling, also annuities would be a common resource in the household. Moreover, as we assume that couples die together, annuities would not redistribute across household members. On the other hand, annuities would give a higher rate of return: This can be thought as a ‘survivors premium’ (see for example Storesletten et al. (Reference Storesletten, Telmer and Yaron2004)). In this case, they would increase the importance of wealth on the household’s budget. But this effect is large only for retired households. As we will show, the limited commitment friction is not important for these households.

13 In our economy, households are formed by agents of the same age cohort. This assumption is reasonable given that one period in the model is five years.

14 Note that could alternatively have written:

and set
$\sum _g \overline{\xi }_g =1$
. This formulation (in terms of properties) is essentially the same as the one in (11).

16 Average hours are reported in the PSID 2007 survey and correspond to the previous year’s work time.

17 We set
$\overline{\xi }_m =28$
and
$\overline{\xi }_f=-28$
. The model predicts an average value of λ equal to 0.45. Our results are not sensitive to the calibration of the initial bargaining position.

18 Note that a model which contains home hours as well as market hours would possibly enable us to set
$\overline{\xi }_m$
and
$\overline{\xi }_f$
equal to zero, because overall hours in the household would reflect specialization in home and market production rather than initial differences in the sharing rule. This is an important extension that we leave to future work.

19 This is not surprising. As discussed previously, our model summarizes in ξ the affection that the spouses feel for one another, the complementarities in home production, the value of staying together to raise children, divorce costs (material or psychological), in other words any reason (beyond mutual insurance) for which married couples want to stay together!

The estimated value of ξ should be of interest, it tells us that in a life cycle model these additional margins are very important to keep couples together.

20 It is well known that with persistent shocks it is particularly difficult to sustain any mutual insurance arrangement, the most likely outcome is autarky.

22 $44,000 is roughly 125% of average earnings in the US economy in 2004. Social security benefits are then computed using the following formula:

where
$\overline{e}(g,\alpha _{g} )$
is average lifetime earnings for group (*g*, α_{
g
}) and
$\overline{\overline{e}}$
is the analogous object for the whole economy. To find *SS*(*g*, α_{
g
}) we proceed as follows: First, we calculate for each *g* and α_{
g
} the average lifetime earnings. This is the average that individuals (single or married) earn from age 25 to 65 adjusted for productivity growth. Note that this calculation is based on model simulations and therefore also includes hours worked and idiosyncratic productivity shocks. These magnify the differences in earnings across different groups. Second, we use the above formula to find for each group the social security benefits and express the benefits as fractions of the benefit level received by men with the lowest α_{
g
} (*SS*(*m*, α_{1})). Third, when the policy changes we keep the tax rate and the benefit ratio constant and vary *SS*(*m*, α_{1}) to balance the social security budget.

A similar approach is followed by Guner et al. (Reference Guner, Kaygusuz and Ventura2012a, Reference Guner, Kaygusuz and Ventura2012b).

23 It is well known that overall wealth inequality in the US is considerably higher. For instance, the wealth Gini for all household types (singles and couples) in the PSID sample is in the order to 0.75. The fact that wealth inequality is lower across married households should not be surprising: As documented by Guner and Knowles (Reference Guner and Knowles2007) an important variable to explain the wealth distribution is marital status (couples hold much more wealth than singles do). But our model is too simplistic to match these features of the data. In the data couples save more to leave bequests to their children or because of they have different preferences than singles and selection into marriage is important; these features are not present in the model.

24 Note that even though we have assumed that the male and the female preferences are the same, the model can generate differences in labor supply elasticities by gender following an argument similar to Alesina et al. (Reference Alesina, Ichino and Karabarbounis2011). Consider the following formula for the elasticity derived from a log linear approximation of the labor supply optimality condition;
${e^g_w} = \frac{\overline{l}_g}{1-\overline{l}_g}$
where
$\overline{l}_g$
denotes the average leisure of gender *g* in the model. According to this expression the spouse which works less hours, has a more responsive labor supply to changes in taxes and productivity. To evaluate the differences in elasticities we simulate a panel of households and estimate the first order condition for hours; our estimates show that the female elasticity in the model is 1.67 and the analogous quantity for married men is 1.3.

26 Note that in our calibration which follows Hansen (Reference Hansen1993), the productivity path of men rises steeply with age; it peaks at age 45–50 with a ratio of the maximum to initial productivity is 1.22. In contrast, female productivity is relatively flat over the life cycle.

27 Rebargaining may occur in retirement even though income is constant, but only for households where the weight of one of the spouses is too high due to a sequence of very good shocks; as wealth is typically run down in retirement, the less favored spouse seeks to rebargain.

28 Though we leave it out of the table, our findings suggest that in response to the reform there is a change in the non-participation pattern. We find that the fraction of married women that do not work increases to roughly 17% when the reform takes place, as opposed to 10.4% in the original steady state. Therefore, the drop in hours for married females represents partly a withdrawal from the labor force. For men, the reduction in hours occurs almost entirely at the intensive margin.

29 In considering the first period of the transition, we study the behavior of λ for a couple that is born right when the change in policy takes place. When the couple is at age 40, it has lived for three periods under the new tax regime. Hence, this family has had enough time to accumulate wealth in response to the drop in capital taxes.

30 In retirement there are no shocks and therefore this calculation does not apply.

31 The derivative of the solid line is roughly two times the derivative of the dashed line.

32 The fact that some of the profiles are concave is a consequence of the fact that the incentive to save for retirement varies with the age of the household. For example, age 60 households with low wealth, save more aggressively than age 45 households with low wealth. Though the initial wealth level is the same for each curve, the next period’s asset stock is a choice variable for households. This obviously influences the variability of λ.

33 We construct average utility as follows:

where μ is the fraction of households populated by couples in our economy (there are 2μ married individuals). Since preferences are the same for all individuals the value of the compensating variation is given by:
$({\frac{U^{ tax} }{ U^{no \, tax}}})^{\frac{1}{\eta (1-\gamma )}}-1$
, where *U*^{tax}
(*U*^{notax}
) is average utility in the steady state with (without) capital taxation. Similarly, when we want to make a welfare assessment for married individuals, we compute expected utility as follows:

Notice that the welfare criterion in (14) is different than the average lifetime utility of married households *W* = *EM*(*a*, *X*, λ, *j*). It holds that

*W* and *U*_{M}
therefore do not coincide because the planner attaches a weight equal to 1/2 to every individual, but households attach weights λ and 1 − λ respectively. In the ergodic distribution Γ_{
M
}, these household weights are generally different from 1/2. Apps and Rees (Reference Apps and Rees1988) show that aggregating the preferences of individuals into a household utility and maximizing over a policy parameter, can be misleading, because the effects of changes in policy are mediated through the household sharing rule λ.

34 See Erosa and Gervais (Reference Erosa and Gervais2002) for an analysis of the effect of life cycle income profiles on the preferences over labor and capital taxes.

35 We calculate in the initial steady state income subsidies from one household member to the other as a fraction of total household resources devoted to finance consumption. We define the transfer as the excess of private consumption over individual income, assuming that each spouse owns half of the household wealth stock each period and finances half of the wealth brought forward to the next period. We find that intrahousehold transfers are roughly 13% of total consumption spending. Moreover, transfers are largest for young households and smaller for retired households due to two reasons: First, because in the model the initial allocation favors the wife (e.g. through the Nash bargaining rule) and second, because labor income does not fluctuate during retirement.

36 The argument is that with high capital taxation more of the household’s resources are made out of risky labor income, which stimulates the demand for precautionary savings. When capital taxes are eliminated households accumulate wealth due to the higher return but lose the strong incentive for precautionary savings since capital income, at least in the model, is riskless.

37 This effect ought to be larger the closer the intrahousehold allocation is to full commitment (in this case transfers are maximized). We have confirmed with numerical simulations from the full commitment model that couples do indeed experience an even stronger increase in wealth accumulation in response to the reform (and slightly larger welfare gains).

38 Since the specification of individual utility is different, the model requires a different value of ξ. As previously we choose this value to give us as much rebargaining as possible.

40 To put this differently if we were to assume ξ is random and with a stochastic process which would enable us to match the divorce rates we see in the data, a smaller fraction of (marginal) families would be infected with the limited commitment friction than in the quantitative model with the constant and low ξ. Another way of saying this is that it is not the shock to ξ which is important, rather the interaction between ξ and the productivity shocks is important as equation (15) reveals.

41 This is so because the rise in labor taxes has an adverse effect on commitment for younger households. It is consistent our previous results.

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