Abstract
Consider spouses who engage in Nash bargaining to allocate resources between them. The person with a higher income when unmarried enjoys a larger share of the joint income, and benefits less from an increase in joint income. This difference can cause spouses who have the same utility functions and the same family incomes to differ in their benefits from governmental tax and spending policies, and to cast opposing votes. In particular, these incentives can generate a gender gap, with women more supportive than men of governmental taxes and spending.
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Notes
Results from the American Community Survey 20082011 show that the husband earns more than the wife in 73 % of the couples. See Bertrand et al. (2015).
When the intertemporal discount factor is 1, the same solution arises with Nash bargaining as in the Rubinstein bargaining game. Some of the results described below can appear under less selfish behavior. Consider warm glow (Andreoni 1990), where a spouse gets utility from giving to the other, and suppose the husband’s income or wealth exceeds the wife’s, so that any intrafamily transfers are made from him to her. A husband who gets a warm glow from giving to his wife would oppose a tax policy that taxes men in his situation, using the revenue to give governmental benefits to women, including his wife. For such an arrangement, though it affects consumption the same way as a voluntary transfer, generates no warm glow. Indeed, he might oppose the policy even if the benefits given his wife are greater than the tax he pays—he wants to be the provider.
http://www.cawp.rutgers.edu/fast_facts/voters/documents/ggapissues, retrieved February 9, 2012.
Table 16, “Labor force characteristics by race and ethnicity, 2012”, U.S. Bureau of Labor Statistics, October 2013.
The function U exhibits nondecreasing relative risk aversion if \(yU''(y){/}U'(y)\) is nondecreasing in y. The following discussion restricts utility functions to satisfy this criterion. Note that for \(U(y)=ln(y),\) relative risk aversion is constant (and thus nondecreasing), and that for \(U(y)=y^{\delta }\) (with \(\delta > 0\)), relative risk aversion is also constant.
Evidence relating to the concern about relative incomes is given by Frank (1985).
The result is clearest for linear utility. Here, each spouse receives his or her income when unmarried plus half the marriage surplus. Suppose initially that the aftertax income of an umarried man is 3/4 and that the aftertax income of an unmarried woman is 1/4. Suppose family income is 1.5. The surplus from marriage is then 1 / 2, and so the wife’s consumption is \(1{/}4+(1{/}2)(1{/}2)=1{/}2\). Now suppose tax policy reduces the income of an unmarried man to 2 / 3, increases the income of an unmarried woman to 1 / 3, and reduces family income to slightly over 4 / 3, say to \(4{/}3+\Delta.\) The surplus from marriage is \(1{/}3+\Delta.\) The married woman’s income thus increases from 1/2 to \(1{/}3+(1{/}3+\Delta )(1{/}2)=1{/}2+\Delta.\)
Commitment can occur early in marriage, e.g. by the choice of buying a house with a large kitchen or instead a large media room and man cave. The choice of where to live—near the husband’s job or near the wife’s job—can also affect future consumption allocations.
The opposite result can occur if G is sufficiently large to make the marginal utility of the woman’s income when single, and when she receives G, decline from a value much larger than the man’s to a value that is close to his. That change would reduce her share of the surplus from marriage.
Abbreviations
 U :

Utility function
 \(Y_M\) :

Income of an unmarried man
 \(1Y_M\) :

Income of an unmarried woman
 \(Y_H\) :

Family income
 \(s_h\) :

1 − tax rate on income of unmarried man
 \(s_w\) :

1 − tax rate on income of unmarried woman
 x :

Share of family income allocated to husband
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Appendix
Appendix
Proposition 1
In the allocation under Nash bargaining, the husband’s consumption is 1 / 2 if and only if \(s_hY_M =s_w(1Y_M );\) that is, if and only if the income of an unmarried man equals the income of an unmarried woman.
Proof

(i)
Let \(x=1{/}2.\) Then the lefthand side of (2) equals 1, and \(U(xY_H )=U((1x)Y_H ).\) Hence \(U(s_hY_M )=U(s_w(1Y_M ))\) and \(s_hY_M =s_w(1Y_M ).\)

(ii)
Let \(s_hY_M =s_w(1Y_M ).\) Clearly, Eq. (2) holds for \(x=1{/}2.\) Now suppose \(x>1{/}2.\) (An equivalent argument holds for \(x<1{/}2\).) Then \(xY_H>(1x)Y_H\). The assumptions that \(U'>0\) and \(U''<0\) imply that \(U(xY_H )>U((1x)Y_H )\) and that \(U\ '(xY_H )<U''((1x)Y_H )\). Thus, the lefthand side of (2) is less than 1 whereas the righthand side of (2) exceeds 1, leading to a contradiction. Hence \(x=1{/}2\).
\(\square\)
Proposition 2
At any Nash bargaining allocation, \(dx{/}dY_M >0\). That is, the husband’s share of consumption within the family increases when his unmarried income increases relative to that of his wife when unmarried.
Proof
The secondorder condition for a maximum of (1) requires that
Differentiating (2) gives
which is positive. \(\square\)
Proposition 3
Let the tax rates be zero, or \(s_h=s_w=1\). and suppose family income \(Y_H\) equals 1. Then the husband’s share of family consumption is \(x=Y_M\), or his income when unmarried.
Proof
Let M be the maximand for the Nash bargaining solution. Under the assumptions of the Proposition,
and
The value of \(\partial M{/}\partial x\) is 0 at \(x=Y_M\). Hence establishing Proposition(3) requires showing that \(\partial ^{2}M{/}\partial x^{2}<0\) at \(x=Y_M\).
We have
At \(x=Y_M\) this expression becomes \(2U'(x)U'(1x)\), which is negative. Thus, \(x=Y_M\) is the Nash bargaining solution for \(Y_H =1\). \(\square\)
Proposition 4
An increase in the tax rate on an unmarried man (holding constant both family income and the aftertax income of the unmarried woman) reduces the husband’s share of consumption within the family.
Implicitly differentiating (2) gives
where D is defined in (3). Because marginal utility is positive, \(U'(s_hY_M )>0\). And because the woman will obtain at least as high utility when married as when unmarried, \((U((1x)Y_H )U(s_w(1Y_M ))>0\). Therefore \(dx{/}d(1s_h)<0\).
Proposition 5
Let \(Y_M >1{/}2\), and let \(Y_H >1\). Then the allocation under Nash bargaining is \(1{/}2<x<Y_M\). That is, the husband’s consumption when married is less than his share of the joint income of the man and woman when unmarried.
Proof
Suppose \(Y_M > 1{/}2\). Without loss of generality, let \(s_w=s_h=1\). Let \(M=(U(Y_H (Y_M \epsilon ))U(Y_M ))(U(Y_H (1(Y_M \epsilon )))U(1Y_M ))\). This expression is the maximand for the Nash bargaining solution, which is maximized at \(x=Y_M \epsilon <Y_M\) if and only if \(dM{/}d\epsilon >0\) at \(\epsilon =0\).
Evaluating at \(\varepsilon =0\) yields
Therefore, \(dM{/}d\epsilon >0\) if and only if \([U(Y_H Y_M )U(Y_M )]U'(Y_H (1Y_M ))>[U(Y_H (1Y_M ))U(1Y_M )]U'Y_H Y_M )\), or if and only if
By the Generalized Mean Value Theorem, (5) is equivalent to:
for some \(\zeta\) such that \(1<\zeta <Y_H\).
By assumption, \(Y_M > 1  Y_M\), thus, a sufficient condition for (6) to hold is that \(d[U'(Y_H \ (1Y_M )){/}U'(Y_H Y_M )]{/}dY_H >= 0\). The derivative is then
The second term in the above equation is zero for constant relative risk aversion, and is positive for increasing relative risk aversion. Thus, at \(\epsilon = 0\), \(dM{/}d\epsilon > 0\). Hence \(1{/}2<x<Y_M\) when \(Y_M >1{/}2\). \(\square\)
Proposition 6
Let \(x>1{/}2\), let \(Y_H \ge 1\). Then \(dx{/}dY_H <0\). That is, an increase in a family’s income reduces the husband’s share of family consumption.
Proof
Use the definition of D from (3) and differentiate (2) to obtain
Rearrange and substitute from Eq. (2) to obtain
which is negative for \(x>1{/}2\). \(\square\)
Proposition 7
Let the equilibrium have the husband enjoy more than half of the family’s consumption (or \(x>1{/}2\) ). Then in a Nash bargaining solution, an increase in a family’s income, \(Y_H\), increases each spouse’s consumption.
Proof
First, by Proposition 6, an increase in \(Y_H\) increases the wife’s share of family consumption \((1x)\) and thus her utility when married must increase. Suppose \(dx{/}dY_H <0\), so that the husband’s share of consumption falls with an increase in \(Y_H\). Let \(Y_H ^{*}\) be slightly larger than \(Y_H\). Let \(x^{*}\) be the allocation under the Nash bargaining solution associated with \(Y_H ^{*}\). Let \(Y_H ^{*}x^{*}<Y_H x\). Then \(U'(Y_H ^{*}x^{*})\ge U'(Y_H x)\) and \(U'(1x^{*})Y_H ^{*}<U'((1x)Y_H )\), because the wife’s consumption unequivocally increases.
Hence, the lefthand side of (2) increases. But because \(U(x{*}Y_H ^{*})U(s_h(1Y_M ))\le U(xY_H )U(s_hY_M )\) and \(U((1x)Y_H ^{*})U(s_w(1Y_M ))>U((1x)Y_H )U(s_w(1Y_M ))\), the righthand side decreases, leading to a contradiction. Thus, \(Y_H^{*}x^{*}>Y_H x\): an increase in the family’s income benefits both the husband and the wife. \(\square\)
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Cohen, L., Glazer, A. Bargaining within the family can generate a political gender gap. Rev Econ Household 15, 1399–1413 (2017). https://doi.org/10.1007/s1115001593176
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DOI: https://doi.org/10.1007/s1115001593176