Bargaining within the family can generate a political gender gap

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.

Appendix

Proposition 1

In the allocation under Nash bargaining, the husband’s consumption is 1 / 2 if and only if $s_hY_M =s_w(1-Y_M );$ 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 left-hand side of (2) equals 1, and $U(xY_H )=U((1-x)Y_H ).$ Hence $U(s_hY_M )=U(s_w(1-Y_M ))$ and $s_hY_M =s_w(1-Y_M ).$

• (ii) Let $s_hY_M =s_w(1-Y_M ).$ Clearly, Eq. (2) holds for $x=1{/}2.$ Now suppose $x>1{/}2.$ (An equivalent argument holds for $x<1{/}2$.) Then $xY_H>(1-x)Y_H$. The assumptions that $U'>0$ and $U''<0$ imply that $U(xY_H )>U((1-x)Y_H )$ and that $U\ '(xY_H )<U''((1-x)Y_H )$. Thus, the left-hand side of (2) is less than 1 whereas the right-hand 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 second-order condition for a maximum of (1) requires that

$$\begin{aligned} D= & {} \frac{U''(Y_H x)}{U'((1-x)Y_H )}+\frac{ U'(Y_H x)}{U((1-x)Y_H )-U((1-Y_M )s_w} \\&-\frac{U''((1-Y_H )x)U'(Y_H x)}{\left( U'((1-x)Y_H )\right) ^{2}}-\frac{U'((1-Y_H )x)\left( U(Y_H x)-U(Y_M s_h)\right) }{\left( U((1-x)Y_H )-U((1-Y_M )s_w\right) ^{2}} \\< & {} \,0. \end{aligned}$$

(3)

Differentiating (2) gives

$$\begin{aligned} \frac{dx}{dY_M }=-\frac{1}{D}U'(xY_H )U'(s_w(1-Y_M ))s_w+U'((1-x)Y_H )U'(s_hY_M)s_h, \end{aligned}$$

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,

$$\begin{aligned} M=(U(x)-U(Y_M ))(U(1-x)-U(1-Y_M )) \end{aligned}$$

and

$$\begin{aligned} \frac{\partial M}{\partial x}=(U(1-x)-U(1-Y_M ))U'(x)+(U(x)-U(Y_M ))U'(1-x)(-1). \end{aligned}$$

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

$$\begin{aligned} \frac{\partial ^{2}M}{\partial x^{2}}=U''(x)(U(1-x)-U(1-Y_M ))+U''(1-x)(U(x)-U(Y_M))-2U'(x)U'(1-x). \end{aligned}$$

At $x=Y_M$ this expression becomes $-2U'(x)U'(1-x)$, 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 after-tax income of the unmarried woman) reduces the husband’s share of consumption within the family.

Implicitly differentiating (2) gives

$$\begin{aligned} \frac{dx}{d(1-s_h)} = D\frac{U'(s_hY_M )}{U((1-x)Y_H )-U(s_w(1-Y_M ))}, \end{aligned}$$

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, $(U((1-x)Y_H )-U(s_w(1-Y_M ))>0$. Therefore $dx{/}d(1-s_h)<0$.

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 $M=(U(Y_H (Y_M -\epsilon ))-U(Y_M ))(U(Y_H (1-(Y_M -\epsilon )))-U(1-Y_M ))$. 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

$$\begin{aligned} \frac{dM}{d\varepsilon }= & {} \left[ U(Y_H (1-Y_M )-U(1-Y_M )\right] U'(Y_H Y_M )(-Y_H ) \\&+\left[ U(Y_H Y_M )-U(Y_M )\right] U'(Y_H (1-Y_M ))Y_H. \end{aligned}$$

(4)

Therefore, $dM{/}d\epsilon >0$ if and only if $[U(Y_H Y_M )-U(Y_M )]U'(Y_H (1-Y_M ))>[U(Y_H (1-Y_M ))-U(1-Y_M )]U'Y_H Y_M )$, or if and only if

$$\begin{aligned} \frac{U'(Y_H (1-Y_M ))}{U'(Y_H Y_M )}>\frac{ U(Y_H (1-Y_M ))-U(1-Y_M )}{U(Y_H Y_M )-U(Y_M )}. \end{aligned}$$

(5)

By the Generalized Mean Value Theorem, (5) is equivalent to:

$$\begin{aligned} \frac{U'(Y_H (1-Y_M ))}{U'(Y_H Y_M )}>\frac{ 1-Y_M }{Y_M }\frac{U'(\zeta (1-Y_M ))}{U'\left( \zeta Y_M \right) } \end{aligned}$$

(6)

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 $d[U'(Y_H \ (1-Y_M )){/}U'(Y_H Y_M )]{/}dY_H >= 0$. The derivative is then

$$\begin{aligned}&\frac{d\left( U'(Y_H (1-Y_M )){/}U'(Y_H Y_M )\right) }{dY_H }\\&\quad =\frac{U'((1-Y_M )Y_H )}{U'(Y_M Y_H )}\left[ \frac{(1-Y_M )U''((1-Y_M )Y_H ))}{ U'((1-Y_M )Y_H )}-\frac{Y_M U''(Y_M Y_H ))}{U'(Y_M Y_H )}\right] . \end{aligned}$$

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

$$\begin{aligned} dx^{*}D= & {} -dY_H \\&\times \,(xU''(xY_H )[U((1-x)Y_H )-U(s_w(1-Y_M ))] \\&+(1-x)U'(xY_H )U'((1-x)Y_H ) \\&-(1-x)U''((1-x)Y_H )[U(xY_H )-U(s_hY_M )] \\&-xU'(xY_H )U'((1-x)Y_H )). \end{aligned}$$

(7)

Rearrange and substitute from Eq. (2) to obtain

$$\begin{aligned} \frac{dx}{dY_H }= & {} -\frac{1}{D} ((1-2x)U'(xY_H )U'((-x)Y_H ) \\&+[U((1-x)Y_H )-U(s_w(1-Y_M ))] \\&U'(xY_H )\left[ \frac{xU''(xY_H )}{U'(xY_H )}\right] -(1-x)\frac{U''(1-x)Y_H }{U'(1-x)Y_H }) \\\le & {} -\frac{1}{D}(1-2x)U'(xY_H )U'((1-x)Y_H ), \end{aligned}$$

(8)

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 $(1-x)$ 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 $U'(1-x^{*})Y_H ^{*}<U'((1-x)Y_H )$, because the wife’s consumption unequivocally increases.

Hence, the left-hand side of (2) increases. But because $U(x{*}Y_H ^{*})-U(s_h(1-Y_M ))\le U(xY_H )-U(s_hY_M )$ and $U((1-x)Y_H ^{*})-U(s_w(1-Y_M ))>U((1-x)Y_H )-U(s_w(1-Y_M ))$, the right-hand 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. $\square$