Husbands, Wives and the Peculiar Economics of Household Public Goods

Abstract

Understanding how household income is allocated among different types of expenditures is of the utmost importance when considering policy targeted at increasing household well-being. This analysis uses the collective household model to determine how household public goods and household public bads are chosen in light of the relative power balance between husband and wife. This article shows both theoretically and empirically that the effect of additional female power on household outcomes depends on the initial level of female power. Moreover, contrary to previous findings, it need not always be positive. The policy implications of this non-monotonic relationship between female decision-making power and household well-being outcomes are important. The effectiveness of aid to households depends not only on current expenditure levels but also on household balance of power. More specifically, there may be situations where enhancing female power (more controversially, and the article shows this theoretically and empirically) may not be good for household well-being.

Comprendre comment les revenus des ménages sont répartis en différentes catégories de dépenses est essentiel dès lors que l’on envisage une politique destinée à améliorer le bien-être des ménages. Cette analyse utilise le modèle collectif de comportement des ménages afin de déterminer comment s’effectuent les arbitrages entre les biens publics et les maux publics des ménages compte tenu de l’équilibre relatif des pouvoirs entre conjoints. Nous montrons théoriquement et empiriquement que l’effet du gain de pouvoir des femmes sur les résultats des ménages dépend de leur niveau de pouvoir initial. Et, contrairement à ce qu’ont montré des études précédentes, il n’est pas toujours nécessaire que cet effet soit positif. Les implications en matière de politiques de cette relation non monotone entre le pouvoir décisionnel des femmes et les résultats par rapport au bien-être des ménages sont importantes. L’efficacité de l’aide aux ménages dépend non seulement des niveaux de dépenses actuels, mais aussi de l’équilibre des pouvoirs dans les ménages. Plus précisément, il y a des situations où l’augmentation du pouvoir des femmes peut ne pas être bonne pour le bien-être des ménages (ce qui est matière à controverse, mais l’article le montre théoriquement et empiriquement).

Appendix

Proofs

Proof of Theorem 1

• The first-order conditions of this household's maximization problem are:

Recall that the optimal levels of each good are functions of θ−x _i ^*(θ), ∀i∈{1,2}, n^*(θ) and m^*(θ). Before we differentiate with respect to θ, to determine how the balance of household power affects the consumption of the HPG and the HPB, let us simplify the above system to make the calculations easier. Combining (A.1) with (A.2) and (A.1) with (A.3) yields (A.4) and (A.5), respectively.

Hence, differentiating (A.1), (A.4) and (A.5) with respect to θ yields the following system of equations (in matrix notation):

where x₂=w−x₁−n−($hskip0.8ptoverline{hskip-0.8ptM}$−m) from the budget constraint.

If we apply Cramer's Rule to solve for ∂n^*/∂θ and ∂m^*/∂θ and simplify, we get:

As b(·) and φ(·) are both strictly concave and c(·) is strictly convex, the denominator, ∣H∣, is always positive. The sign of the numerators are therefore the respective signs of the comparative statics, ∂n^*/∂θ and ∂m^*/∂θ . These depend on θ and γ.

As b′′(·)<0, c′′(·)>0 the sign of ∂n^*/∂θ is the same as the sign of X and the sign of ∂m^*/∂θ has the sign opposite of X. Where X is defined as:

Now we can show that there exists a point at which X>0 for all θ less than some θ̄ and X<0 for all θ greater than some θ̄.

From (A.9), we know that X=0 when . And this implies for all θ>θ̄, by the fact that as θ increases so does the consumption of x₁ from (A.3) and φ′>0, φ′′<0, φ′′′>0. Which means that X<0 for all θ>θ̄. And the opposite is true when θ<θ̄.

This means that ∂n^*/∂θ >0 and ∂m^*/∂θ <0 when θ<θ̄ and ∂n^*/∂θ <0 and ∂m^*/∂θ >0 when θ>θ̄.

Proof of Corollary 1

• With equal concern between agents for HPGs, we know that the sign of ∂n^*/∂θ is the same as the sign of (1−θ)φ′′(x₂)φ′(x₁)−θφ′′(x₁)φ′(x₂). When θ=1/2, we know that x₁=x₂ from (A.1). This means that (1−θ)φ′′(x₂)φ′(x₁)−θφ′′(x₁)φ′(x₂)=0 and so then does ∂n^*/∂θ=0. For θ<1/2, we know that φ′(x₁)>φ′(x₂) by (A.1). This implies that x₁<x₂, which makes sense, as less female power intuitively yields less consumption of the good the female prefers. As long as φ′′′>0 then φ′′(x₁)<φ′′(x₂) and we know that (1−θ)φ′′(x₂)φ′(x₁)−θφ′′(x₁)φ′(x₂)>0 and ∂n^*/∂θ>0. Similarly, if θ>1/2 then ∂n^*/∂θ<0 under the same condition on the third-order derivative of φ. The same logic can be easily applied to ∂m^*/∂θ to show if θ<1/2 then ∂m^*/∂θ<0 and if θ>1/2 then ∂m^*/∂θ>0.

Proof of Corollary 2

• Consider the sign of X at θ=1/2. When household power is perfectly balanced (1−θ)φ′′(x₂)φ′(x₁)=θφ′′(x₁)φ′(x₂), hence we can rewrite Y=(1−θ)φ′′(x₂)φ′(x₁)[1−γ]. Because φ′>0, φ′′<0, the sign of Y is equal to the sign of Z=[γ−1]. The sign of Z is positive when γ>1, so ∂n^*/∂θ>0 and ∂m^*/∂θ<0 at θ=1/2 and the turning point for both functions is at θ̄>1/2 when the woman is relatively more concerned with the HPG and HPB. Similarly, when γ<1, Z<0 so ∂n^*/∂θ<0 and ∂m^*/∂θ>0 at θ=1/2, which makes the turning point at θ̄>1/2.