Tsukuba Economics Working Papers
No. 2015-005

A Note on the Responsiveness Condition of the Household Public Goods
byHisahiro Naito
December 2015

UNIVERSITY OF TSUKUBA
Graduate School of Humanities and Social Sciences
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPANA Note on the Responsiveness Condition of theHousehold Public Goods
Hisahiro Naito∗
Graduate School of Humanities and Social Sciences
University of Tsukuba
December 20, 2015

keyword: Household Public Goods
JEL code: D1, D7∗Address: Tennodai 1-1-1 Tsukuba Ibaraki 305-8573 Japan; e-mail:naito@dpipe.tsukuba.ac.jpAbstract
In this note, I investigate the preferences under which household public goods
such as consumption of children will increase regardless of an initial equilibrium
allocation when the wife’s bargaining power increases and the household resource
allocation is Pareto-efficient. I reformulate the responsiveness condition proposedby Blundell, Chiappori, and Meghir (Journal of Political Economy, 2005) andshow that it is satisfied globally when the marginal utility of the husband’s
consumption is constant and that of the wife’s consumption is diminishing. I
also show other cases in which the consumption of children increases with anincrease of the bargaining power of the wife. This suggests that in a large classof preferences, an increase in bargaining power of the wife will lead to an increasein household public goods regardless of initial equilibrium allocation.

1

Introduction

Consumption of children has the property of public goods in a household. When
the welfare of children increases, it raises the utility of both the mother andfather. When the mother is enjoying such an increase, it is difficult to exclude
the father’s enjoyment of the increased welfare of children.
In the empirical literature, an increase in bargaining power of the motherrelative to that of the father is frequently observed to increase the consumptionof children. In the economics literature, household resource allocation is often
assumed to be Pareto-efficient from the repeated nature of resource allocationand the low cost of negotiation between family members. When the household
resource allocation is assumed to be Pareto-efficient, one might want to know theconditions under which an increase in bargaining power of the mother increases
the consumption of children. Since the welfare of children is considered to havethe property of household public goods, this question might be interpreted as
an attempt to identify the conditions under which household public goods would
increase when the bargaining power of the mother increases relative to that ofthe father.1 One might guess that the fact that a mother has a higher willingness to pay for her children’s welfare is sufficient to show that an increase inbargaining power of the mother increases the consumption of children. However,Bergstrom (1995) and Blundell, Chiappori, and Meghir (2004) correctly pointout that even if the mother has a higher willingness to pay for the children’s welfare than the father, this does not imply that a stronger bargaining power of themother leads to an increase of the consumption of children. Blundell, Chiappori,and Meghir (2005) proposed a responsiveness condition in the collective model
under which an increase in higher weight on the mother’s utility in a Paretoefficient allocation leads to an increase in the consumption of children. Their
responsiveness condition is intuitive and useful. However, sometimes researchers
are interested in ascertaining the type of preferences under which an increase
1

Bergstrom and Cornes (1981) analyze the effect of income distribution on the efficient level of

public goods and show that when preferences have a Gorman form, the Pareto-efficient level of public
goods is independent of income distribution.

1

in bargaining power of the mother will lead to an increase in consumption ofchildren in a model other than the collective model. In addition, researchers areinterested in identifying the conditions under which an increase in bargaining
power of the mother will lead to an increase in consumption of children globally,that is, regardless of the initial equilibrium condition. Because of the empirical
positive correlation between the bargaining power of the mother and consumption of children independent of the initial allocation, this is a natural question.
In this note, I reformulate the responsiveness condition in terms of the utility
function by using the utility possibility frontier (UPF). I show that when themarginal utility of consumption of the father is constant and that of the motheris diminishing, an increase in bargaining power of the wife will lead to an increasein the consumption of children.

22.1Analysis
Main Results

Consider a household consisting of a husband, a wife, and children. I denotethe husband, wife, and children as h, w, and k, respectively. Furthermore, letci be the consumption of member i, where i = h, w, k. The husband and wife
have the utility function denoted as Uj = uj (cj ) + αjk uk (ck ), where uk (ck ) is
the utility of children from the consumption of ck and αjk is j’s altruism on
the utility of children; here, j = h, w. The household uses its income to satisfy
the consumption of the husband, wife, and children and the budget constraintP
j=h,w cj + ck = m. Let τ be a parameter related to the bargaining power ofthe wife. I assume that a household solves the following maximization problem:
main problem
max Uh1−τ Uwτ
Xs.t.
cj + ck = m(1)
(2)

j=h,w

where Uh = uh (ch ) + αhk uk (ck ),(3)

Uw = uw (cw ) + αwk uk (ck ).

(4)2

My interest is in how an increase in τ affects the consumption of children, ck 2 .
Using a collective approach, Blundell, Chiappori, and Meghir (2004) proposed
the following responsiveness condition:
αjk u0k (ck )0X ∂uj (c)j=h,w∂ρj>0(5)where ρj is the share of member j from the expenditure on private goods. The
above condition can be understood as follows. At the Pareto-efficient allocation,P
αjk u0 (c )the amount of public goods is determined so that j=h,w 0 k k = 1. Thus,uj (cj)P
αjk u0k (ck )increases by changing the bargaining power τ , the publicif0j=h,wuj (cj)goods in the household will increase. Several interesting questions arise on theresponsiveness condition. First, what are the preferences under which the above
responsiveness condition is satisfied? Sometimes researchers are interested ina model other than the collective model. In that case, it is useful to examine
the above condition without using the collective model. Second, what are thepreferences under which the above condition is satisfied globally? For example,
when τ decreases, the consumption of the husband increases but that of the wife
decreases. This means that the willingness to pay for the public goods of thehusband increases whereas that of the wife decreases. This implies that even if
the wife’s willingness to pay is initially higher than the husband’s willingness,the latter might become higher than the former as τ decreases. Thus, it isimportant to identify the conditions under which the responsiveness condition
holds globally.
In this paper, I reformulate the responsiveness condition. My results can besummarized by the following propositions.

2

Note that although the objective function in the above problem is Uh1−τ Uwτ , I can replace it with(Uh − Vh )(Uw − Vw ), where Vh and Vw are the threat points of the husband and wife respectively andVh and Vw are affected by τ . As can be seen below, this argument does not change even if I replaceUh1−τ Uwτ with (Uh − Vh )(Uw − Vw ).

3

Proposition 1
If
αhk u0k 00 αwk u0k 00
u − 0 2 uw > 0,
(u0h )2 h(uw )(6)

an increase in τ will increase the consumption of children.
The proof is provided in the next subsection. Several forms of preferences
satisfy (6) globally. One obvious case is that where the marginal utility of consumption of the husband is constant but that of the wife is diminishing. In such
αhk u0αwk u0hw

a case, u00h = 0 and u00w < 0. Thus, we have (u0 )2k u00h − (u0 )2k u00w > 0 and thefollowing proposition:
Proposition 2
If the marginal utility of consumption of the husband is constant and that of thewife is diminishing, an increase in τ will increase the consumption of children
globally.
Proposition 2 suggests that in a large class of preferences, the consumptionof children will increase globally with an increase of the bargaining power of thewife. Another obvious case is when αhk ≤ 0 and αwk > 0. In this case, (6) issatisfied.
Proposition 3
When the wife obtains the positive utility from the welfare of children but thehusband obtains negative utility from the welfare of children, an increase in τ
will increase the consumption of children globally.
The third case is the case where the utility function of the husband andthe wife regarding their own consumption exhibit the same coefficient of theconstant absolute risk aversion (CARA) and αwk /αhk is greater than a certainvalue. Let the coefficient of CARA be γ where γ > 0. Then, (6) becomesn wkohkγuk αuw − αuk > 0. Since γuk > 0, we haveu0wαwk>
u0hαhk(7)0Note that u0w takes the maximum value when cw = 0 and that uh takes theminimum value when ch = m. Thus, if αwk /αhk > u0w (0)/u0h (m), (7) is satisfied
globally.

4Proposition 4
When uh and uw have the same coefficient of the constant absolute risk aversion
and αwk /αhk > u0w (0)/u0h (m), an increase in τ will increase the consumption ofchildren globally.
In the following subsection, I give the proof of the proposition 1.

2.2

Proof of Propositions

For the proof of proposition 1, I first characterize the utility possibility frontier(UPF) of (Uh , Uw ). Next, given this UPF, I assume that the householdchooses Uh and Uw to maximize (1).
(Step 1) To characterize the UPF of (Uh , Uw ), consider the following problem
that calculates the UPF:
UPF
max uh (ch ) + αhk uk (ck )(8)s.t. uw (cw ) + αwk uk (ck ) = U w(9)ch + cw + ck = m(10)

To solve the above UPF, first, from (10), I rewrite cw = m − ch − ck . Then, I
plug into (9) and solve for ch . This implies that ch becomes a function of ck andU w ; I denote this as ch (ck , U w ). I next plug ch (ck , U w ) into (8); now, I have thefollowing unconstrained maximization problem:
Uh (U w ) ≡ max uh (ch (ck ; U w )) + αhk uk (ck ).
{ck }

(11)

The first-order condition becomes0uh∂ch+ αhk u0k (ck ) = 0.
∂ck(12)hNext, I need to calculate ∂c∂ck . From the definition of (9), I haveuw (m − ch (ck ; U w ) − ck ) + αwk uk (ck ) ≡ U w .

(13)

Taking a derivative with respect to ck on both sides, I have
αwk u0k∂ch=− 1.
∂cku0w

5(14)This implies that I have
αhk u0k
αwk u0k+= 1.
u0hu0w(15)

Equation (15) is the famous Samuelson condition for the amount of public goods.
Now, I check the second-order condition (SOC) of (11). The second derivativeof (11) becomes as follows:
00

SOC ≡ uh∂ch 2∂ 2 ch00+ u0h 2 + uk (ck ).
∂ck∂ck(16)2To calculate the SOC, I need to first calculate ∂∂cc2h . From (14), this is equal tok00

wk u u0 − αwk u0 u00 [−1 − ∂ch ]k wk w∂ck
20[uw ]
00
αwk u0αwk uk u0w + αwk u0k u00w [ u0 k ]w=[u0w ]2α
∂ 2 ch=2∂ck(17)(18)< 0.
Therefore, SOC< 0.
(Step 2) For the UPF, it is straightforward to show that it is concave to theorigin. The household maximizes (1) given that (Uh ,Uw ) is on the UPF. Since
the objective function (1) is convex to the origin, maximizing (1) subject to the
UPF gives the unique solution.
(Step 3) Now, I conduct comparative statics on the parameter of the wife’s
bargaining power, τ . Note that when Uh is measured on the horizontal axis
and Uw is measured on the vertical axis, the absolute value of the slope of theindifference curve of the objective function (1) isUh−τdUwUw1−τ
= −(1−τ=.
)dUhUhτUw
Thus, an increase in τ will decrease the absolute value of the slope of the indifference curve when evaluated at the same point of (Uh , Uw ). This implies that
(Uh , Uw ) maximizes the objective function (1) given that the UPF will move tothe north west while Uw will increase and Uh will decrease.
(Step 4) At step 4, I calculate how a move of (Uh , Uw ) on the UPF affects theequilibrium ck . By taking the total derivative of the first-order condition on ck

6(12) with respect to ck and U w , I have∂ch ∂ch∂
SOC × dck + u00hdU w + u0h∂ck ∂U w∂U w
To evaluate (19), I need to first calculate ∂U∂hw∂ch∂cki∂chdU w = 0.
∂ck(19)∂chand ∂U. Note that fromw

αwk u0kh=(15), ∂c∂cku0w − 1. Thus, I have
αwk u0k 00 ∂ch∂∂ch.
=u(u0w )2 w ∂U w
∂U w ∂ck(20)∂chWith regard to ∂U, from (13), I have uw (m−ch (ck ; U w )−ck )+αwk uk (ck ) ≡ U w .
w

From the implicit function theorem, I have1∂ch=− 0 .
uw
∂U w

(21)

Thus, (19) becomes as follows:
SOC × dck + u00h
Note thatαhk u0k
αwk u0+ u0 ku0hw

αwk u0∂ch∂ch ∂chdU w + u0h 0 2k u00w
dU w = 0.
∂ck ∂U w
(uw )
∂U w

h= 1. Thus, ∂c∂ck =αhk u0k
αwk u0ku0w − 1 = − u0h .

(22)

Therefore, IhaveSOC × dck + u00h [−wk 0
αhk u0k ∂ch
0 α uk 00 ∂chdU+uudU w = 0.
]w
hu0h ∂U w(u0w )2 w ∂U w

(23)

This implies that0∂chuh ∂U αhk u0αwk u0dckw{ 0 2k u00h − 0 2k u00w }.
=SOC (uh )
(uw )
dU w

(24)

αhk u0αwk u0hw

∂chdckSince SOC<0 and ∂U< 0, I have dU> 0 if and only if (u0 )2k u00h − (u0 )2k u00w > 0.
w

w

αhk u0αwk u0
Therefore, when (u0 )2k u00h − (u0 )2k u00w , an increase in τ will increase the consumpw
htion of children.

References[1] Theodore Bergstrom, ”A Survey of Theories of the Family,” Handbook ofPopulation and Family Economics. Ed. Mark Rosenzweig and Oded Stark.
Elsevier, 1995. 21-59.

7

[2] Bergstrom, Theodore C. and Richard C. Cornes, ”Gorman and Musgrave aredual : An Antipodean theorem on public goods,” Economics Letters, 1981,
7(4): 371-378.
[3] Blundell, Richard, Chiappori, Pierre-André, and Costas Meghir, “CollectiveLabor Supply with Children,” Journal of Political Economy, 2005, 113(6):1277-12306.

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