Two-agent discrete choice modelwith random coeﬃcient utility functions
for structural analysis
on household labor supply
Tamaki MIYAUCHI
August 2011
KEO Discussion Paper No. 126

Two-agent discrete choice model with random
coeﬃcient utility functions for structural analysis
on household labor supply∗
Preliminary and Incomplete

Tamaki MIYAUCHI†
August 2011

Abstract
This paper discusses a bargaining model on discrete choices of individual household based on two-agent qualitative choice model. The twoagent qualitative choice model describes discrete choices made throughbargaining interactions between two agents. This paper presents a bargaining model of discrete choices on labor supply of husband and wife ofhouseholds.
Contrary to self-employed workers, employees’ hours of work tends tobe assigned by employers. In the case where hours of work is restricted,
the choices the agents make will not be continuous but discrete, i.e., binary choices of whether the agents work or not. This model explicitly
demonstrates utility maximizing behavior of two interacting agents undersuch discrete constraint imposed on hours of work.
As structural equations, an income-leisure preference function of wifeand that of husband are introduced in this paper. These functions haverandom coeﬃcients, which represent taste diﬀerences among wives andhusbands in population.
This paper focuses on labor supply decisions made by Japanese households that consist only one couple of wife and husband with children underﬁfteen years old.

Keywords: binary choice, bargaining model, employee opportunity, incomeleisure preference function, structural estimation
JEL Classification Numbers: C35, C82, J20∗ I would like to thank Professor Daniel McFadden for his valuable discussions on my research, at the University of California, Berkeley. I also thank Professor James Powell, Professor Kenneth Train, and Professor Kenneth Chay for deliberate suggestions at the Qualitative
Choice Analysis Workshop held in UC Berkeley. All errors are naturally mine.
† Keio University, Tokyo, Japan Email:miyauchi@econ.keio.ac.jp1

1

Introduction

In the literature of path breaking analyses on qualitative economic decisions,
McFadden (1973;1981), Hausman and Wise (1978) focus on describing decisionsof individual agents. Most of the models following these contributions, such as
Dubin and McFadden (1984), describe discrete choices based on single-agent
qualitative choice model. On the other hand, this paper discusses a bargaining
model on discrete choices of individual household based on two-agent qualitative
choice model. The two-agent qualitative choice model describes discrete choices
made through bargaining interactions between two agents.
This paper presents a bargaining model of discrete choices on labor supplyof husband and wife of households. The model gives probabilistic distributionsof discrete choices on household labor supply. According to the probabilistic
distributions, the model describes binary choice behavior of whether the agents
accept the employee job opportunity or not.
Contrary to self-employed workers, employees’ hours of work h tends to be
assigned by employers as h = h, where h stands for assigned hours. In the casewhere hours of work is restricted to h, the choices the agents make will not be
continuous but discrete, i.e., binary choices of whether the agents work or not.
This model explicitly demonstrates utility maximizing behavior of two interacting agents under such discrete constraint imposed on hours of work. As
structural equations, an income-leisure preference function of wife and that ofhusband are introduced in this paper. These functions have random coeﬃcients, which represent taste diﬀerences among wives and husbands in population. While indirect utility functions are implicitly introduced in most analyses
on qualitative economic decisions, direct income-leisure utility functions, as wellas income-leisure restrictions are explicitly introduced as structural equationsin this paper, and structural parameters of the utility functions are estimated.
Let μw denote wife’s labor supply probability to her employee job opportunity, and let μh denote husband’s labor supply probability to his employee job
opportunity. Based on the estimated structural parameters, conditional forecasts on μw as well as on μh are performed, given wage rates and assigned hours
of work.
In the literature of quantitative analyses on labor supply, the cross sectional
analysis by Douglas (1934) gave a signiﬁcant evidence that the observed job participation ratios of females are negatively correlated to the observed householdincome levels1 . This ﬁnding implies that labor supply decision made by household members may not be independent, and thus interacting decision making
behavior between household members need to be introduced in theory explicitly.
As for labor supply behavior of females, Mincer (1962) described patternsof female labor supply in a long run by introducing a lifetime hypothesis, andpresented females’ work hour allocation in their lifetime. More than that, Heck1 See Douglas (1934) pp.279-294.

2

man (1974) demonstrated that the notion of “reservation wage” gave a way toprobabilistic analyses on binomial working decisions made by females.
In the literature of household decision making behavior, Mancer and Brown(1980), Bjorn and Vuong (1984) presented theories explicitly describing bargaining interactions between wife and husband in a household. Instead of describing
interactions between household members explicitly, Obi (1969a, 1969b, 1979) introduced an analytical notion of “principal earner” and “non-principal earners”
of household. “Principal earner” is analytically deﬁned as a household member
whose wage rate is the highest among the wages of household members in eachspeciﬁc household. Given the observed wage diﬀerentials between male’s labormarket and female’s labor market, husbands are assigned as principal earners,
and wives are assigned as non-principal earners in most cases. Obi described
the non-principal earners’ labor supply probability to their employee and/or
self-employee job opportunities, given the principal earners’ job participation,and thus given the principal earners’ income level.
This paper focuses on labor supply decisions made by Japanese households
that consist only one couple of wife and husband with children under ﬁfteen
years old2 . This paper explicitly describes work decisions of household throughinteractions between wife and husband in a household, thus both wife’s work
decision and husband’s work decision are described endogenously.
In Section 2, I propose a two-agent discrete choice model of household onlabor supply decisions to employee job opportunities. Based on the model in
Section 2, Section 3 gives a stochastic model of household labor supply. Thismodel introduces random coeﬃcients to income-leisure preference functions representing taste diﬀerence among agents in population. Estimation of structuralparameters and simulation is presented in Section 4. Concluding remarks are
given in Section 5.

2

A two-agent discrete choice model of household on labor supply

This section presents a model of household labor supply of husband and wife,
which is a special case of a two-agent discrete choice model of household onlabor supply. The household described here is supposed to consist of only one
couple of husband and wife, and their children under 15 years3 of age, if any.
Household members, not limited to husbands and wives, generally havechoices among self-employed work opportunities as well as employee work opportunities. The model in this paper exclusively focuses on the decision-making

2 Children under ﬁfteen years old are prohibited to work by law in Japan, and thus theagents that make work decisions are limited to wife and husband in such a household.
3 See footnote 2.

3

behavior concerning employee work opportunities only4 . Contrary to selfemployed workers, employees’ hours of work h tends to be assigned by employers
as h = h, where h stands for assigned hours. In this case, the decision makingof labor supply has characteristics of discrete choice, because what each agent
can choose is, not how long he or she works, but whether he or she works.

2.1

Income-leisure preference function and constraints

Unearned income (in real term) which the ith household gains during a uniti
. The wage rate and assigned hour of work of theperiod is denoted as IA
employee work opportunity, which the husband of the ith household faces, isidenoted by whi and hh respectively. Analogously, the wage rate and assigned
ihour of work, which the wife of the ith household faces, is denoted by ww
and
i
i
i
i
hw respectively. The wage rates, wh and ww , as well as the unearned income IA,
are measured in real term. The total income of the ith household in real termi, and the income
is denoted by X i . X i is the sum of the unearned income, IA
actually earned by the husband and the wife of ith household.
Let the leisure of husband and wife be denoted by Λih and Λiw respectively.
The range of Λih and Λiw should be 0 ≤ Λih ≤ T and 0 ≤ Λiw ≤ T , where T is
the agent’s maximum amount of consumable leisure during a unit period.
The total income, X i is subject to expenditure by the wife as well as by thehusband of ith household. On the contrary, the husband’s leisure, Λih , is not
subject to consumption by the wife, nor the wife’s leisure, Λiw , is not subject
to consumption by the husband. In short, it is reasonably assumed that eachmember of the ith household can exclusively consume his own or her own leisureonly. Therefor the following assumption is introduced in this paper.
Assumption 1:Each of the husband and the wife of ith household has the following utility indicator function, ωh and ωw respectively.
ωhi
i
ωw

=
=

ωh (X i , Λih |Γih )ωw (X i , Λiw |Γiw )

(1)
(2)

where Γih and Γiw are parameter vectors of the utility function, each for the ithhousehold’s husband and wife respectively 5 .

4 See Miyauchi (1992) for a two-agent discrete/continuous choice model of household onlabor supply for employee and/or self-employed job opportunity.
5 Although the case ∂ωh = 0 or ∂ωw = 0 cannot be excluded a priori, a functional form∂Λw
∂Λhsuch that these values are constantly zero is assumed for simplicity in this paper.

4Each husband and wife maximizes his or her own utility indicator function
subject to the constraints of;
Xi

=

i
i i
IA
+ whi hih + ww
hw

Λih

=

T − hihΛiw=

T − hiwi(hih = 0 or hh )i(hiw = 0 or hw )

(3)
(4)
(5)

where each hih and hiw denotes hours of work of husband and wife of ith householdrespectively.
The husband of ith household maximizes the utility indicator function (1)subject to the constraints (3) and (4). Similarly, the wife of ith household
maximizes the utility indicator function (2) subject to the constraints (3) and(5). Inserting these restrictions (3) through (5) into the utility indicator function(1) and (2) yields
ωhi

=

i
ωw

=

i
i i
ωh (IA+ whi hih + ww
hw , T − hih |Γih )i
i i
ωw (IA+ whi hih + ww
hw , T − hiw |Γiw )
hh = 0 or hh = hh
wherehw = 0 or hw = hw

(6)
(7)

Note that both hih and hiw enter the each of husband’s utility function (6),and wife’s utility function (7). For this reason, the attainable maximum utilityof each husband and wife is not independently determined, but aﬀected by thelabor supply choice of the other belonging to the same household.

2.2

Payoﬀs of labor supply choice in terms of preference

As shown in equations (6) and (7), the utility level of husband and wife, ωhand ωw respectively depends on the combination of (hh , hw ). The combinationof (hh , hw ) is exhaustible within the following four cases according to the labor
supply choice made by husband and wife.
(i) (hh , hw ) = (0, 0)
neither works
only husband works(ii) (hh , hw ) = (hh , 0)only wife works(iii) (hh , hw ) = (0, hw )(iv) (hh , hw ) = (hh , hw ) both workwhere the superscript i is suppressed for simplicity.
Let the above (i) through (iv) be noted as the “combination of labor supply
choice.” The husband’s preference order related to the “combination of labor
supply choice” (i) through (iv) is determined by the inequalities among the setof values of ωh ’s, which are obtained by inserting the each value of (hh , hw ) in(i) through (iv) into the right hand side of the husband’s utility function (6). In
other words, the preference order is determined by the shapes of the husband’s
indiﬀerence curves. The wife’s preference order is also determined similarly by5

inserting the each value of (hh , hw ) in (i) through (iv) into her utility function(7), which is equivalent to the statement that the order is dependent on the
shapes of the wife’s indiﬀerence curves.
Income-leisure restrictions, (3) through (5), are graphically depicted in Figure 1. The preference ﬁeld depicted on the right half of Figure 1 is the husband’s,
and that of wife is depicted on the left half. The vertical axis depicts the total
income X, which commonly enters the right hand sides of husband’s and wife’s
utility function, (1) and (2). The horizontal axis depicts the leisure of husband
Λh or that of wife Λw , according to whose ﬁeld it belongs to. Note that Λh andΛw are depicted separately in Figure 1, because Λh enters exclusively into thehusband’s utility function (1), and so does Λw into the wife’s utility function (2).
The length of or, and that of o’r’ shows the husband’s and the wife’s maximum
amount of consumable leisure during a unit period, T , respectively.
The set of the wage rate, wh , and the assigned hours of work, hh , of husband’s
employee work opportunity is depicted by the portion of the line ab or by cd,
depending whether his wife works or not. The portion of the line ab and cd are
depicted so that tan α = wh holds.
On the other hand, the set of the wage rate, ww , and the assigned hours ofwork, hw , of wife’s employee work opportunity is depicted by the portion of theline a’b’ or by c’d’, depending whether her husband works or not. The portionof the line a’b’ and c’d’ are depicted so that tan β = ww holds.
(i) In case neither works [(hh , hw ) = (0, 0)], the husband is located at pointa, and the wife is at point a’. Let the intersection point with the Λh axis
and a vertical line passing point b and d be denoted by H. The length of the
portion of the line rH is the husband’s assigned hours of work. Similarly, let theintersection point with the Λw axis and a vertical line passing point b’ and d’ bedenoted by H’. The length of the portion of the line r’H’ is the wife’s assigned
hours of work. (ii) In case only husband works [(hh , hw ) = (hh , 0)], the husband
is located at point b, and the wife is at point c’. (iii) In case only wife works[(hh , hw ) = (0, hw )], the husband is at point c, and the wife at point b’. Lastly(iv) in case both work [(hh , hw ) = (hh , hw )], the husband is at point d, and thewife at point d’.
Because the total income X, which is common to the husband and the wife,
enters the husband’s and the wife’s utility functions, (1) and (2), simultaneously,
the ordinate of the husband’s position and that of the wife’s position in theirpreference ﬁeld are always the same to each other.
Given the income-leisure constraint equations, (3) and (4), the husband faces
one of the two alternative choice sets {a, b} or {c, d}, depending upon the wife’s
decision of whether she works. If the wife works, the husband faces the choice
set of {c, d}, otherwise {a, b}. The values of utility index can be assigned tothese points a, b, c, and d by inserting the coordinates of these points to theright hand side of the formula (6), and these values are regarded as payoﬀs ofthe husband. Similarly, given the income-leisure constraint equations, (3) and(5), the wife faces one of the two alternative choice sets {a’, b’} or {c’, d’},6

Wife’s income-leisure preference ﬁeld

Husband’s income-leisure preference ﬁeld

hw :Wife’s assigned hours of work
ww :Wife’s wage rate in real termhh :Husband’s assigned hours of work
wh :Husband’s wage rate in real term
tan α = whtan β = ww

X

X

6

6

d’

b’

d

@
@
β@

IA

@
@
β@

o’

H’

S
S
b
S
S S
S
S
S αS c
S
S
αS
a

c’

IA

a’

r’

- Λw

o

r

H

hw

- Λh

hh

T

T

Figure 1: Husband’s and wife’s income-leisure preference ﬁeld and their incomeleisure constraints

7

depending upon the husband’s decision of whether he works. If the husband
works, the wife faces the choice set of {c’, d’}, otherwise {a’, b’}. The values ofutility index at these points a’, b’, c’, and d’ are given by the formula (7), andthese values are regarded as payoﬀs of the wife.
The husband’s payoﬀs varies depending upon the shapes of his indiﬀerence
curves, and so does the wife’s payoﬀs. How many types of payoﬀs possibly exist
as the shapes of indiﬀerence curves vary? The permutation of husband’s set offour points a, b, c, and d or that of wife’s set of a’, b’, c’, and d’ is 4! = 24 in all
respectively. However, only 6 types of payoﬀs out of 24 proved to be plausible
under the conditions of the positivity of marginal utility and the convexity ofindiﬀerence curves6 .
Let the values of husband’s preference index at the points a, b, c, and d be
denoted by ωha , ωhb , ωhc , and ωhd respectively. Similarly let the values of wife’s
a’ , ω b’ , ω c’ ,preference index at the points of a’, b’, c’, and d’ be denoted by ωw
w
w
d’ respectively. Thus the payoﬀs of husband and wife is given as in Tableand ωw
1 and Table 2 respectively.
Table 2: Wife’s payoﬀs
Table 1: Husband’s payoﬀswife hw = 0 hw = hw
wife hw = 0 hw = hw
husbandhusbanda’b’hh = 0ωhaωhc
hh = 0ωw
ωw
c’d’
hh = hhωhbωhd
hh = hhωw
ωw
Let the husband’s payoﬀ matrix of ωh ’s in Table 1 be denoted by πhk (k =1, 2, · · · , 6). Similarly let the wife’s payoﬀ matrix of ωw ’s in Table 2 be denoted( = 1, 2, · · · , 6). The superscript k of πhk indicates the plausible type ofby πw
husband’s payoﬀs, and the superscript  of πw
indicates the plausible type ofwife’s payoﬀs.
The 6 types of husband’s payoﬀ matrix are listed in the left column of Table
3, while the 6 types of wife’s payoﬀ matrix are in the right column7 . Since
are indicatedthe utility index is ordinal, the payoﬀs in the matrices πhk and πw
ordinal number 1, 2, 3, and 4, where 1 is least preferred and 4 is most preferred.

6 See Miyauchi(1991).

7 See Miyauchi(1991) for more precise discussions on the relation between the types of
payoﬀ matrix and the shapes of indiﬀerent curves.

8Table 3: Six types of payoﬀ matrix




1
πh
=
2
πh
=
3
πh
=
4
πh
=
5
πh
=
6
πh
=







1
2

3
4

2
1

3
4

1
2

4
3

2
1

4
3

3
1

4
2

1
3

2
4

1
πw
=



2
πw
=



3
πw
=



4
πw
=



5
πw
=



6
πw
=







1
3

2
4

2
3

1
4

1
4

2
3

2
4

1
3

3
4

1
2

1
2

3
4








The payoﬀ matrices πhk (k = 1, 2, · · · , 6) and πw( = 1, 2, · · · , 6) shown in
Table 3 are relevant and exhaustible for all the plausible cases under the basic
assumption of utility function. In other words, the payoﬀ matrices in Table 3
6should becover whole plausible cases. Note that the wife’s payoﬀ matrix πw
excluded from the plausible set of payoﬀ matrix if and only if wh hh ≥ ww hw
holds, and the husband’s payoﬀ matrix πh6 should be excluded from the plausible
set of payoﬀ matrix if and only if wh hh ≤ ww hw hols8 .

2.3

Guaranteed income level of husband and wife

Let Ih0 denote husband’s guaranteed income, that is the amount of income available to husband even if he does not work. Ih0 is the sum of unearned income ofthe household, IA , and the earned income by his wife, ww hw , as
Ih0 = IA + ww hw(8)

where hw = 0 in case his wife does not work, and where hw = hw in case she
works.
Similarly, let Iw0 denote wife’s guaranteed income, that is the sum of unearned
income of the household, IA , and the earned income by her husband, wh hh , as
Iw0 = IA + wh hh(9)where hh = 0 in case her husband does not work, and where hh = hh in case heworks.

8 See Miyauchi(1991) for more details.

9

2.4

Nash-equilibrium of the two-person game

Combining the husband’s payoﬀ matrices, πhk (k = 1, 2, · · · , 6), and the wife’s( = 1, 2, · · · , 6), payoﬀ table, Πk− (k,  = 1, 2, · · · , 6), is
payoﬀ matrices, πw
obtained. For example, combining the husband’s payoﬀ matrix, π 1 , and thewife’s payoﬀ matrix, π 3 , yields a payoﬀ table Π1−3 .

(1, 1) (3, 2)
1−3=
Π
(2, 4) (4, 3)Some payoﬀ tables have Nash-equilibria, and others not. Among the payoﬀ
tables that have Nash-equilibria, some tables have unique Nash-equilibria andothers have non-unique equilibiria. In the left half of Table 4-1 through Table
4-3, Nash-equilibria corresponding to each payoﬀ table is shown by ◦ or  if
the Nash-equilibria exists. The payoﬀ tables, Π2−3 and Π3−2 have no Nashequilibria. The payoﬀ table, Π3−3 , has two Nash-equilibira, which means only
husband works or only wife works. Other payoﬀ tables have unique Nashequilibria.
Let the husband’s and wife’s utility indicator corresponding to a unique∗
respectively. Moreover, let the husNash-equilibria be denoted by ωh∗ and ωw
band’s and wife’s utility indicator, that they obtain when they change theirchoice on labor supply corresponding to Nash-equilibria simultaneously, be de
respectively. Cooperative equilibrium can exist if and only
noted by ωh and ωw
if the two inequalities
ωh1∗
1∗
ωw

<

ωh2

(10)

<

2
ωw

(11)

hold simultaneously. For the payoﬀ tables, Π2−4 , Π4−2 , and Π4−4 , there exists
cooperative equilibria, as shown in the right half of the Table 4-1 through Table
4-3. (Actually, cooperative equilibria exists only for the payoﬀ tables in Table
4-1.)

10

Table 4-1: Solution of the game (case 1)
Nash-equilibria
cooperative equilibriak − π
πh
w

1–1
1–2
1–3
1–4
1–5
2–1
2–2
2–3
2–4
2–5
3–1
3–2
3–3
3–4
3–5
4–1
4–2
4–3
4–4
4–5
5–1
5–2
5–3
5–4
5–5

hh = 0hw = 0hh = hhhw = 0hh = 0hw = hw

hh = hhhw = hw

hh = 0hw = 0hh = hhhw = 0hh = 0hw = hw

hh = hhhw = hw

no equilibria
◦


◦



no equilibria
◦
◦








◦













Table 4-2: Solution of the game (case 2)
valid only for the case wh hh > ww hw
Nash-equilibria
cooperative equilibriak − π
πh
w

6–1
6–2
6–3
6–4
6–5

hh = 0hw = 0hh = hhhw = 0hh = 0hw = hw

hh = hhhw = hw11

hh = 0hw = 0hh = hhhw = 0hh = 0hw = hw

hh = hhhw = hw

Table 4-3: Solution of the game (case 3)
valid only for the case wh hh < ww hw
Nash-equilibria
cooperative equilibriak − π
πh
w

1–6
2–6
3–6
4–6
5–6

3

hh = 0hw = 0hh = hhhw = 0hh = 0hw = hw

hh = hhhw = hw

hh = 0hw = 0hh = hhhw = 0hh = 0hw = hw

A stochastic model of household labor supplyof husband and wife

A stochastic model of household labor supply is constructed based on the resultspresented in Section 2.

3.1

Threshold income of labor supply (TILS)In this subsection, a concept of “Threshold Income of Labor Supply (TILS)” isdeﬁned.
Given the wage rate, wh , and also given the assigned hours of work, hh , thehusband’s TILS, denoted by Ih∗ , is deﬁned as the income level such that theequation(12)ωh (Ih∗ , T |Γih ) = ωh (Ih∗ + wh hh , T − hh |Γih )holds, where the function, ωh , is given by the formula (1). Solving the equation(12) regarding to Ih∗ yields the formulaIh∗ = Ih∗ (wh , hh |Γih )(13)

indicating that Ih∗ is a function of wh , hh , and parameter vector, Γih .
Analogously, the wife’s TILS, denoted by Iw∗ , is deﬁned as the income levelsuch that the equation
ωw (Iw∗ , T |Γiw ) = ωw (Iw∗ + ww hw , T − hw |Γiw )(14)holds, given the wage rate, ww , as well as given the assigned hours of work,
hw . The function, ωw , is given by the formula (2). Solving the equation (14)regarding to Iw∗ yields
Iw∗ = Iw∗ (ww , hw |Γiw )(15)
indicating that Iw∗ is a function of ww , hw , and parameter vector, Γiw .
12hh = hhhw = hwん

晩 ut t Husbanぜん

s ttLS and MHLS whげe鞘<o hddまA brief explanatiOn on the relatiOn betM′
een labor supply decisiOn and TILS
nlight be necessary Figure 2 describes the relatiOn between husba)nd's decisiOnon iabor supply and his TILS, given the M′ age rateぅ竹ん)aィ
nd also given theassigned hours of wOrk,んん,Of his employee opportunity. Suppose his guarainteed
income level is the amount of the length亜 . In this ca,se, he 11l
H′ accept thecIIlployee Opportunity, because his utility indicatOr at point B,、 he is if
vhere
works‐ is lllgher than his utility indicator at point A,where he is if dOes notM′ork. Next,suppose his guaranteed iIIcollle increases up to the amOunt of thelength TE Colltrarily in this ca)se,he will rttect the emp10yee Opportunity,
becaluse his lltility indicator at point E,where he is if dOes
llOt higllerMァork)isthan his utility indica″
tor at point F,M/here he is if lle wOrks Finall)L suppOSe

his guaranteed incOnie,現,is exactly the amoullt of the length TC.He will beat poillt C if he does nOt Hrork,and he Mrill be at pOint D if he dOes wOrk ln

this illal case, M′
hether he、 vOrks Or not is exactly indirerent to hiln)because
both point C and D are on the same indifFerence curve,(ル序 The guaranteed
incoHle level that ll■
akes it indifFerent whether the husband M′Orks or nOt is thehllsballld's TILS, ィ
文⊃ in Figure 2 The汚)Such as the alnOunt of the length硬、rife's TILS can be explained allalogously

つじ

3.2

Maximum hours of labor supply

In this subsection, a concept of “Maximum Hours of Labor Supply (MHLS)” isdeﬁned.
Given the wage rate, wh , and also given the guaranteed income, Ih0 , husband’s MHLS is deﬁned as the hours of work, hh , such that the equation
ωh (Ih0 , T |Γih ) = ωh (Ih0 + wh hh , T − hh |Γih )(16)

holds, where the function, ωh , is given by the formula (1). Note that hh is avariable and not necessarily equal to hh . Solving the equation (16) regarding tohh yields the formulahxh = hxh (Ih0 , wh |Γih )(17)
where the husband’s MHLS is denoted by hxh . The formula (17) indicates thatthe husband’s MHLS is the function of husband’s guaranteed income, Ih0 , wage
rate, wh , and parameter vector, Γih , of his utility function.
Analogously, wife’s MHLS is deﬁned as the hours of work, hw , such that
ωw (Iw0 , T |Γiw ) = ωw (Iw0 + ww hw , T − hw |Γiw )(18)

holds, given the wage rate, ww , as well as given the guaranteed income, Iw0 . The
function, ωw , is given by the formula (2). Note that hw is also a variable andnot necessarily equal to hw . Solving the equation (18) regarding to hw yields
hxw = hxw (Iw0 , ww |Γiw )(19)

where the wife’s MHLS is denoted by hxw . The formula (19) indicates that thewife’s MHLS is the function of wife’s guaranteed income, Iw0 , wage rate, ww ,and parameter vector, Γiw , of her utility function.
Husband’s MHSL can be shown in Figure 2. Points L, K, J, are the feetof perpendiculars to Λh axis originating points H, D, G, respectively. Suppose
husband’s guaranteed income, Ih0 , is the amount of the length TA. In this case,the husband will accept the employee opportunity of wage rate, wh , and assigned
hours of work, hh . Given his guaranteed income, Ih0 = TA, and also given thewage rate, wh , he will accept any employee opportunity as long as the assigned
hours of work is at most the amount of length TJ. He will reject the employee
opportunity if the assigned hours of work exceeds the length TJ. The husband’s
MHLS is the assigned hours of work that makes it indiﬀerent whether he works
or not. The wife’s MHLS can be shown analogously.
Note that the husband’s MHLS, hxh , decreases as his guaranteed income, Ih0 ,∂hxh< 0 holds in Figure 2. On the contrary, husband’s indifincreases, i.e.,∂Ih0ference curves depicted in Figure 3 make his MHLS increase as his guaranteed∂hxh> 0 holds in Figure 3.
income increases, so that
∂Ih0
14

Figure 3,Husband)s TILS and MHLS where那>oh° ldS・

the regions of his TllreshOld lncOme Of Labor Supplン
らTILS Sinlllar statements
apply tO the ca,se of Mrife.
3。3

The relation between TILS and decision on labor sup―ply

Let the relatiOn bet、A′
e en husband's TILS and ilis decisiOn be described irst.

督 督 押 群t良ξ潜 88督!∬盟器 t牝ど
艦常奉瑞 毛,結群 品鮮 :88:品
蟹岳ん
ん, and let it be denOted
by ωhe does accept the same enlployee
らin case
oppOrtunity The values
Ofりbe given by the fOrmula,
骨and cl,Canω? = りん( 就, T r 先)り, = りん
1
r
ん
万
十
ん
, T
( 就

「先)
ん
万

where ωh is the husband’s utility function (1). The relation between the regionof his TILS, Ih∗ , and the inequalities of ωh0 and ωh1 is∂hxh
Ih0 < Ih∗  ωh0 < ωh1<0(20)
if and only if
0
∗  0
1
Ih > Ihωh > ωh∂Ih0∂hxh
Ih0 < Ih∗  ωh0 > ωh1>0(21)
if and only if
0
∗  0
1
Ih > Ihωh < ωh∂Ih00
For the case of wife, let the wife’s utility index be denoted by ωw
in case shedoes not accept the employee opportunity of ww and hw , and let it be denoted by1
in case she does accept the same employee opportunity, given her guaranteedωw
01
income, Iw0 . The values of ωw
and ωw
can be given by the formula0
ωw= ωw (Iw0 , T |Γiw )1
ωw= ωw (Iw0 + ww hw , T − hw |Γiw )where ωw is the wife’s utility function (2). The relation between the region of0
1
her TILS, Iw∗ , and the inequalities of ωw
and ωw
is

0
1
∂hxw
Iw0 < Iw∗  ωw
< ωw
<0
(22)
if and only if
0
∗  0
1
Iw > Iw
ωw > ωw
∂Iw0

0
1
∂hxw
> ωw
Iw0 < Iw∗  ωw
>0
(23)
if and only if
0
∗  0
1
Iw > Iw
ωw < ωw∂Iw0
The husband’s indiﬀerence curves depicted in Figure 2 correspond to the
∂hxh
case
< 0. In Figure 2, the husband’s TILS, Ih∗ , is shown by the length∂Ih0
TC. Note that as long as his guaranteed income, Ih0 , is equal to his TILS, Ih∗ ,his MHLS, hxh , is equalized to the assigned hours of work, hh , of his employee
opportunity (Ih0 = Ih∗  hxh = hh ). Suppose his guaranteed income, Ih0 , falls
below his TILS, Ih∗ (Ih0 < Ih∗ ), to the level such as the length TA. As long as∂hxh< 0 holds as in Figure 2, hxh > hh holds for the region of Ih0 < Ih∗ . This
∂Ih0
∂hxh
< 0 holds as in Figure 2, the point G is necessarilymeans that as long as∂Ih0
located on the left side of the point B in Figure 2, because hxh > hh holds andbecause the amount of hxh is, by deﬁnition, the distance between TE and point
G, which is the intersection of line AB and the indiﬀerence curve on point A,
ωhA . Considering the convexity of the indiﬀerence curves to the origin, it isconcluded that utility index on point B is necessarily higher than that on point
A , because both point A and G is on the same indiﬀerence curve and because
point B is located between A and G on the line AG. Thus, the relation
Ih0 < Ih∗  ωh0 < ωh1
16

(24)

is obtained.
Next, on the contrary, suppose husband’s guaranteed income, Ih0 , raises
above his TILS, Ih∗ (Ih0 > Ih∗ ), to the level such as the length TE. In this∂hxhcase hxh < hh holds for the region of Ih0 > Ih∗ , as long as< 0 holds as in∂Ih0
Figure 2. By the similar reasoning above, it is concluded that point H is necessarily located on the right side of point F and that the utility index on point H
is necessarily higher than that on point E. Thus, the relation
Ih0 > Ih∗  ωh0 > ωh1(25)

is obtained.
The relation (24) and (25) yield the relation (20).
In the similar manners as described above, it is shown that the relation (21)∂hxhholds under the condition> 0, using Figure 3.
∂Ih0
The relations (20) and (21) shows that the formulas of inequalities between
the guaranteed income and the TILS can be assigned to the payoﬀ matrices.

3.4

Introducing a random coeﬃcient into utility function

In Subsection 3.9, husband’s and wife’s utility functions with random coeﬃcients are introduced. It is assumed that husbands in population share thesame parameters of their utility function except the intercept (constant term)
parameter of marginal utility of leisure, which means that the intercept parameter distributes among households in population. Moreover it is assumed
that wives in population also share the same parameters of their utility function
except the intercept parameter of marginal utility of leisure. This subsection induces the distribution of husband’s or wife’s Threshold Income of Labor Supply(TILS).
It is assumed that the husband’s and wife’s parameter vectors of their utility
function,
t
t

Γh =
Γw =

(γh1 , γh2 , · · · , γhm )(γw1 , γw2 , · · · , γwm )contains random coeﬃcients, γh4 and γw4 , respectively. γh4 is the intercept
parameter of husband’s marginal utility of leisure, and γw4 is the random coefﬁcient of wife’s marginal utility of leisure. The superscript i of the parameter
vector is suppressed for simplicity.
The random coeﬃcients, γh4 and γw4 , are also assumed to follow the joint
probability density function
f (γh4 , γw4 | ζ )(26)
where ζ is the parameter vector of the probability density function f .
17The distribution of husband’s TILS, Ih∗ , and that of wife’s HILS, Iw∗ , areinduced as follows. Solving the formulas (13) and (15) of husband’s and wife’s
TILS, regarding γh4 and γw4 respectively, yields
γh4
γw4

h )
= γh4 (Ih∗ , wh , hh |Γ∗= γw4 (I , ww , hw |Γ
w)

(27)
(28)

w

h and Γwhere Γw are the parameter vectors of husband’s and wife’s including
the common parameters among households in population,th = (γh1 , γh2 , γh3 , γh5 , · · · , γhm )ΓtΓw = (γw1 , γw2 , γw3 , γw5 , · · · , γwm )excluding the random coeﬃcients γh4 and γw4 .
Inserting the formula (27) and (28) into (26) yields
h ), γw4 (I ∗ , ww , hw |Γ
f [γh4 (Ih∗ , wh , hh |Γw) | ζ ]w(29)

The joint probability density function of Ih∗ and Iw∗ is obtained as a productof the formula (29) and the Jacobean
∂γh4
∂Ih∗
∂γw4
∂Ih∗

J≡

∂γh4
∂Iw∗
∂γw4
∂Iw∗

The Jacobean reduces into the formJ=∂γh4 ∂γw4·∂Ih∗ ∂Iw∗

∂γh4
∂γw4
=
= 0 holds. Multiplying the formula (29) by the Jacobean
∂Iw∗
∂Ih∗
J yields the joint distribution density function of Ih∗ and Iw∗ , g(Ih∗ , Iw∗ ), as
becauseg(Ih∗ , Iw∗ ) =h )f [γh4 (Ih∗ , wh , hh |Γ, γw4 (Iw∗ , ww , hw |Γw) | ζ ] ·∂γh4 ∂γw4·∂Ih∗ ∂Iw∗(30)

The formula (30) shows that the shape of the joint distribution density
function of Ih∗ and Iw∗ , g(Ih∗ , Iw∗ ), depends upon the variables wh , hh , ww , hw
h and Γand the parameter vectors, Γ
w , as well as the parameter vector, ζ, ofthe joint distribution density function f .
Let the probability density function of each husband’s TILS, Ih∗ , and wife’s
TILS, Iw∗ , be denoted by gh (Ih∗ ) and gw (Iw∗ ) respectively. The probability density
function gh (Ih∗ ) and gw (Iw∗ ) are the marginal distributions of the joint distribution, g(Ih∗ , Iw∗ ), given by the formula (30).
18

3.5

x

w
The sign of ∂hand observation on labor supply∂I 0w

Let the ﬁtted value of wife’s labor supply probability be denoted by μw . Givenh , Γthe values of wh , hh , ww , hw and the parameter vectors, Γ
w and ζ in theformula (30), the wife’s labor supply probability μw can be calculated by using
the probability density function of wife’s TILS, gw (Iw∗ ). The probability, μw , inthe population where the wife’s guaranteed income is Iw0 is given by the formula
∞

μw

=

0 =I +I j
IwAhgw (Iw∗ )dIw∗

if and only if

∂hxw
<0
∂Iw0

(31)

gw (Iw∗ )dIw∗

if and only if

∂hxw
>0
∂Iw0

(32)

or
μw

=

j
0
Iw
=IA +Ih−∞∂hxw. The formula (31) is justiﬁed by the relation∂Iw0(22), and the formula (32) is justiﬁed by the relation (23). Diﬀerentiating μw
in respect to wife’s guaranteed income, Iw0 , yields
depending upon the sign of

∂ μi
∂Iw0

= −gw (Iw0 ) < 0 if and only if

∂hxw
<0
∂Iw0

(33)

∂ μi
∂Iw0

= gw (Iw0 ) > 0

∂hxw
>0
∂Iw0

(34)

or

again depending upon the sign of

if and only if

∂hxw
.
∂Iw0

∂hxw
∂Iw0
is necessarily negative because it is observed that household income levels andwife’s labor supply ratios are negatively correlated. Thus the equality in formula(34) is inconsistent with observed characteristics of wife’s labor supply ratio.
∂hxw
This concludes that the inequality< 0 must hold as for the wife’s MHLS,
∂Iw0
hxw .
Taking the observations of Douglas(1934) into account, the sign of

3.6

∂hx

x

w
The consistency between sign of ∂I h0 , ∂hand payoﬀ
0∂Iwhmatrices∂hxw< 0, which is clariﬁed in Subsection 3.5, is consistent to∂Iw0
2. This means that thethe wife’s payoﬀ matrices listed in Table 3 except πw
The condition,

19T名= (31)

瑞︒
Figure 4:W掩

's indifFerence curves yielding her pttOr matr破
T名

20

2
whole set of wife’s indiﬀerence curves that generate her payoﬀ matrix, πw
, is
∂hxw
not consistent with the condition,
< 0.
∂Iw0
2
, is depicted in
Indiﬀerence curves that generate wife’s payoﬀ matrix, πw
Figure 4. The wife’s guaranteed income when her husband does not work is00denoted by Iw0
in Figure 4. Iw0= IA holds where IA is the unearned incomeof the household. Next, in Figure 4, the wife’s guaranteed income when her
00, where Iw0= IA + wh hh holds.
husband does work is denoted by Iw1∗
Let the region of wife’s TILS, Iw of the wife’s , be examined if the wife’s
2
payoﬀ matrix is πw. Note that wife’s indiﬀerence curves, which are consistent2
with her payoﬀ matrix, πw, is depicted in Figure 4.
a’ , and simiLet wife’s utility index at point a’ in Figure 4 be denoted by ωwb’ , ω c’ ,larly, let the wife’s utility indexes at point b’, c’, and d’ be denoted by ωw
w
d’ respectively. The relation between the group of ω 0 , ω 1 and the groupand ωww
w
a’ , ω b’ , ω c’ , ω d’ is
of ωw
w
w
w

a’
0
ωw
= ωw
0
if Iw0 = Iw0
b’
1
ωw
= ωw

c’
0
ωw
= ωw
0
if Iw0 = Iw1
ω 1 = ω d’w

w

depending whether her husband works or not.

∂hxw
< 0. Let us∂Iw0
a’ and ω b’ , and the inequality between ω c’
examine the inequality between ωww
w
d’ , according to the relation (22).
and ωw
Now suppose wife’s indiﬀerence curves fulﬁll the condition

0
0
< Iw1
hods,
1. If Iw∗ < Iw0a’b’c’ > ω d’ must follow.
then ωw > ωw and ωw
w
0c’ < ω d’does not apply because the inequality ωw
Nevertheless, Iw∗ < Iw0w
holds in Figure 4.
00< Iw∗ < Iw1
holds,
2. If Iw0a’b’c’ > ω d’ must follow.
then ωw < ωw and ωw
w
0
0
a’ > ω b’ and< Iw∗ < Iw1
does not apply because ωw
Nevertheless, Iw0wd’c’
ωw < ωw holds in Figure 4.
0
0
< Iw1
< Iw∗ holds,
3. If Iw0
a’ < ω b’ and ω c’ < ω d’ must follow.
then ωw
w
w
w
a’ > ω b’ holds in Figure0< Iw∗ does not apply because ωw
Nevertheless, Iw1
w
4.

21

Following the reasoning 1 through 3 above, the region of wife’s TILS, Iw∗ , does2
not exist if her payoﬀ matrix, πw, is assumed. This concludes wife’s payoﬀ∂hxw
2
matrix, πw, is inconsistent with the condition< 0. Thus, the wife’s payoﬀ∂Iw0
2
matrix, πw , is excluded from the further discussion.
Two payoﬀ tables, Π2−3 and Π3−2 , are the two cases where no equilibria∂hxw
exist. (See Table 4-1.) Setting the consistency condition< 0 can preclude∂Iw0
the payoﬀ table Π3−2 , which gives no equilibria, because wife’s payoﬀ matrix,2πw, is no longer relevant.
Following the similar reasoning above, it can be concluded that husband’s∂hxh< 0, because the region
payoﬀ matrix, πh2 , is inconsistent with the condition∂Ih0
of husband’s TILS, Ih∗ , does not exist if his payoﬀ matrix, πh2 , is assumed.
∂hxw
As for the wife’s MHLS, hxw , the condition,< 0, is required for the con∂Iw0∂hxh
sistency with observations. Along with this condition, if the condition,
< 0,
∂Ih0
is assumed for the husband’s MHLS, hxh , we can exclude the payoﬀ table, Π2−3 ,where neither Nash-equilibria nor cooperative equilibria exists, because the husband’s payoﬀ matrix, πh2 , becomes no longer relevant. Thus, combining the
∂hxw
∂hxh
< 0 with the consistency condition< 0 precludes all the
assumption
0
∂Ih
∂Iw0
cases where no equilibria exist.
Assumption 2:

3.7

∂hxh
<0
∂Ih0

The relation between payoﬀ matrix and the region ofTILS

2In this subsection, the relations between the payoﬀ matrices, except πh2 and πw,
listed in Table 3 and the regions of “Threshold Income of Labor Supply (TILS)”
is considered.
Firstly, let the region where husband’s TILS, Ih∗ , exists be considered when
his payoﬀ matrix is πh1 . Husband’s indiﬀerence curves that generate his payoﬀ
matrix πh1 is depicted in Figure 5. Let his guaranteed income when his wife
00
does not work be denoted by Ih0. Now Ih0
= IA holds where IA is the unearned
income of the household. Next, let his guaranteed income when his wife does0000
, where Ih0= IA + ww hw holds. Ih0
and Ih1
are depicted
work be denoted by Ih1
in Figure 5.
Let husband’s utility index at point a in Figure 5 be denoted by ωha , andsimilarly, let the husband’s utility indexes at point b, c, and d be denoted byωhb , ωhc , and ωhd respectively. The relation between the group of ωh0 , ωh1 and the22

／１ ｔ ＼

Ｔ

;:)

現
１

就
０
０

Ftture 5:Husbandうs indi丘
もrence curves yielding his payOr matrix T,

23

group of ωha , ωhb , ωhc , ωhd isωh0 = ωha0
if Ih0 = Ih0ωh1 = ωhbωh0 = ωhc0
if Ih0 = Ih1
ωh1 = ωhd
depending whether his wife works or not.
Let us examine the inequality between ωha and ωhb , and the inequality between ωhc and ωhd . The relation (20) gives the relevant inequalities between these∂hxh< 0 is assumed.
utility indexes because
∂Ih0
1’

2’

3’

0
0  a
< Ih1
ωh > ωhb and ωhc > ωhd holds.
Ih∗ < Ih00
Nevertheless, Ih∗ < Ih0
does not apply because the inequality ωha < ωhbholds in Figure 5.
0
0  a
Ih0
< Ih∗ < Ih1
ωh < ωhb and ωhc > ωhd holds.
0∗0
Nevertheless, Ih0 < Ih < Ih1
does not apply because the inequality ωhc <ωhd holds in Figure 5.
0
0
< Ih1
< Ih∗  ωha < ωhb and ωhc < ωhd holds.
Ih0
The inequalities ωha < ωhb and ωhc < ωhd comply with the preference curves
shown in Figure 5.

The above reasoning 1’, 2’, and 3’ concludes that the region of husband’s TILS,
0< Ih∗ .
Ih∗ = Ih∗ (wh , hh |Γh ), is Ih1
The plausible regions of husband’s TILS for the rest of husband’s payoﬀ
matrices, πh3 , πh4 , πh5 , and πh6 are induced following the similar reasoning above.
1
3
4
5
, πw
, πw
, πw
,
The plausible regions of wife’s TILS for wife’s payoﬀ matrices, πw
6and πw are also induced similarly. The plausible regions of husband’s and wife’s
TILS are shown in Table 5-1 and 5-2 respectively.
Table 5-1: Regions of husband’s TILS corresponding to his payoﬀ matrices
Husband’s payoﬀ matrix
Region of husband’s TILS, Ih∗

k’s index of πhk
1
3
4
5
6

0 = I ,
Note that Ih0
A

0
Ih1
<
0
Ih0 <
0
Ih2
<
0
Ih1
<

Ih∗ (wh , hh |Γh )Ih∗ (wh , hh |Γh )Ih∗ (wh , hh |Γh )Ih∗ (wh , hh |Γh )Ih∗ (wh , hh |Γh )0 =I +w h .
Ih1
w w
A

0 .
See formula (37) for the deﬁnition of Ih2

24

0
< Ih1
0
< Ih0
0
< Ih2

／１ ︲ ＼

打

:;)

域
就

Ftture 6:Husband's indifFerence curves yielding his payOr matrix T島
Table 5‑2: Regions Of H′
ife's TILS corresponding to ller payofF matrixes
Wi免's payofF Inatrix

Resion of wife's TILS,r島's index of T玄で,1
r協1< 二 ぁ(仰り,ん
υ「ω
3
4
5

F枯0<

,υ
r品(取
,ん9tl「ω

r品2<

どw)ん切「ω
二 ぁ(そ

<弓 1
<瑞 。

rぁ(ttlり
ft「切
,ん

<瑞 2

r格1< rぁ (wu,ん切「 切
=rA,
■
Note
lぁ
私。
h=r五 十tl あ
lla(39)偽See bml■r the defln山
おn of略
2
6

0

1n LbL 5‑1,a va五aЫe域 2b° unds the rettOn Of husbandも TILS,whch

0corresponds to his payoﬀ matrix, πh5 . Similarly in Table 5-2, a variable Iw2
5bounds the region of wife’s TILS, which corresponds to her payoﬀ matrix, πw.
00Some description should be made on these variables, Ih2
and Iw2.
0, be described ﬁrst. Husband’s indiﬀerence curves yieldLet the variable, Ih2
ing his payoﬀ matrix, πh5 , is depicted in Figure 6. The coordinates of husband’s
available leisure, Λh , and his available income, X, are indicated by points a, b,
c, and d, given the wage rate, wh , and the assigned hours of work, hh , of his
employee opportunity. Let an auxiliary line passing point a and d be drawn in
Figure 6. The gradient of the line ad to the horizontal axis, θ, is given by theformulawh hh + ww hw
tan θ =
hh
Suppose husband’s wage rate raises up toWh ≡wh hh + ww hw
hh

and his assigned hours of work unchanged as hh . The point d is where husband
would be located if he accepts this employee opportunity of Wh and hh , in casehis wife does not work. Since his utility index at point a is higher than his
utility index at d, he does not accept the employee opportunity of Wh and hh .
Let h denote the subset consisting of Γh ’s such that inequality ωa > ωd holds.
For the Γh ’s in the subset h ,0Ih∗ (Wh , hh |Γh ) < Ih0,∀ Γ h ∈ h(35)

holds according to the relation (20). Note that the argument Wh , instead ofwh , enters the husband’s TILS in formula (35).
For the Γh ’s in the subset h , what will be the region of husbands’ TILS,
Ih∗ , when the wage rate again reduces to wh , with the assigned hours of work
being constant as hh ? Suppose the parameter vector, t Γh= (γh1 , γh2 , · · · , γhm ), of husband’s utility function has constant elements over the population except
γh4 , which is randomly distributed in the population. Solving the equation
0Ih∗ (Wh , hh |Γh ) = Ih0

in respect to γh4 yields the formula0h ), Wh , hh |Γ
γh4 = γh4 (Ih0(36)

Inserting the formula (36) into the element γh4 in the parameter vector, Γh , ofthe formula (13) yields



0
0
h ), · · · , γhm
Ih2= Ih∗ wh , hh | γh1 , γh2 , γh3 , γh4 (Ih0
, Wh , hh |Γ
h )
(37)
= I 0 (I 0 , wh , hh , ww , hw |Γ
h2

h0

26

πh5

A 
A
0
Ih2

πh4

A 
A

πh3

0
Ih0

πh1 , πh6

A 
A

- I∗
h

0
Ih1

Figure 7: Correspondence between husband’s payoﬀ matrixes and the region of
his TILS, Ih∗

5
πw

A 
A
0
Iw2

4
πw

3
πw

A 
A

A 
A

0
Iw0

0
Iw1

1
6
πw
, πw

- I∗

w

Figure 8: Correspondence between wife’s payoﬀ matrixes and the region of her
TILS, Iw∗
Using the formula (37), the region of husbands’ TILS, whose payoﬀ matrix are
πh5 , can be shown as0,Ih∗ (wh , hh |Γh ) < Ih2∀ Γ h ∈ h(38)

0
For the variable, Iw2, bounding the region of wife’s TILS, which corresponds
5
in Table 5-2, a formula
to her payoﬀ matrix, πw
0
0
0
Iw2
= Iw2
(Iw0
, wh , hh , ww , hw |Γ
w)

(39)

can be obtained. Using the formula (39), the region of wives’ TILS, whose payoﬀ
5
matrix are πw, can be shown as0Iw∗ (ww , hw |Γw ) < Iw2(40)

Regions of TILS corresponding to payoﬀ matrices can be shown graphicallyin ﬁgure 7 and 8 9 .
9 For

0 and I 0 , and for the inequalities between I 0
the inequalities between Ih2
w2 and
h0
x
∂h
∂hx
w
0
0
0
0
h
0
Iw0 , it can be proved that< 0  Ih2 < Ih0 and< 0  Iw2 < Iw0 holds. See
0
∂Ih0
∂Iw
Miyauchi(1991) for precise discussion.

27

∗
Iw

j

6k

(5 − 1)(5 − 6)(4 − 1)(4 − 6)(3 − 3)(1 − 3)(6 − 3)(4 − 4)(3 − 4)(1 − 4)(6 − 4)

I0
h2

I0
h0

(4 − 3)m

I0
w0(5 − 4)0n(1 − 1)(1 − 6)(6 − 1)(3 − 1)(3 − 6)I0
w1(5 − 3)

m

i

- Ih∗

I0
h1

n

I0
w2
(5 − 5)(4 − 5)k(1 − 5)(6 − 5)(3 − 5)jiFigure 9: Correspondence between the payoﬀ tables and the region of (Ih∗ , Iw∗ )3.8

Joint distribution of TILS and decision on household
labor supply

In Subsection 3.7, husband’s and wife’s TILS, Ih∗ and Iw∗ respectively, are partitioned into regions corresponding to each of their own payoﬀ matrices. Employing this result, the two dimensional plane of TILS, (Ih∗ , Iw∗ ), can be mapped
to household’s Nash-equilibria and cooperate equilibria clariﬁed in Section 2.
Combining the Ih∗ axis in Figure 7 and the Iw∗ axis in right angle in Figure 8
yields two-dimensional coordinate system of (Ih∗ , Iw∗ ), which is shown in Figure
9. Horizontal axis is of Ih∗ and vertical axis is of Iw∗ . As shown in Figure 9, the(Ih∗ , Iw∗ ) plane is partitioned into regions by orthogonal lines ii, jj, kk passing
0
0
0
, Ih0
, Ih2
on the Ih∗ axis respectively, and also by orthogonal lines , mm, nn
Ih1
0
0
0
passing Iw1 , Iw0, Iw2
on the Iw∗ axis. Note that any region on (Ih∗ , Iw∗ ) planepartitioned by lines ii, jj, kk, , mm, nn can be assigned one of the payoﬀ
tables listed in Table 3. By showing the superscript of payoﬀ tables, Figure 9
assigns the corresponding payoﬀ table to each region on (Ih∗ , Iw∗ ) plane.
Figure 10 assigns the corresponding Nash-equilibria or cooperate equilibria
on husband’s and wife’s labor supply to the region on (Ih∗ , Iw∗ ) plane, based on

28

∗
Iw

j

6k

i

@
@
@@
@
@
@
@
@@
@
@
@
@
Wife Works
Both Work
@
@
@@
@
@
@
@
@
@@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@ 
@
@@
@@
@@
@@

0
Iw1
@
@
@
@ Either@@ Husbandor
Husband Works
@
@
Wife
@
@
@
@ Works
@
@
@
@
@
@
@
@
@
m @@@@@@@@@
m
0
Iw0

6
0
Ih2

0
n

0
Ih0

0
Ih1

- Ih∗
n

0
Iw2
Neither Works

k

j

i

Both Works

Figure 10: Mapping the region of (Ih∗ , Iw∗ ) to household decisions
Figure 9 and Table 4-1 through 4-3. The region in Figure 10 corresponding to
the payoﬀ table Π3−3 (i.e., the region indicated by (3 − 3) in Figure 9) will be of
special interest. Note that the households in population where the coordinateof (Ih∗ , Iw∗ ) belongs to this region yield non unique Nash-equilibria, so that thetheory can only suggest either husband works or wife works.
Integrating the probability density function g(Ih∗ , Iw∗ ) in each region shownin Figure 10 yields probabilities of both work, husband works, wife works, andneither works.

29

3.9

Utility function in quadratic form

Utility functions of husband is speciﬁed in quadratic forms as
ωh =

1
1
γh1 X 2 + γh2 X + γh3 XΛh + γh4 Λh + γh5 Λ2h
2
2

(41)

where γhj (j = 1, · · · , 5) are the parameters of the utility function. γh4 is0+ γh4 · uh , where stochastic variable uh , is assumed todeﬁned as γh4 ≡ γh4
follow logarithmic normal distribution.
loge uh ∼ N (mh , σh2 )(42)

γh4 is the intersection of marginal utility of husband’s leisure, which is assumed
to be a random coeﬃcient distributed in population. Other parameters, γh1 ,0γh2 , γh3 , γh4, γ h4 , and γh5 are assumed to be common over the population. γh1
is normalized as γh1 ≡ −1.
Let u∗h denote the stochastic variable following the standard normal distribution. Standardizing the stochastic variable loge uh in formula (42) yields
loge uh − mh
= u∗h  uh = exp(mh ) · exp(σh · u∗h )σh1where a constraint, mh = − σh2 , is imposed on mh , so that E(uh ) = 1 follows.
2
Thus a formula1
uh = exp(− σh2 ) · exp(σh · u∗h )(43)
2
is obtained.
Given the formula (41) and (43), the formula of husband’s TILS is
Ih∗ = H0h + H2h · exp(σh · u∗h )

(44)

where
H0h

≡

H2h

≡

0
− γh2 wh − γh3 wh (T − hh ) + γh5 (T − 12 hh ) − 12 γh1 wh2 hhγh4γh1 wh − γh3
γ h4
1 2· exp(− σh )γh1 wh − γh32Similarly, utility functions of wife is speciﬁed as quadratic forms as
ωw =

1
1
γw1 X 2 + γw2 X + γw3 XΛw + γw4 Λw + γw5 Λ2w
2
2

(45)

where γwj (j = 1, · · · , 5) are parameters of the utility function. γw4 is deﬁned
0+ γ w4 · uw , where stochastic variable uw , is assumed to follow
as γw4 ≡ γw4
logarithmic normal distribution.
2loge uw ∼ N (mw , σw
)

30

(46)

1 2
where a constraint, mw = − σw
, is imposed on mw , so that E(uw ) = 1 follows.
2
γw4 is the intersection of marginal utility of wife’s leisure, which is assumed to
be a random coeﬃcient distributed in population. Other parameters, γw1 , γw2 ,0, γ w4 , and γw5 are assumed to be common over the population. γw1 is
γw3 , γw4
normalized as γw1 ≡ −1.
Given the formula (45) and (46), the formula of wife’s TILS
Iw∗ = H0w + H2w · exp(σw · u∗w )(47)

is obtained where
H0w

≡

H2w

≡

0
2
− γw2 ww − γw3 ww (T − hw ) + γw5 (T − 12 hw ) − 12 γw1 ww
hw
γw4
γw1 ww − γw3
1 2
γ w4· exp(− σw)γw1 ww − γw32Let ρ denote the correlation coeﬃcient of two dimensional normal distribution of u∗h and u∗w .

3.10

A priori restrictions on structural parameters

A priori restrictions imposed on structural parameters are as follows 10 .
a priori restrictions on parameters of utility function:
∂hxh
<0,
∂Ih0

∂hxw
<0
∂Iw0

∂ωh
>0,
∂X
∂ωh
3.
>0,
∂Λh

∂ωw
>0
∂X
∂ωw
>0
∂Λw

1.
2.

4. Indiﬀerence curves are convex to the origin.
5. H0h > Ih0 max and H0w > Iw0 max must hold, where Ih0 max and Iw0 max are
the maximum values of observed guaranteed income of husband and wife
respectively.
6. H2h < 0 ,H2w < 0a priori restrictions on distribution parameters of uh and uw :a σh > 0b σw > 0c |ρ| < 1
10 See Miyauchi(1991) for detailed discussions on a priori restrictions on structuralparameters.

31

4

Estimation of structural parameters

Applying Japanese data to the model discussed in Section 3, the structuralparameters were estimated. The estimation was performed by minimizing theχ2 for observed cases of household labor supply and these simulated values.
Optimal parameter set was searched within a parameter space that is consistent
with a priori restrictions discussed in Subsection 3.10.
Observations on household labor supply probability are obtained by “Household Labor Status Survey” for the years of 1971, 1974, 1977, 1979, and 1982.
Observations on wage rate and assigned hours of work are obtained by “Wage
Census” for the corresponding years. Because appropriate observations on unearned income, IA , was not available, the value of IA was assumed to be zerofor the ﬁrst attempt. These observations are stratiﬁed by husband’s and wife’s
age classes.
The estimates of the structural parameters are as follows.
γh2
γh3
γ h4
γh5
0
γh4
σh
ρ

= 6540.59= 520.02= 816058.3= −32724.1= 79727.4= 2.534= 0.602

γw2
γw3
γ w4
γw5
0
γw4
σw

= 1150.25= −21.36= 269220.5= −962.0= 26956.1= 1.0474

A comparison between observed and simulated labor supply probabilities ispresented graphically in Figure 11. Although the observed and the simulated
probabilities are obtained for each stratum of husband’s and wife’s age classes,
let the graphical comparison be presented on the basis of aggregating thesestrata for the sake of briefness.

5

Concluding remarks
1. The magnitude in probabilities of husband’s and wife’s labor supply is signiﬁcantly diﬀerent by these patterns of household decision, i.e., both work,
husband works, wife works, and neither works. The model presented inthis paper simulates well the diﬀerence in the magnitude of these probabilities.
2. The model simulates well the time trend observed in these probabilities,
although systematic biases in simulated probabilities are persistent in eachyear.
3. The calculated probability in the region corresponding to the payoﬀ table,
Π3−3
h , where the model fails to give a unique equilibria on household labor
supply, is less than 0.001.
32

(iv) probability

that both work?

(ii)probability that husband works?

(i)probability that neither works(iii)probability that wife works 1.3%

?

?

1971
Observed

16.3%

68.3%

14.1%

18.3%

73.0%

68.7%
0.1%

Simulated(iii)1.6%?

1974
Observed

(iv)

(ii)

(i)

18.6%

65.1%

14.7%

28.2%

64.4%

6
7.2%
0.2%

Simulated(iii)2.1%?

1977
Observed

(iv)

(ii)

(i)

20.8%

62.3%

14.8%

29.1%

63.8%

6
6.9%
0.2%

Simulated(iii)2.0%?

1979
Observed

(iv)

(ii)

(i)

23.3%

61.1%

13.5%

32.6%

60.5%

6
6.7%
0.3%

Simulated(iii)2.4%?

1982
Observed

(iv)

(ii)

(i)

27.2%

57.9%

12.5%

32.1%

61.1%

Simulated

6
6.6%
0.2%

Figure 11: Observed and simulated probabilities of household decision on labor
supply33

4. The model gives systematic biases in simulated probabilities in each year.
The model should be modiﬁed so that the systematic biases can be resolved.

34

References[1] Bjorn, P.A. and Q.H. Vuong (1984); “Simultaneous Equations Models for
Dummy Endogenous Variables: A Game Theoretic Formulation with an
Application to Labor Force Participation.” Social Science Working Paper
no.537, July 1984, California Institute of Technology, Division of the Humanities and Social Sciences.
[2] Douglas, Paul H. (1934); The Theory of Wages, New York: Kelley andMilman Inc.
[3] Dubin, J.A. and D. McFadden (1984); “An Econometric Analysis of Residential Electric Appliance Holdings and Consumption,” Econometrica,
52(2), pp.345-362.
[4] Hausman, J. and D. Wise (1978); “A conditional probit model for qualitative choice: Discrete decisions recognizing interdependent heterogeneous
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[5] Heckman, James J. (1974). “Shadow Prices, Market Wages, and Labor
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[6] Mancer, M. and M. Brown (1980); “Bargaining Analyses of Household
Decisions,” in C.B. Lloyd, E. Andrews and C.Gilroy, eds, Women in The
Labor Market, New York: Columbia University Press.
[7] McFadden, D. (1973); “Conditional Logit Analysis of Qualitative Choice
Behavior,” in P. Zarembka ed., Frontiers in Econometrics, New York: Academic Press.
[8] McFadden, D. (1981); “Econometric Models of Probabilistic Choice,” inStructural Analysis of Discrete Data, Manski,C and D.McFadden eds.,
pp.346-364, Cambridge: MIT Press.
[9] Mincer, Jacob. (1962). “Labor Force Participation and Unemployment: A
Review of Recent Evidence,” in R.A.Gordon and M.S.Gordon, ed. Prosperity Unemployment, New York: John Willey and Sons Inc.
[10] Miyauchi, T. (1991); “An Econometric Model of Household Labor Supply— A Theory and Veriﬁcation —”, Mita Journal of Economics, (84)3, The
Keio Economic Society, Keio University, pp.40–70.
[11] Miyauchi, T. (1992); “A Model of Household Labor Supply Probability —Employee Opportunity Model vs. Employee and/or Self-employed Opportunity Model —”, KEO Discussion Paper, Tokyo: Keio Economic Observatory, Keio University.

35[12] Obi, K. (1969a); “An analysis on labor supply behavior of employee household based on a threshold principal income distribution,” Mita Journal of
Economics, 62(1), Tokyo: The Keio Economic Society, pp.17–45.
[13] Obi, K. (1969b); “On a general model of household labor supply,” Mita
Journal of Economics, Tokyo: Keio Economic Society, 62(8), pp.150–165.
[14] Obi, K. (1979); “A general theory of household labor supply,” Mita Journalof Economics, 72(6), Tokyo: The Keio Economic Society, pp.150–165.

36