Household Members’ Time
Allocation to Daily Activities and
Decision to Hire Domestic Helpers
Donggen WANG and Jiukun LI
Department of Geography
Hong Kong Baptist University
Kowloon Tong, Kowloon
Hong Kong
Outline

Research background and objectives

A model of household time allocation
taking into consideration of hiring
domestic helpers

Model estimation

An empirical study: Hiring domestic
helpers and household time allocation in
Hong Kong

Model prediction and impact analysis

Conclusions
Research Background
Modelling activity and travel interactions
between household members (Bhat and
Pendyala, 2005)
Structural equations models
Interactions in activity participation and travel between
household heads (Golob and McNally 1997)
Time allocation models
Time allocation models of adult household members’
participation in and time allocation to independent and joint
activities (Gliebe and Koppelman 2002; Scott and
Kanaroglou 2002)
Group decision-based models for time allocation to
independent, shared and allocated activities (Zhang et al.
2002; 2004)
Research Background
Discrete choice models
The allocation of maintenance activities to
household members in terms of who and how
(jointly or independently) the maintenance
activities are conducted (Vovsha, et al. 2004;
Srinivasan and associates 2005; 2006)
Tour-based discrete choice models accounting
for interactions between household heads
(Gliebe and Koppelman, 2005)
Research Background
 Existing studies share an underlying assumption:
no external help for household maintenance tasks
 In reality, households may get helps from
members of the extended family, or hire domestic
helpers
 External helps may change activity-travel patterns
of household members
 Consideration of external helps is important for
understanding activity-travel behaviors of
household members
 Hardly any study of this kind is reported in the
literature
Research Objectives
Develop a model to analyze time allocation of
household members taking into consideration
of hiring domestic helpers.
Calibrate the model by empirical data
Apply the model for analyzing the impacts of
wage rate changes on time allocation
Notations
 Ti is the total time available for member i;
j
Ti is the time spent for activity j by member i.
 G is the amount of time that a domestic helper
is hired.
j
 u i is the baseline utility for time invested in
activity j by household member i .
  G is the baseline utility for hiring domestic
helpers
  i is the wage rate of household member i, qmaid
is the wage rate of domestic helpers.
Utility Functions of Household
Members and Domestic Helpers
We define the utility function of a household
member as follows:
ui   uij ln Ti j  1
j
And the utility function of hiring a domestic
helper is:
w

T
i i i  maid G
uG   G ln  G  1 
ln  G  1
w
iTi
i
A Model of Time Allocation
 Households are assumed to maximize the household utility:
Maximize HUF   wi ui  wG uG
i
 T  
 T
w
  wi  uij ln Ti j  1 wG
i
j
i i
maid
i
w
G
ln  G  1
(1)
i i
i
Subject to:
Ti w  Ti m  Ti r  Ti
m
T
 i G  A
(a)
(b)
i
 Where w , i  1, , n and wG are respectively weights of household
i
members and domestic helpers, representing their relative contributions
to household utility.
A Model of Time Allocation
 Lagrangian function
 T  
 T
w
L   wi  uij ln Ti j  1 wG
i
j
i i
maid
i
w
G
ln  G  1 
i i
i




j
m
i i  j Ti  Ti     i Ti  G  A 


 where i and  are respectively the Lagrangian multipliers
associated with the time constraint for member and that
associated with the time constraint of maintenance.
A Model of Time Allocation
The first-order conditions of the Lagrangian
function
wi uiw
1
 wG
Ti w*  1
1
m
i i
wu
Ti
m*
1
i maid G*

w* 
  iTi 
 i

*
ln
G
 1  i  0

2
 i    0
1
w u r*
 i  0
Ti  1
r
i i
wG
G*  1
w*

T
 i i  maid G
i
w*

T
ii
i
wG maid
*

ln
G
 1    0

w*
iTi
i
A Model of Time Allocation
 From the first-order conditions, we may derive the following
interrelations among time allocation to activities and time of hiring
domestic helper:
r
uiw
i maid G*
u
*
i
wi w*
 wG
ln
G

1

w


i
2
r*
Ti  1
T

i 1
w* 
  iTi 
 i

uim
1
wi m*  wG *
Ti  1
G 1
w*
*

T


G
 i i maid
i
w*

T
ii
i
With constraints (a) and (b)
(2)
r
 maid
u
*
i
 wG
ln
G

1

w
(3)


i r*
w*
Ti  1
iTi
i
Specification of the baseline utility
function
 In general the utility function can be expressed as
follows:


ui  vi   i  fi i , xi   i
j
j
j
 i j , j  w, m, r
j
j
j
j
(4)


 Where
are error terms, vi  fi i , xi
the systematic component of utility and assume that it is
continuously differentiable
j
j
j
j
 We assume that
The utility contribution of time allocation is
dependant on socio-economic characteristics of
individuals, the systematic component of utility is
to capture these effects
is
Model Estimation
 Substituting equation (4) into (2)and into (3), we got
the following equation
r
r
w
w

wi vi   i vi   i 
i 
 w*
 r*

wG  Ti  1 Ti  1 
m
m
r
r

wi vi   i vi   i 
i 
 r*
 m*

wG  Ti  1 Ti  1 
Model Estimation
Where
i maid G*
*
i 
ln
G
  1
2

w* 
  iTi 
 i

1
i   *
G 1
w*
*

T


G
 i i maid
i
w*

T
ii
i
 maid
*

ln
G
 1

w*
iTi
i
Model Estimation
Assume that error term  i j , j  w, m, r are
independently and identically standard
normal distributed
we add the following auxiliary random variable:
 i   , i  1,
r
i
,n
based on the property of normally distributed
variables, Then the density function of the
random vector i ,i , i  may be derived as follows:
Model Estimation
f
  , ,   
i
i
i


1
2

3
e
2
2

Ti w* 1  
Tim* 1 
2
w*
m*
 ai  wi Ti 1 i  r*  i    bi  wi Ti 1 i  r*  i    i 
Ti 1  
Ti 1 

2



Ji ,
• Where
wG
wi 
wi
 iw  iw  iw
i i  i
Ti  1 r w
ai  r*
vi  vi
Ti  1
 wi Ti  1
w*
 im  im  im
Ji 
0
i i  i
 ir  ir  ir
i i  i
Ti m*  1 r m
bi  r*
vi  vi
Ti  1
w*
0
0
wi Ti
0
Ti w*  1
Ti r*  1
m*
Ti m*  1
2
 1 r*
   wi  Ti w*  1Ti m*  1
Ti  1
1
Model Estimation
• Therefore the density function of the random
vector i , i  can be derived as follows:
f i i , i  





  wi 
1
2

e
3
2
2

T w* 1  
T m* 1 
2
 ai  wi Ti w* 1 i  i r*  i    bi  wi Tim* 1 i  i r*  i    i 
Ti 1  
Ti 1 

2
 1Ti
2  Ti
w*

m*
 2 
3
ci
 1

 a  w T

i
i
i
w*

    b  w T
1 i
2
i
i
i
m*
 
1 i
2
e
 wi 
2
T
i
w*
 1Ti m*  1 d i
2

di2
4 ci
2
2
w*
m*






T

1
T

1
1
• Where c   i
i



  1
i
r*
r*
2  Ti  1   Ti  1 


Ti w*  1
Ti m*  1
w*
di  r*
ai  wi Ti  1 i  r*
bi  wi Ti m*  1 i
Ti  1
Ti  1




Model Estimation
 The maximum likelihood method may be applied to
estimate the parameters. Introducing the index for
household, we can write the likelihood function as
follows:


n
L iw , im , ir , wi   ln  fi i , i 
i j   i1j ,
d
j
iH

i
,  , i  1, , n, j  w, m, r
 Where
 The sequential number theoretic optimization (SNTO)
algorithm can be used to solve the likelihood function to
find the maximum
likelihood estimation of
j
parameters i and wi . SNTO searches for the global
optimum among points uniformly scattered in the search
space (Fang and Wang, 1994).
An Empirical Study
Hong Kong: A special Administrative
region of China
Area size: 1100 square kilometers
High population density (more than 6,000
per square kilometers)
In 2004, about 218,430 foreign domestic
helpers working in Hong Kong
An Empirical Study
 Hong Kong Travel Characteristics Survey conducted in
2002; only households that have married couples and at
least one of the couple is employed are selected;
 About the sample:
 size: 10,381 households;
 About 9.9% have live-in maids;
 About 15.9% have car available for private use;
 About 17.0% have child aged 1 to 5 and 45.4% have child aged
6-17;
 About 48.2% of the households live in private apartment or
house
 About 76.1% of male heads are employed and 49.2% of female
heads are employed.
An Empirical Study
Variable definitions
Time duration of the following activities:
subsistence, maintenance and recreation
Wage rate: calculated from monthly household
income (divided by duration of subsistence
activity)
If both household heads are working: Male : female
(1.35 :1)
If only one head is working, all household income is
allocated to the person
Domestic helper:
Monthly costs of a domestic helper: around 6000 HK
dollars
Working time: 10 hour/day,
Wage rate: 20 HKD/hour
An Empirical Study
Socio-Economic variables
 household size,
 type of housing (‘1’ for private housing and ‘0’
for public housing),
the presence of child aged 5 or younger and of
child aged 6 to 17,
age of household head (1: 20 years old or
younger, 2: 21 – 40, 3: 41 – 60, 4: older than 60)
car ownership.
Assume the baseline utility is defined as
follows: vij  exp   ihj xihj  , i  1, 2, j  w, m, r

h

Parameter Estimation
Estimation of parameters for subsistence activity
Male head
Variables
Constant
Coefficients
-0.8847
Female head
t
Coefficients
-7.3371 -0.8580
t
-4.9138
Type of housing
-0.6497
-6.7852
0.1502
2.9280
Household size
-0.3326
-6.9505
-0.0942
-3.2963
Age of household head
-0.5415
-6.7453
0.0888
2.5447
Presence of child aged 5 or
lower
-0.6595
-6.5558
-0.3638
-3.5682
0.0233
3.4570
-0.5304
-4.1895
-0.2778
-5.6680
-0.3441
-3.4894
Presence of child aged 6 to 17
Car ownership
Parameter Estimation
Estimation of parameters for maintenance activity
Male head
Variables
Constant
Female head
Coefficients
2.9225
t
Coefficients
5.8414
-2.4992
t
-4.7717
Type of housing
-0.7391
-4.2235
0.9047 3.6105
Household size
-0.6348
-4.9063
-0.6739 -3.3160
Age of household
head
-0.2128
-3.1207
0.0710
Presence of child
aged 5 or lower
0.3654
3.2747
-0.2371
Presence of child
aged 6 to 17
0.7821
4.2799
0.1066
-0.0394
-1.0243
Car ownership
1.0656
-2.2714
1.4720
-0.3398 -2.6312
Parameter Estimation
Estimation of parameters for recreation activity
Male head
Variables
Coefficients
Female head
t
Coefficients
t
Constant
-2.7043
-8.4545 -0.6257
-6.9907
Type of housing
-0.4099
-6.3246 -0.6213
-4.3478
Household size
0.4674
7.2908 0.0218
1.8328
Age of
household head
-0.7797
-0.1273
-2.9048
-0.1447
-2.6463
0.2107
3.2663
-0.8141
-6.7432 0.7342
4.2472
1.8011
59.0389 0.8606
4.6398
-7.1099
Presence of child 0.5436
aged 5 or lower
Presence of child -0.4085
aged 6 to 17
Car ownership
wi
6.3625
-6.3211
Number of observations=10381  2  0.1669
Model Prediction
 The model can be used to predict the time use of household
members and time of hiring domestic helpers for household.
Maximize HUF   wi ui  wG uG
i
 T  
 T
w
  wi  uij ln Ti j  1 wG
i
j
i i
maid
i
w
G
ln  G  1
i i
i
Subject to:
Ti w  Ti m  Ti r  Ti
T
i
i
m
G  A
Ti j  0, G  0
Model Prediction
 Based on the parameters estimation, we can forecast the
time allocation to activities for household members and
the time of hiring domestic helper
 Assume that the socio-economic variables: (type of
housing, household size, Age of household head,
Presence of child aged 5 or lower, Presence of child
aged 6 to 17, Car ownership)=(1, 4, 2, 1, 1,1)
1  65,2  33,maid  20
T1  16, T2  16, A  8
Model Prediction
Optimal time allocation to activities for household
members and time of hiring domestic helper
Member
Activities
Total
Subsistence
Maintenance
Recreation
Male head
6.3416
3.8973
5.7612
16
Female head
5.9019
3.2199
6.8782
16
Helper
2.8828
Impact Analysis
subsistence of male head
recreation of male head
maintenance of female head
hiring time of domestic helper
maintenance of male head
subsistence of female head
recreation of female head
Percentage change of time allocation
40
30
20
10
0
-10
-20
-30
-40
-50
-60
10
15
20
25
30
35
40
45
50
55
60
65
70
75
The wage rate of domestic helper
80
85
90
95 100
Impact Analysis
subsistence of male head
recreation of male head
maintenance of female head
hiring time of domestic helper
maintenance of male head
subsistence of female head
recreation of female head
Percentage change of time
allocation
50
40
30
20
10
0
-10
-20
10
20
30
40
50
60
70
80
90 100 110 120 130 140 150 160 170 180 190 200
The wage rate of male head
Impact Analysis
subsistence of male head
recreation of male head
maintenance of female head
hiring time of domestic helper
maintenance of male head
subsistence of female head
recreation of female head
Percentage change of time
allocation
20
15
10
5
0
-5
-10
10
20
30
40
50
60
70
80
90 100 110 120 130 140 150 160 170 180 190 200
The wage rate of female head
Conclusions
A model of time allocation considering hiring
domestic helpers is developed and successfully
applied calibrated by a large empirical data set.
The model is tested by analyzing the impacts of
wage rate change on time allocation to various
activities and the time of hiring domestic helper.
Future extension of the model should consider
other types of external helps, such as that from
members of the extended family.