Household Members’ Time Allocation to Daily Activities and Decision to Hire Domestic Helpers

Donggen WANG and Jiukun LIDepartment of GeographyHong Kong Baptist UniversityKowloon Tong, KowloonHong 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 modelsThe 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 ObjectivesDevelop a model to analyze time

allocation of household members taking into consideration of hiring domestic helpers.

Calibrate the model by empirical dataApply the model for analyzing the

impacts of wage rate changes on time allocation

Notations is the total time available for member i; is the time spent for activity j by

member i. is the amount of time that a domestic

helper is hired. is the baseline utility for time invested in

activity j by household member i . is the baseline utility for hiring domestic

helpers is the wage rate of household member i,

is the wage rate of domestic helpers.

iTjiT

G

G

jiu

i maidq

Utility Functions of Household Members and

Domestic HelpersWe define the utility function of a

household member as follows:

And the utility function of hiring a domestic helper is:

ln 1j ji i i

j

u u T

ln 1 ln 1

wi i maid

iG G w

i ii

T Gu G G

T

A Model of Time Allocation Households are assumed to maximize the household

utility:

(1)

Subject to: (a) (b) Where and are respectively weights of

household members and domestic helpers, representing their relative contributions to household utility.

ln 1 ln 1

i i G Gi

wi i maid

j j ii i i G w

i j i ii

Maximize HUF wu w u

T Gw u T w G

T

w m ri i i iT T T T

mi

i

T G A

Gw, 1, ,iw i n

A Model of Time Allocation Lagrangian function

where are respectively the Lagrangian multipliers associated with the time constraint for member and that associated with the time constraint of maintenance.

ln 1 ln 1

wi i maid

j j ii i i G w

i j i ii

j mi i i i

i j i

T GL w u T w G

T

T T T G A

i and

A Model of Time AllocationThe first-order conditions of the

Lagrangian function

**

2**

1 ln 1 01

w i maidi i G iw

i wi i

i

Gwu w G

TT

*

1 01

mi i im

i

wuT

*

1 01

ri i ir

i

wuT

*

** * * ln 1 0

1

wi i maid

G i G maidw w

i i i ii i

T Gw w

GG T T

From the first-order conditions, we may derive the following interrelations among time allocation to activities and time of hiring domestic helper:

(2)

(3)

With constraints (a) and (b)

*

*2* *

*

ln 11 1

w ri i maid i

i G iw ri iw

i ii

u G uw w G wT T

T

* *

** * * * *

1 ln 11 1 1

wm ri i maidi i maid i

i G G im w w ri i i i i i

i i

T Gu u

w w w G wT G T T T

A Model of Time Allocation

Specification of the baseline utility function

In general the utility function can be expressed as follows:

(4)

Where are error terms, is the systematic component of utility and assume that it is continuously differentiable

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

,j j j j j j ji i i i i i iu v f x

, , ,ji j w m r ,j j j j

i i i iv f x

Model Estimation

Substituting equation (4) into (2)and into (3), we got the following equation

* *1 1

r r w wi i i i i

i r wG i i

w v vw T T

* *1 1

m m r ri i i i i

i m rG i i

w v vw T T

Model EstimationWhere

*

*2

*

ln 1i maidi

wi i

i

G G

T

* *

** * *

1 ln 11

wi i maid

i maidi w w

i i i ii i

T GG

G T T

Model EstimationAssume that error term

are independently and identically standard normal distributed

we add the following auxiliary random variable:

based on the property of normally distributed variables, Then the density function of the random vector may be derived as follows:

, , ,ji j w m r

, 1, ,ri i i n

, ,i i i

Model Estimation

• Where

2 2* *

2* ** *

1 11 1

1 12

3

1, , ,2

w mw mi i

i i i i i i i i i i ir ri i

T Ta w T b w T

T T

i i i if e J

Gi

i

ww

w

*

*

11

wr wi

i i iri

Ta v vT

*

*

11

mr mi

i i iri

Tb v vT

**

*

*2* * *

*

11 0

1

10 1 1 11

0 0 1

w w w wwi i i i

i i ri i i i

m m m mm w mi i i i

i i i i i iri i i i

r r ri i i

i i i

Tw T

T

TJ w T w T TT

Model Estimation• Therefore the density function of the random

vector can be derived as follows:

• Where

,i i

2 2* *2* *

* *

2 2* * 2

1 11 1

1 12 * *2

3

1 1* *2 2 4

3

1, 1 1 d2

1 1

2

w mw mi i

i i i i i i i i i i ir ri i

w mi i i i i i i i i

i

T Ta w T b w T

T Tw m

i i i i i i i

a w T b w T dw mi i c

i

i

f e w T T

T Tw e

c

2 2* *

* *

1 11 12 1 1

w mi i

i r ri i

T Tc

T T

* *

* ** *

1 11 11 1

w mw mi i

i i i i i i i i ir ri i

T Td a w T b w TT T

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:

Where The sequential number theoretic optimization (SNTO) algorithm can be used to solve

the likelihood function to find the maximum likelihood estimation of parameters . SNTO searches for the global optimum among points uniformly scattered in the search space (Fang and Wang, 1994).

, , , ln ,n

w m ri i i i i i i

d i

L w f

1 , , , 1, , , , ,j j ji i iH i n j w m r

andji iw

An Empirical StudyHong Kong: A special

Administrative region of ChinaArea size: 1100 square kilometersHigh 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 StudyVariable 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 dollarsWorking time: 10 hour/day, Wage rate: 20 HKD/hour

An Empirical StudySocio-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: exp , 1,2, , ,j j ji ih ih

h

v x i j w m r

Parameter Estimation Estimation of parameters for subsistence activity

Male head Female head

Variables Coefficients t Coefficients t

Constant -0.8847 -7.3371 -0.8580 -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

Presence of child aged 6 to 17 0.0233 3.4570 -0.5304 -4.1895

Car ownership -0.2778 -5.6680 -0.3441 -3.4894

Parameter EstimationEstimation of parameters for maintenance activity

Male head Female head

Variables Coefficients t Coefficients t

Constant 2.9225 5.8414 -2.4992 -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 1.0656

Presence of child aged 5 or lower 0.3654 3.2747 -0.2371 -2.2714

Presence of child aged 6 to 17 0.7821 4.2799 0.1066 1.4720

Car ownership -0.0394 -1.0243 -0.3398 -2.6312

Parameter Estimation Estimation of parameters for recreation activity

iw

Male head Female head

Variables Coefficients 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 -7.1099 -0.1273 -2.9048

Presence of child aged 5 or lower

0.5436 6.3625 -0.1447 -2.6463

Presence of child aged 6 to 17

-0.4085 -6.3211 0.2107 3.2663

Car ownership -0.8141 -6.7432 0.7342 4.2472

1.8011 59.0389 0.8606 4.6398

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.

Subject to:

ln 1 ln 1

i i G Gi

wi i maid

j j ii i i G w

i j i ii

Maximize HUF wu w u

T Gw u T w G

T

w m ri i i iT T T T m

ii

T G A 0, 0jiT G

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)

1 216, 16, 8T T A 1 265, 33, 20maid

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

- 60

- 50- 40- 30- 20- 10

010203040

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100The wage rate of domest i c hel per

Perc

enta

ge c

hang

e of

tim

e al

loca

tion

subsi stence of mal e head mai ntenance of mal e headrecreat i on of mal e head subsi stence of f emal e headmai ntenance of f emal e head recreat i on of f emal e headhi r i ng t i me of domest i c hel per

Impact Analysis

- 20

- 10

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200The wage rate of mal e head

Perc

enta

ge c

hang

e of

tim

eal

loca

tion

subsi stence of mal e head mai ntenance of mal e headrecreat i on of mal e head subsi stence of f emal e headmai ntenance of f emal e head recreat i on of f emal e headhi r i ng t i me of domest i c hel per

Impact Analysis

- 10

- 5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200The wage rate of f emal e head

Perc

enta

ge c

hang

e of

tim

eal

loca

tion

subsi stence of mal e head mai ntenance of mal e headrecreat i on of mal e head subsi stence of f emal e headmai ntenance of f emal e head recreat i on of f emal e headhi r i ng t i me of domest i c hel per

ConclusionsA 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.