Seven systems of Engel curves for expenditures on ten commodity groups were estimated using Taiwanese household expenditure data for the period from 1996 through 1998. Results show that the estimated expenditure elasticities are insensitive to the choice of functional forms.Engel curve, Taiwan, Consumer/Household Economics,

Brown, Mark G.

Chung, Rebecca H.

Lee, Jonq-Ying

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
RESEARCH PAPER 2002-misc-1 
 
 
 
An Engel Curve Analysis of Household Expenditure in Taiwan: 
1996-98 
 
 
By 
 
Rebecca Chung, Jonq-Ying Lee, and Mark G. Brown 
 
 
 
 
 
FLORIDA DEPARTMENT OF CITRUS 
Economic and Market Research Department 
P.O. Box 110249 
Gainesville, Florida 32611-2049 USA 
Phone: 352-392-1874 
Fax: 352-392-8634 
Email: jonqying@ufl.edu 
 
 
www.floridajuice.com 
   2
 
 
 
 
 
 
An Engel Curve Analysis of Household Expenditure in Taiwan: 1996-98 
 
 
Rebecca H. Chung, Jonq-Ying Lee, and Mark G. Brown 
 
 
Abstract 
 
Seven systems of Engel curves for expenditures on ten commodity groups 
were estimated using Taiwanese household expenditure data for the period from 1996 
through 1998.  Results show that the estimated expenditure elasticities are insensitive 
to the choice of functional forms. 
 
 
 
 
 
 
 
 
 
 
 
 
 
_______________ 
Rebecca H. Chung is assistant professor in the Institute of Tropical Agriculture at the National 
Pingtung University of Science and Technology, Pingtung, Taiwan; Jonq-Ying Lee, and Mark G. Brown 
are senior economists at Florida Department of Citrus, Lakeland, Florida. 
 
Paper presented at 2002 AAEA annual meetings.An Engel Curve Analysis of Household Expenditure in Taiwan: 1996-98 
 
Introduction 
Since the pioneering work of Engel in 1857, the estimation of Engel curves and 
total expenditure elasticities has been an important part in family budget studies.  An 
Engel curve describes the relationship between a household's expenditure on a particular 
good and the total household expenditure; while the total expenditure elasticity of a good 
measures the percentage change in expenditure of the good of interest relative to a 
percentage change in total household expenditure. 
The traditional approach of estimating the total expenditure elasticities is to select 
an appropriate functional form for each good, estimate it and use the estimated 
parameters to obtain the elasticity at any level of total expenditure.  Early studies (Allen 
and Bowley, 1935; Prais and Houthakker, 1955) used the functional specifications that 
are inconsistent with the adding-up constraint.  As a consequence, there is a theoretical 
bias in the estimates.  In recent studies, there has been a considerable amount of emphasis 
on utility-based approaches that consider the Engel curve as a system of equations.  Such 
a system not only satisfies the adding-up conditions but also predicts non-negative 
component expenditures, and does not preclude saturation.  These models include the 
Working's model (Working, 1943; Leser, 1963), the addilog model (Houthakker, 1960a, 
1960b; Theil, 1965; Bewley, 1982), the almost ideal demand system or AIDS (Deaton 
and Muellbauer, 1980), and the generalized addilog model (Bewley, 1986). 
  A third approach is to derive expenditure elasticities from the estimated 
parameters of the Lorenz curve for total expenditure and concentration curves for 
individual items of expenditure (Iyengar, 1960; Kakwani, 1977, 1978; Tran-Nam & 
Podder, 1992; Podder & Tran-Nam, 1994).  This approach has been shown to be not only   2
better than the traditional approach in terms of goodness of fit and adding-up criteria but 
also more flexible in allowing Engel elasticities to vary non-monotonically with income 
or total expenditure.   
 
Model Specifications 
Numerous functional forms have been suggested and used as the basis for 
estimating Engel curves.  This study attempts to estimate expenditure elasticities based 
upon the last two approaches discussed in the introduction, and then a comparison of 
derived corresponding expenditure elasticity estimates from different models can be 
made.  We wish the estimated expenditure elasticities for Taiwan’s data could be used to 
facilitate policy decision-making processes. 
In general terms, for m households with similar characteristics and facing the 
same price for n consumer goods, the empirical Engel curves are 
(1)   xij = fi(xj),   i = 1, 2, . . . , n; and j = 1, 2, . . . , m. 
where xij is the expenditure by household j on the ith good, xj is the total expenditure by 
the jth household.  If the n consumer goods are mutually exclusive and exhaustive, then 
the adding-up condition implies 
(2)   xj = Σi xij,  j = 1, 2, . . . , m. 
  The set of Engel curves may be obtained from the linear expenditure system 
(Stone, 1954) or the Rotterdam model (Theil, 1965) is 
(3)   xij = αi +βixj + εij,   3
where αi and βi are parameters to be estimated, and εij is the disturbance term.  Adding-
up is assumed by imposing the restrictions Σiαi = 0 and Σiβi = 1.  The total expenditure 
elasticity for the ith good's expenditure is ηij = βi/wij, and wij = xij/xj, the jth household’s 
average budget share for the ith good. 
  Multiplying equation (3) by (1/xj), a set of Engel curves can also be derived: 
(4)   wij = αi + βi/xj + εij, 
note that the parameters and disturbance terms are redefined.  Adding-up is assured if 
Σiαi = 1 and Σiβi = 0.  In (4), the total expenditure elasticity is ηij = αi/wij. 
  Moreover, a set of Engel curves can be derived from the Almost Ideal Demand 
System (Deaton and Muellbauer, 1980) or sometimes called the Working model 
(Working, 1943; Leser, 1963) as 
(5)   wij = αi + βilnxj + εij. 
The adding-up restrictions are Σiαi = 1 and Σiβi = 0; and ηij = [αi + βi(1 + lnxj)]/wij. 
The specification “generalized addilog demand system” proposed by Bewley 
(1982) is derived from (5) and has the advantages of allowing for saturation levels as well 
as ensuring adding-up and non-negativity: 
(6)   ln(wij/wj*) = αi + βilnxj + εij,   4
where lnwj* = Σi lnwij/n, the adding-up restrictions are Σiαi = Σiβi = 0, and ηij = (βi + 1) - 
Σiwij/βi.  In addition, there are two variants of the Addilog model (Bewley, 1982), they 
are 
(7)   ln(wij/wj*) = αi + βixj + εij; and 
(8)   ln(wij/wj*) = αi + βi/xj + εij; 
the adding-up restrictions are similar to (6), and the total expenditure elasticities are ηij = 
1 + xj(βi - Σi βiwij) and ηij = 1 - (βi - Σiβiwij)/xj, respectively, for (7) and (8). 
An alternative to the direct estimation approach is to derive expenditure 
elasticities from concentration curves (Iyengar, 1960; Kakwani, 1977, 1978; Podder and 
Tran-Nam, 1994; Tran-Nam and Podder, 1992).  In this approach, it is assumed that the 
per capita household expenditure, X, is a continuous, non-negative random number.  Let 
f(X) be the probability density function of X, then one can define the following variables: 
  ∫ = =
x
dX X f x F p
0
) ( ) (   is the proportion of households having per capita total 
expenditure up to x; 
∫ =
x
dX X Xf q
0
) ( ) / 1 ( μ is cumulative proportion of X of those households, where μ 
(=E(X)) is the mean value of X; 
) (p L q =   represents the Lorenz function of X.  The Lorenz curve relates the 
proportion of total expenditure to the proportion of persons spending up to a given level 
of per capita total expenditure.   5
 Let  Xi (=gi(X)) denote per capita household expenditure on the ith good and Σi Xi 
= X.  Then  ∫ =
x
i i i dX X f X g q
0
) ( ) ( ) / 1 ( μ  represents the cumulative proportion of Xi by 
those households with per capita total expenditure up to x, where μi (= E(Xi)) is the mean 
value of Xi; and  ) (p C q i i =  represents  the concentration function of Xi.  That is, the 
concentration curve relates the proportion of total consumption of a specific good to the 
proportion of persons spending up to a given level of per capita total expenditure. 
  The first and second-order derivatives of the Lorenz and concentration functions 
are 
(9)  μ / / ) ( ) ( ' x p p L p L = ∂ ∂ = , 
(10)  i i i i x g p p C p C μ / ) ( / ) ( ) ( = ∂ ∂ = , 
(11)  )] ( /[ 1 / ) ( ) ( "
2 x f p p p L p L i μ = ∂ ∂ ∂ = , 
(12)  )]. ( /[ ) / ( / ) ( "
2 x f x g p p C p C i i i i μ ∂ ∂ = ∂ ∂ ∂ =  
Using (10) one can represent the Engel function of the ith good as 
(13) ) ( ' ) ( p C x g i i i μ = . 
Therefore, (13) is the implicit form of the Engel function expressed in terms of the 
concentration function of the ith good.  The following functional forms (Tran-Nam and 
Podder, 1992) was used to estimate the respective individual commodity concentration 
curves and Lorenz curve in the current study: 
 
(14) 
i i i u p
i e e p p C
) 1 ( ) (
− =
β α ; and 
u p e e p p L
) 1 ( ) (
− =
β α ,   6
where ui and u are the disturbance terms.  In general, the above functional form (10) 
satisfies Ci(0) = 0, Ci(1) = 1, and Ci(p) > 0.  With these particular functional forms in (14), 
the total expenditure elasticity of the ith commodity can be expressed as 
(15) 
] ) )[( (
) ]( ) [(
) ( 2
2
α β α β α
β α α β α
η
− + +
+ − +
=
p p
p p
x
i i
i i i
i . 
Note that by the construction of the model (14), adding-up condition may not be satisfied 
under the concentration curve approach. 
 
Data and Results 
  The above model specification (14) and six other models (3)-(8) by Stone (1954), 
Working (1943), Houthakker (1960), and Bewley (1986) were estimated using Taiwanese 
household expenditure survey data for the period from 1996 through 1998 (Executive 
Yuan, ROC).  The survey involved 1,920 households over the study period.  These 
households kept records of their expenditures and incomes for the entire year.  The daily 
expenditures were compiled into annual expenditures and the expenditures of ten 
commodity groups were calculated in this study.  These commodity groups are food, 
clothing, housing and rent, fuel and utilities, furniture and equipment, housekeeping, 
health and medicines, transportation and communication, education and recreation, and 
miscellaneous.  Among the 1,920 households, 11 households had zero expenditure on one 
or more of these ten commodity groups.  Therefore, the expenditure information for these 
11 households was not included in the analysis (Wales and Woodland, 1983; Bewley, 
1986). 
  Table 1 shows the sample’s average expenditure shares and the average 
household annual expenditures.  As shown in Table 1, with the average annual per   7
household total expenditure of NT$218,589, housing expenditure accounted for 26% of 
total expenditure and is the highest among the ten commodity groups studied.  Food 
expenditure ranked the second and accounted for 23% of total expenditure, which is 
followed by education, 15%; health and medicine and transportation, 8% each, and 
clothing, 5%.  Fuel and furniture accounted for 3%, individually.  Housekeeping 
expenditure accounted for only 2% of total expenditure. 
  Since household composition information was not available, per capita total 
expenditure and expenditures for the ten commodity groups were used in the estimation.  
The ordinary least squares method was used to estimate the parameters in (3) through (8) 
and (14).  All parameter estimates are statistically different from zero at α = 0.01 level.  
Table 2 shows the estimated expenditure elasticities calculated at sample means. 
Generally, the estimated expenditure elasticities are insensitive to the choice of 
functional form.  For example, regardless to the functional form used, the expenditure 
elasticities for food, housing and rent, fuel and utilities, and health and medicines are less 
than unity, while the expenditure elasticities for the rest commodity groups are greater 
than unity.  The broad conclusion from Table 2 are that “food”, “housing”, “fuel and 
utilities”, and “health and medicine” emerge as necessities; “clothing”, “transportation”, 
“education and recreation”, “furniture”, and “housekeeping” emerge as luxury 
expenditure groups. 
It is interesting to compare our estimated expenditure elasticities in Table 2 with 
those obtained by previous studies in other countries.  A comparison is made for four 
expenditure groups in Table 3.  Although it should be stressed that with the different time 
periods and countries (an Engel curve is usually deduced from a complete demand   8
system by aggregating the (fixed) price effects into the constant terms, however, the 
prices are not constant across different time periods and countries), estimation methods, 
and types of data involved, there are limits to how much may be drawn from such a 
comparison.  In addition, it should be noted that the estimates reported by Lluch and 
Powell (1975) and by Deaton and Muellbauer (1980) are based on complete demand 
systems, using time-series data rather than on Engel curves. 
In general, our elasticity estimates are of magnitudes similar to those estimated 
from Deaton and Muellbauer (1980) using the AIDS on UK data, i.e., “food” and 
“housing” are necessities, while “clothing” and “transportation” are luxuries.  Our 
estimates are also similar to those obtained by Giles and Hampton (1985) using New 
Zealand data; those estimated by Bewley (1982) using the Addilog model on Australian 
data; and the ones obtained by Lluch and Powell (1975) using the US data.  However, in 
these three previous studies, ‘housing” was found to have close to unity expenditure 
elasticities while the current study found “housing” to be a necessity.  
 
Concluding Remarks 
  In this study we have attempted to analyze some aspects of household expenditure 
behavior in Taiwan by estimating Engel curves on the basis of the 1996-98 household 
expenditure survey conducted by the Executive Yuan of the Taiwanese government.  In 
this study, we find that seven functional specifications each lead to very similar estimates 
of the expenditure elasticities for any given commodity group, at least at the sample mean. 
  In general, our estimated Engel curves appear to be quite well specified and the 
expenditure elasticities generally accord well with our prior expectations and with the   9
results of similar studies for other countries.  Expenditures on food, housing, fuel, and 
health emerge as necessities, while clothing, furniture, housekeeping, education and 
recreation, and transportation are luxury expenditure items.  
  Our work has been limited by the absence of detailed household composition and 
income information.  Given the required survey data, an allowance for differences in 
household composition would be an obvious and interesting extension of this current 
work.   10
 
Table 1.  Sample statistics 
Variable Mean  Maximum  Minimum 
  Expenditure Share 
Food  0.2280 0.6818 0.0305 
Clothing  0.0529 0.2923 0.0003 
Housing  0.2642 0.7315 0.0552 
Fuel 0.0303 0.1602 0.0057 
Furniture  0.0302 0.3237 0.0005 
Housekeeping 0.0221 0.3058 0.0007 
Health  0.0763 0.4173 0.0003 
Transportation 0.0759 0.5827 0.0061 
Education 0.1500 0.5218 0.0001 
Miscellaneous 0.0702 0.5268 0.0038 
     
  Total Expenditures 
NT$/household 218,589 2,388,633 48,123 
 
 
   11
Table 2.  Total expenditure elasticity estimates 
  Total Expenditure Elasticity Estimates 
(3) 
βi/wi 
(4) 
αi/wi 
(5) 
[αi+βi(1+lnx)]/
wi 
(6) 
(βi+1)-Σiβiwi 
(7) 
1-(βi-Σiβiwi)/x 
(8) 
1+x(βi-Σiβiwi) 
Concentration 
curve 
         
Food  0.4215 0.6371 0.5825 0.5853 0.6571 0.6752 0.5815 
Clothing  1.2002 1.3331 1.3737 1.4196 1.2107 1.1642 1.3227 
Housing  0.8606 0.7885 0.7560 0.7544 0.7645 0.8089 0.7189 
Fuel  0.3495 0.4420 0.3964 0.4246 0.4338 0.5025 0.4773 
Furniture  2.5978 1.4673 1.5957 1.5401 1.2930 1.2972 1.6214 
Housekeeping  3.2101 1.6973 1.8452 1.5614 1.2874 1.3399 1.9267 
Health  0.6793 0.9097 0.8801 0.8779 0.8065 0.8068 0.8370 
Transportation  1.5719 1.4249 1.5501 1.3903 1.1541 1.1660 1.5572 
Education  1.2514 1.3598 1.3755 1.5049 1.2963 1.1771 1.1918 
Miscellaneous  1.3436 1.4140 1.4642 1.4265 1.2081 1.1744 1.4927 
 
   12
 
Table 3.  Comparison of estimated total expenditure elasticities 
 
Country 
Taiwan  New 
Zealand
a  Australia
b  United 
Kingdom
c 
United 
States
d 
Food 
 Linear  (3) 
 Linear  (4) 
 W-L  (5) 
 Log-linear 
 Addilog  (6) 
 Addilog  (7) 
 Addilog  (8) 
 (13) 
 
0.42 
0.64 
0.58 
 
0.59 
0.66 
0.68 
0.58 
 
0.73 
0.83 
0.77 
 
0.76 
 
 
 
 
0.42 
 
 
 
0.21 
 
0.32 
Clothing  
 Linear  (3) 
 Linear  (4) 
 W-L  (5) 
 Log-Linear 
 Addilog  (6) 
 Addilog  (7) 
 Addilog  (8) 
 Conc.  (13) 
 
1.20 
1.33 
1.37 
 
1.42 
1.21 
1.16 
1.32
 
1.16 
1.09 
1.12 
 
1.11 
 
 
 
 
0.91 
 
 
 
2.00 
 
1.12 
Housing  
 Linear  (3) 
 Linear  (4) 
 W-L  (5) 
 Log-linear 
 Addilog  (6) 
 Addilog  (7) 
 Addilog  (8) 
 Conc.  (13) 
 
0.86 
0.79 
0.76 
 
0.75 
0.76 
0.81 
0.72 
 
1.05 
1.18 
1.11 
 
1.11 
 
 
 
 
1.16 
 
 
 
0.30 
 
1.02 
    
Transportation 
 Linear  (3) 
 Linear  (4) 
 W-L  (5) 
 Log-linear 
 Addilog  (6) 
 Addilog  (7) 
 Addilog  (8) 
 Conc.  (13) 
 
1.57 
1.42 
1.55 
 
1.39 
1.15 
1.17 
1.56 
 
1.29 
1.23 
1.23 
 
1.23 
 
 
 
 
1.22 
 
 
 
1.23 
 
1.37 
aRefers to Giles and Hampton (data period covers 1981-82) 
bRefers to Bewley (1982, data period covers 1966-68) 
cRefers to Deaton and Muellbauer (data period covers 1954-74) 
dRefers to Lluch and Powell (data period covers 1955-68)   13
 
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A New Approach to Estimating Engel Elasticities from Concentration Curves,&quot;

A New Method of Estimating Engel Elasticities,&quot;

Allocation Models,

An Almost Ideal Demand System,”

An Engel Curve Analysis of Household Expenditure in New Zealand,” Economic Record

Estimation of Consumer Demand Systems with Binding Non-Negativity Constraints,”

Family Expenditure,

Forms of Engel Functions,&quot;

Linear Expenditure Systems and Demand Analysis: an Application to the

On a Method of Computing Engel Elasticities from Concentration Curve,&quot;

On the Estimation of Engel Elasticities from Grouped Observations with Application to Indonesian Data,&quot;

On the Estimation of Total Expenditure Elasticities from Derived Engel Functions with Applications to

An Engel Curve Analysis of Household Expenditure in Taiwan: 1996-98

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