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Economic Research Institute Study Papers Economics and Finance

1994

An Application of the Linear Expenditure Systems to the Pattern

of Consumer Behavior in Taiwan
Tsangyao Chang
Utah State University

Chris Fawson
Utah State University

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Chang, Tsangyao and Fawson, Chris, "An Application of the Linear Expenditure Systems to the Pattern of
Consumer Behavior in Taiwan" (1994). Economic Research Institute Study Papers. Paper 37.
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 May 1994 ERI Study Paper #94-07

AN APPLICATION OF THE LINEAR EXPENDITURE SYSTEMS

TO THE PATTERN OF CONSUMER BEHAVIOR IN TAIWAN

By

Tsangyao Chang
Chris Fawson
 An Application of the Linear Expenditure Systems
to the Pattern of Consumer Behavior in Taiwan

Tsangyao Chang and Chris Fawson

Department of Economics
Utah State University
Logan, Utah 84322

(801) 797-2296

(801) 797-2701 FAX

ABSTRACT

This study reveals certain systematic tendencies in consumer behavior in Taiwan from 1951 to
1990. The method of systems estimation and precision estimates revealed by significant t statistics
and higher R2 make the system of equations a useful tool to characterize broad tendencies in
individual allocation of expenditure behavior. This analysis has important implications for
government production strategies relating to price and subsidy policies and promotion of reasonable
standards of social welfare and health. In general, this study represents our attempt to make a
systematic estimation of consumer demand to identify the broad tendencies of individual demand
behavior in Taiwan.

Ph.d candidate and Assistant Professor, respectively. Department of Economics, Utah State
University. All correspondence concering this paper should be addressed to Chriss Fawson at the
above address.
 I. Introduction

The economy of the Republic of China (ROC) on Taiwan has been undergoing rapid
transformation for the past four decades. The role of consumer behavior in this process of
transformation is extremely crucial. Any changing pattern of demand might have serious
implications for trade and growth strategies in development. The role of prices and appropriate
pricing policies have been given considerable weight in the setting of strategies for economic
development in Taiwan as well as other developing countries. This change motivates an
examination of the structure of demand. We know that commodity composition of the consumers'
basket varies with their income and prices of goods. Thus, it follows that pragmatic policy
decisions, especially with respect to tax and subsidy reforms, require knowledge of price and
expenditure elasticities. The Taiwan (ROC) government in both the past and the present has
pursued various programs in the form of price and income policies to promote consumer welfare
among other objectives of development. In the absence of any systematic study of consumer
behavior, these policies could not be based on any flrtn estimates of price and expenditure
elasticities.
In this paper we attempt to provide a systematic estimate of aggregate consumer-demand behavior
for Taiwan consumers. Elasticity estimates are based on an econometric analysis of time series
budget survey data available for Taiwan from 1951-1990. All of the consumer decisions are
modeled simultaneously using a Linear Expenditure System (LES). This model was originally
developed by Richard Stone as early as 1954. Later it has been shown by Stone-Geary that this
demand system can be derived from the Klein-Rubin utility index of cost of living. According to
Parks (1969), the LES has the defect that for certain values of prices, and income the predicted
expenditure becomes negative. While this is clearly not satisfactory from a theoretical point of
view, the system may still be useful for a rather wide range of price-income points.
The model that we use here specifies seven major categories of expenditures. The specific
categories have been chosen to emphasize the role of basic needs in Taiwan. According to
literature, the LES approach based on time-series data has been extensively used for developed
countries since Stone (1954) first formulated a preliminary version of LES in 1954 and then used
an improved version of LES with British data in 1964. Nevertheless, the application of LES for
developing countries is rather limited owing to a lack of continuous cross-section or long time-
series data. l Previous LES studies for such countries have been based on single-year cross-section
data by making a priori assumptions about at least one of the parameters in the system. Needless
to say, th~se elasticity estimates are unduly restrictive as a result of the arbitrary assumption u~ed

l.Lluch and Powell (1975) (henceforth L-P), in their studies "International Comparisons of
Expenditure Patterns", used time-series data to analyze 19 countries' consumption patterns.
The countries include developed, developing, and less-developed countries. However, the
length of individual time-series ranges from 7 years in the case of Chile to 20 years in the
case of the U.K. Taiwan (1955, 58-68) was also included in their studies.
The commodity groups they picked are the same as what we did in this study .
 in deriving the elasticity parameters. 2 In contrast, we use annual data from the "Statistical
Yearbook of the Republic of China" published by Directorate-General of Budget, Accounting and
Statistics, Executive Yuan, Republic of China (1975, 1985, and 1991). As a result, elasticity
estimates for Taiwan consumers are based on annual estimates of consumption behavior. The paper
is organized as follows. In section II the formulation and estimation of the system demand
equations are discussed. Section III describes the data used for the estimation of the model.
Section IV contains empirical results and section V summarize our research results.

II. Model

The methodology used here in determining the parameters of the demand system is derived from
the neoclassical theory of consumer choice. According to Pollak (1971), this methodology
postulates that a representative consumer's utility function is an additively separable function of
the form U(Xl' x2, ••• , ~) and can be represented, after a monotonic transformation, as the sum of
a set of individual partial utility functions that takes the following form:
n
(1)
U= L P;ln(x;-y;)
;=1

where X; = consumption of the ith good, Pi and Yi are parameters of the utility function with Pi
>- 0 and Xi>- Yi' and the requirement that L Pi= 1 ensures that the preference structure implied by
(1) is well behaved. We can fmd the behavioral relations by maximizing U subject to a budget
constraint. The budget constraint is the total expenditure on all commodities, which takes the
following form:
n
(2)
LPiX;=M
;=1

where Pi = price of the ith good, and M = total expenditure on n goods. We can solve the
constrained maximization to give us a system of behavioral demand equations of the following
form:

2.Deaton (1988), has developed and implemented an appropriate methodology to utilize

information from a single-year cross-section budget survey that permits determination of
price elasticities by utilizing information on the spatial distribution of prices.
 n

P,Xi=PiYi+Pi(M- L PjYj) (3)

j=l

for i = 1, 2, 3, ... n

The adding-up condition demands that parameters Pi are all positive and that the parameters add
up to one to satisfy the budget constraint. Since Pi is positive, the regUlarity condition (that is that
the implied Slutsky substitution matrix is symmetric and negative semidefinite [see Yoshihara
(1969)]) will hold only if (Xi - 'YJ and (Pi - 1) are of opposite sign for every i [see Parks (1969)
and Brown and Deaton (1972)]. This can only be true if~ >- 'Yi' Hence, the regularity condition
suggests that ~ >- 'Yi' A positive Pi implies that the marginal utility of good i is decreasing [see
Goldberger (1987)]. Thus these demand functions satisfy the condition of homogeneity in prices
and total expenditure, the adding-up restriction (that the estimated or predicted expenditure for the
different commodities equal total expenditure in any period), and the regularity condition, which
implies quasi concavity of the utility function.
The parameter 'Yi is called the subsistence parameter. This implies that the individual first
purchases 'Yi units of good i at a cost of Pi'Yi which is called "committed" or "subsistence"
consumption. The total cost of subsistence is LPj'Yj' This leaves M - LPj'Yj as "supernumerary
expenditure". The Pi'S denote how a consumer allocates his supernumerary expenditure over
different commodities. The parameter 'Yi may be positive or negative. Positive 'Yi implies inelastic
demand and negative 'Yi implies elastic demand [see Pollak and Wales (1969)]. The subsistence
parameters 'Yi'S are no longer valid when some of them are negative, hence it is not appropriate to
regard the intercept terms as "subsistence" quantities unless they are assumed to be positive.)
However, if we restrict the 'Yi'S to being positive, then all commodities would be price elastic, a
condition that would not be a realistic assumption empirically. Therefore, I let the signs of the 'Yi
be detennined empirically.
We also can calculate the (noncompensated) own price, cross price, and expenditure elasticities
of commodity i from (3).4 They can be expressed as follows:
J

3.According to Pollak (1971), if the ,),/s are all negative it makes no sense to describe the
individual as purchasing a necessary (negative) collection of goods and dividing his
supernumerary income in constant proportions ({3h {32'" .(3n) among the goods.

4. Compensated Own Price Elasticity and Compensated Cross Price Elasticity can be
calculated as follows:
 ax.p. Y.(1 - P.)
_ "--1+'
11··---- I
u ap.r. x I. (4)
(~,

ax.p. P.y.
n =_'_J=_A~
'Iij ap
Pi P (5)
ri Xi i

(6)

Given that 0 -< Pi -< 1, when Yi is positive, the absolute value of the own price elasticity of
commodity i will always be smaller than one, so commodity i will be price inelastic. Similarly,
when Yi is negative, the absolute value of the own price elasticity will always be greater than one;
hence commodity i will be price elastic. Clearly from (5), we can see.that when commodity j is
price elastic, the cross price elasticity is positive. When the Yi' s are positive, the cross price effects
are negative, a fact implying that the income effects are stronger than the substitution effects.
From (6), we fmd that expenditure elasticity can be obtained on the basis of the equation Wi =
Pi~lM, the budget share of commodity i, and an estimate of the marginal expenditure share
parameter Pi. We can combine them with the additivity restriction of LPi =1 and find that
LlliWi=l, which is a very reasonable restriction stating that the sum of the expenditure elasticities
weighted by budget shares should be equal to one. However, the assumption that Pi >- 0 rules out
the possibility of having inferior goods in the model [see Parks (1969)].
The demand system from (3) has a total of 2n structural parameters, of which 2n-l are
independent parameters in view of the adding-up restriction. The independent parameters are ~-l
P's and ny's. In order to specify the stochastic form of the equation, a disturbance tenn Gi was

y. p.P.
1') .. =(p. -1)(1-~), 1') .. = -(-'_J)(xJ-y.)
n I X IJ P J
i iti
 introduced into each share equation.
n

P,Xi=PiYi+Pi(M- L PjYj)+£i (7)

j=l

for i = 1, 2, 3, ... n
We assume that the e's follow a nonnal distribution with a mean of zero and a variance-
covariance matrix Q for all i. Since these LES equations are highly interrelated, a systematic
approach was used to estimate the parameters of the model. The estimates are obtained by utilizing
Zellner's (1962) method for "iterative seemingly unrelated regression".s

III. Data
We use annual Taiwan time series data for 1951-1990 to estimate the LES model. The primary
source for these data series is the "Statistical Yearbook of the Republic of China" published by
Directorate-General of Budget, Accounting and Statistics, Executive Yuan, Republic of China
(1975, 1985, and 1991). In this study, seven commodity groups are used to estimate the model.
They are as follows :
Group (1): Food, beverage & tobacco.
Group (2): Clothing & footwear.
Group (3): Gross rent, fuel & power, furniture, furnishing & household equipment & operations.
Group (4): Medical care & health expense.
Group (5): Transport & communication.
Group (6): Recreation, entertainment, edu.& cultural services.
Group (7): Miscellaneous goods & services.
The series on expenditure at current prices were deflated by end-year population estimates to
convert them into per capita figures. 6 Following Lluch and Powell (1975) and Hassan and

5. Since the expenditure shares of the seven different groups of commodities always add up to
one, the sum of the disturbances across the seven equations is zero at each observation. This
implies that the covariance matrix of the disturbance tenn will be singular. In order to
ensure a nonsingular covariance matrix, we can drop anyone equation from the system and
then minimize the residual covariance matrix with respect to the parameters and obtain
maximum likelihood estimates of the system.

6.Average per capita income for the past four decades in Taiwan are $ 50,931 National
Taiwan dollars and the average total expenditure on seven commodity groups are $ 27,100
 Johnson (1980), the price series on the groups were obtained by dividing expenditure in current
dollars by expenditure in constant dollars and are called implicit deflators which means that the
weights ofthe various components shift with the changing composition of commodities or services
over time.

IV. Empirical ~esults

The parameter estimates of the Linear Expenditure System are reported in Table 1 along with the
values of the t statistics and R2 coefficient. The t values for all of the estimates are statistically
significant as shown in Table I. The regularity conditions have been satisfied at all observations.
The R2 values for the individual equations have been reported and are all particularly significant
because the data are time series and a large number of parameters are used in the equations [see
Pollak and Wales (1969)]. From Table 1, we see that the negative minimum or committed
expenditure Yi in the LES system can only be measured for commodity group (5) [Transport &
communication] alone. In other words, this is the only commodity expenditure that is price elastic
in the individual budget. This is different from L-P's (1975) fmdings (see Table 2). One
justification for L-P's results may be that their chosen sample periods (1955, 58-68) were based
on data from Taiwan's early development period. During that period, almost all of the transport
& communication were controlled (monopoly) by the government, people had less choice than what
they might have right now.? The Yi values for all other commodities are positive, implying that
they are price inelastic commodities. As expected, the expenditure level for commodity group (1)
[Food, beverage & tobacco] is higher than that for other commodity groups. This result is the
same as those reported in L-P's findings. The parameters ~i'S shown in Table 1 indicate that the
commodity group (3) [such as gross rent, fuel & power, furniture, furnishing & household
equipment & operations] had the highest estimated marginal budget share of 0.237. The second
highest marginal budget share, 0.212, was the commodity group (1) [Food, beverage & tobacco V

National Taiwan dollars. This means that about 53.21 percent of total income spent on these
seven commodity groups. For the past four decades, the average saving rates are 25.5
percent.

7. During this time, Taiwan was still under less-developed status. According to Chang
(1991), Taiwan has ranked as one of the developing countries since 1971.

8.From Table 2, we can see that the marginal budget share for commodity group (1) (Food,
beverage & tobacco) ({31), the L-P's fIDdings ranked as the highest one compared with my
findings is the second high. The commodity group (3) (Gross rent, fuel & power, furniture,
furnishing & household equipment & operations) ({33) jump to become the highest one in my
fIDdings compared with L-P's fIDdings is the second high.
 These results are similar to those found in Hassan and Johnson's (1980) study on U.S. consumer
demand pattern. The third highest is the commodity group (6) [Recreation, entertainment,
educational & cultural services]. Compared with the L-P's [mdings, this commodity group jump
from the lowest (~6=O.047) on L-P's findings to the third high (~6=O.188) on our [mdings. One
justification is that as Taiwan's economy has boomed, spare time for people in Taiwan has also
increased. According to a survey [see Taiwan Sports (1991)] on weekly time allocation, people
in Taiwan now enjoy an average of six hours and twenty minutes of spare time a day, which is
about a quarter of the day. Groups with the lowest marginal budget shares are commodity group
(2) [Clothing & footwear] and commodity group (4) [Medicare care & health expense]. One
reason for low marginal budget shares for these two commodity groups may be that Taiwanese
people are very conservative in their expenditures in these groups. Especially for commodity group
(4), medical care and health expense, since traditional Taiwanese don't like to see the doctors very
often, only when they are very ill. Sometimes they just go to the pharmacy buy some medicine
and take care of their own illness. 9 This low expenditure level for commodity group (4) may have
been the impetus behind a Legislative Branch proposal on Jan 17, 1990, to establish a ministry of
social welfare and health, one major goal of which is to institute a comprehensive national health
insurance program by 1994. The Six-Year National Development Plan (1991-1996) also promotes
this social welfare and health.
Table I:Estimated LES for Taiwanese Consumers. (1951-1990)

Com'modity Minimum required Marginal

Group quantities (Yi) budget share (I3J

Group (1) 14497 (41.764) 0.212 (58.369) 0.9989

Group (2) 957 (16.091) 0.050 (57.125) 0.9973
Group (3) 4238 (11.804) 0.237 (124.59) 0.9995
Group (4) 1080 (12.737) 0.049 (49.393) 0.9967
Group (5) -1003 (-3 .240) 0.179 (53 .649) 0.9933
Group (6) 986 (2.435) 0.188 (64.159) 0.9982
Group (7) 1087 (4.982) 0.085 (29.001) 0.9888
*The number in parenthesis denotes t-statistics.

9.ln Taiwan, people can go to the pharmacy to buy the medicine without any prescription
from doctor. This medical system is totally different from that of U. S.
Table 2: A comparison of L-P and our findings.

Commodity Pi 1i 11ii 11m

Group C-F L-P C-F L-P C-F L-P C-F L-P

Group (1) 0.212 0.327 14497 8280 -0.454 -0.438 0.554 0.577
Group (2) 0.050 0.068 957 427 -0.589 -0.397 1.030 1.230
Group (3) 0.237 0.244 4238 1754 -0.719 -0.534 1.114 1.340
Group (4) 0.0490.089 1080 587 -0.600 -0.493 1.046 1.551
Group (5) 0.1790.052 -1003 75 -1.163 -0.856 1.999 2.962
Group (6) 0.188 0.047 986 257 -0.896 -0.548 1.498 1.835
Group (7) 0.085 0.174 1087 1062 -0.754 -0.601 1.197 1.804
L-P denotes Lluch and Powell's findings.
C-F denotes our own findings.

Elasticity estimates for the LES model such as own price elasticities, both noncompensated and
compensated, cross-price elasticities, and expenditure elasticities have also been calculated on the
basis of the parameters estimates in Table 1 and presented in Table 3 and 4, respectively. All own
price elasticities have the right sign, and all compensated price elasticities are, of course, smaller
than those of the corresponding noncom pen sated price elasticities. The estimated own-price
elasticities indicate relative inelastic demand for all commodity groups, except commodity group
(5) [Transport & communication]. The higher price elasticity of commodity group (5) implies that
the consumption of this commodity is more sensitive to price changes. These results are similar
to those found on L-P's studies except commodity group (5) which is the only one commodity
group has a higher price elasticity (-1,163) in our findings (see Table 2). The estimated
commodity group expenditure elasticities for the LES model indicate that six of the seven
commodities are relative luxurious; that is, commodity group expenditure elasticities are greater
than one for commodity group (2)-(7), except commodity group (1) [Food, beverage & tobacco].
This confinns Engle's Law (decreasing share of expenditure on food when individual's income
goes up). The 11m is only 0.5544 which indicates that the commodity group (1) appear to be
necessities for Taiwanese. The elasticity of near 2 for commodity group (5) [Transport &
communication], implies that as the general level of income rises, demand for such good will
increase rapidly. The results of this analysis suggest that commodity groups such as (2)-(7) are
highly preferred items in the consumer's budget, and their consumption is fairly sensitive to
changes in income. These results may not be surprising for developing countries, where
commodIty group (2)-(7) are rather expensive commodities and may be viewed as relative
luxurious. These results also tend to confirm L-P's fmdings on their studies. However, the 11m
(estimated commodity group expenditure elasticities) are higher from those found by L-P's
fmdings. A justification may be that during the sample period chosen by L-P, the income of
Taiwanese was just taking off, so the income effect at this early period is much more stronger than
what it is right now. This means that as income increases, the demand for such commodities
increase more rapidly at the early period than at the recent period. The cross-price elasticities (l1 ij'
i:;t:j) with respect to the price of each commodity group are presented in Table 4. As we mentioned
before the positive T)ij's indicate that all goods are net substitutes and the negative 11 i/ s indicate
that income effects swamp substitution effects so that all goods are gross complements. Clearly
 from Table 4, the llij's are only positive for the cross-price elasticities of commodity group (1)-(7)
[except (5)] with respect to the price of commodity group (5).10 This indicates that they are net
substitutes. The rest of llij's are all negative which means that they are gross complements.
Table 3:LES Expenditure Elasticities, Own Price Elasticities for Taiwanese Consumers. (1951-1990)

Noncompensated Compensated
Expenditure Own Price Own Price
Commodity Group Elasticities Elasticities Elasticities

Group (1) 0.5544 -0.45395 -0.2232

Group (2) 1.0303 -0.58937 -0.5401
Group (3) 1.1138 -0.71883 -0.4819
Group (4) 1.0465 -0.60006 -0.5510
Group (5) 1.9995 -1.16284 -0.9838
Group (6) 1.4976 -0.89635 -0.7084
Group (7) 1.1974 -0.75358 -0.6960

Table 4:LES Elasticity Matrix for Seven Commodity Groups for Taiwanese Consumers. (1951-1990)

Commodity
Groups (I) (2) (3) (4) (5) (6) (7)

(I) -0.4539 -0.0001 -0.0435 -0.1092 0.04645 -0.0089 -0.0106

(2) -0.2730 -0.5893 -0.0808 -0.0203 0.0183 -0.0165 -0.0197
(3) -0.2951 -0.0233 -0.7188 -0.0219 0.0198 -0.0178 -0.0213
(4) -0.2773 -0.0219 -0.0821 -0.6000 0.0186 -0.0168 -0.0200
(5) -0.5298 -0.0419 -0.1568 -0.0394 -1.1628 0.0320 -0.0382
(6) -0.3968 -0.0314 -0.1174 -0.0295 0.0266 -0.0286 -0.0286
(7) -0.3173 -0.0251 -0.0939 -0.0236 0.0213 -0.0192 -0.7535
*Evaluated at mean expenditure proportion.

V. Conclusion

This study reveals certain systematic tendencies in consumer behavior in Taiwan from 1951 to
1990. The method of systems estimation and precision estimates revealed by significant t statistics
and higher R2 make the system of equations a useful tool to characterize broad tendencies in
individual allocation of expenditure behavior. Evidence from Taiwan aggregate budget data
suggests that the so-called subsistence parameter (Yi'S) are positive for all commodity groups,
except commodity group (5) [Transport & communication]. These imply that all commodity
groups are price inelastic, except commodity group (5) is price elastic. Given the pattern of
demand, the consumption on commodity group (5) will be growing over time; hence lower quality
of these goods produced and supplied at home would entail larger imports of these goods. The
relative higher price elasticities of commodity group (5) also indicate that price interventions for

10. Since the commodity group (5) are price elastic (because the so-called subsistence
parameter is negative), the cross-price elasticity with respect to this commodity is positive.
these products can have a significant impact on consumption, at least in the short run. So it is very
important for the policymakers to consider the effects of price and subsidy policies on these goods.
However, the higher expenditure elasticities of commodity group (5) [11m=1.999] indicate that this
situation might improve automatically as income increases. The low expenditure share (or level)
of commodity group (4) [Medical care & health expense] is also a very important issue that the
policymakers should consider about (This might be the reason that one of the goals of Six-Year
National Development Plan is to promote the social welfare and health of Taiwanese). In sum, our
study produces important insights into consumer behavior in Taiwan. This analysis also has
important implications for government production strategies relating to price and subsidy policies
and promotion of reasonable standards of social welfare and health. In general, this study
represents our attempt to make a systematic estimation of consumer demand to identify the broad
tendencies of individual demand behavior in Taiwan. However, we don't take into account the
income-distribution aspects of consumer behavior. A logical and further extension of this study
would be to test the linearity assumption ofthe Engle curve by specifying an appropriate functional
form where marginal expenditure shares vary with income, and then to study the responses of
prices and expenditure elasticities with respect to the distribution of income in Taiwan.
 Reference
Brown, A., and Deaton, A., "Surveys in Applied Economics:Models of Consumer Behavior,"
Economic Journal, Dec 1972, 1145-1236.
Chang, TV., "Economic Transfonnation In Republic of China on Taiwan," Unpublished
Master's Thesis, East Tennessee State University, 1991.
Council for Economic Planning and Development, Executive Yuan, Republic of China., "The
Six-Year National Development Plan for Taiwan, Republic of China (1991-1196)," Jan
1991.
Deaton, A. S. "The Measurement of Income and Price Elasticities," European Economic
Review, 1975, 261-273.
Deaton, A. S., and Muellbauer, John., "Economics and Consumer Behavior," Cambridge
University Press, 1980. 64-66.
Directorate-General of Budget, Accounting and Statistics, Executive Yuan, Republic of China.,
"Statistical Yearbook of the Republic of China," 1975, 1985, and 1991.
Directorate-General of Budget, Accounting and Statistics, Executive Yuan, Republic of China.
"Taiwan Sports," 1991.
Goldberger, A. S., "Functional Fonn & Utility: A Review of Consumer Demand Theory," 1987
by Westview Press, 43-51, 59, 63-68.
Hassan, Z. A., and Johnson, S. R., "Consumer Demand Parameters for the US:A Comparison
of Linear Expenditures, Rotterdam, and Double-Log Estimates," Quarterly Review of
Economics and Business, 1980, 77-91.
Klein, L. R., and Rubin, H., "A Constant Utility-index of the Cost of Living," Review of
Economic Studies, 1949, 84-87.
Klevmarken, N. A., "A Comparative Study of Complete Systems of Demand Functions,"
Journal of Econometrics, 1979, 165-191.
Lahiri, Supriya., "A Redefinition of Luxuries, Necessities, and Engle Goods:An analysis of
Egyptian Household Budget Data," The Journal of Developing Areas, 25, Oct, 1990, 49-68.
LIuch, C., and Powell, A., "International Comparisons of Expenditure Patterns," European
Economic Review, 1975, 275-303.
Park, Richard W., "Systems of Demand Equations:An Empirical Comparison of Alternative
Functional Fonns," Econometrica, Oct 1969, 629-650.
Pollak, R A., "Additive Utility Functions and Linear Engle curves," Review of Economic
Studies, 1971,401- 414.
Pollak, R, A., and Wales T J., "Estimation of Complete Demand Systems for Household
Budget Data:The Linear and Quadratic Expenditure System," American Economic Review, June
1978, 348-359.
Pollak, R A., and Wales T J., "Estimation of the Linear Expenditure System," Econometrica,
Oct 1969,611-627.
Stone, R., "Linear Expenditure Systems and Demand Analysis:An Application to the Pattern
of British Demand," 1954, Economic Journal, 511-527.
Stone, R., "Models for Demand Projections in Essays in Econometrica and Planning," C.R. Rao
(ed), Pergamon, Oxford, 1964.
Yoshihara, Kunio., "Demand Functions:An Application to the Japanese Expenditure Pattern,"
Econometrica, April 1969, 257- 374.