Income elasticity of demand within individual consumer groups and the level of income elasticity of the entire market demand

Income elasticity of demand within individual
consumer groups and the level of income elasticity
of the entire market demand
Příjmová pružnost poptávky v rámci jednotlivých spotřebitelských
skupin a úroveň příjmové elasticity celé tržní poptávky
P. SYROVÁTKA
Mendel University of Agriculture and Forestry, Brno, Czech Republic
Abstract: The paper is focused on the derivation of the mathematical relationship among the income-elasticity level of the
entire market demand and the income-elasticity values of the demand functions of the consumers’ groups buying on the
defined market. The determination of the mathematical term was based on the linearity of the relevant demand functions.
Under the linearity assumption, the income elasticity coefficient of the entire market demand equals the weighted sum
of the income-demand elasticities of the differentiated consumer groups buying on the given market. The weights in the
aggregation formula are defined as the related demand shares, i.e. as the proportions of the groups’ demands to the entire
market demand. The derived aggregation equation is quite held if no demand interactions (e.g. the snob or fashion effect)
are recorded among differentiated consumers’ groups. The derived formula was examined by using empirical data about the
consumer behaviour of Czech households in the market of meat and meat products (Czech Statistical Office). However, the
application potential of the achieved term for the income-elasticity aggregations is much broader within the consumer-be-
haviour analysis. In addition to the subject aggregations of the demand functions, we can also apply the derived formula for
the analysis and estimations of the income elasticities within the demand-object aggregations, i.e. the multistage analysis of
the income elasticity of consumer demand. Another possibility of the use of the aggregation equation is for the evaluations
and estimations of the income elasticity of the region-demand functions in relation to the subregions’ demands or reversely.
Key words: income elasticity, market demand, consumer groups, group’s demand, income-elasticity aggregation
Abstrakt: Příspěvek se zaměřil na vymezení matematického vztahu mezi příjmovou elasticitou tržní poptávky a hodnotami
příjmových elasticit u poptávkových funkcí jednotlivých spotřebitelských skupin, které se vyskytují na daném trhu. Určení
tohoto vtahu bylo prováděno za předpokladu lineárních aproximací jednotlivých poptávkových funkcí. Odvozený vztah byl
pak vyzkoušen na empirických datech z oblasti chování českých spotřebitelů na trhu s masem a masnými výrobky. Vedle od-
vození a aplikace zkoumaného vztahu jsou v tomto článku rovněž naznačeny některé další možnosti jeho využití při analýze
spotřebitelského chování. Při prováděné analýze bylo zjištěno, že za předpokladu linearity příslušných poptávkových vztahů
lze hodnotu koeficientu příjmové elasticity tržní poptávky určit z váženého součtu dílčích koeficientů příjmové elasticity
poptávky za jednotlivé spotřebitelské skupiny, které se nachází na daném trhu. Váhy v daném součtu jsou definovány jako
podíly příslušné úrovně dílčí poptávky na celkové tržní poptávce. Takto formulovaný vztah ovšem platí pouze v případě, že
mezi poptávkami jednotlivých spotřebitelských skupin neexistují vzájemné interakce, typu módní nebo snobský efekt ap.
Použití odvozeného vztahu je však v rámci analýzy spotřebitelského chování mnohem širší. Vedle agregace poptávkových
vztahů ve smyslu spotřebitelských subjektů je totiž stejně možné získanou rovnici použít při hodnocení příjmové pružnosti
při předmětové agregaci poptávkových vztahů, tedy vícestupňová analýza příjmové elasticity spotřebitelské poptávky. V rámci
zavedených předpokladů lze odvozenou rovnici využít také při hodnocení příjmové elasticity spotřebitelské poptávky po
určitém statku na úrovni určitého územního celku, který je tvořen menšími celky (regiony nebo subregiony).
Klíčová slova: příjmová elasticita, tržní poptávka, spotřebitelské skupiny, poptávka skupiny agregace příjmové elasticity
Supported by the Ministry of Education, Youth and Sports of the Czech Republic (Grant No. MSM 6215648904 – the-
matic direction No. 4 The development tendency of agribusiness, forming of segmented markets within commodity
chains and food networks in the process of integration, globalization and changes of agrarian policy).
412
AGRIC. ECON. – CZECH, 52, 2006 (9): 412–417

---

INTRODUCTION AND AIM OF PAPER
Q � A � B � p � C � m
(1)
The research of the income-demand elasticity gives
Where, Q denotes the total market demand for the
a lot of useful information. For instance, we can use goods, p is their market price and m– represents the
the given information for adjusting the economical y average level of incomes of the consumers buying
effective level of the household-income taxation, see the goods on the target market. With respect to the
Banks et al (1996). First and foremost, the knowledge linear definition of the market-demand model (1), the
of the income-elasticity level of consumer demands is coefficient of income elasticity (η) is given as:
quite essential for the correct analyses and estimations
of price elasticity of the relevant demand functions, it
m
� � C �
(2)
is obvious from the new approach to the construction
Q
of demand models, see Deaton, Muel bauer (1980)
or Pol ak, Wales (1992).
Further, let us suppose that the demand side on
Income elasticity of demand reactions is measured the target market is compound from k consumers’
by the means of the elasticity coefficients in percent- groups. All the consumers’ groups (1, 2, … k) pay
age terms, thus without regard to the original units. the same market price for the given goods (p), but
Due to this property of the elasticity coefficients, it the average income within these groups is different
is possible to compare the income-demand reactions m
–1, m–2, . . m–k as wel as the quantities of the groups’
in the varied consumption fields or among different demands (q1, q2, … qk). For the simulations of these
consumers, respectively among different consumers groups’ demands, the linear models (3-1), (3-2) …
groups. The second possibility of the comparison is (3-k) are sufficiently exact too:
particularly effective in socio-economical researches.
Within these researches, the coefficients of income 1
q � 1
a � 1
b � p � 1
c � 1
m
(3-1)
elasticity could be used for the numerical descrip-
tion of the consumption preferences within studied q � a �b � p � c �m
(3-2)
2
2
2
2
2
consumer subjects. The quantitative analyses and the …………………………
mutual comparisons of the preferences of consumer q � a �b � p � c �m
(3-k)
subjects are possible too, McDowel et al. (1997) or
k
k
k
k
k
Syrovátka (2001).
For the evaluations and estimations of income
Within the introduced system of the linear demand
elasticity of the consumer demands, the analysis of functions (3-1), (3-2) to (3-k), the coefficients of
relationships between the income-elasticity values income elasticities are defined as fol ows:
of the individual demand functions and the level of
income elasticity in their aggregate is very useful as
m

1
1
� � 1
c �
(4-1)
wel . The given relationship may be researched under
1
q
the aggregation by the consumption items or under
m
the aggregation by the consumer subjects. The paper
2
� � c �
(4-2)
2
2
q
was focused on the determination of the mathematical
2
………………
term between the income elasticity level of the entire
mk
market demand and income elasticities of demand �k � ck �
(4-k)
qk
functions of differentiated consumer groups, purchas-
ing on the given market. The formula was derived
If the individual demand functions of consum-
under the linearity assumptions of al the related de- ers’ groups are completely independent, i.e. there
mand functions. The defined formula was applied in are not any mutual relationships among the groups’
the field of consumer behaviour of Czech households demands, we can simply determine the model of the
on the market for meat and meat products.
entire market demand as fol ows:
Q( p, 1
m ,m2,...,mk ) � 1
q ( p, 1
m ) � q2(p,m2) � ...� qk (p, k
m )


METHODOLOGY – DERIVATIO
Q( N
p, 1
m ,m2,...,mk ) � 1
q ( p, 1
m ) � q2(p,m2) � ...� qk (p, k
m )
(5)
OF STUDIED RELATIONSHIPS
Associated with the introduced linear definition of
Let us suppose that the linear model (1) simulates the individual demand functions (3-1), (3-2) … (3-k),
market demand for certain normal (non-inferior) it is possible to write the market-demand model (5)
goods:
as:
AGRIC. ECON. – CZECH, 52, 2006 (9): 412–417
413

---

Q � [a � b � p � c �
1
1
1
1
m ] � [a � b � p � c �
2
2
2 m2] � ... � [ a � b � p � c �
k
k
k mk ] �

� [ 1
a � a2 � ... � ak ] �[ 1
b � 2
b � ... � k
b ]� p � [ 1
c � 1
m � c2 � m2 � ...� ck � k
m ]


(6)
With respect to the aim of this article and the initial coefficients of the income elasticity of the market
assumptions, the notation of the linear model (6) can demand (10-1), (10-2) to (10-k) may also be achieved
be rearranged into the equation (7):
from the coefficients (4-1), (4-2) to (4-k), i.e. from the
coefficients of the income elasticity of the demand
Q � A � B � p � c �
1 m �
1 c �
2 m �
2 ... � ck � mk
(7)
functions of differentiated consumers groups. If we
multiply the elasticity coefficients (4-1), (4-2) to (4-k)
where the partial intercepts (a1, a2, … ak) as wel as the by the related demand shares (q
partial price parameters (b
1/Q, q2/Q, …, qk/Q),
1, b2, … bk) were summed
then we obtain the coefficients of income elasticity
up. The market-demand model in the form (7) that at the level (10-1), (10-2) to (10-k):
reflects the different income-demand functions of
the consumers’ groups may also be obtained by the
1
q
1
m
1
q
1
� �
� 1
c �
�
� � )
1
(
(11-1)
substitution of the fol owing term:
Q
1
q
Q
m
q
m
q
1
m2
m

k
C �
�c �
�
2
2
2
�2 �
� c2 �
�
� (
� )
2
1
c �
2 ...�
�c
(8)

(11-2)
k
m
m
m
Q
q2 Q
……………………………
into the market-demand model (1). Substituting of
qk
mk qk
the found term (8) into the income elasticity coef- �k �
� ck �
�
� (
� k)
(11-k)
Q
qk Q
ficient (2), we achieve the decomposition formula
for the value of this elasticity coefficient (η) into k
Due to the introduced terms (11-1), (11-2) to
elasticity components:
(11-k), the derived equation for the income-elastic-
ity decomposition, respectively, aggregation (9) may

� � � )
1
( � (
� 2) � ... � (
� k)
(9)
consequential y be rewritten as:
The first component in the decomposition term
q1
q2
q

k
� � � �
� � �
�
1
2
...� �k �
(12)
(9) defines the elasticity of the entire market demand
Q
Q
Q
in response to the changes in the average level of
incomes of the 1st consumer group:
RESULTS AND DISCUSSION
m1 m
m
� )
1
( �c1� � �c
1
1�
(10-1)
m Q
Q
The relationship between the level of the income
The second component of the term (9) then mea- elasticity of the entire market demand and the values
sures the elasticity of the entire market demand with of the income elasticities of the demand functions of
respect to the changes in the average level of incomes consumers’ groups buying on the target market was
of the 2nd consumer group:
studied. In accordance with the above-described way,
the fol owing formula was obtained:

m2 m
m
(
� )
2 �c
q1
q2
q
2�
�
�c
2
�
(10-2)
k
m Q
2 Q
� � � �
� � �
�
1
2
... � �
Q
Q
k � Q
Analogical y, we can explicate the kth component of
the decomposition equation (9). Thus, the component
The achieved formula (12) defines that the income
k records the elasticity of the entire market demand elasticity level of the entire market demand equals
in relation to the changes in the level of the average to the weighted sum of the income elasticities of
income of the kth consumer group:
demand functions of the differentiated consumer
groups buying on the given market. The weights
mk m
m
in the sum (12) are defined as the related demand

�(k) � ck �
�
� c
k
�
(10-k)
k
shares, i.e. as the proportions of the groups’ demands
m
Q
Q
to the entire market demand. The aggregation term
With respect to the validity of the term (8), we can (12) was determined under the assumptions of the
natural y determine the coefficients (10-1), (10-2) to linearity of al related demand functions and no de-
(10-k) from the market demand model in the form (7) mand interactions among the differentiated consumer
by a routine method. Furthermore, the introduced groups. Thus, the non-linearity of demands and/or
414
AGRIC. ECON. – CZECH, 52, 2006 (9): 412–417

---

the demand interactions would lead to a difference rmit = average level of the real income with ith the households’
of the obtained results by the equation (12) and the
categories (consumers’ groups) at time t
real values of income elasticity of the studied demand t = time variable.
functions.
In accordance with the specified assumptions, the
The received values of the models’ parameters (Ai),
derived aggregation equation (12) may be used for (B
2
i), (Ci), the determination coefficients (ri ) and the
the estimations of the income elasticity level of the results of F-tests are il ustrated in Table 1.
studied market demand. For these estimations, we
Using the Engel’s demand models displayed in
need to know al the levels of income elasticities of Table 1, we can calculate the levels of real income
the differentiated groups’ demands and their demand elasticities of the investigated groups’ demands for
shares too. Under the analogical conditions, we can meat and meat products between 1994 and 1998
also use the term (12) to the determination of income (ηit). If we do not concentrate on the development
elasticity in the selected group’s demand. The sug- of income elasticity of the studied market demand
gested applications of the defined equation (12) were during the observed period, we can estimate the level
examined in the field of estimations of the income of its income elasticity on basis of the equation (12)
elasticity of the market demand of Czech households from the average values of ηit. These average levels
for meat and meat products. For this purpose, there of the income elasticity of differentiated groups’
the data and some results from the dissertation work, demands in the observed period (1994–1998) were
Syrovátka (1999), were used. In the dissertation work, determined in accordance with the formula for the
the regression models of the Engel’s demand were arithmetic mean:
developed and applied for the behaviour simulation
of four categories of households: employees (i = 1),
20
1

� �
�
i
��it
(i = 1, 2, 3, 4) (14)
farmers (i = 2), self-employed (i = 3) and pension-
20 t�1
ers (i = 4), i.e. for four consumer groups on the tar-
get market. The consumer behaviour of the studied
With respect to the suggested way of the income-
households’ categories was analysed using quarterly elasticity estimation of the studied market demand
data from the Czech Household Budget Survey for (without development of this elasticity coefficient),
the period from 1994 to 1998. The demand models of we also need to calculate the average levels of the
the Engel’s type with the introduced explicit dynamics related demand shares:
were based on the linear construction:
20
20 q
w � 1 �
i
�w � 1 �
it
� it

20
Q
(i = 1, 2, 3, 4) (15)
q
t�1
20 t�1 t
it = Ai + Bi × rmit + Ci × t

(i = 1, 2, 3, 4); (t = 1, 2, . ., 20)
(13)
The achieved values of η–i and w–i are displayed in
where
Table 2.
q
We input the calculated average values (η–
it = quantity of the quarterly purchase of meat and meat
i), (w
–i)
products by the ith households’ categories (consumers’
into the derived aggregation equation (12) and thus
groups) at time t
we determine the total average income elasticity of
Table 1. The Engel’s demand models for the consumers’ groups buying on the market for meat and meat products
Households’
Linear dynamic model
Statistical verification
categories
qit = Ai + Bi × rmit + Ci; (t = 1, 2, …, 20)
r2i
F-test
Employees
(i = 1)
q1t = –0.5125 + 1.0257 × 10–3 × rm1t – 0.1161 × t
0.6384
15.0068
Farmers
(i = 2)
q2t = –2.221 + 1.0447 × 10–3 × rm2t – 1.0537 × 10–2 × t
0.7644
27.5710
Self-employed
(i = 3)
q3t = –1.2468 + 1.0507 × 10–3 × rm3t – 0.1015 × t
0.6144
13.5417
Pensioners
(i = 4)
q4t = +13.6682 + 2.9278 × 10–4 × rm4t + 0.1125 × t
0.5564
10.6598
AGRIC. ECON. – CZECH, 52, 2006 (9): 412–417
415

---

Table 2. The average level of real income-demand elasticity in the derivation process of the studied aggregation
and the average level of demand shares between 1994 and term. In this case, we do not need to differentiate the
1998 within the investigated consumers’ groups
consumer incomes, because the system of n demand
functions of the only one consumer or one consumer
Households’ categories
η
–i
w–i
group is analysed. Thus, the income-elasticity coef-
Employees (i = 1)
1.1409
0.2327
ficients are given as:
Farmers (i = 2)
1.2406
0.1888
m
1
� � 1
c �
(17-1)
Self-employed (i = 3)
1.1938
0.2278
1
q
Pensioners (i = 4)
0.1796
0.3507

m
� � c �
(17-2)
2
2 q2
………………
the entire market demand of Czech households for
m
�n � cn �
(17-n)
meat and meat products (η–):
qn
� �� 1��
� � 1
w �
�
2
� �
w � � 2
w �
�
3
� �
w � � 3
w �
�
4
� �
w � � 4
w �
� w �
With respect to the eventual heterogeneous units

1
1
2
2
3
3
4
4
� 1.1409 � 0 2327
.
�1 2406
.
� 0.1888 �1 1938
.
� 0.2278 � 0
w
1796
.
�it0h.in t
3507he
� aggre
0.8346gation of demanded quantities, the
� � 1
� � 1
w � 2
� � 2
w � 3
� � 3
w � 4
� � 4
w �
� 1.1409 � 0 2327
.
�1 2406
.
� 0.1888 �1 1938
.
� 0.2278 � 0 1796
.
� 0.3507 � 0.8346
individual-demand functions and the aggregate de-
� 1.1409 � 0 2327
.
�1 2406
.
� 0.1888 �1 1938
.
� 0. 2278 � 0 1796
.
� 0.3507 � 0.8346
(16)
mand are investigated in the expenditure terms. The
expenditure analysis of the demand systems requires
Under the introduced assumptions of the linearity the initial transformation of the nominal expenditures
of al demand functions and no demand interactions and the incomes into their real levels. In this context,
among observed consumer groups, in accordance it is possible to bring in the aggregation equation in
with (16), it is possible to say that the average level another form:
of real income elasticity of the market demand for
meat and meat products in the observed time period
x1
x2
xk
1� � �
� � �
�
1
2
...� �
(18)
equals 0.8346. Thus between 1994 and 1998, the 1%
m
m
k � m
rise in the real incomes of Czech households brought
This aggregation formula (18) is based on the as-
the increase in their purchases of meat and meat sumption that the total expenditures for al consumed
products by approximately 0.83%.
goods (x
In addition to the above-mentioned application, the
1), (x2), …, (xn) are equal to the disposal income
of the given consumer subject (m), thus:
derived formula for the aggregation of the income
elasticity coefficients (12) is also useable for the es- m = x
timations of income elasticities of the demand func-
1 + x2 + … + xn = p1 × q1 + p2 × q2 + … + pn × qn
(19)
tions within the regions and their subregions. Thus,
In the theory of consumer’s behaviour, the equation
we can determine the income elasticity of the entire (18) is termed by the Engel’s aggregation condition
region demand in relation to the income elasticities (adding up) and it is thoroughly examined within the
of subregion-demand functions or, analogical y, we development of the theoretical consistent models of
can estimate income elasticity some of the subregion the demand systems. Pursuant to the Engel aggregation
demands on the basis of the relevant values of the condition, the average level of the income-demand
related coefficients of income elasticities. However, elasticity within the income-expenditure well-bal-
the accuracy of the elasticity estimations according anced consumer bundle (19) equals 1, see Maurice
to the equation (12) is also restricted by using the et al. (1998).
linear approximations of the real demand functions
and not taking into account the interactions among
individual demands, which are aggregated, see the CONCLUSION
initial assumptions of this derivation process.
Further, the obtained aggregation principle that is
Under the assumption of the linearity of all re-
defined in the equation (12) can be applied within lated demand functions, the coefficient of income
the object aggregation of demand functions of the elasticity of the entire market demand equals the
individual consumer or the consumer’s group. This weighted sum of the income-demand elasticities of
application of the equation (12) is very useful for the the differentiated consumer groups buying on the
two-stage or the multi-stage analysis of the consumer’s given market. The weights in the aggregation equa-
demand system or the group’s demand system, see tion are defined as the related demand shares, i.e. as
Moschini (2000). However, there are some differences the proportions of the groups’ demands to the entire
416
AGRIC. ECON. – CZECH, 52, 2006 (9): 412–417

---

market demand. The derived aggregation equation Deaton A., Muel bauer J. (1980): Economics and Con-
holds ful y if no demand interactions (e.g. the effect of
sumer Behaviour. Oxford University Press, New
snob and fashion consumption) are recorded among
York, USA, 464 p.; ISBN 0521296765.
the differentiated consumer groups. However, within Maurice S.CH., Phil ips O.R. (1992): Economic Analy-
the analysis of consumer’s behaviour, the application
sis, Theory and Application. 6th Edition. Irwin,
potential of the achieved term for the income-elastic-
Boston, 738 p.; ISBN 0-256-08209-X.
ity aggregations is much broader. In addition to the McDowel D.R., Al en-Smith J.E., McLean-Meyinsse
subject aggregations of the demand functions, we
P.E. (1997): Food expenditures and socioeconomic
can also apply the derived formula for the analysis
characteristics: focus on income class. Ameri-
and estimations of the income elasticities within
can Journal of Agricultural Economics, 79 (5):
the demand-object aggregations, i.e. the multistage
1444–1451.
analysis of the income elasticity of consumer demand. Moschini G. (2000): A Flexible Multistage Demand
Another possibility of the use of the aggregation
System Based on Indirect Separability. Working
equation is within the evaluations and estimations of
Paper 00-WP 265, December 2000. Available online
the income elasticity of the region demand functions
only: www.card.iastate.edu.
in relation to the subregion demands or reversely, Pol ak R.A., Wales T.J. (1992): Demand System Speci-
within the demand income-elasticity evaluations in
fication and Estimation. Oxford University Press,
the some subregion.
USA, 232 p.; ISBN 0195069412.
Syrovátka P. (1999): Vybrané aspekty chování spo-
třebitelů na trhu s masem a masnými výrobky.
REFERENCES
[Doktorská disertační práce.] PEF MZLU, Brno,
133 s.
Banks J., Blundel R., Lewbel A. (1996): Tax reform Syrovátka P. (2001): Preference analysis of meat and
and welfare measurement: Do we need demand
meat products purchases in selected households’
system estimation? Economic Journal, 106, (438):
categories. Agricultural Economics – Czech, 47
1227–1241; ISSN 0013-0133.
(11): 514–520.
Arrived on 13th June 2006
Contact address:
Pavel Syrovátka, Mendel University of Agriculture and Forestry, Zemědělská 1, 613 00 Brno, Czech Republic
e-mail: pavels@mendelu.cz
AGRIC. ECON. – CZECH, 52, 2006 (9): 412–417
417

---