Looking for another relative Engel’s law
Introduction
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The Engel law concerning food expenditures remains an
important tool: for instance for the definition of poverty, or to
discuss the convergence of consumption structures between
nations or social classes.
The relative Engel law which is discussed here is related to Social
Interactions (related to social positioning of the agents within
their reference groups), proved with minimal hypotheses.
It allows to evaluate the influence of income distribution on
consumption structures (an influence which a direct analysis is
not able to evaluate).
Looking for another relative Engel’s law
Introduction
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Engel's law is generally considered as being perfectly shown to
hold empirically, but without clear theoretical foundations.
Furthermore, its simplicity masks uncertainty about its real
meaning: for example, if needs are endogenous, especially with
respect to changes in income, then the intuitive grounds for the
law on the scarcity of goods are not clear.
Secondly, bias in estimating the law using survey data raises
problems about testing it empirically, usually done crosssectionally.
This test of Engel law on food consumption and of the
Duesenberry hypothesis on social interactions is based on an
aggregation of the endogeneity bias of cross-section estimates. It
involves no restriction on the specification of the relative income
effect.
Looking for another relative Engel’s law
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Engel's law is portrayed in the literature as a stable
and timeless relationship between income
changes and certain types of household
consumption: food, clothing, housing and leisure.
The time dimension of the law was put forward
immediately following Engel's work, and refers to
the smaller overall spending levels by rich nations,
that are assumed to follow a universal, historical
trend. It contrasts with the law's empirical proofs
which are always based on cross-section surveys
of household spending.
Engel puts forward his law in his original research :
“The poorer an individual, a family or a people,
the greater the percentage of its income
dedicated to physical upkeep, with spending on
food being the most important”.
Looking for another relative Engel’s law
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The law may generally be translated by the following
hypotheses:
1. A stable relationship exists between certain types of
consumption (spending on personal upkeep for Engel,
spending on clothes and housing, with a unity elasticity, as
well as spending on comfort goods for Wright) on the one
hand, and individuals' or households' standards of living on
the other hand (indeed, Engel was interested, in the first
place, in the link between individual and household
spending, using scales of equivalence).
2. The income-elasticities of these types of spending are ranked,
with spending on personal upkeep being lowest.
As Stigler has pointed out, these hypotheses constitute the
first theoretical generalisation in economics made on the
basis of individual budget data.
Looking for another relative Engel’s law:
Discussing the law
Three questions may be asked regarding the
law:
1. Are the population's needs given and
stable, or do they depend on the socioeconomic changes individuals may
experience? In particular, do they change
with household income?
Looking for another relative Engel’s law:
Discussing the law
2. Does Engel's hypothesis cover change in income and
spending over time (in which case the law would
hold over time and be longitudinal in explaining
consumer choices)? Or does the law just allow for
comparisons in consumption behaviour by
differentiated social groups (it would then be a
cross-sectional law, involving all the social
mechanisms which differentiate the choices of social
groups). The first case, the law would apply to a
society experiencing growth and development, a
typically 18th and 19th century concept. In the
second case, the law would correspond to the usual
tests, on cross-section data, of hypotheses relating to
ranked needs.
Looking for another relative Engel’s law:
Discussing the law
3. Should changes in living standards which
affect the whole of a social group, to
which the individual or household
belongs, be distinguished from personal
changes? In other words, is there a
relative dimension to Engel's and
Wright's laws which would prove the
existence of social interactions within
reference groups which are to be
defined?
1.The endogeneity of needs
To address these vast issues, I draw on two articles
(Gardes-Loisy, 1995; Gardes-Merrigan, JEBO 2007)
which examine empirically the development of French
and Canadian household needs, using surveys and
pseudo-panels of family budgets. The results, common
to both sets of data, indicate an income elasticity of
needs in the order of 0.5 and often more. These
levels hold both for comparisons between poor and rich
households within the same period, as for changes in
income over time. This strong dependence only
partially proves Easterlin's proposition, which assumes
that needs develop at the same pace as growth, thus
cancelling out any contribution to utility derived from
growth. But it demonstrates at least that needs are
endogenous to certain demand explanatory variables.
2.The endogeneity bias in crosssection estimations
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When the parameters estimated from cross-section data differ from
those estimated using time-series data, then there may be an
endogeneity bias in at least one of the two estimations: for example,
the income elasticities of consumer spending on food are about 0.2
for cross-section data and 0.4 for time-series data in the
United States, and respectively 0.5 and 0.8 for Poland. Thus, a
forecast based on a survey estimation will significantly underestimate
changes in food consumption in both countries.
The explanation may lie in the improved quality of food, as Engel
points out in his second article. A way of calculating virtual consumer
prices and income elasticities of these prices is presented in Gardes et
al. (JBES, 2005). Applying this method to American and Polish data
yields elasticities of food price over relative income that are
significantly positive, with levels that are highly comparable
between the two countries: about unity in the United States
and 0.7 for Poland. This income-effect on food prices may be
explained by time constraints, which can be assumed to increase with
household income.
Relative Income Elasticity of Food Expenditures
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PSID (U.S.)
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Period
Polish panel
1987-90
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Income Elasticity
CS
TS
CS
TS
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Food at home
Food away from home
0.19
1.00
0.38
0.39
0.49
1.22
0.76
0.36
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Direct Price Elasticity
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1984-87
-0.19
-0.38
Elasticity of the Shadow (i) F.H. 1.00
Price Relative to Income(ii) F.A. –3.13
0.71
-4.78
Population Size
2430
Prices
by region and social category
3630
no
Reference: Gardes, Duncan, Gaubert and Starzec (JBES, 2005), Tables 1 and 2. Price elasticities are
calibrated, according to Frisch proposal, as minus half of the corresponding T.S. income elasticities.
1. Another relative Engel’s law: Simple
correlation
Table 1.
Correlation between Food Expenditures
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and Relative Income
Survey
1987
0.496 0.495 0.444 0.668 0.526
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Mean y
(my)
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Specific y
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(ys)
(0.036)
1988
(0.039)
1989
(0.034)
1990
(0.039)
average
(0.019)
11??? 0.499 0.443 0.551 0.506
(0.009)
(0.009)
(0.009)
(0.009)
(0.005)
Looking for another relative Engel’s law
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Theory
Consider a model of consumption for individual h
at time t:
zht = Xht  + uht with uht = h + ht
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( 1)
Suppose that the estimation is performed on a
population H of individuals h = 1 to N, surveyed within
the whole population H (H H ). Sub-populations are
defined by crossing characteristics kj, j=1 to J such
that:Hi = {h H / kj(h)=cj(i) for all j} with cj(i)
taking all possible items or values for characteristics kj.
Hi is thus defined as Hi  H.
Looking for another relative Engel’s law
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Suppose that the first explanatory
variable is the logarithmic individual
income yh. The usual assumptions on the
distributions of income and specific
effects for individuals are made:
(H1) h  Hi  yh N(yi,2yi) and
h  N(i,2i), i.i.d.,
with yi = E(yh|hHi), i = E(i |hH i)  .
Looking for another relative Engel’s law
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The average yi in Hi is computed by
regressing yh on the vector of characteristics K:
yHi = Kai + i so that yHi = 1/ni (hHi yh ) with ni
the number of individuals in Hi. So the distribution
of the empirical mean is: yHi  N(yi,2yi/ni). The
specific income (which may be considered as the
relative income of individual h in its reference
population Hi ) is defined as ysh = yh - yHi so that
ysh  N(0, 2yi - 2yi/ni).
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By the same reasoning, Hi = 1/ni ( h )
and Hi  N(i,2i /ni).
hH
i
A
V(y)
Looking for another relative Engel’s law
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The covariance on individual data
between  and some explanatory
variable y (here log-income or total
expenditures) can be decomposed into
the reference population components
and the true individual components:
A=E{(yh-Ey).( h-E)}
= E{ [(yHi-yi) + (yi-y) +ysh].[(Hi-i) +
i + h]}
Looking for another relative Engel’s law
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This expression is shown in Appendix I to
reduce asymptotically to the sum of two of the
nine terms of its decomposition, so that
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A/V(y)=  = (b - w)panel
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= p (b - w)
grouped data
+(1-p) 
where p = V(YHi)/v(Y) and  is the coefficient
resulting from the correlation between the specific
effect h of household h and its specific (relative)
income ysh:  = ∂h/∂ysh.
Looking for another relative Engel’s law
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(i) plim (hH ph.1Hi.(yHi-Eyi).(Hii))=plim (ihHi ph.(yHi-Eyi).(Hi-i))
=iplim pHi.plim (yHi-Eyi).plim
(Hi-i) =0 as plimyHi=Eyi=yi.
(ii) plim (hH ph.(yHi-yi).i)=ipi.plim
((yHi-Eyi).i))(Supii)ipi.plim (yHiEyi)=0.
Looking for another relative Engel’s law
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Thus, this coefficient  and its
standard error can be computed in
terms of the difference between the
estimates of  on individual and
grouped data in the between and
within dimensions:
(h/ysh) =
{(b - w)panel - p.(b - w)grouped data}
Looking for another relative Engel’s law
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Table 2. Income Elasticities and Relative
Income Effects
Food at Home
Food Away
Polish Panel
PSID
PSID

-0.1753
-0.0646
0.0202
σ
(0.0151)
(0.0097)
(0.0137)
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Student for 
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11.60
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2.09
Relative income Elasticity
0.655
0.532
1.36
1.616
Looking for another relative Engel’s law
Results
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For food away, /ys is significantly positive,
which indicates that relatively rich households
have a greater budget share of food away
from home than the relatively poor. This is
another Engel law which describes the
relationship between food expenditure and
the household income measured relatively to
the consumption and income distribution of
its social class.
Looking for another relative Engel’s law
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/ys are negative and significant for food at home,
both in US and Poland. It indicates a negative
Duesenberry effect on food at home consumption: a
household h which is relatively poor in its reference
population P2 (i.e. a household having a negative
specific income ysh) have a greater food budget share,
as a share of its income, than a relatively rich
household-belonging to another reference population
P1-which has the same total income and similar control
variables.
It should be noted that the relative income elasticity
for food at home is similar in both countries, contrary
to the income elasticities which are much greater for
Polish consumers, as would be expected.
A relative rich household and a relative
poor
Looking for another relative Engel’s law
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The relative income elasticities for food at
home are greater than the between
elasticities, which indicates that these two
types of elasticity do not measure exactly the
same effects: the cross-section effect of
income differences does indeed contain
relative income effects, but they also contain
the influence of long term changes in the
average income of the reference populations
which may be recovered by comparing
relative income coefficients and the total
cross-section coefficients.
Section 3. The price effect of relative income changes:
Substitution effects between domestic activities and
market substitutes
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The complete price for food at home and food away
from home depends on the opportunity cost for both
consumptions, so that it imparts a substitution
effect. This substitution is analyzed more generally
to compare the budget share for service in the U.S.
and in Europe.
Suppose that the complete price writes:
i = pmi + ti.
(5)
Optimal allocation of time for food
consumption
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Opportunity cost for time:
ω = k(ymin/Tl) and ymin=K(Zh)yhβ
=> El(ω/y) = β ~ 0.6 if Tl is exogenous.
Consequence: ti = (pmi/){β/El(i/y) -1}
=> El (Tl/y) = - β
Micro-simulation of food budget share
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Change of the French Food budget share for US
inequality:
dwfood = (∂w/∂ln i).El(i/y).dyr/yr = -0.74%
∂w/∂ln i=El(w/i).w = -0.49x0.146
(price-elasticity –O.49 computed by pairing family expenditures and Time use surveys)
El(i/y) = 0.5 (JBES, 2005)
dyr/yr = ln(4.76/3.87) = 0.207
4.76=(D9-D1)/D1 in the US
3.87=(D9-D1)/D1 in France
Change of the US Food budget share with French inequality:
dwfood = +1.22%
Conclusion
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The estimation of the existence of Social Interactions is based on
the endogeneity bias, i.e. on the influence of permanent latent
variables on the household’s relative income (defined as its total
influence minus the common influence to all households
pertaining to the same reference group).
It does not correspond to the overall correlation between relative
income and consumption.
H
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h\H
Wh = α + βyh + Zhγ + uh
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