Looking for another relative Engel’s law Introduction  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  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 cross- sectionally.  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  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  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 socio- economic 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 cross- section estimations  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  PSID (U.S.) Polish panel  Period  Income ElasticityCSTSCSTS  Food at home  Food away from home  Direct Price Elasticity  Elasticity of the Shadow (i) F.H  Price Relative to Income(ii) F.A. –  Population Size  Pricesby region and social categoryno  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 and Relative Income  Survey average  Mean y  (my) (0.036)(0.039) (0.034) (0.039) (0.019)  Specific y11???  (ys) (0.009) (0.009) (0.009) (0.009) (0.005)

Looking for another relative Engel’s law  Theory  Consider a model of consumption for individual h at time t: z ht = X ht  + u ht with u ht =  h +  ht ( 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 k j, j=1 to J such that:H i = {h H / k j (h)=c j (i) for all j} with c j (i) taking all possible items or values for characteristics kj. H i is thus defined as H i  H. 

Looking for another relative Engel’s law  Suppose that the first explanatory variable is the logarithmic individual income y h. The usual assumptions on the distributions of income and specific effects for individuals are made: (H1) h  Hi  y h N(y i, 2 yi ) and  h  N( i, 2 i ), i.i.d., with y i = E(y h |hH i ),  i = E( i |hH i )  .

Looking for another relative Engel’s law  The average y i in H i is computed by regressing y h on the vector of characteristics K: y Hi = Ka i +  i so that y Hi = 1/n i ( hHi y h ) with n i the number of individuals in H i. So the distribution of the empirical mean is: y Hi  N(y i, 2 yi /n i ). The specific income (which may be considered as the relative income of individual h in its reference population H i ) is defined as ys h = y h - y Hi so that ys h  N(0,  2 yi -  2 yi /n i ).   By the same reasoning,  Hi = 1/n i (   h ) and  Hi  N( i, 2 i /n i ). hH i

Looking for another relative Engel’s law  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{(y h -Ey).(  h -E)} = E{ [(y Hi -y i ) + (y i -y) +ys h ].[( Hi - i ) +  i + h ]}

Looking for another relative Engel’s law  This expression is shown in Appendix I to reduce asymptotically to the sum of two of the nine terms of its decomposition, so that   A/V(y)=  = ( b -  w ) panel  = p ( b -  w ) grouped data +(1-p)  where p = V(Y Hi )/v(Y) and  is the coefficient resulting from the correlation between the specific effect h of household h and its specific (relative) income ys h :  = ∂ h /∂ys h.

Looking for another relative Engel’s law  (i) plim ( hH p h.1 Hi.(y Hi -Ey i ).( Hi -  i ))=plim ( i  hHi p h.(y Hi -Ey i ).( Hi - i ))  = i plim p Hi.plim (y Hi -Ey i ).plim ( Hi - i ) =0 as plimy Hi =Ey i =y i.  (ii) plim ( hH p h.(y Hi -y i ). i )= i p i.plim ((y Hi -Ey i ). i ))(Sup i  i ) i p i.plim (y Hi - Ey i )=0.

Looking for another relative Engel’s law  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 /ys h ) = {( b -  w ) panel - p.( b -  w ) grouped data }

Looking for another relative Engel’s law  Table 2. Income Elasticities and Relative Income Effects  Food at Home Food Away  Polish PanelPSID PSID    σ(0.0151)(0.0097) (0.0137)  Student for   Relative income Elasticity

Looking for another relative Engel’s law Results  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  /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 y sh ) 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  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  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 = p m i + t i. (5)

Optimal allocation of time for food consumption  Opportunity cost for time: ω = k(y min /T l ) and y min =K(Z h )y h β => El( ω/y) = β ~ 0.6 if T l is exogenous. Consequence : t i = ( p m i /){β/El( i /y) -1} => El (Tl/y) = - β

Micro-simulation of food budget share  Change of the French Food budget share for US inequality:  dw food = (∂w/∂ln  i ).El( i /y).dy r /y r = -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) dy r /y r = ln(4.76/3.87) = =(D9-D1)/D1 in the US 3.87=(D9-D1)/D1 in France Change of the US Food budget share with French inequality: dw food = +1.22%

Conclusion  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 h\H W h = α + βy h + Z h γ + u h  ||  Latent Variables